Stability and Stabilization of Linear Systems with Saturating Actuators

Transcription

1 Stability and Stabilization of Linear Systems with Saturating Actuators

2 Sophie Tarbouriech Germain Garcia João Manoel Gomes da Silva Jr. Isabelle Queinnec Stability and Stabilization of Linear Systems with Saturating Actuators

3 Sophie Tarbouriech Laboratoire Analyse et Architecture des Systèmes (LAAS) CNRS av. du Colonel Roche Toulouse CX 4 France tarbour@laas.fr Germain Garcia Laboratoire Analyse et Architecture des Systèmes (LAAS) CNRS av. du Colonel Roche Toulouse CX 4 France garcia@laas.fr João Manoel Gomes da Silva Jr. Departamento de Engenharia Elétrica Universidade Federal do Rio Grande do Sul av. Osvaldo Aranha Porto Alegre Rio Grande do Sul Brazil jmgomes@ece.ufrgs.br Isabelle Queinnec Laboratoire Analyse et Architecture des Systèmes (LAAS) CNRS av. du Colonel Roche Toulouse CX 4 France queinnec@laas.fr ISBN e-isbn DOI / Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 To Isabelle, Louise, Julie, Christiane and Pierre with my tenderness Sophie Tarbouriech To Lionel, Pauline, Sophie, Maria-Belen, Maria-Consuelo and Jaime in gratitude and affection Germain Garcia To Silvia and Gabriela. For their love, support and patience during the preparation of this book João Manoel Gomes da Silva Jr. To Sophie, Philippe, Nicole, Yvon. To those who are important for me Isabelle Queinnec

5 Foreword Actuator saturation presents the control system designer with a major challenge. It may cause loss in performance and in some cases unstable behaviour. Control engineers have long been aware of the need to guard against actuators hitting magnitude or rate limits and over the years a number of ad hoc methods have been developed to alleviate unwanted effects. In more recent years, researchers have sought to develop general systematic methods based on rigorous theory to give guarantees on stability and levels of performance. Valuable contributions have been made and there is now a wealth of information, technical results and synthesis techniques available in learned journals and conference proceedings. This book brings together many of these results, particularly those developed by the authors, into a single text. There are three main approaches to tackling saturation problems. In the first, optimal control is used to synthesise a controller which directly takes into account the saturation constraints. This approach is underpinned by a strong theoretical foundation and important results on stability. For example, global stability is only possible for systems whose poles all lie in the closed left-half complex plane; global asymptotic stability with finite-gain stability is not possible using linear control laws if the system contains repeated poles on the imaginary axis; and if a system contains poles in the open right-half complex plain, then the most that can be achieved is local stability, with local gain properties. Part II of this book is devoted to this direct approach and presents the relevant theory leading to optimisation techniques for synthesising controllers that meet performance specifications while respecting the saturation constraints. The optimisation techniques are presented in terms of parameterised Riccati equations and parameterised linear matrix inequalities (LMIs). The second main approach to handling saturation uses anti-windup to compensate an existing controller which has been designed to work well in the absence of saturation. Windup is the term used by early practising control engineers to describe the undesirable behaviour that occurred in integral (or PID) controllers for tracking problems when the control signal reached a physical limit. In these situations, the integrator state would wind up to a large value and the associated energy would then have to be dissipated once the control signal came out of saturation. The result was excessive overshoot and slow settling times. To overcome these problems, an additional compensator, an anti-windup compensator, was included which would modify vii

6 viii Foreword the control signal if saturation occurred but would otherwise do nothing. A number of ad hoc methodologies were developed including high gain anti-windup, reset anti-windup and Hanus anti-windup, all with varying degrees of success depending on the application. A great advantage of this approach, is that the anti-windup compensator works with the existing controller and only becomes active if saturation occurs. The early anti-windup compensators did, however, have a number of weaknesses: there were no stability or performance guarantees, they were mainly restricted to single-input single-output systems, and their design or tuning was imprecise. Over the past decade or so, these weaknesses have been removed, and there are now systematic approaches available for the design of anti-windup compensators, along with reliable algorithms for their synthesis using Riccati equations or LMIs. Part III of this book covers these important developments to the handling of saturation using anti-windup compensation. The third approach to saturation is to use model predictive control strategies. These are not addressed in this book but have been covered well in a variety of other books and so should not be viewed as an omission. The book is completed by two other parts: Part I sets the scene by defining the problems of interest and the Appendices provide useful information on technical topics in the areas of stability theory, robust control, Riccati equations and LMIs. This reference material in the Appendices helps to make the book self-contained. Previous books on this subject matter consist mainly of edited volumes, which are good at bringing the latest research results from key players to a wide readership, but which fall short in connecting the results in a coherent manner suitable for teaching purposes and then, hopefully, for use in real applications. This book has taken on this challenge. To make it work, the authors have had to restrict themselves to certain lines of research and methods. Given the vast array of results available there is no way that they could be exhaustive in their treatment. The authors therefore have been pragmatic and made the sensible decision to focus on what they know best, which is the research they have done themselves during the past ten years. By doing this, the work has a distinctive LAAS flavor; LAAS being the highly regarded Laboratoire d Analyse et d Architecture des Systèmes of CNRS, the French National Centre of Scientific Research, in Toulouse. LAAS is of course home for three of the four authors and the fourth is a frequent visitor to the laboratory, as have been many researchers on saturation control over the years. I congratulate the authors on taking on such a difficult challenge and bringing together so many interesting research results. The inclusion of anti-windup is pleasing to see as in my experience this represents a very practical approach to reducing the undesirable effects of saturation. I wish this book every possible success. Northumbria University Newcastle upon Tyne, UK Ian Postlethwaite

