Robust Control System Design by Mapping

Robust Control System Design by Mapping Specifications into Parameter Spaces Vom Promotionsausschuss der Fakulta¨t fu ¨r Elektrotechnik und Informationstechnik der Ruhr-Universität Bochum zur Erlangung des akademischen Grades Doktor-Ingenieur genehmigte DISSERTATION

von Michael Ludwig Muhler aus Creglingen

Bochum, 2007

iii

Dissertation eingereicht am:

20. Oktober 2006

Tag der m¨ undlichen Pr¨ ufung:

20. März 2007

1. Berichter:

Prof. Dr.-Ing. Jan Lunze

2. Berichter:

Prof. Dr.-Ing. J¨ urgen Ackermann

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Fu ¨ r meine Eltern, Ute und Maria

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Acknowledgments I am indebted to those who have given me the opportunity, support, and time to write this doctoral thesis. It is a pleasure to thank my advisor Professor J¨ urgen Ackermann for his encouragement and advice during my studies. He has always been ready to give his time generously to discuss ideas and approaches, while giving me the freedom to choose the direction of my work. His insights and enthusiasm will have a long-lasting effect on me. I would also like to thank my supervisor Professor Jan Lunze at the Ruhr-University Bochum for his interest in my work. His support and willingness to work with me across the miles and the years is greatly appreciated. I am greatly indebted to my former office-mate, Paul Blue, for creating a friendly and stimulating work atmosphere, and for many discussions, and also to my other colleagues at DLR Oberpfaffenhofen, especially, Dr. Tilman B¨ unte, Dr. Dirk Odenthal, Dr. Dieter Kaesbauer, Dr. Naim Baj¸cinca and Gertjan Looye. Special thanks to Professor Bob Barmish, who initially encouraged me to pursue postgraduate studies. I would like to express my gratitude to Airbus for financial support during a three year grant. Thanks to Dr. Michael Kordt, my contact at Airbus in Hamburg. Financial aid from the DAAD for the conference presentations of parts of this thesis is gratefully acknowledged. The final write-up of this thesis would not have been possible without the support of my supervisor at Robert Bosch GmbH, Dr. Hans-Martin Streib. Finally, special thanks to my parents and to my wife Ute for their continuous encouragement, patience and support. Korntal-M¨ unchingen, March 2007

Michael Muhler

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Contents Nomenclature

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Abstract

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Zusammenfassung

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1 Introduction 1.1 The Control Problem . . . . . . . . 1.2 Background and Previous Research 1.3 Goal of the Thesis . . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . .

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2 Control Specifications and Uncertainty 2.1 Parametric MIMO Systems . . . . . . . . . . . . . . 2.1.1 MIMO Specifications . . . . . . . . . . . . . . 2.1.2 MIMO Properties . . . . . . . . . . . . . . . . 2.2 Symbolic State-Space Descriptions . . . . . . . . . . 2.2.1 Transfer Function to State-Space Algorithm . 2.2.2 Minimal Realization . . . . . . . . . . . . . . 2.2.3 Example . . . . . . . . . . . . . . . . . . . . . 2.3 Uncertainty Structures . . . . . . . . . . . . . . . . . 2.3.1 Real Parametric Uncertainties . . . . . . . . . 2.3.2 Multi-Model Descriptions . . . . . . . . . . . 2.3.3 Dynamic Uncertainty . . . . . . . . . . . . . . 2.4 MIMO Specifications in Control Theory . . . . . . . 2.4.1 H∞ Norm . . . . . . . . . . . . . . . . . . . . 2.4.2 Passivity and Dissipativity . . . . . . . . . . . 2.4.3 Connections between H∞ Norm and Passivity 2.4.4 Popov and Circle Criterion . . . . . . . . . . . 2.4.5 Complex Structured Stability Radius . . . . . 2.4.6 H2 Norm Performance . . . . . . . . . . . . .

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2.5

2.4.7 Generalized H2 Norm . . . . . 2.4.8 LQR Specifications . . . . . . 2.4.9 Hankel Norm . . . . . . . . . Integral Quadratic Constraints . . . . 2.5.1 IQCs and Other Specifications 2.5.2 Mixed Uncertainties . . . . . 2.5.3 Multiple IQCs . . . . . . . . .

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3 Mapping Equations 3.1 Eigenvalue Mapping Equations . . . . . . . . 3.2 Algebraic Riccati Equations . . . . . . . . . . 3.2.1 Continuous and Analytic Dependence . 3.3 Mapping Specifications into Parameter Space . 3.3.1 ARE Based Mapping . . . . . . . . . . 3.3.2 H∞ Norm Mapping Equations . . . . . 3.3.3 Passivity Mapping Equations . . . . . 3.3.4 Lyapunov Based Mapping . . . . . . . 3.3.5 Maximal Eigenvalue Based Mapping . 3.4 IQC Parameter Space Mapping . . . . . . . . 3.4.1 Uncertain Parameter Systems . . . . . 3.4.2 Kalman-Yakubovich-Popov Lemma . . 3.4.3 IQC Mapping Equations . . . . . . . . 3.4.4 Frequency-Dependent Multipliers . . . 3.4.5 LMI Optimization . . . . . . . . . . . 3.5 Complexity . . . . . . . . . . . . . . . . . . . 3.5.1 ARE Mapping Equations . . . . . . . . 3.5.2 Lyapunov Mapping Equations . . . . . 3.5.3 IQC Mapping Equations . . . . . . . . 3.6 Further Specifications . . . . . . . . . . . . . . 3.7 Comparison and Alternative Derivations . . . 3.8 Direct Performance Evaluation . . . . . . . . . 3.9 Summary . . . . . . . . . . . . . . . . . . . . 4 Algorithms and Visualization 4.1 Aspects of Symbolic Computations . . . . 4.2 Algebraic Curves . . . . . . . . . . . . . . 4.2.1 Asymptotes of Curves . . . . . . . 4.2.2 Parametrization of Curves . . . . . 4.2.3 Topology of Real Algebraic Curves 4.3 Algorithm for Plane Algebraic Curves . . .

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4.4

4.5 4.6

4.7

4.3.1 Extended Topological Graph . . . . . . . . 4.3.2 Bézier Approximation . . . . . . . . . . . Path Following . . . . . . . . . . . . . . . . . . . 4.4.1 Common Problems of Path Following . . . 4.4.2 Homotopy Based Algorithm . . . . . . . . 4.4.3 Predictor-Corrector Continuation . . . . . Surface Intersections . . . . . . . . . . . . . . . . Preprocessing . . . . . . . . . . . . . . . . . . . . 4.6.1 Factorization . . . . . . . . . . . . . . . . 4.6.2 Scaling . . . . . . . . . . . . . . . . . . . . 4.6.3 Symmetry . . . . . . . . . . . . . . . . . . Visualization . . . . . . . . . . . . . . . . . . . . 4.7.1 Color Coding . . . . . . . . . . . . . . . . 4.7.2 Visualization for Multiple Representatives

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5 Examples 5.1 MIMO Design Using SISO Methods . . . . . . . . . . . . 5.2 MIMO Specifications . . . . . . . . . . . . . . . . . . . . 5.2.1 H2 Norm . . . . . . . . . . . . . . . . . . . . . . 5.2.2 H∞ Norm: Robust Stability . . . . . . . . . . . . 5.2.3 Passivity Examples . . . . . . . . . . . . . . . . . 5.3 Example: Track-Guided Bus . . . . . . . . . . . . . . . . 5.3.1 Design Specifications . . . . . . . . . . . . . . . . 5.3.2 Robust Design for Extreme Operating Conditions 5.3.3 Robustness Analysis . . . . . . . . . . . . . . . . 5.4 IQC Examples . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Four Tank MIMO Example . . . . . . . . . . . . . . . .

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6 Summary and Outlook 113 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 A Mathematics 115 A.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 A.2 Algebraic Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 116 References

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x

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Nomenclature Acronyms ARE

algebraic Riccati equation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 22

CRB

complex root boundary, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 44

IQC

integral quadratic constraint, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 5

IRB

infinite root boundary, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

KYP

Kalman-Yakubovich-Popov, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 57

LFR

linear fractional representation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 10

LHP

left half plane, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 43

LMI

linear matrix inequality, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 28

LQG

linear quadratic Gaussian, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 36

LQR

linear quadratic regulator, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 36

LTI

linear time-invariant, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 6

MFD

matrix fraction description, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 10

MIMO

multi-input multi-output, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 6

PSA

parameter space approach, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

RHP

right half plane, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 8

RRB

real root boundary, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 44

SISO

single-input single-output, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 8

p. 44

p. 2

xii Symbols ∗

conjugate transpose A(s)∗ = A(−s)T , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 24

ζ

damping factor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 43

=

∼

equivalent state-space realization, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 7

:=

definition, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 22

den

denominator, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 10

diag

diagonal matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 11

Im

image or range space of a matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Im

imaginary part of imaginary number, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 43

⊗

Kronecker product, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 115

⊕

Kronecker sum, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vec

column stacking operator, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 54

lcm

least common multiple, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 10

Lm 2 [0, ∞)

space of square summable functions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 39

Re

real part of imaginary number, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 24

σ ¯

largest singular value, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

σ

singular value, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 22

Λ

eigenvalue spectrum, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

trace

trace of a matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 34

p. 117

p. 54

p. 22

p. 33

Variables G(s)

general transfer matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 7

q

uncertain parameters, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 3

xiii

Abstract Robust controller design explicitly considers plant uncertainties to determine the controller structure and parameters. Thereby, the given specifications for the control system are fulfilled even under perturbations and disturbances. The parameter space approach is an established methodology for systems with uncertain physical parameters. Control specifications, for example formulated as eigenvalue criteria, are hereby mapped into a parameter space. The graphical presentation of admissible parameter regions leads to easily interpretable results and allows intuitive parametrization and analysis of robust controllers. The goal of this thesis is to extend the parameter space approach by new specifications and to broaden the applicable system class. A uniform concept for mapping specifications into parameter spaces is presented for this purpose. This enables the generalized derivation of mapping equations and the identical software implementation for the mapping. Moreover, it allows to extend the parameter space approach by additional specifications which can be mapped. Furthermore, the applicable system class can be broadened. All relevant specifications for linear multivariable systems including the H2 and H∞ norm are covered by this approach. Beyond that, specifications for nonlinear systems can be used in conjunction with the parameter space approach. In particular, the mapping of integral quadratic constraints is introduced. A brief outline of specifications for multivariable systems introduces into the parameter space mapping. All specifications are established using a similar mathematical description that forms the basis for the generalized mapping equations. The mapping equations are then obtained by converting the generalized algebraic specification description into a specialized eigenvalue problem. A symbolic-numerical algorithm is developed to realize the specification mapping. Various graphical means to visualize the results in a parameter plane are explored. This is motivated by specifications which yield a performance index. Various examples demonstrate the extension of the parameter space approach and the new possibilities of the concept.

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Zusammenfassung Beim Entwurf von robusten Reglern werden Unsicherheiten der Regelstrecke explizit ber¨ ucksichtigt, um die Struktur und Parametrierung des Reglers so festzulegen, daß die gestellten Anforderungen an das regelungstechnische System trotz auftretender Störungen und Streckenveränderungen erf¨ ullt werden. Hierzu steht mit dem Parameterraumverfahren eine anerkannte Methodik f¨ ur Systeme mit unsicheren physikalischen Parametern zur Verf¨ ugung. Hierbei werden regelungstechnische Spezifikationen, die zum Beispiel als Eigenwertkriterien formuliert sind, in einen Parameterraum abgebildet. Die grafische Darstellung von zulässigen Gebieten in einer Parameterebene f¨ uhrt zu einfach interpretierbaren Resultaten und ermöglicht die intuitive Parametrierung und Analyse von robusten Reglern. Ziel der Arbeit ist die Erweiterung des Parameterraumverfahrens um Spezifikationen sowie die Vergrößerung der anwendbaren Systemklasse. Hierzu wird ein einheitliches Konzept zur Abbildung von Spezifikationen in Parameterräume vorgestellt. Dieses erlaubt die verallgemeinerte Herleitung von Abbildungsgleichungen und die identische softwaretechnische Realisierung der Abbildung. Neben allen relevanten Spezifikationen f¨ ur lineare Mehrgrößensysteme, wie die H2 und H∞ Norm, erlaubt das vorgestellte Konzept die Anwendung des Parameterraumverfahrens auf nichtlineare Systeme. Insbesondere wird die Abbildung von integral-quadratischen Bedingungen aufgezeigt. Ein kurzer Abriß der Spezifikationen f¨ ur Mehrgrößensysteme f¨ uhrt in die Abbildung in den Parameterraum ein. Alle Spezifikationen werden in einer gleichartigen mathematischen Formulierung dargestellt, die die Basis f¨ ur die verallgemeinerten Abbildungsgleichungen ¨ bildet. Die Abbildungsgleichungen beruhen auf der Uberf¨ uhrung der allgemeinen algebraischen Darstellung f¨ ur die Spezifikationen in ein spezielles Eigenwertproblem. Um die Anwendung des hier vorgestellten Konzeptes zu ermöglichen, wird ein symbolischnumerischer Algorithmus zur Durchf¨ uhrung der Abbildung von Spezifikationen entwickelt. Verschiedene Möglichkeiten zur grafischen Darstellung der Resultate in einer Parameterebene werden vorgestellt, insbesondere f¨ ur Spezifikationen die G¨ utewerte liefern. Mehrere Beispiele stellen die Erweiterung des Parameterraumverfahrens und die neuen Möglichkeiten des Konzeptes dar.

1

1

Introduction

1.1 The Control Problem Why should we use feedback at all? The pure dynamics of a stable plant can be simply modified to the desired dynamics using feedforward control. In the real world every plant is subject to external disturbances. If we want to alter the systems response to disturbances or signal uncertainty we have to use feedback. Another fundamental reason for feedback control arises from instability. An unstable plant cannot be stabilized by any feedforward system. Feedback control is mandatory for these plants, even without signal and model uncertainty. The third fundamental reason for using feedback control is the just mentioned model uncertainty. The term model uncertainty here includes discrepancy between the true system and the model used to design the controller. Reasons for deviations are model imperfections. For example, the modeling of an electric wire as a resistor is known to be perfect up to a certain frequency range. More elaborate models derived from first principles might include a resistor-capacitor chain. But even this model is only valid in a certain frequency range because eventually the encountered physical phenomena reach atomic scale. Thus model uncertainty is not just present in models obtained from measurements and identification. Every model, even a model derived from physical modeling, is only valid to a certain extent. Further model uncertainties can be design imposed, such as limitations on the complexity of the design model or certain model types, for example linear models. Classical control aims at stabilizing a system in the presence of signal uncertainty. Robust control extends this goal by designing control systems that not only tolerate model uncertainties, but also retain system performance under plant variations. While the goal that a feedback control system should maintain overall system performance despite changes in the plant has been around since the early days of control theory, this property is nowadays explicitly called robustness.

2

Introduction

1.2 Background and Previous Research As a reaction to the poor robustness of controllers based on optimization and estimation theory the field of robust control theory emerged, where plant variations play a key role. Several different approaches to deal with plant variations mainly influenced by the uncertainty characterization have evolved [Ackermann 1980, Doyle 1982, Lunze 1988, Safonov 1982]. Central topics in robust control theory common to all approaches are • Uncertainty characterization. • Robustness analysis. • Robust controller synthesis. For systems with real parametric uncertainty, e.g., an unknown or varying system parameter, the parameter space approach (PSA)1 is a well established method for robustness analysis and robust controller design [Ackermann et al. 2002]. The basic idea of the parameter space approach is to map a condition of specification for a system into a plane of parameters, i.e., the set of all parameters for which the specification holds is determined. Initially the PSA considered eigenvalue specifications for linear systems. Its roots can be traced back to the 19th century, where mathematicians such as [Hermite 1856, Maxwell 1866, Routh 1877], inspired by the first mechanical control systems, studied the basic question related to stability of whether a given polynomial p(s) = an sn + . . . + a1 s + a0 = 0,

(1.1)

has only roots with negative real parts. Interestingly these early accounts of stability analysis tried to find a solution that can be expressed using the coefficients of the polynomial, thereby avoiding the explicit computation of roots. Vishnegradsky was the first to visualize the stability condition in a coefficient parameter plane in [1876], analyzing the stability of a third order polynomial p(s) = s3 +a2 s2 +a1 s+1 with respect to varying a1 and a2 . This idea became the building block of the parameter space approach. Based on Hermite’s work, [Hurwitz 1895] reported an algebraic condition in terms of determinants. This stability condition has been extensively used in control theory and extended to robustness analysis. Initially the parameter space method considered stability of a linear system described by the characteristic equation. By mapping the stability condition it originally allowed to analyze robustness with respect to two specific coefficients. 1

Sometimes also referred to as parameter plane method

1.2 Background and Previous Research

3

The parameter space method was then extended to robust root clustering or Γ-stability, by specifying an eigenvalue region [Ackermann et al. 1991, Mitrović 1958]. This allows to indirectly incorporate time-domain specifications and thereby robust performance. The coefficients ai of (1.1) do not directly relate to plant or controller parameters, and therefore hamper the application to control problems. Therefore robust control theory considered polynomials with coefficients that depend on a parameter vector q ∈ Rp : p(s, q) = an (q)sn + . . . + a1 (q)s + a0 (q) = 0.

(1.2)

If q consists of only one parameter q, then robust stability can be evaluated by plotting a generalized root locus [Evans 1948], where q takes the role of the usual linear gain k. Robust, multimodal eigenvalue based parameter space approach is state-of-the-art. The underlying theory is thoroughly understood for linear time-invariant systems with uncertain parameters. In general, the parameter space approach maps a given specification, e.g. a permissible eigenvalue region, into a space of uncertain parameters q ∈ Rp , see Figure 1.1. Usually the specification is mapped into a parameter plane because this leads to understandable and powerful graphical results. Moreover, since we can map several specifications consecutively, this approach actually allows multiobjective analysis and synthesis of control systems. Recently, this approach was extended to the frequency domain for Bode specifications [Besson and Shenton 1997, Hara et al. 1991, Odenthal and Blue 2000] and Nyquist diagrams [B¨ unte 2000]. Static nonlinearities were considered in the parameter space approach in [Ackermann and B¨ unte 1999]. Finally [Muhler 2002] derived mapping equations for multi-input multi-output systems, including H2 , H∞ norms and passivity specifications. During the 1990s there has been considerable interest in design methods such as H∞ and µ-analysis that require only control specifications and yield a controller including the structure and parameters. While this seems to be attractive from the point that the design engineer has not to waste time on thinking of a reasonable control structure and possibly try several different structures. All of these design methods have the disadvantage that they lead to very high controller orders. The direct order reduction of the resulting controllers is a nontrivial task, and often destroys some of the required or desired features of the initial high-order controllers. Using the parameter space approach as a design tool we have to specify a controller structure, e.g., a PID controller, and the parameters of the controller are iteratively tuned until all design specifications are fulfilled. Thus the parameter space approach falls into the category of fixed control structure design methods. Other approaches are given by classical design methods or parameter optimization [Joos et al. 1999].

4

Introduction jω

q2 p(s, q)

σ

q1

Figure 1.1: Mapping stability condition into parameter plane

The clear advantage of fixed structure methods is that the control engineer has full control about the resulting complexity of the control system. This allows to handle implementation issues directly during the design process. We will not consider special feedback structures in this thesis. This approach is backed by the fact that all two degree of freedom configurations have basically the same properties and potentials, although some structures are especially suitable for some design algorithms.

1.3 Goal of the Thesis The main objective of this thesis is to extend the parameter space approach by new specifications and to broaden the applicable system class, e.g., multivariable or nonlinear. The basic idea of the parameter space approach (PSA) is to map control specifications for a given system into the space of defined varying parameters. The boundaries of parameter sets that fulfill the specifications are hereto determined. Usually we consider two parameters at a time and control specifications are mapped into a parameter plane. This allows intuitive interpretation of the graphical results. By mapping we actually mean the identification of parameter regions (or subspaces) for which the specifications are fulfilled. In other words we are interested in the set of all parameters Pgood that fulfill a given specification. The boundary of this good set is characterized by the equality case of the specification. Mathematically this good set is given by a mapping equation. Mapping equations form the mathematical core of the PSA. They combine the control specific system description with specifications that a control design requires to hold for the system.

1.4 Outline

5

This thesis presents a unified approach to consider various control specifications for multivariable systems in the parameter space approach. It is shown how various specifications can be formulated using the same mathematical framework. Since there is no straightforward way to solve the resulting mapping equations a second, but not less important goal is to find and explore computational methods to solve the mapping problem. The results in this thesis can be transferred and applied to time-discrete systems. The required methodology can be directly taken from [Ackermann et al. 2002]. Hence we do not extensively cover the application of the time-continuous results in this thesis to the time-discrete case.

1.4 Outline Chapter 2 serves multiple purposes. We start with some control theoretic background. Subsequently the various specifications are presented. Besides the introduction and some information, the focus lies on a uniform mathematical description of the criteria. This allows uniform treatment and development of mapping equations and finally mapping algorithms. Chapter 3 then presents the mapping equations used to map the specifications into parameter space. Beyond the mapping equations for specific specifications introduced in Section 2.4, we consider mapping equations for general integral quadratic constraint (IQC) specifications [Muhler and Ackermann 2004]. The remaining part of the thesis deals with the practical application of the presented control theory to practical problems. To this end, we take a closer look at algorithms suitable for the mapping equations arising from the various control specifications in Chapter 4. And we explore graphical means to visualize the results in a parameter plane. This is motivated by specifications that can be related to performance. Here not just the fulfillment of a condition, for example stability, is crucial, but we are interested in optimizing the attainable performance level. Therefore contour-like plots with color-coded performance levels in a parameter plane reveal additional insight. The application of the derived mapping equations and the mapping algorithms is demonstrated on various examples in Chapter 5. Concluding remarks and perspectives for further work are given in Chapter 6. For the convenience of the reader, we summarize some mathematical background material and elaborate proofs in Appendix A.

6

2

Control Specifications and Uncertainty


This chapter introduces the system class considered in this thesis, namely multivariable parametric systems. We are mainly concerned with linear time-invariant (LTI) systems throughout the thesis. Nevertheless, some results for nonlinear systems are given that fit into the used framework. After presenting the general multi-input multi-output (MIMO) model, we give a brief overview of important properties of MIMO systems and their limitations. Section 2.3 considers uncertainty structures used to model control systems. The main part of this chapter is Section 2.4, which presents various control system specifications used for MIMO systems. This section presents some arguments why it might be useful to extend the classical eigenvalue based parameter space approach by MIMO specifications mainly derived from the frequency domain. The main goal of Section 2.4 is to present all specifications in a way so that they fit into the same mathematical framework. This formulation makes it possible to derive mapping equations in Chapter 3 and incorporate them into the parameter space approach.

2.1 Parametric MIMO Systems There has been an enormous interest in the design of multivariable control systems in the last decades, especially frequency domain approaches [Doyle and Stein 1981],[Francis 1987],[Maciejowski 1989]. We do not intend to give a comprehensive treatment of all the aspects of multivariable feedback design, and refer the reader to the cited literature. Thus, the scope of this section is limited to the presentation of the basic concepts and some examples. We consider uncertain, LTI systems with parametric state-space realization ˙ x(t) = A(q)x(t) + B(q)u(t),

y(t) = C(q)x(t) + D(q)u(t)

(2.1)

or transfer matrix representation G(s, q), i.e., y(s) = G(s, q)u(s) = (C(q)(sI − A(q))−1 B(q) + D(q))u(s), where u ∈ Rm and y ∈ Rp are vectors of signals.

(2.2)

7

2.1 Parametric MIMO Systems The short-hand notation   A(q) B(q)  G(s, q) ∼ = C(q) D(q)

(2.3)

will be used to represent a state-space realization      ˙ A(q) B(q) x(t) x(t)  =  , y(t) u(t) C(q) D(q)

for a given transfer matrix G(s, q).

The parameters q ∈ Rp are unknown but constant with known bounds. The set of all possible parameters is denoted as Q. If not stated otherwise, we assume upper and lower bounds qi = [qi− ; qi+ ] for each dimension and the operating domain q ∈ Q is also referred to as the Q-box (see Figure 2.1). Since the parameter space approach does not favor controller over plant uncertainties, we will not discriminate these in general equations. Thus, usually q is used for both controller and plant uncertainties. If controller parameters are explicitly mentioned they are also denoted by ki . The mapping plane can be a plane of uncertain parameters for robustness analysis, or a plane of controller parameters in a control design step. Also a mix of both parameter types is useful for the design of gain scheduling controllers. q2 q2+ Q

q2−

q1−

q1+

q1

Figure 2.1: Box-like operating domain Q

We will use the symbol G(s) for general transfer matrices arising in the considered control problem. A specific plant will be denoted P (s), and transfer matrices for controllers are denoted K(s). Thus, G(s) includes arbitrary transfer matrices, general plant descriptions including performance criteria, or even open or closed loop transfer matrices. We use the standard notation for specific transfer matrices such as the sensitivity function S(s) and the complementary sensitivity function T (s).

8


2.1.1 MIMO Specifications The main objective of the parameter space approach is to map specifications relevant for dynamic systems (2.1) and (2.2) into the parameter space or into a parameter plane. Apart from stability, the most important objective of a control system is to achieve certain performance specifications. One way to describe these performance specifications is to use the size of certain signals of interest. For example, the performance of a regulator could be measured by the size of the error between the reference and measured signals. The size of signals can be defined mathematically using norms. Common norms are the Euclidean vector norms v u n n X uX ||x||1 := |xi |, ||x||2 := t |xi |2 , ||x||∞ := max |xi |. i=1

i=1

1≤i≤n

The performance of a control system with input and output signals measured by one of the above norms, not necessarily the same, can be evaluated by the induced matrix norms. The most prominent matrix norms used in control theory are the H∞ and H2 norms, which will be considered in Section 2.4.1 and Section 2.4.6, respectively.

