Multiobjective Control: Linear Matrix Inequality Techniques and

Multiobjective Control: Linear Matrix Inequality Techniques and Genetic Algorithms Approach

Thesis submitted for the degree of Doctor of Philosophy

Arturo Molina Cristóbal

The University of Sheffield Department of Automatic Control and Systems Engineering

Sheffield UK April 2005

© 2005 Arturo Molina-Cristobal In accordance with the copyright legislation no information derived from the dissertation nor quotation from it may be published without full acknowledgment of the source being made nor any substantial extract from the dissertation published without the author’s written consent.

Abstract This thesis addresses some of the open problems in multiobjective control. The aim of this thesis is to compare two emerging techniques in multiobjective control: evolutionary algorithms (EAs) and convex optimisation over linear matrix inequalities (LMIs). In the multiobjective control problem, a trade-off is sought between competing objectives. In such a problem, no single optimal solution exists, rather a set of equally valid solutions, known as the Pareto optimal set. It has been shown that the multiobjective control problem can be tackled with LMI techniques, due to its ability to include convex constraints such as H2 performance, H∞ performance, and poleplacement. The multiobjective control problem is formulated as a semidefinite programming (SDP) problem, which is a single objective convex optimisation problem, solved using interior-point methods. As a novel alternative, multiobjective optimisation using EAs can offer a truly multiobjective treatment of control systems specifications. This approach is described herein and compared with numerical results from the counterpart LMI techniques. This investigation addresses some of the drawbacks of LMI techniques, such as the inability to reduce the order of the controller and the LMI technique's inherent conservatism when tackling multiobjective problems. The truly multiobjective optimisation using an EA is proposed to overcome these problems. Reduction of the order of the controller is achieved for problems such as the H∞ controller design with time-domain specifications. The mixed H2/H∞ control problem is treated as a multiobjective H2/H∞ control problem, and an improvement of the Pareto optimal set is achieved. Both methods are applied to a gas turbine engine controller design problem. Numerical problems with controllers resulting from using the LMI approach are addressed and a solution, based on the EA, is shown to design more numerically robust controllers than the LMI approach.

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The controller design from this application problem is then extended to a gain scheduling controller design with the use of linear parametric-varying (LPV) techniques. The issue of applying LPV modelling and LPV control is addressed, and the EA-based method is proposed to design multiobjective H2/H∞ controllers for two operating points. The results give satisfactory, well-conditioned controllers.

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Acknowledgments I would like to thank my supervisors Professor Peter J Fleming and Professor David H Owens for their advice, guidance, and encouragement throughout the period of this research work. It has been a privilege working with them. I would also like to thank Didier Henrion for introducing me to novel areas of LMI techniques. Thanks also to my colleagues Jorge Escamilla for discussions in related topics of this thesis, Robin Purshouse for his expert advice in genetic algorithms, Ian Griffin for his expert advice in gas turbine engines, and thanks to John Watson and Mike Dewar who proof read this manuscript. This work has been funded by Consejo Nacional de Ciencia y Tecnología (CONACyT) – México. I have also received financial support from the Rolls Royce University Technology Centre (RR-UTC) at the Department of Automatic Control and Systems Engineering, Sheffield University, which facilitated me to attend technical events related to this research work. I am extremely grateful to my parents Eve and Arturo, brothers Felix and Tonatiuh and especially to my sister Esperanza for their love and support. Thanks and my love goes to Helen for her understanding and support for the completion of this project.

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Statement of Originality Unless otherwise stated in the text, the work described in this thesis was carried out solely by the candidate. None of this work has already been accepted for any degree, nor is it concurrently submitted in candidature for any degree. Candidate:

________________________________ Arturo Molina-Cristóbal

Supervisors:

_________________________________ Peter J Fleming

_________________________________ David H Owens

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A mi Mamá y Papá con Cariño (To my Parents with love)

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Contents Chapter 1 ......................................................................................................................1 Introduction..................................................................................................................1 1.1 Motivation of the Research......................................................................................1 1.2 Outline of the Thesis................................................................................................3 1.3 Contributions............................................................................................................4

Chapter 2 ......................................................................................................................6 Review of Linear Matrix Inequalities and Evolutionary Optimisation in Multiobjective Control ................................................................................................6 2.1 Introduction..............................................................................................................6 2.2 Definition of a Linear Matrix Inequality .................................................................7 2.3 Basic properties........................................................................................................7 2.3.1 Convexity……………………………………………………………………...7 2.3.2 Geometry of LMI sets .......................................................................................9 2.3.3 Semidefinite Programming (SDP) ..................................................................10 2.4 Brief History of Linear Matrix Inequalities ...........................................................11 2.4.1 Interior-Point Methods....................................................................................13 2.4.2 Robust Control: Convex approach..................................................................14 2.5 Application and New Trends of LMI Techniques .................................................15 2.6 Quadratic stability of Lyapunov ............................................................................17 2.7 Control Systems problems expressed as LMIs ......................................................19 2.7.1 Definition of the Schur complement...............................................................20 2.7.2 LMI formulation of the LQR optimal problem...............................................21 2.7.3 YKP Lemma ...................................................................................................23 2.7.4 Bounded Real Lemma.....................................................................................23

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2.8 Multiobjective optimisation ...................................................................................25 2.8.1 Pareto Optimality ............................................................................................26 2.9 Multiobjective Control...........................................................................................26 2.10 Genetic Algorithms..............................................................................................31 2.10.1 The population ..............................................................................................32 2.10.2 A Simple Function Optimisation Problem....................................................32 2.10.3 Replication (Selection)..................................................................................33 2.10.4 Crossover (Recombination). .........................................................................34 2.10.5 Mutation 35 2.11 Multiobjective Genetic Algorithms .....................................................................39 2.11.1 Pareto Optimal Ranking................................................................................42 2.11.2 Fitness Sharing and Kernel Density Estimation ...........................................43 2.11.3 Mating Restriction ........................................................................................43 2.12 Conclusions..........................................................................................................44

Chapter 3 ....................................................................................................................45 Multiobjective Optimisation of Controller Structure via YK paramatrization using GAs and LMIs ............................................................................................................45 3.1 Introduction............................................................................................................45 3.2. Problem Statement ................................................................................................47 3.2.1 Youla-Kučera parametrization of stabilizing controllers................................47 3.2.2 LMI optimisation for fixed-order H∞ controller design..................................50 3.3. Optimizing Controller Structure with Genetic Algorithms ..................................52 3.3.1 Evaluation function .........................................................................................52 3.4. Numerical Examples.............................................................................................54 3.4.1 Example 1: Low-order damping mode ...........................................................54 3.4.2 Example 2: Flexible Beam (Doyle et al. 1992):..............................................63

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3.5. Conclusions...........................................................................................................68 Chapter 4......................................................................................................................69 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method. ..................69 4.1 Introduction............................................................................................................69 4.2 Problem statement..................................................................................................71 4.3 LMI Formulation of H2 and H∞ .............................................................................72 4.3.1 Quadratic performance synthesis (Scherer et al. 1997) ..................................72 4.3.2 H2 Performance ...............................................................................................75 4.3.3 H∞ Performance ..............................................................................................78 4.4 Mixed H2/H∞: LMI Approach................................................................................79 4.5 Mixed H2/H∞: MOGA-based method ....................................................................81 4.6 Equivalence of LQG and H2 Optimal Control .......................................................82 4.7 H2 Optimal Control: MOGA-based method ..........................................................84 4.8. Numerical Evaluations of MOGA and LMIs........................................................85 4.8.1 H2 Optimal Control (LQG): LMI Approach. ..................................................85 4.8.2 H2 Optimal Control (LQG): MOGA-based method .......................................88 4.8.3 Mixed H2/H∞: LMI Approach.........................................................................91 4.8.4 Example of Mixed H2/H∞ using MOGA Approach........................................93 4.9. Conclusions...........................................................................................................96

Chapter 5 ....................................................................................................................97 Multiobjective Control Design for the Turbofan Engine.......................................97 5.1 Introduction............................................................................................................97 5.2 Gas Turbine Engine ...............................................................................................97 5.2.1 Control Systems for Gas Turbine Engines....................................................100 5.2.2 Gas Turbine Engine ANTLE. .......................................................................102 5.3 Controller Design for a Linear Point. ..................................................................104 viii

5.3.1 Scaling……………………………………………………………………...104 5.3.2 Design Controller Strategy. ..........................................................................104 5.4 Mixed H2/H∞ Controller Design ..........................................................................107 5.4.1 Numerical Aspects ........................................................................................108 5.5 Multiobjective H2/H∞ Controller Design: MOGA-based method .......................111 5.6 Gain Scheduling...................................................................................................114 5.7 Linear Parameter Varying Techniques ................................................................116 5.8 LPV Modelling ....................................................................................................119 5.8.1 Polytopic Systems .........................................................................................119 5.9 LPV Control.........................................................................................................122 5.10 Gain Scheduling for the Gas Turbine Engine ....................................................124 5.11 LPV Gain-scheduling MOGA-based Method ...................................................131 5.12 Conclusions........................................................................................................136

Chapter 6 ..................................................................................................................137 Conclusions and Future Work................................................................................137 6.1 Multiobjective optimisation .................................................................................137 6.2 Suitability.............................................................................................................138 6.3 Future Work .........................................................................................................141

Appendix A................................................................................................................142 Appendix B ................................................................................................................143 References:.................................................................................................................147

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Chapter 1 Introduction 1.1 Motivation of the Research It is safe to say that any system design involves simultaneous consideration of multiple criteria. In control systems engineering, the design of a single controller which meets a collection of performances or objectives is, in theory and in practice, an open problem. Often some of these objectives will be in conflict, e.g. rise time and overshoot, and might be limited, e.g., the amount of achievable disturbance attenuation in a certain frequency region is limited by constraints on the systems bandwidth, where trying to improve one specification will cause deterioration in another. Thus, a trade-off exists, and is treated by using multiobjective control optimisation, in which a trade-off is sought between competing objectives. In such an optimisation, no single optimal solution exists, rather a set of equally valid solutions, known as the Pareto optimal set. The multiobjective control problem has been under research for some time and several techniques have been proposed to solve this problem. However, research in this area has been initially carried out with single optimisation techniques; this approach makes the multiobjective problem into a formulation of a single objective problem. Moreover, all specifications and constraints of a different nature have to be translated into a unique performance measure (e.g. H2, H∞).

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Chapter 1 Introduction Independent formulation of performance criteria in control theory is well established. One example is the H∞ specification, which ensures that the closed loop system is robustly stable, and the H2 specification, which ensures optimal output performance. However, when these methods are combined, the objectives conflict and the existing methods are not prepared to handle both specifications at the same time. Specifically, theoretical results of the mixed H2/H∞ problem have shown the difficulties that a unified framework involves. Moreover, in practice it is difficult to solve coupled Riccati equations (Doyle et al. 1994; Zhou et al. 1994). Research into this problem has shown that it can be reinterpreted as a convex optimisation problem (Khargonekar and Rotea 1991a; Scherer 1995b). This approach relies on the efficient interior-point algorithm to solve the optimisation problem. Almost a decade ago, this observation introduced a whole new area of research known as semidefinite programming (SDP), which is a convex optimisation problem over linear matrix inequalities (LMIs). The motivation to use LMIs is that several control specifications are naturally formulated as convex constraints. Thus, a unified framework can be proposed (Skelton and Iwasaki 1994). Although advances in control systems with the LMI approach were achieved, this formulation did not overcome open problems, such as conservatism and the low order of the controller. Moreover, this new formulation leads to single objective optimisation problems when multiobjective control problems are addressed, such as the multiobjective H2/H∞ problem. Therefore, this problem is named the mixed H2/H∞ problem. In order to address these problems, the application of an evolutionary approach is considered in this research. The aims were: •

To offer a truly multivariable treatment to problems such as the mixed H2/H∞ problem and H∞ performance with time-domain specifications.

•

To reduce conservatism.

•

To reduce and/or fix the order of controller.

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Chapter 1 Introduction

1.2 Outline of the Thesis Chapter 1 illustrates briefly how LMIs emerge in control systems and provides the necessary background such as the properties of LMIs and the LMI geometry in two dimensions. Fundamental techniques to express control problems in an LMI formulation, (e.g. the YKP Lemma), are provided as well as historical presentation of LMIs in control systems. The state of the art of multiobjective control problem is presented. The multiobjective optimisation in control systems is discussed as well as the motivation to use an evolutionary algorithm (EA) known as the multiobjective genetic algorithm (MOGA). Chapter 3 considers the problem of H∞ performance with time-domain specifications and the synthesis problem of reducing and/or fixing the order of the controller. A different approach to the state-space framework is considered, where by the polynomial methods and the Youla-Kučera parametrization are considered as a framework for the optimisation of the control structure. The MOGA-based method is considered as a way of solving the multiobjective control problem. Two academic control problems are numerically solved and compared with the solutions resulting from LMI techniques. Chapter 4 considers two multiobjective control problems. First, the LQG problem is studied, which is used to find the exact trade-off between process and sensor noise. For this LQG problem, a complete LMI formulation is presented, where the equivalence between LQG and H2 performance is used to formulate the convex optimisation problem over LMIs. The LQG problem was then presented using MOGA-based method in the same way as Fonseca and Fleming (1994). Secondly, the mixed H2/H∞ problem is optimised using the convex optimisation over LMIs, resulting in a trade-off. A discussion of the conservatism of this result is provided. The mixed H2/H∞ problem is then presented using the MOGA-based method and comparative results are shown. Chapter 5 designs a MIMO controller for an operating point of the gas turbine engine model. Both the mixed H2/H∞ controller design: LMI approach and the multiobjective H2/H∞ controller design: the MOGA-based method was considered for this design. The problem of applying gain scheduling control design is studied. Both the linear parametric varying (LPV) model of the gas turbine engine and the LPV controller design are

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Chapter 1 Introduction considered. Finally the LPV gain scheduling: MOGA-based method is proposed and two operating points are considered. Chapter 6 presents a conclusion on the research undertaken and offers suggestions for future research.

1.3 Contributions •

A new approach to optimise controller structures was proposed. This approach contrasts with the traditional implementation of parameters in state-space framework. The new controller structure is based on the new developments on Youla-Kučera parametrization and the optimisation design of weighting functions by Schroder (1998). The design procedure to optimise controller structure involved the use of the multiobjective genetic algorithm (MOGA) as an optimisation tool.

•

The new development on the Youla-Kučera parametrization, used in this thesis, is a result of an international collaboration, and the outcome of this investigation have been reported in Proceedings of the IEEE Conference on Decision and Control (Henrion et al. 2004).

•

Using the optimisation of controller structures, reduction of the order of the controller is achieved for problems such as the H∞ controller with time-domain specifications. In the case of reduction of H∞ controllers with time-domain specifications, the flexibility of a multiobjective genetic algorithm (MOGA) to optimise the order of the controller is demonstrated; these results are compared with the LMI optimisation for fixed-order H∞ controller design (Henrion 2003a). These results will appear in Proceedings of the IFAC World Congress on Automatic Control, Prague, Czech Republic (Molina-Cristobal et al. 2005).

•

Reduction of the conservatism in the mixed H2/H∞ problem is achieved. Although no method to measure the amount of conservatism is available, it is demonstrated that a better approximation to the Pareto optimal can be achieved by using a truly multiobjective optimisation of the H2 performance and H∞ performance. These 4

Chapter 1 Introduction results have been reported in Proceedings of the sixth Portuguese Conference on Automatic Control, CONTROLO, Faro, Portugal (Molina-Cristobal et al. 2004a) and an extended version was submitted to the International Journal of Systems Science (Molina-Cristobal et al. 2004b). •

The multiobjective H2/H∞: MOGA-based method was extended to design MIMO controllers, and applied to the gas turbine engine study case, in one operating point. A comparative study of the controller design between the LMI approach and the MOGA-based method showed that the LMI approach faces numerical problems, when the mixed H2/H∞ trade-off approximates to optimal. However, when using the MOGA-based method, this numerical problem is overcome. The multiobjective H2/H∞: MOGA-based method was extended to design gain scheduling controllers (two operating points).

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Chapter 2 Review of Linear Matrix Inequalities and Evolutionary Optimisation in Multiobjective Control 2.1 Introduction This chapter is devoted to reviewing the two optimisation techniques used in this work for multiobjective control. Although this thesis is related to the topic of multiobjective control, the review will start with the definition of linear matrix inequalities (LMIs), convexity optimisation, and the role of LMIs in control systems. The history of linear matrix inequalities in control systems aims to explain how convex optimisation theory was developed from the linear programming optimisation tool to the interior-point method and to analyse its importance in control systems. Then, the translation of control problems into a set of LMI constraints will be explained using the LQR optimal problem and the H∞ norm problem. The multiobjective optimisation problem is stated with fundamental definitions, as well as the multiobjective control problem. This, in then, will be followed by illustration of how multiobjective control problems can be tackled with LMI techniques. Finally, the evolutionary algorithm approach to multiobjective control problem, known as the multiobjective genetic algorithm (MOGA), is introduced.

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Chapter 2 Review of LMIs and EAs in Multiobjective Control

2.2 Definition of a Linear Matrix Inequality A linear matrix inequality has the form m

F ( x) = FO + ∑ xi Fi > 0,

(2.1)

i =1

where x ∈ ℜ m is the variable, and Fi = FiT ∈ ℜ nxn , i=0,…,m are given. The inequality symbol in (2.1) means that F(x) is positive-definite, i.e., u T F ( x)u > 0 for all non-zero u ∈ ℜ n . An LMI is a set of n polynomial inequalities in x. Multiple LMIs

F1 ( x) > 0,..., Fn ( x) > 0 can be expressed as the simple LMI, ⎛ F1 ( x) 0 ⎜ O ⎜ 0 ⎜ 0 0 ⎝

0 ⎞ ⎟ 0 ⎟>0. Fn ( x) ⎟⎠

(2.2)

2.3 Basic properties The LMIs that arise in control and systems theory can be formulated as convex optimisation problems that are amenable to computer solution. Several developments in convex programming have been able to find solutions for problems where no analytic solution is known and the time to compute is comparable to the time required to evaluate analytical (e.g. Riccati equations) problems (Boyd et al. 1994a).

2.3.1 Convexity Convexity is one of the main geometric ideas underlying much global optimisation theory, and the foundation for analytical tools used in LMI techniques. A set C is convex if the line segment between any two points in C lies in C, i.e., if any x, y ∈ C and any λ with 0 ≤ λ ≤ 1 , thus the line will be λx + (1 − λ) y ∈ C .

(2.3)

In Figure 2.1a, the set is convex because a line between any two points within the set C will be contained in the region, whereas in Figure 2.1b not every line between any two points is contained in the set C.

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x

x

C

C

y

y

b)

a)

Figure 2.1 a) Geometry of a convex set, b) Geometry of a nonconvex set A convex function f : ℜ n → ℜ is convex, if x, y ∈ ℜ n ,0 ≤ λ ≤ 1 then,

f (λx + (1 − λ ) y ) ≤ λf ( x) + (1 − λ ) f ( y ) .

(2.4)

A function f is convex if, for every pair of x, y the graph of f (λx + (1 − λ ) y ) lies on or below a straight line λf ( x) + (1 − λ ) f ( y ) . Geometrically, the inequality means that the line between (x, f(x)), (y, f(y)) lies above the graph of f. (See Figure 2.2).

f(.) λf(x)+(1-λ)f(y)

x

y λx+(1-λ)y

Figure 2.2 Graph of a convex function.

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Chapter 2 Review of LMIs and EAs in Multiobjective Control In a feasibility problem, the issue is to determine if the set C is non-empty. If there exist x ∈ ℜ n which simultaneously satisfies all of the inequalities, such a point is called a feasible point and the set C is then called feasible.

C = {x ∈ ℜ n | Fi ( x) ≥ 0, ∀i = 1,..., m}

(2.5)

2.3.2 Geometry of LMI sets Given symmetrical matrices Fi which characterize the shape in ℜ n of the LMI set,

n ⎧ ⎫ f = ⎨ x ∈ ℜ n | F ( x) = F0 + ∑ xi Fi > 0⎬ , i =1 ⎩ ⎭

(2.6)

the matrix F(x) is positive semidefinite if and only if its diagonal minors fi(x) are nonnegative. Diagonal minors are multivariate polynomials of independent xi. So the LMI set can be described as

{

}

F ( x) = x ∈ ℜ n | f i ( x) > 0, i = 1,L, n

(2.7)

Sets defined by polynomial inequalities and equations are called semialgebraic sets. Moreover, they are convex sets. In order to illustrate feasibility and the convexity of LMIs, a two dimensional subspace (x1, x2) is represented in Figure 2.3. For the sake of illustrating, the geometry of an LMI feasible set, an example was taken from (Parrillo and Lall 2003), ⎛3 ⎜ F ( x) = ⎜ 0 ⎜1 ⎝

0 4 0

1⎞ ⎛ 0 −1 0⎞ ⎛ −1 −1 0 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎟ + x2 ⎜ − 1 − 1 0 ⎟ > 0 0 ⎟ + x1 ⎜ − 1 0 ⎜0 ⎜0 0 0 ⎟⎠ 0 − 1⎟⎠ 0 ⎟⎠ ⎝ ⎝

which is equivalent to, − ( x1 + x 2 ) 1 ⎞ ⎛ 3 − x1 ⎜ ⎟ F ( x) = ⎜ − ( x1 + x 2 ) 4 − x2 0 ⎟ > 0, ⎜ 1 0 − x1 ⎟⎠ ⎝

(2.8)

feasible, if and only if, all principal minors are nonnegative system of polynomial inequalities f i ( x) > 0 .

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Chapter 2 Review of LMIs and EAs in Multiobjective Control •

First order minors f1(x) = 3 – x1 > 0 f2(x) = 4 – x2 > 0 f3(x) = - x1 > 0

•

Second order minors f4(x) = (3 – x1)(4 – x2) – (x1+x2)2 > 0 f5(x) = – x1(3 – x1) – 1> 0 f6(x) = - x1(4 – x2) > 0

•

Third order minors f7(x) = -x1((3-x1)(4-x2)-(x1+x2)2)-(4-x2)>0

The intersection of all these minors of the LMI set (2.8) generates the feasible set, which is the shaded region in Figure 2.3.

Figure 2.3 Intersection of principal minors sets, or LMI sets.

2.3.3 Semidefinite Programming (SDP) The LMI (2.1) is a convex constraint on x, i.e., the set {x ⏐ F(x)>0} is convex and this means that in order to solve LMIs, it is necessary to solve a convex optimisation problem. This problem within convex optimisation theory is known as semidefinite programming (SDP) (Ben-Tal and Nemirovski 2001). SDP is a generalization of linear programming (LP), where the inequality constraints in LP are replaced by generalized inequalities. The SDP in inequality form is conventionally written as

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cT x

minimize

m

subject to

F ( x) = FO + ∑ xi Fi > 0,

(2.9)

i =1

Figure 2.4. Representation of the optimal solution of the SDP problem. where c∈ ℜ m , and F>0 means the matrix F is symmetric and positive semidefinite, also denoted as F f 0 .

2.4 Brief History of Linear Matrix Inequalities Linear matrix inequalities have been used in systems and control for over a century. The first LMI can be attributed to Lyapunov (1892) who showed that the asymptotic stability of the differential equation x& = Ax is related to the matrix inequality,

P > 0.

AT P + PA < 0 ,

(2.10)

where P is positive definite. In the 1940’s; the first engineering application appeared when in the Soviet Union Lur’e and Postnikov, (1944) (cited in (Lur'e 1961; Boyd et al. 1994a)), used a set of three firstorder equations containing nonlinear term to formulate a Lyapunov function. With this work, they solved the problem of stability of a control system with nonlinearity in the actuator (Vidsasagar 1993). Although they did not form LMIs, their stability criteria have the form of LMIs, and were solved by hand for very small systems (Boyd et al. 1994a).

