Constructive Methods for Inequality Constraints in

Linkoping Studies in Science and Technology. Dissertations No. 527

Constructive Methods for Inequality Constraints in Control Mats Jirstrand

Department of Electrical Engineering Linkoping University, S{581 83 Linkoping, Sweden Linkoping 1998

Cover Illustration: The construction of a2 cylindrical algebraic decomposition of R3 for the polynomials x + y2 + z2 - 1 and x - y2 - z2 . The sphere and bowl are the zero sets of the polynomials. The bold lines are vertical tangent projections onto R2 and the dots show some sample points at dierent phases of the construction.

Constructive Methods for Inequality Constraints in Control

c 1998 Mats Jirstrand [email protected] http://www.control.isy.liu.se

Department of Electrical Engineering Linkoping University S{581 83 Linkoping Sweden ISBN 91-7219-187-2

ISSN 0345-7524

Printed in Sweden by Linkopings Tryckeri AB

To Emma

Abstract In practical control problems there are always constraints on inputs and system variables. The constraints are often described by inequalities and there is a need for methods that take them into account. This thesis consists of three parts. In the rst two parts a computer algebra approach is taken and in the third part we use methods based on linear matrix inequalities and convex optimization. A large class of systems can be described by a set of polynomial dierential equations and there are dierent approaches to analyze systems of this kind. In the rst part of the thesis we utilize mathematical tools from commutative and dierential algebra such as Grobner bases and characteristic sets to study input-output relations of systems given in state space form. The state space representation of a system can be described by a nitely generated dierential ideal. We show that this is not always true for the corresponding dierential ideal of input-output relations. However, the input-output relations up to a xed order can be computed and represented using non-dierential tools. Using characteristic sets we also show how the above problem of nite representation of the input-output relations can be resolved. Many problems in control theory can be reduced to nding solutions of a system of polynomial equations, inequations, and inequalities, a so called real polynomial system. The cylindrical algebraic decomposition method is an algorithm that can be utilized to nd solutions to such systems. The extension of real polynomial systems to expressions involving Boolean operators (_; ^; :; !) and quanti ers (9; 8) is called the rst-order theory of real closed elds. There are algorithms to perform quanti er elimination in such expressions, i.e., to derive equivalent expressions without any quanti ed variables. We show how these algorithms can be used to solve problems in control such as stabilization of a system with real parametric uncertainties; feedback design of linear systems; computation of bounds on static nonlinearities in feedback systems that ensure stability; computation of equilibrium points for nonlinear systems subject to constraints on the control and state variables; and curve following. We also consider stabilization of systems by switching among a set of state feedback controllers. Furthermore, for a nonlinear system which is not ane in the control, quanti er elimination can be used to decide if the zero dynamics can be stabilized by switching between dierent controllers. Invariant sets of dynamic systems play an important role for veri cation of control systems, i.e., to check if a designed controller meets performance and safety constraints. In part three of the thesis we have compiled a number of results on representations and computations on polyhedral and quadratic sets, i.e., sets de ned by ane and quadratic inequalities. We also derive criteria for deciding when an ane dynamic system has polyhedral or quadratic invariant sets. These results are utilized to propose a method for computing invariant sets of a class of hybrid systems, i.e., systems that exhibit both continuous and discrete behavior. The methods for invariant set computations and veri cation are based on convex optimization techniques and linear matrix inequalities. Since there are no well established design procedures for hybrid systems, veri cation of heuristically designed controllers is of outmost importance. i

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Preface This thesis consists of three parts. In Part I a brief introduction to commutative and dierential algebra is given. The focus is on constructive methods such as Grobner bases and characteristic sets. The last chapter of this part is an application of these methods to dierential algebraic systems theory. Part II gives an introduction to real algebra and two constructive methods, cylindrical algebraic decomposition and quanti er elimination. A number of applications of quanti er elimination to linear, nonlinear, and switched systems are also presented. Computations on polyhedral and quadratic sets are studied in Part III. We develop a method for computation of invariant sets of piecewise ane dynamic systems, which can be used for veri cation purposes. The work in Part III was carried out during visits to Professor Stephen Boyd at Informations Systems Laboratory, Stanford University. Some of the material in this thesis has been published earlier. A short version of Chapter 4 appears in K. Forsman and M. Jirstrand. Some niteness issues in dierential algebraic systems theory. In Proceedings of 33rd IEEE Conference on Decision and Control (CDC'94), pages 1112{1117, Lake Buena Vista, Florida, December 1994. The material in Section 6.2 on the circle criterion and describing function method are based on M. Jirstrand. Quanti er elimination and application in control. In Proceedings of Reglermote 1996, Lulea, Sweden, June 1996. Parts of the contents of Section 7.2 were presented in M. Jirstrand and S. T. Glad. Computational questions of equilibrium calculation with application to nonlinear aircraft dynamics. In Mathematical Theory of Networks and Systems (MTNS'96), St Louis, Missouri, 1996 and a journal version of this paper has been submitted to Journal of Mathematical Systems, Estimation, and Control. Some of the ideas in Section 7.2.1 appear in M. Jirstrand. Applications of quanti er elimination to equilibrium calculations in nonlinear aircraft dynamics. In Proceedings on The 5th International Student Olympiad on Automatic Control (BOAC'96), St Petersburg, Russia, October 1996. Parts of the material in Section 7.2, 7.3, 7.5, and 7.6 were presented at M. Jirstrand. Nonlinear control system design by quanti er elimination. In IMACS Conference on Applications of Computer Algebra, Linz, Austria, July 1996 iii

and a Journal version of this paper can be found in M. Jirstrand. Nonlinear control system design by quanti er elimination. Journal of Symbolic Computation, 24(2):137{152, August 1997. Some of the results in Section 7.6 appear in M. Jirstrand. Curve following for nonlinear dynamic systems subject to control constraints. In Proceedings of the American Control Conference (ACC'97), Albuquerque, New Mexico, USA, June 1997. Section 7.4 is covered by M. Jirstrand and S. T. Glad. Moving the state between equilibria. In Proceedings of the 4th IFAC Nonlinear Control Systems Design Symposium (NOLCOS'98), Enschede, Netherlands, July 1998. A short version of the material in Section 3.3, 5.4, 5.5 together with some examples from Chapter 7 will appear in S. T. Glad and M. Jirstrand. Using dierential and real algebra in control. In N. Munro, editor, The Use of Symbolic Methods in Control System Analysis and Design, IEE Control Engineering Series. IEE, 1998. The work in Chapter 8 was done together with Dr. Dragan Nesic and parts of it will be presented in D. Nesic and M. Jirstrand. Stabilization of switched polynomial systems. In IMACS Conference on Applications of Computer Algebra (ACA'98), Prague, Czech Republic, August 1998. A condensed version of the material in Section 12.4, 12.5, and 12.6 has been submitted to the conference CDC'98.

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Acknowledgments First of all I would like to thank my supervisor, Professor Torkel Glad, for his pro cient guidance and for many interesting discussions during the evolution of this thesis. I would also like to thank Professor Lennart Ljung for providing me the opportunity to join the group and for creating such a stimulating research atmosphere. The Automatic Control group in Linkoping is really an extraordinary place to work at. Besides the serious research activities I have really enjoyed the often incredible coee room discussions, the junk seminars, the oor ball games on Fridays, and all the fuss about Rainman, to single out a few things. I thank all the people in the group for their contribution. I am also grateful to Dr. Krister Forsman who introduced me to commutative algebra and its applications in control theory. Over the years a number of internal seminar groups called GL(n) (the algebraic group), SyCo (Symbolic Computation), and HybriS (Hybrid Systems) have come into existence. These seminars and their participants have been a great source of inspiration. A special thank to Dr. Hakan Fortell and Dr. Roger Germundsson for inspiring research discussions. Roger initialized the work on polyhedral and quadratic sets during his stay at Stanford University as a postdoc by his preliminary investigations of ellipsoids, polyhedra, and their engineering applications. Thanks also to Dr. Johan Gunnarsson for being a reliable companion in the Mathematica camp, always willing to share his expert knowledge in symbolic computation. I would like to thank Associate Professor Ulf Henriksson, Head of the Electrical Engineering Department, for giving me the opportunity to visit Stanford University at several occasions during my PhD studies. I am grateful to Professor Stephen Boyd at the Information Systems Laboratory (ISL) at Stanford for letting me visit ISL and for many inspiring and enlightening discussions. Thanks also to Professor Lieven Vandenberghe for answering my questions about convex optimization and for giving me pointers to important references. The following people have read parts of the manuscript during various stages of preparation: Valur Einarsson, Lic. Eng. Urban Forssell, Dr. Andrey Sokolov, and Per Spangeus. I thank you all for your valuable remarks and suggested improvements, which I really appreciated. Thanks also to Dr. Alexander Pogromsky for clarifying discussions on stability issues of nonlinear and switched systems. I am also thankful to Lic. Eng. Krister Edstrom, my oce colleague during the rst years in the group, for many clarifying discussions regarding both courses and research. I am indebted to Dr. Dragan Nesic for pointing out the application of quanti er elimination to stabilization of nonlinear systems by discontinuous feedback laws and for valuable correspondence. This work was supported by the Swedish Research Council for Engineering Sciences (TFR), which is gratefully acknowledged. Furthermore, I would like to thank Dr. Hoon Hong at Research Institute for Symbolic Computation (RISC) in Austria for providing me with the qepcad program, which was used to perform quanti er elimination in many of the examples in the thesis. v

I am also grateful to my family for their love and never ending support in whatever I have done. Finally, I would like to thank Emma for all the love, support, and encouragement she has given me, and the enormous patience she has shown during my work with this thesis. Linkoping, March 1998 Mats Jirstrand

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Contents

1 Introduction 1.1 1.2 1.3 1.4 1.5

Dierential Algebraic Systems Theory Quanti er Elimination . . . . . . . . . Polyhedral and Quadratic Sets . . . . Contributions . . . . . . . . . . . . . . Outline of the Thesis . . . . . . . . . .

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I An Introduction to Constructive Algebra 2 Commutative Algebra and Grobner Bases 2.1 Rings and Fields . . . . . . . . . 2.2 Ideals . . . . . . . . . . . . . . . 2.3 Grobner Bases . . . . . . . . . . 2.3.1 Monomial Orderings . . . 2.3.2 Division in k [x1 ; : : : ; xn ] 2.3.3 Grobner Bases . . . . . .

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3 Dierential Algebra and Characteristic Sets

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Contents 3.3.3 Characteristic Sets . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Input-Output Descriptions and Dierential Algebra 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Representations of Dynamical Systems . . . . . The Lowest Order Input-Output Relation . . . The Simplest Description Fails . . . . . . . . . Is the Input-Output Ideal Finitely Generated? . A Non-Dierential Approach . . . . . . . . . . Description of c with Characteristic Sets . . . Conclusions and Extensions . . . . . . . . . . .

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II Quanti er Elimination in Control Theory

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5 Real Algebra and Cylindrical Algebraic Decomposition

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5.1 Real Closed Fields . . . . . . . . . . . . . 5.2 Semialgebraic Sets and Decompositions . 5.3 Tools . . . . . . . . . . . . . . . . . . . . . 5.3.1 Polynomial Remainder Sequences . 5.3.2 Subresultants . . . . . . . . . . . . 5.3.3 The Projection Operator . . . . . 5.3.4 Sturm Chains . . . . . . . . . . . . 5.4 Cylindrical Algebraic Decomposition . . . 5.4.1 The Algorithm . . . . . . . . . . . 5.4.2 An Example . . . . . . . . . . . . 5.5 Quanti er Elimination . . . . . . . . . . .

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6 Applications to Linear Systems

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6.1 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7 Applications to Nonlinear Systems

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Stationarizable Points . . . . . . . . . . . . . . . . . . . . 7.2.1 Equilibrium Computations for Aircraft Dynamics . 7.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Moving the State Between Equilibria . . . . . . . . . . . . 7.5 Output Range . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Curve Following . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Control Law Design . . . . . . . . . . . . . . . . . 7.6.2 Constrained Reachability . . . . . . . . . . . . . . 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Applications to Switched Systems

8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Nonsmooth Systems . . . . . . . . . . . . . . . . . 8.2 Stabilizing Polynomial Systems by Switching Controllers . 8.3 Exact Output Tracking . . . . . . . . . . . . . . . . . . . 8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

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III Computations on Polyhedral and Quadratic Sets

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10 Linear Matrix Inequalities and Optimization

10.1 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . 10.1.1 LMIs in Control . . . . . . . . . . . . . . . . . . 10.2 Convex Optimization . . . . . . . . . . . . . . . . . . . . 10.2.1 Convex Sets and Functions . . . . . . . . . . . . 10.2.2 Some Classes of Convex Optimization Problems 10.2.3 Software . . . . . . . . . . . . . . . . . . . . . . . 10.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Sets De ned by Ane and Quadratic Inequalities

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Representation . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Hyperplanes and Halfspaces . . . . . . . . . . . 11.2.2 Quadratic Sets . . . . . . . . . . . . . . . . . . 11.2.3 Ellipsoids . . . . . . . . . . . . . . . . . . . . . 11.2.4 Quadratic Cones . . . . . . . . . . . . . . . . . 11.2.5 Paraboloids . . . . . . . . . . . . . . . . . . . . 11.3 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Ane Equations and Inequalities . . . . . . . . 11.3.2 Quadratic Inequality . . . . . . . . . . . . . . . 11.3.3 Quadratic Inequality and Ane Equations . . 11.4 Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 S-procedure . . . . . . . . . . . . . . . . . . . . 11.4.2 Convex Hull In Polyhedron . . . . . . . . . . . 11.4.3 Polyhedron In Polyhedron . . . . . . . . . . . . 11.4.4 Quadratic Set In Polyhedron . . . . . . . . . . 11.4.5 Convex Hull In Quadratic Set . . . . . . . . . . 11.4.6 Quadratic Set In Quadratic Set . . . . . . . . . 11.4.7 Quadratic Set and Ane Set In Quadratic Set 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Invariant Sets of Piecewise Ane Systems

12.1 Piecewise Ane Systems . . . . . . . . . . . . 12.2 Invariant Sets . . . . . . . . . . . . . . . . . . 12.2.1 Continuous Systems . . . . . . . . . . 12.2.2 Piecewise Ane Systems . . . . . . . 12.3 Polyhedral Invariant Sets . . . . . . . . . . . 12.3.1 Invariant Halfspaces . . . . . . . . . . 12.3.2 Invariant Polyhedra . . . . . . . . . . 12.4 Invariant Quadratic Sets . . . . . . . . . . . . 12.4.1 Invariant Ellipsoids . . . . . . . . . . . 12.4.2 Invariant Quadratic Cones . . . . . . . 12.4.3 Invariant Paraboloids . . . . . . . . . 12.4.4 A Generalization . . . . . . . . . . . . 12.5 Invariant Sets for Piecewise Ane Systems . 12.5.1 A Computational Procedure . . . . . . 12.5.2 The Choice of Invariant Set . . . . . . 12.6 Examples . . . . . . . . . . . . . . . . . . . . 12.6.1 A Two-Mode System . . . . . . . . . . 12.6.2 A Veri cation Problem . . . . . . . . . 12.6.3 A Cycling System . . . . . . . . . . . 12.7 Invariant Sets for Systems with Disturbances 12.8 Summary . . . . . . . . . . . . . . . . . . . .

13 Conclusions and Extensions Bibliography Notation Index

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1 Introduction Computer algebra or symbolic computation has been recognized as an important tool in many engineering disciplines and continues to nd new elds of application. For control and design of linear and especially non-linear systems there exist toolboxes, user developed packages and algorithms of various sophistication, implemented in computer algebra systems such as Maple and Mathematica, see [66, 113, 159]. In control and systems theory symbolic computation oers solutions to many problems both of theoretical and practical value. For example, it is easy to manipulate systems of equations to reveal connections between dierent variables. Another example is the computation of analytical expressions for system properties and their dependence on parameters together with graphical presentations in various ways. The methods studied in the rst two parts of this document are constructive algorithms for elimination of variables in systems of dierential and non-dierential polynomial equations; nding solutions of systems of polynomial equations, inequations and inequalities; and elimination of variables in formulas containing Boolean operators and quanti ers. We also use the methods to treat some theoretical issues in dierential algebraic systems theory and demonstrate their applicability to a number of problems both in linear and nonlinear control theory. The aim of the rst two parts of the thesis is twofold. On one hand it should serve as an introduction to constructive algebra and symbolic computation and on the other hand present their applicability to some problems in control and systems theory. Recently, linear matrix inequalities and convex optimization have been recog1

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Chapter 1 Introduction

nized to have wide applicability in control theory. For a large class of convex optimization problems there exist ecient algorithms which can handle large problems (100-1000 variables). In combination with polyhedral and quadratic sets, i.e., sets de ned by linear or quadratic inequalities, convex optimization can be used to compute invariant sets of linear and ane dynamic systems. In the third part of this thesis we give a brief introduction to linear matrix inequalities and convex optimization. Furthermore, we present a listing of particular polyhedral and quadratic sets, their representation and how to perform computations on them. These results are applied to develop a procedure for computing invariant sets of a class of hybrid systems, known as piecewise ane systems. This kind of systems are rather common in practice since any linear system controlled by linear controllers and switching devices can be modeled as a piecewise ane system. However, for these rather complex systems there is a lack of systematic methods to investigate properties such as stability and reachability. For such problems the computation of invariant sets is a useful tool. The aim of the third part of the thesis is to provide a number of results for polyhedral and quadratic sets, which together with convex optimization can be used for veri cation of piecewise ane systems.

1.1 Dierential Algebraic Systems Theory A large class of dynamic systems can be described by a set of dierential equations. In the case when all the dierential equations are linear most of the questions about the system can be answered using linear algebra. In the nonlinear case the situation is more complicated and there are no general methods. If we constrain ourselves to the fairly large class of systems which can be described by polynomial dierential equations, the frameworks of commutative algebra and dierential algebra oer insight to some questions. Commutative algebra is a mathematical branch that arose from the study of polynomial equations. Dierential algebra was developed mainly by J.F. Ritt [132] in the forties and is an extension of commutative algebra in the sense that derivation is also taken into account. Dierential algebra made its way into systems theory via the discoveries by Fliess in the mid eighties and has now grown quite extensive. A good survey of the topic is given in [47]. Due to the development of powerful computers and advanced mathematical software for symbolic computations, the treatment of polynomial dierential equations with commutative and dierential algebra has been considerably simpli ed. As examples we here mention Maple [66] and Mathematica [159], which have been of vital importance in this work. In Part I an introduction to commutative and dierential algebra is given together with two constructive methods, Grobner bases and characteristic sets. These tools are then used to investigate some problems that arise when treating polynomial dierential equations with concepts from commutative and dierential algebra.

1.2 Quanti er Elimination

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1.2 Quanti er Elimination A formula containing polynomial equations and inequalities combined with Boolean operators ^ (and) and _ (or) where some of the variables are quanti ed using the quanti ers 9 (there exists) and 8 (for all), constitutes a sentence in the rst-order theory of real closed elds. If all variables in such a formula are quanti ed the problem of deciding its truth value is called a decision problem. Suppose that some variables are bounded by quanti ers but others are not. The search for an equivalent expression in the unquanti ed variables only is called a quanti er elimination problem. In the late forties Tarski showed that there is a procedure that solves the above problems in a nite number of steps [148]. Although Tarski gave a constructive proof, the resulting algorithm is impractical even with the power of todays computers. In the mid seventies Collins invented cylindrical algebraic decomposition which given a set of n-variate polynomials decomposes Rn into regions over which the polynomials have constant sign [33]. Utilizing this method the problem of eliminating quanti ers could be carried out for small examples. Recently, eective algorithms for quanti er elimination have made it possible to solve non-trivial problems on workstations. For real polynomial systems, i.e., systems of polynomial equations and inequalities in real variables, the method of cylindrical algebraic decomposition has a value of its own. Once a decomposition is found, a solution (if there exists one) to any real polynomial system in the de ning polynomials can be found. In Part II we present cylindrical algebraic decomposition, quanti er elimination, subalgorithms, and tools utilized in these methods such as computation of subresultants, principal subresultant coecients, Sturm chains, and representation of real algebraic numbers. We apply the methods to a number of problems in both linear and nonlinear control theory which can be formulated as quanti er elimination problems. To perform quanti er elimination the program qepcad has been used which was developed by Hoon Hong et al. [35] at RISC (Research Institute for Symbolic Computation) in Austria. One of the rst attempts to apply quanti er elimination techniques to problems in control was made by Anderson et al. [4] in the mid seventies, where the possibility of stabilizing an unstable nite-dimensional linear system by static linear output feedback was studied. Since then, both algorithmic improvements and a dramatically increased computer power have made it possible to implement more ecient quanti er elimination algorithms. Recently, some papers treating control related problems have appeared [18, 43, 75, 115, 117, 147, 163] and since the seventies there has been considerable progress in the development of more eective quanti er elimination algorithms starting with Collins [33]. For an extensive bibliography on quanti er elimination see [5] and more recent results [12, 13, 29, 30, 73, 74, 109].

4


1.3 Polyhedral and Quadratic Sets Polyhedral and quadratic sets have many applications. Their simple representations and good approximating properties make them suitable as constraint sets in optimization, building blocks in geometric modeling, and as rst and second order approximations of convex sets. Polyhedral sets and some quadratic sets such as ellipsoids and paraboloids are always convex. This is a very useful property in for example optimization. Hence, the use of these sets is closely tied to the development of linear programming and more general convex optimization. Optimization over particular classes of quadratic sets can be done by using linear matrix inequalities as constraints on the parameterizations of the sets. We utilize such constraints to formulate computations of quadratic invariant sets for dynamic systems as convex optimization problems. Both polyhedral and quadratic sets are natural invariant sets for linear systems. Based on this observation we develop methods for computing invariant sets for piecewise ane systems in Part III. These invariants can be used in for example veri cation of certain system behaviors. Veri cation based on invariants has been used in computer science to tackle the computational complexity of systems with discrete behavior, see [64, 70, 71, 130, 143{145]. However, in cases where continuous dynamics has been taken into account it is of very simple form; essentially only pure integrators. Polyhedral and quadratic sets constitute the tools needed for extending such veri cation procedures to hybrid systems with more complicated continuous dynamics.

1.4 Contributions In Part I the contributions consist of some results regarding the description of the input-output structure of systems represented by state space equations. Although the state space representation of a system obeys a description in terms of a nitely generated dierential ideal, the description of the corresponding input-output relations is a more intricate problem. We show that the dierential ideal describing the input-output relations is, in general, not nitely generated, see Theorem 4.4. We also show how this problem can be resolved in Theorem 4.7. Another possibility is to consider the dierential structures as non-dierential ones in in nitely many variables and then use tools from commutative algebra. Lemma 4.6 and Theorem 4.5 facilitate this approach. In Part II the contributions are the applications of the quanti er elimination method to problems in control theory such as constructive stability investigations using the circle criterion, computation of equilibrium points and their stability properties for systems subject to control constraints, and curve following. In Theorem 7.3 and 7.4 we give some characterizations of when the state of a nonlinear system can be moved between dierent equilibria. The material in Chapter 8 is a joint work with Dr. Dragan Nesic. The contribution of this part is to demonstrate the computational procedures that quanti er elimination oers for testing stabiliz-

1.5 Outline of the Thesis

5

ability of switched systems or zero dynamics. Theorem 8.1 gives a Lyapunov-like result about stabilization of nonlinear systems by switching. The theorem is particularly tailored to suit quanti er elimination. We also observe that many classical problems in control theory can be formulated using the concepts of semialgebraic sets and formulas in the rst-order theory of real closed elds. This implies that these problems can, at least theoretically, be solved in a nite number of steps. The material in Part I and II on mathematical concepts and constructive algorithms in commutative, dierential, and real algebra serves as introductions to the presented applications but contains essentially nothing new. However, most of the material is scattered over many articles and books and a similar collection of introductory material does not exist to the author's knowledge. The contributions of Part III consist of the use of convex optimization techniques to compute invariant quadratic cones and paraboloids for linear and ane dynamic systems. Furthermore, the idea to combine such invariant sets to obtain invariant sets for piecewise ane systems is novel. The conditions for existence of invariant quadratic cones and paraboloids for linear dynamic systems given in Lemma 12.6 and 12.7, have not been reported elsewhere. We also discuss how to utilize invariant sets of these rather complex systems as a tool for veri cation of dierent properties such as stability and reachability. This extends the current methods for hybrid system veri cation using abstractions, to systems with more complicated continuous dynamics.

1.5 Outline of the Thesis The thesis is divided into three parts. The purpose of Part I is to introduce the reader to basic commutative and dierential algebraic terminology together with a number of constructive methods. In Chapter 2 concepts in commutative algebra such as rings, elds and ideals are introduced. Polynomial rings and their ideals play a special role, since there are many algorithms developed and implemented in computer algebra packages that make computations with these objects possible. We introduce Grobner bases as the main constructive tool of commutative algebra and show parallels to multivariate polynomial division. Chapter 3 deals with dierential algebra. We consider dierential rings, elds and ideals and especially dierential rings of polynomials for the same reason as in the non-dierential case. The constructive methods in this chapter concern characteristic sets and Ritt's algorithm. In Chapter 4 the input-output structure of systems represented by state space equations is investigated from a dierential algebraic perspective. An introduction to dierential algebraic representations of dynamic systems is given and input-output relations are discussed. Furthermore, we investigate some niteness issues in dierential algebraic systems theory and point at some problems that arise when treating systems of polynomial dierential equations using the framework of dierential algebra. We also give a non-dierential algebraic approach to the inputoutput description problem and a treatment with characteristic sets that resolve much of the problems. The chapter ends with some extensions and conclusions of

6


the discussed issues. The aim of Part II is to give an introduction to constructive methods in real algebra and investigate their applicability to problems in control. Chapter 5 is devoted to real algebra, cylindrical algebraic decomposition, and quanti er elimination. After a short introduction to real closed elds, semialgebraic sets and decompositions induced by zero sets of polynomials, a detailed introduction to constructive tools in real algebra such as subresultants, Sturm chains and cylindrical algebraic decomposition is given. The last topic in this chapter is quanti er elimination, which is extensively used in the rest of the chapters in this part. Applications of quanti er elimination to linear systems such as stabilization and treatment of model uncertainties are presented in Chapter 6. Here we also give examples of the applicability of quanti er elimination in design of controllers and investigations of stability of feedback systems with static nonlinearities. In Chapter 7 quanti er elimination is utilized to compute descriptions of equilibrium points of (nonlinear) dynamic systems subject to state and control constraints. We give a characterization of when the state of a nonlinear system can be moved between dierent equilibria and present a method for deciding if the system output can be controlled along a given curve despite control constraints. Moreover, a constrained form of computable reachability is also discussed in this chapter. In Chapter 8 we investigate stabilizability of switched nonlinear systems. The stabilization of a system by switching between a number of controllers is treated in a constructive way. We show how to utilize quanti er elimination to estimate the region of attraction of an equilibrium point. Quanti er elimination can also be used to check if the zero dynamics of non control-ane nonlinear systems can be stabilized by switching. A number of examples are given which illustrate the methods. Finally some conclusions and possible extensions of the quanti er elimination approach to problems in control theory are given in Chapter 9. In Part III we utilize polyhedral and quadratic sets to develop methods for computing invariant sets of piecewise ane systems. These invariants can for example be used for veri cation purposes. In Chapter 10 we give a brief introduction to linear matrix inequalities and convex optimization. Chapter 11 deals with polyhedral and quadratic sets such as hyperplanes, half spaces, polytopes, ellipsoids, paraboloids, and quadratic cones. Dierent representations of and operations on sets of this type are covered here. In Chapter 12 we discuss invariant sets and consider the computation of both polyhedral and quadratic invariant sets for linear and ane dynamic systems. These results are used to propose a method for computation of invariant sets of piecewise ane systems based on determinant maximization which is a convex optimization problem. We illustrate the method by a number of examples. Conclusions and extensions of this part of the thesis are given in Chapter 13.

Part I

An Introduction to Constructive Algebra

7

2 Commutative Algebra and Grobner Bases In this chapter we will introduce some concepts and theorems from commutative algebra. We rst study two basic algebraic objects called rings and elds. The reader will probably recognize the properties of these structures and think of them as abstract generalizations of some well known sets. The ring of multivariate polynomials is paid special attention to due to its usefulness in applications and importance later on in the thesis. Next we treat ideals which are subsets of rings with some special properties. One reason for studying ideals is the close relation between ideals in polynomial rings and systems of polynomial equations, which are abundant in engineering applications. The last section treats Grobner bases which is a useful way of describing ideals. The computation of a Grobner basis of an ideal can be seen as a generalization of Gaussian elimination for systems of linear equations to systems of polynomial equations. An excellent rst book on constructive commutative algebra which links commutative algebra and computational algebraic geometry is [39]. Another comprehensive treatment of ideal theory, especially polynomial ideals can be found in [137]. The books [79, 80, 99] and [108] are comprehensive and rigorous but also more abstract than the abovementioned.

2.1 Rings and Fields A ring is an algebraic structure that captures the properties of the integers under the usual operations of addition and multiplication but also rational and complex numbers, polynomials, and rational functions under natural addition and multiplication. Here follows a formal axiomatic de nition. 9

10

Chapter 2 Commutative Algebra and Grobner Bases

De nition 2.1 A ring is a set R together with two binary operations addition \+" and multiplication \". (i) a + (b + c) = (a + b) + c; 8a; b; c 2 R (associative). (ii) a + b = b + a; 8a; b 2 R (commutative). (iii) There is an element denoted 0 in R such that 0 + a = a; 8a 2 R. (iv) For every a 2 R, there is an element -a 2 R such that (-a) + a = 0. (v) a (b c) = (a b) c; 8a; b; c 2 R (associative). (vi) a b = b a; 8a; b 2 R (commutative). (vii) There is an element denoted 1 in R such that 1 = 6 0 and 1 a = a 1 = a; 8a 2 R: (viii) Multiplication is distributive over addition, i.e.,

a (b + c) = a b + a c; 8a; b; c 2 R:

Some authors de ne rings in a slightly dierent manner. The correct name for the structure above would rather be a commutative ring with multiplicative identity. The term commutative originates from the commutativity of the ring multiplication. In this document we will only consider commutative rings and henceforth when referring to a ring we mean a commutative ring. Note that the requirements in the de nition are not independent1 . Observe that the operations themselves are an important part of the structure. However, when the underlying operations are the natural ones we will not state them explicitely.

Example 2.1

The set of integers, Z is a ring. The set of complex numbers, C is a ring. The set of rational numbers, Q is a ring. The set of natural numbers, N is not a ring. We see that a ring is an abstract way of generalizing the properties of some well known sets and it will give us a whole new machinery to deal with such sets. Another more sloppy de nition of a ring could be: A ring is a set where we can add, subtract and multiply the elements according to the \usual" laws of these operations. We then get an element of the ring as the result, i.e., the ring is closed under its de ned operations. Since our main interest will be polynomials we now take a closer look on a special kind of rings called rings of polynomials and their construction. 1 The commutativity of addition (ii) follows from the other postulates: apply the distributive law in two dierent ways to (a + b) (1 + 1).

2.1 Rings and Fields

11

Rings of Polynomials

A polynomial of degree N in the indeterminate or variable x with coecients from the ring R is a formal expression of the form

a0 + a1 x + a2 x2 + : : : + aN xN; where a0 ; a1 ; a2 ; : : : ; aN 2 R and N 2 N . The set of all such expressions is denoted R[x]. Note that two polynomials are equal if and only if they are of the same degree and ai = bi for every i N. Let addition and multiplication in R[x] between two polynomials of the same degree be de ned as usual by N X n=0

!

an xn +

N X n=0

a n xn

!

N X n=0

!

bn xn =

N X n=0

(an + bn )xn ;

! X 2N X n ai bj xr: bn x = r=0 i+j=r n=0 i;j N N X

If the polynomials have dierent degrees we just extend the lower degree polynomial with powers of x with zero coecients. That R[x] is a ring with addition and multiplication de ned as above follows from the ring structure of R. The zero and identity element can be identi ed with the ones in R and it is easy to show that R[x] ful lls the axioms of De nition 2.1. It is now easy to extend a ring of polynomials in a single indeterminate to rings of polynomials in several indeterminates. We start with the ring R[x] and construct a new ring exactly in the same way as we constructed R[x] from the ring R above. The new ring, R[x][y] will be the set of all polynomials in the indeterminate y with coecients from R[x] which is the same as the set of all polynomials in two indeterminates. Another way to denote this set is R[x; y]. Continuing in this way it is possible to construct rings of polynomials in an arbitrary number of indeterminates. This gives us the following de nition.

De nition 2.2 A ring of polynomials in the indeterminates x1; : : : ; xn is the set

of polynomials in x1 ; : : : ; xn with coecients from a ring R together with the usual addition and multiplication of polynomials. This ring is denoted R [x1 ; : : : ; xn ]. When dealing with polynomials in a small number of variables we will usually use the letters x; y, and z instead of indexed variables, e.g., R [x; y; z].

Example 2.2 The polynomial x2y + (1 + 2i)z belongs to the ring C [x; y; z].

12


Fields We have not mentioned anything about division yet. When we now introduce it we will get another algebraic structure which is a special kind of ring.

De nition 2.3 A eld, k is a ring where every element except 0 has a multiplica-

tive inverse, i.e.,

9b : a b = b a = 1 8a 2 k n f0g: From now on we let the symbol k denote an arbitrary eld.

Example 2.3 ; [ x]

Q R, and C are elds. Q is not a eld since

there is no polynomial, f 2 Q [x] such that f x = 1.

Sometimes one needs larger elds. There is a simple way of constructing an extended eld.

De nition 2.4 Let be an indeterminate not in the eld k. Then the smallest eld containing k [ is called the eld obtained by adjoining to k and is denoted k(). This is just a way to introduce a new symbol, which obeys the eld operations, i.e., addition, subtraction, multiplication, and division.

Example 2.4

The eld R(i), where i2 = -1 is the eld of complex numbers, i.e., C . Let k be a eld. The eld k(t) is the eld of all rational functions of t with coecients from k. After de ning the above general algebraic structures we now turn to an important substructure of a ring called ideals.

2.2 Ideals An ideal is an important substructure of a ring. It plays a central role in many applications of ring theory, e.g., nding the zeros of a system of polynomial equations.

De nition 2.5 An ideal I of a ring R is a subset of R satisfying a; b 2 I ) a + b 2 I and r 2 R; a 2 I ) r a 2 I:

2.2 Ideals

13

In the rest of this section we let R denote an arbitrary ring and I an arbitrary ideal in R. From the above de nition it is clear that any element of the form

r1 a1 + : : : + rn an 2 I for any r1 ; : : : ; rn 2 R and a1 ; : : : ; an 2 I. Let P R. We use the notation h P i for the smallest ideal containing P and we call P a generating set or basis for the ideal in question.

Example 2.5 It is easily veri ed that I = a 2 Z j a is even Z is an ideal in Z. Furthermore, I = h 2 i and it can be shown that all ideals in Z can be generated

by a single element.

De nition 2.6 An ideal I of R is said to be nitely generated if there is a nite set P R such that I = h P i.

Note that there are always several dierent generating sets for one ideal. In Example 2.5 we could equally well have chosen f 2; 4 g as a generating set of I.

Ideals in Polynomial Rings

Ideals in polynomial rings appears to be a very fruitful concept in the study of systems of polynomial equations. Therefore, from now on our main interest will be in rings of polynomials and ideals in such rings.

Lemma 2.1 Let f1 ; : : : ; fs be polynomials in k [x1; : : : ; xn]. The ideal generated by these polynomials is

h f1 ; : : : ; fs i =

s

X i=1

gi fi gi 2 k [x1 ; : : : ; xn ] :

Proof The proof follows almost immediately from the de nition of an ideal, see [137]. 2 Later on we need some results concerning a special kind of polynomial ideals, so called monomial ideals. A monomial is an arbitrary product of the variables of the ring, e.g., x y2 z 2 k[x; y; z].

De nition 2.7 An ideal I in k[x1; : : : ; xn] is a monomial ideal if there is a generating set for I consisting only of monomials. Example 2.6 Consider the ideal I = h x - y; y i 2 k[x; y; z]. By de nition I consists of all polynomials f in k[x; y; z] which can be expressed as

f = g1 (x - y) + g2 y = g1 x + (g2 - g1 )y

where gi 2 k[x; y; z]. The expression after the last equality sign is by de nition an element in h x; y i, hence I h x; y i. The other inclusion can be proved similarly which shows that I = h x - y; y i = h x; y i is a monomial ideal.

14


Algebra-Geometry Connection

Above we have only considered symbolic or algebraic properties of polynomials. Next we will give a de nition which links polynomials with geometry. This link is one reason for the power of the algebraic framework for systems theory.

De nition 2.8 Let k be a eld and F k [x1; : : : ; xn] a set of multivariate polynomials. Then we de ne

V(F ) , (a1 ; : : : ; an) 2 kn j f(a1; : : : ; an) = 0; 8 f 2 F : We call V(F ) the variety de ned by F .

This is nothing else but the set of solutions of a system of polynomial equations. In applications one is often more interested in the solutions of a system of equations than the equations de ning the system. This leads us to search for a representation of the solution set of a system of equations where the de ning equations is not that central. We will see that this has a close relation to the ideal generated by the equations of the system. There is a close connection between the ideal h f1 ; : : : ; fs i and the corresponding system of polynomial equations. Given f1 ; : : : ; fs 2 k [x1 ; : : : ; xn ], we can form the following system of equations

f1 = 0 .. .

fs = 0 From these equations, one can derive others by algebraic manipulations. For example, if we multiply the rst equation by g1 , the second by g2 , etc. where gi 2 k [x1 ; : : : ; xn] and then add the resulting equations, we obtain

g1 f1 + : : : + gs fs = 0; which is a consequence of our original system. Notice that the left hand side of this equation is exactly an arbitrary element of the ideal h f1 ; : : : ; fs i according to Lemma 2.1. Thus, we can think of h f1 ; : : : ; fs i as being the set of all polynomials corresponding to those equations that can be formed by multiplying the original equations with arbitrary polynomials and then sum such products. The important observation here is that any solution of the original system is a solution to all equations derived in this way.

Lemma 2.2 The set of common zeros of the polynomials f1; : : : ; fs 2 k[x1 ; : : : ; xn] is the same as the set of common zeros of all polynomials in the ideal h f1 ; : : : ; fs i or in other words

V(ff1; : : : ; fsg) = V(h f1; : : : ; fs i):

2.2 Ideals

15

Proof By de nition any polynomial in h f1; : : : ; fs i can be written as Psi=1 gifi where gi 2 k [x1 ; : : : ; xn ] and f1 = : : : = fs = 0

)

s X i=1

gi fi =

s X i=1

gi 0 = 0:

Hence, a common zero of f1 ; : : : ; fs is a zero of any polynomial in h f1 ; : : : ; fs i. A zero which is common to all polynomials in h f1 ; : : : ; fs i is of course also a zero of the generators, f1 ; : : : ; fs and the lemma follows. 2 The connection between ideals and systems of polynomial equations will be shown to be very important. It lets us explore the inherent structure of polynomial systems with the power and strength of commutative algebra.

Properties of Ideals

Ideals can be classi ed in a number of dierent ways according to their properties. In this document we only need two such classi cations; prime and radical ideals.

De nition 2.9 The ideal I R is prime if for all a; b 2 R a b 2 I ) a 2 I or b 2 I: We have the convention that R itself is not a prime ideal. Clearly an ideal in k [x1 ; : : : ; xn] generated by a single polynomial is prime if and only if its gener-

ator is irreducible , i.e., it cannot be written as a product of two polynomials in k [x1 ; : : : ; xn]. In polynomial rings it depends on the coecient ring if an ideal is prime or not.

Example 2.7 The ideal h x2 + y2 i is prime in R [x; y] but not in C [x; y], since

x2 + y2 = (x + iy)(x - iy).

The proposition that an ideal with irreducible generators is prime only holds for ideals that can be generated by a single element.

Example 2.8 The ideal I = h x2 + 1; y2 + 1 i 2 R[x; y] is not prime, since 1 (x2 + 1) - 1 (y2 + 1) = (x + y)(x - y) 2 I but x + y 62 I and x - y 62 I. Next we introduce an operation on ideals that may give a larger ideal as result.

De nition 2.10 Let I be an ideal. The radical ideal of I is p I = a j 9m 2 N such that am 2 I :

16


p

p

It is easy to show that I is an ideal. Clearly I I for any ideal I.

Example 2.9 Consider the ideal I = h (x + 1)2 ; y3 i. The radical ideal of I is p I = h x + 1; y i: p De nition 2.11 An ideal I is a radical ideal if I = I. Notice that the operation of taking the radical of a polynomial ideal does not change p the solution set or variety of the corresponding equations, i.e., V(I) = V( I). Furthermore, a prime ideal is always radical but the reversed statement is not true in general. The following property of chains of sets is useful to ensure that algorithms terminate. The sets Ai are said to satisfy the ascending chain condition if

A1 A2 A 3 : : :

)

9i : Aj+1 = Aj for all j i:

(2.1)

De nition 2.12 A ring is Noetherian2 if its ideals satisfy the ascending chain condition. The following example of a non-Noetherian ring can be found in [48].

Example 2.10 Let R be the ring of all real-valued functions de ned on the

positive real axis:

R = f j f : R+ ! R : For every j 2 N

Ij = f j f(x) = 0; 8x 2 [0; 1j ]

is an ideal, and

I1 I2 I3 : : : is a strictly ascending, in nite chain of ideals. Hence R is not Noetherian.

Example 2.11 The rings Z; Q; R , and C can all be shown to be Noetherian. In fact, any eld, k is Noetherian since the only ideals are the so called improper ones h 0 i = f 0 g and h 1 i = k. 2

Emmy Noether (1882-1935), one of the founders of abstract algebra.

2.3 Grobner Bases

17

Theorem 2.1 The ring R is Noetherian if and only if all ideals of R are nitely generated.

Proof See [108].

2

Theorem 2.2 (Hilbert's basis theorem) If the ring R is Noetherian then so is

R [ x1 ; : : : ; x n ] .

Proof See [99].

2

This tells us that the ideals in a polynomial ring R [x1; : : : ; xn] are nitely generated if R is Noetherian. R could for example be the ring of integers Z or any eld k, e.g., Q ; R , or C .

2.3 Grobner Bases In the previous section we noted that systems of polynomial equations can be linked to algebraic objects called ideals. Here we will study a particularly useful set of generators (or basis) of an ideal called a Grobner basis, which lets us solve many important problems concerning ideals and systems of polynomial equations algorithmically. The presentation in this section leans heavily on the one in [39]. A calculation of a Grobner basis of an ideal almost immediately solves the following problems: (i) The Ideal Description Problem: Does every ideal I k [x1 ; : : : ; xn ] have a nite generating set? (ii) The Ideal Membership Problem: Given f 2 k [x1 ; : : : ; xn ] and an ideal I = h f1 ; : : : ; fs i, determine if f 2 I. (iii) The Problem of Solving Polynomial Equations: Find all common solutions in kn of a system of polynomial equations: f1 = f2 = : : : = fs = 0. (iv) The Elimination Problem: Given an ideal I 2 k [x1 ; : : : ; xn ], determine a generating set of the ideal I \ k [xr ; : : : ; xn ], where 1 r n. To understand the need for Grobner basis we will study a generalization of the division algorithm for polynomials in one variable to the multivariable case. We start this section by considering orderings of monomials which are essential to the division algorithm.

18


2.3.1 Monomial Orderings

In the division algorithm for polynomials in one variable (or indeterminate) ordering of terms , plays an important role. It is essential that the terms of the involved polynomials are written in decreasing degree for the algorithm to work properly. The ordering of terms becomes even more important when we consider the analog operations on polynomials in several variables. This is the reason for introducing an ordering, denoted >, on the monomials in k [x1 ; : : : ; xn ]. First some convenient notations.

De nition 2.13 Let 2 Nn . For monomials in k [x1; : : : ; xn] we use the notation:

x = x1 1 : : : xnn ; = mdeg(x ); jj =

n X i=1

i :

We will refer to the mapping from the monomials of k [x1 ; : : : ; xn ] to N n as the multidegree and jj as the total degree.

Example 2.12 Let f = x21x2 x33 2 k [x1; x2 ; x3; x4], then

f = x(2;1;3;0) ; mdeg(f) = (2; 1; 3; 0); j(2; 1; 3; 0)j = 6:

The multidegree de nes a one-to-one mapping between the elements of N n and the monomials in k [x1 ; : : : ; xn]. Due to this fact, every ordering > we establish on N n automatically carries over to an ordering of the monomials in k [x1 ; : : : ; xn ]. If > we will also say that x > x . However, an ordering on N n has to obey some special conditions to be a useful ordering on the monomials in k [x1 ; : : : ; xn ]. If we multiply a polynomial (with its terms written in decreasing order) with a monomial, we do not want the terms to \jump around". A critical fact in the division algorithm is that the highest ordered term still is the highest ordered after multiplication with a monomial. How to de ne the highest ordered term or the leading term of a polynomial in k [x1 ; : : : ; xn ] will be shown later. If x > x and x is any monomial we require that x x > x x . This prevents the terms from \jumping around" as mentioned above.

De nition 2.14 A monomial ordering on Nn is any relation > between elements in N n or equivalently, any relation between the monomials x; 2 N n , which satis es: (i) > is a total ordering on N n , i.e., exactly one of > , = , > should be true.

(ii) If > and 2 N n , then + > + (prevents from \jumping around").

2.3 Grobner Bases

19

(iii) > is a well ordering on N n . This means that every nonempty subset of N n has a smallest element under >. There are many ways of de ning orderings of monomials in several variables. We only consider the lexicographic order (or lex order, for short). For other orderings see, e.g., [39] or [1].

De nition 2.15 The lexicographic term ordering >lex is de ned by >lex , 9j : j > j ; 8i < j : i = i

It can be shown that this actually de nes a monomial ordering. See for example [39, page 56].

Example 2.13

x1 x22 x33 >lex x42 x53 x1 x22 x33 >lex x1 x22 x23 (1; 0; : : : ; 0) >lex : : : >lex (0; 0; : : : ; 1) , x1 >lex : : : >lex xn : (1; 2; 3) >lex (0; 4; 5) (1; 2; 3) >lex (1; 2; 2)

, ,

To decide if >lex we just compare the elements in and from left to right. The rst non-equal pair will determine the order. Sometimes the letters x; y; z are used instead of x1 ; x2 ; x3 . The above example then shows that x >lex y >lex z and the lexicographic ordering will be the same as the one used in dictionaries, which explains its name. P Given a polynomial f = a x in k [x1 ; : : : ; xn ] and a monomial ordering we can arrange the terms of f in an unambiguous way.

Example 2.14 Let f = xy2 z3 + 5y4 z5 - 3x2 . If we order the terms in decreasing order using lex order we will get f = -3x2 + xy2 z3 + 5y4 z5 .

We close this subsection by de ning a few notations concerning ordering of the terms in polynomials.

De nition 2.16 Let f = P ax 2 k [x1; : : : ; xn] n f0g and let > be a monomial

ordering. (i) The multidegree of f is mdeg(f) , max( 2 N n : a 6= 0). (ii) The leading coecient of f is lc(f) , amdeg(f) 2 k. (iii) The leading monomial of f is lm(f) , xmdeg(f) . (iv) The leading term of f is lt(f) , lc(f) lm(f).

Example 2.15 Let f = -3x2 z + xy2 z3 + 5y4 z5 , then using lex order

mdeg(f) = (2; 0; 1); lc(f) = -3; lm(f) = x2 z; lt(f) = -3x2 z:

20


2.3.2 Division in k [x1; : : : ; xn]

As a prelude to Grobner bases we introduce a generalization of the division algorithm for polynomials in a single variable to the multivariate case. We want to divide f 2 k [x1 ; : : : ; xn ] by f1 ; : : : ; fs , which analogously to the single variable case means that f can be expressed as

f = g1 f1 + : : : + gs fs + r; where the \quotients" gi and the remainder r belong to k [x1 ; : : : ; xn ]. The basic idea is the same as in k[x]: In each step of the algorithm we try to cancel the leading term of the latest achieved polynomial by multiplying some fi

by an appropriate monomial and subtract. As in the single variable case we have to order the terms in the polynomials in decreasing order using lex order.

Example 2.16 We will divide f = x2y + y2 by f1 = xy - 1 and f2 = y - 1 where

the terms of the polynomials are ordered using lex order with x > y. The divisors f1 ; f2 and the quotients g1 ; g2 are listed vertically, so we start with g1 : g2 : xy - 1 qx2 y + y2 y-1 Analogously to the single variable case we should try to cancel the leading term of f. Here this can be done in two ways, since lt(f) is divisible by both lt(f1 ) and lt(f2 ). We use f1 which gives

g1 : x g2 : r x2 y + y2 xy - 1 x2 y - x y-1 x + y2 Now, it is not possible to cancel the leading term of the partial remainder since neither lt(f1 ) = xy nor lt(f2 ) = y divides x. Despite this x + y2 can be simpli ed further since lt(f2 ) = y divides y2 . This never occurs in the single variable case since the algorithm stops when the leading term of the divisor no longer divides the leading term of the dividend. We now create a new column where we put terms that belongs to the remainder, in this case x.

g1 : x g2 : r x2 y + y 2 xy - 1 x2 y - x y-1 x + y2 y2

r

!

x

2.3 Grobner Bases

21

Now we use f2 to cancel y2 .

g1 : x g2 : r y x2 y + y2 xy - 1 x2 y - x y-1 x + y2 y2 2 y -y y

r

After the last two steps we get

g1 : x g2 : r y + 1 x2 y + y2 xy - 1 y-1 x2 y - x x + y2 y2 2 y -y y y-1 1 0

x

!

r

!

x

!

x+1

The algorithm stops when we get zero under the line. Hence f = x2 y + y can be written in the form f = x(xy - 1)+(y + 1)(y - 1)+ x + 1. Notice that the remainder is a sum of monomials, none of which is divisible by the leading terms of f1 and f2 . In the rst step above we made a choice between using f1 or f2 . This freedom will in general lead to dierent nal expressions. In this case we would have ended up with the representation f = 0(xy - 1) + (x2 + y + 1)(y - 1) + x2 + 1 of f if we had used f2 instead of f1 . We notice the important fact that the remainder is not uniquely determined as in the single variable case. The division algorithm for multivariate polynomials presented in the above example always terminates since the multidegree of the partial remainder drops in every step and according to property (iii) in De nition 2.14 this cannot go on for ever. A natural question is whether this algorithm solves the ideal membership problem? In the single variable case this is true. By repeated use of the division algorithm, a single generator of any ideal in k [x] can be found (the greatest common divisor of the original generators). A polynomial belongs to an ideal in k [x] if and only if the remainder on division by this single generator is zero (the remainder is unique in this case). Do we always get a remainder equal to zero in the multivariate case too? Clearly, if the polynomial can be written f = g1 f1 + : : : + gs fs then it belongs to the

22


ideal h f1 ; : : : ; fs i. The question is whether f 2 h f1 ; : : : ; fs i implies r = 0 when the division algorithm is used. The answer is no which we demonstrate with an example.

Example 2.17 Let f = x2y - xy2 ; f1 = y2 - 1 and f2 = xy - 1. Dividing f by f1 ; f2 , using lex order, gives the result f = -x(y2 - 1) + x(xy - 1) which implies that f 2 h f1 ; f2 i. On the other hand, by cancelling terms in another order during

the division we obtain f = (x - y)(xy - 1) + x - y where the remainder is not equal to zero and we cannot draw the above conclusion from this calculation.

The example points at a big disadvantage with the division algorithm for polynomials in several variables. It does not solve the membership problem. Since we are studying ideals one can ask whether there might be another generating set for a given ideal, such that the division algorithm always gives a zero remainder if f is a member of the ideal? The answer to this question is yes and we will see that Grobner bases solve the problem.

2.3.3 Grobner Bases

According to Hilbert's basis Theorem 2.2 every ideal I k [x1 ; : : : ; xn] has a nite basis. In this subsection we introduce an especially useful basis of a polynomial ideal called a Grobner basis. We start with a de nition, where lt(I) denotes the set of all leading terms of the polynomials in the ideal I.

De nition 2.17 A nite subset G = fg1; : : : ; gtg of an ideal I is said to be a Grobner basis if

h lt(g1 ); : : : ; lt(gt ) i = h lt(I) i: Equivalently, a set G = fg1 ; : : : ; gt g I is a Grobner basis of I if and only if the leading term of any element of I is divisible by one of the lt(gi ).

If we impose some further conditions on what we have de ned as a Grobner basis it can be shown that every ideal, except h 0 i, has an unique Grobner basis obeying these conditions.

De nition 2.18 A reduced Grobner basis of a polynomial ideal I is a Grobner basis G for I such that: (i) lc(p) = 1 8p 2 G. (ii) No monomial of p lies in h lt(G n fpg) i; 8p 2 G. Theorem 2.3 Fix a monomial ordering. Then every polynomial ideal I has a unique reduced Grobner basis.

Proof See [39].

2

2.3 Grobner Bases

23

Hence, this gives us a unique description of an ideal once a monomial ordering has been chosen. Unless otherwise is stated, we mean a reduced Grobner basis whenever talking about Grobner bases in the rest of this document. A reduced Grobner basis of an ideal I is denoted GB(I). One important property of Grobner bases is presented in the following theorem.

Theorem 2.4 Let I k [x1 ; : : : ; xn] be a polynomial ideal, GB(I) = fg1; : : : ; gtg and f 2 k [x1 ; : : : ; xn ]. Then there is a unique r 2 k [x1 ; : : : ; xn ] with the following properties: (i) No term of r is divisible by any of the lt(gi ). (ii) There is a g 2 I such that f = g + r.

Proof See [39].

2

This theorem tells us that we get the same remainder in the division algorithm irrespectively of how the elements in GB(I) are listed. Hence we can speak of the remainder r of a polynomial f modulo the ideal h f1 ; : : : ; fs i. In the previous section we saw that even if a polynomial was a member of an ideal the remainder, when using the division algorithm, did not have to be zero. Grobner bases removes this ambiguity.

De nition 2.19 We write rem(f; F) for the remainder on division of f by the stuple F = (f1 ; : : : ; fs ) using lex order. We observe that if F = GB(I), then we can regard F as a set without any particular order according to Theorem 2.4. Theorem 2.5 Let GB(I) = fg1; : : : ; gtg and f 2 k [x1; : : : ; xn]. Then f 2 I , rem(f; GB(I)) = 0: Proof See [39].

2

This theorem gives a solution to the membership problem once we have a Grobner basis of the ideal. Just compute rem(f; GB(I)) and check if it is zero.

Buchberger's Algorithm

Given a set of generators, fi of an ideal I = h f1 ; : : : ; fs i. How does one compute a Grobner basis GB(I) = fg1 ; : : : ; gt g? In the previous subsection we noted that the remainder of f on division with a sequence of polynomials (f1 ; : : : ; fs ) is not uniquely determined. This was not the case if the sequence is a Grobner basis according to Theorem 2.4. The ambiguity is introduced in the steps of the division algorithm where more than one lt(fi ) can

24


be used to cancel lt(f). This observation together with the insight that it may be possible to construct polynomials in I with lt(p) of lower order than lt(fi ) gives a hint on how to enlarge ff1 ; : : : ; fs g to a Grobner basis of h f1 ; : : : ; fs i. The key ingredient is so called S-polynomials.

De nition 2.20 Let f; g 2 k [x1; : : : ; xn] and ; ; 2 Nn , where = mdeg(f),

= mdeg(g), i = max(i ; i ). We call x the least common multiple of lm(f) and lm(g), denoted x = lcm(lm(f); lm(g)).

Example 2.18 Let f = 3x2 yz2 - xy2 and g = 2x3 y + xy. Then lm(f) = x2 yz2 and lm(g) = x3 y which gives = (2; 1; 2) and = (3; 1; 0). Hence = (3; 1; 2) so lcm(lm(f); lm(g)) = x3 yz2 = x lm(f) = lm(g) z2 . In other words, the least common multiple is the monomial of least multidegree which contains lm(f) and lm(g) as factors, not necessarily both at the same time.

De nition 2.21 Let f; g 2 k [x1 ; : : : ; xn]. The S-polynomial of f and g is S(f; g) = lcm(lmlt(f()f;)lm(g)) f - lcm(lmlt((fg);)lm(g)) g

We observe that if f; g 2 I are two generators of the ideal I then S(f; g) 2 I. The S-polynomial is constructed to cancel the leading terms of f and g. Hence it may have a leading term of lower order than f and g but not necessarily. This is maybe best understood by an example.

Example 2.19 Let f and g be the same as in the previous example. Then, 3 2

3 2

x yz (3x2 yz2 - xy2 ) - x yz (2x3 y + xy) S(f; g) = 3x 2 yz2 2x3 y 2 2 2 2 = x3 3x2 yz2 - x 3y - z2 2x3 y - xyz 2 1 1 = - 3 x2 y2 - 2 xyz2 :

Notice the cancelation of the underlined terms. The S-polynomial of f and g is constructed to produce cancelation of the leading terms of f and g. Let h = 6x3 y2 z2 +1 and divide h by (f; g). In the rst step of the division algorithm we can use either f or g to cancel lt(h)

h - 2xy f = 6x3 y2 z2 + 1 - 2xy(3x2 yz2 - xy2 ) = 2x2 y2 + 1 h - 3yz2 g = 6x3 y2 z2 + 1 - 3yz2 (2x3 y + xy) = -3xy2 z2 + 1

The ambiguity introduced in this step is

2 (h - 2xy f) - (h - 3yz2 g) = -6y( x3 f - z2 g) = -6y S(f; g):

2.3 Grobner Bases

25

It can be shown that this is the only way ambiguity can be introduced in the division algorithm. To remove this problem it is natural to include the S-polynomials (or in fact reduced versions of them) among the divisors. The S-polynomials let us decide if an ideal basis also is a Grobner basis.

