Mathematical Methods for Robust and Nonlinear Control: EPSRC Summer School (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma, M. Morari

367

Matthew C. Turner, Declan G. Bates (Eds.)

Mathematical Methods for Robust and Nonlinear Control EPSRC Summer School

ABC

Series Advisory Board F. Allgöwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis

Editors Matthew C. Turner Department of Engineering University of Leicester University Road LE1 7RH Leicester UK Email: [email protected]

Declan G. Bates Department of Engineering University of Leicester University Road LE1 7RH Leicester UK Email: [email protected]

Library of Congress Control Number: 2007932607 ISSN print edition: 0170-8643 ISSN electronic edition: 1610-7411 ISBN-10 1-84800-024-3 Springer Berlin Heidelberg New York ISBN-13 978-1-84800-024-7 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007  MATLAB and Simulink are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, U.S.A. http://www.mathworks.com and Scilab is a trademark of INRIA, Domaine de c 1989Voluceau, Rocquencourt - B.P. 105, 78153 Le Chesnay cedex, France. www.scilab.org Copyright  2007. INRIA. The use of general descriptive names, 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 protective laws and regulations and therefore free for general use. Typesetting: by the authors and SPS using a Springer LATEX macro package Printed on acid-free paper

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Preface

The underlying theory on which much modern robust and nonlinear control is based can often be difficult for the student to grasp. In particular, the mathematical aspects can be problematic for students from a standard engineering background. The EPSRC sponsored Summer School which was held in Leicester in September 2006 attempted to “fill the gap” in students’ appreciation the theory relevant to several important areas of control. This book is a collection of lecture notes which were presented at that workshop and consists of, broadly, two parts. The first nine chapters are devoted to the theory behind several areas of robust and nonlinear control and are aimed at introducing fundamental concepts to the reader. The last six chapters contain detailed case studies which aim to demonstrate the use and effectiveness of these modern techniques in real engineering applications. It is hoped that this book will provide a useful introduction to many of the more common robust and nonlinear control techniques and serve as a valuable reference for the more adept practitioner.

Leicester, May 2007

Matthew C. Turner Declan G. Bates

Contents

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV

Part I Theory of Robust and Nonlinear Control 1 H∞ Control Design Declan G. Bates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Design Specifications and Fundamental Trade-offs . . . . . . . . . . . . . . . . . . 1.2.1 Linear Design Specifications for Robust Control Systems . . . . . 1.2.2 Frequency Domain Design Specifications and Fundamental Trade-offs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mixed-sensitivity H∞ Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Formulating the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Weighting Function Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Solution of the H∞ Control Problem . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Design Example: Control Law Design for the Bell 205 Helicopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 H∞ Loop-shaping Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Fundamental Trade-offs in Terms of L . . . . . . . . . . . . . . . . . . . . . 1.4.2 The H∞ Loop-shaping Design Procedure . . . . . . . . . . . . . . . . . . 1.4.3 Advantages of H∞ Loop-shaping . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Design Example: Control Law Design for the Harrier V/STOL Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Structural Methods for Linear Systems: An Introduction Nicos Karcanias and Efstathios Milonidis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Classification of System Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 State Space Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Polynomial Models: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Transfer Function Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Background on Polynomial Matrices and Matrix Pencils . . . . . . . . . . . . .

3 3 5 6 6 8 8 11 11 13 24 24 26 29 33 43 47 48 50 50 51 51 52

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2.3.1 Matrix Divisors and Minimal Bases . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Strict Equivalence Invariants of Matrix Pencils [13] . . . . . . . . . . 2.4 Dynamics, Stability, Controllability and Observability . . . . . . . . . . . . . . . 2.4.1 Solution of State Space Equations . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Internal-External and Total Stability . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Controllability and Observability . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 System Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Poles and Zeros of State Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Eigenvalues, Eigenvectors and Free Rectilinear Motions . . . . . . 2.5.2 Forced Rectilinear Motions and Frequency Transmission . . . . . 2.5.3 Frequency Transmission Blocking and State Space Zeros . . . . . 2.5.4 Right Regular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Properties of Zero Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 Right Singular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.7 Frequency Transmission Blocking for Infinite Frequencies . . . . 2.5.8 Zero Structure and System Transformations . . . . . . . . . . . . . . . . 2.5.9 The Zero Pencil of Strictly Proper System . . . . . . . . . . . . . . . . . . 2.5.10 Decoupling Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Poles and Zeros of Transfer Function Models . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Dynamic Characterisation of Transfer Function Poles and Zeros 2.6.2 Smith–McMillan Form Characterisation of Poles and Zeros . . . 2.6.3 Matrix Fraction Descriptions, and Poles and Zeros . . . . . . . . . . . 2.6.4 Infinite Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Smith–McMillan Form at Infinity: Infinite Poles and Zeros . . . 2.6.6 Impulsive Dynamics and Properties of Infinite Poles and Zeros [57] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 System Algebraic Functions and Generalised Nyquist and Root Locus . 2.7.1 Characteristic Gain, Frequency Functions . . . . . . . . . . . . . . . . . . 2.7.2 Poles and Zeros of the System Algebraic Functions . . . . . . . . . . 2.7.3 Root Locus and the Output Zeroing Problem . . . . . . . . . . . . . . . 2.8 The Feedback Configuration and Structural Properties . . . . . . . . . . . . . . . 2.8.1 Structural Properties of the Feedback Configuration . . . . . . . . . 2.8.2 Closed-loop Performance and the Return Ratio Difference and Sensitivity Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Determinantal Assignment Problems: Exterior Algebra-Algebraic Geometry Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Determinantal Assignment Problems . . . . . . . . . . . . . . . . . . . . . . 2.9.2 The General Determinantal Assignment Problem . . . . . . . . . . . . 2.9.3 Grassmann- Plucker Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Invariants and Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B List of Symbols, Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 55 55 55 57 58 60 61 61 62 63 65 65 66 67 69 70 71 72 72 73 74 74 75 76 77 77 78 79 80 80 83 84 85 88 89 92 92 94 94

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3 Modelling and Model Reduction—State-Space Truncation† David J. N. Limebeer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.2 State-space Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.2.1 The Truncation Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.2.2 Singular Perturbation Approximation . . . . . . . . . . . . . . . . . . . . . . 104 Main Points of the Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.3 Balanced Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.3.1 Model Reduction Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.3.2 Balanced Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Main Points of the Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.4 Balanced Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.4.2 Error Bound for “one-step” Truncation . . . . . . . . . . . . . . . . . . . . 112 3.4.3 The Error Bound for Balanced Truncation . . . . . . . . . . . . . . . . . . 113 Tightness of the Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Frequency Dependence of the Error . . . . . . . . . . . . . . . . . . . . . . . 115 Main Points of the Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.5 Balanced Singular Perturbation Approximation . . . . . . . . . . . . . . . . . . . . . 116 Main Point of the Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4 Linear Matrix Inequalities in Control Guido Herrmann, Matthew C. Turner and Ian Postlethwaite . . . . . . . . . . . . . . . 123 4.1 Introduction to LMI Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.1.1 Fundamental LMI Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1.2 Systems of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.1.3 Types of LMI Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 LMI Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Linear Objective Minimization Problems . . . . . . . . . . . . . . . . . . 127 Generalized Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2 Tricks in LMI Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.2.1 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.2.2 Congruence Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.2.3 Schur Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2.4 The S-procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.2.5 The Projection Lemma and Finsler’s Lemma . . . . . . . . . . . . . . . 132 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.3.1 Lyapunov Stability for Continuous-time Systems . . . . . . . . . . . . 134 4.3.2 L2 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.3.3 Lyapunov Stability for Discrete-time Systems . . . . . . . . . . . . . . 135 4.3.4 l2 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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4.3.5 Sector Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 A Slightly More Detailed Example . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 138 141 141

5 Anti-windup Compensation and the Control of Input-constrained Systems Matthew C. Turner, Guido Herrmann and Ian Postlethwaite . . . . . . . . . . . . . . . . 143 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.1.1 Input Constraints in Control Systems . . . . . . . . . . . . . . . . . . . . . . 143 5.1.2 Constrained System Description . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.1.3 Constrained Control and Anti-windup . . . . . . . . . . . . . . . . . . . . . 146 5.2 Problems Due to Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.2.1 Clues From Classical Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.3 Stability of Systems with Input Saturation . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.3.1 Definitions of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.3.2 Saturation Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 An Equivalent Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Sector Bounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.3.3 The Multivariable Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . . 157 5.4 Anti-windup Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.5 An Anti-windup Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.5.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.5.2 Full Order Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.6 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.6.1 Simple 2nd-order Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 The Nominal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 The Constrained System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 The Constrained System and Anti-windup . . . . . . . . . . . . . . . . . . 166 5.6.2 Lockheed Martin F104 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 168 The Nominal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.6.3 The Constrained System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.6.4 The Constrained System and Anti-windup . . . . . . . . . . . . . . . . . . 169 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.7.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6 Output Feedback H∞ Loop-shaping Controller Synthesis Emmanuel Prempain and Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.2.1 LMI Formulation of Performance Specifications . . . . . . . . . . . . 176 H∞ Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 H2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

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6.2.2 Normalized Left Coprime Factorization for LTI Systems . . . . . 177 H∞ Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 H∞ Loop-shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.4.1 LMI Formulation of the H∞ Loop-shaping Controller Synthesis 181 6.4.2 Controller Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.4.3 Design Procedure for a Static H∞ Loop-shaping Controller . . . 184 6.4.4 Static H∞ Flight Control System Design for the Bell 205 Helicopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Plant Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Static H∞ Helicopter Controller Design . . . . . . . . . . . . . . . . . . . 184 6.5 H∞ Loop-shaping for Polytopic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.5.1 Left Coprime Factors for Polytopic Systems . . . . . . . . . . . . . . . . 187 6.6 LMI Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.6.1 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.3 6.4

7 Stability and Asymptotic Behaviour of Nonlinear Systems: An Introduction Hartmut Logemann and Eugene P. Ryan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.2 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.3 Background Concepts in Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.4 Initial-value Problems: Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . 199 7.4.1 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.4.2 Autonomous Differential Inclusions . . . . . . . . . . . . . . . . . . . . . . . 201 7.4.3 ω -limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.5 Barb˘alat’s Lemma, LaSalle’s Invariance Principle, and Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.6 Generalizations of Barb˘alat’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.7 Nonautonomous Ordinary Differential equations . . . . . . . . . . . . . . . . . . . . 211 7.8 Autonomous Differential Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8 Sliding-mode Observers† Christopher Edwards, Sarah K. Spurgeon, Chee P. Tan and Nitin Patel . . . . . . . 221 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.2 A Discontinuous Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.3 Observers with Linear and Discontinuous Injection . . . . . . . . . . . . . . . . . 225 ˙ Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.4 The Walcott and Zak 8.4.1 Synthesizing the Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.5 A Convex Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.6 A Case Study: Road Tyre Friction Estimation . . . . . . . . . . . . . . . . . . . . . . 232 8.6.1 Tyre/Road Friction and Vehicle Modelling . . . . . . . . . . . . . . . . . 232 8.6.2 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

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8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

238 240 240 240

9 Sliding-mode Control in Systems with Output Time Delay Alan S.I. Zinober, G. Liu and Yuri B. Shtessel . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.2 Sliding-mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.2.1 Regulator System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.2.2 Model-Following Control System . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.2.3 Sliding-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.2.4 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 9.2.5 Second-order Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.3 Application Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 9.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.3.2 Pad´e Approximations and Time Delay Systems . . . . . . . . . . . . . 252 9.3.3 System Centre Method and Sliding-mode Control . . . . . . . . . . . 254 9.3.4 Numerical Example and Simulations . . . . . . . . . . . . . . . . . . . . . . 255 9.3.5 Feedback by yˆ and Describing Function . . . . . . . . . . . . . . . . . . . . 257 9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Part II Applications of Robust and Nonlinear Control 10 Control Engineering and Systems Biology Burton W. Andrews and Pablo A. Iglesias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 10.2 Negative Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10.2.1 Negative Feedback: Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10.2.2 Negative Feedback: Sensitivity and Robustness . . . . . . . . . . . . . 273 10.3 Positive Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 10.3.1 Positive Feedback: Amplification . . . . . . . . . . . . . . . . . . . . . . . . . 277 10.3.2 Positive Feedback: Switching and Memory . . . . . . . . . . . . . . . . . 278 10.3.3 Positive Feedback: Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 11 Robust Control of a Distillation Column Da-Wei Gu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 11.2 Dynamic Model of the Distillation Column . . . . . . . . . . . . . . . . . . . . . . . . 290 11.3 Uncertainty Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 11.4 Closed-loop System Performance Specifications . . . . . . . . . . . . . . . . . . . . 296 11.5 Open-loop and Closed-loop System Interconnections . . . . . . . . . . . . . . . . 300

Contents

11.6

Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Loop Shaping Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 µ -synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Nonlinear System Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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301 301 307 314 318 319

12 Robust Control of a Hard-disk Drive Da-Wei Gu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 12.1 Hard Disk Drive Servo System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 12.2 Derivation of Uncertainty Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.3 Closed-loop System Design Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 340 12.4 System Interconnections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 12.5 Controller Design in Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 12.5.1 µ -design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 12.5.2 H∞ Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 12.5.3 H∞ Loop-shaping Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 12.6 Comparison of Designed Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 12.7 Controller Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 12.8 Design of Discrete-time Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 12.9 Nonlinear System Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 12.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 13 Modelling and Control of Railway Vehicle Suspensions Argyrios C. Zolotas and Roger M. Goodall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 13.1 Overview of Railway Vehicle Dynamics and Control . . . . . . . . . . . . . . . . 373 13.1.1 Railway Vehicles: Conventional Configuration . . . . . . . . . . . . . . 373 13.1.2 Suspension Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 374 13.1.3 Modelling of Suspensions (for Applying Control) . . . . . . . . . . . 375 13.1.4 Control Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Tilting Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Active Secondary Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Active Primary Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 13.2 Case Study: Control of Secondary Suspensions - Tilting Trains . . . . . . . 381 13.2.1 Historical Facts on Tilt Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 13.2.2 Tilting Vehicle Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 13.2.3 Tilt Control Requirements and Assessment Approach . . . . . . . . 385 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Tilt Control Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Track Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 13.2.4 Conventional Tilt Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Classical Nulling Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . 387 Command-driven with Precedence Control . . . . . . . . . . . . . . . . . 390 13.2.5 Nulling-type Tilt via Robust Control Techniques . . . . . . . . . . . . 394

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LQG/LTR Nulling-type Tilt Control . . . . . . . . . . . . . . . . . . . . . . . 13.2.6 Multi-objective H∞ /H2 Nulling-type Control via LMIs . . . . . . . 13.2.7 Case Study Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Appendix A- Tilting Train Parameter Values and Notation . . . . . . . . . . . . 13.4 Appendix B- H∞ Based Controllers: Preliminaries . . . . . . . . . . . . . . . . . . 13.4.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Frequency Domain Spaces and Norms . . . . . . . . . . . . . . . . . . . • Linear Fractional Transformations . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

394 399 406 407 407 407 408 409 411

14 Case Study on Anti-windup Compensation - Micro-actuator Control in a Hard-disk Drive Guido Herrmann, Matthew C. Turner and Ian Postlethwaite . . . . . . . . . . . . . . . 413 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 14.2 The Micro-actuator Control loop and Windup Problems . . . . . . . . . . . . . 415 14.3 Anti-windup Compensation for Discrete Linear Control Systems . . . . . . 419 14.4 Anti-windup Compensation for the Micro-actuator . . . . . . . . . . . . . . . . . . 424 14.5 The Micro-actuator Control Loop as Part of a Hard-disk-drive Servo-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 15 Enhancing Immune System Response Through Optimal Control Robert F. Harrison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 15.2 Lotka-Volterra Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 15.2.1 Analysis of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 15.3 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 15.3.1 Linear, Time-varying Quadratic Optimal Control . . . . . . . . . . . . 436 15.4 Immune System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 15.4.1 Optimal Enhancement of the Immune Response . . . . . . . . . . . . . 440 15.4.2 Some Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 15.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

List of Contributors

Declan G. Bates University of Leicester University Road Leicester, LE1 7RH [email protected]

Matthew C. Turner University of Leicester University Road Leicester, LE1 7RH [email protected]

Nicos Karcanias City University Northampton Square London, EC1V 0HB [email protected]

Ian Postlethwaite University of Leicester University Road Leicester, LE1 7RH [email protected]

Efstathios Milonidis City University Northampton Square London, EC1V 0HB [email protected]

Emmanuel Prempain University of Leicester University Road Leicester, LE1 7RH [email protected]

David J.N. Limebeer Imperial College London South Kensington Campus London, SW7 2AZ [email protected]

Eugene P. Ryan Mathematical Sciences University of Bath Bath, BA2 7AY [email protected]

Guido Herrmann University of Leicester University Road Leicester, LE1 7RH [email protected]

Hartmut Logemann Mathematical Sciences University of Bath Bath, BA2 7AY [email protected]

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List of Contributors

Christopher Edwards University of Leicester University Road Leicester, LE1 7RH [email protected] Sarah K. Spurgeon University of Leicester University Road Leicester, LE1 7RH [email protected] Nitin Patel University of Leicester University Road Leicester, LE1 7RH [email protected] Chee P. Tan Monash University Malaysia 46150 Petaling Jaya Malaysia [email protected] Alan S.I. Zinober University of Sheffield Mappin Street Sheffield, S10 2TN [email protected] G. Liu University of Sheffield Mappin Street Sheffield, S10 2TN [email protected] Yuri B. Shtessel The University of Alabama in Huntsville Huntsville, AL 35899 USA [email protected]

Pablo A. Iglesias The Johns Hopkins University Baltimore, MD 21218 USA [email protected] Burton W. Andrews The Johns Hopkins University Baltimore, MD 21218 USA [email protected] Da-Wei Gu University of Leicester University Road Leicester, LE1 7RH [email protected] Argyrios C. Zolotas Loughborough University Leicestershire Loughborough, LE11 2TJ [email protected] Roger M. Goodall Loughborough University Leicestershire Loughborough, LE11 2TJ [email protected] Robert F. Harrison The University of Sheffield Mappin Street Sheffield, S1 3JD [email protected]

1 H∞ Control Design Declan G. Bates

Summary. This chapter summarises the basic ideas behind H∞ control theory.

1.1 Introduction Classical design and analysis techniques, many of which date back to the 1950s, are still widely used in industry for the design and analysis of automatic control systems. The con­ tinued success and popularity of these techniques is particularly impressive considering the radical advances in systems and electronics made over this period. Clearly, an understand­ ing of both the advantages and limitations of these methods is necessary in order to properly evaluate the more modern techniques for the design and analysis of robust control systems de­ scribed in this chapter. We therefore begin by briefly summarising the philosophy, advantages and limitations of the classical approach to control system design and analysis. Philosophy •

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Classical control system design is based on the philosophy of successive loop closure. This means that even when more than one plant output is required to be controlled simul­ taneously, single-input-single-output (SISO) design techniques such as root-locus, Bode and Nyquist plots are used in a sequential design process, with individual controllers for each feedback loop being designed one at a time. As will be clear from the above, classical control systems are highly structured. Indeed, the actual structure of the control system may be completely determined in advance, with the control law designer only left to determine the values of particular gains and filters. Although the reasons for adopting a particular controller structure are generally quite com­ plex, at least three important factors can be identified. The first is the limitations imposed by the various hardware components required to implement the controller on the actual system. The second is the natural tendency to exploit previous design experience and ‘lessons learned’ by basing each successive generation of control systems on the previous one, with only those modifications which are shown to be absolutely necessary being im­ plemented. The third reason is to assist in the process of certification of the overall system it is easier to demonstrate the correct operation (in normal and failure cases ) of the con­ trol system in the case of highly structured systems, where every element of the control system has a high level of visibility and functionality. Classical design and analysis techniques are predominantly linear, and are based around linearised models of the plant generated at particular operating conditions. Modifications

M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 3-46, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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•

D.G. Bates to the control law to cope with non-linear effects such as actuator rate and magnitude saturation are usually added after the initial linear design is complete, based on non linear closed loop simulations. Extension of the control law to cover the full non linear operating envelope is usually achieved by scheduling the gains of the controller as functions of one or more parameters such as speed, dynamic pressure or angle of attack. Finally, analysis of the control system is usually carried out using linear stability metrics such as Gain and Phase margins, before final checks are conducted using non linear simulations. Classical control techniques make use of predominately frequency domain tools (Bode and Nyquist plots, Root locus design, Gain and Phase margins, etc.) to achieve designs which satisfy closed loop specifications which are themselves often given in the time as well as the frequency domain. Despite this apparent gap between specifications and design tools, there is clearly a consensus that frequency domain tools offer a level of design transparency and intuition that cannot be matched by purely time domain techniques. This fact, together with the questionable robustness properties of some time-domain ‘optimal’ control techniques, has probably been responsible for the relative lack of success of these optimal techniques in real applications to date.

A key theme of this chapter is that the robust controller design and analysis methods it describes are direct and natural developments of the classical frequency domain control techniques still in common use in industry today. This fact is often obscured, for two main reasons. The first is largely a historical one. Before the advent of robust control theory in the early 1980s, much attention and publicity was given to so-called ‘optimal’ or modern control theory and its associated methods - state-space equations and linear quadratic optimisation for controller synthesis, [1]. These concepts were thought of, and taught as, time-domain methods. This fact, together with the introduction of ‘optimisation’ as a tool for controller design, represented a complete break from the traditional classical methods. Unfortunately, however, the fundamental problem of system uncertainty was largely ignored by modern control theorists. The not altogether surprising result was that supposedly ‘optimal’ controllers were often found to give poor performance, or even instability, in real-world applications because of high sensitivity to modelling errors. The well known and well-publicised 60-degree guaranteed phase margins of LQR full state feedback, for example, were found to disappear when Kalman filters were used to estimate unmeasurable states in the LQG control method [2]. Although certain industries have successfully applied modern control methods to control problems through careful attention to robustness margins [3], modern control techniques are still not widely used. The resulting lack of successful applications of modern control techniques has seriously hampered their acceptance by industry, with the result that many industrial control system designers today still rely on classical methods for both analysis and design. Robust control theory was born out of a belated realisation of the flaws inherent in the time domain modern control methods. In particular, the lack of a sensible way of handling system uncertainty, and the loss of design transparency associated with time-domain methods were the starting point for the development of H∞ control, [4]. Thus, robust control should properly be seen as a return to the frequency domain philosophy of classical control, with the primary development being to allow for truly multivariable design in the presence of system and signal uncertainty. The second issue obscuring the link between robust and classical control techniques is the relative complexity of the mathematical machinery required to solve H∞ optimisation problems. The generation of efficient and elegant solutions to these challenging problems has been the subject of intensive research over the past twenty years, and therefore it is not surprising that much of the robust control literature seems from the outside to have been written by, and for,

H

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Control Design

5

system theorists and mathematicians, rather than practising control engineers. With many of these computational problems now settled, however, more attention is being devoted to the development and application of robust control methods as practical engineering tools, rather than as mathematical concepts, [38]. The recent significant increase in successful implementations of robust controllers in a wide variety of applications strongly suggests that these techniques are reaching maturity and have finally begun to achieve widespread acceptance in industry.

1.2 Design Specifications and Fundamental Trade-Offs In this section we will describe the basic set of design specifications that must be satisfied by robust control systems. Since many of these specifications are to a greater or lesser extent contradictory, we will also analyse the various fundamental trade-offs which exist in all multivariable control systems. Note that the following development is based on the assumption that many of the most important specifications for control systems can be addressed in a linear controller synthesis framework. In this approach, any additional non-linear specifications (e.g., limiting of safety critical signals, mode-switching and scheduling logic, anti-windup for actuator saturation, etc.) are addressed via conditioning and scheduling schemes, after the initial linear controller (or set of controllers) has been designed. This approach follows exactly the approach used in classical control system design, i.e., linear controller synthesis followed by the addition of scheduling, anti-windup and safety-limiting schemes. The fundamental difference between the two approaches lies in the additional complexity of the nonlinear schemes needed in the case of multivariable linear controllers. In particular, scheduling and anti-windup schemes for classical SISO controllers are relatively easy to design and implement - in the case of multivariable (especially dynamic multivariable) linear controllers these two tasks become more difficult. However, significant progress has been made in these areas in recent years, with the result that systematic and powerful techniques are now available to solve these problems. An alternative approach to that outlined above is to employ non-linear controller design methods which would allow non-linear controllers to be synthesised directly, and indeed much attention has been devoted to developing such techniques in recent years. Two of the most important of these techniques are the so-called Sliding Mode, [5], and Non-Linear Dynamic Inversion (NDI), [6,7], approaches. In addition, non-linear versions of the H∞ controller design techniques discussed in this text have also been developed, [8, 9] and are starting to be applied to real applications. Although all these methods show great promise, they also have some inherent disadvantages. NDI techniques, for example, provide no robustness guarantees, and it is not even clear how to ‘tune’ an NDI design to improve its robustness (although see [10] for some significant progress in this direction). Indeed, a fundamental problem with all non-linear controller synthesis methods is that corresponding non-linear robustness analysis methods are in general much less well developed. Sliding Mode and non-linear H∞ techniques do offer sensible robustness guarantees, but employ mathematical techniques of a complexity far beyond what is generally used in industry. Finally, it can be argued that attempts to address simultaneously linear and non-linear aspects of the controller synthesis process actually lead to additional complexity and reduced visibility in the final design - one step at a time can be faster and simpler. The above discussion motivates our approach to the problem of synthesising controllers as set out in the following sections.

6

D.G. Bates

1.2.1 Linear Design Specifications for Robust Control Systems We summarise the most important linear design specifications for a robust control system below. Note that in the following we consider the design of a single linear controller at a particular point in the plant’s operating envelope. 1. Robust stability to unstructured system uncertainty: the linear controller must provide adequate levels of stability against unstructured uncertainty (as defined by some robustness measure) for the nominal linearised model of the plant. 2. Robust stability to unstructured and structured system uncertainty: the linear controller must provide adequate levels of stability against unstructured uncertainty (as defined by some robustness measure) for all linear models of the plant in some set defined by a structured uncertainty model. 3. Nominal performance: the linear controller must provide adequate attenuation of disturbance signals (at the plant sensors and actuators), adequate attenaution of measurement noise, and adequate tracking and decoupling of reference commands, for the nominal linearised model of the plant. 4. Robust performance: the linear controller must provide adequate attenuation of disturbance signals (at the plant sensors and actuators), adequate attenaution of measurement noise, and adequate tracking and decoupling of reference commands, for all linear models of the plant in some set defined by a structured uncertainty model. Items 3 and 4 above are, in effect, specifications on signal uncertainty (recall that good tracking of reference demands amounts to attenuating the effect of the uncertain command signal on the tracking error signal). The specifications must be satisfied nominally (in the absence of system uncertainty) and robustly (in the presence of system uncertainty). For our current purposes we will adopt the classical approach and consider adequate performance with respect to reference demands to imply well damped responses with appropriate rise/settling times, zero steady-state error and minimal coupling into other controller variables. In the following subsection , we will show how each of the design specifications given above can be formulated in terms of bounds on the minimum or maximum singular values of various open and closed-loop transfer function matrices.

1.2.2 Frequency Domain Design Specifications and Fundamental Trade-Offs Consider the multivariable feedback control system shown in Figure 1.1. Assuming that the system is internally stable, the following fundamental equations hold: y = TO r + SO GdI + SO dO − TO m,

u = KSO r − KSO dO + SI dI − KSO m

(1.1) (1.2)

Thus, nominal specifications on signal uncertainty (i.e., specifications in the absence of system uncertainty) can be defined in terms of the maximum singular values of the input and output sensitivity and complementary sensitivity functions SI , SO , TI and TO . In particular, we can formulate the following design objectives: For attenuation of output disturbance signals at the plant output, σ (SO ) should be small. For attenuation of output disturbance signals at the plant input, σ (KSO ) should be small.

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Control Design

dI

r

�� �

� K

7

dO

�� � � G u

� ��

�y ��

m

Fig. 1.1. A typical feedback control system For attenuation of input disturbance signals at the plant input, σ (SI ) should be small. For attenuation of input disturbance signals at the plant output, σ (SO G) should be small. For attenuation of measurement noise signals at the plant output, σ (TO ) should be small. For attenuation of measurement noise signals at the plant input, σ (KSO ) should be small. For good reference tracking σ (TO ) and σ (TO ) should be ≈ 1. Thus, good reference track­ ing is equivalent to attenuation of output disturbances at the plant output, since forcing SO to 0 forces TO to the identity. For avoidance of large control signals due to reference demands (and hence actuator sat­ uration), σ (KSO ) should be small. Requirements for robust stability in the presence of unstructured system uncertainty can also be stated in terms of the maximum singular values of these sensitivity functions: For robust stability to input multiplicative uncertainty, σ (TI ) should be small. For robust stability to output multiplicative uncertainty, σ (TO ) should be small. For robust stability to additive uncertainty, σ (KSO ) should be small. Now, since SI + TI = I

(1.3)

SO + TO = I

(1.4)

it is obvious that all of the above requirements cannot be satisfied simultaneously. Feedback controller design is therefore a problem of managing trade-offs between conflicting design objectives. In practice, this problem is made easier by the fact that different design objectives are often important over different frequency ranges. For example, disturbance attenuation is generally required at low frequencies, while the attenuation of measurement noise becomes important at high frequencies. Finally we note that the constraints outlined above do not constitute a complete list of the limitations on the achievable performance of a feedback control system. In particular, right-half-plane zeros in the plant, and constraints on control signal rates and amplitudes can seriously limit closed-loop performance - see [38] for a full discussion. As we shall see in the next section, the so-called mixed sensitivity H∞ controller design method provides a powerful technique for synthesising controllers which optimally satisfy the various design objectives described above. Before examining this method in detail, we note at this stage that the singular value design objectives given above correspond to the case of unstructured plant uncertainty. For control design problems where the plant uncertainty is

8

D.G. Bates

structured (e.g., diagonal), the structured singular value µ , and its associated synthesis techniques, [38], may be used to obtain less conservative designs - see [19] for further details.

1.3 Mixed-Sensitivity H∞ Controller Design In this section we describe the mixed sensitivity H∞ controller design method, and illustrate its application via a control law design for a model of the Canadian NRC’s Bell 205 variable stability helicopter.

1.3.1 Formulating the Problem Consider the general feedback configuration shown in Figure 1.2. The various signals in the figure are: u the control variables, v the signals which the controller has access to, w the exogenous signals such as disturbances, reference demands, measurement noise, etc., and z the

� �

w

z� P

u K

v

�

Fig. 1.2. A general configuration for H∞ controller design

controlled variables, typically error signals and control signals which are to be minimised in some sense to meet the control objectives. Now, if we partition the plant P compatibly with K, the closed-loop transfer matrix from w to z is given by a lower linear fractional transformation (LFT): z = {P11 + P12 K(I − P22 K)−1 P21 } w

(1.5)

= Fl (P, K) w

(1.6)

The standard H∞ optimal control problem is then the minimisation of the H∞ -norm of Fl (P, K) over all stabilising controllers, i.e. inf k Fl (P, K)( jω ) k∞ = inf max σ (Fl (P, K)( jω )) = inf max K

K

ω

K

w= 6 0

k z( jω ) k2 k w( jω ) k2

(1.7)

In practice, it is usually not necessary to compute the optimal H∞ controller, and it is com­ putationally (and theoretically) simpler to design a sub-optimal controller via the following iterative procedure. Let γmin be the minimum value of k Fl (P, K)( jω ) k∞ over all stabilising

H

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Control Design

9

controllers K. Then the H∞ sub-optimal control problem is: given a γ > γmin , find all stabilising controllers K such that k Fl (P, K)( jω ) k∞ < γ

R This problem can be solved efficiently using standard MATLAB software which uses the algorithm of Doyle et al., [20] - see below. By iteratively reducing the value of γ an optimal solution is approached. So far, we have described the H∞ control problem in its most general setting. We now need to show how the various singular value control objectives discussed in the previous subsection can be represented under the H∞ control framework. This is the task of formulating the H∞ control problem: for the configuration of Figure 1.2, how do we construct the generalised plant P and which signals do we include in z and w, in order to reflect our design objectives on SI , SO , TI and TO ? This task clearly depends on which particular design objectives we are trying to satisfy, and is best illustrated by example.

Example 1.1. S/KS Mixed-sensitivity design Consider the problem of synthesising a con­ troller K to satisfy the following design objectives: • • •

Attenuation of low frequency disturbances at the plant output Minimisation of high frequency actuator usage Robust stability to additive uncertainty at high frequencies

These control objectives correspond to a regulation problem, where we simply want to maintain the output of the plant at some set-point in the presence of signal and system uncertainty. Now, the first two design objectives given above amount to requiring σ (SO ) to be small at low frequencies and σ (KSO ) to be small at high frequencies, since as we have seen, SO is the transfer matrix between dO and the output, and KSO is the transfer matrix between dO and the control signals. Note that minimisation of KSO at high frequencies will also satisfy the third objective of providing good robustness to additive uncertainty. We are thus interested in shaping the closed-loop transfer matrices SO and KSO as a function of frequency. We can do this by selecting frequency dependent weighting functions W1 (s) and W2 (s) and computing a stabilising controller K which minimises the cost function

 

W1 SO

W2 KSO ∞

where W1 (s) is a low pass filter with a bandwidth equal to that of the disturbance and W2 (s) is a high pass filter with a crossover frequency approximately equal to that of the desired closed-loop bandwidth. Note that in general the weighting functions will be (usually diagonal) transfer matrices. The S/KS mixed sensitivity minimisation problem given above can be put into the standard H∞ control configuration as shown in Figure 1.3. In this set-up, the output disturbance signal dO is chosen as the single exogenous input w, while the vector of controlled variables is given by z = [zT1 zT2 ]T , where z1 = W1 y and z2 = −W2 u. It is then easy to show that z1 = W1 SO w and z2 = W2 KSO w. Thus, the generalised plant P is given by the relation      z1 w  z2  = P11 P12 (1.8) P21 P22 u v

10

D.G. Bates

P

w=d

�

G

u

�� � +

� W1

�

z1

� −W2

�

z2

-

+

y

�� + �

v

r=0

K

�

Fig. 1.3. S/KS mixed sensitivity optimisation in standard form (regulation) where P11 = and



   W1 W1 G , P12 = , P21 = −I, P22 = −G 0 −W2 Fl (P, K) =



W1 S W2 KS



(1.9)

(1.10)

The S/KS mixed sensitivity optimisation can also be formulated in terms of a tracking prob­ lem, as shown in Figure 1.4. In this formulation, the exogenous input signal w is a reference command r, and the vector of controlled variables z is made up of z1 = −W1 e = W1 (r − y) and z2 = W2 u. It is left to the reader to verify that for this set-up we again have that z1 = W1 SO w and z2 = W2 KSO w. Example 1.2. Consider the following set of controller design objectives: Attenuation of low frequency disturbances at the plant output Attenuation of high frequency measurement noise Tracking of low frequency reference commands Robust stability to output multiplicative uncertainty at high frequencies Formulate a mixed sensitivity H∞ control problem to satisfy these objectives. Construct the generalised plant P required to cast this problem in the standard H∞ control configuration. Note that when performing numerical calculationsthe generalised plant P can be easily calcu­ R lated, either directly by using the appropriate functions in MATLAB or by constructing the R R plant as a Simulink block diagram and then using the MATLAB function linmod.

1.3.2 Weighting Function Selection As will by now be apparent, the process of designing controllers using the H∞ mixed sensitiv­ ity method essentially revolves around choosing (and usually iteratively modifying) weighting

H

P

w=r

� u

G

�� �

∞

Control Design

11

� W1

�

z1

� W2

�

z2

+

y

-

K

v

�

Fig. 1.4. S/KS mixed sensitivity optimisation in standard form (tracking) functions in order to ‘shape’ certain closed-loop transfer matrices over frequency. In general, weighting function selection becomes difficult when the ‘stacked’ cost functions contain more than two terms. For two terms, e.g., S/KS or S/T, the process is relatively simple, however, since the bandwidth requirements on each term are usually complementary, and simple, stable low-pass and high-pass filters are sufficient to carry out the required shaping over frequency. Note that the weights Wi in mixed sensitivity H∞ control must all be stable and proper (this is due to some assumptions made about the generalised plant by the algorithm to compute the H∞ controller - see the next section). Thus, for example, if we wish to provide zero steadystate tracking error by weighting S with a term including integral action, we have to approx­ imate 1s by s+1 ε , where ε 1, then 1 1 ≤ σ (SI ) σ (KG) + 1 σ (KG) − 1 1 1 ≤ σ (SO ) ≤ σ (GK) + 1 σ (GK) − 1

(1.24) (1.25)

This gives the following set of equivalencies:

σ (SO ) > 1 σ (SI ) > 1

(1.26) (1.27)

Also, from the fact that TI = KG(I + KG)−1 and TO = GK(I + GK)−1 , it follows that

σ (TI ) ≈ σ (KG) at frequencies where σ (KG) is 1

(1.31)

σ (SO G) ≈ 1/σ (K) at frequencies where σ (GK) is >> 1

(1.30)

From the above relationships, we can formulate the following equivalent set of design objec­ tives to those listed in Section 2.1 in terms of the input and output open-loop transfer matrices KG and GK:

H •

•

• • • •

• •

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Control Design

25

For attenuation of output disturbance signals at the plant output, σ (GK) should be large; valid for frequencies at which σ (GK) >> 1 For attenuation of output disturbance signals at the plant input, σ (K) should be small; valid for frequencies at which σ (GK) > 1, atten­ uation of output disturbance signals at the plant input requires 1/σ (G) to be small, i.e., it is a limitation imposed by the plant and cannot be set by the designer. For attenuation of input disturbance signals at the plant input, σ (KG) should be large; valid for frequencies at which σ (KG) >> 1 For attenuation of input disturbance signals at the plant output, σ (K) should be large; valid for frequencies at which σ (GK) >> 1 For attenuation of measurement noise signals at the plant output, σ (GK) should be small; valid for frequencies at which σ (GK) 1, attenu­ ation of measurement noise signals at the plant input requires 1/σ (G) to be small, i.e., it is a limitation imposed by the plant and cannot be set by the designer. For good reference tracking σ (GK) should be large; valid for frequencies at which σ (GK) >> 1, and equivalent to attenuation of output disturbances at the plant output. For avoidance of large control signals due to reference demands (and hence actuator satu­ ration), σ (K) should be small; valid for frequencies at which σ (GK) > 1, avoidance of large control signals due to reference demands requires 1/σ (G) to be small, i.e., it is a limitation imposed by the plant and cannot be set by the designer.

Recalling the requirements for robust stability in the presence of unstructured system uncer­ tainty given in Section 2.1, we have the following additional objectives: • • •

For robust stability to input multiplicative uncertainty, σ (KG) should be small; valid for frequencies at which σ (KG) γmin , is then given by K∞ =



Ak Bk Ck Dk



Ak = A + BF + γ 2 (L∗ )−1 ZC∗ (C + DF) 2

∗ −1

Bk = γ (L )

(1.40)

(1.41) (1.42)

ZC

Ck = B∗ X

(1.43)

Dk = −D

(1.44)

where F = −S−1 (D∗C + B∗ X) 2

L = (1 − γ )I + XZ

(1.45) (1.46)

For γ < 4 (i.e., allowable coprime factor uncertainty of at least 25%) it can be shown theoretically, [39], that the controller K∞ (s) does not significantly change the shapes of the open-loop singular values. Thus robust stability is achieved without significant degradation in the origi-

H

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Control Design

29

nal performance characteristics specified by the plant weighting functions. If γ is greater than 4, this indicates that the chosen loop shapes are incompatible with robust stability (e.g., the roll-off rate is too big around crossover), and further adjustment of the weighting functions is then required.

The final step of the design procedure is to add the constant prefilter K∞(0)W2 in order to ensure zero steady state tracking error, assuming integral action in W1 . Note that the K∞ con­ troller block is a dynamic compensator of order equal to that of the shaped plant - thus the order of the complete H∞ loop shaping controller is equal to the order of the original plant plus twice the order of the weighting functions.