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Chapter 2 Review of LMIs and EAs in Multiobjective Control In the 1960’s; Popov (1962), Yakubovich (1962) and Kalman (1963) developed graphical techniques for solving certain families of LMIs. Nowadays, their results are known as the KYP lemma. In the late 1960’s; LMIs were intensively studied in control theory by Yakubovich (1964; 1965a; 1965b), who developed the method of matrix inequalities in the theory of nonlinear control and found solutions to certain matrix inequalities. In the 1970’s; Willems (1971) found that analytical treatment of quadratic optimal control problems leads to a series of matrix relations and frequency domain inequalities, such as the linear matrix inequality: ⎛ AT K + KA + Q KB + C T ⎞ ⎟ ≥ 0, F ( K ) = ⎜⎜ T ⎟ + B K C R ⎝ ⎠

(2.11)

and the algebraic Riccati equation (ARE) was considered for its solution. Anderson and Vongpanitlerd (1973) found a solution for a simplified version of the general form of the quadratic matrix inequality and found that solving the general form is difficult. Willems recognised that LMIs must be solved by computational algorithms, and he quoted: “The basic importance of the LMI seems to be largely unappreciated. It would be interesting to see whether or not it can be exploited in computational algorithms, for example.” Although graphical solutions for special forms of LMI systems which appear in engineering applications were available, the major problem faced was to solve more general forms of LMI systems. In the early 1980’s; two separate major advances happened; The first, an interior-point algorithm was proposed by Karmarkar (1984). This polynomial-time linear programming method was faster than the simplex method (developed by Dantzing in 1948 (Dantzing 1963)). The second major advance was the observation that LMIs in control and in systems theory could be formulated as convex optimisation problems (Horisberger and Belanger 1976). Pyatnitskii and Skorodinskii (1982) reduced the problem of Lur’e to a convex optimisation problem involving linear matrix inequalities, which they solved

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Chapter 2 Review of LMIs and EAs in Multiobjective Control using an ellipsoid algorithm (developed in the 1970’s to prove that linear programs can be solved in polynomial-time (Boyd et al. 1994b)). At this point, the interior-points algorithm and convex optimisation involving LMIs in control systems were two separate areas. In the late 1980’s; other authors started formulating control systems as a convex optimisation problem involving LMIs, such as the quadratic stabilisation of uncertain systems and solved using linear programming (Bernussou et al. 1989). The work of Boyd et al. (1988) showed that convex optimisation was emerging as a technique for controller design. In this work, objectives or control specifications, such as overshoot, settling time, classical single-loop gain/phase margins, were translated into standard convex programming, using Youla parametrization of all stable controllers. Nesterov and Nemirovski (1988), in Russia, extended the interior-point method for convex optimisation over LMI constraints, but it is not until the next decade that their work was spread and widely recognised. The reader is referred to Tokhomirov (1996) for a more detailed Russian version of the history of convex optimisation.

2.4.1 Interior-Point Methods In the early 1990’s; it was well-known that robust control problems led to LMIs (Packard et al. 1991; Boyd et al. 1993). However, current algorithms for solving the convex problem, such as the ellipsoid algorithm, were not efficient. For example, in the cuttingplane algorithm for the quadratic robust state feedback (Bernussou et al. 1989), the size of the convex programming problem is approximately exponential in terms of the numbers of uncertain parameters. Therefore, the need for more efficient algorithms to solve general forms of LMIs directed the researchers to develop interior point methods for solving LMI-based problems (Boyd et al. 1993; Boyd and El Ghaoui 1993). It is then that, the interior point algorithm for convex programming, developed by Nesterov and Nemirovski (1988), was recognised almost in every paper which used LMI techniques, as the algorithm that “hold great promise”, and it works extremely efficient in practice (Boyd et al. 1993). The culmination of this recognition came with the very popular monograph of Nesterov and Nemirovski (1994). This development of convex programming theory dramatically extended the ability to process convex programs numerically, while creating attractive areas of theoretical 13

Chapter 2 Review of LMIs and EAs in Multiobjective Control research, like linear programming (LP), conic quadratic programming (CQP) and semidefinite programming (SDP) (Vandenberghe and Boyd 1996; Boyd et al. 1997; Vandenberghe and Balakrishnan 1997; Forsgren et al. 2002). In control systems, several problems related to convex optimisation involving linear matrix inequalities can be solved efficiently using interior-point algorithms. Numerical experiments shows that these algorithms solve LMI problems with a polynomial-time complexity (Gahinet and Nemirovski 1997). These algorithm is also known as the Projective method and the first implementation, as a piece of commercial software, is the LMI Control Toolbox for use with MATLAB® (Gahinet et al. 1995). Another popular solver is SeDuMi, developed by Sturm (1999). Both of them can be interfaced with YALMIP, developed by Lofberg (2004); this software provides a more friendly way to write LMIs in MATLAB. The interested reader in more theoretical developments of convex optimisation is referred to the books: “Lectures notes on modern convex optimisation” by Ben-Tal and Nemirovski (2001) and “Convex optimisation” by Boyd and Vandenberghe (2004).

2.4.2 Robust Control: Convex approach An extensive list of LMI problems in systems and control without analytical solutions were studied and solved numerically by the new interior-point algorithm. For example, the relationship between the upper bound for μ (complex structured singular value) and LMIs were studied by Packard and Doyle (1993). LMIs arise naturally in both μ analysis and synthesis. They also claim that “LMIs will replace Lyapunov and Riccati equations, which are both special cases of LMIs, as the central computational problems in robust control”. A survey of the LMI approach in control systems can be found in the very popular monograph of Boyd et al. (1994b). One of the areas more affected by the new LMI approach in the 1990’s was H∞ Control Theory: H∞ Control Theory: One of the researchers who intensively studied H∞ optimisation was

Scherer (1990). His work was based on the previous work of Willems (1971), Willems studied the solvability of Riccati inequalities and the bounded real lemma which are the regular tools for H∞ optimisation. Other researchers (Lu and Doyle 1992; Packard 1994) formulated the state-space H∞ control problem as a convex constraint using linear fractional transformation (LFT). Iwasaki and Skelton (1994a) gave necessary and sufficient conditions for the existence of an H∞ controller, in terms of LMIs. The first

14

Chapter 2 Review of LMIs and EAs in Multiobjective Control solvability conditions of H∞ control involving the LMI approach was given by Gahinet (1994), Gahinet and Apkarian (1994), where the H∞ synthesis is performed in two steps (find a Lyapunov matrix and obtain the controller); here, the Riccati equations are replaced by Riccati inequalities. This inequality formulation provided a parametrization of all H∞ suboptimal controllers, and the computation of the controller reduces to solving LMIs. In 1997, a one step procedure to overcome the expensive numerical optimisation was proposed by Gahinet 1996). Other lines of research studied the flexibility of LMI techniques in combining several constraints with H∞ performance. Chilali and Gahinet (1996) and Chilali et al. (1999) gave the LMI formulation to combine Pole-Placement objectives, H2 performance and H∞ performance. This formulation together with the mixed H2/H∞ formulation by Scherer (1995b) were the first investigations of multiobjective optimisation with LMI techniques in control systems, a topic which will be reviewed later in this chapter. Some researchers have proposed general frameworks in order to obtain a unified LMI approach. Skelton and Iwasaki (1995) have combined the Covariance Control approach with the LMI approach, and it has resulted in added computational efficiency. This approach reduces a large class of single problems to a single problem in linear algebra. It is also important to mention that all advances of LMI techniques mentioned up to now have been developed under the framework of state-space techniques. However, LMI techniques have also been studied under the framework of polynomial systems. A survey of these developments is found in Henrion (2000); Henrion et al. (2003). Other advances on robust stability and performance analysis of uncertain systems using LMIs and convex methods for robust H2 analysis are found in Balakrishnan and Kashyap (1999) and Paganini (1999a) respectively. For a more complete survey of advances in linear matrix inequalities up to the year 2000, the reader is referred to the monograph of Ghaoui and Silviu-lulian (1999) as well as the book: “Robust Control: a convex approach” by Dullerud and Paganini (2000).

2.5 Application and New Trends of LMI Techniques A brief list of applications relevant to this work is presented as follows. As a practical example, Niewoehner and Kaminer (1996) applied LMI techniques in aircraft design. Their work has developed a new methodology that provides a numerical framework for 15

Chapter 2 Review of LMIs and EAs in Multiobjective Control the integrated aircraft-controller design. They consider a problem of combining the design of some aircraft parameters with the control system development. They also show that many flying qualities can be expressed as LMIs, as well as maneuverability requirements considered by other researchers. The controller design for an F-16 aircraft automatic landing system has been tackled using H∞ design with pole placement constraints: LMI approach (Seto et al. 2000). A MIMO control, for a compact disc player with multiple norm specifications, was developed with the mixed H2/H∞ controller design (Dettori and Scherer 2002). In July 2004, participants of the Workshop on Linear Matrix Inequalities in Control, Toulouse, France, LAAS-CNRS, reported on the latest achievements in the area of linear matrix inequality methods in control systems. In this workshop the following applications were presented: •

The mixed H2/H∞ controller design by Scherer et al. (1997) was applied to a space Launch Vehicle (Clement 2004) and to a satellite controller (Zasadzinski and Frapard 2004).

•

The fly by wire helicopter controller (Prempain and Postlethwaite 2004) was designed using a static H∞ loop shaping controller, derived from nonlinear inequalities and the determination of the controller reducing to an optimisation problem over LMIs. This controller was designed low-order and was successfully flight-tested and high levels of performance were achieved.

•

A LMI approach was used for robustness analysis of fighter aircraft control laws (Biannic 2004). Because of the need not only for linear time invariant uncertainties, but also for time varying uncertainties, the integral quadratic constraint (IQC) approach (Megretski and Rantzer 1997) was also implemented.

•

In robotics, an LMI approach to a multicriteria visual servo for a mobile robot (Danes 2004).

16

Chapter 2 Review of LMIs and EAs in Multiobjective Control As a summary of this workshop, the practical applications showed that LMI techniques improve results over classical methods, but still conservative, low-order controllers are required, and in some cases there is a numerical limitation (number of variables) with the state of the art LMI solvers. In this workshop, theoretical achievements in the area of LMI techniques were reported. The problems found that real applications, when LMI techniques are used, have motivated further investigations on convex optimisation (Nesterov and Nemirovski 1994). In convex optimisation, SDP has been bringing new ways to formulate control theory as SDP problems (Vandenberghe 2004), i.e. minimization of a linear objective subject to LMI constraints. Among these formulations, the ones that stand out are: duality theory, and LMI relaxations. Duality theory provides systematic and unified proofs of necessary and sufficient conditions for solvability of LMIs. Recently it has been used to obtain lower bounds of the optimal value (Balakrishnan and Vandenberghe 2003). In robust control, the system descriptions are affected by either time-varying parametric or dynamic uncertainties. The area of LMI relaxations is attempting to deal with these descriptions and give new LMI formulations, e.g. in gain scheduling design, linear parametric varying (LPV) can be formulated to handle rate of variations and naturally capture the parameter dependency of the control system (Scherer 2003; 2004). Recent improvements in SDP solvers and developments of polynomial optimisation have been due to the application of the so-called sum of squares (SOS) techniques in control. These relaxations are based on the decomposition of certain polynomials into a sum of squares (Parrillo 2000). This approach allows a “certification” of whether or not an SDP problem is feasible by analyzing its structure (Parrillo and Lall 2003). Another development of SOS is the possibility of optimising non-convex problems (Hol and Scherer 2004).

2.6 Quadratic stability of Lyapunov The stability analysis of a linear time invariant (LTI) system, continuous time, can be T studied with Lyapunov stability. The requirement P > 0 , A P + PA < 0 is called the

17

Chapter 2 Review of LMIs and EAs in Multiobjective Control Lyapunov inequality. A representation of all the matrices P satisfying Lyapunov’s LMI is represented in Figure 2.5 (Henrion 2003b),

1 ⎞ ⎛0 Figure 2.5. Lyapunov’s LMI with A = ⎜⎜ ⎟⎟ , ⎝ − 1 − 2⎠

Lyapunov also showed that this first LMI could be explicitly solved, choosing any value of Q where Q = QT > 0 . The linear equation AT P + PA = −Q for the matrix P is guaranteed to be positive-definite if the system x& = Ax is stable. The same problem can be formulated as a SDP problem or LMI optimisation problem, thus, consider the system x& (t ) = Ax(t ) ,

(2.12)

with an initial value x(0) = x 0 and the criterion function ∞

J = ∫ x(t ) T Qx(t )dt ,

(2.13)

0

where Q = Q T is positive definite. Assume that the system is asymptotically stable, then all solutions of (2.12) are bounded so that J < ∞ . In order to show this, consider a quadratic Lyapunov function V ( x(t )) = x T (t ) Px(t ) to establish a bound on J. Using the second method of Lyapunov, for some P = P T > 0 , the condition is

(

)

dV d ( x(t )) = x(t ) T Px(t ) ≤ − x T (t )Qx(t ) dt , dt dt

18

(2.14)

Chapter 2 Review of LMIs and EAs in Multiobjective Control which is negative definite, for all trajectories and all t. Integrating this inequality from t=0 to t=T , the equation (2.14) becomes T

x(T ) Px(T ) − x(0) T Px(0) ≤ − ∫ x(t ) T Qx(t )dt , T

0

since x(T )T Px(T ) ≥ 0 , and this holds for t → ∞ , then, ∞

J = ∫ x(t ) T Qx(t )dt ≤ x(0) T Px(0) .

(2.15)

0

The criterion function is bound as follows, T

J ≤ xo Pxo .

For any solution P = P T of (2.14), the differentiation,

(

)

d x(t ) T Px (t ) = x(t ) T ( AT P + PA) x(t ) ≤ − x(t ) T Qx(t ) , dt

(2.16)

provides that the condition (2.14) is equivalent to AT P + PA + Q ≤ 0 .

Thus, finding a P that provides a bound J can be done by solving an LMI feasibility problem. Moreover, it can be optimized over the Lyapunov function P by searching for the best (i.e. smallest) bound, the optimisation problem being written as follows, xoT Pxo

minimize subject to

P > 0, AT P + PA + Q ≤ 0

(2.17)

Since xoT Pxo is a linear function of the variable P, this is an SDP in the variable P = PT, and is numerically tractable by using interior-point methods.

2.7 Control Systems problems expressed as LMIs In SDP problems, there is a wide list of approaches for control systems (Vandenberghe and Balakrishnan 1997). Examples include: optimal system realization, robust stability, inverse problem of optimal control (Boyd et al. 1994b), regional pole placement (Chilali et al. 1999), RMS gain attenuations and gain scheduled controller design expressed as LMIs (Packard 1994). For very few, there are analytical solutions to SDPs (via Riccati equations, e.g. H2, H∞) but in general they can be solved numerically.

19

Chapter 2 Review of LMIs and EAs in Multiobjective Control In this Section some of the fundamental tools for expressing control problems as LMI optimisation problem will be presented. This procedure involves changes of variables and the use of Schur complements, and the use of the KYP lemma.

2.7.1 Definition of the Schur complement Nonlinear (convex) inequalities are converted to LMI form using Schur complements. The Lemma of Schur has the basic idea as follows: the block matrix ⎛Q ⎜⎜ T ⎝S

S⎞ ⎟ is positive definite, R ⎟⎠

(2.18)

if and only if, Q > 0 and

R − S T R −1 S > 0 ,

(2.19)

R > 0 and

Q − SR −1 S T > 0 ,

(2.20)

if and only if,

Proof: I ⎛ ⎜⎜ T −1 ⎝− S Q

0 ⎞⎛ Q ⎟⎜ I ⎟⎠⎜⎝ S T

S ⎞⎛ I − Q −1 S ⎞ ⎛ Q 0 ⎞ ⎟ = ⎜⎜ ⎟⎟⎜⎜ ⎟ T −1 ⎟ . ⎟ R ⎠⎝ 0 I ⎠ ⎝0 R−S Q S⎠

(2.21)

Thus, the set of nonlinear inequalities (2.19) and (2.20) can be represented as the LMI (2.18). An example to illustrate the use of Schur complement, is the quadratic matrix inequality AT P + PA + PBR −1 B T P + Q < 0 , R>0,

(2.22)

where A, B, Q=QT, R=RT>0 are given matrices of appropriate size, and P=PT is the variable. Note that this is a quadratic inequality matrix in the variable P. It can be expressed as the linear matrix inequality using Schur complements, see equation (2.23). ⎛ − AT P − PA − Q PB ⎞ ⎟ > 0. ⎜ T ⎟ ⎜ B P R ⎠ ⎝

20

(2.23)


2.7.2 LMI formulation of the LQR optimal problem Consider the linear system with input u, given by

x& (t ) = Ax(t ) + Bu(t ),

x ( 0) = x o ,

(2.24)

and the feedback gain K, i.e., u (t ) = Kx(t ) , so the closed-loop system is described by

x& (t ) = ( A + BK )x (t ) ,

x ( 0) = x o .

(2.25)

The linear quadratic regulator (LQR) can be used to solve a family of regulator design problems in which the state is accessible and regulation and actuator effort are each measured by mean-square deviation. The LQR cost function has the form,

J =

∫ (x(t )

∞ 0

T

)

∞

(

)

Qx (t ) + u (t ) T Ru (t ) dt = ∫ x(t ) T Q + K T RK x(t )dt ,

(2.26)

0

where Q and R are positive definite. This cost depends on the trajectory, of x(t ) , taken, so the worst trajectory will correspond to the worst cost. The control problem is to find a state feedback gain K and a quadratic Lyapunov function P that minimizes the bound

xoT Pxo on the worst cost of J . With the LMI formulation from Section 2.6, this problem can be solved by designing K and finding a Lyapunov function that gives a guaranteed performance bound. Proceeding in the same way as Section 2.6, the problem can be translated into an optimisation problem as follows, minimize xoT Pxo subject to P > 0, ( A + BK )T P + P( A + BK ) + Q + K T RK ≤ 0 .

(2.27)

As a result and contrary to the SDP problem, this optimisation problem cannot be considered an SDP because the constraint involves a quadratic term K and products between the variables P and K. However, this problem can be formulated as an SDP with the following transformations (Boyd et al. 1997). Define new matrices Y and W as,

W = KP −1 ,

Y = P −1 ,

21

Chapter 2 Review of LMIs and EAs in Multiobjective Control since P > 0, hence Y > 0, so that

K = WY −1 .

P = Y −1 ,

Substituting of the variables Y and W instead of P and K in equation (2.27), and then multiplying the left and the right hand sides by Y, the inequality above becomes

YAT + W T B T + AY + BW + YQY + W T RW ≤ 0 .

(2.28)

The matrix inequality (2.28) can be expressed as the LMI,

⎛ − (YAT + W T BT + AY + BW ) Y W T ⎞ ⎟ ⎜ ⎜ Y Q −1 0 ⎟ ≥ 0 , ⎟ ⎜⎜ 0 R −1 ⎟⎠ W ⎝

(2.29)

where it is assumed that Q and R are invertible. Once again, using the Schur complement, the cost, xoT PxoT = xoT Y −1 xo ≤ γ ,

is expressed as the LMI

⎛γ ⎜ ⎜x ⎝ o

xoT ⎞ ⎟≥0. Y ⎟⎠

(2.30)

Then, the SDP problem can be formulated in terms of LMIs (2.29) and (2.30) minimize γ ⎛ − (YAT + W T BT + AY + BW ) Y W T ⎞ ⎜ ⎟ −1 ⎜ ⎟≥0, 0 Y Q subject to ⎜⎜ −1 ⎟ 0 R ⎟⎠ W ⎝

⎛γ ⎜ ⎜x ⎝ o

(2.31)

xoT ⎞ ⎟≥0. Y ⎟⎠

This is an SDP problem that can be solved by interior-point methods. The reader is referred to Feron et al. (1992) for the stochastic interpretation of the LQR optimal problem and its LMI formulation. In the stochastic case, the cost function (2.26) is formulated as the expected value of the quadratic function and the state space equation (2.24) contains unit intensity white noise.

22


2.7.3 YKP Lemma The so-called Kalman-Yacubovich-Popov (YKP) lemma is one of the most fundamental tools of systems theory. It first originates from the criterion of absolute stability of Popov (1962), which gives frequency conditions for stability of nonlinear systems. Then, Yakubovich (1962) and Kalman (1963) complement these results and state the YKP lemma, sometimes also referred to as the positive real lemma. The following statement of this lemma follows the lines of Rantzer (1996) and Scherer (1990), which also contain its proof. YKP lemma. Given A ∈ ℜ nxn , B ∈ ℜ nxm , M=MT ∈ ℜ ( n+ m ) x ( n+ m ) , with det(iωI − A) ≠ 0 for ω ∈ ℜ and (A,B) controllable, the following two statements are equivalent.

•

The frequency domain inequality (FDI) ∗

⎡(iωI − A)−1 B ⎤ ⎡(iωI − A)−1 B ⎤ ⎥ M⎢ ⎥0,

(2.39)

and the FDI for all ω ∈ ℜ with det(iωI − A) ≠ 0 is

[C (iωI − A)

−1

][

]

∗

B + D C (iωI − A) −1 B + D − γ 2 I < 0 ,

which is equivalent to the left hand side of the equality (2.36), i.e. C ( sI − A) −1 B + D

∞

0,

AT P + PA < 0.

such that

This LMI characterization relies on the assumption that it can be solved using the same Lyapunov function, at the expenses of conservatism. Otherwise this approach becomes a multiobjective controller design problem written as a system of bilinear matrix inequalities (BMIs). Bilinear matrix inequalities are affine in two different sets of variables but not in all together. Unfortunately, no reliable numerical algorithms are available to solve BMIs (Arzelier and Peaucelle 2002; Dettori and Scherer 2002). In Chapter 4 of this thesis, the mixed H2/H∞ control problem is studied using the general quadratic synthesis procedure developed by Scherer et al. (1997). The numerical results of this LMI technique are compared with numerical results resulting from the MOGAbase method (in Section 2.11).

29


Another approach resulted from the fact that the multiobjective H2/H∞ control problem is equivalent to the generally infinite-dimensional convex optimisation problem (Scherer 1995b; Elia and Dahleh 1997). The infinite-dimensional formulation can be shown by using the Youla parametrization of all internally stabilizing controllers, T j1 + T j 2QT j 3 with Q varying in norm space H ∞ℜ qxp ,

where Tj is obtained from an arbitrary initial stable controller and H ∞ℜqxp denotes the algebra of real-rational proper and stable transfer matrices of dimension q x p. Thus, the multiobjective H2/H∞ control problem can be stated as follows: T1,1 + T1, 2 QT1,3

2

< γ1 ,

T2,1 + T2, 2 QT2,3

∞

< γ2 .

(2.45)

This problem is then translated into a convex optimisation problem in the parameter Q, which varies in the infinite-dimensional space of all transfer functions which are in the space H ∞ℜqxp . An approximation sketched by Scherer and Weiland (1999) showed that a Ritz-Galerkin approximation scheme leads to finite-dimensional problem by using the following sequence of finite-dimensional subspaces, and a fixed real a > 0 , ⎧ s−a ( s − a) 2 (s − a) υ qxp ⎫ L Q L Q + Q2 + + ∈ ℜ Q ξ υ = ⎨Q0 + Q1 | , ⎬. υ υ υ , s+a (s + a) 2 ( s + a) υ ⎩ ⎭

(2.46)

For a more comprehensive discussion, the reader is referred to Scherer (1995b). The advantage over the single Lyapunov approach method by Scherer et al. (1997) is that the Youla parametrization gives a better approximation to the optimal result at the expense of increasing the order of the controller (Scherer 1995a; Hindi et al. 1998; Scherer 2000). Another approximation to the mixed H2/H∞ problem was proposed by Arzelier and Peaucelle (2002). They proposed an iterative method which relies on the “smart” choice of an initial seed.