Theorem 2.6 Let I = h F i be a polynomial ideal, where F = ff1; : : : ; fsg. Then F = GB(I) , rem(S(fi ; fj ); F) = 0; i =6 j: Proof See [39].

2

Example 2.20 Let I = h f1 ; f2 i, where f1 = x2 - x and f2 = x - y. We

want to decide if ff1; f2 g is a Grobner basis of I. Computing the corresponding S-polynomial gives 2

2

S(f1 ; f2 ) = xx2 (x2 - x) - xx (x - y) = xy - x

and the remainder on division by f1 and f2 is rem(S(f1 ; f2 ); ff1 ; f2 g) = y2 - y: Since we get a nonzero remainder Theorem 2.6 tells us that ff1 ; f2 g is not a Grobner basis of I. We can now describe an algorithm due to Buchberger for computing the Grobner basis of an ideal. Let I = h F i, where F = (f1 ; : : : ; fs ). Calculate the S-polynomial for all pairs fi ; fj and their remainders, rem(S(fi ; fj ); F) until there is a nonzero remainder. Add this remainder to the generating set and continue. Eventually the process stops and the generating set ful lls the conditions in Theorem 2.6. The algorithm terminates since the corresponding sequence of monomial ideals generated by the leading monomials of the successively enlarged basis forms an ascending chain. According to the ascending chain condition (2.1), this sequence of monomial ideals saturates and the algorithm terminates.

Example 2.21 We continue the previous example and calculate a Grobner basis for I. We include the remainder f3 = y2 - y in the generating set. Now ff1 ; f2 ; f3 g gives us three S-polynomials to investigate S1 = S(f1 ; f2 ) = xy - x; rem(S1 ; F) = 0; S2 = S(f1 ; f3 ) = x2 y - xy2 ; rem(S2 ; F) = 0; S3 = S(f2 ; f3 ) = xy - y3 ; rem(S3 ; F) = 0:

Hence, fx2 - x; x - y; y2 - yg is a Grobner basis of I, but it is not reduced.

26


Discussions of the computational complexity of Grobner bases can be found in for example [15, 40, 100, 103]. The next theorem solves the problem of elimination, i.e., given a system of polynomial equations, what conditions does it impose on a given subset of the variables?

Theorem 2.7 (The Elimination Theorem) Let I k [x1 ; : : : ; xn ] be an ideal and GB(I) a Grobner basis of I with respect to lex order where x1 > x2 > : : : > xn . Then for every 1 r n I \ k [xr ; : : : ; xn ] = h GB(I) \ k [xr; : : : ; xn] i: Proof See [39].

2

Example 2.22 Consider the following system of equations z2 - 5xz - xy2 + 2x2 y + 3y = 0; xy2 - 2x2 y + x2 - 2x + 1 = 0; xz2 - 5x2 z + 3xy = 0

(2.2)

and let f1 ; : : : ; f3 denote the corresponding polynomials. We compute a Grobner basis of the ideal h f1 ; f2 ; f3 i w.r.t. the order z > y > x using the function gbasis in Maple GB(h f1 ; f2 ; f3 i) = f z2 - 5zx + 3y; y2 - 2xy; x2 - 2x + 1 g: According to Lemma 2.2 the polynomials in the Grobner basis have the same common zeros as fi , which means that the following system is equivalent to (2.2).

z2 - 5xz + 3y = 0; y2 - 2xy = 0; x2 - 2x + 1 = 0:

(2.3)

The simple \triangular" structure of system (2.3) follows from Theorem 2.7 and makes it easy to solve by back substitution. We have seen that Grobner bases solve the problems we posed in the beginning of the section. The ideal description problem is solved by computing the unique reduced Grobner basis of an ideal. The ideal membership problem is solved by applying polynomial division w.r.t. a Grobner basis of the ideal. We get a zero remainder if and only if the polynomial belongs to the ideal. Solving systems of polynomial equations and elimination can be carried out according to Theorem 2.7.

3 Dierential Algebra and Characteristic Sets Many physical systems can be modeled by a set of polynomial dierential equations. We have seen that Grobner bases are an excellent tool to deal with ideals in rings of polynomials and hence systems of polynomial equations. Is it possible to treat systems of polynomial dierential equations in a similar way as in the non-dierential case? What diers is the concept of derivation, which we will see leads to a more complex theory. In this chapter we give a brief introduction to this subject, called dierential algebra. The standard references are [132] and [96] but for most of our needs the material on the subject in [47, 48] is enough. The chapter is organized as follows. In Section 3.1 we de ne the formal derivation operator, dierential rings and elds. Dierential ideals and some of their properties are treated in Section 3.2. Finally, characteristic sets and division in dierential rings are presented in Section 3.3.

3.1 Rings and Fields Two main concepts of dierential algebra are dierential rings and dierential elds, respectively. To see how they relate to the de nitions of rings and elds in commutative algebra we need to de ne a formal derivation.

De nition 3.1 Let R be a ring. A derivation is a operator @ : R ! R such that 8x; y 2 R : @(x + y) = @x + @y; @(xy) = (@x)y + x(@y): 27

28

Chapter 3 Dierential Algebra and Characteristic Sets

Example 3.1 Consider the ring R = k[x]. Then we could de ne @ ,

d dx as

the usual derivative w.r.t. x. This operator satis es De nition 3.1 and is thus a derivation. For example @(x2 + 3x) = 2x + 3 = @(x2 ) + @(3x). Another way of d. de ning a derivation on R is to consider x as a function of t and de ne @ , dt dx Then @(x2 + 3x) = 2x dx dt + 3 dt = x @x + @x x + 3@x. Notice the dierence between this formal algebraic de nition of derivation and the one in calculus. In the algebraic de nition no limit process is involved.

De nition 3.2 The elements a 2 R for which @a = 0 are called constants. Example 3.2 From the de nition of a derivation we have @1 = @(1 1) = @(1) 1 + 1 @(1) = @1 + @1 , @1 = 0: De nition 3.3 A ring R is said to be a dierential ring if there is a derivation

de ned on R and R is closed under derivation. A dierential ring can also be described as a set where we can add, subtract, multiply and dierentiate elements. The results of these operations will still be elements of the set. After renaming the derivatives, a dierential ring can be considered as a ring in in nitely many variables. Since there are in nitely many variables there is also in nite, strictly ascending chains of ideals, e.g.,

h z1 i h z1 ; z2 i h z1 ; z2 ; z3 i : : : Dierential rings are sometimes referred to as non-Noetherian.

De nition 3.4 A eld k is said to be a dierential eld if there is a derivation de ned on k and k is closed under derivation.

Example 3.3 The elds Q ; R , and C are dierential elds where all elements are

constants.

A ring of polynomials in x1 ; : : : ; xn can be extended to a dierential ring if the variables are considered as functions of some variable t and the derivation is de ned d . This dierential ring is called the ring of dierential polynomials in by @ , dt x1 ; : : : ; xn and is denoted k fx1; : : : ; xn g. In many applications the dot notation is used to denote derivations w.r.t. time, i.e.,

d x = x_ ; @2 x = d2 x = x and @ x = d x = x(); > 2: @x1 = dt 1 1 1 dt2 1 1 1 dt 1 1

3.2 Ideals

29

Example 3.4 The dierential polynomials

p1 = 3x_ 1 x2 + x33 ; p2 = x(24) x2 x23 + 2x1 + 1 are examples of elements in Q fx1 ; : : : ; xn g. Derivatives of elements in Q fx1 ; : : : ; xn g is easily calculated using the chain rule, e.g., p_ 1 = 3x1 x2 + 3x_ 1 x_ 2 + 3x23 x(33) :

As in the non-dierential case it is possible to consider extensions of dierential elds.

De nition 3.5 Let be an indeterminate not in the dierential eld k. Then the smallest dierential eld containing k [ is called the dierential eld obtained by adjoining to k and is denoted khi. Example 3.5 The dierential eld extension Q hui of the rational numbers, where u 62 Q is an indeterminate, consists of all rational functions in u and its derivatives. The dierential ring Q hui fx; yg is the set of all dierential polynomials in x and y with coecients that are quotients of dierential polynomials in u, e.g., 2u_ + 1 _ 2 _

1 + u2 xy - u x + 1 2 Q hui fx; yg:

3.2 Ideals As for rings and elds there are objects in dierential algebra corresponding to ideals called dierential ideals.

De nition 3.6 A dierential ideal is an ideal, which is closed under derivation. We now want to describe dierential ideals in a similar way as we did with ideals in the non-dierential case, namely with generating sets. Consider a set of dierential polynomials P k fx1; : : : ; xn g. We use the notation [ P ] for the smallest dierential ideal containing P and we call P a generating set for the dierential ideal in question. The dierential ideal [ P ] is formed in the same way as h P i except that we also allow derivation of elements as well. We saw above that the ring of polynomials k [x1 ; : : : ; xn ] considered as functions d , was a dierential ring, k fx1 ; : : : ; xn g. of t together with the derivation @ , dt Let P 2 k fx1 ; : : : ; xn g be a set of dierential polynomials. Then the elements of [ P ] is the same as the elements of [ P ] = h p; p;_ p ; : : : j p 2 P i 2 k [x1 ; : : : ; xn ; x_ 1 ; : : : ; x_ n ; : : : ]; i.e., a dierential ideal can be seen as a non-dierential ideal in a non-dierential ring in in nitely many variables.

30


Example 3.6 Consider the dierential ideal [ x2 ] = h x2 ; 2x @x; 2x @2x + 2(@x)2 ; : : : i: d is denoted by a dot, we get If x = x(t) and @ = dt

[ x2 ] = h x2 ; 2x x_ ; 2x x + 2(x_ )2 ; : : : i:

We can classify ideals in an analogous way of what we did in the non-dierential case.

De nition 3.7 A prime dierential ideal is a prime ideal, which is closed under derivation. We adopt the convention that a dierential ring R is not considered to be a prime dierential ideal itself. De nition 3.8 A radical dierential ideal is a radical ideal, which is closed under

derivation. In Section 2.2 we saw that to every system of polynomial equations we could associate a polynomial ideal. This allowed us to use the framework of commutative algebra to draw conclusions about for example solutions of systems, equivalent systems, and elimination problems. In the same way we now associate a dierential ideal to every system of polynomial dierential equations and hope that dierential algebra will give us answers to similar questions as posed above. So far we have not seen any major dierences between commutative and dierential algebra. Unfortunately, dierential ideals in dierential rings do not behave as \well" as ideals in non-dierential rings.

Theorem 3.1 The dierential ideals in k fx1; : : : ; xng do not satisfy the ascending chain condition.

Proof According to [91, page 45] and [122] the chain [ x2 ] = I0 I1 I2 : : : ; where

Ij = [ x2 ; (@x)2 ; (@2 x)2 ; : : : ; (@j x)2 ]; j > 0 is an in nite strictly ascending chain of dierential ideals. The dierential ideal [ x2 ] is also treated in [102] and [110].

2

3.3 Characteristic Sets

31

This theorem tells us that the dierential ring k fx1; : : : ; xn g is non-Noetherian, which leads to the fact that there are dierential ideals which are not nitely generated. We will explore this fact in Chapter 4 where we also examine other messy properties of dierential ideals, especially those ideals arising from state space descriptions of dynamic systems.

3.3 Characteristic Sets In this section we present a way of describing sets of dierential polynomials, especially dierential ideals. The method of characteristic sets, developed by J. F. Ritt1 is such a description and can be found in [132]. The material in this section mainly comes from [58], which is a good introduction to the subject. We will consider variables which are time dependent and denote derivation d . Higher order derivatives is denoted with with respect to time with a dot or dt superscripts within parentheses in the usual way, i.e., x; x_ ; x; x(3) ; x(4) ; : : :

3.3.1 Dierential Polynomials and Rankings

In dierential algebra we often consider systems of polynomial dierential equations, that is

p1 = 0; : : : ; ps = 0; where p1 ; : : : ; ps are dierential polynomials . By a dierential polynomial we mean a polynomial in a nite number of variables and their derivatives. In the framework of dierential algebra a dierential polynomial is an element of the differential ring k fx1; : : : ; xn g. In this context we sometimes use the term polynomial but still mean dierential polynomial.

Example 3.7 The following are examples of dierential polynomials in x; y; z d (x_ y2 + z) = xy2 + 2x_ yy_ + z_ ; 5x_ yz + 17 2 Q fx; y; zg: x_ y2 + z; dt

In the non-dierential case an ordering on the variables and monomials was essential to the development of algorithms such as the division algorithm. To develop a similar algorithm for dierential polynomials we also need some kind of ordering. In the dierential case the conventional term is ranking . A dierential equation is often considered more complex if it involves high order derivatives. For one variable it seems natural to rank the derivatives according to x < x_ < x < x(3) < x(4) < : : : 1

Joseph Fels Ritt (1893-1951), American mathematician.

32


If we have two variables we can choose between, e.g.,

x < y < x_ < y_ < x < y < : : : or

x < x_ < x < : : : < y < y_ < y < : : : In the algorithms we will study, it turns out that the ranking has to obey two conditions: (i) x() < x(+) ; ; 2 N

(ii) x() < y() ) x(+) < y(+) ; ; ; 2 N The reason for imposing these conditions are similar to the one in the non-dierential case. It prevents terms from \jumping around" after the derivation operator has been applied to a dierential polynomial.

Example 3.8 If we consider three variables x; y; z we have many possible rankings, e.g.,

x < y < z < x_ < y_ < z_ < x < y < z < : : :

(3.1)

x < x_ < x < : : : < y < y_ < y < : : : < z < z_ < z < : : :

(3.2)

x < x_ < x < : : : < y < z < y_ < z_ < y < z < : : :

(3.3)

In algorithms one often generates sequences of derivatives of strictly descending order. The next result is very useful when showing that such an algorithm eventually terminates.

Theorem 3.2 A sequence of derivatives, each one ranked lower than the preceding one, can only have a nite length.

Proof See [58].

2

The ranking of variables is easy to extend to a ranking of dierential polynomials.

De nition 3.9 The leader of a dierential polynomial is the highest ranked derivative. The corresponding variable is called the leading variable.


33

Example 3.9 Let the ranking be given according to (3.2) and consider the differential polynomials f = xy + z_ ; ld(f) = z_ and lv(f) = z; g = xx_ + (y )3 ; ld(g) = y and lv(g) = y; where we have introduced the notation ld and lv for the leader and leading variable, respectively.

De nition 3.10 Let f and g be two dierential polynomials. (i) If f and g has dierent leaders, the one with highest ranked leader is ranked highest. (ii) If f and g has the same leader, the one with the highest power of the leader is ranked highest.

Example 3.10 Let the ranking be given according to (3.2). Then, xy + z_ is higher ranked than xx_ + (y )3 ; since y < z_ xx_ + (y )3 is higher ranked than x2 x + y(y )2 ; since (y)3 has a higher power than (y )2 xx_ + (y )3 has the same rank as x + (y )3 , since neither of the conditions of De nition 3.10 is ful lled.

For dierential polynomials there is a result similar to Theorem 3.2.

Theorem 3.3 A sequence of dierential polynomials, each one ranked strictly lower than the preceding one, can only have a nite length.

Proof See [58].

2

Later on we want to compare two dierential polynomials in the following way.

De nition 3.11 Let f; g 2 k fx1; : : : ; xng and let the leader of g be x(k). Then f is said to be reduced with respect to g if

(i) f contains no higher derivative of xk than x(k) and

(ii) f is a polynomial of lower degree than g in x(k) .

Notation (non-standard): f / g.

34


Example 3.11 Let the ranking be given according to (3.2) and consider the same dierential polynomials as in the previous example xy + z_ / xx_ + (y)3 and xx_ + (y )3 / xy + z_ ; xx_ + (y )3 is not reduced w.r.t. x2 x + y(y )2 but x2 x + y(y )2

/

xx_ + (y )3 :

Observe that two dierential polynomials can be mutually reduced with respect to each other. Given two polynomials not reduced w.r.t. each other, we need an algorithm to carry out the reduction. This algorithm will be similar to the division algorithm in the non-dierential case with one important dierence, we also allow division by the derivatives of g.

3.3.2 Division in k fx1; : : : ; xng

The concepts from the preceding section make it possible to develop a division algorithm in k fx1 ; : : : ; xn g. In the non-dierential case the division algorithm for multivariate polynomials lets us write a polynomial f 2 k [x1 ; : : : ; xn ] on the form

f = g1 f1 + : : : + gs fs + r; where f1 ; : : : ; fs ; r 2 k [x1 ; : : : ; xn ] and r is not divisible by any of lt(f1 ); : : : ; lt(fs ). We use the notation

f = h f1 ; : : : ; fs i + r

or

f - r 0 mod h f1 ; : : : ; fs i

using the modulo operation. In the dierential case the problem becomes a bit more tricky. A corresponding representation could be

f = [ f1 ; : : : ; f s ] + r

or

f - r 0 mod [ f1 ; : : : ; fs ];

(3.4)

where r should be a remainder in some sense. We notice that we also have to allow dierentiation in the division process. The following example shows how a division algorithm in k fx1 ; : : : ; xn g can be constructed.

Example 3.12 Let the ranking be x < x_ < x < : : : < y < y_ < y < : : : and consider the dierential polynomials f = x_ (y(3) )2 + yy ; g = xy2 : We observe that f is not reduced w.r.t. g and we try to nd a representation of f in the form f = [ g ] + r. Following the non-dierential case we should somehow use g and its derivatives to eliminate the leader of f. If we dierentiate g we get a polynomial g_ = 2xyy(3) +x_ y2


35

which has the same leader as f. Now, consider f and g as polynomials in their common leader, i.e., ld(f) = ld(g) = y(3) : f = a(y(3) )2 + b; g_ = Sg y(3) + c; where a = x_ ; b = yy; Sg = 2xy and c = x_ y2 are expressions in variables ranked lower than y(3) . We use the division algorithm for polynomials in one variable to determine a remainder on division of f by g_ . This gives us

ac2 + b _ ) g + f = ( Sa y(3) - ac S2 S2 g

g

g

)

S2g f = q1 g_ + r1

(3.5)

where q1 and r1 are polynomials. We observe that according to the division algorithm for polynomials in one variable r1 will be a polynomial of lower degree in y(3) than g_ , i.e., r1 is reduced w.r.t. g_ . Carrying out the calculations we get r1 = ac2 + bS2g = x_ 3 y4 + 4x2 yy3 which has the leader y, i.e., the same leader as g. We observe that r1 is not reduced w.r.t. g. Applying the division algorithm again to r1 = dy4 + ey3 and g = Ig y 2 considered as polynomials in their common leader we get

r1 = ( Id y2 + Ie y)g + 0 g

g

)

Ig r1 = q2 g + r2 ; r2 = 0;

(3.6)

where r2 is of lower degree in y than g, i.e., r2 is reduced w.r.t. g. Combining (3.5) and (3.6) we get the expression (3.7) Ig S2g f = 1 g + 2 g_ + r; where r = 0 in our case. We see that this is almost an expression of the proposed form (3.4). The procedure to clear denominators above can be avoided if f is premultiplied by a positive power of the \leading coecient" of g and g_ considered as polynomials in their leaders. In this case S2g and Ig , respectively. This way of avoiding fractional coecients is sometimes called pseudo-division, see [112] for an extensive treatment of this algorithm. If we start with two dierential polynomials f and g, where f is not reduced w.r.t. g following the same calculations as in the example we will always get an expression

Ig Sg f = 1 g + : : : + s g(s) + r; where r is reduced w.r.t. g. We notice that 1 g + : : : + s g(s) 2 [g], which motivates the notation

Ig Sg f = [g] + r:

The expressions Ig and Sg turns out to be of such importance that we give them special names.

36


De nition 3.12 The initial Ig of a dierential polynomial g is the coecient of the highest power of the leader.

Example 3.13 For the ranking (3.2) g1 = xy2 has the leader y; hence Ig1 = x g2 = x_ y3 + y_ y3 + xy2 has the leader y; hence Ig2 = x_ + y:_

De nition 3.13 The separant Sg of a dierential polynomial g is the partial derivative of g w.r.t. the leader.

Example 3.14 For the ranking (3.2) 1 g1 = xy2 has the leader y ; hence Sg1 = @g @y = 2xy 1 2 2 g2 = x_ y3 + y_ y3 + xy2 has the leader y ; hence Sg2 = @g @y = 3x_ y + 3y_ y + 2xy :

Notice that the initial and separant are the \leading" coecients for g and g(i) considered as polynomials in their leaders.

3.3.3 Characteristic Sets

The division algorithm in k fx1 ; : : : ; xn g can be used to construct an algorithm for calculating a useful description, called a characteristic set, of a possibly in nite set of dierential polynomials. The details of the complete algorithm for constructing a characteristic set is rather involved and we will not reproduce it here. The interested reader may consult [96, 132] or [58] for a full algorithmic treatment. To de ne characteristic sets we need some auxiliary concepts.

De nition 3.14 A set of dierential polynomials, all of which are reduced with respect to each other, is called an autoreduced set. Example 3.15 Let the ranking be given according to (3.2). The polynomials xx_ + x2 ; x + xy; y2 y_ + z form an autoreduced set.


37

Next example shows an interesting fact about state space equations of dynamic systems.

Example 3.16 Let f1 ; : : : ; fn; h 2 k [x1; : : : ; xn; u] and consider x_ 1 - f1 (x1 ; : : : ; xn ; u) = 0 .. .

x_ n - fn (x1 ; : : : ; xn ; u) = 0 y - h(x1 ; : : : ; xn ; u) = 0 which is the state space equations for some system. These polynomials form an autoreduced set under the ranking u < u_ < : : : < x1 < : : : < xn < y < x_ 1 < : : : < x_ n < y_ < : : : Notice that the equations have the leaders x_ 1 ; : : : ; x_ n ; y, respectively.

Theorem 3.4 In an autoreduced set the polynomials must have dierent leading

variables.

Proof Suppose that two dierential polynomials in an autoreduced set have leaders which are derivatives of the same variable. Then we have two cases: (i) One of the leaders is a higher derivative of the leading variable than the other. Then the polynomial whose leader is of higher order is not reduced w.r.t. the other one. (ii) The leaders are of the same order. Then one of them is of equal or higher power in the leader than the other. Hence, this polynomial is not reduced w.r.t. the other one. Both cases contradicts the fact that the polynomials belong to an autoreduced set, since they exclude mutual reducedness. 2 There is a way to generalize the division algorithm in k fx1; : : : ; xng to the case when we want to calculate the remainder of a dierential polynomial w.r.t. an autoreduced set of dierential polynomials, (cf. Grobner bases). Given an autoreduced set A and a dierential polynomial f then one can show that f can be written: Y j j SAj IAj f = [ A ] + r; (3.8) Aj A where r is called the remainder of f with respect to the autoreduced set A. The remainder r is reduced with respect to the elements of A and is lower ranked than f. We now introduce a common Q notation, which allow us to write (3.8) in a more convenient way. Let HA = Aj A SAj IAj , where SAj and IAj are the separant and initial of Aj , respectively. 2

2

38


De nition 3.15 Let [ A ] : H1A = f j HsAf 2 [ A ] for some s 2 N . Example 3.17 Consider the polynomials f and g in Example 3.12 and let A = fgg. Since A only consists of one element it is autoreduced. In Example 3.12 we deduced the expression

Ig S2g f = 1 g + 2 g_ 2 [ A ]

)

I2g S2g f 2 [ A ]:

Using De nition 3.15 we can now write this as

f 2 [ A ] : H1 g: A = [ g ] : H1

Continuing the generalization of ranking, beginning with variables and polynomials we now de ne a ranking of autoreduced sets.

De nition 3.16 Let A = fA1; : : : ; Arg and B = fB1; : : : ; Bsg be two autoreduced sets of dierential polynomials, where the polynomials are ordered in increasing rank. We say that A is higher ranked than B if (i) there is a j min(r; s) such that for all i < j Ai and Bi are equally ranked and Aj is higher ranked than Bj or (ii) if s > r and for all i r Ai and Bi are equally ranked. Example 3.18 Let the ranking be given by (3.2) and consider the autoreduced

sets

A = f xx2 + x_ ; xy_ + y; z_ - x g; B = f x2 + 17; yy_ - x_ ; z - x_ g: A is ranked higher than B since the two rst polynomials in each set are equally ranked but the last polynomial in A is higher ranked than the last one in B. Let C = f x2 + x; y_ - xy g; which is an autoreduced set. Then, C is higher ranked than both A and B since the polynomials in C are equally ranked as the two rst polynomials in A and B but C has fewer elements. As for variables and dierential polynomials there is a theorem concerning niteness of descending sequences.


39

Theorem 3.5 A sequence of autoreduced sets, each one ranked strictly lower than the preceding one, can only have a nite length. Proof See [58].

2

Compare the above theorem with the ascending chain condition (2.1) and Definition 2.12. Although dierential ideals do not satisfy the ascending chain condition, chains of autoreduced sets have a similar saturating property. We next de ne the concept of characteristic sets, which is a nite description of a possibly in nite set of dierential polynomials, e.g., an ideal.

De nition 3.17 Let be a set of dierential polynomials, not necessarily nite. If A is an autoreduced set, such that no lower ranked autoreduced set can be formed in , then A is called a characteristic set of . Theorem 3.6 Every set of dierential polynomials has a characteristic set. Proof Follows from Theorem 3.5.

2

This theorem tells us that every dierential ideal, which is an in nite set of dierential polynomials has a characteristic set.

Ritt's algorithm There is an algorithm due to Ritt, see [58, 132], for nding the characteristic set of a prime dierential ideal [ ] generated by a nite set of dierential polynomials . In short the algorithm consists of extracting a characteristic set A from a nite set of dierential polynomials . Then one computes the remainders of the polynomials in w.r.t. A. The set is now extended with the nonzero remainders and one picks a new characteristic set A from . The above steps are repeated and the process continues until all elements in produces zero remainders on division by A, (cf. the calculation of remainders of S-polynomials in the Grobner basis algorithm). The algorithm terminates according to Theorem 3.5. It can be shown that if = [ ] is a dierential ideal, not necessarily prime and A is the set of dierential polynomials obtained by running Ritt's algorithm then

[ A ] [ A ] : H1 A:

(3.9)

We will examine this expression a bit closer in Chapter 4. An implementation of Ritt's algorithm is described in [57].

40


3.4 Summary We have given a brief introduction to concepts and tools in dierential algebra. Dierential rings, elds, and ideals have been treated and we have pointed at differences between the dierential and non-dierential case. Dierential rankings of variables make it possible to construct a polynomial division algorithm for differential polynomials. This algorithm is a part of Ritt's algorithm for computing characteristic sets, which is a constructive method with applications in dierential algebraic systems theory. We saw that a characteristic set is a nite representation of a possibly in nite set of dierential polynomials.

4 Input-Output Descriptions and Dierential Algebra In this chapter a few problems relating to dierential algebraic systems theory are addressed. We will point out some problems that arise when treating systems of polynomial dierential equations using the language of dierential algebra. In particular we will see what happens when we search for the input-output equations, which are possible to derive from the original system. Most of the problems are related to computational diculties, for example the existence of a nite set of dierential equations describing the system. We have mentioned that dierential rings may be considered as non-dierential rings in in nitely many variables. The problem is that most of the methods and theorems we have presented so far, e.g., Grobner bases, are limited to the case of nitely many variables. The exception is the method of characteristic sets. Dierential algebra made its way into systems theory via the discoveries by Michel Fliess in the mid eighties and has now grown quite extensive. A very good survey is given in [47]. Several authors have studied how the algorithms of dierential algebra, mostly elimination theory, can be used in systems theory. Let us here mention [42, 55, 56, 111, 124]. The results in this part of the thesis can be found in condensed form in [49]. This chapter is organized as follows. In Section 4.1 we review som basic facts from dierential algebraic systems theory. Section 4.2 consider the problem of computing the input-output dierential equation of lowest order for a system. In Section 4.3 and 4.4 we investigate the relation between the input-output equation of lowest order and the dierential input-output ideal of the system. We discover some properties of this system, which are probably rather surprising and counterintuitive to the non-expert. Section 4.5 and 4.6 explain how most of the problems 41

42

Chapter 4 Input-Output Descriptions and Dierential Algebra

that arose can be resolved. Section 4.7 describes some problems that are still open and summarizes the results in the chapter.

4.1 Representations of Dynamical Systems Consider the class of single-input, single-output (SISO) systems which can be represented by the following classical state space description: x_ 1 = f1 (x; u) .. . (4.1) x_ n = fn (x; u)

y = h(x; u); where x = (x1 ; : : : ; xn) and f1 ; : : : ; fn ; h 2 k [x; u], in other words, we consider

a set of polynomial dierential equations. This represents a rather large class of systems and it can be shown that systems where the nonlinearities are not originally polynomial may be rewritten on this polynomial form if the nonlinearities themselves are solutions to algebraic dierential equations. For more details on this we refer the reader to [103, 135]. One way of considering the state space equations is as generators of a dierential ideal. This is the motivation for the following de nition.

De nition 4.1 To a given state space description (4.1) of a system we associate a dierential ideal = [ x_ 1 - f1 (x; u); : : : ; x_ n - fn (x; u); y - h(x; u) ];

(4.2)

called the state space dierential ideal of system (4.1). We now turn our interest to which dierential equations in only y and u that belong to , i.e., which input-output equations we can derive from the state space description (4.1). Mathematically speaking this means that we look for c = \ k fu; yg, where superscript c stands for contraction [137], a mathematical operation we will not discuss here. The intersection of two dierential ideals can be shown to be a dierential ideal [96]. Thus the set c of all input-output polynomials or input-output relations , derivable from system (4.1), is a dierential ideal.

De nition 4.2 The dierential ideal c = \ k fu; yg is called the input-output

ideal of system (4.1). The input-output equations of a system are important in problems concerning, e.g., identi ability, observability [54, 60], and when one tries to decide if two systems of state equations are equivalent. To simplify notation we often let subscripts indicate dierentiation w.r.t. time for input and output variables, i.e., y0 = y, y1 = y;_ : : : ; yj = y(j) .

4.1 Representations of Dynamical Systems

43

Example 4.1 Consider the state space equations x_ 1 - x2 x1 - u = 0 x_ 2 = 0 y - x1 = 0:

(4.3)

If we dierentiate the last equation and substitute into the rst one we get

y_ - x2 y = u:

(4.4)

This is a rst order linear dierential equation with the constant coecient -x2 , since x_ 2 = 0. If we dierentiate y twice

y = x1 ; y_ = x_ 1 = x1 x2 + u; y = x_ 1 x2 + x1 x_ 2 + u_ = x1 x22 + x2 u + u_ and eliminate x1 and x2 from these equations we get yy - y_ (y_ - u) - yu_ = 0

or with the subscript notation

y0 y2 - y1 (y1 - u0 ) - y0 u1 = 0: The left hand side dierential polynomial, denoted p, is an example of an element in c, which we found by elimination. Let A; B and C denote the left hand side dierential polynomials of system (4.3). Then p can be rewritten as _ - yu_ p = yy - y_ 2 + yu _ _ - (x2 y + y_ )C_ + yC 2 [ A; B; C ] = _ + yA + x1 yB + x2 yC = -yA which again shows that p 2 . A physical interpretation of the state equations in the above example could be a model of a DC-motor where the input is an applied voltage and the output is the angular velocity of the motor shaft. Since x_ 2 = 0, we can consider x2 as a xed parameter. Equation (4.4) can be recognized as a rst order system with time constant - x12 . This corresponds to the transient behavior for zero input signal of the angular velocity modeled as a rst order system. More general, the equations (4.3) can be seen as a state space description of all systems, which is described by a linear, rst order dierential equation with constant coecients. In Example 4.1 we found a dierential polynomial of dierential order two in kfu; yg. This is the dierential polynomial in kfu; yg of lowest order that can be derived from system (4.3). It can be shown [48] that starting from the state space representation (4.1) we can always nd an element in c of the same (or possibly lower) dierential order than the number of state variables. In the next section we will see how this can be accomplished in the general case.

44


The aim of this part of the thesis is to study the input-output ideal c. Is c = [ p ], where p is the dierential polynomial of lowest order in c ? If the

answer to this question is negative, can we then compute a nite set of generators for c at all? In the non-dierential case all ideals are nitely generated according to Theorem 2.2. Theorem 3.1 tells us that dierential ideals do not have this nice property in general and a valid question is: Does c always has a nite set of generators? One may anticipate this since is always nitely generated by construction. We will use both non-dierential algorithmic tools as Grobner bases and dierential algebraic techniques such as characteristic sets from Chapter 2 and 3 to answer these questions and to reveal some of the structure of the inputoutput ideal, c.

4.2 The Lowest Order Input-Output Relation The computation of the input-output relation of lowest order for a given state space description, as in Example 4.1, can be generalized by the use of Grobner bases [48]. We will brie y describe this method. Let us rst consider the case of two state variables, x = (x1 ; x2 ): x_ 1 = f1 (x; u) x_ 2 = f2 (x; u) Dierentiating y gives

y = h(x; u):

@h + x_ @h + u_ @h : y_ = x_ 1 @x 2 @x @u 1 2

By use of the state equations the state derivatives can be eliminated giving

@h + f @h + u_ @h : y_ = f1 @x 2 @x @u 1 2 We notice that y_ is a function of x1 ; x2 ; u, and u_ . With the subscript notation @h + f2 @h + u_ @h . We now dierentiate y_ = for the output y we have y1 = f1 @x @x2 @u 1 y1 (x; y; u; u_ ) and repeat the above process 1 + x_ @y1 + u_ @y1 + u @y1 = f @y1 + f @y1 + u_ @y1 + u @y1 : y = x_ 1 @y @x1 2 @x2 @u @u_ 1 @x1 2 @x2 @u @u_ This is again an expression in x1 ; x2 ; u, and the derivatives of u. We observe that each derivation can be considered as applying the operator

N X f1 @x@ + f2 @x@ + u(i) @u(@i-1) 1 2 i=1 to the former derivative of y, where N is taken big enough. The generalization of this operator to the case with n state variables is imme-

diate.

4.2 The Lowest Order Input-Output Relation

45

De nition 4.3 The extended Lie-derivative operator is de ned by 1 n X @ @ X Lf =

i=1

fi @x + i

i=1

u(i) @u(i-1) :

The key observation now is that all derivatives of y can be obtained by repeated use of the Lie-derivative operator and that the result is a function of the state variables x, the input u, and the derivatives of u. After applying the extended Lie-derivative operator n times we have n + 1 polynomials

y0 = y = h(x; u); y1 = y_ = Lf h(x; u); : : : ; yn = y(n) = Lnf h(x; u): Using Grobner bases it is always possible to eliminate the n variables x1 ; : : : ; xn from the above n + 1 equations giving a polynomial in y; u and its derivatives, i.e.,

a dierential equation which describes the input-output behavior of the system.

De nition 4.4 Let hj = Ljf h(x; u), where x = (x1; : : : ; xn) and u = (u0 ; : : : ; uN), i.e., u and its N rst derivatives. We de ne the ideal

Oj , h y0 - h0 (x; u ); : : : ; yj - hj (x; u ) i k [y0 ; : : : ; yj ; x; u ]:

Theorem 4.1 On \ k [y0 ; : : : ; yn ; u ] contains at least one input-output relation

for (4.1).

Proof See [48].

2

To nd this input-output equation we choose an ordering with y0 ; : : : ; yn and u lower than x1 ; : : : ; xn and compute a lex Grobner basis for On . We then get

at least one input-output equation in the Grobner basis according to Theorem 2.7 and 4.1. To nd the input-output equation of lowest degree the above procedure is carried out repeatedly for Oj ; j = 1; 2; : : : terminating when the Grobner basis contains an input-output equation. It can be shown that when the process stops we get one polynomial, which is unique [48]. Note that the input-output equation is just a dependency relation over khui of h; Lf h; : : : ; Lnf h.

Example 4.2 We look at the same system as in Example 4.1. Then h0 = x1 ; h1 = x1 x2 + u0 ; h2 = x1 x22 + x2 u0 + u1 and we get

O2 = h y0 - x1 ; y1 - x1 x2 - u0 ; y2 - x1 x22 - x2 u0 - u1 i:

46


Let x2 > x1 > y2 > y1 > y0 > u1 > u0 and calculate a Grobner basis w.r.t. this ordering, which can be made in, e.g., Maple with the command gbasis.

GB(O2 ) = f-y2 - x2 y1 + u1 ; -y1 + x2 y0 + u0 ; - y0 + x1 ; y0 y2 - y21 + y1 u0 - y0 u1 g: Among the elements in the Grobner basis we recognize the last one as p = y0 y2 y21 + y1 u0 - y0 u1 .

4.3 The Simplest Description Fails Having a method to compute the input-output polynomial of lowest order, denoted p, from a classical state space description, we can pose the question: Is c = [ p ]? If the answer is negative, then how many generators do we need? Is c even nitely generated? We start our investigation by considering a particular system.

Example 4.3 Consider the same system as in Example 4.1 but without an input x_ 1 = x1 x2 ; x_ 2 = 0; y = x1 :

(4.5)

The equations correspond to the following state space dierential ideal = [ x_ 1 - x1 x2 ; x_ 2 ; y - x1 ] and the (input-)output dierential equation of lowest order in is yy - y_ 2 = 0: Using subscripts to denote derivation we let p = y0 y2 - y21 . The dierential polynomial p = y0 y2 - y21 has a special property that we will exploit. The following de nitions can be found in [96].

De nition 4.5 Let M =Pym11 y22 : : : ymm 2 k fyg be a dierential monomial. Then,

M is said to have weight i=1 i i . Notation: wt(M). De nition 4.6 A dierential polynomial p 2 k fyg is said to be isobaric of weight w if all its monomials have the same weight w. We then use the notation wt(p) = w.

Example 4.4 The dierential polynomials

g1 = y0 y1 y3 - y22 ; g2 = y1 y2 y3 - y23 + y61 ; g3 = y0 y2 - y21

are all isobaric and wt(g1 ) = 4; wt(g2 ) = 6; wt(g3 ) = 2:

4.3 The Simplest Description Fails

47

Lemma 4.1 The derivative p_ of an isobaric polynomial p is isobaric and wt(p_ ) = wt(p) + 1

1 2 m Proof Consider Pm an arbitrary dierential monomial M = y1 y2 : : : ym of weight

wt(M) = i=1 i i . Use the rule for derivation of a product and dierentiate M: M_ = (1 y11 -1 y1 +1 )y22 : : : ymm +

y11 (2 y22-1 y2 +1 ) : : : ymm + : : : + y11 y22 : : : (m ymm-1 ym +1 ):

We see that M_ is an isobaric polynomial and wt(M_ ) =

m X i=1

i i + 1 = wt(M) + 1

If a polynomial p is isobaric of weight wt(p) it consists of monomials of equal weight. The derivative of such a polynomial is a sum of derivatives of the monomials, all being isobaric polynomials of weight wt(p) + 1 and the lemma follows. 2

Lemma 4.2 Let f and g 2 k fyg be isobaric polynomials, then f g is isobaric with weight wt(f) + wt(g). Proof Consider two monomials

12 1m 21 22 2n M1 = y11 11 y12 : : : y1m and M2 = y21 y22 : : : y2n : 12 1m 21 22 2n The product M1 M2 = y11 11 y12 : : : y1m y21 y22 : : : y2n of these is again a

monomial. The weight of this polynomial is wt(M1 M2 ) =

m X i=1

1i 1i +

n X i=1

2i 2i = wt(M1 ) + wt(M2 ):

The product of two isobaric polynomials f and g is a sum of monomial products where each monomial product consists of one monomial from f with weight wt(f) and one from g with weight wt(g), hence every monomial product has weight wt(f) + wt(g) and thus the total product is isobaric with weight wt(f) + wt(g). 2 The input-output polynomial of lowest degree, y0 y2 - y21 from Example 4.3 is isobaric. Consider a dierential ideal [ p ], where p is an isobaric polynomial. The isobaric property of the generator of this principal dierential ideal1 can help us solve the membership problem for such dierential ideals, which is shown in the following theorem. We introduce the notation Pj , h p; p;_ : : : ; p(j) i k [y0 ; : : : ; ywt(p)+j ]: 1

An ideal that can be generated by a single ring element.

48


Theorem 4.2 Let [ p ] k fyg, where p is isobaric with weight wt(p) and f 2 k fyg. If the monomials of f have weight less than or equal to w and w wt(p) then f 2 [ p ] , f 2 Pw-wt(p) : If w < wt(p) then f 62 [ p ]. Proof Clearly f 2 Pw-wt(p) implies f 2 [ p ], since Pw-wt(p) [ p ]. Going the other direction we know f 2 [p]

)

f=

N X i=0

i p(i) ;

where N is big enough but nite (f is a polynomial) and i 2 kfyg. Considering each term in this sum we notice that i p(i) consists of monomials of weight equal to or greater than wt(p(i) ). From this it immediately follows that if f 2 [ p ] it cannot contain monomials of weight less than wt(p). Furthermore, the monomials in f of weight wt(p) must be equal to 00 p, where wt(00 ) = 0, since all the other terms in the sum de ning f consist of monomials of weight greater than wt(p). In the same way we conclude that the monomials in f of weight wt(p) + 1 must be _ wt(10 ) = 1; wt(11 ) = 0. Continuing in this written in the form 10 p + 11 p; way we see that the monomials of weight w in f have to be written in the form

w-wt(p);0 p + w-wt(p);1 p_ + : : : + w-wt(p);w-wt(p) p(w-wt(p)) ; where wt(w-wt(p);i ) = w - wt(p) - i; i = 1; : : : ; w - wt(p): Higher derivatives of p only contribute with monomials of weight larger than w. This means that if f 2 [ p ] and the monomials of f have weight less than or equal to w then f 2 Pw-wt(p) . 2 As already mentioned this theorem solves the membership problem in the case of dierential principal ideals with isobaric generator. We will now use this to show the main result of this section: the dierential ideal [ y0 y2 - y21 ], constructed from the system in Example 4.3, is not prime. We carry out the calculations in terms of an example.

Example 4.5

A dierential ideal, A is not prime if there exists some f 62 A and g 62 A but fg 2 A. We claim that f = y1 and g = y1 y3 - y22 are such elements for [ y0 y2 - y21 ]. The dierential polynomial f = y1 62 [ y0 y2 - y21 ] since any member of [ y0 y2 - y21 ] has monomials of at least weight 2.

4.3 The Simplest Description Fails

49

To decide if the dierential polynomial g = y1 y3 - y22 2 [ y0 y2 - y21 ] we use Theorem 4.2. Thus, we have to check if g 2 P2, i.e., if

y1 y3 - y22 2 h y0 y2 - y21 ; y0 y3 - y1 y2 ; y0 y4 - y22 i: Similarly, the question if fg 2 [ y0 y2 - y21 ] can be answered by checking if fg 2 P3 since the weight of the monomials in fg are 5 and wt(y0 y2 - y21 ) = 2.

The membership problem for non-dierential ideals was treated in Section 2.3 in Chapter 2. According to Theorem 2.5 we only have to compute the remainder of a polynomial w.r.t. a Grobner basis of an ideal to decide membership. A Maple session for doing this looks like follows:

> with(grobner): > vars:=[seq(y[i],i=0..5)]; vars := [y[0], y[1], y[2], y[3], y[4], y[5]] > P3:=[ y[0]*y[2]-y[1]^2, y[0]*y[3]-y[1]*y[2], y[0]*y[4]-y[2]^2, y[0]*y[5]+y[1]*y[4]-2*y[2]*y[3] ]: > GB:=gbasis(P3,vars): > f:=y[1]: > g:=y[1]*y[3]-y[2]^2: > normalf(f,GB,vars); y[1] > normalf(g,GB,vars); y[1] y[3] - y[0] y[4] > normalf(f*g,GB,vars); 0

Since the remainder is non-zero for f and g we conclude that neither of these polynomials lies in [ y0 y2 - y21 ]. On the other hand fg 2 [ y0 y2 - y21 ] since the remainder in this case becomes zero. In the above Maple session we have checked for membership in P3 for all polynomials, without loss of generality since P1 P2 P3. We have found an element fg 2 [ p ] but f 62 [ p ] and g 62 [ p ], which implies that [ p ] is not prime! Searching for generators of the input-output dierential ideal c , the inputoutput polynomial of lowest degree p is a natural candidate. Above we have given an example of a system whose input-output polynomial p generates a dierential ideal [ p ], which is not prime. The non-primeness of a dierential ideal describing the input-output properties of a system implies that there are input-output relations not in the ideal but whose product belongs to the ideal. Hence, there may be non-unique solutions to the corresponding dierential equations, since a solution only has to satisfy one of the relations. The \uniqueness of solutions" is often a desired property for models of physical systems. With the approach in Section 4.5 we will be able to show that and c, derived from classical state space descriptions, always are prime dierential ideals. Hence,

50


this property of c together with the observation in the above example shows there are systems for which the input-output ideal c cannot be generated by the input-output polynomial p of lowest degree only, i.e.,

[p]

(

c

in general and we have to include more generators. Can we compute a nite set of generators for c at all?

4.4 Is the Input-Output Ideal Finitely Generated? In this section we will show the somewhat surprising result that there are systems described by the nitely generated dierential ideal that have an input-output ideal c which is not (dierentially) nitely generated. This means that we need in nitely many elements in the usual description of c as a dierential ideal. In other words this particular dierential ideal corresponds to in nitely many dierential equations, which all are needed to describe the input-output behavior of the system. As in the previous section we consider the system in Example 4.3 x_ 1 = x1 x2 ; x_ 2 = 0; y0 = x1 and its dierential algebraic representation = [ x_ 1 - x1 x2 ; x_ 2 ; y0 - x1 ]: In the rest of this section c denotes the input-output ideal for this speci c system. If we dierentiate the last generator w.r.t. time and substitute x_ 1 and x_ 2 by the rst two equations or equivalently use the extended Lie-derivative operator we get y1 = x1 x2 . Repeating this a number of times results in the expression ys = x1 xs2 . This expression can now be used to determine if a polynomial f 2 k fyg is a member of c. Since the elements in c is formed by elimination in the dierential ideal , which includes the elements ys - x1 xs2 , the substitution ys 7! x1 xs2 in an element of c must yield zero. We will now study how an arbitrary element of c can be constructed.

De nition 4.7 A monomial M 2 k [y0 ; y1 ; : : : ] is said to be of type (i; j), where

i = tdeg(M) is the total degree and j = wt(M) is the weight of M. Notation (non-standard): type(M). By the notation tdeg(M) we here mean the sum of the powers of all variables in M.

Example 4.6 Let M = y0 y23 y4 then

tdeg(M) = 4; wt(M) = 10 , type(M) = (4; 10):

4.4 Is the Input-Output Ideal Finitely Generated?

51

Lemma 4.3 Let M 2 k [y0 ; y1 ; : : : ] be a monomial of type(M) = (i; j). Then M 7! xi1 xj2 under the substitution rule ys 7! x1 xs2 . 1 2 m Proof P The monomial Pm M can be written as M = y1 y2 : : : ym and stype(M) = m (i; j) = ( i=1 i ; i=1 i i ). We perform the substitution ys ! 7 x1 x2 in M and

get

Pm Pm M 7! (x1 x2 1 )1 : : : (x1 x2 m )m = x1 i=1 i x2 i=1 i i = xi1 xj2 ;

which proves the lemma.

2

The lemma tells us that monomials of the same type in k [y0 ; y1 ; : : : ] will be reduced to the same monomial in k [x1 ; x2 ] under the substitution ys 7! x1 xs2 . Now let f 2 c . A consequence of the discussion just before De nition 4.7 and of Lemma 4.3 is that the coecients of monomials of same type in f have to sum up to zero.

Example 4.7 Consider the following dierential polynomial

f = y0 y2 - y21 + y20 y6 - 21 y1 y2 y3 - 12 y21 y4 ;

where the rst two monomials are of type (2,2) and the three last monomials are of type (3,6). The substitution ys 7! x1 xs2 gives Hence, f 2 c .

f = (1 - 1)x21 x22 + (1 - 12 - 12 )x31 x62 = 0:

We now generalize these ideas. Every polynomial in k [y0 ; y1 ; : : : ] can be divided into homogeneous polynomials, i.e., polynomials whose monomials are of the same total degree. Furthermore, every polynomial can be divided into isobaric polynomials. Combining these two observations implies that we can always obtain the following decomposition of a polynomial P 2 k [y0 ; y1 ; : : : ]

P=

N X i=0

Pi =

Li N X X Pj ; i=0 j=0

i

where Pij is a homogeneous and isobaric polynomial, i.e., all its monomials are of type (i; j). N is the largest total degree of any monomial in P and Li is the highest weight of any monomial in Pi. An explicit expression for Pij is

Pij =

Sij X ij Mij ;

s=1

s

s

52


where Mijs are monomials of type (i; j), Sij are the number of monomials in Pij and ijs 2 k. From the above discussion we know that P 2 c if and only if the coecients of the monomials of P that are of the same type sums up to zero, i.e., Sij X ij = 0;

s=1

s

8i; j:

Now concentrate on the pieces Pij in which we can decompose P if P 2 c . From which monomials can then Pij be constructed? We consider a number of dierent cases. type(Mijs ) = (1; j) Then Mijs = yj and hence there is only one monomial of each type. Thus, the coecient of the monomial of each type have to be zero if P 2 c , i.e., P contains no monomials of type (1; j). type(Mijs ) = (2; 1) Then Mijs = y0 y1 and again this monomial cannot appear in a P 2 c . type(Mijs ) = (2; j) If j 2 then we have the monomials y0 yj ; y1 yj-1 ; : : : ; y2j=2 if j is even and y1 yj ; y1 yj-1 ; : : : ; y j-21 y j+21 if j is odd.

We consider the case when j is even. Then any P2j 2 c can be written

P2j =

j=2 X

s=0

s ys yj-s = 0 y0 yj + 1 y1 yj-1 + : : : + j=2 y2j=2

= -1 (y0 yj - y1 yj-1 ) - 2 (y0 yj - y2 yj-2 ) - : : : - j=2 (y0 yj - y2j=2 ) +

j=2 X

s y0 yj s=0 = -1 (y0 yj - y1 yj-1 ) - 2 (y0 yj - y2 yj-2 ) - : : : - j=2 (y0 yj - y2j=2 ); where the last equality follows from the assumption Pij 2 c, since the s then have to sum up to zero. Hence, we observe that P2j can be expressed as a linear combination of the isobaric polynomials q2j s , y0 yj - ys yj-s , i.e., P2j =

j=2 X

s=0

-s q2js ; q2js = y0 yj - ys yj-s :

In the same way we construct isobaric polynomials q3j s ; q4j s ; : : : such that Pij can be written as a linear combination of qijs . These polynomials are given by

qijs , y0i-1 yj - yl1 yl2 : : : yli ;

i X r=1

lr = j;


53

where ylr are written in ascending order, i.e., l1 : : : li .

Example 4.8 Here we list q35 s . y0 y0 y5 - y0 y1 y4 ; y0 y0 y5 - y0 y2 y3 ; y0 y0 y5 - y1 y1 y3 ; y0 y0 y5 - y1 y2 y2 The part of a polynomial, P 2 c that consists of monomials of type (3; 5), i.e., P35 can be written as a linear combination of these polynomials. We summarize the above discussion in the following theorem, which gives us a useful representation of the elements in c.

Theorem 4.3 Let Pij be a linear combination of the isobaric polynomials qijs = y0i-1 yj - yl1 yl2 : : : yli :

where the index s is an enumeration of the index sets fl1 ; : : : ; li g satisfying Pi l r=1 r = j; 0 l1 : : : li j. Then

P 2 c

,

P=

Li N X X Pj ; i=2 j=2

i

where N and Li are de ned as before.

Note that the polynomials qijs can be considered as a basis of a nite dimensional vector space with coecients ijs as coordinates. In Chapter 3 we mentioned that dierential ideals are not always nitely generated. Though, there are some ideals that are generated by an in nite set of polynomials but still can be generated by a nite number of polynomials.

Lemma 4.4 Consider the following set of polynomials

S = f 2 k fyg j f =

N X i=1

i fi ; i 2 k; f1 ; : : : ; fN 2 k fyg :

Then,

[ S ] = [ f 1 ; : : : ; f N ]:

Proof An element h in [ S ] can be written as h=

X 0 X 1 0 X f + f + : : : + M f(M) ;

f

f

f

f 2 S ; jf 2 k fyg;

54


where each sum only ranges over nitely many f 2 S since h has a nite number of terms. Using the representation of an element in S we can rewrite the above sum according to

h=

N N N X X X (M)

0i fi + 1i fi0 + : : : + M i fi ; i=1

i=1

i=1

i.e., by de nition an element in [ f1 ; : : : ; fN ]. Hence, [ S ] [ f1 ; : : : ; fN ]. The other inclusion is trivial since ff1 ; : : : ; fN g [ S ] and the theorem is proved.

2

The above lemma tells us that if we take the set of all linear combinations (over k) of a nite number of polynomials, then the dierential ideal generated by this (in nite) set is equal to the ideal generated by the original nitely many polynomials. We can now attack the question: Is c (dierentially) nitely generated? The answer is negative despite the fact that is generated by a nite number of polynomials (the state equations).

Theorem 4.4 There are systems whose input-output ideal c is not (dierentially) nitely generated.

Proof The proof consists of an example of a system whose input-output ideal is not nitely generated. We consider the following system and its corresponding input-output ideal = [ x_ 1 - x1 x2 ; x_ 2 ; y - x1 ];

c = \ k fyg;

which is the same system as in Example 4.3. We also introduce the notation

cwt L , P 2 c j The monomials of P have weight L : Observe that cwt L is not a dierential ideal, since it is not closed under derivation. We will prove that c is not nitely generated by contradiction. Suppose that c is nitely generated. This means that there are polynomials g1 ; : : : ; gK 2 c such that c = [ g1 ; : : : ; gK ]. Since K is nite there is a mono

mial in one of the generators which has higher or equal weight than all the other monomials of the generators. This implies that fg1 ; : : : ; gK g 2 cwt L if L is chosen big enough. Fix such an L. Now,

fg1 ;

: : : ; gK g cwt L

)

[ cwt L ] = c:


55

If we can show that there exists a polynomial in c that is not in [ cwt L ] we have the sought contradiction. To succeed with this task we utilize the special representation of the polynomials in c derived in Theorem 4.3. We claim that

fL+1 , y L-2 1 y L+2 3 - y2L+2 1 2 c but fL+1 62 [ cwt L ]; where L is an odd number, i.e., fL+1 is the kind of polynomial we are searching for. That fL+1 2 c holds is easy to see since fL+1 maps to zero under the substitution ys 7! x1 xs2 . By Theorem 4.3 we know that an element in cwt L always can be written on the P form P = M r=1 r f~r since

Sij Li Li X N X N X M X X X j ij ij P= Pi = s qs = r f~r ; i=2 j=2

i=2 j=2 s=1 r=1 where r and f~r are new names for all the ijs and qijs , respectively. According to Lemma 4.4 we have [ cwt L ] = [ f~1 ; : : : ; f~M ], i.e., [ cwt L ] is nitely generated by the isobaric polynomials f~r = qijs where wt(f~r ) maxi (Li ).

Hence,

[ cwt L ] = [ y0 y2 - y21 ; y0 y3 - y1 y2 ; : : : ; y0 yL - y1 yL-1 ; : : : ; y0 yL - y L-2 1 y L+2 1 ; y20 y2 - y0 y21 ; y20 y3 - y0 y1 y2 ; y0 y3 - y31 ; : : : ];

where the last dots denote a nite number of generators of weight less than or equal to L. We observe that the total degree of a generator remains constant under derivation. This means that if fL+1 2 [ cwt L ] then it has to be constructed of generators of total degree two and their derivatives. Since fL+1 is an isobaric polynomial of weight L + 1 and all the generators are isobaric, fL+1 has to be constructed of generators of weight L+1. The only polynomials which can be used to construct fL+1 are the rst derivative of the generators of weight L, i.e.,

d dt (y0 yL - yj yL-j ) = y0 yL+1 + y1 yL - yj yL-j+1 - yj+1 yL-j ; where j = 1; 2; : : : ; L-2 1 and L is odd. The polynomial fL+1 has to be a linear combination over k of these derivatives since this is the only construction that

56


keeps the weight and total degree constant. We now try to express fL+1 as a linear combination of these derivatives, i.e.,

y L-2 1 y L+2 3 - y2L+2 1 = 1 (y0 yL+1 + y1 yL - y1 yL - y2 yL-1 ) + 2 (y0 yL+1 + y1 yL - y2 yL-1 - y3 yL-2 ) + 3 (y0 yL+1 + y1 yL - y3 yL-2 - y4 yL-3 ) + : : : + L-2 1 (y0 yL+1 + y1 yL - y L-2 1 y L+2 3 - y2L+2 1 )

We now introduce the notation

z0 = y0 yL+1 + y1 yL ; z1 = y1 yL ; z2 = y2 yL-1 ; : : : ; z L+2 1 = y2L+2 1 to simplify solving for i in the above expression. These new variables are linearly independent over k. Consider the resulting system z L-2 1 - z L+2 1 = 1 (z0 - z1 - z2 ) + 2 (z0 - z2 - z3 ) + : : : + L-2 1 (z0 - z L-2 1 - z L+2 1 ) Since z1 appears only in the rst parenthesis and not on the left hand side of the expression, 1 = 0. If 1 = 0, then z2 appears only in the second parenthesis hence

2 = 0. Continuing in this way we get 1 = : : : = L-2 3 = 0 and L-2 1 can not be determined to get a valid equality. Hence, fL+1 62 [ cwt L ] and we have a contradiction. This proves the theorem. 2

The conclusions to be drawn from the results in this section is that the representation of the input-output ideal is a non-trivial question. Since some systems do not have a nite set of generators for their input-output ideals the input-output behavior cannot explicitely be represented on this form in, e.g., a computer program. Compare with the situation in commutative algebra where the ideals behave much nicer. Then all ideals are nitely generated, even those that are constructed by intersection, see Theorem 2.7.