7 Preface This book addresses both the problems of stability analysis and stabilization of linear systems when subject to control saturation. It is well known that actuator saturation is present in practically all control systems. The magnitude of the signal that an actuator can deliver is usually limited by physical or safety constraints. This limitation can be easily identified in the most common devices used in the process industry, such as proportional valves, heating actuators, power amplifiers, and electromechanical actuators. Common examples of such limits are the deflection limits in aircraft actuators, the voltage limits in electrical actuators and the limits on flow volume or rate in hydraulic actuators. While such limits obviously restrict the performance achievable in the systems of which they are part, if these limits are not treated carefully and if the relevant controllers do not account for them appropriately, peculiar and pernicious behaviors may be observed. In particular, actuator saturation both in magnitude and rate have been implicated in various aircraft crashes [8] and the meltdown of the Chernobyl nuclear power station [330]. Hence, an in-depth understanding of the phenomena caused by saturation is key to solve and avoid problems in industrial control. Actually, the operation problems which can be induced by actuator saturations are well known in industry. This is the reason why, frequently, the actuators are oversized with respect to the energy necessary to control the system and the control gains are limited with respect to what could be accepted while preserving the stability. The engineering solutions intend to avoid, as much as possible, occurrence of saturation during operation. Then, the study of saturated systems is of a real interest for engineers in various domains due to its potential to provide, for example: Reduction of validation costs of control laws; Better use of actuator (and/or sensor) capacity; The possibility for control engineers to become involved in the design process at a much earlier stage in order to help choose actuators/sensors which have a reduced size, mass etc. Motivated by these practical issues, the actuator saturation problem has deservedly received the attention of many researchers since the second part of the twentieth ix

8 x Preface century. An excellent overview on saturated systems has been published in 1995 by Bernstein and Michel [25]. More recently, great advances were made on this subject. This is attested by the large number of publications on this topic that can be found in important conferences and journals of control systems and applied mathematics. Although an extensive literature in the form of articles can be found, only a small number of books devoted to this subject is available. Examples are the books edited by two of ourselves [346, 361] and the one edited by Kapila and Grigoriadis [208]. The former three books are presented in the form of a collection of chapters, written by different authors, with the particular aim of presenting new trends and specific advanced topics on the analysis and synthesis of control systems subject to actuator saturation. We believe now it is timely to write a textbook on the problem of control saturation, but given the diversity of approaches, it is certainly a challenge. To write a book addressing all the approaches and aspects related to this problem is a nearly impossible task. The main challenge then is to choose a set of specific topics or approaches and present them in a coherent and self-contained way. We think that this has been successfully achieved, for instance, in the book of Saberi et al. [310], where the main focus concerns the structural conditions for the solvability of many problems involving systems with input constraints. Other examples are the books [127, 188]. Roughly speaking, there are two approaches which one could adopt to avoid saturation problems in systems which are known to have actuator limits (i.e. the vast majority of practical systems). One approach, which we shall refer to as the one step approach is, as its name implies, an approach to controller design where a (possibly nonlinear) controller is designed from scratch. This controller then attempts to ensure that all nominal performance specifications are met while also handling the saturation constraints imposed by the actuators. Let us emphasize that Part II of this book is dedicated to present several solutions in the scope of this approach in both analysis and synthesis frameworks. An alternative approach to the above is to perform some separation in the controller such that one part is devoted to achieving nominal performance and the other part is devoted to control constraint handling. This is the approach taken in antiwindup compensation, which is the subject of Part III of this book. In anti-windup compensation, a linear controller which does not explicitly take into account the saturation constraints is first designed, usually using standard linear design tools. Then, after this controller has been designed, a so-called anti-windup compensator is designed to handle the saturation constraints. The anti-windup compensator is designed to ensure that stability is maintained (at least in some region near the origin) and that less performance degradation occurs than when no anti-windup is used. The primary goal of the present work is to provide basic concepts and tools that we consider fundamental for the analysis and synthesis of linear systems subject to actuator saturation. Additionally, we aim to present in a formal, but also didactic way, some recent developments on the subject. As pointed out above, we do not intend to cover exhaustively all the approaches and the results on the subject. Naturally, we mainly focus on topics directly related to our own research in the field. In this sense, we consider a state space approach and we focus on the problems of