30

Chapter 2 Review of LMIs and EAs in Multiobjective Control The remaining sections of this chapter are devoted to introducing a novel approach to the multiobjective control problem; this is the multiobjective genetic algorithm. This approach is fundamentally different to traditional optimisation techniques because it is not deterministic and it can be applied to a wide range of applications in engineering, medicine, social science, etc. First, the basis of this approach will be presented with the standard genetic algorithm.

2.10 Genetic Algorithms Genetic algorithms (GAs) are a search procedure based on Darwinian “survival of the fittest” theory (Darwin 1859). GAs were developed to solve optimisation problems based on the mechanics of natural selection and genetics. The artificial implementation of the natural selection and reproduction into genetic operations have been shown to optimize design problems (Goldberg 1989; Coello Coello 1998; Veldhuizen and Lamont 2000; Fleming and Purshouse 2002). GAs optimize by evolving or generating successive populations from an initial random population of individuals (binary strings) to improved populations. GAs are part of a broad class of search techniques within the field of evolutionary algorithms (EAs) or evolutionary computing (EC) (Fonseca 1994; Fleming and Purshouse 2002). The genetic algorithm concept was developed by Holland (1975) and the computational implementation and mathematical foundations were developed by Goldberg, who first investigated the application of GAs to the control of gas pipeline transmission for his PhD dissertation (Investigation co-chaired by J. Holland). In his book, “Genetic Algorithms in Search, Optimisation, and Machine Learning” (Goldberg 1989), the computational code of the so-called standard genetic algorithm (SGA) is introduced, and it constitutes of three genetic operators: reproduction, crossover, and mutation. These operators perform the evolution using probabilistic rules, which manipulate the “genetic code”. Goldberg identifies fundamental differences with traditional search methods (deterministic): •

GAs work from a population; many other methods work from a single point.

•

GAs only require an evaluation function or objective function to assess the fitness of the individuals, this objective function is the only information from the

31

Chapter 2 Review of LMIs and EAs in Multiobjective Control problem. Other search methods need assumption on the existence of derivatives, such as the hill-climbing method. •

GAs use probabilistic rules to guide their search, not deterministic rules.

2.10.1 The population First, an initial population (decision space) of individuals is needed, where the individuals in the population are the candidate solution made to compete with each other for survival. Each individual is equivalent to the design parameters, e.g. in a PID controller design problem the gains are the parameters, and the set of parameters are encoded as a chromosome. The chromosome or genetic code is the one used for reproduction, crossover, and mutation. Traditionally, a chromosome is encoded as a concatenated binary string. When starting a GA, a genotypic domain is initially generated at random. An individual’s genotype is a representation of its phenotype at a lower level, analogous to the genetic sequences contained in the biological chromosomes. Then, they are decoded or mapped to phenotypes or the decision variables, which, in turn, will be evaluated and fitness assigned.

2.10.2 A Simple Function Optimisation Problem To explain the basic idea of how the genetic operators work, the same optimisation example from the book of Goldberg is used. The simple function optimisation example consists of maximizing the function f(x) = x2, where x ∈ [0, 31]. This optimisation problem with just one parameter might be treated by the genetic operators in the following way: consider a binary string of length five, where each of the bits will be generated randomly by using successive coin flips (head = 1, tail = 0), and consider a population of four genotypes (small by genetic algorithm standards), see Table 2.1. Individual number

genotypic (binary)

phenotypic (decimal)

Fitness f(x)= x2

% total

1 2 3 4 Total

10111 10000 01001 00111

23 16 9 7

529 256 81 49 915

57.8% 27.9% 8.8% 5.3% 100%

Table 2.1 Simulation by hand: population, assign fitness.

32


2.10.3 Replication (Selection) Replication or selection is the process of choosing the best individuals to participate in the production of offspring. This process is carried out stochastically and proportionally to their fitness. Typical selection operators are: •

Sampling with replacement or roulette wheel selection (RWS) (Goldberg 1989).

•

Stochastic universal sampling (SUS).

•

Tournament selection.

For the simple optimisation problem, the roulette is constructed with slots sized according to their fitness. With the percentage of fitness taken from Table 2.1, the roulettes, in Figure 2.7, are divided in four slots each of one is proportional to their fitness in percentage. If the roulette in Figure 2.7a spins, individual two (Table 2.1) has a probability 0.279 of been selected by the pointer. In this way, individuals with height fitness have more probability of being selected for the succeeding generation, by making an exact replica of the selected individual. This new individual is then entered into a mating pool for further manipulations of its code. This kind of selection is known as roulette wheel selection (RWS). Unfortunately, the RWS approach allows large selection errors to occur (Fonseca 1994), and introduces a large variance in its realizations (Deb 2001). An alternative selection approach is the stochastic universal sampling (SUS), which (Fonseca 1994) visualizes this selection as follows: SUS has the same principia of RWS but with multiple, equally spaced pointers.

5.3%

5.3% 8.8%

57.8%

8.8%

57.8%

27.9%

a) (RWS)

27.9%

b) (SUS)

Figure 2.7 Reproduction using a roulette wheel with slots sized according to fitness.

33


Notice that in the roulette Figure 2.7b, individual one will have four copies (see Table 2.1), the individual two will have two copies and the remainder only one copy each. These proportionate selection methods have a scaling problem (Deb 2001), for example in the simple problem, Table 2.1, individual one has a probability tending to 1. This might dominate the mating pool with its copies, and it might not necessarily be the optimal solution. The scaling problem can also be avoided by using a ranking selection method. A third selection operation known as the tournament selection works by comparing a pair of individuals, where the better of those two is declared winner of the tournament. In summary, the primary objective of reproduction is to make duplicates of good solutions by performing the following tasks (Deb 2001): •

Identify good solutions in a population.

•

Make multiple copies of good solutions

•

Eliminate bad solutions from the population, while keeping the population size constant.

2.10.4 Crossover (Recombination). After selection, the next task is to create new individuals or solutions by mixing couples from the matting pool. The aim of recombination is to exchange genetic information with one another by bringing into a single individual, the genetic features of two or more parents. Overall, this operation significantly accelerates the search process (Fonseca 1994). Typically the reproduction is created by picking two strings (solutions) from the mating pool and performing a crossover operation between them (Goldberg 1989).

34


Crossover operation

Parents

Offspring

0 1 0 0 1

0 1 0

1

1

0 0 1 1 1

0 0 1

0 1

0 1 0 1 1 0

0 1 0 1

Crossover point Figure 2.8 Recombination: Single point Crossover. Each string is split in two by the crossover point, which is chosen at random. The second part of each string is in a crossover position with the second. As a result, two new offspring are created, see Figure 2.8. Other typical recombination operators are: •

Double-point crossover (Booker 1987). The above concept of exchanging genetic information is extended to perform the recombination with two crossover points.

•

Uniform crossover (Syswerda 1989). One offspring is constructed by choosing every bit with a probability from either parent.

•

Shuffle crossover (Caruana et al. 1989). Performs shuffle before single-point crossover is applied, according to probability. After recombination, the bits in the offspring are unshuffled.

•

Reduced surrogate crossover (Booker 1987). The crossover point is restricted to the non-identical bits in the chromosome, and then one of the above crossover operators is applied.

•

Shuffle crossover with reduced surrogate (Chipperfield et al. 1994). Performs shuffle crossover with reduced surrogates between pairs of individuals contained in the current population according to the crossover probability, standard probability 0.7.

2.10.5 Mutation This genetic operation causes individuals’ genotypes to be changed according to some probabilistic rule. In other words it is a random alteration of some bits in individuals. In artificial genetic algorithms, the mutation operator protects against an irrecoverable loss

35

Chapter 2 Review of LMIs and EAs in Multiobjective Control of some potentially useful genetic material (Goldberg 1989). The probability is calculated in different ways, the probability pm of mutation according to Fonseca (1994) is p m = 1 − σ −1 / l

(2.47)

where l = length of the chromosome, σ is the selective pressure, recommended value 1.8.

Initialize population

gen = 0

stop

False

gen < maxgen True

Evaluation

Assign Fitness

Replication

gen =gen +1

Crossover

Mutation Figure 2.9 Flowchart of the Standard Genetic Algorithm (SGA).

36


Some applications have found difficulties with the binary representation of the population. Research in GAs has brought continuous search space representation. This has advantages, such as achieving any arbitrary precision in the optimal solution. The replication operator remains the same, but the crossover and mutation operator are applied in a different way; the approaches used in this work are, simulated binary crossover (SBX) (Deb and Agrawal 1995) and polynomial mutation (Deb and Goyal 1996) respectively. The SGA consists of replication, crossover and mutation, and it works as follows (see Figure 2.9): it starts with a random population, while a condition is true that the following will be executed; each individual will be evaluated and the assignment of fitness will be performed, and then replication crossover and mutation is executed as explained above. Traditionally by practitioners of GAs, a simulation of a SGA is conditioned to be executed for a finite number of generations. This number changes according to the application; in this work, the problems treated had a range of generations between 100 and 250. Recall that the aim of this section and the next one is to introduce an alternative approach to the multiobjective control problem. Before completing this section, two observations by researchers in the field of multiobjective control and in the field of genetic algorithms will be presented. The field of multiobjective control has been developed though techniques with the aims of finding a unified framework to include a variety of control engineering specifications and overcome conservatism, high order controllers, nonlinearities, etc. Moreover, solving a truly multiobjective control problem is desired instead of mixed or aggregating approaches. The benefits will be to get a better approximation to the optimal Pareto-front, and consequently a deeper understanding of the trade-offs that a particular problem might have. The reason for applying GAs to the multiobjective control problem is linked with the following two quotes:

37

Chapter 2 Review of LMIs and EAs in Multiobjective Control Quote one: first, consider the population, strings and fitness, generated for the simple

optimisation problem. In the book by Goldberg (1989, page 18), a similar population was presented with the following explanation of ideas: String Fitness 10111 10000 01001 00111

529 256 81 49

“What information is contained in this population to guide a directed search for improvement? On the face of it, there is not very much: four independent samples of different strings with their fitness values. As we stare at the page, however, quite naturally we start scanning up and down the string column, and we notice certain similarities among the strings. Exploring these similarities in more depth, we notice that certain strings patterns seem highly associated with good performance. The longer we stare at the strings and their fitness values, the greater is the temptation to experiment with these high fitness associations. It seems perfectly reasonable to play mix and match with some of the substrings that are highly correlated with the past success.” Quote two: Let us think that the strings are controllers and the fitness is their

performance in controlling a plant. The following quote was written in a paper by Khargonekar and Rotea (1991c), where they explain the challenges of the multiobjective control problem. “For example, suppose we design a finite number of controllers each of which meets a certain design specification. One might wonder if it is possible that these controllers could somehow be glued together to meet some or all of the objectives simultaneously. This remains a mostly unexplored area.”

Although research on multiobjective control has been carried out up to today, this challenge of the multiobjective control problem has not been entirely solved. The next

38

Chapter 2 Review of LMIs and EAs in Multiobjective Control section introduces a GA-based multiobjective optimisation technique, which offers alternative approach and is method undertaken in this thesis.

2.11 Multiobjective Genetic Algorithms The main difference between GAs and multiobjective GAs is that the latter explore the relevant trade-offs between multiple objectives. Engineering problems do not usually have a unique, perfect solution. Instead, they admit a set of equally valid, or a nondominated, alternative solution, which is known as the Pareto optimal. As it was pointed it out before, GAs work with a population, and support several different solutions simultaneously. This ability has been exploited, with the minimal of effort, to set up GAs to search for Pareto optimal (Fonseca and Fleming 1995). This field of research is known as multiobjective evolutionary algorithms (MOEAs). There exist numerous MOEAs available in literature, the interested reader is referred to (Coello Coello 1998; Veldhuizen and Lamont 2000; Deb 2001) for a survey of the most popular approaches. In this work the one used is the Multiobjective Genetic Algorithm (MOGA) by Fonseca and Fleming (1993). MOGA can be applied to multiobjective control problems by encoding the control problem, defining the vector spaces and the objectives. If a successful design is easily located, further improvements such as reduced controller order can be incorporated. Alternatively, if acceptable solutions are not readily obtained, the designer may choose to relax the design goals. MOGA has been successfully applied to control systems problems, e.g. the MIMO controller structure and parameter optimisation for a gas turbine engine (Chipperfield and Fleming 1996), the H∞ design of an electromagnetic suspension (EMS) control system for maglev vehicle (Dakev et al. 1997), the optimisation of the low-pressure spool speed governor of a Rolls-Royce Pegasus gas turbine engine (Fonseca and Fleming 1998b). MOGA was also used in conjunction with an H∞ loop-shaping design procedure (LSDP) for the ALSTOM benchmark problem (Griffin et al. 2000). MOGA has been applied in combination with other approaches, such as the Fuzzy Logic approach to the scheduling controller design of a gas turbine engine (Chipperfield et al. 2002), and a multiobjective optimisation approach with DK-Iteration (Griffin and Fleming 2003).

39

Chapter 2 Review of LMIs and EAs in Multiobjective Control MOGA is an evolutionary algorithm that uses standard genetic algorithm operators (selection, crossover and mutation), Pareto optimal ranking, fitness sharing and mating restriction (see Figure 2.10). The design philosophy of MOGA is to develop a population of Pareto optimal or near Pareto optimal solutions whilst maintaining the independence of the objectives throughout the optimisation process (Fonseca and Fleming 1998a).

Initialize population

gen = 0

stop

False

gen < maxgen True

Multiobjective Evaluation

Pareto Optimal Ranking Fitness Assignment

Kernel density Estimation

Fitness Sharing gen = gen+1

σ share = σ mate

Replication

Mating Restriction

Crossover

Mutation

Figure 2.10 Flowchart of the Multiobjective Genetic Algorithm (MOGA).

40

Chapter 2 Review of LMIs and EAs in Multiobjective Control Within MOGA, the initial population is randomly generated within a defined range and then decoded (in case of a non-real chromosome) to produce the corresponding vectors of decision variables. A set of objective function values is then evaluated for each individual within the population (see Figure 2.11a). The sequence of genetic operators is then applied, resulting in the subsequent generation of further potential solutions. Note that fitness assignment is a more elaborate process (see shaded region in Figure 2.10).

0

Dominated solutions

5

2 2

F2

0

1

F2

0

nondominated solutions

0

attainment surface F1

F1

a)

b)

approximation to optimality F2

F2 diversity density estimation

F1

F1

c)

d)

Figure 2.11 MOGA approach: a) Pareto Ranking, b) Nondominated and dominated solutions, c) Kernel Density estimation, d) Diversity and approximation to optimality.

41


2.11.1 Pareto Optimal Ranking In the absence of any information regarding the relative importance of design objectives, Pareto-dominance (Goldberg 1989) is the only method of determining the relative performance of solutions estimates. Nondominated individuals are all therefore, considered the best performers and are thus assigned the same fitness, e.g. zero. However, determining a fitness value for dominated individuals is a more subjective matter (Fonseca and Fleming 1995). Individuals in a population dominate a given individual. In Figure 2.11a, the Pareto ranking approach adopted, for a two dimension problem, is to assign a cost proportional to how many individuals in a population dominate a given individual (see equation 2.43). MOGA employs multiobjective preference articulation extensions to the standard GA. Individuals are ranked, based on the objective vector and the designer’s preference (goal, priority). The consideration of goal and priority selectively excludes objectives according to their priority and whether or not they meet their goals, the decision support tool was developed and mathematically formalized by Fonseca and Fleming (1998a). Priority and goal information can often be extracted directly from the problem description. Priorities are integer values that determine in which order objectives are to be optimized, according to their importance. Frequently in control systems, closed-loop stability has the highest priority and is minimized first. Goals values indicate the desired level of performance in each objective dimension. The goal vector delineates the region of the trade-off where MOGA concentrates its computational effort. Once rank is sorted, this genetic operator will assign fitness to individuals by interpolating from the best to the worse, according to an exponential rule. Then a single value of fitness is derived for each group of individuals with the same cost, through averaging. In the Pareto set, interpolating between the solutions to obtain a smooth representation of the Pareto set is not generally correct. Solutions corresponding to intermediate nondominated individuals, even if they exist, are unknown. A more convenient representation will be to draw a boundary separating those points in objective space or the attainment surface, see Figure 2.11b. This concept was introduced by Fonseca and Fleming (1993).

42


2.11.2 Fitness Sharing and Kernel Density Estimation Although the population is potentially able to search many local optima, a finite population will tend to evolve towards a small region of the search space even if other equivalent optima exist. This phenomenon is known as genetic drift. A remedy to this problem has been proposed by Fonseca and Fleming (1995) with Fitness Sharing. This is a technique involving the estimation of the population density, niche size σ share , at the points defined by each individual. An appropriate niche size was developed by Fonseca and Fleming by using a remarkable similar density estimation method known as Epanechnikov kernel (Silverman 1986), and is used to penalize individuals according to the proximity of other individuals (see figure 2.11c).

2.11.3 Mating Restriction Mating restriction specifies how close individuals should be in order to mate and it is used before crossover. The issue is to keep diversity along the trade-off surface, see Figure 2.11d. This avoids an arbitrary combination of pairs outside a given distance σ mate that might conduct to the formation of a large number of unfit offspring. The distance is commonly σ mate = σ share . In addition, population diversity is encouraged by applying a mutation operator to a small number of the existing individuals. The genetic operators presented in this section might be implemented with the SGA as in Figure 2.10. The MOGA has been recognized as a good approach, efficient and easy to implement (Coello Coello 1998). The approximation to the Pareto optimal is particularly difficult because the Pareto optimal is unknown. The traditional way is to experiment and decide either by using prior knowledge or by comparing with the results of other methods if the current Pareto-front is improving towards the Pareto optimal. In general, caution must be taken when using evolutionary algorithms. Even if a good choice of parameters is found for a particular application; this set will be sub-optimal for many other problems. Making the wrong choice of parameters can produce exceedingly poor results (Fleming and Purshouse 2002).

43


2.12 Conclusions LMI formulations can mix various specifications and objectives, which is an improvement over classical LQG or H∞ techniques. Semidefinite program (SDP) has an efficient numerical solution using recent interior-point methods for convex optimisation. This brings a numerical solution to problems when no analytical solution is known. Some problems in control systems will reduce to convex problems, in general, due to the existence of convex constraints. According to researches Ghaoui and Silviu-lulian (1999), in the future, LMIs will play the same central role in post-modern theory as the Lyapunov function and Riccati equations played in modern, theory and in turn, the role various graphical techniques such as Bode, Nyquist and Nichols plots played in the classical. There is not a general method that converts a control specification into LMI constraints and even if this is achieved it might be non-convex as in the case of the reduced order controller. Alternatively, MOGA can deal with non-convex constraints and provide truly multiobjective optimisation treatment to multiobjective control problems. The following chapters will present multiobjective control problems and their solution with both LMI techniques and the MOGA-based method.

44

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs 3.1 Introduction This chapter studies the open problem of reduced- and fixed-order synthesis, which is known to be a rank minimization problem subject to a linear matrix inequality (LMI) constraint in H∞ state-space techniques. The optimisation involved is difficult to solve due to the non-convexity of the objectives. In order to address this problem, the application of an evolutionary approach is considered. A controller of optimal order is selected from a set of feasible controller structures. Two controller design problems are presented. A comparison of time and frequency domain responses shows that low order controllers resulting from the evolutionary optimisation perform as well as those low order controllers designed using the latest achievements in LMI techniques. Within the field of embedded control systems, it is desirable to implement a low order controller. When a state-space H∞ technique is used, the order of the controller must be at least the same as the order of the plant. There are three alternatives to reduce the order of such controllers:

45

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs 1. A reduced order approximation of the plant can be generated before designing the controller. 2. A reduced order approximation of the controller can be generated after controller design. However, the resulting reduced-order controller may not stabilize the plant. 3. The order of the controller can be constrained during the design. This work examines the third alternative. Using LMIs, the intention is to include a constraint dim (Ac) ≤ k for some k that is less than or equal to the dimension of A where A and Ac are the state matrices of the plant and controller respectively. The corresponding LMIs can be derived. rank

⎧⎪⎡ X ⎨⎢ ⎪⎩⎣ I

I ⎤ ⎫⎪ ⎬ ≤ n + nk Y ⎥⎦ ⎪⎭

where n is the order of matrix Ac and nk is the order of the matrix Ac, X>0 and Y>0 are n x n matrices. However, the rank constraints are non-convex and difficult, if not impossible,

to treat by current optimisation techniques (Scherer et al. 1997). However, recent research has developed some solutions to this problem; e.g. an algorithm in state space realization (Iwasaki and Skelton 1994b; Apkarian et al. 2003; Xin 2004) and a sufficient LMI condition in polynomial systems (Henrion 2003a) have been proposed to overcome this non-convexity and can be used to fix the order of the controller. Alternatively, the multiobjective genetic algorithm (MOGA) has been shown capable of optimizing controller structure and controller parameters (Chipperfield and Fleming 1996; Schroder 1998). A controller is designed by optimising the controller structure (order of the controller) and multiple design objectives in both the time domain (overshoot, undershoot and settling time) and the frequency domain. In Section 3.4, two examples of flexible structures have been solved numerically to illustrate the proposed procedure. The results and the procedure are compared with those obtained by LMI techniques.

46

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs

3.2. Problem Statement The mathematical framework employed in this work is polynomial systems. Consider the plant P of order n where P=

bn an

(3.1)

and the SISO feedback configuration shown in figure 3.1,

v r +

K

u

-

P

+ +

z

Figure 3.1 Feedback system. with coprime polynomials an and bn, and where K is the controller of order m with coprime polynomials yn and xn,

K=

yn xn ,

(3.2)

r is the reference signal and v is the disturbance and z is the control system output. Given

a plant, P, the design of the feedback control system consists of designing the controller, K, such that the resulting feedback system exhibits the desired performance.

