Rosario Toscano

Structured Controllers for Uncertain Systems A Stochastic Optimization Approach

Springer

Preface

This monograph is concerned with the problem of designing low-order/fixedstructure feedback controllers for uncertain dynamical systems. Feedback controllers are required to reduce the effect of unmeasured disturbance acting on the system and to reduce the effect of uncertainties related to the system dynamics. This requires some knowledge about the system in order to design an effective feedback controller. The available knowledge is generally expressed as a mathematical description of the real system which is called the model of the system. However the model thus obtained is always an approximation of the reality and is thus subject to some uncertainties. Therefore, the design of an effective feedback controller for the real system must take into account model uncertainties. A controller is said to be robust if the stability and/or the desired performance of the closed-loop system are not affected by the presence of bounded modelling uncertainties. Uncertainty and robustness Since the early 1940s, robustness has been recognized of paramount importance when designing a feedback controller. Indeed, in the classical control theory of Black-Nyquist-Bode, plant uncertainties are taken into account via gain and phase margins which are measures of stability in the frequency domain. These stability margins confer to the closed-loop system fairly good robustness properties. However, classical control is mainly concerned with single-input single-output systems. The transition to the so called modern control theory was done in the early 1960s by the development of optimal control. The basic principle is to determine the controller that minimizes a given performance index. This approach applies to multivariable systems represented in the time domain by a set of first order ordinary differential equations called a state space model. In the context of linear systems, the main result of this theory is certainly the Wiener-Hopf-Kalman optimal regulator, also known as the Linear Quadratic Gaussian (LQG) regulator. An inherent limitation of LQG control is that

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uncertainties are considered only in the form of exogenous stochastic disturbances having known statistical properties, while the system is assumed to be perfectly described by a linear, possibly time-varying, state space representation. These assumptions are so strong that the LQG control leads to poor performance when applied to systems for which no precise model is available. The need of a control theory capable of dealing with modelling errors and disturbance uncertainty thus became very clear. A major step toward a robust control theory was taken in 1981 when Zames introduced the optimal H∞ control problem. It was soon recognized that the H∞ norm can be used to quantify not only disturbance attenuation but also robustness against modelling errors. Since then, many contributions in robust control theory have been made such as the introduction of structured singular value, the two-Riccati-equation method, the H∞ loop-shaping and the linear matrix inequality approach, to cite only the most important advances. Nowadays optimal robust control is a mature theory and can be applied to a number of industrial problems which were beyond the scope of both classical control theory and LQG control theory. This is due to the fact that the robust control theory provides a systematic treatment of robustness against modelling errors and disturbance uncertainty for both scalar and multivariable systems. Limitations A weakness of traditional robust control is that the controller obtained is of full order, in other words, the order of the controller is always greater than or equal to the dimension of the process model itself which can be very high. This is a serious limitation especially when the memory and computational power available are limited, in embedded controllers. Moreover, traditional robust control is unable to incorporate constraints into the structure of the controller. This is also a strong limitation especially when the control law must be implemented on commercially available controllers that have inherently a fixed structure such as PID or lead-lag compensators. All these reasons justify the need for designing robust reduced-order/fixedstructure controllers. Unfortunately, the problem of designing a robust controller with a given fixed structure (e.g. a PID) remains an open issue. This is mainly due to the fact that the set of all fixed-order/structure stabilizing controllers is non-convex and disconnected in the space of controller parameters. This is a major source of computational intractability and conservatism. Nevertheless, due to their practical importance, some new approaches for structured control have been proposed in the literature. Most of them are based on the resolution of Linear Matrix Inequalities LMIs or Riccati equations. However, a major drawback with this kind of approach is the use of Lyapunov variables, whose number grows quadratically with the system size. For instance, if we consider a system of order 70, this requires, at least, the introduction of 2485 unknown variables whereas we are looking for the pa-

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rameters of a fixed-order/structure controller which contains a comparatively very small number of unknowns. It is then necessary to introduce new techniques capable of dealing with the non-convexity of optimization problems arising in automatic control without introducing extra unknown variables. Stochastic optimization via HKA The main optimization tool used in this book to tackle the problem of nonconvexity is the so-called Heuristic Kalman Algorithm (HKA). The main characteristic of HKA is the use of a stochastic search mechanism to solve a given optimization problem. From a computational point of view, the use of a stochastic search procedure appears essential for dealing with non-convex problems. The HKA method falls into the category of the so-called “population-based stochastic optimization techniques”. However, its search heuristic is entirely different from other known stochastic algorithms such as genetic algorithm (GA) or particle swarm optimization (PSO). Indeed, HKA explicitly considers the optimization problem as a measurement process designed to give an estimate of the optimum. A specific procedure, based on the Kalman estimator, is utilized to improve the quality of the estimate obtained through the measurement process. HKA shares with GA and PSO interesting features such as: ease of implementation, low memory and CPU speed requirements, a search procedure based only on the values of the objective function, and no need for strong assumptions such as linearity, differentiability, convexity etc, to solve the optimization problem. In fact it could be used even when the objective function cannot be expressed in an analytic form; in this case, the objective function is evaluated through simulations. The main advantage of HKA compared to other stochastic methods, lies in the small number of parameters that need to be set by the user (only three). In addition these parameters have an understandable effect on the search procedure. These properties make the algorithm easy to use for non-specialists. Structure of the book This book focuses on the development of simple and easy to use design strategies for robust low-order/fixed-structure controllers. HKA is used to solve the underlying constrained non-convex optimization problems. Chapter 1 introduces some basic definitions and concepts from the classical optimization theory and indicates some limitations of the classical optimization methods. The class of convex optimization problems is also briefly presented with an emphasis on semi-definite programs. Some aspects related to the optimization in engineering design are also introduced. After that, the main objectives of the book are presented. Chapter 2 introduces some basic materials related to signal and systems norms. Many control objectives can be stated in terms of the size of some