1.4.3 Advantages of H∞ Loop-Shaping The H∞ loop shaping design method has a number of important advantages, which are sum­ marised below. •

•

•

No gamma iteration - the solution of the H∞ loop shaping controller synthesis equations is particularly attractive in that the optimal γ can be found without recourse to the γ -iteration which is normally required to solve H∞ control problems. Provides robustness to unstable perturbations, and uncertainty in the location of lightly damped resonant poles, by maximising robustness to normalised coprime-factor uncertainty. No pole-zero cancellations - cancellation of the stable plant poles by the controller is a common phenomonon in mixed sensitivity H∞ designs, [35, 40]. Consider, for example, the controller designed for the Bell 205 helicopter in the previous section. The poles of the helicopter model G are given by:

Poles = {−0.67 ± 0.60 j, +0.40, −0.13 ± 0.44 j, +0.19, −0.79, −0.40 The zeros of the controller K are given by Zeros = {−0.67 ± 0.60 j, −0.13 ± 0.44 j, −0.79, −0.14 ± 0.12 j, −0.40, −0.5, −0.5, −0.5, −0.5

which includes all the stable poles of G. A physical interpretation of this phenomenon is that in the S/KS mixed sensitivity problem formulation, only disturbances at one point in the loop (i.e., the plant output) are considered by the controller. The optimisation exploits the fact that there is no disturbance at the plant input which can excite plant poles, and tries to cancel the feedback path from the output disturbance to the plant input with ze­ ros (see [40] for a more complete discussion of this phenomenon). Cancellation of stable plant poles by the controller presents a particular problem for systems with slow stable poles compared with the bandwidth of the system, e.g., lightly damped resonances. This is because the controller will only have cancelled these poles in the closed-loop trans­ fer function between certain exogenous inputs and the controlled outputs; these lightlydamped poles will then still appear in other important closed-loop transfer functions. A second problem with the cancellation of such poles is that the closed-loop system will ex­ hibit poor robust performance properties. This is because, as shown in Section 3.4.5, even small amounts of system uncertainty can allow significant movement in the locations of lightly damped poles. Thus, the controller may fail to cancel the poles exactly, leading to poor robust performance. This problem can be avoided in mixed sensitivity designs by introducing extra weighting functions to allow for disturbances at more than one point in

30

D.G. Bates the loop - the choice of the resulting weights is then more complicated, however, since the various trade-offs become much less clear. In contrast, complete pole/zero cancellations do not occur in controllers p roduced using the H∞ loop shaping method. The reason for this becomes clear on noting that the coprime factor singular value robustness measure γ defined as

γ =



 K (I + GK)−1 M −1 k∞ I

can also be written as, (see [4], pp. 320),

   

w1 K −1 γ = (I + GK) [I G] k∞ =

w2 I

(1.47)



 z1

z2 ∞

(1.48)

where w1 and w2 are disturbances at the output and input of the shaped plant as shown in Figure 1.22. The H∞ loop shaping controller takes account of disturbances at more than one point in the loop, and thus does not attempt to cancel all the stable plant poles. z1

�

w2

� ��

w1

�

�� �

Gs

−K

z2

�

Fig. 1.22. Alternative H∞ loop shaping problem formulation •

•

Clear management of conflicting specifications around crossover - in mixed sensitivity optimisation separate weights are chosen for S and T , or S and KS. At low and high frequencies, it is relatively easy to make sure that the weights do not conflict with each other, and indeed it can be argued that one of the great advantages of the mixed sensi­ tivity approach is that the designer can directly shape many different closed-loop transfer functions. Around the desired loop bandwidth frequency, however, it is not always easy to manage the trade-offs between conflicting objectives. This problem does not arise when selecting weighting functions in the H∞ loop shaping procedure, because shaping the open-loop transfer matrix L simultaneously shapes both S and T , thus removing the pos­ sibility of choosing conflicting weights. Provides balanced robustness and performance properties at the plant input and output ­ the coprime factor singular value robustness measure γ provides bounds on all input and output closed-loop transfer functions, since, from Theorem 3.1 in [36] we have that     K G γmin = inf k (I + GK)−1 [I G] k∞ = inf k (I + KG)−1 [K I] k∞ (1.49) I I K K

H

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Control Design

31

In particular, we have that, ( [36] Theorem 5.4), −1 σ ((I + GK)−1 ) = σ (SO ) ≤ εmax σ (Ms )c(W2 ) −1

σ ((I + KG)

) = σ (SI ) ≤

σ (GK(I + GK)−1 ) = σ (TO ) ≤ σ (KG(I + KG)−1 ) = σ (TI ) ≤

σ (K(I + GK)−1 ) = σ (KSO ) ≤

−1 εmax σ (Ms )c(W1 ) −1 εmax σ (Ns )c(W2 ) −1 εmax σ (Ns )c(W1 ) −1 εmax σ (Ms )σ (W1 )σ (W2 )

(1.50) (1.51) (1.52) (1.53) (1.54)

where

σ 2 (W2 GW1 ) 1 + σ 2 (W2 GW1 )

σ (Ns ) = 

1 1 + σ 2 (W2 GW1 ) σ (W1 ) c(W1 ) = σ (W1 ) σ (W2 ) c(W2 ) = σ (W2 )

σ (Ms ) =

•

!1 2

1

≤1

(1.55)

≤1

(1.56)

2

(1.58)

From the above we can see that γmin provides balanced performance and robustness guar­ antees at the plant input and output. This is in sharp contrast to mixed sensitivity H∞ optimisation which generally only provides robustness guarantees at the plant input or output. Guaranteed simultaneous gain/phase margins - From [37], we have that the coprime factor singular value robustness measure γ provides the following guaranteed symmetric multi­ variable gain and phase margins: r r 1+ε 1+ε −20log10 dB ≤ GMcp ≤ +20log10 dB (1.59) 1−ε 1−ε − sin−1 ε degrees ≤ PMcp ≤ + sin−1 ε degrees

•

(1.57)

(1.60)

These gain and phase variations are allowed simultaneously at each input and output of the plant. In the SISO case the allowable gain and phase variation are double those given above. Exact observer implementation - In [40], it was shown that the controller resulting from the H∞ loop shaping procedure can be written as an exact plant observer plus state feed­ back. Assuming, purely for notational convenience, a strictly proper shaped plant, with a stabilisable and detectable state-space realisation   As Bs Gs = (1.61) Cs 0 the relevant equations are x˙ˆs = As xˆs + Hs (Cs xˆs − ys ) + Bs us

us = Ks xˆs

(1.62) (1.63)

32

D.G. Bates

where xˆs is the observer state, us and ys are respectively the input and output of the shaped plant, and

Hs = −ZsCs∗

(1.64)

Ks = −B∗s [I − γ −2 I − γ −2 Xs Zs ]−1 Xs

(1.65)

where Zs and Xs are the appropriate solutions to the generalised algebraic Riccati equations for Gs given previously. In general, H∞ controllers cannot be written as exact plant state observers, as there will be a worst-case disturbance term entering the observer state equation, [20]. However, for the controllers produced by the H∞ loop shaping method it is possible, and this clear structure lends itself to gain-scheduling in that the controller matrices Ks and Hs can be simply scheduled as a function of one or more aircraft parameters. Figure 1.23 shows the implementation structure of the H∞ loop shaping controller in observer form.

r

� K0

�+ � + �

us

�

u

�y

� G

kW1

� W2

�

Hs

Bs + � � �+ � R + �

�

ys

� �+ �

� Cs

As � Ks �

Fig. 1.23. Observer-form implementation of H∞ loop shaping controller

Note that the observer is for the shaped plant Gs and that the constant prefilter must be recalculated to take account of the different structure of the observer-form implementa­ tion. As well as greatly simplifying the task of controller scheduling, the observer/state feedback structure gives a clearer (more ‘classical’) functionality to the different ‘blocks’ of the H∞ loop shaping controller - this extra transparency can be of great help in the controller certification process.

H

∞

Control Design

33

1.4.4 Design Example: Control Law Design for the Harrier V/STOL Aircraft A significant advance in aerospace engineering in recent years has seen the development of actuator technologies which allow much greater flexibility in the use of propulsion generated thrusts as primary flight control effectors. Pitch, roll and yaw moments necessary for attitude control, as well as various thrust and lift forces necessary for trim and manoeuvring control, can now all be partially or exclusively generated by vectoring nozzles and various propulsive lift systems such as ejectors, remote augmented lift systems (RALS), and various techniques of boundary-layer control using engine air, [41]. The development of these new technologies means that there are now many combinations of aerodynamic and propulsion effectors which can be employed to enhance control authority during STOVL and conventional operations, to minimise fuel consumption and pilot workload throughout different flight modes, and to permit reconfiguration in the event of malfunction of one or more components. The use of propulsion effectors for aerodynamic control, however, creates significant interaction between the two (heretofore independent) airframe and engine subsytems, necessitating the use of robust multivariable control methods to design integrated flight and propulsion control (IFPC) systems. In [41] a methodology for IFPC design was developed, based on a ‘centralised’ approach to the problem. This approach consists of first designing a centralised controller considering the airframe and propulsion subsystems as one integrated system, and then partitioning the centralised controller into decentralised subcontrollers with a specified coupling structure. The subcontrollers will in general be of lower order than the centralised controller, and will thus be easier to implement. They can also be independently tested and validated at subsystem level. Independent integrity of the subcontrollers is necessary not just for control theoretic reasons (different types of control to be applied to systems with different dynamics, uncertainties, nonlinearities etc. ) but for practical and ‘political’ reasons as well. Since different subsystems may be designed and built by different and independent suppliers, design accountability and commercial issues will often dictate that each manufacturer retains a high degree of control over the particular subsystem (and its controller) for which it is responsible. The example described in this section investigates the application of the H ∞ loop shaping design method to the problem of designing a centralised integrated flight and propulsion control system for the VAAC Harrier aircraft. As well as integrating the airframe and engine control problems, the longitudinal and lateral/directional control problems are also addressed simul­ taneously. This is to ensure minimal coupling between longitudinal and lateral dynamics for manoeuvres at high angles of attack. Results of piloted simulations with an IFPC system de­ signed using this approach, as well as more details on stategies for partitioning centralised IFPC systems can be found in [17, 42–44]. The aircraft model used in this example is based on the Harrier Wide Enve lope Model (WEM): a full envelope non-linear lateral plus longitudinal representation of the aerodynamics, engine and actuator characteristics of the VAAC Harrier aircraft. The WEM software incorporates a full thermodynamic powerplant simulation of a Pegasus engine, which is integrated with the airframe system dynamics: thus in principle all interactions between these two systems are captured in this model. Representative actuation systems, including rate and saturation limits, have been placed on all control motivators. Saturation limits for control motivators are as follows: elevator (etad): -10.25 to 11.25 deg, aileron (xid): +/- 14 deg, rudder(zetad): +/- 15 deg, nozzle angle (thejd): 0 to 98.5 deg, throttle position (pthtp): 0.26 to 1 dimensionless.

34

D.G. Bates

R The WEM software is coded in fortran and run under the Simulink simulation environment. The overall software package represents a highly detailed model of the aircraft, producing 22 aerodynamic outputs and 12 engine outputs. The package was configured to allow the extrac­ tion of linear state space models at various points over the flight envelope for the purposes of control system design. These (lower order) linear models are of the form

x˙ = Ax + Bu; y = Cx + Du, where the state vector are given by x = [θ , φ , ψ , q, p, r, v f , ve, vd, f np, hnp, qe f ] where v f , ve and vd are forward, east and down velocities respectively, f np is engine fan speed, hnp is compressor speed, qe f is fuel flow and the other symbols have their usual mean­ ings. The control inputs are given by u = [etad, xid, zetad,the jd, ptht p] where etad is elevator angle, xid is aileron angle, zetad is rudder angle, the jd is thrust vector angle and ptht p is thrust magnitude. All of the states defined above are available as outputs. Both the linear and non-linear WEM models cover the full flight envelope for the VAAC, from fully airborne flight at 200 knots down to hover. In this example, the linear model corresponding to the 80 knot transition flight phase is used for controller design. At this point in the flight envelope, the aircraft is longitudinally unstable and propulsion system generated forces and moments are taking over control of the aircraft from the aerodynamic effectors. The controlled variables z are given by z = [Qq, Pp, r, v f , vd] where Qq = q + 0.3θ , Pp = p + 0.3φ , r = yaw rate, v f = velocity forward and vd = velocity down. The above choice of controlled variables was based on control configuration mode 4 of the NASA V/STOL systems research aircraft [45], and provides response types that are generally desirable for good handling qualities in transition flight, [41]. Control of Qq and Pp corresponds to tracking of transient rate commands and steady state attitude commands - a so-called rate command-attitude hold system. The corresponding demands on q and θ for a demand on Qq are shown in Figures 1.24 and 1.25 respectively (this can be verified by solving the first order differential equation θ˙ + 0.3θ = Qq). Performance specifications, based on [46], for the closed-loop system are given as:

1. 90 per cent of demanded rate/attitude to be reached within 2 seconds for the pitch, roll and yaw outputs. 2. Bandwidth of the pitch, roll and yaw channels to be approximately 7 rads/sec. 3. 90 per cent of demanded velocity to be reached within 4 seconds. 4. Maximise decoupling of command tracking over all channels, ensuring: (a) 10 degrees attitude demand causes less than 1 degree change in other attitudes, and less than 1 knot change in velocities,

H

∞

Control Design

35

(b) 10 knots velocity demand causes less than 1 knot change in other velocities, and less than 1 degree change in attitudes. 5. Control signals to stay within rate/saturation limits at all times. The structure of the closed-loop system with a one degree-of-freedom H ∞ Loop-Shaping controller K(s) is shown in Figure 1.20. As discussed previously, the design parameters in the H ∞ Loop-Shaping procedure are the transfer function matrix W1 , and the two constant matrices W2 and k. For this example all three matrices are chosen to be diagonal. An essential initial step in the design procedure is the proper scaling of the plant inputs and outputs, [14]. In this example, input scaling was applied to the plant to approximately normalise the actuator signals by their maximum allowable values. Cross coupling between all outputs was considered of equal importance, so output scaling was used only to convert the units of the first three outputs from radians to degrees. Note that the scaled plant G(s) in this example also includes linearised models of the dynamics of the plant actuators - the singular values of the scaled plant are shown in Figure 1.26. The design procedure itself then consists of two basic steps. First the weighting matrices W1 , W2 , and k are chosen to shape the singular values of the open-loop plant so that the given performance specifications are satisfied. Then the feedback controller K∞ (s) is calculated so that the shaped plant is robustly stabilised against normalised coprime factor uncertainty. For this example, initial values for the weighting matrices were chosen as:

W1 (s) =

s+5 I , W2 = I5×5 , k = diag [0.1 0.1 0.1 0.1 0.1] s 5×5

W1 was chosen to ensure good tracking properties with zero steady state error, good distur­ bance rejection at low frequencies, and a moderate roll-off rate around crossover. The matrix W2 is generally used to reflect the relative importance of the outputs to be controlled and was therefore chosen as the identity matrix for this example. Finally, the matrix k was used to set the bandwidth of the open-loop singular values, and to adjust the relative magnitudes of the various actuator signals. The singular values of the shaped plant are shown in Figure 1.27. The implementation structure shown in Figure 1.20 has the advantage that reference signals do not directly excite the dynamics of K∞ (s) - since this block has been designed for stabilisation and not performance purposes, its presence in the forward loop can result in large amounts of overshoot (classical derivative kick). The constant prefilter K∞ (0)W2 is included to ensure zero steady state tracking error, assuming integral action in W1 . The second step in the design procedure is the calculation of the H ∞ robust stabilisation controller K∞ (s). For the choice of weighting matrices given above, a robust stabilisation controller K∞ (s) was calculated which returned an εmax of only 0.233, indicating poor robust stability properties for the system. Thus, although the performance of the controller in time domain simulation was quite acceptable (see Figures 1.28 and 1.29), adjustment of the weighting functions is required in order to improve the robustness of the IFPC system. In order to improve the robust stability of the closed-loop, we need to decrease the rate of roll-off of the open-loop singular values of the shaped plant around cross-over. This can easily be achieved in the H∞ loop shaping provedure by adjusting the weighting matrix W1 so that

36

D.G. Bates

the zero at 5 rads/sec takes effect at a slightly lower frequency, say 1 rads/sec, thus reducing the roll-off of the singular values around crossover. For the new weighting matrix W1 (s) =

s+1 I , s 5×5

with W2 and k unchanged, a new robust stabilisation controller K∞ (s) was calculated which returned an εmax of 0.279, indicating much improved robust stability properties for the system, i.e., allowable coprime factor uncertainty of > 27%. Analysis of the closed-loop system in time domain simulation, however, revealed that the tracking response for pilot demands was now a little sluggish, with increased coupling between the different controlled variables. In particular the response to demands on Qq (Figure 1.30) no longer meets the specification of achieving 90% of the reference demand within 2 seconds. In order to improve the closed-loop performance, the first element of the W1 weighting matrix was changed to s+s 5 (to improve the Qq tracking response at low frequencies) while the gain matrix k was set to

k = diag [0.4 0.2 0.2 0.4 0.2] in order to emphasis the use of the elevator and nozzle angle (for improved control of Qq), and to increase the closed-loop bandwidth slightly in all loops. Robust stabilisation of this shaped plant gave a (still quite acceptable) εmax of 0.269, with improved tracking response for all controlled variables, and Qq in particular (Figure 1.31). The responses for all other variables now also meet command tracking, decoupling and control usage specifications - see for example Figures 1.32 to 1.35. Finally, the achieved open-loop singular values after robust stabilisation (i.e., including K∞ ) are compared with those specified for the shaped plant in Figure 1.36. As shown by the figure, and as expected from the ‘good’ value of γmin /εmax , the controller K∞ has not significantly altered the desired loop shapes. As with the mixed sensitivity H∞ synthesis example, the design described above is not complete, and is intended primarily to illustrate the process by which performance and robustness trade-offs can be managed via adjustment of the weighting functions in the design process. It does, however, demonstrate some of the more appealing features of the H∞ loop shaping method - in particular, the classical nature of the approach, the transparency of the relationship between the design specifications and the plant weighting functions, and the robustness of the controllers produced by the H∞ robust stabilisation optimisation.

H

∞

Control Design

Qqdemand 12

10

magnitude

8

6

4

2

0

0

1

2

3

4

5 time(secs)

6

7

8

9

10

Fig. 1.24. Pilot demand on Qq

20

theta_c and theta

15

magnitude

10

5 q_c and q

0

−5 0

1

2

3

4

5 time(secs)

6

7

8

9

10

Fig. 1.25. Corresponding demands on q and θ - rate-command attitude-hold response

37

38

D.G. Bates

Nominal plant 80

60

40

mag (dB)

20

0

−20

−40

−60

−80

−100 −3

−2.5

−2

−1.5

−1

−0.5 log10(w)

0

0.5

1

1.5

2

1.5

2

Fig. 1.26. Open-loop singular values of the scaled plant

Shaped plant 150

100

mag (dB)

50

0

−50

−100

−150 −3

−2.5

−2

−1.5

−1

−0.5 log10(w)

0

0.5

1

Fig. 1.27. Open-loop singular values of the shaped plant

H

−3

10

6

8

6

4

−3

Pp

0

5

Design

r

10

4

2

−2

10

x 10

15

5

0

0

2

0

x 10

degrees/sec

8

magnitude

magnitude

Qq 12

∞ Control

0

5

vf

10

−5

0

5 time(secs)

10

vd

80.5

0.35

0.3

80.4

0.25

0.2

ft/s

ft/s

80.3

0.15

80.2

0.1

80.1

80

0.05

0

5 time(secs)

0

10

0

5 time(secs)

10

Fig. 1.28. Responses for a demand on Qq - initial weighing functions

−3

5

degrees (away from trim)

degrees (away from trim)

1

0

−1

−2

−3

−4

0

5

10

x 10

XID

0

−5

−10

−15

ZETAD 0.015

degrees (away from trim)

ETAD 2

0

THEJD

5

10

0.01

0.005

0

−0.005

−0.01

0

5 time(secs)

10

PTHTP

40

magnitude (away from trim)

degrees (away from trim)

0

20

0

−20

−40

−60

0

5 time(secs)

10

−0.05

−0.1

−0.15

−0.2

−0.25

−0.3

−0.35

0

5 time(secs)

10

Fig. 1.29. Actuator responses for a demand on Qq - initial weighing functions

39

40

D.G. Bates

Pp

8

0.008

6

4

2

0

r 0.015

degrees/sec

0.01

magnitude

magnitude

Qq 10

0.006

0.004

0.01

0.005

0.002

0

5

0

10

0

5

vf

0

5 time(secs)

10

vd 0.4

80.8

0.3

80.6

0.2

ft/s

81

ft/s

0

10

80.4

0.1

80.2

0

80

0

5 time(secs)

−0.1

10

0

5 time(secs)

10

Fig. 1.30. Degraded responses for a demand on Qq

−3

8

3

6

4

x 10

−3

Pp 6

r

x 10

5

degrees/sec

4

magnitude

magnitude

Qq 10

2

1

4

3

2

1

2

0

0

−1

0

5

10

0

0

vf

5

10

−1

0

5 time(secs)

vd 0.3

80.35

80.3

0.2

0.1

80.2

ft/s

ft/s

80.25

80.15

0

80.1

−0.1

80.05

80

0

5 time(secs)

10

−0.2

0

5 time(secs)

10

Fig. 1.31. Improved responses for a demand on Qq

10

H

−3

8

−0.2

6

4

2

−4

Pp 2

0

5

−0.4

−0.6

−1

10

x 10

r

0

−0.8

0

−2

x 10

degrees/sec

0

magnitude

magnitude

Qq 10

Control Design

∞

−2

−4

−6

0

5

vf

10

−8

0

5 time(secs)

10

vd

80.4

12

10

80.3

8

ft/s

ft/s

80.2

6

4

80.1

2

80

79.9

0

0

5 time(secs)

−2

10

0

5 time(secs)

10

Fig. 1.32. Responses for a demand on vd −3

ETAD 4

0

−5

−10

−15

0

5

10

x 10

XID

2

0

−2

−4

−6

−8

−10

ZETAD 0.015

degrees (away from trim)

degrees (away from trim)

degrees (away from trim)

5

0

THEJD

5

10

0.01

0.005

0

−0.005

−0.01

0

PTHTP magnitude (away from trim)

degrees (away from trim)

0

40

20

0

−20

−40

−60

0

5 time(secs)

10

−0.05

−0.1

−0.15

−0.2

−0.25

−0.3

−0.35

0

5 time(secs)

10

Fig. 1.33. Actuator responses for demand on vd

5 time(secs)

10

41

D.G. Bates

−3

x 10

Qq

Pp

r

12

0.3

4

10

0.2

3

8

magnitude

magnitude

5

2

1

6

4

0

2

−1

0

0

5

10

degrees/sec

42

1

0

−0.1

−0.2

−0.3

0

5

−4

vf 80.0002

0.1

x 10

10

−0.4

0

5 time(secs)

10

vd

0

80.0001

−1

ft/s

ft/s

80.0001

−2

−3

80

−4

80

79.9999

−5

0

5 time(secs)

−6

10

0

5 time(secs)

10

Fig. 1.34. Responses for a demand on Pp

ETAD

XID

0.03

0.02

0.01

0

5

10

10

5

0

−5

magnitude (away from trim)

degrees (away from trim)

0

20

0

−20

−40

−60

0

5 time(secs)

0

5

−4

THEJD 40

1

degrees (away from trim)

degrees (away from trim)

degrees (away from trim)

0.04

0

ZETAD

15

0.05

10

x 10

10

0.5

0

−0.5

−1

−1.5

0

PTHTP

−0.2

−0.4

−0.6

−0.8

−1

−1.2

0

5 time(secs)

10

Fig. 1.35. Actuator responses demand on Pp

5 time(secs)

10

H

∞

Control Design

43

Loop gain frequency response 100

50

dB

0

Specified loop gain − − −50

Achieved loop gain −

−100

−150 −2 10

−1

10

0

10 rad/s

1

10

2

10

Fig. 1.36. Comparison of achieved versus specified open-loop singular values

References 1. F.L. Lewis, Optimal Control, Wiley, 1986. 2. J.C. Doyle, “Guaranteed margins for LQG regulators”, IEEE Transactions on Automatic Control, 23(4), pp. 756-757, 1978. 3. J.D. Blight, R.L. Dailey and D. Gangsaas, “Practical control law design for aircraft using multivariable techniques”, International Journal of Control, 59(1), pp. 93-137, 1994. 4. K. Zhou and J.C. Doyle Essentials of robust control, Prentice Hall, 1998. 5. C. Edwards and S.K. Spurgeon Sliding mode control: theory and applications, Taylor and francis, London, 1996. 6. D. Enns, D. Bugajski, R. Hendrick and G. Stein “Dynamic inversion: an evolving method­ ology for flight control design”, International Journal of Control 59(1), pp. 71-79, 1994. 7. J. Reiner, G. Balas and W.L. Garrard “Robust dynamic inversion for control of highly maneuverable aircraft”, AIAA Journal of Guidance Control and Dynamics, 18(1), pp. 18­ 24, 1995. 8. C.D. Yang and C.C. Kung, “Nonlinear H∞ flight control of general six-degree-of-freedom motions”, AIAA Journal of Guidance, Control and Dynamics, 23(2), pp. 278-288, 2000. 9. C.S. Wu, B.S. Chen and Y.W. Jan “Unified design for H2 , H∞ , and mixed control of spacecraft”, AIAA Journal of Guidance, Control and Dynamics, 22(6), pp. 884-896, 1999. 10. G. Papageorgiou and R. Hyde “Analysing the stability of NDI-based flight controllers with LPV methods”, Proc. of the AIAA Conference on Guidance, Navigation and Control, Montreal, 2001.

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11. G.E. Cooper and R.P. Harper “The use of pilot rating in the evaluation of aircraft handling qualities”, NASA TN D-5153, April 1969. 12. S. Bennani and G. Looye “Design of flight control laws for a civil aircraft using µ ­ synthesis”, Proc. of the AIAA Conference on Guidance, Navigation and Control, pp. 314­ 324, 1998. 13. I. Fialho, G. Balas, A. Packard, J. Renfrow and C. Mullaney “Linear Fractional Transfor­ mation Control of the F-14 Aircraft Lateral-Directional Axis during Powered Approach Landing”, Proc. of the Amrican Control Conference pp. 128-132, 1997. 14. R.A. Hyde H∞ Aerospace Control Design - A VSTOL Flight Application, Springer Verlag, 1995. 15. A. Smerlas, I. Postlethwaite, D.J. Walker, M.E. Strange, J. Howitt, R.I. Horton, A.W. Gubbels, and S.W. Baillie “Design and Flight Testing of an H∞ Controller for the NRC Bell 205 experiemntal Fly-By-Wire Helicopter”, Proceedings of the AIAA Conference on Guidance, Navigation and Control, Boston, 1998. 16. I. Postlethwaite, A. Smerlas, D.J. Walker, A.W. Gubbels, S.W. Baillie, M.E. Strange and J. Howitt, “H∞ control of the NRC Bell 205 fly-by-wire helicopter”, Journal of the Amer­ ican Helicopter Society, 44(4), pp. 276-284, 1999. 17. D.G. Bates, S.L. Gatley, I. Postlethwaite and A.J. Berry “Design and Piloted Simulation of a Robust Integrated Flight and Propulsion Controller”, AIAA Journal of Guidance, Navigation and Control, 23(2), pp. 269-277, 2000. 18. K. Glover and J.C. Doyle “State-space formulae for all stabilizing controllers that satisfy an H∞ norm bound and relations to risk sensitivity”, Systems and Control Letters, 11, pp. 167-172, 1988. 19. G. J. Balas, J. C. Doyle, K. Glover, A. Packard and R. Smith, µ -Analysis and Synthesis Toolbox User’s Guide, The Mathworks, 1995. 20. J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis “State-space solutions to stan­ dard H2 and H∞ control problems”, IEEE Transactions on Automatic Control, AC-34(8), pp. 831-847, 1989. 21. G. Meinsma “Unstable and non-proper weights in H∞ control”, Automatica, 31(1), 1655­ 1658, 1995. 22. J.S. Freudenberg and D.P. Looze “An analysis of H∞ optimization design methods”, IEEE Transactions on Automatic Control, AC-31(3), pp. 194-200, 1986. 23. A. Smerlas, I. Postlethwaite and D.J. Walker “H∞ loop shaping: the Bell 205 helicopter case study”, Proceedings of the AIAA Conference on Guidance, Naviagation and Control, pp. 1605-1612, 1999. 24. A. Smerlas Robust multivariable control of helicopters: from mathematical models to flight tests, Ph.D. Thesis, Department of Engineering, University of Leicester, March 1999. 25. R.K. Heffley, W.F. Jewell, J.M. Lehman and R.A. Van Winkle A compilation and analysis of helicopter handling qualities data, NASA Contractor Report 3144, 1979. 26. M. Strange, J. Howitt Configuration of the DERA HELISIM model to represent the flight dynamics of the NRC Bell 205 fly-by-wire helicopter, DERA Technical Report TR97459­ 1, 1997. 27. D.G. Bates, S.L. Gatley, I. Postlethwaite and A.J. Berry “Integrated Flight and Propulsion Control Design using H∞ loop-shaping Techniques”, Proc. IEEE Conference on Decision and Control, pp. 1523-1528, 1999. 28. S. Le Ballois and G. Duc “ H∞ Control of an Earth Observation Satelite”, AIAA Journal of Guidance Navigation and Control, 19(3), pp. 628-635, 1996. 29. G. Ferreres, and M. M’Saad “Parametric Robustness Analysis of a Missile Autopilot”, AIAA journal of Guidance Control and Dynamics, 19(3), pp. 621-627, 1996.

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30. R. Samar Robust Multi-Mode Control of High Performance Aero-Engines, Ph.D. Thesis, Department of Engineering, University of Leicester, UK, 1995. 31. G. Papageorgiou and K. Glover “Design of a robust gain scheduled controller for the high incidence research model”, Proceedings of the AIAA Conference on Guidance, Naviagation and Control, pp. 1613-1623, 1999. 32. G. Papageorgiou, K. Glover, G. D’Mello, G. and Y. Patel “Taking robust LPV control into flight on the VAAC Harrier”, Proc. of the IEEE Conference on Decision and Control, Sydney, 2000. 33. H.W. Bode Network analysis and feedback amplifier design, D. Van Nostrand Co. New York, 1945. 34. Doyle, J., C. and Stein, G., “Multivariable feedback design: concepts for a classical/modern synthesis”, IEEE Transactions on Automatic Control, AC-26(1), pp. 4-16, 1981. 35. I. Postlethwaite, M.-C. Tsai and D.-W. Gu “Weighting function selection in H∞ design”, in proceedings of the IFAC World Congress on Automatic Control, 1990. 36. K. Glover, J. Sefton, J. and D.C. Mcfarlane “A tutorial on loop shaping and H-infinity robust stabilization”, Proc. of the IFAC World Congress on Automatic Control, pp. 94103, 1990. 37. K. Glover, G. Vinnicombe, G. and G. Papageorgiou “Guaranteed multi-loop stability margins and the gap metric”, Proc. of the IEEE Conference on Decision and Control, Sydney, 2000. 38. S. Skogestad and I. Postlethwaite Multivariable feedback control, Wiley, 1996. 39. D. McFarlane and K. Glover “A Loop Shaping Design Procedure using H ∞ Synthesis”, IEEE Trans. on Automatic Control, AC-36, pp. 759-769, 1992. 40. J. Seftonand K. Glover “Pole-zero cancellations in the general H∞ problem with reference to a two block design”, Systems and Control Letters, 14, pp. 295-306, 1990. 41. S. Garg “Robust Integrated Flight/Propulsion Control Design for a STOVL Aircraft using H∞ Control Design Techniques”, Automatica 29(1), pp. 129-145, 1993. 42. S.L. Gatley, D.G. Bates, and I. Postlethwaite “A Partitioned Integrated Flight and Propulsion Control System with Engine Safety Limiting”, IFAC Journal of Control Engineering Practice, 8, pp. 845-859, 2000. 43. S.L. Gatley, and D.G. Bates and I. Postlethwaite “Partitioning and re-design of H∞ loop shaping integrated flight and propulsion control systems”, Proc. of the AIAA Conference on Guidance, Naviagation and Control, Montreal, 2001. 44. S.L. Gatley, and D.G. Bates and I. Postlethwaite “An engine-limiting and anti-windup scheme for a partitioned integrated flight and propulsion control system”, Proc. of the AIAA Conference on Guidance, Naviagation and Control, Montreal, 2001. 45. J.A. Franklin, M.W. Stortz, P.F. Borchers and E. Moralez III, Flight evaluation of advanced controls and displays for transition and landing on the NASA V/STOL systems research aircraft, NASA TP 3607, 1996. 46. J.A. Franklin, J., A. “Design criteria for integrated flight/propulsion control systems for STOVL fighter aircraft”, proceedings of Piloting Vertical Flight Aircraft: A conference on flying qualities and human factors, San Francisco, CA, pp. 37-57, 1993.

2 Structural Methods for Linear Systems: An Introduction Nicos Karcanias and Efstathios Milonidis

Summary. This paper assumes familiarity with the basic Control and Dynamics, as covered in undergraduate courses. It introduces the different alternative system representations for lin ear systems and provides a quick review of the fundamental mathematical tools, which are essential for the treatment of the more advanced notions in Linear Systems. The paper focuses on some fundamental concepts underpinning the study of linear systems and dynamics and which play a crucial role in the analysis and design of control systems; thus, the paper deals with notions such as those of Controllability, Observability, Stability, poles and zeros and re­ lated dynamics and their properties under different compensation schemes. There is a concrete flavour in the current approach which runs through this presentation and this is that of the un­ derlying algebraic structure. The term “structure” refers to aspects of the state space/transfer function description, which remain invariant under a variety of transformations. The set of transformations considered here are of the compensation type and include state feedback and output injection, dynamic compensation, as well as of the representation type transformations that include state, input, and output co-ordinate transformations. This structure stems from the system description and defines the nature of the dynamics and the related geometric proper­ ties and these in turn define what it is possible to achieve under feedback; such an approach is known as a structural approach. Central to our analysis are the notions of poles and zeros. The poles of a system are crucial characteristics of the internal system dynamics, characterise system free response, stability and general aspects of the performance of a system. The poles of a system are affected by the different compensation schemes and their assignment is the subject of many design methodologies aiming at shaping the internal system dynamics under different compensation schemes. The notion of zeros is more complex, since they express the interaction between internal dynamics and the effort to control and observe the system and they are thus products of overall system design, that apart from process synthesis involves selection of actuation and measurement schemes for the system. The significance of zeros is mainly due to that they remain invariant under a large set of compensation schemes, as well as that they define limits of what can be achieved under compensation. This makes zeros crucial for design, since they are part of those factors characterising the potential of a given system to achieve certain design objectives under compensation. The invariance of zeros implies that their design is an issue that has to be addressed outside the traditional control design; this requires understanding of the zero formation process and involves early design stages mech­ anisms such as process instrumentation. Poles and zeros are conceptually inverse concepts (resonances, antiresonances) and such mechanisms are highlighted throughout the paper. The M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 47-98, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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N. Karcanias and E. Milonidis

role of system structure in characterising different system properties is central to this paper and it is defined by a set of discrete and continuous invariants; these invariants characterise a vari­ ety of key system properties and their type/values define the structure of canonical forms and determine somehow the potential of a given system for compensation. Invariants and canoni­ cal forms under the general transformation group are linked to compensation theory, whereas those associated with representation transformations play a key role in system identification. The emphasis in this article is to provide an overview of the fundamentals concepts, the back ­ ground mathematical tools, explain their dynamic significance and link them to problems of control and systems design.

2.1

Introduction

The study of problems of Control Analysis, Control Synthesis – Design and Model Identi­ fication, heavily relies on the Theory of Linear Systems. The notion of “system structure” is dominant in describing system properties and the term implies those aspects of the sys­ tem model, which remain invariant under a set of transformations that may be applied on the system. The system structure is a generic term that refers to system aspects such as intercon ­ nection graph for system components and functions that may be defined on the system model and are referred to as invariants, which are further classified into discrete (integer, real num­ bers) and continuous (polynomials, rational functions etc.). The notion of structure is referred to both input-output and state space models. The sets of transformations are clearly different for the two cases, but relationships between the structure of state space and transfer function models exist. The notion of system structure is intimately related to that of canonical form, which corresponds to a system description, derived under the given group of transformations, expresses a specialisation of the system invariants in the set of considered models and involves the minimal number of parameters. System structure is essential for the Model Identification problem and canonical forms, corresponding to representation transformations, provide a ve­ hicle for Model Identification. For the Control Analysis and Synthesis, aspects of the structure (as it is expressed by the system invariants) characterise the presence, or absence of certain system properties and the type and values of invariants provide criteria for solvability of a number of control synthesis problems. In the area of control design, the types and values of invariants frequently impose limitations in what it is possible to achieve. Although the link be­ tween system structure and achievable performance, under certain forms of compensation is not explicitly known, system structure expresses in a way the potential of a system to provide certain solutions to posed control problems. This paper considers the link between fundamental system notions, which are defined in a dynamic sense and the system structure, as this is expressed by invariants of the state space, or transfer function description. We consider different state space representations and trans­ fer function representations which lead to unifying descriptions, such as the matrix pencils, polynomial and rational matrices which provide the set up for studying system properties and their link to system structure. The unifying operator for all these state space descriptions is that of a matrix pencil. The theory of invariants and canonical forms of matrix pencils under the general group of the strict equivalence transformations is known as the Kronecker theory and underpins the theory of invariants and canonical forms under any combination of state space transformations. For the case of polynomial and rational models, the theory of Smith and Smith–McMillan forms provide the natural vehicle for defining structure and then link­ ing it the fundamental system properties. The paper focuses on some fundamental concepts

Structural Methods for Linear Systems: An Introduction

49

underpinning the study of linear systems and their dynamics and which play a crucial role in the analysis and design of control systems; thus, we focus on fundamentals such as the notions of Controllability, Observability, Stability, poles and zeros and related dynamics and try to establish their links to the underlying system structure. It is these links which allow the establishment of connections between structural tools and methodologies and control design. It is assumed that the fundamentals of Control and State Space Analysis are known from the undergraduate courses. The structural characterisation of system properties allows the developmentof an understand ­ ing on the effect of feedback on them and allows the means for the development of com ­ pensation schemes. Simple structural tests for Controllability and Observability properties are given, which allow their characterisation with algebraic means and enable the establishment of simple tests for their invariance under different forms of feedback. The concepts of pole, zero take a considerable attention. These notions have emerged as the key tools of the classi ­ cal methods of Nyquist-Bode and root locus for the analysis and design of linear, single-input, single –output (SISO) feedback systems. The development of the state space S(A, B,C, D) de ­ scription and transfer function G(s) description for linear multivariable systems has led to a variety of definitions for the zeros and poles for the multivariable case. Loosely speaking, multivariable poles and zeros are resonant and antiresonant frequencies respectively, that is to say they are frequencies whose transmission explodes with time, or whose transmission is completely blocked. The inversion of roles of poles and zeros suggested by their classical complex analysis definition (resonance, antiresonance) motivates the dynamic (in terms of tra ­ jectories) properties of zeros. The physical problem used to define multivariable zeros is the “output zeroing problem”, which deals with defining appropriate non­ zero input exponential signal vectors and initial conditions which result in identically zero output. Such a problem is the dual of the “zero input” problem defining poles, which deals with defining appropriate initial conditions, such that with zero input the output is a nonzero exponential vector signal. Those two physical problems emphasize the duality of the roles of poles and zeros. Apart from its natural dynamic appeal such definitions for poles and zeros have the addi ­ tional advantages that they reveal the geometric dimension of such concepts, as well as their link with fundamental structural invariants of the system. The poles-eigenvalues have a well­ defined geometry introduced by the eigenvectors and the corresponding spaces (the A-invariant spaces of the state space). Similarly, the geometry of zeros is linked to generalised eigenvalue ­ eigenvector problems and corresponding spaces (types of (A, B)-invariant spaces). The Jordan form of the state matrix reveals the invariant structure of poles in the state space set-up. The Smith form and in some more detail the Kronecker form of the state space system matrix in ­ troduce the zero structure of the state space models; for transfer function models the pole zero structure is introduced by the Smith–McMillan form. Such links reveal the poles as invariants of the alternative system representations under a variety of representation and feedback trans ­ formations. The strong invariance of zeros (large set of transformations) makes them critical structural characteristics, which strongly influence the potential of systems to achieve perfor­ mance improvements under compensation. The dynamic characterisation of zeros leads to algebraic characterisations, which reveal them as byproducts of the interaction of the internal dynamics and the model, input, output struc ­ ture. This is contrary to the pole characterisation, which shows that they express the internal dynamics. Such observations lead to that zero design is a task associated with overall selection of inputs, outputs and thus belongs to the earlier system design stage of process instrumen ­ tation. In the paper we consider both finite and infinite zeros and examine them in both state space and transfer function context. The relationships between the corresponding notions for

50

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state space and transfer function models reveal interesting links to the fundamental system properties of controllability and observability. Every square system (same number of input and outputs) has zeros (finite and/or infinite); however, nonsquare systems generically do not have zeros and this is an important difference with the poles which exist independent from input, output dimensionalities. Closing feedback loops creates square systems and this involves creation of zeros; such phenomena are within the area of designing zeros and the mechanisms for zero formation are examined. The effect of feedback and other forms of transformations on the pole and zero structure of the system is finally considered. We examine the general feedback configuration under dynamic pre-compensation (or feedback compensation) and this leads to a pole assignment (stabilization) problem linked to the solution of a matrix Diofantine equation. The study of such equations over “relevant” for control rings is briefly examined, their solvability is linked to Minimality of representations and the solution of such equations is given in parametric forms. Finally, an alternative structural framework for generalised design is introduced through the formulation of the Determinantal Assignment Problem (DAP). DAP provides a unifying formulation to all frequency assignment problems (pole, zero) and introduces new types of algebraic structure. This alternative approach provides a powerful machinery for studying problems of “structure assignment” based on tools from exterior algebra and classical algebraic geometry. The zero assignment by a constant squaring down may be studied and solved within this framework. Throughout the paper we will use the following notation: R, C are the fields of real, complex numbers respectively, R[s] is the ring of polynomials in s with coefficients in R, R(s) is the field of rational functions in s with coefficients in R, Rmxn is the set of m × n matrices with real coefficients Rmxn (s) is the set of m × n rational matrices, V denotes a vector space over R, or C, or R(s), by V we denote a basis matrix and by v a vector of V . We denote by deg the degree of a polynomial, det{A} is the determinant of a square matrix A, which is also denoted by |A|. Finally, if i takes values from the set {1, 2, . . . , n}, we shall denote this by i ∈∼n.