2.1.2 MIMO Properties MIMO systems exhibit some properties not known for single-input single-output (SISO) systems. These differences make it difficult to apply standard SISO design guidelines to MIMO systems, e.g., for eigenvalues or loop shapes. At least care has to be taken when simply using these rules. While for a SISO system the behavior can be characterized by the gain and phase for a single channel, these entities depend on the direction of the input for MIMO systems. The same applies to eigenvalues. While eigenvalues can be used to describe the behavior of a SISO system effectively, for MIMO systems the directionality of the associated eigenvectors becomes important. This can be also seen from a design point of view. The eigenvalues of a controllable system with available state information can be moved to any desired location using Ackermann’s formula [Ackermann 1972]. For multivariable systems there are additional degrees of freedom that can be used to shape the closed-loop eigenvectors or other design specifications. It is a well known fact that right half plane (RHP) zeros impose fundamental limitations on control of SISO systems. While these zeros can be found by inspection of the numerator of the transfer function of a SISO system, this does not hold for MIMO systems. Although all elements of a transfer matrix G(s) are minimum-phase, RHP zeros may exist for the overall multivariable system.

2.2 Symbolic State-Space Descriptions

9

The role of RHP zeros is further emphasized by some design methods, e.g., successive loop closure, where zeros can arise during intermediate steps. Nevertheless sometimes we can take advantage of the additional degree of freedom found in MIMO systems to move most of the deteriorating effect of an RHP zero to a particular output channel.

2.2 Symbolic State-Space Descriptions All methods and algorithms presented in this thesis require a symbolic state-space description. In particular the mapping equations for control system specifications presented in Chapter 3 are based on a parametric, linear state-space description of the considered system as in (2.1). The purpose of this section is to present an algorithm that calculates a symbolic state-space description from a given symbolic transfer function, because this is essential for the methods developed in this thesis. Such a system description can be obtained by first-principle modeling such as Lagrange functions or balance equations, where it might be necessary to symbolically linearize the equations. Note that this linearization preserves the parametric dependency on the uncertain parameters q. The references Otter [1999], Tiller [2001] give a good introduction to object oriented modeling where the used software symbolically transforms and modifies the system description. While a parametric transfer matrix is easily obtained from a state-space description by evaluating the symbolic expression G(s, q) = C(q)(sI −A(q))−1 B(q)+D(q), the opposite is much more involved. For SISO systems or systems with either a single input or output, a canonical form provides a minimal realization. Particular variants are the controllable and observable canonical form [Chen 1984, Kailath 1980]. These canonical forms can be easily obtained in symbolic form from the coefficients of a transfer function. Consider a multivariable transfer matrix G(s). One way to obtain a state-space description is to form canonical forms of all transfer functions gij (s) and combine them to get a model with input-output relation equivalent to the considered transfer matrix G(s). This model will be nonminimal, i.e., it contains spurious states, which are non-controllable or nonobservable, or both. For MIMO systems the dimension of a minimal realization is exactly the McMillan degree [Chen 1984]. There exist standard methods to determine a minimal realization for a numerical state-space model. Unfortunately these algorithms are not transferable to symbolic transfer matrix descriptions.

10


2.2.1 Transfer Function to State-Space Algorithm Besides system representations in state-space and transfer function form, a matrix fraction description (MFD) is another useful way of representing a system. Actually these models are the keystone of all linear fractional representation (LFR) based robust control methods, where the idea is to isolate the uncertainty from the system inside a single block. The aim here is to present a symbolic algorithm. It will be formulated using right coprime factorization. A dual left coprime version is possible, but does not provide any advantages over the presented one. Any transfer matrix G(s) can be written as a right or left matrix fraction of two polynomial matrices, G(s) = Nr (s)Mr (s)−1 ,

(2.4a)

G(s) = Ml (s)−1 Nl (s).

(2.4b)

The numerator matrices Nr (s) and Nl (s) have the same dimension as the transfer matrix G(s), whereas the denominator matrices Mr (s) and Ml (s) are square matrices of matching dimension. Special variants are coprime factorizations, which will be discussed later in Section 2.2.2. A right (left) MFD for a given transfer matrix G(s) ∈ Rl, m is easily obtained as follows. Determine the polynomial denominator matrix M (s) as a diagonal matrix, where the entries mii are the least common multiple of all denominator polynomials in the i-th column (row) of G(s), i.e., mii = lcm(den g1i , den g2i , . . . , den gli ),

i = 1, . . . , m,

(2.5a)

i = 1, . . . , l.

(2.5b)

and for left MFDs we use mii = lcm(den gi1 , den gi2 , . . . , den gil ),

The fraction-free numerator matrices Nr (s) and Nl (s) are then determined by simply evaluating Nr (s) = G(s)Mr (s),

(2.6a)

Nl (s) = Ml (s)G(s).

(2.6b)

Having found a column-reduced MFD a state-space realization can be determined using the so-called controller-form [Kailath 1980]. The algorithm presented here will work for proper and strictly proper systems.

11


Given a right MFD G(s) = Nr (s)Mr (s)−1 , the input-output relation y(s) = G(s)u(s) can be rewritten as Mr (s)ξ(s) = u(s),

(2.7a)

y(s) = Nr (s)ξ(s),

(2.7b)

where ξ(s) is the so-called partial state. The polynomial matrices Nr (s) and Mr (s) are now decomposed as Mr (s) = Mhc S(s) + Mlc Ψ(s),

(2.8a)

Nr (s) = Nlc Ψ(s) + Nf t Mr (s).

(2.8b)

The decomposition matrices are computed as follows. Let the highest degree of all polynomials in the i-th column of Mr (s) be denoted as ki . The matrix S(s) is diagonal with S(s) = diag[sk1 , . . . , skm ]. Then the matrix Mhc is the highest-column-degree coefficient matrix of Mr (s). The term Mlc Ψ(s) contains the lower-column-degree terms of Mr (s), where Mlc is a coefficient matrix and Ψ(s) a block diagonal matrix: 

   T Ψ(s) =    

1 s · · · sk1 −1 0

0 1 s ··· s

0

0

0

0

k2 −1

···

0

···

0

· ···

0

··

1 s · · · skm −1



   .   

Output equation (2.8b) is obtained by first computing the direct feedthrough matrix as Nf t = lim G(s) = lim Nr (s)Mr (s)−1 . s→∞

s→∞

The remaining task is to compute the trailing coefficient matrix Nlc by columnwise coefficient evaluation of Nlc Ψ(s) = Nr (s) − Nf t Mr (s) for orders of s up to degree ki for the i-th column. Having found the decomposition (2.8), a state-space description is now easily obtained by assembling m integrator chains with ki integrators in the i-th chain. The total order nt of the system will be given by nt =

m X i=1

ki .

12


A basic state-space realization of S(s) is given by A0 = block diag[Ak1 , . . . , Akm ],

(2.9a)

B

0

= block diag[Bk1 , . . . , Bkm ],

(2.9b)

C

0

= I nt .

(2.9c)

where An is an n × n Jordan block matrix with corresponding input matrix Bn ,

An



0

h

0 ···

   =    

BnT =

1 ··

·



   · ,  0 1   0 i 0 1 ,

··

An ∈ Rn, n ,

Bn ∈ Rn, 1 .

The correct input-output behavior is then achieved by closing a feedback loop around the core integrator chains. The final state-space realization is given by −1 A = A0 − B 0 Mhc Mlc ,

(2.10a)

−1 B = B 0 Mhc ,

(2.10b)

C = Nlc ,

(2.10c)

D = Nf t .

(2.10d)

2.2.2 Minimal Realization A state-space realization is minimal, if it is controllable and observable, and thus contains no subsystems that are not controllable or observable, or both. State-space representations are in general not unique. Nevertheless, minimal state-space realizations are unique up to a change of the state-space basis. More important, the number of states is constant and minimal. This minimality is especially important for the symbolic parameter space approach methods presented in Chapter 3, since they minimize the computational burden in handling and solving the symbolic equations. Since the 1960s the minimal realization problem has attracted a lot of attention and a wide variety of algorithms have emerged, e.g., Gilbert’s approach based on partialfraction expansions [Gilbert 1963] or Kalman’s method, which is based on controllability and observability and reduces a nonminimal realization until it is minimal [Kalman 1963]. Note that an input-output description reveals only the controllable and observable part of a dynamical system.


13

Rosenbrock [Rosenbrock 1970] developed an algorithm, which uses similarity transformations (elementary row or column operations), to extract the controllable and observable, and therefore minimal subpart of a state-space realization. Variants of this algorithm are now implemented in Matlab and Slicot. We will use a similar approach based on MFDs, which directly fits into the results presented in Section 2.2.1. Consider a right MFD, G(s) = Nr (s)Mr (s)−1 , with polynomial matrices Nr (s) and Mr (s). We now determine the greatest common right divisor Rgcd (s) of Nr (s) and Mr (s), such that ˜r (s) = Nr (s)Rgcd (s)−1 , N ˜ r (s) = Mr (s)Rgcd (s)−1 , M

(2.11) (2.12)

and we obtain the right coprime factorization ˜r (s)M ˜ r (s)−1 . G(s) = N

(2.13)

The greatest common right divisor of two polynomial matrices can be found by consecutive row operations, or left-multiplication with unimodular matrices, until the stacked matrix [Mr Nr ]T is column reduced. Since all steps in finding Rgcd are either multiplication or addition of polynomials the algorithm is fraction free and can be easily applied to parametric matrices. Note that we are not interested in special coprime factorization, e.g., stable factorization over RH∞ . So we can symbolically compute Rgcd (s), e.g., using Maple.

2.2.3 Example The above algorithm is illustrated by a small MIMO transfer matrix. Consider the following plant [Doyle 1986], which is an approximate model of a symmetric spinning body with constant angular velocity for a principal axis, and two torque inputs for the remaining two axes,   2 s−a a(s + 1) 1 .  (2.14) G(s) = 2 2 s + a −a(s + 1) s − a2 For this example a right MFD is easily obtained by inspection or using (2.5a) and (2.6a),     2 2 2 s−a a(s + 1) s +a  , Nr (s) =  . (2.15) Mr (s) =  −a(s + 1) s − a2 s2 + a 2

14


It is obvious that if by simply following the algorithm given in Section 2.2.1, we would end up with a state-space description of order four. To minimize the order, we use the minimal realization procedure of Section 2.2.2. It turns out that Nr (s) is actually a greatest common right divisor of Mr (s) and Nr (s), such that Rgcd (s) = Nr (s), and using (2.11), we immediately determine a right coprime factorization with ˜ r (s) = M

−1 Nr (s), 1 + a2

˜r (s) = I. N

(2.16)

We can then proceed with the decomposition (2.8) and using (2.9) and (2.10) we finally obtain a minimal state-space realization of order two,   0 a 1 a    −a 0  −a 1   G(s) ∼ = .   1 0 0 0   0 1 0 0

2.3 Uncertainty Structures The main aim of control is to cope with uncertainty. System models should express this uncertainty, but a very precise model of uncertainty is an oxymoron. George Zames This section contains material on the description of uncertainty for models used in control system design. While in general the material can be applied to models from different application areas, e.g., chemical reactors or power plants, the exposition is especially suited for models of mechanical systems arising in vehicle dynamics. Control engineers, unless only experiments are used, are working with mathematical models of the system to be controlled. These models should describe all system responses vital for the considered system performance. That is, other system properties are neglectable. Apart from matching the responses of the true plant as close as possible, models should be simple enough to facilitate design. By the term uncertainty we will summarize 1. all known variations of the model, 2. differences or errors between models and reality.

15

2.3 Uncertainty Structures

The first type of uncertainty refers to variations of the model due to parameter changes. For parametric LTI systems a mathematical description is given in (2.1) and (2.2). As mentioned earlier these system descriptions are easily obtained from first principle modeling. The parameters q are unknown but constant. That is, we assume that the parameters do not change during a regular operation of the system; or the change is so slow that they can be treated as quasi-stationary. For parameters which change rapidly or whose rate of change lies well within the system dynamics special care has to be taken. In this case the results obtained by treating the parameters as constant or quasi-constant might be false or misleading. Representing the changing parameters through an unstructured uncertainty description might be a remedy. Another approach is to utilize the mapping equations derived in Section 3.4. The stability of a system with an arbitrary fast varying parameter is analyzed in Example 3.1. The uncertain parameters might be states of a full-scale model describing the plant behavior with respect to input and output variables. For example, the mass of an airplane can be considered as a fixed parameter for directional control while it is actually a state of a model and decreasing with fuel consumption used for full flight evaluation. From a frequency point of view the system knowledge generically decreases with frequency. While there are system models which are accurate in a specific frequency range, e.g., for flexible mechanical systems, at sufficiently high frequencies precise modeling is impossible. This is a consequence of dynamic properties which occur in physical systems. We propose the following modeling philosophy: Use the physical knowledge about the plant to include parametric uncertainties as real perturbations with known bounds. Additional uncertainties are modeled by (unstructured) dynamic uncertainties. Any information, e.g., directionality, although not expressible as parametric uncertainties about these uncertainties should be incorporated in analysis and design. Thus the overall model which includes an uncertainty description is denoted as G = G(q, ∆),

(2.17)

where ∆ represents unstructured uncertainty considered in detail in Section 2.3.3. The term unstructured is not to be interpreted literally. The uncertainty description might actually contain some degree of structure. But here it is used as uncertainty which cannot be described as real parameter variation. The uncertainty description used here is an extension of pure parametric models used in the classical parameter space approach to include unstructured uncertainty.

16


The frequency domain approaches to robust control, i.e., the popular H∞ and µ control paradigms, use a different representation of uncertainty. For H∞ robust controllers all uncertainties have to be captured by a norm-bounded uncertainty description. The model is described by a nominal plant and H∞ norm-bounded stable perturbations. Thus even parametric uncertainty, i.e., a detailed parametric model, has to be approximated by a norm-bounded uncertainty description. Furthermore all uncertainty has to be lumped in a single uncertainty transfer function or uncertainty block, see Figure 2.2. ∆

u

w

v

G

z

Figure 2.2: General lumped uncertainty description

While the structured singular value or µ approach tries to alleviate problems associated with the unstructured uncertainty description by using structured and unstructured uncertainties, the single uncertainty block ∆ remains. Parametric uncertainties can be rendered in this single block uncertainty, although some conservativeness is associated with this approach. See Section 2.3.1 for some comments regarding the transformation of parametric models into the single block form of Figure 2.2. Robustness of a control system is not only affected by the plant uncertainty. There are many other aspects which have to be taken care of, when it comes to successful and robust operation of a control system. This includes failure of sensor and actuators, fragility of the implemented controller, physical constraints. Furthermore the opening and closing of individual loops for a multivariable system can be crucial, especially during manual start-up or tuning. Nevertheless, robustness refers to robustness with respect to model uncertainty in this thesis. We will usually try to find a fixed linear controller that robustly satisfies all design specifications. Apart from robust stability, for real control systems some performance specification has to be achieved robustly.

2.3.1 Real Parametric Uncertainties Real parametric uncertainties are lumped into a single vector q ∈ Rp . The general models for parametric LTI systems were given in (2.1) and (2.2). Since the concept of real parametric uncertainties is pretty straightforward we will consider some special

17


variants and the transformation of a parametric model into a model where all parametric uncertainty is lumped into a single block. State perturbations State perturbations are important in the analysis of robust stability. The most general state perturbation model is given by the following state-space description ˙ x(t) = (A + Aq (q)) x(t),

(2.18)

where Aq (q) is a matrix whose entries are polynomial fractions and Aq (0 ) = 0 , i.e., A is the nominal state matrix for q = 0 . Usually only pure polynomial matrices are considered, since a fractional matrix can be transformed into a polynomial matrix by multiplication with the least common multiple of all denominators. Of special interest are representations where all perturbations are combined in a single block, such as in the general lumped uncertainty description of Figure 2.2. Actually, for polynomial state perturbations we can find a representation of the form ˙ x(t) = A + U ∆(I − W ∆)−1 V x(t), (2.19) where ∆ is a diagonal matrix of the form ∆ = diag[q1 Im1 , q2 Im2 , . . . , qp Imp ]. The integers mi are the multiplicities with which the i-th parameter appears in ∆. Figure 2.3 shows a block diagram for state perturbation (2.19). U

u

v

∆

V

W

R

x

A

Figure 2.3: State perturbation block diagram

If no product terms of parameters are present in (2.18) we can write the system as ! p X ˙ x(t) = A+ Ai (qi ) x(t), (2.20) i=1

where Ai (qi ) is the perturbation matrix depending on the parameter qi .

18


Affine state perturbations A further specialization is obtained, if the perturbations are affine in the unknown parameters, i.e., Ai (qi ) = Ai qi ,

i = 1, . . . , p.

Real structured perturbations An even more special variant of affine state perturbations are the so-called real structured perturbations. Thus (2.20) can be written as ˙ x(t) = (A + U ∆V ) x(t),

(2.21)

where ∆ is a diagonal matrix with real uncertain parameters, ∆ = diag[q1 , . . . , qp ] ∈ Rp, p . Lumped real parametric uncertainty In this section we revisit the general state perturbation representation (2.18) ˙ x(t) = (A + Aq (q)) x(t), and we investigate how to obtain a lumped real parametric uncertainty description (2.19), ˙ x(t) = (A + U ∆(I − W ∆)−1 V ) x(t), where all perturbations are inside a single block ∆. For fractional, polynomial parameter dependency such a representation can be found by extracting all non-reducible factors and representing them using a diagonal uncertainty matrix with the individual factors as elements, e.g., using a tree decomposition [Barmish et al. 1990a]. Another technique uses Horner factorization [Varga and Looye 1999]. See [Magni 2001] for an overview of realization and order reduction algorithms. Representation (2.19) is shown as a block diagram in Figure 2.3. From this block diagram it becomes obvious that the uncertainty block ∆ can be pulled out and the system can be put into the lumped uncertainty form of Figure 2.2. This is shown in Figure 2.4. The transfer function for the uncertainty block from output u(s) to input v(s) is Guv (s) = (I − W ∆)−1 V (sI − A)−1 U.

(2.22)

19


∆ u

v

W

R

U

x

V

A

Figure 2.4: Lumped state perturbation

Example 2.1 Consider the following system with state perturbation:     0 1 −q + q1 q2 q1 q2 + 1  x(t) ˙ x(t) =  −1 −2 q2 q1 − q 2

(2.23)

This representation can  q1   q1  ∆=  q2 

be reduced to the following minimal form:    0       0    W = , , −1 1 0      0 q2     1 0 1 0 −1 0 0 1 ,  U = VT =  0 1 0 1 0 1 0 −1

2.3.2 Multi-Model Descriptions A multi-model description consists of a finite number of fixed model descriptions of the form ˙ x(t) = Ai x(t) + Bi u(t),

y(t) = Ci x(t) + Di u(t),

i ∈ 1, . . . , p,

(2.24)

20


where p is the number of individual models. Thus a multi-model description usually does not contain any parameters. Multi-model descriptions are easily treated in the parameter space approach by consecutively mapping a specification for each model.

2.3.3 Dynamic Uncertainty The term dynamic uncertainty might be a bit misleading in the sense that all other uncertainty structures presented in Section 2.3 are describing uncertainties of dynamic systems. By dynamic uncertainty we refer to uncertainties whose underlying dynamics are not precisely known and are possibly varying within known bounds. Dynamic uncertainty operators are often associated with modeling errors that are not efficiently described by parametric uncertainty. Unmodeled dynamics and inaccurate mode shapes of aero-elastic models [Lind and Brenner 1998] are examples of modeling errors that can be described with less conservatism by dynamic uncertainties than with parametric uncertainties. These dynamic uncertainties are typically complex in order to represent errors in both magnitude and phase of signals. The set of unstructured uncertainties ∆ is given as all stable transfer functions (rational or irrational) of appropriate dimension that are norm bounded: ∆ := {∆ ∈ RH∞ ,

||∆|| < l(ω)}.

(2.25)

We will use the H∞ norm throughout the thesis (see Section 2.4.1 for a review of the H∞ norm) and usually the following normalization condition holds: ||∆||∞ ≤ 1. This normalization can always be enforced by using suitable weighting functions. There are several possibilities how to describe plant perturbations using unstructured uncertainties ∆. The most prominent and most intuitive are the (output) multiplicative and additive perturbations: Gp (s) = G(s) + Wa (s)∆(s),

(2.26)

Gp (s) = (I + ∆(s)Wo (s)) G(s),

(2.27)

where Wa (s) and Wo (s) are weights such that ||∆(s)||∞ ≤ 1. See Figure 2.5. Similar plant perturbations are inverse additive uncertainty, inverse multiplicative output and (inverse) multiplicative input uncertainty. Another common form is coprime factor uncertainty: Gp (s) = (Ml + ∆M )−1 (Nl + ∆N ),

(2.28)

where Ml , Nl is a left coprime factorization of the nominal plant model. This uncertainty description, suggested by McFarlane and Glover [1990], is mainly used in an H∞ norm

21


∆

Wa

G

Wo

∆

G

Figure 2.5: Additive and multiplicative output uncertainty

loop-shaping procedure, where the open-loop shapes are shaped by weights and the robustness of the resulting plant to this type of uncertainty is maximized. Usually, no problem-dependent uncertainty modeling is used in this approach, see [Skogestad and Postlethwaite 1996] for a thorough treatment. For plants with different physically motivated perturbations, e.g., input and output multiplicative uncertainty, it is possible to lump all uncertainties into a single perturbation, see Figure 2.2. Unfortunately even for unstructured perturbations the resulting overall uncertainty ∆ is block-diagonal and therefore structured. A straightforward application of the small-gain theorem (see Theorem 2.2 in Section 2.4.1) will be obviously conservative, because the system is checked for a much larger set of uncertainties which actually cannot appear in the real system. Back in 1982, Doyle and Safonov introduced simultaneously equivalent entities to measure the robustness margin of a system with respect to structured uncertainties. For a general system as in Figure 2.2 the so-called structured singular value µ is defined as µ∆ (G(s)) :=

1 , min{¯ σ (∆) | det[I − G(s)∆] = 0}

where σ ¯ is the maximal singular value (see Section 2.4.1). The value µ∆ (G(s)) is a simple scalar measure of the smallest structured uncertainty which destabilizes the feedback system. The µ approach has been extended to the synthesis of robust controllers and there are several related toolboxes. Yet, the exact computation of µ is not possible except for special cases. Thus all available software tries to compute meaningful bounds for µ. This emphasizes the goal of this thesis to treat real parameter variations directly and only represent them into a lumped uncertainty ∆, if the parameters are changing fast or their number is large and we want to refrain from gridding. Another rational of this approach comes from the fact that µ with respect to pure real uncertain parameters is discontinuous in the problem data [Barmish et al. 1990b]. That is, for small changes of the nominal system, maybe due to a neglected parameter, the stability margin might be subject to large, discontinuous changes. Put in other words,

22


for a specific model the real µ is not a good indicator of robustness because it might deteriorate for an infinitesimal perturbation of the considered plant.

2.4 MIMO Specifications in Control Theory This section reviews the various specifications and objectives relevant for design and analysis of multivariable control systems. All specifications will be formulated using algebraic Riccati equations (AREs) or Lyapunov equations. See Section 3.2 for an introduction to AREs. While there will be no special notation for parametric dependencies, the considered systems might depend on several real parameters q ∈ Rp . The specifications are briefly motivated from a general control theoretic point of view. Special attention is given to reasons why it might be advantageous to include these specifications into the parameter space approach. For motivation of the specifications presented in this section see for example [Boyd et al. 1994] or [Scherer et al. 1997]. Apart from the introduction of the specifications, the main aim of this section is to present them in a mathematical formulation, that is tractable for the mapping equations developed in Chapter 3.

2.4.1 H∞ Norm Probably the most prominent norm used in control theory to date is the H∞ norm. The H∞ norm of a transfer function G(s) is defined as the peak of the maximum singular value of the frequency response ||G(s)||∞ := sup σ ¯ (G(jω)),

(2.29)

ω

where σ ¯ is the largest singular value or maximal principal gain of an asymptotically stable transfer matrix G(s). Note that (2.29) defines the L∞ norm, if the stability requirement is dropped. There are several interpretations of the H∞ norm. A signal related interpretation is given by ||G||∞ = sup w6=0

||Gw||2 . ||w||2

Consider a scalar transfer function G(s), then the infinity norm can be interpreted as the maximal distance of the Nyquist plot of G(s) from the origin or as the peak value of the Bode magnitude plot of |G(jω)|. In that sense, frequency response magnitude specifications [Odenthal and Blue 2000] can be recast as scalar H∞ norm problems.

2.4 MIMO Specifications in Control Theory

23

For SISO systems the H∞ norm is simply the maximum gain of the transfer function, whereas for MIMO systems it is the maximum gain over all directions. Thus the H∞ norm takes the directionality of MIMO systems into account. For MIMO systems, the H∞ norm describes the maximum amplitude of the steady state response for all possible unit amplitude sinusoidal input signals. In the context of stochastic input signals, the H∞ norm can be interpreted as the square root of the maximal energy amplification for all input signals with finite energy. Note that unlike the induced matrix norms ||A||p, which are related to vector norms ||x||p , the norms used for matrix functions are not directly related to the namesake signal norms. For example, the L1 norm is another norm, frequently used for LTI systems, ||G||1 := sup w6=0

||Gw||∞ . ||w||∞

The H∞ norm can be used to evaluate nominal stability of a system without uncertainty. By evaluating the H∞ norm of special transfer functions, e.g., a weighted sensitivity function, performance and robustness of a control system can be assessed. We will show how to incorporate the latter feature into the PSA in the next chapter. Based on the control theoretic useful mathematical properties the so-called H∞ problem was defined by Zames [1981]. Using the general control configuration of Figure 2.6, the standard H∞ optimal control problem is to find all stabilizing controllers1 K which minimize ||Fl (P, K)||∞ ,

(2.30)

where Fl (P, K) := P11 + P12 K(I − P22 K)−1 P21 is a lower linear fractional transformation. Often one is content with a suboptimal controller which is close to the optimal. Then the H∞ control problem becomes: given a γ > γmin , where γmin is the minimal, achievable value, find all stabilizing controllers K such that ||Fl (P, K)||∞ < γ.