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particular signals. Therefore, a quantitative treatment of the performance of control systems requires the introduction of appropriate norms, which give measurements of the sizes of the signals considered. Another concept closely related to the size of a signal, is the size of an LTI system. The latter concept is of great practical importance because it is at the basis of H∞ control as well as the robustness analysis. Chapter 3 introduces how a control design problem can be formulated as an optimization problem. To this end, the standard control problem as well as the notion of stabilizing controllers are first briefly reviewed. After that, some closed-loop system performance measurements are presented, they are essential to evaluate the quality of a given controller. These performances measurements are then used to formulate the optimal controller design problem which is multi-objective in nature. The resulting multi-objective problem is scalarized using the notion of ideal point. The last part of the chapter is dedicated to the case of structured controllers i.e., structural constraints have to be taken into account in the optimization problem. These structural constraints make the resulting optimization problem non-smooth and non-convex, which results in intractability. This is why the use of stochastic optimization methods is suggested to find an acceptable solution. The robustness issue is also briefly discussed. It is pointed out that the optimal robust control problem in addition to being non-smooth and non-convex is also semi-infinite. This means that the optimization problem has an infinite number of constraints and a finite number of optimization variables. Chapter 4 introduces the notion of acceptable solution; after that, a brief overview of the main stochastic methods which can be used to solve continuous non-convex constrained optimization problems is presented i.e., Pure Random Search Methods, Simulated Annealing, Genetic Algorithm, and Particle Swarm Optimization. The last part is dedicated to the problem of robust optimization, i.e., optimization in the presence of uncertainties in the problem data. Chapter 5 introduces a new optimization method, called Heuristic Kalman Algorithm (HKA). This algorithm is proposed as an alternative approach for solving continuous non-convex optimization problems. The performance of HKA is evaluated in detail through several non-convex test problems, both in the unconstrained and constrained cases. The results are compared to those obtained via other metaheuristics. These various numerical experiments show that the HKA has very interesting potentialities for solving non-convex optimization problems, especially with regard to the computation time and the success ratio. Chapter 6 deals with the concept of uncertain system. This is a key notion when designing a robust feedback controller. The objective is indeed to determine the controller parameters ensuring acceptable performance of the closed-loop system despite the unknown disturbances affecting the system as well as the uncertainties related to the plant dynamics. To this end, it is necessary to be able to take into account the model uncertainties during the

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design phase of the controller. In this chapter, we briefly describe some basic concepts regarding uncertain systems and robustness analysis. The last part of this chapter is dedicated to structured robust control for which a specific stochastic algorithm is developed. In chapter 7 we consider the design of fixed structure controllers for uncertain systems in the H∞ framework. Although the design procedures presented apply for any kind of structured controller, we focus mainly on the most widely used of them, that is the PID controller structure. Two design approaches will be considered: the mixed sensitivity method and the H∞ loop-shaping design procedure. Using these methods, the resulting PID design problem is formulated as an inherently non-convex optimization problem. The resulting tuning method is applicable both to stable and to unstable systems, without any limitation concerning the order of the process to be controlled. Various design examples are presented to give some practical insights into the methods presented. Chapter 8 is concerned with the design of structured controller for uncertain parametric systems in the H2 and mixed H2 /H∞ framework. We restrict ourselves to the case of static output feedback (SOF) controllers; this is not restrictive because any dynamical controller can be reformulated as SOF for an augmented plant. Some design examples are presented to illustrate the design methods proposed. Chapter 9 is devoted to the design of a nonlinear structured controller for systems that can be well described by uncertain multi-models. In a first part, the concept of multi-model is introduced and some examples are given to show how this works. After that, the problem of designing a nonlinear structured controller for a given uncertain multi-model is considered. A characterization of the set of quadratically stabilizing controllers is first introduced. This result is then used to design a nonlinear structured controller that quadratically stabilizes the uncertain multi-model, while satisfying a given performance objective. Some design examples are presented to illustrate the main points introduced in this chapter. Finally, chapter 10 concludes the book by recalling the general philosophy behind the approach developed from chapter to chapter as well as the difficulty we can encounter when designing a structured controller and thus the development that needs to be done. Acknowledgments Place(s), month year

Firstname Surname Firstname Surname

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 General Formulation of an Optimization Problem . . . . . . . . . . . 1.1.1 Design Variables, Constraints and Objective Function . 1.1.2 Global and Local Minimum, Descent Direction . . . . . . . 1.1.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Convex Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Semidefinite Programming . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Optimization in Engineering Design . . . . . . . . . . . . . . . . . . . . . . . 1.4 Main objectives of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Uncertain System G . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Structured Controller K . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Interconnection [G, K] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Performance Specifications . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Algorithms for Finding an Acceptable Solution . . . . . . . 1.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 7 9 12 14 15 17 18 19 19 20 22 23

2

Signal and System Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Signal Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 L1 -space and L1 -norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 L2 -space and L2 -norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 L∞ -space and L∞ -norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Extended Lp -space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 RMS-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 System Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Controllability, Observability . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 System Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Definition of the H2 -norm and H∞ -norm of a system . 2.3.2 Singular values of a transfer matrix . . . . . . . . . . . . . . . . .

25 25 26 26 27 27 28 29 30 32 34 35 36 37

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2.3.3 Singular Values and H2 , H∞ -norms . . . . . . . . . . . . . . . . 39 2.3.4 Computing norms from the state space equation . . . . . . 40 2.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3

4

Optimization in Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Notion of System and Feedback Control . . . . . . . . . . . . . . . . . . . 3.2 The Standard Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Standard Control Problem as an Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Extended System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Closed-loop System and Stabilizing Controllers . . . . . . . 3.3 Performance Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Control Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Time-domain Specifications . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Optimal Controller design and Multi-objective Optimization . 3.4.1 Scalarization of the Multi-Objective Function . . . . . . . . 3.4.2 Limits of a Convex Formulation of the Optimal Control Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Structured Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Important Examples of Structured Controllers . . . . . . . 3.5.2 Difficulties in Solving the Structured Control Problem 3.5.3 Robustness issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Motivations and Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Notion of Acceptable Solution . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Some Characteristics of Stochastic Methods . . . . . . . . . 4.2 Pure Random Search Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Non-localized Search Method . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Localized Search Method . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Metropolis algorithm and simulated annealing . . . . . . . 4.3.2 Simulated annealing algorithm . . . . . . . . . . . . . . . . . . . . . 4.4 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The main steps of a Genetic Algorithm . . . . . . . . . . . . . . 4.4.2 The standard genetic algorithm . . . . . . . . . . . . . . . . . . . . 4.5 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Dynamics of the particles of a swarm . . . . . . . . . . . . . . . 4.5.2 The standard PSO algorithm . . . . . . . . . . . . . . . . . . . . . . 4.6 Robust Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Worst-Case Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Average Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50 51 52 53 54 63 65 66 66 70 73 74 75 76 78 79 80 83 83 85 86 88 88 91 92 92 93 96 96 99 100 100 102 103 104 105

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4.7 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5

Heuristic Kalman Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Principle of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Gaussian Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Measurement Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Kalman Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Equation of the Kalman Estimator . . . . . . . . . . . . . . . . . . . . . . . 5.4 Heuristic Kalman Algorithm and Implementation . . . . . . . . . . . 5.4.1 Updating Rule of the Gaussian Distribution . . . . . . . . . 5.4.2 Initialization and Parameter Settings . . . . . . . . . . . . . . . 5.4.3 Stopping Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 The Feasibility Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Unconstrained Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Constrained Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 112 113 114 115 116 117 118 119 120 120 121 121 122 127 128

6

Uncertain Systems and Robustness . . . . . . . . . . . . . . . . . . . . . . . 6.1 Notion of uncertain system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Parametric and dynamic uncertainty . . . . . . . . . . . . . . . . 6.1.2 General representation of an uncertain linear system . . 6.2 Parametric uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Affine parameter-dependent model . . . . . . . . . . . . . . . . . . 6.2.2 Polytopic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 LFT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Dynamic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Multiplicative Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Additive Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Coprime Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Mixed Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Structured Robust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Robust Stability Condition . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Robust Performance Condition . . . . . . . . . . . . . . . . . . . . . 6.5.3 General Stochastic Algorithm . . . . . . . . . . . . . . . . . . . . . . 6.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 134 134 137 138 139 140 142 143 145 147 148 149 152 158 160 161

7

H∞ 7.1 7.2 7.3

165 165 167 171 173 175

Design of Structured Controllers . . . . . . . . . . . . . . . . . . . . . General Formulation of the Structured H∞ Control Problem . Mixed Sensitivity Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H∞ Loop Shaping Design Approach . . . . . . . . . . . . . . . . . . . . . . 7.3.1 The H∞ Loop-Shaping Design Procedure . . . . . . . . . . . 7.3.2 Robustness and ν-Gap Metric . . . . . . . . . . . . . . . . . . . . . .