2.2 2.2.1

Classification of System Representations State Space Descriptions

Linear time invariant multivariable systems are represented in the time domain by state variable model S(A, B,C, D) : x˙ = Ax + Bu, y = Cx + Du (2.1) where x is an n-vector of the state variables, u is a p-vector of inputs and y is an m-vector outputs. A, B,C, D are respectively n × n, n × p, m × n, m × p matrices. The above description may be represented in an autonomous or implicit form as:       x   x˙ A B O   I OO   u (2.2) u˙ = S(Φ , Ω ) : C D −I OOO | {z } y | {z } y˙ | {z } | {z } ,Φ

,ξ˙

,Ω

= [xt , ut , yt ]t

,ξ

is the composite vector, or implicit where Φ , Ω are the coefficient matrices and ξ vector of the state space description. The vector ξ contains the state, input and output vectors

Structural Methods for Linear Systems: An Introduction

51

and makes no distinction between them. The S(Φ , Ω ) description belongs to the general class of generalised autonomous differential descriptions [29], [31] S(F, G) : F z˙ = Gz

(2.3)

where F, G are r × k matrices and z is a k-vector. The above system is characterised by the matrix pencil pF − G, where p = d/dt denotes the derivative operator; pF − G completely characterises the state space description and it is referred to as the implicit system pencil . An alternative matrix pencil form for the state space description is obtained by taking Laplace transforms of (2.1), which lead to the s-domain description sx(s) ˜ − x(0) = Ax(s) ˜ + Bu(s) ˜ y(s) ˜ = Cx(s) ˜ + Du(s) ˜ where x(s), ˜ u(s), ˜ y(s) ˜ denote the Laplace transforms of x(t), u(t), y(t) vectors respectively and x(0) the initial value of x(t). We may express the above in a matrix form as        sI − A −B x(s) ˜ x(0) sI − A −B = , P(s) = (2.4) −C −D u(s) ˜ −y(s) ˜ −C −D and the matrix coefficient P(s) is a matrix pencil entirely characterising the state space model and it is known as the Rosenbrock System Matrix Pencil [58].

2.2.2

Polynomial Models

The state space description of a linear system assumes that the system is described in terms of first order differential equations; however, this is not the most general internal description for linear systems. For a number of processes the most natural description is that defined by the general differential system [58], [6]:

Σp

¯ ¯ d A(p)v(t) = B(p)u(t) p= ¯ ¯ y(t) = C(p)v(t) + D(p)u(t) dt

(2.5)

¯ ¯ ¯ ¯ where A(p), B(p), C(p), D(p) are polynomial matrices in p of dimensions n¯ × n, ¯ n¯ × p, ¯ m × n¯ respectively, and v(t) is a vector valued function with values in Rn¯ known as pseudo-state vector [6]. The above description is known as Polynomial Model Description (PMD) and may be also represented as        ¯ ¯ A(p) −B(p) 0 v(t) 0 ξ (t) = (2.6) = ⇔ T (p) ¯ ¯ −y(t) −C(p) −D(p) u(t) −y(t) where T (p) is known as the Rosenbrock’s System matrix. The relationship between PMDs and state space models in extensively treated in [58]. Such models are very important for electromechanical systems, or models derived by the Lagrange methodology.

2.2.3

Transfer Function Descriptions

The input-output, or transfer function model is described by y(s) ˜ = G(s)u(s), ˜ G(s) = C(sI − A)−1 B + D

(2.7)

52

N. Karcanias and E. Milonidis

where G(s) is an m× p rational matrix. The transfer function may also be described in a matrix fraction description form as G(s) = Nr (s)Dr (s)−1 = Dℓ (s)−1 Nℓ (s)

(2.8)

where Nr (s), Nℓ (s) are the m × p right, left polynomial matrix numerators respectively and Dr (s), Dℓ (s) are the p × p, m × m polynomial matrix denominators correspondingly. It will be assumed that Dr (s), Nr (s) are right coprime and Dℓ (s), Nℓ (s) are left coprime (see background on polynomial matrices below) [1], [6], [18]. Using (2.7) and the factorisation (2.8) we can readily obtain the following description ˜ = Nℓ (s)u(s) ˜ Dℓ (s)y(s)

(2.9)

˜ = Dℓ (s)−1 u(s) ˜ the description and by introducing the vector h(s) ˜ ˜ y(s) ˜ = Nr (s)h(s), u(s) ˜ = Dr (s)h(s) The above two lead to the following input output type representations for the system   y(s) ˜ = 0, Tℓ (s) = [Dℓ (s)Nℓ (s)] [Dℓ (s), Nℓ (s)] −u(s) ˜       y(s) ˜ Nr (s) ˜ Nr (s) = h(s), Tr (s) = u(s) ˜ Dr (s) Dr (s)

(2.10)

(2.11)

(2.12)

The first, based on Tℓ (s), is referred to as Kernel input-output description, whereas the second based on Tr (s) as a parametric input-output description. The matrices Tℓ (s), Tr (s) based on coprime MFDs will be called right-, left- composite matrices.

2.3

Background on Polynomial Matrices and Matrix Pencils

The study of algebraic structure of linear systems represented by state space, or transfer function models heavily relies on the theory of polynomial matrices [13], [18] and matrix pencils [13]; here we review some of the fundamentals of their structure and introduce some useful notation. We consider matrices T (s) of dimension q × r with elements from the field of rational functions R(s), or the ring of polynomials R[s]; such matrices are called respectively rational, polynomial. The rank of T (s) over R(s) is denoted by ρ = rank{T (s)} and will be called the normal rank of T (s). T (s) may be viewed as a function of the complex variable s and thus for some s = z, rank{T (s)} = ρz < ρ ; such values s = z are called zeros of T (s) and ρz is called the local rank of T (s). The structure of zeros of T (s) is linked to study of certain form of equivalence defined on such matrices, which reveals the zeros as roots of invariant polynomials. Let T1 (s), T2 (s) be q × r polynomial matrices. These matrices are called R[s] -unimodular equivalent [13], or simply R[s]-equivalent, if there exist polynomial matrices Uℓ (s),Ur (s) of dimension q × q, r × r respectively with the property |Ur (s)| = c1 6= 0, |Uℓ (s)| = c2 6= 0 ( | · | denotes determinant) and called R[s]-unimodular such that: T1 (s) = Uℓ (s)T2 (s)Ur (s)

(2.13)

Structural Methods for Linear Systems: An Introduction

53

The above relation introduces an equivalence and for any matrix T (s) we have an equivalence class and associated invariants which are described by the following result. Smith Form Theorem [13]: If T (s) is a q × r polynomial matrix with normal rank ρ ≤ min(q, r) there exist unimodular matrices Uℓ (s),Ur (s) such that   0 0 f1 (s)   ..   .     (2.14) .. Uℓ (s)T (s)Ur (s) =   = S(s)   .    0 fρ (s)  0 0

where S(s) is q × r polynomial matrix f1 (s), . . . , fρ (s) are uniquely defined and f1 (s)/ f2 (s)/ . . . / fρ (s) denotes the successive divisibility (i.e., f1 (s) divides f2 (s) etc.).



The polynomials fi (s) are called invariant polynomials of T (s) and the set { fi (s), i = 1, . . . , ρ } is a complete invariant under R[s]-equivalence. The roots of fi (s) (including multiplicities) define finite zeros of T (s). The structure of these zeros (multiplicities and groupings) is defined by factorising the fi (s) into irreducible factors over the real, or complex numbers; for every zero z we define the set of z-elementary divisors by grouping all factors with root at z. The set of elementary divisors (for all zeros) is also a complete invariant under R[s]-equivalence. Note that although the Smith form under R[s]-equivalence defines the finite zero structure (finite frequencies), it does not convey any information on the structure at infinity; an alternative form is required and will be considered in a later section.

2.3.1

Matrix Divisors and Minimal Bases

Definition 2.1. For a matrix P(s) ∈ R p×m [s], rank{P(s)} = m we define [39], [18], [1]: (i) A matrix R(s) ∈ Rm×m [s] such that P(s) = P′ (s)R(s) is called a right matrix divisor (RMD) of P(s). If R(s) is any other RMD and R(s) = W (s)R(s) then R(s) is called a right greatest matrix divisor (RGMD) of P(s). If rank{P(s)} = p, the notions of left matrix divisors (LMD) and greatest left matrix divisors (LGMD) are defined similarly. (ii) Let P(s) = [p1 (s), . . . , pm (s)] ∈ R p×m [s] and rank{P(s)} = m. The set IP = {δi : δi = ∂ (pi (s)), i ∈ m} ˜ of the degrees of the columns pi (s) is defined as the set of column degrees and cP = ∑m i=1 δi as the column complexity of P(s). Row degrees and row complexity are defined in a similar manner. If pi (s) = pi,h sδi + . . . + pi,0 , then we may express it as

ˆ P(s) = [p1,h , . . . , pm,h ]diag{sδ1 , . . . , sδm } + P(s)

(2.15)

ˆ have degrees less than δi and Ph = [p1,h , . . . , pm,h ] = [P(s)]h ∈ R p×m The columns of P(s) is referred to as the high column coefficient matrix of P(s). If rank{Ph } = m, then P(s) is called column reduced. The high row coefficient matrix and row reducedness notions are defined similarly. (iii) A matrix P(s) ∈ R p×m [s] with rank{P(s)} = m is called right irreducible, or least degree, if all RMDs are R[s]-unimodular. A left irreducible matrix is defined in a similar manner. P(s) ∈ R p×m [s] with rank{P(s)} = m (or p) is called a minimal basis [12] if it is: (a) Right (left) irreducible (b) Column reduced.

54

N. Karcanias and E. Milonidis

If Pr , {Pi (s) ∈ Rki ×m [s], i ∈ ν˜ } is a set of matrices, then the matrix Pr   P1 (s) v   Tpr (s) ,  ...  ∈ Rk×m [s], k = ∑ ki Pv (s)

i=1

is called a matrix representative of Pr . Pr is right regular, if ρ (Tpr (s)) = m. If Pr is right regular, then a right common matrix divisors (RCMD) and a right greatest common matrix divisor (RGCMD) of Pr is defined as a RMD and a RGMD of Tρr (s) respectively. The set Pr is called right coprime (RC), if it is right regular and Tρr (s) is right irreducible. For a set of matrices with the same number of rows, the notions of a matrix representative, left regularity, left common matrix divisors (LCMD), left greatest common matrix divisor (LGCMD) and left coprimeness are defined in a similar manner. The above definitions for matrices over R[s] have their counterparts for matrices over R pr (s), R p (s) the ring proper and Ω −stable rational functions respectively with the appropriate changes [76], [77], [80]. Proposition 2.1. For an m × p rational matrix G(s) consider the matrix fraction description G(s) = Nr (s)Dr (s)−1 = Dℓ (s)−1 Nℓ (s) where Nr (s), Nℓ (s) are the m × p right, left polynomial matrix numerators respectively and Dr (s), Dℓ (s) are the p × p, m × m polynomial matrix denominators correspondingly. Then, (i) The pair Dr (s), Nr (s) is right coprime, iff the matrix Tr (s) = [Nr (s)t , Dr (s)t ]t has full rank and no zeros. (ii) The pair Dℓ (s), Nℓ (s) is left coprime, iff the matrix Tℓ (s) = [Dℓ (s), Nℓ (s)] has full rank and no zeros. With a polynomial, or rational matrix T (s) we may associate two important rational vector spaces (vector spaces of rational vectors and with scalars the rational functions R(s)) Nr (T ) = {x(s) : T (s)x(s) = 0, x(s) ∈ Rr [s]}

(2.16)

Nℓ (T ) = {yt (s) : yt (s)T (s) = 0, y(s)t ∈ R1×q [s]} Nr (T ), Nℓ (T ) are called respectively right-, left-rational vector spaces, where dimNr (T ) = r − ρ , dimNℓ (T ) = q − ρ ; with such spaces we can always define polynomial bases. If X(s) is an (r − ρ ) × r polynomial basis for Nr (T ), or of any rational vector space X with dimX = r − ρ , then it is called least degree if it has no zeros [60]. A polynomial basis X(s) = [x1 (s), . . . , xr−ρ (s)] with column degrees {d1 , . . . , dr−ρ } is said to be of least complexity, if Σ di = δ (X). Note that δ (X) denotes the degree of X(s), which is defined as the maximal of the degrees of all maximal order minors of X(s). A least degree and least complexity polynomial basis of Nr (T ) is a minimal basis and the ordered set of degrees {d1 , . . . , dr−ρ } are called right minimal indices and δr (T ) = Σ di as the right-order of T (s) [12], [60]. The notion of left minimal indices and left order are defined similarly on Nℓ (T ). Note that the sets of minimal indices are invariants of the corresponding rational vector spaces. A special case of a polynomial matrix is that of a matrix pencil [13], sF − G, where F, G are q × r real (or complex) matrices and s is an independent complex variable taking values on the compactified complex plane (that includes the point at infinity). For such matrices we define the notion of strict equivalence in the following way: Two pencils sF − G, sF ′ − G′ of dimension q × r are strict equivalent, if there exist real matrices Q, R of dimension q × q, r × r respectively such that

Structural Methods for Linear Systems: An Introduction sF ′ − G′ = Q(sF − G)R, |Q|, |R| 6= 0

55 (2.17)

The above introduces the notion of strict equivalence of matrix pencils and the equivalence classes are characterised by a set of invariants which will be defined subsequently. Pencils may ˆ ′ where s, sˆ are independent complex varibe represented in a homogenous form as sF ′ − sG ables. Frequencies on the compactified complex plane are represented as ordered pairs (α , β ), where at least one of the α , β is 6= 0. Pairs (α , β ) : β 6= 0 correspond to finite frequencies. With the homogeneous pencil sF − sG ˆ we associate the single variable pencils sF − G, sF − sG. ˆ The sets of invariants that may be defined under the strict equivalence are known as Kronecker Invariants and are summarised below:

2.3.2

Strict Equivalence Invariants of Matrix Pencils [13]

Elementary Divisors: The Smith form of the homogeneous pencil sF − sG ˆ defines a set of ˆ τ , sˆq . The set of elementary divisors elementary divisors of the following type: s p , (s − α s) ˆ τ , are called finite elementary divisors (fed) of sF − G, whereas those of the sˆq type s p , (α − s) are called infinite elementary divisors (ied) of sF − G.

Minimal Indices: A matrix pencil sF − G, where at least one of Nr {sF − G}, or Nℓ {sF − G} are non trivial {6= 0} are called singular, otherwise they are called regular. If Nr {sF − G} = 6 0, then the minimal indices of this rational space are denoted Ic (F, G) = {εi , i = 1, . . . , µ } and are referred to as column minimal indices (cmi) of the pencil. Similarly, if Nℓ {sF − G} then the minimal indices of this rational vector space are denoted by Ir (F, G) = {η j , j = 1, . . . , ν } and referred to as row minimal indices. This set of invariants are complete [36] for the strict equivalence of matrix pencils, that is they uniquely characterise the strict equivalence class of a matrix pencil. There is a uniquely defined element by the invariants which is referred to as Kronecker canonical form [13]. Kronecker Canonical Form of a matrix Pencil: Consider a matrix pencil sF − G and assume that its Kronecker invariants are of the types: elementary divisors of the type: {(s − α )τ , . . . ; sˆq } column minimal indices: {ε1 = . . . = εt = 0, ε j > 0, j = t + 1, . . . µ }, row minimal indices: {η1 = . . . = ησ = 0, ηi > 0, i = σ + 1, . . . , ν }. There always exists a pair of strict equivalence transformation Q, R such that Q(sF − R)R = blockdiag{Oσ ,t ; . . . Lε (s), . . . , Lη (s), . . . ; sFw − Gw }     I 0 : (η +1)× η block (2.18) Lε (s) = s[Iε 0]−[0, Iε ] : ε ×(ε +1) block, Lη (s) = s η − 0 Iη sFw − Gw = block − −diag{sI − J(a); . . . ; sHq − Iq ; . . .}

where J(a) is the τ × τ Jordan block associated with (s − α )τ and Hq is a q × q nilpotent block (1s on the first super diagonal and the rest zero).

2.4 2.4.1

Dynamics, Stability, Controllability and Observability Solution of State Space Equations

The eigenvalues and eigenvectors of the matrix A define the internal dynamics of the system. The set of eigenvalues of the matrix A will be referred to as the system poles and with every

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eigenvalue λ of A we have two eigenvalue-eigenvector problems Au = λ u, vt A = vt λ , vt u = 1

(2.19)

where u is a right - and vt a left-eigenvector and the triple (λ , u, vt ) is called a system mode. If φ (A) is the set of distinct eigenvalues then the structure of λ ∈ φ (A) is defined by the λ -Segre characteristic, S(λ ) = {νi , i ∈ q} ˜ that is the dimensions of λ -Jordan blocks in the Jordan form of A. Alternatively, S(λ ) is defined by the set of degrees of the (s − λ )ν type of the Smith form q of sIn −A, ∑i=1 νi = ρ is called the algebraic multiplicity and q the geometric multiplicity of λ . The maximal of all geometric multiplicities of the eigenvalues of A is referred to as the Segre index of A. The structure of eigenvalues defines the type of free motions and has implications on internal stability. For the state space model S(A, B,C, D), which is excited by an initial condition x(0) = x0 and a control input u(t) the corresponding solutions for the state and output trajectories are given by [1] Z x(t) = eAt x0 +

y(t) = CeAt x0 +

Z t 0

t

0

eA(t−τ ) Bu(τ )d τ ,

CeA(t−τ ) Bu(τ )d τ + Du(t)

(2.20)

The solutions above consist of two parts: The integral parts of (2.20) defining the forced response, which are contributed from inputs (and disturbances) and the remaining part of the solutions, which are the free response contributions where the matrix eA(t−t0 ) is called the state transition matrix. Taking Laplace transforms of (2.1), the solutions are expressed as: x(s) = (sI − A)−1 x0 + (sI − A)−1 Bu(s)

y(s) = C(sI − A)−1 x0 + {C(sI − A)−1 B + D}u(s)

(2.21)

where G(s) = C(sI − A)−1 B + D is the transfer function matrix of the system. Assuming for the shake of simplicity that A = U Λ V, where Λ is diagonal, and using the dyadic expansion of eAt , the spectral form of the state trajectory x(t) for a zero input response can be written as x(t) = eAt x(0) = eλ1 t u1 v1 x0 + . . . + eλn t un vtn x0 ⇒ x(t) =

n

∑ eλ t hvi , x0 iui 1

(2.22)

i=1

where h·, ·i denotes the inner product. The spectral form of the state trajectory when x(0) = 0 (initial condition) and for a nonzero input is x(t) =

Z t 0

eA(t−τ ) Bu(τ )d τ =

Z t n

∑ ui eλ (t−τ ) vti Bu(τ )d τ i

(2.23)

0 i=1

and if ∗ denotes convolution, then the total output response is then n

n

y(t) =

∑ γi eλ t hvi , x0 i + ∑ γi eλ t ∗ hβi , u(t)i

i=1

where

i

i

(2.24)

i=1

 t  vt1 β1  ..   ..  CU = C[u1 . . . u1 ] = [γ1 . . . γn ], V B =  .  B =  .  βnt vtn 

(2.25)

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57

Internal-External and Total Stability

For linear, time invariant systems the notions of stability, which are more frequently used are defined next. We consider stability of equilibrium points, whereas stability of motion is always reduced to the previous case. Note that the origin (x = 0) is always an equilibrium point for S(A, B,C, D) models. Definition 2.2 ([1). [7]] The state space model S(A, B,C, D) will be called: (i) Internally stable in the sense of Lyapunov (LIS), if for any initial x(0) the zero input response (free motion, u(t) = 0) remain bounded for all t ≥ 0. (ii) Asymptotically internally stable, if for any initial state x(0) the zero input response remains bounded for all t ≥ 0 and tends to zero as t → ∞. This property will be referred to in short as internal stability (IS). (iii) Bounded Input Bounded Output Stable (BIBO), if for any bounded input the zero state output response x(0) = 0 is bounded. (iv) Totally stable (TS) if for any initial state x(0) and any bounded input u(t), the output, as well as all state variables are bounded. The notion of BIBO stability refers to the transfer function description and may also be referred to as external stability. A number of criteria for these properties, based on eigenvaluespoles, are summarised below [7]. Theorem 2.1. Consider the system S(A, B,C, D) with G(s) transfer function and let {λi = σi + jωi , i ∈∼n}, {p j = σ j + jω j , j ∈∼}v be the sets of eigenvalues, poles respectively. The system has the following properties:

(i) Lyapunov internally stable, iff σi ≤ 0, all i ∈∼n, and those with σi = 0 have a simple structure (algebraic multiplicity is equal to the geometric multiplicity). (ii) Asymptotically internally stable, iff σi < 0, all i ∈∼n. (iii) BIBO stable, iff σ¯ j < 0 all i ∈∼υ. (iv) Totally stable, if it is Lyapunov internally stable and BIBO stable. Note that IS implies BIBO-S and thus TS. BIBO stability does not always implies IS, since transfer function and state space are not always equivalent. If the two representations are equivalent (when system is both controllable and observable), then BIBO-stability is equivalent to IS and thus TS. Remark 2.1. Eigenvalues and poles are indicators of stability. Equivalent tests for stability, without computing the eigenvalues, poles are defined on the characteristic, pole polynomial by the Routh-Hurwitz conditions and equivalent tests.  Tests for BIBO stability and corresponding indicators may be formulated on the impulse response of the system. For discrete time systems, the stability definitions and criteria are similar to those discussed here with the obvious changes (open closed right half place, becomes open, closed unit circle etc.). A stronger notion of stability is the notion of Finite Settling Time Stability (reaching the steady state in finite number of discrete time steps), which does not occur in continuous time system and it is related to Dead-Beat control [39].

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2.4.3

Controllability and Observability

Some of the most fundamental concepts in control theory characterising the coupling of internal mechanism to its environment are those of controllability and observability [19], [20], [1], [7], [65]. These concepts are defined below. Definition 2.3. Consider the state space model S(A, B,C, D) and let X be its state space (Rn ). Then the system is called: (i) State controllable, or simply controllable, if there exists of finite time T > 0, such that for initial state x(0) ∈ X and any x1 ∈ X , there exists an input u(t), defined on [0, T ] that will transfer x(0) to x1 at time T (x(T ) = x1 ). Otherwise, it is called uncontrollable. (ii) State observable, or simply observable, if there exists of finite time T > 0, such that for initial state x(0), the knowledge of the input u(t) and output over the time interval [0, T ] suffices to determine the state x(0); otherwise, the system is called unobservable. (iii) Output controllable, if for any output vector y1 there exists T > 0 and an input u(t) defined over [0, T ] that transfers the output y(0) = 0 to y(T ) = y1 . Furthermore, if it is output controllable and the output can be steered over any interval of time on a pre-assigned curve, it will be called output function controllable. Note that strictly speaking, transferring to the zero state any initial state x(0) is referred to as controllability, whereas transferring the origin x(0) = 0 to any final state in the state space is referred to as reachability. These differences become essential in more general families of models than those considered here. The problem of observability has to do with reconstruction of initial conditions (measurement) and thus reconstruction of state trajectories; thus, we may refer to observability as state reconstructibility. A summary of tests for the above properties is given below. (see for instance [1], [7], [58], [22], [34] etc.) Theorem 2.2. (Controllability/Observability criteria): For the state space model S(A, B,C, D) with n, m, p number of states, outputs, inputs and with (λ , u, vt ) as the triple of eigenvalues, right and left-eigenvector respectively we have the following results: (a) Controllability : The system is state controllable, iff either of the equivalent conditions hold true: (i) All rows of e−At B are linearly independent on [0, +∞) over C. (ii) All row of (sI − A)−1 B are linearly independent over C. R tτ (iv) The controllability Grammian, Wc = 0T eAτ BBt eA d τ is nonsingular for all T (iii) If A has distinct eigenvalues then for every λ , vt B 6= 0t , any T > 0. (v) The n × (np) controllability matrix Qc = [B, AB, . . . , An−1 B] has rank n. (vi) The controllability pencil , Pc (s) = [sI − A, −B] has rank n for all s ∈ C, or equivalently it has no fed. (vii)The restricted controllability pencil, Rc (s) = sN − NA ( N is a left annihilator of B) has rank n for all s ∈ C, or equivalently, it has no fed. (b) Observability : The system is state observable, iff either of the equivalent conditions hold true. (i) All columns of CeAt are linearly independent on [0, +∞) over C. (ii) All columns of C(sI − A)−1 are linearly independent over C. (iii) If A has distinct eigenvalues then forR every λ , Cu 6= 0. t (iv) The observability Grammian, Wo = 0T eA τ Ct CeAτ d τ is nonsingular for any T > 0.

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(v) The nm × n observability matrix, Qo = [Ct , At Ct , . . . , (At )n+1Ct ]t has rank n. (vi) The observability pencil, Pot (s) = [sI − At , −Ct ]t has rank n for all s ∈ C, or equivalently, it has no fed. (vii)The restricted observability pencil, Ro (s) = sM − AM (M is a right annihilator of C), has rank n for all s ∈ C, or equivalently, it has no fed. (c) Output Controllability: Output controllable, iff either of the equivalent conditions hold true: (i) All rows of G(s) are linearly independent over C. (ii) The matrix Qoc = [D,CB,CAB, . . . ,CAn−1 B] has rank m. (iii) The system is output function controllable, iff rank of G(s) is equal to m, over R(s). The above tests define equivalent indicators for the controllability, observability properties and demonstrate that both controllability, observability properties express the interaction of internal mechanism with the environment represented by the inputs, outputs and are shaped at the instrumentation stage of the process design. Remark 2.2. A generic system is always state controllable and observable. Thus, if all parameters in S(A, B,C, D) with fixed dimensions, are free the cases of uncontrollability, unobservability are nongeneric. Note however, that because of the process interconnections, we frequently deal with S(A, B,C, D) families with a fixed underlined graph and it is then the graph that may determine the controllability, observability properties. These graph based notions of controllability, observability are referred to as “structural”.  Remark 2.3. Controllability, observability tests are based on the notions of rank of matrices, which generically is full. The degree of nonsingularity, singularity, measured by the smallest singular value, or the condition number is important indicators of “how well” the system is controllable, observable. Such issues are essential in design problems when either there are restriction on the size of gains in feedback matrices, or when there are limitation in the energy of control signals etc.  The questions of controllability, observability may be equivalently interpreted as questions of controlling, or observing the system eigenvalues. This interpretation highlights the internal structure, illustrates certain aspects of “mode connectibility” to inputs, outputs and provides some structural tests for controllability, observability. Using the Jordan description of state equations (A is in Jordan canonical form) alternative tests for controllability, observability may be stated and are found in [7]. Some important implications of these tests are. Corollary 2.1. Let q be the Segre index of A (the maximal of all geometric multiplicities of the eigenvalues) and let Σ be the family of n, m, p fixed dimension systems, having the same q, but with otherwise arbitrary parameters. (i) Necessary condition for every S(A, B,C, D) ∈ Σ to the controllable and observable is that p ≥ q and m ≥ q respectively. (ii) Every system in Σ for which p < q, m < q is uncontrollable, unobservable respectively. Note that the value of q is frequently a property that may be inferred from the structure (diagram, graph) of the process and the nature of subprocesses and thus q may serve as a prime indicators on the necessary minimum number of inputs and outputs.

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2.4.4

System Minimality

The concepts of uncontrollability, observability are essential in the establishment of the relationships between internal and external descriptions. This is illustrated by the following important result [20]. Theorem 2.3. For the system S(A, B,C, D), there is a coordinate transformation x′ = Ux such that the corresponding state space description S′ (A′ , B′ ,C′ , D′ ) has the form  ′   ′  ′   ′  x˙co Aco A′12 A′13 A′14 xco Bco ′  ′ ′ ′   B′   x˙co   xco 0  co   ′  =  0 Aco A23     (2.26)  x˙   0 0 A′ 0   x′  +  0  u co co co ′ ′ ′ x˙co 0 0 0 Aco xco 0

′ ′ y = [0,Cco ,Cco , 0]x′ + Du ′ , x′ , x′ , x′ are subvectors of x′ which are: controllable-unobservable, controllablewhere xco co co co observable, uncontrollable-observable and uncontrollable-unobservable states. Furthermore, the systems S(A, B,C, D) and S′ (A′ , B′C′ , D′ ) have the same transfer function which is expressed as: ′ G(s) = Cco (sI − A′co )−1 B′co + D (2.27)

This result expresses the Kalman decomposition of the state space, which shows that the transfer function represents only the controllable and observable subsystem, but not in general the whole S(A, B,C, D) system. Remark 2.4. The transfer function and the state space descriptions are completely equivalent, iff the system is both controllable and observable. The dimension of the controllable and observable subsystem is defined as the McMillan degree of G(s) and it is denoted by δM (G). We will see later on that δM (G) is the degree of the pole polynomial of G(s). Furthermore, any realization S(A, B,C, D) is observable and controllable iff its dimension is equal to δM (G).  Two important concepts related to controllability, observability are those of stabilizability, detectability [74]: Definition 2.4. The system S(A, B,C, D) will be called: (i) Stabilizable, if the unstable eigenspace of A is contained in the controllable subspace of the system. (ii) Detectable , if the unobservable subspace of the system is contained in the stable eigen space A. Corollary 2.2. The system S(A, B,C, D) is: (i) Stabilisable, iff its uncontrollable eigenvalues (in the modal sense) are stable. (ii) Detectable, iff its unobservable eigenvalues (in the modal sense) are stable. Those two more relaxed conditions (than controllability, observability) have implications as far as using transfer functions as design models. In fact, if the system is both stabilisable and detectable, then the transfer function may be used for feedback design, but not otherwise. The uncontrollable, unobservable, uncontrollable and unobservable eigenvalues are also referred to as input-output, input-output decoupling zeros (idz, odz, i-odz) [58] and the corresponding sets, including multiplicities, will be denoted by ZID , ZOD , ZIOD respectively. These sets are computed by the following property [58], [36].

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61

Corollary 2.3. For the sets of decoupling zeros we have the properties: (i) ZID is defined by the roots of the f-ed of Pc (s) = [sI − A, −B], or Rc (s) = sN − NA. (ii) ZOD is defined by the roots of f-ed of Po (s) = [sI − At , −Ct ]t , or Ro (s) = sM − AM. (iii) ZIIOD = ZID ∩ ZOD . Remark 2.5. The system is stabilisable and detectable iff the f-ed of Pc (s) (or Rc (s)) and Po (s) (or Ro (s)) are stable.  Output controllability is only concerned with getting to y(T ) = y1 final value, but nothing in the definition states that the output will stay at y(t) = y1 value, or track a specified function c(t). Output function controllability addresses this property, but the required u(t) might contain impulses. Output function Controllability with a smooth input u(t) will be referred to as smooth output controllability [60].

2.5 2.5.1

Poles and Zeros of State Space Model Eigenvalues, Eigenvectors and Free Rectilinear Motions

For a single input, single output (SISO) system represented by a rational transfer function g(s) where g(s) = n(s)/d(s), where n(s), d(s) are coprime polynomials with deg{n(s)} = r and deg{d(s)} = n we define as finite poles the roots of d(s) and as finite zeros the zeros of n(s). If r < n we say that g(s) has an infinite zero of order n − r, and if r > n then g(s) has an infinite pole with order n − r. Finite and infinite poles have the property that the gain of the transfer function becomes unbounded (tends to infinity) and finite and infinite zeros are those frequencies for which the gain vanishes. In this sense, the notion of poles and zeros are dual since the first characterises resonance and the second antiresonances. It is this basic property that motivates a number of the definitions and problems that relate to multivariable poles and zeros. For a state space model the internal natural dynamics are defined by the zero input response (u(t) = 0) which in turn is characterised by the eigenvalues and eigenvectors of the state matrix A (see System Description in Time Domain), which determine the solution space of the autonomous system. Using S(Φ , Ω ) description for u(s) ≡ 0 we get the zero input differential description.       A 0 x(t) I O x(t) ˙ S(H, Θ ) : (2.28) = C −I y(t) OO y(t) ˙ which of course is equivalent to

x(t) ˙ = Ax(t), y = Cx(t)

(2.29)

Definition 2.5. We define as poles of the state space model the eigenvalues of A and as pole directions the corresponding eigenvectors. If (λ , u) is a pair of an eigenvalue and eigenvector of A, then for every initial condition x(0) = cu (c constant) the corresponding solution of S(H, Θ ) is

62

N. Karcanias and E. Milonidis x(t) = exp(λ t)cu, y(t) = exp(λ t)Cu

(2.30)

which are λ exponential trajectories in the state and output spaces along the constant direction xλ = u and yλ = Cu. Such straight line exponential motions are frequently called rectilinear motions and xλ , yλ are called state pole and output pole directions correspondingly. The fundamental operator describing the pole structure is the pencil W (s) = sI − A

(2.31)

which is called the state pole pencil. The eigenstructure of A is defined by its Jordan form and this is defined algebraically by the elementary divisors of the Smith form of W (s) [13]. The presence of elementary divisors with degree higher than one implies the existence of Jordan blocks for the corresponding eigenvalues. The dynamic characterisation of such multiple eigenvalues is in terms of generalised rectilinear motions (involving terms of the type exp(λ t)t i , i = 1, 2, . . .). The pole-zero duality motivates the definition of the output zeroing problem, that is investigation of the type of solutions of the system for which the output is identically zero (the output now is zero instead of the input for the case of pole). When y(t) ≡ 0 the S(Φ , Ω ) description is reduced to the system       I 0 x(t) ˙ A B x(t) = (2.32) S(Γ , ∆ ) : 0 0 u(t) ˙ C D u(t) the solutions of which describe the zero dynamics of the original system. The fundamental pencil of S(Γ , ∆ ) is the Rosenbrock system matrix pencil [58] and its structure describe the zero structure of the system [46]. The dynamic characterisation of the zero structure benefits by extending first the notion of free rectilinear motion to that of the forced rectilinear motion.

2.5.2

Forced Rectilinear Motions and Frequency Transmission

Free rectilinear motions are characterised by frequencies, which are fixed and are the eigenvalues of the matrix A. We consider now the generation of rectilinear motions for arbitrary complex frequencies λ , but in a forced manner. This is described next [46]: Proposition 2.2. For the state space model S(A, B) a rectilinear motion in the state x(t) = exp(λ (t))xλ , t ≥ 0, where x(0) = xλ and λ a given complex number can be generated, if and only if the input is also rectilinear of the same type, i.e. u(t) = exp(λ t)uλ , t ≥ 0 and the triple ( λ , xλ , uλ ) satisfy the condition (λ I − A)xλ = Buλ

(2.33)

This result establishes the property that every complex frequency can be transmitted in a simple rectilinear way through the input and state space of the system. The totality of xλ vector solutions of (2.33) is denoted by T (λ ) and it is called the transmission space of λ . Note that (2.33) may also be written as:   x [λ I − A, −B] λ = 0 (2.34) uλ

and this demonstrates that the triple (λ , xλ , uλ ) satisfies a generalised eigenvalue eigenvector problem on the pencil Pc (s) = [sI − A, −B] which is known as input-state space pencil. A number of interesting properties are listed below [22], [26].

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63

Proposition 2.3. For every λ ∈ C we have the properties: (i) The set of xλ solutions by T (λ ) define a p-dimensional linear vector space. (ii) If λ is not an eigenvalue of A then for any arbitrary uλ the corresponding xλ is defined by xλ = (λ I − A)−1 uλ . (iii) If N is an (n − p) × n left annihilator of B (NB = 0 and N is a basis matrix for left null space of B) and B† is a p × n left inverse of B (B† B = I p ), then the computation of ( λ , xλ , uλ ) may be reduced to the decoupled set of conditions (λ N − NA)xλ = 0, uλ = B† (λ I − A)xλ

(2.35)

(iv) For any triple (λ , xλ , uλ ) there exists a p × n matrix L, which may be interpreted as a state feedback such that uλ = Lxλ . Under this condition equation (2.34) becomes (λ I − A − BL)xλ = 0

(2.36)

and thus (λ , xλ ) becomes a pair of closed loop eigenvalue and eigenvector under an appropriate state feedback. The triples (λ , xλ , uλ ) defines closed loop eigenvalues and eigenvector and the space Tx (λ ) defines the family of closed loop eigenvectors for a given λ which may be generated under suitable state feedback. The pencil Rc (s) = sN − NA that provides the decoupled characterisation in (2.35) is referred to as restricted input state pencil. The structure of Pc (s), Rc (s) describe the controllability properties of the system and are instrumental in generating the closed loop eigenstructure. The result for the frequency transmission may now be extended to a general frequency transmission from input to state and then output spaces [46], [32]: Theorem 2.4. For the state space model S(A, B,C, D) a pair of forced rectilinear motions in the state x(t) = exp(λ t)xλ ,t ≥ 0, where x(0) = xλ and output y(t) = exp(λ t)yλ ,t ≥ 0, where λ is a given complex frequency can be generated in a forced mode, if and only if the input is also rectilinear of the same type, i.e. u(t) = exp(λ t)uλ ,t ≥ 0 and the quadruple (λ , xλ , uλ , yλ ) satisfies the condition      λ I − A −B xλ 0 = (2.37) −C −D uλ −yλ For every complex λ the above condition is always satisfied for a triple (xλ , uλ , yλ ). The family of solutions of all xλ is the space Tx (λ ) and the corresponding set of solutions yλ is denoted by Ty (λ ) and called the output transmission space of λ . Note that the transmission in the input space Tu (λ ) covers the entire input space. A diagram illustrating the general frequency transmission is shown below. It is readily seen that condition (2.37) represents a generalised eigenvalue eigenvector problem for λ and vector [xtλ , utλ , ytλ ]t which is defined on the implicit system pencil sΦ − Ω .