4.5 A Non-Dierential Approach To be able to apply some non-dierential algebraic results, such as Grobner bases, to state space dierential ideals in dierential rings, we have to develop some preliminary results concerning rings of in nitely many variables. As pointed out in Chapter 3 a dierential ring can be considered as a ring in in nitely many variables. In order to use non-dierential tools it is useful to consider the subset of a dierential ideal which consists of all elements up to a speci c dierential order. This subset can be seen as an ideal in a non-dierential ring and will be called a truncation of the dierential ideal. To compute truncations we introduce the concept of graph ideals.

4.5 A Non-Dierential Approach

57

Graph Ideals By a graph of a mapping one usually means the set of points de ned by the expression y = f(x); x 2 R or equivalently (x; y) j y - f(x) = 0; x 2 R . If we allow more variables and equations we can generalize the meaning of a graph to be the point set (x; z1 ; : : : ; zj ) j z1 - f1 (x) = 0; : : : ; zj - fj (x) = 0; x 2 Rn . In Chapter 2 we saw that there is a correspondence between a set of polynomial equations and the ideal generated by the left hand sides of these equations. This motivates the name of the kind of ideal we de ne now.

De nition 4.8 A graph ideal is an ideal I of the form I = h z1 - f1 ; : : : ; zs - fs i; where zi are variables and fi 2 k [x1 ; : : : ; xn ]. The aim now is to study relations of dependency between f1 ; : : : ; fs , which are supposed to be known polynomials. By dependency we mean algebraic dependency . The polynomials f1 ; : : : ; fm 2 k [x1 ; : : : ; xn ] are said to be algebraically dependent if there is a non-zero polynomial P 2 k [w1 ; : : : ; wm ] such that P(f1; : : : ; fm ) = 0. We recall Theorem 2.7, which tells us how to eliminate the xi . If we compute a Grobner basis of the graph ideal w.r.t. the ordering

x1 > : : : > x n > z1 > : : : > z s ; then we immediately get a generating set for the ideal I \ k [z1 ; : : : ; zs ]. From the discussion of ideals we know that this ideal is the subset of all polynomials in the variables z1 ; : : : ; zs only, which can be \formed" by the polynomials that generate I.

Example 4.9 Consider the graph ideal I = h z1 - x1 + x2 ; z2 - (x1 - x2 )2 + x1 - x2 i; where f1 = x1 - x2 and f2 = x21 - 2x1 x2 + x22 - x1 + x2 . If we calculate a Grobner basis for I w.r.t. the ordering x2 > x1 > z2 > z1 we get GB(I) = fz1 - x1 + x2 ; z2 - z21 + z1 g: According to Theorem 2.7, fz2 - z21 + z1 g is a Grobner basis of I \ Q [z1 ; z2 ], that is I \ Q [z1 ; z2 ] = h z2 - z21 + z1 i. Hence, we have the dependency relation

f2 - f21 + f1 = 0

between f1 and f2 .

58


We will now consider chains of graph ideals and will therefore need a notation for rings of variable size. Let

Rj , k [x1 ; : : : ; xn ; z1 ; : : : ; zj ]; (where n is xed) and

Gj , fz1 - f1 ; : : : ; zj - fj g; where as before zi are variables and f1 ; : : : ; fj 2 k [x1 ; : : : ; xn ].

De nition 4.9 Let Ij;N be the graph ideal generated by Gj in RN, i.e., X Ij;N = h z1 - f1 ; : : : ; zj - fj i = f 2 RN j f = p : RN p Gj 2 2

The ideals Ij;N ; 1 j N form a strictly ascending chain of ideals

I1;N I2;N : : : IN;N ;

since a new variable is introduced in each new ideal of the chain.

Truncations of Graph Ideals

The concept of graph ideals is very useful when considering dierential ideals like (4.2), i.e., dierential ideals generated by the state space equations. These dierential ideals can be considered as graph ideals of a ring in in nitely many variables. First we present a preliminary lemma.

Lemma 4.5 Let p 2 RN and f1; : : : ; fj 2 k [x1 ; : : : ; xn], then the substitution zi ! 7 fi ; 1 i j maps p 7! 0 if and only if p 2 Ij;N , i.e., zi ! 7 fi ) p ! 7 0 , p 2 Ij;N : Proof Consider a polynomial h 2 k [x] in one variable. Using the binomial theorem it is easy to see that h(x + c) = h(c) + g(x; c) x = h(c) + q(x); for some polynomial g 2 k [x] or q 2 h x i. This is easily generalized to the multivariate case, i.e., if h 2 k [w1 ; : : : ; wm ] then h(w + c) = h(c) + where h(c) do not contain any wi . Hence

m X i=1

h(w + c) 2 h w1 ; : : : ; wm i

gi (w; c) wi ; ,

h(c) = 0:

4.5 A Non-Dierential Approach

59

Now, take a polynomial p 2 RN and observe that the identity zi = (zi - fi ) + fi is an expression of the above form. Suppressing the x1 ; : : : ; xn dependence we can rewrite p as follows:

p(z1 ; : : : ; zN ) = p((z1 - f1 ) + f1 ; : : : ; (zj - fj ) + fj ; zj+1 ; : : : ; zN ) = p(f1 ; : : : ; fj ; zj+1 ; : : : ; zN ) + q(z1 ; : : : ; zN ); where q 2 h z1 - f1 ; : : : ; zj - fj i RN . As above we have p(z1 ; : : : ; zN ) 2 h z1 - f1 ; : : : ; zj - fj i , p(f1 ; : : : ; fj ; zj+1 ; : : : ; zN ) = 0 2

and the lemma follows.

The following lemma applied to graph ideals in in nitely many variables projects the problems onto problems in a nite number of variables. Hence tools from commutative algebra as Grobner bases can be utilized.

Lemma 4.6 Let Ij;N be the graph ideal de ned in De nition 4.9, then (i) (ii) (iii)

Ij;N is a prime ideal. Ij;N \ k [x1 ; : : : ; xn ] = f0g. For all j; s; N where j s N we have that IN;N \ k [z1 ; : : : ; zj ] = Is;N \ k [z1 ; : : : ; zj ]:

Proof We start by showing that Ij;N is a prime ideal. Let p 2 RN and p~ denote that the substitution zi 7! fi ; 1 i j has been performed. According to Lemma 4.5 we have

p 2 Ij;N

,

p~ = 0:

Now, suppose that f g 2 Ij;N . Then this is equivalent to

f~g~ = 0: Both f~ and g~ are polynomials in the ring k [x1 ; : : : ; xn ; zj+1 ; : : : ; zN ], since fi 2 k [x1 ; : : : ; xn]. The product of two polynomials cannot be zero unless one of them is zero. This forces at least one of f~ or g~ to be zero, i.e., at least one of f or g belongs to Ij;N according to Lemma 4.5. Hence, we have that f g 2 Ij;N ) f 2 Ij;N _ g 2 Ij;N ; which is the de nition of that Ij;N is prime and we have shown (i).

60


Suppose that p 2 Ij;N . We know that this is equivalent to p~ = 0. On the other hand if p 2 k [x1 ; : : : ; xn ] then p = p~. Hence, if p 2 Ij;N \ k [x1 ; : : : ; xn ] then p = 0 and (ii) follows. To show (iii) we observe that Is;N IN;N , since Ik;N ; 1 k N is an ascending chain of ideals. Hence

Is;N \ k [z1 ; : : : ; zj ] IN;N \ k [z1 ; : : : ; zj ]: Now suppose p 2 IN;N . Then p(x1 ; : : : ; xn ; f1 ; : : : ; fN ) = 0 according to Lemma 4.5. If in addition p 2 k [z1 ; : : : ; zj ] then

p(x1 ; : : : ; xn ; f1 ; : : : ; fj ; zj+1 ; : : : ; zN ) = p(x1 ; : : : ; xn ; f1 ; : : : ; fN ) since p is free of zj+1 ; : : : ; zN . Hence p(x1 ; : : : ; xn ; f1 ; : : : ; fj ; zj+1 ; : : : ; zN ) = 0, i.e., p 2 Ij;N Is;N . This shows that

Is;N \ k [z1 ; : : : ; zj ] IN;N \ k [z1 ; : : : ; zj ] and the lemma is proved.

2

In words (ii) means that there is no polynomial in any graph ideal in the variables x1 ; : : : ; xn only. Number (iii) is even more useful. It tells us that if we are interested in relations (i.e., polynomial equations) between z1 ; : : : ; zj or equivalently between f1 ; : : : ; fj we do not have to care about zs+1 ; : : : ; zN or equivalently fs+1 ; : : : ; fN , where s j.

The Relation Between c and Oj

How much information does one get about the truncated dierential ideal c \ k [y0 ; : : : ; yj ; u ] by considering truncations of the graph ideal Oj which was introduced in Section 4.2?

Theorem 4.5 c \ k [y0 ; : : : ; yj ] = Oj \ k [y0 ; : : : ; yj] Proof By changing our point of view we know that we can consider a dierential ring as a ring of in nitely many variables, i.e.,

k fx1 ; : : : ; xn ; yg ! k [x1 ; : : : ; xn ; y; x_ 1 ; : : : ; x_ n ; y;_ : : : ]; where the superscripts together with the subscripts are now nothing more than labels for the variables. A nitely generated dierential ideal as [ A; B; C ] then be_ : : : ; B; B; _ : : : ; C; C; _ : : : i, i.e., an ideal generated by an in nite number comes h A; A; of generators.

4.6 Description of c with Characteristic Sets

61

Consider the state space dierential ideal = [ x_ 1 - f1 ; : : : ; x_ n - fn ; y - h ]. If we now rename the variables according to

z0 = y0 ; : : : ; z j = yj ; zj+1 = x_ 1 ; : : : ; zj+n = x_ n ; zj+1+n = yj+1 ; zj+2+n = x1 ; : : : ; zj+1+2n = xn ; zj+2+2n = yj+2 ; : : : we get the ideal

h z0 - h0 ; : : : ; zj - hj ; zj+1 - f1 ; : : : ; zj+n - fn ; : : : i This is a graph ideal in in nitely many zi and x1 ; : : : ; xn . Furthermore hi is de ned as in De nition 4.4. Since the proofs of Lemma 4.5 and 4.6 are completely independent of N, they hold even for N = 1. With the non-dierential description of it now follows from (iii) of Lemma 4.6 that

h z0 - h0 ; : : : ; zj - hj ; zj+1 - f1 ; : : : i \ k [z0 ; : : : ; zj ] = h z0 - h0 ; : : : ; zj - hj i \ k [z0 ; : : : ; zj ] or

\ k [y0 ; : : : ; yj ] = h y0 - h0 ; : : : ; yj - hj i \ k [y0 ; : : : ; yj ] that is

\ k [y0 ; : : : ; yj ] = Oj \ k [y0 ; : : : ; yj ] which completes the proof.

2

Observe that the input u does not appear anywhere in the proof. The theorem holds equally well if we adjoin u and its derivatives to k, i.e., if we let khui be the eld of coecients instead of k, see De nition 3.5. The above theorem tells us that all elements in c up to order j are the same as those elements in Oj which contain only yi ; 0 i j. The main advantage is that Oj is a non-dierential ideal which can be treated with methods from commutative algebra presented in preceding chapters. Observe the interpretation of as a graph ideal in the above proof. This tells us that is prime, a property we utilized to show that c 6= [ p ] in Section 4.3.


In Section 4.2 we showed that if p is a polynomial of lowest order in c then in general c 6= [ p ]. An interesting question now is: Are there cases when we have equality and in that case which conditions do we then have to pose on p?

62


Consider a system described by state space equations of the form (4.1) but without an input, i.e., u = 0. These equations correspond to the dierential ideal

= [x_ 1 - f1 (x); : : : ; x_ n - fn (x); y - h(x)] kfx1 ; : : : ; xn ; yg (4.6) As before we want to know which dierential equations the output y satis es, i.e., we are interested in c = \ k fyg. We are especially interested in the dierential equation of lowest order p = 0 that y satis es. The dierential polynomial p can be found using the method in [48] reviewed in Section 4.2. The polynomial p has some interesting properties. Since p 2 is the polynomial of lowest order in the single variable y in it is also of lowest order in c. Now c is a set of dierential polynomials in one variable and since p is a polynomial of lowest order it has to be a characteristic set of c (two dierential

polynomials in one variable cannot be reduced w.r.t. each other so the characteristic set has only one element).

Theorem 4.6 If A is a characteristic set of a prime dierential ideal I then I = [ A ] : H1 A: Proof See [96].

2

From the preceding sections we know that can be interpreted as a graph ideal and hence is a prime dierential ideal. That is prime implies that c is prime. Furthermore, we have a characteristic set fpg of the prime dierential ideal c and consequently the theorem gives

c = [ p ] : H1 p:

(4.7)

[ p ] c

(4.8)

We notice that since p 2 c we have

and from the expressions (4.7) and (4.8) we get the following limits on c :

[ p ] c = [ p ] : H1 (4.9) p: We will now examine the relationship between [ p ] and the dierential ideal [ p ] : H1 p. If f 2 [ p ] then f = f H0p 2 [ p ] which shows that [ p ] [ p ] : H1 (4.10) p: Suppose that [ p ] is prime and that fpg is a characteristic set of [ p ]. Let f 2 [ p ] : H1 p , then fHrp 2 [ p ] for some r. Since [ p ] is prime either f or Hrp has to be in [ p ]. According to the de nition of Hp it has lower order than p. Since fpg is a characteristic set of [ p ] there are no element of lower order than p in [ p ], hence Hp 62 [ p ]. The product H2p = Hp Hp 62 [ p ], since [ p ] is prime and none of the


63

factors of H2p are in [ p ]. By induction one realizes that Hrp 62 [ p ]. Thus we have a product fHrp 2 [ p ], where the factor Hrp 62 [ p ]. Since [ p ] is prime then f has to be in [ p ]. Thus we have showed that if p is the characteristic set of the prime dierential ideal [ p ] then

[ p ] : H1 p [ p ]:

(4.11)

Combining the inclusions (4.10) and (4.11) gives the equality

[ p ] = [ p ] : H1 p;

(4.12)

where p is the characteristic set of the prime dierential ideal [ p ]. This discussion is in fact a proof of Theorem 4.6 in the case when the characteristic set consists of only one element. We now observe the fact that since fpg is a characteristic set of c it is also a characteristic set of [ p ], since [ p ] c . Notice that if c = [ p ] then [ p ] has to be prime since c is prime. The discussion above is summed up in the following theorem:

Theorem 4.7 If p 2 k fyg is the polynomial of lowest order in c then [ p ] is prime

,

c = [ p ]:

Despite this characterization of c the theorem can be of limited use, since it can be a very dicult task to decide if [ p ] is prime or not. However, there is one case when it is easy, namely if the initial Ip and separant Sp of p belong to k. Then Hp 2 k and [ p ] = [ p ] : Hp , which is prime. We also note that c can be nitely generated without being equal to [ p ].

Example 4.10 The dierential ideal [y1 2 - y0 ] has a simple interpretation in

mechanics and it has also been discussed by Pommaret [129]. Dierentiating the generator once we obtain the dierential equation 2 y1 y2 - y1 = 0 which has two solutions y1 = 0 and y2 = 21 . The last solution corresponds to the following state equations

x_ = 12 ; y = x2 ;

(4.13)

where x could be the velocity of an object and y its kinetic energy. One can show that the (input-)output ideal c of (4.13) is 2 [ y1 2 - y0 ] : y1 1 = [ y1 - y0 ; 2y2 - 1 ]:

64


4.7 Conclusions and Extensions We have considered systems described by polynomial dierential equations, i.e., x_ 1 - f1 (x; u) = 0 .. . (4.14) x_ n - fn (x; u) = 0

y - h(x; u) = 0; where x = x1 ; : : : ; xn are the state variables, u is the input, y is the output, and f1 ; : : : ; fn are polynomials. In the language of dierential algebra we can formulate

this system description as a dierential ideal generated by the left hand sides of (4.14), i.e., = [ x_ 1 - f1 ; : : : ; x_ n - fn ; y - h ]: We have studied the set c = \ k fu; yg, i.e., all input-output dierential equations that the systems input and output variables have to satisfy. In particular we have considered the case of no input, that is k fx; yg and c k fyg, which corresponds to the transient behavior of the system. There is a method for obtaining the unique input-output equation of lowest order of the system represented by (4.14). Denoting the corresponding dierential polynomial by p the results can be summarized as follows: (i) c 6= [ p ] in general. (ii) c might not even be generated by a nite number of dierential polynomials. (iii) c = [ p ] : H1 p in general and c = [ p ] if and only if [ p ] is prime. (iv) c is always prime but [ p ] does not have to be prime. These results were mainly obtained by considering a parameterized version of a rst order system, namely x_ 1 - x1 x2 = 0; x_ 2 = 0; y - x1 = 0 (4.15) so they are not anomalies from some non-physical system. These results show that despite the fact that the structure of the original system of polynomial dierential equations is captured in the nitely generated dierential ideal , we do not always have a simple dierential algebraic description of the input-output ideal c . This means that one has to be careful in the study of such ideals and that the ideal description problem for them is highly non-trivial. It is also an indication that it could be more convenient to consider a system as described by solutions (time-trajectories) rather than equations. This is one of the ideas in the behavioral framework of Willems, see e.g., [157]. There is also a close analogy with the duality between algebraic geometry and commutative algebra, i.e., zero manifolds vs. equations, as in the spirit of [111].

4.7 Conclusions and Extensions

Extensions

65

The results were obtained by examining a system written in polynomial state space form without an input. It would be interesting to examine the possibility to construct an example with nonzero input, which has similar consequences as the one treated. The input-output dierential ideal c 2 k fu; yg is then a dierential ideal in two variables. How \badly" can the ideal [ p ] behave? Are there examples of systems, where [ p ] is not even a radical dierential ideal? If this is the case, then there are elements in the ideal with the property f 62 [ p ] but fr 2 [ p ] for some r. Translated into dierential equations this means that [ p ] only contains the dierential equation fr = 0; r > 1 and not the simpler description f = 0. A candidate of a non-radical ideal is P = [ y1 2 - y0 ]. It can be shown that y3 2 2 P. To show that P is not radical, one have to show that y3 62 P. The dierential ideal [ y0 y2 - y1 ] considered as an ideal in in nitely many variables has some similarities with a graph ideal. Is it prime? If this is the case we have an example of a prime dierential ideal generated by a single dierential polynomial whose initial and separant are nontrivial. To use the method of Grobner bases in dierential algebraic calculations we had to interpret the dierential algebraic problem in algebraic terms. This interpretation is possible due to the fact that we found a restriction of the problem to a ring of nitely many variables. In the general case this is not trivial or even possible, since there are dierential ideals, which is not (dierentially) nitely generated as was pointed out in Chapter 3. An interesting subject is the concept of dierential Grobner bases. Some references on this topic are [45, 106, 107, 123].

66


Part II

Quanti er Elimination in Control Theory

67

5 Real Algebra and Cylindrical Algebraic Decomposition The formulation of problems in terms of real numbers have an advantage over the often implicit assumption that the variables are complex. We can use not only equations to describe objects but also inequalities to form so called real polynomial systems , i.e., systems of polynomial equations and inequalities. The set of solutions to such systems has a much more descriptive power than the solution sets of systems of polynomial equations. Another advantage is the possibility of visualizing objects described in this way. It is easier to imagine or draw pictures of shapes and forms when all coordinates are real numbers. However, the price we have to pay seems to be more complex algorithms. Real algebra, roughly speaking, is the study of \real objects" such as real rings, real elds and real varieties. The constructive parts of real algebra consists of, e.g., computation of bounds on the real zeros of a polynomial, sign determination of a polynomial evaluated over a real zero of another polynomial, and computation of the number of real distinct zeros of a polynomial in an interval. The main objective of this chapter is to serve as a an introduction to two constructive methods in real algebra; so called cylindrical algebraic decomposition (CAD) and quanti er elimination (QE). Given a set of multivariate polynomials the CAD algorithm decomposes Rn into components over which the polynomials have constant signs. This decomposition can then be utilized to decide if any real polynomial system determined by the given polynomials has a solution. If this is the case the algorithm also provides a point in each connected solution set of the system. Brie y the QE method can be said to be an algorithm for eliminating variables, quanti ed by 9 (there exists) and 8 (for all), from formulas consisting of polynomial equations and inequalities combined by Boolean operators, e.g., ^ 69

70

Chapter 5 Real Algebra and Cylindrical Algebraic Decomposition

(and), _ (or), and ! (implies). The objective is to nd an equivalent formula without quanti ers in the unquanti ed or free variables only. The chapter is organized as follows. In Section 5.1 some basic de nitions and concepts of real algebra is reviewed. Section 5.2 presents some of the terminology needed in connection with CAD such as semialgebraic sets and decompositions. A detailed presentation of CAD relies on a number of technical tools such as resultants, subresultants, Sturm chains, projection operators, real algebraic numbers etc., which are treated in Section 5.3. In Section 5.4 we present the CAD algorithm and give a number of examples illustrating the dierent steps of the algorithm. Finally, an introduction to QE and its connections to CAD is given in Section 5.5.

5.1 Real Closed Fields One of the underlying basic structures in real algebra is so called real closed elds. This is a specialization of the eld concept such that an ordering of the eld elements can be introduced, i.e., we can always compare eld elements w.r.t. the ordering and use terms like greater/less than and between to describe how eld elements are related. Formally we have the following axiomatic de nition.

De nition 5.1 An ordered eld is a eld, k together with a subset P k, the set of positive elements, such that (i) 0 62 P. (ii) If a 2 k, then either a 2 P; a = 0 or a 2 -P. (iii) If a; b 2 P, then a + b 2 P and ab 2 P, i.e., P is closed under addition and multiplication. Hence, an ordered eld is a eld that can be decomposed into three disjoint subsets: the positive elements, zero and the negative elements. The negative elements are the additive inverses of the positive elements. Given an ordered eld we can introduce an ordering, > by de ning a > b if a - b 2 P; which can be shown to satisfy the properties usually associated with the inequality sign and the real numbers such as a > b ) a + c > b + c and a > b; c > 0 ) a c > b c: By a b we mean a > b or a = b. Using the concept of an ordering we can de ne closed and open intervals, respectively: [a; b] , x 2 k j a x b ; and (a; b) , x 2 k j a < x < b : The absolute value jaj of a eld element a is de ned in the obvious way and by construction jaj 2 P.

5.1 Real Closed Fields

71

De nition 5.2PAn eld k is called formally real if -1 cannot be written as a sum of squares, i.e., ni=1 a2i 6= -1 for any ai . A motivation for this de nition is given by the following observations. In any ordered eld k we have 1 > 0 > -1, since if -1 > 0, i.e., -1 2 P then (-1)2 2 P and the positive elements is closed under addition. Hence, 0 = (-1) + (-1)2 > 0, which is impossible. Furthermore, a 6= 0 ) a2 = (-a)2 = jaj2 > 0 > -1: p P Hence (-1) 62 k and ai 6= 0 ) ni=1 a2i > 0 > -1. The second observation

follows from

Pn

i

i

n n n X X X a2 = ja2 j a2 > 0 i=1

i

i=1

i

1

=

and i=1 a2i = 0 if and only if ai = 0; i = 1; : : : ; n. Hence, every ordered eld is formally real.

Example 5.1 The rational numbers Q is a formally real eld. The complex numbers C is not a formally real eld, since i2 = -1. The set of rational functions in one variable over Q , Q (t) is a formally real eld.

To get an algebraically more complete structure the following concept can be introduced.

De nition 5.3 An ordered eld k is called real closed if

(i) Every positive element of k has a square root in k. (ii) Every polynomial f(x) 2 k [x] of odd degree has a root in k.

Example 5.2 The real numbers p R is a real closed eld. The rational numbers Q is not a real closed eld since 2 62 Q . A real algebraic number, is a real number that satis es a polynomial equation f() = 0, where f(x) 2 Q [x]. The set of real algebraic numbers is a real closed eld. The real algebraic numbers are important in constructive algorithms in real algebra for two reasons. They obey an exact representation on computers and are the natural ground eld in many computations. Much more about algebraic numbers can be found in [32]. Real algebraic numbers are used in the CAD algorithm. This introduction to some basic concepts in real algebra is only a scratch on the surface of this mathematical branch. Further topics in real algebra concerns quadratic forms, ideals, Artin-Schreier theory etc. The interested reader may consult [16, 37, 93, 98, 112].

72


5.2 Semialgebraic Sets and Decompositions The notions of semialgebraic sets and decompositions play a key role when studying algorithms in real algebra and geometry. Here we will introduce the reader to semialgebraic sets and some other basic concepts needed to understand the CAD method. The de nitions essentially come from [6]. A comprehensive treatment of CAD is given in [112]1 and some briefer presentations can be found in [40, 72]. The ground eld is assumed to be the real numbers R in the rest of this section, which allows us to draw pictures.

De nition 5.4 A subset of Rn is semialgebraic if it can be constructed by nitely

many applications of union, intersection, and complementation operations on sets of the form

x 2 Rn j f(x) 0 ;

where f 2 R[x1 ; : : : ; xn ]. We use the Boolean operators ^ (and) and _ (or) to combine inequalities of the above form to represent semialgebraic sets. In the sequel we use calligraphic letters such as S to denote these sets. The corresponding de ning formula is denoted S (x), i.e., S = x 2 Rn j S (x) .

Example 5.3 Using Boolean operators we have S = x 2 Rn j f(x) 0 [ x 2 Rn j g(x) 0 = x 2 Rn j f(x) 0 _ g(x) 0 : Here S (x) = f(x) 0 _ g(x) 0, i.e., a Boolean formula which for each x evaluates

to True or False.

An interesting property of semialgebraic sets is that they are closed under projection, i.e., the projection of a semialgebraic set to some lower dimensional space is again a semialgebraic set. In general, this is not true for varieties or algebraic sets, i.e., sets de ned by a system of polynomial equations. For example, the projection of the unit circle onto one of the axis is an interval, which is not a zero set of any polynomial. We also observe that the set of points which satis es a system of equalities and inequalities is a semialgebraic set.

Example 5.4 An example of a semialgebraic set S 2 R2 , S = x 2 R2 j x21 + x22 - 1 0 ^ x21 - x2 = 0 _ (x - 1)2 + (x - 1)2 - 1 0 ^ (x - 2)2 + (x - 2)2 - 1 0 : 1 2 1 2 1 Unfortunately there are some errors in the description of the algorithm, but a correct version can be found in the errata.

5.2 Semialgebraic Sets and Decompositions

73

x2

x1 PSfrag replacements

Figure 5.1 A semialgebraic set (black region) and its projection onto the

x1 -axis (gray region). The dashed lines correspond to the zero sets of the de ning polynomials.

The projection of the set onto the x1 -axis

Sx1 = x 2 R2 j x41 + x21 - 1 0

_

1 x1 2 ;

is again a semialgebraic set, see Figure 5.1. We now introduce some terminology for describing how algebraic curves and surfaces partition Rn .

De nition 5.5

(i) A region, R is a connected subset of Rn . (ii) The set Z(R) = R R = (; x) j 2 R; x 2 R is called a cylinder over R. (iii) Let f; f1 ; f2 be continuous, real-valued functions on R. A f-section of Z(R) is the set (; f()) j 2 R and a (f1 ; f2 )-sector of Z(R) is the set (; ) j 2 R; f1 () < < f2 () . (iv) Let X Rn . A decomposition of X is a nite S collection of disjoint regions (or components) whose union is X, i.e., X = ki=1 Xi ; Xi \ Xj = ;; i 6= j. (v) A stack over R is a decomposition which consists of fi -sections and (fi ; fi+1 )sectors, where f0 (x) < : : : < fk+1 (x) for all x 2 R and f0 = -1; fk+1 = +1. Notice that an algebraic equation in n variables implicitly de nes a set of realvalued, piecewise continuous functions over Rn-1 .

74

Chapter 5 Real Algebra and Cylindrical Algebraic Decomposition x3

Z(R)-cylinder

f2 -section f1 ; f2 )-sector

(

PSfrag replacements x1

f1 -section x2

R-region

Figure 5.2 A geometrical interpretation of the de nitions of region, cylinder, sections, sectors and stack. The stack consists of three sectors and two sections in this particular case. Observe the strict inequalities in De nition 5.5. Geometrically this means that the graphs of dierent functions do not intersect each other over R. The above concepts are visualized in Figure 5.2.

De nition 5.6 A decomposition D of Rn is cylindrical if n = 1 D is a partition of R1 into a nite set of numbers, and the nite and in nite

open intervals bounded by these numbers. n > 1 D = F1 [ : : : [ Fm is a cylindrical decomposition of Rn-1 and over each Fi there is a stack which is a subset of D. From the de nition of a cylindrical decomposition it is clear that any cylindrical decomposition of Rn induces a cylindrical decomposition of Rn-1 etc. down to R1 . The property that a polynomial has constant sign over a region appears to be important. 0

De nition 5.7 Let X Rn and f 2 R [x1 ; : : : ; xn]. Then f is invariant on X if one of f(x) > 0, f(x) = 0 or f(x) < 0 holds for all x 2 X. The set F = ff1 ; : : : ; fr g 2 [x1 ; : : : ; xn ] of polynomials is invariant on X if each fi is invariant on X. We also say that X is F -invariant if F is invariant on X.

R

5.2 Semialgebraic Sets and Decompositions

75

Example 5.5 Let F = fx21 + x22 - 1; (x1 - 1)2 + x22 - 1g. Then the set I is F -invariant but J is not, see Figure 5.3. J PSfrag replacements

x2

I

x1

Figure 5.3 The zero set of F in Example 5.5 (dashed lines), the set J and the F -invariant set I . De nition 5.8 A decomposition is algebraic if each of its components is a semialgebraic set.

Example 5.6 Let f(x) = (x1 - 2)2 + (x2 - 2)2 - 1. Then x j f(x) > 0 [ x j f(x) = 0 [ x j f(x) < 0 is an algebraic decomposition of R2 . Notice that when the decomposition is de ned by a set of polynomials it is algebraic since all boundaries are zero sets of the de ning polynomials.

De nition 5.9 A Cylindrical Algebraic Decomposition (CAD) of Rn is a decom-

position which is both cylindrical and algebraic. The components of a CAD are called cells.

Example 5.7 Let F = f(x1 - 2)2 + (x2 - 2)2 - 1; (x1 - 3)2 + (x2 - 2)2 - 1g. The

CAD consists of the distinct black \dots", \arcs" and \patches of white space" in Figure 5.4 whose union is R2 . The induced CAD of R1 consists of the gray dots and the intervals on the x1 -axis.

76

Chapter 5 Real Algebra and Cylindrical Algebraic Decomposition x2

PSfrag replacements

x1

Figure 5.4 A CAD of R2 and the induced CAD of R1 . One of the cells may be semialgebraically characterized as f x 2 R2 j (x1 - 2)2 + (x2 - 2)2 - 1 > 0 ^ (x1 - 3)2 + (x2 - 2)2 - 1 < 0 ^ x1 < 3 ^ x2 < 2 g:

Which one? ;-)

5.3 Tools In this section we present some constructive tools which are used in dierent steps of the CAD-algorithm. These are polynomial remainder sequences, subresultants, principal subresultant coecients, Sturm chains, projection operators, and representations of algebraic numbers. These are the main tools for other constructive methods in real algebra and real algebraic geometry, see [16, 19, 93, 112]. Most of the concepts in this chapter are also treated in [112]. Readers interested in the CAD algorithm but not the underlying details may at this stage proceed to Section 5.4.

5.3.1 Polynomial Remainder Sequences

An important step in the CAD algorithm is to nd projections of \signi cant points" of higher dimensional real zero sets such as vertical tangents and selfcrossings. These projections are strongly related to how the multiplicity of the zeros of a polynomial depends on its coecients. This dependency can be captured by studying common factors of multivariate polynomials. We start with the univariate case and then generalize the ideas to multivariate polynomials. Given two polynomials f; g 2 k[x] we can ask how many common zeros f and g have and how many distinct zeros f has? These two questions can be answered knowing the greatest common divisor (gcd) of two polynomials.

5.3 Tools

77

Lemma 5.1 Let f; g 2 k[x]. Then the number of common zeros of f and g is equal

to the degree of gcd(f; g). Furthermore, the number of distinct zeros of f is equal to the dierence between the degree of f and the degree of gcd(f; f ). 0

Proof The rst assertion follows from the de nition of gcd and the fact that the number of zeros (counting multiplicity) of a univariate polynomial is equal to its degree. The second assertion follows from the observation that the multiplicity of a zero drops by one when dierentiating a polynomial. Hence, all multiple zeros can be removed by dividing f by gcd(f; f ) and the lemma follows. 2 0

How do we compute the gcd of two univariate polynomials? This is a classical question and the answer is provided by the Euclidean algorithm for polynomials. By repeated polynomial division we end up with the gcd of the two original polynomials. The process may be described as follows:

f1 = q1 f2 + f3 ; f2 = q2 f3 + f4 ; .. .

deg(f3 ) < deg(f2 ) deg(f4 ) < deg(f3 )

fp-2 = qp-2 fp-1 + fp ; deg(fp ) < deg(fp-1 ) fp-1 = qp-1 fp + 0; where deg(f2 ) deg(f1 ) and fp = gcd(f1 ; f2 ), see [94] or [39] for a detailed treat-

ment. The algorithm terminates since the degree decreases in each step. The sequence (f1 ; f2 ; : : : ; fp ) is called a polynomial remainder sequence (PRS).

Example 5.8 Let f1 = (1 + x)(3 + x)(1 + x + x2 ) and f2 = (1 + x)(2 + x)2 . The PRS of f1 ; f2 in expanded form is

f1 (x) = 9 + 24x + 31x2 + 23x3 + 8x4 + x5 f2 (x) = 4 + 8x + 5x2 + x3 f3 (x) = 9 + 12x + 3x2 f4 (x) = 1 + x; where f4 (x) = 1 + x = gcd(f1 ; f2 ). The above ideas can be extended to multivariate polynomials. Let f; g 2 k [x1 ; : : : ; xn-1 ][xn ], i.e., f and g is considered as polynomials in xn with polynomial coecients in the x1 ; : : : ; xn-1 . Hence the coecients of f and g is no longer xed numbers but depends on over which point = (1 ; : : : ; n-1 ) 2 Rn-1 they

are evaluated. This implies that both the number of zeros and their location varies for dierent 2 Rn-1 .

78


Example 5.9 Let f = (x21 + x22 - 1)x23 + (x2 - 1)x3 + x22 and consider f as a

polynomial in x3 . Then the degree of f is 2 everywhere except on the cylinder x21 + x22 - 1 = 0. On the cylinder the degree is 1 except on the line (x1 ; x2 ) = (0; 1) where it is zero.

For some 2 Rn-1 there might be common factors of f and g but not for others, i.e., the gcd(f; g) and hence its degree depends on . To compute a gcd of two polynomials we use the Euclidean algorithm with so called pseudo-division in each step since the polynomial coecients no longer belongs to a eld. Here is a brief exposition of pseudo-division.

Lemma 5.2 Let f; g 2 k[x1 ; : : : ; xn-1][xn], be written in the form f = fp xpn + : : : f0 and g = gm xm n + : : : g0 ; where m p, i.e., fi ; gj are polynomials in k[x1 ; : : : ; xn-1 ]. Then gsm f = qg + r; where q; r 2 k [x1 ; : : : ; xn-1 ][xn ] are unique, s = p - m + 1 and the degree of r w.r.t. xn is less than m.

Proof A proof can be found in [112].

2

The remainder r of the pseudo-division is called the pseudo-remainder of f and g and is denoted prem(f; g). If we allow denominators pseudo-division can be seen as ordinary polynomial division for polynomials in xn with coecients from the rational function eld k(x1 ; : : : ; xn-1 ), followed by clearing denominators. The only term which needs to be inverted in the division is the leading coecient, gm of g. Hence we get the equation gsm f = qg + r. Compare with the division algorithm in k fx1 ; : : : ; xn g treated in Section 3.3. In general, a factorization of a non-invertible ring element into irreducible elements does not need to be unique2 . A ring whose non-invertible elements can be uniquely factorized is henceforth referred to as a unique factorization domain (UFD) [112]. The polynomial rings we have seen so far is UFDs.

De nition 5.10 Let R be a UFD. Two polynomials in R[x] are similar if there exist a; b 2 R such that a f(x) = b g(x). Notation: f(x) g(x). Using the notation of similarity we can now generalize the concept of polynomial remainder sequences. 2

The element 9 in Z[

p

5 can be factorized according to (2 +

- ]

p

5 2

- )( -

p

5

- )=

9 = 3 3.

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De nition 5.11 Let R = k[x1; : : : ; xn-1] and f1 ; f2 2 R[xn] with deg(f1 ) deg(f2 ), where deg denotes the degree w.r.t. xn . The sequence f1 ; f2 ; : : : ; fk is a PRS for f1 and f2 if: (i) For all i = 3; : : : ; k fi prem(fi-2 ; fi-1 ). (ii) The sequence terminates with prem(fk-1 ; fk ) = 0. Since we only claim similarity there are in nitely many PRSs to every pair f1 ; f2 2 k [x1 ; : : : ; xn-1 ][xn ] of polynomials. Two such sequences are: (i) Euclidean Polynomial Remainder Sequence (EPRS): fi = prem(fi-2 ; fi-1 ) 6= 0; prem(fk-1 ; fk ) = 0: (ii) Primitive Polynomial Remainder Sequence (PPRS):

fi = pp(prem(fi-2 ; fi-1 )) 6= 0; prem(fk-1 ; fk ) = 0: where pp stands for primitive part , i.e., the remaining part of the polynomial when we have removed all common factors of the coecients. We observe that a PRS is unique up to similarity since pseudo-division is unique. Furthermore, gcd(f1 ; f2 ) : : : gcd(fk-1 ; fk ) fk ; i.e., PRS essentially computes the gcd of two polynomials up to similarity. From a computational point of view both EPRS and PPRS suer from complexity problems [94]. EPRS has an exponential coecient growth and the PPRS algorithm involves calculations of gcd of the coecients, which in our case again are polynomials. However, this computational complexity is not inherent in the problem. The Subresultant Polynomial Remainder Sequence (SPRS) oers a tradeo between coecient growth and the cost of gcd calculations of the coecients. We will now take a closer look at subresultants and SPRS.

5.3.2 Subresultants

To extend the Euclidean algorithm for computing the gcd of multivariate polynomials the SPRS seems to be the most attractive alternative. The basic idea of the construction of this PRS is that the coecients of all polynomials of a PRS can be computed as minors of a special matrix. In this subsection we present one way of computing the SPRS, see [94, 104, 112]. The process of pseudo-division can be organized as row operations on a matrix containing the coecients of the involved polynomials.

80


Example 5.10 Pseudo-division applied to f = x3 + 2x2 + 3x + 1 and g = 2x2 + x + 2 gives

22 f = (2x + 3)g + 5x - 2 : One way of organizing the calculations is

21 2 3 13 22 1 2 03 ~ M=4 2 1 2 0 54 0 2 1 2 5=M 0 0 5 -2

0 2 1 2

~ is obtained from M by row operations. Observe the correspondence bewhere M ~ . In fact, multiplying tween the pseudo-remainder coecients and the last row of M ~ can be obtained by just subtracting integer multhe rst row of M with 22 , M tiples of the other rows from the rst row. Finally, permuting the rows gives the ~ the pseudo-remainder triangular form. According to the triangular structure of M coecients may be found, modulo a common factor, by determinant calculations ~ augmented by either column on a matrix consisting of the rst two columns of M three or four. ~ only diers by row operations their minors are strongly related. Since M and M Hence the coecients of a polynomial similar to the pseudo-remainder may be calculated as minors of the original matrix M. In fact, by computing minors of matrices whose elements are the coecients of the polynomials f and g we can construct a whole PRS of f and g. To see this we rst need a couple of de nitions [104, 112]. Throughout the rest of this subsection we let R denote a UFD.

De nition 5.12 Let fi = Pnj=i0 fijxj 2 R[x]; i = 1; : : : ; k. The matrix associated with f1 ; : : : ; fk is

mat(f1 ; : : : ; fk ) = fi;l-j ; where l = 1 + max1 i k (ni ). In other words the matrix associated with f1 ; : : : ; fk is a matrix where each row contains the coecients of one of the polynomials and each column corresponds to decreasing powers of x from left to right.

De nition 5.13 Let M 2 Rk l; k l. The determinant polynomial of M is

detpol(M) = det(M(k) )xl-k + : : : + det(M(l));

where M(j) = M ;1 : : : M ;k-1 M ;j .

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Example 5.11 In Example 5.10 we have M = mat(f; x g; g), detpol(M) = 5x - 2 ~ ) = 22 (5x - 2). and detpol(M

De nition 5.14 Let f; g 2 R[x] and deg(f) = m; deg(g) = n; m n. (i) The kth subresultant of f and g is subresk (f; g) = detpol(Mk ); where Mk = mat(xn-k-1 f; : : : ; f; xm-k-1 g; : : : ; g). (ii) The subresultant chain of f and g is fSj gjn=+01 , where

Sn+1 = f; Sn = g; Sn-1 = subresn-1 (f; g); .. .

S0 = subres0 (f; g): Observe that M0 is the so called Sylvester matrix of f and g and subres0 (f; g) = detpol(M0 ) = det(M0 ) is the resultant of f and g [39]. Furthermore, the Mk matrices are obtained by deleting rows and columns of M0 . The importance of the de nition of subresultants and subresultant chains lies in the fact that all PRS of f and g is embedded in the subresultant chain of f and g up to similarity. We illustrate the de nitions with a numerical example.

Example 5.12 Let f = x5 - 2x4 + 3x3 - 4x2 + 5x - 6 and

g = 3x3 + 5x2 + 7x + 9. Then 2 1 -2 3 -4 5 -6 0 0 3 66 0 1 -2 3 -4 5 -6 0 77 66 0 0 1 -2 3 -4 5 -6 77 6 7 M0 = 66 03 35 57 79 90 00 00 00 77 66 0 0 3 5 7 9 0 0 77 64 0 0 0 3 5 7 9 0 75 0 0 0 0 3 5 7 9

21 66 0 6 M1 = 66 03 64 0

3

-2 3 -4 5 -6 0 2 1 -2 3 -4 5 -6 77 5 7 9 0 0 0 77 ; M2 = 66 4 3 5 7 9 0 0 77 0 3 5 7 9 05 0 0 0 3 5 7 9

3

1 -2 3 -4 5 -6 3 5 7 9 0 0 77 : 0 3 5 7 9 05 0 0 3 5 7 9

82


Observe how M1 and M2 are obtained by deleting rows and columns of M0 . The M(kl) matrices are formed by the left part of the partitioned matrices and one column from the right part. Now, the subresultants are the determinant polynomials of M0 ; M1 and M2 , i.e.,

S4 = x5 - 2x4 + 3x3 - 4x2 + 5x - 6 S3 = 3x3 + 5x2 + 7x + 9 S2 = det(M2(4) )x2 + det(M(25) )x + det(M(26) ) = 263x2 - 5x + 711 S1 = det(M1(6) )x + det(M(17) ) = -2598x - 11967 S0 = det(M0 ) = 616149 = 32 223 307 Compare the subresultant chain with the EPRS and PPRS for f and g: f1 = x5 - 2x4 + 3x3 - 4x2 + 5x - 6 f~1 = x5 - 2x4 + 3x3 - 4x2 + 5x - 6 f2 = 3x3 + 5x2 + 7x + 9 f~2 = 3x3 + 5x2 + 7x + 9 f3 = -263x2 + 5x - 711 f~3 = -263x2 + 5x - 711 f4 = -70146x - 323109 f~4 = -866x - 3989 f5 = -38 223 2632 307 f~5 = -1 where f4 = 81 f~4 = 3 S1 . Notice the quick growth of the coecients in the EPRS compared with the modest growth in the subresultant chain (the dierence is more obvious for longer chains). In this example there is a one-one correspondence between the PRS and the subresultant chain. This is due to the fact that the degree drops by one for each pseudo-division. If the degree drops with more than one, some of the subresultants becomes similar. What is the connection between a polynomial in a PRS and the determinant polynomial of Mk ? By row operations on Mk a number of pseudo-divisions may be carried out consecutively. The process is described by the following example which is a generalization of Example 5.10.

Example 5.13 Let f = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0 and

g = b3 x3 + b2 x2 + b1 x + b0 . The corresponding M1 matrix becomes 2a a a a a a 3 2b b b b 5 4 3 2 1 0 66 a5 a4 a3 a2 a1 a0 77 66 3 b23 b12 b01 b0 6 77 (1) 66 b3 b2 b1 M1 = 66 b3 bb23 bb12 bb01 b0 77 66 b3 b2 64 5 4 b3 b2 b1 b0 a~2 a~1 b3 b2 b1 b0 a~2

3 77 77 b0 b1 b0 77 5 a~0

a~1 a~0

2

( )

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2

( )

2b b b b 66 3 b23 b12 b01 b0 66 b3 b2 b1 b0 66 a~2 a~1 a~0 4 a~2 a~1 a~0 b~ 1 b~0

3 77 77 = M~ 77 1 5

Row operations in detail: (1) Multiply the rst row with b33 and subtract multiplicities of row 3; 4 and 5 to eliminate a5 ; a4 and a3 . A similar treatment of the second row followed by interchanging rows gives the second matrix. The resulting matrix elements a~i are the coecients of prem(f; g). (2) The same process as in (1) repeated on row 4; 5 and 6. The matrix elements b~ i are the coecients of prem(g; prem(f; g)). As pointed out earlier the only operations needed to get from Mk to the last ~ k are row operations. It is then clear that the minors det(M(kl)) triangular matrix, M and the corresponding minors of the triangular matrix only dier by a common factor. Our original motivation for calculating PRSs was to determine the gcd of two polynomials and especially its degree, since the degree was strongly related to the number of common zeros. Hence, we are interested in the coecient of the highest power of a gcd.

De nition 5.15 Let f; g 2 R[x] and deg(f) = m; deg(g) = n; m n. The kth

principal subresultant coecient of f and g is psck (f; g) = det(M(kk)); 0 k n: In other words psck (f; g) is the coecient of xk in subresk (f; g). The following lemma, which is not hard to believe in knowing the connection between a PRS and the corresponding subresultant chain, tells us that knowing the psc chain of two polynomials we know the degree of their gcd.

Lemma 5.3 Let f; g 2 R[x] where deg(f) = m, deg(g) = n and 0 < i min(m; n). Then f and g have a common factor of degree i if and only if pscj (f; g) = 0; j = 0; : : : ; i - 1 and psci (f; g) 6= 0:

Proof See Corollary 7.7.9. in [112].

2

In this subsection we have worked with polynomials in R[x] where R is an UFD. For simplicity the examples were done for polynomials over the integers but any UFD will do, e.g., multivariate polynomials. In the following we use R = R[x1 ; : : : ; xn-1 ], i.e., polynomials in R[x1 ; : : : ; xn-1 ][xn ].

84


5.3.3 The Projection Operator

The key idea in the so called projection phase of the CAD algorithm is to nd regions over which a given set of polynomials have a constant number of real roots. This is formalized by the concept of delineability. Let f 2 R[x1 ; : : : ; xn-1 ][xn ] be a real polynomial in n-variables:

f(x1 ; : : : ; xn-1 ; xn ) = fd (x1 ; : : : ; xn-1 )xdn + + f0 (x1 ; : : : ; xn-1 ) and = (1 ; : : : ; n-1 ) 2 Rn-1 . Then we write f(xn ) , f(1 ; : : : ; n-1 ; xn ) for the univariate polynomial obtained by substituting for the rst n - 1 variables. We also introduce the reductum , f^k of a polynomial. Let f^k (x1 ; : : : ; xn-1 ; xn ) = fk (x1 ; : : : ; xn-1 )xkn + + f0 (x1 ; : : : ; xn-1 ); where 0 k d. Thus, if fd () = = fk+1 () = 0 and fk () 6= 0, then

f (xn ) = f^k (; xn ) = fk ()xkn + + f0 ():

De nition 5.16 Let F = ff1; f2 ; : : : ; frg R[x1 ; : : : ; xn-1][xn] be a set of multivariate real polynomials and C Rn-1 a region. We say that F is delineable on

C, if it satis es the following invariant properties: (i) For every 1 i r, the total number of complex roots of fi; (xn ) (counting multiplicity) remains invariant as varies over C. (ii) For every 1 i r, the number of distinct complex roots of fi; (xn ) remains invariant as varies over C. (iii) For every 1 i < j r, the total number of common complex roots of fi; (xn ) and fj; (xn ) (counting multiplicity) remains invariant as varies over C. Observe that the set of polynomials de ning the CAD in Figure 5.4 is delineable over each cell of the decomposition of R1 . Consider one of the polynomials fi 2 F . Since it has real coecients its complex roots have to occur in conjugate pairs. Let and be two points in C Rn-1 . The only way for a pair of complex conjugated roots of fi; to become real when varies to is to coalesce into a real double root, i.e., the number of distinct roots of fi drops. Hence a transition from a non-real root to a real root is impossible over C. Similar arguments holds for the reversed problem. 0

0

Lemma 5.4 Let F R[x1 ; : : : ; xn-1][xn] be a set of polynomials and let F be delineable over C Rn-1 . Then the total number of distinct real roots of F is invariant over C. Proof See Lemma 8.6.3. in [112]. We are now able to formulate the main theorem of this subsection.

2

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Theorem 5.1 Let F R[x1 ; : : : ; xn-1][xn] be a set of polynomials and let C Rn-1 be a connected maximal F -delineable set. Then C is semialgebraic. Proof A complete proof may be found in [33] or in the errata of [112]. Here we

only sketch the ideas. The idea is to show that the three invariant properties of De nition 5.16 have semialgebraic characterizations. Let pscxi n denote the ith principal resultant coecient w.r.t. xn and Dxn denote the formal derivative operator w.r.t. xn . Let F = ff1 ; : : : ; fr g. (i) Total number of complex roots of fi; remains invariant over C. This corresponds to the claim that

(81 i r)(90 ki di ) h i (8k > ki ) fki (x1 ; : : : ; xn-1 ) = 0 ^ fki i (x1 ; : : : ; xn-1 ) 6= 0 holds for all 2 C. (ii) The number of distinct complex roots of fi; remains invariant over C, which corresponds to

(81 i r)(90 < ki di )(90 li ki - 1) h (8k > ki ) fki (x1 ; : : : ; xn-1 ) = 0 ^ fki i (x1 ; : : : ; xn-1 ) 6= 0 ^ h i (8l < li ) pscxl n (f^ki i (x1 ; : : : ; xn ); Dxn f^ki i (x1 ; : : : ; xn)) = 0 ^

i

pscxlin (f^ki i (x1 ; : : : ; xn ); Dxn f^ki i (x1 ; : : : ; xn)) 6= 0 :

holds for all 2 C. (iii) The total number of common complex roots of fi; and fj; (counting multiplicity) remains invariant over C, which is characterized by

(81 i < j r)(90 < ki di )(90 < kj dj )(90 mi;j min(di ; dj )) h (8k > ki ) fki (x1 ; : : : ; xn-1 ) = 0 ^ fki i (x1 ; : : : ; xn-1 ) 6= 0 ^ (8k > kj ) fkj (x1 ; : : : ; xn-1 ) = 0 ^ fkj j (x1 ; : : : ; xn-1 ) 6= 0 ^ (8m < mi;j ) pscxmn (f^ki i (x1 ; : : : ; xn); f^kj j (x1 ; : : : ; xn )) = 0 ^ pscxn (f^ki (x ; : : : ; x ); f^kj (x ; : : : ; x )) 6= 0 i; n j 1 n mi;j i 1 holds for all 2 C.

2

In summary, given a set of polynomials F R[x1 ; : : : ; xn-1 ][xn ] we can compute another set of (n - 1)-variate polynomials proj(F ) R[x1 ; : : : ; xn-1 ], which characterizes the maximal connected F -delineable subsets of Rn-1 . Hence, we have a projection operator.

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De nition 5.17 Let F = ff1; f2 ; : : : ; frg, where fi 2 R [x1 ; : : : ; xn-1][xn]. The projection operator is de ned as

proj(F ) , proj1 (F ) [ proj2 (F ) [ proj3 (F ) where

proj1 , fki (x1 ; : : : ; xn-1 ) j 1 i r; 0 k di ; proj2 , pscxl n f^ki (x1 ; : : : ; xn ); Dxn f^ki (x1 ; : : : ; xn ) j

1 i r; 0 l < k di - 1 ; proj3 , pscxmn f^ki i (x1 ; : : : ; xn ); f^kj j (x1 ; : : : ; xn ) j 1 i < j r; 0 m ki di ; 0 m kj dj :

Suppose that the set proj(F ) is invariant over C Rn-1 . The invariance of the set proj1 (F ) implies that the degree w.r.t. xn of the polynomials is constant over C and hence the number of roots of each polynomial is constant over C (condition (i) in De nition 5.16). The invariance of proj2 (F ) implies that the gcd of each polynomial and its derivative has constant degree according to Lemma 5.3. Together with the invariance of proj1 (F ) the number of distinct zeros of each polynomial in F is constant, cf. Lemma 5.1 (condition (ii) in De nition 5.16). The invariance of proj3 (F ) together with the invariance of proj1 (F ) implies that the number of common zeros of each pair of polynomials in F is constant, cf. Lemma 5.3 (condition (iii) in De nition 5.16). According to Lemma 5.4 the set proj(F ) characterizes the sets over which there are a constant number of real zeros of the polynomials in F . The projection operator is technically the most involved part of the CAD algorithm. However, we repeat that the basic idea of the operator is to project the \signi cant points" of an n-dimensional zero set of a number of polynomials to an n - 1-dimensional zero set of another set of polynomials.

5.3.4 Sturm Chains

The number of distinct real zeros of a univariate polynomial is important in both the so called base and extension phase of the CAD algorithm. In these phases we need to isolate the real zeros of univariate polynomials. Sturm chains [51, 112] are the appropriate tool to count the number of real zeros of a polynomial in an interval. Knowing the number of real roots it is easy to compute isolating intervals for them. In this subsection we present Sturm chains and comment on how to represent and make arithmetic operations with algebraic numbers.

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De nition 5.18 Let f1; : : : ; fr 2 R[x] have the following properties in the interval

(a; b): (i) fk (x) = 0 ) fk-1 (x)fk+1 (x) < 0; a < x < b; k = 2; : : : ; r - 1 (ii) fr (x) 6= 0; a < x < b Then f1 ; : : : ; fr is called a Sturm chain in the interval (a; b). If each polynomial of the Sturm chain is multiplied by the same arbitrary polynomial d(x) the chain

obtained is called a generalized Sturm chain.

Example 5.14 Let f1 2 R[x]. A Sturm chain whose rst two elements are f1 and

f1 gives information about the number of real zeros of f1 in (a; b). Let f2 = f1 and 0

0

consider the following PRS:

f1 = q1 f2 - f3 f2 = q2 f3 - f4 .. .

fr-2 = qr-2 fr-1 - fr fr-1 = qr-1 fr + 0; where fr = gcd(f1 ; f2 ). This sequence is a Sturm chain in (a; b) by construction if fr 6= 0 in (a; b) or a generalized Sturm chain in (-1; 1), since f1 ; : : : ; fr may be seen as a Sturm chain multiplied by the gcd of all fi . Observe the reversed sign of each remainder, which guarantees property (i) in De nition 5.18.

De nition 5.19 Let faigri=1 be a nite sequence of real numbers. Then let

var(a1 ; : : : ; ar ) denote the number of sign variations in the sequence.

Example 5.15 var(-1; 0; -2; 0; 0; 4; 3; -1) = 2 and var(3; 5; 0; -1; 2; 0; -2; 1) = 4:

Theorem 5.2 Let f1; : : : ; fr be the chain obtained in Example 5.14 and V (x) = var(f1 (x); : : : ; fr(x)). Then the number of distinct real zeros of f1 in the interval3 (a; b) is

V (a)- V (b):

Proof See [40, 112]. 3

Here a may be -1 and b may be 1.

2

88


Example 5.16 Let f1 = (x2 + x + 1)(x + 1)(x - 1)(x - 3). Then f2 = f1 = 0

5x4 - 8x3 - 9x2 - 2x + 2 and the Sturm chain from Example 5.14 of f1 and f2 becomes:

f1 f2 f3 f4 f5 f6

= = = = = =

x5 - 2 x4 - 3 x3 - x2 + 2 x + 3 5 x4 - 8 x3 - 9 x2 - 2 x + 2 46 x3 + 33 x2 - 36 x - 79 25 25 25 25 6825 2625 37875 2 - 2116 x + 1058 x + 2116 50784 - 321632 29575 x - 4225 - 2142075 190969

According to Theorem 5.2 we can compute the number of distinct real zeros of f1 in (-2; 2). We have

V (-2) = var(-; +; -; +; +; -) = 4;

V (2) = var(-; -; +; +; -; -) = 2

where only the signs of the evaluated Sturm chain is presented. Hence the number of real zeros of f1 in (-2; 2) is 2. The total number of distinct real zeros of f1 is 3 since

V (-1) = var(-; +; -; -; +; -) = 4;

V (1) = var(+; +; +; -; -; -) = 1:

f1 (x) x PSfrag replacements

Figure 5.5 A plot of f1(x). We now have a method for counting the number of distinct real zeros of a univariate polynomial in an interval. Together with a bound on the modulus of the zeros it is easy to state an algorithm which separates or isolates the real zeros of a polynomial to any desired accuracy, e.g., using some bisection method, see [40] and [112].