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7.3.3 Loop-Shaping Design with Structured Controllers . . . . 7.4 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Design Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Design Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Design Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Design Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 179 180 184 192 198 200

8

H2 and Mixed H2 /H∞ Design of Structured Controllers . . 8.1 H2 Design of Structured Controllers . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Formulation of the robust H2 design problem . . . . . . . . 8.1.2 Set of robustly stable SOF controllers . . . . . . . . . . . . . . . 8.1.3 Worst-Case Performance and Average Performance . . . 8.1.4 Guaranteed LQ Cost with Time Varying Parameters . . 8.2 Mixed H2 /H∞ Design of Structured Controllers . . . . . . . . . . . . 8.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Set of robustly stable SOF controllers . . . . . . . . . . . . . . . 8.3 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Design Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Design Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 204 206 209 211 216 216 218 218 219 223 230

9

Nonlinear Structured Control via the Multi-Model Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Multi-Model Representation of a Given Nonlinear System . . . 9.1.1 Global Representation of a System from Local Models 9.1.2 Some Considerations for Building a Multi-model . . . . . 9.2 Design of a Nonlinear Structured Controller . . . . . . . . . . . . . . . 9.2.1 The System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 The Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Closed-loop Quadratic Stability . . . . . . . . . . . . . . . . . . . . 9.3 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Design Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Design Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Design Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 237 239 243 252 252 255 258 260 260 265 268 273

10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 The necessity of the uncertain model . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Certainty Prevision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Uncertain prevision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The closed-loop interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 277 277 279 280 280

A

Convergence Properties of the HKA . . . . . . . . . . . . . . . . . . . . . . 283

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Notation and Acronyms

Sets R Rn Rn×m C C− Cn Cn×m H ⊂ Rn×n Sn PH L1 L2 L∞ RHn×m 2 RHn×m ∞

set of real numbers set of real column vectors with n entries set of real matrices of dimension n × m set of complex numbers open left-half plane set of complex column vectors with n entries set of complex matrices of dimension n × m set of real Hurwitz matrices set of real symmetric matrices of size n, i.e., Sn = {S ∈ Rn×n : S = S T } set of Hurwitz polynomials set of absolute-value integrable signals set of square integrable signals set of signals bounded in amplitude set of strictly proper and stable transfer matrices of dimension n × m set of proper and stable transfer matrices of dimension n × m

xix

xx

Notation and Acronyms

Relational Operators = ≈ < ≤  > ≥  ≺,  0 is a given positive real number. 3

6

1 Introduction

Obviously, it is desirable to look for a global minimizer since this gives the assurance that nothing better can be found. However, the global minimizer is generally very difficult to find, because f is usually known locally only. Most algorithms are able to find a local minimizer only, which is a point that achieves the smallest value of f in its neighborhood. Such a point can be reached using successive descent directions. Descent direction. Usually, an optimization problem is solved via an iterative algorithm. This mean that, given an initial point x0 , a sequence of feasible points {xk } is generated by repeating application of a transition rule. In other words, two successive points of the sequence are such that xk+1 = A(xk )

(1.5)

where A is the transition rule by means of which the new point xk+1 is calculated according to the information gained at point xk . The available information at point xk can be the values of the objective function and the constraints at xk , the values of the first-order derivatives of these functions at xk and, possibly, the values of the second-order derivatives of these functions at xk . To solve a minimization problem, it seems reasonable to require that the transition rule A must be such that xk+1 ∈ F, and f (xk+1 ) < f (xk ). Under these conditions, we can expect that by successive applications of (1.5), we finally obtain a local minimum point. This is the principle of the so called descent methods. For this kind of algorithm, the transition rule can be written as follows: xk+1 = A(xk ) = xk + αk dk (1.6) where dk is a vector in Rnx called the search direction, and the scalar αk ≥ 0 is called the step size or step length at iteration k. The new point xk+1 is thus generated by adding to the current point xk an appropriately chosen vector αk dk . Among all the feasible search directions4 we are interested in finding a direction dk along which the objective function f is decreasing. Such a direction is called a feasible descent direction. More formally, a vector dk ∈ Rnx is said to be a feasible descent direction for f at xk if there exists δ > 0 such that:  xk + αk dk ∈ F , for all αk ∈ (0, δ] (1.7) f (xk + αk dk ) < f (xk ) If the objective function f is differentiable, we can use the first order Taylor series to expand f at xk , thus we can write: f (xk + αk dk) = f (xk ) + αk ∇f (xk )T dk + h.o.t 4

(1.8)

A search direction is said to be a feasible direction at the point xk , if there exists δ > 0 such that xk + αk dk ∈ F , for all αk ∈ [0, δ]

1.1 General Formulation of an Optimization Problem

7

where the high order terms (h.o.t) tend to zero as αk → 0+ , consequently, the quantity ∇f (xk )T dk , called directional derivative of f at xk along dk , can be defined as: ∇f (xk )T dk = lim + αk →0

f (xk + αk dk) − f (xk ) αk

(1.9)

From (1.9), we can see that a vector dk is a descent direction (i.e. f (xk + αk dk ) − f (xk ) < 0 for some αk ), if the directional derivative of f at xk along dk is negative: ∇f (xk )T dk < 0 (1.10) This inequality means that dk is a descent direction if the angle between dk and the direction of the gradient at xk is obtuse. Therefore, the anti-gradient −∇f (xk ) is a descent direction. Note that if ∇f (xk )T dk > 0, then dk is an ascent direction, i.e., the function f is increasing along dk , more precisely we have f (xk + αk dk ) > f (xk ) for a sufficiently small positive value of αk . In this case, the angle between ∇f (xk ) and dk is acute (see figure 1.3).

Fig. 1.3: The angle between ∇f (x) and d is obtuse i.e., ∇f (x)T d < 0, d is then a descent direction.