9 Preface xi stability analysis and synthesis of stabilizing control laws both in local and global contexts. In particular, different ways of modeling the saturation term and the behavior of the nonlinear closed-loop system are explored. Also, different kinds of Lyapunov functions such as polyhedral, quadratic and Lure type, are considered in order to present different stability and stabilization conditions. Results considering uncertain systems and performance in the presence of saturation are also presented. Associated with the different theoretical results, we propose methods and algorithms for computing estimates of the basin of attraction and for designing control systems taking into account, a priori, the control bounds and the saturation possibility. These methods and algorithms are mainly based on the use of linear programming and convex optimization problems with LMI constraints. Thus, they can easily be implemented with widely known mathematical software packages. We believe we address some aspects that are not yet covered, or not discussed in-depth, in the books currently available on this subject. Finally, we hope this book will be a valuable reference for engineers (mainly aeronautical, electrical, mechanical and chemical), working on control applications, as well as graduate students and researchers interested in the development of new tools and theoretical results concerning systems with saturation. Due to its structure, it can also be used as a textbook in a specific course on saturation or as a complementary text in classical courses on control systems. The background required for the reader consists of basic concepts of linear and nonlinear systems, including linear algebra and matrix theory, differential equations, state space methods, and robust control theory, which can be found, for example, in classics like the books of Chen [64], Kailath [205], Khalil [215], Luenberger [252], O Reilly [277], Sontag [327], Vidyasagar [385], Colaneri et al. [74], Zhou et al. [414], Skogestad and Postlethwaite [323]. Some familiarity with linear programming, convex analysis and LMIs is also desirable. As additional references for all these topics we can recommend the books of Maciejowski [255], Boyd et al. [45] and Zaccarian and Teel [406]. Toulouse, France Toulouse, France Porto Alegre, Brazil Toulouse, France Sophie Tarbouriech Germain Garcia João Manoel Gomes da Silva Jr. Isabelle Queinnec

10 Contents Part I Generalities 1 Linear Systems Subject to Control Saturation Problems and Modeling Introduction The Open-Loop System TheClosed-LoopSystem The Region of Linearity TheRegionofAttraction ProblemsConsidered Asymptotic Stability Analysis AsymptoticStabilization External Stability Analysis ExternalStabilization The Anti-windup Problem Models for the Saturation Nonlinearity Polytopic Models Sector Nonlinearity Models Regions of Saturation Models Equilibrium Points Conclusion Part II Stability Analysis and Stabilization 2 Stability Analysis and Stabilization Polytopic Representation Approach Introduction Asymptotic Stability Analysis Ellipsoidal Sets of Stability Polytopic Approach I Polytopic Approach II xiii

11 xiv Contents Polytopic Approach III OptimizationProblems External Stability Amplitude Bounded Exogenous Signals Energy Bounded Exogenous Signals Stabilization State Feedback Stabilization Observer-Based Feedback Stabilization Dynamic Output Feedback Stabilization Global Stabilization Uncertain Systems Stability Analysis Extensions Discrete-TimeCase Ellipsoidal Regions of Asymptotic Stability Polyhedral Regions of Asymptotic Stability Extensions Conclusion Stability Analysis and Stabilization Sector Nonlinearity Model Approach Introduction Asymptotic Stability Analysis Quadratic Lyapunov Function Lure Lyapunov Function Computational Burden OptimizationProblems External Stability Amplitude Bounded Exogenous Signals Energy Bounded Exogenous Signals OptimizationIssues Stabilization State Feedback Stabilization Observer-Based Feedback Stabilization Dynamic Output Feedback Stabilization Uncertain Systems Discrete-TimeCase Extensions NestedSaturations Nested Nonlinearities Nonlinear and/or Hybrid Systems Conclusion Analysis via the Regions of Saturation Model Introduction Polyhedral Regions of Stability...186