5.4 Cylindrical Algebraic Decomposition

89

Real Algebraic Numbers In the so called extension phase of the CAD-algorithm multivariate polynomials are evaluated over algebraic numbers, i.e., zeros of univariate polynomials over Z. Hence, we need both representations and algorithms for arithmetic operations on these numbers. A real algebraic number, , may be represented in a number of dierent ways. Perhaps the most intuitive way is by a polynomial and an isolating interval. A common representation is an irreducible polynomial, which has as a zero:

: [p (x); I ]; where p 2 Z[x], I = (a; b) 2 Q 2 and I contains no other real zero of p than .

The main tool for arithmetic operations with algebraic numbers is resultants. We will not go into details here but there are algorithms for, e.g., addition, multiplication, inverse, and sign evaluation of algebraic numbers. More about representation and computation in connection with algebraic numbers can be found in [32, 38, 40, 112]. There is also software developed for these calculations. For example saclib [27], which is a library of C routines oering algebraic number arithmetic. Recently, such facilities have also been built into Mathematica.

5.4 Cylindrical Algebraic Decomposition In this section we will describe the CAD algorithm and consider some examples of its use. The input of the algorithm is a set, F of n-variate polynomials and the output is a representation of a decomposition of Rn into regions over which the polynomials of F are invariant. The representation also includes a sample point of each region that can be used to determine the actual signs of the polynomials.

5.4.1 The Algorithm

The algorithm can be divided into three phases: projection, base and extension. The projection phase consists of a number of steps, each in which new sets of polynomials is constructed. The zero sets of the resulting polynomials of each step is the projection of \signi cant" points of the zero set of the preceding polynomials, i.e., self-crossings, isolated points, vertical tangent points etc. In each step the number of variables is decreased by one and hence the projection phase consists of n - 1 steps. The base phase consists of isolation of real roots, i 2 R1 of the monovariate polynomials, which are the outputs from the projection phase. Each root and one point in the each interval between two roots are chosen as sample points of a decomposition of R1 . The purpose of the extension phase is to construct sample points of all cells of the CAD of Rn . The extension phase consists of n - 1 steps. In the rst step a

90


sample point, (i ; j ) 2 R2 of each cell of the stack over the cells of the base phase is constructed. In the following steps the above procedure is repeated until we have sample points in all cells of the CAD of Rn . Observe that to determine if a real polynomial system has a solution, it is enough to determine the signs of F in a sample point of each cell since F is invariant in each cell by construction. The algorithm can be summarized as: Input: F = ff1 ; : : : ; fr g R[x1 ; : : : ; xn ]. Output: An F -invariant CAD of Rn .

Projection: F0 = F R[x1 ; : : : ; xn ]; F1 R[x1 ; : : : ; xn-1 ]; : : : ; Fn-1 R[x1 ]: Base: Root isolation and sample point construction for a CAD of R1 . Extension: Sample point construction for CADs of Rk ; k = 2; : : : ; n. We following subsections give a detailed description of each phase.

Projection Phase

In the projection phase the proj-operator is applied recursively n - 1 times. Let F = ff1 ; : : : ; fr g, proj0 (F ) = F and proji (F ) = proj(proji-1 (F )) for 1 i n - 1. Then

F R[x1 ; : : : ; xn ] proj(F ) R[x1 ; : : : ; xn-1 ] proj2 (F ) R[x1 ; : : : ; xn-2 ] .. .

projn-1 (F ) R[x1 ]; where the corresponding zero sets are the successive projections of the \signi cant" points of the original zero set.

Example 5.17 Consider the 3D-sphere of radius 1 centered at (2; 2; 2) which is given by the set of real zeros of f = (x1 - 2)2 + (x2 - 2)2 + (x3 - 2)2 - 1:


91

3

2


x2

1

4 3

2 1

0 0

1

x1 2

3 4

Figure 5.6 The set of real zeros of f and the projections of its \signi cant" points.

Now, the projection polynomials are proj(f) = 4((x1 - 2)2 + (x2 - 2)2 - 1); (x1 - 2)2 + (x2 - 2)2 + 3 ; proj2 (f) = 4(x1 - 1)(x1 - 3)(x21 - 4x1 + 19);

16(x41 - 8x31 + 30x21 - 56 + 113); 4(x21 - 4x1 + 7); x21 - 4x1 + 11; 256(x1 - 1)(x1 - 3) : The only real zeros of the polynomials in proj(f) correspond to the circle in the x1 x2 -plane in Figure 5.6 and the only real zeros of the polynomials in proj2 (f) is 1 and 3, i.e., the gray dots on the x1 -axis in Figure 5.6.

Base Phase

The zeros of the monovariate polynomials in projn-1 (F ) de ne a sign invariant decomposition of R1 . Enumerate the real zeros of the polynomials in projn-1 (F ) according to

-1 < 1 < 2 < < s < +1:

The projn-1 (F )-invariant decomposition of R1 consists of these zeros and the intermediate open intervals. The purpose of the base phase is to isolate the above zeros and nd sample points for each component in the decomposition, i.e., the zeros and a sample point in each interval. For an open interval we may choose a rational sample point but for a zero we must store an exact representation of an algebraic number, see Subsection 5.3.4. The base phase may thus be described brie y as:

92

Chapter 5 Real Algebra and Cylindrical Algebraic Decomposition Input: A set of monovariate polynomials, projn-1 (F ). Output: Sample points of each component of the decomposition of R1 .

Real root isolation: 1 2 (u1 ; v1 ]; : : : ; s 2 (us ; vs ], where ui ; vi 2 Q . Choice of sample points: 1 = u1 ; 2 = 1 ; 3 = u2 ; : : : ; 2s+1 = vs + 1.

Example 5.18 Given the proj2(f) set from the projection phase of Example 5.17

we isolate the real roots of its polynomials. The number of real roots of each polynomial may be calculated using Theorem 5.2. This gives the real roots 1 and 3 with isolating intervals ( 12 ; 2] and (2; 27 ], respectively. The x1 -axis is decomposed into 5 cells with the following sample points: 1 7 2 ; 1; 2; 3; 2 :

Extension Phase

In the extension phase we \lift" a sign invariant decomposition, Di-1 of Ri-1 to a sign invariant decomposition, Di of Ri using the technique of the base phase repetitively. Consider the \lift" from R1 to R2 . According to how projn-2 (F ) is constructed it is delineable over each cell of D1 . We will construct the sample points of those cells of D2 which belong to the stack over a cell C D1 . Evaluate the polynomials in projn-2 (F ) over the sample point, of C. We then get a set of monovariate polynomials in x2 corresponding to the values of projn-2 (F ) on the \vertical" line x1 = . These monovariate polynomials are then treated in the same way as the polynomials in the base phase, i.e., root isolation and choice of sample points. Hence the \lift" from R1 to R2 corresponds to the construction of the second component of the sample points of Dn . Having sample points for all cells of D2 the above process may be repeated to construct sample points in the cells of D3 ; : : : ; Dn . A geometrical picture of the steps in the extension phase consists of so to speak raising vertical lines over each sample point of the lower dimensional decomposition and calculate the intersections between these lines with the zero set of the next higher dimensional set of polynomials. The result of the extension phase is a list of cells (indexed in some way) and their sample points. Hence, the decomposition may be represented as a tree structure where the rst level of nodes under the root corresponds to the cells of R1 , the second level of nodes represents the cells of R2 , i.e., the stacks over the cells of R1 etc., see Figure 5.8. The leaves represents the cells of the CAD of Rn . In each node or leaf a sample point of the corresponding cell is stored. To each level of the tree there are a number of projection polynomials projk (F ) whose signs when evaluated over a sample point de nes a cell.


93

Notice that the projection polynomials projk (F ); k = 0; : : : ; n - 1 together with the sample points describe the set of solutions of any system of polynomial equations and inequalities de ned by the original polynomials. The signs of the projection polynomials gives us information about the restrictions on the solution set imposed by the original system on smaller and smaller sets of variables.

Example 5.19 We now extend the CAD of R1 from Example 5.18 to a CAD of R2

and R3 . The base phase gives a decomposition of the x1 -axis into ve cells and sample points for each cell, see the top left plot in Figure 5.7. Specializing the polynomials of proj(f) over each sample point gives ve univariate polynomials in x2 . For each specialized polynomial we use the same algorithms as in the base phase to construct sample points of the cells of R2 . We may choose the following sample points

( 21 ; 2); (1; 1); (1; 2); (1; 3); (2; 12 ); (2; 1); (2; 2); (2; 3); (2; 27 ); (3; 1); (3; 2); (3; 3); ( 72 ; 2); see the top right plot in Figure 5.7. In the same way we specialize f over each sample point of R2 and construct the 25 sample points of the corresponding CAD of R3 . In Figure 5.8 the structure of a tree representation of the CAD is given. Observe that even for this trivial example the number of cells become quite large. In general the number of cells of a CAD grows very quickly as the number of variables increases.

5.4.2 An Example

The following example show how the CAD-algorithm can be used to nd solutions to a system of polynomial inequalities. Once a sample point of each sign invariant cell of the decomposition has been computed, we only have to evaluate the original polynomials over these sample points to determine if there are any solutions.

Example 5.20 Given the following polynomials f1 = x22 - 2x1 x2 + x41 f2 = (2431x1 - 3301)x2 - 2431x1 + 2685; determine a corresponding CAD of R2 .

(5.1)

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Chapter 5 Real Algebra and Cylindrical Algebraic Decomposition 3

PSfrag replacements

x3

3

2

x2

1

PSfrag replacements

4

x3

2

x2

1

3

1

0

1

0

0 1

4 3

2

2

x1

0 1

x1 2

2

3 4

3 4

3

2

x3 1

PSfrag replacements

x2

4 3

2 1

0 0

1

x1

2 3 4

Figure 5.7 The sample points of each step of the extension phase. proj2 (F ) PSfrag replacements

proj(F )

F

Figure 5.8 A tree representation of the CAD.


95

3

x2

2

1

-2

0

-1

PSfrag replacements

0

1

x1

2

3

-1

-2

Figure 5.9 A plot of the real zero set of the polynomials in (5.1). Projection:

proj1 (f1 ) = -2 x1 ; 1; x1 4 proj1 (f2 ) = -2431 x1 + 2685; 2431 x1 - 3301

proj2 (f1 ) = 4 x1 2 (x1 - 1) (x1 + 1) proj2 (f2 ) =

proj3 (ff1 ; f2 g) = - x1 -4862 x1 + 5370 + 2431 x1 4 - 3301 x1 3 ;

(17 x1 - 15) (13 x1 - 5) 26741 x1 4 - 38742 x1 3 8854 x1 2 - 51552 x1 + 96123

Base: The real roots of proj(ff1 ; f2 g) are 2685 3301 -1; 0; 135 ; 15 17 ; 0:93208; 1; 2431 ; 2431 ; 1:59982 where and are the real zeros of 2431x41 - 3301x31 - 4862x1 + 5370. It turns out that we only need ve of these roots to determine a CAD of R2 . These are 15 ; 1; 3301 : -1; 0; 135 ; 17 2431

We also need sample points from each interval between the above roots, e.g.,

-2; - 21 ; 14 ; 12 ; 109 ; 54 ; 2: Hence the base phase produces 13 sample points.

96


Extension: We compute the sample points for the stack over cell 7 (x1 = 12 ) of the decomposition of R1 to illustrate the procedure. We have

with real roots

f1 ( 21 ; x2 ) = x2 2 - x2 + 161 2939 f2 ( 21 ; x2 ) = - 4171 2 x2 + 2 ; 1 1 p3 2 4

2939 and 4171 ; respectively. Together with four points in the intervals we get seven sample points, see Table 5.1.

Sample point (x1 ; x2 ) sign(f1 ) sign(f2 )

( 12 ; 0) p ( 12 ; 21 - 14 3) ( 21 ; 12 ) 2939 ) ( 12 ; 4171 ( 21 ; 34 ) p ( 12 ; 21 + 14 3) ( 12 ; 2)

+ 0 0 +

+ + + 0 -

Table 5.1 Sample points and signs of f1 and f2 over cell 7. The information in one row of Table 5.1 is typically what is stored in a leaf of a tree representation of a CAD, i.e., the sample point and the signs of the polynomials. The whole CAD of R3 consists of 63 cells, see Figure 5.10. Given the sign of f1 and f2 in all cells we can \solve" any real polynomial system de ned by f1 and f2 . The point x1 = 12 ; x2 = 34 is one solution to f1 < 0; f2 < 0. In the above example we were able to avoid evaluating polynomials over irrational algebraic numbers. This is one of the most time consuming parts in an implementation of the CAD algorithm. However, computations with algebraic numbers can be avoided if we only are interested in open solution sets of systems of polynomial strict inequalities [146]. Observe that given a CAD generated by a set of real polynomials, it is possible to solve any real polynomial system de ned by these polynomials. A CAD can also be utilized in connection with quanti er elimination, which is the main tool in Chapter 6, 7, and 8. This is the reason for the detailed treatment of CAD in this chapter.


97

Figure 5.10 The CAD, where the dashed line and the corresponds to x1 = 12 and the sample points over cell 7 of the decomposition of R1 . The gray region corresponds to row ve in Table 5.1 where f1 < 0; f2 < 0.

5.5 Quanti er Elimination Many problems can be formulated as formulas involving polynomial equations, inequalities, quanti ers (9; 8) and Boolean operators (e.g., _; ^). Formally these expressions constitute a sentence in the so called rst-order theory of real closed elds. The problem of quanti er elimination consists of nding an equivalent expression without any quanti ed variables. In the late forties Tarski [148] gave a general algorithm for quanti er elimination in expressions of the abovementioned type, but this algorithm had a very high complexity and is impractical for most problems. In the mid seventies Collins [33] presented a method that decreased the algorithmic complexity and since then the algorithmic development has proceeded. Recently, more eective algorithms have made it possible to solve non-trivial quanti er elimination problems on workstations [35]. Introductions to quanti er elimination are given in, e.g., [40, 112] and a survey over the use of CAD for QE is given in [34]. For an extensive bibliography on the subject, see [5]. A recent collection of articles describing the state-of-the-art of CAD and QE is presented in the book [30]. The aim of this section is to give a brief introduction to QE and its connection to CAD.

Formulas

By a quanti er free Boolean formula we mean a formula consisting of polynomial equations (fi (x) = 0) and inequalities (fj (x) < 0) combined using the Boolean operators ^ (and), _ (or), : (not), and ! (implies).

98


De nition 5.20 A formula in the rst-order theory of real closed elds is an expression in the variables x = (x1 ; : : : ; xn ) of the following type (Q1 x1 : : : Qs xs )F (f1 (x); : : : ; fr (x));

where Qi is one of the quanti ers 8 (for all) or 9 (there exists) and F (f1 (x); : : : ; fr(x)) is a quanti er free Boolean formula. The term sentence is often used synonymously for formula. Note that there is always an implicit assumption that all variables are considered to be real. A quanti er free Boolean formula F (f1 (x); : : : ; fr (x)) de nes a semialgebraic set, cf. De nition 5.4. Even though the quanti ers give expressive power they do not enlarge the class of sets de ned by quanti er free Boolean formulas, i.e., formulas including quanti ers still de ne semialgebraic sets. This implies that given a formula including quanti ers there is always a logically equivalent quanti er free Boolean formula, a result rst shown by Tarski [148].

Example 5.21 The following expression is a formula in the rst-order theory of real closed elds:

9y 8x x2 + y2 > 1

^

(x - 1)2 + (y - 1)2 > 1 ;

which can be read as: There exists a real number y such that for all real numbers x we have that x2 + y2 > 1 and (x - 1)2 + (y - 1)2 > 1. This particular formula is True since both inequalities are satis ed for all real x if y = 3, for instance. The above problem is called a decision problem since all variables are bounded by quanti ers and the problem consists of deciding if the formula is True or False. If there are some free or unquanti ed variables as well, one has a general quanti er elimination problem. As mentioned above, it is always possible to eliminate quanti ed variables.

Example 5.22 The following is a general quanti er elimination problem. 8x x > 0 ! x2 + ax + b > 0 ; which can be read as: For all real numbers x, if x > 0 then x2 + ax + b > 0. Performing quanti er elimination we obtain the following equivalent expression

4b - a2 > 0 _ [ a 0 ^ b 0 ] ;

which are the conditions on the coecients of a second order polynomial so that it only takes positive values for x > 0.


99

The Boolean operator ! (implies) is used to get more structured formulas but can be expressed using other Boolean operators. Let A and B be two formulas. Then

A

!

B

,

:A _

B:

Many properties of semialgebraic sets can be exploited by eliminating quanti ers in formulas, e.g., set inclusion, projection and set adjacency.

Example 5.23 Consider the semialgebraic sets A = (x; y) 2 R2 j x2 + y2 < 1 and B = (x; y) 2 R2 j y > x4 - 2 : Is A B? What is the projection of B onto the y-axis? The corresponding formulas and their equivalent quanti er free counterparts are

8x 8y x2 + y2 < 1 ! y > x4 - 2 , True; 4 9x y > x - 2 , [ y > -2 ] : Hence, A B and the projection of B onto the y-axis is y 2 R j y > -2 . The quanti ed variables are implicitly assumed to range over all real numbers. Often we need to express that some of the bounded variables are constrained to a semialgebraic set. Then the following equivalences turns out to be useful

9(u 2 U ) X (x; u) 8(u 2 U ) X (x; u)

, ,

9u X (x; u) ^ U (u) ; 8u U (u) ! X (x; u) :

(5.2) (5.3)

More about Boolean formulas and logic can be found in, e.g., [152].

Relations between QE and CAD

For a decision problem the truth value of a formula can be decided once a CAD de ned by the involved polynomials has been computed. The quanti er free Boolean part of the original formula F (f1 (x); : : : ; fr (x)) is evaluated in each sample point and depending on how a variable is quanti ed F (f1 (x); : : : ; fr (x)) has to be True for some or all sample points. For the quanti er elimination problem we evaluate F (f1 (x); : : : ; fr(x)) over the sample points. The cells corresponding to the sample points for which this formula is True can be characterized by the sign of the polynomials from the projection phase of the CAD algorithm. The solution formula can then be constructed by combining such partial formulas, see [74].

100


Example 5.24 Consider the polynomials in Example 5.20 and their correspond-

ing CAD shown in Figure 5.10. Suppose that we have the following decision problem

9x1 9x2 [ f1 < 0 ^ f2 < 0 ] : (5.4) This particular CAD consists of 63 cells and by construction we know that f1 and f2 are invariant over each cell. Since x1 and x2 are bounded by existential quanti ers

we only have to evaluate them in each sample point and check if for some point the inequalities in (5.4) are satis ed. In this particular case we know that the formula is True according row ve in Table 5.1. Now, suppose we have the formula

8x2 [ f1 > 0

^ f2 < 0 ] ;

(5.5)

which is a quanti er elimination problem. Consider the cells of the induced decomposition of the x1 -axis in Figure 5.10. The formula is True for those cells over which there is a stack with f1 > 0 and f2 < 0 for all its sections and sectors. Hence, we have to nd the cells with this property (by sign evaluation in sample points) and describe them in terms of their de ning polynomials, i.e., the polynomials from the projection phase. In Figure 5.10 it is easy to see that the only stack in which f1 > 0 and f2 < 0 is the rightmost vertical line. This stack is described by the zero of the projection polynomial 2431x1 - 3301. Hence, a quanti er free equivalent formula to (5.5) is

2431x1 - 3301 = 0:

Software

A systematic treatment of real polynomial systems, i.e., systems of real polynomial equations and inequalities has not yet been implemented as a standard feature in any of the major computer algebra systems. The main reason for this is the algorithmic complexity of such solvers which is a subject of intensive research [12, 13, 67, 131]. However, there are solvers under construction, e.g., in [146] an implementation in Mathematica of a solver for systems of polynomial strict inequalities is described. To perform quanti er elimination in the non-trivial examples of this thesis we have used the program qepcad, developed by Hoon Hong et al. at RISC in Austria, see [35]. An interface between Mathematica and qepcad has been written by the author to be able to work in the more integrated computer algebra environment of Mathematica. In the remaining chapters of this part of the thesis we use QE and demonstrate its applicability to problems in control theory.

6 Applications to Linear Systems To investigate the usefulness of quanti er elimination in linear control theory we have chosen two types of problems. Stabilization, where the objective is to choose parameters in a controller such that the closed-loop system becomes stable; and the more general problem of feedback design, where the objective is not only to stabilize the system but also to ful ll a number of speci cations. The methods are combinations of a variant of the Routh criterion and Nyquist techniques together with quanti er elimination. We also use quanti er elimination to investigate stability of feedback systems including static nonlinearities. We only treat SISO-systems but many of the ideas have straightforward generalizations to MIMO-systems. However, the computational complexity of these algorithms grows quickly with the number of variables, which makes them less attractive for larger examples. Some recent applications of quanti er elimination to linear control system design is presented in [43, 147, 163]. In Section 6.1 stabilization problems are treated. An algebraic approach to feedback design is given in Section 6.2. We also use the circle criterion to compute bounds on static nonlinearities ensuring closed-loop stability and the describing function method to avoid instability or limit cycles.

6.1 Stabilization To investigate stability of linear systems, inequality criteria by Routh [134], Hurwitz [77] et al. have been extensively used. These stability criteria give conditions on the coecients of the characteristic equation of a linear time invariant dieren101

102

Chapter 6 Applications to Linear Systems

tial equation in terms of a number of inequalities. A dierential equation which results from modeling or design of a controller for a linear system often includes parameters (e.g., physical design constants and/or parameters of a controller) to be determined so that the system becomes stable. If the coecients of the dierential equation are rational functions of those parameters the Routh criterion is equivalent to a system of inequalities or possibly a union of such systems. In robust control a common objective is to design a controller which stabilizes a system even though some parameters are only known to belong to certain intervals. These uncertainties impose additional constraints on the solution to the stabilization problem but can equally well be treated with the method presented in this section.

Stability Conditions It is well known that stability of a linear time invariant system is determined by its poles. Stable systems have pole polynomials whose zeros all have strictly negative real parts and such polynomials have been given a special name.

De nition 6.1 A polynomial is said to be Hurwitz if all its zeros have strictly negative real parts.

A matrix is said to be Hurwitz if its characteristic polynomial is Hurwitz. The criterion by Routh states that a polynomial is Hurwitz if and only if its coecients satisfy a system of polynomial inequalities. There are many equivalent criteria and here we choose to present the Lienard-Chipart criterion, which has the advantages over the Routh criterion that it involves only about half the number of determinantal inequalities and hence gives a system of inequalities of lower degree.

Theorem 6.1 Let f(s) = a0sn +a1sn-1 +: : :+an-1s+an; a0 > 0 be a given polynomial with real coecients. De ne the Hurwitz determinant of order 1 i n as

a a1 00 Di = 0 ...

a3 a2 a1 a0 .. .

a5 a4 a3 a2 .. .

::: ::: ::: a4 .. .

...

; ai

ak = 0 for k > n:

Then f is a Hurwitz polynomial if and only if

an > 0; an-2 > 0; an-4 > 0; : : : ; D2 > 0; D4 > 0; D6 > 0; : : :

Proof See Parks and Hahn [126] or Gantmacher [51].

2

6.1 Stabilization

103

To illustrate the use of quanti er elimination in stability analysis of linear systems we apply this theorem in some examples. In all examples the program qepcad [35] has been used to perform quanti er elimination.

Example 6.1 Consider the unstable linear system G(s) = s2 - 42s + 2 ;

which has poles in 1 i. We use a phase-lead compensator

b F(s) = N ss++Nb

to stabilize the system, where b > 0 and 1 < N < 10, e.g., due to physical limitations in the implementation of the controller. The resulting closed-loop system becomes

(s + b) Gc (s) = 1 +GFGF = s3 + (Nb - 2)s2 4N + (2 + 4N - 2Nb)s + 6Nb :

The application of Theorem 6.1 to the pole polynomial of the closed-loop system gives the following conditions on N and b which ensure stability

6Nb > 0; Nb - 2 > 0; (Nb - 2)(2 + 4N - 2Nb) - 6Nb > 0: (6.1) The rst inequality implies that N and b have equal sign and a plot of the semi-

algebraic set de ned by the other two inequalities is shown in Figure 6.1. The rst question we pose is: For which values of b > 0, exists a value of N 2 (1; 10) such that the closed-loop system is stable? The second half of this question can be formulated as

9N 1 < N < 10 ^ b > 0 ^ 6Nb > 0 ^ Nb - 2 > 0 ^ (Nb - 2)(2 + 4N - 2Nb) - 6Nb > 0 ; where we have used equivalence (5.2) since the quanti ed variable N is constrained to a semialgebraic set. Performing quanti er elimination gives

50b2 - 100b + 21 < 0 ^ b > 0 ;

which approximately constrains b to the interval (0:24; 1:76). Compare with the projection onto the b-axes of the gray-shaded region in Figure 6.1. We can also nd the projection of the gray-shaded region onto the N-axis by quantifying b instead, which gives

N2 - 4N - 2 > 0 ^ N > 1 ^ N - 10 < 0 :

104

Chapter 6 Applications to Linear Systems 2 1.8 1.6 1.4

b

(Nb - 2)(2 + 4N - 2Nb) -6Nb > 0

Nb - 2 > 0

1.2 1 0.8

PSfrag replacements

0.6 0.4 0.2

2

4

N

6

8

10

Figure 6.1 Parameter values in the gray-shaded region given by the in-

equalities (6.1) correspond to a stable closed-loop system. The dashed rectangle illustrates the interval constraints from (6.2) and (6.3).

Hence, N is approximately constrained to the interval (4:45; 10:0). Since the values of N and b cannot be chosen independently the interval constraints only give necessary conditions for stability. To get a sucient condition we can pose the question: For which values of b > 0, hold that for all N 2 (5; 10) the closed-loop system is stable? The second half of this question can be formulated as

8N [ 5 < N < 10 ] ! [ b > 0 ^ 6Nb > 0 ^ Nb - 2 > 0 ^ (Nb - 2)(2 + 4N - 2Nb) - 6Nb > 0 ] ; (6.2) where we have used equivalence (5.3) since the quanti ed variable N is constrained.

Eliminating the quanti er gives

25b2 - 50b + 22 0 ;

(6.3)

i.e., approximately the interval constraint b 2 (0:66; 1:34). Compare with the dashed rectangle in Figure 6.1. Model uncertainties are easy to incorporate in the stability analysis, which is shown in the following example.

6.1 Stabilization

105

Example 6.2 Suppose that the system is described by 1

; (s - 12 )(s2 + 2!0 s + !20 ) where the damping ratio and frequency !0 are uncertain but constant parameters. Furthermore, assume that 2 ( 14 ; 13 ) and !0 2 (3; 4). The objective is G(s) =

to stabilize the system with a proportional compensator, F(s) = K despite the uncertainties. The closed-loop system becomes

Gc (s) = 1 +GFGF = 2s3 + (4! - 1)s2 + (2K 2!20 - 2!)s + 2K - !20 : 0

According to the stability criterion in Theorem 6.1 the closed-loop system is stable if and only if

2K - !20 > 0; 4!0 - 1 > 0; (4!0 - 1)(2!20 - 2!) - 2 (2K - !20 ) > 0

and in addition the following inequalities have to be satis ed

(6.4)

1 < < 1 ; 3 < !0 < 4; 4 3

(6.5) due to the uncertainties. We can now formulate the stabilization problem as follows: There exists a K such that for all and !0 satisfying (6.5), the conditions (6.4) hold. The corresponding formula becomes

9K 8!0 8 [ 41 < < 13 ^ 3 < !0 < 4 ] ! [ 2K - !20 > 0 ^ 4!0 - 1 > 0 ^ (4!0 - 1)(2!20 - 2!0 ) - 2 (2K - !20 ) > 0 ] ;

where we have used equivalence (5.3) since the quanti ed variables are constrained. Performing quanti er elimination on this decision problem the formula can be shown to be True. If we leave out the rst quanti er and eliminate the others, we obtain 8 K 514 which guarantees stability of the closed-loop system. Given these stabilizing values of K we can compute the region over which !0 and are allowed to vary without loss of stability. The formula becomes

8K 8 K 514 ! [ 2K - !20 > 0 ^ 4!0 - 1 > 0 ^ (4!0 - 1)(2!20 - 2!0 ) - 2 (2K - !20 ) > 0 ] ;

which is equivalent to

!0 4 ] ^ 82 !20 - 8!30 - 2!0 + 51 0: The part of this region with !0 > 0 is displayed in Figure 6.2. [ -4 !0 - 12

_ 12

We observe that the region is larger than the speci cations on the uncertainties.

106

Chapter 6 Applications to Linear Systems 4

3

2

1

PSfrag replacements 0 0

1

2

!0

3

4

5

Figure 6.2 The gray-shaded region corresponds to values of !0 and for which any 8 < K < 514 stabilizes the system. (Dashed line = 41 ). Generalized stability

The stability conditions in terms of inequalities as presented in Theorem 6.1 can be generalized to conditions of what is known as generalized stability [105]. According to properties of linear fractional transformations [31]

+b z = as cs + d

the half plane Re(s) < 0 can be mapped onto a new half plane or the interior/exterior of an arbitrary circle1 . This together with Theorem 6.1 can be used to derive necessary and sucient conditions on the coecients of a transformed polynomial such that its zeros lie in the transformed half plane. We illustrate the idea by an example.

Example 6.3 Suppose that the plant is given by

2s + 1 G(s) = s2 - 2as + 1 + a2 ;

where a is a constant but unknown uncertainty satisfying 43 < a < 54 . The plant is controlled using a proportional controller F(s) = K; K > 0 such that the closed-loop system becomes

+ 1) Gc(s) = 1 +GFGF = (s2 - 2as +K1(2s + a2 ) + K(2s + 1) :

1 Only transformations where a; b; c; d are real numbers can be used since the transformed pole polynomial must be a real rational function for Theorem 6.1 to apply.

6.1 Stabilization

107

The objective is to choose K such that the poles of the closed-loop system are located inside a disk of radius 2, centered at -3 on the real axis, regardless of the uncertainty. The linear fractional transformation

s = 5z1 --z1

z = ss ++ 15 ; maps the disk

(Re(s) + 3)2 + Im(s)2 < 4

(6.6)

onto the left half plane Re(z) < 0. The zeros of the pole polynomial p of Gc are mapped to the zeros of the transformed pole polynomial, which is a rational function, i.e.,

f(z) = p( 5z1 --z1 ) = (10a - 9K + 26 + a2 )z2 + (10K - 12 - 2a2 - 12a)z + 2 + a2 + 2a - K : (6.7) (-1 + z)2 Now, Theorem 6.1 applied to the numerator of this rational function, gives necessary and sucient conditions for the zeros of f(z) to have negative real parts, i.e., for the zeros of p(s) to belong to the disk (6.6). Since the numerator is a polynomial of degree two, a necessary and sucient condition for stability is that all its coecients have the same sign. The stabilization problem may thus be formulated as: For which values on K > 0, hold that for all a 2 ( 34 ; 54 ), the coecients of the numerator of (6.7) all have the same sign? Utilizing equivalence (5.3) we get the following formula describing the conditions on K that guarantee stability.

8a

h3 5 4 0 ^ (10K - 12 - 2a2 - 12a) > 0 ^ 2 + a2 + 2a - K > 0 ] _ [ (10a - 9K + 26 + a2 ) < 0 i 2 + a2 + 2a - K < 0 ] :

^

(10K - 12 - 2a2 - 12a) < 0

^

545 Performing quanti er elimination results in the stability condition K 2 ( 241 80 ; 144 ). The root locus is shown in Figure 6.3 where the roots are located inside the dashed 545 circle for K 2 ( 241 80 ; 144 ) (3:02; 3:78).

108

Chapter 6 Applications to Linear Systems 4

3

2

Imag Axis

1

0

−1

−2

−3

−4 −6

−5

−4

−3

−2 −1 Real Axis

0

1

2

Figure 6.3 The root locus of p(s; K) = 0; K > 0. Starting points: . End points: . The uncertainty corresponds to a locus in the gray-shaded region.

6.2 Feedback Design In this section we will apply quanti er elimination to problems in feedback design of linear time-invariant systems. The con guration of the closed-loop system is shown in Figure 6.4. + +

PSfrag replacements

F

G

-

Figure 6.4 The closed-loop system. A common approach to feedback design is to shape the \open-loop" return ratio

G(s)F(s; ) by choosing a number of parameters, in the controller, e.g., PID or

lead-lag parameters. The objectives of the design are often given in the following form [105]:

6.2 Feedback Design

109

(i) jG(i!)F(i!; )j > L ( 1); 0 < ! < !1 .

(ii) jG(i!)F(i!; )j < ( 1); !1 < ! < !2 . (iii) Gain margin > and phase margin > . (iv) The graph of G(i!)F(i!; ) is to remain outside a neighborhood of the point -1 + i 0. It is well known that robustness and performance speci cations on the closed-loop system can be translated to the above requirements. More about this subject can be found in any textbook on control systems, see e.g., [105]. The above objectives can all be formulated as logical combinations of equality and inequality constraints on the real and imaginary part of the return ratio. Let

x1 (!; ) = Re[G(i!)F(i!; )] and x2 (!; ) = Im[G(i!)F(i!; )]; where we assume that x1 and x2 are rational expressions in ! and . In what follows we suppress the !- and -dependence of x1 and x2 . The requirements (i) and (ii) are both constraints on the modulus of the complex number x1 + ix2 for

certain frequencies and hence correspond to the exterior and interior of two circles, respectively. This can be formalized as

0 < ! < !1 ! > !2

! !

x21 + x22 > L2 ; x21 + x22 < 2 :

(6.8) (6.9)

We notice that the denominator polynomial on the left hand side of these equations always is positive. Hence, we can clear fractions and get equivalent polynomial inequalities. The gain and phase margin are de ned as in Figure 6.5. Im

1= PSfrag replacements

-1

Re

Figure 6.5 The de nition of phase margin, and gain margin, .

110


The gain margin requirement can be stated as Am > or in terms of x1 and

x2 as

x2 = 0

!

x1 > -1=:

(6.10)

The constraint on the phase margin can be formulated as follows: if the modulus of the return ratio equals one and its real and imaginary part are negative then the angle to the negative real axis should exceed . On a more compact form we have (0 < < =2)

x2 + x2 = 1 ^ x < 0 ^ x < 0 1 2 1 2

!

x2 > tan(): x1

(6.11)

The \forbidden" area de ned in requirement (iv) is often taken as the interior of an M-circle which imposes some minimum degree of damping on the closed-loop poles [105]. The interior of an M-circle is

z z 2 C j 1 + z > M; M > 1 ;

where the condition M > 1 ensures that the M-circle encloses -1 + i0. Requirement (iv) corresponds to

x21 + x22 < M2 : (1 + x1 )2 + x22

(6.12)

Investigating the rational expressions in (6.8) to (6.12) one realizes that we can clear fractions without having to keep track on the directions of the inequality signs except in (6.11) where the inequality sign switches direction due to the assumption x1 < 0. We now have a number of formulas where the components are polynomials in x1 and x2 .

x2 + x2 = 1 ^ 1

2

[ 0 < ! < !1 ] ! x21 + x22 > L2 ; ! > !2 ! x21 + x22 < 2 ; x2 = 0 ! x1 > -1=; x1 < 0 ^ x2 < 0 ! x2 < x1 tan(); x21 + x22 < M2 (1 + x1 )2 + x22 ;

(6.13)

where !1 ; L; ; ; and M are prede ned constants chosen by the designer. The real and imaginary parts of the return ratio are rational functions in ! and but it is easy to rewrite the above formulas such that all inequalities become polynomial. If we now require the resulting formulas to be true for all real positive values of ! we get a quanti er elimination problem which can be solved using the algorithms in Section 5.5. We do not write down the general formula but instead give an example of the method.

6.2 Feedback Design

111

Example 6.4 Consider the following second order system and PI-controller G(s) = (s + 1)(2 s + 2) ;

F(s; kp ; ki ) = kp + ksi ;

which result in the open-loop return ratio

G(i!)F(i!; kp ; ki ) = s(2s(+kp1s)(+s k+i )2) : Suppose that the objective is to nd kp and ki such that the following speci cations are ful lled (i) jG(i!)F(i!; kp ; ki )j > 2; 0 < ! < 43 , (ii) jG(i!)F(i!; kp ; ki )j < 41 ; ! > 3, p (iii) The Nyquist curve is to remain outside the M-circle with M = 2. Following the previous discussion we rst derive rational expressions for the real and imaginary part of the return ratio, x1 and x2 , respectively 2

p ! + 4kp - 6ki x1 = Re[G(i!)F(i!; kp ; ki )] = -2k (1 + !2 )(4 + !2 ) ; p )!2 - 4ki : x2 = Im[G(i!)F(i!; kp ; ki )] = (!2k(i1 -+ 6k 2 ! )(4 + !2 )

Using these expressions the formulas corresponding to (6.13) become

8! 0 < ! < 43 ! k2p !2 + k2i > !2 (1 + !2 )(4 + !2 ) ; 8! ! > 3 ! 4k2p !2 + 4k2i < 412 !2 (1 + !2 )(4 + !2 ) ; 8! !6 + (5 - 4kp )!4 + (2k2p + 8kp - 12ki + 4)!2 + 2k2i > 0 : If we perform quanti er elimination on each of the above formulas, then the following conditions on kp and ki can be established

2304k2p + 4096k2i > 16425; 288k2p + 32k2i < 585; which are the exterior and interior of two ellipsoids and

ki 6= 0

^

8k6p - 8k5p - 48k4p ki + 56k3p k2i - 407k4p + 864k3p ki - 966k2p k2i + 432kp k3i - 27k4i - 1240k3p + 4116k2p ki - 4080kp k2i + 1188k3i - 940k2p + 2064kp ki - 898k2i - 144kp - 24ki + 36 < 0;

112


288k2p + 32k2i = 585

4

3

ki

(kp ; ki )

2

PSfrag replacements

p 2

2304k2p + 4096k2i = 16425

1

0

! j GFGF+1 j
-1. Observe that the solution set in this example is a non-convex set, which means that the problem cannot be treated successfully with a convex programming approach.

The Circle and Popov Criterion Conditions given on the Nyquist curve also appear in stability analysis of feedback schemes where one of the blocks is a static nonlinearity, see Figure 6.8. The small gain theorem and its implications for these schemes such as the circle and Popov criterion give various sucient conditions on the linear system to ensure stability, see [156].

k2 x

+

PSfrag replacements

+

G(s)

-

f()

f(x) k1 x

Figure 6.8 Left: The system con guration for the circle and Popov criterion. Right: Limits on the static nonlinearity (k1 = 0 in the Popov criterion.) The circle criterion states that the closed-loop system is stable if the Nyquist curve G(i!) does not enclose or enter the disk

k + k k - k 2 1 2 1 : D , z 2 C z + 2k 1 k2 2k1 k2

(6.14)

This is a condition of the same type as the requirement that the Nyquist curve is to remain outside an M-circle which was treated previously. We give two examples that show how quanti er elimination can be used to compute either bounds on a static nonlinearity or bounds on controller parameters to ensure stability.

114


Example 6.5 Suppose we are interested in bounds (k1 ; k2) on a static nonlinearity f() such that the linear system described by G(s) = s3 + 2s21+ 3s + 1 remains stable under feedback according to Figure 6.8. The real and imaginary part of the Nyquist curve become 2

x1 = (1 - 2!21)2-+2! (3! - !3 )2 ; + !3 x2 = (1 - 2!-2 )3! 2 + (3! - !3 )2 : and the Nyquist curve remains outside the disk D if the following inequality is satis ed for all ! k1 + k2 2 2 x1 + 2k k + x22 > (k24k-2 kk21 ) ; 1 2 1 2 where the !-dependence is suppressed. Clearing denominators and simplifying this expression give the following equivalent formula

8! !6 - 2!4 + (5 - 2k1 - 2k2 )!2 + k1 + k2 + k1 k2 + 1 > 0 ^ k1 > 0 ^ k2 > k1 ; (6.15) where the original conditions on k1 and k2 also are included. However, we also have to ensure that the Nyquist curve does not enclose D. This can be prevented in various ways, e.g., if for all ! > 0 x2 (!) > 0 ! x1 (!) 6= - k1 2 which corresponds to the formula

h

8! [ ! > 0 ] ! [ !3 - 3! > 0 ]

!

[ !6 - 2!4 + (5 - 2k2 )!2 + 1 + k2 6= 0 i ^ k1 > 0 ^ k2 > k1 ] : (6.16)

Performing quanti er elimination in (6.15) and (6.16) gives

0 < k < k 1 2

-27k21 k22 + 32k31 + 114k21 k2 + 114k1 k22 + 32k32 - 179k21 - 560k1 k2 - 179k22 + 390k1 + 390k2 - 575 < 0 (6.17) ^ k2 < 5 ^

6.2 Feedback Design

115

6

5

4

k2 PSfrag replacements

3

2

1

G(i!)

Re Im

0

0

1

2

k1

3

5

4

1

6

Im

0.5

Re -1.5

-1

-0.5

00

0.5

1

PSfrag replacements

k1 k2

-0.5

G(i!) -1

Figure 6.9 Top: Region of (k1; k2)-pairs that guarantees closed-loop stability (gray-shaded). Bottom: The Nyquist curve and a number of circles corresponding to dierent (k1; k2)-pairs just ful lling the conditions.

116


as sucient conditions for the bounds (k1 ; k2 ) on a static nonlinearity to ensure closed-loop stability. The semialgebraic set in the k1 k2 -plane described by (6.17) is shown in Figure 6.9. We also show the Nyquist curve and a number of limiting circles. The vertical line in this gure corresponds to k1 = 0; k2 = 115 32 . In the above example the system G(s) was xed. Additional \parameters to be determined" in the linear system as well as xed bounds on the nonlinearity can also be taken into consideration in various ways.

Example 6.6 Consider the feedback system in Figure 6.8 where

G(s) = KP + KsI s2 +1s + 1 and the bounds on the static nonlinearity f() is k1 = 1 and k2 = 2. Suppose that we are interested in bounds on the PI-controller parameters (KP ; KI ) that ensure stability of the feedback system. The real and imaginary part of the Nyquist curve G(i!) are 2 x1 = -K!P !4 -+!K2 P+-1 KI ;

KI - KP )!2 - KI : x2 = (! (!4 - !2 + 1)

The Nyquist curve remains outside the disc D, corresponding to the given (KP ; KI ) values, if the following inequality is satis ed for all !

2 3 x1 + 4 + x22 > 412 :

Clearing denominators and simplifying this expression gives the following equivalent formula

8! ! > 0 ! !6 - (3KP + 1)!4 + (1 + 3KP + 2K2P - 3KI )!2 + 2K2I > 0 (6.18) ^ KP > 0 ^ KI > 0 ; where we also have included two additional conditions on KP and KI . Performing quanti er elimination in (6.18) gives the following equivalent formula

K > 0 ^ K > 0 ^ P I

108K4I - 324KP K3I - 216K3I + 315K2P K2I + 414KP K2I + 127K2I - 36K4P KI - 198K3P KI - 294K2P KI - 162KP KI - 30KI - 4K6P + 12K5P + 71K4P + 108K3P + 74K2P + 24KP + 3 > 0 :

(6.19)

6.2 Feedback Design

117

The solution set to this system of inequalities is shown in Figure 6.10. Observe that we have not given any conditions that exclude the case when the Nyquist curve encloses D. The upper of the two disjoint parts of the solution set corresponds to parameter values for Nyquist curves that enclose D. This can be prevented in various ways, e.g., by the additional constraints x1 > -1 _ x2 6= 0. Including these in (6.18) before QE results in the additional constraints KP > 0 ^ KI < KP + 1, i.e., only the lower of the disjoint parts of the original solution set corresponds to a Nyquist curve satisfying the circle criterion. To the right in Figure 6.10 the disc D and two Nyquist curves are shown, one from each part of the original solution set. 6

G (i!)

5

PSfrag replacements

KI

KI = KPreplacements +1 PSfrag

4

KP KI

3

Re Im

G (i!) G (i!) D

KI = KP + 1

2

-3

-2

-1

Im Re

D

G (i!) -1

1

-2 0

0

1

2

3

KP

4

5

6

Figure 6.10 Left: Region of (KP ; KI )-values that guarantees closed loop stability (gray-shaded). Right: The disc D and two Nyquist curves.

To infer stability using the Popov criterion, one has to show that the so called Popov curve lies to the right of a straight line, whose location depends on the nonlinearity, see [156]. This condition can easily be given as a formula involving quanti ed variables and stability can be proved by deciding if the formula is True.

The Describing Function Method

The describing function method [9] is another approach to detect instability or limit cycles in feedback systems with static nonlinearities. Conditions on design parameters to avoid instability can be calculated using QE as in the above example. According to the describing function method the negative reciprocal of the describing function : Yf-(1C) ; C 2 [0; 1] corresponds to -1 in the Nyquist criterion [9]. Hence, the system is stable if the Nyquist curve do not enclose or cross this curve. The curve can be embedded in a disc, ellipsoid, or any region in the complex plane that can be described by a semialgebraic set. As in the previous example

118


conditions on design parameters that guarantees stability can be computed using QE. We demonstrate this in the following example.

Example 6.7 Consider the same linear system G(s) as in Example 6.6 and let

the nonlinearity be a saturation at the input of G(s). Suppose we want to calculate conditions on KP and KI that guarantees stability of the closed loop system. The describing function for a saturation 8 >
1 f(e) = > e; jej 1 : -1; e < -1 is given by

q

Yf(C) = 2 arcsin( C1 ) + C1 1 - C12 ;

C > 1:

The curve for this nonlinearity corresponds to x1 -1 ^ x2 = 0, where x1 and x2 are the real and imaginary part, respectively. Hence, to avoid crossing the real and imaginary part of the Nyquist curve have to satisfy x1 > -1 _ x2 6= 0, i.e., the same condition we imposed on the Nyquist curve in the previous example. Performing QE then gave us the conditions KP > 0 ^ KI < KP + 1.

6.3 Summary In this chapter we have investigated a number of applications of quanti er elimination to linear control systems. We have shown how to compute feasible values for parameters of a stabilizing controller, even in the case of uncertainties. Classical feedback design with gain and phase margin constraints together with additional constraints on the open loop gain have been put into the quanti er elimination framework. We have also studied the in uence of static nonlinearities in the feedback loop using the circle criterion and describing function method. Both of these methods can be applied to feedback systems with static nonlinearities in the loop using quanti er elimination.

7 Applications to Nonlinear Systems In this chapter we discuss some applications of quanti er elimination to nonlinear control theory. Since the basic framework is real algebra and real algebraic geometry we consider dynamic systems described by dierential and non-dierential equations and inequalities in which all nonlinearities are of polynomial type. This represents a rather large class of systems and it can be shown that systems where the nonlinearities are not originally polynomial may be rewritten in this polynomial form if the nonlinearities themselves are solutions to algebraic dierential equations. For more details on this, see [103, 135] In the control community there is a growing interest to use inequalities in modeling and control of dynamic systems, see e.g., [158]. In optimal control it is common to have inequality constraints both on the control and system variables, see [26, 101]. However, the existence of algorithms for symbolic computation with systems of polynomial equations and inequalities have still not yet been fully recognized. The author believes that there will be an increased interest for this kind of algorithms in the control community since many problems, especially in non-linear control theory, can be formulated as real polynomial systems combined with logical operations. The chapter is organized as follows. In Section 7.1 we give motivations for the work in this chapter. In Section 7.2 we show how to compute the states that can be made stationary by use of an admissible control. The results are applied in a computation of stationary orientation of an F-16 aircraft and for investigations of pitch control of a missile. Section 7.3 deals with the same problem but also require asymptotic stability. In Section 7.4 we temporarily abandon the polynomial framework and derive some results on when it is possible to move the state of 119

120

Chapter 7 Applications to Nonlinear Systems

a nonlinear system between equilibria. These results are then utilized together with quanti er elimination in Section 7.5 to compute the range of possible output set-points for a polynomial system subject to control constraints. Finally, curve following and constrained reachability are presented in Section 7.6. A summary of the chapter is given in Section 7.7.

7.1 Introduction Given a classical state space description of a dynamic system and constraints on the states as well as control signals we consider a number of problems which can be solved by eliminating quanti ers: Which states correspond to equilibrium points of the dynamic system for some admissible control signal and which stability properties do these equilibrium points have? Which output set-points do these equilibrium points correspond to? Given two equilibrium points, when is it possible to transfer the state between them? Given a parameterized curve in the state space of the system, is it possible to follow the curve by use of available control signals? Given a set of parameterized curves, which states can be reached by following one of these curves? Stationary (equilibrium, critical) points play an important role in both analysis and design of dynamic systems and for synthesis of control laws. These points are the possible operating points of the system and often a control law is designed such that the state of the system will return to such a point after moderate disturbances. The set of equilibrium points of a dynamic system is parameterized by the available control signals and the rst and second problem address the construction of this set for polynomial systems. The range of possible output set-points for a system subject to control constraints is an important design issue. We will give conditions on when it is possible to transfer the state and hence the output between dierent set-points. Utilizing the symbolic nature of the methods, these results can be used for design since we can investigate how the output range depends on design parameters. The second last problem is a natural question in many control situations where the objective is to steer the dynamic system from one point to another along a certain path. Observe that the prescribed path belongs to the state (or output) space, which implies that the whole (or a part of the) system dynamics is speci ed. Hence this is an extension of the motion planning problem also taking into account the system dynamics. The last problem is a generalization to a constrained form of computable reachability.

7.2 Stationarizable Points

121

7.2 Stationarizable Points It will be assumed that the dynamic system is described by a nonlinear dierential equation written in state space form x_ = f(x; u) (7.1)

y = h(x) where x is an n-vector, u an m-vector, y a p-vector and each component of f and h is a real polynomial, fi 2 R[x; u]; hj 2 R[x]. The x; u, and y vectors will be

referred to as the state, control, and output of the system, respectively. Suppose also that the system variables have to obey some additional constraints x 2 X and u 2 U ; (7.2) where X and U are semialgebraic sets which de ne the constraints on the state and control variables. We call x 2 X the admissible states and u 2 U the admissible controls. Restrictions on amplitudes of control signals are common in real applications but hard to take into account in many classical design methods other than by simulation studies. A variety of constraints can be represented by semialgebraic sets, e.g., amplitude and direction constraints.

Example 7.1 Let F be a two dimensional thrust vector which can be pointed in any direction, and whose magnitude, jFj can be varied between 0 and Fmax . Let u1 = cos(); u2 = sin() and u3 = jFj: Then the semialgebraic set describing these constraints becomes

U = u 2 R3 u21 + u22 = 1

^

0 u3 Fmax

Similarly, constraints on the states may originate from speci cations on the system outputs, e.g., jh(x)j . Using Boolean combinations of polynomial equations and inequalities it is possible to represent discontinuous and non-smooth static input-output maps such as relays and saturations.

Example 7.2 Consider the following saturation function 8 < -1; u1 -1 u2 = : u1 -1 < u1 < 1 1; u1 1

The set of admissible controls can then be described by the following semialgebraic set

U = f u 2 R2 j ( u 1 - 1 ! u 2 = - 1 ) ^ ( -1 < u1 < 1 ! u2 = u1 )

^

( u1 1 ! u2 = 1 ) g:

122


The main question in this section concerns equilibrium or stationary points of a dynamic system, i.e., solutions of (7.1) which correspond to constant values of the admissible states and controls. In other words, we are interested in those admissible states for which the system can be brought to rest by use of an admissible control. The conditions for a point, x0 to be stationary are easily seen to be f(x0 ; u0 ) = 0, where x0 2 X and u0 2 U . For the class of dynamic systems considered here, this set of stationarizable states turns out to be semialgebraic.

De nition 7.1 The stationarizable states of system (7.1) subject to the constraints (7.2) is the set of states satisfying the formula

h

S (x) , 9u f(x; u) = 0

^

X (x)

^

i

U (u) :

(7.3)

The computation of a \closed form" of the set of stationarizable states, i.e., an expression not including u, is a quanti er elimination problem and hence this set is semialgebraic.

Example 7.3 Consider the following system x_ 1 = -x1 + x2 u x_ 2 = -x2 + (1 + x21 )u + u3 subject to the constraints

(7.4)

- 21 u 21 :

According to De nition 7.1 the stationarizable set is described by the formula

h

9u -x1 + x2 u = 0

^

-x2 + (1 + x21 )u + u3 = 0

^

i

- 21 u 21 ;

which after quanti er elimination becomes

h

x42 - x31 x22 - x1 x22 - x31 = 0

^

[ x2 + 2x1 0

_ x2 - 2x1

i

0] :

In this case the stationarizable set is easy to visualize, see Figure 7.1. As an example of a speci c application of the stationarizability result we will consider nonlinear dynamics for an aircraft and a missile.

7.2.1 Equilibrium Computations for Aircraft Dynamics

Nonlinear aircraft dynamics for conventional aircrafts is usually described by the so called 6-DOF (degrees of freedom) equations [142], which is a classical state space description of a nonlinear system, written on the form x_ = f(x; u); (7.5)


123

x2 1

0.5

x1 0.2

0.4

-0.5

PSfrag replacements

-1

Figure 7.1 The stationarizable set (bold curve) of system (7.4) subject to the control constraints juj 12 . where the state vector x has 12 components and comprise the velocity vector, the vector of Euler angles, the angular rate vector, and the position vector of the aircraft; all states are measured in some convenient coordinate system. The control surface de ections are represented by the control variables u. The function f contains the rigid body dynamics and the aerodynamic forces and moments. The latter quantities are often given by tables of values initially measured in wind tunnel experiments and later on updated during real ight. For computational reasons these quantities are often represented by polynomial approximations. The question of nding an equilibrium of the system under constraints on the actuators then results in a system of polynomial equations and inequalities. If restrictions are put on the linearized dynamics at the equilibrium, further polynomial constraints are produced. Solving the problem purely by numerical methods is made complicated by the fact that multiple solutions might exist. The aim of this section is to show how quanti er elimination and cylindrical algebraic decomposition, can be used to solve the problem. We consider the computation of possible stationary orientations (; ) of the aircraft w.r.t. its wind xed coordinate system, see Figure 7.2. The orientation angles are called angle of attack and sideslip angle, respectively. An investigation of the state equations reveals that this problem is equivalent to keeping the angular rate components of the state vector constant and equal to zero. The derivatives of the angular rate components of the state vector are given by the so called moment

124


R

y-axis (body)

el

at

α

iv

e

w

in

d

z-axis (body)

β x-axis (wind)

x-axis (body) x-axis (stability)

Figure 7.2 The orientation of an aircraft with respect to the air ow. equations of the aircraft model (7.5) P_ = (c1 R + c2 P)Q + c3 L + c4 N Q_ = c5 PR - c6 P(P2 - R2 ) + c7 M R_ = (c8 P - c2 R)Q + c4 L + c9 N;

(7.6)

where P; Q and R are the components of the angular velocity vector of the body xed coordinate system. The ci are constants determined by the moments and cross-products of inertia of the aircraft and L; M, and N are the rolling, pitching, and yawing moment, respectively. From the equations (7.6) we observe that the condition for the aircraft to have stationary orientation, P = Q = R 0, is equivalent to L = N = M = 0 for all values of c3 ; c4 ; c7 ; and c9 satisfying c3 c9 -c24 6= 0 and c7 6= 0. The moments L; M, and N are proportional to the dimensionless aerodynamic coecients Cl ; CM , and CN , respectively, which usually are measured and given by look-up tables and interpolation routines [142]. These quantities are functions of the angle of attack , the sideslip angle , and the aileron, elevator, and rudder de ections. Now, if the aerodynamic coecients are approximated by polynomial functions, the set of and values corresponding to a stationary orientation of the aircraft can be described by the solution set of the following system of polynomial equations and inequalities

Cl (; ; u) = 0; CM (; ; u) = 0; CN (; ; u) = 0; jui j 1; i = 1; 2; 3;

(7.7)

where the inequalities are normalized constraints for the aileron, elevator, and rudder de ections, respectively. Using quanti er elimination we can now eliminate the ui from (7.7) and obtain closed form expressions in and describing the possible stationary orientations of the aircraft.


125

To demonstrate the application of quanti er elimination in computations regarding aircraft dynamics we give two examples. The rst example is the computation of stationarizable states of the F-16 ghter aircraft and the second example deals with the maneuvering of a missile. In both cases the dynamics is subject to constraints on the control surface de ections.