1.1.3 Optimality Conditions To solve an optimization problem, it is essential to give the conditions that a local minimum point must satisfy. For a constrained optimization problem, we can reasonably say that a point x∗ ∈ F is a local minimum point if the objective function f cannot be decreased for sufficiently small positive displacements from x∗ along any feasible direction d. In other words, x∗ is a

8

1 Introduction

local minimum point if there is no feasible d satisfying ∇f (x∗ )T d < 0 and so we can only find feasible d such that ∇f (x∗ )T d ≥ 0

(1.11)

This is indeed a necessary condition that a local minimum of a constrained optimization must satisfy. Figure 1.2 gives a geometric interpretation of the optimality condition (1.11). The feasible set F is shown shaded. We can infer that from x∗ it is not possible to find a feasible descent direction, thus x∗ satisfies the necessary condition of local minimum. In the case of an unconstrained optimization problem, any direction d ∈ Rnx is a feasible direction. Thus if we have ∇f (x∗ )T d ≥ 0 for all d ∈ Rnx , this necessarily means that ∇f (x∗ ) = 0 (1.12) we recognize the well-known necessary condition that a local minimum must satisfy in the case of unconstrained optimization problems. A point x∗ is said to be a stationary point if it satisfies ∇f (x∗ ) = 0. Therefore, for an unconstrained optimization problem, a necessary condition to be satisfied to get a local minimizer is that it must be a stationary point. It is interesting to note that the set of stationary points can be determined by solving the set of equations ∇f (x) = 0 (1.13) In the same way, it is important to know the set of equations that a candidate solution of a constrained optimization problem must satisfy to be a local minimum point. This can be done using the Karush-Kuhn-Tucker conditions.  KKT Necessary Optimality Condition. Let x∗ be a local minimizer

of the constrained optimization problem (1.3), and assume that x∗ is a regular5 point of the constraints. Then there exist uniquely defined vectors λ∗ , µ∗ of Lagrange multipliers such that vectors x∗ , λ∗ , µ∗ are a solution of the KKT system of equalities and inequalities with unknowns x, λ, µ: ∇x L(x, λ, µ) = 0

(1.14)

∇λ L(x, λ, µ) = 0

(1.15)

T

µ g(x) = 0

(1.16)

g(x)  0

(1.17)

µ0

(1.18)

where L(x, λ, µ), called the Lagrangian function, is defined as L(x, λ, µ) = f (x) + µT g(x) + λT h(x)

(1.19)

A point x∗ is said to be regular for the system of constraints if the gradient of the equality constraints ∇hi (x∗ ), i = 1, · · · , nh and the gradient of the active inequality constraints ∇gi (x∗ ), i ∈ {j : gj (x∗ ) = 0} are linearly independent. 5

1.2 Convex Optimization Problems

9

Equation (1.14) represents the nx stationarity conditions, (1.15) represents the nh equality constraints, (1.16) represents the ng complementary slackness, i.e. the fact that µi = 0 if gi is non active and µi 6= 0 if gi is active, (1.17) represents the inequality constraints and (1.18) impose that the multipliers µi must be non negative. Generally speaking, the KKT system cannot be solved analytically and therefore only a numerical method can be employed to find a local minimum point. It can be shown that the above KKT conditions are necessary and sufficient conditions of optimality in the case of a convex optimization problem.

1.2 Convex Optimization Problems Convex programming is a special case of nonlinear programming of great practical importance for at least two reasons. First, it includes a broad class of commonly addressed optimization problems such as linear programming, quadratic programming, geometric programming, etc. Moreover, many practical engineering problems can be formulated as a convex optimization problem. Secondly, what is more important, is that a convex optimization problem can be very efficiently solved because any local optimum is also the global optimum. In fact, it is now well recognized that the frontier between those problems that are tractable and those that are intractable relies on convexity. In other words, if we are able to formulate an optimization problem that is convex, it is guaranteed that we can find the global optimum very efficiently, i.e., in polynomial time in the the number of decision variables (see notes and reference for a short discussion on complexity). In contrast, a nonconvex optimization problem can be very hard to solve even for a small number of decision variables. Roughly speaking, a convex optimization problem is formulated as the problem of minimizing a convex function over convex set. In what follows, we introduce some basic definitions for the convexity of sets and functions. Convex Sets. A set S ⊂ Rn is said to be convex if it contains the line segment between any two points in S. In other words, for any x1 ∈ S, x2 ∈ S and any α satisfying 0 ≤ α ≤ 1, we have αx1 + (1 − α)x2 ∈ S

(1.20)

Figure 1.4 gives an example sets in R2 . A point PN of convex and nonconvex PN written in the form x = i=1 αi xi with αi ≥ 0 and i=1 αi = 1 is called a convex combination of the points x1 ,...,xN . It can be shown that a set is convex if and only if it contains all the convex combinations of its points.

10

1 Introduction

Fig. 1.4: Example of convex and nonconvex sets. The set on the left is convex whereas the V-shaped set is not convex because the line segment between the two points in the set is not contained in the set.

Fig. 1.5: Convex hull of the V-shaped set of the figure 1.4, and convex hull of a finite set of points, i.e., a polytope.

The convex hull of S, denoted by Co(S) is defined as the set of all convex combinations of points in S, Co(S) is then a convex set even if S is not convex. A polytope is defined as the convex hull of a finite set of points {x1 , · · · , xN } ⊂ Rn , i.e., (N ) N X X Co({x1 , · · · , xN }) = αi xi : αi ≥ 0, αi = 1 (1.21) i=1

i=1 2

Figure 1.5 gives an example of convex hulls in R . Convex Functions. A function f : Rn → R is convex if its definition domain, denoted domf is a convex set and if for all x, y ∈ domf , and α satisfying 0 ≤ α ≤ 1, we have f (αx + (1 − α)y) ≤ αf (x) + (1 − α)f (y)

(1.22)

The geometric interpretation of this inequality is that line segment between any two point on the graph of f is above the graph (see figure 1.6). A function f is said to be concave if −f is convex. A function both convex and concave is affine. A necessary and sufficient condition of convexity of a differentiable function f is that f (y) ≥ f (x) + ∇f (x)T (y − x), ∀x, y ∈ domf (1.23)

1.2 Convex Optimization Problems

11

Fig. 1.6: The function represented by this graph is convex because the line segment between any two points on the graph is above the graph.

In other words, f is a convex function, if and only if the affine approximation of f near any x ∈ domf , obtained via the first-order Taylor expansion, i.e., fˆ(y) = f (x) + ∇f (x)T (y − x), is always a global underestimator of f : f (y) ≥ fˆ(y). This is illustrated figure 1.7.

Fig. 1.7: A differentiable function f is convex if and only if the first-order Taylor approximation of f in any x ∈ domf , expressed as fˆ(y) = f (x) + ∇f (x)T (y − x), is such that f (y) ≥ fˆ(y).

This property shows that from a local evaluation of f , we can get global information about it. In particular, according to the first order optimality condition, i.e. ∇f (x) = 0, it follow from (1.23) that f (y) ≥ f (x) for all y ∈ domf , and thus x is the global minimizer of f .