2.5.3

Frequency Transmission Blocking and State Space Zeros

The general frequency transmission problem enables the formulation of “frequency transmission blocking” [46], [32] as shown below:

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N. Karcanias and E. Milonidis

INPUT SPACE U

ue u(

D y(

)

OUTPUT SPACE Y

t

)

B

ye

C

t

STATE SPACE X

( )

xe

t

A

Fig. 2.1. General Frequency Transmission Theorem 2.5. For the state space model S(A, B,C, D) there exists a forced rectilinear motion in the state x(t) = exp(z(t))xz ,t ≥ 0, where x(0) = xz such that the output is identically zero, i.e. y(t) ≡ 0, if and only if the input is also rectilinear of the same type, i.e. u(t) = exp(zt)uz ,t ≥ 0 and the triple (z, xz , uz ) satisfies the condition    zI − A −B xz =0 (2.38) −C −D uz This result establishes a link of the output zeroing problem with the existence of complex numbers z for which the generalised eigenvalue eigenvector problem has a solution; such solutions exist as long as the right null space of P(z), Nr {P(z)} = 6 0 for some complex number z. The nature of solutions depends on the existence of the rational vector space defined by Nr = Nr {P(s)} = {ξ (s) polynomial vector : P(s)ξ (s) = 0}

(2.39)

and referred to as right system space. Systems may be classified as follows: Definition 2.6. A system S(A, B,C, D) is called right regular if Nr = 0 and it is called right singular if Nr 6= 0. A right regular system implies that the rational space Nr is the 0 space; however, Nr {P(z)} may be nonzero for some specific complex number z. For right singular systems, condition (2.38) is satisfied for any complex z. Remark 2.6. Note that if ρ = rankR(s) {P(s)} ≤ min(m + n, p + n) is the normal rank of P(s) we have the following characterisation: System is Right Regular ⇔ ρ = n + p,

(2.40)

System is Right Singular ⇔ ρ < n + p

(2.41) 

Structural Methods for Linear Systems: An Introduction

2.5.4

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Right Regular Systems

For this class of systems the Output Zeroing Problem establishes the link with the Smith structure of P(s) as indicated below: Corollary 2.4. If S(A, B,C, D) is right regular, then the Output Zeroing Problem has a solution for some complex number z, if and only if z is a zero of the Smith form of P(s). Definition 2.7. For any system S(A, B,C, D) we define as its set of state space zeros the roots of the invariant polynomials of the Smith form of P(s) (including multiplicities). For right regular systems to every zero z there corresponds a pair (xz , uz ) which are called respectively state and input zero directions. For the case of elementary divisors with degree higher than one, higher order rectilinear motions exist (based on P(s) pencil) which characterise such multiplicities. In the following we shall assume for the sake of simplicity that all elementary divisors of P(s) are linear (degree one). The properties of the zeros and zero directions are summarised below.

2.5.5

Properties of Zero Directions

Theorem 2.6. [32] For a right regular system having a set of distinct zeros Λz = {z1 , . . . , zk } the following properties hold true: (i) For every zero z the pair (xz , uz ) is uniquely defined modulo c, constant. The triple (z, xz , uz ) satisfies the generalised eigenvector condition and the output zeroing condition Axz = zxz + Buz , Cxz + Duz = 0

(2.42)

(ii) The set of 1-dimensional spaces {xzi }, i = 1, 2, . . . , k are linearly independent and their direct sum Vz = {xzi }, ⊕ . . . ⊕ {xzk }, is a maximal dimension space for which the equivalent conditions hold true: (a) AVz ⊂ Vz + B, Vz ∩ B = 0, B = Im {B} (2.43)

(b) For any n × k basis matrix V for V there exists a p × k matrix U and a k × k matrix A¯ z , where A¯ z has a Jordan form uniquely defined by the set of elementary divisors of the Smith form of P(s) : AV = V A¯ z +U (2.44)

The above results stated for distinct zeros also generalises to the general case with the difference that the 1-dimensional spaces {xz } now become Jordan structure type spaces [17]. The space Vz associated with zeros is referred to as the Fixed spectrum output Nulling space.

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2.5.6

Right Singular Systems

The class of right singular systems includes those with more inputs than outputs (p > m), as well as those for which ρ < min(m + n, p + n). The latter condition always implies right singularity and such systems are called degenerate; if ρ = min(m + n, p + n) the system is called nondegenerate. Degeneracy always implies right singularity but not vice versa. For right singular systems the rational vector space Nr defined in (2.39), we may always define minimal polynomial bases, i.e., matrices with no zeros and full rank high coefficient matrix [31], [23]. Such matrices are referred to as Right Nulling Minimal Bases and if r = dimNr {P(s)}, any such a basis matrix may be represented as:   X(s) Tζ (s) = , X(s) = [x1 (s), . . . , xr (s)], U(s) = [u1 (s), . . . , ur (s)] (2.45) U(s) where xi (s), ui (s) are n-, p-dimensional polynomial vectors respectively with corresponding degrees deg{xi (s)} = εi , deg{ui (s)} = εi + 1. Any pair defined by the partitioning of the columns of Tζ (s) (x(s), u(s)) is called a right pair and if x(s) = a0 + a1 + . . . + sε aε , u(s) = b0 + sb1 + . . . + sε bε + sε +1 bε +1 , then {ai }, {bi } are the supporting sets of x(s), u(s) respectively and the linear space Rε = span{a0 , a1 , . . . , aε } is the supporting linear space of xε (s). The significance of right pairs in characterising the zero dynamics of systems is discussed next: Theorem 2.7. [22], [23] Let R be a subspace of the state space of the right singular system S(A, B,C, D) and let δ i (t) denote the i-th order impulse. If x(0− ) ∈ R, necessary and sufficient condition for a control input u(t) = δ (0) (t)b0 + δ (1) (t)b1 + . . . + δ (k) (t)bk to result in y(t) ≡ 0 is that the state trajectory is of the type x(t) = δ (0) (t)a0 + . . . + δ 1 (t)a1 + . . . δ k−1 (t)ak−1 , the space span{a0 , a1 , . . . , ak−l } ⊆ R, x(0− ) ∈ span{a0 , . . . , ak−1 } and the Laplace transforms x(s), ˜ u(s), ˜ satisfy the conditions    sI − A −B x(s) ˜ =0 (2.46) −C −D u(s) ˜ The above result establishes an alternative form of output zeroing based on impulsive trajectories. Such characterisation is intimately linked to the minimal basis theory and provides an alternative, impulse based dimension, to the output nulling properties. The geometry of right nulling minimal bases has a number of properties, which are summarised below.

Geometric Properties of Right Nulling Minimal Bases Theorem 2.8. [23], [32], [70] For the minimal basis in (2.45) with the state vectors {x1 (s),. . .,xr (s)}, deg{xi (s)} = εi , {a j } is the supporting set and Rε i supporting space, the following properties hold true: (i) The vectors in the εi+1 set {a j }i are linear independent and {a j }i is a basis for the εi+1 dimensional space Rε i , which is a minimal dimension controllability subspace with the output nulling property. (ii) The set of subspaces {Rε1 , . . . , Rεr } is linearly independent and the subspace R = Rε1 ⊕ . . .⊕Rεr is the maximal output nulling controllability subspace of the system and dimR = ∑ri=1 εi + r.

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(iii) Each subspace Rε satisfies the conditions ARε ⊆ Bε + B and Rε ∩ B 6= 0; furthermore, for every ε + 1 symmetric complex set Λ there exists a basis matrix Rε , a n × (ε + 1) matrix U and a (ε + 1) × (ε + 1) matrix A with spectrum Λ such that ARε = Rε A +U

(2.47)

The subspaces V satisfying the condition AV ⊆ V + B are known as (A, B)-invariant subspaces [3], [74]; for such spaces, for any initial condition x(0) there exists an input u(t) such that the trajectory is restricted in V for all t ≥ 0. Controllability subspaces [74] are special classes of (A, B)-invariant subspaces which have the additional property that any two points in the subspace may be joined in finite time by a trajectory (generated by some appropriate input) that lies entirely in the subspace. An important property of controllability subspaces is the assignment of the spectrum of the restriction map A in (2.47). This latter property is behind the following alternative form of the output zeroing problem. Corollary 2.5. Consider the right singular system S(A, B,C, D) and let λ be an arbitrary complex number. There always exists a control input u(t) = exp(λ t)uλ such that the state trajectory is x(t) = exp(λ t)xλ and y(t) ≡ 0. Furthermore, the triple (λ , xλ , uλ ) satisfies the conditions    λ I − A −B xλ =0 (2.48) −C −D uλ The vectors xλ are always vectors of the maximal output nulling controllability subspace R and this suggests the following characterisation of zeros as unique solutions of output zeroing. Theorem 2.9. [32] For the right singular system S(A, B,C, D) there exists an output nulling solution x(t) = exp(zt)xz , u(t) = exp(zt)uz , t ≥ 0 such that x(0) = xz ∈ / R and y(t) ≡ 0, iff z is a zero of an invariant polynomial of P(s) and the triple (z, xz , uz ) satisfies condition (2.48). The above result states that although for initial conditions in the space R we have output zeroing problems with arbitrary frequencies, additional initial conditions may be found outside the space R that are linked to specific frequencies. These are the zeros of the Smith form and the analysis given for right regular systems applies to them. With such solutions a space V f , the output nulling fixed spectrum space, is defined which is independent of R. The space W = V f ⊕ R is the maximal output nulling (A, B)-invariant subspace of the system characterised by a finite spectrum and W .

2.5.7

Frequency Transmission Blocking for Infinite Frequencies

Frequency transmission blocking has been considered so far for finite frequencies. Such problems however, may be posed also for frequencies which tend to infinity. There are two ways such problems may be posed. The first is to use the notion of blocking of impulsive input and state trajectories (similarly to the treatment given for right singular systems) and the second is to use duality between 0 and infinite frequencies (described by the w = 1/s bilinear transformation). The second approach is used here, whereas the impulsive approach will be demonstrated on the input-output model formulation of frequency blocking later on. The essence of this duality may be described on the autonomous differential system S(F, G) in (2.4) as follows:

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Differentiator – Integrator duality [29] : If F, G are r × k real matrices, we may associate the following two autonomous differential systems S(F, G) : F pz = Gz, p = d/d ˆ G) : F zˆ = pGˆz S(F,

(2.49) (2.50)

Those two systems have associated pencils (sF − G) and (F − sG) ˆ where s, sˆ are independent complex variables and they are special cases of the homogeneous matrix pencil sF − sG. ˆ As such the pencils are called algebraically dual, the systems are characterised by differentiator integrator duality and the following relationships hold between the set of their elementary divisors (ed) [29], [31]. Lemma 2.1. [29]: For the pencils (sF − G) and (F − sG) ˆ the following properties hold true: (i) A zero ed of sF − G, s p , is an infinite ed of F − sG. ˆ (ii) An infinite ed of sF − G, sˆq is a zero ed of F − sG. ˆ (iii) A finite non zero ed of sF − G, (s − α )τ is a finite nonzero ed (sˆ − α −1 )τ of F − sG. ˆ The above properties suggest that infinite frequency behaviour of S(A, B,C, D) may be studied as an equivalent zero frequency problems on the algebraically dual system ˆ B,C, D) : {x(t) ˙ˆ + Bu(t), ˙ˆ y(t) ˙ˆ + Du(t) ˙ˆ S(A, ˆ = Ax(t) ˆ = Cx(t)

(2.51)

ˆ B,C, D) be its algebraically Theorem 2.10. [30] Let S(A, B,C, D) be right regular and S(A, dual. If {xˆ2 , . . . , xˆk } is a chain of vectors, then for all x(0) ˆ = xˆi+1 , i = 1, 2, . . . , k − 1, there ˙ˆ = uˆi + t uˆi−1 + ti−1 uˆ1 ,t ≥ 0 such that x(t) ˙ˆ = xˆi + t xˆi−1 + . . . + t i−2 x2 exists a control input u(t) and y(t) ˆ ≡ 0, if and only if the vectors xˆi , uˆi satisfy.       I 0 xˆi A B xˆi−1 = , i = 1, 2, . . . , k, xˆ0 = 0, uˆ0 = 0 (2.52) 0 0 uˆi C D uˆi−1 The significant of the above, as far as the characterisation of the infinite structure of P(s) is indicated by the following result: Corollary 2.6. The set of conditions (2.52) have a nontrivial solution if P(s) has an infinite elementary divisor sˆq , q > 1, where k ≤ q − 1. Furthermore, q − 1 is the maximal length chain of independent vectors {xˆ2 , . . . , xˆk } for which conditions (2.52) are satisfied. The set of uˆi and xˆi vectors are then defined by:    D 0 ··· 0 uˆ1    CB D 0   uˆ2   =0    .. . . . .   ..    . uˆi−1 CAi−3 CAi−4 · · · D and xˆi = Ai−2 Buˆ1 + Ai−3 Buˆ2 + . . . + Buˆi−1

(2.53)

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The above results suggest that every infinite elementary divisor sˆq of P(s) is characterised by a q − 1 chain of vectors {xˆ2 , . . . , xˆq } linked to an output zeroing problem. The number q¯ = q − 1 is called an order of an infinite zero of the system. The vectors uˆi , xˆi are called input-state-infinite zero directions. The subspace Vq = span{xˆ2 , . . . , xˆq } has dimension q¯ = q − 1 [78] and it is called an output-zeroing asymptotic subspace [32], [17]. Such subspaces behave as the limits of transmission subspaces where the associated frequency λ becomes unbounded ( |λ | → ∞); for this reason this class is referred to as asymptotic, or sliding subspaces [71]. The set of subspaces Vqi associated with nonlinear ied are linearly independent and V∞ = Vq1 ⊕ . . . ⊕ Vqr is the maximal asymptotic output nulling space and the space Z = V f ⊕ P ⊕ V∞ is the general output nulling space of the system [32], [17].

The results in this section reveal that the invariants of the system matrix pencil such as finite and infinite elementary divisors and row minimal indices characterise different aspects of the dynamics of a system. The set of row minimal indices may have an interpretation similar to those of row minimal indices, but on the transposed dual system S′ (At , Bt ,Ct , Dt ).

2.5.8

Zero Structure and System Transformations

The significance of zeros is due to that they do not change under a number of transformations that may be applied on a state space model. The general set of such transformations is described in Figure (2.2) where the following notation is used: T : m × m Output Co-ordinate transformation, |T | 6= 0; Q, Q−1 : n × n pair of State Co-ordinate transformations, |Q| 6= 0; F : m × p Constant Output Feedback matrix; L : n × p State Feedback matrix; K : n × m Output Injection matrix. The above set of transformations (R, T, Q, Q−1 , L, K) when applied on the original system described by P(s), produce a system S′ (A′ , B′ ,C′ , D′ ) described by P′ (s) system matrix where P(s), P′ (s) are strict equivalent since  −1     sI − A −B QO Q K P′ (s) = (2.54) −C −D L R O T Theorem 2.11. [34], [68], [52] The set of finite and infinite elementary divisor, column and row minimal indices of the system matrix pencil P(s) is invariant under the group of transformations that involves: state, input, output co-ordinate transformations, state feedback, output injection and output feedback. The zero structure as defined by the above set of invariants is therefore unaffected by this group of transformations which is usually referred to as Kronecker transformation group. This is why the state space zeros are also referred to as invariant zeros. The set of column, row minimal indices of P(s) are called respectively right -, left indices of the system. The significance of such invariants as far as the fundamental properties of controllability and observability under feedback transformations is described by the following result [34], [52], [74]. Theorem 2.12. Consider the system S(A, B,C). The following properties hold true: (i) (a) The controllability properties remain invariant under state feedback. (b) If the system has finite zeros and/or left indices then there exists output injection that can make the system maximally uncontrollable.

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F

K

D

+

uc

~ u

+

u R

B +

+

+

+

+

~ x

Q

x

z

x Q-1

~ x

+

y

C

T

~ y

A L

Fig. 2.2. The general set of State Space Transformations (ii) (a) The observability properties remain invariant under output injection. (b) If the system has finite zeros and/or right indices, then there exists state feedback that can make the system maximally unobservable. The nature of the above results stems from that the presence of finite zeros and/or right indices implies the existence of a subspace W = V f ⊕ R which has to be in Nr (C). Given that W is (A, B)-invariant there exists state feedback L such that W becomes an eigenspace of (A + BL); moreover W is in Nr (C) which implies that the system becomes maximally unobservable. Of course, for a generic feedback L we have preservation of observability, but this property is not true for all L.

2.5.9

The Zero Pencil of Strictly Proper System

The computation of the invariants of the pencil P(s) is a problem reduced to finding the Kronecker form. Such computations may be considerably reduced for the case of strictly proper system by defining the zero pencil, which has considerably smaller dimensions. If D = 0 then the condition y(t) = Cx(t) = 0 implies that the differential system S(Γ , ∆ ) may be reduced to the following equivalent description S(NM, NAM) : NM v(t) ˙ = NAMv(t), x(t) = Mv(t), u(t) = B† {x(t) ˙ − Ax(t)}

(2.55)

where N, M are (n − p) × n, n × (n − m) respectively base matrices for Nℓ {B}, Nr {C} and B† is a p × n left inverse of B. The matrix pencil Z(s) = sNM − NAM associated with S(NM, NAM) has dimensions (n − p) × (n − m) (much smaller than (n + m) × (n − p)P(s)) and it is called the zero pencil of the system [32]. Z(s) characterises the zero dynamics of the system in an autonomous form (that does not involve the inputs) and its structure is related to that of P(s) as shown below [32], [44], [36].

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′

Corollary 2.7. The pencils P(s) and Z(s) have the same sets of finite e.d. If sˆq is an infinite ′ e.d of Z(s), then q′ = q − 2, where sˆq is an infinite e.d. of P(s) for which q ≥ 3. If ε ′ , η are ′ row minimal indices of Z(s), then ε = ε − 1, η ′ = η − 1, where ε , η are the corresponding indices of P(s). The direct link between the invariants of the two pencils allows the computation of the zero structure from the Kronecker form of the smaller pencil Z(s). A direct link between properties of the state space parameters and the number of s, ˆ and sˆ2 infinite elementary divisors of P(s) [78], [30] allows the recovery of the full zero structure from Z(s). The smaller dimensions pencil Z(s) allows the study of properties related to the number, as well as computation of zeros and the results are summarised below [32]: Corollary 2.8. For the strictly proper system S(A, B,C) the number of finite zero σ f has the following properties: (i) If p 6= m and the system S(A, B,C) is generic, then σ f = 0. Nongeneric systems with p 6= m have a number of zeros σ f such that σ f ≤ min(n − p, n − m). (ii) If p = m, S(A, B,C) always has zeros (finite and infinite) which have the following properties (a) The zeros are defined as the roots of the zero polynomial z(s) = |Z(s)|. (b) The number of zeros σ f ≤ n − p and equality holds when CB has full rank. If CB is rank deficient, then some of the finite zeros migrate to infinity.

2.5.10

Decoupling Zeros

The term zero has been also used in the literature for values of complex numbers, which are not linked to output zeroing problems. The most common term used is that of decoupling zeros which are defined below: Definition 2.8. For the System S(A, B,C, D) we define: (i) The set of Input Decoupling Zeros as the set of eigenvalue (including multiplicities) of A, which are uncontrollable. (ii) The set of Output Decoupling zeros, as the set of eigenvalues (including multiplicities) of A, which are unobservable. (iii) The set of Input-Output Decoupling zeros, as the set of eigenvalues (including multiplicities) of A, which are both uncontrollable and unobservable. The above notions may be characterised algebraically in terms of the structure of sets of elementary divisors of the Smith form of matrix pencils. If M is an n × (n − m) right annihilator of C and N is an (n − p) × n left annihilator of B then [34]: Input Decoupling zeros: The structure of input decoupling zeros is defined by the set of elementary divisors of either of the pencils Pc (s) = [sI − A, −B], Rc (s) = sN − NA

(2.56)

Output Decoupling Zeros: The structure of output decoupling zeros is defined by the set of elementary divisors of either of the pencils

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N. Karcanias and E. Milonidis P0 (s) = [sI − At , −Ct ]t , R0 (s) = sM − AM

(2.57)

The characterisation of input-output decoupling zeros structure may be achieved using the Kalman decomposition of the state space [20], [34]. There are certain links between decoupling zeros and invariant zeros and these are described below. Corollary 2.9. For the strictly proper system S(A, B,C) the following relationship hold between zeros and decoupling zeros: (a) If m > p, then every output decoupling zero is also an invariant zero; in general, input decoupling zeros are not invariant zeros. (b) If m < p, then every input decoupling zero is also an invariant zero; in general, output decoupling zeros are not invariant zeros. (c) If m = p, then every input and every output decoupling zero is also an invariant zero. (d) There always exists state feedback that makes all invariant zeros output decoupling zeros. Similarly, there always exists output injection that makes all invariant zeros input decoupling zeros.

2.6

Poles and Zeros of Transfer Function Models

The state space characterisation of poles links them with the eigenvalues, a subset of which appear as poles of the transfer function matrix G(s). Thus, in an input-output set-up we expect poles to be the frequencies for which the gain explodes in elements of G(s). On the other hand, the characterisation of zeros (right regular systems) implies that a zero s = z has an associated direction xz = (zI − A)−1 Buz and that G(z)uz = 0, G(s) = C(sI − A)−1 B + D

(2.58)

Thus, transfer function zeros correspond to frequencies where G(s) loses rank beyond its normal rank, θ = rankR(s) {G(s)}. The finite poles and zeros of transfer functions may also be characterised by corresponding dynamic problems.

2.6.1

Dynamic Characterisation of Transfer Function Poles and Zeros

Dynamic problems similar to those used for characterising state space poles and zeros may be used for transfer function matrix models as shown below [9]: Theorem 2.13. Given a transfer function matrix G(s), then a complex number p is a pole of G(s) if and only if there is an input u(t) = ∑ki ui δ (i) (t), δ (i) (t) is the ith order impulse, such that with zero initial condition the output is y(t) = y p exp(pt), t > 0

(2.59)

The above implies that p is a pole, if there exists an impulsive input that transfers the 0 state to an appropriate state at t = 0+ (point on an eigenvector) such that the resulting output is rectilinear. The relationship with the state space pole characterisation is clear.

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Theorem 2.14. Given a right regular transfer function matrix G(s) (Nr {G(s)} = 0), then the complex number z is a zero of G(s), if and only if we can find an input u(t) = uz exp(zt) + ∑ki ui δ (i) (t), δ (i) (t) is the i-th order impulse, such that with zero initial condition the output y(t) ≡ 0 for t > 0. The above result expresses an output zeroing problem, where the impulsive part of the input transfer the zero state to an appropriate initial condition for t = 0+ (a state zero direction) which together with the exponential input make the output identically zero for t > 0. This clearly provides links with the state space output zeroing problem. The dynamic characterisation given here in terms of Theorems (13) and (14) can be satisfied as long as p is a pole of an entry of G(s) and z is a frequency where G(z) loses rank beyond the value of the normal rank. The analysis of the pole, zero structure is in terms of the Smith–McMillan form is considered next.

2.6.2

Smith–McMillan Form Characterisation of Poles and Zeros

The Smith–McMillan form of a rational matrix is a canonical form under R[s]-unimodular equivalence which has as invariants the polynomials characterising the finite pole and zero structure of the rational transfer function [58]. Theorem 2.15. Let G(s) be an m× p rational matrix and let θ = rankR(s) {G(s)} ≤ min(m, p). There exist R[s]-unimodular matrices Uℓ (s),Ur (s) of m × m, p × p dimensions respectively and an m × p matrix M(s) such that  ∗  M (s) 0 G(s) = Uℓ (s)M(s)Ur (s), M(s) = 0 0 M ∗ (s) = diag{ε1 (s)/ψ1 (s), . . . , εθ (s)/ψθ (s)} = E(s)Ψ −1 (s) = Ψ −1 (s)E(s)

(2.60)

E(s) = diag{εi (s)}, Ψ (s) = diag{ψi (s)} where εi (s) divides εi+ j (s) and each ψi (s) divides ψi− j (s). The set of polynomials {εi (s), i = 1, . . . , θ }, {ψi (s) : i = 1, . . . , θ } completely characterise the R[s] equivalence class of G(s). The canonical matrix M(s) is called the Smith–McMillan form of G(s) and the polynomials εi (s), ψi (s) are the elementary zero, pole polynomial respectively and z(s) = ∏θi=l εi (s), p(s) = ∏θi=l ψi (s) are correspondingly the zero, pole polynomial of G(s). The roots of the εi (s), ψi (s) are respectively the zeros, poles of G(s) and they are the frequencies which satisfy the conditions of Theorems (13), (14) correspondingly. We define the number δM (G) = deg{p(s)} as the McMillan degree of G(s); this defines the number of states required for a minimal realisation of G(s) [20]. The zeros and poles defined through the Smith–McMillan form of G(s) are called transmission zeros, poles respectively. Their computation, apart from the derivation of the canonical form, may be achieved as described below [46]: Corollary 2.10. The polynomial p(s) is the least common denominator of all non-identically zero minors of all orders of G(s), z(s) is the greatest common divisors of the numerators of all non identically zero minors of maximal order of G(s), which have been adjusted to have p(s) as their common denominator.

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2.6.3

Matrix Fraction Descriptions, and Poles and Zeros

An alternative input–output description for a system is that provided by coprime MFDs, or their corresponding composite representation, that is: G(s) = Nr (s)Dr (s)−1 = Dℓ (s)−1 Nℓ (s) (2.61)   Nr (s) Tr (s) = , Tℓ (s) = [Dℓ (s), Nℓ (s)] (2.62) Dr (s) Such representations have a number of important properties and they provide alternative characterisations of poles, zeros which is equivalent to the Smith–McMillan characterisation [72], [18]. Theorem 2.16. Let G(s) be a rational transfer function matrix represented as in (2.61), (2.62). (i) Let (Nr (s), Dr (s)), (N˜ r (s), D˜ r (s)) (respectively (Dℓ (s), Nℓ (s)), (D˜ ℓ (s), N˜ ℓ (s))) be two right (left) coprime MFD pairs with the same dimensions. These two pairs correspond to the same transfer function matrix G(s), iff there exist Ur (s),Uℓ (s) R[s] unimodular matrices of appropriate dimensions such that T˜r (s) = Tr (s)Ur (s), T˜ℓ (s) = Uℓ (s)Tℓ (s)

(2.63)

(ii) If (Nr (s), Dr (s)) is a right coprime MFD pair and (Dℓ (s), Nℓ (s)) is a left coprime MFD pair of G(s) then, Nr (s), Nℓ (s) have essential parts of Smith form defined by E(s) = diag{ε1 (s), . . . , εθ (s)} and Dr (s), Dℓ (s) have essential parts of Smith form defined by Ψ (s) = diag{ψ1 (s), . . . , ψθ (s)}. (iii) Every transfer function matrix G(s) has a minimal factorisation. G(s) = N¯ r (s)Zr (s)Dr (s)−1 = Dℓ (s)−1 Zℓ (s)N¯ ℓ (s)

(2.64)

where (N¯ r (s)Zr (s), Dr (s)) are right coprime, (Dℓ (s), Zℓ (s)N¯ ℓ (s)) are left coprime Zr (s), Zℓ (s) are p × p, m × m matrices with Smith form defined by E(s), and Dr (s), Dℓ (s) are p × p, m × m matrices with Smith form defined by Ψ (s) and the matrices N¯ r (s), N¯ ℓ (s) have no zeros. Part (i) expresses the property that coprime MFDs of the same transfer function are right, or left R[s]-equivalent and thus define representations of the same system with respect to different co-ordinate frames. Part (ii) provides a characterisation of the pole, zero structure in terms of Smith forms of denominators, numerators respectively. Finally, part (iii) defines a factorisation of G(s) where the p × p, or m × m transfer functions Zr (s)Dr (s)−1 , Dℓ (s)−1 Zℓ (s) define completely the pole, zero structure and N¯ r (s), N¯ ℓ (s) are least degree bases of the column, or row space respectively of the system.

2.6.4

Infinite Poles and Zeros

The structure of Smith–McMillan form defined over R[s] [58] provides information on the finite poles and zeros. However, R[s]-equivalence does not preserve the structure of poles and zeros at infinity and another form of equivalence is required [79], which is based on the ring of proper rational functions R pr (s) [18], [76]. For state space models, properties at infinity were examined before based on the duality between zero and infinite elementary divisors. The current treatment provides also an alternative characterisation of the structure at infinity of state space models.

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2.6.5

75

Smith–McMillan Form at Infinity: Infinite Poles and Zeros

The set of proper rational functions R pr (s) under the standard operations of addition and multiplication is a Euclidean ring with a degree function introduced by the notion of discrete valuation introduced below: Definition 2.9. [76]: (Discrete Valuation) Let t(s) = n(s)/d(s) ∈ R(s), where n(s), d(s) ∈ R[s], d(s) 6= 0 and define the map δ∞ (·) : R(s) → Z ∪ {∞} via:

δ∞ (t(s)) := degd(s) − degn(s),t(s) 6= 0, δ∞ (t(s)) := +∞,t(s) ≡ 0

(2.65)

This function satisfies the conditions: δ (xy) = δ (x) + δ (y) and δ (x + y) ≥ min{δ (x) + δ (y)} and it is referred to as discrete valuation of R(s). For proper rational functions, δ∞ (t(s)) ≥ 0, and thus the function introduces a Euclidean degree. The set of elements of R pr (s) with the property δ∞ (t(s)) = 0 are the biproper rational functions and they are the units (their inverse is proper) of R pr (s). Every t(s) ∈ R(s) can be written as: 1 n (s) t(s) = ( )q∞ 1 s d1 (s)

(2.66)

where q∞ := δ∞ (t(s)) and n1 (s), d1 (s) ∈ R[s] with degn1 (s) = degd1 (s). If q∞ > 0 we say that t(s) has a zero at s = ∞ of order q∞ and if q∞ < 0, then we say that t(s) has a pole at s = ∞ of order |q∞ |. Clearly, elements of R pr (s) have no poles at s = ∞. For a transfer function matrix G(s) ∈ Rmxp (s), the structure at infinity is studied under equivalence over the ring R pr (s) and it is defined by the Smith–McMillan form at infinity. We note that G(s) is called proper if lim G(s), as s → ∞, exists, strictly proper if the limit is zero and nonproper if the limit tends to infinity. The m × m rational matrix U(s) ∈ Rmxm pr (s) is said to be biproper, or R pr (s) unimodular, if |U(s)| = u(s) ∈ R pr (s) is a unit. Two m × p rational ˜ matrices G(s), G(s) are said to be R pr (s)-equivalent, if there exists Uℓ (s),Ur (s) such that ˜ G(s), G(s) are related as in (2.13). The matrices Uℓ (s),Ur (s) have no poles, or zeros at s = ∞, since their limit at s = ∞ are constant full rank matrices and thus they preserve the infinite pole zero structure of G(s). This structure is described by the following result [76], [79]. Theorem 2.17. Let G(s) ∈ Rmxp (s) with rankR(s) {G(s)} = θ . There exist biproper rational matrices Uℓ (s),Ur (s) such that  ∗  M∞ 0 G(s) = Uℓ (s)M∞ (s)Ur (s), M∞ (s) = (2.67) 0 0 M∞∗ (s) = diag{sq1 , . . . , sqθ }, where q1 ≥ q2 ≥ . . . ≥ qk ≥ 0 ≥ qk+1 ≥ . . . ≥ qθ

The above result yields an immediate characterisation of poles and zeros at infinity of the rational matrix. If π∞ is the number of qi ’s in (2.67) with qi > 0 then G(s) has π∞ poles at infinity, each having order qi . Similarly, if τ∞ is the number of qi ’s in (2.67) with qi < 0 then G(s) has τ∞ zeros at infinity, each having order |qi |. Similar results on the characterisation of poles and zeros at infinity based on R pr (s) coprime MFDs may be given as for the case of finite poles and zeros. The above definition is equivalent to the definition of the structure at infinity based on bilinear transformation and use of the standard Smith–McMillan form. In fact, the infinite pole/zero structure of G(s) may be equivalently defined as the finite pole/zero structure of G(1/w) at w = 0. For state space models the structure at infinity is described by the infinite elementary divisors; the relationship between them and the infinite structure is summarised next [76], [78].

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Corollary 2.11. If P(s) is the system matrix pencil of the systems S(A, B,C, D), then the following properties hold true: (i) P(s) has no infinite poles. (ii) P(s) has an infinite elementary divisor sˆq , q > 1, if and only if P(s) has an infinite zero with order q¯ = q − 1. (iii) The infinite elementary divisors s, ˆ i.e., with degree 1 have no dynamic significance and do not correspond to infinite poles, zeros. The notion of the Smith–McMillan form over R(s), or over R pr (s) may be generalised with respect to the region P = Ω ∪ {∞}, where Ω is any symmetric region with respect to the real axis that excludes at least one point of the real axis. The corresponding equivalence is over the ring of proper and Ω -stable rational functions, which is also a Euclidean ring with a degree function of the total number of zeros in the region P [77], [80].

2.6.6

Impulsive Dynamics and Properties of Infinite Poles and Zeros [57]

The infinite pole zero structure of a rational matrix may be also characterised dynamically. Two such results are given next. Theorem 2.18. Let G(s) ∈ Rmxp (s) and assume that it has a pole at infinity of order q. Then, there exists an input consisting solely of exponential modes u(t) = ∑ j ∑k u jk t k−1 exp(λ j t), where u jk are constant vectors such that the output is given by q−1

y(t) =

∑ yi δ (i) (t) + ∑ ∑ yg exp(λ j t), yq−1 6= 0 k

i=0

(2.68)

g k

The above suggests that the presence of infinite pole causes the transmittance to contain a linear combination of impulses, which were not present in the input. The property that yq−1 6= 0 implies that the presence of a q order infinite pole generates an impulse of order δ (q−1) (t). For infinite zeros we have the result: Theorem 2.19. Let G(s) ∈ Rmxp (s) and assume that it has a zero at infinity of order q. Then q−1 there exists an input of the form u(t) = ∑i=0 ui δ (i) (t) + ∑g ∑k u jk t k−1 exp(λ j t), where ui , u jk are constant vectors uq−1 6= 0, such that the corresponding output y(t) contain no impulses. The above suggest that the presence of an infinite zero in the transfer function causes the nontransmittance of the impulsive part of certain inputs. The condition that uq−1 6= 0 indicates that the presence of an infinite zero of order q does ensure the absorption of a particular impulse of order q. Remark 2.7. Although the standard Smith–McMillan form cannot be directly used for system compensation, due to that R[s]-equivalence transformation are not physically realisable, the Smith–McMillan form at infinity, derived under R pr (s)-equivalence has a compensation interpretation. For any G(s), condition (2.67) may also be expressed as Uˆ ℓ (s)G(s)Uˆ r (s) = M∞ (s), where Uˆ ℓ (s), Uˆ r (s) denote respectively the inverses of Uˆ ℓ (s), Uˆ r (s) which are also biproper matrices and thus they can be realised in terms of (A, B,C, D) parameters. In this case Uˆ ℓ (s)Uˆ r (s) express respectively proper post-, pre-compensation and M∞ (s) is the resulting transfer function. 

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M∞ (s) is the simplest transfer function that can be obtained under biproper post and pre compensation and the infinite poles and zeros represent the “strongest” set of invariants and is linked to the Kronecker canonical form of the state space system [36], [44], [34].

2.7

System Algebraic Functions and Generalised Nyquist and Root Locus

The pole, zero structure of a linear system plays a crucial role in Control analysis and design and it is particularly important in the study of Nyquist and Root Locus of Control Systems. The study of poles and zeros so far has been based on the algebraic properties of the structure of the system. In the case of Nyquist and Root Locus, however, the complex nature of the representation becomes dominant and this gives rise to alternative characterisations of poles and zeros which differs in a number of aspects from the algebraic nature. A brief introduction to the basics of multivariable Nyquist [47], [48], [56] and Root Locus [38], [47] is given next.

2.7.1

Characteristic Gain, Frequency Functions

We consider the general rectilinear motion problem for a frequency s and for the case of square systems m = p. If we define the scalar output feedback law y(t) = ku(t), k real, 6= 0, or equivalently u(t) = gy(t), g = k−1 and apply it on the system description S(Φ , Ω ) in (2.2) this leads to the closed-loop differential system       I 0 x(t) ˙ A B x(t) S(Γ , ∆ (g)) : = (2.69) 0 0 u(t) ˙ C gI − D u(t) the solutions of which characterise the scalar output feedback solutions of the original system. The study of rectilinear motions according to the complex frequency s, i.e. [x(t)t , ut (t)]t = [xt , ut ]t exp(st) implies that condition (2.38) has to be satisfied, which leads to the eigenvalueeigenvector problem.      sI − A −B x sI − A −B = 0, P(s, g) = (2.70) −C gI − D u −C gI − D where P(s, g) is known as the closed-loop system net [47]. The two variable nature of P(s, g) suggests that we may consider: (i) s as a function of g : For a fixed g, then u = (gI − D)−1Cx and {sI − S(g)}x = 0, S(g) = A + B(gI − D)−1C

(2.71)

(ii) g as a function of s : For a fixed s, then x = (sI − A)−1 BU and {gI − G(s)u = 0, G(s) = D +C(sI − A)−1 B

(2.72)

By (2.71) and (2.72) it is seen that the eigenvalue-eigenvector problems on S(g) and G(s) define the Root-locus and Nyquist diagrams respectively for the system S(A, B,C, D). S(g) is referred to as characteristic gain matrix and defines the closed loop eigenvalues for a scalar

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output feedback k = g−1 . G(s) is the transfer function, which is now seen as the generator of Nyquist gains and directions and thus it is also called characteristic gain matrix. The characteristic equations for G(s) and S(g) are defined as:

∆ (g, s) = det{gI − G(s)] = 0, ∇(g, s) = det{sI − S(g)] = 0

(2.73)

and they are a pair of algebraic equations relating the complex variables s and g. ∆ (g, s) = 0 and ∇(s, g) = 0 define a pair of algebraic functions g(s) and s(g) known as characteristic gain and characteristic frequency functions [47], [48], [56] respectively g(s) and s(g) are the natural means of generalising Nyquist and Root-Locus theory to multivariable case correspondingly; the eigenvalue-eigenvector problems defined by (2.71), (2.72) demonstrate that both Root Locus and Nyquist in the multivariable case has an eigenframe dimension, which expresses the closed-loop eigenvectors and Nyquist directions respectively. Definition 2.10. For the system described by S(A, B,C, D), or G(s) we define: (i) If s takes values on the Nyquist contour (i.e., s = jω ), then the m branches gi ( jω ) of ∆ (g, s) are defined as the Nyquist diagrams, or characteristic gain loci of the system and the eigenvalue eigenvector decomposition (defined frequency by frequency). G( jω ) , W ( jω )Λ ( jω )W ( jW )−1 , Λ ( jω ) = diag{gi ( jω )}

(2.74)

expresses the characteristic gain decomposition of G( jω ). The columns of W ( jω ) are eigenvectors for the corresponding gi ( jω ) and they are called the Nyquist directions, or characteristic gain vectors. (ii) If g, (or k) takes values on the real axis including g = ∞(k = o), then the n branches si (k) of ∇(s, g) are defined as characteristic frequency loci or Root loci and the corresponding eigenvectors are the characteristics frequency vectors, or closed loop eigenvectors under the scalar negative feedback scheme.

2.7.2

Poles and Zeros of the System Algebraic Functions

The functions ∆ (g, s) and ∇(g, s) that define the two system algebraic equations are not independent. Expansion of the determinant of |P(s, g)| leads to the following identity det[sI − A]∆ (g, s) = deg[gI − D]∇(s, g)

(2.75)

/ σ (A) and g ∈ / σ (D) If σ (A), σ (D) denote the spectra of A, D respectively, then for all s ∈

∆ (g, s) = 0 ⇔ ∇(s, g)

(2.76)

Hence, a knowledge of characteristic gain as a function of frequency for G(s) is equivalent to a knowledge of characteristic frequency as a function of gain for S(g). If G(s) = N(s)D(s)−1 is a right coprime MFD, then ∆ (g, s) · |D(s)| = |gD(s) − N(s)| = ∆˜ (g, s) and

∆˜ (g, s) = |gD(s) − N(s)| = am (s)gm + am−1 (s)gm−1 + . . . + a0 (s) = 0

(2.77)

where am (s) = |D(s)| and a0 (s) = (−1)m |N(s)|. Clearly, the total number of poles and zeros of G(s) are given by the roots of am (s) and a0 (s) respectively ∆˜ (g, s) may be factorised as

∆˜ (g, s) = e(s)∆1 (g, s) . . . ∆υ (g, s)c(g)

(2.78)

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where ∆i(g,s) are irreducible polynomials over C[g, s]. The terms e(s), c(g) are the products of all irreducible factors which are independent of g, s respectively. The roots of e(s), c(g) are called fixed poles, fixed gains correspondingly. In the following, ∆˜ (g, s) is assumed irreducible and that g(s) is the associated algebraic function. Equation (2.77) is called the characteristic gain-frequency equation. Remark 2.8. (McMillan Structure and Characteristic Equation) :If{εi (s), i =1, . . . , m},{ψi (s), i = 1, . . . , m} are the zero, pole elementary polynomials then the coefficients am (s),. . ., a0 (s) of the characteristic equation are expressed as: am (s) = ψ1 (s)ψ2 (s) . . . ψm (s) am−1 (s) = ε1 (s)ψ2 (s) . . . ψm (s)hm−1 (s), hm−1 (s) ∈ R[s] .. .