12 Contents xv Positive Invariance Contractivity Compact Case Determination of Stability Regions Ellipsoidal Regions Test Condition Test Condition Discrete-TimeCase Positive Invariance Contractivity Compact Case Unbounded Sets of Stability Unbounded Polyhedra Unbounded Ellipsoidal Sets Conclusion Synthesis via a Parameterized ARE Approach or a Parameterized LMI Approach Introduction The Parameterized ARE Approach Preliminaries The Single-Input Case The Multi-variable Case ControlComputationandImplementation A Parameterized LMI Approach Multi-objective Control: Eigenvalues Placement and Guaranteed Cost Preliminaries The Single-Input Case The Multi-variable Case Disturbance Tolerance Preliminaries τ -ParameterizedControl ControlLawComputationandImplementation Nonlinear Bounded Control for Time-Delay Systems ProblemStatement Preliminaries Riccati Equation Approach LMI Approach Conclusion Part III Anti-windup 6 An Overview of Anti-windup Techniques Introduction Philosophy General Anti-windup Architecture ABitofHistory FormulationofProblems...274

13 xvi Contents 6.5 Regional (Local) Versus Global Strategies A Quick Overview in the Regional (Local) Context A Quick Overview in the Global Context Mismatch-Based Anti-windup Synthesis Conclusion Anti-windup Compensator Synthesis Introduction ProblemsSetup Direct Linear Anti-windup Model Recovery Anti-windup Direct Linear Anti-windup Design PreliminaryElements DLAW Schemes with Global Stability Guarantees DLAW Schemes with Regional Stability Guarantees SomeAlgorithms Model Recovery Anti-windup Design PreliminaryElementsontheArchitecture SomeAlgorithms Anti-windup Algorithms Summary SomeExtensions RateSaturation Sensor Saturations NestedSaturations Time-DelaySystems Anti-windup for Nonlinear Systems Conclusion Applications of Anti-windup Techniques Introduction Static Anti-windup Examples Dynamic Anti-windup Examples Toward More Complex Nonlinear Actuators Actuator with Position and Rate Saturations Dynamics Restricted Actuator Electro-HydraulicActuator Pseudo Anti-windup Strategies when the Dead-Zone Element is notaccessible Anti-windup Scheme Using a Fictitious Linear Element Observer-Based Anti-windup Scheme Conclusion Appendix A Some Concepts Related to Stability Theory A.1 Introduction A.2 Stability of Linear Autonomous Systems A.3 Stability for Nonlinear Systems A.3.1 Stability Definition (Lyapunov Stability)

14 Contents xvii A.3.2 Lyapunov s Stability Theorem A.3.3 The Invariance Principle A.3.4 RegionofAttraction A.3.5 Set of Equilibrium Points A.4 Application of Lyapunov Stability A.4.1 Absolute Stability A.4.2 Positive Real Function Concept A.4.3 CircleCriterion A.4.4 Popov Criterion A.4.5 Quadratic Stability Appendix B Quadratic Approach for Robust Control B.1 Introduction B.2 Uncertain Models and Quadratic Stability B.2.1 Uncertain Models B.2.2 Quadratic Stability B.3 Quadratic Stabilizability B.3.1 P-Uncertainty B.3.2 NB-Uncertainty B.4 GuaranteedCostControl B.4.1 P-Uncertainty B.4.2 NB-Uncertainty B.5 Pole Placement B.5.1 Pole Placement in a Disk B.5.2 Pole Placement in LMI Regions Appendix C Linear Matrix Inequalities (LMI) and Riccati Equations C.1 Introduction C.2 Algebraic Lyapunov Equation (ALE) C.2.1 Continuous Algebraic Lyapunov Equation (CALE) [414] C.2.2 Discrete Algebraic Lyapunov Equation (DALE) [414] C.3 Algebraic Riccati Equation (ARE) C.3.1 Continuous Algebraic Riccati Equation (CARE) [414] C.3.2 Discrete Algebraic Riccati Equation (DARE) [414] C.4 Linear Matrix Inequalities (LMI) C.5 Schur s Complement C.6 Bilinear Matrix Inequalities (BMI) C.6.1 Eigenvalues and Generalized Eigenvalues Problems C.7 The Elimination Lemma and the S-Procedure C.7.1 The Elimination Lemma [45] C.7.2 The S-Procedure [45] C.8 Ellipsoids and Polyhedral Sets C.8.1 Ellipsoid and Invariant Ellipsoid [291, 346] C.8.2 Convex Polyhedron and Invariant Polyhedron [135, 136, 344]...406