12

1 Introduction

General Form of a Convex Optimization Problem. An optimization problem is said to be convex if it can be written as follows minimize subject to

f (x) gi (x) ≤ 0, hi (x) = 0,

i = 1, · · · , ng i = 1, · · · , nh

(1.24)

where f and gi are convex functions, and the equality constraints are affine: hi (x) = aTi x + bi . Since the functions gi of inequality constraints are convex and the functions hi of the equality constraints are affine, the resulting feasible domain is a convex set. Due to the convexity of the objective function and of the feasible set, the optimality condition given in (1.11) is necessary and sufficient. Convex programming includes as a special case the so called semidefinite programming under which a broad class of control problem can be formulated and efficiently solved.

1.2.1 Semidefinite Programming A semidefinite program (SDP) has the following general form minimize subject to

cT x F (x)  0 Ax = b

(1.25)

where c ∈ Rnx , Fi ∈ Sk , i = 0, · · · , nx , A ∈ Rnh ×nx and b ∈ Rnh , are the problem data. In our notations, Sk represents the set of symmetric matrices of size k, i.e., Sk = {S ∈ Rk×k : S = S T }. The constraint F (x)  0 is a linear matrix inequality (LMI), which is defined as F (x) = F (x)T = F0 +

n X

xi Fi  0

(1.26)

i=1

where Fi ∈ Sk , i = 0, · · · , nx are given symmetric matrices. The notation F (x)  0 means that the inequality is satisfied if x ∈ Rnx is such that F (x) is a positive semidefinite matrix, i.e., z T F (x)z ≥ 0 for all z ∈ Rk . Equivalently, this means that the eigenvalues of F (x) are nonnegative6 . As a result, it can be easily verified that the constraint F (x)  0 is convex, i.e., the set {x : F (x)  0} is convex. Therefore, a SDP is a convex optimization problem. Multiple LMI’s F (1) (x)  0, · · · , F (ng ) (x)  0 can be rewritten as a single LMI by using a block-diagonal matrix, i.e., diag(F (1) (x), · · · , F (ng ) (x))  0. 6

Since F (x) is symmetric its eigenvalues are necessarily real numbers.

1.2 Convex Optimization Problems

13

By using a block-diagonal matrix, we can also incorporate linear inequality constraints in a single LMI. Indeed, the set of constraints {aT1 x ≤ b, · · · , aTng x ≤ bng , F (x)  0} can be rewritten as diag(b1 − aT1 x, · · · , bng − aTng x, F (x))  0. Note that SDP is reduced to a linear program when the matrices Fi are all diagonal, indeed, in this case F (x)  0 defines a set of linear inequality constraints. Note however that, in general, a LMI is a nonlinear constraint, the term linear matrix inequality come from the fact that F (x) is affine in x. We can also encounter strict LMI, denoted F (x)  0, which means that the inequality is satisfied if x ∈ Rnx is such that F (x) is a positive definite matrix, i.e., z T F (x)z > 0 for all non-zero z ∈ Rk (or equivalently the eigenvalues of F (x) are positive). Note that if the non strict LMI F (x)  I, with  > 0, is satisfied then the strict LMI F (x)  0 is satisfied. Matrix Variable. Most LMIs are not formulated in the standard form (1.26) as for instance the Lyapunov inequality AT P + P A ≺ 0,

P = PT  0

(1.27)

where A is a given matrix and P is the matrix variable. In this case (1.27) is said to be a LMI in P . As shown in the example 1.2, LMIs that are formulated with matrix variables can be rewritten into the standard form (1.26). Example 1.2. For simplicity, assume that the matrix variable P belongs to S2 , then any P ∈ S2 can be expressed as       00 01 10 = x1 P 1 + x2 P 2 + x3 P 3 + x3 + x2 P (x) = x1 01 10 00

(1.28)

Note that a symmetric matrix variable of size k, represents k(k + 1)/2 decision variables. A symmetric matrix variable is often called a Lyapunov variable. The Lyapunov inequality (1.27) can then be written as   X3 Pi 0 F (x) = xi Fi , Fi = ∈ S4 (1.29) 0 −AT Pi − Pi A i=1 which is a LMI in the standard form (1.26) with F0 = 0.

Schur Complement Lemma. The Schur complement lemma states that for any given matrices Q = QT , R = RT  0 and S of appropriate dimension, the LMI   Q S L= 0 (1.30) ST R is equivalent to R  0, −1

Q − SR−1 S T  0

(1.31)

T

The matrix Q − SR S is called the Schur complement of R in L. This result is very often used to convert a convex quadratic constraint into a LMI. For instance, the Riccati inequality AT P + P A + P BR−1 B T P + Q ≺ 0,

P = PT  0

(1.32)

14

1 Introduction

where A, B, Q = QT , R = RT  0 are given matrices and P is the variable, can be rewritten as   −AT P − P A − Q P B 0 (1.33) BT P R by application of the Schur complement lemma.

1.2.2 Dual Problem The optimization problem (1.24), also referred to as the primal problem, can be converted into a particular form called the dual problem. The interesting fact, is that the solution of the dual problem, denoted d∗ , provides a lower bound to the optimal value of the primal problem, denoted f ∗ . Since any feasible point x ˜ of the primal problem satisfies f ∗ ≤ f (˜ x), it is guaranteed that the optimal value is such that d∗ ≤ f ∗ ≤ f (˜ x). Therefore, the solution of the dual problem can be used to provide a guarantee of optimality. The dual problem can be obtained by associating to the optimization problem (1.24) the Lagrangian L(x, λ, µ) L(x, λ, µ) = f (x) +

ng X

µi gi (x) +

i=1

nh X

λi hi (x)

i=1

where λ and µ are the Lagrange multiplier vectors also called the dual variables. The dual function Ld (λ, µ) is defined as the minimum over x ∈ D of the Lagrangian ! ng nh X X Ld (λ, µ) = min L(x, λ, µ) = min f (x) + µi gi (x) + λi hi (x) (1.34) x∈D

x∈D

i=1

i=1

Note that the dual function is concave since it is defined as the minimization of a family of affine functions in the variables λ and µ. The main interest of introducing the dual function is that it gives lower bounds of the optimal objective function value f ∗ of the problem (1.24). For any λ and µ  0, we have Ld (λ, µ) ≤ f ∗ (1.35) Indeed, for any feasible point x ˜, we have λT h(˜ x) + µT g(˜ x) ≤ 0 because h(˜ x) = 0 and g(˜ x)  0. Therefore L(˜ x, λ, µ) = f (˜ x)+λT h(˜ x)+µT g(˜ x) ≤ f (˜ x), and thus we have Ld (λ, µ) = min L(x, λ, µ) ≤ L(˜ x, λ, µ) ≤ f (˜ x) x∈D

(1.36)

given that Ld (λ, µ) ≤ f (˜ x) for all x ˜ ∈ F, we have in particular Ld (λ, µ) ≤ f ∗ .