3.2.1 Youla-Kučera parametrization of stabilizing controllers Another desired specification is that the system be internally bounded-input boundedoutput (BIBO) stable. This means that for any bounded exogenous input in r and v, the internal signal u will be bounded too. This can be achieved by using the Youla-Kučera (YK) parametrization of stabilizing controllers (Kučera 1979; Kučera 1993). For a given continuous-time plant, P=b/a, a stabilizing controller exists in a feedback configuration and all the controllers that stabilize the given plant are generated by all pairs of xˆ , yˆ that solve the Bézout equation (3.3).

axˆ + byˆ = 1

47

(3.3)

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs This particular Diophantine equation (Bézout equation) has been useful in the area of control systems because it provides the parametrization of a family of all stabilizing controllers for P and it was proposed by Kučera (1975); Youla et al. (1976a); Youla et al. (1976b) (see equation 3.4):

K=

yˆ − aq xˆ + bq

(3.4)

where q is an arbitrary proper stable rational parameter. Traditionally, controller design with YK parametrization leads to high order controllers; however, recent research has shown that this problem can be overcome in the scalar case. The following results in this section are a brief description of this achievement on YK parametrization and the reader is referred to Lemma 1 in Henrion et al. (2004) for a more formal statement. The procedure starts by calculating an initial proper controller of order m. Such a controller can be obtained by placing the poles at arbitrary locations and

solving the Diophantine equation in equation (3.5):

an xˆn + bn yˆ n = ad xd

(3.5)

where the degree of the polynomials ad and xd are n and m=n-1 respectively and xˆ n and yˆ n

define the initial controller. Defining

a=

an , ad

b=

bn , ad

xˆ =

xˆ n , xd

yˆ =

yˆ n , xd

(3.6)

equation (3.5) can be transformed into the Bézout equation (3.3). Defining the YoulaKučera polynomial q =

ad qn , and substituting (3.6) in (3.4), the parametrization can be xd qd

expressed with the controller polynomials xn and y n in the arrangement of equation (3.7). ⎡ x n ⎤ ⎡ bn x d ⎥=⎢ ⎣ y n ⎦ ⎣− a n x d

(a d x d )2 ⎢

48

a d xˆ n ⎤ ⎡ a d q n ⎤ , ⎥ a d yˆ n ⎦ ⎢⎣ x d q d ⎥⎦

(3.7)

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs where xdqd is an arbitrary stable polynomial and adqn is an arbitrary polynomial of equal or lower degree. Equation (3.7) can be simplified in equation (3.8) indicating that the vector N = [x n

yn

qn

q d ]T belongs to the null-space of a given polynomial matrix

A, ⎡ xn ⎤ ⎢ 0 − bn − xˆ n ⎤ ⎢ y n ⎥⎥ ⎡a d x d =0 ⎥ ⎢ ad xd a n − yˆ n ⎦ ⎢ q n ⎥ 0 ⎣14 4444244444 3⎢ ⎥ A ⎣q d ⎦

(3.8)

Assuming that q n , q d are polynomials and noting also that polynomial matrix A has full row rank, the dimension of the null-space will be equal to two. Let ⎡ x1n ⎢y N = ⎢ 1n ⎢ q1n ⎢ ⎣q1d

x 2n ⎤ y 2 n ⎥⎥ q 2n ⎥ ⎥ q 2d ⎦

(3.9)

be a minimal basis for the null-space of matrix A, i.e. such that AN = 0 and the column degree of N are minimal among all possible null-space bases. By extracting a minimal polynomial basis N for the null-space of the polynomial matrix A defined in (3.8), all the stabilizing controllers with denominator and numerator polynomials xn and yn can be generated with the parametrization shown in equation. (3.10), ⎡ xn ⎤ ⎡ x1n ⎢y ⎥ = ⎢y ⎣ n ⎦ ⎣ 1n

x2 n ⎤ ⎡ λ1 ⎤ , y 2 n ⎥⎦ ⎢⎣ λ2 ⎥⎦

(3.10)

where λ1 , λ2 are polynomials such that the YK denominator polynomial, qd, is stable and the corresponding YK numerator polynomial is given by

⎡q d ⎤ ⎡q1d ⎢q ⎥ = ⎢q ⎣ n ⎦ ⎣ 1n

q 2 d ⎤ ⎡ λ1 ⎤ . q 2 n ⎥⎦ ⎢⎣ λ2 ⎥⎦

(3.11)

Then a controller, K, of fixed order m can be found if there exist polynomials λ1 and λ2 of order m. The remainder of the controller design procedure using LMI techniques is fully

49

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs explained in Henrion et al. (2004) and used later in this chapter to find the LMI controllers.

3.2.2 LMI optimisation for fixed-order H∞ controller design In Henrion et al. (2003a), results on positive polynomials and strictly positive real (SPR) transfer functions were used to produce an LMI formulation of fixed-order robust controller design, in the algebraic, or polynomial framework. The key ingredient in the design procedure resides in the choice of a central polynomial. The continuous-time H ∞ design problem for scalar transfer functions can be stated as follows. Given a set of polynomials ni (s) , d i (s) for i = 1,2,... Ni as well as the positive real number γ , seek polynomials xi ( s ) of given degrees such that.

∑ ni ( s ) x i ( s )

< γ,

i

∑ d i (s) xi ( s) i

In the above inequalities:

S

∞

(3.12)

∞

= sup S ( s ) s =iω ,ω∈ℜ

denotes the peak value of the magnitude of rational transfer function S evaluated along the imaginary axis. If c(s) is defined as a central polynomial, the degree of this polynomial is equal to the highest degree of polynomials ni (s) , d i (s) plus the degree of the desired polynomial xi ( s ) , which is the desired controller. Similarly to standard H∞ techniques, this LMI design technique is iterative, and a trial-and-error approach cannot be avoided in choosing appropriate the central polynomial. A general rule-of-thumb is that open-loop stable poles must be mirrored in the central polynomial, completed by sufficiently fast additional dynamics. ⎛ ⎞ n ( s ) = γ⎜⎜ ∑ d i ( s ) xi ( s ) ⎟⎟ + ∑ ni ( s ) xi ( s ) , ⎝ i ⎠ i ⎛ ⎞ d ( s ) = γ⎜⎜ ∑ d i ( s ) xi ( s ) ⎟⎟ − ∑ ni ( s ) x i ( s ) . ⎝ i ⎠ i

50

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs Let us define the 2 by 2 polynomial matrices ⎡ n( s ) N (s) = ⎢ ⎣ 0

0 ⎤ , n( s )⎥⎦

⎡d ( s ) D( s ) = ⎢ ⎣ 0

⎡ c( s ) c( s ) ⎤ C ( s) = ⎢ ⎥, ⎣ − c( s ) c( s ) ⎦

0 ⎤ , d ( s )⎥⎦

whose coefficient matrices corresponding to increasing powers of indeterminate s are gathered in block matrices,:

[

N = N0

N1

...

]

Nδ ,

[

D = D0

D1

...

]

Dδ ,

C = [C 0

C1 ... Cδ ] .

Note that matrices N and D are linear in coefficients of polynomials xi ( s ) , where δ is the highest degree arising in polynomials ni (s) , d i (s) and c(s). With these notations, results on positive polynomial matrices are invoked in Henrion (2003a) to derive the following solution. Theorem 3.1 (Henrion 2003a) Given polynomials ni (s) , d i (s) , positive scalars γ for and

central polynomial c( s ) , there exist polynomials xi ( s ) solving H ∞ specifications (3.12) if matrix inequalities ( N ) T C + C T N − H ( Pn ) > 0 ,

(3.13)

( D) T C + C T D − H ( Pd ) > 0 ,

(3.14)

are feasible. This is a convex LMI problem in coefficients of polynomials xi ( s ) and symmetric matrices Pn and Pd . The linear mapping

H ( P) = Π T (H ⊗ P )Π where, ⎡I 2 ⎢ ⎢ ⎢ Π=⎢ ⎢0 ⎢M ⎢ ⎢⎣ 0

O I2 I2 O

51

0⎤ M ⎥⎥ 0⎥ ⎥ ⎥ ⎥ ⎥ I 2 ⎥⎦

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs is a matrix of size {4δ x 2(δ+1)}, ⎡0 P=⎢ ⎣1

⎡− 1 0⎤ P=⎢ ⎥. ⎣ 0 1⎦

1⎤ , 0⎥⎦

for the left half-plane and the unit disk respectively.

3.3. Optimizing Controller Structure with Genetic Algorithms As an alternative to the LMI techniques, YK parametrization can be used to structure the controller (3.18), which is then optimised with the MOGA-based method. The suitability of MOGA is due to a trade-off between the controller order and the performance of the closed-loop system (Middleton 1991; Chipperfield and Fleming 1996; Astrom 2000). Hence, rather than a single optimal solution, this multiobjective problem results in a family of non-dominated or Pareto optimal solutions.

3.3.1 Evaluation function To define the control problem, an evaluation function is encoded, which contains all the objectives functions that the controller, K, has to satisfy. The performance evaluation function used in this work has seven objectives. The first two objectives are the H∞ norm in polynomial systems (Kwakernaak 1990) of the sensitivity function and the complementary sensitivity function. In the feedback system, Figure 3.1, the sensitivity of the control system output z to disturbances υ is characterized by the sensitivity function:

S=

an xn 1 = , b y a n x x + bn y n 1+ n n an xn

then the H∞ norm is: S

∞

=

a n xn a n x n + bn y n

∞

bn y n a n x n + bn y n

∞

(3.15)

and the complementary sensitivity function:

T =1− S =

bn y n , a n x n + bn y n

then the H∞ norm is:

T

∞

=

(3.16)

If the controller design requires limiting the actuator effort, the transfer function KS should be included as another objective. In this work this objective was not required and hence not studied.

52

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs Because of the nature of the problems in this Chapter (flexible structures), a third objective called Total Variation (TV) (Owens and Chotai 1981) was included. TV provides a measure of the oscillations in the time response to a step change in reference signal. The TV of a function f, is defined by TV ( f ) = max

N

∑

0≤ti ≤t N i =1

f (t i +1 ) − f (t i )

(3.17)

where ti+1 and ti corresponds to a peak and valley or vice verse. The function f used in this chapter is the step response of the complementary transfer function T. Another way to compute the total variation is by using the truncated integral of absolute area (L1 norm) of the derivative of f(t) Boyd and Barratt (1991, page 98). tN

TV = ∫ | f& (t ) |dt. 0

Another suitable function would be the integral of the step response of the sensitivity function S (Whidborne and Liu 1993, Chapter 8), as this would give a bound on the maximum tracking error. In order to ensure good time-response performance, three time-domain objectives were also included: •

Overshoot is the maximum amplitude of the step response of T(s).

•

Undershoot is minimum amplitude of the step response of T(s).

•

Settling time is the time required for the step response of T(s) settle and stay within 5% of its final value.

An additional objective was included that measured the order of the controller. The chromosome structure shown in Figure 3.2 was implemented along the lines of Schroder (1998), where he used for the optimisation of the weighting function structure. C

α0

α1

β0

α2

α3

α4

β1

β2

α5

α6

α7

α8

β3

β4

β5

Fig. 3.2. Structure of the chromosome.

53

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs The variable C shown in Figure 3.2 selects the controller structure, and then the controller K(λ1,λ2) is decoded according to equations (3.18). Finally, the controller K=yn/xn can be

obtained using equation (3.10). ⎧ λ1 = α 0 + α1 s ⎪ 0 ≤ C < 1, λ2 = β 0 + s ⎪ ⎪⎪ λ1 = α 2 + α 3 s + α 4 s 2 K ( λ1 , λ2 ) = ⎨1 ≤ C < 2, λ2 = β1 + β 2 s + s 2 ⎪ ⎪ λ1 = α 5 + α 6 s + α 7 s 2 + α8 s 3 ≤ ≤ 2 C 3 , ⎪ ⎪⎩ λ2 = β 3 + β 4 s + β 5 s 2 + s 3

(3.18)

3.4. Numerical Examples The numerical examples were carried out with the help of the MATLAB® Genetic Algorithm Toolbox with MOGA extension (Fonseca and Fleming 1998a) for the optimisation of controller structure, the Polynomial Toolbox for MATLAB® (PolyX 2001) to solve polynomial equations, and the semidefinite programming problems were solved with SeDuMi 1.05 (Sturm 1999) interfaced with YALMIP (Lofberg 2004).

3.4.1 Example 1: Low-order damping mode

P=

bn 1 = 2 an s ( s + s + 10)

(3.19)

The control problem is to design a controller, K, so that the resulting feedback system exhibits a step response with no undershoot and the minimal possible overshoot and settling time as well as a high damping signal response. The first step is to obtain an initial controller, for example by solving the Diophantine equation (3.5) with arbitrary placement of the poles at s = -1, giving a d x d = ( s + 1) 5 .

The result is an initial controller with order m=2 and polynomials

yˆ n 1 + 45s − 26s 2 = . xˆ n − 4 + 4s + s 2

54


Step Response 10

8

Amplitude

6

4

2

0

-2

0

5

10

15

20

25

30

Time (sec)

Figure 3.3 Flexible mode controlled by initial controller. Figure 3.3 shows an unacceptable overshoot as well as a small undershoot (inverse response). The initial controller is unstable and non-minimum phase. However, the only requirement of the initial controller is to be stabilizing. In fact this controller has to satisfy the Bézout equation (3.3) (Kučera 1993). The minimal polynomial basis N for the nominal matrix A is given by 0 ⎡ ⎢ −1 N=⎢ ⎢− 4 + 4 s + s 2 ⎢ −1 ⎣

⎤ ⎥ ⎥ − 103 + 149s ⎥ ⎥ − 26 + 10s + s 2 + s 3 ⎦ 1 − 26

By solving equations (3.10) and (3.11), all the stabilizing controllers can be computed from polynomials λ1 and λ 2 in equation. (3.20) and it may be possible to find polynomials x n and y n (controllers K) that have low degree with a stable YK polynomial qd (3.11),

x n = λ2 ⎧ K =⎨ ⎩ y n = − λ1 + 26λ2 55

(3.20)

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs LMI Approach The H∞ design procedure was used to find polynomials x n and y n (controllers K). The

formulation to solve the control problem is as follows: Given the polynomials an and bn from plant (14) and the bound γ s , the sensitivity function with YK controller (3.4) is: S=

1 b 1+ n an

⎛ yˆ − aq ⎞ ⎜⎜ ⎟⎟ ⎝ xˆ + bq ⎠

= axˆ + abq

This transfer function can be simplified by substituting q =

ad qn , equation (3.8), and xd qd

using equations (3.6). The sensitivity function is transformed, thus: S = axˆ + abq =

a n (xˆ n q d + bn q n ) a n x n = ad xd qd qd

The frequency domain specification of the sensitivity function is then expressed as follows:

S

∞

=

a n xn a n xn + bn y n

= ∞

a n xn qd

< γs

(3.21)

∞

where polynomial xn satisfies the algebraic constraint

a d x d x n = bn q n + xˆ n q d

(3.22)

This frequency domain specification (3.21) can be expressed as an LMI constraint, using specification 3.12 and theorem 3.1, and then solved simultaneously with the linear constraint. The H ∞ design algorithm was set up to solve the optimisation control problem, and then the polynomials xn, qd and qn of a given degree were sought such that (3.21) and (3.22) are satisfied. In each single run, a controller was found by using the concept of the central polynomial, c(s), which is the key design step. The H∞ design procedure consists of iteratively

adjusting roots of c(s), while lowering the upper bound γ s , see Table 3.1. After a series of trials a set of controllers of orders m=1, 2 and 3 were obtained and the closed-loop step responses of the best controllers are shown in Figure 3.3.

56


λ1

trial num c(s)

λ2

1 (s+4)(s+0.5)(an/s)

-54.618-26.177s

2 (s+4)(s+0.5)(an/s)

-92.906-26.57s

3

1.909+s

-1915.9-376.3648s-108.8049s

4 (s+4)(s+4)(s+3)(an/s)

-1081.9-307.0037s-55.9931s

33.3574+11.5561s+s

2

1.2970 70

2

1.1918 1.5

5 (s+2)(s+2.5)2(s+1.5)(an/s) -299.92-750.64s-254.36s2-45.409s3 8.2823+23.11s+9.0767s2+s3 1.2946 2

2

2

6 (s+2) (s+2.5) (an/s)

-370.45-929.48s-296.19s -50.108s

3

γs

1.1490 1.3

59.3898 + 11.9056s +s

2

∞

1.1642 70

3.362+s 2

3 (s+5) (an/s)

S

2

3

9.5269+27.50s+10.186s +s 1.2662

2 1.5

Table 3.1 Family of controllers resulting from the H∞ design procedure: LMI Approach.

Step Response 1.2

K (2nd) 1

Amplitude

0.8

K (1st)

0.6

0.4

K (3rd)

0.2

0 0

5

10

15

Time (sec)

Figure 3.4 Step response of: first-order controllers (‘dotted line‘), second-order controllers (‘dashed line’) and third order controllers (‘solid line’): LMI approach. MOGA-based method The controller design problem was then tackled using the MOGA-based method. The

decision variables used were the controller parameters defined in (3.18) in the chromosome structure from Figure 3.2. An evaluation function was encoded with seven objectives defined in Section 3.3.1. The MOGA parameters used were: •

16 bit resolutions Gray coding and linear scaling,

•

Crossover operator employed was shuffle with reduced surrogate with probability 0.7. The mutation operator was modified such that only decision variables that are currently active are mutated. 57

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs •

The probability of applying mutation is 7e-4.

•

A population of 100 individuals.

The algorithm was iterated for 100 generations. The results are illustrated in Figures 3.5 and 3.6. It can be seen that one single solution does not exist, rather a family of solutions. The trade-off of the objectives is shown in Figure 3.5, which is a graphical format known as the method of parallel coordinates. Each line represents a Pareto optimal solution. The x-axis represents the objective and the y-axis shows the normalized performance in the

interval [0, 1]. Table 3.2 shows objective functions values (the second and the third row) of the cost for each objective in Figure 3.5 as well as the goal vector (the fourth row).

Cost

MOGA Trade-off

1

2

3

4 Objective no.

5

6

7

Figure 3.5. Trade-off graph for the controllers KMOGA

1

2

3

4

5

6

7

Objective

||T||∞

||S||∞

TV

Normalized Cost [1] Normalized Cost [0] Goal vector

1.1

2

1.8

Overshoot (amplitude) 1.04

Settling time (sec) 22

K nth-order 4

Undershoot (amplitude) 0.1

0.9

1

0.9

0.94

0

0

-0.5

2

2

4

1.001

25

4

0

Table 3.2 Normalization of the design objectives in Figure 3.5 and goal vector. For comparison, the LMI controller, shown in equation (3.24) using trial number 4 as shown in Table 3.1, was included with bold lines and marked (o) in Figure 3.5. The objective 2 H∞ norm of the sensitivity function, the objective 3 TV, the objective 4 overshoot and objective 5 settling time appear to compete quite heavily, since the improvement in one objective is the detriment in the other. Here is where the control

58

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs engineer can select candidate solutions depending on the specifications of the applications. Figure 3.6 illustrates the most representative step responses. Most of the step responses that have large settling times are first-order controllers. The second-order and third order controllers show better settling time, and this is confirmed by looking at the Trade-off graph in Figure 3.5. Step Response 1.2

1

Amplitude

0.8

0.6

0.4

0.2

0 0

2

4

6

8

10 12 Time (Sec)

14

16

18

20

Figure 3.6. Family of preferable step responses. By changing the values of the goals, the search is forced to examine other areas of the trade-off surface. Note for example, different values of TV produce a range of responses from smooth to oscillatory. Taking advantage of the initial information, some of the goals (||T||∞, ||S||∞, TV and settling time) were tightened. The new goal vector is shown in Table 3.3. The new results are displayed in Figure 3.7, which shows the performance of the controllers that meet the new design requirements. The order of the objectives was rearranged in order to facilitate the visualisation of the trade-offs among the objectives, such as TV, ||S||∞ and settling time. Figure 3.8 illustrates the “best” closed-loop step responses obtained by the MOGA. The MOGA controller (3.23) was chosen and compared with the LMI controller (3.24). The first comparison is in Figure 3.7, where the MOGA controller is marked (◊) and the LMI controller is marked (o); both solutions are in bold lines. The selected MOGA

59

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs controller offers the best compromise among the Pareto front. For example, it offers best compromise between settling time and ||S||∞. The step response produced by the selected MOGA controller is compared with the “best” step response produced by the LMI controller (Henrion et al. 2004). The objective measures of those controllers are shown in Table 3.4.

Cost

MOGA Trade-off

1

2

3

4 Objective no.

5

6

7

Figure 3.7. New trade-off graph for the controllers KMOGA.

1

2

3

4

5

6

7

Objective

||T||∞

TV

||S||∞

Cost [1] Cost [0] Goal vector

2 0 1.5

Overshoot (amplitude) 0.008 0 0.001

1.2 0.99 1.2

1.4 1.1 1.5

Settling time (sec) 4.3 2 4

K nth-order 4 0 4

Undershoot (amplitude) 0.1 -0.99 0

Table 3.3 Normalization of the design objectives in Figure 3.7 and goal vector.

The selected MOGA controller and the LMI controller are:

yn 1311.4 + 49.513s + 191.81s 2 = xn 167.47 + 13.465s + s 2

(3.23)

yn 214.59 + 6.5461s + 29.993s 2 = = xn 33.357 + 11.556s + s 2

(3.24)

K ( s ) MOGA =

K ( s) LMI

60


Objective

KLMI

KMOGA

||S||∞ ||T||∞ TV Overshoot Undershoot Settling time (sec) K nth-order

1.1918 (1.5241db) 1 (0 db) 1.0092 0.003 0 3.8182 2

1.1185 (0.97272db) 1 (0 db) 1.0078 0 0 3.8 2

Table 3.4 Objective measures of the final controllers.

Step Response 1 0.9 0.8

KLMI

0.7

KMOGA Amplitude

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

Time (sec)

Figure 3.8. Flexible mode closed-loop step responses of a MOGA controller (‘solid line’) and the best LMI controller (‘dashed line’). In Figure 3.9, the closed-loop frequency domain performance characteristics of the MOGA controller are ||T||∞= 0 dB and ||S||∞ ≈0.9 dB and those of the LMI controller are ||T||∞= 0 dB and ||S||∞ ≈1.5 dB The closed-loop bandwidth of the MOGA controller is ωb1 ≈0.7079 rad /s (||S||∞ crosses 3db from below, dashed grey line) whereas the bandwidth of the LMI controller is ωb 2 ≈0.52481 rad/s. Since ωb1 > ωb 2 , the MOGA controller will have effective control for

61

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs a larger frequency range, and after the frequency ωb1 , the tracking performance (see ||T||∞ ) will degrade and reach higher magnitude of resonance than the LMI controller.

Bode Diagram 5

0

0

-5

-5

-10

-10 |T(jw)| (dB)

|S(jw)| (dB)

Bode Diagram 5

-15 -20

-15 -20

-25

-25

-30

-30

-35

-35

-40 -2 10

0

10

-40 -2 10

2

10

Frequency (rad/sec)

0

10

2

10

Frequency (rad/sec)

Figure 3.9 Flexible mode closed-loop frequency-domain characteristics of the MOGA controller (‘solid line’) and the LMI controller (‘dashed line’). Some experiments showed that MOGA with YK parametrization enhances the population of closed-loop stable controllers providing a reduction in the computational burden more so than the same formulation without YK parametrization. This experiment was performed with a fixed-order controller (3.25)

K=

α 0 + α1 s + α 2 s 2 β0 + β1 s + s 2

(3.25)

Using the settings of Example 1 and the controller (3.25), MOGA was iterated for 100 generations. Figure 3.10 illustrates a family of preferable closed-loop step responses obtained by the MOGA-base method and LMI techniques. This figure shows that for the same number of generations in Example 1, the MOGA results are still inferior to the LMI results or the MOGA with YK parametrization results. The YK parametrization changes

62

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs the shape of the decision space; the benefit is that it might change the shape of narrow decision space. Step Response 1.2 LMI:Approach

1

Amplitude

0.8

0.6

MOGA-base method Without YK Parametrization

0.4

0.2

0 0

2

4

6

8

10 12 Time (Sec)

14

16

18

20

Figure 3.10 Step responses of controllers resulting from MOGA-base method without YK parametrization.

3.4.2 Example 2: Flexible Beam (Doyle et al. 1992): P=

− 6.475s 2 + 4.0302 s + 175.77 s (5s 3 + 3.5682 s 2 + 139.5021s + 0.0929)

(3.26)

The control problem is to design a controller, K, so that the resulting feedback system exhibits a step response with overshoot no greater than 10% and settling time approximately 8 sec. The procedure is the same as in Example 1. A new initial controller was calculated by solving the Diophantine equation (3.5) with arbitrary placement of the poles at s=-1, then a d x d = ( s + 1) 7 . The controller design problem was then tackled using the MOGA-based method. All the settings of the MOGA were the same as for Example 1. The goal vector was set up to search for controllers of first, second and third order and Table 3.5 shows the objective function values (the second and the third row) of the cost for each objective in Figure 3.11 as well as the initial goal vector (the fourth row). In this way a Pareto optimal front was found. This information allows the control engineer to visualise which specification can be achieved and if a reduction in the order of the

63

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs controller will achieve good performance. (see Figure 3.11). For comparison, the LMI controller, shown in equation (3.28) was included with bold line and marked (o) in Figure 3.11.