F-16 Aircraft

The following expressions are scaled versions of the aerodynamic coecients of an F-16 aircraft. The polynomial approximations have been obtained from look-up tables and interpolation routines in [142]. Cl = CM = CN =

38x2 - 170x1 x2 + 148x21 x2 + 4x32 2 3 2 2 +u1 (-52 - 2x1 + 114x1 - 79x1 + 7x2 + 14x1 x2 ) 4 2 2 2 2 2 3 4 +u3 (14 - 10x1 + 37x1 - 48x1 + 8x1 - 13x2 - 13x1 x2 + 20x1 x2 + 11x2 ) 2 3 2 -12 - 125u2 + u2 + 6u2 + 95x1 - 21u2 x1 + 17u2 x1 2 2 3 -202x1 + 81u2 x1 + 139x1 139x2 - 112x1 x2 - 388x21 x2 + 215x31 x2 - 38x32 + 185x1 x32 2 3 2 2 +u1 (-11 + 35x1 - 22x1 + 5x2 + 10x1 - 17x1 x2 ) 2 2 3 2 4 2 2 4 +u3 (-44 + 3x1 - 63x1 + 34x2 + 142x1 + 63x1 x2 - 54x1 - 69x1 x2 - 26x2 ) -

Here x1 and x2 are normalized angle of attack and sideslip angle, respectively and u1 , u2 , and u3 the aileron, elevator, and rudder de ections. The angles of attack and sideslip angles corresponding to constant orientation of the aircraft can now be described by a formula using quanti ers and Boolean operators together with (7.7) as follows

9u1 9u2 9u3 Cl = 0 ^ CM = 0 ^ CN = 0 ^ u2i 1 : Elimination of u1 and u3 is easy in this case since they appear linearly in the expressions. To eliminate u2 we utilize qepcad and get the result [ [ p 1 0 ^ p2 0 ] _ [ p3 0 ^ p2 0 ] ] ^ (7.8) [ p4 0 ^ p5 0 ^ p6 0 ^ p7 0 ] ; where

p1 = 139x31 - 283x21 + 133x1 + 108; p2 = 139x31 - 121x21 + 91x1 - 130 p3 = 11698571x61 - 33047468x51 + 27950104x41 - 51263858x31 + 114876460x21 - 27878444x1 - 46912705 and p4 ; : : : ; p7 are polynomials in x1 and x2 of total degree 7 of about 40 terms

each. However, although the output is quite complicated, the elimination of the control variables reduce the dimension of the problem and the two dimensional solution set (7.8) can easily be visualized. The solution set is shown in Figure 7.3.

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Chapter 7 Applications to Nonlinear Systems 1

0.5

x2

0

-0.5

PSfrag replacements

-1 0

0.2

0.4

x1

0.6

0.8

1

Figure 7.3 The possible stationary orientations if juij 1 (white). In Figure 7.3 we observe that for zero sideslip angle there are no constraints (in the valid region of approximation) on the angle of attack, i.e., the aircraft can be kept stationary at any angle of attack between 0 and 1 (normalized values). Furthermore, the maximal constant sideslip angle can be obtained at an angle of attack of approximately 0:4. A number of other questions related to equilibrium calculations can also be posed. What happens if some actuators are further constrained, e.g., jammed or damaged? In which regions of the x1 x2 -plane can we guarantee that certain moments can be generated by the control surfaces? How does the computed equilibrium region depend on a design parameter or an uncertainty? As an example we investigate the following problem. Suppose that the de ection of the elevator has been further constrained to ju2 j 21 . For what angles of attack can the elevator generate pitching moments corresponding to an aerodynamic coecient of magnitude greater than 40? The QE formula becomes

9u2 [ C2M > 1600 ^ 4u22 1 ]: Eliminating u2 in this formula gives 0 < x1 < a, where a 0:739182 is the unique root between 0 and 1 of 556x3 - 646x2 + 355x - 134 = 0: The answer to this question is a region (in this case an interval) where we have some guaranteed robustness against disturbances, since we can generate compensating moments of a prescribed magnitude.


127

Note also that it is easy to incorporate symbolic design parameters or parameters representing system uncertainties in the above formulas and investigate their eect on the computed regions, since QE is a symbolic method. A prediction of possible, stationary orientations of an aircraft is usually carried out by non-symbolic techniques, typically simulation studies and test ights. The advantage of the above approach is that we can get closed form expressions for stationary orientations in terms of design parameters of the aircraft. Furthermore, these expressions can then be utilized to choose optimal values of these parameters.

A Missile

Here we consider the aerodynamics for pitch-axis control of a missile, see Figure 7.4.

CM PSfrag replacements

u1

x1 CZ

Figure 7.4 Normal force and moment on a missile. The quantities that determine the motion in the vertical plane of the missile are the normal force and pitching moment, which in turn are functions of the angle of attack, sideslip angle, and elevator. The following is a polynomial expression for the normal force (neglecting gravity)

CZ (x1 ; x2 ; u1 ) = -13x1 - 120x21 + 390x31 - 100x1 x22 1350x51 - 1400x31 x22 + 875x1 x42 - 4u1 and the pitching moment

CM (x1 ; x2 ; u1 ) = -13x1 - 10x21 + 4x22 + 460x31 - 520x1 x22 2600x51 + 3000x31 x22 + 680x1 x42 - 35u1 ; where x1 is the angle of attack, x2 is the sideslip angle, and u1 is the elevator angle.

These polynomial approximations come from [44]. An analysis of how quick the missile can turn in the vertical plane can be carried out by QE methods. The missile describes a circular motion if the normal force is constant and the pitching moment is zero, the larger force the smaller radius. This implies that limits on what normal forces that can be generated, limit the possible turn radii. The question how quick the missile can turn, subject to elevator de ection constraints, can thus be formulated as follows: What normal

128


forces can be generated, for constant speed of rotation (i.e., zero pitching moment), if the maximum elevator angle is limited? The analysis is carried out in the vertical plane so the sideslip angle is x2 = 0. Adding some constraints on the angle of attack, elevator, and normal force gives a quanti er formula in the variables f, x1 , u1

9x1 9u1

C (x ; 0; u ) = f ^ C (x ; 0; u ) = 0 ^ Z 1 1 M 1 1 0 x1 103

^ -401

u1 201

^

-8 f 0 :

Quanti er elimination via cylindrical algebraic decomposition shows that the

f-axis is divided into 51 cells. An investigation of the truth value of the above formula over each cell (interval) shows that there exist x1 and u1 satisfying the formula for f in the intervals [f1 ; f2 ]; [f3 ; 0] where f1 = -7:5993, f2 -3:148712 is the unique real root between -13=4 and -25=8 of 38932892876800000x5 + 1718497027686400000x4 +24272290834718720000x3 + 138926900972696832000x2 +326132672903058884000x + 250374264550106371777 = 0; and f3 -1:504850 is the unique real root between -13=8 and -3=2 of 38932892876800000x5 + 1718497027686400000x4 +24272290834718720000x3 + 138926900972696832000x2 +326132672903058884000x + 250374264550106371777 = 0: The intervals correspond to two disjoint \turn regions" shown to the left in Figure 7.5. To the right in Figure 7.5 the moment and force functions for u1 = 0 are shown. Here we can clearly see that the constraint on the elevator makes it impossible to compensate for large negative moments. This is the reason for the limits on the turn radius. The region of smaller turn radii in Figure 7.5 is due to the shape of the moment curve. However, it is probably hard to control the missile to perform turns corresponding to the region of smaller radii due to the intermediate region of non-realizable turns.

7.3 Stability In stability theory for nonlinear dynamic systems one is often interested in the character of the solution in the neighborhood of a stationary point. If all solutions starting in a small neighborhood of a stationary point x0 stays within this neighborhood for all future times, then the stationary point is called stable. If in

7.3 Stability

129 CM

1

x1

0.3

0.4

x1

0.3

0.4

0

0.1

0.2

0

0.1

0.2

0

-1

-2

-3

-4

0 -2 -4

PSfrag replacements

-6 -8 -10 -12

CZ

Figure 7.5 Left: Possible turns. Right: The normal force and pitching moment (u1 = 0).

addition the solutions converge towards x0 , the stationary point is called (locally) asymptotically stable. For an extensive treatment of stability of dynamic systems see [63]. The following theorem gives a sucient condition for asymptotic stability of a stationary point.

Theorem 7.1 Let x0 be a stationary point of system (7.1) corresponding to u = u0 . Then x0 is asymptotically stable if all eigenvalues of fx (x0 ; u0 ) have strictly negative real parts.

Proof See [63].

2

Here fx denotes the functional matrix (or Jacobian) of f(x; u) regarding u as a constant. Since the eigenvalues of a matrix are the zeros of its characteristic polynomial we are interested in determining if all the zeros of this polynomial have strictly negative real parts. This question can be answered in a number of dierent ways by examining the coecients of the characteristic polynomial, e.g., by the criteria of Hurwitz, Routh or Lienard-Chipart. In Theorem 6.1 we presented the Lienard-Chipart criterion. This gives a polynomial criterion for testing the stability of a stationary point. The characteristic polynomial of fx (x0 ; u0 ) in Theorem 7.1 is det (In - fx (x0 ; u0 )), i.e., a polynomial

130


in with coecients that are polynomials in x0 and u0 . Utilizing Theorem 6.1 we get n polynomial inequalities in x0 and u0 , which are sucient conditions for the stationary point x0 to be asymptotically stable. We summarize the above discussion in the following theorem.

Theorem 7.2 The stationarizable states of system (7.1) subject to the constraints (7.2) that are asymptotically stable are given by the formula

h

i

AS (x) , 9u f(x; u) = 0 ^ X (x) ^ U (u) ^ Re(eig(fx (x; u))) < 0 ; (7.9) where Re(eig(fx (x; u))) < 0 denotes the set of inequalities corresponding to Theorem 6.1.

Example 7.4 Consider the following system

x_ 1 = -x31 + x2 x_ 2 = -x21 - x2 - x32 + u

subject to the constraints We get the functional matrix

(7.10)

u2 1:

21 1 fx (x; u) = --3x 2x1 -1 - 3x22

and its corresponding characteristic polynomial

2 + 3 x1 2 + 1 + 3 x2 2 + 3 x1 2 + 9 x1 2 x2 2 + 2 x1 : The inequalities Re(eig(fx (x0 ; u0 ))) < 0 become 3 x1 2 + 1 + 3 x2 2 > 0; 3 x1 2 + 9 x1 2 x2 2 + 2 x1 > 0; where the rst inequality is trivially satis ed for all real x1 and x2 . The asymptotically stable stationarizable points of system (7.10) are given by formula (7.9)

h

AS (x) = 9u - x31 + x2 = 0 ^ -x21 - x2 - x32 + u = 0 u2 1 ^ i 2x1 + 3x21 + 9x21 x22 > 0 ;

^

which after quanti er elimination becomes

h

AS (x) = - x31 + x2 = 0 ^ - x21 - x2 - x32 - 1 0 ^ x21 + x2 + x32 - 1 0 x > 0 _ 3x + 9x x2 + 2 < 0 i; 1 1 1 2

see Figure 7.6.

^

(7.11)

7.4 Moving the State Between Equilibria

131

x2

2

1

-2

PSfrag replacements

-1

0

x1 0

1

2

-1

-2

Figure 7.6 The set of states of system (7.10) which is stationarizable and

asymptotically stable (bold curve). Any point on the cubic which is not in the dark gray region is a stationarizable state. The part of the cubic in the light gray region corresponds to stationary points which are not asymptotically stable.

Observe that points which are stationarizable but not asymptotically stable equilibrium points can be chosen as operating points in applications as well, but the control in this case has to be active which in general is a harder problem (e.g., the stabilization of an inverted pendulum).

7.4 Moving the State Between Equilibria In many industrial applications the controlled system is often operated at several dierent equilibrium points, corresponding to dierent set-points. It is then necessary to transfer the system state from one equilibrium (or stationarizable) point to another. In this section we will investigate conditions under which it is possible to do this. We will concentrate on the situation where two equilibrium points in the x u-space are connected via a curve consisting of equilibrium points. Some problems related to the ones treated here have been studied in the literature. In [65] a single input nonlinear system is uniformly approximated such that it agrees to rst order on the set of equilibrium points and the approximate system is input-to-state linearizable in a neighborhood of the equilibrium points. Then a control law for the feedback linearized system is used to control the system in a neighborhood of the equilibrium points. A time-varying linear quadratic controller is derived in [95] for changing the set-point of a nonlinear system.

132


In [141] a robust controller design method is proposed. The system is linearized around all stationary points and a robust controller is designed for each such system. The set-point can then be changed by switching among the controllers.

Stationary State Transfer

Consider the following nonlinear dynamic system x_ = f(x; u); f : Rn Rm ! Rn ; (7.12) where f(x; u) is a twice continuously dierentiable function of the state x and control u. Observe that we do not impose any particular structure on f or constraints on the state or control. The stationary points or equilibria are given by the set E = (x; u) 2 Rn Rm j f(x; u) = 0 : (7.13) Among these the asymptotically stable points are given by Eas = (x; u) 2 Rn Rm j f(x; u) = 0

Re(eig(fx (x; u))) < 0 : (7.14) The stationarizable points of (7.12) are given by the projection of the set E onto the state space, i.e., S = x 2 Rn j 9u 2 Rm s.t. f(x; u) = 0 : (7.15) If m = 1 we have a curve of stationarizable points and if m > 1 we get a hypersurface implicitly parameterized by the control u. Similarly the set of asymptotically stable stationarizable points of (7.12) are given by

^

AS = x 2 Rn j 9u 2 Rm s.t. f(x; u) = 0 ^ Re(eig(fx (x; u))) < 0 : (7.16) The set AS is important since it consists of all operating points of the system (7.12),

which can be used without a stabilizing controller. Observe that both S and AS are semialgebraic sets according to De nition 7.1 and Theorem 7.2 if the state equations (7.12) have polynomial right hand sides. Here we will investigate the following problem: Given two points xa 2 AS and xb 2 AS , is it possible to transfer the state from xa to a neighborhood of xb in nite time? For a linear system, x_ = Ax + Bu, we know that this is possible for any two points in AS if A is Hurwitz since the system then is globally asymptotically stable for any constant u u0 , which corresponds to the stationary point x0 = -A-1 Bu0 . For nonlinear systems the situation is much more intricate. First, it is usually hard to determine global asymptotic stability of the system. Second, E or Eas may consist of several non connected components even if S or AS are connected. An example of this is the scalar system x_ = (u2 + 4u - x + 3)(u2 - 4u + x + 3)(x - u): (7.17)


133

5

4

3

2

x

1

0

−1

−2

−3

PSfrag replacements

−4

−5 −5

−4

−3

−2

−1

u

0

1

2

3

4

5

Figure 7.7 Equilibrium points of (7.17). Solid lines: asymptotically stable linearizations, dashed lines: unstable linearizations. The set E consists of two parabolas and one straight line in the u x-space, see Figure 7.7. The linearizations along the parabolas are asymptotically stable, while the linearizations along the straight line are unstable. The set Eas is thus not connected, while its projection AS on the x-axis is. In practice it is often possible to transfer the system state between two asymptotically stable stationary points on the same component by changing the control input slowly between its corresponding initial and nal values. This suggests that if we can only keep the state suciently close to the set of asymptotically stable stationarizable points during the transfer we will succeed. In this section we will show that given a curve AS entirely in one component of AS , starting at xa and ending at xb , it is possible to transfer the state from xa to a neighborhood of xb in a nite amount of time. To estimate the region of attraction of a curve of asymptotically stable stationarizable points we need the following lemma.

Lemma 7.1 Let As : ! Rn n be a continuous matrix valued function where

R is a closed interval. Furthermore, assume that As is Hurwitz for all s 2 .

Then there exist uniform bounds on the solution of the following Lyapunov equation ATs Ps + PsAs = -I; PsT = Ps; (7.18) i.e., there exists constants m < M < 1 such that mI Ps M I; 8s 2 : (7.19)

134


Here R S means that S - R is a positive semide nite matrix.

Proof Since As is Hurwitz, it is well known that that equation (7.18) has a

unique positive de nite solution, see [140]. Now, (7.18) is a system of linear equations and the uniqueness of its solution implies that the corresponding determinant is nonzero. This means that the entries of Ps are continuous functions of the entries of As since the solution of a nonsingular system of linear equations can be expressed as rational functions of the entries of the system matrix and the right hand side vector (this follows from the formula for the inverse of a matrix, M-1 = det(M)-1 adj(M)). The eigenvalues of a matrix are continuous functions of the matrix entries, see [140]. This means that the smallest and largest eigenvalue, min(Ps) and max(Ps) respectively, of a symmetric matrix also are continuous functions of the entries of Ps. Using that compositions of continuous functions is continuous we have that min(Ps) and max(Ps) are continuous functions of s. Now, min(Ps) and max (Ps) are continuous functions over a closed interval . Hence, they attain nite minimum and maximum values on , i.e., there exists m < M < 1 such that

min(Ps) m and max(Ps) M

8s 2 :

To show the matrix inequalities in the lemma we use the fact that the symmetric matrix Ps can be written as Ps = QTs s Qs , where QTs Qs = I and s is diagonal. This gives

Ps - mI = QTs (s - m I)Qs 0 since m is less than or equal to all entries of s . The other inequality is shown similarly and the lemma follows. 2

Theorem 7.3 Let xa and xb be two points in AS that can be joined by a continuous curve AS . Also assume that can be chosen to be the projection of a continuous curve Eas on AS . Then the state of system (7.12) can be transferred from xa to an arbitrary neighborhood of xb in a nite amount of time. Proof Let the curve be parameterized as u = us = (s) and x = xs = (s)

where we can assume, without loss of generality, that xa = (0) and xb = (1). Let xs denote an arbitrary point on and us the corresponding control (we have f(xs ; us ) = 0). Let be the set = x j dist(x; ) where dist is the Euclidean distance function. Since f is twice continuously dierentiable, the linearization of (7.12) on (keeping us xed) can be written d dt (x - xs ) = As (x - xs ) + gs (x); As = fx (xs ; us )

(7.20)


135

where jgs (x)j Kjx - xs j2 ; x 2 and K = supdist(x; Now, de ne the following positive de nite function

jfxx (x; us )j, see [136].

)

Vs(x) = (x - xs )T Ps(x - xs ); PsT = Ps 0: (7.21) Using Lemma 7.1 it is easy to show that Vs(x) is a local Lyapunov function for system (7.12) at xs . Here As = fx (xs ; us ) = fx ((s); (s)) ful lls the conditions of Lemma 7.1 and Ps can be uniformly bounded on . Furthermore, we have V_ s(x) = (x - xs )T (ATs Ps + PsAs )(x - xs ) + 2(x - xs )Psgs (x) = -jx - xs j2 + 2(x - xs )T Psgs (x): Here we used (7.18) to get the second inequality. Now, using the bounds on Ps and gs (x) from (7.19) and (7.20) we get V_ s(x) -jx - xs j2 + 2M Kjx - xs j3 = -jx - xs j2 (7.22) where = (1 - 2M Kjx - xs j). From here on we assume that jx - xs j r1 , r1 = min(; 3M1 K ), which gives 13 and V_ s(x) - 31 jx - xs j2 .

Now,

jx - xs j2 = (x - xs )T I(x - xs ) 1m (x - xs )T Ps (x - xs ) = 1m Vs(x)

(7.23)

which gives

V_ s(x) - 31m Vs(x): Integrating both sides of the inequality between t0 and t and using Gronwall's inequality [155] gives the following estimate of the rate of convergence to the state

xs 2

1

Vs(x(t)) Vs(x(t0 ))e- 3m (t-t0)

(7.24)

which, using (7.23), gives the following estimate 1 (t0 )) - 3 (t-t0 ) jx(t) - xs j2 Vs(xm e m

(7.25)

Now, using the upper bound on the Lyapunov function, we nally get the following estimate of the rate of decrease of the distance between x(t) and xs 1

3m (t-t0) : jx(t) - xs j2 jx(t0 ) - xs j2 M me

(7.26)

The condition jx - xs j r1 q restricts the region of states for which inequality (7.26) is valid to jx(t0 ) - xs j < r1 Mm = r r1 . This gives a uniform estimate on both the region of attraction and rate of convergence for points on .

136


We can de ne a nite number of points x1 ; : : : ; xN on in such a way that dist(xj ; xj+1 ) < r=3, j = 1; : : : ; N - 1, x1 = xa and xN = xb . Let uj ; j = 1; : : : ; N be the corresponding control values on . We introduce the notation Bd (x0 ) for an open ball of radius d centered at x0 . We now describe how to transfer the state from Br=3 (xj ) to Br=3 (xj+1 ). Assuming that the state belongs to Br=3 (xj ), choose the control u = uj+1 . The state then also belongs to the ball Br (xj+1 ). After at most T = 3m log( 9mM ) units of time (the time it takes to reduce the radius to r 3 ) we know that the state belongs to Br=3 (xj+1 ) from the estimate (7.26). The same procedure can now be used to reach Br=3 (xj+2 ) etc. Hence, an estimate of the total time for transferring the state between xa and Br=3 (xb ) is NT . Using again the estimate (7.26) we conclude that the state can come arbitrarily close to xb in a nite amount of time. 2

Remark: From the proof it is clear that the state can also be transferred from an initial point in a neighborhood of xa to a neighborhood of xb in nite time.

Example 7.5 Consider system (7.17) whose stationary points are shown in Fig-

ure 7.7. It follows immediately from Theorem 7.3 that it is possible to move between arbitrary states in (-1; 1] or in [-1; 1) by moving along one of the parabolas. Actually one can show that it is possible to move e.g. between x = -5 and x = 5 by applying Theorem 7.3 twice. The theorem shows that it is possible to move from x = -5 to x = 0 along the right parabola. The control can then be momentarily switched to move to the neighborhood of an equilibrium on the left parabola. Theorem 7.3 and the above remark can then be used to show that the state can be moved along the left parabola to x = 5. In the next example we apply Theorem 7.3 to a DC-motor controlled by the eld current, which is a bilinear system.

Example 7.6 Consider the following model of a DC-motor

v0 = L dtd i + Ri + K1 u! (7.27) J!_ = -B! + K2 ui where i is the current, the control signal u is the eld current, ! is the angular velocity, and v0 the applied voltage. We let the state variables be x1 = i - vR0 ; x2 = ! and rewrite the equations (7.27) in state space form

x_ 1 = - L1 x1 - KLR1 ux2 2 v0 u: x_ 2 = - BJ x2 + KJ2 ux1 + KJR

(7.28)


137

The stationary points of (7.28) are parameterized by u according to

K1 K2 v0 u2 ; x = K2 v0 u ; x1 = - R(BR + K K u2 ) 2 BR + K K u2 1 2

1 2

which corresponds to an ellipse (-1 < u < 1) in the state space. Let u = s, then the matrix for a linearization of the system is

"

#

- 1 - K1 s As = K2Ls -LR B : J J Using the Routh stability criterion we get the following inequalities

J + BL > 0; BR + K1 K2 s2 > 0; JL JLR which are satis ed for all s since the physical values of the parameters are always positive. Hence, the linearization is globally asymptotically stable for all s. The conditions of Theorem 7.3 are satis ed for all points on the curve of stationarizable points and hence the state can be transferred between any two points on this curve in nite time (except the point corresponding to u = s = 1). Note that the system in this example is globally asymptotically stable for any xed u since its linearized dynamics (for xed u) corresponds to the nonlinear dynamics (the system is bilinear). If the linearization of the system is not asymptotically stable, there is the possibility of active stabilization by state feedback. We rst give a technical lemma.

Lemma 7.2 Let the matrices As and Bs depend continuously on the parameter s which belongs to a closed interval , and assume that As ; Bs is stabilizable for all s 2 . Also assume that the linear dependence structure of the controllability matrix [Bs ; As Bs ; : : : ; Asn-1 Bs ] is the same for all s 2 (i.e. if some columns are linearly independent (dependent) for some s 2 they are so for all s 2 ). Then

it is possible to choose Ls such that

As - Bs Ls is Hurwitz; s 2

and Ls depends continuously on s.

(7.29)

Proof Since the dependence structure of the controllability matrix is independent

of s, it is possible to choose an invertible transformation matrix Ts that depends continuously on s such that

2;s -1 Ts 1As Ts = A01;s A A3;s ; Ts Bs = B1;s 0 -

(7.30)

138


where the block matrices A1;s , B1;s form a controllable pair and A3;s corresponds to the uncontrollable modes. For a description of this canonical form see e.g. [90]. It follows from (7.29) that A3;s is Hurwitz. By choosing A1;s , B1;s to be in controller form, it is easy to see that the feedback coecients will depend continuously on s if the desired eigenvalue locations for A1;s depend continuously on s. 2 We can now present a strengthened form of Theorem 7.3.

Theorem 7.4 Let xa and xb be two points in S that can be joined by a continuous curve 2 S . Assume that can be chosen to be the projection on S of a continuous curve 2 E , parameterized by some s belonging to a closed interval . Let the linearization As = fx (x(s); u(s)), Bs = fu (x(s); u(s)) satisfy the technical conditions of Lemma 7.2. Then the state of system (7.12) can be transferred from xa to an arbitrary neighborhood of xb in a nite amount of time.

Proof From Lemma 7.2 it follows that there exists a continuous feedback matrix

Ls such that As -Bs Ls is Hurwitz. Now, the rest of the proof of Theorem 7.3 applies with As substituted by As - Bs Ls since this matrix satis es the requirements of Lemma 7.1. However, the procedure (control law) for transferring the state from Br=3 (xj ) to Br=3 (xj+1 ) has to be modi ed as follows. Assuming that the state belongs to Br=3 (xj ), choose the control u = uj+1 - Lj+1 (x - xj+1 ) and we get the same estimate of the nite transfer time between Br=3 (xj ) and Br=3 (xj+1 ) as in the proof of Theorem 7.3 and the theorem follows. 2 We illustrate the application of Theorem 7.4 by two examples, an invented nonlinear system and cruise control of an aircraft.

Example 7.7 Consider the following system x_ 1 = -x31 + x2 x_ 2 = -x21 - x2 - x32 + u

(7.31)

The stationarizable points are the cubic x2 = x31 and the system matrices of a linearization around a point (x01 ; x02 ) on this curve is

A=

-3(x0 )2 1 -2x01

1 0 -1 - 3(x02 )2 ; B = 1

It can be shown that the linearized system is asymptotically stable for all points on the cubic except for < x01 < 0, where -0:59 is a root of a polynomial of degree 7. However, the linearized system is controllable everywhere and hence the state can be transferred between any two points on the cubic in nite time according to Theorem 7.4.

7.5 Output Range

139

Example 7.8 Consider a very simpli ed model for velocity control of an aircraft x_ 1 = x2 - f(x1 ) x_ 2 = -x2 + u

(7.32)

Here x1 is the velocity, x2 is the engine thrust and f(x1 ) = x21 - 2x1 + 2 is the aerodynamic drag. The expression for the drag is typical for some ghter aircrafts operating at high angle of attack and low speed. The mass is normalized to 1 and we have assumed a rst order system response from pilot command to engine thrust. The stationarizable points are given by the parabola

x21 - 2x1 + 2 = x2 : (7.33) A linearization of the system around (x01 ; x02 ) on the parabola is given by 2 - 2x0 1 0 x_ = 0 1 -1 x + 1 u where x = (x1 - x01 ; x2 - x02 ) and u = u - u0 . We observe that the linearization is asymptotically stable if x01 > 1. However, since the linearization is controllable

(and hence stabilizable) for all points on the parabola we can always nd a linear state feedback that stabilizes the linearization around each stationarizable point. According to Theorem 7.4, it is possible to transfer the state between points on the parabola (7.33). Let u = us - L(x - xs ), where L = 8 2 . This expression together with a parameterization of the parabola gives the control law

u = -8x1 - 2x2 + 6 + 2s + 3s2 ; which can be used to follow the parabola by changing s in small steps, see Figure 7.8.

In this section we have investigated the problem of moving the state from one equilibrium point to another. We have shown that it is possible to transfer the state between two such points in nite time if they can be joined by a continuous curve of equilibria and the linearization of the system at each point on the curve is stabilizable. In the next section we utilize these results for polynomial systems.

7.5 Output Range The concept of controllability of dynamic systems is an important issue in control theory. There are a number of dierent ways of de ning this concept depending on the context. In this section we specialize to single output systems and devise a method for calculating an interval on which the output is controllable in the following sense: the output can be controlled to take any value in the interval and

140


4.5

4

3.5

x2

3

2.5

2

1.5

PSfrag replacements

1

0.5

0

0.5

1

x1

1.5

2

2.5

3

Figure 7.8 Transfer between equilibrium points of (7.32). Solid line: trajectory corresponding to small increments of s, dashed line: equilibrium points. be kept constant at that value. This interval of set-points for the output will be called the output range. The outputs corresponding to the asymptotically stable, stationarizable states are easily calculated as the projection of these states onto the output, i.e.,

h

9x y = h(x)

^

i

AS (x) :

From this information we only know that there is an admissible control, u such that the output, y may be kept at a constant level despite small disturbances. What happens with y when we change u by a small amount? Is it possible to change u such that y increases or decreases to a new constant level? According to Theorem 7.3 in the previous section we know that this is possible if there is a curve in AS that joins two points in the state space corresponding to the initial and nal output levels. Furthermore, u has to be changed in suciently small steps so that the state does not escape from the region of attraction of AS . Here we will study how to compute the output range of a polynomial system, given control and state constraints. An examination of the output map, y = h(x) gives no information about the in uence of u on y since u does not appear explicitely in this expression. However, since we assume that u eects y in some way, u has to appear explicitely in some of the time derivatives of y. The lowest order of the time derivative of y where

7.5 Output Range

141

u appears explicitely is usually called the relative degree of the dynamic system,

see [78]. Let y(r) denote this derivative. For a stationary state the output y is constant and hence all derivatives are zero. If it is possible to change u such that y(r) > 0, all lower derivatives become positive after an in nitesimal amount of time and y increases. The subset of the asymptotically stable, stationarizable states for which this is possible is described by the formula

h

i

AS + (x) , 9u AS (x) ^ y(r) > 0 ^ U (u) : (7.34) The formula for states in AS (x) corresponding to decreasing y is obtained in the

same way and becomes

h

AS - (x) , 9u AS (x)

^

y(r) < 0

^

i

U (u) :

(7.35)

Combining these formulas we get the states for which both an increase and decrease of the output is possible CS (x) , AS + (x) ^ AS - (x): (7.36) The corresponding output range is the projection of this set onto y

h

CO(y) , 9x y = h(x)

^

i

CS (x) ;

(7.37)

which we will call the controllable output range of the dynamic system.

Example 7.9 Consider the system in Example 7.4 and let y = x1, where the asymptotically stable, stationarizable set is given by (7.11). Since y_ = x_ 1 = -x31 + x2 ; y = -3x21 x_ 1 + x_ 2 = -3x21 (-x31 + x2 ) - x21 - x2 - x32 + u; the relative degree of this system is 2 and the asymptotically stable, stationarizable states for which y can be increased or decreased becomes

h

i

AS + (x) = 9u AS (x)

^

-x21 + 3x51 - 3x21 x2 - x2 - x32 + u > 0

^

u2 1 ;

AS - (x) = 9u AS (x)

^

-x21 + 3x51 - 3x21 x2 - x2 - x32 + u < 0

^

u2 1 ;

h

CS (x) = AS + (x)

^

i

AS - (x):

In this case it can be shown that the semialgebraic set described by CS (x) is the same as AS (x) except for some points on the border of AS (x). The controllable output range of this system is

h

i

CO(y) = h9x y = h(x) ^ CS (x) i = [y + 1 > 0 ^ 9y7 + 3y + 2 < 0] _ [y > 0 ^ y9 + y3 + y2 - 1 < 0] h i = [-1 < y < -0:591 : : : ] _ [0 < y < 0:735 : : : ] :

142


0.5

u=0.75 0

u = −0.05 −0.5

PSfrag replacements

u

u= −0.05 u= −0.5

−1 −1

−0.5

0

0.5

1

Figure 7.9 The phase portrait corresponding to a number of dierent con-

trols.

Compare the controllable output range with the projection on the x1 -axis of the states in Example 7.4 which are both stationarizable and asymptotically stable. The character of solutions to system (7.31) with initial values near the points in CS is shown in Figure 7.9 where the phase portrait for a number of dierent admissible controls is shown. Observe that an output interval in CO might be composed of subintervals, which correspond to projections of several disjoint parts of the state space, cf. Figure 7.7. If this is the case, it might happen that we cannot steer the output from a point on one subinterval to a point on another subinterval.

7.6 Curve Following In many control applications one of the main objectives is to track or follow a given trajectory in the state or output space of a dynamic system. There are several approaches to control law design for such problems, e.g., adaptive control or local control laws based on linearizations of the system around the trajectory. However, these methods do not take control constraints, such as amplitude and rate limitations, into account. Usually, one has to verify the design afterwards by simulations to check that the constraints are not violated. For nonlinear systems it is usually very hard to show that the simulations have covered all possibilities. Hence, there is a need for more systematic procedures to verify that a given system can follow a prescribed trajectory in the state-space using only admissible values of the control signals.

7.6 Curve Following

143

A slight variation of the curve following problem can be used to compute approximations of the set of states that is reachable from a given set of possible initial states. Not only control constraints but also constraints on the states can be taken into account. Consider system (7.1) subject to some semialgebraic control constraints, u 2 U . Let be a rationally parameterized curve in the output space Rp , i.e.,

= y 2 Rp j y = g(s); g : R ! Rp ; s 2 [; ] ; where the orientation of is de ned by increasing values of s. First, suppose that g(x) = x, that is is a curve in the state space. To steer the system along , there has to be an admissible control u at each point on the curve such that the solution trajectory tangent vector f(x; u) points in the same direction as a forward pointing tangent vector of , i.e.,

f(g(s); u) = dsd g(s); > 0;

8s 2 [; ]:

(7.38)

The more general case when is a curve in the output space is treated in the following theorem. Note that if the number of outputs is less than the number of states, a curve in the output space corresponds to a manifold of dimension greater than one in the state space.

Theorem 7.5 There exists an admissible control u 2 U such that the output y of system (7.1) follows the curve if and only if the formula

8s 2 [; ] 9x 9u 2 U 9 > 0 h hx (x)f(x; u) = dsd g(s)

^

g(s) = h(x)

i

(7.39)

is True. Here hx (x) denotes the Jacobian matrix of h(x) w.r.t. x.

Proof The curve in the output space can be followed if and only if the output

trajectory tangent y_ (t) can be chosen parallel to the tangent vector of , at each point of , by an admissible choice of u. Now

y_ (t) = hx (x) x_ = hx(x) f(x; u) and a tangent of is given by dsd g(s). Hence, the parallel condition becomes

hx (x)f(x; u) = dsd g(s) for some > 0; u 2 U and the theorem follows.

2

144


Now, quanti er elimination can be used to eliminate all quanti ed variables. In (7.39) all variables are quanti ed and the quanti er elimination algorithm gives True or False as a result, i.e., either there is an admissible control that steers the system along the curve or not. If the equations that specify the dynamic system or the curve contain additional parameters, then the quanti er elimination gives necessary and sucient conditions on these free parameters such that an admissible control law exists.

Example 7.10 Consider the following system x_ 1 = -ax2 + 21 x1 x2 ; x_ 2 = -x1 - x2 + (1 + x21 )u; y = (x1 ; x2 )T

(7.40)

where -2 u 2 and is the unit circle. Here we have one free design parameter a. The unit circle can be rationally parameterized according to

= x 2 R2 j x1 = ss22 +-11 ; x2 = s-22s ; s2R : +1

In this example the output map is the identity. Hence we can eliminate x directly by substitution. Clearing denominators, the parallel condition in formula (7.39) becomes

s + 2 a s - s3 + 2 a s3 = 4 s ^ 1 + 2 s + 2 s3 - s4 + 2 u + 2 s4 u = 2 (s2 - 1)

Next follow a few lines of Mathematica code:

In[1]:= EliminateQuantifiers

Out[1]=

8s 9u 9 s + 2 a s - s3 + 2 a s3 = 4 s ^ 1 + 2 s + 2 s3 - s4 + 2 u + 2 s4 u = 2 (s2 - 1) > 0 ^ -2 u ^ u 2 - 1 + 2 a 0 ^ -4409 + 2548 a + 790 a2 452 a3 - 73 a4 + 24 a5 + 4 a6 0

q

^

In[2]:= InequalitySolve[ %, a ] Out[2]=

1 2

a -1 + 1 + Root[ -4409 + 1274 #1 - 121 #12 + 4 #13 &; 1 ]

In[3]:= N[ % ] Out[3]=

0:5 a 1:8819

7.6 Curve Following

145

The function EliminateQuantifiers calls the external program qepcad and returns the output from this program. InequalitySolve is a built-in Mathematica function for simpli cation of Boolean combinations of univariate inequalities. Hence, there is an admissible control such that the output (states) of system (7.40) follows the unit circle if and only if 21 a , where 1:8819 is given by a real zero of a sixth order polynomial.

7.6.1 Control Law Design

How do we construct a control law that steers the system along the curve once the quanti er elimination in formula (7.39) gives an armative answer?

Open Loop Control Law

To construct an open loop control law, i.e., a control law of the form u = u(t), where u(t) is a given function of time, we only have to eliminate quanti ers in

i

h

(7.41) 9 > 0 9x hx (x)f(x; u) = dsd g(s) ^ g(s) = h(x) : The result is an implicit expression in u and s. Solving for u in this expression gives an open loop control law, where s corresponds to time. A time scaling s = '(t), where '() is a monotonically increasing function can then be used to gain some additional freedom without aecting the alignments of hx (x)f(x; u) and d ds g(s). However, after this operation we have to check that u is still admissible.

State Feedback Control Law

A state feedback control law can be computed by rst eliminating quanti ers in

h

8s 2 [; ] 9 > 0 hx (x)f(x; u) = dsd g(s)

^

i

g(s) = h(x) ;

(7.42)

which gives the implicit form of a state feedback control law. However, for the computation of a state feedback law the use of quanti er elimination algorithms leads to unnecessarily complex computations. Next we describe an alternative approach. We start with computing the implicit form of the set in the state space corresponding to the parameterized curve in the output space, i.e., eliminating s from g(s) = h(x). This implicitization problem can be solved using for example Grobner bases, see [39]. If the number of outputs p is less than the number of states n the implicitization procedure gives an implicit description of a manifold C in the state space (algebraic variety of dimension greater than one) whose map under h(x) include as a subset. Now, let the equations describing the manifold C be denoted by ci (x) = 0; i = 1; : : : ; s, or by vector notation

c(x) = 0:

146


The state trajectory tangent, f(x; u) has to be an element of the tangent space of C at each point on the curve for the output to remain on . This condition can be formulated as follows

cx (x)f(x; u) = 0;

(7.43)

which is an implicit expression for the desired state feedback control law and we have to solve for u to get the nal form. However, this is not the most general implicit form of an algebraic state feedback control law. Since the polynomials ci (x) are zero on C a more general implicit control law is given by

cx (x)f(x; u) = v; (7.44) vj = Psi=1 qij (x) ci (x); qij 2 R[x]: These control laws give identical system behavior on C but the extra freedom (free choice of the polynomials qij (x)) can be used to tune the system behavior in Rn n C . Note that (7.44) is equivalent to cx (x)f(x; u) 2 h c1 ; : : : ; cs i. We demonstrate the application of the results on control law design by two examples.

Example 7.11 Consider system (7.40) again. The implicit description of curve can be computed as outlined above. However, in this simple case we already know the implicit form: x21 + x22 - 1 = 0. The orthogonality condition (7.44) becomes 2x1 (-ax2 + 12 x1 x2 ) + 2x2 (-x1 - x2 + (1 + x21 )u) = q11 (x)(x21 + x22 - 1); where q11 (x) 2 R[x]. Solving for u gives 2 - x21 + q11 (x)(x21 + x22 - 1) : u = (2 + 2a2)(x11++x2x 2) 2x2 (1 + x21 ) 1 A simulation of the closed loop system behavior for q11 = 0, a = 0:55, and a = 1:88,

respectively, is shown in Figure 7.10. The computed control law does not force the state to converge to the unit circle if the initial state is outside the curve. However, one can show that using the modi ed control law 2 - x21 - 2x (x2 + x2 - 1); u = - (2 + 2a2)(x11++x2x 2 1 2 2) 1

convergence for initial conditions in a neighborhood of the unit circle is obtained. In Figure 7.11 a simulation of system (7.40) for a = 1 and dierent initial conditions is shown.

7.6 Curve Following

147 States

States

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

0

20

40

60

0

2

Control 2

1

1

0

0

−1

−1

20

6

4

6

Control

2

−2 0

4

40

−2 0

60

2

Figure 7.10 The states and control behavior for a = 0:55 (left) and a = 1:88 (right), respectively.

1.5

1

x2

0.5

0

−0.5

−1

PSfrag replacements −1.5 −1.5

−1

−0.5

x1 0

0.5

1

1.5

Figure 7.11 A phase portrait of the system (7.40) when the modi ed control law is used.

148


Example 7.12 Consider the following system x_ 1 = -x1 + 2 x_ 2 = -x2 - x21 + 4u y = (x1 ; x2 )T ; subject to the constraints

-1 u 1: We want to decide if it is possible to follow the curve

: x = g(s) = 3s2 -s 2s3 ; s 2 [0; 1]; using an admissible control. According to Theorem 7.5 we get the following formula.

8s 2 [0; 1] 9u 2 [-1; 1] 9 > 0 h i -s + 2 = ^ -(3s2 - 2s3 ) - s2 + 4u = (6s - 6s2 ) ;

which can be shown to be True. We now compute the control laws that steer the system along . An implicit description of is x2 - 3x21 + 2x31 = 0 and the orthogonality condition (7.44) gives

(-6x1 + 6x21 )(-x1 + 2) - x2 - x21 + 4u 2 h x2 - 3x21 + 2x31 i:

In general one has to check that the chosen control law steers the system in the right direction along . In this example we know that there exists a control law which steers the system in the right direction on but there is also only one way of choosing u modulo h c i on . Hence any of the above u can be chosen, e.g.,

u = 23 x31 - 174 x21 + 3x1 + 14 x2 ; which is a state feedback control law that steers the system along in the right direction. Observe that the symbolic nature of the quanti er elimination method makes it possible to have symbolic parameters undetermined during the whole computation, from system description (7.1) to the nal expression of the control law. Hence the control law is valid for all parameters satisfying the algebraic constraints that result from quanti er elimination in (7.39).

7.6 Curve Following

149

7.6.2 Constrained Reachability

The construction of the reachable set of a dynamic system, w.r.t. a set of initial states, is not solvable by algebraic methods in general1. The reason is that generically the solution of a system of dierential equations such as (7.1) is not a semialgebraic set. However, a more restricted form of reachability can be investigated using semialgebraic tools. The ideas on curve following can be generalized to give approximations of the set of reachable states of system (7.1) subject to some control constraints. Let X0 be a given semialgebraic set of initial states and G a family of rationally parameterized curves

G = x = g(s; x0 ; x1 ; ) 2 Rn s 2 [0; 1]; 2 ; g(0; x0 ; x1 ; ) = x0 2 X0 ; g(1; x0 ; x1 ; ) = x1 ; where is a semialgebraic set of shape parameters. We also want to take state constraints into account. Let X be a semialgebraic set of admissible states.

De nition 7.2 We say that a curve, G 2 G is admissible if (i) All points on G belong to the admissible states, X .

(ii) There is an admissible control u such that the solution trajectory of (7.1) follows G.

De nition 7.3 A point x1 in the state space of system (7.1) is G -reachable from X0 if there is an admissible curve 2 G such that x1 is reachable along using an admissible control. The set of all G -reachable states is called the G -reachable set of system (7.1) w.r.t. X0 . We can formulate those states that are reachable from X0 along one of the admissible curves in G as a formula including quanti ers similar to formula (7.39). Theorem 7.6 The G -reachable states of system (7.1) is given by the following formula

9 2 9x0 2 X0 8s 2 [0; 1] h 9u 2 U 9 > 0 f(g; u) = dsd g

^

i

X (g) ;

(7.45)

where we have omitted the arguments of the rational function g.

Proof The proof is almost identical to the proof of Theorem 7.5.

2

1 For nite discrete event systems, reachability can be treated completely algebraically, see [53, 61, 128].

150


Performing quanti er elimination gives a formula in x1 describing the G -reachable set w.r.t. X0 . The rst two quanti ers select a curve in G , the other quanti ers are similar to the ones in formula (7.39). Note that this is formulated as a curve following problem in the state space, i.e., here h(x) is the identity mapping. Using a family, G , of parameterized curves which are very exible, (e.g., Bezier curves, see e.g. [39]) the G -reachable set w.r.t. some set of initial states should be a good approximation of the true reachable set. We give two examples to illustrate the concept of G -reachable set.

Example 7.13 Consider the following system x_ 1 = x1 + u x_ 2 = x22

(7.46)

subject to the control constraints

-1 u 1: Which states are reachable along straight lines from the point (x1 ; x2 ) = (0; 1)? The set of initial states is

X0 = x0 2 R2 x01 = 0

^ x02

=1

and a parameterization of straight lines is given by

G : x = g(s; x0 ; x1 ) = s Formula (7.45) becomes

h

" x1 - x0 # " x0 # 1 1 1 x12 - x02

+

x02

; s 2 [0; 1]:

9x0 8s 9u 9 x01 = 0 ^ x02 = 1 ^ 0 s 1 ! s(x1 - x0) + x0 + u = (x1 - x0) ^ 1 1 1 1 1 i 2 1 0 1 0 0 s(x2 - x2 ) + x2 = (x2 - x2 ) ^ > 0 ^ u2 1 and eliminating quanti ers we get the semialgebraic set

-x1(x1)2 + x1x1 - x1 - x1 + 1 0 ^ 1 2

1 2

2

1 1 1 x1 (x2 )2 - x11 x12 - x12 + x11 + 1 0 ;

(7.47)

see Figure 7.12. A control law that steers the system along a straight line can be computed as in Example 7.12 noting that a curve in G with slope k is a part of the zero set of

7.6 Curve Following

151

x2

5

u = -1

u = +1

4

3

2

PSfrag replacements

1

0

-1

x1 1

0

Figure 7.12 The semialgebraic set de ned by (7.47) (gray shaded region) and the reachable set of system (7.46) (the region above the solutions corresponding to u = +1 and u = -1). c(x) = x2 - 1 - kx1 . The orthogonality condition (7.44) with the choice q(x) = 0

gives

-k(x1 + u) + x22 = 0

!

u = k1 x22 - x1 ; k 6= 0:

The cases k = 0 and k = 1 cause no problems since the line x2 = 1 does not belong to the set of reachable states and for k = 1 the control law simply becomes u = -x1 . Furthermore, this control law steers the system in the right direction.

Example 7.14 Consider the following system

x_ 1 = -x1 + 2x2 - x22 + 1; x_ 2 = -2x2 + u

(7.48)

where -3 u 3. We compute the set of states that can be reached from (0; 0) along straight lines

x = g(s; x1 ) = (sx11 ; sx12 )T ; s 2 [0; 1]:

Quanti er elimination on a formula corresponding to (7.45) gives the set of states shown in Figure 7.13.

152

Chapter 7 Applications to Nonlinear Systems 1.2

1

0.8

x2

0.6

0.4

0.2

0

PSfrag replacements

−0.2

−0.4 0

0.2

0.4

0.6

0.8

x1 1

1.2

1.4

1.6

1.8

2

Figure 7.13 The set of states reachable along straight lines from the origin for system (7.48) using -3 u 3.

Once we know the set of reachable states from a point along straight lines an obvious generalization is to let this set be possible initial states of a new calculation of reachability. The resulting set would be an even better approximation to the real reachable set of states.

7.7 Summary In this chapter we have presented a number of applications of quanti er elimination to problems in nonlinear control. A closed form of the set of stationarizable points for a polynomial system subject to state and control constraints can be computed using quanti er elimination. This is also the case for the subset of the stationarizable states which corresponds to asymptotically stable states. As an application we have computed the set of stationarizable angles of attack and sideslip angles for an F-16 aircraft subject to actuator constraints. An investigation of the pitch-axis control of a missile has also been carried out. Suppose we have a manifold of equilibrium states of a dynamic system corresponding to dierent control signals. We have proved that it is possible to move the state between two points on such a manifold if there is a continuous curve on the manifold joining the points such that the linearization of the system is controllable at all points on the curve. This result together with quanti er elimination can be combined to give a constructive method of computing the range of possible output set-points for a polynomial system subject to control and state constraints. We have put the curve following problem for dynamic systems subject to control

7.7 Summary

153

constraints in an algebraic framework. It has been shown how this problem can be formulated as a formula including quanti ers and quanti er elimination can then be used to synthesize admissible control laws. If the system description includes design parameters to be determined, the procedure also gives necessary and sucient conditions on these parameters such that the curve can be followed. A slight generalization of the curve following problem gives approximations of reachable sets of systems subject to control constraints.

154


8 Applications to Switched Systems Switching between dierent controllers to achieve a required behavior is rather common in practical applications. A common way of implementing switching is for example by the use of PID controllers with selectors [7]. This can be used to guarantee that certain process variables are kept within speci ed bounds. Simple simulation studies and results from optimal control theory such as bang-bang control show that performance can be enhanced by switching between a number of state-feedback laws. A sucient condition to be able to design control laws based on switching between dierent controllers, is that we can choose a controller that makes the time derivative of a positive de nite function negative along all trajectories in a neighborhood of the desired set-point [139]. Here we give a computational test of this requirement, formulated as a quanti er elimination problem. For nonlinear systems, switched controllers can be used for stabilization. Brockett has given a necessary condition for when a system with continuously dierentiable right hand side can be stabilized by a continuously dierentiable state feedback [25, 140]. However, there are many examples of systems that do not satisfy this condition but can be stabilized by a discontinuous state feedback [119]. In output regulation and tracking problems, the zero dynamics [28, 78] plays an important role. Stable zero dynamics is necessary for proper regulation or tracking, since otherwise there are internal variables that exhibit unbounded behavior. There are nonlinear systems whose zero dynamics cannot be stabilized by continuous state feedback but can be stabilized by switched controllers [118, 119]. In this chapter we approach the problem of analysis and design of discontinuous control laws in a constructive way. Quanti er elimination is used to answer a number of questions. Is it possible to stabilize a system by switching between a number of so called 155

156

Chapter 8 Applications to Switched Systems

basic controllers ? Can we obtain stable zero dynamics for the output regulation or tracking problem using a switched controller? Can we obtain an invariant set of states using a switched controller? The chapter is organized as follows. In Section 8.1 we review some basic concepts and de nitions related to stability. Since we focus on switched systems we also need to de ne what we mean by a solution to a system of dierential equations with discontinuous right hand sides. Section 8.2 deals with the problem of stabilizing a polynomial system by switching between a number of continuous controllers. These results are further applied in Section 8.3 where we show how zero dynamics of a nonlinear system which is not ane in the control can be stabilized by switching between dierent output zeroing controllers. A number of examples, where we use quanti er elimination software, is presented in Section 8.4, and Section 8.5 gives a summary of the chapter. Throughout the chapter we utilize quanti er elimination to deal constructively with the computational questions that arise.

8.1 Preliminaries Consider a nonlinear system of the following form

x_ = f(x; u) (8.1) y = h(x); where x 2 Rn ; u 2 R, and f : Rn R ! R and h : Rn ! R are continuous functions with f(0; 0) = 0 and h(0) = 0. Since we are interested in whether a system can be

stabilized by switching between dierent controllers we introduce a set of so called basic controllers of the form

ui (x) = Ki (x);

i = 1; : : : ; N (8.2) where Ki : Rn Xi ! R is continuous, Ki (0) = 0; i = 1; : : : ; N, and Xi is closed. To determine when to use a basic controller we also introduce a switching function I(x) which maps the set of state measurements into the set of integers f1; : : : ; Ng. Hence, the switching function induces a partition of the state space with dierent enumerated components. We use the following notation

u(x) = KI(x) (x)

(8.3)

for the control law obtained by choosing basic controller with respect to the switching function I. In the sequel we assume that (8.1) is controlled by the basic controllers (8.2) with respect to the switching function I. Such systems may arise in cases when the controller consists of a number of relay nonlinearities which can be used to change the structure of the system (8.1). Notice that the right hand side of the closed loop system

x_ = f(x; KI(x) )

(8.4)

8.1 Preliminaries

157

is bounded as long as the state belongs to some compact subset X of [i Xi . Hence, jx_ j is bounded on X . Sometimes it can be desired to implement a switching controller using some hysteresis elements, for example to avoid sliding modes. Observe that this kind of controllers cannot be described by a switching function since the hysteresis element introduce a discrete state. We will use the term switching rule to distinguish these cases. Here we state a number of de nitions and results that we need for investigating stability and stabilizability of switched systems, see for example [92, 97].

De nition 8.1 A function V is said to be positive de nite if V (0) = 0 and V (x) > 0 for x 6= 0. A function W is called negative de nite if -W is positive de nite. De nition 8.2 A positive de nite function V is radially unbounded if lim V (x) = 1: x !1 j j

De nition 8.3 Consider the system x_ = f(x); f(0) = 0 and let x(t; x0) denote the solution that starts from the point x0 . The origin is (i) stable if for any > 0 there exists > 0 such that jx0 j < =) jx(t; x0 )j < ; 8t 0;

(ii) (globally) attractive if there exists > 0 ( = 1) such that jx0 j < =) tlim !1 jx(t; x0)j = 0; (iii) (globally) asymptotically stable if it is stable and (globally) attractive. If the origin is asymptotically stable it has a region of attraction , i.e., a set of initial states such that x(t) ! 0 for all x(0) 2 .

De nition 8.4 For a time invariant dynamic system, an invariant set M Rn is a set such that if x(t0 ) 2 M then x(t) 2 M for all t t0 .

Note that sometimes the term positively invariant is used to describe the above property. However, we do not adopt that terminology here.

De nition 8.5 The system (8.1) is stabilizable by switching the basic controllers (8.2)

if there exists a switching function I such that the origin of the closed loop system (8.4) is asymptotically stable. The main problem we consider in this chapter is: What are the conditions for existence and how is it possible to design a (globally) stabilizing switching controller (8.3). Much of the stability analysis of smooth systems can be generalized to nonsmooth systems.

158


8.1.1 Nonsmooth Systems

If we use a piecewise continuous state feedback controller u = KI(x)(x) the right hand side of the closed system (8.4), f~(x) , f(x; KI(x) (x)) becomes discontinuous. There are several ways to de ne what we mean by a solution to such a system. For example it is possible to de ne solutions according to Filippov [46]. De nition 8.6 A function x() is called a solution of x_ = f~(x) on [t0; t1] if x() is absolutely continuous on [t0 ; t1 ] and for almost all t 2 [t0 ; t1 ] x_ 2 F(x); (8.5) where \ \ ~ co f(B (x) n N): (8.6) F(x) , >0 N=0

Here co f~(x) denotes the convex closure of all limiting T values of f~(x), B (x) is an open hyperball centered at x with a radius , and N=0 is the intersection over all sets N of Lebesgue measure zero. The de nition can be interpreted as follows: the tangent vector to a solution, where it exists, must lie in the convex closure of the limiting values of the vector eld in progressively smaller neighborhoods around the solution point. Note that F(x) in (8.5) is a set-valued function but corresponds to a singleton in the domains of continuity of f~(x). Solutions to dierential inclusions of the type (8.5) can be shown to exist under rather general conditions [46]. It can be shown that there always exists a solution of the system (8.1) if we switch among the controllers (8.2) in a suciently regular way. Here we have no intention of going into detail of existence issues. In the sequel we will assume that the chosen switching function or switching rule guarantee that a solution always exist. Investigations of Lyapunov stability of nonsmooth systems can be found in [46, 138].

8.2 Stabilizing Polynomial Systems by Switching Controllers First, we present a general approach to the design of stabilizing switched controllers for systems (8.1) which are based on the basic controllers (8.2). It turns out that if we always can choose a basic controller such that a positive de nite function decreases along trajectories then the system can be stabilized by switching basic controllers. To be able to formulate the problem of stabilizing a system by switching basic controllers in a way suitable for quanti er elimination, we will use the following (x) theorem. In the sequel we let Vx(x) = ( @V@x(1x) ; : : : ; @V @xn ) denote the gradient of V w.r.t. x.

8.2 Stabilizing Polynomial Systems by Switching Controllers

159

Theorem 8.1 Let V: Rn ! R be a continuously dierentiable positive de nite function, and = x 2 Rn j V (x) a compact sublevel set of V . Suppose that for any x 2 n f0g there exists i 2 f1; : : : ; Ng such that Vx(x) f(x; Ki (x)) < 0:

(8.7) Then the system (8.1) is stabilizable by switching the basic controllers (8.2) and

is a region of attraction.

Proof Introduce the sets Si = x 2 Rn j Vx(x)f(x; Ki (x)) < 0 ;

i = 1; : : : ; N: Since Vx(x)f(x; Ki(x)) is continuous, these sets are open. Let 0 > 0 and consider

n B0 (0). According to the assumptions, any x 2 n B0 (0) belongs to some Si . Since Si is open, there is an i 2 f1; : : : ; Ng and > 0 such that B (x ) Si . The collection fB=2 (x )gx B0 (0) forms an open cover of n B0 (0). Since

n B0 (0) is compact, there exists a nite subcover f B=2 (xj ) gM j=1 ; xj 2 n M B0 (0). The collection fB=2 (xj )gj=1 is also a cover. Now, in each B=2 (xj ) we have Vx(x)f(x; Ki(x)) < 0 for at least one of the functions, which also attains its

2

n

maximum in this set. Hence -1 , max min max Vx(x)f(x; Ki (x)); j i

x B 2 (xj) where 1 > 0. We conclude that for any 0 > 0 there exists 1 > 0 such that for any x 2 n B0 (0) there is an i 2 f1; : : : ; Ng such that 2

Vx(x)f(x; Ki (x)) -1 < 0:

(8.8)

Let x(t; x0) denote a solution starting at x0 2 n B0 (0) at time 0 of the system (8.1) controlled by switching basic controllers such that (8.7) is satis ed. Furthermore, let V_ (x(t; x0 )) denote the time derivative of V (x) along a solution. According to (8.8) we have V_ (x(t; x0 )) -1 < 0 for all points on the trajectory in

n B0 (0). Integrating this inequality from 0 to t gives V (x(t; x0)) V (x0) - 1 t - 1 t: (8.9) This shows that x(t; x0 ) can only stay in n B0 (0) for a nite time, since V is bounded from below. Since is invariant (V is decreasing along solutions) the trajectory has to enter B0 (0). Hence, limt!1 x(t; x0 ) = 0 since 0 > 0 was arbitrary, i.e., the origin is attractive. To prove stability we observe that ; is invariant. Moreover, there exists d1 ( ) > 0 and d2 ( ) > 0 such that Bd1 () ( ( Bd2 () , since V is positive de nite, see for example [92]. We conclude that the origin is an asymptotically stable solution and is a region of attraction of the system (8.1). 2

160


The assumptions in the theorem implies that the domains of de nition of the control laws cover , i.e., [i Xi . Notice that each of the basic controllers may yield an unstable closed loop system but there may still exist a switched stabilizing strategy. The following corollary follows easily from the proof of Theorem 8.1 since all sublevel sets of a function, which is radially unbounded, are compact.