1.3 Optimization in Engineering Design

15

Thus, for any λ and µ  0, the dual function provides a lower bound of the optimal value of the corresponding primal problem. Therefore, the greatest lower bound is solution of the following optimization problem maximize subject to

Ld (λ, µ) µ0

(1.37)

called the Lagrange dual problem associated to the problem (1.24). The dual problem is convex because of the maximization of a concave objective function over convex constraints. This property is true even when the primal problem is non-convex. The determination of the best lower bound is thus guaranteed and can be determined very efficiently using available convex solvers. The difference between the optimal value of the dual problem and the primal problem p∗ − d∗ is known as the optimal duality gap. For convex problems, the optimal duality gap is zero under very mild assumptions7 . In general, this is not true for non-convex problems.

1.3 Optimization in Engineering Design Many problems encountered in the engineering sciences can be formulated as optimization problems. For instance, a design problem can be defined as ¯} such that the set of performance follows: find x ∈ D = {x ∈ Rnx : x  x  x specifications fi (x, θ) ≤ αid , i = 1, · · · , nf (1.38) are satisfied, where the positive real numbers αid are the design objectives that are chosen by the designer, x = (x1 , · · · , xnx )T is the vector of design variables, fi (., .) are real positive functions, and θ = (θ1 , · · · , θnθ )T is the parameters vector, which depends on the design problem. Usually, θ is assumed to be perfectly known; in this case, we denote by θ = θ0 the nominal parameters vector. The nominal design problem is then be defined as: find x ∈ D subject to fi (x, θ0 ) ≤ αid , i = 1, · · · , nf (1.39) Any x satisfying (1.39) is a solution of the design problem considered. In fact, the set of specifications (1.39) define the set of solutions F of the design problem F = {x ∈ D | f (x, θ0 )  αd } (1.40) where f (x, θ0 ) = (f1 (x, θ0 ), · · · , fnf (x, θ0 ))T and αd = (α1d , · · · , αnd f )T . A design problem thus formulated is not, strictly speaking, an optimization 7 If the primal problem is convex and if there exists a strictly feasible point i.e., a point x ˜ satisfying g(˜ x) ≺ 0 and A˜ x = b, then the optimal duality gap is zero. The condition of existence of a strictly feasible point is known as the Slater’s constraint qualification.

16

1 Introduction

problem because all x of F are equally valid with respect to the design objective specified by αd . However, it can be extremely interesting to find x∗ ∈ F such that all the performance values fi (x∗ , θ0 ) are as small as possible with each fi (x∗ , θ0 ) satisfying the corresponding design objective  T fnf (x∗ ,θ0 ) f1 (x∗ ,θ0 ) ∗ ∗ T d , · · · , αd i.e., (f1 (x , θ0 ), · · · , fnf (x , θ0 ))  α ⇔  1, αd nf

1

where 1 is the unity vector. This can be done by solving the following optimization problem minimize

α

subject to

fi (x, θ0 )/αid ≤ α, i = 1, · · · , nf ¯} x ∈ {x ∈ Rnx : x  x  x

α, x

(1.41)

Note that the additional scalar variable α has been introduced to formulate the optimization problem. A solution (x∗T , α∗ )T of problem (1.41) satisfies fi (x∗ , θ0 ) ≤ α∗ αid . Since α∗ is minimal, the quantity α∗ αid represents the best possible achievable performance. In the context of control system design, the functions fi (x, θ0 ) given in (1.39) can be various performance measurements taken from the system step response, say the overshoot and the settling time, and the design variables xi are the tuning parameters of the controller. In this case, solving the optimization problem (1.41), means finding the controller parameters ensuring that the system step response has the smallest overshoot and the lowest settling time while satisfying the design goals. Robust Design. Until now it has been assumed that the parameters (i.e., the problem data) which enter in the formulation of the design problem are precisely known. However, in many practical applications some of these parameters are subject to uncertainties. It is then important to be able to calculate solutions that are insensitive to parameters uncertainties; this leads to the notion of optimal robust design. We say that the design is robust, if the various specifications (i.e., the constraints) are satisfied for a set of values of the parameters uncertainties. Usually, it can be realistically assumed that θ lies in a bounded set Θ defined as follows:  Θ = θ ∈ Rnθ : θ  θ  θ¯ , (1.42) where the vectors θ = (θ1 , · · · , θnθ )T , and θ¯ = (θ¯1 , · · · , θ¯nθ )T are the uncertainty bounds of the parameters vector θ. Thus, the uncertain vector belong to the nθ -dimensional hyperrectangle Θ also called the parameter box. The robust version of the optimization problem (1.41) is then written as follows minimize

α

subject to

fi (x, θ)/αid ≤ α, i = 1, · · · , nf , ¯} x ∈ {x ∈ Rnx : x  x  x

α, x

∀θ ∈ Θ

(1.43)

1.4 Main objectives of the book

17

The formulation (1.43) is quite general and can be applied to a broad class of engineering problems including the synthesis of robust structured controllers, which is the subject of this book. Unfortunately, there are several obstacles to solving this kind of problem efficiently. The main obstacle is that most of the optimization problems are N P-hard (see the Notes and References for a short introduction on complexity). Therefore the known theoretical methods cannot be applied except possibly for some small size problems (i.e., small number of decision variables). Other difficulties arise when some of the functions fi are not differentiable and/or non-convex. In this case, the set of methods requiring the derivatives of the functions fi cannot be used and the achievement of a global optimum cannot be guaranteed in a reasonable computation time. Another obstacle is when some qualities of a design cannot be expressed in an analytic form, in this case, these qualities can only be evaluated through simulations. In these situations, probabilistic approaches seem to be a good way for solving this kind of optimization problem. By probabilistic approach, we mean a computational method employing experimentations, evaluations and trial-and-errors procedures in order to obtain an approximate solution for computationally difficult problems. This is the approach adopted in this book whose main objectives are presented in the next section.

1.4 Main objectives of the book The main objective of this book is to develop simple and user-friendly design methods for robust structured controllers satisfying multiple performance specifications. We call this the multi-objective robust structured control design problem, which can be stated as follows. Given an uncertain system, denoted by G, and a structured controller K(κ) parameterized by κ ∈ D = {κ ∈ Rnκ : ¯ }, find an appropriate setting κ = κ∗ such that the interconnection κ≤κ≤κ between G and K(κ), denoted [G, K(κ)], forms a satisfactory design, i.e., satisfies a given set of performance specifications φi ([G, K(κ)]) ≤ αid ,

i = 1, · · · , nφ , κ ∈ D

(1.44)

As seen above, the optimal version of this design problem is formulated as follows minimize α α, κ (1.45) subject to φi ([G, K(κ)])/αid ≤ α, i = 1, · · · , nφ κ ∈ D = {κ ∈ Rnκ : κ ≤ κ ≤ κ ¯} To solve this problem, we have to define: • the uncertain system G, • the structured controller K(.),

18

1 Introduction

• the interconnection [G, K], • the performance measures φi (.), • and an appropriate algorithm so as to find an acceptable solution to the problem (1.45). These various points are considered in some details in this book, and are now briefly presented. For better reading, the parts of the book dedicated to these issues are also given.