(2.31)

Following [Zames 1981] a number of different formulations and solutions were developed. One successful approach to solve H∞ control problems involves AREs, see [Doyle et al. 1989, Petersen 1987]. The ARE based algorithm of Doyle et al. [1989] is summarized in [Skogestad and Postlethwaite 1996, p. 367]. The formulation based on AREs will become important in Chapter 3 when we derive mapping equations for H∞ norm specifications. 1

We use K for controllers to avoid ambiguity with the output-state matrix C

24


w

z P

u

v

K

Figure 2.6: General control configuration

Hence we do not pursue the automatic solution of (2.31), since we are trying to incorporate H∞ criteria into the PSA. Nevertheless the achievable level γ is of interest when mapping an H∞ specification. The following theorem, which is known as the bounded real lemma [Boyd et al. 1994], provides an important link between H∞ control problems and AREs and will therefore become important in Chapter 3. This theorem, besides its theoretical significance, is often used as a preparation for the solution of the H∞ problem. Theorem 2.1 Bounded real lemma Consider a linear system with transfer function G(s) and corresponding minimal state-space realization G(s) = C(sI − A)−1 B + D. Then the following statements are equivalent: (i) G(s) is bounded-real, i.e., G(s)∗ G(s) ≤ I, (ii) G(s) is non-expansive, i.e., Z τ Z τ T u(t)T u(t)dt, y(t) y(t)dt ≤ 0

0

∀ Re s > 0;

τ ≥ 0;

(iii) the H∞ norm of G(s) with A being stable, σ¯ (D) < γ, and γ = 1 satisfies ||G(s)||∞ ≤ γ; (iv) the algebraic Riccati equation 1

1

1

1

γXBSr B ∗ X + γC ∗ Sl C − X(A−BSr D ∗ C) − (A−BSr D ∗ C)∗ X = 0 , (2.32) with γ = 1 has a Hermitian solution X0 such that all eigenvalues of the matrix A − BB ∗ X0 lie in the open left half-plane, where Sr = (D ∗ D − γ 2 I) and Sl = (DD ∗ − γ 2 I).

25


Note: The equivalence of (iii) and (iv) in Theorem 2.1 was stated using the parameter γ such that we can map different performance levels γ for an H∞ norm specification into parameter space in Chapter 3. All robust stability conditions for uncertain systems using the H∞ norm can be based on the following rather general result [Zhou et al. 1996]. Theorem 2.2 Small gain theorem Consider the feedback system of Figure 2.2, with stable G(s). Then the closed-loop system is stable for all ∆ ∈ RH∞ with 1 ||∆||∞ ≤ if and only if ||G||∞ < γ. γ Small gain theorems have a long history in control theory, starting with [Sandberg 1964]. The above printed version is a norm based gain version [Zhou et al. 1996]. There are even more general versions for nonlinear functionals. Note that the small gain theorem can be very conservative. For example, unity feedback of stable first-order systems with gain greater than one is not covered. Owen and Zames [1992] make the following observation which is quoted: The design of feedback controllers in the presence of non-parametric and unstructured uncertainty . . . is the raison d’être for H∞ feedback optimization, for if disturbances and plant models are clearly parametrized then H∞ methods seem to offer no clear advantages over more conventional state-space and parametric methods. Next, consider an SISO control specification, which can be formulated using the H∞ norm. Nyquist stability margin An important measure of robustness for SISO transfer functions is the so-called Nyquist stability margin. The Nyquist stability margin is defined as the minimal distance of the Nyquist curve from the critical point (−1, 0), ρ := min |1 + G0 (jω)|, ω

(2.33)

where G0 (s) is the open-loop transfer function. Observe that the Nyquist stability margin is related to the sensitivity function S(s) by 1 ρ= , (2.34) ||S(s)||∞ where S(s) = 1/(1 + G0 (s)).

26


2.4.2 Passivity and Dissipativity The roots of passivity as a control concept can be traced back to the 1940’s, when researchers in the Soviet Union applied Lyapunov’s methods to stability of control systems with a nonlinearity. But it took up to 1971, when Willems [1971] formulated the notion of passivity in a system theoretic framework. The most striking feature of passivity is that any interconnection of passive systems is passive. Figure 2.7 illustrates some connections of passive subsystems which comprise a passive system. Passive System 1

y1

u

w1

u1

Passive System 1

y1

y Passive System 2

y2

y2

Passive System 2

u2

w2

Figure 2.7: Interconnection of passive systems

This fact can be used to design robust controllers by subdividing the complete control system into passive subsystems and designing a passive controller. If the plant is not passive, a suitable approach is to fix a controller which leads to a passive controlled subsystem. On top of this, additional performance enhancing controllers can be determined which preserve passivity. This approach is similar to the classical feedback - feedforward filter design steps of many control design approaches. Since passivity is also defined for nonlinear systems this concept can be applied to control systems with either nonlinear plant or controller. This approach is easily extended to parametric robustness by checking or guaranteeing that a system is passive under all parameter variations. We will consider quadratic MIMO systems, i.e., the dimension of the input equals the output dimension. This is a mandatory assumption for passivity. For dissipativity, which can be seen as the generalization of passivity, this is not necessary, see Definition 2.1 on page 28. Nevertheless, commonly used dissipativity definitions, e.g., (2.40), assume that the system is quadratic. For a linear system passivity is equivalent to the transfer matrix G(s) being positive-real, which means that G(s) + G(s)∗ ≥ 0 ∀ Re s > 0.

(2.35)

2.4 MIMO Specifications in Control Theory In the time-domain a system is said to be passive if Z τ u(t)T y(t) dt ≥ 0, ∀ τ ≥ 0, x(0) = 0 .

27

(2.36)

0

Passivity can be interpreted for physical systems, if the term u(t)T y(t) is a power, e.g., current and voltage for electrical and co-located force and velocity for mechanical systems. Equation (2.36) then says that the difference between supplied and withdrawn energy is positive. For SISO transfer functions passivity can be checked graphically by plotting the Nyquist diagram. If the resulting curve lies in the right half plane then the system is passive. The following lemma [Anderson 1967] translates the frequency-domain condition (2.35) into a matrix condition which will lead to mapping equations. Lemma 2.3 Positive Real Lemma Consider a linear, time-invariant system G(s) = C(sI − A)−1 B + D, with (A, B) stabilizable, (A, C) observable and D + D ∗ nonsingular. Then G(s) is positive real or passive, if there are matrices L, W , and X = X ∗ > 0, such that A∗ X + XA = −L∗ L,

XB − C ∗ = −L∗ W, D+D

∗

∗

= W W.

(2.37a) (2.37b) (2.37c)

Using elementary matrix operations the unknown matrices L and W can be eliminated to give the ARE A∗ X + XA + (XB − C ∗ )(D + D ∗ )−1 (XB − C ∗ )∗ = 0 .

(2.38)

Condition (2.35) is equivalent to the following statement: There exists X = X ∗ satisfying the ARE (2.38). This equivalence can be found in [Willems 1971]. Using Theorem 3.1 the equivalence between (2.35) and the non-existence of pure imaginary eigenvalues of a Hermitian matrix can be established. This was first done in [Lancaster and Rodman 1980]. Equation (2.38) will be used to obtain mapping equations for passivity in Chapter 3. In the preceding formulations a non-zero feedthrough D with D + D ∗ nonsingular was assumed. It turns out that for strictly-proper systems with D + D ∗ singular matters become more difficult. Equation (2.38) disintegrates into: A∗ X + XA < 0,

(2.39a)

XB − C ∗ = 0 .

(2.39b)

28


The linear matrix inequality (LMI) (2.39a) has to hold, while the constraint (2.39b) is satisfied. Because this system of equations and inequalities does not fit into the ARE framework we will use the more general dissipativity in Chapter 3 in order to develop mapping equations for passivity. A system is said to have dissipation η if Z τ (u(t)T y(t) − ηu(t)T u(t)) dt ≥ 0, 0

∀ τ ≥ 0, x(0) = 0 .

(2.40)

Thus passivity corresponds to non-negative dissipation. A system has dissipativity η, if the following ARE has a Hermitian solution A∗ X + XA + (XB − C ∗ )(D + D ∗ − 2ηI)−1 (XB − C ∗ )∗ = 0 .

(2.41)

The AREs (2.38) and (2.41) will be used later to derive algebraic mapping equations for LTI systems. Thus these AREs allow to incorporate passivity and dissipativity for LTI systems into the parameter space approach. As mentioned earlier, the concept of dissipativity carries over to nonlinear systems. In order to do this we need a more general definition of dissipativity [Willems 1971]. ˙ Definition 2.1 Let x(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) be a system, with state x(t), input u(t) and output y(t). This system is dissipative with respect to a supply rate S, if there is a (storing) function V (x), such that Z τ V (x(τ )) − V (x(0)) ≤ S(u(t), y(t)) dt, (2.42) 0

is satisfied.

Note that the latter definition is formulated in terms of the internal state x(t), while the formulation in (2.40) is related to the input-output behavior of a system. For quadratic systems, where the input and output dimensions are equal, this definition can now be specialized using particular supply rates [Sepulchre et al. 1997]. ˙ Definition 2.2 Let x(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) be a system, with state x(t), input u(t) and output y(t). This system is α−input dissipative, β−output dissipative, if (2.42) is satisfied with S(u, y) = uT y − α||u||2 , S(u, y) = uT y − β||y||2 , respectively. Definition 2.2 defines the general α-input and β-output dissipativity. There is no assumption on the values of these parameters. In comparing Definition 2.2 with (2.40) we conclude that (2.40) corresponds to η-input passivity. After (2.40) we noted that passivity corresponds to non-negative dissipation η. While the interconnection properties of passive systems are well-known, we are now interested in feedback interconnections of dissipative systems used in Definition 2.2.

29

2.4 MIMO Specifications in Control Theory Theorem 2.4 [Sepulchre et al. 1997]

Let two systems be feedback interconnected as in the right part of Figure 2.7. The closed loop system is asymptotically stable, if System 1 is α-input dissipative, System 2 is β-output dissipative, and α + β > 0 holds. This theorem assures that the feedback interconnection of two purely passive systems, i.e., α = β = 0, is stable. We will use Theorem 2.4 and Definition 2.2 to apply the passivity results to famous results for nonlinear control systems in Section 2.4.4. Remark 2.1 Quadratic Constraints Both H∞ norm and dissipativity specifications fit into the more general framework of quadratic constraints of the form  T    Z τ y(t) Q S y(t)      dt ≤ 0, ∀ τ ≥ 0, x(0) = 0 . (2.43) 0 u(t) ST R u(t)

We will consider an even more general set of specifications, which includes (2.43) in Section 2.5.

2.4.3 Connections between H∞ Norm and Passivity For SISO systems both the H∞ norm and the passivity condition resemble classical gain and phase conditions. They are even more related and can be even translated into each other. Using the bilinear Cayley transformation ˆ G(s) = (I − G(s))(I + G(s))−1 ,

with

det [I + G(s)] 6= 0,

(2.44)

the following equivalence holds [Anderson and Vongpanitlerd 1973, Haddad and Bernstein 1991]. Theorem 2.5 (i) G(s) is positive-real, i.e., G(s) + G(s)∗ ≥ 0 ∀ Re s > 0;

ˆ (ii) ||G(s)|| ∞ < 1.

30


By Theorem 2.5 a positive realness condition can be recast as an H∞ norm condition and vice versa. The application of this theorem in the time-domain is facilitated by the following conversion. Let G(s) have the state-space realization   A B , G(s) ∼ = C D ˆ then a realization for G(s) is given by   −1 −1 A − B(I + D) C B(I + D) ∼ ˆ . G(s) = −1 −1 −2(I + D) C (I − D)(I + D)

(2.45)

If we want to apply the results in the opposite direction, we can exchange the symˆ since the Cayley transformation is bilinear, i.e., the converse of transforbols G and G, −1 ˆ ˆ ˆ mation (2.44) is simply G(s) = (I − G(s))(I + G(s)) with det[I + G(s)] 6= 0. For systems with a static feedthrough matrix D which satisfies det[I + D] = 0 the conversion based on (2.45) fails. In this case, the transformation from a positive-real to an H∞ norm condition can be done by using a positive realness preserving congruence ˜ transformation G(s) → G(s) = V ∗ G(s)V :   A BV ∼ ˜ , G(s) = ∗ ∗ V C V DV where V is a nonsingular matrix such that det[I + V ∗ DV ] 6= 0. For the converse transformation from an H∞ norm to a positive-real condition we can use a sign matrix transfor˜ mation G(s) → G(s) = G(s)S, where S is a sign matrix such that det[I + DS] 6= 0 [Boyd and Yang 1989].

2.4.4 Popov and Circle Criterion Although this thesis is mainly concerned with specifications for LTI systems, the presented theory can be applied to criteria for nonlinear systems. Absolute stability theory allows to analyze the stability of an LTI system in the feedforward path interconnected with a static nonlinearity in the feedback path. There are two well-known variants, namely the Popov and Circle criterion, which can be formulated as the feedback of two passive systems [Khalil 1992, Kugi and Schlacher 2002]. Both criteria assume a sector nonlinearity. Figure 2.8 shows the general feedback structure for absolute stability, where Ψ (y) represents the nonlinearity.

31


Circle criterion Consider the feedback structure given in Figure 2.8 where the multivariable static nonlinearity Ψ (y) satisfies the sector condition (Ψ (y) − K1 y)T (Ψ (y) − K2 y) ≤ 0,

(2.46)

with K1 , K2 matrices which satisfy K2 − K1 > 0. Thus the nonlinearity is contained in the sector K1 , K2 . Note that apart from the positive definiteness of K2 − K1 there are no further assumptions such as SISO.

A C

B 0

y

u Ψ (y)

Figure 2.8: Feedback structure for absolute stability

Using an equivalence transformation the feedback loop in Figure 2.8 can be put in the form Figure 2.9. It can be shown that the nonlinear System 2 in Figure 2.9 is β-output dissipative, with β = 1. From Theorem 2.4 then follows the Circle criterion, see [Khalil 1992, Vidyasagar 1978]:

Theorem 2.6 Circle Criterion

Consider the feedback structure in Figure 2.8 with a sector nonlinearity Ψ (y) which satisfies (2.46). The closed loop is absolutely stable, if the LTI system ˙ x(t) = (A − BK1 C)x(t) + Bu(t) y(t) = (K2 − K1 )Cx(t)

(2.47)

is α-input dissipative with α > −1.

32

Control Specifications and Uncertainty System 1

K1

A C

B 0

Ψ (y)

K

K −1

K1 System 2

Figure 2.9: Equivalent feedback loop for circle criterion

Theorem 2.6 can now be formulated as an ARE using the µ-input dissipativity ARE (2.41) with µ = −1, XBB ∗ X + X(2A − BKs C) + (2A − BKs C)∗ X + C ∗ Kd∗ Kd C = 0 ,

(2.48)

where Ks = K1 + K2 , and Kd = K2 − K1 . The condition for absolute stability given in Theorem 2.6 is sufficient. Because this does not imply necessity there might be considerable conservativeness. If we sacrifice the generality of the multivariable sector condition (2.46), we can derive sufficient conditions which can be much tighter. This leads to the Popov criterion considered next. Popov criterion For the Popov criterion the considered nonlinearity Ψ (y) is a simple decentralized function Ψj = Ψj (yj ) with 0 ≤ kj ≤ Ψj (yj ), i.e., K = diag(ki ) ∈ Rm, m .

j = 1, . . . , m,

(2.49)


33

Using an equivalence transformation similar to the one considered for the circle criterion the nonlinearity Ψ (y) can be embedded into a dissipative subsystem. The decentralized constraint on the nonlinearity offers some freedom, which can be used to include the factor (µs + 1)−1 in the nonlinear subsystem while still maintaining dissipativity. This additional degree of freedom µ can be chosen arbitrarily. Theorem 2.7 Popov criterion Consider the feedback structure in Figure 2.8 with a stationary, sector nonlinearity Ψ (y) which satisfies (2.49). The closed loop is absolutely stable, if there is a µ ∈ R such that the LTI system ˙ x(t) = Ax(t) + Bu(t), y(t) = KC((I + µA)x(t) + µBu(t)),

(2.50)

is α-input dissipative with α > −1. Let C˜ = KC(I + µA), then the ARE related to Theorem 2.7 is given by A∗ X + XA + (XB − C˜ ∗ )(2I + µKCB + µB ∗ C ∗ K ∗ )−1 (XB − C˜ ∗ )∗ = 0 .

(2.51)

2.4.5 Complex Structured Stability Radius The complex structured stability radius of the system ˙ x(t) = (A + U ∆V )x(t)

(2.52)

is defined by n o rC = inf ||∆|| : Λ(A + U ∆V ) ∩ C+ 6= ∅ ,

(2.53)

where ∆ is a complex matrix of appropriate dimension, C+ denotes the closed right half plane, and ||∆|| is the spectral norm of ∆. Lemma 2.8 Let G(s) = V (sI − A)−1 U . Then rC (A, U, V ) = ||G||−1 ∞.

(2.54)

34


A proof can be found in [Hinrichsen and Pritchard 1986]. Thus the determination of the complex structured stability radius is equivalent to the computation of the H∞ norm of a related transfer function. Note that not all possible perturbations are expressible with a single block perturbation (2.52), e.g., 

˙ x(t) = 

−1

0

2 −2





+

0

δ1

δ2 δ3



 x(t).

2.4.6 H2 Norm Performance The H2 norm is a widely used performance measure that allows to incorporate timedomain specifications into control design. The H2 norm of a stable transfer matrix G(s) is defined as ||G(s)||2 :=

1 trace 2π

Z

∞ ∗

G(jω) G(jω)dω −∞

1/2

.

The above norm definition can be used for generic square integrable functions on the imaginary axis and is then called L2 norm. Thus, strictly speaking without the stability condition, we are mapping the L2 norm instead of the H2 norm. The H2 norm arises, for example, in the following physically meaningful situation. Let the system input be zero-mean stationary white noise of unit covariance. Then, at steady state, the variance of the output is given by the square of the H2 norm. This can be seen from the general definition of the root mean square response norm for systems driven by a particular noise input with power spectral density matrix Sw : ||G||rms,s :=

1 trace 2π

Z

∞ ∗

G(jω) Sw (jω)G(jω)dω −∞

1/2

.

The H2 norm is only finite, if the transfer matrix G(s) is strictly proper, i.e., the direct feedthrough matrix D = 0 (or D(q) = 0 ). Hence we assume D = 0 in this subsection, which is a valid assumption for almost any real physical system. By Parseval’s theorem, the H2 norm can be expressed as ||G||2 =

trace

Z

∞ T

H(t) H(t)dt 0

1/2

,

with H(t) = CeAt B being the impulse matrix of G(s) = C(sI − A)−1 B.

(2.55)

35

2.4 MIMO Specifications in Control Theory From this follows ||G||22

= trace [B

T

Z

∞

T

eA t C T CeAt dt B] 0

= trace [B T Wobs B],

(2.56)

where Wobs :=

Z

∞

T

eA t C T CeAt dt 0

is the observability Gramian of the realization, which can be computed by solving the Lyapunov equation AT Wobs + Wobs A + C T C = 0 .

(2.57)

Alternatively a dual output-controllable formulation exits for ||G||22 , which involves the controllability Gramian Wcon , ||G||22 = trace [CWcon C T ],

(2.58)

where Wcon can be determined from AWcon + Wcon AT + BB T = 0 .

(2.59)

The H2 norm is different from the specifications presented so far in that a specification ||G||2 ≤ γ cannot be expressed by an ARE. In that sense the H2 norm does not really fit into the ARE framework. But this specification can be formulated by means of the more special Lyapunov equation, which is affine in the unknown Wobs .

2.4.7 Generalized H2 Norm In the scalar case, the H2 norm can be interpreted as the system gain, when the input are L2 functions and the output bounded L∞ time functions. Thus the scalar H2 norm is a measure of the peak output amplitude for energy bounded input signals. Low values for this quantity are especially desirable if we want to avoid saturation in the system. Unfortunately this interpretation does not hold for the H2 norm in the vector case. Following [Wilson 1989], the so-called generalized H2 norm [Rotea 1993] is defined by Z ∞ 1/2 G(t)T G(t)dt, if ||y||∞ = sup ||y(t)||2 , ||G||2,gen = λmax 0≤t≤∞

0

or

||G||2,gen =

d1/2 max

Z

∞

G(t)T G(t)dt, 0

if ||y||∞ = sup ||y(t)||∞ , 0≤t≤∞

36


depending on the type of L∞ norm chosen for the vector valued output y. Here, λmax and dmax denote the maximum eigenvalue and maximum diagonal entry of a non-negative matrix, respectively. The generalized H2 norm can also be expressed as T ||G||2,gen = λ1/2 max [B Wobs B],

or T ||G||2,gen = d1/2 max [B Wobs B],

depending on the L∞ norm chosen, where Wobs is the observability Gramian.

2.4.8 LQR Specifications The classical linear quadratic regulator (LQR) problem, which aims to minimize the objective function Z 1 ∞ (x(t)T Qx(t) + u(t)T Ru(t))dt (2.60) J= 2 0 for a state-feedback controller u(t) = −Kx(t), was introduced back in 1960 [Kalman and Bucy]. A core advantage is the easily interpretable time-domain specification, which allows a transparent tradeoff between disturbance rejection and control effort utilization. Another feature is the fact that the easily solvable optimization problem produces gains that coordinate the multiple controls, i.e., all loops are closed simultaneously. The classical LQR problem is to find an optimal input signal u(t) which drives a given ˙ system x(t) = Ax(t) + Bu(t) by minimizing the performance index (2.60). The optimal solution for the constant gain matrix K is given by K = R−1 B T X, where X is the unique positive semidefinite solution of the ARE AT X + XA − XBR−1 B T X + Q = 0 .

(2.61)

This ARE is another example for the wide-spread use of AREs in control theory. The signal-oriented formulation of LQR is the linear quadratic Gaussian (LQG) control problem. In LQG, input signals are considered stochastic and the expected value of the output variance is minimized. Mathematically this is equal to the 2-norm of the stochastic output.

37


If frequency dependent weights on the signals are included, we arrive at the so-called Wiener-Hopf design method, which is nothing else than the H2 norm problem considered in Section 2.4.6. The classical LQR problem can be cast as an H2 norm problem [Boyd and Barratt 1991]. Consider a linear time-invariant system described by the state equations ˙ x(t) = Ax(t) + Bu(t) + w(t) ,    1 u(t) R2 0  , z(t) =  1 x(t) 0 Q2

(2.62a) (2.62b)

where u(t) is the control input, w(t) is unit intensity white noise, and z(t) is the output signal of interest. 1

1

The square root W 2 of a square matrix W is defined as any matrix V = W 2 which satisfies W = V T V or W = V V T . The matrix square root exists for symmetric, positive definite matrices. One possible algorithm to obtain the matrix square root is to use lower or upper triangular Cholesky decompositions. The LQR problem is then to design a state-feedback controller u(t) = −Kx(t), which minimizes the H2 norm between w(t) and z(t). From this follows that the performance index J is given as J = ||Gw→z (s)||22 , where Gw→z (s) is the transfer function of (2.62) from w(t) to z(t), with state-space realization   A − BK I     1 Gw→z ∼ =  −R 2 K 0  .   1 2 0 Q The parametric LQR control design allows us to explicitly incorporate control effort specifications into a robust controller design, which is not possible with pure eigenvalues specifications. It also applies to parametric SISO systems. The LQG problem provides another example why it can be advantageous to combine classical, non-robust methods and the PSA. Back in 1978 [Doyle] showed that there are LQG controllers with arbitrary small gain margins. The PSA based LQR and H2 norm design described in this thesis allows to combine the transparent robustness of PSA with the easy tunability of LQR. The author considers the

38


combination of the invariance plane based pole movement [Ackermann and T¨ urk 1982] and LQR performance evaluation in the PSA as a very promising method to design robust state-space controllers.

2.4.9 Hankel Norm The Hankel norm of a system is a measure of the effect of the past system input on the future output. It is known [Glover 1984] that the Hankel norm is given by ||G||hankel = λ1/2 max [Wobs Wcon ], where the controllability Gramian Wcon is the solution of

AWcon + Wcon AT + BB T = 0 . The Gramian Wobs measures the energy that can appear in the output and Wcon measures the amount of energy that can be stored in the system state using an excitation with a given energy. The Hankel norm is extensively used in connection with model reduction. For example, reduced state-space models with minimal deviations in the input-output behavior can be achieved by neglecting modes with the smallest Hankel singular values, which are defined as the positive square roots of the eigenvalues of the product of both Gramians σi =

p λi [Wobs Wcon ].

(2.63)

2.5 Integral Quadratic Constraints This section presents a recently developed, unified approach to robustness analysis of general control systems. Namely we consider integral quadratic constraints (IQCs) introduced by Megretski and Rantzer [1997]. In general, IQCs provide a characterization of the structure of a given operator and the relations between signals of a system component. An IQC is a quadratic constraint imposed on all possible input-output pairs in a system. The IQC framework combines results from three major control theories, namely inputoutput, absolute stability, and robust control. Using IQCs specifications from all these research fields can be formulated by the same mathematical language. Actually it has

2.5 Integral Quadratic Constraints

39

been shown that some conditions from different theories lead to identical IQCs and are therefore equivalent. Since all specifications expressible as IQCs share the same mathematical formulation we can use the same computational methods to map them into parameter space. In a system theoretical context the following general IQC is widely used. Two bounded l signals w ∈ Lm 2 [0, ∞) and v ∈ L2 [0, ∞) satisfy the IQC defined by the self-adjoint multiplier Π(jω) = Π(jω)∗ , if Z

∞ −∞

 

v(jω)

∗





v(jω)  Π(jω)   dω ≥ 0, w(jω) w(jω)

(2.64)

holds for the Fourier transforms of the signals. Consider the bounded and causal operator ∆ defined on the extended space of square integrable functions on finite intervals. If the signal w is the output of ∆, i.e., w = ∆(v), then the operator ∆ is said to satisfy the IQC defined by Π, if (2.64) holds for all signals v ∈ Ll2 [0, ∞). We use the shorthand notation ∆ ∈ IQC(Π) for operator-multiplier pairs (∆, Π) for which this property holds. Thus the multiplier Π gives a characterization of the operator ∆. The operator ∆ represents the nonlinear, time-varying, uncertain or delayed components of a system. For example, let ∆ be a saturation w = sat(v) then the multiplier 

Π=

1

0

0 −1



,

defines an IQC which holds for this nonlinear operator. Note, that this multiplier is not necessarily unique. Actually there might be an infinite set of valid multipliers. See [Megretski and Rantzer 1997] for a summarizing list of important IQCs, and [Jönsson 2001] for a detailed treatment. We consider the general configuration of a causal and bounded linear time-invariant (LTI) transfer function G(s), and a bounded and causal operator ∆ which are interconnected in a feedback manner v = Gw + e2 , w = ∆(v) + e1 , where e1 and e2 are exogenous inputs. See Figure 2.10. The stability of this system can be verified using the following theorem.