(2.79)

a1 (s) = ε1 (s)ε2 (s) . . . εm−1 (s)ψm (s)h1 (s), h1 (s) ∈ R[s] a0 (s) = ε1 (s)ε2 (s) . . . εm−1 (s)εm (s)h, h ∈ R 6= 0

Remark 2.9. The characteristic gain-frequency equation defines the characteristic gain algebraic function g(s) which defines the Nyquist diagrams; by re-arranging with respect to s (polynomials in g) the characteristic frequency algebraic function s(g) is obtained. The characterisation of poles and zeros of algebraic functions may be introduced by using their power series, and this is treated in [67]. 

2.7.3

Root Locus and the Output Zeroing Problem

The relationships between the current Root Locus formulation and zero structure are considered here. For square nondegenerate systems the system matrix pencil P(s) is characterised only be finite and infinite elementary divisors. For the asymptotic case when k → ∞, that is g = 0 the differential system S(Γ , ∆ (0)) defines the asymptotic solutions. Note S(Γ , ∆ (0)) = S(Γ , ∆ ) which is the output zeroing problem of Equation (2.22) and thus we are led to the following results on asymptotic properties [38], [17]. Theorem 2.20. (Asymptotic Properties of Root Locus)For square nondegenerate systems and for the unity feedback configuration with scalar gain k, the following hold true: (i) As k → ∞ the closed-loop poles of the system tend to the finite zeros, with the structure defined by the set of fed and the infinite zeros, with a structure defined by the orders of infinite zeros of P(s) (degrees of nonlinear ∞- e.d. minus one). (ii) The closed loop eigenstructure as k → ∞ defined by the structure of the decomposition of the maximal output nulling subspace Z , which is expressed as Z = Vz1 ⊕ . . . ⊕ Vzk ⊕ Vq1 ⊕ . . . ⊕ Vqµ

(2.80)

where Vz corresponds to (s − z)d , dimVz = d and Vq corresponds to sˆq , dimVq = q − 1. Conditions (2.80) define the structure of closed-loop eigenvectors as k → ∞. The rates at which closed-loop eigenvalues tend to infinity is defined by the order of the “Butterworth Patterns” of

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the Asymptotic Root Locus; such rates are also referred to as Root-Locus Infinite Zeros and in general they differ from the algebraic definition previously given. A proper definition of such infinite zeros is given through the Laurent expansions of the Root Locus algebraic function and this lead to orders, which may be fractional (see [38], [56], [67]). The properties of root loci may be inferred from the Markov parameters (CB,CAB, ...,CAk B, . . .) [38] and the basic rules described there reveal the explicit relationships between the system structure (expressed in terms of finite, infinite zeros) and their asymptotic properties. The finite zeros define the terminal locations of root loci under high gain, whereas the orders of infinite zeros define the generic values of the asymptotic Butterworth pattern. For the Nyquist diagrams, the infinite zeros describe the generic values of the terminal phases of the Nyqusit diagrams. Apart from stability, the characteristic gain loci may reveal further closed-loop properties such as ability to track, reject signals under the assumption of relative normality of G(s) [56].

2.8

The Feedback Configuration and Structural Properties

In this section we consider the general Feedback Configuration for a multivariable system and establish its general structural properties and fundamental operators used in control design.

2.8.1

Structural Properties of the Feedback Configuration

The most commonly used feedback configuration in the context of algebraic synthesis method is that denoted in Figure (2.3), where P(s) ∈ Rm×p (s),C(s) ∈ R p×m (s) are the plant, controller transfer functions and w1 , w2 are external control, or disturbance signals, e1 , e2 are plant controller inputs and y1 , y2 are the plant, controller outputs. For the sake of simplicity we drop (s) in the descriptions of C(s), P(s) and their MFDs, as well as from the Laplace transforms of vector signals. Such a configuration is quite versatile and may accommodate several control problems. For instance, in a tracking problem, w1 would be a reference signal to be tracked by plant output y2 . In a disturbance rejection, or desensitisation to noise, w1 would be the disturbance/noise. Depending on whether w1 , or w2 is the externally applied control signal (as opposed to noise etc.) the configuration can represent either feedback, or cascade compensation. Such a configuration will be referred to as complete feedback configuration. The system equations are defined by [39], [80], [6]:            e1 w1 0 P e1 y1 C 0 e1 = − , (2.81) −C 0 0 P e2 e2 w2 e2 y2 or, by



     e1 w1 y , w= , y= 1 , e2 w2 y2     C 0 0 I G= ,F = 0 P −I 0

e = w − FGe, y = Ge, e =

(2.82)

The feedback configuration is well formed if |I + FG| = |I + PC| = |I + CP| = 6 0. Then we may define the transfer functions H(P,C),W (P,C) by

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81

w2

+

w1 +

y1 +

C

e2

y2

P

-

controller

plant

Fig. 2.3. Complete Feedback Configuration e = (I + FG)−1 w = H(P,C)w, y = G(I + FG)−1 w = W (P,C)w,

where

W (P,C) = F{H(P,C) − I}

(2.83)

 C(I + PC)−1 −CP(I +CP)−1 , W (P,C) = PC(I + PC)−1 P(I + PC)−1   (I + PC)−1 −P(I +CP)−1 H(P,C) = C(I + PC)−1 (I + PC)−1 

Furthermore, if we also define the transfer functions         y1 w y w = T (P,C) 1 , 2 = R(P,C) 2 e1 w2 e2 w1

(2.84)

(2.85)

then T (P,C), R(P,C) are expressed as :   C(I + PC)−1 −C(I +CP)−1 T (P,C) = , −1 −1 (I + PC) (I + PC) P R(P,C) =



P(I + PC)−1 P(I +CP)−1 (I +CP)−1 (I +CP)−1C



(2.86)

In the above expressions a number of common terms appear which have particular significance for the study of feedback systems. Thus, we define Q = PC, Q′ = CP, F = I + PC, F ′ = I +CP, L = I + (PC)−1 , S = (I + PC)−1 , S′ = (I +CP)−1

(2.87)

where Q, Q′ are referred to as return ratio matrices, F, F ′ as return difference matrices, L as inverse-return difference matrix and S, S′ as sensitivity matrices. Remark 2.10. All transfer function matrices, W (P,C), H(P,C), T (P,C), R(P,C) associated with the complete feedback configuration have the same pole polynomial. 

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The above implies that any one of them may be used for the study of pole assignment and stabilisation of the feedback configuration. We shall use in the following, the simplest, i.e. ,H(P,C). −1 −1 Theorem 2.21. [39], [80] If P = A−1 B1 , = B2 A−1 2 ,C = D1 N1 = N2 D2 are left, right coprime MFDs (over R[s], or R p (s), then

(i) A left , right coprime MFD for H(P,C) (over R[s], or R p (s)) is defined by  −1     −1 A1 B1 A1 0 D2 0 D2 B2 H(P,C) = = −N1 D1 0 D1 0 A2 −N2 A2

(2.88)

(ii) The pole polynomial (for R[s] MFDS), or pole function (for R p (s) MFDs) of H(P,C) is f ∼ |U1 | ∼ |U2 | , U1 = A1 D2 + B1 N2 , U2 = D1 A2 + N1 B2

(2.89)

where ∼ means equal modulo units (for R[s] or R p (s) cases). An important issue for the feedback system is that of well posedness i.e., properness of all transfer function associated with the feedback system. Proposition 2.4. [39], [80] If C, P are proper transfer functions, then the completed feedback configuration is well posed, iff |I +C(∞)P(∞)| = |I + P(∞)C(∞)| = 6 0

(2.90)

Remark 2.11. Note that C(∞), P(∞) are constant matrices, since C, P are proper and 0 matrices if they are strictly proper. The complete feedback configuration with P,C proper is generically well posed and if at least one of P,C is strictly proper, then this always holds true.  Let S p , Sc be the plant, controller state space model and S f be the state space description of the complete feedback configuration. Theorem 2.22. [7], [80] : For the well posed complete feedback system S f . The following properties hold true: (i) S f is controllable, observable, iff both S p , Sc are controllable, observable. (ii) S f is stablizable, detectable, iff both S p , Sc are stablizable, detectable. (iii) If S p , Sc are both stabilizable and detectable, then S f is internally stable, iff H(P,C) is BIBO stable. The above result is fundamental, since it provides the means for studying internal stability in terms of BIBO stability of H(P,C) and it provides the basis for the study of stabilization and pole assignment of the feedback configuration. Corollary 2.12. Consider the well posed feedback system S f with S p , Sc stabilisable and detectable. −1 −1 (i) If P = A−1 B1 = B2 A−1 2 ,C = D1 N1 = N2 D2 are R p (s)-coprime MFD’s the S f is internally stable, iff either of the equivalent condition hold true

A1 D2 + B1 N2 = U1 , U1 : R p (s) − unimodular, D1 A2 + N1 B2 = U2 ,U2 : R p (s) − unimodular

(2.91)

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−1 −1 (ii) If S p , Sc are controllable and observable and P = A−1 B1 = B2 A−1 2 ,C = D1 N1 = N2 D2 are R p (s)-coprime MFD’s, then the characteristic polynomial of S f is defined by sI − A f ∼ |A1 D2 + B1 N2 | ∼ |D1 A2 + N1 B2 | (2.92)

Furthermore, if F = I + PC, F ′ = I +CP are the return difference matrices and A p , Ac are the plant controller state matrices, then |I + PC| ∼ |I +CP| ∼ |sI − A f |/|sI − A p | · |sI − Ac |

(2.93)

In (2.91) we may assume that U1 ,U2 are identity matrices; such equation are known as Diofantine Equations (for solvability and parametrisation of family of solutions see [39]) and (A1 , B1 ), (D2 , N2 ) (also (A2 , B2 ), (D1 , N1 ) are referred to as mutually stabilising pairs. For a given P, these equations define the whole family of C compensators which stabilize the system; additional performance characteristics may then be assigned by selecting appropriate elements from the stabilising family. If C is given, the solution of these equations defines the family of plants which are stabilised by C; this has important implications in the parametrisation of those plants that may be stabilised by a given type controller, as well as in the study of robustness. Note that (2.91) may also be expressed as a Diofantine equations, where now U1 or U2 are polynomial matrices with given set of zeros; this provides the means for an algebraic formulation of the pole assignment problem. Finally, condition (2.93), establishes the important property of the determinant of the return difference as the ratio of closed-loop to open-loop pole polynomials of the feedback system; it is this property that underlies the theory of both Nyquist and root locus.

2.8.2

Closed-Loop Performance and the Return Ratio Difference and Sensitivity Matrices

An enlarged version of the complete feedback configuration frequently used in design problems is shown in Figure (2.4) and it is referred to as a Control Design Configuration, P,C, F are proper transfer function matrices of the plant, precompensator, feedback compensator respectively (with appropriate dimensions), and r, di , do , n are the reference, plant input disturbance, plant output disturbance, sensor noise vector signals respectively. We assume that P,C, F are stabilisable and detectable and that C, F stabilise the feedback configuration, and thus all transfer function matrices from any external signal to the output y are stable. Let us define the transfer functions from all external signals to the output by: Hr : r → y, Hod : do → y, Hid : di → y, Hn : n → y

Hr = PC(I + FPC)−1 = (I + PCF)−1 PC, Hod = (I + PCF)−1 Hid = (I + PCF)−1 P, Pn = (I + PCF)−1 PCF and because of linearity we have that y = Hr r + Hod d0 + Hid di + Hn n

(2.94) (2.95) (2.96)

For external signals with given bandwidth, the above equation describes their effect on the overall system response in terms of the frequency domain transfer functions Hr ( jω ), Hod ( jω ), Hid ( jω ), Hn ( jω ), which are expressed in terms of return ratio PCF, return difference (I + PCF) and sensitivity matrix (I + PCF)−1 . The frequency domain study of tracking, disturbance, noise rejection, sensitivity to plant parameter variations, robustness etc. involves the notion of gain of transfer function matrices which is defined in terms of the singular values [66], [11], [56].

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d1

d0

+

r

+ -

e1

C

y1 +

e2

P

y2

+

y

+ +

n +

F Fig. 2.4. Control Design Configuration

2.9

Determinantal Assignment Problems: Exterior Algebra-Algebraic Geometry Methods

For the study of problems of linear feedback synthesis which are of the determinantal type (such as pole zero assignment, stabilisation) a specific framework has been developed which is specially suited to tackle such problems and it is referred to as algebrogeometric approach, since it relies on tools from algebra and algebraic geometry. The problems faced in this area are of a multilinear nature and the number of design parameters is not necessarily large. The main difficulty of the determinantal problems arises from that they are equivalent to finding real solutions to sets of nonlinear and linear equations; in the case of stabilisation, this is equivalent to determining solutions of nonlinear equations and nonlinear inequalities. The first of the two problems naturally belongs to the intersection theory of complex algebraic varieties, whereas, the latter belongs to the intersection theory of semialgebraic sets. Additional difficulties arise due to the need of studying design problems with specific dynamic structure (centralised or decentralised) which make the varieties involved not standard, and the compactification issues quite prominent. Determining real intersections is not an easy problem; furthermore, it is also important to be able to compute solutions whenever they exist. The DAP approach [27] has been developed for determinantal problems which are of multilinear nature and thus may be naturally split into a linear and multilinear problem (decomposability of multivectors). The final solution is thus reduced to the solvability of a set of linear equations (characterising the linear problem) together with quadratics (characterising the multilinear problem of decomposability). The approach heavily relies on exterior algebra and this has implications on the computability of solutions (reconstruction of solutions whenever they exist). New sets of invariants (of a projective character) are introduced [27] which, in turn, characterise the solvability of the problem. The distinct advantages of the DAP approach are that it provides the means for computing the solutions [42], [43] and can handle both generic and exact solvability investigations.

2.9.1

Determinantal Assignment Problems

Consider the linear system described by equations (2.1), or (2.8). We may define the following cluster of problems:

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(I) Pole assignment by state feedback : Consider L ∈ Rn×p , where L is a state feedback applied on system (2.1). If we denote by Pc (s) = [sI − A, −B], L˜ = [In , Lt ]t , then the closed loop characteristic polynomial is given by ˜ pL (s) = det{sI − A − BL} = det{Pc (s) − L}

(2.97)

(II) Design of an n-state observer : Consider the problem of designing an n-state observer for the system of (2.1). If T ∈ Rn×m is a feedback, T˜ = [In , T ] and P0 (s) = [sI − At , −Ct ]t , then the characteristic polynomial of the observer is defined by pT (s) = det{sI − A − TC} = det{T˜ P0 (s)}

(2.98)

(III) Pole assignment by constant output feedback: Consider the system described by (2.2) under an output feedback K ∈ Rm×p . The closed loop characteristic polynomial pK (s) is given by [18]: pK (s) = det{Dℓ (s) + Nℓ (s)K} = det{Dr (s) + KNr (s)} (2.99) By defining the matrices Tℓ (s) = [Dℓ (s), Nℓ (s)] ∈ Rm×(m+p) [s], Tr (s) =

then,



 Dr (s) ∈ R(m+p)×p [s] Nr (s)

K˜ ℓ = [Im , K t ]t ∈ R(m+p)×m , K˜ r = [I p , K] ∈ R p×(m+p)

(2.100)

pK (s) = det{Tℓ (s)K˜ ℓ } = det{K˜ r Tr (s)}

(2.101)

(IV) Zero assignment by squaring down [37], [32], [28], [61]: The characterisation of zeros in terms of the properties of P(s) system matrix pencil, or the transfer function matrix G(s), clearly demonstrate that zeros express the interaction between internal dynamics and the effort to control and observe the system. Earlier stages than Control Design affect directly the zero formation and typical problems that may be posed are: (i) Given a system described by either the (A, B), or (A,C) pairs, define C, or B matrices respectively such that the resulting triple (A, B,C) corresponds to a square system with zeros which can be arbitrarily assigned. (ii) Given a system described by the triple (A, B,C, D), or the m× p transfer function G(s), m > p, define a constant p × m matrix K, such that the resulting KG(s) p × p transfer function has a resulting set of zeros. The first class, and its variations (where C, or B are not entirely free but selected from a given set) are referred to as zero assignment by input, output structure design. Such problems may be studied either in a geometric, or algebraic setup and correspond to problems of transforming the Kronecker structure of the corresponding pencils; this is expressed for instance as: Problem 2.1. Given Rc (s) = sN − NA of (n − p) × n dimension, find a matrix M n × (n − p) constant, such that Z(s) = (sN − NA)M has a given zero structure (defined by the set of ed), or for a given φ (s) ∈ R[s] det{Z(s)} = |sNM − NAM| (2.102)

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The second class represents problems of squaring down the system by selecting an appropriate linear combination of the outputs and it is an integral part of the process of closing feedback loops, or defining sets of outputs which may be controlled independently. In fact, for a system with m > p we can expect to have independent control over at most p linear combinations of m outputs. If c ∈ R p is the vector of the variables which are to be controlled, then c = Hy, where H ∈ R p×m is a squaring down post-compensator, and G′ (s) = HG(s) is the squared down transfer function matrix [37], [28]. This may be expressed formally as: Problem 2.2. Given the m × p transfer function G(s), m > p, expressed as the factorised MFD G(s) = N¯ r (s)Zr (s)Dr (s)−1 , define K constant, p × m such that the p × p square down transfer function Gσ (s) = KG(s) has a given zero structure, or for a given φ (s) ∈ R[s] det{Nσ (s)} = |K N¯ r (s)Zr (s)| = |K N¯ r (s)| · |Zr (s)| = φ (s)

(2.103)

The latter formulation clearly demonstrates that the existing zeros of the nonsquare system remain as zeros of the squared down system; the process of squaring down adds new zeros which are created by the matrix K N¯ r (s), and thus a simpler formulation is the study of K N¯ r (s) zero structure formation. A family of problems of determinantal type are associated with dynamic compensation [43] and are considered next. (V) Dynamic Compensation Problems: Consider the standard feedback configuration of Figure (2.3). If G(s) ∈ R pr (s)m×p ,C(s) ∈ R(s) p×m , and assume coprime MFD’s and C(s) = Aℓ (s)−1 Bℓ (s) = Br (s)Ar (s)−1

The closed loop characteristic polynomial may be expressed as [7], [39], [80]    Ar (s) f (s) = det [Dℓ (s), Nℓ (s)] , Br (s)    Dr (s) or f (s) = det [Aℓ (s), Bℓ (s)] Nr (s)

(2.104)

(2.105)

(i) if p ≤ m, then C(s) may be interpreted as feedback compensator and we will use the expression of the closed loop polynomial. (ii) if p ≥ m, the C(s) may be interpreted as precompensator and we will use the expression of the closed loop polynomial. The above general dynamic formulation covers a number of important families of C(s) compensators as: (a) Constant, (b) PI, (c) PD, (d) PID, (e) Bounded degree. In fact, (a) Constant Controllers: If p ≤ m, Aℓ = I p , Bℓ = K ∈ R p×m , then (100) expresses the constant output feedback case, whereas if p ≥ m, Ar = Im , Br = K ∈ R p×m expresses the constant precompensation formulation. (b) Proportional plus Integral Controllers: Such controllers are defined by 1 C(s) = K0 + K1 = [sI p ]−1 [sK0 + K1 ] s

(2.106)

where K0 , K1 ∈ R p×m and the left MFD for C(s) is coprime, iff rank(K1 ) = p. From the above the determinantal problem for the output feedback PI design is expressed as:       sDr (s)   Dr (s) f (s) = det [sI p , sK0 + K1 ] = det [I p , K0 , K1 ]  sNr (s)  (2.107) Nr (s)   Nr (s)

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(c) Proportional plus Derivative Controllers: Such controllers are expressed as C(s) = sK0 + K1 = [I p ]−1 [sK0 + K1 ]

(2.108)

where K0 , K1 ∈ R p×m and the left MFD for C(s) is coprime for finite s and also for s = ∞ if rank(K0 ) = p. From the above the determinantal output PD feedback is expressed as:     Dr (s) Dr (s) f (s) = det{[I p , sK0 + K1 ] } = det{[I p , K1 , K0 ]  Nr (s) } (2.109) Nr (s) sNr (s)

(d) PID Controllers: These controllers are expressed as

1 C(s) = K0 + K1 + sK2 = [sI p ]−1 [s2 K2 + sK0 + K1 ] s

(2.110)

where K0 , K1 ∈ R p×m and the left MFD is coprime with the only exception possibly at s = 0, s = ∞ (coprimeness at s = 0 is guaranteed by rank(K1 ) = p and at s = ∞ by rank(K2 ) = p). From (2.109), the determinantal output PID feedback is expressed as :   Dr (s) 2 } f (s) = det{[sI p , s K2 + sK0 + K1 ] Nr (s)   sDr (s)  sNr (s)   } = det{[I p , K0 , K1 , K2 ]  (2.111) Nr (s)  2 s Nr (s)

(e) Observability Index Bounded Dynamics (OBD) Controllers: These are defined by the property that their McMillan degree is equal to pk, where k is the observability index [74], [18] of the controller. Such controllers are expressed as [Aℓ (s), Bℓ (s)] = Tk sk + . . . + T0

(2.112)

Tk , Tk−1 , . . . , T0 ∈ R p×(p×m) and Tk = [I p , X]. Note that the above representation is not always coprime, and coprimeness has to be guaranteed first for McMillan degree to be pk; otherwise, the McMillan degree is less than pk. The dynamic determinantal OBD output feedback problem is expressed using as Tr (s) = R(s) in the form 

sk R(s)



 sk−1 R(s) 

 f (s) = det{(Tk sk + . . . + T0 )R(s)} = det{[Tk , Tk−1 , . . . , T0 ]  

.. . R(s)

 } 

(2.113)

Remark 2.12. The above formulation of the determinantal dynamic assignment problems is based on the assumption that p ≤ m and thus output feedback configuration is used. If p ≥ m, we can similarly formulate the corresponding problems as determinantal dynamic precompensation problems and use right coprime MFDs for C(s). 

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2.9.2

The General Determinantal Assignment Problem

All the problems introduced in the previous sections belong to the same problem family, i.e., the determinantal assignment problem (DAP) [28]. This problem is to solve the following equation with respect to polynomial matrix H(s) : ¯ det(H(s)M(s)) = f (s)

(2.114)

where f (s) is the polynomial of an appropriate degree d. The difficulty for the solution of DAP is mainly due to the multilinear nature of the problem as this is described by its determinantal character. We should note, however, that in all cases mentioned previously, all dynamics can ¯ be shifted from H(s) to M(s), which in turn transforms the problem to a constant DAP as it has been shown by the analysis in the previous section. This equivalent formulation of the problem may be described as follows: Let M(s) ∈ R[s] p×r [s], r ≤ p, such that rank(M(s)) = r, f (s) be a real polynomial of an appropriate degree d, and let H be a family of full rank r × p constant matrices having a certain structure. Solve with respect to H ∈ H the equation: fM (s, H) = det(HM(s)) = f (s)

(2.115)

Remark 2.13. The degree of the polynomial f (s) depends firstly upon the degree of M(s) and secondly, upon the structure of H. However, in most of our problems the degree of p(s) is equal to the degree of M(s).  The determinantal assignment problem has two main aspects. The first has to do with the solvability conditions for the problem and the second, whenever this problem is solvable, to provide methods for constructing these solutions. We classify the solutions to two classes: exact and generic solvability conditions. Notation [50]: Let Qk,n denote the set of lexicographically ordered, strictly increasing sequences of k integers from 1, 2, . . . , n. If {xi1 , . . . , xik } is a set of vectors of V , ω = (i1 , . . . , ik ) ∈ Qk,n , then xi1 ∧ . . . ∧ xik = xω ∧ denotes the exterior product and by ∧r V we denote the r-th exterior power of V . If H ∈ F m×n and r ≤ min{m, n}, then by Cr (H) we denote the r-th compound matrix of H.  If hti , mi (s), i = 1, . . . , r, we denote the rows of H, columns of M(s) respectively, then Cr (H) = ht1 ∧ . . . ∧ htr = ht ∧ ∈ R1×σ Cr (M(s)) = m1 (s) ∧ . . . ∧ mr (s) = m(s)∧ ∈ Rσ [s], σ = and by Binet-Cauchy theorem [50] we have that [28]: fM (s, H) = Cr (H)Cr (M(s)) = hh∧, m(s)∧i =

∑

  p r

hω mω (s)

(2.116)

ω ∈Qr,p

where h, i denotes inner product, ω = (i1 , . . . , ir ) ∈ Qr,p , and hω , mω (s) are the coordinates of h∧, m(s)∧ respectively. Note that hω is the r × r minor of H which corresponds to the ω set of columns of H and thus hω is a multilinear alternating function of the entries hi j of H. The multilinear, skew symmetric nature of DAP suggests that the natural framework for its study is that of exterior algebra [49]. The study of the zero structure of the multilinear function fM (s, H) may thus be reduced to a linear subproblem and a standard multilinear algebra problem as it is shown below.

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(i) Linear subproblem of DAP: Set m(s)∧ = p(s) ∈ Rσ [s]. Determine whether there exists a k ∈ Rσ , k 6= 0, such that fM (s, k) = kt p(s) = ∑ ki pi (s) = f (s), i ∈ ∼σ, f (s) ∈ R[s]

(2.117)

(ii) Multilinear subproblem of DAP: Assume that K is the family of solution vectors k of (2.118). Determine whether there exists H t = [h1 , . . . , hr ], where H t ∈ R p×r , such that h1 ∧ . . . ∧ hr = h∧ = k, k ∈ K

(2.118)

Polynomials defined by Equation (2.118) are called polynomial combinants [28] and the zero assignability of them provides necessary conditions for the solution of the DAP. The solution of the exterior Equation (2.118) is a standard problem of exterior algebra and it is known as decomposability of multivectors. Note that notions and tools from exterior algebra play also an important role in the linear subproblem, since fM (s, k) is generated by the decomposable multivector m(s) ∧ . The essence of the approach is projective, that is we use a natural embedding for determinantal problems to embed the space of the unknown, H, of DAP, into an appropriate projective space. In this way we can see our problem as search for common solutions of some set of linear equations and another set of second order polynomial equations. This also allows us to compactify H into H and then use algebraic geometric, or topological intersection theory methods to determine existence of solutions for the above sets of equations. The characteristic of the current framework is that it allows the use of algebraic geometry and topological methods [15] for the study of solvability conditions but also computations. Central to the latter is the solution of the linear system derived by (2.118) with the quadratics characterising the solvability of (2.118), which are known as Quadratic Pl¨ucker Relations (QPR) [49], [16].

2.9.3

Grassmann- Plucker Invariants

The importance of the DAP framework is that it uses the natural embedding of a Grassmannian into a projective space and this in turn defines new sets of invariants characterising the solvability of the different DAP problems [28]. We may summarise the results relating to the new invariants as follows: Let T (s) ∈ R p×r (s), T (s) = [t1 (s), . . . ,tr (s)], p ≥ r, rankR(s) {T (s)} = r and let Xt = spanR(s) (T (s)). If T (s) = M(s)D(s)−1 is a RCMFD of T (s), then M(s) is a polynomial −1 , where ˜ basis for Xt . If Q(s) is a greatest right divisor of M(s) then T (s) = M(s)Q(s)D(s) ˜ M(s) is a least degree polynomial basis of Xt [15]. A Grassmann Representative (GR) for Xt is defined by t(s)∧ = t1 (s) ∧ . . . ∧ tr (s) = m˜ 1 (s) ∧ . . . ∧ m˜ r (s) · zt (s)/pt (s)

(2.119)

where zt (s) = det{Q(s)}, pt (s) = det{D(s)}  are  the zero, pole polynomials of T (s) and p ˜ m˜ 1 (s) ∧ . . . ∧ m˜ r (s) = m(s)∧ ˜ ∈ Rσ [s], σ = , is also a GR of Xt . Since M(s) is a least r degree polynomial basis for Xt , the polynomials of m(s)∧ ˜ are coprime and m(s)∧ ˜ will be ˜ then δ is the referred to as a reduced polynomial GR (R - R[s]-GR) of Xt . If δ = deg(m(s)∧), Forney dynamical order [12] of Xt . The polynomial vector m(s)∧ ˜ may be expressed as

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m(s)∧ ˜ = p(s) = p0 + sp1 + . . . + sδ pδ = Pδ eδ (s), Pδ ∈ Rσ ×(δ +1)

(2.120)

˜ and eδ (s) = [1, s, . . . , sδ ]t . It can be readily shown that all where Pδ is a basis matrix for m(s)∧ R[s]-GRs of Xt differ only by a nonzero scalar factor a ∈ R. By choosing an m(s)∧ ˜ for which ||pδ || = 1, a monic reduced Grassman representative (C-R[s]-GR) of Xt is defined, and shall be denoted by g(Xt ) and is referred to as canonical R[s]-GR; the basis matrix Pδ of g(Xt ) is defined as the Pl¨ucker matrix of Xt [28]. The following properties hold true [28]: Theorem 2.23. g(Xt ), or the associated Pl¨ucker matrix Pδ , is a complete (basis free) invariant of Xt . Corollary 2.13. Let T (s) ∈ R p×r (s), p ≥ r, rankR(s) {T (s)} = r, zt (s), pt (s) be the monic zero, pole polynomials of T (s) and let g(Xt ) = p(s) be the C-R[s]-GR of the column space Xt of T (s). The vector t(s)∧ may be uniquely decomposed as t(s)∧ = c · p(s) · zt (s)/pt (s), where c ∈ R − {0}

(2.121)

˜ ˜ If M(s) ∈ ≥ r, rankR(s) {M(s)} = r, then M(s) = M(s)Q(s), where M(s) is a least degree basis and Q(s) is a greatest right divisor of M(s) and thus R p×r [s], p

m(s)∧ = m(s) ˜ ∧ ·det{Q(s)} = p(s) · zm (s) = Pδ eδ (s) · zm (s)

(2.122)

The linear part of DAP is thus reduced to fM (s, k) = kt p(s)zm (s) = kt Pδ eδ (s) · zm (s)

(2.123)

Corollary 2.14. The zeros of M(s) are fixed zeros of all combinants of m(s) ∧ . ˜ k) = The zeros of fM (s, k) which may be freely assigned are those of the combinant f M(s, ˜ km(s)∧, ˜ where m(s)∧ ˜ is reduced. Given that the zeros of f M(s, k) are not affected by scaling with constants, we may always assume that m(s)∧ ˜ = Pδ eδ (s). In the following, the case of combinants generated by reduced m(s)∧ ˜ will be considered. If a(s) ∈ R[s] is the polynomial which has to be assigned, then max(deg(a(s)) = δ , where δ is the Forney dynamical order of Xt . If a(s) = atδ eδ (s) = a0 + a1 s + . . . + aδ sδ , where a ∈ Rδ +1 , then the problem of finding k, k ∈ Rσ , such that fM˜ (s, k) = a(s) is reduced to the solution of   p Pδt k = a, Pδt ∈ R(δ +1)×σ , σ = (2.124) r The matrix M(s) ∈ R p×r [s] generating DAP will be called linearly assignable (LA), if (2.124) has a solution for all a; otherwise, it will be called linearly nonassignable (LNA). M(s) will be called completely assignable (CA), if it is LA and (2.118) has a solution for at least a solution of the linear problem defined by (2.124). An important family of nonassignable M(s) matrices are those for which there is no k such that fM (s, k) = c, c ∈ R; such M(s) are called strongly nonassignable (SNA) and they imply that they cannot assign all zeros at s = ∞. Some results characterising the above properties are stated below: Remark 2.14. Necessary condition for M(s) to be LA is that M(s) is a least degree matrix.  Corollary 2.15. Let M(s) ∈ R p×r [s] be a least degree matrix, Pδ be the Pl¨ucker matrix of Xm and let π = rank(Pδ ). Then,

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  p (i) Necessary and sufficient condition for M(s) to be LA is that π = δ + 1 (i.e., ≥ δ +1 r and π = δ + 1). (ii) Let M(s) be LA and let P˜ † be the right inverse of Pδt and P˜δ⊥ a basis for Nr (Pδt ). For every a ∈ Rδ +1 , the solution of (eq:118) is given by k = P˜ † a + P˜δ⊥ c, where c ∈ Rσ −δ −1 arbitrary.

(2.125)

(iii) Let Pδ = [p0 , p1 , . . . , pδ ] = [p0 , . . . , pδ ] ∈ Rσ ×(δ +1) and let π = rank{Pδ }, π¯ = rank{P¯δ }. M(s) is strongly nonassignable, iff π¯ = σ . A number of Pl¨ucker type matrices associated with a linear system are: ¨ (a) Controllability Plucker Matrix [33] : For the pair (A, B), b(s)t ∧ denotes  the exterior n+ p basis matrix product of the rows of Pc (s) = [sI − A, −B] and P(A, B) is the (n + 1) × n of b(s)t ∧ . P(A, B) will be called the Controllability Pl¨ucker matrix and its rank properties characterise the system controllability as shown below: Theorem 2.24. [33]: The system S(A, B) is controllable iff P(A, B) has full rank. ¨ (b) Observability Plucker Matrix [33] : For the pair (A,C), c(s)∧denotes the exterior prodn+m t t t uct of the columns of Po (s) = [sI − A , −C ] and P(A,C) is the × (n + 1) basis n matrix of c(s) ∧ . P(A,C) will be called observability Pl¨ucker matrix and its rank properties characterise system observability. Corollary 2.16. [33]: The system S(A,C) is observable, iff P(A,C) has full rank. ¨ (c) Transfer Function Matrix Plucker Matrices: For the transfer function matrix G(s) represented by the RCMFD, LCMFD we define by tr (s)∧, tℓ (s)t ∧ the  exterior  product of the m+ p columns of Tr (s), rows of Tℓ (s) respectively. By P(Tr ) we denote the × (n + 1) basis p   m+ p matrix of tr (s)∧, and by P(Tℓ ) the (n + 1) × basis matrix for tℓ (s)t ∧ . P(Tr ), P(Tℓ ) p will be referred to as right, left fractional representation Pl¨ucker matrices respectively. Such matrices provide the prime indicators for the solution of the output feedback, or constant precompensation problem. Theorem 2.25. [41], [42]: For a generic system with mp > n, then the corresponding Pl¨ucker matrices P(Tr ), P(Tℓ ) have full rank. The full rank of these matrices is a necessary condition for the solvability of pole assignment problems and their singular values characterise the norm properties of the corresponding solutions. Given that Tr (s), Tℓ (s) uniquely characterise (modulo unimodular equivalence) the transfer function G(s), we may also refer to P(Tr ), P(Tℓ ) as the right-, left- transfer function Pl¨ucker matrices and we denote them simply by Pr (G), Pℓ (G). ¨ (d) Column, Row Plucker Matrices: For the transfer function G(s), m ≥ p, n(s)∧ denotes the exterior product of the columns of the numerator Nr (s), of a RCMFD and by P(N) the

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 m × (d + 1) basis matrix of n(s) ∧ . Note that d = δ , the Forney order of Xg , if G(s) has ℓ no finite zeros and d = δ + κ , where κ is the number of finite zeros of G(s), otherwise. If Nr (s) is least degree (has no finite zeros), then Pc (N) will be called the column space Pl¨ucker matrix of the system. For this case the row space Pl¨ucker matrix may be similarly defined and it is Pr (N) = 1. For systems with m ≤ p and full rank transfer functions Pc (N) = 1, whereas Pr (N) is a nontrivial matrix. Such matrices play a key role in problems such as the squaring down problem. Corollary 2.17. For a generic system with m > p, for which p(m − p) > δ + 1, where δ is the Forney order, Pc (N) has full rank. The above classes of problems are typical formulations amongst a very large class of problems of integrated system design [24], [25] which involve the design, or redesign of input, output structure of the system aiming at producing square systems with prescribed zero structure and thus good potential for control design. Such problems are referred to as zero structure assignment problems and may be studied with algebraic means (polynomial matrix theory, or matrix pencils) geometric theory (dynamic cover problems for invariant spaces), or exterior algebra and algebraic geometry tools (determinantal formulation and intersection theory of varieties) [4], [28]. The study of generic solvability of such problems and a method for computing solutions may be found in [41], [42], [43].

2.10

Conclusions

This overview paper provides an introduction to the algebraic structural aspects of Linear Systems with emphasis on the system structure and related system properties. Our viewpoint has been that the algebraic structure of the system model underpins the nature of dynamic and feedback properties. Poles and zeros and system invariants have thus been central to this approach. Poles express the internal dynamics of the system and they are directly related to stability and other aspects of performance and they are affected by the different types of feedback transformations. On the other hand, zeros are measures of the interaction between internal dynamics and their coupling to inputs and outputs. They are invariant under fundamental feedback transformations and their alternation may be achieved under dynamic compensation, or by design or re-design of the input, output system structure. The significance of poles and zeros for control design is discussed in the references cited in the bibliography. The theory of invariants and canonical forms of state space systems can be considered from the perspective defined by the theory of matrix pencils [13], [22], [44] and the theory of minimal bases of rational vector spaces [12], [75]. The role and significance of invariants for Control Synthesis and Control Design problems may be found in the literature [7], [12], [18], [43], [58], [72], [74] and references therein.

A Invariants and Canonical Forms The development of theory of matrix pencil invariants and canonical forms requires some basic results that may be found in algebra books, as well as some definitions on a number of important notions [36].

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Definition 2.11. Let X be a set, E be an equivalence relation on X and let x ∈ X ; the equivalence class, or orbit of x under E is defined by E(x) = {y : y ∈ X : xEy}. The set of all equivalence classes is called the quotient set, or orbit set and it is denoted by X /E. Theorem 2.26. If E is an equivalence relation on a set X , then the family of all E-equivalence classes forms a partition of X , i.e.,there exist {xi ∈ X : i = 1, 2, . . .} such that X = E(x1 ) ∪ . . . ∪ E(xi ) ∪ . . . , E(xi ) ∩ E(x j ) 6= 0/

(2.126)

The set of elements {xi , i = 1, 2, . . .} for which conditions (2.126) hold true is called a system of distinct representatives for E and it is a subset D of X that contains precisely one element from each of the E- equivalence classes. Definition 2.12. Let X , T be sets, E an equivalence relation defined on X we define: (i) A function f : X → T is called an invariant of E, when ∀x, y ∈ X : xEy implies f (x) = f (y). (ii) f : X → T is called a complete invariant for E, when f (x) = f (y) implies xEy. (iii) A set of invariants { fi : fi : X → Ti , i = 1, 2, . . . , k} is a complete set for E, if the map f , f : X → T1 × . . . × Tk , where x → ( f1 (x), . . . , fk (x)) is a complete invariant for E on X. A complete invariant defines a one to one correspondence between the equivalence classes E(x) and the image of f . If f : X → T1 × . . . × Tk where x → ( f1 (x), . . . , fk (x)) is a complete invariant for E on X , then the set ( f1 (x), . . . , fk (x)) characterises uniquely E(x). The values fi (x) are often called invariants and this definition has been used in the paper. If we “specialise” the invariant f such that its image T ⊂ X expresses f in the “simplest” possible way, then we define the notion of a canonical element, or canonical form. Definition 2.13. A set of canonical forms for E equivalence on X is a subset C of X such that ∀x ∈ X there is a unique c ∈ X for which xEc. Equivalence relations may be introduced by the action of groups on sets and in particular by the action of transformation groups. If X is a set and G (X ) is the group of permutations of X , any subgroup H (X ) of G (X ) is called a transformation group of X . A transformation group is therefore a set H of mappings of X into X , which are bijective. If (G , ⋆) is a group operating on a set X , where ⋆ denotes the group operation and ◦ the operation of G on X , then the relation: there exists s ∈ G such that y = s ◦ x, is an equivalence relation and it is called the induced equivalence relation of G on X . G ◦ x is then the equivalence class of x under this relation.