15 xviii Contents C.8.3 Maximum Volume Ellipsoid Contained in a Symmetric Polytope C.8.4 Smallest Volume Ellipsoid Containing a Symmetric Polytope References Index...429

16 Notation Main notation 1 R The set of real numbers C The set of complex numbers N The set of natural numbers R + The set of nonnegative real numbers R n The n-dimension real space R n m The set of real matrices of dimensions n m A The transpose of matrix A A (i) The ith row of matrix A A (i,j) The element of the ith row and jth column of matrix A A B>0 Means that matrix A B is positive definite A B 0 Means that matrix A B is semi-positive definite λ max (A) The maximal eigenvalue of matrix A λ min (A) The minimal eigenvalue of matrix A Re(λ i (A)) The real part of the eigenvalue λ i of matrix A σ(a) The spectrum of matrix A Diag(A 1,A 2,...,A p ) Denotes the block-diagonal matrix for which the block-diagonal matrices are the A i, j = 1,...,p The symbol stands for symmetric blocks in the expression of a matrix det(a) The determinant of matrix A trace(a) The trace of matrix A rank(a) The rank of matrix A Ker(A) The kernel of matrix A, that is, the set Ker(A) ={x R n ; Ax = 0} A #, A R m n, rank(a) = m The pseudo-inverse of matrix A such that AA # = I m I n The identity matrix of dimensions n n.the subscript may be removed when there is no ambiguity 1 Throughout the book, the notation used is quite standard. The main elements are described below. xix

17 xx Notation 1 n The vector in R n defined by 1 n =[1...1] 0 Denotes the null scalar or the null matrix of appropriate dimensions diag(x) The n-order diagonal matrix generated by the components of x R n. For example for n = 2: diag(x) = [ x (1) 0 ] 0 x (2) x y with x,y R n Means that x (i) y (i) 0, i = 1,...,n Co{x j,j= 1,...,r} x The convex hull defined by the vertices x j The vector composed by the absolute values of the components of x. The same notation will be used for a matrix A, i.e., A x The Euclidean norm of vector x, i.e., x = x x x 2 The L 2 -norm of vector x, i.e., x 2 = 0 x xdt sign(x) The sign function of x R sat(z), z R m The saturation function of the vector z defined by sat(z (i) ) = sign(z (i) ) min(u 0(i), z (i) ), i = 1,...,m x (j) Denotes the jth time-derivative of x n! Denotes the factorial function, i.e. n!=n(n 1)...1, n N Denotes the Kronecker product Particular Notation Vectors and matrices are defined with brackets as follows For x R n one gets x = x (1).. x (n) For A R n m one gets A (1,1)... A (1,m) A =... A (n,1)... A (n,m) S(R,ρ 1,ρ 2 ), with R R m n, ρ 1,ρ 2 R m, is the polyhedral set defined by S(R,ρ 1,ρ 2 ) = { x R n } ; ρ 2 Rx ρ 1 S( R,ρ), with R R m n, ρ R m +, is the symmetrical polyhedral set defined by S ( R,ρ ) = { x R n ; ρ Rx ρ } S(R,ρ), with R R m n, ρ R m, is the polyhedral set defined by S(R,ρ) = { x R n ; Rx ρ }

18 Notation xxi E(P, η), with P = P > 0, P R n n, and η>0, is the ellipsoid defined by E(P, η) = { x R n ; x Px η 1} For two sets S 1 and S 2, S 1 \S 2 denotes the set S 1 deprived of S 2 For a set S, S and int S denote the boundary and the interior of S, respectively For a linear operator H, H X is a restriction of H on the subspace X Acronyms ALE Algebraic Lyapunov Equation ARE Algebraic Riccati Equation AW Anti-windup BIBO Bounded Input Bounded Output BMI Bilinear Matrix Inequality CALE Continuous Algebraic Lyapunov Equation CARE Continuous Algebraic Riccati Equation DALE Discrete Algebraic Lyapunov Equation DARE Discrete Algebraic Riccati Equation DLAW Direct Linear Anti-windup EVP EigenValue Problem GAS Global Asymptotic Stability GES Global Exponential Stability GEVP Generalized EigenValue Problem HPPD Homogeneous Polynomial Parameter-Dependent IMC Internal Model Control LMI Linear Matrix Inequality MIMO Multiple Input Multiple Output MPC Model Predictive Control MRAW Model Recovery Anti-windup SISO Single-Input Single-Output PID Proportional Integral Derivative RAS Region of Asymptotic Stability RES Regional Exponential Stability SDP Semi-definite Programming