40


e1

w

G

∆

v

e2

Figure 2.10: General IQC feedback structure

Theorem 2.9 [Megretski and Rantzer 1997] l×m Let G(s) ∈ RH∞ , and let ∆ be a bounded causal operator. Assume that

(i) for τ ∈ [0; 1], the interconnection (G, τ ∆) is well-posed, (ii) for τ ∈ [0; 1], the IQC defined by Π is satisfied by τ ∆, (iii) there exists ε > 0 such that  ∗   G(jω) G(jω)  ≤ −εI,  Π(jω)   I I

∀ω ∈ R.

(2.65)

Then, the feedback interconnection of G(s) and ∆ is stable.

Note that the considered feedback interconnection uses positive feedback only.

2.5.1 IQCs and Other Specifications Many specifications presented in Section 2.4 can be cast as IQCs. In particular, we can find a multiplier for all specifications which can be formulated as AREs. For example, for the prominent condition that H∞ norm is less than one (small gain theorem), the IQC multiplier is given by   I 0 , Π= 0 −I and



Π=

0

I

I

0



,

defines a valid multiplier for passivity.

2.5 Integral Quadratic Constraints

41

2.5.2 Mixed Uncertainties For multiple, mixed uncertainties with different descriptions, e.g., LTI and time-varying, the individual multipliers Πk can be combined into a single multiplier. The IQC can be then used to verify stability with respect to both uncertainties occurring simultaneously. A typical example for a system with mixed uncertainties is a system which has a saturation actuator nonlinearity and where the plant is modeled using a multiplicative output uncertainty. Let a mixed uncertainty ∆ be given as,   ∆1 0 , ∆= 0 ∆2

(2.66)

with associated IQC multipliers Π1 and Π2 , which characterize the uncertainties ∆1 and ∆2 . Then the overall IQC multiplier Π is given as the chess board like block (transfer) matrix   Π1(1,1) 0 Π1(1,2) 0   0 Π2(1,1) 0 Π2(1,2)    (2.67) Π= .   Π∗ 0 Π 0 1 (2,2)   1(1,2) ∗ Π2(2,2) 0 Π2(1,2) 0

2.5.3 Multiple IQCs As mentioned previously, IQC multipliers are not unique. In fact, sometimes there are not just IQCs with different parameters, but there are IQCs with fundamental differences. In order to reduce the conservatism associated with the IQC uncertainty description, convex parametrizations have been proposed [Jönsson 2001]. If ∆ ∈ IQC(Πk ), k = 1, . . . , n, P then the convex combination of multipliers satisfies ∆ ∈ IQC( nk=1 λk Πk ), where λk ≥ 0. In the PSA a dual and most times more practical approach is to utilize different IQCs by iteratively mapping the individual IQCs. A good approximation of the uncertainty set, e.g., using numerical LMI optimization, alleviates the increased conservativeness which results from the reduced optimization variables used in the IQC stability test.

42

3

Mapping Equations

Mapping Equations

This chapter presents the mapping equations for control specifications considered in Chapter 2. In particular we will present mapping equations based on algebraic Riccati and Lyapunov equations. This approach is then extended in Section 3.4 to the uniform IQC framework which allows to cover a large set of specifications and to broaden the system classes and uncertainty descriptions. Before we consider the new results for mapping additional specifications, we briefly present some known results about eigenvalue based mapping equations.

3.1 Eigenvalue Mapping Equations The roots of the parameter space approach stem from robust stability and have been extended to eigenvalue specifications [Ackermann 1980]. A review of eigenvalue based mapping equations is given in the following section. Many important properties of control systems are characterized by eigenvalue specifications. This section briefly reviews the associated mapping equations. This serves two purposes. First the mapping equations presented in this chapter will be compared to the well-known equations for eigenvalue specifications. Second, as discussed in Section 2.4.7 and 2.4.9, some norm specifications lead to eigenvalue problems. Furthermore, it will be shown that mapping several specifications considered in Section 2.4 involves a stability condition, for example the H∞ norm. Generally, the PSA allows to separately map different specifications. It seems to be always advantageous to map eigenvalue specifications, especially since this can be done very efficiently. In that sense the results in this thesis do not replace but extend the mapping of eigenvalue specifications. Consider the characteristic polynomial pc (s) of a control system (SISO or MIMO). Mathematically pc (s) is calculated as the determinant of the matrix [sI − A], where A is the closed loop system matrix, so that the roots of pc (s) coincide with the eigenvalues of A. Without loss of generality, let the characteristic equation of a parametric system with n states and coefficients ai be given as pc (s, q) =

n X i=0

ai (q)si .

(3.1)

43

3.1 Eigenvalue Mapping Equations

As shown in Ackermann et al. [1991], many control system relevant specifications can be expressed by the condition, that the eigenvalues lie within an eigenvalue region Γ. Any root s = sr of pc (s) satisfies pc (sr ) = 0. Thus, if we are interested in roots lying on the boundary ∂Γ of a region Γ, e.g., the left half plane (LHP) for asymptotic stability, we have to check if pc (s) becomes zero for any s lying on ∂Γ. A simple condition is that both the real and imaginary part of pc (s) equal zero. Therefore mapping equations for eigenvalue specifications are given by e1 (q, α) = Re pc (s = s(α), q) = 0,

(3.2a)

e2 (q, α) = Im pc (s = s(α), q) = 0,

(3.2b)

where s(α) is an explicit parametrization of the boundary ∂Γ with the running parameter α. Some parametrizations for common specifications are given in Table 3.1. The parametrization of a Γ boundary influences the order of mapping equations for an actual system. In turn, this order determines the required complexity when these mapping equations are solved. Thus we will discuss the order of mapping equations for Γ specifications. All of the parametrizations given in Table 3.1 are either affine in α or contain fractional terms that are quadratic in α. The resulting mapping equations with s = s(α) are algebraic equations in α of order n for affine dependency, and 2n for parametrizations with quadratic terms. Both, the number of states present in the control system and the parametrization of the given specification determine the degree of the polynomial pc (s) in the variable α. Table 3.1: Parametrization of Γ boundaries Hurwitz stability:

Re s < 0

s = jα

Real part limitation:

Re s < σ

s = σ + jα ζ s = α + j 1−ζ 2α

Damping: ζ Absolute value (circle):

2

2α 1−α s = r( 1+α 2 + j 1+α2 )

|s| < r

Parabola: Re s + aIm2 s = 0 Ellipses:

1 Re2 s a2

Hyperbola:

1 Re2 s a2

− +

1 Im2 s b2

=1

1 Im2 s b2

=1

s = −aα2 + jα 2

2α s = −a 1−α + jb 1+α 2 1+α2 2

1+α 2α s = −a 1−α 2 + jb 1−α2

The preceding paragraph discussed the complexity of the mapping equations with respect to the parametrization variable α. We will now focus on the parameters q. The parametric dependence of e1 (q, α) and e2 (q, α) on the parameters q only depends on the way those parameters enter into the coefficients ai (q) of pc (s). Thus, if these coefficients are affine

44

Mapping Equations

in q, the resulting mapping equations will be affine in q. This affine dependence is of great impact, since for this case the mapping equations can be easily solved for two parameters. For the synthesis of robust controllers the affine dependence of pc (s) on two controller parameters can be enforced by choosing an appropriate controller structure for SISO systems. However, for MIMO systems additional conditions have to hold. If two parameters are affinely present only in a single row or column of A, we cannot get any terms resulting from products of parameters present in A, since pc (s) is calculated as det[sI − A]. For MIMO systems with static-gain feedback, this is the case, if two parameters related to the gains of the same input or output are considered. The mapping equation (3.2) determines critical parameters for general complex eigenvalues s. The critical parameters obtained by (3.2) are thus called complex root boundary (CRB). Furthermore two special cases exist. These are the so-called real root boundary (RRB) and infinite root boundary (IRB). The first corresponds to roots being purely real which can be mapped by using (3.2a) solely. The latter is characterized by roots going through infinity. The characteristic polynomial pc (s) has a degree drop for an IRB, i.e., one or more leading coefficients vanish. The RRB and IRB conditions will mathematically reappear for MIMO specifications, although their interpretation is less intuitive there. The mathematical theory required for the mapping equations based on Riccati equations is more involved. We will present the relevant properties in the next section. See Appendix A for more details.

3.2 Algebraic Riccati Equations This section gives an overview of algebraic Riccati equations (AREs). The general ARE for the unknown matrix X is given by XRX − XP − P ∗ X − Q = 0 ,

(3.3)

where P , R, and Q are given quadratic complex matrices of dimension n with Q and R Hermitian, i.e., Q = Q∗ , R = R∗ . Although in most applications in control theory P , R, and Q will be real, the results will be given for complex matrices where possible. An ARE is a matrix equation that is quadratic in an unknown Hermitian matrix X. It can be seen as a matrix extension to the well-known scalar quadratic equation ax2 +bx+c = 0 that obviously has two, not necessarily real solutions. An ARE has in general many solutions. Real symmetric solutions, and especially the maximal solution, play a crucial role in the classical continuous time quadratic control

3.2 Algebraic Riccati Equations

45

problems [Kwakernaak and Sivan 1972]. Numerical algorithms for finding real symmetric solutions of the ARE have been developed (see, e.g., [Laub 1979]). Many important problems in dynamics and control of systems can be formulated as AREs [Boyd et al. 1994], [Zhou et al. 1996]. The importance of Riccati equations and the connection between frequency domain inequalities, e.g., ||G(s)||∞ ≤ 1, has been pointed out by Willems [1971]. Later on it was shown, that the famous and long sought solution to the state-space H∞ problem can be found using AREs [Petersen 1987]. In general, the symbolic solution of AREs is not possible due to the rich solution structure. Despite that, a successive symbolic elimination of variables using Gröbner bases is considered in Forsman and Eriksson [1993]. Although this symbolic elimination might reveal some insight into the structure and parameter dependency of the solution set, the explicit solution is obtainable only for degenerate cases. The authors of the latter report also conclude, that the computational complexity of the required symbolic computations can be quite large. For R = 0 the general ARE reduces to an affine matrix equation in X. These so-called Lyapunov equations have proven to be very useful in analyzing stability and controllability [Gajić and Qureshi 1995], while the design of control systems usually involves AREs. Since an ARE is in general not explicitly solvable, there is no direct way to obtain mapping equations from AREs. These mapping equations will be derived using some special properties of AREs. Associated with the general ARE (3.3) is a 2n × 2n Hamiltonian matrix:   −P R . H :=  ∗ Q P

(3.4)

The matrix H in (3.4) can be used to obtain the solutions to the equation (3.3), see Theorem A.1 for a constructive method. For the PSA these solutions are not relevant. This is equivalent to the fact that the PSA does not compute actual eigenvalues when mapping Γ specifications. Nevertheless we will use a particular property of (3.4). Namely the set of all eigenvalues of H is symmetric about the imaginary axis. To see that, introduce   0 −I . J :=  I 0

It follows that J −1 HJ = −JHJ = −H ∗ . Thus H and −H ∗ are similar and λ is an ¯ is. eigenvalue of H if and only if −λ The following well-known theorem [Zhou et al. 1996] provides an important link between solutions of AREs and the Hamiltonian matrix H.

46

Mapping Equations

Theorem 3.1 Stabilizing solutions Suppose that R ≥ 0, Q = Q∗ , (P, R) is stabilizable, and there is a Hermitian solution of (3.3). Then for the maximal Hermitian solution X + of (3.3), P − RX + is stable, if and only if the Hamiltonian matrix H from (3.4) has no eigenvalues on the imaginary axis. Note: A Hermitian solution X + (resp. X − ) of (3.3) is called maximal (resp. minimal ) if X − ≤ X ≤ X + for all X satisfying (3.3), where X1 ≤ X2 means that X2 − X1 is non-negative definite. A proof of Theorem 3.1 is given in the Appendix in Section A.2. Theorem 3.1 shows that the non-existence of pure imaginary eigenvalues is a necessary and sufficient condition that the ARE (3.3) has a maximal, stabilizing, Hermitian solution. Since we have seen in Chapter 2 that the adherence of many control specifications is equivalent to the existence of a maximal, stabilizing, Hermitian solution of an ARE, we can test this adherence by checking if the associated Hamiltonian matrix H has no pure imaginary eigenvalues. The PSA deals with uncertain parameters q. The purpose of the next subsection is to extend the previous results for invariant matrices to matrices with uncertain parameters.

3.2.1 Continuous and Analytic Dependence So far we considered AREs with constant matrices. Suppose now that the matrices P, Q, and R are analytic functions of a real parameter q ∈ R, i.e., P = P (q), Q = Q(q), and R = R(q). We are thus concerned with the parametric ARE X(q)R(q)X(q) − X(q)P (q) − P (q)∗ X(q) − Q(q) = 0 ,

(3.5)

and associated Hamiltonian matrix H(q) defined analogously to (3.4). Before we turn to the analytic dependence of maximal, stabilizing solutions and the important equivalence of the eigenvalue properties of the Hamiltonian matrix H similar to Theorem 3.1, we study the continuity of all maximal solutions of (3.5) with respect to q. If an ARE has Hermitian solutions, then there is a maximal Hermitian solution X + for which Λ(P − RX + ) lies in the closed left half-plane. See Theorem A.2 for a rigorous statement of this fact. The behavior of X + as a function of P, Q, and R will be characterized subsequently.

3.2 Algebraic Riccati Equations

47

Lemma 3.2 (Lancaster,Rodman) The maximal Hermitian solution X + of (3.5) is a continuous function of the matrices (P, Q, R) ∈ Cn, n . Although Lemma 3.2 is concerned with the continuity of maximal Hermitian solutions, these solutions are in general not differentiable. That is, the maximal parametric solution X + = X + (q) is not necessarily an analytic function of the real parameter q. The analyticity of maximal Hermitian solutions is ensured by the invariance of the number of pure imaginary eigenvalues of H. The following theorem provides the link between control system specifications and the Hamiltonian matrix H(q). This theorem forms the corner stone of ARE based mapping equations. It appeared partly in a mathematical context in [Ran and Rodman 1988]. See [Lancaster and Rodman 1995] for a detailed exposition. Theorem 3.3 (Lancaster,Rodman) Let P = P (q), Q = Q(q), and R = R(q) be analytic, complex n × n matrix functions of q on a real interval [q − ; q + ], with R(q) positive semidefinite Hermitian, Q(q) Hermitian, and (P (q), R(q)) stabilizable for every q ∈ [q − ; q + ]. Assume that for all q ∈ [q − ; q + ], the Riccati equation (3.5) has a Hermitian solution. Further assume that the number of pure imaginary or zero eigenvalues of   −P (q) R(q)  H(q) :=  (3.6) Q(q) P (q)∗

is constant. Then the maximal solution X + (q) of (3.5) is an analytic function of the parameter q ∈ [q − ; q + ]. Conversely, if X + (q) is an analytic function of q ∈ [q − ; q + ], then the number of pure imaginary or zero eigenvalues of H(q) is constant.

Proof: The proof of this theorem is rather involved and provides no insight for the successful application, i.e., the derivation of the mapping equations. It is therefore omitted for brevity. See [Lancaster and Rodman 1995] for a sketch of the proof.

48

Mapping Equations

Theorem 3.3 is applicable to AREs, where the matrices P, Q and R are real. We get necessarily real maximal solutions X + for this control theory relevant case. The previous results can be generalized to the case when P (q), Q(q) and R(q) are analytic functions of several real variables q = (q1 , . . . , qp ) ∈ Q, where Q is an open connected set in Rp .

3.3 Mapping Specifications into Parameter Space In this section we present the mapping equations for the specifications given in Section 2.4 for systems with uncertain parameters. For eigenvalue specifications the boundary of the desired region in the eigenvalue plane is mapped into a parameter plane by the characteristic polynomial. Using the real and imaginary part of the characteristic polynomial, two mapping equations are obtained which depend on a generalized frequency α and the uncertain parameters q. The mapping equations presented in this section will have a similar structure. Actually, all mapping equations presented in this thesis will consist of pe individual equations with uncertain parameters q ∈ Rp and pe − 1 auxiliary variables. Usually pe is either 1 or 2. Thus if the vector of uncertain parameters q is of dimension p = 1, we get either a single equation or a regular system of equations that can be solved for q. This allows to explicitly determine the critical parameter values of q for which the specification is marginally fulfilled. Related to this case is the dual problem of direct performance evaluation considered in Section 3.8. For p > 1 we get an underdetermined set of equations. The case p = 2 is not only important for the easily visualized plots, but also because it admits tractable solution algorithms considered in Chapter 4. For p > 2 gridding of p − 2 parameters is necessary to determine the resulting solution sets.

3.3.1 ARE Based Mapping While we provided the definition of H∞ norm, dissipativity and complex stability radius specifications, we pointed out that all of these specifications are equivalent to the existence of a maximal, Hermitian solution of an ARE. Using Theorem 3.1 we can in turn formulate the adherence of the given specifications as the non-existence of pure imaginary eigenvalues of an associated Hamiltonian matrix. This is a well-known fact used in standard numerical algorithms [Boyd et al. 1989].

49

3.3 Mapping Specifications into Parameter Space 2

1.5

1

0.5

ℑ{λ}

0

−0.5

−1

−1.5

−2 −3

−2

−1

0

ℜ{λ}

1

2

3

Figure 3.1: Appearance of pure imaginary eigenvalues

Consider now the uncertain parameter case. Using Theorem 3.3 we can extend this equivalence to systems with analytic dependence on uncertain parameters. Given a specific parameter q ∗ ∈ Rp for which a maximal, Hermitian solution X + (q ∗ ) exists, we know from Theorem 3.1 that the Hamiltonian matrix (3.6) has no pure imaginary eigenvalues. Using Theorem 3.3 we can extend this property as long as the number of eigenvalues on the imaginary axis is constant. In other words, having found a parameter for which a specification described by an ARE holds, the same specification holds as long as the number of imaginary eigenvalues of the associated Hamiltonian matrix (3.6) is zero. Hence, the boundary of the region for which the desired specification holds is formed by parameters for which the number of pure imaginary eigenvalues of (3.6) changes. A new pair of imaginary eigenvalues of (3.6) arises, if either two complex eigenvalue pairs become a double eigenvalue pair on the imaginary axis, or if a double real pair becomes a pure imaginary pair. Note: Another possibility is a drop in the rank of H, which corresponds to eigenvalues going through infinity. The appearance of pure imaginary eigenvalues is depicted in Figure 3.1. Let us first discuss the appearance of pure imaginary eigenvalues through a double pair on the imaginary axis. The matrix H(q) has an eigenvalue at λ = jω if det [jωI − H(q)] = 0.

(3.7a)

In general a polynomial f (ω) has a double root not only if f (ω) = 0, but the derivative of f (ω) with respect to the argument ω has to vanish also ∂f (ω) = 0. ∂ω

50

Mapping Equations

Since the characteristic polynomial of H(q) (3.7a) is a polynomial, we obtain ∂ det [jωI − H(q)] = 0, (3.7b) ∂ω as the second condition for a double eigenvalue at λ = jω. Equations (3.7a) and (3.7b) define two polynomial equations that can be used to map a given specification into parameter space. A necessary condition for a real eigenvalue pair to become a pure imaginary pair through parameter changes is det [jωI − H(q)] ω=0 = det H(q) = 0.

(3.8)

Additionally the opposite end of the imaginary axis has to be considered det [jωI − H(q)] ω=∞ .

(3.9)

Equation (3.9) is just the coefficient of the term with the highest degree in ω of the determinant det[jωI − H(q)]. Equation (3.8) is not sufficient, since it determines all parameters for which (3.6) has a pair of eigenvalues at the origin. This includes real pairs that are just interchanging on the real axis. To get sufficiency we have to check all parameters satisfying (3.8), if there are only real eigenvalues. Figure 3.2 shows the two possible eigenvalue paths that are determined by (3.8). The same check has to be performed for solutions of (3.9), because here eigenvalues can interchange at infinity and (3.9) is not sufficient as well.

ℑ{λ}

2

2

1.5

1.5

1

1

0.5

0.5

ℑ{λ}

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 −3

−2

−1

0

ℜ{λ}

1

2

3

−2 −3

−2

−1

0

ℜ{λ}

1

2

3

Figure 3.2: Possible eigenvalue paths for real root condition

From the eigenvalue spectrum of a Hamiltonian matrix follows that (3.7a) is purely real and does not contain any imaginary terms. This fact is interesting in the following comparison and we will revisit this property in Section 3.5, where we evaluate the complexity of the mapping equations.

3.3 Mapping Specifications into Parameter Space

51

Furthermore, using this eigenvalue property the following simplifications can be made for (3.7a) and (3.7b). Equation (3.7b) contains the factor ω, which can be neglected since the solution ω = 0 is independently mapped using (3.8). After dropping this factor, both equations contain only terms with even powers of ω. The mapping equations (3.7), (3.8) and (3.9) have a similar structure like the well-known equations for pole location specifications presented in Section 3.1, where (3.7) can be interpreted as the condition for the CRB, (3.8) as the RRB equivalent, and (3.9) as the IRB condition. Actually, using the above approach and Lyapunov’s famous matrix ˙ equation for Hurwitz stability of an autonomous system x(t) = P x(t) [Boyd et al. 1994], P T X + XP = −Q, Q = QT > 0, leads to mapping equations for the CRB, RRB and IRB, which have the same solution set as the equations (3.2) derived from the characteristic polynomial. The only difference is that the ARE based mapping equations for Hurwitz stability are factorizable containing the same term squared. This can be easily seen from the Hamiltonian matrix H (3.4), where P appears twice on the diagonal with R = 0 .

3.3.2 H∞ Norm Mapping Equations We will use the H∞ norm as an example for ARE based mapping equations. Many important properties for this particular example are equivalent for other specifications discussed in Chapter 2, for example passivity. Using Theorem 3.1 and the ARE (2.32) of the bounded real lemma, we get the following well-known theorem, see [Boyd et al. 1989], where this result is used to derive a numerical bisection algorithm to compute the H∞ norm. Theorem 3.4 Let A be stable, γ > σ ¯ (D) and define the matrices Sr = (D ∗ D − γ 2 I) and Sl = (DD ∗ − γ 2 I). Then ||G||∞ ≥ γ if and only if   A − BSr−1 D T C −γBSr−1 B T  (3.10) Hγ =  T T −1 T T −1 −A + C DSr B γC Sl C has pure imaginary eigenvalues, i.e., at least one.

52

Mapping Equations

This theorem provides the Hamiltonian matrix needed in the general mapping equations (3.7), (3.8) and (3.9). In order to get the mapping equations, we have to compute the determinant of the partitioned matrix det[jωI − H]. For a 2 × 2 block matrix   M11 M12  M = M21 M22 the determinant is

−1 det M = det M11 det M22 − M21 M11 M12 ,

with M11 nonsingular. For ease of presentation let D = 0 , then det[jωI − Hγ ] can be written as det [jωI − Hγ ] = det [jωI − A] det jωI + AT + γ 2 C T C(jωI − A)−1 BB T . (3.11)

The first factor of the latter equation is just the Hurwitz stability condition. This is in line with the fact that the H∞ norm requires the transfer function G(s) being stable. When we directly compute det[jωI − Hγ ] the stability related factor det[jωI − A] is canceled. Hence, stability has to be mapped separately. Without this additional condition we are actually mapping an L∞ norm condition. Concluding, apart from the H∞ norm condition det[jωI − Hγ ], the Hurwitz stability condition has to be mapped additionally using the mapping equations for Hurwitz stability described in Section 3.1. The next point to notice about the H∞ norm mapping is that the static feedthrough condition σ ¯ (D) < γ in Theorem 3.4 is implicitly mapped by (3.8) and (3.9). This can be shown using a frequency domain formulation. See Section 3.7, which considers alternative derivation of mapping equations for some specifications. Note that the mapping equations for the H∞ norm are not just characterizing the parameters for which ||G||∞ = γ. Actually, all parameters for which G(s, q) has a singular value σ(G) = γ are determined. Thus, we might get boundaries in the parameter space for which the i-th biggest singular value σi has the specified value γ. This is similar to eigenvalue specifications, where we get boundaries for each eigenvalue crossing of the Γboundary. Analogously to the Γ-case, the resulting regions in the parameter space have to be checked, if the specifications are fulfilled or violated (possibly multiple times).

If the critical gain condition on the static feedthrough matrix σ ¯ (D) = γ is fulfilled, e.g., if D = γI, then Sl and Sr are singular. A remedy is to either consider a different performance level γ or to use a high-low frequency transformation and map the −1 ˜ ˜ ˜ condition ||G(s)|| ∞ = γ, where G(s) = G(s ), since ||G||∞ = ||G||∞ . A state-space ˜ representation of G(s), which retains stability if A is stable, is given as   −1 −1 A −A B ∼ ˜ . G(s) (3.12) = CA−1 D − CA−1 B


53

To make the presented theory more clear we will consider a very simple exemplary system in Example 3.2 on page 71.