B List of Symbols, Abbreviations R:

field of real numbers

C:

field of complex numbers

R[s] : ring of polynomials in s with coefficients in R

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R(s) : field of rational functions in s with coefficients in R R p (s) : ring of proper and Ω -stable rational functions R pr (s) : ring of proper rational functions Rm×n : the set of m × n matrices with real coefficients Rm×n : the set of m × n rational matrices V :

vector space over R, or C, or R(s)

deg :

degree of a polynomial

det{A} : determinant of a square matrix A, also denoted by |A| f ed :

finite elementary divisors

ied :

infinite elementary divisors

cmi :

column minimal indices

rmi :

row minimal indices

∧:

exterior product of vectors

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3 Modelling and Model Reduction—State-Space Truncation† David J.N. Limebeer

3.1 Introduction The approximation of high-order plant and controller models by models of lower-order is an integral part of control system design and analysis. Until relatively recently model reduction was often based on physical intuition. For example, chemical engineers often assume that mixing is instantaneous and that packed distillation columns may be modelled using discrete trays. Electrical engineers represent transmission lines and the eddy currents in the rotor cage of induction motors by lumped circuits. Mechanical engineers remove high-frequency vibration modes from models of aircraft wings, turbine shafts and flexible structures. It may also be possible to replace high-order controllers by low-order approximations with little sacrifice in performance. We will develop elementary procedures which seek to automate the model reduction process. Suppose a high-order, linear, time-invariant model G is given, then the prototype b of G such that kG − Gk b ∞ L∞ model reduction problem is to find a low-order approximation G is small. Truncation methods of model reduction seek to remove, or truncate, unimportant states from state-space models. If a state-space model has its A-matrix in Jordan canonical form, statespace truncation will amount to classical modal truncation. For example, one may truncate all those states that correspond to “fast modes”—eigenvalues with a large negative real part. One’s interpretation of “fast” will obviously depend on the application, but this could mean modes outside the control system bandwidth. Since the poles of the truncated model are a subset of the poles of the original high-order model, any low-order modal approximation of a stable high-order model is guaranteed to be stable. As we will show later, it is also possible to b ∞ for modal truncation. Because any transfer function can be realized get a bound on kG − Gk in terms of an infinite number of state-space models, there are, in principle, also an infinite number of candidate truncation schemes. For a truncation scheme to be useful, it must preserve stability and carry with it a guaranteed error bound. The aim of these notes is to develop the balanced truncation method of model reduction, which satisfies an infinity norm bound on the absolute approximation error. † This

chapter is based on Chapter 9 of “Linear Robust Control” [1]. This book, together with the solutions manual, can be obtained from the authors in .pdf form on request.

M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 99 -122, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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It is well known that the modes of a realization that are either uncontrollable or unobservable do not appear in the corresponding system transfer function matrix. It is therefore natural to conjecture that the modes that are almost uncontrollable and unobservable can be omitted from the realization with little effect on the input-output characteristics of the model. The balanced realization has the property that mode i is equally controllable and observable, with these properties measured in terms of a number σi ≥ 0. As σi increases, the corresponding level of controllability and observability increases; the σi ’s are the singular values of the Hankel operator associated with the transfer function matrix G. The model reduction method that applies the truncation operation to a balanced realization is known as balanced truncation. For this algorithm the absolute error is guaranteed to satisfy the twice-the-sum-of-the-tail bound b ∞ ≤ 2(σr+1 + . . . + σn ), kG − Gk

b in which n is the McMillan degree of G and r is the McMillan degree of G. Example 3.1. The transfer function

g=

1 (s + 1)(s + 2)

has modal realization g = Cm (sI − Am )−1 Bm , in which     −1 0 1 Am = , Bm = , 0 −2 −1   Cm = 1 1 .

(3.1) (3.2)

If we truncate the fast mode we obtain the reduced-order system gm =

1 . s+1

The norm of the error is

1 1 k∞ = . s+2 2 The transfer function g also has realization g = Cb (sI − Ab )−1 Bb , in which     −0.40859 −0.970143 0.492479 Ab = , Bb = , 0.970143 −2.59141 −0.492479   Cb = 0.492479 0.492479 .

kg − gm k∞ = k

(3.3) (3.4)

This realization is balanced because the controllability and observability gramians are both equal and diagonal:   0.296796 0 Σ= , 0 0.0467961

so that σ1 = 0.296796 and σ2 = 0.0467961. If we truncate this realization, we obtain the reduced-order system (0.492479)2 gb = , s + 0.40859 which is stable and the norm of the error is kg − gb k∞ = 0.0935921 = 2σ2 .

(3.5) (3.6)

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The point to note is that, in this case, the error associated with balanced truncation is considerably smaller than that associated with modal truncation. The generally good performance of balanced truncation has led to its widespread popularity and enduring usefulness. Another well known model order reduction method is the singular perturbation approximation. This technique is usually associated with a fast-slow decomposition of the state space, with the approximation achieved by setting the “fast” states to their steady-state values. Since the singular perturbation approximation is related to state-space truncation by the frequency inversion transformation s → 1/s, it is also considered to be a truncation method. Our main interest is in balanced singular perturbation approximation, in which the singular perturbation method is applied to a balanced realization. In this method, states corresponding to small σi ’s are set to their steady-state values. Although the error bounds for balanced truncation and balanced singular perturbation approximation are identical, the resulting models have different high- and low-frequency characteristics. Direct truncation gives a good model match at high frequency, while singular perturbation methods have superior low-frequency properties. We will develop most of our theory for direct truncation because of its notational simplicity.

3.2 State-Space Truncation Consider a linear, time-invariant system with the realization x(t) ˙ = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t)

x(0) = x0 ,

(3.7)

and divide the state vector x into components to be retained and components to be discarded:   x (t) x(t) = 1 . (3.8) x2 (t) The r-vector x1 (t) contains the components to be retained, while the (n − r)-vector x2 contains the components to be discarded. Now partition the matrices A, B and C conformably with x to obtain     A11 A12 B1 A= , B= , A21 A22 B2 (3.9)   C = C1 C2 . By omitting the states and dynamics associated with x2 (t), we obtain the lower-order system p(t) ˙ = A11 p(t) + B1 u(t) q(t) = C1 p(t) + Du(t).

p(0) = p0 ,

The rth -order truncation of the realization (A, B,C, D) is given by Tr (A, B,C, D) = (A11 , B1 ,C1 , D).

(3.10)

In general, very little can be said about the relationship between x and p, y and q or the transfer b associated with (A, B,C, D) and (A11 , B1 ,C1 , D). In particular, the function matrices G and G

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truncated system may be unstable even if the full-order system is stable, and the truncated system realization may be nonminimal even if the full-order realization is minimal. One thing that clearly does hold is b G(∞) = G(∞), which means that all reduced-order models obtained by truncation have perfect matching at infinite frequency.

Example 3.2. (Modal Truncation). The truncation of modal realizations is common in engineering practise, because it is often the case that high-frequency modes may be neglected on physical grounds, or because the phenomena resulting in such modes only play a secondary role in determining the model’s essential characteristics. Let G be a transfer function matrix with an asymptotically stable modal realization     λ1 0 0 B1     A =  0 ... 0 , B =  ...  , (3.11) 0 0 λm Bm   C = C1 . . . Cm (3.12)

For simplicity, we assume that each of the eigenvalues λi of the A-matrix has a simple Jordan structure. If the modes with a fast decay rate may be omitted from the model, the eigenvalues should be ordered so that |Re (λi )| is nondecreasing with increasing i. Alternatively, if one is to omit the high-frequency modes, the eigenvalues should be ordered so that |Im (λi )| is nondecreasing with increasing i. As a combination of these two, one might order the modes so that |λi | is nondecreasing with increasing i, so that those modes with the highest “natural” frequency are deleted. The error incurred in modal truncation depends not only on the λi ’s, but also on the size of the b we have residues Ci Bi . If modes labeled r + 1 to n are omitted by truncation to obtain G, and therefore that

b= G−G

n

Ci Bi s i=r+1 − λi

b ∞≤ kG − Gk

∑

n

kCi Bi k . i=r+1 |Re λi |

∑

Since the error associated with deleting a mode depends on the ratio kCi Bi k/|Re λi | and not |Re λi | alone, the modal decay rate is not a reliable guide as to whether a particular mode should be included in the reduced-order model. The main features of modal truncation are: 1. It is conceptually simple. 2. The poles of the reduced-order model are a subset of the poles of the original model. In addition, the poles of the reduced-order model retain their physical interpretation, because one knows, for example, that certain vibration modes are being retained while others are being omitted. 3. It is computationally cheap, because the main calculation is an eigenvalue decomposition of A.

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3.2.1 The Truncation Error To help us with later work, in which we develop bounds for the error incurred by model reduction based on truncation, we develop some of the basic properties of the error system. Lemma 3.1. Suppose (A, B,C, D) is a realization of order n partitioned as in (3.9) and let b = D +C1 (sI − A11 )−1 B1 . Define G = D +C(sI − A)−1 B and G ˜ A(s) = A22 + A21 (sI − A11 )−1 A12 ˜B(s) = B2 + A21 (sI − A11 )−1 B1 ˜ C(s) = C2 +C1 (sI − A11 )−1 A12 .

(3.13)

Then: ˜ 1. det(sI − A) = det(sI − A11 ) det(sI − A(s)). 2. The truncation error system satisfies −1 ˜ b = C(s)(sI ˜ ˜ G(s) − G(s) − A(s)) B(s). (3.14)   P 0 3. If AP + PA′ + BB′ = 0 and P = 1 , in which the partitioning is conformable with 0 P2 (3.9), then

A11 P1 + P1 A′11 + B1 B′1 = 0 (3.15) ∼ ∼ ˜ ˜ ˜ ˜ A(s)P2 + P2 A (s) + B(s)B (s) = 0. (3.16)   ˜ ˜ If (A, B) is controllable, then sI− A(s) B(s) has full row rank for all s.  Q1 0 ′ ′ 4. If A Q + QA +C C = 0 and Q = , in which the partitioning is conformable with 0 Q2 (3.9), then A′11 Q1 + Q1 A11 +C1′ C1 = 0 ˜ + C˜ ∼ (s)C(s) ˜ A (s)Q2 + Q2 A(s) = 0.   ˜ sI − A(s) If (A,C) is observable, then has full column rank for all s. ˜ C(s) ˜∼

Proof. Write Φ (s) = (sI − A11 )−1 and note that    0 I 0 sI − A11 sI − A = ˜ 0 sI − A(s) −A21 Φ (s) I   I −Φ (s)A12 . × 0 I

(3.17) (3.18)

(3.19)

1. This follows directly from (3.19). 2. From (3.19), we have C(sI − A)−1 B      I Φ (s)A12 Φ (s) 0 = C1 C2 −1 ˜ 0 I 0 (sI − A(s))    I 0 B1 × B2 A21 Φ (s) I −1 ˜ ˜ ˜ = C1 Φ (s)B1 + C(s)(sI − A(s)) B(s),

which proves (3.14).

(3.20) (3.21) (3.22) (3.23)

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3. Equation (3.15) is immediate from the assumed block diagonal structure of P. To prove (3.16), write BB′ = (sI − A)P + P(−sI − A)′ . Now use (3.19) to obtain    B1  ′ ˜ ′ B1 B (−s) ˜ B(s)    sI − A11 −A12 P1 P1 Φ ′ (−s)A′21 = ˜ 0 P2 0 sI − A(s)    −sI − A′11 0 P1 0 + . ′ A21 Φ (s)P1 P2 −A′12 −sI − A˜ (−s)

(3.24) (3.25) (3.26)

The (2, 2)-block of this equation is (3.16). We also note that if x is such that   ˜ ˜ B(s) =0 x∗ sI − A(s)

for some s, then

  I 0  sI − A B = 0 A21 Φ (s) I   ˜ ˜ B(s) and we see that (A, B) controllable implies x = 0. Hence sI − A(s) has full row rank when (A, B) is controllable. 4. This follows from calculations which are dual to those given in Item 3. 

0 x∗





3.2.2 Singular Perturbation Approximation The steady-state error associated with state-space truncation is given by −1 b G(0) − G(0) = C1 A−1 11 B1 −CA B.

In applications requiring good low-frequency models this may be unacceptably large. In these cases, it is appropriate to use a singular perturbation approximation in preference to state-space truncation because of its greatly improved low-frequency model reduction characteristics. Consider the full-order model given by x(t) ˙ = Ax(t) + Bu(t),

x(0) = x0 ,

y(t) = Cx(t) + Du(t),

(3.27) (3.28)

which is partitioned as in (3.8) and (3.9). If x2 (t) represents the fast dynamics of the system, we may approximate the low-frequency behavior by setting x˙2 (t) = 0. This gives 0 = A21 x1 (t) + A22 x2 (t) + B2 u(t), which yields the quasi-steady-state solution x2 (t) = −A−1 22 (A21 x1 (t) + B2 u(t))

(3.29)

provided A22 is nonsingular. Eliminating x2 from the remaining equations using (3.29) yields

Modelling and Model Reduction State-Space Truncation −1 p(t) ˙ = (A11 − A12 A−1 22 A21 )p(t) + (B1 − A12 A22 B2 )u(t)

q(t) =

−1 (C1 −C2 A−1 22 A21 )p(t) + (D −C2 A22 B2 )u(t).

105 (3.30) (3.31)

The rth -order singular perturbation approximation (SPA) is given by

in which

b11 , Bb1 , Cb1 , D), b Sr (A, B,C, D) = (A

(3.32)

b11 = A11 − A12 A−1 A21 , Bb1 = B1 − A12 A−1 B2 , A 22 22 −1 b Cb1 = C1 −C2 A−1 A , D = D −C A 2 22 B2 . 22 21

(3.33)

The following result shows that SPA is equivalent to: (a) setting H(w) = G(w−1 ); (b) performb ing a state-space truncation of H(w) to obtain H(w); and (c) defining the reduced-order model b = H(s b −1 ). as G(s)

Lemma 3.2. Let G(s) = D + C(sI − A)−1 B, in which A is nonsingular, and let H(w) = G(w−1 ). Then: 1. H(w) = D −CA−1 B −CA−1 (wI − A−1 )−1 A−1 B. 2. The realizations of G(s) and H(w) have the same controllability and observability gramians (when they exist). 3. Suppose that A22 is nonsingular, that Gr (s) is the rth -order SPA of G(s) and that H r (w) is the rth -order system obtained by truncation of the realization of H(w) defined in Item 1. Then Gr (s) = H r (s−1 ).

Proof. 1. This follows from the identity (w−1 I − A)−1 = −A−1 − A−1 (wI − A−1 )−1 A−1 . 2. Suppose P and Q are the controllability and observability gramians of the realization of G satisfying AP + PA′ + BB′ = 0

(3.34)

A′ Q + QA +C′C = 0.

(3.35)

Multiplying by A−1 and (A−1 )′ gives A−1 P + P(A−1 )′ + (A−1 B)(A−1 B)′ = 0 QA 3. Writing

−1

+ (A

−1 ′

) Q + (CA



I A12 A−1 22 A= 0 I gives A−1 =



I −A−1 22 A21

Now truncate the realization of H(w):

0 I

 

−1 ′

) (CA

b11 0 A 0 A22



b−1 0 A 11 0 A−1 22

−1

) = 0.

I 0 A−1 A I 21 22



(3.36)



 I −A12 A−1 22 . 0 I

(3.37)

106

D.J.N. Limebeer 

   I b−1 I 0 A−1 =A 11 0   −1 b−1 Bb1 I0 A B=A 11   I b−1 −CA−1 = −Cb1 A 11 0

b−1 Bb1 . b − Cb1 A D −CA−1 B = D 11

(3.38) (3.39) (3.40) (3.41)

Applying Item 1 to the realization of Gr given in (3.32), H r (w) = Gr (w−1 ), which is equivalent to Gr (s) = H r (s−1 ). Since the singular perturbation and truncation operations are related in a straightforward way, it suffices to develop all our theoretical results for state-space truncation. When the lowfrequency fidelity of the approximation is important, the singular perturbation approximation is the method of choice. Conversely, direct truncation should be preferred when good highfrequency modelling is the central concern.

Main Points of the Section 1. State-space truncation is a simple but general procedure for generating reducedorder models. The properties of the reduced-order model will depend on the realization selected for truncation. For example, reduced-order models obtained from the truncation of balanced and modal realizations (of the same full-order system) will generally be quite different. 2. State-space truncation produces zero error at infinite frequency. 3. Since the singular perturbation method of model reduction is related to statespace truncation by the bilinear transform s → 1/s, singular perturbation approximations have zero steady-state error.

3.3 Balanced Realization The aim of this section is to introduce balanced realizations, which are of interest because they have good absolute-error truncation properties.

3.3.1 Model Reduction Motivation b∈ Suppose we are given a G ∈ RH ∞ and our aim is to produce a reduced-order model G RH ∞ that approximates G. A natural criterion with which to measure the absolute error is b ∞. kG − Gk b G with the same input u, we get If we drive G and y = Gu,

and therefore that

b yb = Gu

Modelling and Model Reduction State-Space Truncation

b ∞ = sup kG − Gk

u∈L2

If G = D +C(sI − A)−1 B, then

ky − ybk2 . kuk2

107

(3.42)

x˙ = Ax + Bu

(3.43)

y = Cx + Du.

(3.44)

b ∞ to be small, the We assume that A is asymptotically stable (i.e., Re λi (A) < 0). For kG − Gk identity given in (3.42) suggests we should delete those components of the state-vector x that are least involved in the energy transfer from the input u to the output y. This observation leads us to consider two closely related questions: 1. What is the output energy resulting from a given initial state x(0) = x0 ? 2. What is the minimum input energy required to bring the state from zero to the given initial state x(0) = x0 ? The solutions are well known and closely related to the theory of Hankel operators [1]: 1. Suppose x(0) = x0 is given and that u(t) = 0 for t ≥ 0. By standard theory [1], the L2 [0, ∞) norm of y is given by kyk22 = x0′ Qx0 , in which Q is the observability gramian. 2. Consider the LQ problem min

Z 0

u∈L2 (−∞,0] −∞

u′ (t)u(t)dt

subject to x˙ = Ax + Bu with x(0) = x0 . This is equivalent to min

Z ∞

v∈L2 [0,∞) 0

v′ (τ )v(τ ) d τ

subject to d p(τ ) = −Ap(τ ) − Bv(τ ), p(0) = x0 , dτ with τ = −t, p(τ ) = x(t) and v(τ ) = u(t). By standard LQ theory [1], the optimal control is v(τ ) = B′ X p(τ ) and Z ∞

min v

0

v′ (τ )v(τ ) d τ = x0′ Xx0 ,

in which X is the solution to

−XA − A′ X − XBB′ X = 0 such that −A − BB′ X is asymptotically stable. If (A, B) is controllable and P is the controllability gramian satisfying AP + PA′ + BB′ = 0, then P is invertible and −P−1 A − A′ P−1 − P−1 BB′ P−1 = 0. Furthermore −A − BB′ P−1 = PA′ P−1 , which is asymptotically stable. Hence X = P−1 and we conclude that the optimal control is u(t) = B′ P−1 x(t) and that min

Z 0

u∈L2 (−∞,0]:x(0)=x0 −∞

u(t)′ u(t) dt = x0′ P−1 x0 .

108

D. J. N. Limebeer

Combining the answers to our two questions we get R∞ ′ 0 y (t)y(t) dt = R0

max

u∈L2 (−∞,0]:x(0)=x0 −∞ u′ (t)u(t) dt

=

x0′ Qx0 ′ x0 P−1 x0 1

(3.45) 1

α ′ P 2 QP 2 α , α ′α

1

x0 = P 2 α .

b ∞ small, the state-space for the trunThese calculations suggest that in order to keep kG − Gk cated system should be the space spanned by the eigenvectors corresponding to the larger 1

1

1

1

eigenvalues of P 2 QP 2 . That is, we should truncate a realization in which P 2 QP 2 is diagonal, with the eigenvalues ordered in descending order.

3.3.2 Balanced Realization In the last section we argued that an appropriate realization for absolute-error model reduction 1 1 is one in which P 2 QP 2 is diagonal. We now show that these realizations, known as a balanced realizations, always exist for asymptotically stable minimal realizations. The following is the most commonly used definition of a balanced realization: Definition 3.1. A realization (A, B,C) is balanced if A is asymptotically stable and

in which



AΣ + Σ A′ + BB′ = 0

(3.46)

A′ Σ + Σ A +C′C = 0,

(3.47)

 0 σ1 Ir1 0   Σ =  0 ... 0 , 0 0 σm Irm

σi 6= σ j , i 6= j and σi > 0 ∀i.

(3.48)

Note that n = r1 +· · ·+rm is the McMillan degree of C(sI −A)−1 B and that ri is the multiplicity of σi . We say that the realization is an ordered balanced realization if, in addition, σ1 > σ2 > · · · > σm > 0. In a balanced realization, the basis for the state space is such that each basis vector is equally controllable and observable, with its “degree” of controllability and observability given by the corresponding diagonal entry of Σ . Suppose (A, B,C) is a balanced realization and the initial condition x0 is partitioned as x0 =   ′ ′ ′ with x an r × 1 vector. It follows from (3.45) that x1 · · · xm i i max

R∞ ′ m 0 y (t)y(t) dt = ∑ σi2 xi′ xi . R0 ′

u∈L2 (−∞,0]:x(0)=x0 −∞ u

σi2

(t)u(t) dt

i=1

This shows that is a measure of the extent to which the corresponding ri dimensional subspace of the state space is involved in the transfer of energy from past inputs to future outputs. These ideas can be related to the norm of the Hankel operator (with symbol G) [1]. The next result is concerned with the existence and uniqueness of balanced realizations.

109

Modelling and Model Reduction State-Space Truncation

Lemma 3.3. A given realization (A, B,C) can be transformed by a state transformation to a balanced realization if and only if it is asymptotically stable and minimal. Furthermore, a balanced realization obtained from such an (A, B,C) is unique up to: (a) the ordering of the σi ’s and (b) an orthogonal matrix S satisfying SΣ = Σ S. When the σi have multiplicity one (i.e., ri = 1 for all i), S is a diagonal matrix with diagonal elements ±1. When (A, B,C)is asymptotically stable and minimal, (TAT −1 , T B,CT −1 ) is balanced if T = 1 Σ 2 U ′ R−1 . When defining T , P = RR′ is a Cholesky factorization of P and R′ QR = U Σ 2U ′ is a singular value decomposition of R′ QR, in which P and Q are the controllability and observability gramians, which satisfy AP + PA′ + BB′ = 0

(3.49)

A′ Q + QA +C′C = 0.

(3.50)

Proof. To begin we note that if P and Q satisfy (3.49) and (3.50), then for any nonsingular T , (TAT −1 )(T PT ′ ) + (T PT ′ )(TAT −1 )′ + (T B)(T B)′ = 0 and

(T ′ )−1 QT −1 (TAT −1 ) + (TAT −1 )′ (T ′ )−1 QT −1 + (CT −1 )′ (CT −1 ) = 0.

If (A, B,C) is balanced, it is asymptotically stable by assumption and Σ > 0 implies minimality. If (A, B,C) is asymptotically stable and minimal is has positive definite controllability and ob1 servability gramians P and Q satisfying (3.49) and (3.50) respectively. Setting T = Σ 2 U ′ R−1 gives 1 1
T PT ′ = (Σ 2 U ′ R−1 )RR′ (R′ )−1U Σ 2 = Σ

and

1

1

(T ′ )−1 QT −1 = (Σ − 2 U ′ R′ )Q(RU Σ − 2 ) = Σ .

Clearly, we may re-label the state components in a balanced realization to obtain another balanced realization. To determine the nonuniqueness that is possible while maintaining the same Σ , let S be a transformation that preserves the ordering of the σi in some balanced realization. Under this assumption we have Σ = SΣ S′ and Σ = (S−1 )′ Σ S−1 . This gives SΣ 2 = Σ 2 S, which implies that SΣ = Σ S since σi > 0. It now follows that Σ = SΣ S′ = Σ SS′ , so that I = SS′ as required.

Main Points of the Section 1. A balanced realization is an asymptotically stable and minimal realization in which the controllability and observability gramians are equal and diagonal. 2. Any stable transfer function matrix has a balanced realization. The balanced realization is unique up to ordering of the numbers σi and an orthogonal transformation that commutes with Σ . 3. An analysis of the extent to which states are involved in energy transfer from past inputs to future outputs motivates the consideration of the balanced realization as an appropriate realization for absolute-error model reduction.

110

D.J.N. Limebeer

3.4 Balanced Truncation Model reduction by balanced truncation simply applies the truncation operation to a balanced realization (A, B,C, D) of a system G. Suppose (A, B,C) is a balanced realization as described in Definition 3.1 and partition Σ as   Σ1 0 (3.51) Σ= 0 Σ2 with



 σ1 Ir1 0 0   Σ1 =  0 . . . 0  , 0 0 σl Irl

 σl+1 Irl+1 0 0   .. Σ2 =  . 0 . 0 0 0 σm Im 

(3.52)

We never “split” states corresponding to a σi with multiplicity greater that one. If (A, B,C) b with realization (A11 , B1 ,C1 , D), is partitioned as in (3.9) conformably with Σ , we obtain G which is a balanced truncation of G. b is We will show that (A11 , B1 ,C1, D) is itself a balanced realization, which implies that G stable, has McMillan degree r = r1 + . . . + rl and that the approximation error satisfies the twice-the-sum-of-the-tail infinity norm bound b ∞ ≤ 2(σl+1 + . . . + σm ). kG − Gk

3.4.1 Stability Lemma 3.4. Suppose (A, B,C) is a balanced realization as described in Definition 3.1 and that (A11 , B1 ,C1 , D) is a balanced truncation of (A, B,C, D). Then (A11 , B1 ,C1 ) is a balanced realization. In particular, A11 is asymptotically stable and (A11 , B1 ,C1 ) is minimal. Note that by a trivial re-ordering argument (A22 , B2 ,C2 , D) is also a balanced realization. Proof. From (3.46) and (3.47) we have A11 Σ1 + Σ1 A′11 + B1 B′1 = 0

(3.53)

A′11 Σ1 + Σ1 A11 +C1′ C1 = 0.

(3.54)

If we can show that Re λi (A11 ) < 0, then it is immediate that (A11 , B1 ,C1 ) is a balanced realization because Σ1 > 0. Since Σ1 > 0, we have Re λi (A11 ) ≤ 0, but we still need to show that there can be no imaginary axis eigenvalues. Suppose, to obtain a contradiction, that there is a real ω such that jω I − A11 is singular. Let V be a basis for the kernel of jω I − A11 : ( jω I − A11 )V = 0.

(3.55)

Multiplying (3.54) on the left by V ∗ and on the right by V , and then multiplying just on the right by V , we obtain C1V = 0, ( jω I + A′11 )Σ1V = 0.

Modelling and Model Reduction State-Space Truncation

111

Multiplying (3.53) on the left by V ∗ Σ1 and on the right by Σ1V , and then multiplying just on the right by Σ1V , we obtain B′1 Σ1V = 0,

( jω I − A11 )Σ12V = 0.

Therefore Σ12V is also a basis for the right nullspace of jω I − A11 and hence 2

Σ12V = V Σ 1 2

for some matrix Σ 1 , which will have eigenvalues which are a subset of the eigenvalues of Σ12 . 2

(In fact, since ( jω I − A11 )(V T ) = 0 and Σ12 (V T ) = (V T )(T −1 Σ 1 T ) for any nonsingular T ,

we can assume V is such that those in Σ12 .)

2 Σ1

is diagonal—it will have diagonal entries that are a subset of

Now consider A21 Σ1 + Σ2 A′12 + B2 B′1 = 0

(3.56)

A′12 Σ1 + Σ2 A21 +C2′ C1

(3.57)

= 0,

which come from the (2, 1)-blocks of (3.46) and (3.47). Multiplying (3.56) on the right by Σ1V and multiplying (3.57) on the left by Σ2 and on the right by V , we obtain A21 Σ12V + Σ2 A′12 Σ1V = 0

Σ22 A21V

+ Σ2 A′12 Σ1V

(3.58)

= 0.

(3.59)

Subtracting these gives 2

Σ22 A21V = A21 Σ12V = A21V Σ 1 , which we may write as

"

2

Σ1 0 0 Σ22

#

I





I



2

Σ 1. A21V ′  2 Since Σ 1 and Σ22 have no eigenvalues in common, I (A21V )′ must be a basis for the " # 2 2 Σ1 0 eigenspace of corresponding to the eigenvalues Σ 1 . That is, we have 0 Σ22 

A21V

I A21V



=

=

  I , 0

which amounts to A21V = 0. Combining this with (3.55) we obtain   V ( jω I − A) = 0, 0 which contradicts the asymptotic stability of A. Notice that Lemma 3.4 does not assume that the balanced realization is ord ered in any way. The only assumption is that the partitioning of Σ does not split the states associated with a multiple σi .

112

D.J.N. Limebeer

3.4.2 Error Bound for “one-step” Truncation Our next result determines the infinity norm of the error that occurs when deleting the state(s) associated with one of the σi ’s. ˜ ˜ ˜ Lemma 3.5. Let A(s), B(s) and C(s) be proper real rational transfer function matrices, with˜ jω ) − jω I is nonsingular for all real ω . Suppose out poles on the imaginary axis, such that A( that ˜ + A˜∼ (s)) + B(s) ˜ B˜ ∼ (s) = 0 σ (A(s) ∼ ˜ + A˜ (s)) + C˜ ∼ (s)C(s) ˜ σ (A(s) =0

(3.60) (3.61)

−1 B(s) ˜ ˜ ˜ ˜ for some σ > 0. Then E(s) = C(s)(sI − A(s)) satisfies kEk∞ = 2σ . Moreover, if A(s)
has odd dimension, then σ E(0) = 2σ .

Proof. The proof is divided into two steps: first we show that kEk
∞ ≤ 2σ and then that equal˜ ity holds. In establishing the equality, we observe that σ E(0) = 2σ when A(s) has odd

˜ dimension. In the proof, we shall assume that the number of rows of C(s)( jω ) is greater than ˜ or equal to the number of columns of B(s)( jω ); when this is not the case, we may consider a dual argument based on E ∗ rather than E. Choose any real ω . From (3.60) and (3.61), we have ˜ jω ), ˜ jω )B˜ ∗ ( jω ) = C˜ ∗ ( jω )C( B( ˜ jω ) such that so there exists a matrix U( ∗ ˜ jω ) = σ 2 I, U˜ ( jω )U(

∗

˜ jω ) = 0. ˜ jω ) + C˜ ( jω )U( σ B(

(3.62)

(see Problem 3.1.) Now note that

˜ jω ) + E( jω ) ∗ U( ˜ jω ) + E( jω ) U(

∗ ∗ ˜ −∗C˜ ∗ U˜ + C( ˜ −1 B˜ ˜ jω I − A) = U˜ + B˜ ( jω I − A)

∗ ˜ −∗ ( jω I − A) ˜ + ( jω I − A) ˜ ∗ = σ I − σ B˜ ( jω I − A)
∗ ˜ −1 B˜ +(A˜ + A˜ ) ( jω I − A) 2

= σ 2 I.

(3.63) (3.64) (3.65) (3.66) (3.67)

Hence
kEk∞ = sup σ E( jω )

(3.68)

ω


˜ jω ) + E( jω ) − U( ˜ jω ) = sup σ U( ω

˜ jω ) + E( jω ) + sup σ U( ˜ jω ) ≤ sup σ U( ω

ω

= 2σ .


We now show that there is a frequency ω0 such that σ E( jω0 ) = 2σ . Define ˜ jω ) − jω I, Φ˜ ( jω ) = A(

∗

˜ ( jω ) − Φ ˜ ( jω ) X( jω ) = Φ

(3.69) (3.70) (3.71)

Modelling and Model Reduction State-Space Truncation

113

and note that X( jω ) is skew-Hermitian since X( jω ) + X ∗ ( jω ) = 0.

(3.72)

+ λ )x∗ x

If X( jω )x = λ x, x 6= 0, it follows from (3.72) that (λ = 0, so that λ + λ = 0. That is, every eigenvalue of X( jω ) is on the imaginary axis. Now X(0) is real, so it has an equal number of eigenvalues with positive and negative imaginary part. For sufficiently large ω , all the eigenvalues of X( jω ) have negative imaginary part, since X( jω ) → − j2ω I as ω → ∞. Since the eigenvalues of a matrix are continuous functions of the matrix entries, it follows that there is a frequency, ω0 , such that X( jω0 ) is singular (i.e., has a zero eigenvalue). If X has odd dimension, X(0) must be singular, since it is a real matrix of odd dimension with all its eigenvalues on the imaginary axis, so we may take ω0 = 0 in this case. Let x 6= 0 and ω0 be selected so that X( jω0 )x = 0. Then 0= = = = =

1 X( jω0 )x 2
1 ˜ ∗ Φ ( jω0 ) − Φ˜ ( jω0 ) x 2 1 ∗ Φ˜ ( jω0 )x − (Φ˜ ( jω0 ) + Φ˜ ( jω0 ))x 2 1 ˜ ∗ Φ˜ ( jω0 )x + by (3.60) B( jω0 )B˜ ( jω0 )x 2σ ˜ jω0 )u, Φ˜ ( jω0 )x + B(

(3.73) (3.74) (3.75) (3.76) (3.77)

1 ˜∗ 2σ B ( j ω0 )x.

˜ ( jω0 )x = 0, in which u = Note that u 6= 0, since otherwise we would have Φ which is banned by assumption. Now using (3.62) we have ˜ jω0 )x, ˜ jω0 )u = C( −2U( giving



Hence ˜∗

˜ jω0 Since U ( jω0 )U(

˜ jω0 ) Φ˜ ( jω0 ) B( ˜ 0 C( jω0 )

    x 0 = ˜ jω0 )u . u −2U(

˜ jω0 )u, E( jω0 )u = −2U( u 6= 0.
we see that σ E( jω0 ) = 2σ .

) = σ 2 I,

Consider the situation in which the state(s) associated with one σi are deleted by balanced ˜ ˜ ˜ truncation and define A(s), B(s) and C(s) as in Lemma 3.1. Then by Lemma 3.4 the assumptions of Lemma 3.5 are satisfied and we conclude that the infinity norm of the error associated with this special “one-step” balanced truncation is exactly 2σ . Moreover, if the associated multiplicity r of σ is odd, then the maximum error (as measured by the maximum singular value) occurs at ω = 0.

3.4.3 The Error Bound for Balanced Truncation Lemma 3.5 provides an infinity norm bound on the absolute error associated with the truncation of the state(s) associated with a single σi in a balanced realization. To determine a bound that is applicable when the states associated with several σi ’s are deleted, we simply remove the σi ’s one at a time and allow the error to accumulate. This procedure yields the twice-the-sum-of-the-tail error bound.

114

D.J.N. Limebeer

Theorem 3.1. Let G = D + C(sI − A)−1 B, in which (A, B,C, D) is a balanced realization b = partitioned as in (3.51). Let r = r1 + · · · + rl , (A11 , B1 ,C1 , D) = Tr (A, B,C, D) and G D +C1 (sI − A11 )−1 B1 . Then b ∞ ≤ 2(σl+1 + · · · + σm ). kG − Gk
b In the case that l = m − 1 equality holds, and σ G(0) − G(0) = 2σm if rm is odd.

(3.78)

Proof. Truncating the states associated with σl+1 , . . . , σm may be achieved by a succession of one-step truncation operations. At the kth step we delete the states associated with σm−k+1 , k = 1 . . . m − l, to obtain Gk from Gk−1 where G0 = G. Each truncation step preserves the balanced realization and each step incurs an error of E k = Gk−1 − Gk with kE k k∞ = 2σm−k+1 . b Now write The last of these steps gives Gm−l = G. b = (G0 − G1 ) + · · · + (Gm−l−1 − Gm−l ) G−G = E 1 + · · · + E m−l

(3.79)

and observe that the triangle inequality gives b ∞ ≤ kE 1 ||∞ + · · · + kE m−l k∞ . kG − Gk

Since kE k k∞ = 2σm−k+1 , we obtain the error bound (3.78).

By considering the case when all the states are deleted, the bound (3.78) yields kG − G(∞)k∞ ≤ 2(σ1 + · · · + σm ). Hence, kGk∞ = kG − G(∞) + G(∞)k∞

≤ kG(∞)k + 2(σ1 + · · · + σm ).

(3.80) (3.81)

Tightness of the Bound We have already established that the infinity norm error bound for one-step truncation is tight. In the case of multi-step truncation the situation is less clear cut—one can find examples for which the error bound is close to the true error and there are also cases for which it is conservative. It is natural to suspect that because the twice-the-sum-of-the-tail bound arises by repeated application of the triangle inequality, it gets weaker and weaker as more and more states are deleted. Although this is usually the case, the bound may remain tight however many states are deleted. The next example illustrates this point. Example 3.3. Consider the transfer function n

gn =

i=1

which may be realized as

αi

∑ s + αi

α > 0,

α 6= 1

115

Modelling and Model Reduction State-Space Truncation



 −α 0 0   A =  0 ... 0  0 0 −α n

√  α  .  B =  ..  √ αn

C = B′ .

It is easily verified that the controllability and observability gramians are equal and the (i, j) entry of the controllability gramian is √ α i− j Pi j = i− j i, j = 1, . . . , n. α +1 Since P = Q, we have

σi = giving

p λi (PQ) = λi (P),

i = 1, . . . , n,

2(σ1 + · · · + σn ) = 2 × trace(P) n = 2× 2 = n.

(3.82) (3.83) (3.84)

Therefore (3.79) implies kgn k∞ ≤ n. Since g(0) = n, equality must hold.

Systems of the form given in the above example have interlaced poles and zeros on the negative real axis (Problem 3.4). The bound is tight for such systems. At the other extreme, for systems that have interlaced poles and zeros on the imaginary axis, the bound exceeds the true error by a factor that is approximately twice the number of states that are deleted—see Enns [2].

Frequency Dependence of the Error The error bounds given in Theorem 3.1 says nothing about the way the true error varies as a b match function of frequency. Since the full-order system G and the reduced-order system G each other exactly at infinite frequency, one would expect good high-frequency fidelity from the reduced order model. Apart from the guaranteed satisfaction of the error bound, little can be said about the variation of the error at low and intermediate frequencies. In most cases, the σi ’s will have unit multiplicity and the truncation of n − r states will be achieved via n − r truncations of one state. Since one is odd, each of these truncations incurs a maximum error at zero frequency. One would therefore expect the largest truncation error to occur at low frequency.

Main Points of the Section 1. Any truncation of a balanced realization that does not “split” the states associated with a single singular value is called balanced truncation. 2. Balanced truncation preserves stability and minimality, and the approximation error satisfies the twice-the-sum-of-the-tail infinity norm error bound. 3. The infinity norm of a strictly proper transfer function matrix is bounded above by twice the sum of the σi ’s in its balanced realization.

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3.5 Balanced Singular Perturbation Approximation The fact that balanced truncation generally incurs the greatest approximation error in the lowfrequency region is undesirable in many applications. An algorithm which produces zero error at zero frequency may be obtained via an easy modification to the basic balanced truncation algorithm. The idea is to simply replace s with 1/s as follows: 1. Set H(w) = G(1/w). 2. Let H r (w) be an rth -order balanced truncation of H(w). 3. Set Gr (s) = H r (1/s). This algorithm will have exact matching at zero frequency, thereby leading to prefect steadystate performance; it is a singular perturbation version of balanced truncation, which we call balanced singular perturbation approximation (BSPA). Since singular perturbation and truncation are connected by the simple fre quency inversion s → 1/s, the infinity norm error bounds will still hold. Secondly, since frequency inversion preserves stability, both algorithms will produce reduced-order models with the same stability properties. Finally, since frequency inversion preserves the controllability and observability gramians and hence a balanced realization, we obtain the following result: b =D b11 )−1 Bb1 , b + Cb1 (sI − A Theorem 3.2. Assume the hypotheses of Theorem 3.1 and define G b11 , Bb1 , Cb1 , D) b = Sr (A, B,C, D). Then: in which (A b11 , Bb1 , Cb1 , D) b is balanced with its controllability and observability 1. The realization (A b11 is stable, (A b11 , Bb1 ) is gramians given by Σl = diag(σ1 Ir1 , . . . , σl Irl ). In particular, A b b controllable and (C1 , A11 ) is observable. b ∞ ≤ 2(σl+1 + · · · + σm ), and equality holds if l = m − 1. 2. kG − Gk b 3. G(0) = G(0). 4. kG − G(0)k∞ ≤ 2(σ1 + · · · + σm ).

Proof. Since (A, B,C) is balanced, A is stable and therefore nonsingular. By Lemma 3.4, A22 b11 , Bb1 , Cb1 , D) b given by (3.33) are well defined is also stable and hence is nonsingular. Thus (A and the assumptions of Lemma 3.2 hold. All the items now follow directly from their balanced truncation counterparts by using Lemma 3.2.