1.4.1 The Uncertain System G The design of a controller requires a mathematical description, called a model, of the system to be controlled. This model is always obtained at the price of approximations and idealizations and so we have an approximate description of the system behavior. For instance, the mathematical description of the system is practically always assumed to be linear and time invariant8 . However, linear models often constitute a gross oversimplification of the real system, and so we have to take into account the uncertainties resulting from this simplification. Two kinds of model uncertainties are usually considered: the parametric uncertainties, also called structured uncertainties 9 and dynamical uncertainties, also known as unstructured uncertainties 10 . Formally, an uncertain system can be seen as set of systems defined as G = {G(∆) : ∆ ∈ D∆ }

(1.46)

where G(.) represents a linear input/output mapping, obtained from a mathematical modeling of the system under consideration. This mapping is parameterized by ∆ ∈ D∆ , where D∆ can be a set of parameter vectors reflecting the uncertainty of the system parameters, and/or a set of stable systems reflecting the neglected dynamics. Thus, for every ∆ ∈ D∆ , we have a particular system model denoted G(∆). The so called nominal model is given by G(θ0 ), where θ0 is the nominal parameter vector. The details of the elaboration of the uncertain model G are discussed in chapter 6. 8

This is very usual in almost all areas of applied sciences and engineering because the theory of linear systems is well developed, and is usually very good for sufficiently small variations of the system variables around an operating point. 9 Structured uncertainties can be due to a change of the system operating point or to the fact that the values of certain physical parameters (such as resistance, inductance, mass values etc) are not precisely known. 10 Non parametric uncertainties are essentially due to unmodelled or neglected fast dynamics. The fast dynamics are often neglected to simplify a model of high order by retaining only the slow dynamics also called the dominant modes of the system.

1.4 Main objectives of the book

19

Another source of uncertainties comes from the undesirable but inevitable effects of the environment to the system. This kind of uncertainty is called disturbance. For example, the ambient temperature of a thermal process is a disturbance because it affects the temperature in the system due to imperfect insulation. At this point, it must be stressed that the main objective of a control system is to make the system relatively insensitive to disturbances and to the model uncertainties. This can be done by designing an appropriate controller for the system. In this book, we focus on the design of structured controllers (see chapter 3 section 3.5).

1.4.2 Structured Controller K In this book, the controllers considered are themselves linear time invariant (LTI) systems except in chapter 9 where we will consider the design of a non linear controller obtained by aggregation of linear controllers. The fact of imposing that the controller is a LTI system is in itself a structural constraint, i.e., a constraint on the mathematical form of the controller. This constraint is justified because we want to remain in the context of linear systems for which the design problems are usually much more tractable than nonlinear design problems. In this book, a linear controller is called structured when its order and its mathematical form are imposed by the designer (see chapter 3 section 3.5). This requirement is very important for many practical reasons. For example, it can be imposed that the controller has a specific form (e.g., a PID); the controller can be distributed, i.e., not localized at a single place; the available memory and computational power of the processor utilized to implement the controller is limited, therefore this imposes the design of a low order controller, i.e., of low complexity, etc. A more detailed presentation of structured controllers is given in chapter 3 section 3.5, some design examples of robust structured controllers are presented in chapters 7 and 8. In the sequel, we will denote by K a structured controller and by κ = vect(K) the controller parameter vector.

1.4.3 Interconnection [G, K] The basic goal of a control system is to ensure that the system output yS follows a reference input r as closely as possible. To this end, there are two possible interconnections between the system and the controller, that are of practical interest: the open-loop interconnection, presented figure 1.8-a, and the closed-loop interconnection shown in figure 1.8-b.

20

1 Introduction

Fig. 1.8: The two basic interconnections: the open- and closed-loop control. The open-loop control is unable to make the system less sensitive to disturbances. The closed-loop control utilizes the system output yS to elaborate the control signal u. This result in a control system that is much less sensitive to disturbances and model uncertainties.

In the case of an open/closed-loop interconnection, the controller K(.) is called an open/closed-loop controller, and the resulting control scheme is called an open/closed-loop control. Open-loop control is the simplest form of control; the control signal u is elaborated by the controller by using only the reference signal r, which represents the desired system output (see figure 1.8a). The real situation of the system is not taken into account with this kind of control. As a result, the open-loop control system is very sensitive to the effect of disturbances d affecting the system and to its model uncertainties. An open-loop controller is often used in simple processes because of its simplicity and low cost, especially in systems where feedback is not critical. In cases where high performance is required, the use of a closed-loop controller is inevitable. Indeed, only the feedback control is able to ensure that the controlled system behaves as desired despite the inevitable presence of model uncertainties and disturbances. This is possible because the control signal is elaborated not only from the reference signal but also from the actual system output yS (see figure 1.8-b). Consequently, if appropriately designed, the controller is able to compensate the effect of disturbances and is not too much sensitive to the uncertainties of the model used to make its design. In this book, only closed-loop control is considered, and a more general framework, known as the standard control problem is presented in chapter 3 section 3.2.

1.4.4 Performance Specifications A fundamental requirement of any control system is the stability. Stability means that in the absence of external input signals, all system variables tend

1.4 Main objectives of the book

21

to zero (see chapter 2 section 2.2.1). In the case of instability, the system variables diverge to infinity causing thus damages to the system. Therefore the stability of the closed-loop system is an absolute necessity. It must be stressed that due to the model uncertainties, the stability must hold for every system in G defined by (1.46). In other words, the closed-loop system must be stable over the uncertainty set D∆ . This is called the robust stability11 (see chapter 6 section 6.5.1). Robust stability is an essential requirement but is insufficient for practical applications. It is also required that the closed-loop system must satisfy some additional design goals including performances specifications and constraints on the structure of the controller. We denote by φi ([G, K]) a given performance measure of the closed-loop system [G, K]. This function allows evaluating a certain quality of the closed-loop such as disturbance rejection, tracking error, stability degree etc. In chapter 3 some detailed examples of performance measure functions are presented. We denote by φi ([G, K]) ≤ αid a performance specification, where αid is the specification fixed by the designer. The quantity αid can be seen as the maximum tolerated value of the performance measure φi . According to what was mentioned above, the multiobjective robust structured control design problem is then formulated as follows minimize

α

subject to

φi ([G, K])/αid ≤ α, K ∈ KG

α, κ

i = 1, · · · , nφ

(1.47)

where KG denote the set of structured controllers that stabilize the uncertain system, that is KG = {K : [G, K] is stable, κ = vect(K) ∈ D}

(1.48)

Note that unlike to (1.45), this formulation makes explicit the constraint relative to the robust stability. This constraint is actually not negotiable and must absolutely be satisfied. Problem (1.47) can be also rewritten as follows minimize κ

subject to

max {φi ([G, K])/αid }

1≤i≤nφ

K ∈ KG

(1.49)

The nominal design of the structured controller corresponds to the case where G is reduced to the nominal system that is G = {G(θ0 )}. Therefore, the nominal design can be seen as a relaxed version of the robust design problem, and is thus more tractable. 11

The necessity of robust stability has been recognized as a fundamental requirement from the beginning of automatic control, and has been obtained via the, now classical, gain and phase margins.