3.3.3 Passivity Mapping Equations The Hamiltonian matrix for dissipativity follows from the ARE (2.41) as   ∗ A + BSC −BSB , Hη =  C ∗ SC −(A + BSC)∗

(3.13)

where S = (2ηI − D − D ∗ )−1 . The specific mapping equations are then easily formed using (3.7), (3.8), and (3.9). In Section 2.4.2 the ARE for passivity was only valid for D 6= 0 . The following derivation therefore relies on the more general dissipativity to obtain mapping equations for the not uncommon case D = 0 . For D = 0 , S reduces to S = 

Hη = 

A+

1 BC 2η

1 C ∗C 2η

1 I, 2η

and the Hamiltonian matrix Hη becomes

1 − 2η BB ∗

−(A +

1 BC)∗ 2η



,

(3.14)

The mapping equations for passivity can be obtained by evaluating the limit of the general equations with H = Hη where η goes to zero, after extracting factors which only depend on η. Since in general, det[jωI − Hη ] is either fractionless with respect to η or a fraction with denominator η, we get the relevant factor by simply evaluating the numerator. More specifically the two mapping equations for the CRB condition are given as, lim num det [jωI − Hη ] = 0,

(3.15)

∂ lim num det [jωI − Hη ] = 0. ∂ω η→0

(3.16)

η→0

and

The RRB condition is given by lim num det Hη = 0,

η→0

(3.17)

and the IRB follows as the leading coefficient of (3.15) with respect to ω lcω lim num det [jωI − Hη ] = 0. η→0

(3.18)

54

Mapping Equations

For all four mapping equations (3.15)–(3.18) the limit has to be taken after the determinant is calculated. A simple passivity example is presented in Example 5.6. A dual approach to obtain mapping equations for passivity would be to use the results from Section 2.4.3 and transform the passivity or positive realness condition into an H∞ norm problem. Not surprisingly, we run into exactly the same singularity for D = 0 , which arises if we use the passivity ARE (2.38) directly. Using Theorem 2.5 and the transformation (2.44) the ˆ condition that G(s) = C(sI − A)−1 B is positive real can be formulated as ||G(s)|| ∞ < 1, where   A − BC 2B ∼ ˆ . G(s) = −C I

The resulting matrices Sr and Sl for the associated Hamiltonian matrix Hγ in Theorem 3.4 thus become zero.

3.3.4 Lyapunov Based Mapping We now turn to special variants of AREs. If the quadratic term R of the general ARE (3.3) vanishes, a so-called Lyapunov equation is obtained: P ∗ X + XP + Q = 0 ,

(3.19)

where Q = Q∗ . Apply the Kronecker stacking operator (see Section A.1) to get (P ∗ ⊕ P ) vec(X) = − vec(Q).

(3.20)

¯ j (P ) 6= 0, ∀ i, j. This equation has a unique solution X if and only if λi (P ) + λ In control theory, equation (3.19) is commonly used to test stability, controllability, and observability. While stability is not of interest here, because it can be easily handled by the classical PSA, Chapter 2 presented several specifications, e.g., the H2 norm, that were formulated with observability and controllability Gramians Wobs and Wcon , and the associated Lyapunov equations A∗ Wobs + Wobs A + C ∗ C = 0 ,

(3.21a)

AWcon + Wcon A∗ + BB ∗ = 0 .

(3.21b)

We shall first consider the parametric solution of a Lyapunov equation. Equation (3.19) is an affine equation in the unknown matrix elements xij of X = X T . Thus (3.19) constitutes a system of 21 n(n + 1) linear equations, where n is the dimension of the quadratic


55

matrix P ∈ Cn, n . This system can be readily solved for X(q) using a computer algebra system for any parametric dependency of P (q) and Q(q). Note that the symmetry of (3.19) should be used to minimize the computational effort, instead of using the full system (3.20). In comparing (3.19) and (3.21), we see that the symbolic observability and controllability Gramians Wobs (q) and Wcon (q) are obtained as the solution of a system of 21 n(n + 1) linear equations. We will now present the mapping equation for the H2 norm. Using equation (2.56), a specification on the H2 norm like ||G(s, q)||2 ≤ γ can be mapped into the parameter space. In order to use (2.56) as a mapping equation, we need the parameter dependent observability Gramian Wobs (q). This matrix can be obtained as described above. Substituting this parametric solution Wobs (q) into (2.56), the parametric mapping equation is obtained as ||G(s)||22 = trace B(q)T Wobs (q)B(q) = γ 2 ,

(3.22)

where γ 2 specifies the desired performance level.

The dual output controllable formulation is given by ||G(s)||22 = trace [C(q)Wcon (q)C(q)] = γ 2 .

(3.23)

Equation (3.22) is an implicit equation in the uncertain parameters q. Since the desired or achievable performance level is not known a priori, it is recommended to vary γ and determine the set of parameters P2,good for which ||G(s)||22 = γ 2 holds for multiple values γ. Also, a gray-tone or color coding of the different sets P2,good (γ) is useful. The visualization of parameter space results will be considered in greater detail in Section 4.7. The H2 norm mapping equation (3.22) is a single equation, that depends only on the system parameters q. This is in line with the fact that the general definition of the H2 norm includes an integral over all frequencies. Thus there is no auxiliary variable α or frequency ω in the mapping equations. This fact makes (3.22) useful on its own, as opposed to being solely used for mapping H2 norm specifications into a parameter plane, especially to analyze and design control systems for more than two parameters. Furthermore (3.22) will be used to directly evaluate H2 norms in Section 3.8. If the parameters q enter in a polynomial fashion into A(q), B(q), C(q), the mapping equation (3.22) is a polynomial equation. There may be some special cases, when (3.22) is affine in one or more parameters, but in general this equation is polynomial in q, even if A(q), B(q), and C(q) are affine in q.

56

Mapping Equations

In general there is no difference in using either (3.22) or (3.23). Nevertheless depending on the complexity of B(q) and C(q) one or the other mapping equation might be easier to solve in isolated cases.

3.3.5 Maximal Eigenvalue Based Mapping We conclude this section with the remaining specifications presented in Chapter 2 for which no mapping equations have been derived yet. Both the Hankel and the generalized H2 norm can be expressed as a function of a parametric matrix. These associated matrices can be computed using the solution of parametric Lyapunov equations. To get mapping equations for the Hankel and generalized H2 norms, we apply standard results for mapping eigenvalue specifications. Namely a condition λmax (M ) = γ, where M is a non-negative matrix leads to the mapping equation det [γI − M ] = 0. Accordingly the condition dmax (M ) = γ, M ≥ 0 leads to the system of mapping equations mii = γ,

i = 1, . . . , n.

3.4 IQC Parameter Space Mapping Our goal is to use the unifying framework of IQCs to find mapping equations which allow to incorporate an even larger set of specifications. We will show that mapping IQCs extends the parameter space beyond the ARE mapping equations presented in the previous section. Having found the mapping equations for general IQC specifications enables us to consider specifications from the input-output theory, absolute stability theory and the robust control field. Specifications from all these research fields can be used in conjunction with the parameter space approach. Using the same mathematical formulation the same computational methods can be used for these different specifications. IQCs have been introduced in Chapter 2. A brief treatment of IQCs was given in Section 2.5, where the basic stability condition (iii) of Theorem 2.9 is given as  ∗   G(jω) G(jω)  ≤ −εI, ∀ω ∈ R.  Π(jω)   (3.24) I I In the current section we state the main result, the mapping equations for IQCs. We will first consider fixed, frequency-independent multipliers. Section 3.4.4 will then show

57

3.4 IQC Parameter Space Mapping

how to map specifications based on IQCs with frequency-dependent multipliers Π(jω) into parameter space. As an example a nonlinear system is analyzed in Section 3.4.4 using a frequency-dependent multiplier. For many uncertainties not just a single multiplier but sets of multipliers exist to characterize the uncertainty structure. Often these sets can be described by parametrized multipliers. We will show how to utilize these additional degrees of freedom in order to minimize the conservativeness inherent in mapping only a single multiplier in Section 3.4.5. An exemplary problem that utilizes LMI optimization is given in Example 3.1, Section 3.4.5.

3.4.1 Uncertain Parameter Systems l×m Consider an uncertain LTI system G(s, q) ∈ RH∞ interconnected to a bounded causal operator ∆ as shown in Figure 2.10. The parameters q ∈ Rp are uncertain but constant parameters with possibly known range. The operator ∆ might represent various types of uncertainties, including constant uncertain parameters not contained in q. Let Π be a constant multiplier that characterizes the uncertainty ∆ with partition   Π11 Π12 . Π= (3.25) ∗ Π12 Π22

Conditions (i) and (ii) of Theorem 2.9 are parameter-independent. Hence, the parameterdependent stability condition (3.24), which can be written as G(jω)∗ Π11 G(jω) + Π22 + Π∗12 G(jω) + G(jω)∗ Π12 ≤ −εI,

∀ω ∈ R,

(3.26)

has to be fulfilled by all parameters in Pgood . The avenue to determine mapping equations will be the application of the KalmanYakubovich-Popov (KYP) lemma. Previously known results on how to map specifications expressible as AREs are then easily applied to IQC specifications.

3.4.2 Kalman-Yakubovich-Popov Lemma In this section, the well known Kalman-Yakubovich-Popov (KYP) lemma is discussed. The KYP lemma relates very different mathematical descriptions of control theoretical properties to each other. In particular, it shows close connections between frequencydependent inequalities, AREs and LMIs. Nowadays a very popular application of the KYP lemma is to derive LMIs for frequency domain inequalities, since efficient numerical algorithms for the solution of LMI problems exist.

58

Mapping Equations

Theorem 3.5 (Kalman, Yakubovich, Popov)

Let (A, B) be a given pair of matrices that is stabilizable and A has no eigenvalues on the imaginary axis. Then the following statements are equivalent. (i) R > 0 and the ARE Q + XA + AT X − (XB + S)R−1 (XB + S)T = 0

(3.27)

has a stabilizing solution X = X T , (ii) the LMI with unknown X  

T

XA + A X XB T

B X

0

has a solution X = X T ,





+

Q

S

T

R

S



>0

(3.28)

(iii) for a spectral factorization the condition  

−1

(jωI − A) B I

∗   

Q

S

ST R

 

−1

(jωI − A) B I



>0

(3.29)

holds ∀ω ∈ [0; ∞), (iv) there is a solution to the general LQR problem min J = u

Z

∞

(x(t)T Qx(t) + 2x(t)T Su(t) + u(t)T Ru(t)) dt

(3.30)

0

˙ with x(t) = Ax(t) + Bu(t), x(0) = x0 and limt→∞ x(t) = 0. (v) R > 0 and the Hamiltonian matrix 

H =

−1

T

BR B

−1

T

T

A − BR S Q − SR S

−1

T

−1

−A + SR B

has no eigenvalues on the imaginary axis.

T

 

(3.31)

Proof. See, for example, [Rantzer 1996] or [Willems 1971]. In order to apply the KYP lemma we will need the following remark.

59


Remark 3.1 The spectral factorization condition (3.29) of the KYP lemma can be extended to the case where a transfer function G(s) = C(sI − A)−1 B + D appears in the outer factors similar to the IQC condition (2.65):  ∗  ∗      Π11 Π12 (jωI − A)−1 B G(jω) G(jω) (jωI − A)−1 B  , (3.32) =  M     ∗ I I I I Π12 Π22

where



M =

Q

S

ST R





=

C D 0

I

T   

Π11 Π12 Π∗12

Π22

 

C D 0

I



.

Remark 3.2 The minimum of J in (3.30) equals x0 X + x0 , where X + is the largest symmetric solution of the ARE (3.27). The optimal input u is then given by u(t) = −Kx(t), where K = R−1 (XB + S).

3.4.3 IQC Mapping Equations We are now ready to derive the main result, the mapping equations for IQC specifications. Let G(s, q) have a state-space realization A(q), B(q), C(q), D(q), i.e., G(s, q) = C(q)(sI − A(q))−1 B(q) + D(q). We will not express the parametric dependence of matrices in the remainder for notational convenience. Using Remark 3.1 the basic IQC condition (2.65) in Theorem 2.9 for a constant multiplier Π with partition (3.25) can be transformed into the condition  ∗   −1 −1 (jωI − A) B (jωI − A) B   M   ≤ −εI, (3.33) I I

60

Mapping Equations

where the multiplier is transformed into   C T Π11 C C T (Π12 +Π11 D) . M = T ∗ T ∗ T (D Π11 +Π12 )C D Π11 D+Π12 D+D Π12 +Π22 Since we are interested in mapping equations describing the boundaries of a parameter set Pgood , we consider marginal satisfaction of (3.33), i.e., ε = 0. Now, use statements (3.29) and (3.31) of the KYP lemma to get the equivalent condition that the Hamiltonian matrix ∗      T C (Π12 + Π11 D) B A 0 ˜ −1   , Π − (3.34) H= 22 T T T −B C (Π12 + Π11 D) C Π11 C −A ˜ 22 = Π22 + D T Π12 + Π∗ D + D T Π11 D has no eigenvalues on the imaginary axis. with Π 12

We have now formulated the adherence of a given IQC specification as the non-existence of pure imaginary eigenvalues of an associated Hamiltonian matrix. Using Theorem 3.3 and the same arguments and methods as in Section 3.3.1, we can substitute (3.34) into the ARE based mapping equations (3.7a), (3.8) and (3.9) to get mapping equations for IQC based specifications. Thus the KYP lemma established the important relationship which allows to derive mapping equations for IQC conditions. The consequence is that the statements about the properties of ARE based mapping equations can be directly transferred to the IQC mapping equations, e.g., the non-sufficiency of (3.8). We extend the previous IQC results by considering frequency-dependent multipliers in the next section.

3.4.4 Frequency-Dependent Multipliers Consider the case when the multiplier Π is frequency-dependent, i.e., Π = Π(jω). The common frequency domain criterion used in conjunction with IQCs for frequencydependent multipliers is  ∗   G(jω) G(jω)   Π(jω)   < 0, ∀ω ∈ [0; ∞), (3.35) I I l×m l+m×l+m where G(jω) ∈ RH∞ and Π(jω) ∈ RH∞ .

A particular example of a frequency-dependent multiplier is the strong result by Zames and Falb [1968] for SISO nonlinearities1 . Put into the IQC framework, an odd nonlinear 1

The results in [Zames and Falb 1968] can be only applied to MIMO nonlinearities using additional restrictions, see [Safonov and Kulkarni 2000].

61

3.4 IQC Parameter Space Mapping operator, e.g., saturation, satisfies the IQC defined by 

Π(jω) = 

0

1 + L(jω)

1 + L(jω)

∗

−2 − L(jω) − L(jω)

∗



,

where L(s) has an impulse response with L1 norm less than one. Following [Megretski and Rantzer 1997], any bounded rational multiplier Π(jω) can be factorized as Π(jω) = Ψ(jω)∗ Πs Ψ(jω),

(3.36)

where Ψ(jω) absorbs all dynamics of Π(jω) and Πs is a static matrix. Actually factorization (3.36) is known as a J-spectral factorization, which plays an important role on its own in H∞ and H2 control theory [Green et al. 1990], or as a canonical Wiener-Hopf factorization in operator theory [Bart et al. 1986]. We will now derive mapping equations for frequency-dependent multipliers by simple transformations and equivalence relations. Rewrite (3.36) as 

 ∗   Π 0 s Ψ(jω) Ψ(jω)    , Π(jω) =  I I 0 0

(3.37)

and apply a transformation similar to Remark 3.1, where Ψ(jω) has state-space representation Ψ(jω) = Cπ (jωI − Aπ )−1 Bπ + Dπ to get 

Π(jω) = 

−1

(jωI − Aπ ) Bπ I

∗  

CT Π C  π s π



(jωI CπT Πs Dπ 

DπT Πs Cπ DπT Πs Dπ {z } | Mπ

−1

− A π ) Bπ I



.

(3.38)

Next, partition the input matrix Bπ of Ψ(jω) according to the signals in the general IQC feedback loop (see Figure 2.10) as Bπ = [Bπ,v Bπ,w ], which was suggested in the derivation of a linear quadratic optimal control formulation of (3.35) in [Jönsson 2001], and use (3.38) to write condition (3.35) as 

−1

∗



−1



(jωI − Aπ ) (Bπ,v G(jω) + Bπ,w ) (jωI − Aπ ) (Bπ,v G(jω) + Bπ,w )      < 0, (3.39)  Mπ   G(jω) G(jω)     I I for all ω ∈ [0; ∞).

62

Mapping Equations

Let G(jω) = C(jωI − A)−1 B + D, then the following state-space representation for the factor to the right of Mπ in (3.39) can be deduced:







Aπ Bπ,v C Bπ,v D + Bπ,w

  0 A (jωI − A π ) (Bπ,v G(jω) + Bπ,w )        =   G(jω) 0  I     0 I C  0 0   ˜ A˜ B . =  ˜ C˜ D −1

B 0 D I

         

(3.40)

(3.41)

Using this state-space representation, condition (3.39) becomes ˜ ˜ −1 B ˜ + D) ˜ ∗ Mπ (C(jωI ˜ ˜ −1 B ˜ + D) ˜ < 0, (C(jωI − A) − A)

∀ω ∈ [0; ∞),

(3.42)

or  

˜ −1 B ˜ (jωI − A) I

   h i (jωI − A) ˜ −1 B ˜ C˜ T  < 0,    Mπ C˜ D ˜  T ˜ I D {z } | M ∗ 

∀ω ∈ [0; ∞).

(3.43)

As a result, we have obtained an IQC condition with a frequency-independent multiplier M , where     T h i ˜ Q S C , ˜ = (3.44) M =  Mπ C˜ D ˜T D ST R ˜ B) ˜ represents an augmented system composed of both the LTI system and where (A, dynamics and the multiplier dynamics. The multiplier matrices can be computed as,   CπT Πs Cπ CπT Πs Dπ,v C , (3.45) Q =  T T T T C Dπ,v Πs Cπ C Dπ,v Πs DΠ,v C   T Cπ Πs (Dπ,v D + Dπ,w ) , S =  (3.46) T C T Dπ,v Πs (Dπ,v D + Dπ,w ) h i T R = (Dπ,v D + Dπ,w ) Πs (Dπ,v D + Dπ,w ) . (3.47)

As in the frequency-independent case discussed in the previous section, we can now apply the KYP lemma in order to obtain the Hamiltonian matrix which will lead to the mapping equations. Namely, use statements (3.29) and (3.31) of the KYP lemma to get the

3.4 IQC Parameter Space Mapping equivalent condition that the Hamiltonian matrix   ˜ −1 S T ˜ −1 B ˜T A˜ − BR BR  H= −1 T T −1 ˜ T ˜ Q − SR S −A + SR B

63

(3.48)

has no eigenvalues on the imaginary axis.

Hence frequency-dependent bounded rational multipliers Π(jω) can be mapped into parameter space using basic matrix transformations and the results from the previous section.

3.4.5 LMI Optimization For a system with fixed parameters, all multipliers considered so far led to a simple stability test, which could be evaluated by computing the eigenvalues of a Hamiltonian matrix (3.31). For systems with uncertain parameters q ∈ Rp , we showed how to map an IQC specification into a parameter plane. But in general, the uniqueness of a multiplier is not given. While the main idea behind the IQC framework is to find a suitable multiplier for an uncertainty, for many uncertainties a set of possible multipliers exists. Especially for nonlinearities and time-delay systems there is an enormous list of publications involving different multipliers. See [Megretski and Rantzer 1997] for some references. Depending on the considered LTI system one or the other multiplier might be advantageous and yield less conservative results. For example, consider the following multiplier from [Jönsson 1999] for a system ˙ x(t) = (A + U ∆V )x(t), with slowly time-varying uncertainty ∆ and known rate bounds   T T Z Y − jωW . Π(jω) =  Y + jωW −X

(3.49)

(3.50)

Jönsson [1999] derives a set of LMI conditions to check stability involving real matrices W, X = X T , Y and Z = Z T . These matrices can be easily obtained solving a convex optimization problem. The result of the optimization is not only a binary stability check, but also an optimal multiplier Π(jω). There are two different possibilities to exploit the degrees of freedom in the multiplier formulation during the mapping process.

64

Mapping Equations

One approach would be to use a limited set of parameter points (q1 , q2 ) for which we obtain optimal multipliers and subsequently determine the set of good parameters Pgood for each individual multiplier. The actual overall set of uncertain parameters which fulfill the specification is then given as the union of all individual good sets. The second approach could be denoted as adaptive multiplier mapping. Hereby we obtain successive multipliers as we actually determine the boundary by moving along the boundary of the set Pgood . Thus we adaptively correct the optimal multiplier on the way as we generate the boundary by solving an underlying optimization problem. While the first approach needs to solve a limited and predefined number of optimization problems, the adaptive multiplier mapping requires a possibly large number of optimization, which is not known a priori. Nevertheless the second approach gives the actual set Pgood directly and there is no need to determine the union of individual sets. Furthermore, when the actual mapping is expensive, it might be favorable to use a single adaptive mapping run.

Example 3.1 The following example shows how the IQC results of the current section can be used to extend the parameter space approach to stability checks with respect to varying parameters. Consider the following parametric system, which fits into the setup of problem (3.49), 

˙ x(t) = 

q1 + q 2

1 + q2

−3

− 21 − q1





+

0 2 5

 

2 5

0

∆ x(t),

(3.51)

where ∆ is a diagonal matrix containing an arbitrarily fast varying parameter δ. The parameter δ is assumed to be bounded in the interval [0; 1]. Note that if we treat δ as a constant parameter, then the set of good parameters is given by δ = 0 since all stability sets for δ > 0 contain the set for δ = 0. So from the perspective of constant parameters we do not lose anything, if we allow δ to deviate from δ = 0 to values in the interval [0; 1]. We assume arbitrary fast variations of the parameter δ, therefore the matrix W , which appears in (3.50), equals zero. Following [Jönsson 1999], system (3.49) is stable with respect to ∆, if a feasible solution for the following LMI problem exists.

65


2

1

0

q

2 −1

−2

−3 −3

−2

−1

0

1

2

q

1

Figure 3.3: Stability regions using adaptive LMI optimization

Find X = X T , Y, Z = Z T , P = P T such that,

 

X ≥ 0, 

(3.52)

Z > 0,

(3.54)

Z + Y + Y T − X > 0.

(3.55)

AT P + P A + V T ZV

PU + V TY T

UT P + Y V

−X

 < 0,

(3.53)

We determine the stability boundaries in the q1 , q2 parameter plane. In the first step we calculate the maximal stability boundary using adaptive multiplier mapping. In each step optimal values for the X, Y, and Z matrices, which determine the multiplier (3.50), are computed. The resulting stability boundaries are shown in Figure 3.3 as a solid curve. The figure also shows the common real root boundary, assuming uncertain, but constant δ values, as a dotted line, and the complex root boundaries for δ = 0, and δ = 1 as dashed lines. Figure 3.4 shows the stability regions depicted as solid curves, which are obtained using optimal multipliers for the points (0, 0), (1, 0), and (−1, −0.5). For comparison the stability region obtained using adaptive multiplier mapping is shown as a dash-dotted

66

Mapping Equations

2

1

0

q

2 −1

−2

−3 −3

−2

−1

0

1

2

q

1

Figure 3.4: Stability regions using multiple fixed IQC multipliers

curve. The figure shows that a small number of reference points might suffice to get a rather accurate approximation of the true stability region. It should be noted that the LMI problem (3.52) is only a feasible problem. Although the resulting multiplier guarantees stability for the reference point, i.e. A(q1 , q2 ), the resulting size of the stability region in the (q1 , q2 ) parameter plane is not necessarily maximal. Thus treating the parameters q1 and q2 as additional uncertainty to the ∆ uncertainty and augmenting the LMI problem might lead to even larger stability regions. For q1 = 0 and q2 > 0 the true stability limit is q2 = 0.23018, and instability occurs near the origin for a switching behavior of δ. Whereas the optimal multiplier yields a guaranteed stability for q2 = 0.067, which exemplifies the conservativeness of this IQC condition. Nevertheless Figure 3.3 and Figure 3.4 show that the IQC condition gives very accurate results for stability regions, where the varying parameter does not affect the stability region. And the simplicity of the used multiplier (frequency-independent) suggests that a more complicated multiplier might even improve the shown results.

3.5 Complexity

67

3.5 Complexity This section discusses the complexity of the mapping equations derived above. In particular we will look at the order of the equations as a function of the number of states in the system and as a function of the input and output dimensions. Furthermore the complexity with respect to the uncertain parameters q is considered.

3.5.1 ARE Mapping Equations The complexity of ARE based mapping equations is determined by the Hamiltonian matrix H defined in (3.4). The corresponding Hamiltonian matrix for all ARE expressible specifications in this thesis, i.e., H∞ norm (2.32), passivity (2.38), dissipativity (2.41), circle criterion (2.48), Popov criterion (2.51) and complex structured stability radius (2.54), can be written in the following form:   A + BS1 C −BS2 B T , (3.56) H= T T C S3 C −(A + BS1 C) where Si , i = 1, . . . , 3, are specification dependent, nonsingular matrices, which depend on D. For uncertain parameters q in the plant description, all matrices A, B, C and Si can be parameter dependent.

The dimension of H depends only on the size of A ∈ Rn, n , where n equals the number of states in the system, H ∈ R2n, 2n . The dimension of H does not increase with the number of inputs and outputs. Thus complexity does not increase when we consider MIMO instead of SISO systems. The order of det[jωI − H] with respect to ω is 2n and corresponds directly to the number of states. Note that (3.7a) has only terms with even powers of ω. Thus after we eliminate the factor ω, the second, double root condition (3.7b) has order 2(n − 1) with respect to ω. The order of the mapping equations with respect to a single parameter q or qi can be determined by studying a general determinant. The determinant of an arbitrary matrix M ∈ Rn,n can be calculated as the algebraic sum of all signed elementary products from the matrix. An elementary product is given by multiplying n entries of an n × n matrix, all of which are from different rows and different columns: X det M = sgn(π)m1π1 m2π2 · · · mnπn , (3.57) π∈Sπ

where the index π in the above sum varies over all possible permutations Sπ of {1, . . . , n}. The total number of possible permutations is n!, which therefore equals the number of terms in the defining sum of determinant.