Cost

MOGA Trade-off

1

2

3

4 Objective no.

5

6

7

Fig. 3.11. Trade-off graph. 1

2

3

4

5

6

7

Objective

||S||∞

||T||∞

TV

Cost [1]

1.9 1.2 2

1.5 0.9 2

2 1 2


Undershoot (amplitude) 0 -0.025 0

Settling time (sec) 20 0 20

K nth-order 4 1 4

Cost [0] Goal vector

Table 3.5 Normalization of the design objectives and goal vector. The objectives 4 to 6 appear to compete whereas the objectives 1 to 4 exhibit poor competence. Here the control engineer might draw some preliminary conclusions, such as: First order controllers will have an excessive settling time, whereas third order controllers will have the minimum settling time, and second order controllers have a good compromise between settling time and undershoot. According to the time domain specifications and the information from Figure 3.10, all the goals were tightened except the objective 5 undershoot and the new goal vector is shown in Table 3.6. The new tradeoff results are shown in Figure 3.12. The MOGA controller 3.27 was chosen and compared with the LMI controller 3.27. The first comparison is in Figure 3.12, where the MOGA controller is marked (◊) and the LMI controller is marked (o); both solutions are in bold lines and cannot be distinguish because they overlap.

64


Cost

MOGA Trade-off

1

2

3

4 Objective no.

5

6

7

Figure 3.12. New trade-off graph for the controllers KMOGA for example 2. 1

2

3

4

5

6

7

Objective

||S||∞

||T||∞

TV

Cost [1]

1.5 1.25 1.5

1.04 1 1.5

1.17 1 1.5


Undershoot (amplitude) -0.004 -0.011 0

Settling time (sec) 7.5 3.5 8

K nth-order 3 1 2

Cost [0] Goal vector

Table 3.6 Normalization of the design objectives and goal vector. The selected MOGA controller offers the best compromise among all the objectives, for example, it offers best compromise among overshoot, undershoot and settling time. Figure 3.13 illustrates the “best” closed-loop step responses obtained by the MOGA and LMI techniques (Henrion 2003a). Both controllers produced very similar step responses. Step Response 1.2

1

Amplitude

0.8

KLMI 0.6

0.4

KMOGA

0.2

0 0

2

4

6

8

10

12

14

Time (sec)

Figure 3.13. Flexible Beam closed-loop step responses of MOGA controller (‘solid line’) and the best LMI controller (‘dashed line’). 65

Chapter 3 Multiobjective Optimisation of Controller Structure via YK parametrization using GAs and LMIs The selected MOGA controller and the LMI controller are: K MOGA =

K LMI =

0.56229⋅ 10−2 + 1.3052s − 1.3691⋅10−2 s 2 3.851+ 3.73s + s 2

7.7489⋅10−5 + 0.016572s + 0.36537s 2 0.041025+ 1.0437 + s 2

(3.27)

(3.28)

Although is impossible to have zero undershoot since the plant is non-minimum phase, and thus the plant has a right-half-plane (rhp) zero that will appear in T(s). In MOGA, the goal was set up to be zero in order to enforce minimization of the undershoot. The final MOGA controller is non-minimum phase. However, the rhp zero is far in the rhp, and could be ignored. Thus a final controller of K MOGA =

0.56229⋅10−2 + 1.3052s 3.851+ 3.73s + s 2

would remove the additional rhp zero, and the performance remain almost unaffected Objective

KLMI

KMOGA

||S||∞ ||T||∞ TV Overshoot Undershoot Settling time (sec) K nth-order

1.2704 (2.0788 db) 1.0093 (0.0804 db) 1.1108 0.039858 -0.0097915 4.4496 2

1.3554 (2.6413 db) 1.0018 (0.0156 db) 1.0554 0.021325 -0.0060244 4.5104 2

Table 3.7 Objective measures of the final controllers. In Figure 3.13 the closed-loop frequency domain performance measures of the MOGA controller in equation. (3.25) are ||T||∞ = 0 dB and ||S||∞ ≈2.07 dB and those the LMI controller are ||T||∞ = 0 dB and ||S||∞ ≈1.5 dB. It is important to mention that similar specifications were achieved with H∞ state space techniques. However, the controller was of order eight (Doyle et al. 1992). The closed-loop bandwidth of the MOGA controller and LMI controller are ωb =0.3 rad/s. The LMI controller has slightly better bandwidth. However, the tracking performance (see ||T||∞) will reach slightly higher magnitude of resonance than the MOGA controller.

66


Bode Diagram 10

0 -3

0 -3

-10

-10

-20

-20

-30

-30 |T(iw)| (dB)

|S(iw)| (dB)

Bode Diagram 10

-40 -50

-40 -50

-60

-60

-70

-70

-80

-80

-90 -5 10

-90

0

10

-2

10

Frequency (rad/sec)

0

10

2

10

Frequency (rad/sec)

Figure 3.14 Flexible Beam closed-loop frequency-domain characteristics of the MOGA controller (‘solid line’) and the LMI controller (‘dashed line’). The algorithms were run on a standard PC with Pentium IV, 1.7GHz processor. The computational cost of solving the examples with MOGA was CPU time 15min to 25min. A single run of LMI algorithms was CPU time 15 sec. The MOGA-based method can deal with MIMO systems, but because of the controller structure, equation (3.18) relies on the preliminary results of YK paramatrization by Henrion et al. (2004), and only applies to the SISO case.

67


3.5. Conclusions The LMI optimisation for fixed-order H∞ controller design can be computed quickly. However, the whole procedure for designing a controller requires several attempts. Multiple trials changing the central polynomial, bound on the frequency domain specification and the order of the controller have to be carried out until the specification seems to be satisfied or no progress is noticed. Despite the computational cost of the MOGA, the Pareto surface between controller order and closed-loop performance can be investigated in a single run. Thus, the control engineer has a choice from among the family of Pareto optimal solutions and design time is saved. The YK parametrization of stabilizing controllers in conjunction with the MOGA-base method has shown to be successful in finding a controller of optimal order. The non-YK parametrization experiment performed on Example 1 shows that a faster convergence of MOGA might be achieved when using YK parametrization. In general, it is rather difficult with LMI techniques to cope with time-domain specifications (non-convex specifications), whereas the MOGA approach affords a straightforward way of the use of them. Moreover, a mixture of objectives with a considerable mathematical complexity can be included in the evaluation function of MOGA with excellent flexibility.

68

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method. 4.1 Introduction This chapter deals with two of the most popular design specifications, H2 (LQG case) performance and H∞ performance, which have been shown to have competing objectives (Bernstein and Haddad 1989). This chapter highlights the relative flexibility offered by a multiobjective genetic algorithm (MOGA) and linear matrix inequalities (LMIs) in addressing complex problems. Research into LMIs reveals an extensive list of possible design specifications (Boyd et al. 1994b; Ghaoui and Silviu-lulian 1999), which can be expressed in an LMI formulation. However, the application usually requires transformation of the controller parameters, more about this later in section 4.3. This chapter illustrates briefly how the H2 norm case can be cast as an LMI optimisation problem. Alternatively, MOGA can be applied to multiobjective control problems by encoding the control problem, defining the vector spaces and the objectives. If a successful design is easily located, further improvements such as reduced controller order can be incorporated. Alternatively, if acceptable solutions are not readily obtained, the designer may choose to relax the design goals.

69

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method These techniques are applied to two control problems. Firstly, the LQG design problem, which has been used to find the trade-off between process and sensor noise for a simple but realistic controller design problem (see Section 4.8.1 and 4.8.2). Secondly, these techniques have also been applied to the mixed H2/H∞ problem (see Section 4.8.3 and 4.8.4). The LQG problem is an example from Barratt and Boyd (1989), they computed an exact trade-off involving noise sensitivity and robustness measurements. Such trade-off can serve as the limit of performance bounds against which controllers of any complexity, obtained by any design method, can be computed. This chapter shows how this problem can be formulated as a convex program (LMI formulation) using the equivalence between LQG control problem and the H2 optimal control theory. This formulation is presented in continuous-time. However, because the Barratt and Boyd (1989) example is given in discrete-time, the LMI constraints are derived in discrete-time. The LMI technique and the MOGA-based method are used to find the same trade-off as Barratt and Boyd (1989) and the advantages and disadvantages of using each method are addressed. The mixed H2/H∞ problem has been extensively studied and it remains an open problem (Bernstein and Haddad 1989; Khargonekar and Rotea 1991a; Kaminer et al. 1993; Doyle et al. 1994; Zhou et al. 1994). The LMI techniques for multiobjective control design have been proposed to handle the mixed H2/H∞ control problem (Boyd et al. 1994b). The approaches can be divided in two. The first approach (Scherer 1995b; Scherer 2000) expresses the mixed H2/H∞ control problem as LMI conditions by using the Youla parametrization of the controller. This approach truly solves the multi-objective problem but at the expense of an infinite dimensional problem. In order to alleviate this problem, the Youla parameter is approximated (see Section 2.9). The accuracy of the approximation compromises the order of the controller. The second approach (Scherer 1995a; Scherer et al. 1997) leads to finite dimensional controllers but at the expense of introducing dependence (see Section 4.4) between objectives. It also makes the problem a single-objective problem. In this chapter, when using the LMIs, the second approach is studied.

70

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method

4.2 Problem statement Consider a linear time invariant (LTI) plant P: ⎛ x& ⎞ ⎛ A ⎜ ⎟ ⎜ ⎜ z1 ⎟ ⎜ C1 ⎜ z ⎟ = ⎜C ⎜ 2⎟ ⎜ 2 ⎜ y⎟ ⎜C ⎝ ⎠ ⎝

B1

B2

D11

D12

D21

D22

F1

F2

B ⎞⎛ x ⎞ ⎟⎜ ⎟ E1 ⎟⎜ w1 ⎟ E2 ⎟⎜ w2 ⎟ ⎟⎜ ⎟ 0 ⎟⎠⎜⎝ u ⎟⎠

(4.1)

where, x is the state vector, u is the control input, y is the measured output available for the controller, and the channels from wi (disturbance inputs) to zi (controlled outputs) serve to specify performance objectives. A controller, K, is a LTI system described as:

⎛ x& c ⎞ ⎡ Ac ⎜⎜ ⎟⎟ = ⎢ ⎝ u ⎠ ⎣Cc

w1 w2

Bc ⎤⎛ xc ⎞ ⎜ ⎟. Dc ⎥⎦⎜⎝ y ⎟⎠

(4.2)

z1 z2

P u

y

K Figure 4.1 Generalized control structure for a two objective control design problem. The controller is parameterized by the matrices (Ac, Bc, Cc, Dc). The closed loop system in figure 4.1 is denoted in equation (4.3),

⎡ Acl z j = Tj wj = ⎢ ⎣C cl , j

Bcl , j ⎤ w , Dcl , j ⎥⎦ j

(4.3)

where the index, j, refers to a given channel. Then Tzw = Dcl + C cl ( sI − Acl ) −1 Bcl with the

notation:

71


⎛ Acl ⎜ ⎜ Ccl , j ⎝

⎛ A + BDC C

BC C

BC C ⎝ C j + E j DC C

AC

Bcl , j ⎞ ⎜ ⎟= Dcl , j ⎟ ⎜ ⎠ ⎜

E j CC

B j + BDC F j ⎞ ⎟ BC F j ⎟. D j + E j DC F j ⎟⎠

(4.4)

This chapter deals with the computation of a dynamic output-feedback control law

u = Ky .

4.3 LMI Formulation of H2 and H∞ In Section 4.3.1 a general procedure to design a controller that satisfies a performance specification (e.g. H2, H∞) is presented. In Sections 4.3.2 and 4.2.3 the LMI formulation for H2 and H∞ will be presented. These sections will show that their inequalities are in general nonlinear, hence non-convex, since products among the variables appear. The purpose of section 4.3.1 is to show the procedure that transforms the synthesis problem into convex. The procedures illustrated in this sections 4.3 follow the lines of Scherer and Weiland (1999).

4.3.1 Quadratic performance synthesis (Scherer et al. 1997) The following proposition provides an equivalent condition in terms of a frequency domain inequality and equivalent linear matrix inequality for this. These results are a consequence of the YKP lemma (see Section 2.7.3) and it is used to derive the controller parametrization. Consider the design of a controller that achieves stability and quadratic performance in the channel wi to zi and suppose have the following performance index: ⎛Q j P j = ⎜⎜ T ⎝S j

Sj ⎞ ⎟, R j ⎟⎠

Rj ≥ 0 .

A characterization of linear dissipative systems have revealed (Scherer et al. 1997) that the controller (4.2) renders (4.3) internally stable and leads to

∞

T

∞

⎛ w (t ) ⎞ ⎛ w j (t ) ⎞ ⎟ P j ⎜ j ⎟dt ≤ −ε w j (t ) T w j (t )dt ⎜ ⎜ z j (t ) ⎟ ⎜ z j (t ) ⎟ ⎠ ⎝ ⎠ 0⎝ 0

∫

∫

for some ε>0 if and only if

72


1) All the eigenvalues of A have negative real parts. 2) The Frequency Domain Inequality (FDI) T

⎛ I ⎞ ⎛ I ⎞ ⎜ ⎟ Pj ⎜ ⎟ ⎜ T j (i ω ) ⎟ ⎜ T j (iω) ⎟ < 0 for all ω ∈ ∞ ⎝ ⎠ ⎝ ⎠

With these results, it can be stated that the closed loop system (4.3) is asymptotically stable and has quadratic performance with index Pj if and only if there exists a matrix

x

such that,

x>0,

x xA xB 0 x

⎛ AclT + ⎜ ⎜ BclT ⎝

cl

cl

⎞ ⎛ 0 ⎟+⎜ ⎟ ⎜C ⎠ ⎝ cl

(4.5) T

I ⎞ ⎟ Dcl ⎟⎠

⎛Qj ⎜ T ⎜Sj ⎝

S j ⎞⎛ 0 ⎟⎜ R j ⎟⎠⎜⎝ Ccl

I ⎞ ⎟ < 0. Dcl ⎟⎠

(4.6)

Since Acl , Bcl , Ccl , Dcl are in terms of the state variables Ac , Bc , Cc , Dc of the controller and the inequality (4.6) turns into nonlinear after substitution, a transformation is necessary. A partition of

x is proposed: x = ⎛⎜⎜UX ⎝

U⎞ ⎟ * ⎟⎠

T

x

−1

⎛Y = ⎜⎜ T ⎝V

V⎞ ⎟ * ⎟⎠

with the auxiliary variables: ( U , V ) and I − XY nonsingular, there exists a nonsingular U , V with I − XY = UV T . Define

⎛Y

y = ⎜⎜V T ⎝

Using the matrix

y

I⎞ ⎟ , 0 ⎟⎠

⎛I

z = ⎜⎜ X ⎝

0⎞ ⎟ to get U ⎟⎠

yT x = z

the following two transformations can be performed

y

T

⎛Y xy = ⎜⎜ I ⎝

I⎞ ⎟ = X (v) , X ⎟⎠

(4.7)

where ν denotes the new Lyapunov matrices (X,Y), and the second transformation is T

⎛ y 0 ⎞ ⎛ x 0 ⎞⎛ Acl ⎜ ⎟ ⎜ ⎟⎜ ⎜ 0 I ⎟ ⎜ 0 I ⎟⎜⎝ C cl ⎝ ⎠ ⎝ ⎠

Bcl ⎞⎛ y 0 ⎞ ⎛ y T xAcl y ⎟ =⎜ ⎟⎜ Dcl ⎟⎠⎜⎝ 0 I ⎟⎠ ⎜⎝ C cl y

substituting (4.4) into transformation (4.8), 73

y T xBcl ⎞⎟ Dcl

⎟, ⎠

(4.8)


⎛ y T xAcl y ⎜ ⎜ C y cl ⎝

y xBcl ⎞⎟ ⎛⎜ zAcl y = Dcl ⎟⎠ ⎜⎝ C cl y T

⎛0 B ⎞ ⎜ ⎟ ⎡⎛U I 0 ⎜ ⎟ ⎢⎜⎜ + ⎜⎜ ⎟⎟ ⎣⎢⎝ 0 ⎝0 E j ⎠

⎛ AY ⎜ =⎜ 0 Dcl ⎟⎠ ⎜⎜ ⎝C jY

0 ⎞⎤⎛ I 0 ⎟⎥⎜ 0 ⎟⎠⎦⎥⎜⎝ 0 C

0 ⎞⎟ ⎛ XAY +⎜ I ⎟⎠ ⎜⎝ 0

BC ⎞⎛V T ⎟⎜ DC ⎟⎠⎜⎝ CY

XB ⎞⎛ AC ⎟⎜ I ⎟⎠⎜⎝ C C

0 Bj ⎞ ⎟ XA XB j ⎟ + ⎟ C j D j ⎟⎠

zBcl ⎞⎟

0⎞ ⎟ F j ⎟⎠ .

(4.9)

The term inside the square brackets contains all the non-linear products. By introducing ⎛K

the parameters ⎜⎜ ⎝M

L⎞ ⎟ , the expression (4.9) becomes affine in the new set of variables N ⎟⎠

if: ⎛K ⎜⎜ ⎝M

BC ⎞⎛ V T ⎟⎜ DC ⎟⎠⎜⎝ CY

XB ⎞⎛ AC ⎟⎜ I ⎟⎠⎜⎝ CC

L ⎞ ⎛U ⎟=⎜ N ⎟⎠ ⎜⎝ 0

0 ⎞⎟ ⎛ XAY +⎜ I ⎟⎠ ⎜⎝ 0

0⎞ ⎟, 0 ⎟⎠

(4.10)

then the transformation is ⎛ ⎛ AC ⎜ ,⎜ ⎜ ⎜C ⎝ ⎝ C

x

⎛ ⎛K v = ⎜⎜ X , Y , ⎜⎜ ⎝M ⎝

BC ⎞ ⎞ ⎟⎟ → DC ⎟⎠ ⎟⎠

L ⎞⎞ ⎟⎟ . N ⎟⎠ ⎟⎠

where ν now denotes the new Lyapunov matrices (X,Y) and the new controller variables (K, L, M, N). This nonlinear transformation

v relies on a simple mapping of all LMIs,

(e.g. H2 performance) into a set of affine constraints on the new controller variables,

⎛Y X (v) = ⎜⎜ ⎝I ⎛ y T xAcl y ⎜ ⎜ C y cl ⎝

y T xBcl ⎞ ⎛⎜ A(v) ⎟= Dcl ⎟⎠ ⎜⎝ C j (v)

I⎞ ⎟ X ⎟⎠ ,

⎛ B j (v) ⎞ ⎜ AY + BM ⎟=⎜ K D j (v) ⎟⎠ ⎜

⎝ C jY + E j M

A + BNC XA + LC C j + E j NC

B j + BNF j ⎞ ⎟ XB j + LF j ⎟ . D j + E j NF j ⎟⎠

(4.11)

Then the LMIs (4.5) and (4.6) are transformed as follows:

y T xy > 0 and T ⎛ y 0 ⎞ ⎡⎛ AclT x + xAcl ⎜ ⎟ ⎢⎜ ⎜ 0 I ⎟ ⎢⎜ BclT x ⎝ ⎠ ⎣⎝

xBcl ⎞⎟ + ⎛⎜ 0 ⎟⎠

0 ⎜C ⎝ cl

I ⎞ ⎟ Dcl ⎟⎠

T

⎛Q j ⎜ T ⎜S j ⎝

S j ⎞⎛ 0 ⎟⎜ R j ⎟⎠⎜⎝ C cl

I ⎞⎤⎛ y 0 ⎞ ⎟ < 0, ⎟⎥⎜ Dcl ⎟⎠⎥⎜⎝ 0 I ⎟⎠ ⎦

as a result the LMIs (4.5) and (4.6) are transformed into a set of affine variables v .

X (v ) > 0 ,

74

(4.12)

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method ⎛ A(v) T + A(v) B j (v) ⎞ ⎛ 0 I ⎞ ⎜ ⎟+⎜ ⎟ T ⎜ ⎟ ⎜ C j (v ) D j (v ) ⎟ B v ( ) 0 j ⎠ ⎝ ⎠ ⎝

Theorem 4.1 (Scherer and Weiland 1999) There exists a controller ( Ac , Bc , Cc , Dc ) and

T

x

⎛Q j ⎜ T ⎜S j ⎝

S j ⎞⎛ 0 I ⎞ ⎟⎜ ⎟ < 0. R j ⎟⎠⎜⎝ C j (v) D j (v) ⎟⎠

(4.13)

satisfying (4.17) if only if there exists

v that solves the inequalities (4.13). If v satisfies (4.13), then I − XY is nonsingular and there exist nonsingular U , V T with I − XY = UV T . After having solved the synthesis inequalities for v , one factorizes I − XY into square and non-singular blocks U and V T using SVD and the controller by the following equations: ⎛ AC ⎜⎜ ⎝ CC

BC ⎞ ⎛U ⎟=⎜ DC ⎟⎠ ⎜⎝ 0

and the Lyapunov matrix

XB ⎞ ⎟ I ⎟⎠

−1

⎛ K − XAY ⎜⎜ ⎝ M

L ⎞ ⎛V T ⎟⎜ N ⎟⎠ ⎜⎝ CY

−1

0⎞ ⎟ , I ⎟⎠

(4.14)

x as,

x

−1

⎛Y V ⎞ ⎛ I 0 ⎞ ⎟⎟ ⎜⎜ ⎟⎟ . = ⎜⎜ ⎝ I 0 ⎠ ⎝0 U ⎠

(4.15)

Proof. The necessity part has been proven in the construction of the theorem. The

sufficiency part of the proof leads to an Ac that has the same size as A.

4.3.2 H2 Performance This section shows the formulation of H2 performance specifications as an LMI constraint on the design variables. The Frobenius norm is used to obtain the H2 norm of G(s) and is defined by integrating over frequency.

G( s)

2

=

1 ∞ trace(G(iω) H G(iω))dω ∫ 2π −∞

(4.16)

where G(s) must be strictly proper. By Parseval’s theorem, G (s ) 2 is equal to the H2 norm of the impulse response,

75

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method ∞

G ( s ) 2 = g (t ) 2 =

∫ trace(g

T

)

( τ ) g ( τ ) dτ ,

0

and the impulse response matrix is:

0 t0) and thus has full rank n. P may also be obtained as the solution to the Lyapunov equation: AP2 + P2 AT = − BB T ,

(4.17)

where P2 denotes the controllability Gramian for the H2 norm. The controllability Gramian is the unique positive solution to the Lyapunov equation. This is equivalent to saying that there exists P2>0 such that A is stable and G ( s )

2 2

0, I ⎟⎠

⎞ C clT ⎟>0, T ⎟ Z − Dcl Dcl ⎠

(4.22)

trace( z ) < γ .

(4.23)

Observe that the matrix inequalities in equations (4.22) and (4.23), are non-linear, as they involve multiplication of variables, and the variables ( Acl , Bcl , C cl , Dcl ) are expressed in terms of the state controller variables ( Ac , Bc , Cc , Dc ). Hence, they cannot be handled by LMI optimisation and do not seem easily tractable numerically. In order to alleviate this problem, two transformations have to be carried out. The first is a simple change of variable

x=P

2

−1

(Boyd et al. 1994b). The LMIs in (4.22) have to be transformed by

multiplying dig ( P2−1 , I ) on the left and right hand side, the change of variable must be applied as well as the Schur complement for the terms

x=P

−1 2

x − xAx−1A x T

and

Z − Dcl DclT . The results are LMI (4.24) and LMI (4.25).