Corollary 8.1 If in addition to the assumptions in the previous theorem, V is radially unbounded, then system (8.1) is globally stabilizable by switching the basic controllers. If the assumptions in Theorem 8.1 or Corollary 8.1 are satis ed then the following controller can be used to stabilize the system u(x) = KI(x) (x) I(x) = argmin Vx(x) f(x; Ki (x)): i f1;:::;Ng

(8.10)

2

With this choice we get V_ (x) = Vx(x) f(x; KI(x) (x)) < 0 for all x 2 n f0g. Observe that the switching function I might be multiple valued since the minimum in (8.10) might be obtained by several controllers simultaneously. This is the case along switching surfaces. For states corresponding to multiple values the behavior of the controller has to be further speci ed. Either techniques from sliding mode control [151] can be used or a slight variation of controller (8.10) where hysteresis is introduced. Due to the fact that the sets Si (introduced in the proof of Theorem 8.1) are open, the switching surfaces of controller (8.10) can be substituted by hysteresis zones. This will prevent sliding modes but chattering solutions may appear instead depending on the direction of the vector elds along the switching surface, see Example 8.3. The conditions in Theorem 8.1 and Corollary 8.1 can be checked by quanti er elimination if the functions that appear in the theorems can be described by semialgebraic sets. This means that we are able to handle feedback control laws which are implicitly de ned as solutions to multivariate polynomial equations. In connection to stabilization of the zero dynamics of a nonlinear system, which is not ane in the control, this turns out to be very useful, see Section 8.3. The decision problem that has to be solved to verify the conditions of Theorem 8.1 is h i (8.11) 9r 8x x 6= 0 ^ V (x) r2 ! V_ 1(x) < 0 _ : : : _ V_ N(x) < 0 and for Corollary 8.1 we get

h

8x x 6= 0

!

V_ (x) < 0 _ : : : _ V_ (x) < 0 i: 1 N

(8.12)

If Ki is an algebraic function of x, i.e., if the relation between ui and x is a polynomial equation ki (ui ; x) = 0 and ui is one of the roots, the above formulas have to be modi ed. We have to quantify u and V_ i(x) < 0 is changed to Vx(x) f(x; ui ) < 0 ^ ki (ui ; x) = 0, see Example 8.4.


161

Furthermore, given a family of parameterized positive de nite functions V (x; ) we can use quanti er elimination to determine if there exists a function in this family such that it satis es the conditions in Theorem 8.1 or Corollary 8.1. For instance, in order to check global stabilizability using the family of quadratic positive de nite functions, V (x) = xT P x, we consider the following decision problem

h

9P 8x PT = P ^ Re(eig(P)) > 0 ^ x 6= 0 ! V_ (x) < 0 _ : : : 1

_

V_ N(x) < 0

i;

(8.13)

where P is a matrix variable and Re(eig(P)) > 0 denotes a set of inequalities which guarantee that P is positive de nite, e.g., the positiveness of the principal diagonal minors of P or inequalities similar to those in Theorem 6.1. This procedure is much more general than when we just use one xed function V (x). The main drawback of this approach is the large number of variables in the formula. The number of variables is equal to the sum of the number of states, n, and the number of parameters we need to describe V (x). In the quadratic case we get n + n(n + 1)=2 - 1 variables. To reduce the required computations, we can search for a quadratic Lyapunov function for the linearized switched dynamics and then use this in the decision problem (8.11). The following results from [119, 139] are then useful.

Lemma 8.1 Consider a linear system x_ = A x + B u

(8.14)

and a set of linear basic controllers

ui (x) = Ki x; i = 1; 2; : : : ; N P If there exist numbers i 0; i i > 0 such that the matrix N X i=1

(8.15)

i (A + B Ki )

is Hurwitz, then the system (8.14) is stabilizable by switching the basic controllers (8.15).

Proof See [118, 139].

2

Lemma 8.2 Consider the system (8.1) with basic controllers (8.2) and assume that the system has been rewritten in the following form

x_ = @f(x;@xKi(x)) jx=0 x + gi (x) = Fi x + gi (x);

(8.16)

162


where gi (x) denotes higher order terms satisfying jgi (x)j lim = 0; jxj!0 j xj

i = 1; : : : ; N:

P

If there exist numbers i 0; i i > 0 such that the matrix N X i=1

i Fi

(8.17)

is Hurwitz, then the system (8.1) is locally stabilizable by switching the basic controllers (8.2).

Proof See [119].

2

The results above and in particular the linearization result of Lemma 8.2 can be used to considerably reduce the required computations of the stabilizability test. Indeed, in order to test stabilizability it is easier if one uses linearizations but the design of a stabilizing controller and the estimation of the domain of attraction can be done directly by using quanti er elimination. We propose the following procedure: 1. Rewrite system (8.1) in the form (8.16) for dierent basic controllers (8.2). Consider the matrix (8.17). Find the set of inequalities which follow from the Lienard-Chipart criterion, see Theorem 6.1. The resulting polynomial inequalities in i are denoted Re(eig(1 F1 + : : : + N FN ) < 0. Check whether the following decision problem is True

h

9 1 0

^

:::

^ N 0 ^ 1 + : : : + N > 0 ^

i

Re(eig(1 F1 + : : : + N FN ) < 0 :

(8.18)

Note that the problem can be normalized by xing one of the i to 1, which reduces the number of variables. 2. If the above formula is True, the system (8.1) is stabilizable by switching the basic controllers (8.2). A numerical instantiation of can be extracted from the quanti er elimination procedure. Take any such solution . Since the P matrix i i Fi is Hurwitz, it satis es the Lyapunov matrix equation. Given any symmetric and positive de nite matrix Q = QT ; Q > 0, there exists a unique solution P to

(

X

i

i Fi )T P + P(

X

i

i Fi ) = -Q;

(8.19)

which is symmetric and positive de nite. Fix the matrix Q and compute P.


163

3. With the computed P consider the quanti er elimination problem

h

8x x 6= 0 ^ xT P x r2 xT P f(x; K (x)) < 0 1

! _

:::

_ xT P f(x; KN (x)) < 0

i

:

(8.20)

Performing quanti er elimination gives an estimate of the domain of attraction, since the resulting constraints on r correspond to invariant ellipsoids such that all trajectories starting in such an ellipsoid converge to the origin. The numbers i and the matrix Q in the above procedure are design parameters which can be chosen dierently in order to obtain other estimates of the domain of attraction. The method proposed above cannot be used if the linearization matrices Fi do not satisfy the conditions of Lemma 8.2.

Example 8.1 Consider the two nonlinear systems x_ = fi(x); i = 1; 2 where

x1 2 f1 = 2 +-xx22+ x2 ;

The linearization matrices are

f2 = -xx21--xx32 : 1 3

1 A1 = 02 -11 ; A2 = -01 -11 3 which both are unstable. However, A1 + A2 is Hurwitz and we can switch between

the two systems to stabilize the system locally according to Lemma 8.2. It is easy to see that P = I solves the Lyapunov equation (8.19). To estimate the domain of attraction we utilize formula (8.20) and get 0 < r < 1:0552. Hence, within a circle of radius at least 1:055 we can control the state to the origin.

Lemma 8.2 may only be used in situations when the linearizations of the continuous dynamics contain enough structure. However, these conditions are not satis ed in general and quanti er elimination is the only tool, which we are aware of, that can handle these cases.

Example 8.2 Lemma 8.2 cannot be used for the system x_ = fi(x); i = 1; 2 where

2 f1 = --xx3 1--xx2 x1 2 ; 2 1 2

5 4 2 f2 = -xx2 1--xx4 x1 x2 2++xx2 ;2 : 1 1 2

since the linearizations do not contain enough information about the system behavior near the origin. We have to use Theorem 8.1 or Corollary 8.1 directly instead. We try with the Lyapunov candidate function V (x) = x21 + x22 . Performing quanti er elimination in formula (8.12) gives True, which shows that the system is globally stabilizable by switching basic controllers.

164

Chapter 8 Applications to Switched Systems 4

4

2

2

0

0

-2

-2

-4

-4 -4

-2

0

2

4

-4

-2

0

2

4

Figure 8.1 The regions (gray) where V_ 1(x) < 0 (left) and V_ 2(x) < 0 (right). In Figure 8.1 we illustrate the regions where V_ 1(x) < 0 and V_ 2(x) < 0. The union of the gray-shaded regions covers the whole state space. The curve x2 = 3x21 =2 - 1 can be shown to lie in the interior of the overlap of the regions in Figure 8.1 and can hence be used to switch between the systems to obtain global asymptotical stability.

8.3 Exact Output Tracking In this section we show how some of the results presented on stabilizability of switched systems can be applied to investigate the problem of exact output tracking and minimum phase properties of a class of polynomial systems, which are not ane in the control. The material in this section is based on [119]. Consider the following class of non control-ane systems

x_ = f(x; u) (8.21) y = h(x) where x 2 Rn ; u 2 R; y 2 R and f and h are analytic vector valued functions of their

arguments. The system (8.21) is a generalization of the nonlinear systems usually investigated in the nonlinear literature [78, 121], which is ane in the control. A good discussion on the motivation for considering the zero dynamics of the system (8.21) can be found in [28]. Note that the usual way of investigating systems of the form (8.21) is to introduce an integrator at the plant input [121], which transforms it into a control-ane system. However, the new augmented system may have the following undesirable properties according to [28]:

8.3 Exact Output Tracking

165

(i) Stabilizability of the new system with static feedback implies stabilizability of (8.21) by dynamic feedback. (ii) The relative degree of the new system is higher than that of (8.21). (iii) The transformation may introduce singularities.

De nition 8.7 A state x 2 Rn is termed an output zeroing stationarizable state

for system (8.21) if it is stationarizable and h(x) = 0. Without loss of generality, it can be assumed that the origin is an output zeroing stationarizable state. We use the notation x(t; x0; u()) to denote the solution of (8.21) with initial value x0 .

De nition 8.8 A closed set S, S Rn , is said to be a viable set of the system (8.21), if there exists a (continuous) feedback control law u = u(x) de ned on S such that there exists a solution x(t; x0 ; u()) of the system (8.21) which satis es: x(t; x0; u()) 2 S; 8x0 2 S 0 t < T 6 ; is where either T = 1 or limt!T jx(t)j = 1. Suppose that SO h-1 (0); SO = -

a viable set. Any control law, u = u(x), which is de ned on SO is called an output zeroing controller. Hence, a viable set is a subset of the state space that can be made invariant by a suitable choice of state feedback.

De nition 8.9 Consider the following sets

L0 = h-1 (0) S = fS L0 : S viableg [ M = S

(8.22)

S S 2

If M 6= ;, a zero dynamics is said to exist for the system de ned by (8.21). In other words, there exists a (continuous) feedback control law u(x) such that for all x0 2 M it follows that x(t; x0; u()) 2 M for t 2 [0; T ]; T > 0, i.e., M is an invariant set of system (8.21) if u(x) is used. De nitions 8.8 and 8.9 are taken from [28], where continuity of the output zeroing controllers is required. We drop this assumption in the sequel. The following de nition of minimum phase is due to [119] and diers from the usual de nitions found in [78, 121]. Observe that continuity of output zeroing controllers is not required and the problem of minimum phase is that of stabilizability and not of stability of the zero dynamics.

De nition 8.10 The system (8.21) is termed minimum phase at x if its zero

dynamics is stabilizable at x .

166


In the sequel we will assume that there exists zero dynamics and an a priori known output zeroing stationarizable state x 2 M at which we wish to investigate the minimum phase property.

De nition 8.11 Suppose that there exists zero dynamics for the system (8.21) and an output zeroing stationarizable state x . Then the zero dynamics is stabilizable at x if there exists an output zeroing control law u = u(x) with the following properties: (i) for any > 0, there exists > 0 such that

x0 2 M \ B (x )

x(t; x0; u(x)) 2 M \ B (x ); 8t 0:

=)

(ii) there exists > 0 such that

x0 2 M \ B (x )

=)

tlim !1 jx(t; x0; u(x)) - x j = 0

and we have x(t; x0; u(x)) 2 M for all t 0. Here Bd (x) denotes an open ball with radius d, centered at x 2 Rn . Any control law which satis es the above given conditions is referred to as a minimum phase controller. In order to analyze the minimum phase property it is very useful if we transform the system into a normal form [78]. Suppose that the system (8.21) has a relative degree r n at an output zeroing stationarizable state x . Then there exists a locally invertible coordinate transformation z = (x) such that the system (8.21) is transformed into the following form [121]

z_ 1 = z2 z_ 2 = z3 .. .

(8.23)

z_ r = g(; ; u) _ = F(; ; u) y = z1 where = (z1 : : : zr )T and = (zr+1 : : : zn )T . We say that the zero dynamics is

de ned by

0 = g(0; ; u) _ = F(0; ; u)

(8.24)

which corresponds to De nition 8.9. If we suppose that g(0; ; u) and F(0; ; u) are vector valued polynomial functions, we can use the result in the previous section to investigate minimum phase

8.4 Examples

167

properties. Since u is implicitly de ned by the equation 0 = g(0; ; u) there might be several solutions u for a given x, which cannot happen in the control-ane case. Analysis of minimum phase properties for non control-ane nonlinear systems which is based on linearization was considered in [118, 119]. A number of issues arise when investigating the output zeroing control laws and the stability of the corresponding zero dynamics, which are not incorporated into the known de nitions of minimum phase. We will illustrate these issues in a number of examples and at the same time demonstrate how quanti er elimination can be applied to analyze the stabilizability of the zero dynamics. To be able to use quanti er elimination we specialize our study to polynomial systems.

8.4 Examples In the following examples we illustrate how one can obtain discontinuous zero dynamics by using the switched controller ideas. Moreover, we illustrate how incorporating bounded controls induces \shrinking" of the region of attraction for the zero dynamics. There may not exist continuous output zeroing controllers which yield stable zero dynamics, whereas discontinuous output zeroing controllers which achieve this may exist. This was illustrated in the paper [119]. We can also incorporate control and state constraints in the design in a straightforward manner. Hence, the approach is practical and we can construct genuine target sets which need to be reached for the exact output tracking problem.

Example 8.3 Consider the system

x_ 1 = (x2 - 5x3 - u)(-x2 x3 - x2 - u) x_ 2 = x3 x_ 3 = u y = x1

(8.25)

The continuous output zeroing controllers are u1 (x) , x2 - 5x3 and u2 (x) , -x2 x3 - x2 : They give the following (continuous) zero dynamics

x_ 2 = x3 x_ 2 = x3 x_ 3 = x2 - 5x3 x_ 3 = -x2 x3 - x2 It is easily seen that 0 is an unstable stationary point of both systems. Hence,

by applying a continuous output zeroing controller we do not obtain stable zero dynamics. Introduce the Lyapunov function

V (x) = x22 + x2 x3 + x23 :

(8.26)

168


The derivatives of V (x) along solutions to the two systems become V_ 1(x) = 2x2 x3 + x23 + x2 (x2 - 5x3 ) + 2x3 (x2 - 5x3 ) and

V_ 2(x) = 2x2 x3 + x23 + x2 (-x2 x3 - x2 ) + 2x3 (-x2 x3 - x2 ): To test if it is possible to nd a discontinuous (switched) control law which yields stable zero dynamics with respect to the chosen Lyapunov function, we check the decision problem (8.12). We get the following formula

h

8x x 6= 0 ! V_ 1(x) < 0

_

i

V_ 2(x) < 0 ;

which can be shown to be False and hence it is impossible to prove global stability with the given Lyapunov function. However, performing quanti er elimination in

h

8x x 6= 0

^

V (x) 5

! V_ (x) < 0 _ V_ (x) < 0 i 1

2

gives True and we havean estimate of the region in which there exists a minimum phase controller V5 = (x2 ; x3 ) j V (x) 5 . See Figure 8.2 for regions where V_ i(x) < 0. 2

2

1

1

0

0

-1

-1

-2 -2

-1

0

1

2

-2 -2

-1

0

1

2

Figure 8.2 The regions (gray) where V_ 1(x) < 0 (left) and V_ 2(x) < 0 (right). A switched controller that stabilizes the zero dynamics can be implemented using the following switching rule. Initially, choose controller according to argmin V_ i(x0 ) i f1;2g 2

8.4 Examples

169

where x0 denotes the initial state. In the sequel, use the current controller as long as V_ (x) < 0, when V_ (x) hits zero, switch to the other controller. Due to the overlap between regions de ned by V_ 1(x) < 0 and V_ 2(x) < 0, we will always switch to a controller that makes V_ (x) < 0 and the state converges to the origin. In Figure 8.3 we show some state trajectories and a time response for the zero dynamics of system (8.25), when the above control law has been used. 3

2 1.5

1

1

0.5

-3

-2

-1

1

2

3 0

−0.5

-1

−1

-2 −1.5

−2

-3

0

1

2

3

4

5

6

7

8

9

10

Figure 8.3 Left: Estimate of the region of attraction, switching surfaces, and some trajectories. Right: A time response for initial state x0 = (-1:3; -1:3)T . It is straightforward to take control constraints into account. Suppose that we have the constraint juj 1. Then by considering the following decision problem

h

9r 8x x 6= 0 ^ V (x) r2 ! [ V_ (x) < 0 ^ jx - 5x j 1 ] 1 2 3

_

[ V_ 2(x) < 0

^ j x2 x3 + x2 j 1 ]

i

we see that the region ofattraction is much smaller than without control constraints. In fact, V0:24 , (x2 ; x3 ) j x22 + x2 x3 + x23 0:24 is an estimate of the region of attraction and for only slightly larger values of r2 the control constraints prevent controlled invariance of the set Vr2. It may happen that the neither of the solutions which keep the output identically equal to zero is de ned on a neighborhood of the equilibrium of interest. However, by combining the dierent solutions, we may achieve that the domain of de nition is a neighborhood of the origin. This is a purely algebraic constraint and we are not aware of any techniques in the literature which could be used for this kind of stability analysis. The constructive power of quanti er elimination is very useful in this case.

170


Example 8.4 The switched dynamic system considered in this example can be seen as the zero dynamics of the following system

x_ 1 = (x2 - 6x3 - u)(x22 + x33 - u2 ) x_ 2 = x3 x_ 3 = u y = x1

(8.27)

We get three dierent systems which correspond to the continuous output zeroing controllers

x_ 2 = x3 x_ 3 = x2 - 6x3

x_ 2 = q x3 x_ 3 = x22 + x33

x_ 2 = x3q x_ 3 = - x22 + x33

Can we switch between these systems to achieve stability of the zero dynamics? Observe that the domain of de nition of the state feedback control laws that give the last two systems are not the whole state space. Furthermore, the rst system is unstable so the zero dynamics cannot be stabilized without switching. We will show that the zero dynamics can be stabilized by switching despite these observations. The regions where V_ i(x) < 0 are shown in Figure 8.4. 2

2

2

1

1

1

0

0

0

-1

-1

-1

-2 -2

-1

0

1

2

-2 -2

-1

0

1

2

-2 -2

-1

0

1

2

Figure 8.4 The regions (gray) where V_ i(x) < 0; i = 1; 2; 3. Let V (x) = x22 + x2 x3 + x23 . The derivatives of V (x) along solutions to the systems are

V_ i(x) = 2x2 x3 + x23 + (x2 + 2x3 )ui ; where ui is given by one of the solutions to (x2 - 6x3 - u)(x22 + x33 - u2 ) = 0. The modi cation of the quanti er elimination problem (8.11) due to the implicit

8.4 Examples

171

relation between x and u becomes

h

9r 8x2 8x3 9u x 6= 0 ^ V (x) r2 ! 2x x + x2 + (x + 2x )(x - 6x ) < 0 _ 2 3 3 2 3 2 3 i 2 2x2 x3 + x3 + (x2 + 2x3 )u < 0 ^ x22 + x33 - u2 = 0 : Performing quanti er elimination we can prove that V (x) 169 describes a control

invariant set in which the zero dynamics can be stabilized by switching.

We investigate the example from [119] to show how quanti er elimination can be used to obtain a parameterized set of solutions for the problem { in [119] only numerical solutions were available. We will follow the procedure outlined in Section 8.2.


x_ 1 = a1 (x) + u + u2 x_ 2 = x3 (8.28) x_ 3 = a2 (x) + u - u3 y(x) = x1 where a1 (x) = -x1 + 2:1x2 - 2x3 and a2 (x) = 3x2 - 3:8x3 . The control objective is

to design a control law which would keep the output identically equal to zero and for which the zero dynamics is stable. There are two output zeroing control laws

p

u (x) = -1 1 - 42(2:1x2 - 2x3 ) ;

(8.29)

which are well de ned on the set S = x 2 R3 j 1 - 4(a1 (x)) 0; x1 = 0 . The zero dynamics is de ned by y = x1 0 and hence we obtain x_ 2 = x3 (8.30) x_ 3 = 3x2 - 3:8x3 + u - u3 :

After linearization of (8.30) at the origin with the choice u+ we obtain that the linearized zero dynamics has its eigenvalues at f 0:41; -2:21 g. Similarly, the linearized zero dynamics with u- has as eigenvalues 0:10 1:09i. Hence, for both control laws (8.29) the corresponding zero dynamics is unstable. The Jacobians obtained by linearizing the zero dynamics (8.30) is denoted by F1 and F2 . We compute the Routh-Hurwitz (or Lienard-Chipart) inequalities for 1 F1 + 2 F2 which are parameterized by 1 and 2

-(321 - 1 2 - 422 ) > 0; 91 - 2 > 0:

172


Performing quanti er elimination, we rst check the existence of solution (this was veri ed numerically in [119])

h

91 92 -(321 - 1 2 - 422 ) > 0

^

91 - 2 > 0

^ 1

>0

^ 2

i

>0 :

This decision problem can be shown to be True and the system can be stabilized at least locally. If we only eliminate the quanti ed variable 1 in above formula we obtain 2 > 0. Hence, for any positive 2 there exists 1 for which the positive combination of matrices F(1 ; 2 ) = 1 F1 + 2 F2 is Hurwitz. By xing any pair of 1 ; 2 which yields stability, the matrix F(1 ; 2 ) satis es a Lyapunov matrix equation. That is, for any matrix Q = QT 2 Rn n with Q 0 there exists a matrix P = PT 2 Rn n with P 0 which is the solution of

FT (1 ; 2 ) P + P F(1 ; 2 ) = -Q:

Note that Q introduces another degree of freedom and can be regarded as a design parameter. The following P is a solution of the above matrix equation

2 2

P= 2 3 : With this P we estimate the domain of attraction of the zero dynamics, which is the target set we want to reach if we want exact tracking of constant outputs. By using quanti er elimination we obtain that the ellipsoid de ned by xT Px < 0:057735 is a domain of attraction for the zero dynamics. In fact, the plane 1 - 4a1 (x) = 0 which is the boundary of the domain of de nition of the switched control laws, is a tangent to this ellipsoid. The zero dynamics is usually described by nonlinear dierential equations and yet this fact is not incorporated suciently into the de nition of minimum phase. For instance if the zero dynamics has a globally stable limit cycle, the system would be termed non-minimum phase despite that the behavior of the systems when tracking constant outputs may be satisfactory. In this sense the known de nition of minimum phase is often misleading. This was tried to overcome in [11] where global stability of an invariant set was suggested as the de nition of minimum phase. We believe that the local equilibrium stability and global set stability de nitions are just two ends of a wide spectrum of possible situations. In the following example we show how to use quanti er elimination to estimate the region of attraction of an invariant set for a given zero dynamics.


x_ 1 = (x2 x23 - x22 - u)(x22 - x2 - u)(x2 - 4x3 - u) x_ 2 = x3 x_ 3 = u y = x1

(8.31)

8.5 Summary

173

The question of invariance of a given set can be formulated as follows: Is it possible to choose the control such that the solution trajectory tangent always points inwards along the boundary of the given set, i.e., Vx(x) f(x; Ki(x)) < 0. Using the Lyapunov function V (x) = x22 + x2 x3 + x23 , we can show that the following formula is True

h 8x2 8x3 21 V (x) 61 ! 2x x + x2 + (x + 2x )(x x2 - x2 ) < 0 _ 2 3 3 2 3 2 3 2 2 2x2 x3 + x3 + (x2 + 2x3 )(x22 - x2 ) < 0 _ i 2x2 x3 + x23 + (x2 + 2x3 )(x2 - 4x3 ) < 0 which means that the invariant set V (x) 1=2 has a domain of attraction de ned by V (x) 61. If we change V (x) 1=2 to V (x) = 6 0 the formula is False. Further analysis is needed to decide how the state behaves inside the ellipsoid V (x) 1=2

(asymptotically stable, limit cycles, etc).

Some of the issues treated here were raised in [119] but we show that for a large class of systems, quanti er elimination provides a tool to test these conditions. The use of switched output zeroing controllers allow for more exibility when control and state constraints have to be taken into account. If the goal is to exactly track constant outputs, then xing the equilibrium around which we wish our zero dynamics to be stable, does not seem to be appropriate. Indeed, it may happen that the zero dynamics system has several dierent equilibria with (perhaps) disjoint regions of attractions. In order to exactly track the output and have bounded states and control, reaching any of the basins of attraction would satisfy the control objective. Quanti er elimination could be used for such computations.

8.5 Summary In this chapter we have investigated the application of quanti er elimination to the analysis of stabilizability of polynomial systems by switching between a number of controllers. We have shown how to check if a polynomial system is stabilizable by switching but also how to estimate the region of attraction in such cases. In exact output tracking, the zero dynamics plays an important role. For control-ane systems the zero dynamics is uniquely determined and stability of the zero dynamics or, equivalently, the concept of minimum phase, is usually used as a feasibility condition for exact output tracking problems. For more general nonlinear systems the zero dynamics or the output zeroing controller are no longer unique. We argue that stabilizability of the zero dynamics is a more appropriate condition than just stability. In a number of examples we show how to stabilize the zero dynamics of dierent systems by switching between dierent output zeroing controllers, where no continuous stabilizing controller exists. Quanti er elimination is used to carry out the necessary computations, where we also can take control and state constraints into account in a direct manner.

174


9 Conclusions and Extensions In this part of the thesis we have formulated a number of problems in control theory as formulas in the rst-order theory of real closed elds and then applied quanti er elimination to solve them. Many problems in control theory seem to t into this framework and it is not hard to come up with other ideas worth investigation. An appealing property of this approach is the often close connection between the original problem formulation and the initial formula containing quanti ed variables. It is also possible to take into account constraints on both the control and state variables in a direct manner which is not a common feature of other approaches.

Applications to Linear Systems We have singled out two problems in linear control theory to show the applicability of the quanti er elimination approach. First we addressed the problem of stabilization in presence of uncertainties. Given a suitable controller structure and a description of the uncertainties, we end up with a number of conditions on the controller parameters such that the system is stabilized regardless of the actual values of the uncertain parameters. The second problem was feedback design. Here the objectives were to shape the return ratio of the closed-loop system by an appropriate choice of controller parameters, where model uncertainties also were taken into consideration. Speci cations on the Nyquist curve of the return ratio were translated into formulas and quanti er elimination was performed to get conditions on the controller parameters. A common approach to deal with model uncertainties in design of robust controllers and stability analysis is to use tools such as convex programming, linear matrix inequalities (LMIs), -analysis, and quantitative feedback theory [14, 21, 175

176

Chapter 9 Conclusions and Extensions

69, 76, 105]. When considering real parametric uncertainties the problems often become non-convex and most of these methods fail. The quanti er elimination method does not utilize any information about possible convexity of the problem and hence does not suer from this drawback. On the other hand, for convex problems the above approaches are known to be very ecient. The conditions on the controller parameters in both problems are in general given by a system of equations and inequalities, or a union of such systems, produced by the quanti er elimination algorithm. If the controller parameters are bounded by existential quanti ers one may also get speci c values on the controller parameters, i.e., get a numerical solution to the problem (a sample point from a cell in the CAD for which the formula is true).

Applications to Nonlinear Systems For nonlinear systems we have restricted the discussion to mainly two problems, computation of stationary points and curve following in the state space. Quanti er elimination provides a way of constructing a real polynomial system in the state variables, whose solution consists of those states that can be made stationary by an appropriate choice of available control signals, the so called stationarizable states. The subset of these stationarizable points that corresponds to asymptotically stable equilibrium points, can also be computed in a similar fashion. The output range corresponding to a computed set of stationarizable states can be computed as a projection. We have also considered the problem of computing the subset of the output range in which we can control the output to take any constant value. Curve following in the state space gives an answer to the question if it is possible to steer the state between two points along a given curve using an admissible control. This is a kind of motion planning problem where not only the output but also the state of the system is speci ed at every time instant. We have also given a generalization of this problem where the curve to follow is to be chosen in a parameterized set of curves which starts in a set of initial states and ends in a set of nal states. If we do not put any restrictions on the set of nal states and choose a exible set of curves we can consider the curve following problem as a constructive method to compute global reachability from a given set of initial points. However, due to the many variables of this problem the computational complexity becomes very high.

Applications to Switched Systems There are nonlinear systems that cannot be stabilized by a continuous state feedback control law but can be stabilized by a controller that switches among a number of state feedback laws. We show how to use quanti er elimination to decide if a set of basic controllers can be used to stabilize a system. The method is based on Lyapunov techniques. To cut down the computational complexity, a candidate

177 Lyapunov function can rst be computed from linearizations of the closed loop systems corresponding to dierent controllers. If the system can be stabilized by switching, a control law can easily be designed using the Lyapunov function as a control Lyapunov function [50]. For example, a pure minimization or hysteresis approach can be utilized. In exact output tracking the zero dynamics plays an important role. For systems which are not ane in the control, the control that keeps the output identically to zero is not uniquely determined in general. The non-uniqueness can be exploited to design stabilizing switched controllers for the zero dynamics. We have given a number of examples and pointed at some issues where quanti er elimination can be utilized.

General Remarks and Complexity The method of quanti er elimination seems attractive due to its expressive power, e.g., we can easily incorporate control and state constraints. These kind of constraints are known to be very common in practice but hard to take into consideration in classical methods. The main problem is to pin point the speci cations and objectives in terms of (semi-)algebraic relations. To write down the question or formula, i.e., the input to the solution algorithm is then often a minor problem. The expressive power is not only a good thing. It implies that problems which are known to be very hard to solve should also be solvable by a general quanti er elimination algorithm. This means that any algorithm for quanti er elimination has to be complex, by its own nature. The bad time-complexity w.r.t. the number of variables at present, limits its usefulness for large scale control applications. However, recently improved complexity bounds and a new algorithm have appeared [12, 13] that probably will enlarge the set of tractable problems. Many classical problems in control theory can be formulated using the concepts of semialgebraic sets and formulas in the rst-order theory of real closed elds. An important observation is that this implies that these problems can, at least theoretically, be solved in a nite number of steps.

Extensions

Nothing prevents us from working with systems given in implicit form, f(x_ ; x; u) = 0 or more general mixed state and control constraints, UX (x; u) but we have chosen a simpler setting to demonstrate the ideas. The problem of simultaneous stabilization addresses the design of a controller that stabilizes a number of dierent linear systems simultaneously. An application of this can be design of simple controllers for hybrid systems, i.e., systems including switching devices that change the system properties drastically. According to the methods given here, the problem can be reduced to nding a common solution of a number of real polynomial systems. Veri cation of control laws and further stability analysis of nonlinear systems, e.g., by Lyapunov theory are other interesting subjects to study. Once the system,

178


the controller, and the control objectives are de ned in terms of (semi-)algebraic relations, the veri cation procedure can be described formally according to

8z [ control law(z)

^

system description(z) ] ! [ system behavior(z) ] ;

i.e., if the above formula can be decided to be true the control law implies a certain behavior of the closed-loop system. One way to circumvent the problem of complexity of the quanti er elimination method can be to combine several approaches. This can be done rst by trying to solve the linear part of the problem using linear programming techniques. Some of the nonlinear constraints may then become linear because the values of certain coecients were determined. The solver continues until no more simpli cations can be done. This approach has been proposed in [36]. Now, either the problem is solved completely or we hope that the number of variables has been decreased substantially, to allow a solution by quanti er elimination in a reasonable amount of time. Another way to deal with the complexity problem is to utilize the sparse structure of multivariate polynomials which seems to be a common property in many applications. Recent developments along these lines have been made by Canny [29]. If it is possible to decide if a part of a problem is convex one could use convex programming to solve the problem as far as possible. Quanti er elimination would then be a nal step in the solution algorithm where many of the original variables already have got their values determined in the preceding convex programming step. For many problems in control the quanti er elimination method is unnecessarily general. It would be interesting to examine if it is possible to utilize some of the subalgorithms of the quanti er elimination method for some speci c control problems and in this way reduce the complexity.

Part III

Computations on Polyhedral and Quadratic Sets

179

10 Linear Matrix Inequalities and Optimization Optimization plays an important role in many engineering applications. For large optimization problems, i.e., problems with many variables it is often of outmost importance to exploit structure to deal with the computational complexity. Convexity is an example of a structural property which reduce computational complexity considerably. Recently, a number of results and algorithms have emerged for problems where the optimization constraints are given by linear matrix inequalities. Many constraints in engineering problems can formulated in terms of linear (or ane) matrix inequalities, see for example [21, 153, 154] and references therein. A linear matrix inequality is a positive (semi-)de nite constraint on a anely parameterized symmetric matrix. In the subsequent chapters we will formulate many problems as convex optimization problems. The eciency of available implementations implies that these problems scale to much larger size than the samples treated in this thesis. In this chapter we give a brief introduction to linear matrix inequalities in Section 10.1. In Section 10.2 we review some basic facts about convexity and list a number of convex optimization problems. The chapter ends with a summary in Section 10.3.

10.1 Linear Matrix Inequalities A linear matrix inequality can be seen as a sign constraint on the eigenvalues of a symmetric matrix. We recall the standard de nitions of de niteness for symmetric matrices and linear matrix inequalities. 181

182

Chapter 10 Linear Matrix Inequalities and Optimization

De nition 10.1 Let P 2 Rm (i) (ii) (iii) (iv)

m and x 2 Rm . A matrix P is called positive de nite, denoted P 0, if it is symmetric and xT P x > 0; 8x 6= 0. positive semide nite, denoted P 0, if it is symmetric and xT P x 0; 8x. negative de nite, denoted P 0, if it is symmetric and xT P x < 0; 8x 6= 0. negative semide nite, denoted P 0, if it is symmetric and xT P x 0; 8x.

De nition 10.2 A linear matrix inequality (LMI) is an ane matrix-valued function

F(x) = FT (x) = F0 +

m X i=1

xi Fi

(10.1)

constrained by a de nite or semide nite inequality constraint. The variables x 2 Rm is called decision variables and the matrices Fi = FT i 2 Rn n are symmetric. A more appropriate name for these kind of inequalities would be ane matrix inequalities. However, we adopt the traditional name here. LMIs de ne semialgebraic sets. The de niteness of a matrix can be de ned in terms of the sign of some of its minors [52] and the minors are polynomial functions of the matrix entries. For example, a matrix is positive de nite if and only if all its successive principal minors are positive. The solution set of an LMI is always convex, see Example 10.1. This property can be exploited to get very ecient numerical methods for optimization purposes and for computing feasible points. The most ecient numerical algorithms for solving LMIs are based on so called interior point methods [120].

10.1.1 LMIs in Control

There are many problems in systems and control theory which can be formulated in terms of LMIs or as optimization problems over sets de ned by LMIs. For an introduction we refer the reader to the excellent book [21] and the references therein. Here we only give a brief example. For a linear time invariant system the search for a quadratic positive de nite Lyapunov function can be formulated as an LMI. In fact, V (x) = xT P x is a Lyapunov function for the system x_ = A x if P is symmetric and satis es the following LMIs

AT P + P A 0;

P 0;

where the entries of P can be chosen as decision variables and both inequalities can be written in the form (10.1). It is well known that this particular problem has an analytical solution. Observe that any matrix expression where a matrix variable appears anely can be rewritten in the form (10.1). Furthermore, a number of LMIs can always be written as one LMI, since we can list them as blocks along the diagonal of a large matrix.

10.2 Convex Optimization

183

10.2 Convex Optimization Convex optimization is the problem of minimizing a convex function over a convex set. There are several classes of convex optimization problems such as linear and quadratic programming, semide nite programming and determinant maximization problems. For all these optimization problems there exist ecient interior point methods and there are several implementations available. Here we give a brief introduction to convexity, list some important problem classes, and give pointers to some implementations.

10.2.1 Convex Sets and Functions

The geometrical interpretation of a convex set is that a line segment between any two points in C lies in C . Formally we have the following de nition.

De nition 10.3 A set C is convex if for all such that 0 1 x1 2 C ; x2 2 C =) x1 + (1 - ) x2 2 C : We refer to 1 x1 + 2 x2 as a positive combination of x1 and x2 if 1 0; 2 0. If in addition 1 + 2 = 1 we call it a convex combination .

Example 10.1 It is easy to see that an LMI de nes a convex set. Consider the set de ned by F(x) 0 and let x1 and x2 be two elements of this set. Then F(x1 + (1 - )x2 ) = F(x1 ) + (1 - ) F(x2 ) 0 where the equality follows from the fact that F is an ane function. The convex combination of F(x1 ) and F(x2 ) is also positive de nite, since the sum of two positive de nite matrices is positive de nite.

De nition 10.4 A function f : ! R is convex if is convex and for all such that 0 1 x1 2 ; x2 2 =) f( x1 + (1 - ) x2 ) f(x1 ) + (1 - ) f(x2 ): A geometrical interpretation of this condition is that a line between any two points on the graph of the function lies above the graph. Convexity is a one-dimensional property: a function is convex if and only if it is convex when restricted to any line that intersects its domain of de nition. If a function is twice dierentiable then it is convex if and only if its Hessian (second derivative) is positive semide nite for all points in its domain of de nition.

184


Example 10.2 The function log det P-1 is convex over the set of positive de nite

matrices. The set of positive de nite matrices is convex since any positive combination (and in particular a convex combination) of positive de nite matrices is positive de nite. Consider the function g(h) , log det(P + h V )-1, where P 0, V is any symmetric matrix, and h is a real parameter. We have

g(h) = - log det P - log det(I + h P-1=2 V P-1=2) = Y X - log det P - log (1 + h i ) = - log det P - log(1 + h i ); i

i

where i denotes the eigenvalues of P-1=2 V P-1=2. Dierentiating twice w.r.t. h gives

g (h) = 00

1 ( 1 + h i )2 > 0; i

X

and we conclude that log det P-1 is convex.

10.2.2 Some Classes of Convex Optimization Problems The general form of a convex optimization problem is minimize f(x) subject to x 2 C

(10.2)

where f : Rn ! R is a convex function and C Rn is a convex set. The function f is called the objective function or cost function and C is called the feasible set or constraint set . Introductions to convex optimization can be found in for example [17, 22, 133]. Linear programming (LP) can be seen as a very special class of convex optimization problems, where the objective function is linear and the feasible set is a polyhedron, i.e., a nite intersection of a number of halfspaces. We will use the form minimize cT x (10.3) subject to Ax b; where x 2 Rm , c 2 Rm , A 2 Rn m , b 2 Rn and the inequality sign denotes componentwise inequality. If the feasible set is de ned by an LMI instead we get a semide nite programming (SDP) problem

minimize subject to

cT x F(x) 0;

(10.4)

10.3 Summary

185

where x 2 Rm , c 2 Rm , and F(x) is given by (10.1). A survey of the theory and applications of semide nite programming can be found in [153]. A determinant maximization (MAXDET) problem is an optimization problem of the form minimize cT x + log det G(x)-1 subject to G(x) 0 (10.5)

F(x) 0;

where x 2 Rm , G : Rm ! Rl l and F : Rm ! Rn n are of the form (10.1). In [154] an overview of applications where the MAXDET problem appears are given and an interior point method for its solution is described.

10.2.3 Software

In the examples given in Chapter 11 and 12 the C-program sdpsol [160{162] has been used. An interface between Mathematica [159] and sdpsol has been written by the author to provide an integrated environment for MAXDET computations and visualization. There exist several other semide nite programming implementations, see for example [2, 20, 68, 149].

10.3 Summary In this chapter we have given a brief introduction to linear matrix inequalities and some convex optimization problems such as linear programming and determinant maximization problems. In Chapter 11 and 12 we will encounter many problems where convex optimization applies.

186


11 Sets De ned by Ane and Quadratic Inequalities 11.1 Introduction Sets de ned by ane and quadratic inequalities have been extensively studied for a very long time. There is a large collection of results related to them which has been exploited in many areas such as optimization, geometric modeling, approximation theory, etc. In Chapter 12 we will utilize these sets for computation of invariant sets of dynamic systems. A very nice presentation of polyhedra and systems of linear inequalities is given in [164]. Sets de ned by quadratic inequalities are treated in for example [10, 22]. There are many dierent ways of representing sets in Rn and the best choice of representation depends on the problem at hand. Hence, it is useful to have a gallery of dierent representations to choose between. In Section 11.2 we list a number of dierent representations for hyperplanes, halfspaces, polyhedra, ellipsoids, etc. The feasibility problem of a set of equations and inequalities is treated in Section 11.3 which lets us determine if the corresponding set is empty. In Section 11.4 we give conditions on when a set is a subset of another set.

11.2 Representation

We can use systems of ane and quadratic inequalities to de ne subsets of Rn . The generic description of the sets we will treat in this chapter is

x = f(z) j g(z) 0 187

^

h(z) = 0 ;

(11.1)

188

Chapter 11 Sets De ned by Ane and Quadratic Inequalities

where f is an ane function, g(z) 0 denotes a number of ane or quadratic inequalities and h(z) = 0 is a number of ane equalities. We call this a constrained parameter representation since z can be thought of as a parameter which is mapped into Rn by f and whose value is constrained by equations and inequalities. There are a number of special cases when some of f, g, and h are trivial or are missing. A set of the form

x = f(z) j z 2 Rm

is said to be described by a free parameter representation since there are no constraints on z. The following is said to be a constraint representation

x 2 Rn j g(x) 0

^

h(x) = 0

(11.2)

since x is constrained by equations and inequalities.

11.2.1 Hyperplanes and Halfspaces

Hyperplanes and halfspaces are very simple geometric objects. It is easy to construct more complicated geometric sets using these basic sets as building blocks.

De nition 11.1 A hyperplane in Rn is a set of the form HP = x 2 Rn j aT(x - x0 ) = 0 or equivalently

(11.3)

(11.4) HP = x = x0 + Az j z 2 Rn-1 : The representations are related as follows: A = aT and aT = A . We use M to denote any matrix whose columns form a basis of N (M), i.e., the null space of M. The matrix M exists only if N (M) =6 f0g. By A we mean any matrix of maximal full rank such that A A = 0. Note that A exists only if A has linearly ?

N

N

N

?

?

?

dependent rows. Here (11.3) and (11.4) are the constraint representation and free parameter representation of the hyperplane HP , respectively. We will also use aT x = b to represent a hyperplane. To rewrite this in the form (11.3) we only need to nd a point x0 in the plane, e.g., x0 = (aT ) b, where M denotes the Moore-Penrose inverse or the generalized inverse of the matrix M. y

y

De nition 11.2 A halfspace is a set of the form H = x 2 Rn j aT (x - x0 ) 0 or equivalently

(11.5)

H = x = x0 + Az + at j z 2 Rn-1 ; t 0 ; where a 62 R(A).

(11.6)

11.2 Representation

189

Here R(M) denotes the range space of the matrix M. We have the relations A = aT and aT = - sign(A a)A . Representation (11.5) can be interpreted as a sublevel set of aT x or as all points x 2 Rn such that a and x - x0 form an obtuse ?

N

?

angle. The intersection of a nite number of hyperplanes forms a so called ane set.

De nition 11.3 An ane set is a set of the form A = x 2 Rn j C(x - x0 ) = 0 or equivalently

(11.7)

A = x = x0 + Dz j z 2 Rn-m :

(11.8)

Here m is the number of equalities in (11.7), i.e., the number of rows of C. We have the relations D = C and C = D . If the ane set is represented by Cx = d we can choose x0 = C d to convert to the above representations. The intersection of a nite number of halfspaces is called a polyhedron. N

?

y

De nition 11.4 A polyhedron is a set of the form PH = x 2 Rn j Ax b or equivalently

PH = x = V + W j

X

i

(11.9)

i = 1; 0; 0 :

(11.10)

The inequalities in (11.9) and (11.10) should be interpreted componentwise. Representation (11.10) is the Minkowski sum of the convex hull of the columns of V and the conic hull of the columns of W . In the sequel we will use the more compact notation Cofv1 ; : : : ; vp g to denote the convex hull of a set of points fv1 ; : : : ; vp g. Conversions between these two representations is rather straightforward in theory, but in practice it is non-trivial due to the computational complexity. The so called double description method can be used for the conversion, see e.g., [114, 125]. A polytope is a bounded polyhedron, i.e., it contains no rays of the form

x0 + tw j t 0

^

Ax0 b :

De nition 11.5 A polytope is a set of the form PT = x 2 Rn j AT x b or equivalently

PT = x = V j

X

i

(11.11)

i = 1; 0 :

(11.12)

190


Given a set of inequalities Ax b it is easy to see that a necessary and sucient condition for the set of solution to be a polytope is that there is no vector w such that Aw 0. This is a feasibility problem of a set of linear inequalities which is treated in Section 11.3. Representation (11.12) is the convex hull of a number of points, i.e., the conic hull part in (11.10) is missing. We will also encounter intersections of ane sets and polyhedra. Suppose that the intersection is given by the following constraint representation

x 2 Rn j Ax b

^

Cx = d :

(11.13)

In some cases (e.g., when testing for non-emptiness) it is more convenient to rewrite (11.13) using a constrained parameter representation such that the parameters are only constrained by inequalities. We can rewrite (11.13) using the free parameter representation of the ane set de ned by Cx = d. This gives the following representation of the intersection of an ane set and polyhedron

x = x0 + C z j A~ z b~; z 2 Rn-m ;

(11.14)

N

where m is the number of rows of C, A~ = AC and b~ = b - AC d. N

y

11.2.2 Quadratic Sets We have seen that a linear inequality de nes a halfspace and more complicated geometric objects can be described in terms of intersections of these basic sets. A large number of geometric objects can be represented by considering only one quadratic polynomial (total degree two) inequality. Some of these have special names such as ellipsoids, paraboloids, quadratic cones, etc. We will use the term quadratic sets for sets de ned by one quadratic polynomial inequality. The situation is clearly more complex than in the linear case. Quadratic sets can be either convex or non-convex, connected or non-connected, bounded or non-bounded. Quadratic sets have very compact descriptions and can be used to approximate more complicated sets. A polytope can for example be approximated rather well by an ellipsoid. Here we have singled out a small number of quadratic sets with some useful properties and whose extensions to higher dimensions still are easy to work with. A quadratic polynomial in n-variables can always be parameterized as follows

xT A bx T T x Ax + 2b x + c = 1 bT c 1 ;

(11.15)

where AT = A 2 Rn n ; b 2 Rn , and c 2 R.

De nition 11.6 A quadratic set is a set of the form Q = x 2 Rn j xT Ax + 2bT x + c 0 :

(11.16)

11.2 Representation

191

Observe that certain conditions on A; b, and c have to be satis ed for Q to be non-empty, see Section 11.3. It is not possible to give one general constrained parameter representation of a quadratic set, i.e., it is not possible to represent Q according to (11.1) where f, g, and h have similar structures for all quadratic sets. Instead there are dierent representations depending on the properties of A; b, and c. The intersection of an ane and quadratic set can be represented as follows A \ Q = x 2 Rn j xT Ax + 2bT x + c 0 ^ Cx = d : (11.17) To check non-emptiness, a constrained parameter representation of A \ Q turns out to be more convenient

T T T A \ Q = x = x0 + Dz j 1z D0 x10 bAT bc D0 x10 1z 0 ; (11.18) where D = C and x0 = C d. N

y

We will now take a closer look at ellipsoids, quadratic cones, and paraboloids which all are special instances of quadratic sets.

11.2.3 Ellipsoids

Ellipsoids come up naturally in many engineering and scienti c applications. In our case we need them since they are natural invariant sets of stable linear systems, see Chapter 12.

De nition 11.7 An ellipsoid is a set of the form E = x 2 Rn j (x - x0 )T P(x - x0 ) 1 ; where PT = P 0, or equivalently E = x = Bu + x0 j uT u 1 ; where BT = B 0.

(11.19) (11.20)

Here the relation between P and B is P = B-2 . Furthermore, P = A=(bT A-1 b - c) and x0 = -A-1 b if we use the generic representation (11.16) of a quadratic set to describe the ellipsoid. The additional constraint bT A-1 b - c > 0 has to be imposed on the parameters to ensure that E does not reduce to a point or the empty set. A number of geometric properties of an ellipsoid can be computed from these parameterizations.

Center The center of the ellipsoid is x0 . T be a real orthonormal eigenvalue decomposition of P, Semi-axes Let P = UU where U = u1 : : : un ; = diag(1 ; : : : ; n ); i > 0. This is alwaysppossible for symmetric matrices. Then u is the semi-axes of the ellipsoid and 1= their i

lengths, see Figure 11.1. Observe that B = U-1=2 UT .

i

192


p

1= 1u1

p

1= 2u2


Figure 11.1 The center and semi-axes for an ellipsoid. Volume The volume is

p

vol(E ) = n det(B) = n = det(P);

(11.21)

where n is the volume of the unit-sphere in Rn . This follows immediately from representation (11.20) (which is an ane transformation of the unit-sphere) and the corresponding change of variables in the multiple integral for the volume of the ellipsoid. We observe that the logarithm of the volume of an ellipsoid can be written as log(vol(E )) = - log det(B-1 ) or log(vol(E )) = + 12 log det(P-1):

(11.22) (11.23)

Now, log det(M-1 ) is a convex function of M, see Example 10.2, which means that (11.22) is a concave function of B and (11.23) is a convex function of P. This observation can be used for optimization purposes, see [21, 154].

11.2.4 Quadratic Cones

A Quadratic cone is an unbounded set, which consists of two convex components1 connected only by a single point, the vertex . Quadratic cones have a very compact description compared to ane cones (the intersection of a number half spaces through the origin) since we only need one inequality to describe them. Quadratic cones are invariant sets of a class of linear systems, see Chapter 12.

1 The term quadratic cone is a little abuse of notation since the set is not a cone according to the usual de nitions. However, either of the two convex components of the set is a cone.

11.2 Representation

193

De nition 11.8 A quadratic cone is a set of the form QC = x 2 Rn j (x - x0 )T P(x - x0 ) 0 ; (11.24) where PT = P 2 Rn n and P has n - 1 positive eigenvalues and one negative

eigenvalue. After an ane change of coordinates it is always possible to write a quadratic cone as z~ (11.25) 2 Rn j ~zT P~ z~ z2 ; QC =

zn

n

where P~ 2 Rn-1 n-1 ; P~ T = P~ 0. Note that for each xed non-zero value of zn the inequality describes an ellipsoid. For quadratic cones we list a few geometric properties.

Vertex The vertex of the quadratic cone is x0. The size Since a quadratic cone is an unbounded set its volume cannot be used as a measure of its size (its in nite!). One way to measure the size of a quadratic cone is to compute the volume of the ellipsoid obtained by intersecting QC by a hyperplane which has the cone axis as a normal and does not contain the vertex of the cone, see Section 12.4.2 for more details on this.

Axis and semi-axes LetP = UUT be a real orthonormal eigenvalue decom-

position of P, where U = u1 : : : un ; = diag(1 ; : : : ; n ); i > 0; i = 1; : : : ; n - 1; n < 0. The axis of the quadratic cone is given by un and ui ; i = 1; : : : ; n - 1 are semi-axes of the ellipsoid p obtained by intersecting p QC by the hyperplane de ned by uTn (x - x0 ) = 1= jn j. Furthermore, 1= i ; i = 1; : : : ; n - 1 are the lengths of these semi-axes, see Figure 11.2.

11.2.5 Paraboloids

A paraboloid is a convex and unbounded quadratic set. Paraboloids are invariant sets of a class of linear systems, see Chapter 12.

De nition 11.9 A paraboloid is a set of the form P = x 2 Rn j (x - x0 )T P(x - x0 ) qT (x - x0 ) ; (11.26) where PT = P 2 Rn n and P has n - 1 positive eigenvalues and one eigenvalue that is zero2. Furthermore, qT v = 6 0 where Pv = 0; v =6 0. This condition ensures that P does not reduce to a single point.

2

This implies that we can have an additional term, qT v, on the right hand side of (11.26).

194


x0

x0

PSfrag replacements

Figure 11.2 Two dierent views of a quadratic cone and p its vertex, axis and semi-axes. The plane is de ned by uT3 (x - x0 ) = 1= j3 j. After a change of coordinates it is always possible to write a paraboloid as

z~

(11.27) P = zn 2 Rn j z~T P~z~ zn where P~ 2 Rn-1 n-1 ; P~ T = P~ 0. Note that for each xed positive value of zn the

inequality describes an ellipsoid. For paraboloids we have the following geometric properties.

Base point The base point of the paraboloid is x0 . The geometrical interpretation of this point is that there exists a v 2 Rn such that the ray x0 + tv; t 0 contains all rays which also are included in P . The size The paraboloid is also an unbounded set. To measure the size we can compute the volume of the ellipsoid obtained by intersecting P by a hyperplane

which has the paraboloid axis as a normal and gives a nonempty intersection, see Section 12.4.3 for more details on this.

11.3 Feasibility

195

Axis and semi-axes LetP = UUT be a real orthonormal eigenvalue decom-

position of P, where U = u1 : : : un ; = diag(1 ; : : : ; n ); i > 0; i = 1; : : : ; n - 1; n = 0. The axis of the quadratic cone is given by un and the ui ; i = 1; : : : ; n - 1 are semi-axes of the ellipsoid obtainedpby intersecting P by the hyperplane de ned by qT U(x - x0 ) = 1. Furthermore, 1= i ; i = 1; : : : ; n - 1 are the lengths of these semi-axes, see Figure 11.3.

x0

x0

PSfrag replacements

Figure 11.3 Two dierent views of a paraboloid and its base point, axis and semi-axes for a paraboloid. The plane is de ned by qT U(x - x0 ) = 1.

11.3 Feasibility How can we determine if a set de ned by a number of ane inequalities or by a quadratic inequality is empty? In other words we are interested in the feasibility of a system of ane inequalities or conditions such that there exists a solution of a quadratic inequality. It turns out that the feasibility problem for ane inequalities is easily solved by linear programming. For a quadratic inequality there exists an explicit test in terms of the parameters A, b, and c if the parameterization xT Ax + 2bT x + c 0 is used.

196


11.3.1 Ane Equations and Inequalities

How can we determine if a system of linear inequalities Ax b has a solution? Consider the following linear program in the variables x 2 Rn and t 2 R.

t Ax - b tI; t -1:

minimize subject to

(11.28)

It is trivial to nd a feasible point for this problem (take x = 0 and t > max(-b)). Now Ax b has a solution if and only if the solution t of (11.28) is negative. The additional inequality t -1 makes the problem bounded from below. If we have a system of both linear equations and inequalities it can always be reduced to a system of only inequalities according to the equivalence between (11.13) and (11.14).

11.3.2 Quadratic Inequality

The following lemma answers the question when a quadratic set is empty. It turns out that we have an explicit characterization of solvability in terms of the parameterization.

Lemma 11.1 Let AT = A 2 inequality

Rn ,

b 2 Rn , and c 2 R. Then the quadratic

xT Ax + 2bT x + c 0

(11.29)

has no solution x 2 Rn if and only if

A 0; (I - AA )b = 0 and bT A b - c < 0: y

y

(11.30)

Proof We split the proof into two cases: A has at least one negative eigenvalue and A 0. Observe that the two cases are mutually exclusive and that a symmetric matrix belongs to either of them. First, suppose that A has at least one negative eigenvalue. Since AT = A it can be written as A = TTT, where T TT = T T T = I and is a diagonal matrix containing the eigenvalues of A. The non-singular coordinate transformation x = Tz gives zT z+2bT Tz+c 0, where at least one diagonal element of is negative. It is easy to see that the inequality holds if we chose the corresponding entry of z suciently large and put the other entries of z equal to zero. Hence, inequality (11.29) has always a solution if A has at least one negative eigenvalue. Now, suppose that A 0 and that the eigenvalues in is sorted in decreasing order, i.e., ~ 0 T A = TT = T 0 0 T T; where ~ 0:

11.3 Feasibility

197

If A 0 we let = ~ . Now, rewrite inequality (11.29) as follows

(x + A b)T A(x + A b) + 2bT (I - A A)x - bT A b + c 0; y

y

y

y

where A is the pseudo-inverse of A. If A 0 the linear term disappears and the following nonsingular coordinate change Tz = x + A b gives the inequality y

y

zT z - bT A b + c 0; which has a solution if and only if bT A b - c 0 since zT z is a positive sum of squares. If A has at least one zero eigenvalue the same change of coordinates gives y

y

zT T TATz + 2bT (I - A A)(Tz - A b) - bT A b + c = 0 T ~ z~z~ + 2b T z - bT A b + c; y

y

y

y

where z = z~ z T and the dimension of z corresponds to the number of zero eigenvalues. Observe the separation of quadratic terms in z~ and linear terms in z. We easily see exists a solution to (11.29) if the term involving z is present, 0that0 there T i.e., if b T 0 I 6= 0 or equivalently bT (I - A A) 6= 0. However, if the linear m term vanishes, i.e., if bT (I - A A) = 0 then we have the same situation as when A 0 (but with a positive sum of squares in z~) and we must have bT A b - c 0 for a solution to exist. Hence, if A 0 there exists no solution to (11.29) if and only if bT (I - A A) = 0 and bT A b - c < 0 which concludes the proof. 2 y

y

y

y

y

Note that bT (I - A A) = 0 is trivially satis ed if A 0 since then A = A-1 . Condition (11.30) is very similar to the conditions for a block matrix to be negative semide nite, see [21]. In fact an alternative proof can be given based on this observation. Compare condition (11.30) with the condition given in subsection 11.2.3 that ensures that a parameterized ellipsoid does not reduce to the empty set or a single point. Sometimes a quadratic inequality does not impose any constraints on the variables, i.e., the set of solutions of it is the whole Rn . Here follows the converse of Lemma 11.1. y

y

Lemma 11.2 Let A 2 Rn n , b 2 Rn , and c 2 R. Then

xT A bx n 1 bT c 1 0; 8x 2 R

Proof

,

A b bT c 0:

() follows directly from the de nition of negative semi-de niteness.