68

Mapping Equations

For n = 3 the position of the individual factors for the 3! elementary products is shown below:        ∗ ∗ ∗  ∗ ∗  ∗               . ∗ ∗ ∗ ∗  ∗ ∗        ∗ ∗ ∗ ∗ ∗ ∗ The matrix A appears twice on the diagonal of H. Because an entry aij appears both at hij and hn+j,n+i, there is always an elementary product that contains both hij and hn+j,n+i. Thus the general determinant det[jωI − H], depends quadratically on the entries of A. Hence, even if the entries of A depend affinely on qi we will get quadratic mapping equations in qi . This holds for special canonical representations, which yield a characteristic polynomial with affine parameter dependency. The quadratic dependency of the mapping equations makes it clear that, even for really simple parameter dependence, we have to deal with mapping equations that are more complicated than the mapping equations obtained for eigenvalue specifications. For parametric entries of the input and output matrices B and C it is obvious that we get quadratic terms for each entry, since the Hamiltonian matrix H already has quadratic terms due to BS2 B T and C TS3 C. The appearance of entries from the D matrix in the final mapping equations depends on the considered specification. Thus, there is no obvious way to show the dependence of the mapping equations with respect to parameters in D. Nevertheless, evaluating the dependence for each specification expressible as an ARE or by looking at the general IQC ˜ 22 = Π22 + D T Π12 + Π∗ D + D T Π11 D, we can deduce Hamiltonian matrix (3.34), with Π 12 that entries of D will reappear as quadratic terms in the mapping equations.

3.5.2 Lyapunov Mapping Equations Lyapunov based mapping equations can be obtained by evaluating one of the two equations given in (3.21). In order to determine the mapping equations we first have to solve a system of linear equations, where the parametric coefficient matrix has size 21 n(n + 1), where n is the number of states of the considered transfer matrix. The resulting Gramian has the same dimensions as A. In the final step, we have to compute a parametric matrix product including the just obtained Gramian, and determine the trace. Thus the main obstacle to compute Lyapunov mapping equations is the symbolic solution of a linear system.

69

3.6 Further Specifications

3.5.3 IQC Mapping Equations The order of the IQC mapping Hamiltonian matrix is given by the number of plant states plus the number of states required to realize the multiplier. Thus, only for frequencydependent multipliers there is a multiplier state-space augmentation, and the complexity increases. Hence, the accuracy gained by using a higher order multiplier has to be paid for by an increased Hamiltonian matrix complexity and therefore increased computational requirements.

3.6 Further Specifications So far we covered a list of specifications that includes almost the entire list of specifications commonly used in control engineering. Nevertheless some candidates of unequal importance are missing: the structured stability radius µ, the entropy, and the L1 norm. The structured stability radius µ has been introduced to cope with parametric uncertainties in the H∞ norm framework. To date there are only approximate methods to estimate its value. So, there is no way to map µ specifications into a parameter space. Since the structured stability radius tries to deal with mixed uncertainties it considers a similar problem as the methods presented in this thesis. Having the definition of µ in mind, we do not expect advantageous insights of mapping a specification for parametric uncertainties into a parameter plane. Another specification considered in the control literature is the entropy of a system. This entity is actually not a norm, but it is closely related to the H2 and H∞ norm. The γ-entropy [Mustafa and Glover 1990] of a transfer matrix G(s) is defined as  2 Z ∞   −γ ln det I − γ 2 G(jω)G(jω)∗ dω, if ||G||∞ < γ, 2π −∞ Iγ (G) :=   ∞, otherwise.

(3.58)

The γ-entropy has been used as a performance index in the context of so-called risk sensitive LQG stochastic control problems, and reappeared in the H∞ norm framework in [Doyle et al. 1989] as the so-called central controller, where the selected controller satisfying an H∞ norm condition minimizes the γ-entropy. The γ-entropy of G(s), when it is finite, is given by Iγ (G) = trace [B T XB],

(3.59)

where X is the positive definite solution of the ARE AT X + XA + C T C +

1 XBB T X = 0 . γ2

(3.60)

70

Mapping Equations

Although the familiar ARE might suggest that mapping equations for this specifications can be derived, this is not the case. The existence of a solution to (3.60) can be tested with the familiar Hamiltonian matrix (3.6). But in order to compute and map Iγ we actually need this solution. Therefore the entropy does not fit into the presented framework and no algebraic mapping equations can be derived. Another example of a specification, that cannot be easily mapped, is the L1 or peak gain norm: ||G(s)||1 :=

||Gw||∞ . ||w||∞ 6=0 ||w||∞ sup

(3.61)

For a given system, the total variation of the step response is the peak gain of its transfer function, see [Lunze 1988]. This specification is not computable by algebraic equations. Boyd and Barratt [1991] suggest to use numerical integration to calculate it. A remedy might be to use lower and upper bounds given in [Boyd and Barratt 1991], and map H∞ and Hankel norm specifications. Namely, if G(s) has n poles, then the L1 norm can be bounded by the Hankel and H∞ norm: ||G||∞ ≤ ||G||1 ≤ (2n + 1)||G||hankel ≤ (2n + 1)||G||∞.

(3.62)

3.7 Comparison and Alternative Derivations A particular strength of the approach pursued in this thesis is the generality. All specifications expressible as an ARE or Lyapunov equation can be considered. Nevertheless there are alternative derivations of mapping equations for some specifications. Most authors to date have considered either eigenvalue or frequency domain specifications for SISO systems [Besson and Shenton 1999, B¨ unte 2000, Hara et al. 1991, Odenthal and Blue 2000]. Reasons for this might be the elegant derivation of eigenvalue mapping equations, which are computationally very attractive for practically relevant problems. For frequency domain specifications the extension to MIMO systems is nontrivial and the complexity associated with SISO mapping equations might have hindered the application to MIMO systems.

3.8 Direct Performance Evaluation The PSA maps specifications into a parameter plane. We are neither interested in the direct, numerical evaluation of a specification, e.g., ||G(s, q ∗ )||∞ , where q ∗ is a fixed parameter vector, nor in the solution of an optimal control problem traditionally considered

71

3.8 Direct Performance Evaluation

in the H2 and H∞ literature. Nevertheless the presented mapping equations can be used for the direct performance evaluation for some of the presented specifications. Although in a different mathematical framework, Schmid [1993] devoted a large portion of his work to the direct evaluation of the H∞ norm, see also [Kabamba and Boyd 1988]. The mapping equations establish conditions, that allow the direct computation of the H∞ norm. Thus, instead of using the numerically attractive bisection algorithm widely used in control software, the H∞ norm can be computed by solving the two algebraic equations (3.7a) and (3.7b) in the two unknowns ω and γ. These positive solutions provide candidate values γ for the H∞ norm. Additionally, the solutions of (3.8) and (3.9) are computed and the H∞ norm of the system is given by the maximal value over all candidate solutions. As a byproduct, we get the frequency ω, for which the maximal singular value occurs. For the H2 norm the direct evaluation is simply possible by solving the linear equation (3.21b) and substituting the solution into the mapping equation (3.22). Example 3.2 We use the mapping equations to directly compute the H∞ norm for the open-loop transfer function   2 −2s 1 .  (3.63) G(s) = (s + 1)(s + 2) s 3s + 2

A state-space representation for G(s) is given by 

−2

0

  0 −1  ∼ G(s) =   −2 2  2 −1

1 2



 1 1   . 0 0   0 0

(3.64)

The mapping equations (3.7a) and (3.7b) become e1 = γ 4 ω 4 + (5γ 4 − 14γ 2 )ω 2 + 4γ 4 − 8γ 2 + 4,

e2 = ω(4γ 4 ω 2 + 10γ 4 − 28γ 2 ).

This polynomial system of equations has only one single relevant solution ω = 1, γ = with ω, γ > 0. The DC-gain condition (3.8) is det Hγ = γ 4 − 2γ 2 + 1,

√

2

72

Mapping Equations

which has the positive solution γ = 1. Note that this can be easily observed by the fact that G(j0) = I, and thus the singular values are given by σ1 = σ2 = 1. There is no solution for (3.9) and we can conclude that the maximal singular value of G(s) √ is given for ω = 1 and ||G(s)||∞ = 2.

3.9 Summary The main contribution of this chapter is the presentation of a uniform framework that allows to derive mapping equations for parametric control system specifications expressible by AREs. Besides ARE expressible specifications, previously unknown mapping equations for IQC based stability tests are determined. Using the results in this section we can draw from the vast number of available IQCs and incorporate them into the parameter space approach. Furthermore, application of standard parameter space methods allows to include an even larger list of specifications into control system analysis and design. The resulting equations are similar to well-known Γ-stability mapping equations. This allows similar computational methods for the mapping, although the complexity is in general higher, due to the quadratic nature of the specifications.

73

4

Algorithms and Visualization The purpose of computing is insight, not numbers. Richard Hamming

The main contribution of this chapter is the presentation of algorithms which solve the mapping problem, i.e., they plot the critical parameters in a parameter plane. Thus these algorithms are a prerequisite to the successful application of the presented MIMO control specifications in the parameter space context. Geometrically, the mapping is either a curve plotting or surface-surface intersection problem. These can be approached with a variety of different techniques, including numerical, analytic, geometric, and algebraic methods, and using various methods such as subdivision, tracing, and discretization. While classic algorithms used a single method, recently algorithms which combine multiple methods or so-called hybrid algorithms appeared. We will follow this line and combine previous known results such that the new combination of basic building blocks forms an efficient algorithm for the mapping problem. The robustness of the algorithm has to be considered. It is easy to generate a specialized, fast algorithm for special curve intersections. This is actually the topic of solid geometry, vision or computer-aided design, where the complexity of the geometrical objects considered is known beforehand. For practical experimentation the algorithms were implemented using Maple and Matlab. The Lyapunov based mapping equations in Section 3.3.4, e.g., for the H2 norm, directly lead to an implicit polynomial equation f (x, y) =

n X

aij xi y j = 0,

(4.1)

i,j=0

where x and y represent the two parameters of the parameter plane. In order to plot the parameters, which satisfy a control system specification, we have to determine the curve C of real solutions satisfying (4.1). General properties of algebraic curves will be presented in Section 4.2.

74

Algorithms and Visualization

For ARE based mapping equations we get two polynomial equations f1 (x, y, ω) = 0, f2 (x, y, ω) = 0,

(4.2)

and we are interested in the plane curve of real solutions (x, y) ∈ R2 which can be obtained for all positive ω. Mathematically (4.2) defines a spatial curve in R3 . Thus we are interested in the plane curve C given as the projection of the spatial curve (4.2) onto the plane ω = 0. Since (4.2) is a generalization of (4.1) we will first treat plane parameter curves, and consider the extension to (4.2) in Section 4.5.

4.1 Aspects of Symbolic Computations This section considers general aspects of symbolic computation and compares them to numerical computations. This gives rationals for designing hybrid algorithms which employ both symbolic and numerical computations, such that the overall results are obtained fast and robust. Symbolic computation is considerably different from numerics, as new aspects such as internal data representation, memory usage, and coefficient growth arise, while difficult aspects of numerical algorithms, e.g., rounding errors and numerical instability, disappear [Beckermann and Labahn 2000]. The main advantage of symbolic computations is the fact that general equations with unknown variables can be solved. Furthermore the exact arithmetic avoids common problems of numerical algorithms introduced by finite precision. A common symbolic algebra package such as Maple can handle 216 −1 individual symbolic expressions. Compare this to the determinant of a matrix A with unknown entries aij . The determinant of a matrix with size n has n! expressions. Thus even the modest size of n = 9 exceeds the capacities of available software. And since the factorial grows much faster than polynomial, even with more memory and computer power, the memory usage will be always a limiting factor in the application of symbolic algorithms to real world problems. These limitations impose some restrictions on the application of the presented control specifications. Although the mapping equations given in Chapter 3 can be computed symbolically, treating all parameters as variable complicates the computations drastically. A remedy is to substitute all parameters not part of the parameter plane into which the specifications are mapped with their numerical values. For example, when we are mapping the H2 norm specification into a q1 , q2 parameter plane, all remaining parameters in the state space model should be replaced by their numerical values prior to the mapping

4.2 Algebraic Curves

75

equation computation. Thus, if we are gridding a parameter q3 , we will generate the mapping equations for each grid point. This is much more attractive than computing universal mapping equations, where q3 appears as a free parameter. The computational complexity of symbolic algorithms cannot be measured by operational complexity. The main reason is the varying cost associated with operations such as addition or multiplication. The cost here depends on the size of the components and the size of the result. This is related to the way computer algebra systems such as Maple store expressions. Here all expressions are stored in an expression tree or more precise a direct acyclic graph. The following example shows the possible explosion on the number of expressions for a very small problem: p1 (s) = s2 + q1 s + q2 , p2 (s) = s2 + (q1 − q2 )s + q1 ,

p1 (s)p2 (s) = s4 + (2q1 − q2 )s3 + (q12 − q1 q2 + q1 + q2 )s2 + (q12 − q22 + q1 q2 )s + q1 q2 . So in order to evaluate the computational complexity for symbolic algorithms we need bit complexity. The cost of intermediate operations might be especially large if the coefficients of expressions are in the field of quotients Q. Here the cost might grow exponentially. Thus fraction-free algorithms which avoid expressions lying in Q are very attractive from a computational cost view.

4.2 Algebraic Curves This section presents some basic facts about algebraic curves which will be important for the algorithms presented in the subsequent sections. The content presented here is rather self contained, since in general algebraic geometry and curves are not covered in basic engineering courses. While algebraic curves can be generally represented in explicit, implicit and parameter form, we will only consider curves defined in implicit form f (x, y) = 0 as in (4.1), because this is the natural description of the mapping equations. For each summand aij xi y j of (4.1), the degree is defined as the sum of the individual powers dij = i + j. The total degree of the polynomial is then given as the maximum deg f = max dij . For systems of polynomials as in (4.2), the complexity is measured by its total degree, which is the product of the total degrees of all individual polynomiQ als deg f = deg fi . We will distinguish two different classes of points or solutions of f (x, y) = 0, which arise in the following definition.

76


Definition 4.1 We call (x0 , y0 ) ∈ C a singular point of curve (4.1), if both partial derivatives of f vanish at that point: fx (x0 , y0 ) = fy (x0 , y0 ) = 0, If a point is not singular, then it is called regular. A curve of degree d has at most 12 d(d−1) singular points. A curve C is regular, if every point of C is regular. The singular points are not only important for the topology of a curve, e.g. number of self intersecting points, but are also vital to numerical algorithms. In most cases numerical algorithms will become inaccurate or show slow convergence in the vicinity of singular points. Furthermore, phenomena such as the birth of new branches (or more formal bifurcations) and branch switching can happen at singular points. A remedy for singular points is to develop algorithms which either can handle singular points or compute and detect them and switch to specialized methods to handle singular points. If point (x0 , y0 ) is a regular point of C then C has a well-defined tangent direction at (x0 , y0 ), with tangent line equation (x − x0 )fx (x0 , y0 ) + (y − y0 )fy (x0 , y0 ) = 0.

(4.3)

For singular points the tangent lines (possibly multiple) have to be computed using the higher derivatives. A singular point is said to have multiplicity m, if all partial derivatives up to degree m vanish. The fundamental theorem of algebra tells that a polynomial of degree n has n roots in C. The number of possible intersection points of two curves, defined implicitly by two polynomials, is bounded by Bézout’s theorem. Applied to curves, it states that in general two irreducible curves of degree m and n have exactly mn common points in the complex plane C2 , i.e., they intersect in at most mn real points [Coolidge 2004]. Besides the singular points, there are other special points of curves. First consider socalled critical points with horizontal or vertical tangent, i.e., a partial derivative f x (x0 , y0 ) or fy (x0 , y0 ) equals zero. Other important points are inflection points, or flexes. A regular point of a plane curve is an inflection point, if the tangent is parallel to the curve, i.e., the curvature is zero. A regular point (x0 , y0 ) of f (x, y) = 0 is an inflection point, if and only if   f f f  xx xy x    det fxy fyy fy  = 0. (4.4)   fx fy 0


77

4.2.1 Asymptotes of Curves Another property of curves are asymptotes. These are lines which are asymptotically approached. In general, points on an asymptote do not belong to the solution of (4.1). An easy and mathematical complete description of asymptotes can be obtained by homogenization of the polynomial. A homogeneous polynomial contains only irreducible components with the same degree n. Any algebraic curve is homogenized by introducing the auxiliary variable z and multiplying each term in fi with z m such that the new term is of order n. The obtained polynomial p(x, z) contains the original problem since p(x, 1) = f (x). Furthermore for z = 0 we get the solutions at infinity. The asymptotes of the homogenized polynomial are thus given by the solution of p(x, 0). The actual asymptotes of a curve might possess an offset cx x + cy y + c0 = 0, which is easily computed by substituting this equation into the curve equation. If an offset c0 6= 0 exists, the resulting equation has to be zero for all values of x, y respectively. Thus, we can for example solve the coefficient f (c0 ) of the highest degree of x. Example 4.1 Consider the curve y 3 −y 2 −yx2 +2xy+x2 +1. The homogenized polynomial is p(x, y, z) = y 3 − y 2 z − yx2 + 2xyz + x2 z + z 3 . Setting z = 0 the terms relevant for the asymptotes are y 3 − yx2 . This polynomial can be easily factored into (y − x)(y + x)y. The actual asymptotes are then computed as y − x + 1 = 0, y + x − 1 = 0 and y = 1.

4.2.2 Parametrization of Curves Parametric curves of the form [x(α), y(α)],

α ∈ R,

(4.5)

where x(α) and y(α) are rational polynomials in α, are very easily plotted by evaluating the polynomials for sufficiently many values of parameter α. This form can be always transformed into an implicit curve (4.1) by eliminating α, e.g., using resultants. The opposite transformation, called parametrization, is not necessarily possible. The theory of algebraic curves states that an implicit curve can be parametrized if and only if the curve is rational. Rationality of a curve is given, if the so-called genus of a curve equals zero [Walker 1978]. The genus is characterized by the degree and properties of the singular points. Let C be an irreducible curve of degree d which has δ double points and κ cusps. Then 1 genus(C) = (d − 1)(d − 2) − (δ + κ). (4.6) 2

78


Note that the curve is reducible if genus(C) < 0. As a consequence of (4.6), linear and quadratic curves can be always parametrized. For cubic curves (d = 3), at least one singularity (double point or cusp) has to be present. So in general for curves with degree d > 2, it is not possible to derive a rational parametrization. See [van Hoeij 1997] for a method to compute parametrizations for rational curves.

4.2.3 Topology of Real Algebraic Curves The set of real zeros Cf = {(x, y) ∈ R2 | f (x, y) = 0} of a bivariate rational polynomial is usually referred to as a real algebraic curve. Such a curve might have special points or singularities, where the tangent is not well defined, e.g., isolated points, self-intersections, or cusps. Figure 4.1 depicts the quartic curve f (x, y) = (x2 + 4y 2 )2 − 12x3 + 96xy 2 + 48x2 − 12y 2 − 64x,

(4.7)

which has three singularities, two self-intersections and a cusp. 3

2

1

y 0

−1

−2

−3 −3

−2

−1

0

1

2

3

4

5

x

Figure 4.1: A quartic curve with exactly three singularities

Numerous papers, e.g., [Arnon and McCallum 1988, Sakkalis 1991], consider the problem of determining the topology of real algebraic curves defined by a polynomial f (x, y) = 0. The topology is approached by means of an associated graph, that has the same critical points as vertices, and where an edge of the graph represents a curve segment connecting two vertices. Figure 4.2 shows the topological graph of curve (4.7).


79

The common steps to compute the topological graph of f (x, y) are (see e.g., [GonzálezVega and Necula 2002]): 1. Determine the x-coordinates of the critical points by computing the discriminant of f (x, y) with respect to y, and determine its real roots x1 , . . . , xm . Each vertical line x = xi contains at least a critical point of the curve. 2. Compute the vertices of the graph, by computing the y-coordinates yij , i.e., determine the real roots of f (xi , y) = 0. 3. For each vertex (xi , yij ) compute the number of branches emanating to the left and right. 4. Construct the graph by appropriately connecting the vertices. This can be simply done by ordering the vertices in terms of the coordinate y. Note that the connected graph is uniquely determined, since any incorrect branch between two vertices leads to at least one intersection of two edges at a non-critical point. Some published algorithms precede this scheme by an initial step involving a linear change of coordinates, which ensures that there are no vertical lines that contain two critical points. In step 1, the discriminant is used to reduce the system of equations f (x, y) = fy (x, y) = 0, to a univariate polynomial. Numerically calculating the real roots of a univariate polynomial is standard, and software packages such as Matlab and Maple have no difficulty to determine accurate solutions efficiently. Other approaches to solve a system of n polynomials in n unknowns are for example interval analysis, homotopy methods [Allgower and Georg 1990, Morgan 1987], elimination theory and Gröbner bases. Note that there can be multiple branches between two vertices, see for example the connection between vertices V1 and V3 in Figure 4.2. The usual approach to perform step 3 is to compute additional solutions of f (x, y) in the vicinity of critical vertical points and singular points or in between them. We propose to evaluate a Taylor series expansion of f (x, y) for a critical vertical point. With f (x0 , y0 ) = fy (x0 , y0 ) = 0, we get the quadratic approximation 1 1 (4.8) fx (x0 , y0 )∆x + fxx (x0 , y0 )∆x2 + fxy (x0 , y0 )∆x∆y + fyy (x0 , y0 )∆y 2 = 0. 2 2 For a singular point with multiplicity m, we can obtain the direction of tangents by solving m X m fxi ym−i (x0 , y0 )∆xi ∆y m−i = 0, (4.9) i i=1 evaluated at (x0 , y0 ).

80


Figure 4.2: Topology of a quartic curve with exactly three singularities

4.3 Algorithm for Plane Algebraic Curves A scheme for a general algorithm which determines the plane algebraic curve of an implicit bivariate polynomial f (x, y) is: 1. Preprocessing 2. Determine the topological graph 3. Approximate all curve segments Phase 1 usually involves factorization and possible coordinate changes which alleviate the subsequent computations. There are numerous approaches to perform phase 3, e.g., path-following, Bézier curve approximation and piecewise-linear approximations. Using an extended topological graph, we aim to get a very robust and efficient algorithm, which allows to use predictor-corrector based path following for individual regular curve segments. The extended topological graph divides the curve into a number of easily traceable curve segments. From the extended topological graph we already know the behavior of a curve in the vicinity of a singular point. This allows to determine the curve close to a singular point, avoiding numerical problems. Before the final curve is plotted, a Bézier approximation can be displayed. This allows fast response times to user inputs and gives the user a first impression of the final results.

81

4.3 Algorithm for Plane Algebraic Curves

4.3.1 Extended Topological Graph The basic topological graph, described in Section 4.2.3 is now extended, such that each segment is convex inside a triangle. This allows to get a robust algorithm for plotting the overall curve. As a first step, we divide segments with critical vertical points on both vertices. The tangents of both vertices never cross because they are parallel. Second, we include all inflection points. These can be computed by solving f (x, y) = 0, and (4.4) using a resultant method. The extended topological graph, has the property that all curve segments are convex and lie inside a triangle formed by the vertices and the intersection of the tangents at both vertices.

T12 V1 V2 Figure 4.3: Convex segment of rational curve

Consider a curve segment with vertices V1 and V2 . Since there are no singular, critical vertical or inflection points, the gradient of the tangent monotonically changes from the angle at V1 to the angle at V2 . Thus, the curve segment is convex and it has to lie in the triangle formed by V1 , V2 , and the intersection of both tangents labeled T12 . See Figure 4.3 which shows a convex segment inside the bounding triangle. Thus, we not only have a topological graph with convex curve segments, but we know the triangular area in which all points on a curve segment have to lie. The individual triangles can intersect each other, although this happens only for very degenerate curves. The intersection of triangles can be eliminated by introducing additional points into the extended topological graph until no intersection occurs. Figure 4.4 shows the extended topological graph of the curve defined by f (x, y) = x4 + y 4 − x3 − y 3 − x2 y + xy 2 ,

(4.10)

and the actual dotted curve. This curve has a triple multiplicity at V6 = (0, 0) and two flexes at V2 and V3 .

82


V1 V2 V3 V4

V6 V5

Figure 4.4: Extended topological graph

4.3.2 B´ ezier Approximation The Bézier curve is a parametric curve important in computer graphics [Farin 2001]. A quadratic Bézier curve is the path traced by the function p(α) = (1 − α)2 p0 + 2α(1 − α)p1 + α2 p2 ,

α ∈ [0; 1].

(4.11)

The curve passes through the end points p0 and p2 with tangent vectors p1 −p0 and p2 −p1 . See Figure 4.5 for a simple quadratic Bézier curve. The functions (1 − α)2 , 2α(1 − α), and α2 are degree two Bernstein polynomials that serve as blending functions for the control points p0 , p1 , and p2 . The Bernstein polynomials are non-negative and add to one. Thus p(α) is an affine combination of the points p0 , p1 , and p2 contained in the triangle p0 p1 p2 . Geometrically quadratic Bézier curves are parabolas. p1

p0

p2 Figure 4.5: Quadratic Bézier

We will use information from the extended topological graph to sketch the curve. Using simple quadratic Bézier curves for each curve segment, a good approximation of the

83

4.4 Path Following

true curve with bounded error can be sketched. The highest computational burden is associated with computing the support point for a Bézier spline involving a singular point. The branch tangents of the singular point are hereto calculated. See Figure 4.6 for a Bézier based approximation of curve (4.10). Note the small deviation of the Bézier approximation from the true dotted curve. p1

p0 p2

Figure 4.6: Bézier approximation of quadratic curve

4.4 Path Following In this section we will consider the approximation of a single continuous curve segment using path following or curve tracing. While there are several approaches to path following, we particularly treat predictor-corrector continuation methods [Allgower and Georg 1990]. In virtue of the polynomial equation defining a curve, the required numerical calculations can be performed with high precision, and thus this approach is very suitable. Before we present high-fidelity predictor-corrector algorithms, we will first consider common problems of gradient based path following algorithms in Section 4.4.1. We then present a very easily implementable formulation of the path following problem, using homotopy in Section 4.4.2, before we extend this to a full-scale predictor-corrector algorithm in Section 4.4.3.

84


(x1 , λ1 )

(x2 , λ2 )

(x0 , λ0 )

Figure 4.7: Branch skipping

4.4.1 Common Problems of Path Following A common problem is branch skipping, i.e., while the path of a single branch is followed, the algorithm misleadingly converges to a point which is on a separate branch not continuously connected to the branch currently followed. See Figure 4.7 for an example, where (x2 , y2 ) is wrongly connected to (x1 , y1 ). Branch skipping might lead to missed curve segments, or incorrect segment connection. In extreme cases consecutive branch skipping might lead to branch looping, where the algorithm enters an infinite loop, while connecting points from different branches, see Figure 4.8.