⎛ x ⎜ T ⎜ Acl x ⎜ BT ⎝ cl

x

⎛ ⎜ ⎜ 0 ⎜⎜ C ⎝ cl

xA x

cl

0 0 I Dcl

xB

⎞ ⎟ 0 ⎟ > 0, I ⎟⎠ cl

(4.24)

C clT ⎞⎟ DclT ⎟ > 0 . ⎟ Z ⎟ ⎠

(4.25)

The second more sophisticated transformation (see Section 4.3.1) has been proposed by (Scherer et al. 1997). The idea is to rewrite the H2 norm inequalities involving leads to inequalities in the blocks y T xy ,

77

y T xAcl y , y T xBcl , C cl y , Dcl .

y , which

Multiplying

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method the LMI (4.24) by diag ( y, y, I ) T on its left hand side and diag ( y , y , I ) on its right hand side gives the LMI (4.26). Multiplying the LMI (4.25) by diag ( y , I , I ) T on its left hand side and with diag ( y , I , I ) on the right hand side gives the LMI (4.27). ⎛ X (v ) ⎜ T ⎜ A (v ) ⎜ B T (v ) ⎝ j

A(v) B j (v) ⎞ ⎟ X (v ) 0 ⎟ >0, I ⎟⎠ 0

(4.26)

⎛ X (v ) C Tj (v) ⎞ 0 ⎜ ⎟ I D Tj (v) ⎟ > 0 . ⎜ 0 ⎜C v D v Z ⎟⎠ j( ) ⎝ j( )

(4.27)

where X(v), A(v), Bj(v), Cj(v), Dj(v) are linear and affine on the controller variables K, L, M and N. Using equation (4.11) the LMIs (4.26 and 4.27) can be transformed into LMIs (4.28 and 4.29). Then the LMI optimisation is tractable numerically and the controller can be recovered by reversing the transformation using Theorem 4.1.

Y ⎛ ⎜ I ⎜ ⎜ ( AY + BM ) T ⎜ ⎜ ( A + BNC ) T ⎜ B + BNF T j) ⎝( j

I

AY + BM

A + BNC

X

K

XA + LC

Y

I

I

X

0

0

KT ( XA + LC ) ( XB j + LF j )T T

B j + BNF j ⎞ ⎟ XB j + LF j ⎟ ⎟ > 0, 0 ⎟ 0 ⎟ ⎟ I ⎠ (4.28)

⎛ Y ⎜ I ⎜ ⎜ 0 ⎜ ⎜C Y + E M j ⎝ j

I

0

X 0

0

C j + E j NC

D j + E j NF j

I

(C j Y + E j M )T ⎞ ⎟ (C j + E j NC )T ⎟ >0. ( D j + E j NF j )T ⎟⎟ ⎟ Z ⎠

(4.29)

4.3.3 H∞ Performance The H ∞ performance has been formulated as an LMI constraint (Scherer and Weiland 1999). Using the Kalman-Yacubovich-Popov lemma together with a specific quadratic supply function reduces frequency domain inequalities to equivalent systems of LMIs

78

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method and this is known as the Bounded Real Lemma (Scherer 1990). Using this lemma, let A be asymptotically stable and T j

∞

< γ ∞ if and only if there exists a solution P∞=(P∞)T

with the LMI constraint:

⎛ AclT P∞ + P∞ Acl ⎜ BclT P∞ ⎜ ⎜ C cl ⎝

P∞ Bcl − γ∞ I Dcl

C clT ⎞ ⎟ DclT ⎟ < 0 , − γ ∞ I ⎟⎠

P∞ > 0

(4.30)

where P∞ denotes the positive definite matrix for the H∞ norm. Using the LMI approach general procedure from section 4.3.1, the LMI in equation (4.30) can be turned into equation (4.31), which is affine in v , (see equation (4.11)).

⎛ A(v) T + A(v) B (v) ⎜ − γ∞ I B (v ) T ⎜ ⎜ C (v ) D (v ) ⎝

C (v ) T ⎞ ⎟ D (v ) T ⎟ < 0 , − γ ∞ I ⎟⎠

X ∞ (v) > 0 .

(4.31)

The smallest possible upper bound of the L2-induced gain of the system can be computed by minimizing γ ∞ > 0 over all variables in v satisfying equation (4.31).

4.4 Mixed H2/H∞: LMI Approach The multiobjective H2/H∞ control problem is to keep bounds on the H2 norm of, say, the first channel w1 → z1 and on the H∞ norm of, say, the second channel w2 → z 2 of the controlled system:

TZ1W

2

< γ 2 and TZ 2W

∞

< γ∞

(4.32)

Despite the recent advances in robust control theory, the robust performance problem formulated in the multiobjective H2/H∞ framework largely remains an open problem. First of all, this problem is an infinite-dimensional optimisation problem. Approximations to the finite dimensional problem using Youla parameter Q have been developed (Scherer 1995b; Hindi et al. 1998). System impulse responses were truncated to a finite horizon but LMIs have shown that when a state-space description is available, then many of the

79

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method infinite horizon cost and constraints can be represented as LMIs and minimized efficiently as semidefinite programs (SDPs). In this way, errors by truncations are eliminated. Another alternative is to introduce dependence among the different objectives, which leads to finite dimensional controllers of the same order of the generalized plant (Scherer 1995a; Scherer et al. 1997). This last approach is known as the mixed H2/H∞ problem and it is used in this work. Mixed H2/H∞ problems can be motivated in different ways. As a matter of fact, there are many different mixed H2/H∞ problems. The mixed H2/H∞ problem studied in this work is to find an internal stabilizing controller K(s) which:

min TZ1W

2

subject to TZ 2W

∞

< γ∞ .

(4.32)

A controller that solves this mixed H2/H∞ problem will ensure that the closed-loop system is robustly stable to all finite-gain stable perturbations Δ, and at the same time TZ1W

2

represents the steady-state variance of the output z1 when w2 = 0 and w1 is white noise with unit intensity. Investigating the trade-off between H∞ norm and the H2 norm constraints consist of plot a

[

]

curve of optimal values, by varying γ ∞ in some interval γ ∞l ,..., γ ∞u , where the lower bound γ ∞l could be taken close to the smallest achievable H∞ norm of transfer function TZ 2W

∞

.

In order to solve the problem (4.32), the mixed H2/H∞ problem is reformulated in terms of the solvability of LMIs. In fact, Acl is stable and (4.32) holds if only if there exist symmetric P1, P2, Z such that trace(Z) < γ2 and

⎛ AclT P∞ + P∞ Acl ⎜ ⎜ BclT ,1 P∞ ⎜ ⎜ C cl ,1 ⎝

P∞ Bcl ,1 − γ∞ I Dcl ,1

80

C clT ,1 ⎞⎟ DclT ,1 ⎟ < 0 , ⎟ − γ∞ I ⎟ ⎠

P∞ > 0 ,

(4.33)


D cl , 2 = 0 ,

⎛ AclT P2 + P2 Acl ⎜ ⎜ BclT , 2 P2 ⎝

P2 Bcl , 2 ⎞⎟ 0 . (4.34) Z 2 ⎟⎠

This reformulation includes two different LMI constraints (4.33, 4.34), making the problem non-convex (bilinear matrix inequality). In order to recover convexity, a common Lyapunov function is used by equating the Lyapunov matrices P2 from H2 performance to the P∞ from H∞ performance:

P = P2 = P∞ .

(4.35)

This leads to a conservative result, which will be illustrated in section 4.8.3. The controller can be recovered by using the general procedure (see Section 4.3.1)

4.5 Mixed H2/H∞: MOGA-based method Typically the problems of H2 optimal control and H∞ optimal control have been studied independently when using EAs. For example the H∞ design has been assisted with the MOGA in order to satisfy a number of conflicting design criteria (Dakev et al. 1997; Griffin et al. 2000; Schroder et al. 2001). Other researchers (Hunt 1992; Patton and Liu 1994; Chen and Cheng 1998) have been directly optimized over the mixed sensitivity functions (S, T, KS). Their GA has been used to search for a controller which minimized the following criteria: ⎧⎡ S ⎤ ⎫ ⎪ ⎪ min sup σ ⎨⎢⎢ T ⎥⎥ ⎬ , K ω∈Ω ⎪⎢ KS ⎥ ⎪ ⎩⎣ ⎦ ⎭

where Ω denotes the frequency range and σ denotes the maximum singular value. The demand of controllers with multiple criteria like robustness and performance has motivated the research into mixed H2/H∞ problems. This problem is originally a multiobjective control problem that has been found difficult to solve because of the nonconvexity in the controller parameters. Recently, this problem has been tackled with GAs (Chen et al. 1995; Herreros et al. 2002; Takahashi et al. 2004). The idea of optimising the norm of transfer functions and the previous work of Fonseca and Fleming (1994) have motivated, in this thesis, the study of the multiobjective H2/H∞ problem. Details are shown in Section 4.8.4. 81


4.6 Equivalence of LQG and H2 Optimal Control The equivalence derived in this section is used in section 4.8.1. It provides the formulation of the linear quadratic Gaussian (LQG) control problem into the H2 optimal control: LMI approach. It is known that the LQG problem is a special case of H2 optimal control (Boyd and Barratt 1991; Skogestad and Postlethwaite 1996; Burl 1999). The LQG controller overcomes the need to measure the entire state by estimating the state, using a Kalman filter. The estimated state is then used in the linear quadratic regulator (LQR). Thus, the LQG controller is a combination of a LQR and a Kalman filter, known as the separation principle. The LQG control refers to an optimal control problem where the plant model is linear, the cost function is quadratic, and the test conditions consist of random initial condition w , a white noise disturbance input wd , and a white measurement noise wn . The plant is described by the following linear state equation: x& = Ax + Bu + wd ,

y = Cx + wn . The cost function ∞ ⎤ 1 ⎡ T J ( x (t ), u (t )) = E ⎢ ∫ x (t )Qx (t ) + u T (t ) Ru (t ) dt ⎥ . 2 ⎣0 ⎦

{

}

(4.36)

This cost function is equal to the cost functions specified for the Stochastic LQR version. A reasonable controller design for the LQG control problem can be obtained by using the LQR feedback gain matrix K operating on the state estimate generated by the Kalman filter u (t ) = − Kxˆ (t ) ,

The cost function (4.36) is rewritten as follows:

J=

∞ 1 ⎡ E ⎢ ∫ (Q1 / 2 x(t ))T ( R1 / 2 u (t ))T 2 ⎢⎣ 0

[

82

x (t ) ⎤ ⎤ ⎥dt ⎥ 1/ 2 R u t ( ) ⎣ ⎦ ⎥⎦

]⎡⎢Q

1/ 2

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method 0 ⎤ ⎡ x⎤ ⎥⎢ ⎥ , R1 / 2 ⎦ ⎣u ⎦

⎡Q1 / 2 z=⎢ ⎣ 0

Defining an error signal z as

∞ ⎤ 1 ⎡ T J = E ⎢ ∫ z (t ) z (t )dt ⎥ , 2 ⎣0 ⎦

T ⎡ ⎤ 1 1 J = E ⎢lim z T (t ) z(t )dt⎥ = trE z(t ) z T (t ) = T →∞ 2T ∫ 2π −T ⎣ ⎦

{

}

∞

∫ tr (G ( jω )

H

G ( jω ))dω .

(4.37)

−∞

2

In 4.37 Parseval’s theorem was applied and it is equal to G (s ) 2 , which is H2 norm. In

order to finish with the stochastic interpretation of H2 norm and LQG, the input w has to be the white noise of the unit intensity. That is

{

}

E w(t ) w(τ ) T = S wδ (t − τ ) .

With the scalar spectral density of each element S w = I , ⎧ ⎡ w (t ) ⎤ E ⎨⎢ d ⎥ wd (τ ) T ⎩⎣ wn (t ) ⎦

[

⎫ ⎡W wn (τ ) T ⎬ = ⎢ ⎭ ⎣0

]

0⎤ δ (t − τ ) . V ⎥⎦

In summary, by minimizing the H2 norm, the output power of the system, due to a unit intensity white noise input, is minimized; we are minimizing the root-mean-square (RMS) value of z. Thus, the LQG can be cast as an H2 optimisation in the following general framework: Stochastic inputs wd , wn as

Output signal regulation

⎡Q1 / 2 z=⎢ ⎣ 0

⎡ wd ⎤ ⎡W 1 / 2 0 ⎤ ⎥w , ⎢w ⎥ = ⎢ V 1/ 2 ⎦ ⎣ n⎦ ⎣ 0

0 ⎤ ⎡ x⎤ ⎥⎢ ⎥ . R1 / 2 ⎦ ⎣u ⎦

Looking at these two stochastic conditions, the input and the output signal of the LQG problem leads to a control configuration of the H2 optimal problem. See Figure 4.2 and equation (4.38).

83


⎧ w ⎨ ⎩

W1/2

process noise

+ B

+

R1/2

x

(sI-A)-1

⎫ ⎬ z ⎭

Q1/2

C

u

1/2

V

+

measurement noise

y

+

Figure 4.2 The LQG problem formulated in the H2 optimal control problem. In Figure 4.2, w is a white noise process of unit intensity. Finally the state-space description of the LQG problem is x& = Ax + (W 1 / 2

0)w + Bu ,

⎛ Q1 / 2 ⎞ ⎛ 0 0 ⎞ ⎛ 0 ⎞ ⎟ x + ⎜⎜ ⎟⎟ w + ⎜⎜ 1 / 2 ⎟⎟u , z = ⎜⎜ ⎟ ⎝R ⎠ ⎝ 0 ⎠ ⎝0 0⎠

(4.38)

y = Cx + (0 V 1 / 2 )w + (0)u . It was shown that LQG problem can be cast as an H2 optimal control problem. Thus, under this framework, the LQG problem can be cast using linear matrix inequalities. This is shown in Section 4.8.1.

4.7 H2 Optimal Control: MOGA-based method The H2 optimal controller design can be tackled with GAs, applying an indirect methodology. For example, Mei and Goodall (2000) developed an H2 optimal controller for active steering and which is fine-tuned using GAs. The separation principle is adopted in the development of the LQG controller. First, an optimal controller is designed as if full state feedback is available to optimize the design. GAs are used to fine-tune the controller parameters, and thus, they are used to search for the best values of the weighting factor matrix Q in the cost function (4.39). 84


[

]

J = E y T (∞ )Qy (∞ ) + u T (∞ ) Ru (∞ ) ,

(4.39)

where the infinity symbol is used as an argument to indicate that the given quantity has reached steady state, y is the output regulation, and u is the control effort. The polynomial LQG problem (Hunt 1992) is another approach where the optimisation problem aims to select the regulator in such a way that the cost function J is minimal. In the genetic algorithm, J serves as the evaluation function in the search. Hunt used the cost function J in the frequency domain and introduces another parameter to check stability. Fonseca and Fleming (1994) tackled the LQG problem from a multiobjective point of view. LQG controller design was used to find a trade-off between steady-state output and actuator variances due to the presence of process and measurement noise. The aim was to find the minimum of the linear combination of the two variances, in equation (4.39). Fonseca and Fleming used the MOGA-based method to minimize the cost function J by encoding the controller parameters (poles zeros and gain). The closed-loop system evaluation of RMS(u) and RMS(y) served as multi-evaluation functions. The MOGA was used to search for a Pareto optimal set or non-dominated solutions, these results are recalculated in section 4.8.2 and compared with the Pareto optimal resulting from the LMI approach.

4.8. Numerical Evaluations of MOGA and LMIs In this section, computational examples of the H2 optimal control and the mixed H2/H∞ problem were taken in order to illustrate how both LMI techniques and the MOGA-based method work for the same problem.

4.8.1 H2 Optimal Control (LQG): LMI Approach. This section studies a problem already considered by Barratt and Boyd (1989) and Fonseca and Fleming (1994). It consists of designing a regulator for the plant with a double integrator and a non-minimum phase (NMP) zero shown in equation (4.40).

85

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method (4 − s)

P(s) =

(4.40)

s 2 (4 + s)

The objective is to investigate the trade-offs in the design of a regulator, K(s), for this plant, where the closed loop system shown in figure 4.3 is disturbed by an input process noise, wP. The only signal available to the controller is yp=y+vs , where vs is sensor noise and y is the output, u

r =0

wP +

+ K

-

+

y

P

+ +

vS

Figure 4.3 Closed-loop regulator system with noise injection wP and vs. The process and sensor noise variances are zero mean, white noise, with root-meansquare (RMS) values of wP = 10 for and vs = 1. The trade-off under study is between the steady-state

output

variance

{ }

lim k →∞ E y k2

and

the

steady

state

actuator

{ }

variance lim k →∞ E u k2 . This trade-off can be computed finding the LQG optimal regulator for different penalty values on the control effort, ρ , which minimises (over all stabilizing regulators) the linear combination, JLQG, of actuator and output variance.

{

}

{

J LQG = lim E y k2 + ρ u k2 = lim E x kT Qx k + u kT Ru k k→∞ k→∞

}

(4.50)

Alternatively, this trade-off can be found using H2 performance measures by the fact that LQG is a special case of H2 optimal control and it can be solved via LMI optimisation. The smallest possible upper bound of the H2 norm of the transfer function can be calculated by minimizing the criterion trace (z) over the variables X, Y, K, L, M and N that satisfy LMIs (4.28 and 4.29). The trade-off was computed as follows. The NMP plant (4.49) was discretized using a zero-order hold at 10 Hz and its state space realization, equation (4.51), was used.

86

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method ⎛ 0.6703 0 ⎜ ⎛ A B ⎞ ⎜ 0.0824 1 ⎜⎜ ⎟⎟ = ⎝ C D ⎠ ⎜ 0.0044 0.1 ⎜ −1 ⎝ 0

0 0.0824 ⎞ ⎟ 0 0.0044 ⎟ . 1 0.0002 ⎟ ⎟ 4 0 ⎠

(4.51)

The H 2 optimal problem under consideration can be stated using the state-space description of the plant for the LQG problem (4.38). One could obtain the LTI plant (4.52) with some H2 performance γ>0 to find an LTI control law u = K ( z ) y such that: Twz < γ where Twz (z ) denotes the closed-loop transfer function from w to z. The LTI

controller K (z ) can be represented with the state space realization (4.2) and the generalized state-space description of the plant in equation (4.51) for the LQG problem is derived from equation (4.38) with Q = CT C , R = ρ and the stochastic inputs are wp=[W1/2 0]w and vs=[0 V1/2]w. The resulting realization is state-space realization (4.52). ⎛ 0.6703 0 0 ⎞ ⎛ 1 0 ⎞ ⎛ 0.0824 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x& = ⎜ 0.0824 1 0 ⎟ x + ⎜ 0 0 ⎟ w + ⎜ 0.0044 ⎟u , ⎜ 0.0044 0.1 1 ⎟ ⎜ 0 0 ⎟ ⎜ 0.0002 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

0 0 ⎞ ⎛0 ⎛0 ⎜ ⎟ ⎜ 0 0 . 2425 0 . 9791 ⎜ ⎟ ⎜0 z =⎜ x+ 0 0.9701 3.8806 ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜0 0 0 ⎟⎠ ⎜⎝ 0 ⎝

0⎞ ⎛0⎞ ⎟ ⎜ ⎟ 0⎟ ⎜0⎟ , w + ⎜ ⎟u ⎟ 0 0 ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ 0⎠ ⎝ρ⎠

(4.52)

y = (0 1 − 4)x + (0 − 1)w + (0)u .

The optimal LQG cost, JLQG, was calculated through the specification G (s) 2 ≤ γ , which is achievable if and only if γ2≥ JLQG (Boyd and Barratt 1991). In order to cast the H2 optimal problem with LMI techniques, the H2 performance can be formulated as the linear objective minimization problem. min γ subject to Acl P2 AclT − P2 + B cl BclT < 0 , C cl P2 C cl + Dcl DclT < Z .

87

(4.53)

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method where P2 = P2T > 0 and also, G ( s )

2 2

= trace(C T PC ) < γ 2 .

The optimisation problem was solved implementing the LMIs (4.28 and 4.29) in the MATLAB® LMI Control Toolbox (Gahinet et al. 1995). Single runs changing ρ from 0.0001 to 10 (marked ♦) were computed to search for the Pareto optimal surface, which represents the trade-off between noise sensitivities, as shown in figure 4.4. The result obtained was the same as that in Barratt and Boyd (1989). Note that this LMI technique found third order controllers. In general, the output-feedback technique applied will always find a controller with the same order as the plant.

Trade-off between noise sensitivites

1

10

rho=10

(RMS) value of y

rho=1

rho=0.1

rho=0.01 rho=0.001 rho=0.0001

0

10 0 10

1

10 (RMS) value of u

2

10

Figure 4.4 Trade-off between noise sensitivities achieved with LMI techniques.

4.8.2 H2 Optimal Control (LQG): MOGA-based method Next, the problem used by Barratt and Boyd (1989) was tackled with MOGA. The plant (4.40) and the closed-loop system shown in figure 4.3 were used in the feedback configuration again. The decision variables for this problem were the controller parameters, which have the form given in equation (4.54). This discrete-time controller was parameterized in terms of its roots (poles and zeros) and its gain. Two parameters

88

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method were used, the average, α , and the deviation, β , of each pair of roots (Kristinsson and Dumont 1992).

C=

K (z + a1 )( z + a 2 ) ( z + a3 )( z + a 4 )

(4.54)

where a1, 2 = α1 ± β1 and a3, 4 = α 2 ± β 2 . A positive deviation indicates real roots and negative deviation indicates complex conjugate roots. A pair of zeros and a pair of poles were set up to search in the interval [-1, 1], and the remainder were set up to search the interval [0, 1]. The gain was set up to search the interval [0, 100]. The objectives under consideration were the RMS values of noise sensitivity for the output y and actuator u. These variances RMS(y) and RMS(u) for a given discrete-time controller plant were computed as two separated objectives. In order to avoid unstable closed-loop systems, another objective was added. The maximum of the magnitudes of the poles of the corresponding close loop systems were evaluated minimizing the degree of instability of the closed-loop system. Since closed-loop stability is an absolute design requirement, it was set as a high priority objective, or constraint, with an associated priority level of 1. The region of the trade-off curve to be evolved was delimited with the goal vector

u rms = 100

and y rms = 10 . A population of 100 individuals were generated as real

parameters. The recombination operator was simulated binary crossover (SBX) for continuous search space and the mutation operator was a polynomial mutation (Deb 2001). In figure 4.5 the evolution of the individuals (family of controllers) is shown. Notice how the solution evolves at generation 50, 100 and 250 and how they have found a preliminary trade-off. The Pareto surface at generation 250 (marked o) is close to the desired trade-off (solid line). Fonseca and Fleming’s (1994) results used 100 generations, but in order to improve the diversity, SBX was used, which requires a population size equal to 100 and 250 generations as a standard setting. Note from Table 4.1 that MOGA was fixed to find second order controllers; see equation (4.54). In Table 4.1, six controllers, resulting from the LMI optimisation, have been chosen. They are third order controllers. For each of these controllers, a controller resulting from MOGA optimisation 89

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method has been chosen by finding the closest MOGA controller (marked ♦ in figure 3) to the LMI controller in the Pareto front. 10

Trade-off between noise sensitivities

1

rho = 10

Generation 250 Generation 100

(RMS) value of y

rho = 1

Generation 50

rho = 0.1

“MOGA Approach” rho = 0.01 rho = 0.001 rho = 0.0001

“LMI Approach”

10

0 0

10

1

10 (R M S) value of u

10

2

Figure 4.5 Trade-off between input and output RMS noise sensitivities achieved with MOGA and LMIs. ρ (rho) 10

1

0.1

0.01

0.001

0.0001

LMI 3rd order controllers

.