(11.31)

198


)) Multiply the inequality by the square of an arbitrary but nonzero zn

Then it can be rewritten as

2 R.

z xT A bz x n n n zn bT c zn 0; 8x 2 R ; zn 2 R: Since we can get arbitrarily close to any z = z~ z T in Rn by an admissible choice of x and zn the lemma follows.

n

2

Note that the equivalence in the lemma above is no longer valid if the inequalities is replaced by their strict counterparts.

11.3.3 Quadratic Inequality and Ane Equations

If we have a system of both linear equations and a quadratic inequality it can always be reduced to a single quadratic inequality according to the equivalence between (11.17) and (11.18). Then Lemma 11.1 can be applied to determine feasibility. Alternatively, Lemma 11.2 can be used to check if the quadratic inequality does not impose any constraints.

11.4 Inclusion Given sets de ned by ane and quadratic inequalities we want to decide if one set is a subset of another. Here we will list a number of results on this topic.

11.4.1 S-procedure

The constraint that some quadratic function should be negative whenever some other quadratic functions are negative can be converted to an LMI. This conversion is known as the S-procedure. The feasibility of the obtained LMI is a sucient condition for the inclusion of an intersection of quadratic sets in a quadratic set. The reader is referred to [21, 150] for an extensive list of references and history of the S-procedure.

Lemma 11.3 Let F0 ; : : : ; Fp be quadratic functions of x 2 Rn , i.e., Fi (x) = xT Ai x + 2bTi x + ci ; i = 0; : : : ; p

where ATi = Ai . Consider the following condition F0 (x) 0 for all x such that Fi (x) 0; i = 1; : : : ; p: A sucient condition for (11.32) to be true is

9 1 0; : : : ; p 0 such that for all x 2 Rn ; F0 (x)

p X i=1

i Fi (x):

(11.32) (11.33)

11.4 Inclusion

199

If p = 1 and there exists an x0 such that F1 (x0 ) < 0 condition (11.33) is also necessary.

Proof See [150].

2

According to Lemma 11.2 condition (11.33) can be reformulated as

A b X p 0 0 - i Ai bi 0; i 0; i = 1; : : : ; p; bTi ci bT0 c0

(11.34)

i=1 which is an LMI in A0 ; b0 ; c0 , and i . If the functions Fi are ane this is Farkas'

lemma which states that (11.32) and (11.33) are equivalent [164]. Hence, this lets us formulate an LMI describing all half spaces (de ned by an ane inequality) that contain a polyhedron (de ned by a number of ane inequalities). Using linear programming it is easy to determine if a polyhedron is a subset of a half space as we will see later. Since the S-procedure is both necessary and sucient for two quadratic functions we have a computational procedure to determine if one quadratic set is a subset of another.

11.4.2 Convex Hull In Polyhedron

Given a polytope de ned as the convex hull of a number of points

PT = Cofv1 ; : : : ; vp g

and a polyhedron in constraint form

PH = x 2 Rn j Ax b it is easy to determine if PT PH. Due to convexity we only have to check if the vertices of PT belongs to PH, i.e., if the inequalities Avi b; i = 1; : : : ; p are satis ed.

11.4.3 Polyhedron In Polyhedron

Consider the polyhedron PH = x 2 Rn j Ax b and the halfspace H = x 2 Rn j cT x d . We can determine if PH H by linear programming. Let x be the solution to the following linear program maximize cT x (11.35) subject to Ax b: Then it is easy to see that a necessary and sucient condition for PH H is that cT x d. If the solution is unbounded then PH 6 H. This result can be extended to check inclusion of two polyhedra, i.e., PH1 PH2 represented in constraint form. Then we have to solve as many linear programs as there are inequalities that de ne PH2 .

200


11.4.4 Quadratic Set In Polyhedron

Lemma 11.4 Let Q = x 2 Rn j xTPx + 2qT x + r 0 be a quadratic set and PH = x 2 Rn j ^i cTi x d a polyhedron. Then Q PH if and only if 9i 0 such that

0 T

P q ci =2 ci =2 -d - i qT r 0; i = 1; : : : ; m (11.36)

Proof This lemma is a straightforward application of the S-procedure. Each LMI corresponds to the inclusion of Q in a halfspace de ned by cTi x d. 2 The above criterion for inclusion is a feasibility problem of a number of LMIs. However, if the block matrix containing P, q, and r is non-singular then feasibility can be determined by eigenvalue computations. This is shown in the next subsection. If the quadratic set is an ellipsoid there are much simpler conditions to determine if the ellipsoid is contained in the polyhedron. Lemma Let E = x 2 Rn j (x - x0 )T P(x - x0 ) 1 be an ellipsoid and 11.5 H = x 2 Rn j cT x d a halfspace. Then E H if and only if p (11.37) cT x0 + cT P-1c d:

Proof We change coordinates z = P1=2(x - x0) where P1=2 is any symmetric matrix such that P1=2P1=2 = P. Note that P1=2 always exists since P 0. The

ellipsoid is mapped onto the unit ball and the hyperplane is described by c~T x d~ ; where c~ = P-1=2 c and d~ = d - cT x0 : (11.38)

Now a necessary and sucient condition for the unit ball to be included in the halfspace (11.38) is max c~T x d~ : (11.39) T z z 1

Now the maximal value p T of-1 c~T x on the unit sphere is attained for x = c~=jc~j and the maximum is jc~j = c P c. Hence, in the original coordinates condition (11.39) becomes

p cT x0 + cT P-1c d

and the proof is completed.

2

To check if an ellipsoid is contained in a polyhedron we only have to evaluate (11.37) for each halfspace de ning the polyhedron.

11.4 Inclusion

201

Example 11.1 Consider the ellipsoids Ei = x 2 R2 j (x - xi)T Pi(x - xi) 1 , where

-0:32 P1 = -0:81 0:32 0:44 ; -1:54 ; P2 = -1:33 1:54 3:11 0:25 0 ; P3 = 0 4:0

x1 = 0:3 0 ; x2 = -0:2 0 ; x3 = 0:3 0 :

and the polyhedron de ned by

PH = x 2 R2 j - 5x1 + 4x2 6

^

9x1 + 7x2 28 :

Evaluating two inequalities of the form (11.37) for each ellipsoid shows that E1 PH and E2 PH but E3 6 PH. 4

3

2

1

-2

-1

1

2

3

-1

Figure 11.4 Three ellipsoids and a polyhedron.

11.4.5 Convex Hull In Quadratic Set

In general it is hard to give conditions for inclusion of the convex hull of points Cofv1; : : : ; vp g in a quadratic set x 2 Rn j xTAx + 2bTx + c 0 . However, if the points belong to the quadratic set and we can decide that they belong to a convex component of the set, then the convex hull also belongs to the quadratic set.

202


For convex quadratic sets, such as ellipsoids and paraboloids, we only have to check if the vertices of the polytope belong to the quadratic set, i.e.,

vTi Avi + 2bT vi + c 0;

i = 1; : : : ; p:

(11.40)

Since a quadratic cone is the union of two convex sets a necessary condition for a convex hull of points to be a a subset is that all points belong to either of the two sets. If the axis and vertex of the quadratic cone are given all points should belong to one of the two halfspaces through the vertex which have the axis as normal. If in addition inequality (11.40) is satis ed then the convex hull is a subset of the quadratic cone.

11.4.6 Quadratic Set In Quadratic Set

To determine if a quadratic set is a subset of another quadratic set we can use the S-procedure. Recall that the S-procedure gives both a necessary and sucient condition for a quadratic function to be negative whenever another quadratic function is negative.

Lemma 11.6 Let Qi = x 2 Rn j xTAix + 2bTi x + ci 0 ; i = 1; 2 be two quadratic sets. Then Q1 Q2 if and only if 9 0 such that

A b A b 1 1 2 2 bT2 c2 - bT1 c1 0:

(11.41)

Proof Note that Q1 Q2 if xTA2 x + 2bT2x + c2 0 for all x such that xTA1x + 2 2bT1 x + c1 0. Apply the S-procedure and the lemma follows. Let H; Ei ; QC , and P ; Pi denote a halfspace, ellipsoids, a quadratic cone, and paraboloids, respectively. Then Lemma 11.6 can be used to determine if e.g.,

E1 H ; E1 P ;

E1 E2 ; E1 QC ; P QC ; P1 P2 : To determine if there exists a 0 in the above lemma, is an LMI feasibility

problem. However, for some special cases there are simpler solutions. Let

M1 = AbT1 bc11 ; M2 = AbT2 bc22 1 2 and suppose that M1 is non-singular. This is the case for e.g., ellipsoids and quadratic cones. Then the question if there exists 0 such that the pencil M2 - M1 is negative semide nite can be determined by eigenvalue computations. The pencil M2 - M1 is singular if and only if M-1 1 M2 - I is singular, i.e., when is an eigenvalue of M-1 1 M2 . Now the eigenvalues give a decomposition of the -axis in at most n + 1 open intervals where M-1 1 M2 is nonsingular for all

11.4 Inclusion

203

in an interval. Furthermore the number of positive and negative eigenvalues are xed over each interval. This implies that we only have to test if M2 - M1 is negative semide nite for the endpoints and an arbitrary in each interval (e.g., the midpoint) to determine if Q1 Q2 . These observations are also applicable to the LMIs in Lemma 11.4.

Example 11.2 Consider the two ellipsoids E1 , E2 and the quadratic cone QC

that are shown in Figure 11.5. These quadratic sets can be represented by the inequality

xT

x

1 M 1 0;

where

2 8:61 6 0:15 ME1 = 6 4-4:34 -8:55

Q

2 3 8:61 4:34 -8:55 6 -0:26 0:84 7 6 0:15 7 3:6 1:47 5 ; ME2 = 4 -4:34 -10:71 1:47 13:16

0:15 1:94 -0:26 0:84 -

-

2 1:13 -0:8 6 -0:8 2:5 6 MQC = 4-0:88 0:33 -0:09 0:03

0:88 0:33 -0:63 -0:06 -

0:15 1:94 -0:26 0:7 -

4:34 -0:26 3:6 3:27 -

3 10:71 0:7 7 7 3:27 5 ; 15:53

-

3 0:09 0:03 7 7: -0:065 -0:01 -

Figure 11.5 Left: E1 and QC. Right: E2 and QC . It is easy to verify that M - M 1 0 for = 2. Hence, E1 QC . The real eigenvalues of M-21 M are f0:408238; 0:00102009g. Checking the eigenvalues of M - M 2 for = 0:0005; 0:00102009; 0:25; 0:408238; 0:5 shows that this is always an inde nite matrix. Hence, E2 6 QC . QC

E

QC

E

QC

E

204


11.4.7 Quadratic Set and Ane Set In Quadratic Set

To determine if the intersection of a quadratic set and an ane set is a subset of another quadratic set we can rst reduce the number of variables and then use the S-procedure. We give the following lemma.

Lemma 11.7 Consider the quadratic sets Qi = x 2 Rn j xT Ai x + 2bTi x + ci 0 , n i = 1; 2 and the ane set A = x 2 R j Cx = d . Then A \ Q1 Q2 if and only if there exists 2 R such that C C dT A b A b C C d 2 2 1 1 0 1 0 1 0; 0: (11.42) bT c2 - bT c1 y

N

2

N

1

y

Proof First, note that A \ Q1 Q2 is equivalent to A \ Q1 A \ Q2 . The intersection A \ Qi can be represented as in (11.18), i.e.,

T T x = C z + C d j 1z C0 C1d bATj bcjj C0 C1d 1z 0 : (11.43) j To determine if A \ Q1 A \ Q2 is true we only have to apply the S-procedure to the inequality constraints of (11.43), j = 1; 2, since subset relations are preserved under ane mappings. This concludes the proof. 2

N

y

y

N

y

N

Example 11.3 Consider the ellipsoid E2 and the quadratic cone QC in Example 11.2 again. The two planes and

HP 1 = x 2 R3 j x1 + x3 = 3:3

HP 2 = x 2 R3 j x1 + x3 = 4

both intersect E2 . We can verify this using Lemma 11.5. The inequality (11.37) is not satis ed for any of the four halfspaces de ned by HP 1 and HP 2 . To check if HP i \ E2 QC for i = 1; 2 we have to check feasibility of the LMI (11.42). For HP 1 \ E2 QC we get the LMI

2 2:26 4 1:13

3

2

1:13 -2:87 20:89 2:50 -0:755 - 1 4-0:11 -2:87 -0:75 -3:94 5:71 and for HP 2 \ E2 QC we get 2 1:13 -0:80 -2:473 210:44 4-0:80 2:50 0:91 5 - 2 4 0:08 -2:47 0:91 -5:65 2:80

3

-0:11 5:71 1:94 0:025 0; 1 0 0:02 0:59

(11.44)

3

0:08 2:80 1:94 0:12 5 0; 2 0: 0:12 -0:11

(11.45)

Since all matrices in (11.44) and (11.45) are nonsingular we can decide if there exist 1 or 2 according to the method described after Lemma 11.6. The LMI (11.44) has no solution and 2 = 2:5 is a solution of (11.45). Hence, HP 1 \ E2 6 QC and HP 2 \ E2 QC . The sets are shown in Figure 11.6.

11.5 Summary

205

Figure 11.6 Left: E2 , HP 1, and QC. Right: E2, HP 2 , and QC.

11.5 Summary In this chapter we have reviewed de nitions and representations of sets de ned by ane or quadratic inequalities. Due to the variety of quadratic sets in higher dimensions we have focused the presentation on ellipsoids, quadratic cones and paraboloids since their properties are easily generalized to higher dimensions. We have shown how tests for inclusion can be formalized in terms of inequalities for parameters de ning the sets. Both ordinary inequalities and linear matrix inequalities have been used. Observe that all results in this chapter scales to any dimension. The choice of two and three dimensional examples were only for the sake of visualization.

206


12 Invariant Sets of Piecewise Ane Systems Invariant sets of dynamic systems play an important role in many situations when the dynamic behavior is constrained in some way. Knowing that a set in the state space of a system is invariant means that we have bounds on the behavior. We can verify that prespeci ed bounds which originate from for example safety restrictions, physical constraints, or state-feedback magnitude bounds are not violated. This veri cation becomes subset inclusion tests, i.e., we have to check if the invariant set is a subset of the admissible part of the state space. For piecewise ane systems, i.e., systems whose dynamics is given by an ane dierential equation in dierent polyhedral regions of the state space, the computation of invariant sets is harder than for systems de ned by state equations with continuous right hand sides, since both continuous and discrete dynamics have to be taken into account. This kind of systems appear e.g., when a linear system is controlled by a PID-controller equipped with selectors [7, 8], if there are saturations present in the system, or as approximations of nonlinear systems obtained by system identi cation using hinging hyperplanes [24]. In this chapter we will use the results presented in Chapter 11 to compute invariant sets of piecewise ane systems. We will also give examples on how such invariant sets can be used for veri cation of system properties. Controller design and veri cation based on invariant sets have been studied in [64, 70, 71, 130, 143{145]. However, the continuous dynamics in the studied systems are often of very simple form (constant right hand sides or rectangular differential inclusions). If more complex dynamics are considered the papers are of more theoretical than practical interest, since non-constructive methods are used. This contribution is an attempt to allow veri cation of hybrid systems with more 207

208

Chapter 12 Invariant Sets of Piecewise Ane Systems

advanced continuous dynamics without sacri cing the constructive part. In [62] an algorithm is presented for nding the maximal invariant set for a discrete-time linear system with polyhedral constraints on the states and control. The chapter is organized as follows. In Section 12.1 we give a de nition of piecewise ane systems and point at some earlier work in the literature. Section 12.2 rst gives the de nition of invariant sets for continuous systems and then extends this notion to piecewise ane systems. Polyhedral invariant sets are treated in Section 12.3. We show how to compute the smallest invariant polyhedron of a particular structure such that a given set of initial states is included. Section 12.4 parallels the section on polyhedral invariant sets but here we use quadratic sets such as ellipsoids, quadratic cones, and paraboloids instead. In Section 12.5 we use the results on polyhedral and quadratic sets to compute invariant sets of piecewise ane systems. A summary of the chapter is given in Section 12.8.

12.1 Piecewise Ane Systems By a piecewise ane system we mean a dynamic system which has dierent ane dynamics in dierent polyhedral regions of a continuous state space. To describe the state of the system we require two distinct sets of states. The set of continuous states x 2 Rn and the set of discrete states (or equivalently modes ) s 2 S = f1; : : : ; Ng. For each discrete state the continuous state evolves according to an ane dierential equation

x_ (t) = As(x(t)) x(t) + bs(x(t)); (12.1) where As(x(t)) 2 Rn n and bs(x(t)) 2 Rn . The evolution of the discrete state s is determined by a number of transition sets which we will describe below. We may think of the state space of (12.1) as a collection of N enumerated copies of Rn . A

simple generalization is to let the continuous part of the state space have variable dimension, i.e., x 2 Rns(t) . This is needed in some applications, especially if so called reset maps or valuations is allowed at discrete state transitions, see e.g. [3]. The discrete and continuous dynamics are coupled through a number of transition sets Tij 2 Rn . We use the convention that Tii = ;. A discrete state transition from mode i to mode j occurs when the continuous state enters a transition set, i.e., x(t) 2 Tij. Then the continuous state is mapped into the continuous state space of mode j. For simplicity we let this mapping be the identity mapping. We assume that each transition set is de ned by a halfspace

Tij = x 2 Rn j cTij x dij

(12.2)

and the boundaries @Tij are denoted switching planes . The above description of the coupling between the continuous and discrete dynamics will be referred to as the transition mechanism . A piecewise ane system can be thought of as a nite state machine with a continuous system at each discrete state, see Figure 12.1. In [3, 41, 130] the

12.2 Invariant Sets

209

term hybrid automaton is used for system models similar to the ones studied here. The transitions between the discrete states are triggered by conditions of the type x(t) 2 Tij. The notion (12.2) of transition sets de ned by halfspaces and not only hyperplanes allows for rather complicated behaviors. For example, it might be the case that the continuous state enters a transition set immediately after a discrete state transfer. This means that the system will only be in this discrete state for an in nitesimal instant of time before entering the next discrete state. To prevent nondeterministic behavior of the system some kind of order of priority between the transition sets of the same mode has to be introduced. This priority order should specify which transition that should occur if a state belongs to two or more transition sets simultaneously. We will not elaborate more on these questions but will assume that these kind of multiple transitions does not occur for the systems and initial sets we consider. PSfrag replacements

x_ = A1 x + b1 1

x 2 T12

x 2 T32 x_ = A4 x + b4 4

x 2 T34

2 x_ = A2 x + b2 x 2 T23 3 x_ = A3 x + b3

Figure 12.1 A nite state machine with piecewise ane dynamics at each node.

Stability is related to invariant sets since it is a sucient condition for invariance. Stability of piecewise linear and ane systems has been investigated in [23, 88, 89, 127].

12.2 Invariant Sets An invariant set of a dynamic system is a subset of the state space such that once the state enters this set it will remain in it for all future times, see [92, 97, 156]. Sometimes the term positive invariant set is used to stress that the invariance only holds for future times and not backwards in time. Formally we have the following de nition.

210


De nition 12.1 Let z(t) be the state of a dynamic system at time t and M a subset of the state space. Then M is an invariant set w.r.t. the dynamic system if z(t1 ) 2 M =) z(t) 2 M for all t > t1 :

(12.3)

It is easy to give conditions on invariance for continuous time dynamic systems. Based on these conditions we can then derive conditions for invariant sets for piecewise ane systems.

12.2.1 Continuous Systems

Consider the following dynamic system

x_ = f(x);

(12.4)

where f : Rn ! Rn is a continuously dierentiable function. A sucient condition for invariance of a set M is that the ow of the dierential equation is directed into the set at each point on the boundary, @M. For a set with smooth boundary (i.e., there exists a normal, nM (x) at each point x 2 @M) this can be expressed as follows

nM (x)T f(x) < 0

8x 2 @M:

If the invariant set is described by an inequality

M = x 2 Rn j V (x) c ;

(12.5) (12.6)

i.e., as a sublevel set of a continuously dierentiable function V : Rn ! R, a normal is given by the gradient of V , denoted Vx. In this case condition (12.5) becomes

Vx(x)f(x) < 0

8x such that V (x) = c:

(12.7)

Many invariant sets can be described as sublevel sets of a parameterized class of functions V (x; ) under certain restrictions on the parameters. Inequality (12.7) can be used to derive these restrictions. In this chapter we will develop results based on the assumption that M is a polyhedral or quadratic set, which were treated in Chapter 11. Observe that sublevel sets of a Lyapunov function of a dynamic system are invariant sets since the ow of the vector eld is always directed into the set according to the de nition of a Lyapunov function (V_ (x) = Vx(x)f(x) < 0; x 6= 0). The following lemma turns out to be useful for computations of invariant sets. It states the rather natural result that invariance of a set w.r.t. a dynamic system is a property which is not lost by changing coordinates.

Lemma 12.1 Let V : Rn

and f : Rn ! Rn be continuously dierentiable functions. An invariant set M = x 2 Rn j V (x) c of the dynamic system x_ = f(x), remains invariant under dieomorphisms. !

R

12.2 Invariant Sets

211

Proof We show that condition (12.7) for invariance is invariant under dieomor-

phisms [78]. Let : Rn ! Rn be a dieomorphism and y = (x). For this nonlinear change of coordinates we get y_ = f~(y) and the invariant set is de ned by V~ (y) c where f~(y) = x -1 (y) f -1 (y) and V~ (y) = V -1 (y) : Now V~ y(y) f~(y) = Vy -1 (y) x -1 (y) f -1 (y)

= Vx -1 (y) -y 1 (y) x -1 (y) f -1 (y) = Vx(x)f(x);

where we use the chain rule in the second equality and the identity

-y 1 (y) x -1 (y) = I

in the last equality. Since the scalar product between a normal to the boundary of the set and the

ow of the dierential equation is preserved under dieomorphisms the invariance follows. 2 We will apply a special case of this lemma later which states that an invariant set remains invariant under a linear change of coordinates y = T x, where T 2 Rn n is nonsingular. For a linear (or ane) system there are some particularly simple types of invariant sets which are closely related to the eigenvalues and eigenvectors of the system matrix. These are invariant sets described by linear inequalities such as invariant halfspaces, slabs, and polyhedra. Examples of invariant sets described by quadratic inequalities are invariant ellipsoids, cylinders, cones, and paraboloids. The following lemma turns out to be very useful for computing some invariant sets. The change of coordinates transforms the dynamics into two parts that do not in uence each other.

Lemma 12.2 Suppose that A 2 Rn n and that there are v; w 2 Rn and 2 R suchthat Av = v; wT A = wT . Then there exists a similarity transformation T = wT v 2 Rn n such that

A~ 0 1 T AT = 0 :

?

-

(12.8)

The rows of w form an orthonormal basis of the orthogonal complement of w. ?

T Proof Let T -1 = YyT . Then

TwT Y Tv I = T -1T = YyT w T yT v =) Y TwT = I; Y Tv = 0; yT wT = 0; yT v = 1 ?

?

?

?

(12.9)

212


Furthermore, y has to be a left eigenvector of A. This follows from yT wT = 0, that is, y 2 (wT ) and the orthogonal complement of wT is spanned by w. Now Y TAwT Y TAv A~ Y Tv A~ 0 -1 ?

?

?

?

T AT = yT AwT yT Av = yT wT yT v = 0 : In the second equality we use that v and y are right and left eigenvectors, respectively. The last equality follows from the identities in (12.9). 2 ?

?

?

If A is nonsingular, the ane system x_ = Ax + b can always be transformed to a linear system by the coordinate change z = x + A-1 b. Here -A-1 b is the stationary point of the ane system. If A is singular and b 62 R(A) then there is no stationary point. However, if b 2 R(A) then there exists a stationary subspace. In the sequel of this chapter we will assume that A is nonsingular and can without loss of generality concentrate on only linear systems. Observe that if A 2 Rn n is nonsingular and satis es the assumptions in Lemma 12.2 we can use the ane transformation z = T -1 (x + A-1 b); T = wT v (12.10) to rewrite system x_ = Ax + b as ~ z_ = A0 0 z: (12.11)

?

In the sequel the described methods for computing invariant sets assume that the ane system has been put in the form (12.11). Once an invariant set has been computed, we can make the inverse transformation to get the appropriate invariant set. The stability degree of a linear time invariant system is a useful bound on how fast the state converges to the origin. Later on we will see that comparing the stability degrees of two subsystems of a linear system, we can decide if there exists invariant quadratic sets of a certain kind.

De nition 12.2 The stability degree of a linear dynamic system x_ = Ax is the negative of the maximum real part of the eigenvalues of A.

Lemma 12.3 The stability degree of a linear dynamic system is the maximal value of such that the LMI AT P + PA + 2P 0; P 0 (12.12) has a solution. Proof The characteristic polynomial of A is given by det(I - A). A necessary

and sucient condition for all eigenvalues of A to be in the open left half plane is that the LMI AT P + PA 0; P 0 (12.13)

12.2 Invariant Sets

213

has a solution, see [90]. Now, let = ~ - ; 2 R, i.e., a shift of the imaginary axis. Then det(I - A) = det(~I - (A + I)). This shows that a necessary and sucient condition for the eigenvalues of A to have real parts less than is that the eigenvalues of A + I belongs to the open left half plane. We substitute A with A + I in (12.13) to get the corresponding LMI

(A + I)T P + P(A + I) = AT P + PA + 2P 0; P 0: We relax this to a nonstrict LMI since the maximal value of belongs to the boundary of the feasible set and the lemma follows. 2 The decay rate (or largest Lyapunov exponent) of a dynamic system is de ned to be the largest such that t tlim !1 e jjx(t)jj = 0

is true for all trajectories. Using a quadratic Lyapunov function V (x) = xT Px, P 0 it is easy to show that if dV (x)=dt -2V (x) for all trajectories then V (x(t)) V (x(0))e-2t. Hence, jjx(t)jj e-t (P)1=2 jjx(0)jj where (P) is the condition number of P. Therefore the decay rate is at least . The condition that dV (x)=dt -2V (x) for all trajectories is equivalent to AT P + PA + 2P 0 which is the same as (12.12). Hence, for linear time invariant systems the stability degree is equivalent to the decay rate.

12.2.2 Piecewise Ane Systems

For a piecewise ane system, an invariant set is a subset of Rn f1; : : : ; Ng. This set can be considered as a collection of N indexed sets Mi Rn ; i = 1; : : : ; N, i.e.,

( x; s ) j x 2 Ms ; s 2 f1; : : : ; Ng :

We will call the sets Mi ; i = 1; : : : ; N the mode invariant sets of system (12.1). Observe that a mode invariant set Mi does not have to be connected due to the switching behavior of the system. Two components of a mode invariant set might correspond to continuous states mapped from dierent discrete states. Furthermore, inequality (12.5) does not have to be valid on the boundary of Mi . However, either inequality (12.5) is valid at a point x 2 @Mi or x 2 Tij, where Tij is the switching set de ning transitions between the discrete states i and j. Furthermore, the de nition of a piecewise ane system implies that if x(t-) 2 Tij then x(t+ ) 2 Mj , where t is the switching time. The concept of an invariant set of a piecewise ane system is illustrated in Figure 12.2. The modes and transition sets correspond to the hybrid automaton depicted in Figure 12.1. Note that if a component of a mode invariant set does not intersect any of the transition sets of the mode the state will remain in this component (and in the corresponding mode) for all future times.

214


T12 M1 PSfrag replacements

T23

s=1

M4

M2 s=2

T32 M3

s=4

T34

s=3

Figure 12.2 An invariant set of a piecewise ane system described by the hybrid automaton in Figure 12.1.

12.3 Polyhedral Invariant Sets A polyhedral invariant set of a dynamic system is an invariant set which is a polyhedron, i.e., it can be described as the intersection of a number of invariant halfspaces. In this section we will derive conditions for a linear system to have invariant polyhedra and show how to compute the smallest such polyhedron that includes a given set of possible initial conditions. The results are easily generalized to ane systems with nonsingular system matrices.

12.3.1 Invariant Halfspaces

The real eigenvalues and their eigenvectors of a system matrix A correspond to linear functionals of the state which depending on the initial state is either increasing or decreasing w.r.t. time. As we will see, these functionals de ne hyperplanes in the state space and their initial values give the boundary of an invariant set. The following lemma characterizes the connection between invariant halfspaces and left eigenvectors of a system matrix A of a linear system x_ = Ax.

12.3 Polyhedral Invariant Sets

215

Lemma 12.4 Let v 2 Rn be a left eigenvector of A 2 Rn n with a real eigenvalue 2 R, i.e., vT A = vT . Then the state space can be decomposed into three

invariant sets given by vT x < 0, vT x = 0, and vT x > 0.

Proof The derivative w.r.t. time of vT x(t) along a trajectory of the linear system can be written as

d T T T T dt (v x(t)) = v x_ (t) = v Ax(t) = v x(t):

The solution of this dierential equation is

vT x(t) = et vT x(0)

(12.14) and together with the properties of the exponential function, the invariance of the sets in the lemma follows. 2 In fact, we can give even smaller invariant sets if the sign of is known. We have the following characterizations If > 0 then vT x > a is an invariant halfspace if a > 0 and vT x < a is an invariant halfspace if a < 0. If = 0 then vT x = a is an invariant hyperplane. If < 0 then 0 < vT x < a is an invariant slab if a > 0 and a < vT x < 0 is an invariant slab if a < 0. These observations follow immediately from the properties of the exponential function and (12.14).

12.3.2 Invariant Polyhedra

Computing the left eigenvectors corresponding to real eigenvalues gives a number of invariant slabs for negative eigenvalues, half spaces for positive eigenvalues, and hyperplanes for zero eigenvalues. Since intersections of invariant sets are invariant, the (real) left eigenvectors de ne invariant polyhedra. If the system matrix is diagonalizable (i.e., there exists n independent left eigenvectors) and all eigenvalues are real and negative, the described invariant sets are polytopes.

Example 12.1 Consider the following linear system

2 -1 2:5 0:9 3 x_ = 4-1:5 -1 -0:55 x:

0

0 -0:8

(12.15)

Here v = 0 0 1 T is a left eigenvector of the system matrix and the corresponding eigenvalue is = -0:8. The slab given by 0 < x3 < 2 is an invariant set. See Figure 12.3 for a number of trajectories in this slab.

216

Chapter 12 Invariant Sets of Piecewise Ane Systems -1 0

2

1 1.5

1

0.5

0 -1 0 1

Figure 12.3 Trajectories of system (12.15) in the invariant slab given by

0 < x3 < 2.

Now, given a set of initial states, what is the smallest invariant set of a given form (slab, hyperplane, etc.)? To compare the size of sets of the same form we use set inclusion, i.e., a set S1 is said to be smaller than S2 if S1 S2 . This makes it possible to compare two halfspaces or slabs de ned by translated versions of the same hyperplane.

Theorem 12.1 Let v 2 Rn be a left eigenvector of A 2 Rn n with a real eigenvalue 2 R, i.e., vT A = vT . Given the set X0 Rn and the bounds

a = xmin0 (vT x); 2X

b = xmax0(vT x); 2X

(12.16)

then the smallest invariant halfspace or slab that includes X0 can be described as follows. If > 0 then

0 < a b =) vT x a a b < 0 =) vT x b a < 0 < b =) Rn is invariant.

< 0 =) min(0; a) vT x max(0; b). = 0 =) a vT x b.

12.3 Polyhedral Invariant Sets

217

Proof From the proof of Lemma 12.4 we have vTx(t) = etvT x(0); t 0. Now

jvT x(t)j

is increasing, constant, or decreasing if is positive, zero, or negative, respectively. From this observation each of the above cases follows immediately. Suppose for example that < 0 and 0 < a < b. Then 0 < a < vT x(0) < b T and vT x(t) is monotonically decreasing towards 0. This means that 0 < v x(t) < n T b; t 0. Hence X0 x 2 R j 0 < v x < b , which is the smallest invariant slab containing X0 . 2 Let v1 ; : : : ; vr be r independent left eigenvectors of A with eigenvalues 1 ; : : : ; r . For each eigenvector vi we have the bounds from (12.16) ai = xmin (vTi x); bi = xmax (vTi x): (12.17) 0

0

2X

2X

According to Theorem 12.1 there is an invariant halfspace, slab, or hyperplane corresponding to each of the left eigenvectors vi . The intersection of these sets is again an invariant set and hence we get an invariant polyhedron. A compact description of this polyhedron is

PH = x 2 Rn j

^

i:i =0

^

ai vTi x bi

j:j >0; 0 2Re[~], where ~ denotes any other eigenvalue of A. Then there exist invariant paraboloids.

Proof We can assume that A = diag(A~ ; ) since this can always be obtained after a change of variables according to Lemma 12.2. Consider the paraboloids described by

P = xx~n 2 Rn j ~xTP~x~ xn

(12.35)

where P 2 Rn n ; PT = P 0. Inequality (12.7) where V (x) = x~T Px~ - xn gives

2x~T P -1 A~ x~ = x~T A~ TP + PA~ x~ - x < 0; n xn

(12.36)

where the equality follows by symmetrization. For P to be invariant this inequality has to be satis ed for all x on the boundary of P , i.e., xn = x~T Px~. With this substitution (12.36) becomes

x~T A~ T P + PA~ - P x~ < 0; which should be satis ed for all x~ since any x~ is a projection of a point on the boundary of the invariant paraboloid. We get the corresponding LMI in P A~ T P + PA~ - P 0: This LMI has a positive de nite solution if and only if the real parts of the eigenvalues of A~ is less than =2 according to Lemma 12.3. Hence, QC is invariant.

2

226


Using this lemma we can now state the optimization problems which give invariant paraboloids. We use to distinguish two dierent invariant paraboloids, given by the inequalities x~T P~ x~ xn . One of them contains the positive xn -axis and the other one the negative xn -axis. Note that the projection onto the xn -axis of an initial set cannot include both positive and negative values of xn since neither P+ nor Phas this property. In the inequalities below the upper of the two signs should be used if the initial set has positive projection onto the xn axis.

Initial Point Let the initial state be x(0) = v1. Then the invariant paraboloid w.r.t. (12.24) which contains v1 and whose intersection with the plane xn = 1 has smallest volume can be computed by minimize log det P-1 subject to A~ T P + PA~ - P 0; P 0; v~T1 Pv~1 (v1 )n ;

(12.37)

where the sign in front of (v1 )n is determined by the sign of the last component in

v1 .

Initial Polytope Let PT = Cofv1 ; : : : ; vp g be a polytope of initial states such

that (vi )n ; i = 1; : : : ; p all have the same sign. If this assumption is not satis ed, then there are no invariant paraboloids for this initial polytope, see the discussion in Section 11.4.5. We get the following optimization problem minimize log det P-1 (12.38) subject to A~ T P + PA~ - P 0; P 0; T v~i Pv~i (vi )n ; i = 1; : : : ; p; where the sign is determined by the sign of the last component in vi .

Initial Ellipsoid Let the set of initial states be given by the ellipsoid E0 parameterized as in the case of invariant ellipsoids. The S-procedure is used to get an LMI which corresponds to the constraint E0 P , see Lemma 11.6. minimize subject to

log det P-1 ~T ~ A 2 P + PA - P 30; P 0; 0

(12.39) P P 0 0 P x 0 0 0 40 0 1=25 - -xTP xTP x - 1 0: 0 0 0 0 0 0 1=2 0 Note that the initial ellipsoid E0 has to belong to either xn > 0 or xn < 0 for an invariant paraboloid to exist. This can be veri ed by Lemma 11.6. If this is the case the sign is determined by checking on which side of xn = 0 the center of the ellipsoid is placed. If (x0 )n > 0 the minus sign should be used and vice versa.

12.4 Invariant Quadratic Sets

227

Initial Quadratic and Ane Set Intersection The description of and additional assumptions on the initial set Q0 \ A0 are the same as in the case of an invariant quadratic cone. The optimization problem becomes minimize log det P-1 subject to A~ T P0+2PA~ - P 0; P3 0; 0; 1

C

A b C 0 C d T @4P0 00 1=2 5 - bT c A 0 0 1 0 1=2 0 y

N

C d 0: 1 y

N

(12.40)

Example 12.4 Consider the linear system in Example 12.1 again. Let the set of initial states be a degenerated ellipsoid given by a quadratic cone and a plane QC 0 \ H0 , where

2x - 1:253T 2 3:87 -1:3 -1:953 2x - 1:253 1 1 QC 0 = x 2 R3 j 4x2 + 3:055 4 -1:3 3:5 -2:255 4x2 + 3:055 0 ;

x3 - 2:7 -1:95 -2:25 1:62 x3 - 2:7 3 H0 = x 2 R j (x1 - 1:95) + (x2 + 2:35) + 2(x3 - 4:1) = 0 :

Solving the optimization problem (12.40) gives the invariant quadratic cone shown as a wire-frame in Figure 12.6. We also show trajectories for some initial states at the boundary of the degenerate ellipsoid.

4 3 2

x3

1 -4

0 -2

1 x1

0

0

-1 -2 2

-3

x2

Figure 12.6 An invariant paraboloid and some trajectories.

228


12.4.4 A Generalization

Consider the function V (x) = x~T Px~ - x n , where x = ( x~ xn )T , PT = P 2 Rn n 0 and > 0. It is easy to derive conditions such that the sublevel set with P x 2 Rn j V (x) 0 is an invariant set of the linear system (12.24). For = 1 and = 2 the sublevel set is a paraboloid and a quadratic cone, respectively. Note that for even values of the set consists of two components as in the case of quadratic cones but for other values there are only one component as in the case of paraboloids. Following the steps in the proof of Lemma 12.6 we get the LMI

A~ T P + PA~ - P 0:

(12.41)

Comparing (12.41) with LMI (12.12) in Lemma 12.3 we conclude that this LMI has a positive de nite solution if and only if the stability degree of A~ is greater than - =2. If we let denote the maximal real eigenvalue of a system matrix A, then there are invariant sets de ned by the sublevel set of V above if and only if 2 Re[~]= for all other eigenvalues ~ of A. Using LMI (12.41) we can parallel almost all results for invariant paraboloids and cones. However, note that for 0 < < 1 the sublevel sets of V are not convex. Hence, the optimization problems that use the convexity of the invariant set, i.e., the computation of an invariant set for a convex hull of points, cannot be generalized.

12.5 Invariant Sets for Piecewise Ane Systems The computations of invariant sets for linear systems presented in Section 12.3 and 12.4 together with the results in Chapter 11 can be used for computations of invariant sets of piecewise ane systems. We rst note that all results on invariant sets for linear systems also hold for ane systems with nonsingular system matrices since such systems can always be transformed to a linear system by a trivial change of coordinates. In the sequel we will assume that the encountered ane systems have nonsingular system matrices.

The Problem Suppose that we have a set of initial states XS 0 RN f1; : : : ; Ng of a piecewise ane system, such that for all ( x; s ) 2 XS 0we have that s is xed to some value, say i 2 f1; : : : ; Ng. Furthermore, let X0 = x j ( x; s ) 2 XS 0 . In other words we have an initial set of continuous states X0 in mode i. We want to

compute an invariant set of the piecewise ane system that contains this set of initial states. Using the notion for mode invariant sets introduced in Section 12.2.2, we have the condition

XS 0 ( x; s ) j x 2 Ms ; s 2 f1; : : : ; Ng :

12.5 Invariant Sets for Piecewise Ane Systems

229

12.5.1 A Computational Procedure

We will now describe an iterative procedure for computing an invariant set of a piecewise ane system. Note that this procedure is not guaranteed to terminate. We utilize an abstraction of the piecewise ane system for computing invariant sets. Instead of considering speci c trajectories in the continuous state space of each mode, we focus on invariant sets w.r.t. the ane dynamics in the mode. The evolution of the abstracted system can be described by a nondeterministic nite state machine, which means that several transitions are possible from each mode simultaneously. Transitions from a mode of the abstracted system occur when the currently computed invariant set of the mode intersects one or more transition sets. The idea of using abstractions for veri cation of hybrid systems is not new, see e.g. [3, 130]. Using abstractions is also a standard procedure in computer science for handling the large number of states in discrete systems. We rst give a brief description of the computational procedure and then formalize the steps as an iterative application of an update function on a set of mode objects. Given a set of initial continuous states in a mode, we compute an invariant set that contains this initial set. This is in fact an outer approximation of the set of reachable states for these initial states. We can then compute the intersections of the invariant set and the transition sets of the mode, which give an outer approximation of the continuous states that will be mapped into other modes by the transition mechanism. These mapped sets become initial sets for new invariant set computations and the procedure is repeated. The computations terminate if there are no new nonempty intersections with transition sets or if the intersections are mapped into already computed invariant sets. The computations performed in each step of the iterative procedure can be described as a function, the update function, operating on a nite set of mode objects f Modei gN i=1 which correspond to the modes or discrete states of the piecewise ane system. To formulate the procedure we need some additional notation. Let Mki Rn denote a mode invariant set component of mode i. A mode invariant set component is an invariant set w.r.t. the ane dynamics in the mode if we neglect the transition mechanism. The following relation holds between the mode invariant set and mode invariant set component

1 [

N \ k Mi = Mi Tijc ; k=1 j=1

(12.42)

where Tijc denotes the closure of the complement of the transition set Tij. The in nite union of mode invariant set components is in fact nite if the iterative procedure terminates. A mode initial set, Ii Rn , is a set of continuous states in mode i which is used as input to a computation of a mode invariant set component. Examples of mode initial sets are the initial set of continuous states X0 but also sets of continuous

230


states that enters a mode due to the transition mechanism. Suppose that Mki \ Tij is nonempty. According to the transition mechanism the states in this set are mapped into the continuous state space of mode j. This set will be a mode initial set in mode j in the next iteration. We now have the terminology to introduce the mode object.

The Mode Object The mode object is a data structure for storing information

about a mode of a piecewise ane system such as the continuous dynamics, the transition sets, the mode invariant set components, and the mode initial sets. The structure of the mode object is 8 f Ai ; bi g > > > > > < f Ti1; : : : ; TiN g Modei = > > f M1i ; : : : ; Mli g > > > : f I1 ; : : : ; Im g i i

Ane dynamics List of transition sets List of mode invariant set components Current list of mode initial sets

Initially only one mode object has a nonempty mode initial set, X0 . The iterative procedure terminates when all lists of mode initial sets in the mode objects are empty.

The Update Function The update function, which we denote F , operates on the current set of mode objects and produces an updated version of it. We have

F : f Modei gNi=1 ! f Modei gNi=1 : Termination of the iterative procedure corresponds to a x-point of the update function. Let f Modei gN i=1 denote the current set of mode objects and f Modei gN i=1 the updated set of mode objects. When the update function is applied to the current set of mode objects, the new set of mode objects are computed as follows. First, every mode object Modei is given the same value as Modei but with an empty list of mode initial sets. For each mode object Modei with a nonempty list of mode initial sets we proceed as follows. 1. For every mode initial set, decide if it belongs to some of the mode invariant set components, i.e., check if Ipi Mki ; p = 1; : : : ; m; k = 1; : : : ; l. Let P be the set of values of p for which this is not the case. If P = ; then skip rest of the items in this list. 2. Compute a mode invariant set, Mki 0 such that Ipi Mki 0 ; p 2 P. 0

0

3. Append Mki 0 to the list of mode invariant set components of Modei . 0

4. Decide if Mki 0 \ Tij; j = 1; : : : ; N is empty (trivial if Tij = ;). Let J denote the set of values of j which correspond to nonempty intersections. 5. Append Mki 0 \ @Tij to the list of mode initial sets of Modej for j 2 J. 0

12.5 Invariant Sets for Piecewise Ane Systems

231

The last item in the list corresponds to the mapping of the continuous states from mode i to mode j due to the transition mechanism. Observe that the mode invariant set computed in step two should be as small as possible to reduce the number of iterations. There are two reasons why the number of mode invariant set components in the list of each mode object may not grow during an iteration. First, there may not be any mode initial sets. This means that no transitions to this mode were detected in the previous iteration and we do not have to compute any mode invariant set component. Second, the mode initial sets might already belong to the elements of the list of mode invariant set components. This corresponds to the fact that all the continuous states, that are mapped from other modes by the transition mechanism in the previous iteration, enter subsets of the continuous state space of this mode whose future behavior is already calculated. This is an indication of that some kind of cyclic behavior between the discrete states can occur. However, since the computation of invariant sets is only an outer approximation of the set of reachable states, there need not be a cyclic behavior in the piecewise ane system. We illustrate the iterative procedure in Figure 12.7 by lining up all the discrete states of the piecewise ane system in Figure 12.1 and show the transitions that occur in each iteration. The modes at position k correspond to the mode objects after iteration k and the arrows show which transitions that have occured. We assume that the initial set of continuous states X0 belongs to mode 1 and that the process terminates after 6 iterations. PSfrag replacements

1

1

1

1

1

1

1

2

2

2

2

2

2

2

3

3

3

3

3

3

3

4

4

4

4

4

4

4

0

1

2

3

4

5

6

k

Figure 12.7 The iterative procedure. Here X0 belongs to mode 1 and the

arrows correspond to possible transitions. The dashed arrow indicates that the continuous states of mode 2 are mapped into an existing mode invariant set component of mode 3 and the procedure terminates.

232


If the process had terminated after 6 iterations because the computed mode invariant set components in iteration 5 did not intersect any transition set, then we would have no dashed arrow. The invariant set illustrated in Figure 12.2 corresponds to termination of the procedure after four iterations. In iteration 3 no mode initial set for mode 2 is generated since the computed mode invariant set component does not intersect the transition set to mode 2, i.e., M13 \ T32 is empty. There are many procedures that are similar to the one outlined above for computing invariant sets of piecewise ane systems. We can for example compute a mode invariant set component for each mode initial set in a mode. This would mean that the list of mode invariant set components of a mode object would grow with more than one element in each iteration. Note that we have not prescribed what kind of invariant set that should be computed in each iteration. According to Section 12.3 and 12.4 we have a number of dierent types to choose among. Another alternative is to organize the computations in a more multi threaded manner. This means that we only follow one possible path or thread in the behavior of the nite state machine. If it terminates we follow another path, utilizing the already computed invariant set components. The process terminates if there are no more threads to follow. In Section 12.6 we will present a number of examples to illustrate the outlined procedure.

12.5.2 The Choice of Invariant Set

In the iterative process of computing invariant sets for piecewise ane systems, we have to choose the type of invariant set each time we compute a mode invariant set component. Recalling the conditions for existence of invariant quadratic cones and paraboloids, see Lemma 12.6 and Lemma 12.7, we observe that the existence of an invariant quadratic cone implies existence of an invariant paraboloid, for stable systems. For unstable systems the opposite implication holds. The intersection between a mode invariant set component and the transition sets are outer approximations of the sets of continuous states that will be mapped into new modes. Hence, it is important to keep the size of these intersections as small as possible. For stable systems which have both invariant paraboloids and quadratic cones, one should choose invariant quadratic cones. This will give a smaller intersection between the invariant set and any hyperplane that separates the initial set and the stationary point (i.e., the vertex of the cone or paraboloid). If the system is unstable, we should choose a paraboloid instead of a quadratic cone. Polyhedral invariant sets can be used in combination with the quadratic sets to get smaller invariant sets. This is particularly useful for paraboloids and quadratic cones since these sets are unbounded. Bounded invariant sets described by ane and quadratic inequalities can be used to derive upper and lower bounds for any linear function of the states using SDP, see Section 10.2. If an invariant set is unbounded, an optimization problem over this set may be unbounded.

12.6 Examples

233

12.6 Examples In this section we will give a number of examples where we compute invariant sets for piecewise ane systems. We choose continuous dynamics in R3 since trajectories and sets are then easy to visualize. However, this is only for clarity of presentation. Convex optimization computations, which the computational procedures are based on, are very ecient in practice [21] and are scalable to much larger problems.

12.6.1 A Two-Mode System

Consider a two-mode system described by the hybrid automaton in Figure 12.8. ThePSfrag systemreplacements matrix A, constant vector b and the stationary points of the ane

T12 : x3 2 x_ = Ax + b 1

2 x_ = Ax - b T21 : x3 -2

Figure 12.8 A hybrid automaton for a two-mode system. systems are

2-2 -5 0 3 223 A = 4 5 -2 0 5 ; b = 4-55 ; 0

0 -1

3

213 2-13 x01 = 405 ; x02 = 4 0 5 : 3

-3

Matrix A ful lls the conditions for invariant quadratic cones to exist. Hence, there exist invariant quadratic cones for each ane system. We now use the iterative procedure described in the previous section to compute an invariant set of the whole system. Suppose that the initial state is the origin. The iterative process in this example is very simple since we switch between two states in each iteration. We also note that since the stationary points are located in the transition sets, i.e., outside the region in which the continuous states evolve, the system exhibits a cyclic behavior. In each iteration we compute the smallest invariant quadratic cone that includes the intersection of the invariant quadratic cone from the previous iteration and one of the switching planes. of the transition The switching planes are the boundaries sets, hence HP 1 = x 2 R3 j x3 = 2 and HP 2 = x 2 R3 j x3 = -2 . The procedure terminates after a few iterations with two quadratic cones QC 1 and QC 2 satisfying the following criteria:

QC 1 \ HP 1 QC 2 QC 2 \ HP 2 QC 1 QC 1 and QC 2 are invariant sets for system 1 and 2, respectively.

234


Hence, the set

c (x; s) j x 2 QC 1 \ T12

is invariant. We have

^

s=1

_

x 2 QC 2 \ T21c

^

s=2

(12.43)

QC 1 = x 2 R3 j 4(x1 - 1)2 + 5:9517x22 (x3 - 3)2 QC 2 = x 2 R3 j 4(x1 + 1)2 + 5:9517x22 (x3 + 3)2

and the projection of the invariant set into R3 is shown in Figure 12.9. 2 1 0 -1 -2 2

0

-2

-2 0 2

Figure 12.9 The invariant set. In Figure 12.10 we show some trajectories for a number of initial states in the invariant set together with a wire frame of the invariant set. There also exist invariant slabs for the continuous dynamics in each mode. Taking these into account give an even smaller invariant set than the set (12.43).

12.6.2 A Veri cation Problem

Consider the hybrid automaton shown in Figure 12.1. Suppose that mode 4 corresponds to a forbidden system state due to some safety or performance reasons. The aim of this example is to show that given a speci c set of initial states, the trajectories will never enter the forbidden mode. Let the transition sets be given by

T12 = x 2 R3 j x1 + x2 + x3 9:6 ; T23 = x 2 R3 j x1 6:5 ; T32 = x 2 R3 j x3 10 ; T34 = x 2 R3 j x1 11

12.6 Examples

235 -2

-1

0

1

2

2

0

-2

-2 0 2

Figure 12.10 Some trajectories in the invariant set. and let the system matrices and constant vectors be

2-2: 5 -4:93 A1 = 4 -5 -2 5:1 5 ;

27:63 b1 = 47:65 ;

2-1:86 0:79 0:18 3 A2 = 4-0:79 -1: -3:925 ;

2 9:66 3 b2 = 4 22:78 5 ;

2-0:7 -7 4:7 3 A3 = 4 7 -0:7 -9:35 ;

212:53 b3 = 4-265 ;

20:1 -7 4:73 A4 = 4 7 0:1 -95 ;

2 10 3 b4 = 4-205 :

0

0 -1:9

0:18 3:92 -1:04 0

0

0

0 -3

-3

7:6

-11:32

9

10

Let the set of initial states be a regular icosahedron. Now, examining the eigenvalues of the system matrices, we choose to compute an invariant cone QC , paraboloid P , and ellipsoid E for system 1, 2, and 3, respectively. We have the following mode initial sets: the icosahedron X0 , the intersection between the quadratic cone QC and the plane @T12 and the intersection between the paraboloid P and the plane @T23. The computation of an invariant set in each case corresponds to the optimization problems (12.32), (12.40), and (12.28), respectively. We get

236

Chapter 12 Invariant Sets of Piecewise Ane Systems 3 32 0:18 -8:66 x1 - 4 8:84 -8:665 4x2 - 45 0 ; -8:66 14:32 x3 - 4 3 3T 2 3 2 32 x1 - 7 -0:98 x1 - 7 0 0:38 1:99 -0:015 4x2 - 3:55 4 0 5 4x2 - 3:55 ; x3 - 3:5 0:2 -0:01 1:87 x3 - 3:5 3 32 0:01 -0:02 x1 - 8 0:14 -0:095 4x2 - 35 1 : x3 - 3 -0:09 0:5

3 2 2 x1 - 4 T 8:84 QC = x 2 R 4x2 - 45 4-0:18 -8:66 x3 - 4 3 2 2 x1 - 7 T 0:08 3 4 P = x 2 R x2 - 3:55 4 0 0:38 x3 - 3:5 3 2 2 x1 - 3 T 0:14 E = x 2 R3 4x2 - 35 4 0:01 -0:02 x3 - 3

3

-

j

j

j

These invariant sets, together with the initial icosahedron and the switching planes are shown in Figure 12.11. x2

6 4

2 0

6

x3

4 2 0

0 5 x1 10

Figure 12.11 The invariant sets QC , P , E , the initial set X0 , and the boundaries of the transition sets.

Using Lemma 11.5 it is easy to verify that E\T32 = ; and E\T34 = ;. Hence, we have shown that mode 4 cannot be reached and that there is no cyclic behavior for any initial state in the given icosahedron. In this case this is also easy to verify by visual inspection. In Figure 12.12 we show the projection into R3 of the invariant set of the piecewise ane system as a wire frame. Furthermore, the intersections of the mode invariant sets and the switching planes (i.e., the mode initial sets) and the trajectories starting at the vertices of the icosahedron are also shown. A time-response for the system is shown in Figure 12.13. In this particular simulation we notice that x1 (t) < 11, which we know is true for all trajectories starting in the icosahedron.

12.6 Examples

237 x2

6 4

2 0

6

x3

4 2 0

0 5 x1 10

Figure 12.12 The mode invariant sets, the initial icosahedron in mode 1, switching planes, and a number of trajectories.

x(t)

x1

10 8

PSfrag replacements 6 4

x3 x2

2

2

4

6

8

t

Figure 12.13 The time-response for the initial state x(0) = (-1:5; 0; 0:3)T .

12.6.3 A Cycling System In this example we will compute an invariant set of a piecewise ane system which illustrates how cycles between several discrete states can occur. The hybrid automaton is shown in Figure 12.14. The three ane systems only dier by a change of coordinates using an orthonormal transformation matrix. The transition sets and ane systems are given

238


PSfrag replacements

T12 2 x_ = A2 x + b2

x_ = A1 x + b1 1 T31

3 x_ = A3 x + b3

T23

Figure 12.14 A hybrid automaton of a cycling system. by

T12 = x 2 R3 j x3 2:5 ; T23 = x 2 R3 j x1 2:5 ; T34 = x 2 R3 j x1 0:5 ;

2-1:96 -4:9 0:09 3 A1 = 4 4:9 -2: -1:015 ;

2-0:283 b1 = 4 3:02 5 ;

2-1:5 -3:54 -0:53 A2 = 4 3:54 -2: 3:54 5 ;

2 4:5 3 b2 = 4-10:615 ;

2-1:02 0:6 0:4 3 A3 = 4 -0:4 -1:99 -4:965 ;

203 b3 = 405 :

0:29 0:97 -1:04

-0:5 -3:54 -1:5 -0:6

3:12 1:5

4:94 -1:99 0 The eigenvalues of all three systems are -2 5i and -1 which means that there

exists invariant quadratic cones for each system with vertices at the stationary points

203 233 203 x01 = 405 ; x02 = 405 ; x03 = 405 : 3

0

0

Suppose that the initial set is a sphere in mode 1 centered at (-1:5; 0; 0)T with radius 0:3. Following the procedure described in Section 12.5.1 we rst compute an invariant set component for mode 1, which is a quadratic cone QC 1 . This cone intersects the transition set T12 to mode 2 and we compute a mode invariant set component QC 2 for mode 2 which contains QC 1 \ T12. Now QC 2 intersects the

12.7 Invariant Sets for Systems with Disturbances

239

transition set T23 to mode 3 and we compute a mode invariant set component QC 3 . This set intersects T31 but the intersection is not contained in QC 1 . Hence, we cannot conclude that QC i ; i = 1; 2; 3 form an invariant set. However, after a few

more iterations the computational procedure terminates and we have an invariant set of the piecewise ane system. Using the subset tests presented in Section 11.4 it is easy to show that QC i ; i = 5; 6; 7 form an invariant set and that the initial set is a subset of this set. In Figure 12.15 we show the projection of the invariant set into R3 , the intersections with the transition set boundaries and a number of trajectories starting in the initial sphere. -2 2

x2 1 0

x1 0 2

-1 -2

2 x3

0

-2

Figure 12.15 The projection of an invariant set into R3 and some trajectories.