Main Point of the Section The stability and infinity norm properties of any model reduction procedure will be preserved by a change of variables that maps the left-half plane into itself. The balanced singular perturbation approximation, which is equivalent to balanced truncation of the system obtained by setting w = 1/s, has the same infinity norm error as balanced truncation, but has zero steady-state error. In general, its performance at low frequencies is superior to that of balanced truncation.

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3.6 Example We illustrate the model reduction methods described above with an eighth-order model of a flexible structure [3]. The model is 4

g=

ω2

∑ ki s2 + 2ζi ωi i s + ω 2 ,

i=1

i

in which i 1 2 3 4

ωi 0.56806689746895 3.94093897440699 10.58229653714164 16.19234386986640

ζi 0.00096819582773 0.00100229920475 0.00100167293203 0.01000472824082

ki 0.01651378989774 0.00257034576009 0.00002188016252 0.00027927762861

The controllability and observability gramians of a balanced realization for this system are   4.26819 4.25994 0.641754 0.640469 Σ = diag . 0.069856 0.0697163 0.00546623 0.00545529 By making use of the error bound given in Theorem 3.1, we see that if two states are eliminated, the infinity norm of the error is less than 0.0218431. If four states are eliminated, the error bound increases to 0.300988 and so on. Eliminating four states by balanced truncation gives   −0.000550222 −0.568066 0.000993291 0.00134457 0.068534  0.568066 −0.000549776 0.00134607 0.000994008 −0.0684399   s  0.000993291 −0.00134607 −0.00394544 −3.94093 −0.0711619  b g=  .  −0.00134457 0.000994008 3.94093 −0.00395455 0.0711725  0.068534 0.0684399 −0.0711619 −0.0711725 0 Figure 3.1 shows the gain of the full-order model g, the fourth-order balanced truncation model b g and the fourth-order balanced singular perturbation model. The gains of the error incurred by each method, along with the error bound, are shown in Figure 3.2. The solid line is the error bound, the dashed line is the balanced truncation error and the dash-dot line is the balanced singular perturbation error. Notice that the actual error is an order of magnitude less than the error bound. This is not unexpected for this type of system because its poles and zeros are interlaced and close to the imaginary axis. It is also worth noting that the balanced truncation reduced-order model is virtually identical to the fourth-order model obtained by deleting the two high-frequency modes. The main difference is that the balanced truncation reduced-order model has an additional zero at −3980.19.

3.7 Notes and References Balanced realizations first appeared with the work of Mullis and Roberts [4], who were interested in realizations of digital filters that are optimal with respect to round-off errors in the state update calculations. These issues are developed extensively in the book by Williamson [5].

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10 1 10 0

10 -1

gain

10 -2

10 -3

10 -4

10 -5

10 -6

10 -7

10 -8 10 -1

10 0

10 1

10 2

10 3

frequency (rad/s)

Fig. 3.1. Flexible structure models: full-order g (solid), balanced truncation b g (dashed) and balanced singular perturbation (dash-dot) 10 0 10 -1

10 -2

gain

10 -3

10 -4

10 -5

10 -6

10 -7

10 -8 10 -1

10 0

10 1

10 2

10 3

frequency (rad/s)

Fig. 3.2. Model reduction errors: error bound (solid), balanced truncation (dashed) and balanced singular perturbation (dash-dot)

The balanced truncation method of model reduction is due to Moore, who argued that the method is sensible on system theoretic grounds. He also proved a weak version of the stability result. The asymptotic stability result, Lemma 3.4, is due to Pernebo and Silverman [6]. The twice-the-sum-of-the-tail infinity norm error bound for balanced truncation is due to Enns [2],

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119

who also proposed a frequency weighted generalization He introduced formula (3.14) for the truncation error system and proved that the error for “one-step” truncation is 2σ when the multiplicity of σ is one. Our new proof shows that no matter what the multiplicity of σ is, the infinity norm of the error is exactly 2σ . The twice-the-sum-of-the-tail error bound was also proved by Glover [7] using a different approach. The balanced singular perturbation approximation method was introduced by Fernando and Nicholson [8]. Liu and Anderson [9] showed that it was related to balanced truncation by the transformation s → 1/s. Numerical algorithms for computing balanced realizations have been considered by Laub, Heath, Page and Ward [10]. Tombs and Postlethwaite [11], and Safonov and Chiang [12] have developed various algorithms that compute balanced truncation approximations without computing balanced realizations. None of these procedures eliminate the need to compute the controllability and observability gramians, which is a serious problem when large-scale models need to be reduced. Jaimoukha, Kasenally and Limebeer [13] have developed algorithms for computing approximate solutions to Lyapunov equations using Krylov subpace methods that can be effective even for models containing many hundreds of states. A relative error based truncation method of model reduction is the balanc ed stochastic truncation method of Desai and Pal [14]. Green derived an infinity norm bound for the relative error associated with this method [15]; the bound was subsequently improved by Wang and Safonov [16]. Another feature of the algorithm is that it preserves the closed-right-half-plane zeros of the model and will therefore produce a minimum phase approximation of a minimum phase system [17]. Discrete-time versions of the results in these notes (see Problem 3.6) hav e been proved by Al-Saggaf and Franklin [18].

3.8 Problems Problem 3.1. Let B and C be complex matrices such that BB∗ = C∗C. 1. If the number of rows of C is greater than or equal to the number of columns of B, show that for any nonzero real number σ , there exists a matrix U such that σ B + C∗U = 0 and U ∗U = σ 2 I. (Hint: Write a singular value decomposition of C and infer the form of the singular value decomposition of B; see Lemma 3.5 of [7].) 2. Show that for any nonzero real number σ , there exists a matrix U such that σ B +C∗U = 0 and U ∗U ≤ σ 2 I. Show further that the nonzero singular values of U are all equal to σ . Problem 3.2. Suppose (Aii , Bi ,Ci ), i = 1, 2 are two balanced realizations with controllability/observability gramians Σ1 and Σ2 respectively. Suppose also that Σ1 and Σ2 have no eigenvalues in common. Construct the unique matrices A21 and A12 such that     A11 A12 B1 A= , B= , (3.85) A21 A22 B2   C = C1 C2 (3.86)   Σ1 0 is a balanced realization with controllability/observability gramian . 0 Σ2

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Problem 3.3. Assume the hypotheses of Lemma 3.5. Using the bounded real lemma [1], prove that
˜ jω ))−1 B( ˜ jω )( jω I − A( ˜ jω ) ≤ 2σ . sup σ C( ω

Problem 3.4. Show that the poles and zeros of the system gn defined in Example 3.3 have an interlacing property—between any two poles there is exactly one zero. If gn is the impedance of a linear circuit, explain why this interlacing property implies that the circuit is passive. Problem 3.5. Let (A, B,C, D) be any realization such that A is asymptotically stable and let P and Q satisfy AP + PA′ + BB′ = 0

(3.87)

A′ Q + QA +C′C = 0.

(3.88)

Show that the following procedure computes the rth -order balanced truncation approximation to G = D +C(sI − A)−1 B: a. Let P = UP SPUP′ and Q = UQ SQUQ′ be ordered singular value decompositions of P and Q. 1

1

b. Let SQ2 UQ′ UP SP2 have (ordered) singular value decomposition V Σ U ′ . c. Partition the matrices V , Σ and U as       Σ1 0 , U = U1 U2 , V = V1 V2 , Σ = 0 Σ2

in which Σ1 is r × r. d. Define

1

−1

L = UQ SQ2 V1 Σ1 2 ,

1

−1

M = UP SP2 U1 Σ1 2 .

e. Define Ar = L′ AM, Br = L′ B and Cr = CM. b = D +Cr (sI − Ar )−1 Br . f. Define G

Problem 3.6. [18] A realization (A, B,C, D) of a discrete-time system G(z) = D + C(zI − A)−1 B is balanced if |λi (A)| < 1 for all i and there exists a positive definite diagonal matrix of the form given in (3.48) such that AΣ A′ − Σ + BB′ = 0 ′

′

A Σ A − Σ +C C = 0.

(3.89) (3.90)

1. Show that a given realization of a discrete-time system can be transformed into a discretetime balanced realization if and only if it is stable and minimal. 2. Suppose that (A, B,C, D) is a discrete-time balanced realization which is partitioned as in (3.9) and (3.48). Show that A11 is asymptotically stable (|λi (A11 )| < 1 for all i). 3. Define ˜ θ ) = A22 + A21 (e jθ I − A11 )−1 A12 A( ˜ θ ) = B2 + A21 (e jθ I − A11 )−1 B1 B( ˜ C(θ ) = C2 +C1 (e jθ I − A11 )−1 A12 .

(3.91) (3.92) (3.93)

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121

Show that ˜ θ )Σ2 A˜∗ (θ ) − Σ2 + B( ˜ θ )B˜ ∗ (θ ) = 0 A( ∗ ˜ θ ) = 0. ˜ θ ) − Σ2 + C˜ ∗ (θ )C( A˜ (θ )Σ2 A(

(3.94) (3.95)

4. Suppose that A is asymptotically stable |λi (A)| < 1 for all i and that AA∗ − I + BB∗ = 0

A∗ A − I +C∗C = 0.

(3.96) (3.97)

Show that σ [C(e jθ I − A)−1 B] < " 2. 1 # h i 2 1 5. Show that (A11 , A12 Σ 2 B1 , A21 Σ2 ) is a balanced realization. 2 C1

6. Suppose that G(z) = D + C(zI − A)−1 B, in which (A, B,C, D) is a discrete-time balanced realization. Let (A11 , B1 ,C1 , D) be an rth -order balanced truncation of (A, B,C, D) and let b = D +C1 (zI − A11 )−1 B1 . Show that G(z) b ∞ < 2(σr+1 + . . . + σn ). kG − Gk

Problem 3.7. [19, 20] Let (A, B,C, D) be a balanced realization of a continuous-time system and let α be a nonnegative number. Define b = A11 + A12 (α I − A22 )−1 A21 A Bb = B1 + A12 (α I − A22 )−1 B2 Cb = C1 +C2 (α I − A22 )−1 A21

b = D +C2 (α I − A22 )−1 B2 , D

(3.98) (3.99) (3.100) (3.101)

in which (A, B,C, D) are partioned as in (3.9) conformably with a partioning of Σ in (3.52). The generalized singular perturbation approximation (GSPA) is defined by G S r (A, B,C, D) = b B, b D). b C, b (A,

1. Show that the GSPA reduced-order model results from replacing the dynamics associated with x2 by the exponential system x˙2 = α x2 . 2. Show that the GSPA approximant has zero error at s = α . +s , which 3. Suppose that 0 < α < ∞ and define the linear fractional transformation z = αα −s maps the left-half plane to the unit circle. Suppose also that we consider the equivalent α (z−1) ˜ − A) ˜ −1 B, ˜ in discrete-time system defined by F(z) = G( z+1 ). Show that F = D˜ + C(zI which A˜ = (α I + A)(α I − A)−1 = (α I − A)−1 (α I + A) √ B˜ = 2α (α I − A)−1 B √ C˜ = 2α C(α I − A)−1

D˜ = D +C(α I − A)−1 B.

(3.102) (3.103) (3.104) (3.105)

Show that this realization of F is a discrete-time balanced realization. 4. Show that for 0 < α < ∞, GSPA is equivalent to: (a) mapping to discrete-time via +s ; (b) discrete-time balanced truncation (see Problem 3.6) and (c) mapping back z = αα −s α (z−1)

to continuous-time via s = z+1 . 5. For 0 ≤ α ≤ ∞, show that reduced-order models obtained by GSPA enjoy the same stability and infinity norm error bound properties as balanced truncation and SPA.

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References 1. M. Green and D. Limebeer, Linear Robust Control. Englewood Cliffs, N.J.: PrenticeHall, 1995. 2. D. Enns, “Model reduction with balanced realizations: An error bound and frequency weighted generalization,” in Proceedings of the IEEE Conference on Decision and Control, Las Vegas, 1984, pp. 127–132. 3. H. V. et al., “Flexible structure control laboratory development and technology demonstration,” California Institute of Technology, JPL Publication 88-29, 1987. 4. C. Mullis and R. Roberts, “Synthesis of minimum roundoff noise fixed point digital filters,” IEEE Transactions of Circuits and Systems, vol. 23, pp. 551–562, 1976. 5. D. Williamson, Digital Control and Implementation : Finite Wordlength Considerations. Englewood Cliffs, N.J.: Prentice-Hall, 1990. 6. L. Pernebo and L. Silverman, “Model reduction by balanced state space representations,” IEEE Transactions on Automatic Control, vol. 27, pp. 382–387, 1982. 7. K. Glover, “All optimal hankel-norm approximations of linear multivariable systems and their L∞ -error bounds,” International Journal of Control, vol. 39, no. 6, pp. 1115–1193, 1984. 8. K. Fernando and H. Nicholson, “Singular perturbational model reduction of balanced systems,” IEEE Transactions on Automatic Control, vol. 27, pp. 466–468, 1982. 9. Y. Liu and B. Anderson, “Singular perturbation approximation of balanced systems,” International Journal of Control, vol. 50, pp. 1379–1405, 1989. 10. A. Laub, M. Heath, C. Page, and R. Ward, “Computation of balacing transformations and other applications of simultaneous diagonalization algorithms,” IEEE Transactions on Automatic Control, vol. 32, pp. 115–122, 1987. 11. M.S.Tombs and I. Postlethwaite, “Truncated balanced realizations of a stable nonminimal state-space system,” Rpt. No. OUEL 1647/86, Engineering Dept., Oxford., 1986. 12. M. Safonov and R. Chiang, “A Schur method for balanced-truncation model reduction,” IEEE Transactions on Automatic Control, vol. 43, no. 7, pp. 729–733, 1989. 13. I. Jaimoukha, E. Kasenally, and D. Limebeer, “Numerical solution of large scale lyapunov equations using krylov subspace methods,” in Proceedings of the IEEE Conference on Decision and Control, Tuscon, 1992, pp. 1927–1932. 14. U. Desai and D. Pal, “A transformation approach to stochastic model reduction,” IEEE Transactions on Automatic Control, vol. 29, no. 12, pp. 1097–1100, 1984. 15. M. Green, “A relative error bound for balanced stochastic truncation,” IEEE Transactions on Automatic Control, vol. 33, no. 10, pp. 961–965, 1988. 16. W. Wang and M. Safonov, “A tighter relative-error bound for balanced stochastic truncation,” Systems and Control Letters, vol. 14, no. 4, pp. 307–317, 1990. 17. M. Green, “Balanced stochastic realizations,” Linear Algebra and its Applications, vol. 98, pp. 211–247, 1988. 18. U. Al-Saggaf and G. Franklin, “An error bound for a discrete reduced order model of a linear multivariable system,” IEEE Transactions on Automatic Control, vol. 32, pp. 815– 819, 1987. 19. P. Heuberger, “A familily of reduced order models based on open-loop balancing,” in Selected Topics in Identification, Modelling and Control, 1990, vol. 1, pp. 1–10. 20. U. Al-Saggaf and G. Franklin, “Model reduction via balanced realization: An extension and frequency weighting techniques,” IEEE Transactions on Automatic Control, vol. 33, pp. 681–692, 1988.

4 Linear Matrix Inequalities in Control Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite

Summary. This chapter gives an introduction to the use of linear matrix inequalities (LMIs) in control. LMI problems are defined and tools described for transforming matrix inequality problems into a suitable LMI-format for solution. Several examples explain the use of these fundamental tools.

4.1 Introduction to LMI Problems The design of a controller should in general satisfy four basic requirements: 1. Closed-Loop Stability 2. Robustness: The closed-loop controller has to remain stable despite uncertainty in the mathematical plant description or disturbances. 3. Performance: The controller has to have certain dynamical or steady state characteristics such as rise time, overshoot, controller bandwidth or steady state error etc. 4. Robust Performance: The controller has to remain well performing and certainly stable, although disturbances and uncertainties affect the plant. 1 These control design requirements are usually best encoded in the form of an optimization criterion subject to some constraints e.g., terminal constraints or practical constraints on plant variables. Control design optimization criteria were initially based on the idea of Linear Quadratic Gaussian Control [1, 1, 8] which was later generalized to the idea of H2 -controller design [20]. The development of small gain theory [10, 17] layed the foundations of robust H∞ -control [3, 20]. For linear systems and suitable optimization criteria, such as H2 - and H∞ , the solution to the optimization problem is readily found solving Riccati-equations [3, 19, 20]. Many of these optimal control problems can be stated in terms of linear matrix inequalities and their existence can be traced back over 100 years to the work of Lyapunov. Linear matrix inequalities are matrix inequalities which are linear (or affine) in a set of matrix variables. 1

It is obvious that robust performance implies robustness and performance but it is a stronger requirement than considering robustness and performance each as a single entity.

M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 123-142, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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ItisinterestingtonotethattheequivalencebetweentheRiccati-equationandLMIformulations of the control problem was found at an early point in the seventies [19]. This created the possibility of solving optimal control problems using LMI-methods, as the numerical LMIoptimization problem was found to be convex. However, only during the past 10-15 years has the development of sophisticated numerical routines, i.e., semi-definite programming [11,18], made it possible to solve LMIs in a reasonably efficient manner. From a control engineering perspective, one of the main attractions of LMIs is that they can be used to solve problems which involve several matrix variables, and, moreover, different structures can be imposed on these matrix variables. Another attractive feature of LMI methods is that they are flexible, so it is often relatively straightforward to pose a variety of problems as LMI problems amenable to LMI methods. Furthermore, in many cases the use of LMIs can remove restrictions associated with conventional methods and aid their extension to more general scenarios. Often LMI methods can be applied in instances where conventional methods either fail, or struggle to find a solution [15]. In actual fact, the flexibility of LMIs has created a much wider scope for controller design. They allow the efficient consideration of H2 and H∞ -constraints for performance, robustness and robust performance in the design of one single controller [13, 14]. Thus an advantage of LMIs, also in a pedagogical sense, is that they are able to unite many previous results in a common framework. This can also enable one to obtain additional insight into established areas. Some of the contents of this chapter are available as a University of Leicester Technical report [16] which draws heavily on the material contained within the book in [2] and the R MATLAB LMI control toolbox in [6]. A similar treatment can be found in [12]. In order to convey the main points, the presentation is somewhat condensed and the interested reader should consult the work in [2] for a more complete exposure to LMIs.

4.1.1

Fundamental LMI Properties

A notion central to the understanding of matrix inequalities is definiteness. In particular, a matrix Q is defined to be positive definite if xT Qx > 0 ∀x 6= 0

(4.1)

Likewise, Q is said to be positive semi-definite if xT Qx ≥ 0

∀x

(4.2)

It is common practise to write Q > 0 (Q ≥ 0) to indicate that it is positive (semi-) definite. In particular, we are interested in positive definite matrices which are also symmetric, i.e., Q = QT . A symmetric, positive definite matrix has two key features: it is square and all of its eigenvalues are positive real. A symmetric, positive semi-definite matrix shares the first attribute, but the last is relaxed to the requirement that all of its eigenvalues are positive real or zero. A matrix P = −Q is said to be negative (semi) definite if Q is positive (semi) definite. To indicate negative (semi) definiteness we write P < 0 (P ≤ 0). In fact, once the notation Q > 0 (Q ≥ 0) or P > 0 (P ≥ 0) is used, we usually also require Q and P to be symmetric and we will do so in the rest of this chapter, although from a mathematical point of view this is not necessary. Nevertheless, numerical solution routines for LMIs have enforced this fact as this simplifies the computational process and it is therefore a common assumption for LMIs [2].

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125

The basic structure of an LMI is m

F(x) = F0 + ∑ xi Fi > 0

(4.3)

i=1

where x ∈ Rm is a variable and F0 , Fi are given constant symmetric matrices. The basic LMI problem - the feasibility problem - is to find x such that inequality (4.3) holds. Note that F(x) > 0 describes an affine relationship in terms of the matrix x. Normally the variable x, which we are interested in, is composed of one or many matrices whose columns have been ‘stacked’ as a vector. That is, F(x) = F(X1 , X2 , . . . , Xn )

(4.4)

where Xi ∈ Rqi ×pi is a matrix, ∑ni=1 qi × pi = m, and the columns of all the matrix variables are stacked up to form a single vector variable. Hence, from now on, we will consider functions of the form

F(X1 , X2 , . . . , Xn ) = F0 + G1 X1 H1 + G2 X2 H2 + . . .

(4.5)

n

= F0 + ∑ Gi Xi Hi > 0

(4.6)

i=1

where F0 , Gi , Hi are given matrices and the Xi are the matrix variables which we seek.

4.1.2

Systems of LMIs

In general, we are frequently faced with LMI constraints of the form

F1 (X1 , . . . , Xn ) > 0 .. .>0

(4.7)

Fp (X1 , . . . , Xn ) > 0

(4.9)

(4.8)

where n

Fj (X1 , . . . , Xn ) = F0 j + ∑ Gi j Xi Hi j

(4.10)

i=1

However, it is easily seen that, by defining F˜0 , G˜ i , H˜ i , X˜i as F˜0 = diag(F01 , . . . , F0p ) G˜ i = diag(Gi1 , . . . , Gip ) H˜ i = diag(Hi1 , . . . , Hip )

(4.12)

X˜i = diag(Xi , . . . Xi )

(4.14)

(4.11) (4.13)

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we actually have the inequality n

Fbig (X1 , . . . , Xn ) := F˜0 + ∑ G˜ i X˜i H˜ i > 0

(4.15)

i=1

That is, we can represent a (big) system of LMIs as a single LMI. Therefore, we do not distinguish a single LMI from a system of LMIs; they are the same mathematical entity. We may also encounter systems of LMIs of the form:

F1 (X1 , . . . , Xn ) > 0

(4.16)

F2 (X1 , . . . , Xn ) > F3 (X1 , . . . , Xn ).

(4.17)

Again, it is easy to see that this can be written in the same form as inequality (4.15) above. For the remainder of the chapter, we do not distinguish between LMIs which can be written as above, or those which are in the more generic form of inequality (4.15).

4.1.3

Types of LMI Problems

The term ‘LMI problem’ is rather vague and in fact there are several sub-groups of LMI problems. These will be described below in the same way that they are separated in the R MATLAB LMI toolbox. Note that by ‘LMI problem’ we normally mean solving an optimization problem or an eigenvalue problem with LMI constraints.

LMI Feasibility Problems These are simply problems for which we seek a feasible solution {X1 , . . . , Xn } such that F(X1 , . . . , Xn ) > 0

(4.18)

We are not interested in the optimality of the solution, only in finding a solution, which may not be unique. Example 4.1. (Determining stability of a linear system). Consider an autonomous linear system x˙ = Ax

(4.19)

then the Lyapunov LMI problem for proving asymptotic stability of this system is to find a P > 0 such that AT P + PA < 0 (4.20) This is obviously an LMI feasibility problem in P > 0. However, given any P > 0 which satisfies this, it is obvious that any matrix from the set P = {β P : scalar β > 0}

(4.21)

also solves the problem. In fact, as will be seen later, the matrix P forms part of a Lyapunov function for the linear system.

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Linear Objective Minimization Problems These problems are also called eigenvalue problems. They involve the minimization (or maximization) of some linear scalar function, α (.), of the matrix variables, subject to LMI constraints:

min α (X1 , . . . , Xn ) s.t.

(4.22)

F(X1 , . . . , Xn ) > 0

(4.23)

where we have used the abbreviation ‘s.t.’ to mean ‘such that’. In this case, we are therefore trying to optimize some quantity whilst ensuring some LMI constraints are satisfied. Example 4.2. (Calculating the H∞ norm of a linear system). Consider a linear system

x˙ = Ax + Bw

(4.24)

z = Cx + Dw

(4.25)

then the problem of finding the H∞ norm of the transfer function matrix Tzw from w to z is equivalent to the following optimization procedure (see, for example the work in [5]):

 

min γ AT P + PA BT P C

s.t. 

(4.26)

CT

PB −γ I DT  < 0 D −γ I P>0

(4.27)

Note that although γ > 0 is unique, the uniqueness of P > 0 is, in general, not guaranteed.

Generalized Eigenvalue Problems The generalized eigenvalue problem, or GEVP, is slightly different to the preceding problem in the sense that the objective of the optimization problem is not actually convex, but quasiconvex. However, the methods used to solve such problems are similar. Specifically a GEVP is formulated as

min λ

s.t.

(4.28)

F1 (X1 , . . . , Xn ) + λ F2 (X1 , . . . , Xn ) < 0

(4.29)

F2 (X1 , . . . , Xn ) < 0

(4.30)

F3 (X1 , . . . , Xn ) < 0

(4.31)

The first two lines are equivalent to minimizing the largest ‘generalized’ eigenvalue of the matrix pencil F1 (X1 , . . . , Xn ) + λ F2 (X1 , . . . , Xn ). In some cases, a GEVP problem can be reduced to a linear objective minimization problem, through an appropriate change of variables.

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Example 4.3. (Bounding the decay rate of a linear system). A good example of a GEVP is given in [2]. Given a stable linear system x˙ = Ax, the decay rate is the largest α such that kx(t)k ≤ exp(−α t)β kx(0)k ∀x(t)

(4.32)

where β is a constant. If we choose V (x) = xT Px > 0 as a Lyapunov function for the system and ensure that V˙ (x) ≤ −2α V (x) it is easily shown that the system will have a decay rate of at least α . Hence the problem of finding the decay rate could be posed as the optimization problem

T

min −α

s.t.

(4.33)

A P + PA + 2α P < 0

(4.34)

−P < 0

(4.35)

This problem is a GEVP with the functions F1 (P) := AT P + PA

(4.36)

F2 (P) := −2P

(4.37)

F3 (P) := −I

4.2

(4.38)

Tricks in LMI Problems

Although many control problems can be cast as LMI problems, a substantial number of these need to be manipulated before they are in a suitable LMI problem format. Fortunately, there are a number of common tools or ‘tricks’ which can be used to transform problems into suitable LMI forms. Some of the more useful ones are described below.

4.2.1

Change of Variables

Many control problems can be posed in the form of a set of nonlinear matrix inequalities; that is, the inequalities are nonlinear in the matrix variables we seek. However by defining new variables it is sometimes possible to ‘linearise’ the nonlinear inequalities, hence making them solvable by LMI methods. Example 4.4. (State feedback control synthesis problem). Consider the problem of finding a matrix F ∈ Rm×n such that the matrix A + BF ∈ Rn×n has all of its eigenvalues in the open left-half complex plane. By the theory of Lyapunov equations (see [20]), this is equivalent to finding a matrix F and a positive definite matrix P ∈ Rn×n such that the following inequality holds (A + BF)T P + P(A + BF) < 0

(4.39)

AT P + PA + F T BT P + PBF < 0

(4.40)

or

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This problem is not in LMI form due to the terms which contain products of F and P these terms are ‘nonlinear’ and as there are products of two variables they are said to be ‘bilinear’. If we multiply on either side of equation (4.40) by Q := P−1 (which does not change the definiteness of the expression since rank(P) = rank(Q) = n) we obtain QAT + AQ + QF T BT + BFQ < 0

(4.41)

This is a new matrix inequality in the variables Q > 0 and F, but it is still nonlinear. To rectify this, we simply define a second new variable L = FQ giving QAT + AQ + LT BT + BL < 0

(4.42)

We now have an LMI feasibility problem in the new variables Q > 0 and L ∈ Rm×n . Once this LMI has been solved we can recover a suitable state-feedback matrix as F = LQ−1 and our Lyapunov function as P = Q−1 . Hence, by making a change of variables we have obtained an LMI from a nonlinear matrix inequality. The key facts to consider when making a change of variables is the assurance that the original variables can be recovered and that they are not over-determined.

4.2.2

Congruence Transformation

For a given positive definite matrix Q ∈ Rn×n , we know that, for another real matrix W ∈ Rn×n such that rank(W ) = n, the following inequality holds W QW T > 0

(4.43)

In other words, definiteness of a matrix is invariant under pre and post-multiplication by a full rank real matrix, and its transpose, respectively. The process of transforming Q > 0 into equation (4.43) using a real full rank matrix is called a ‘congruence transformation’. It is very useful for ‘removing’ bilinear terms in matrix inequalities and is often used, in conjunction with a change of variables, to make a bilinear matrix inequality linear. Often W is chosen to have a diagonal structure. Example 4.5. (Making a bilinear matrix inequality linear). Consider  T  A P + PA PBF +CT V Q= 0,V > 0 and F (definiteness not specified) are the matrix variables and the remaining matrices are constant. The ⋆ in the bottom left entry of the matrix denotes the term required to make the expression symmetric and will be used frequently hereafter. Notice that this inequality is bilinear in the variables P and F which occur in the (1,2) and (2,1) elements of the matrix Q ∈ R(n+p)×(n+p) . However, if we choose the matrix  −1  P 0 W= ∈ R(n+m)×(n+m) (4.45) 0 V −1 which is full rank (rank(W ) = n + m) by virtue of the inverses of P and V (which exist as the matrices are positive definite), then calculating W QW T gives

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W QW T =



P−1 AT + AP−1 BFV −1 + P−1CT ⋆ −2V −1



0. This inequality can be used to minimize the cost function J=

Z ∞ 0

(xT Qx + uT Ru)dt

(4.49)

for the computation of the state feedback controller u = Kx controlling x˙ = Ax + Bu with an initial condition of x(0) = x0 . We know that the solution to this problem is ˜ K = −R−1 BT P, and J = min u

˜ − PBR ˜ −1 BT P˜ + Q = 0 AT P˜ + PA

Z ∞ 0

˜ 0 (xT Qx + uT Ru)dt = x0T Px

However, the solution can be also obtained from a linear matrix inequality problem. Using a congruence transformation for S = P−1 , it follows from (4.48)

Linear Matrix Inequalities in Control SAT + AS − BR−1 BT + SQS < 0

131 (4.50)

We can now define

Φ11 := SAT + AS − BR−1 BT

(4.51)

Φ12 := S

(4.52)

Φ22 := −Q−1

(4.53)

and use the Schur complement identities. Thus, we can transform our Riccati inequality into  T  SA + AS − BR−1 BT S x0T Px0 and (4.54). Using the Schur complement again for x0T Px0 − σ < 0, it follows for Φ11 := −σ , Φ12 := x0T and Φ22 := −S:   −σ x0T 0 WIT (AQ + QAT )WI < 0, Q > 0

(4.70)

(4.71)

However, as WI is a matrix whose columns span the null space of the identity matrix which is N (I) = {0} the above equation simply reduces to WBT (AQ + QAT )WB < 0,

Q>0

which is an LMI problem. Alternatively, using Finsler’s Lemma we get  AQ + QAT − σ BBT < 0, Q > 0 AQ + QAT − σ I < 0, Q > 0

(4.72)

(4.73)

However, we can neglect the second inequality because if we can find a σ satisfying the first inequality, we can always find one which satisfies the second. Notice that the use of both the projection lemma and Finsler’s Lemma effectively reduces our original LMI problem into two separate ones: the first LMI problem involves the calculation of Q > 0; the second ivolves the back-substitution of Q into the original problem in order for us to find L (and then F). The reader is, however, cautioned against the possibility of illconditioning in this two-step approach. For some problems, normally those with large numbers of variables, X can be poorly conditioned, which can hinder the numerical determination of Λ from equation (4.65).

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4.3

Examples

4.3.1

Lyapunov Stability for Continuous-Time Systems

The stability of a nonlinear systems is generally more difficult to ascertain than that of a linear systems. A sufficient (but not necessary) condition was given by Lyapunov; see, for example, [7]. Theorem 4.1 (Lyapunov’s Theorem for continuous-time systems). Given an unbounded positive definite function V (x) > 0 ∀x 6= 0 and an autonomous system x˙ = f (x), then the system x˙ = f (x) is asymptotically stable if

∂V V˙ (x) = f (x) < 0 ∂x

∀x 6= 0

(4.74)

For linear systems described by x˙ = Ax this condition (4.74) is necessary and sufficient. A suitable Lyapunov function is V (x) = xT Px,

P > 0.

This implies that the linear system is asymptotically stable if and only if xT PAx + xT AT Px < 0 ∀x 6= 0 This is equivalent to what has was previously said in Example 1.

4.3.2 L2 Gain In linear systems, the H∞ norm is equivalent to the maximum RMS energy gain of the system and is also called the H∞ -gain of the linear system. The equivalent measure for nonlinear systems is the so-called L2 gain, which is a bound on the RMS energy gain. Specifically a nonlinear system with input w(t) and output z(t) (see Figure 4.1) is said to have an L2 gain of γ if (4.75) kzk2 < γ kwk2 + β where β is a positive and k(.)k2 denotes the standard 2-norm-in-time (L2 norm) of a pRconstant ∞ T vector, i.e. kxk2 = t=0 x (t)x(t)dt. Thus the L2 gain of a system can be taken as a measure of the size of its output relative to the size of its input. For a linear system

Fig. 4.1. Nonlinear System

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135

x˙ = Ax + Bw

(4.76)

z = Cx + Dw

(4.77)

this fact can be derived from the LMIs (4.26)-(4.27) as this is equivalent to: " # AT P + PA + 1γ CT C PB + 1γ CT D 0. Writing this matrix inequality in terms of the vector xT wT , it follows that we need to find the minimum of γ so that #   T " T h iT A P + PA + 1γ CT C PB + 1γ CT D x x < 0, ∀ xT wT 6= 0 (4.79) 1 1 T T T w w B P + γ D C −γ I + γ D D or 1 1 xT AT Px + xT PAx + xT CT Cx + xT (PB + CT D)w γ γ 1 1 +wT (BT P + DT C)x + wT DT Dw − γ wT w γ γ 1 = xT AT Px + xT PAT x + 2xT PBw + zT z − γ wT w < 0 γ

(4.80)

Defining now V = xT Px, it is easily derived from V˙ = xT AT Px + xT PAx + 2xT PBw and (4.80) that: 1 V˙ + zT z − γ wT w < 0 γ Integration in the interval [0, ∞) implies V (t = ∞) −V (t = 0) +

Z ∞ Z ∞ 1 T z (s)z(s)ds − γ wT (s)w(s)ds < 0 t=0

γ

t=0

Shifting terms, taking the square root and using the triangle inequality, it is easily shown that γ is indeed the L2 -gain of the linear system (4.76) as (4.75) follows.

4.3.3

Lyapunov Stability for Discrete-Time Systems

Discrete-time systems are often treated in less detail than continuous-time systems. However, digital technology for controller implementation makes discrete controller design and analysis a pertinent subject of interest. As for Lyapunov stability of continuous-time systems a similar stability theorem can be stated. Theorem 4.2 (Lyapunov’s Theorem for discrete systems). Given an unbounded positive definite function V (x) > 0 ∀x 6= 0 and an autonomous system x(k + 1) = f (x(k)), then the system x(k + 1) = f (x(k)) is asymptotically stable if

∆ V (x(k + 1)) = V (x(k + 1)) −V (x(k)) < 0 ∀x 6= 0

(4.81)

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As before, for linear systems described by x(k + 1) = Ax(k) this condition (4.81) is necessary and sufficient and a Lyapunov function is given by V (x) = xT Px,

P > 0.

Thus, the linear system is asymptotically stable if and only if xT AT PAx − xT Px < 0 ∀x 6= 0 or AT PA − P < 0.

4.3.4 l2 Gain Similar to a continuous-time system, the computation of the l2 -gain can be very helpful to the design and analysis of discrete-time control systems. As for continuous time (4.75), a discrete-time system with input w and output z has a finite l2 -gain, γ , if kzk2 < γ kwk2 + β

(4.82)

The significant difference is the definition of the l2 -norm k(.)k2 which is now given by an q T infinite sum kxk2 = ∑∞ k=0 x (k)x(k). For linear systems x(k + 1) = Ax(k) + Bw(k)

(4.83)

y(k) = Cx(k) + Dw(k)

(4.84)

the value of the finite l2 -gain, γ , is the same as the H∞ norm or the maximum RMS energy gain of the system. The computation of the l2 -gain is easily achieved by considering the following matrix inequality problem: "

min γ AT PA − P + 1γ CT C BT PA + 1γ DT C

s.t. #

AT PB + 1γ CT D −γ I + BT PB + 1γ DT D

(4.85) 0. This readily follows from the analysis of 1 ∆ V (x(k + 1)) + yT (k)y(k) − γ wT (k)w(k) < 0 γ

(4.88)

for a Lyapunov function V = xT Px. To complete the analysis for the l2 -gain, the sum of the inequality of (4.88) over k in the interval [0, ∞) is considered (rather than the integral as in Section 4.3.2). The problem with matrix inequality (4.86) is that it is not linear in γ . This is easily amended by using the Schur complement so that the l2 -gain computation can be equivalently expressed in a convex formulation:

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min γ s.t.  AT PA − P AT PB CT T T T  B PA −γ I + B PB D  < 0 C D −γ I −P < 0

(4.89)



(4.90) (4.91)

The matrix P > 0 and the scalar γ > 0 are variables of this LMI-problem. Another approach of creating a linear matrix inequality from (4.86) is using the change of variable approach from Section 4.2.1. The first step is to multiply all elements of (4.86) by γ so that 

min γ s.t.  γ AT PA − γ P +CT C γ AT PB +CT D 0 are variables of the LMI√ problem and the l2 -gain is readily computed with γ = µ .

4.3.5

Sector Boundedness

The saturation function is defined as sat(u) = [sat1 (u1 ), . . . , sat2 (um )]T

(4.98)

and sati (ui ) = sign(ui ) × min {|ui |, u¯i } , u¯i > 0 ∀i ∈ {1, . . . , m}, where u¯i is the i’th saturation limit. From this, the deadzone function can be defined as Dz(u) = u − sat(u)

(4.99)

It is easy to verify that the saturation function, sati (ui ) satisfies the following inequality ui sati (ui ) ≥ sat2i (ui )

(4.100)

or sati (ui )[ui − sati (ui )]wi ≥ 0 for some wi > 0. Collecting this inequality for all i we can write sat(u)T W [u − sat(u)] ≥ 0

(4.101) (4.102)

for some diagonal W > 0. Similarly, it follows that Dz(u)T W [u − Dz(u)] ≥ 0

(4.103)

for some diagnal W > 0. We will make use of this inequality in the next section when computing the L2 -gain of a linear system with input and output constraints.

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4.3.6

A Slightly More Detailed Example

Suppose we consider the computation of the L2 -gain for the linear SISO system with saturated input signal u: x˙ = Ax + bsat(u), x ∈ Rn (4.104) which is subject to a limitation in the measurement range of the output y (see Figure 4.2): y = sat(cx + dsat(u)).