4.4.2 Homotopy Based Algorithm The earliest account of a continuation method can be found in [Poincaré 1892]. The idea of using a differential equation to solve a system on nonlinear equations was first explicitly reported in [Davidenko 1953]. Davidenko’s approach is a subset of homotopic methods, which can be used to the curve segment approximation problem. Two functions y = f (x) and y = g(x) are homotopic, if one function can be continuously deformed into the other, in other words, if there is a homotopy between them: a continuous function y = h(α, x), with h(0, x) = f (x) and h(1, x) = g(x). The easiest homotopy is given by the affine interpolation h(α, x) = (1 − α)f (x) + αg(x).

(4.12)

The solution of parametrized nonlinear equations and algebraic mapping equations in particular can be formulated as a homotopy. To this end, one variable of f (x, y) = 0 is

85

4.4 Path Following

(x4 , λ4 ) (x3 , λ3 ) (x5 , λ5 ) (x2 , λ2 )

(x1 , λ1 )

Figure 4.8: Branch looping

used as a homotopic parameter. For example, let x be the homotopic parameter. We then try to obtain an explicit solution of y as a function of x. Hereto, the total differential of f (x, y) is determined as fx dx + fy dy = 0.

(4.13)

Now write this equation to get the ordinary differential equation dy fx =− , dx fy

(4.14)

known as Davidenko’s equation. We can now merely use a numerical initial value problem solver to solve (4.14) . While this method has been successfully used by some authors, the exploitation of the contractive properties of the curve by Newton type predictors is preferable. The main difference in using (4.14) for path following and the solution of nonlinear equations by homotopic methods is the intermediate accuracy. A homotopy (4.12) has to simply track all paths approximately and the ultimate requirement is only the solution for t = 1. Whereas the curve segment approximation for the parameter space approach requires an acceptable solution for intermediate steps too.

86


4.4.3 Predictor-Corrector Continuation An efficient and robust predictor-corrector method possesses the following features [Allgower and Georg 1992]: 1. efficient higher order predictor, 2. fast converging corrector, 3. effective step size adaptation, 4. detection and handling of special points such as bifurcation or turning points. These properties are important if we want to successfully approximate complicated or difficult curves, e.g., arising from H∞ norm specifications for systems with high-order or polynomial parameter dependency, or both. Yet, the precision should be high enough in order to make sure that the build up of numerical error does not lead to an increase in the number of required iterations or prevents convergence of the corrector step at all. If necessary, e.g., in the vicinity of singularities, some software packages have the ability to explicitly change the precision. Note: Although predictor-corrector methods are commonly used to integrate ordinary differential equations, these methods are considerable different than the methods described in this section. While we can use the contractive properties of the solution set following a solution path, this is not the case for initial value problem solvers. Actually the corrector converges in the limit only to an approximate point for differential equations.

Predictors During the predictor step a point close to the curve with some distance from the current point (xk , yk ) is determined. Very commonly and sufficient for the parameter space curve approximation is an Euler predictor, where the predictor step uses the tangent to the curve, xk+1 = xk + hk t(xk ),

(4.15)

with current step length hk > 0, and tangent vector t(xk ) at the curve point xk = (xk , yk ). An even more simple predictor step can be performed using a secant predictor, which uses two previous points to approximate the current direction xk+1 = xk + hk (xk − xk−1 ).

(4.16)

87

4.5 Surface Intersections

Correctors A straightforward corrector is given by a Newton type iteration, xk+1,n+1 = xk+1,n + ∆xk+1,n ,

n = 0, 1, . . .

(4.17)

where ∆xk+1,n is the solution of the linear equation [fx (xk+1,n ) fy (xk+1,n )] ∆xk+1,n = −f (xk+1,n ).

(4.18)

The n-th iteration of the corrected point (xk+1 , yk+1 ) is denoted xk+1,n . Due to the convex nature of the curve segments, low iteration numbers n are sufficient to get close to the real curve. Step length control Any efficient path following algorithm has to incorporate a step length control mechanism because the local properties of the followed curves vary largely on the curve. Of course, any step length adaption will depend on the desired curve tracing accuracy. Furthermore, a robust step length control will prevent path skipping illustrated in subsection 4.4.1. Due to the extended topological graph and the subdivision into convex curve segments, we can employ a very simple step length control, e.g., by using the function value at the final point (˜ xk , y˜k ) of the corrector step as an error model. Prevention of path skipping Using the bounding triangle of a convex segment, the path skipping can be avoided robustly by checking that the predicted point, and points computed during the correction phase are inside this triangle.

4.5 Surface Intersections This section describes the algorithms proposed for solving the implicit surface-surface intersection problem which is essential to the parameter space approach. Both polynomial equations in (4.2) define a surface in R3 . The intersection of these two surfaces forms a spatial curve in R3 . If a parametrization exists, which is in general not true, this curve can be written as [x(α), y(α), ω(α)].

(4.19)

88


Since we are only interested in the parameters x and y which satisfy (4.2), the generalized frequency ω can be treated as an auxiliary variable. Thus, the critical boundaries in the parameter plane are obtained as the projection of the space curve onto the plane ω = 0. The projection of the intersection onto this plane, is mathematically described by the resultant of both polynomials r(x, y) = res(f1 (x, y, ω), f2 (x, y, ω), ω).

(4.20)

Actually, for all mapping boundaries presented in Section 3.3.1 and Section 3.4, we have the additional constraint, that f2 is the derivative of f1 with respect to ω, ∂f1 (x, y, ω) f2 (x, y, ω) = . ∂ω Thus, r(x, y) in (4.20) can be written as ∂f1 (x, y, ω) r(x, y) = res(f1 (x, y, ω), , ω), ∂ω which is the discriminant of f1 with respect to ω. And we obtain r(x, y) = disc(f1 (x, y, ω), ω).

(4.21)

Finally, we can show that (4.21) contains not just the CRB condition (3.7a), but this polynomial has additional factors which are equivalent to the RRB (3.8) and IRB condition (3.9). Utilizing this property, the critical boundaries can be determined by evaluation the discriminant (4.21), and factorizing this polynomial, with additionally eliminating double factors. Subsequently, all critical boundaries can be plotted by consecutively plotting the resulting curves in the (x, y) plane using the algorithm developed in Section 4.3. As an alternative we can evaluate the CRB projection separately, after eliminating the factor ω, using the resultant equation (4.20), while the RRB and IRB conditions are already implicit equations. For this approach, (4.20) will contain a term simply squared. Thus, it suffices to consider only the argument of the square.

4.6 Preprocessing The computational burden of generating the solution curves of algebraic equations can be alleviated in many cases by symbolic preprocessing. By preprocessing we mean any transformation of the mapping equation system which preserves the solution structure and alleviates the determination of the actual solution set. For some systems this symbolic preprocessing is actually mandatory. An intuitive preprocessing step is to scale the equations. Furthermore, using a computer algebra system, e.g., Maple, we can use factorization.

4.6 Preprocessing

89

4.6.1 Factorization If a polynomial f (x, y) is factorizable into individual polynomial factors, then the solution set can be determined by treating the individual factors separately. A polynomial can be possibly symbolically factorized, if the coefficients are integers. We therefore assume that the coefficients are integers, prior to factorization. This can be done by normalizing all rational coefficients. In a separate step, the integer coefficients can be factored such that the polynomial is primitive over the integers. All algorithms presented for plotting the curve of an implicit polynomial are faster when factors are considered consecutively. Therefore, assume that the polynomial is irreducible.

4.6.2 Scaling Scaling transforms the equations such that the coefficients are not extremal. The purpose of scaling is to make a problem involving equations numerically more tractable. This is a pretty vague goal and it should be clear that depending on the algorithm which is used for the problem there is no theoretical best scaling. Thus, we have to use common sense in choosing a scaling. In general any algorithm will benefit from a problem which has coefficients with absolute values close to one and only small variations within equations. There are two types of scaling. The multiplication of the equation by common factor is called equation scaling, while the transformation of a variable of the form x = constant · x˜ is referred to as variable scaling. The scaled form of a univariate polynomial equation an xn + . . . + a1 x + a0 = 0 with equation scaling factor 10c1 and variable scaling x = 10c2 x˜ is given by 10c1 +nc2 +log10 an xñ + . . . + 10c1 +c2 +log10 a1 x˜ + 10c1 +log10 a0 = 0.

(4.22)

The coefficients can now be centered about unity and the variation of coefficients minimized by solving linear equations. See Chapter 5 of [Morgan 1987] for an implementable algorithm.

4.6.3 Symmetry The successful exploitation of symmetry can not only lead to a much more efficient algorithm but also to a more robust one. Having identified the axis of symmetry we have to follow the path on one side only. The path on the opposite side can be mirrored. Thus, symmetry will immediately lead to reduction of the computational cost by a factor of 1/2.

90


4.7 Visualization Visualization is concerned with exploring data and information graphically - as a means of gaining understanding and insight into the data [Earnshaw and Wiseman 1992]. Nowadays it is common understanding, that a robust control toolbox should provide a user-friendly, possibly graphical interface, as suggested in [Boyd 1994]. While all computations are done in the background and only the results are finally visualized. See [Muhler et al. 2001, Sienel et al. 1996] for a toolbox which enables plant descriptions via block diagrams and allows to graphically specify Γ-stability eigenvalue specifications. As proposed in [Muhler and Odenthal 2001] this approach can be extended to frequency domain specifications. We will explore several ways how to visualize specifications in a parameter plane. A straightforward approach is to simply plot the critical boundaries of parameter region, which fulfill a specification. This is equivalent to plots generated for Γ-stability specification, see Figure 3.3 for an IQC stability example. Varying line styles might be used to distinguish different specifications, e.g., Γ-stability and H∞ norm. Further improvements of the visualization can be achieved by using the following methods: • Overlay • Complementary Colormaps • Slave Cursor

4.7.1 Color Coding Most frequency domain specifications, e.g., Nyquist stability margin, yield a scalar value for fixed parameters. Thus, it is possible to determine regions with performance in a specific range. Color coding these regions according to their performance level allows immediate assessment of performance satisfaction. Note that critical boundary lines resulting from eigenvalue specifications can be overlaid on top of color coded contour plots. Hence, multiple objectives can be represented simultaneously in a parameter plane. The Nyquist stability margin is used to explain the color coding scheme. Annuli in the Nyquist plane are color coded according to their distance from the critical point (-1,0). In order to make the color coded plots intuitively evident, we propose a traffic light color coding scheme. This color map resembles the colors of a traffic light ranging from green to red. We use colors close to red to visualize regions with poor performance and colors

91

4.7 Visualization

close to green for good performance. Thus, the plots are readily understandable by the control designer. Alternatively gray-scale color coding could be employed with black for poor and white for good performance, if the use of colors is not possible. Figure 4.9 shows the used color coding scheme for the Nyquist stability margin. Stability margin

ρ

m

1

1.0

0.5

Im(G(jω))

0.75 0

0.5 −0.5

0.25 −1

−2.5

−2

−1.5

−1

−0.5

0

0.5

0

Re(G(jω))

Figure 4.9: Color coding for the Nyquist stability margin

Nyquist performance can now be visualized in the parameter plane by determining parameter sets which lead to performance in a specific range. These sets are color coded according to their performance level. Thus, we seek to determine the set of parameters Ki with Nyquist stability margin in a specific range + Ki := {k1 , k2 : ρ− i ≤ ρ(k1 , k2 ) ≤ ρi }. + Using different values of ρ− i , ρi , we can exactly determine the sets Ki . The boundary lines obtained for each value of ρ represent the contour lines, which are used to color code the sets in the parameter space according to their performance level.

4.7.2 Visualization for Multiple Representatives In this section we consider the visualization of admissible sets and frequency domain performance for multiple representatives. Eigenvalue specifications Similar to the nominal design case, the eigenvalue specifications are mapped into the controller parameter plane for each representative. Intersection of sets leads to a set K Γ of

92


admissible controller parameters, for which the Γ-specifications are simultaneously fulfilled for all representatives [Ackermann et al. 1993], see Figure 4.10 where superscripts (1) and (2) denote two different representatives.

!

# $ ! ! # $ % # $

"!

Figure 4.10: Mapping of ∂Γ for multiple representatives

Appropriate visualization enables the designer not only to identify the admissible set, but to identify specifications and operating points which constrain the admissible set. Color coding for multiple representatives For multiple representatives a scalar function which is to be maximized can be visualized by worst case color coding. For several representatives only the lowest value is relevant. Therefore only the minimal value over all representatives is color coded and visualized. Visualizing the Nyquist stability margin for several representatives in the controller parameter plane can be done by plotting the lowest value obtained for all representatives. Figure 4.11 shows a simple example for two representatives.

Rep 1

Rep 2

worst case for Rep 1+2

Figure 4.11: Worst case color coding example

93

5

Examples

This chapter presents applications of the derived mapping equations. Various practical examples demonstrate the usability for robust control system design. The model description is given as a parametric state-space model as in (2.1) or as a transfer matrix representation (2.2). For ease of presentation, we consider only systems with the same number of inputs and outputs, i.e., m = p. Nevertheless, all results are valid for non-square systems with m 6= p.

5.1 MIMO Design Using SISO Methods For SISO control systems, classical gain and phase margins are good measures of robustness. Furthermore, loop-shaping techniques provide a systematic way to attain good robustness margins and desired closed-loop performance. The methods introduced in [Ackermann et al. 2002, Sections 5.1−5.4] facilitate such a design for the PSA. However, the classical gain and phase margins are not reliable measures of robustness for multivariable systems. The simplest approach to multivariable design is to ignore its multivariable nature and just look at one pair of input and output variables at a time. Sometimes this approach is backed up by decoupling, although robust decoupling is in general difficult to achieve. A classical design procedure using this idea for multivariable systems is the sequential loop closing method, where a SISO controller is designed for a single loop. After this design has been done successfully, that loop is closed and another SISO controller is designed for a second pair of variables, and so on. Example 5.1 Consider the following plant   2 −2s 1  . G(s) = (s + 1)(s + 2) s 3s + 2

(5.1)

In the first step, we design a constant gain controller k11 (s) = k1 . The transfer function seen by this controller is g11 (s) = 2/((s + 1)(s + 2)). Setting k1 = 1 leads to a stable

94

Examples

transfer function. After closing this loop, the transfer function seen by a controller from output 2 to input 2 is g˜22 (s) = g22 (s) −

g12 (s)k11 (s)g21 (s) 3s + 4 = 2 . 1 + k11 (s)g11 (s) s + 3s + 4

A stabilizing controller for this transfer function is k22 (s) = 1. The resulting decentralized controller is thus given by the identity matrix K(s) = I. We will come back to this example with a better solution in Example 5.3 and Example 5.4. This method has a number of weaknesses. During the controller design, the resulting scalar transfer function for the i-th step might be nonminimum-phase, although all members of G(s) are minimum-phase transfer functions. This might pose a severe constraint for the control design, since nonminimum-phase transfer functions limit the maximal usable gain.

5.2 MIMO Specifications 5.2.1 H2 Norm Example 5.2 Consider the attitude control of a satellite for one axis. The transfer function is given by Y (s) =

1 U (s), Iz s2

where Iz is the moment of inertia for the z-axis. We are now designing a state-feedback controller u(t) = −(k1 x1 (t) + k2 x2 (t)) which minimizes the objective Z 1 ∞ J= (x1 (t)2 + x2 (t)2 + u(t)2 ) dt. 2 0 The resulting matrices for this problem are 

˙ x(t) =  

0

1

− kIz1 − kIz2



 x(t) + w(t),



 −k1 −k2    z(t) =  1 0  x(t).   0 1

95

5.2 MIMO Specifications Using (3.21a) the observability Gramian becomes   2 2 2 2 k (k + 1)I + k + k I (k + 1)k 1  1 1 z z 1 2 1 2 . Wobs = 2 2 2k1 k2 Iz (k + 1)k2 Iz (k + 1)Iz + k1 k 2 + k1 1

1

2

And we get J2 =

(k12 + 1)Iz2 + (k1 + k2 )(k12 + k1 k2 + 2)Iz + k12 + k22 2k1 k2

as the mapping equation which is quadratic in k2 . Figure 5.1 shows the resulting parameter sets Q2 for J 2 = 12 and two different moments of inertia Iz = 1 and Iz = 2. It can be seen that the operating point with the highest moment of inertia is the limiting case for the LQR specification. As an additional specification the set QΓ for which the damping ζ of the closed-loop system is at least ζ = 0.9 is determined in Figure 5.1, and shown by a dotted line. The figure shows that a robust controller can be determined from the set of parameters which satisfy both specifications, marked as the light shaded area. 14 12

Iz = 1

10

k2

8 6

Iz = 2

4 2 0 0

1

2

3

4

5

6

k1 Figure 5.1: LQR and Γ-stability boundaries for attitude control example

Example 5.3 Revisit Example 5.1. We will design a decoupled constant-gain outputfeedback controller u(t) = −Ky(t) with   k1 0 , K= (5.2) 0 k2

96

Examples

which minimizes the following LQR-like performance index 1 J= 2

Z

∞

(y(t)T y(t) + α u(t)T u(t)) dt.

(5.3)

0

This performance index treats both outputs equally, which is reasonable since the openloop plant has similar gains for these outputs. The parameter α provides an adjustable design knob, which allows an intuitive tradeoff between the integral error of the commanded output and the actuator effort. For this specific example, we assume α = 1. The open-loop plant has pure real eigenvalues at {−1, −2}. Thus, we additionally require the rather stringent specification that all closed-loop eigenvalues should have at least a minimum damping of ζ = 1.0. We will solve this problem by mapping the design requirements into the k1 , k2 controller parameter plane. To this end, we formulate the LQR output problem (5.3) in the H2 norm framework by employing the results of Section 2.4.8. Using the fact that y(t) = Cx(t), the LQR weight matrices in (2.62b) for this problem are given by Q = C T C,

R = I.

In order to apply the algebraic mapping equations (3.23), we need a state-space description of the system. A minimal realization of the system (5.1) is given by 

   G(s) ∼ =  

−2

0 −2 −4



 0 −1 −2 −2   . 1 −1 0 0   −1 1/2 0 0

(5.4)

We incorporate the controller u(t) = −Ky(t) in parametric form into (2.62a) and (2.62b) to get the state-space system G(s) defined in the H2 norm mapping equation (3.23). For the particular problem considered in this example, these equations are given by

˙ x(t) = (A − BKC)x(t) + w(t),   −K  Cx(t). z(t) =  I

(5.5) (5.6)

97

5.2 MIMO Specifications And the parametric transfer function G(s)w→z becomes 

G(s)w→z

      ∼ =     

2(k1 − 2k2 − 1) 2(k1 − k2 ) −k1 k2 1 −1

−2(k1 − k2 )

1 0

1/2

0 0



 −2k1 + k2 − 1 0 1    k1 0 0   . − 21 k2 0 0    −1 0 0  

The controllability Gramian for this problem is obtained from (3.21b) as 1 Wcon (k1 , k2 ) = 6(k1 + 1)(k2 + 1)2   2 2 4k − 5k1 k2 + 5/2k2 + 3k1 + 3/2 (k1 − k2 )(4k1 − 5k2 − 1)  1 . (k1 − k2 )(4k1 − 5k2 − 1) 4k12 − 11k1 k2 + 10k22 − 3k1 + 9k2 + 3 Finally, the resulting performance index can be computed as J=

(4k1 + 5k2 + 9)(2k12 k2 + 2k12 + k1 k22 + k22 + k1 + 2k2 + 3) . 24(k1 + 1)(k2 + 1)2

(5.7)

For a given performance level J = J ∗ , (5.7) provides an implicit mapping equation in the unknowns k1 and k2 . The minimal achievable performance level J for the decentralized output feedback (5.2) is bounded from below by the performance level Jf ull obtainable with a dense static-gain feedback controller. Using classical LQR theory, Jf ull is easily calculated as Jf ull = 0.824. In general J > Jf ull , therefore, we will map J = 1 into the parameter plane. Figure 5.2 shows the resulting parameter set Q2 for J = 1. The region satisfying both LQR and Γ-stability requirements is shaded in the figure. The figure actually shows that we can set k1 = 0, while still obtaining reasonable controllers. Note that without the damping specification on the closed-loop eigenvalues, we would still need to map the Hurwitz-stability requirement to get the correct LQR set. For this example, stability is assured if k1 > −1

∧

k2 > −1.

The actual boundary values k1 = k2 = −1 appear in the denominator of (5.7). The minimal obtainable performance level J for the decentralized controller can be computed using the algebraic equation (5.7). For k1 = 0.2074, k2 = 0.9329, we get J = 0.8421, which is only slightly higher than Jf ull . Compare this to J = 1.125 for the controller designed in Example 5.1.

98

Examples 3 k2 2

J < 1 1

–3

–2

–1

0

1

2

k1

3

–1

–2

–3

Figure 5.2: LQR boundaries (solid) and Γ-stability (dashed) for MIMO control example

The numerical values of J change with the state-space representation considered. Thus, the actual values of J should be only considered to measure the relative performance of a controller compared to a reference controller, e.g., the dense optimal controller or zerogain controller (open-loop system). This becomes apparent when one is computing J for the state-space representations (5.4) and (3.64), which both lead to the same input-output behavior but different quantitative values of J.

5.2.2 H∞ Norm: Robust Stability We use robust stabilization as a classical control problem that fits into the H∞ framework to motivate the mapping of H∞ norm specifications. Different from the traditional literature about H∞ control theory [Zhou et al. 1996],[Francis 1987] we will treat structured (parametric) and unstructured uncertainties. The well-known small-gain theorem (see Theorem 2.2 in Section 2.4.1 or [Zhou et al. 1996]) states that a feedback system composed of stable operators will remain stable, if the H∞ norm of the product of all operators is smaller than unity. As an example, consider a plant G(s) with multiplicative, unknown uncertainty ∆(s) at the output as in Figure 2.5 and associated weighting function W0 . The block diagram for the closed feedback loop with controller K(s) is shown in Figure 5.3.

99

5.2 MIMO Specifications

Wo

K

∆

G

Figure 5.3: Feedback system including plant with multiplicative uncertainty

The problem is, how large can ||∆||∞ be, so that internal stability is preserved? Using simple loop transformations, we can isolate the uncertainty ∆, which can be seen in Figure 5.4. W0 (I + GK)−1 GK

∆

Figure 5.4: Feedback system including plant with isolated multiplicative uncertainty

Using the small-gain theorem (Theorem 2.2), we get the following sufficient condition for internal stability with respect to unstructured multiplicative output uncertainty: ||∆||∞ ≤

1 . ||W0 (I − GK)−1 GK||∞

(5.8)

Example 5.4 We analyze the robust stability of the plant given in Example 5.1 for decentralized static-gain controllers (5.2) with respect to unstructured multiplicative output uncertainty. Consider the weighting function Wo (s) =

3s + 12 I. s+3

This implies a moderate relative uncertainty of up to 16% for low frequencies, which increases to higher frequency reaching 100% at about 1 [rad/sec] and finally going to 300% in the high frequency range. Figure 5.5 shows the gray-tone coded sets of parameters that correspond to different tolerable uncertainty sizes. Dark areas correspond to poor robustness, whereas areas

100

Examples

Robust stability for multiplicative uncertainty 2

1.5

1

k

2 0.5

0

−0.5

−0.5

0

0.5

1

k

1.5

2

1

Figure 5.5: Stability with respect to unstructured multiplicative uncertainty

with lighter colors indicate good robustness. The controller designed in Example 5.3 with k1 = 0.2074, k2 = 0.9329 yields better robustness than the initial controller from Example 5.1 with k1 = k2 = 1. Comparing the results in Figure 5.2 and Figure 5.5, we see that by varying k2 there is a tradeoff between robustness and performance.

5.2.3 Passivity Examples Example 5.5 Consider a general strictly proper second order system in pole-zero factorized form s − z1 . (5.9) G(s) = (s − p1 )(s − p2 ) Using a controllable canonical form state-space representation we obtain the Hamiltonian Hη as   0 1 0 0   1  −p p − z1 p + p + 1 0 − 1 2 1 2   2η 2η 2η Hη =  . z1 z1 z1   − 0 p p + 1 2   2η 2η 2η z1 1 1 −1 −p1 − p2 − 2η − 2η 2η

101

5.2 MIMO Specifications The resulting CRB (p1 + p2 − z1 )ω 2 + p1 p2 z1 = 0, 2(p1 + p2 − z1 )ω = 0,

has no valid solution for ω 6= 0 and dismantles into the RRB and IRB conditions: RRB: IRB:

p 1 p2 z1

= 0,

p1 + p2 − z1 = 0.

Evaluating eigenvalues of Hη with z1 = p1 + p2 for η → 0 we can deduce the simple rule that a general second order transfer function (5.9) is passive, if z1 > p 1 + p 2 ,

∀p1 , p2 < 0.

(5.10)

Example 5.6 Consider the following robust passivity problem. Let s2 + a 1 s + a 0 G(s) = 3 . s + 8s2 + 17s + 10

(5.11)

Then a controllable state-space realization in canonical form is given by   0 1 0 0      0 0 1 0  ∼  . G(s) =   1   −10 −17 −8   0 a0 a1 1 The passivity boundaries in the (a0 , a1 ) parameter plane and the resulting good parameter set Pgood are shown in Figure 5.6. Obviously the parameter set Pgood is contained in the first quadrant which corresponds to minimum phase systems. Furthermore it can be easily seen from this plot that there is no weakly minimum phase passive system, i.e., a1 = 0. To illustrate the robust passivity, the Nyquist plots for three exemplary passive systems are shown in Figure 5.7, which correspond to the three circular markers on the edges of Pgood in Figure 5.6.