12.7 Invariant Sets for Systems with Disturbances A natural question to ask is if an invariant set is still invariant for a system in uenced by disturbances? Can we give bounds on how large disturbances we can have and still guarantee invariance of some sets? It is possible to extend some of the results on invariant sets in Section 12.2 to systems with norm bounded disturbances. It is also possible to take uncertainties into account using polytopic dierential inclusions, i.e., when the system matrix is a convex combination of matrices, or norm-bounded linear dierential inclusions. The reader is referred to [21] for more results on these topics.

240


For a given invariant set computed for the disturbance free case it is also possible to check how large disturbances we can allow without losing the invariance. Using the S-procedure it is possible to get sucient bounds on the disturbances such that a given set remains invariant.

12.8 Summary In this chapter we have considered how to compute invariant sets for piecewise ane systems. Such sets can for example be used for veri cation purposes or to show that all solutions for initial states in some set are bounded. Depending on the eigenstructure of the system matrix of a system there are invariant sets of dierent type. We have investigated two classes of invariant sets: polyhedral and quadratic sets. Invariant polyhedral sets can be computed by eigenvector computations and linear programming techniques. Note that polyhedral invariant sets only take the part of the behavior corresponding to real eigenvalues into account. Quadratic invariant sets such as ellipsoids, quadratic cones, and paraboloids on the other hand utilize the whole eigenstructure. Invariance of quadratic sets can be formulated in terms of LMIs and they can be computed by convex optimization methods. To extend the invariant sets computed in each discrete mode of a piecewise ane system to an invariant set of the whole system, we utilize the results in Chapter 11 on inclusions of sets. An iterative procedure was described that produce an invariant set of a piecewise ane system.

13 Conclusions and Extensions In this part of the thesis we have considered sets de ned by ane and quadratic inequalities. For linear or ane dynamic systems there are invariant sets of this form. With this observation as a starting point we have proposed a method to compute invariant sets for piecewise ane systems. This kind of system is rather common in practice since any linear system controlled by linear controllers and switching devices can be modeled as a piecewise ane system. Switching devices and logic elements are often used in control system design to handle plant start-ups, shut downs, and failures. The control system design for complex plants is often divided into two steps. First, continuous controllers are designed for proper operation of the plant in a number of dierent modes. Second, logic and switching devices are used to transfer the plant between the modes. Satisfactory performance of the system in each mode is no guarantee for good performance or even stability of the overall system. Hence, there is a need for tools for veri cation of dierent properties such as stability, state constraints, and faulty behavior. Invariant sets for piecewise ane systems can be used to carry out such veri cations. A computational engine for sets de ned by ane and quadratic inequalities is needed to perform the operations involved in veri cation. We utilize linear matrix inequalities and convex optimization together with dierent parameterizations of these sets and algebraic results formulated in terms of these parameterizations.

Linear Matrix Inequalities and Determinant Maximization Linear matrix inequalities provide a very convenient way of describing a large class of convex sets, which are easy to use in computations. In control they have found 241

242


many applications. We use LMIs as constraints on parameterizations of invariant sets. Semide nite programming and determinant maximization are convex optimization problems which can be solved eciently using interior point methods. The constraint set in these problems are LMIs and this provides a way of computing invariant sets which are optimal in some sense, for example as small as possible but still containing an initial set of states.

Sets De ned by Ane and Quadratic Inequalities Sets de ned by ane and quadratic inequalities are also called polyhedral and quadratic sets. There are two dierent ways of representing this kind of set; the constrained parameter representation and the constraint representation. It is important to be able to convert between these representations since dierent operations can be carried out more easily in one representation than in the other. Geometrically, polyhedral sets can be de ned as the intersection between a nite number of halfspaces. Hence hyperplanes, halfspaces, polyhedra, and polytopes are examples of polyhedral sets. Examples of quadratic sets are ellipsoids, quadratic cones, and paraboloids. The operations on these sets that we need for veri cation purposes are intersection, test of emptiness, and inclusion. These operations can be carried out using the dierent representations of the sets. Some operations only require evaluation of an algebraic constraint in terms of the parameters of the representations, while others require feasibility computations of an LMI. To illustrate the results, sets in R2 or R3 are used. The reason for this is that we want to be able to visualize the result and has nothing to do with computational complexity. In fact, the numerical methods which the operations are based on are known to scale to very large problems (100-1000 variables). We use polyhedral and quadratic sets for computations of invariant sets of dynamic systems but this kind of sets are natural approximators in many other situations too.

Invariant Sets for Piecewise Ane Systems It is well known that sublevel sets of Lyapunov functions are invariant sets. Hence, ellipsoidal and polyhedral invariant sets for linear and ane systems are easily constructed using well known results from stability theory. Invariant quadratic cones and paraboloids do not correspond to Lyapunov functions in a direct way and they do not appear frequently in the literature. For piecewise ane systems there exist methods to construct piecewise quadratic Lyapunov functions [88] which give another class of invariant sets. Boundedness of solutions might be more important to verify than asymptotic stability, for systems that contain switching devices. Invariant sets of the form considered in this thesis can then be useful.

243 Knowledge of invariant sets for piecewise ane systems can be useful in applications. Linear functionals of the states can be optimized over the invariant sets which give bounds on the behavior of the system. Hence, we can guarantee that for any state in a given initial set these bounds will never be exceeded during operation of a plant even if the system exhibits very complicated or even chaotic behavior. We can also show that certain modes of operation never will be obtained since the transition conditions to these modes never will be satis ed, i.e., the invariant set does not intersect any transition sets to these modes. Note that for veri cation purposes, the set of reachable states is what we really want to consider. However, invariant sets are outer approximations of the true reachable set and give a trade-o between computational complexity and accuracy. Moreover, it is easier to extract information such as intersection with other sets, boundedness, and representation, from the polyhedral and quadratic sets than from the true reachable sets. There are several advantages of the proposed method for computing invariant sets. It scales to much larger problems than the examples we have given in this thesis. Furthermore, even if the number of switching devices in the system to be veri ed are large, we only have to handle the modes that are reachable from the initial mode. In fact, we actually get an outer approximation of the reachable modes, since the mode invariant sets are outer approximations of the continuous system behavior in the modes. The disadvantage of using invariant sets for veri cation is that if a veri cation fails we cannot say if it is due to the outer approximation involved or an unsatisfactory behavior of the true system.

Extensions There are many more operations on polyhedral and quadratic sets that can be implemented than the ones treated here. A number of approximating operations such as inner and outer approximations of intersections or unions of polyhedral and quadratic sets should be possible to formulate as convex optimization problems. The computed invariant polyhedral or convex quadratic sets can be used as constraint sets for veri cation purposes. Convex sets of this type can be described as LMIs in the state variables which let us use them as constraint sets in convex optimization problems. It would be interesting to investigate the implications of this observation. To further evaluate the presented approach to piecewise ane system veri cation, a large scale application is needed. In the chemical industry and process industry there are many nonlinear processes that could be approximated by piecewise ane dynamics and hence would be suitable for this purpose. Plant start-ups and shut downs, as well as dierent modes of operation during a production cycle are common and our framework is well suited to deal with these behaviors. The search for large scale applications and evaluation of the proposed veri cation methods are topics of future investigations. An obvious generalization of the invariant set computations is to take norm

244


bounded disturbances into account. Can we nd an optimal invariant set even in the presence of disturbances? How large disturbances can be allowed without destroying the invariance of a computed set? Suppose that a system consists of three modes: start-up, normal operation, and failure. Then it is interesting to decide for which initial states we can guarantee that a failure will not occur. Starting in the normal operation mode we compute the largest invariant set not intersecting the transition set to the failure mode. This invariant set should have a nonempty intersection with the image of the transition set from the start-up mode, for the normal operation mode to be reachable from the start-up mode. All initial states corresponding to trajectories entering this set will not cause a failure. This kind of backwards reachability questions can be handled by slight modi cations of the tools presented in this part of the thesis. Another interesting question is how to utilize invariant sets for control system design of hybrid systems? Invariant sets such as paraboloids, cylinders, and quadratic cones with a distinguished direction (the axis) along which the dynamics evolves, could be used to plan the behavior in the state space. Sets of this form could be \glued" together to give an invariant set for the overall system which satis es given state and input constraints. Furthermore, many properties of these invariant sets can be abstracted such as maximal state transfer time between two hyperplanes in a cone/paraboloid, state constraints, sensitivity to disturbances etc. The abstracted properties make it possible to use hierarchical design procedures and construct controllers for very complex mode switching systems, where the invariant sets are utilized as building blocks at the lowest level and their abstracted properties are used at higher levels. This is a very interesting direction for future research.

Bibliography [1] W. W. Adams and P. Loustaunau. An Introduction to Grobner Bases, volume 3 of Graduate Studies in Mathematics. American Mathematical Society, 1994. [2] F. Alizadeh et al. SDPpack user's guide. Technical report, New York University Computer Science Department, June 1997. Version 0.9 BETA for Matlab 5.0. [3] R. Alur, C. Courcoubetis, N. Halbwachs, T. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138:3{34, 1995. [4] B. Anderson, N. Bose, and E. Jury. Output feedback stabilization and related problems - solution via decision methods. IEEE Transactions on Automatic Control, AC(20):53{65, 1975. [5] D. S. Arnon. A bibliography of quanti er elimination for real closed elds. Journal of Symbolic Computation, 5(1-2):267{274, Feb-Apr 1988. [6] D. S. Arnon, G. E. Collins, and S. McCallum. Cylindrical algebraic decomposition I: The basic algorithm. SIAM Journal of Computation, 13(4):865{877, November 1984. strom and T. Hagglund. PID Controllers: Theory, Design, and Tun[7] K. J. A ing. The International Society for Measurement and Control, Research Triangle Park, NC, second edition, 1995. 245

246

Bibliography

[8] K. J. Astrom and B. Wittenmark. Computer-Controlled Systems : Theory and Design. Information and System Sciences Series. Prentice Hall, third edition, 1997. [9] D. P. Atherton. Nonlinear Control Engineering. Van Nostrand Reinhold Company Ltd., 1982. [10] A. I. Barvinok. Feasibility testing for systems of real quadratic equations. Technical Report 1991/92:3, Institut Mittag-Leer, Djursholm, Sweden, 1992. [11] G. Bastin, F. Jarachi, and I. M. Y. Mareels. Dead beat control of recursive nonlinear systems. In Proceedings of the 32rd IEEE Conference on Decision and Control, pages 2965{2971, San Antonio, TX, 1993. [12] S. Basu. An improved algorithm for quanti er elimination over real closed elds. In Proceedings of 38th Annual Symposium on Foundations of Computer Science, pages 56 { 65, October 1997. [13] S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quanti er elimination. Journal of the ACM, 43(6):1002{1045, 1996. [14] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty. Nonlinear Programming Theory and Algorithms. Wiley, second edition, 1993. [15] T. Becker and V. Weispfenning. Grobner Bases. A Computational Approach to Commutative Algebra, volume 141 of GTM. Springer, 1993. [16] R. Benedetti and J. Risler. Real Algebraic and Semi-Algebraic Sets. Hermann, Paris, 1990. [17] D. P. Bertsekas. Nonlinear Programming. Athena Scienti c, 1995. [18] V. D. Blondel and J. N. Tsitsiklis. NP-hardness of some linear control design problems. In ECC'95, European Control Conference, volume 3, pages 2066{ 2071, Roma, Italy, 1995. [19] J. Bochnak, M. Coste, and M.-F. Roy. Geometrie Algebrique Reelle. Springer, 1987. [20] B. Borchers. CSDP, a C library for semide nite programming. Technical report, Department of Mathematics New Mexico Tech, Socorro, NM, March 1997. [21] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics. SIAM, Philadelphia, Pennsylvania, 1994.

Bibliography

247

[22] S. Boyd and L. Vandenberghe. Introduction to convex optimization with engineering applications. Course Notes, EE364, 1997. Information Systems Laboratory, Stanford University. [23] M. Branicky. Stability of switched hybrid systems. In Proceedings of the 33rd IEEE Conference on Decision and Control, Lake Buena Vista, FL, December 1994. [24] L. Breiman. Hinging hyperplanes for regression, classi cation and function approximation. IEEE Transactions on Information Theory, 39(3), May 1993. [25] R. W. Brockett. Asymptotic stability and feedback stabilization. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Dierential Geometric Control Theory, volume 27 of Progress in Mathematics, pages 181{191. Birkhauser, Boston, 1983. [26] A. E. Bryson and Y.-C. Ho. Applied Optimal Control. Ginn and Company, 1969. [27] B. Buchberger, H. Hong, R. Loos, G. Collins, J. Johnson, A. Mandache, M. Encarnacion, W. Krandick, A. Neubacher, and H. Vielhaber. Saclib 1.1 user's guide. Technical Report 93-19, RISC, Johannes Kepler University, A-4040 Linz, Austria, 1993. [28] C. I. Byrnes and X. Ho. The zero dynamics algorithm for general nonlinear system and its application in exact output tracking. Journal of Mathematical Systems, Estimation, and Control, 3:51{72, 1993. [29] J. Canny. A toolkit for non-linear algebra. Technical report, Computer Science Division, University of California, Berkeley, CA, 94720, August 1993. [30] B. F. Caviness and J. R. Johnson, editors. Quanti er Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer-Verlag, Berlin, 1998. [31] R. V. Churchill and J. W. Brown. Complex Variables and Applications. Mathematics Series. McGraw-Hill, fourth edition, 1984. [32] H. Cohen, editor. A Course in Computational Algebraic Number Theory. Springer, 1993. [33] G. E. Collins. Quanti er elimination for real closed elds by cylindrical algebraic decomposition. In Second GI Conf. Automata Theory and Formal Languages, Kaiserslauten, volume 33 of Lecture Notes Comp. Sci., pages 134{183. Springer, 1975. [34] G. E. Collins. Quanti er elimination by cylindrical algebraic decomposition { twenty years of progress. In B. F. Caviness and J. R. Johnson, editors, Quanti er Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation. Springer-Verlag, Berlin, 1998.

248

Bibliography

[35] G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quanti er elimination. Journal of Symbolic Computation, 12:299{328, September 1991. [36] A. Colmerauer. Naive solving of non-linear constraints. In A. Colmerauer and Frederic Benhamou, editors, Constraint Logic Programming - Selected Research, chapter 6, pages 89{112. The MIT Press, 1993. [37] M. Coste, L. Mahe, and M.-F. Roy, editors. Real Algebraic Geometry, volume 1524 of Lecture Notes in Mathematics. Springer, 1992. Proceedings of the Conference held in Rennes, France, June 24-28, 1991. [38] M. Coste and M.-F. Roy. Thom's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets. Journal of Symbolic Computation, 5:121{129, 1988. [39] D. Cox, J. Little, and D. O'Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer-Verlag, second edition, 1996. [40] J. H. Davenport, Y. Siret, and E. Tournier. Computer Algebra. Systems and Algorithms for Algebraic Computation. Academic Press, 1988. [41] A. Deshpande and P. Varaiya. Viable control of hybrid systems. In P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, editors, Hybrid Systems II, volume 999 of Lecture Notes in Computer Science, pages 128{147. Springer-Verlag, Berlin, 1995. [42] S. Diop. Elimination in control theory. Math. Control Signals Systems, 4(1):17{32, 1991. [43] P. Dorato, W. Yang, and C. Abdallah. Robust multi-objective feedback design by quanti er elimination. Journal of Symbolic Computation, 24(2):153{ 159, August 1997. [44] P. G. Eriksson and T. Finnstrom. Exact linearization and analysis of nonlinear systems. Master's thesis LiTH-ISY-EX-1199, Department of Electrical Engineering, Linkoping University, 1992. [45] G. Carra Ferro. Grobner bases and dierential algebra. In L. Huguet and A. Poli, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, volume 356 of Lecture Notes Comp. Sci., pages 129{140. Springer, 1989. Proceedings of AAECC-5, Menorca. [46] A. F. Filippov. Dierential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, Dordrecht, 1988.

Bibliography

249

[47] M. Fliess and S.T. Glad. An algebraic approach to linear and nonlinear control. In H.L. Trentelman and J.C. Willems, editors, Essays on Control: Perspectives in the Theory and its Applications, pages 223{267. Birkhauser, Boston, 1993. [48] K. Forsman. Constructive Commutative Algebra in Nonlinear Control Theory. PhD thesis 261, Department of Electrical Engineering, Linkoping University, 1991. [49] K. Forsman and M. Jirstrand. Some niteness issues in dierential algebraic systems theory. In Proceedings of 33rd IEEE Conference on Decision and Control (CDC'94), pages 1112{1117, Lake Buena Vista, Florida, December 1994. [50] R. A. Freeman and P. V. Kokotovic. Robust Nonlinear Control Design { State Space and Lyapunov Techniques. Birkhauser, 1996. [51] F. R. Gantmacher. Matrix Theory, volume II. Chelsea, 1960. [52] F. R. Gantmacher. Matrix Theory, volume I. Chelsea, 1977. [53] R. Germundsson. Symbolic Systems { Theory, Computation and Applications. PhD thesis, Dept. of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden, 1995. [54] R. Germundsson and K. Forsman. A constructive approach to algebraic observability. In Proceedings of the 30th Conference on Decision and Control, Brighton, UK, 1991. IEEE. [55] S. T. Glad. Nonlinear state space and input output descriptions using dierential polynomials. In J. Descusse, M. Fliess, A. Isidori, and D. Leborgne, editors, New Trends in Nonlinear Control Theory, volume 122 of Lecture Notes in Control and Information Sciences, pages 182{189. Springer, 1988. Proceedings of the International Conference Nonlinear Systems, Nantes, France, June 13-17, 1988. [56] S. T. Glad. Nonlinear regulators and Ritt's remainder algorithm. In Colloque international sur l'analyse des systemes dynamiques controlles, volume 2, July 1990. Lyon, France. [57] S. T. Glad. Implementing Ritt's algorithm of dierential algebra. In M. Fliess, editor, Proceedings of NOLCOS'92, pages 610{614, Bordeaux, France, 1992. IFAC. [58] S. T. Glad. Algebraic Methods in Systems Theory - Dierential Algebra. Technical report, Dept. of Electrical Engineering, Linkoping University, 1996. [59] S. T. Glad and M. Jirstrand. Using dierential and real algebra in control. In N. Munro, editor, The Use of Symbolic Methods in Control System Analysis and Design, IEE Control Engineering Series. IEE, 1998.

250

Bibliography

[60] S. T. Glad and L. Ljung. Model structure identi ability and persistence of excitation. In Proceedings of the 29th Conference on Decision and Control, pages 1069{1073, Honululu, Hawaii, December 1990. IEEE. [61] Johan Gunnarsson. Symbolic Methods and Tools for Discrete Event Dynamic Systems. PhD thesis 477, Department of Electrical Engineering, Linkoping University, May 1997. [62] P.-O. Gutman and M. Cwikel. An algorithm to nd maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states. IEEE Transactions on Automatic Control, 32(3):251{254, March 1987. [63] W. Hahn. Stability of Motion. Springer, 1967. [64] N. Halbwachs, P. Raymond, and Y.-E. Proy. Veri cation of linear hybrid systems by means of convex approximations. In Proceedings of the First International Static Analysis Symposium, pages 223{237, Namur, Belgium, 1994. Springer-Verlag. [65] J. Hauser. Nonlinear control via uniform system approximation. Systems & Control Letters, 17:145{154, 1991. [66] K. M. Heal, M. Hansen, and K. Rickard. Maple V Learning Guide. Springer Verlag, 1996. [67] J. Heintz, M.-F. Roy, and P. Solerno. On the theoretical and practical complexity of the existential theory of reals. The Computer Journal, 36(5):427{ 431, 1993. [68] C. Helmberg, F. Rendl, R. J. Vanderbei, and H. Wolkowicz. An interiorpoint method for semide nite programming. SIAM Journal on Optimization, 6(2):342{61, May 1996. [69] A. Helmersson. Methods for Robust Gain Scheduling. PhD thesis 406, Department of Electrical Engineering, Linkoping University, Sweden, 1995. [70] T. A. Henzinger and P.-H. Ho. A note on abstract interpretation strategies for hybrid automata. In P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, editors, Hybrid Systems II, volume 999 of Lecture Notes in Computer Science, pages 253{264. Springer-Verlag, Berlin, 1995. [71] T. A. Henzinger and P.-H. Ho. HyTech: The cornell hybrid technology tool. In P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, editors, Hybrid Systems II, volume 999 of Lecture Notes in Computer Science, pages 265{293. Springer-Verlag, Berlin, 1995. [72] J. Hollman and L. Langemyr. Algorithms for non-linear algebraic constraints. In F. Benhamou and A. Colmerauer, editors, Constraint Logic Programming - Selected Research, pages 113{131. The MIT Press, 1993.

Bibliography

251

[73] H. Hong. Improvements in CAD-based Quanti er Elimination. PhD thesis, The Ohio State University, 1992. [74] H. Hong. Simple solution formula construction in cylindrical algebraic decomposition based quanti er elimination. In P. S. Wang, editor, ISAAC'92, International Symposium on Symbolic and Algebraic Computation, pages 177{188, Berkely, California, 1992b. ACM Press. [75] H. Hong and R. Liska. Special issue on applications of quanti er elimination. Journal of Symbolic Computation, 24(2), August 1997. [76] I. Horowitz. Quantitative feedback theory. In Proceedings of the Institution of Electrical Engineers, volume 119, pages 215{226, 1982. [77] A. Hurwitz. U ber die bedingungen unter welchen eine gleichung nur wurzeln mit negativen reellen teilen besitzt. Matematische Annalen, 46:273{284, 1895. [78] A. Isidori. Nonlinear Control Systems. Communications and Control Engineering Series. Springer-Verlag, third edition, 1995. [79] N. Jacobson. Basic Algebra, volume 1. Freeman, New York, second edition, 1985. [80] N. Jacobson. Basic Algebra, volume 2. Freeman, New York, second edition, 1985. [81] M. Jirstrand. Applications of quanti er elimination to equilibrium calculations in nonlinear aircraft dynamics. In Proceedings on The 5th International Student Olympiad on Automatic Control (BOAC'96), St Petersburg, Russia, October 1996. [82] M. Jirstrand. Nonlinear control system design by quanti er elimination. In IMACS Conference on Applications of Computer Algebra, Linz, Austria, July 1996. [83] M. Jirstrand. Quanti er elimination and application in control. In Proceedings of Reglermote 1996, Lulea, Sweden, June 1996. [84] M. Jirstrand. Curve following for nonlinear dynamic systems subject to control constraints. In Proceedings of the American Control Conference (ACC'97), Albuquerque, New Mexico, USA, June 1997. [85] M. Jirstrand. Nonlinear control system design by quanti er elimination. Journal of Symbolic Computation, 24(2):137{152, August 1997. [86] M. Jirstrand and S. T. Glad. Computational questions of equilibrium calculation with application to nonlinear aircraft dynamics. In Mathematical Theory of Networks and Systems (MTNS'96), St Louis, Missouri, 1996.

252

Bibliography

[87] M. Jirstrand and S. T. Glad. Moving the state between equilibria. In Proceedings of the 4th IFAC Nonlinear Control Systems Design Symposium (NOLCOS'98), Enschede, Netherlands, July 1998. [88] M. Johansson and A. Rantzer. Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Transactions on Automatic Control, 43(4):555{559, April 1998. [89] Mikael Johansson and Anders Rantzer. On the computation of piecewise quadratic Lyapunov functions. In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, California, December 1997. [90] T. Kailath. Linear Systems. Information and System Sciences Series. Prentice-Hall, Englewood Clis, NJ, 1980. [91] I. Kaplansky. An Introduction to Dierential Algebra. Hermann, 1957. [92] H. K. Khalil. Nonlinear Systems. Prentice-Hall, Upper Saddle River, NJ, second edition, 1996. [93] M. Knebusch and C. Schneiderer. Einfuhrung in die reelle Algebra. Vieweg, Braunschweig, 1989. [94] D. E. Knuth. The Art of Computer Programming. Vol. 2: Seminumerical Algorithms. Addison-Wesley, second edition, 1981. [95] H. Kobayashi and E. Shimemura. Set-point changing for nonlinear systems. International Journal of Control, 50(6):2397{2406, 1989. [96] E. R. Kolchin. Dierential Algebra and Algebraic Groups., volume 54 of Pure and Applied Mathematics. Academic Press, 1973. [97] M. Krstic, I. Kanellakopoulos, and P. Kokotovic. Nonlinear and Adaptive Control Design. Adaptive and Learning Systems for Signal Processing, Communications, and Control. John Wiley & Sons, 1995. [98] T. Y. Lam. An introduction to real algebra. Rocky Mountain J. of Math., 14(4):767{814, Fall 84 1984. [99] S. Lang. Algebra. Addison-Wesley, second edition, 1984. [100] D. Lazard. Systems of algebraic equations (algorithms and complexity). Notes from a tutorial at the ISAAC'91, Bonn, July 1991. [101] G. Leitmann. The Calculus of Variations and Optimal Control - An Introduction. Plenum Press, New York, 1981. [102] H. Levi. On the structure of dierential polynomials and on their theory of ideals. Trans. Amer. Math. Soc., 51:532{568, 1942.

Bibliography

253

[103] P. Lindskog. Methods, Algorithms and Tools for System Identi cation Based on Prior Knowledge. PhD thesis 436, Department of Electrical Engineering, Linkoping University, Sweden, May 1996. [104] R. Loos. Generalized polynomial remainder sequences. In B. Buchberger, G.E. Collins, and R. Loos, editors, Computer Algebra. Symbolic and Algebraic Computation, pages 115{137. Springer, second edition, 1983. [105] J. M. Maciejowski. Multivariable Feedback Design. Electronic Systems Engineering Series. Addison-Wesley, 1989. [106] E. Mans eld. Dierential Grobner Bases. PhD thesis, University of Sidney, 1991. [107] E. L. Mans eld. Digrob2: A symbolic algebra package for analysing systems of PDE using Maple. Technical report, Department of Mathematics, University of Exeter, Exeter, U.K., 1993. [108] H. Matsumura. Commutative Ring Theory., volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1986. [109] S. McCallum. Solving polynomial strict inequalities using cylindrical algebraic decomposition. The Computer Journal, 36(5):432{438, 1993. [110] D. G. Mead. Dierential ideals. Proceedings of Amer. Math. Soc., 6:430{432, June 1955. [111] J. Michalik and J. C. Willems. The elimination problem in dierential algebraic systems. In Mathematical Theory of Networks and Systems (MTNS'93), Regensburg, Germany, 1993. [112] B. Mishra. Algorithmic Algebra. Texts and Monographs in Computer Science. Springer-Verlag, 1993. [113] M. Monagan, K. Geddes, K. Heal, G. Labahn, and S. Vorkoetter. Maple V Programming Guide. Springer Verlag, 1996. [114] T. S. Motzkin, H. Raia, G. L. Thompson, and R. M. Thrall. The double description method. In Contributions to the Theory of Games, volume II of Annals of Mathematics Study. Princeton University Press, Princeton, New Jersey, 1953. [115] D. Nesic. Dead-Beat Control for Polynomial Systems. PhD thesis, The Australian National University, 1996. [116] D. Nesic and M. Jirstrand. Stabilization of switched polynomial systems. In IMACS Conference on Applications of Computer Algebra (ACA'98), Prague, Czech Republic, August 1998.

254

Bibliography

[117] D. Nesic and I. M. Y. Mareels. Dead beat controllability of polynomial systems: Symbolic computation approaches. IEEE Transactions on Automatic Control, 43(2):162{175, 1998. [118] D. Nesic, E. Ska das, I. M. Y. Mareels, and R. J. Evans. Minimum phase properties for input non-ane nonlinear systems. To appear in IEEE Transactions on Automatic Control. [119] D. Nesic, E. Ska das, I. M. Y. Mareels, and R. J. Evans. Analysis of minimum phase properties for non-ane nonlinear systems. In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, 1997. [120] Y. Nesterov and A. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics. SIAM, Philadelphia, Pennsylvania, 1994. [121] H. Nijmeijer and A. J. van der Schaft. Nonlinear Dynamical Control Systems. Springer-Verlag, New York, 1990. [122] K. B. O'Keefe. A property of the dierential ideal [yp ]. Trans. Amer. Math. Soc., 94:483{497, March 1960. [123] F. Ollivier. Standard bases of dierential ideals. In S. Sakata, editor, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, volume 508 of Lecture Notes Comp. Sci., pages 304{321. Springer, 1990. Proceedings of AAECC-8, Tokyo. [124] F. Ollivier. Generalized standard bases with applications to control. In Proceedings of the First European Control Conference, volume 1, pages 170{ 176, Grenoble, France, July 1991. Hermes. [125] M. Padberg. Linear Optimization and Extensions, volume 12 of Algorithms and Combinatorics. Springer Verlag, Berlin, Heidelberg, 1995. [126] P. C. Parks and V. Hahn. Stability Theory. Prentice Hall international series in systems and control engineering. Prentice Hall, 1993. [127] S. Pettersson and B. Lennartson. Stability and robustness for hybrid systems. In Proceedings of the 35th IEEE Conference on Decision and Control, pages 1202{1207, Kobe, Japan, December 1996. [128] J. Plantin. Algebraic Methods for Veri cation and Control of Discrete Event Dynamic Systems. Licentiate thesis, Dept. of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden, 1995. [129] J. F. Pommaret. Dierential Galois Theory. Gordon and Breach, 1983. [130] A. Puri and P. Varaiya. Veri cation of hybrid systems using abstractions. In P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, editors, Hybrid Systems II, volume 999 of Lecture Notes in Computer Science, pages 358{369. SpringerVerlag, Berlin, 1995.

Bibliography

255

[131] J. Renegar. On the computational complexity and geometry of the rst-order theory of the reals. Parts I{III. Journal of Symbolic Computation, 13:255{ 352, 1992. [132] J. F. Ritt. Dierential Algebra. Dover, 1950. [133] R. T. Rockafellar. Convex Analysis. Princeton University Press, 1970. [134] E. J. Routh. Stability of a given state of motion. In A.T. Fuller, editor, Stability of Motion - A collection of early scienti c papers by Routh, Cliord, Sturm and B^ocher. Taylor and Francis, London, 1877. [135] L. A. Rubel and M. F. Singer. A dierentially algebraic elimination theorem with applications to analog computability in the calculus of variations. In Proceedings of Amer.Math.Soc., volume 94, pages 653{658. Springer, 1985. [136] W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, third edition, 1976. [137] R. Y. Sharp. Steps in Commutative Algebra, volume 19 of London Mathematical Society Student Texts. Cambridge University Press, 1990. [138] D. Shevitz and B. Paden. Lyapunov stability theory of non-smooth systems. IEEE Transactions on Automatic Control, 39:1910{1914, 1994. [139] E. Ska das, R. J. Evans, A. V. Savkin, and I. R. Petersen. Stability results for switched controller systems. Submitted to Automatica, 1998. [140] E. D. Sontag. Mathematical Control Theory. Springer-Verlag, 1990. [141] A. L. Stevens and B. Wigdorowitz. A systematic methodology for the analysis and design of controllers for nonlinear processes. The Transactions of the South African Institute of Electrical Engineers, 3:125{136, 1995. [142] B. L. Stevens and F. L. Lewis. Aircraft Control and Simulation. Wiley, 1992. [143] J. Stiver, P. J. Ansaklis, and M. D. Lemmon. Interface and controller design for hybrid systems. In P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, editors, Hybrid Systems II, volume 999 of Lecture Notes in Computer Science, pages 462{492. Springer-Verlag, Berlin, 1995. [144] J. Stiver et al. Digital control from a hybrid perspective. In Proceedings of the 33rd Conference on Decision and Control, pages 4241{4246, Lake Buena Vista, FL, December 1994. [145] J. A. Stiver, P. J. Antsaklis, and M. D. Lemmon. An invariant based approach to the design of hybrid control systems containing clocks. In R. Alur, T. Henzinger, and E. D. Sontag, editors, Hybrid Systems III { Veri cation and Control, volume 1066 of Lecture Notes in Computer Science, pages 464{ 474. Springer-Verlag, Berlin, 1996.

256

Bibliography

[146] A. W. Strzebonski. An algorithm for systems of strong polynomial inequalities. The Mathematica Journal, 4(4):74{77, 1994. [147] V. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis. Static output feedback - A survey. Automatica, 33(2):125{137, February 1997. [148] A. Tarski. A Decision Method for Elementary Algebra and Geometry. University of California Press, second edition, 1948. [149] K. C. Toh, M. J. Todd, and R. H. Tutuncu. SDPT3 { a Matlab software package for semide nite programming. Technical Report No. 735, Department of Mathematics, National University of Singapore, February 1998. [150] F. Uhlig. A recurring theorem about pairs of quadratic forms and extensions: A survey. Linear Algebra and Its Applications, 25(219-237), 1979. [151] V. I. Utkin. Variable structure systems with sliding modes: A survey. IEEE Transactions on Automatic Control, 22(2):212{222, 1977. [152] D. van Dalen. Logic and Structure. Springer-Verlag, second edition, 1980. [153] L. Vandenberghe and S. Boyd. Semide nite programming. SIAM Review, 38(1):49{95, March 1996. [154] L. Vandenberghe, S. Boyd, and S.-P. Wu. Determinant maximization with linear matrix inequality constraints. SIAM Journal on Matrix Analysis and Applications, April 1998. To appear. [155] F. Verhulst. Nonlinear Dierential Equations and Dynamical Systems. Springer-Verlag, 1985. [156] M. Vidyasagar. Nonlinear Systems Analysis. Prentice Hall, second edition, 1993. [157] J. C. Willems. Paradigms and puzzles in the theory of dynamical systems. IEEE Transactions on Automatic Control, AC-36(3):259{294, March 1991. [158] J. C. Willems. A new setting for dierential algebraic systems. In A. Isidori, S. Bittanti, E. Mosca, A. De Luca, M. D. Di Benedetto, and G. Oriolo, editors, ECC'95, European Control Conference, volume 1, pages 218{223, Roma, Italy, 1995. [159] S. Wolfram. The Mathematica Book. Wolfram Media, Cambridge University Press, Champaign, IL, USA, third edition, 1996. [160] S.-P. Wu and S. Boyd. Design and implementation of a parser/solver for SDPs with matrix structure. In Proceedings of the IEEE Conference on Computer Aided Control System Design, 1996.

Bibliography

257

[161] S-.P. Wu and S. Boyd. sdpsol: A Parser/Solver for Semide nite Programming and Determinant Maximization Problems with Matrix Structure { User's Guide. Information Systems Laboratory, Stanford University, May 1996. [162] S.-P. Wu and S. Boyd. sdpsol: Parser/solver for semide nite programs with matrix structure. In L. El Ghaoui and S.-I. Niculescu, editors, Recent Advances in LMI Methods for Control. SIAM, 1998. To appear. [163] W. Yang, P. Dorato, and C. Abdallah. Robust multiobjective feedback design via combined quanti er elimination and discretization. In Proceedings of the American Control Conference (ACC'97), volume 3, pages 1843 { 1847, Albuquerque, New Mexico, USA, June 1997. [164] G. M. Ziegler. Lectures on Polytopes. Graduate Texts in Mathematics. Springer-Verlag, New York, 1994.

258

Bibliography

Notation Symbols N , Z, Q , A R, C

k [ x1 ; : : : ; x n ]

Ip I h f1 ; : : : ; fs i V(f1 ; : : : ; fs) k fx1 ; : : : ; xn g [ f1 ; : : : ; fs ] c

A ^ , _ , :, 9, 8 S , S (x) , Bd (x) ;

!

The sets of natural, integer, rational, and algebraic numbers. The sets of real and complex numbers. Polynomial ring in the variables x1 ; : : : ; xn , coecients from a eld k, usually Q , R, or C . Ideal or identity matrix of appropriate dimension. Radical ideal. Ideal generated by the polynomials f1 ; : : : ; fs . Variety (zero set) of the polynomials f1 ; : : : ; fs . Dierential polynomial ring in the variables x1 ; : : : ; xn , coecients from a eld k, usually Q , R or C . Dierential ideal generated by the polynomials f1 ; : : : ; fs . State space dierential ideal. Input-output dierential ideal. Autoreduced set of dierential polynomials. Boolean conjunction, disjunction, negation, and implication. Existential and universal quanti er. Semialgebraic set and its de ning Boolean formula. The left side is de ned by the right side. An open ball with radius d, centered at x. The empty set.

259

260

HP H A Q PH PT E QC P @M

Notation Hyperplane. Halfspace. Ane set. Quadratic set. Polyhedron. Polytope. Ellipsoid. Quadratic cone. Paraboloid. Boundary of set M.

Operators and Functions AB A(B A\B A[B AnB > gcd(f; g) lcm(f; g) mdeg(f) tdeg(f) lc(f) lm(f) lt(f) rem(f; F) S(f; g) GB(I) @ x_ ; x; x(i) yi ; u i Lf wt(p) type(p) ld(f) lv(f) If Sf Hf HA [ A ] : H1 A

A is a subset of B. A is a proper subset of B. Intersection of sets A and B. Union of sets A and B. The set fx j x 2 A and x 62 Bg.

Monomial ordering or dierential ranking of variables. Greatest common divisor of two polynomials f and g. Least common multiple of two polynomials f and g. Multi-degree of a multivariate polynomial f. Total degree of a multivariate polynomial f. Leading coecient of a polynomial f. Leading monomial of a polynomial f. Leading term of a polynomial f. Remainder of a polynomial f on division by F = ff1 ; : : : ; fs g. S-polynomial of polynomials f and g. Reduced Grobner basis of the ideal I. Derivation operator. First, second, and ith derivative of x w.r.t. t. The ith derivatives w.r.t. t of input and output variables. Extended Lie-derivative operator computed with respect to f. Weight of an isobaric dierential polynomial p. Type of an isobaric dierential polynomial p. Leader of a dierential polynomial f. Leading variable of a dierential polynomial f. Initial of a dierential polynomial f. Separant of a dierential polynomial f. The product IQf Sf . The product Ai A IAi SAi , where A is an autoreduced set. Set of dierential polynomials whose remainder w.r.t. A is zero. 2

Notation prem(f; g) subresi (f; g) psci (f; g) proj(f1 ; : : : ; fs ) var(a1 ; : : : ; as )

f/g dist(x; A) P 0, P 0 P 0, P 0 MT M-1 M det(M) log det(M) M N (M) R(M) M min(M) max (M) argminx f(x) vol(E ) Cofv1; : : : ; vp g sign(a) Vx(x) j xj y

?

N

261 Pseudo-remainder of a polynomial f on division by g. The ith -subresultant of two polynomials f and g. The ith -principal subresultant coecient of f and g. Projection operator w.r.t. the set ff1 ; : : : ; fr g of polynomials. Number of sign variation in a sequence of real numbers. The dierential polynomial f is reduced w.r.t. g. The Euclidean distance function, dist(x; A) , inf y A jx - yj. Positive (semi-)de nite matrix. Negative (semi-)de nite matrix. Transpose of matrix M. Inverse of matrix M. Generalize (or Moore-Penrose) inverse of matrix M. Determinant of matrix M. Logarithm of the determinant of matrix M. Full rank matrix such that M M = 0. Null space of matrix M. Range space of matrix M. A matrix whose columns form a basis of N (M). Smallest singular value of matrix M. Largest singular value of matrix M. Minimizing argument of f(x). Volume of an ellipsoid E . Convex hull of the vectors fv1 ; : : : ; vp g. Sign of a 2 R n f0g. Row vector of partial derivatives of V (x) w.r.t. x1 ; : : : ; xn . Euclidean norm of a vector x. 2

?

Acronyms CAD QE PRS EPRS PPRS SPRS MIMO SISO UFD LMI LP SDP MAXDET

Cylindrical Algebraic Decomposition. Quanti er Elimination. Polynomial Remainder Sequence. Euclidean Polynomial Remainder Sequence. Primitive Polynomial Remainder Sequence. Subresultant Polynomial Remainder Sequence. Multiple Input Multiple Output. Single Input Single Output. Unique Factorization Domain. Linear Matrix Inequality. Linear Programming. Semide nite Programming. Determinant Maximization.

262

Notation

Index A

controller basic . . . . . . . . . . . . . . . . . . . . . . . 156 minimum phase. . . . . . . . . . . . .166 output zeroing . . . . . . . . . . . . . . 165 stabilizing . . . . . . . . . . . . . . . . . . 157 convex combination . . . . . . . . . . . . . . . . 183 function . . . . . . . . . . . . . . . . . . . . 183 hull. . . . . . . . . . . . . . . . . . . . . . . . .190 optimization . . . . . . . . . . . . . . . . 183 set. . . . . . . . . . . . . . . . . . . . . . . . . .183 cost function . . . . . . . . . . . . . . . . . . . . 184 critical point . . . . . . . . . . . . . . . . . . . . 120 curve following . . . . . . . . . . . . . . . . . . 142 cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 73

admissible controls. . . . . . . . . . . . . . . . . . . . .121 curve . . . . . . . . . . . . . . . . . . . . . . . 149 states. . . . . . . . . . . . . . . . . . . . . . .121 ane set . . . . . . . . . . . . . . . . . . . . . . . . 189 algebraic decomposition . . . . . . . . . . 75 algebraic numbers . . . . . . . . . . . . . . . . 89 algebraic dependency . . . . . . . . . . . . .57 ascending chain condition . . . . . . . . 16 attractive . . . . . . . . . . . . . . . . . . . . . . . 157 autoreduced set . . . . . . . . . . . . . . . . . . 36

B

base phase . . . . . . . . . . . . . . . . . . . . . . . 91 basic controllers . . . . . . . . . . . . . . . . . 156 Boolean formula. . . . . . . . . . . . . . . . . .97 Buchberger's algorithm . . . . . . . . . . . 25

D

decay rate. . . . . . . . . . . . . . . . . . . . . . .213 decision problem . . . . . . . . . . . . . . . . . 98 decision variables.. . . . . . . . . . . . . . .182 decomposition . . . . . . . . . . . . . . . . . . . .73 delineable set. . . . . . . . . . . . . . . . . . . . .84 derivation . . . . . . . . . . . . . . . . . . . . . . . . 27 describing function method. . . . . .117 determinant maximization. . . . . . .185 dieomorphism. . . . . . . . . . . . . . . . . .210

C

CAD. . . . . . . . . . . . . . . . . . . . . .69, 75, 89 cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 characteristic set . . . . . . . . . . . . . . . . . 39 circle criterion. . . . . . . . . . . . . . . . . . .113 constrained reachability .. . . . . . . . 149 constraint set . . . . . . . . . . . . . . . . . . . 184 263

264

Index

dierential eld . . . . . . . . . . . . . . . . . . 28 dierential eld extension . . . . . . . . 29 dierential ideal . . . . . . . . . . . . . . . . . . 29 prime . . . . . . . . . . . . . . . . . . . . . . . .30 radical. . . . . . . . . . . . . . . . . . . . . . .30 dierential polynomial. . . . . . . . . . . .31 dierential ring. . . . . . . . . . . . . . . . . . .28

E

ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . .191 semi-axes . . . . . . . . . . . . . . . . . . . 191 volume . . . . . . . . . . . . . . . . . . . . . 192 equilibrium point. . . . . . . . . . . . . . . .120 extended Lie-derivative operator. .45 extension phase . . . . . . . . . . . . . . . . . . 92

F

feasibility ane inequalities . . . . . . . . . . . 196 quadratic inequality . . . . . . . . 196 feasible set . . . . . . . . . . . . . . . . . . . . . . 184 feedback design . . . . . . . . . . . . . . . . . 108 eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Filippov type solution . . . . . . . . . . . 158 nite state machine. . . .208, 229, 232 nitely generated . . . . . . . . . . . . . . . . .13 formally real eld . . . . . . . . . . . . . . . . 71 functional matrix. . . . . . . . . . . . . . . .129

G

generalized inverse . . . . . . . . . . . . . . 188 generalized stability . . . . . . . . . . . . . 106 generating set . . . . . . . . . . . . . . . . . . . . 13 Grobner basis . . . . . . . . . . . . . . . . . . . . 22 graph ideal . . . . . . . . . . . . . . . . . . . . . . . 57 greatest common divisor. . . . . . . . . .76

H

halfspace . . . . . . . . . . . . . . . . . . . . . . . . 188 Hilbert's basis theorem . . . . . . . . . . . 17 Hurwitz determinant . . . . . . . . . . . . . . . . 102 matrix. . . . . . . . . . . . . . . . . . . . . .102 polynomial . . . . . . . . . . . . . . . . . 102 hybrid automaton . . . . . . . . . . . . . . . 209 hyperplane . . . . . . . . . . . . . . . . . . . . . . 188

I

ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 prime . . . . . . . . . . . . . . . . . . . . . . . .15 radical. . . . . . . . . . . . . . . . . . . . . . .15 ideal basis. . . . . . . . . . . . . . . . . . . . . . . .13 inclusion . . . . . . . . . . . . . . . . . . . . . . . . 198 indeterminate . . . . . . . . . . . . . . . . . . . . 11 initial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 input-output ideal . . . . . . . . . . . . . . . . 42 input-output polynomials. . . . . . . . .42 input-output relations . . . . . . . . . . . . 42 interior point methods . . . . . . . . . . 182 invariant . . . . . . . . . . . . . . . . . . . . . . . . . 74 ellipsoid . . . . . . . . . . . . . . . . . . . . 219 half space. . . . . . . . . . . . . . . . . . .215 hyperplane . . . . . . . . . . . . . . . . . 215 paraboloid .. . . . . . . . . . . . . . . . .225 polyhedra . . . . . . . . . . . . . . . . . . 215 quadratic cone. . . . . . . . . . . . . .221 quadratic set . . . . . . . . . . . . . . . 219 slab . . . . . . . . . . . . . . . . . . . . . . . . 215 invariant set . . . . . . 157, 207, 209, 210 irreducible . . . . . . . . . . . . . . . . . . . . . . . 15 isobaric . . . . . . . . . . . . . . . . . . . . . . . . . . 46

J

Jacobian . . . . . . . . . . . . . . . . . . . . . . . . 129

L

leader . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 leading coecient . . . . . . . . . . . . . . . . 19 leading monomial . . . . . . . . . . . . . . . . 19 leading term . . . . . . . . . . . . . . . . . . . . . 19 leading variable . . . . . . . . . . . . . . . . . . 32 least common multiple . . . . . . . . . . . 24 lexicographic ordering . . . . . . . . . . . . 19 Lie-derivative operator . . . . . . . . . . . 45 Lienard-Chipart criterion. . . . . . . .102 linear matrix inequality . . . . . . . . . 182 linear programming . . . . . . . . . . . . . 184 LMI. . . . . . . . . . . . . . . . . . . . . . . . . . . . .182 LP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Lyapunov exponent . . . . . . . . . . . . . 213

M

M-circle . . . . . . . . . . . . . . . . . . . . . . . . . 110

Index

265

matrix negative de nite . . . . . . . . . . . . 181 negative semide nite. . . . . . . .181 positive de nite. . . . . . . . . . . . .181 positive semide nite . . . . . . . . 181 MAXDET . . . . . . . . . . . . . . . . . . . . . . 185 minimum phase . . . . . . . . . . . . . . . . . 165 minimum phase controller . . . . . . . 166 Minkowski sum . . . . . . . . . . . . . . . . . 189 mode initial set . . . . . . . . . . . . . . . . . . . 229 invariant set component . . . . 229 object . . . . . . . . . . . . . . . . . 229, 230 mode invariant set . . . . . . . . . . . . . . 213 modes. . . . . . . . . . . . . . . . . . . . . . . . . . .208 monomial . . . . . . . . . . . . . . . . . . . . . . . . 13 monomial ordering . . . . . . . . . . . . . . . 18 Moore-Penrose inverse. . . . . . . . . . .188 multidegree . . . . . . . . . . . . . . . . . . .18, 19

N

negative de nite. . . . . . . . . . . . . . . . .157 Noetherian . . . . . . . . . . . . . . . . . . . . . . . 16 nonsmooth systems . . . . . . . . . . . . . 158

O

objective function . . . . . . . . . . . . . . . 184 ordered eld. . . . . . . . . . . . . . . . . . . . . .70 ordering of terms . . . . . . . . . . . . . . . . . 18 outer approximation .. . 229, 231, 232 output range . . . . . . . . . . . . . . . . . . . . 140 controllable .. . . . . . . . . . . . . . . .141 output zeroing controller . . . . . . . . 165

P

paraboloid . . . . . . . . . . . . . . . . . . . . . . 193 axis . . . . . . . . . . . . . . . . . . . . . . . . 194 base point . . . . . . . . . . . . . . . . . . 194 semi-axes . . . . . . . . . . . . . . . . . . . 194 size . . . . . . . . . . . . . . . . . . . . . . . . .194 piecewise ane system . . . . . 208, 213 polyhedron. . . . . . . . . . . . . . . . . . . . . .189 polytope . . . . . . . . . . . . . . . . . . . . . . . . 189 Popov criterion. . . . . . . . . . . . . . . . . .113 positive combination . . . . . . . . . . . . 183

positive de nite . . . . . . . . . . . . . . . . . 157 primitive part . . . . . . . . . . . . . . . . . . . . 79 principal subresultant coecient. .83 projection operator . . . . . . . . . . . . . . .86 projection phase. . . . . . . . . . . . . . . . . .90 PRS . . . . . . . . . . . . . . . . . . . . . . . . . . 76{79 Euclidean. . . . . . . . . . . . . . . . . . . .79 primitive . . . . . . . . . . . . . . . . . . . . 79 subresultant . . . . . . . . . . . . . . . . . 79 pseudo-division .. . . . . . . . . . . . . . . . . .78 pseudo-remainder . . . . . . . . . . . . . . . . 78

Q

QE . . . . . . . . . . . . . . . . . . . . . . . . . . . 69, 97 quadratic cone . . . . . . . . . . . . . . . . . . 193 axis . . . . . . . . . . . . . . . . . . . . . . . . 193 semi-axes . . . . . . . . . . . . . . . . . . . 193 size . . . . . . . . . . . . . . . . . . . . . . . . .193 vertex . . . . . . . . . . . . . . . . . . . . . . 193 quadratic set . . . . . . . . . . . . . . . . . . . . 190 quanti er elimination . . . . . . . . . 69, 97 quanti er elimination problem . . . . 98

R

radially unbounded. . . . . . . . . . . . . .157 ranking . . . . . . . . . . . . . . . . . . . . . . . . . . 31 reachability constrained . . . . . . . . . . . . . . . . . 149 real polynomial system . . . . . . . . . . . 69 real closed eld. . . . . . . . . . . . . . . . . . .71 reduced Grobner basis. . . . . . . . . . . .22 reductum. . . . . . . . . . . . . . . . . . . . . . . . .84 region .. . . . . . . . . . . . . . . . . . . . . . . . . . . 73 region of attraction. . . . . . . . . . . . . .157 relative degree . . . . . . . . . . . . . . . . . . 141 representation .. . . . . . . . . . . . . . . . . .187 constrained parameter . . . . . . 188 constraint . . . . . . . . . . . . . . . . . .188 free parameter . . . . . . . . . . . . . . 188 ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 ring of polynomials . . . . . . . . . . . . . . . 11

S

S-polynomial . . . . . . . . . . . . . . . . . . . . . 24 sdpsol . . . . . . . . . . . . . . . . . . . . . . . . . 185

266

Index

S-procedure . . . . . . . . . . . . . . . . . . . . . 198 SDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 section . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 semialgebraic set . . . . . . . . . . . . . . . . . 72 semide nite programming . . . . . . . 184 separant. . . . . . . . . . . . . . . . . . . . . . . . . .36 similar . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 stability degree. . . . . . . . . . . . . . . . . .212 stabilizable system . . . . . . . . . . . . . . 157 stable . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 asymptotically . . . . . . . . . . . . . .157 stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 state space dierential ideal . . . . . . 42 state transition. . . . . . . . . . . . . . . . . .208 stationarizable states . . . . . . . . . . . . 122 output zeroing . . . . . . . . . . . . . .165 stationary point . . . . . . . . . . . . . . . . . 120 Sturm chain . . . . . . . . . . . . . . . . . . . . . . 87 subresultant . . . . . . . . . . . . . . . . . . . . . . 81 subresultant chain . . . . . . . . . . . . . . . . 81 switched systems . . . . . . . . . . . . . . . . 155 switching function . . . . . . . . . . . . . . . . . . . . 156 planes . . . . . . . . . . . . . . . . . . . . . . 208 rule. . . . . . . . . . . . . . . . . . . . . . . . .157 set. . . . . . . . . . . . . . . . . . . . . . . . . .213

T

total degree . . . . . . . . . . . . . . . . . . . . . . 18 transition mechanism. . . . . . . . . . . .208 transition sets . . . . . . . . . . . . . . . . . . .208 truncation. . . . . . . . . . . . . . . . . . . . . . . .56 type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50

U

UFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 unique factorization domain . . . . . . 78 update function . . . . . . . . . . . . 229, 230

V

variety . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 veri cation . . . . . . . 207, 229, 241, 243 vertex. . . . . . . . . . . . . . . . . . . . . . . . . . .192 viable set . . . . . . . . . . . . . . . . . . . . . . . 165

W

weight. . . . . . . . . . . . . . . . . . . . . . . . . . . .46

Z

zero dynamics . . . . . . . . . . . . . . . . . . .165

PhD Dissertations, Division of Automatic Control, Linkoping University M. Millnert: Identi cation and control of systems subject to abrupt changes. Thesis no. 82, 1982. ISBN 91-7372-542-0.

A.J.M. van Overbeek: On-line structure selection for the identi cation of multivariable systems. Thesis no. 86, 1982. ISBN 91-7372-586-2. B. Bengtsson: On some control problems for queues. Thesis no. 87, 1982. ISBN 91-7372-593-5. S. Ljung: Fast algorithms for integral equations and least squares identi cation problems. Thesis no.

93, 1983. ISBN 91-7372-641-9. H. Jonson: A Newton method for solving non-linear optimal control problems with general constraints. Thesis no. 104, 1983. ISBN 91-7372-718-0. E. Trulsson: Adaptive control based on explicit criterion minimization. Thesis no. 106, 1983. ISBN 91-7372-728-8. K. Nordstrom: Uncertainty, robustness and sensitivity reduction in the design of single input control systems. Thesis no. 162, 1987. ISBN 91-7870-170-8. B. Wahlberg: On the identi cation and approximation of linear systems. Thesis no. 163, 1987. ISBN 91-7870-175-9. S. Gunnarsson: Frequency domain aspects of modeling and control in adaptive systems. Thesis no. 194, 1988. ISBN 91-7870-380-8. A. Isaksson: On system identi cation in one and two dimensions with signal processing applications. Thesis no. 196, 1988. ISBN 91-7870-383-2. M. Viberg: Subspace tting concepts in sensor array processing. Thesis no. 217, 1989. ISBN 91-7870529-0. K. Forsman: Constructive commutative algebra in nonlinear control theory. Thesis no. 261, 1991. ISBN 91-7870-827-3. F. Gustafsson: Estimation of discrete parameters in linear systems. Thesis no. 271, 1992. ISBN 91-7870-876-1. P. Nagy: Tools for knowledge-based signal processing with applications to system identi cation. Thesis no. 280, 1992. ISBN 91-7870-962-8. T. Svensson: Mathematical tools and software for analysis and design of nonlinear control systems. Thesis no. 285, 1992. ISBN 91-7870-989-X. S. Andersson: On dimension reduction in sensor array signal processing. Thesis no. 290, 1992. ISBN 91-7871-015-4. H. Hjalmarsson: Aspects on incomplete modeling in system identi cation. Thesis no. 298, 1993. ISBN 91-7871-070-7. I. Klein: Automatic synthesis of sequential control schemes. Thesis no. 305, 1993. ISBN 91-7871-090-1. J.-E. Stromberg: A mode switching modelling philosophy. Thesis no. 353, 1994. ISBN 91-7871-430-3. K. Wang Chen: Transformation and symbolic calculations in ltering and control. Thesis no. 361, 1994. ISBN 91-7871-467-2. T. McKelvey: Identi cation of state-space models from time and frequency data. Thesis no. 380, 1995. ISBN 91-7871-531-8. J. Sjoberg: Non-linear system identi cation with neural networks. Thesis no. 381, 1995. ISBN 91-7871-534-2. R. Germundsson: Symbolic systems { theory, computation and applications. Thesis no. 389, 1995. ISBN 91-7871-578-4. P. Pucar: Modeling and segmentation using multiple models. Thesis no. 405, 1995. ISBN 91-7871627-6. H. Fortell: Algebraic approaches to normal forms and zero dynamics. Thesis no. 407, 1995. ISBN 91-7871-629-2. A. Helmersson: Methods for robust gain scheduling. Thesis no. 406, 1995. ISBN 91-7871-628-4. P. Lindskog: Methods, algorithms and tools for system identi cation based on prior knowledge. Thesis no. 436, 1996. ISBN 91-7871-424-8. J. Gunnarsson: Symbolic methods and tools for discrete event dynamic systems. Thesis no. 477, 1997. ISBN 91-7871-917-8.

268

Notes

Notes

269

270

Notes

Notes

271

272

Notes

Notes

273

274

Notes

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Constructive Methods for Inequality Constraints in

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