RMS (u)

RMS (y)

0.85715( z + 0.004227 )( z − 0.6511)( z − 0.9715) 2 ( z − 0.5464)( z − 1.57 z + 0.6831)

3.3125

8.4555

1.8014 ( z + 0.01199)( z − 0.6094 )( z − 0.9589 ) 2 ( z − 0.4448)( z − 1.558 z + 0.7054 )

4.8383

4.0011( z + 0.01609)( z − 0.6393)( z − 0.9449 ) 2 ( z − 0.415)( z − 1.502 z + 0.7137 )

8.7479

9.2423( z − 0.9318)( z − 0.657 )( z + 0.001701) 2 ( z − 0.3651)( z − 1.374 z + 1.7106)

16.027

21.3759( z − 0.9209)( z − 0.6617 )( z + 0.01372) 2 ( z − 0.2566)( z − 1.11z + 0.6779)

26.962

45.8816( z − 0.914 )( z − 0.663)( z + 0.05016) 2 ( z − 0.1311)( z − 0.6331z + 1.6107 )

72.593

MOGA . 2nd order controllers 1.8398( z − 0.9547 )( z − 0.1961) ( z − 0.7181)( z − 0.4166 )

5.6014

1.6811( z − 0.8057 )( z − 0.3795) ( z − 0.6143)( z − 0.5449 )

3.6898

1.728( z − 0.9772)( z − 0.2097 ) ( z + 0.1479)( z − 1.204 )

2.8678

7.8333( z − 0.9371)( z − 0.0482) ( z + 2.462)( z − 3.69 )

2.4089

1.728( z − 0.9773)( z − 0.2096) ( z + 0.1349)( z − 1.195)

2.0798

11.6629( z − 0.8942)( z − 0.542) ( z + 2.442)( z − 3.844)

Table 4.1 LMI and MOGA Controllers from the Pareto front for the Trade-off between input and output RMS noise sensitivities. 90


4.8.3 Mixed H2/H∞: LMI Approach The second problem is the mixed H2/H∞ output-feedback problem and the plant considered is the following LTI system, which was studied by Herreros et al. (2002), ⎛ − 21 − 120 − 100 ⎜ 0 0 ⎛ A B⎞ ⎜ 1 ⎜⎜ ⎟⎟ = ⎜ C D 0 1 0 ⎝ ⎠ ⎜ ⎜ 0 0 150 ⎝

1⎞ ⎟ 0⎟ 0⎟ ⎟ 0 ⎟⎠

.

(4.55)

The control system under consideration is the output-feedback problem for the generalized plant in its state-space realization form shown in equation (4.56), I B⎞ ⎟⎛ x ⎞ 0 D ⎟⎜ ⎟ , ⎜ w⎟ 0 0 ⎟⎜ ⎟ ⎟ u 0 D ⎟⎠⎝ ⎠

⎛ x& ⎞ ⎛ A ⎜ ⎟ ⎜ ⎜ z1 ⎟ ⎜ C ⎜z ⎟ = ⎜A ⎜ 2⎟ ⎜ ⎜ y ⎟ ⎜C ⎝ ⎠ ⎝

(4.56)

where x is the system state vector, u is the control input vector and w is the exogenous disturbance vector, z1 is the regulated output that satisfies H2 performance and z2 is the regulated output that satisfies H∞ performance. An assumption is: w1 = w2 , which is a necessary condition to obtain a convex approximation of the problem (Khargonekar and Rotea 1991a). Mixed H2/H∞ output-feedback has a trade-off that is interesting to investigate because the H∞ norm and the H2 norm are objectives that mutually compete. In order to find such a trade-off, the mixed H2/H∞ problem seeks an internal controller, K(s), as shown in equation (4.57). min TZ1W

2

subject to TZ 2W

∞

< γ2 .

(4.58)

The trade-off between the H∞ norm and the H2 norm constraints consists of plotting the l u curve of values of γ 2 in some interval, [ γ 2 ,..., γ 2 ], versus optimal values of

TZ1W

2

obtained from equation (4.57). For this problem γ 2 = [10…600], where γ 2l was

chosen to be close to the smallest achievable TZ W 2

91

∞

. As was discussed in section 4.4, this

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method problem can be cast as LMI constraints. The LMI Algorithm was set up with the generalized plant (4.56) and single optimisation for fifty values within the range of γ 2 was computed and the resulting trade-off is plotted in figure 4.6. The synthesis values obtained by the LMI approach (marked ◊) are just an upper bound. The trade-off curve looks non-convex. This might be a result of controller computation, the existence of the controller depends on the non-singularity of the matrix I-XY as I-XY=UVT with nonsingular U and V. A simple test shows that the minimal eigenvalues of matrix XY of the fifty LMI controllers are close to one, which means that the controllers are illconditioned; this issue is studied in Chapter 5, Section 5.4.1.

Trade-off Mixed H2/Hinf 600

500

"Analysis values: LMI Approach"

||Tz2w ||inf

400

300 "Synthesis values: LMI Approach" 200

100

0 20

30

40

50

60

70 80 ||Tz1w (s) ||2

90

100

110

120

130

Figure 4.6 Trade-off Mixed H2/H∞ achieved with LMI techniques. In order to illustrate the conservatism of this approach, a second curve is plotted and labelled “Analysis values: LMI Approach” (marked ∆). For example, considering the following problem, see equation (26), for γ 2 =600, and solving this problem with LMI optimisation, gives 26.5376 as the best constrained H2 performance, which was achieved with the following LMI controller:

92

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method 2 KLMI= − 2.9598e12(s + 21.09s + 117 ) .

(s + 4.225e5)(s + 3.995e4)(s + 1124 )

The loop with the plant (see equation (4.56)) is then closed, and the H2 norm computed. The result will be 22.2081 (16% lower than the LMI optimal value). When computing the H∞ norm, the result will be 492.6874 (18% lower than the LMI constraint 600). In this way, the second curve, labelled “Analysis values: LMI Approach”, was computed. This conservatism of the results is due to the introduction of the common Lyapunov matrix (see Section 4.4). It is clear that LMI optimal values are only an upper bound on the closed-loop H2 performance.

4.8.4 Example of Mixed H2/H∞ using MOGA Approach Next, the mixed H2/H∞ output-feedback problem was tackled with MOGA. The generalized plant in equation (4.56) was used again and the mixed H2/H∞ problem was encoded in MOGA to search for internally stabilizing controllers, K, with canonical controller form as follows. ⎛ Ac ⎜⎜ ⎝ Cc

⎛ − a1 ⎜ Bc ⎞ ⎜ 1 ⎟= Dc ⎟⎠ ⎜ 0 ⎜ ⎜ b ⎝ 1

− a2 0 1

− a3 0 0

b2

b3

1⎞ ⎟ 0⎟ 0⎟ ⎟ 0 ⎟⎠

(4.59)

The lower and upper boundaries on the decision variables were -5e+15 and -5e+4 respectively and the binary resolution was 32 bit Gray coding with exponential scaling in each of the six decision variables space. Objectives were taken into account: The first objective was Closed-Loop Stability. It was evaluated to minimise the degree of instability of the closed-loop system and it was set up as a high priority objective, with an associated goal of 0. The second objective was min ||Tz1w||2 norm with an associated goal of 120. TZ1W

2

=

1 2π

∞

∫ tr (T

Z1W

( jω ) H TZ1W ( jω ))dω ,

(4.60)

−∞

⎡ A + BDC C ⎢ where TZ 1W = ⎢ BC C ⎢C1 + E1 DC C ⎣

93

BC C AC E1C C

B j + BDC F j ⎤ ⎥ BC F j ⎥. D1 + E1 DC F j ⎥⎦

(4.61)

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method The third objective was min ||Tz2w||∞ norm with an associated goal of 600, where

TZ 2W ( s)

∞

= max TZ 2W ( jω ) ,

(4.61)

ω

⎡ A + BDC C ⎢ where TZ 2W = ⎢ BC C ⎢C 2 + E 2 DC C ⎣

BC C AC E 2 CC

B j + BDC F j ⎤ ⎥ BC F j ⎥. D2 + E 2 DC F j ⎥⎦

(4.62)

MOGA used mating restriction and crossover shuffle with reduced surrogate with probability 0.7. A mutation rate of 0.09 was applied to all individuals after crossover. The population size was 200 candidate solutions and the result after 250 generations is presented in Figure 4.7 (marked o) to compare with the corresponding LMI results.

Trade-off Mixed H2/ H i n f 600

"Analysis values: LMI Approach"

500

|| Tz2w ( s ) || i n f

400

“Synthesis values: LMI Approach”

300

200

100 “MOGA Approach” 0 10

20

40

60 80 || Tz1w ( s ) || 2

100

120

130

Figure 4.7 Trade-off Mixed H2/H∞ achieved with MOGA and LMIs. Results obtained from 5 independent runs of the algorithm provide a clear estimate of how likely the various regions of the objective space were to be attained in a run of GA. The Pareto fronts are presented in Figure 4.8. The Pareto front obtained using MOGA is better than the one obtained with the LMI approach. Another achievement was that this

94

Chapter 4 Mixed H2/H∞ problem with LMIs Approach and MOGA-based method Pareto front is better than the one obtained by Herreros et al. (2002). Note the way in which both problems were solved. When the LMI approach is used, the multi-objective problem is reduced to a problem that optimises a single objective function. In the case of the MOGA approach, there is no need for such a reduction and the original objectives are optimised retaining the multiobjective formulation. For the LMI approach, the difference between the synthesis values and analysis values only shows the conservatism of the approach. A good estimation can be achieved only through lower bound computation (Scherer 1999) and the use of Youla techniques to approximate the optimal value (Scherer 2000).


500

500

400

400 ||Tz2w (s) ||inf

||Tz2w (s) ||inf


300

200

200

100

0

300

100

0

20

40

60 80 ||Tz1w (s) ||2

100

120

0

140

0

20

a) 25 generations

40

Trade-off Mixed H2/Hinf

120

140


500

500

400

400 ||Tz2w (s) ||inf

||Tz2w (s) ||inf

100

b) 100 generations

600

300

200

300

200

100

0

60 80 ||Tz1w (s) ||2

100

0

20

40

60 80 ||Tz1w (s) ||2

100

120

0

140

c) 150 generations

0

20

40

60 80 ||Tz1w (s) ||2

100

120

d) 250 generations

Figure 4.8 Evolution of the Multiobjective H2/H∞ Trade-off achieved with MOGA (5 runs). 95

140


4.9. Conclusions In conclusion, solving control design problems with LMI techniques lacks full flexibility. There is not a general method that converts a control specification into LMI constraints and even if this is achieved it might be non-convex as in the case of the reduced order controller. A significant issue with using MOGA is that the search space is not always known. For complex problems where there is no indication where the solutions are, MOGA will need more prior knowledge until a set of Pareto optimal solutions can be found. However, once the results are found the design requirements can be refined, by introducing more constraints into the design, such as minimising the order of the controller. The order of the controllers is of great relevance in control design. The flexibility of MOGA permits the structure of the controller to be changed and a reduction in the order of the controller to be found. Using the LMI framework, the problem of controller order reduction is still a hard constraint on the optimisation and is often not tractable numerically. This comparison was illustrated by the problem trade-off of noise sensitivities. The example, involving noise sensitivity and robustness measurements, demonstrated that H2 optimal control using the LMI approach finds the exact trade-off of noise sensitivities. However, as soon as any other norm or constraint is introduced (eg: H∞) there is not an exact solution, and approximations and bounds must be required. Using LMI techniques, the multiobjective problems were formulated with upper bounds, whereas the MOGA-based method truly formulates the multiobjective H2/H∞ problem. Thus, it is clear that LMI techniques are limited and they do not represent a true multiobjective design methodology. Despite the emergence of the LMI approach as a powerful optimisation technique, it introduces conservatism into the design by using a single Lyapunov function. The second problem studied illustrates that the LMI solutions are still far from optimal. In contrast, MOGA offers flexibility and a truly multiobjective treatment of the original objectives.

96

Chapter 5 Multiobjective Control Design for the Turbofan Engine 5.1 Introduction Both the multiobjective H2/H∞ control design: evolutionary approach and mixed H2/H∞ control design: LMI approaches are applied to the optimisation of the Turbofan engine model MIMO controller. In this chapter, a numerical analysis on the Pareto optimal result is carried out. Numerical issues are assessed and a solution based on the evolutionary approach is proposed. The gain scheduling controller design is considered and practical issues are addressed. A multiobjective H2/H∞ gain scheduling controller design is proposed, for two operating points.

5.2 Gas Turbine Engine The gas turbine engine (GTE) is an internal combustion engine employing a continuous combustion process; for aviation applications it is called a jet engine. Jet engines produce propulsive thrust from the thermal energy of fuel and are most effective when they can be operated at or near their mechanical limits (flow or pressure limitation such as rotor speeds, turbine temperatures, internal pressures, etc.) Three basic types of jet engines are in current use. The turbojet is the best for high subsonic and supersonic flight speeds. The turbofan is chosen for subsonic commercial

97

Chapter 5 Multiobjective Control Design for the Turbofan Engine airplanes. The turboprop or turboshaft is the best for power plants, helicopters and for small lower speed aircraft. This chapter is concerned with the turbofan engine whose core consists of a compressor, combustor and turbine which drive the compressor. It has a fan in front of the core compressor and a second power turbine behind the core turbine to drive the fan as shown in Figure 5.1a.

Combustor

B

Exhaust

C

E Fuel flow Core Compressor

Fan

Core Turbine

Turbine

Double shaft

Inlet, A

(a) Combustion (heat energy added)

Pressure

B

C

Expansion (output work to run compressor)

Compression (pressure energy added)

D Output work for Thrust

Pressure is increased through compressor as volume is reduced Inlet

A

Ambient Air Exhaust Heat

E

Volume (b) Figure 5.1 (a) Gas engine schematic (b) Brayton cycle pressure-volume diagram.

98

Chapter 5 Multiobjective Control Design for the Turbofan Engine The turbofan operates on the working cycle as depicted in Figure 5.1b, which is the thermodynamic Brayton cycle. This cycle is based on the relation between pressure and volume. The cycle can be described as a sequence of five stages: 1. Inlet (A): the air is taken from the ambient air. 2. Compression (A-B): the pressure increases through the compressor as the volume of the air is reduced. 3. Combustion (B-C): fuel is burned in the combustor at a slight pressure drop. 4. Expansion (C-D): the resulting product of combustion is expanded in the turbines and this work provides the power to drive the compression system and the fan. 5. Thrust (D-E): the remainder of output work is used for thrust. In the compression system, one of the most critical mechanical problems occurs. Two modes of mechanical instability can occur: “surge” which is a longitudinal flow oscillation over the length of the compressor and turbine and “stall” which is the lack of pressure rise between the compressor and blades. Stall occurs at low rotor speeds and surge at high rotor speeds. Both surge and stall generate violent axial oscillations of the internal air column which can cause substantial damage to both the compressor and the engine. Operating characteristics of a typical compressor are shown in terms of corrected parameters in the compressor map in Figure 5.2.

Surge line Surge avoidance line

Pressure Ratio

Surge zone

Compressor Characteristics

Zone of poor compression efficiency

Mass flow Figure 5.2 Compressor map.

99

Chapter 5 Multiobjective Control Design for the Turbofan Engine

5.2.1 Control Systems for Gas Turbine Engines Controlling at, but not exceeding a limit, is a very important aspect of jet engine control. Therefore, both regulation and limit management are important. Minimum control requirements include a main fuel control for setting and holding steady-state thrust with fuel acceleration and deceleration schedules to provide limit protection. Other controls provide fan and booster stall protection and control variable parasitic engine flows (Spang and Brown 1999). Also, the control system must be designed to avoid the compressor becoming too close to the surge line and to maintain a sufficient design margin, for example a surge avoidance line, see Figure 5.2. A design parameter to avoid surge is known as surge margin and is based on the total pressure ratio between outlet pressure P0 and inlet pressure Pi at the compressor. The surge margin (SM) is defined as follows (Tavakoli et al. 2004):

SM =

⎛ Po ⎜⎜ ⎝ Pi

⎞ ⎛P ⎞ ⎟⎟ − ⎜⎜ o ⎟⎟ ⎠ surge ⎝ Pi ⎠ surge ⎛ Po ⎜⎜ ⎝ Pi

⎞ ⎟⎟ ⎠ surge

avoidance

.

avoidance

The overall function of the engine controller is to provide thrust in response to throttle position. It must achieve the requested thrust with the lowest specific fuel consumption. It must also ensure that limits are not exceeded, such as maximum compressor speed, maximum turbine temperature, fan stall and compressor stall. The overall control must control each limiting parameter so that none of the limits are exceeded. Part power engine operation should occur below all limits at the lowest specific fuel consumption for the thrust requested. Because of the nature of thrust, commercial jet engines use an indirect measurement called integrated engine pressure ratio (IEPR) to control it. However, the only closed loop control in commercial engines is the main fuel control. On modern turbo engines, thrust is more accurately controlled by setting the fan speed or the engine pressure ratio, which is defined as the ratio of low pressure turbine to inlet pressure (Spang and Brown 1999).

100

Chapter 5 Multiobjective Control Design for the Turbofan Engine Commercial engines are generally controlled by a single closed loop main fuel control. There is minimal interaction between each open loop control and the main fuel control. Consequently, single input single output design techniques are usually adequate (Spang and Brown 1999). Multivariable control has been developed for military engines or turbojets and much of the literature on multivariable engine control is focused on the regulator problem, which maintains the engine near the desired operating trajectory. In 1978, an international forum on alternatives for linear multivariable control showed the use of several multivariable techniques using a linear model of the F100 engine (DeHoff and W. Eral Hall 1979). Three different multivariable techniques have been used to develop controllers: multivariable transfer function, inverse Nyquist array and characteristic root locus. Each strategy used some of the measurable variables: fan speeds, compressor exit speed, exhaust pressure, and turbine inlet temperature. Polley et al. (1989) found a nonlinear controller by using appropriate engine corrected parameters to schedule multivariable linear compensator gains designed at selected operating points of the flight envelope. A KQ (K matrix compensator, Q-desired response) multivariable control design technique was used to design the compensator using linear state-space models. The KQ compensator design methodology is a frequency domain technique, which uses closed-loop Nyquist and Bode plots. Parameters of the matrix compensator K are estimated by minimizing the mean square error between the desired closed-loop response Q and the actual closed-loop response. This method is known as Edmunds’ model matching design technique, and is described in Edmunds (1979). Frederick et al. (2000) points out that the use of Edmunds’ design technique is easy to use. However, it does not guarantee stability and does not take into account actuator rates. They proposed using H∞ controller design to include weights on actuator rates and enhance robustness as well as a simplified scheduling scheme to cover a range of power levels from idle to maximum. They addressed the following areas of future research: design of a low-order robust H∞ controller, reduction of gain scheduling effort and complexity. Balas et al. (1998) have proposed applying linear parameter varying (LPV) techniques for the design of gas turbine controllers. They demonstrated that LPV

101

Chapter 5 Multiobjective Control Design for the Turbofan Engine techniques provide a methodology for designing H∞ gain scheduling controllers with low design effort and complexity. These new techniques have been developed as a result of recent improvements in the area of LMI in control systems. LMI was first introduced in another related area, aircraft design, by Niewoehner and Kaminer (1996). They developed a new methodology that provided a numerical framework for the integrated aircraft-controller design. They considered a problem of combining the design of some aircraft parameters with the control system development using linear matrix inequalities. The remainder of this chapter investigates multivariable control strategies using LMI techniques and evolutionary algorithms. The following section will describe the GTE undertaken for this study.

5.2.2 Gas Turbine Engine ANTLE. The turbofan engine model used in this investigation was the Affordable Near Term Low Emissions (ANTLE) technology demonstrator engine from Rolls Royce.

Bypass duct Fan (XNL)

Variable inlet Guide vanes (ZIGV)

High Pressure Compressor (HPC) Speed (ZXNH)

Intermediate Pressure Compressor (IPC)

High Pressure Turbine (HPT)

Combustor Fuel flow (ZWFE)

Intermediate Pressure Turbine (IPC)

Figure 5.3 Diagram of a three spool ANTLE gas turbine engine.

102

Low Pressure Turbine (LPT)

Nozzle area

Chapter 5 Multiobjective Control Design for the Turbofan Engine The engine comprises of 3 spools; the fan being driven by the low pressure turbine (LPT), the intermediate pressure compressor (IPC) which is driven by the intermediate pressure turbine (IPT) and the high pressure compressor (HPC) which is driven by the high pressure compressor (HPC) (see Figure 5.3). The engine has separated ducts; the outermost part of the flow enters from the fan to the bypass duct which exhausts through the cold nozzle. The combustor is selected as staged or conventional type. Variable inlet guide vanes (ZIGV) are provided at IPC entry. The flight envelope was constrained to sea level, zero Mach number and an outside environment of 15° C and 14.696 lb/in2. The model is a nonlinear, thermodynamic type. Linear models were obtained for nine steady-state operating points. The linearisation of the nonlinear plant was carried out by perturbation of the states or Jacobian linearisation using the software MATRIXx. The operating point was changed by varying the specified HPC speed (ZXNH), see Appendix A. Based on the literature of control for gas turbine engines (Postlethwaite et al. 1995; Spang and Brown 1999; Frederick et al. 2000), the selected inputs and outputs are proposed for control; Inputs: u1 is the specified engine fuel flow (ZWFE) and u2 is the variable guide vane angle in the compressor (ZIGV). Outputs: y1 is the fan speed (XNL), y2* is the surge margin IP (SMI) and y3* is the surge margin HP (SMH).

Work based on Griffin (2004) and consideration of functional controllability suggests that a square plan is required for control, e.g. 2 input, 2 output. The two surge margins SMI and SMH are combined into a measure describing the overall surge risk to the

engine. For this reason the second output, y2, is changed to the mixed surge margin (SMmix). The state-space realization for the ANTLE RRAP model is:

•

⎛ x1 ⎞ ⎛ x1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎛ u1 ⎞ ⎜ x 2 ⎟ = Ai ⎜ x 2 ⎟ + Bi ⎜⎜ ⎟⎟ , ⎝ u2 ⎠ ⎜x ⎟ ⎜x ⎟ ⎝ 3⎠ ⎝ 3⎠

⎛ x1 ⎞ ⎜ ⎟ ⎛u ⎞ ⎛ y1 ⎞ ⎜⎜ ⎟⎟ = C i ⎜ x 2 ⎟ + Di ⎜⎜ 1 ⎟⎟ , ⎝u2 ⎠ ⎝ y2 ⎠ ⎜x ⎟ ⎝ 3⎠

103

(5.1)

Chapter 5 Multiobjective Control Design for the Turbofan Engine i = 1 L 9.

where the states: x1 is the specified HP speed (ZXNH) , x2 is the specified IP speed (ZXNI) and x3 is the specified LP speed (ZXNL).

5.3 Controller Design for a Linear Point. In this section, the controller design strategies are applied to one operating point, at 80% of power (see Appendix A). In section 5.10 the results are extended to two operating points.

5.3.1 Scaling For controller design purposes, the plant was scaled as suggested in Skogestad and Postlethwaite (1996). A condition number was obtained from the ratio between the maximum and the minimum singular values of the plant; this condition number was reduced by a factor of 34.261 (from 3866.7 to 112.86) by pre- and post-multiplying the plant with diagonal scaling matrices. The inputs were scaled in a way that the maximum allowed fuel flow (ZWFE) and the maximum allowed angle of the variable guide vine in the compressor (ZIGV) correspond to the unit at the operating point. The outputs were scaled in such a way that 2.2% of the fan speed corresponds to the unit and 5% of the surge margin corresponds to the unit.

5.3.2 Design Controller Strategy. The literature in robust control design for turbofan engines shows that the H∞ loopshaping design procedure (Postlethwaite et al. 1995) and the mixed sensitivity H∞ optimisation (Frederick et al. 2000; Balas 2002; Bruzelius et al. 2002) have been used as control design strategies. In this research, the second design method has been investigated. The control configuration of Figure 5.4 shows the tracking problem in S/KS mixed-sensitivity standard form with integrator. The exogenous input is a reference command r, and the error signals are Z1= W1 S w and Z2=W2 KS w.