102

Examples

10

8

6

a1

4

Pgood

2

PSfrag replacements

0

−2 −2

0

2

4

6

8

10

12

14

16

18

20

a0 Figure 5.6: Robust passivity

0.5

0.4

0.3

0.2

Imag

0.1

ω=−∞ 0

ω=0

ω=+∞

ω=0

ω=0

−0.1

−0.2

−0.3

−0.4

−0.5 −0.2

0

0.2

0.4

0.6

0.8

Real

Figure 5.7: Nyquist curves for passive systems

1

103

5.3 Example: Track-Guided Bus

5.3 Example: Track-Guided Bus We demonstrate the application of the presented methods by designing a controller for a track-guided bus introduced in [Ackermann et al. 1993, Ackermann and Sienel 1990], see also [Muhler and Ackermann 2000]. The task is to minimize the distance of the bus from a guideline. We investigate automatic steering based on feedback of the lateral displacement. An actuator which commands the front wheel steering angle δf is used. The lateral displacement y is measured by a sensor at the front bumper. Figure 5.8 shows a sketch of the bus with front wheel angle as input and the displacement sensor. δf CG

Front Sensor

y

Guideline

Figure 5.8: Track-guided bus

After an appropriate controller has been found for the nominal operating point the next step is to design a controller which simultaneously satisfies the specifications for several operating conditions. Thus, we repeat the previous design for the four vertices of the operating domain Q and try to find a simultaneous solution and choose the controller parameters accordingly. A controller which stabilizes the four vertex conditions is likely to successfully stabilize the whole operating domain. But satisfaction of the specifications for the four vertices is not sufficient for all parameters q ∈ Q. Hence, as a final step we have to do a robustness analysis in the (q1 , q2 )-plane to check for satisfaction of performance specifications in the entire operating domain Q. Note that we understand the analysis for the whole operating domain Q as an essential step in the robust controller design process. If initially the performance criteria are not satisfied for the entire operating domain, then the design is repeated with further representatives of Q in addition to the vertices. The transfer function of the bus with uncertain parameters mass m (normalized by the friction coefficient between tire and road) and velocity v is given by G(s) =

a0 v 2 + a1 vs + a2 mv 2 s2 . s3 (b0 + b‘0 mv 2 + b1 mvs + m2 v 2 s2 )

104

Examples

The operating domain is given by m ∈ [9.95; 32]t and v ∈ [3; 20]m/s, which represents all possible operating conditions relevant for a city bus. Figure 5.9 shows the operating domain for the bus with the four vertex points.

Figure 5.9: Operating domain for bus

The following controller structure was used in [Ackermann et al. 1993]: K(s) =

k1 + k 2 s + k 3 s2 . d0 + d 1 s + d 2 s2 + d 3 s3

The coefficients k1 , d0 . . . d3 are fixed. A root locus plot shows that the controller parameters k2 , k3 , which determine the zeros of K(s), are most crucial for the design step. Therefore we design the controller in the k2 , k3 -plane.

5.3.1 Design Specifications The specifications for the closed loop system can be expressed through Γ-stability. All roots should lie left to the hyperbola σ 2 ω 2 − = 1. (5.12) 0.35 1.75

This guarantees a maximal settling time T = 2.9 s and a minimal damping ζ = 0.196.

A suitable controller which simultaneously stabilizes the four vertices of the operating domain for these specifications was determined in [Ackermann et al. 1993] as K(s) =

253 (0.15s2 + 0.7s + 0.6) . (s + 25)(s2 + 25s + 625)

We are are extending these specifications by trying to maximize the Nyquist stability margin.

5.3 Example: Track-Guided Bus

105

5.3.2 Robust Design for Extreme Operating Conditions We design a simultaneous controller for the four vertex operating conditions. The task is to tune k2 , k3 such that the roots lie left of (5.12) and the worst Nyquist stability margin is maximized for the four representatives. By mapping the Nyquist stability margin using (2.34) and the Γ-stability boundaries for the four vertices Figure 5.10 is generated. The plots are arranged as in Figure 5.9 with minimal v and m in the lower left corner.

Figure 5.10: Color coded ρ and Γ-stability boundaries

To make the joint analysis for all four vertices easier we generate the worst-case overlay by determining the set of simultaneous Γ-stabilizing controllers through intersection and plot the worst-case Nyquist stability margin for the four vertices. Figure 5.11 shows the worst case overlay for four representatives in the (k2 , k3 )-plane. From this plot we can choose k2 , k3 values from the admissible set with maximal worst-case ρ. For a controller with values k2 = 0.7, k3 = 0.15 Γ-stability is guaranteed for vertex conditions, but we could only achieve a poor Nyquist stability margin.

5.3.3 Robustness Analysis The controller resulting from the previous design process satisfies the given specifications at least for the extremal operating conditions. As a final step we verify the Γ-stability and Nyquist stability margin specifications for the whole range of operating conditions. We therefore map the eigenvalue and Nyquist stability margin specifications into the v, mplane. Figure 5.12 shows the Γ-stability boundaries and the color coded Nyquist stability

106

Examples

0.25

0.2

k3 = 0.15

0.15

0.1

0.05 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k2

selected controller

k2 = 0.7

Figure 5.11: Worst-Case Overlay

margin in the (v, m) parameter plane. The Γ-stability boundaries do not intersect the operating domain, depicted by a rectangle. In addition the color coded Nyquist stability margin in Figure 5.12 is sufficient for the entire operating domain. Hence, the designed controller guarantees robust satisfaction of the given specifications. Instead of using Figure 5.12 as a mere robust stability check, we get valuable information about the performance for different operating conditions. The most critical operating condition regarding Nyquist margin occurs for maximal velocity and minimal mass, whereas for the Γ-stability margin the worst case is maximal velocity and maximal mass.

5.4 IQC Examples Example 5.7 Consider the following nonlinear control example depicted in Figure 5.13 with a PI controller, a dead zone which models the actuator, and a linear plant G(s). The transfer function of the controller is given by KP I (s) = k1 + ks2 . The plant is given by G(s) =

qs + 1 , s2 + s + 1

where q ∈ R is an uncertain parameter. We aim at analyzing the robustness of the system with respect to variations in q. Furthermore we want to tune the controller such that robustness to parameter variations is achieved.

107

5.4 IQC Examples Γ−stability and Frequency−Domain Analysis 50

45

40

35

30

m

25

20

15

10

5

0

5

10

15

20

25

30

v Figure 5.12: Analysis in the operating domain

The given feedback interconnection is called critical since the worst case linearization is at best neutrally stable. Note that the transfer function KP I (s)G(s) is unbounded which prevents the application of standard stability criteria for nonlinear systems which require bounded operators. r

PI

G(s)

y

Figure 5.13: Dead zone PI controller example We use the Zames-Falb IQC derived in [Jönsson and Megretski 1997], where it was shown that an integrator and a sector bounded nonlinearity can be encapsulated in a bounded operator that satisfies the following IQC   ∗ 0 1 − H(jω) , Π(jω) =  (5.13) 1 − H(jω) − k2 Re(1 − H(jω) − kF (jω)) where

H(s) − H(0) , s where H(s) is a stable transfer function with L1 norm less than one, and the parameter k equals the static gain of the open loop linear part k = k2 G(0). This IQC corresponds to the Zames-Falb IQC for slope restricted nonlinearities [Zames and Falb 1968]. F (s) =

108

Examples

Let the integral gain k2 = 2/5 and H(s) = 1/(s + 1). For our particular example the parameter k equals the proportional gain k = k2 . We map the stability condition into the (k1 , q) parameter plane. This allows to evaluate robustness with respect to q, while we can select the controller gain k1 to maximize the robustness. Since the multiplier Π(jω) in (5.13) is frequency-dependent, we use the method described in Section 3.4.4 to reformulate the IQC stability problem with a constant multiplier. The multiplier evaluates to 

 jω 0  jω − 1    Π(jω) =  .  jω 5ω 2 + 2  − 2 jω + 1 ω +1

(5.14)

This multiplier is not positive definite, so that most algorithms for spectral factorization fail. A remedy is to use a constant offset matrix Π0 which makes the remainder positive definite: Π(jω) = Π0 + Πp (jω). The transfer matrix Πp (jω) can now be factorized. So we alter (3.36) from Π(jω) = Ψ(jω)∗ Πs Ψ(jω), into Π(jω) = Ψ(jω)∗ Πs Ψ(jω) + Π0 , to get the spectral factorization ∗    Π 0 s Ψ(jω) Ψ(jω)    , Π(jω) =  I I 0 Π0 

which again has the form (3.36). ˜ B) ˜ in (3.43) is of forth order: For this particular example the augmented system (A, 

−1

 0  A˜ =  0  0

0

1 − q2 − 25 k1 1 − 52 k1 q2

−1

0

0

0

0

1

0

−1

−1



   ,  

  0    1  ˜= B  ,  0   1

and the submatrices Q, S and R of the multiplier M relevant for the mapping equations are given as:

109

5.4 IQC Examples



M = 



Q

S

T

R

S

 

4

−2

  −2 3   5 =   −2 + 2q2 + 5k1 1 − q2 − 2 k1   −2 + 5k1 q2 1 − 25 k1 q2  0 0

−2 + 2q2 + 5k1 −2 + 5k1 q2 1 − q2 − 25 k1

1 − 52 k1 q2

0

0

0

0

1 − q2 − 25 k1

1 − 52 k1 q2

0



    1 − q2 − 25 k1  .  1 − 25 k1 q2   −5 0

The corresponding mapping equations are of eighth order containing k1 and q2 as parameters. The resulting stability boundaries are shown in Figure 5.14. The set of stable parameters Pgood contains the origin. 10 8

q

6 4

Pgood

2

PSfrag replacements 0 −2 −2

0

2

4

6

8

10

k1 Figure 5.14: Stability boundaries To evaluate the conservativeness of the results numerical simulations were performed using the nonlinear system. The simulations showed that the upper line shown in Figure 5.14 is far from the real boundary, while the lower boundary is very close to the actual boundary. Figure 5.15 not only shows the nonlinear boundaries (solid) but also the stability boundaries for a linear system (dashed) which lacks the nonlinear dead zone actuator. The results show that the nonlinear stability region is only slightly smaller than the linear counterpart. Although the mathematical description of the curves is different, however.

110

Examples

2 1.5 1

Pgood

q

0.5 0

−0.5

PSfrag replacements

−1 −1.5 −2 −1

0

1

k1

2

3

Figure 5.15: Comparison of linear and nonlinear system

5.5 Four Tank MIMO Example A multivariable laboratory process is considered to show a practical application of the robust control analysis and synthesis methods presented in this thesis. The process is the level control of a system of four interconnected water tanks. The process has been described in [Johansson 2000]. The system is shown in Figure 5.16. The system not only shows considerable cross-couplings, but has an adjustable multivariable zero. Furthermore it has static and dynamic nonlinearities. The task is to control the water levels of the first and second tank by varying the flows generated by the two pumps. The inputs are the control voltages for the pumps v1 and v2 and the outputs are level measurement voltages y1 and y2 . The nonlinear system model can be derived from mass balances and energy conservation in the form of Bernoulli’s law for flows of incompressible, non-viscous fluids a1 p 2gh1 + h˙ 1 = − A1 a2 p h˙ 2 = − 2gh2 + A2 a3 p h˙ 3 = − 2gh3 + A3 a4 p 2gh4 + h˙ 4 = − A4

a3 p γ1 k1 2gh3 + v1 A1 A1 a4 p γ2 k2 v2 2gh4 + A2 A2 (1 − γ2 )k2 v2 A3 (1 − γ1 )k1 v1 , A4

(5.15)

where Ai is the cross-section of tank i, ai the cross-section of outlet i, and hi the water level of the i-th tank. The output signals for the measured levels are proportional to the water level y1 = kc h1 and y2 = kc h2 .

111

5.5 Four Tank MIMO Example

Tank 3

Tank 4

(1 − γ1)v1

(1 − γ2)v2 γ1 v1

v1

γ2 v2

Tank 1

Tank 2

Pump 1

v2 Pump 2

Figure 5.16: Schematic diagram of the four-tank process

The transfer matrix linearized for a given static operating point is 

  G(s) =  

γ1 c11 (T1 s + 1) (1 − γ1 )c21 (T2 s + 1)(T4 s + 1)

 (1 − γ2 )c12 (T1 s + 1)(T3 s + 1)   .  γ2 c22

(5.16)

(T2 s + 1)

Let the ratio of water diverted to tank one rather than tank three be γ1 = 0.33 and the corresponding ratio of pump two is set to γ2 = 0.167. All other nominal parameter values are given in Table 5.1. The transfer matrix (5.16) then has two multivariable zeros at s = −0.197 and s = 0.118. The multivariable RHP zero limits the achievable performance for this system. Table 5.1: Parameter values for four tank example A1 , A2 , A3 , A4

[cm2 ]

[30.0, 30.0, 20.0, 20.0]

a1 , a2 , a3 , a4

[cm2 ]

[0.1, 0.067, 0.067, 0.1]

k1 , k2 kc

[cm3 /Vs]

3.0

[V/cm]

1.0

112

Examples

We consider decentralized PI control with input-output pairing y1 − u2 and y2 − u1 suggested by relative gain array analysis. Here, we link both Kp and Ki parameters of the individual loops, i.e. Kp = Kp1 = Kp2 and Ki = Ki1 = Ki2 and map a real part limitation of Re s < 0.004 and a nominal performance ||Wp S||∞ specification, where S is the sensitivity and Wp = (250s + 10)/(1000s + 1) a performance weighting function. The resulting plot is shown in Figure 5.17 where good performance is represented by light colors and the eigenvalue specification is depicted by dashed lines. Taking an optimal value from Figure 5.17 the resulting multivariable controller can be fine-tuned, for example by considering the individual loops or by evaluating robustness with respect to changes in the nominal parameter values. Four tank PI controller design

0.018

0.016

0.014

0.012

K

i

0.01

0.008

0.006

0.004

0.002 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

K

p

Figure 5.17: PI controller design for four tank example

113

6

Summary and Outlook

We conclude with a brief summary of the work presented, and remarks on future directions for related research.

6.1 Summary Robust control of systems with real parameter uncertainties is a topic of practical importance in control engineering. In this thesis we considered the mapping of new specifications. Although well-known and often used, some of these specifications have not been used in the parameter space context. Furthermore, the considered specifications can be mapped for multivariable systems. Special criteria for nonlinear systems are presented. To this end, not only standard specifications like the Popov criterion are considered, but the mapping of versatile integral quadratic constraints is introduced. Starting point of the thesis is the observation that many practical specifications, for example H∞ norm and passivity, can be formulated using the same mathematical framework. Namely algebraic Riccati equations (AREs). An important link forms the KYP lemma, which translates different mathematical formulations of specifications into each other. A corresponding set of specifications can be formulated using Lyapunov equations, a special case of an ARE. Important representatives are the H2 norm and LQR specifications. These allow to express performance specifications in parameter space. These specifications directly lead to a single implicit mapping equation, which is in general at least quadratic in the uncertain parameters. Mathematical results on the analytic dependence of solutions for parametric AREs led to the derivation of mapping equations. Hereby control specifications formulated as AREs can be converted into a specific eigenvalue problem using properties of associated Hamiltonian matrices. The introduction of mapping equations for IQCs broadens the applicable system class even further by enabling to consider specifications for input-output theory, absolute stability theory, and the robust control field. The exploitation of additional degrees of freedom provided by variable uncertainty characterization is shown.

114

Summary and Outlook

Mapping equations for ARE and Lyapunov equation based specifications are similar in structure to root locus specifications. Nevertheless, due to the quadratic nature of the specifications, which shows in the AREs, the mapping equations are in general more complex and nontrivial to solve. Practical aspects of the thesis therefore include the presentation of a hybrid symbolicnumerical algorithm. This algorithm exploits the properties of the algebraic mapping equations to determine characteristic points on the resulting curves in a parameter plane. The topology of the curve then allows to either numerically connect the individual curve points by curve following or to approximate the curve using a Bézier approximation. The uniform mathematical description of the mapping equations allows to employ a single mapping algorithm for all specifications. The mapping equations are based on a parametric state-space realization. A symbolic algorithm is hereto presented, which calculates a state-space realization for a given transfer function. This thesis shows that classical eigenvalue criteria and modern norm and nonlinear specifications can be combined in the parameter space approach to yield an efficient control engineering tool that takes all practical aspects like stability, performance and robustness into account.

6.2 Outlook We note related future research topics. In this thesis, a comprehensive methodology to map control specifications into parameter spaces is presented. The successful and wide-spread use of these methods necessitates an efficient and robust software implementation of the given algorithms. Furthermore, a user-friendly front-end facilitates the practical application of the thesis results. It can be used to specify the considered system and corresponding specifications and to evaluate the graphical results. One possible way is a computer toolbox with graphical user interaction. While the new possibilities have been demonstrated on some practical examples, one line of research is to apply the methods to a large-scale real-world problem and to investigate the limits and computational burdens. There are numerous ways to characterize uncertainty. While the approach pursuit in this thesis is to deal explicitly with real uncertain parameters, one research topic is to evaluate different uncertainty characterizations.

115

A

Mathematics

This appendix reviews some mathematical topics used in the thesis, that are not necessarily treated in graduate level engineering courses. Particularly all theorems needed to prove the fundamental theorems in Chapter 3 are stated here. We shall always work with finite dimensional Euclidean spaces, where Rn = {(x1 , . . . , xn ) : x1 , . . . , xn ∈ R},

Cn = {(x1 , . . . , xn ) : x1 , . . . , xn ∈ C}.

A point in Rn is denoted by x = (x1 , . . . , xn ) and the coordinate or natural basis vectors are written as ei :       1 0 0       0  1  0       e1 =  .  , e2 =  .  , . . . , e n =  .  .  ..   ..   ..        0 0 1 All arguments of functions are either given in parenthesis or omitted for brevity.

A.1 Algebra This section reviews basic facts from tensor algebra. The use of tensors facilitates the notation of matrix derivatives used for some algorithms presented in Chapter 4. Furthermore tensors are supported by symbolic and numerical software packages such as Matlab and Maple and therefore allow an easy implementation of the presented algorithms. See [Graham 1981] for a good reference on tensor algebra. Given two matrices A ∈ CnA , mA , B ∈ CnB , mB , the Kronecker product matrix of A and B, denoted by A ⊗ B ∈ CnA ∗nB , mA ∗mB , is defined by the partitioned matrix   a11 B a12 B · · · a1mA B      a21 B a22 B · · · a2mA B   . A ⊗ B :=  .  .. .. . ·  .  . ·· .   a nA 1 B a nA 2 B · · · a nA mA B

116

Mathematics

The Kronecker power of a matrix is defined similar to the standard matrix power as X ⊗,2 = X ⊗ X,

X ⊗,3 = X ⊗ X ⊗,2 ,

X ⊗,i = X ⊗ X ⊗,i−1 ,

(A.1)

where the superscript ⊗, i denotes the i-th Kronecker power. The Kronecker sum of two matrices A ∈ CnA , nA B ∈ CnB , nB is defined as A ⊕ B := A ⊗ InB + InA ⊗ B, where In is the identity matrix of order n. Furthermore, let vec(X) denote the vector that is formed by stacking the columns of X into a single column vector: h i vec(X) := x11 x21 . . . xm1 x1n x2n . . . xmn T . Using this stacking operator the following properties hold for complex1 matrices with matching dimensions: vec(AXB) = (B T ⊗ A) vec(X), and vec(AX + XB) = (B T ⊕ A) vec(X).

A.2 Algebraic Riccati Equations This section will review important facts about algebraic Riccati equations (AREs). In order to make this section rather self-contained we will state theorems presenting basic properties of AREs. The general algebraic matrix Riccati equation is given by XRX − XP − P ∗ X − Q = 0 ,

(A.2)

where R, P and Q are given n×n complex matrices with R and Q Hermitian, i.e., R = R ∗ and Q = Q∗ . Together with (A.2) we will consider the matrix function R(X) = XRX − XP − P ∗ X − Q. 1

Note that

T

denotes the transpose, while

∗

(A.3) is used for the complex conjugate transpose

A.2 Algebraic Riccati Equations

117

Associated with (A.2) is a 2n × 2n Hamiltonian matrix: 

H := 

−P

R

Q P

∗



.

(A.4)

The following theorem [Zhou et al. 1996] gives a constructive description of all solutions to (A.2). Theorem A.1 ARE solutions Let V ⊂ C2n be an n-dimensional invariant subspace of H, and let X1 , X2 ∈ Cn×n be two complex matrices such that   X1 V = Im   . X2

If X1 is invertible, then X = X2 X1−1 is a solution to the Riccati equation (A.2) and Λ(P − RX) = Λ(H|V ), where H|V denotes the restriction of H to V.

Proof: We only show the first part of the theorem that proofs the construction of solutions. The following equation holds because V is an M invariant subspace:      −P R X X    1  =  1  Λ. Q P∗ X2 X2 Premultiply the above equation by [X −I ] and then postmultiply with X1−1 to get h

   i −P R I    = 0, X −I  X Q P∗ XRX − XP − P ∗ X − Q = 0 .

This shows that X = X2 X1−1 is actually a solution of (A.2).

118

Mathematics

Theorem A.1 shows that we can determine all solutions of (A.2) by constructing bases for those invariant subspaces of H. For example, the invariant subspaces can be found by computing the eigenvectors v i and corresponding generalized eigenvectors v i+1 , . . . , v i+ki −1 related to eigenvalues λi of H with multiplicity ki . Taking all combinations of these vectors, which have at least one actual eigenvector v i and are therefore H invariant, we can calculate the solutions X = X2 X1−1 from   h i X  1  = v i v j , i 6= j. X2 Before we turn to the important stabilizing solutions of an ARE, we consider maximal solutions. The following theorem will be used in the proof of Theorem A.3 also. Theorem A.2 Maximal solutions Suppose that R = R∗ ≥ 0, Q = Q∗ , (P, R) is stabilizable, and there is a Hermitian solution of the inequality R(X) ≤ 0. Then R(X) = 0 has a maximal Hermitian solution X + for which X + ≥ X for every Hermitian solution X of R(X) ≤ 0 holds. Furthermore the maximal solution X + guarantees that all eigenvalues of P − RX + lie in the closed left half-plane. See [Kleinman 1968], [Lancaster and Rodman 1995, p. 232] or [Zhou et al. 1996] for a proof. The proof is constructive, i.e., it not only proofs the preceding statements, but it also gives an iterative procedure to determine the maximal solution X + . A Newton procedure can be derived to solve the equation R(X) = 0. Decompose R into R = BB ∗ . Since the pair (P, R) is stabilizable, it can be seen from the controllability matrix that the pair (P, B) is also stabilizable and there is a stabilizing feedback matrix F0 for which P0 = P − BF0 is stable. Then we determine X0 as the unique, Hermitian solution of the Lyapunov equation X0 P0 + P0∗ X0 + F0∗ F0 + Q = 0. In order to apply the Newton procedure to the matrix function R(X), we need the first Fréchet derivative [Ortega and Rheinboldt 1970]: dRX (H) = −(H(P − RX) + (P − RX)∗ H). The Newton procedure is than given as dRX (Xk+1 − Xk ) = −R(Xk ),

k = 0, 1, 2, . . .

(A.5)

119

A.2 Algebraic Riccati Equations This can be written as the following Lyapunov equation, Xk+1 (P − RXk ) + (P − RXk )∗ Xk+1 = −Xk RXk − Q,

k = 0, 1, 2, . . .

(A.6)

We can now finally turn to the important Theorem 3.1 which forms the basis for the mapping equations, because it converts conditions for an ARE into an eigenvalue problem. The theorem is restated here, see [Lancaster and Rodman 1995, p. 196]. The actual mapping equations can be derived using the analytic extension given by Lancaster and Rodman in 1995. Theorem A.3 Stabilizing solutions Suppose that R ≥ 0, Q = Q∗ , (P, R) is stabilizable, and there is a Hermitian solution of (A.2). Then for the maximal Hermitian solution X + of (A.2), P −RX + is stable, if and only if the Hamiltonian matrix H defined in (A.4) has no eigenvalues on the imaginary axis. Proof: Since the pair (P, R) is only required to be stabilizable, we start with a decomposition into a controllable and stable part, such that the matrices P , R, Q, and X can be written as         P11 P12 R 0 Q Q X X12 , R =  11 , Q =  11 12 , X =  11 , (A.7) P = ∗ ∗ 0 P22 0 0 Q12 Q22 X12 X22

where the pair (P11 , R11 ) is controllable and P22 is stable. This decomposition leads to a Riccati equation in standard form for the controllable pair (P11 , R11 ) ∗ X11 R11 X11 − X11 P11 − P11 X11 − Q11 = 0 ,

(A.8)

a Sylvester equation in X12 , and a Lyapunov equation in X22 . Furthermore the term P − RX can be written as   P11 − R11 X11 P12 − R11 X12 , P − RX =  (A.9) 0 P22 and the ARE (A.8) has the associated Hamiltonian H11 .

We are now ready to proof that stability of P − RX + follows from H having not a single pure imaginary eigenvalue. To this end, suppose that H has no eigenvalues on the imaginary axis. The decomposition above leads to ∗ Λ(H) = Λ(H11 ) ∪ Λ(−P22 ) ∪ Λ(P22 ),

(A.10)

120

Mathematics and any solution X11 of (A.8) has the property Λ(P11 − R11 X11 ) ⊆ Λ(H11 ). Using + + Theorem A.2 it follows that (A.8) has a maximal solution X11 , and Λ(P11 − R11 X11 ) is stable. From the triangular decomposition and in particular (A.9) then follows that P − RX + holds for the corresponding solution X + of (A.2). To proof the converse, assume that X + is a maximal solution of (A.2) and P − RX + is stable. Let X + be decomposed similar to X in (A.7). From (A.9) then follows + + that P11 − R11 X11 is stable and that X11 is a maximal solution of (A.8). Therefore H11 has no pure imaginary eigenvalues and the proof is concluded, since (A.10) implies that the same holds for H.

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129

Lebenslauf Michael Ludwig Muhler geboren am 3. Februar 1973 in Creglingen verheiratet, ein Kind September 1983 - Mai 1992

Gymnasium Weikersheim

Oktober 1993 - Juli 1997

Studium der Technischen Kybernetik an der Universität Stuttgart

August 1997 - September 1998

Graduate Studies in Chemical and Electrical Engineering University of Wisconsin, Madison

Oktober 1998 - Januar 1999

Studium der Technischen Kybernetik an der Universität Stuttgart

Januar 1999 - August 2002

Wissenschaftlicher Mitarbeiter beim Deutschen Zentrum f¨ ur Luft- und Raumfahrt Oberpfaffenhofen Institut f¨ ur Robotik und Mechatronik

seit September 2002

Mitarbeiter der Robert Bosch GmbH Stuttgart

Korntal-M¨ unchingen, im März 2007