22

1 Introduction

1.4.5 Algorithms for Finding an Acceptable Solution Problem (1.49) is in general very difficult to solve because even in the nominal case this is a non-convex and non-smooth optimization problem (see chapter 3 section 3.5). If the robust design is considered, the problem is in addition semi-infinite. This means that the optimization problem has a finite number of optimization variables and an infinite number of constraints. Therefore, the robust design is much more difficult to solve than the nominal case, which is itself very difficult to solve when constraints on the structure of the controller must be satisfied. In this book, to deal with these various difficulties, we make use to the stochastic (or probabilistic) optimization methods. As will be seen later on, this kind of approach is well adapted indeed to find an acceptable solution12 to problems that are non-convex non-smooth and semi-infinite. Several stochastic methods, also called metaheuristics, have been developed in the last decades, which have demonstrated a strong ability to solve problems that were previously difficult or impossible to solve. These metaheuristics include Simulated Annealing (SA), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO), to quote only the most widely used in the framework of continuous optimization problems (see chapter 4). The main characteristic of these approaches is the use of a stochastic mechanism to seek a solution. From a general point of view, the use of a stochastic search procedure seems in fact unavoidable to find a promising solution of non-convex and non-smooth optimization problems. In this book, an alternative probabilistic optimization method, called Heuristic Kalman Algorithm (HKA), will be introduced (see chapter 5). Throughout the book, the HKA will be used to find an acceptable solution to the robust structured control design problem (see chapter 7, 8, and 9). This is not a limitation because any other probabilistic approach could be used to solve the problems considered in this book. However, a significant advantage of HKA against other stochastic methods lies mainly in the small number of parameters which have to be set by the user. This property results in a user-friendly algorithm.

12

By acceptable solution, we mean a setting that satisfies the design goals but which is not necessarily the global optimum of the optimization problem.

1.5 Notes and references

23

1.5 Notes and references The existence of optimization can be traced back to Newton, Lagrange, and Cauchy13 . The invention of differential and integral calculus by Fermat, Newton and Leibnitz has made possible the development of methods for solving optimization problems. Constrained optimization was first considered by Lagrange and the notion of steepest descent was introduced by Cauchy. The real development of the numerical optimization has been possible only from the middle of the twentieth century with the advent of digital computers that have made possible the implementation of algorithms for solving optimization problems. This period also marks the transition from classical control to the modern control theory characterized by the state space approach and the use of optimization techniques. General non-linear optimization Many excellent textbooks can be found on the subject of optimization, the references hereafter, do not claim to be exhaustive. Local methods for nonlinear programming are covered in the books by Gill, Murray, and Wright [60], Luenberger [88], Nocedal and Wright [102], Bertsekas [22], Fletcher [51], Sun and Yuan [128]. Global optimization is considered in the books by Pinter [108], Horst, Pardalos and Thoai [71], Hendrix and Toth [67]. Optimization with a strong orientation to engineering design is covered in the books by Rao [112], Deb [43] and Arora [6]. Computational complexity Consider a given optimization problem, denoted P , and an algorithm, denoted A, designed to solve the problem P . We say that an algorithm runs in polynomial time (or also is a polynomial-time algorithm) if it can solve the optimization problem P with a number of operations that grows no faster than a polynomial of the problem dimensions. Polynomial-time algorithms are considered efficient, and problems for which polynomial time algorithms exist are considered “easy”. The theory of computational complexity classifies problems according to their inherent tractability or intractability, i.e., whether they are “easy” or “hard” to solve. This classification includes the well-known classes14 P and N P. 13

Some optimization problems were considered by the ancient Greek mathematicians. For instance, Euclid (300 bc) considers the minimal distance between a point and a line, and proves that a square has the greatest area among the rectangles with given total edge lengths. 14 The notation P stands for polynomial time algorithm; the notation N P stands for non-deterministic polynomial time algorithm. This means that the algorithm consists in two phases; in the first phase, a candidate solution is generated in a non-

24

1 Introduction

The class P is defined as the set of problems for which there exists a polynomial-time algorithm. Therefore, P is considered as the set of problems that are easy to solve. The class N P is defined as the set of problem for which the validity of a candidate solution can be verified in polynomial time. If a problem is at least as hard as the hardest problem in the class N P then we say that this problem is N P-hard. A N P-hard problem is usually considered as an“intractable problem”. The problem of complexity is covered in the books by Garey and Johnson [57], Papadimitriou [104]. It as been shown that many problems arising in automatic control are N P-hard see for instance the paper by Blondel and Tsitsiklis [24]. In particular, it has been proved that the static output feedback stabilization problem is N P-hard if one constrains the coefficients of the controller to lie in prespecified intervals. This is precisely the case when designing a structured controller. Convex Optimization The theory of convex optimization is much more developed and complete than the theory of general nonlinear optimization. Nesterov and Nemirovski [101] were the first to show that interior-point methods can solve efficiently, i.e., in polynomial time, convex optimization problems15 . This result confirm the observation made by Rockafellar [113], that: “the great watershed in optimization is not between linearity and nonlinearity, but convexity and non-convexity”. The interesting thing is that a large variety of problems of practical interest can be formulated as a convex optimization problem [14]. Convex optimization is very well covered in the book by Boyd and Vandenberghe [18], and includes many applications of great practical interest. Applications of convex optimization in control theory can be found in the books by Boyd and Barratt [16], Boyd, El Ghaoui, Feron, and Balakrishnan [17]. Stochastic optimization Simulated annealing, genetic algorithms, and particle swarm methods represent a new class of mathematical programming techniques that have gained importance over the last two decades. Simulated annealing is analogous to the physical process of solid annealing. The genetic algorithms are search techniques based on the principle of natural selection. Particle swarm is a search technique based on the principle of collective behavior. A survey of deterministic way, while the second phase consists of a deterministic algorithm which verifies or rejects the candidate solution as a valid solution to the problem. 15 This is in contrast to the situation of non-convex optimization problems, for which the known algorithms require, in the worst case, a number of operations that is exponential in the problem dimensions.

1.5 Notes and references

25

these approaches can be found in [138], see also the Notes and Reference for chapter 4.