104


Paug

W1

w=r

z1

z2 W2

+

G(s)

1 s

-

u

ym K(s)

Figure 5.4 S/KS scheme. The performance specification ||W1S||∞ γ2,opt

Analysis W2 KS ∞

Analysis W1 S 2

minimal eigenvalues of matrix (X·Y)

0.17 0.2 0.22 0.25 0.3 0.35 0.4 0.45 0.5 1 4 6 8 10

9.2838 6.7082 5.7872 4.8891 4.7 4.5 4.4 4.3 4.2 4.1 4 4 4 4

0.14924 0.16132 0.16784 0.17963 0.19285 0.20632 0.21983 0.23236 0.24462 0.33793 0.57789 0.6589 0.73322 0.80508

3.8933 3.2018 2.9196 2.6077 2.4934 2.3779 2.2959 2.2178 2.1489 1.8759 1.4902 1.3704 1.2708 1.1869

2.868 2.5617 2.3713 2.2794 2.6341 2.8914 3.1401 3.3102 3.409 4.7245 7.3621 9.1723 11.403 14.08

Table 5.2 Numerical analysis of the LMI trade-off with β condition. Trade off Mixed H2/Hinf 10 9 8 7

||W 1 S||2

6

"Synthesis values"

5 4 "Analysis values"

3

"well conditioned controllers"

2 "ill-conditioned controllers" 1 0

0

1

2

3

4

5 6 ||W 2 KS||inf

7

8

9

Figure 5.8 Trade-off curves for problem in equation 5.4 with β constraint.

110

10

Chapter 5 Multiobjective Control Design for the Turbofan Engine Trade off Mixed H2/Hinf 10 9 8 "Synthesis values with beta constraint"

7

||W 1 S||2

6 5 4 "Analysis values with beta constraint" 3 "ill-conditioned controllers"

2 1 0

0

0.1

0.2

0.3

0.4

0.5 0.6 ||W 2 KS||inf

0.7

0.8

0.9

1

Figure 5.9 Magnification of the trade-off curves for problem in equations 5.4 with β constraint.

5.5 Multiobjective H2/H∞ Controller Design: MOGA-based method Based on the results of Chapter 4, the MOGA-based method was used to tackle the multiobjective H2/H∞ design problem. In this case the system is MIMO, so a different structure for the controller was proposed. In Chapter 4, the canonical form was proposed for the scalar case, but in the MIMO case there is not a unique canonical form. However, some special structures (5.6) have shown good results for control systems purposes (Kailath 1980).

⎛ Ac ⎜⎜ ⎝ Cc

⎛ 0 ⎜ ⎜ 0 ⎜ 0 ⎜ ⎜a Bc ⎞ ⎜ 1 ⎟⎟ = ⎜ 0 Dc ⎠ ⎜ 0 ⎜ ⎜ a8 ⎜c ⎜⎜ 16 ⎝ c 23

1

0

0

0

0

0

0

0 0

1 0

0 1

0 0

0 0

0 0

0 0

a2 0

a3 0

a4 0

a5 0

a6 1

a7 0

1

0

0

0

0

0

1

0

a9

a10

a11

a12

a13

a14

0 d 30 0

c17

c18

c19

c 20

c 21

c 22

c 24

c 25

c 26

c 27

c 28

c 29

0

0 ⎞ ⎟ 0 ⎟ 0 ⎟ ⎟ b15 ⎟ ⎟ 0 ⎟ 0 ⎟ ⎟ 1 ⎟ 0 ⎟ ⎟ 0 ⎟⎠

(5.6)

MOGA used mating restriction and crossover shuffle with reduced surrogate with probability 0.7. A mutation rate of 0.09 was applied to all individuals after crossover. The 111

Chapter 5 Multiobjective Control Design for the Turbofan Engine population size was 200 candidate solutions and the Pareto front after 200 generations is presented in Figure 5.10 and 5.11 (marked o) to compare it with the corresponding LMI results. Trade off Mixed H2/Hinf 10 9 8 7

"Synthesis values with beta constraint"

||W 1 S||2

6 5 4 3

"Analysis values with beta constraint"

MOGA

"ill-conditioned controllers"

2 1 0

0

0.1

0.2

0.3

0.4

0.5 0.6 ||W 2 KS||inf

0.7

0.8

0.9

1

Figure 5.10 Trade-off curves. Comparison between MOGA-based method and LMI Approach. A comparison of step responses resulting from the LMI synthesis and the MOGA-based method is shown in Figure 5.12. The LMI controller (“dashed line”) is the controller 4 in Tables 5.1 and 5.2 and it has analysis values W2 KS ∞ = 0.33794 and W1 S 2 =1.8759. The controller resulting from the multiobjective H2/H∞: MOGA-base method (“solid line”) has synthesis values γ∞= 0.41946 and γ2 = 1.0072. It was demonstrated that the H∞ controller design technique and the mixed H2/H∞ controller design technique find ill-conditioned controllers for this gas turbine case study. Although a remedy has been proposed by Scherer et al. (1997), the procedure to obtain the Pareto-front set requires a lot of trial and errors to fix an appropriate optimal value or bound γ. This procedure is impractical and time-consuming. On the other hand, the multiobjective H2/H∞ controller design technique tackled by the MOGA-based method overcomes these problems.

112

Chapter 5 Multiobjective Control Design for the Turbofan Engine Trade off Mixed H2/Hinf 10 9 8 7 "Synthesis values with beta constraint" ||W 1 S||2

6 5 4 "Analysis values with beta constraint"

3 2 1 MOGA 0

0

0.1

0.2

0.3

0.4

0.5 0.6 ||W 2 KS||inf

0.7

0.8

0.9

1

Figure 5.11 Trade-off curves, comparison between MOGA-based method and LMI Approach with beta constraint. Step Response From: ZWFE

From: ZIGV

1.5

To: XNL

1 0.5

Amplitude

0 -0.5 1.5

To: SMmix

1 0.5 0 -0.5 -1

0

1

2

3

4

5 0

1

2

3

4

5

Time (sec)

Figure 5.12 Step responses resulting form the Mixed H2/H∞: LMI Synthesis (“Dashed line”) and MOGA-base method (“Solid line”). 113


5.6 Gain Scheduling In the previous section, the linear controller design was applied to a single operating point. Now the question will be whether or not this control can be extended to a different operating point while keeping stability and performance. Since the gas turbine engine is a nonlinear system, gain scheduling control seems to be an attractive approach, and will now be studied. A three-step gain scheduling control design procedure has been widely applied (Shamma and Athans 1990; Rugh 1991): 1) Consider the nonlinear plant x& = f ( x, u ) y = h( x, u ),

(5.7)

where x is the state, u is the input. It is known that the dynamical behaviour of a nonlinear system (5.7) changes with the operating regions, which are physical variables, such as speed, temperature, altitude, etc. The designer selects several operating points which cover the range of the dynamics of the plant, then the model is linearised about one or more operating points (also called set points, equilibrium points, or operating conditions). In the literature for nonlinear systems, the equilibrium point is more frequently used. An equilibrium point is a point such that f ( xo , u o ) = 0 , Associated with this equilibrium point is the output operating point y o = h( x o , u o ) . For the purpose of this investigation, an equilibrium point will be referred to as an operating point. For the system (5.7), suppose an operating point x o , u o , and y o is known. The difference between the equilibrium vector function and some slightly perturbed function x(t ) , u (t ) , and y (t ) can be defined by

δx = x(t ) − xo δu = u (t ) − u o δy = y (t ) − y o and the equation (5.7) is written as

114

(5.8)

Chapter 5 Multiobjective Control Design for the Turbofan Engine ⎡ ∂f ⎤ ⎡ ∂f ⎤ x& o + δx& = f ( x o + δx, u o + δu ) = f ( x o , u o ) + ⎢ ⎥ δx + ⎢ ⎥ δu + higher − order terms ⎣ ∂u ⎦ o ⎣ ∂x ⎦ o ⎡ ∂h ⎤ ⎡ ∂h ⎤ y o + δy = h( x o + δx, u o + δu ) = h( x o , u o ) + ⎢ ⎥ δx + ⎢ ⎥ δu + higher − order terms ⎣ ∂x ⎦ o ⎣ ∂u ⎦ o

where

[ ]o means

the derivates are evaluated at the operating point. Since the

operating point solution satisfies equation (5.7), the first terms in the preceding Taylor series expansions cancel. For sufficiently small δx , δu and δy perturbations, the higher-order terms can be neglected, leaving the linear equations

⎡ ∂f ⎤

⎡ ∂f ⎤ ⎡ ∂f ⎤ δx& = ⎢ ⎥ δx + ⎢ ⎥ δu , ⎣ ∂u ⎦ o ⎣ ∂x ⎦ o

(5.9)

⎡ ∂h ⎤ ⎡ ∂h ⎤ δy = ⎢ ⎥ δx + ⎢ ⎥ δu , ⎣ ∂x ⎦ o ⎣ ∂u ⎦ o

(5.10)

⎡ ∂f ⎤

⎡ ∂h ⎤

⎡ ∂h ⎤

where A = ⎢ ⎥ , B = ⎢ ⎥ , C = ⎢ ⎥ , D = ⎢ ⎥ . ⎣ ∂x ⎦ o ⎣ ∂u ⎦ o ⎣ ∂x ⎦ o ⎣ ∂u ⎦ o When the system has more than one operating point, the linearization family will require a family of operating points and they may need to be parametrized by the scheduling variable θ , and then the operating points will be defined as x o (θ ) , u o (θ ) and y o (θ ) as well as the model, δx& = A(θ )[ x(t ) − x o (θ )] + B (θ )[u (t ) − u o (θ )], δy = C (θ )[ x(t ) − x o (θ )] + D(θ )[u (t ) − u o (θ )],

(5.11)

where δy = y (t ) − y o (θ ) . 2) After linearisation, one arrives at a set of linear models at each operating point. In order to arrive at a set of linear feedback control laws, linear design methods are applied performing satisfactorily when the closed-loop systems is operated near the respective operating points. Consider the linear controller family: δx& c = Ac (θ )[ x c (t ) − x o ,c (θ )] + Bc (θ )[ y (t ) − y o (θ )] ,

δu = C c (θ )[ x c (t ) − x o ,c (θ )] + Dc (θ )[ y (t ) − y o (θ )] .

115

(5.12)

Chapter 5 Multiobjective Control Design for the Turbofan Engine The objective of the controller is that the operating point of the plant matches the same operating point of the corresponding closed-loop system. 3) The final step is the gain scheduling. This is achieved either by interpolating the controller parameters between operating points or by using non-smooth switching strategy between the linear controllers (Astrom and Wittenmark 1989). The result is a global compensator. Qualities such as stability and robust performance are then evaluated through exhaustive simulations, under the assumption that the scheduling variable should very slowly and that the scheduling variable should capture the nonlinear plant behaviour (Shamma and Athans 1990). Traditionally, gain scheduling techniques formulated relatively simple structure control laws and gains were selected for each operating condition by using classic SISO control methods. As process systems and aerospace applications such as GTE became more complex, and performance capabilities more demanding, modern control techniques have to be applied. The resulting controllers became more complex with many more gains. Consequently, evaluating gain scheduling controllers using exhaustive simulations results is impractical.

5.7 Linear Parameter Varying Techniques Linear parameter varying (LPV) techniques, first introduced by Shamma and Athans (1991), have been developed to provide a natural method for scheduling controllers (Biannic et al. 1997; Marcos and Balas 2004). By using the LPV techniques, the traditional three-step gain scheduling control procedure, described in Section 5.6, is reduced to a two-step procedure. First step, LPV modelling will capture the whole nonlinear behaviour of the plant, within the range of interesting operating points; this is discussed in the next Section. Second step, LPV control design will replace the following procedure: controller synthesis at each operating point, interpolation of resulting controllers, extensive closed-loop simulation to ensure stability and robust performance. The closed-loop LPV control structure is represented in Figure.5.13. LPV has been applied to high performance aircraft systems such as the F-16 aircraft (Shin et al. 2002), helicopters (Sun and Postlethwaite 1998), turbofan engines (Wolodkin et al. 1999; Balas 2002; Bruzelius et al. 2002), missiles autopilots (Wu et al. 1995; Biannic and Apkarian

116

Chapter 5 Multiobjective Control Design for the Turbofan Engine 1999), robots (Kajiwara et al. 1998), inductor motors (Prempain et al. 2002) and power systems (Qiu et al. 2004).

r

+

u

K(s,θ) -

G(s,θ)

y

θ(t)

Figure 5.13 LPV control structure The LPV structure incorporates the parameter measurement θ in real time. This technique “automatically” produces gain scheduling controllers which simultaneously ensure global stability over the operating domain, and linear time invariant robustness properties (Biannic et al. 1997). It has been shown that the illustrated LPV control problem formulated in Figure 5.13 can be formulated as a convex optimisation problem (Gahinet et al. 1995). This formulation is explained as follows: Consider the LPV plant

x& = A(θ ) x + B(θ )u y = C (θ ) x + D(θ )u

(5.13)

where θ := θ (t ) is a time-varying parameter describing the range of possible dynamics of the plant. LPV-based one step gain scheduling control follows three main approaches: •

Polytopic Systems / quadratic H∞ performance

Based on the notion of quadratic H∞ performance (Becker and Packard 1994), solvability conditions are obtained for continuous- and discrete-time systems. In both cases the synthesis problem reduces to solving a system of linear matrix inequalities (Apkarian et al. 1995). A drawback of this quadratic H∞ performance design is that it allows arbitrary fast parameters variations, thus conservatism will be present for slowly varying parameters.

117


•

Linear Fractional Transformation (LFT) dependence

This approach treats gain-scheduling controllers as a single entry, with the gain scheduling being affected entirely using linear fractional transformations (LFT). This approach relies on the small gain theorem (Packard 1994). The small gain theorem states that a feedback loop composed of stable operators will certainly remain stable if the product of all operator gains is smaller than unity. The gain scheduled H∞ controllers are linear fractional functions of θ and the synthesis problem is solved using LMI techniques and the bounded real lemma (Apkarian and Gahinet 1995) or using the necessary and sufficient conditions in the form of three affine matrix inequalities (AMIs) (Packard 1994). A drawback of the LFT formulation is that the variations of θ are allowed to be complex, thus introducing some conservatism when parameters are known to be real. •

Parameter-dependent Lyapunov Functions

The previous techniques make use of a fixed Lyapunov function. A significant improvement over such techniques can be obtained by exploiting the concept of parameter-dependent Lyapunov functions,

V ( x cl , θ ) = x clT P (θ ) x cl ,

(5.14)

where x cl represents the state of the closed-loop system. The parameterdependent Lyapunov function allows the incorporation of knowledge on the rate of variations in the analysis and synthesis technique, and therefore is less conservative, see Scherer and Weiland (1999, Chapter 5) and Dettori (2001, Chapter 3). However, this problem is infinite-dimensional and infinitely constrained. A remedy for turning this problem into a finite set of LMIs is to “grid” the value set of θ. This is another design step where the density of the grid has to be checked until the constraints are satisfied (Apkarian and Adams 1998; Shamma and Xiong 1999).

118


5.8 LPV Modelling A condition for applying LPV control is to transform the nonlinear model of the system into an LPV model. However, this area of research turns out to be not well established and is not considered as an independent research topic (Marcos and Balas 2004). The aim of developing an LPV model is to capture the actual non-linear behaviour of the plant in a global model, although a family of linearized models of a plant can be parametrized by some signal, and therefore could be presented by an LPV system. Traditionally, control designers use Jacobian linearisation to obtain a family of linear time-invariant plants at different operating points. Therefore, this will be the starting point to globally reconstruct nonlinear systems from families of linear systems into a LPV model. This task is not straightforward and the method can change depending on the application. However, different LPV modelling methods have been proposed; such as A hybrid method which derives from a LPV model by using linear fits (Sun and Postlethwaite 1998), stochastic methods and system identification methods (Fujisaki et al. 2003; Wu et al. 2004), and a velocity-based linearisation method (Leith and Leithead 1998a; 1998b). Motivated by the hybrid LPV modelling method (Sun and Postlethwaite 1998), and the LPV modelling method for the longitudinal control of a high-performance aircraft by (Biannic et al. 1997), the LPV modelling and control with polytopic systems was selected to tackle the gas turbine engine control problem.

5.8.1 Polytopic Systems Provided that the plant satisfies the following requirements, the system can be mapped to a LPV system with Polytopic form. 1) The state-space matrices A(·), B(·), C(·), D(·) depend affinely on the time-varying parameter θ , which is a vector of physical variables (velocity, angle, temperature, etc) in the following structure:

119

Chapter 5 Multiobjective Control Design for the Turbofan Engine i= j

A(θ ) = A0 + ∑ θ i Ai , i =1

i= j

B (θ ) = B0 + ∑ θ i Bi , i =1

i= j

C (θ ) = C 0 + ∑ θ i C i , i =1

i= j

D(θ ) = D0 + ∑ θ i Di .

(5.15)

i =1

2) A jth-dimensional parameter vector θ = (θ1 L θ j ) with its variation defined on

{(

) (

)}

(

)

the domain Ρ ⊆ θ 1 , θ 1 ,..., θ j , θ j , where θ j , θ j are the lower/upper bounds of θ j , can be put into an rth order polytopic form, where r = 2 j . The measurements

θ are available in real time and vary in a polytope Ξ of vertices Θ1 , Θ 2 ,...,Θ r , that is, r

θ (t ) ∈ Ξ := Co{Θ1 , Θ 2 ,..., Θ r } := ∑ ρi Θ i , i =1

where

ρi ≥ 0 ,

(5.16)

r

∑ ρi = 1 . i =1

These vertices represent the extreme values of the parameters. Though not fully general, this description encompasses many practical situations. The Polytopic system is defined as follows: ⎧⎛ A ⎛ A(θ ) B(θ ) ⎞ ⎜⎜ ⎟⎟ ∈ Co⎨⎜⎜ i ⎝ C (θ ) D(θ ) ⎠ ⎩⎝ C i

r ⎫ Bi ⎞ ⎛ A(Θ i ) B(Θ i ) ⎞ ⎛A ⎟⎟ := ⎜⎜ ⎟⎟, i = 1,..., r ⎬ := ∑ ρi ⎜⎜ i Di ⎠ ⎝ C (Θ i ) D(Θ i ) ⎠ ⎭ i =1 ⎝ C i

Bi ⎞ ⎟ (5.17) Di ⎟⎠

From this characterization, it is clear that the state space matrices evolve in a polytope of matrices whose vertices are the images of the vertices Θ 1 , Θ 2 ,..., Θ r . For an illustration of this mapping, see Figure 5.14.

120

Chapter 5 Multiobjective Control Design for the Turbofan Engine In Figure 5.14, the variation of two parameters are represented in a polytope of r = 2 2 = 4 vertices, where the vertices are the following polytopic parameters: Θ1 = (θ 1 , θ 2 ) ,

(

)

Θ 2 = (θ 1 , θ 2 ) , Θ 3 = (θ 1 , θ 2 ) and Θ 4 = θ 1 , θ 2 . Mapping the polytopic parameters into a

polytopic of matrices creates a set of state-space systems which evolve into a polytope of matrices with four vertices.

θ2

Θ4

S(θ2)

Θ3

⎛ A(Θ 3 ) B(Θ 3 ) ⎞ ⎜⎜ ⎟⎟ ⎝ C (Θ 3 ) D (Θ 3 ) ⎠

⎛ A(Θ 4 ) B (Θ 4 ) ⎞ ⎜⎜ ⎟⎟ ⎝ C (Θ 4 ) D (Θ 4 ) ⎠

⎛ A(Θ1 ) B (Θ1 ) ⎞ ⎜⎜ ⎟⎟ ⎝ C (Θ 1 ) D (Θ 1 ) ⎠

Ξ Θ1

Θ2

⎛ A(Θ 2 ) B (Θ 2 ) ⎞ ⎜⎜ ⎟⎟ ⎝ C (Θ 2 ) D (Θ 2 ) ⎠

θ1 Polytopic of parameters

S(θ1) Polytopic of matrices

Figure 5.14 Mapping polytopic systems The LPV system is extended to augmented plants. Consider an augmented LPV plant, mapping exogenous inputs w and control inputs u to controlled outputs z and measured outputs y,

x& = A(θ ) x + B1 (θ ) w + B 2 u , z = C1 (θ ) x + D11 (θ ) w + D12 u , y = C 2 x + D 21 w + D 22 u .

(5.18)

Assumptions concerning the plant are as follows: A1) D22=0 A2) B2, C2, D12, D21 are parameter independent. A3) The pairs ( A(θ ) ,B2) and ( A(θ ) ,C2) are quadratically stabilizable and quadratically detectable over Ξ respectively.

121

Chapter 5 Multiobjective Control Design for the Turbofan Engine If the assumption A2 is not satisfied, the computation of a LPV controller (next Section) requires the solution of a problem with an infinite number of constraints. However, this difficulty can be alleviated by pre- and/or post-filtering of the control inputs u and/or the measured outputs y. In real world problems, the plant will include actuator and sensor dynamics, and then the control and measurement matrices are parameter independent.

5.9 LPV Control The LPV plant might be controlled with a LPV controller under the same polytopic framework. The bounded real lemma for LTI systems can be extended to LPV systems in conjunction with the notion of quadratic H∞ performance. This approach is valid only because a single Lyapunov function is used over the entire operating range (Apkarian et al. 1995). The resulting polytopic LPV controller enforces stability and H∞ performance over the entire parametric polytope. The parameter dependent controller is defined as follows:

x&c = Ac (θ ) xc + Bc (θ ) y u = Cc (θ ) xc + Dc (θ ) y

(5.19)

Following the vertex property, the controller system can be mapped into an LPV system with polytopic form: ⎧⎪⎛ Ac,i ⎛ Ac (θ ) Bc (θ ) ⎞ ⎜⎜ ⎟⎟ ∈ Co⎨⎜⎜ ⎪⎩⎝ C c,i ⎝ C c (θ ) Dc (θ ) ⎠

⎫⎪ r ⎛ Ac, i Bc ,i ⎞ ⎛ Ac (Θ i ) Bc (Θ i ) ⎞ ⎟ := ⎜⎜ ⎟ i r , 1 ,..., = ⎬ := ∑ ρi ⎜⎜ Dc ,i ⎟⎠ ⎝ C c (Θ i ) Dc (Θ i ) ⎟⎠ ⎪⎭ i =1 ⎝ C c ,i

Bc , i ⎞ ⎟. Dc ,i ⎟⎠

(5.20) The controller state-space matrices at any operating point θ(t) are obtained by convex interpolation of the LTI vertex controllers, in the following way, ⎛ A (Θ ) B c (Θ 1 ) ⎞ ⎛ A (Θ ) Bc (Θ 2 ) ⎞ ⎛ A (Θ ) Bc (Θ r ) ⎞ ⎟⎟ + ρ 2 ⎜⎜ c 2 ⎟⎟ + L + ρ r ⎜⎜ c r ⎟⎟, (5.21) K (θ ) := ρ1 ⎜⎜ c 1 ⎝ C c (Θ 1 ) D c (Θ 1 ) ⎠ ⎝ C c (Θ 2 ) Dc (Θ 2 ) ⎠ ⎝ C c (Θ r ) Dc (Θ r ) ⎠

where ρ1 ≥ 0 , ρ 2 ≥ 0 , …, ρ r ≥ 0 and ρ1 + ρ 2 + L + ρ r = 1 .

122

Chapter 5 Multiobjective Control Design for the Turbofan Engine The existence of an LPV controller depends on the assumptions (A1)-(A3) and the following two statements. 1) Given some positive scalar γ, there exists a kth-order LPV controller solving the quadratic H∞ problem where the Lyapunov function V ( x cl ) = xclT X cl xcl ensures asymptotic stability, and the L2 gain of the input/output map is bounded by γ. This is y

2

Multiobjective Control: Linear Matrix Inequality Techniques and

Multiobjective Control: Linear Matrix Inequality Techniques and

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