(4.105)

Fig. 4.2. Linear SISO System with input/output saturation

This is usually the case for many practically controlled systems. For instance actuator inputs u can be constrained due to mechanical limits (e.g., valves) or due to digital-to-analogue converter voltage signal limits, while sensor signals are possibly constrained due to sensor voltage range limits or simply by analogue-to-digital converter limits. Hence, the analysis of such systems is vital to practical control systems and will be pursued in greater detail in a later chapter. An upper bound of theL2 -gain of this system can be computed by considering sector constraints for the saturation functions. Furthermore, the Projection Lemma will be used to show that the L2 -gain of (4.104)–(4.105) is in actual fact not larger than the L2 -gain of the system without any of the saturation limits. For analysis, we may define s = sat(u). Hence, similar to (4.103), it follows that sw1 (u − s) ≥ 0,

(4.106)

where w1 > 0. In the same way, it can be shown that for the output signal y we have: yw2 (cx + ds − y) ≥ 0

(4.107)

for an arbitrary diagonal matrix w2 > 0. We know from Section 4.3.2 that from 1 V˙ + y2 − γ u2 ≤ 0 γ it follows that our system of interest (4.104-4.105) has the L2 -gain γ . However, in our case, this analysis is conducted under the condition (4.106–4.107). Hence, from the S-procedure of Section 4.2.4, it follows that 1 V˙ + y2 − γ u2 + 2sw1 (u − s) + 2yw2 (cx + ds − y) < 0 γ

(4.108)

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139

  for xT s u y 6= 0 implies similarly an L2 -gain of γ . This is also easily derived when integrating the inequality over the interval [0, ∞] above and considering a similar procedure as in Section 4.3.2 under the assumptions of (4.106–4.107). By computing the expression for V˙ , it follows that: xT AT Px + xPAx + xT Pbs + sbT Px 1 + y2 − γ u2 + 2sw1 (u − s) + 2yw2 (cx + ds − y) < 0 γ Rewriting this inequality, it is shown that:  T  T A P + PA Pb x  s   bT P −2w1    u  0 w1 w2 c w2 d y

for



(4.109)

  0 cT w2 x s  w1 dw2    < 0  u  0 −γ 1 0 −2w2 + γ y

 6 0. xT s u y =

This is equivalent to the matrix inequality of  T A P + PA Pb  bT P −2w1   0 w1 w2 c w2 d

 0 cT w2 w1 dw2   < 0.  0 −γ 1 0 −2w2 + γ

(4.110)

The target is now to minimize the value of γ > 0 to find the L2 -gain of our system of interest (4.104–4.105). However, the variable γ appears twice in the matrix inequality, once in the denominator of 1γ . This results in a non-linear and non-convex matrix inequality and it is not easily possible to avoid this non-linearity. Furthermore, this matrix inequality looks rather complicated as the positive definite matrix P, γ , w1 and w2 are variables. The Projection Lemma allows us to derive a significantly simpler matrix inequality which delivers the L2 -gain of our system. The first step is to rewrite the matrix inequality of (4.110) in the same format as for (4.65). Hence,   T A P + PA Pb 0 cT w2 T  b P 0 0 dw2      0 0 0 −γ w2 c w2 d 0 −2w2 + 1γ     0 0 1    −1       + (4.111)  0  w1 0 −1 1 0 +  1  w1 0 1 0 0 < 0. 0 0

Defining the matrices  T     A P + PA 0 0  bT P 1  −1      g1 =   0  , h1 =  1  , Ψ1 =  0 w2 c 0 0

allows us to write

Pb 0 0 w2 d

 0 cT w2 0 dw2  ,  −γ 0 1 0 −2w2 + γ

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Ψ1 + g1 w1 hT1 + h1 wT1 gT1 < 0.

(4.112)  T  Thus, the null space matrices Wg1 and Wh1 for g1 and h1 need to be found so that Wg1 g1 and h i WhT1 h1 are full rank and Wg1 g1 = 0, Wh1 h1 = 0. This implies: 

   I 000 I 000 Wg1 =  0 0 1 0  , Wh1 =  0 1 1 0  0001 0001

Hence, it follows from (4.66) that both  T  A P + PA 0 cT w2  u¯i u |ui | ≤ u¯i sati,u¯i (ui ) =  i −u¯i ui < −u¯i

(5.4)

(5.5)

(5.6)

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Note that each i’th saturation function depends on two parameters: the i’th control signal, ui (t), which will vary with time and the i’th saturation-limit, u¯i > 0, which determines the maximum magnitude of the output of the saturation, i.e., |sati,u¯i (ui )| ≤ u¯i ∀ui (t). A more succinct way of expressing equation (5.6) is sati,u¯i (ui ) = sign(ui ) min(|ui |, u¯i )

(5.7)

Note also that this description of the saturation function is symmetric; both upper and lower limits are the same. Non-symmetric saturation functions can be considered with little extra difficulty (in fact this is trivial) but they complicate the notation so we do not consider them here. In fact, to simplify things further, for the remainder of the chapter we drop the subscript u¯ and simply write sat(u) to indicate saturation of a vector u(t). An interesting feature of the saturation function defined as such is that if we construct a hyperbox defined as U = [−u¯1 , u¯1 ] × [−u¯2 , u¯2 ] × . . . × [−u¯m , u¯m ] ⊂ Rm (5.8) we can see that if the control signal stays within this box, that is u(t) ∈ U ∀t, then sat(u) = u i.e., the saturation function behaves as the identity. This is an obvious fact as it essentially says that providing that our control signal is sufficiently small, we do not have any saturation problems. However, it gives us a convenient mathematical way of capturing this feature.

5.1.3

Constrained Control and Anti-windup

r(t)

κ( r,y,t )

G(s)

y(t)

u(t)

Fig. 5.2. Architecture for “one-shot” constrained control design

Let us now consider the control problem. We want to control the system y = Gsat(u)

(5.9)

using the control signal u(t), such that the system is stable and performs well. For nonlinear systems such as the one above, the words “stable” and “performance” are not as precise as for linear sytems. However it suffices to say that we want non-oscillatory and non-divergent behaviour and we want to make sure that some performance objectives are met. Given these goals, one is very tempted to attempt to design a controller - be it linear or nonlinear - directly. That is we could design a controller u = K (r, y,t)

(5.10)

where K (., ., .) is a nonlinear operator which closes the loop using y(t) and also takes information from the reference signal, r(t) ∈ Rnr . This approach is shown in Figure 5.2 and for

Anti-windup Compensation and the Control of Input-Constrained Systems

147

want of a better name we call this the constrained control approach. This seems a very obvious approach to take and indeed in some cases it may be appropriate. However, in this chapter we are going to argue for a different approach: a two-step design procedure characterised by the following two steps. 1. Design a linear controller u = K[r′ y′ ]′ such that for all u(t) ∈ U our closed loop system is stable and behaves in a desirable manner, meeting linear design criteria. (Note that when u(t) ∈ U , we have linear behaviour). 2. Design an extra (normally linear) element, Θ , such that when u(t) ∈ / U , Θ becomes active and improves the performance and stability properties of the system. So what the above procedure is saying is (i) design a linear controller to yield good small signal behaviour and (ii) design an extra element to improve the large signal behaviour of the system when saturation occurs. At this stage, the second part of our two-step design procedure has been left deliberately vague; this will be fleshed out shortly. The two-step design procedure has become known as the anti-windup (AW) approach to the control of constrained systems. Although something of a misnomer, due to its links with integrator windup, the term “antiwindup” is now associated with a two-step design procedure for coping with input saturation. There is still some disagreement in the literature on the minutae of the definition, but generally the two-step approach outlined above can be considered an anti-windup approach. A schematic of the anti-windup approach is shown in Figure 5.3. Here Θ represents the AW compensator which becomes active when sat(u) − u 6= 0 (i.e., u ∈ / U ), and injects two signals, θ1 ∈ Rm into p the control signal and θ2 ∈ R into the input of the controller. We now have two approaches

d r K(s)

+ −

u

um

G(s)

y

θ1 + − + +

θ2

Θ1 Θ= Θ 2

Fig. 5.3. Architecture for anti-windup compensator approach

to controller design for constrained systems which can be “in-formalised” as follows.

INFORMAL CONSTRAINED CONTROL PROBLEM Given the linear plant G(s), design a controller K such that the system in Figure 1 is stable and yields desirable performance for all u(t).

M.C. Turner, G. Herrmann , and I. Postlethwaite

148

INFORMAL ANTI-WINDUP CONTROL PROBLEM Given the linear plant G(s) and the linear controller K(s), design an anti-windup compensator Θ such that (i) when u(t) ∈ U ∀t, Θ (s) is not active and when u(t) ∈ / U ∀t, the system in Figure 2 is stable and yields performance as close to nominal as possible. Again note that these are only informal problems; the precise nature of the stability and performance objectives we seek has not yet been characterised. The key observation to make is that the constrained control problem does not make any requirements on small signal performance, whereas the anti-windup problem imposes a structure on the small signal performance through the local controller K(s) (compare Figures 5.2 and 5.3). Therefore the anti-windup problem may be conceptually more difficult than the constrained problem as we require our overall control strategy to coincide with a local (linear) control strategy providing saturation does not occur. Despite this apparent difficulty there are a number of features which make the anti-windup approach more appealing than the constrained control approach. Some of these are listed below: •

•

•

•

Given a nominal linear controller K(s), small signal performance is not altered by the anti-windup compensator Θ . This is particularly important if nominal linear performance is desirable and saturation problems occur only occasionally. With a good anti-windup compensator, linear performance would then be disrupted rarely. Linear controller design can be done independently of the anti-windup compensator design. Thus there is no restriction in the linear controller design, making this step fairly easy to accomplish. Anti-windup compensators can be introduced into control loops once a saturation problem is identified. Thus they can be introduced into systems which have legacy controllers which are known to function well in most situations and only encounter difficulties during saturation. Thus anti-windup compensators can be useful in industry where much time and effort has been used to assess a given controller and the anti-windup compensator can be used to improve the system’s performance only where necessary without requiring a complete controller re-design. Conceptually, it is appealing to think of designing the nominal controller for linear performance and then an additional control element (the anti-windup compensator) for nonlinear stability and performance.

For these reasons, most industrial control systems which have saturation alleviating properties have an anti-windup like structure and indeed, the controller design is typically carried out in the two-step anti-windup procedure. It is worth mentioning that recent work on one-shot constrained controller design has made very good progress in recent years and, in particular, the constraint handling abilities of model predictive control (MPC) make it a very good candidate for the constrained control problem. The interested reader is urged to consult [18] for details of this. For the remainder of the chapter, we focus on the anti-windup approach.

5.2

Problems Due to Saturation

Before we look into how we might alleviate saturation problems in systems which are otherwise linear, it makes sense to have some idea of the types of instances in which problematic

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behaviour might occur. Although the nonlinear behaviour of the saturation function makes this difficult, it turns out that some indication of performance degradation can be inferred by looking at the dynamics of the nominal linear system (both linear plant and linear controller). First we look at some hints from classical control and then we move on to more formal statements.

5.2.1 Clues from Classical Control As saturation may be responsible for some very peculiar behaviour we shall start with very simple systems and consider how saturation may contribute to performance degradation during saturation. It therefore makes sense to start with single-input-single-output systems of the form in equation (5.9) but where the control and measured signals are purely scalar, viz u(t) ∈ R and y(t) ∈ R. In this case we can simplify our saturation to um (t) = sat(u) ∈ R Note that if we consider the input to the saturation block and the output of the saturation block, we can describe a nonlinear “gain” as the ratio of the output to the input. This is defined as  u¯ u > u¯ sat(u)  u 1 |u| ≤ u¯ β (u) = = (5.11)  u¯ u u < −u¯ −u sat(u)

sat(u)

Note that u ∈ (0, 1) when u > u¯ and likewise − u ∈ (0, 1) when u < −u. ¯ Therefore the “gain” of the saturation has the very interesting property that

β (u) ∈ (0, 1]

(5.12)

So it can be thought of as a function which represents a loop gain in the system. When β (u) = 1, and thus the saturation behaves as the identity, nominal performance is preserved. When β (u) < 1 and saturation occurs, then the system loses gain and behaves in a more “open-loop” manner. Finally when the gain β (u) ≈ 0 we can expect behaviour similar to open-loop. This extra gain term is depicted in Figure 5.4. The classical root-locus approach to control system

r(t)

β( u)

K(s) u(t)

um(t)

G(s)

y(t)

Fig. 5.4. [Overly] Simple approach to modelling saturation design tells us how the closed-loop poles move as the loop gain is varied. We know that the closed-loop poles start off (at zero gain) on the open-loop poles and then migrate along the root-locus to the open-loop zeros as gain is increased. At infinite gain, the closed-loop poles would sit on the open-loop zeros. The root-locus can therefore also be used to explain, in an approximate fashion, how the closed-loop poles move for varying amounts of saturation. We have the following interpretation: first for no saturation the closed-loop poles of the saturated system sit on the closed-loop poles of the nominal linear system. When saturation starts and as it gets more severe the poles move off the poles of the nominal closed-loop and head back

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M.C. Turner, G. Herrmann , and I. Postlethwaite

down the root-locus towards the open-loop poles. For very severe saturation, the poles of the saturated system effectively sit on the poles of the open-loop system. Some examples where the root-locus allows us to anticipate poor behaviour are shown overleaf. From the plots overleaf, it is plausible to suggest that we can compile several properties about single-loop saturated systems just from looking at the root locus. To aid us in this analysis we define the term spec to represent the poles of a transfer function (or equivalently the eigenvalues of a system A-matrix) and we define the open-left half complex plant, the imaginary axis and the open-right half complex plane respectively as C− , C0 and C+ . We define the open-loop transfer function L(s) = G(s)K(s) and draw the following conclusions (guidance): • •

•

If a system L(s) is such that at least one pole is in C +, then for sufficiently large state, reference or disturbance, saturation will always cause stability problems (divergence). If a system L(s) is such that spec(L(s)) ∈ C− but part of the root locus between the nominal closed-loop poles and the open-loop poles is in C+ , saturation will cause stability problems (limit cycling) for sufficiently large states, references or disturbances. If a system L(s) is such that spec(L(s)) ∈ C− but the part of the root locus between the nominal closed-loop poles and the open-loop poles is in a region corresponding to light damping, one can expect oscillatory responses for certain references or disturbances.

The root-locus gives us a guide to anticipating problems with saturation given a linear plant G(s) and a linear controller K(s). However it must be remembered that as we are dealing with a nonlinear problem, this only gives us a guide as we are essentially treating the saturation function as a linear gain, which is not mathematically correct. Also, as saturated systems are very “input-dependent”, predictions made on the basis of the root locus may only apply to certain classses of reference or disturbance inputs. Furthermore, it must be remembered that the root-locus only gives us information about SISO systems. If we now move to the case where again u(t) ∈ Rm and y(t) ∈ R p , it is more difficult to predict a system’s behaviour accurately. This is particularly true in systems which feature strong coupling. Recall that in multi-input multi-output (MIMO) systems the control signals are vectors and hence have directions associated with them. Saturation may not only limit the magnitude of this vector but can also alter the direction of the control vector causing closedloops which, in the absence of saturation were decoupled, to loose their decoupling. Figure 5.5 shows an example of this change in direction. u2 _ u2

un−saturated control vector saturated control vector

_ u1

_ −u 1

u1

_ −u 2

Fig. 5.5. Control direction corruption by saturation

Anti-windup Compensation and the Control of Input-Constrained Systems

ROOT-LOCUS AND WINDUP SUSCEPTABILITY

Conditionally stable system Root Locus

6

5

40

4

30

4

3 20

Output [non−dimensional]

Imaginary Axis

Closed−loop poles 1

Gain = 0.042

0

−1

Control [non−dimensional]

2

2

0

−2

−2

10

0

−10

"bang−bang" actuator switching

−4 −20

−3 High−amplitude limit cycle

−6

−30

−4

−5 −10

−8

−6

−4 Real Axis

−2

0

−8

2

0

5

10 Time [sec]

15

−40

20

0

5

10 Time [sec]

15

20

System root-locus and corresponding responses with (−) and without (−−) saturation. System with lightly damped plant poles Root Locus 2.5

4

25

20

3 2

15

2

Closed−loop poles

0

−1

Output [non−dimensional]

Output [non−dimensional]

1

Imaginary Axis

1.5

1

0.5

10

5

0

−5

−2 −10

0

−3

−15

−4 −6

−5

−4

−3 Real Axis

−2

−1

−0.5

0

0

5

10 Time [sec]

15

20

−20

0

5

10 Time [sec]

15

20

System root-locus and corresponding responses with (−) and without (−−) saturation. System with one unstable plant pole Root Locus

Output [non-dimensional]

0.8

3

Output [non-dimensional]

14000

1

12000

0.6

Imaginary Axis

0.4

0.2

0

−0.2

Critical gain point −0.4

−0.6

−0.8

−1 −8

Divergent instability

10000

−7

−6

−5

−4

−3 Real Axis

−2

−1

0

1

8000

6000

1

0

−1

4000

−2

2000

0

2

Actuator lock−up

−3

0

5

10

15

Time [sec]

20

−4

0

5

10

15

20

Time [sec]

System root-locus and corresponding responses with (−) and without (−−) saturation

151

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The conclusion of this section is that although classical control can give us a useful guide to predicting saturation problems, it does not tell us the full story and we need something more powerful to guarantee stability and performance. In fact some of the predictions made by classical control turn out to be true (e.g., if G(s) has unstable poles) but this cannot be verified by classical control and we must turn to more systematic methods.

5.3

Stability of Systems with Input Saturation

The very first thing which it is desirable to do for systems with input saturation is to establish whether they are stable. Again, remember that we are considering nonlinear systems so that this may not be a trivial task and that most stability conditions will be necessary or sufficient but not both. For instance it can be verfied that G(s) having no poles in C+ is a necessary condition for global asympotic stability of any closed-loop containing it and input saturation, but it is far from sufficient. Converesely, it is easy to prove that a system of the form M(s) =

α s+α

is always stable when put in a unity negative feedback loop with saturation1 , but this is hardly a necessary condition. This chapter will introduce some techniques for establishing stability of linear systems interconnected with saturation elements. The techniques will only provide sufficient conditions for stability and will be expressed as linear matrix inequalities.

5.3.1

Definitions of Stability

Throughout this work we try to keep things as simple as possible and so we will confine our attention to the minimum mathematical baggage possible. However, we shall introduce concepts of stability in a fairly general but brief manner in the interests of rigour and clarity. First we consider zero input stability of state-space systems i.e., the stability of autonomous systems with no forcing term. Definition 5.1. Consider the system x˙ = f (x)

(5.13)

then we say the origin of the system is i) Globally asymptotically stable (GAS) if limt→∞ x(t) = 0 for all x ∈ Rn ii) Globally exponentially stable (GES) if there exist scalars, η1 > 0 and η2 > 0 such that kx(t)k ≤ η1 exp[−η2t]kx(0)k for all x ∈ Rn iii) Locally asymptotically stable (LAS) if limt→∞ x(t) = 0 for all x(t = 0) ∈ X ⊂ Rn and 0 ∈ intX . 1

This follows as both the saturation and M(s) are passive.

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153

iv) Locally exponentially stable (LES) if there exist scalars, η1 > 0 and η2 > 0 such that kx(t)k ≤ η1 exp[−η2t]kx(0)k for all x(t = 0) ∈ X

⊂ Rn

and 0 ∈ intX .

Note that exponential stability is a stronger form of stability than merely asymptotic as it stipulates that the state must decay at a certain rate. The local versions of stability will not be dealt with in this chapter, although they play a crucial role in the assessment of saturated unstable plants (see [1, 6, 12, 22]). For systems with a forcing term, w we take a slightly different approach. We normally use a relationship between the forcing (input) signal, w ∈ Rnw and an output signal z ∈ Rnz . It is common to assume that signals belong to certain vector spaces. A vector space is a space which is defined in terms of its norm. In control, the L p norm is a popular way of measuring the size of signals. It is defined as kxk p :=

Z

0

∞

kx(t)k p dt

1

p

The space L p is then defined as the space of all signals with finite L p norm, that is  L p := x : kxk p < ∞

It follows that if a signal belongs to L p it is “well-behaved” in some sense. This allows us to define more concrete notions on input-output stability Definition 5.2. Consider the system S : w 7→ y then we say the system is

i) L p stable if w ∈ L p implies z ∈ L p ii) Finite gain L p stable if kzk p ≤ γ kwk p + β ,

β >0

Note that the finite gain form of input-output stability is stronger as it forces a linear relationship between the L p norm of the input and the L p norm of the output. The smallest such quantity γ is sometimes called the induced L p norm or the L p gain of the system S and is denoted kS ki,p . It might at first seem that the two definitions of stability are unrelated, but there are connections between them and some which we shall use later on. One connection is summarised in the following theorem which is effectively paraphrased from [15]. Theorem 5.1. Consider the state-space system  x˙ = f (x, w) S ∼ z = h(x, w)

(5.14)

If the state equation x˙ = f (x, 0) is GES, f (x, w) is Lipschitz in w and kh(x, w)k ≤ η1 kxk + η2 kwk then the system S is finite gain L p stable for all integers p ∈ [1, ∞]. This says that, if our state equation is globally exponentially (and therefore globally asymptotically) stable, then providing our output equation z = h(x, w) is sufficiently “nice” we get L p stability for free. A local version of this theorem can also be stated. Hence, at least for the time being, it makes sense to try and prove exponential stability of a state-space system.

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How can we prove exponential stability of a nonlinear system? Well the approach used throughout the control community is to find a Lyapunov function. Although this is too large a topic to be considered here it roughly boils down to finding a positive definite scalar function of the state, V (x) > 0 and then proving that the time derivative of this is negative along the dV (x) ∂ V (x) trajectories of the closed loop systems, i.e., dt = ∂ x f (x) < 0. In particular, we have the following theorem 2 Theorem 5.2. If there exists a function V (x) such that k1 kxkc ≤ V (x) ≤ k2 kxkc ∂ V (x) f (x) ≤ −k3 kxkc ∂x then the origin of x˙ = f (x) is globally exponentially stable.

(5.15) (5.16)

It is this Lyapunov argument which will be used extensively in the remainder of the chapter.

5.3.2

Saturation Modelling

Although it is possible to use the saturation function directly in proving stability using a Lyaponov function, it may be difficult. This is certainly the case with MIMO systems and to a lesser extent with SISO systems. Despite the saturation function’s simple description, it is nonlinear, and this complicates things. To make the saturation function more tractable, we will do two things. The first is a simple transformation and the second is known as sector bounding.

An Equivalent Representation Given the system y = Gsat(u), where the control signal u(t) is fed directly into the saturation element, it is useful to consider instead y = G(u + ∆ (u)) The system is now driven by the nominal control signal plus a nonlinear perturbation term, ∆ (.). This allows us to analyse the stability of a perturbed system, which is often conceptually simpler. Clearly ∆ (u) = sat(u) − u, but even more remarkably, ∆ (u) = −Dz(u), where Dz(.) is the deadzone function defined as   Dz1 (u1 )   .. (5.17) Dz(u) =   . Dzm (um )

where Dzi (ui ) = sign(ui ) max {0, |ui | − u¯i }. Equivalently it can be defined through the identity sat(u) = u − Dz(u).

(5.18)

A graph of a scalar version of the deadzone is shown in Figure 5.6. Thus if we have a system M(s) which is in a feedback loop with a saturation element, as shown in Figure 5.7, this can equivalently be represented by a feedback loop with [M(s) − I]−1 M(s) and a deadzone element. Effectively this lets us represent any system containing saturation as the nominal linear closed-loop plus a perturbation term involving the deadzone. Note that for all u(t) ∈ U , Dz(u) = 0 and thus the system behaves in a nominal way. 2

Essentially a simple version of Corollary 3.4 in [15] .

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155

Dz(u)

_ −u

_ u

u

Fig. 5.6. The scalar deadzone function −1

M(s) um

(M−I) M

u

~

u

u

"−∆ " Fig. 5.7. Representation of linear and nonlinear elements

Sector Bounding

N(v) slope =β slope =α

v

Fig. 5.8. Graphical representation of Sector[α , β ] Let us consider a static (memoryless) nonlinear z = N (v) which is continuous in its argument (for simplicity). Consider the graph in Figure 5.8. This shows a nonlinear function which is quite irregular, but notice that the graph of this function always lies between the gradient α and the gradient β lines. It is thus said that the function lies within the Sector[α , β ] and in fact it can be seen that the following inequality holds.

α v2 ≤ vN (v) ≤ β v2

β >α

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M.C. Turner, G. Herrmann , and I. Postlethwaite

It is easy to see that this implies v(α v − N (v)) ≤ 0

(5.19)

v(N (v) − β v) ≤ 0

(5.20)

(N (v) − β v)(N (v) − α v) ≤ 0

(5.21)

Hence it follows that

Thus any nonlinearity N (.) ∈ Sector[α , β ] will satisfy the inequality (5.21). Note that there are an infinite number of nonlinearities which belong to this sector and hence “modelling” a particular nonlinearity in this way introduces some conservatism. If we let N (.) = sat(.), or N (.) =Dz(.) it follows that the lower bound on the sector is α = 0 and the upper bound, is β = 1 for both nonlinearities. Hence, strangely, both the saturation and the deadzone nonlinearities inhabit the same sector. This idea can be easily extended to multivariable static nonlinearities. For simplicity we will only consider so-called decentralised nonlinearities in which N (.) : Rm 7→ Rm is a nonlinearity, with each element satisfying a scalar sector bound. Formally, we assume that N can be described as   N1 (u1 )   .. N (u) =  (5.22)  . Nm (um )

Ni ∈ Sector[αi , βi ]

(5.23)

In this case, using a similar process to the above, we say that a decentralised static nonlinearity N is in the Sector[A , B] if the following inequality holds (N (v) − Bv)(N (v) − A v) ≤ 0

(5.24)

where A = diag(α1 , . . . , αm )

(5.25)

B = diag(β1 , . . . , βm )

(5.26) (5.27)

Now, if we assume that the saturation and deadzone nonlinearities are multivariable, it follows that as each i’th component is within the sector[0, 1], that the full multvariable nonlinearity is such that sat(.) ∈ sector[0, I] Dz(.) ∈ sector[0, I] (5.28)

In turn this means that for both nonlinearities, the following inequality holds N (u)′ (N (u) − u) ≤ 0

(5.29)

N (u)′ X(N (u) − u) ≤ 0

(5.30)

or, more generally, that

where X is a positive definite diagonal matrix and N (u) is either Dz(u) or sat(u).

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157

Remark 5.1. A very interesting feature of bounding both the saturation and deadzone nonlinearities in this way is that the sector bounds are not functions of the limits u. ¯ That means that no matter what u¯ is -for either saturation or deadzone - it plays no role in determining the sector bound. Thus henceforth, the subscript u¯ will be dropped as it plays no further role in the development of the results. However, it should be noted that if we wish to develop local versions of these results, the limits u¯ actually play a crucial role in determining the local stability properties. ⊓ ⊔

5.3.3

The Multivariable Circle Criterion

z

w ~

T(s)

u

u

"−∆ " Fig. 5.9. The nonlinear stability problem We are now ready to state the main result of the section. This result is a very well-known result which effectively is going to be presented in a slightly different form to that in which it is often presented. However, this form will be of direct relevance to the results to be introduced later. We consider the system     w u (5.31) = T (s) u˜ z

where T (s) is a linear system with state-space realisation   x˙ = Ax + Bw w + Bu˜ T (s) ∼ u = Cx  z = Cz x + Dzw w + Dz u˜

(5.32)

where x ∈ Rn is the system’s state, u˜ = Dz(u) ∈ Rm is the “deadzoned” control signal 3 , w ∈ Rnw is the exogenous input and z ∈ Rnz is the performance output. Remark 5.2. We have deliberately restricted the feedback term u = Cx to be strictly proper to avoid questions of well-posedness (i.e., issues of existence and uniqueness of the loop equations). It is actually possible to use the more general feedback u = Cx + Du, ˜ at the expense of extra analysis. This chapter will not consider this additional analysis in detail, although the topic of well-posedness will be touched upon later. ⊓ ⊔ 3

This actually does not need to be the case - it could be any nonlinearity in the Sector [0, I]. . .

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M.C. Turner, G. Herrmann , and I. Postlethwaite

We wish to examine the conditions under which, the closed-loop system with w = 0 is globally exponentially stable. We proceed using a Lyapunov argument and attempt to establish some linear matrix inequalities. The following is the main result of the section. Theorem 5.3. The system described by equation (5.32) with u = Dz(u) satisfies the following properties • •

When w = 0 the origin is globally exponentially stable. There exist β , γ > 0 such that kzk2 ≤ γ kwk2 + β

if the following linear matrix inequality is satisfied  ′  A P + PA PB +C′W M := 0 ∀x 6= 0 (5.34)

where P > 0 is a postitive definite symmetric matrix. Note that V (x) is radially unbounded so that if we can prove V˙ (x) < 0 ∀x 6= 0, then global asymptotic stability will follow. Therefore, let us consider V˙ (x) V˙ (x) = 2x′ P(Ax + Bu) (recall w = 0) (5.35) Next note that as u˜ = Dz(u) satisfies the sector condition it follows that V˙ (x) ≤ x′ (A′ P + PA)x + 2x′ PBu + 2 u˜′W (u − u) ˜ | {z } sector bound ′ ′

= x′ (A′ P + PA)x + 2x′ (PB +C W )u˜ − 2u˜ W u˜ =: V˙1 (x)

(5.36) (5.37) (5.38)

where W > 0 is a positive definite diagonal matrix. Note that the adjoining of the sector bounds to the Lyapunov function to create the augmented Lyapunov function is a simple application of the S-procedure. Note further that it follows that if V˙1 (x) < 0, then obviously V˙ < 0, although the converse is not true (this has introduced conservatism). The expression above can be majorised to get  ′  ′   x A P + PA PB +CW x V˙1 (x) = (5.39) u˜ ⋆ −2W u˜ | {z } =:−M

Thus if LMI (5.33) as stated in Theorem 3 is negative definite it follows that

2

x

V˙1 (x) ≤ −λmin (M)

u˜ ≤ −λmin (M)kxk2

(5.40) (5.41)

This proves that the origin is asymptotically stable. Note now that this implies V˙ (x) ≤ λmin (M)kxk2 . Furthermore as P > 0, it follows that

λmin (P)kxk2 ≤ V (x) ≤ λmax (P)kxk2

(5.42)

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159

Thus, applying Theorem 2 we can see that the origin is not only asymptotically stable, but exponentially stable too. Our proof is almost complete. What remains is to prove that a finite L2 gain exists between w and z. To do this simply note that in our case we have f (x, w) = Ax + BDz(Cx) + Bw w

(5.43)

h(x, w) = Cx + DDz(Cx) + Dw w

(5.44)

where Dz(x) is Lipschitz in x. From this it follows that f (x, w) and h(x, w) are both certainly Lipschitz and hence that they satisfy the conditions of Theorem 1 and hence we have a finite L2 gain between w and z, which completes the proof. ⊓ ⊔⊓ ⊔ Remark 5.3. The so-called “Multivariable Circle Criterion” is the workhorse of many modern anti-windup strategies. Its main advantage is that through an appropriate change of variables it can be used for convex synthesis of anti-windup compensators. ⊓ ⊔

5.4

Anti-windup Problem Definition

The anti-windup problem is more than just ensuring stability of a constrained input system; as mentioned earlier anti-windup stipulates that large signal stability and performance goals must be achieved while preserving a local controller structure - i.e., that dictated by the linear controller. This objective is more complicated than a simple constrained control problem, where no constraints are placed on the structure and small-signal performance of the system. As mentioned before we would like the anti-windup compensator to become active only during periods of saturation and to interfere with the linear control loop as little as possible. Furthermore, when a saturation event has passed we would like linear behaviour to be recovered, at least asymptotically. In this section we formalise the anti-windup problem which we would like to solve 4 . Definition 5.3. Let ulin ∈ Rm denote the nominal linear control signal; and yd ∈ R p be the difference between the linear output ylin and the actual output y (i.e., yd = ylin − y) which results from actuator saturation.

• •

Then the anti-windup compensator Θ (s) is said to solve the anti-windup problem if the closed loop system in Figure 5.3 is internally stable, well-posed and the following hold: 1. If dist(ulin , U ) = 0, ∀t ≥ 0, then yd = 0, Θ (s)). 2. If dist(ulin , U ) ∈ L2 , then yd ∈ L2 .

∀t ≥ 0 (assuming zero initial conditions for

The anti-windup compensator Θ (s) is said to solve strongly the anti-windup problem if, in addition, the following condition is satisfied: 3. The operator T p : ulin 7→ yd is well-defined and finite gain L2 stable. 4

This definition is heavily influenced by that proposed in [24].

M.C. Turner, G. Herrmann , and I. Postlethwaite

160

Some comments on this problem definition are appropriate. •

•

•

•

Note that stability and well-posedness are part of the anti-windup problem but are not the limit of it. A system is well-posed if unique solutions exist to the closed-loop equations. In general, for saturated systems, it requires a detailed analysis; existence of solutions is relatively easy to prove, but uniqueness is somewhat harder and requires an argument similar to that found in [26] or the use of an implicit function theorem as advocated in [7] The first condition of the definition involves the use of the distance function. This function effectively gives a measure of the “distance” of a vector from a set. For example assume we are measuring the distance of a vector z from a set Z , then the distance function is the Euclidean norm from the vector z to the nearest point in Z i.e., dist(z, Z ) = infv∈Z kv − zk. If z ∈ Z , then dist(z, Z ) = 0. In this work we assume that our sets are connected and, moreover, convex. The first condition of the definition says that if the linear control signal ulin remains in U for all time, then the deviation from the linear output, ylin to the actual output, y is zero for all time; that is the anti-windup compensator does not impede linear behaviour if saturation is not encountered. The second item also makes use of the distance function, but it looks at the L2 norm of the distance function. It effectively says that if the signal which can be constructed by measuring the distance between ulin and the set U is square integrable (i.e., well behaved) then the deviation from linear behaviour will also be well behaved, and importantly, decay to zero asymptotically (this can be proved using Barbalat’s lemma). Note this ensures that linear behaviour will at least be recovered asymptotically, providing saturation does not occur indefinitely 5 . The third item is the “strong” version of the anti-windup problem and it requires there to be a finite L2 gain between our linear control signal, ulin and the deviation from linear behaviour, yd . Ideally this should be as small as possible to represent small (in an “energy gain” sense) deviation from linear behaviour. This means that if our control signal has a certain energy, then the signal representing the difference between linear and nonlinear behaviour has an energy bounded by a linear function of the energy in the linear control signal.

This definition has now “fleshed” out the problem we would like to solve. The next section offers a certain solution to that problem.

5.5

An Anti-windup Solution

There are a number of ways to solve the above problem. The one we describe here is a method which we think is particularly attractive as it gives a nice graphical interpretation of the problem (we now have words, maths and figures!). It also gives a convex formulation of the problem which can be solved using linear matrix inequalities which we shall describe shortly.

5.5.1

Architecture

In order to appreciate the strength of the method, it is fruitful to consider Figure 5.10, which shows the anti-windup problem in a reasonably generic form (compare to Figure 5.3), but 5

It is important to realise that the key point in establishing this is that there is no direct feedthrough term in the loop.

Anti-windup Compensation and the Control of Input-Constrained Systems

161

d r K

u lin + − ud

y lin

M−I + +

y

G

+ −

yd

G2 M

~ u

Fig. 5.10. A special parametrisation of the AW problem

M−I

ud

Disturbance Filter

Nonlinear Loop

− ~ u

+

r

d K

G

u lin

yd

G 2M

+

−

ylin

y

Nominal Linear Transfer Function Fig. 5.11. An equivalent “decoupled” representation of the AW problem

with a special choice of our AW compensator, Θ . This architecture allows the anti-windup compensator Θ (s) to inject signals, which modify the controller’s behaviour, at the controller output and also one of its inputs. Here G(s) = [G1 (s) G2 (s)] is the finite-dimensional linear time invariant (FDLTI) plant which we have assigned the following state-space realisation  x˙ p = A p x p + B p um + B pd d (5.45) G(s) ∼ y = C p x p + D p um + D pd d where x p ∈ Rn p is the plant state, um ∈ Rm is the control input to the plant, y ∈ R p is the plant (measurement) output and d ∈ Rnd is the disturbance. The FDLTI controller K(s) is assumed to be implemented as K(s) = [K1 (s) K2 (s)] and is assigned the following state-space realisation

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M.C. Turner, G. Herrmann , and I. Postlethwaite K(s) ∼



x˙c = Ac xc + Bc ylin + Bcr r ulin = Cc xc + Dc ylin + Dcr r

(5.46)

where xc ∈ Rnc is the controller state, ulin ∈ Rm is the linear controller output r ∈ Rnr is the disturbance on the controller, normally the reference. The overall controller output is given by u = ulin + ud ∈ Rm where ud is a signal produced by the anti-windup compensator; if the anti-windup compensator is inactive, u = ulin . The signal ylin ∈ R p is the linear input to the controller and is given by y + yd ∈ R p ; if the anti-windup compensator is inactive ylin = y. Note that in terms of Figure 5.3, the signals ud and yd play the roles of θ1 and θ2 respectively. As it stands, the scheme depicted in Figure 5.3 does not have any illuminating features. However, consider a particular case of the generic scheme, as shown in Figure 5.10 (introduced in [29]), where the anti-windup compensator Θ (s) has been parameterised in terms of a transfer function M(s) ∈ RH ∞ and a copy of the plant G2 (s). Using the identity (5.18), Figure 5.10 can be re-drawn as the decoupled scheme in Figure 5.11. Notice that this system exhibits an attractive decoupled structure: 1. Nominal linear system. This represents the linear system which would have resulted if no saturation was present. 2. Nonlinear loop. This contains the nonlinear stability problem and assuming that the linear loop is stable and the linear operator G2 M is also stable, the entire nonlinear stability problem is contained within this part of the system 3. Disturbance filter. This part of the system determines how the system recovers after a saturation event has stopped. This part is responsible for both the speed and manner of recovery after saturation has ceased. In [30], it was shown that most anti-windup schemes can be interpreted as certain choices of M(s) and therefore schemes such as the Hanus conditioning scheme ( [8]) and the high gain approach ( [4, 14]) can be analysed in terms of Figure 5.11. The advantages of viewing antiwindup in terms of Figure 5.11 is that the nominal linear performance is separated from the nonlinear part of the scheme. Moreover, the stability of the scheme is dependent on the stability of the nonlinear loop, assuming stability of the nominal linear closed loop and stability of the plant. This leads to the following assumption.

Assumption 5.5.1 • •



I −K2 (s) The poles of −G2 (s) I

plane. The limit lims→∞ (I − K2 (s)G2 (s))−1 exists.

−1

are in the open left-half complex

The first point ensures that all nominal closed loop transfer functions are asymptotically stable (in RH ∞ ) and if r = 0 and d = 0, then limt→∞ [x′p (t) xc′ (t)]′ = 0. The second assumption ensures that the nominal linear system is well-posed i.e., unique solutions exist to the feedback equations. These assumptions will be satisfied in most circumstances. From Figure 5.11, it can be seen that the performance of the overall closed loop system, including the anti-windup compensator, is closely related to the mapping T p : ulin 7→ yd . This mapping represents the deviation from nominal linear behaviour in response to a saturation

Anti -windup Compensation and the Control of Input-Constrained Systems

163

event and can be used as a measure of the anti-windup compensator’s performance. If some appropriate norm of this mapping is small, then the anti-windup compensator is successful at keeping performance close to linear (which we assume is the desired performance). In fact, as the AW compensator is parameterised by M(s), the choice of M(s) dictates the system’s stability properties under saturation. This chapter proposes choosing M(s) such that performance during saturation is improved. In [26] (see also [9, 25]), the L2 gain of T p was minimised using a system of linear matrix inequalities and, furthermore, M(s) was chosen such that it corresponded to static or low order anti-windup compensators. The result of [26] demonstrated, using suitable examples, that direct minimisation of T p was central to good anti-windup performance, and compensators designed according to the ideas in [26] seemed to perform at least as well, and often better, than most other anti-windup compensators. This idea was extended to more general cases in [10].

5.5.2

Full Order Compensators

Full-order anti-windup compensators are the class of AW compensators which are of order equal to the plant; that is they have n p states. They are attractive because they always exist for any stable plant, G2 (s), and can be designed in a relatively straightforward manner, through judicious choice of M(s). In fact, an appealing choice of M is as a part of a right comprime factorisation of G2 (s) = N(s)M(s)−1 where M, N ∈ RH ∞ . This means that the disturbance filter is given as yd (s) = N(s)u(s) ˜ and is hence a stable transfer function. The central reason for the appeal of this choice of M(s) is that this means that the anti-windup compensator   M(s) − I (5.47) Θ (s) = N(s) has order equal to that of the plant, providing the coprime factors share the same state-space. A state-space realisation of the operator T p is then easily calculated (using [33], for example) as 6      Ap + BpF Bp   x˙2 M(s) − I x ∼  ud   (5.48) F 0  2 N(s) u˜ yd Cp + D p F D p

where we have used the state x2 ∈ Rn p to represent the anti-windup compensator’s state and to distinguish it from the plant state. Another endearing property of this choice of AW compensator is that there is no direct-feedthrough from u˜ to ud which prevents algebraic loops (and thus does not require us to deal with well-posedness issues). The following is the main result of the section: Theorem 5.4. Under Assumption 5.5.1, there exists a full anti-windup compensator Θ (s), described by equations (5.47) and (5.48), which solves strongly the anti-windup problem if there exist matrices Q > 0, U = diag(µ1 , . . . , µm ) > 0,L ∈ R(m×n) and a positive real scalar γ such that the following LMI is satisfied 6

We could in fact use a more general coprime factorisation as discussed in [27] and [10], although it is not clear that this gives us much advantage over the simple factorisation described above.

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

 QA′p + A p Q + L′ B′p + B p L B pU − L′ 0 QC′p + L′ D′p ′   ⋆ −2U I UD p   0 which satisfies J=

d x2 (t)′ Px2 (t) + kyd k2 − γ 2 kulin k2 < 0 ∀[x2′ dt

u˜′

u′lin ]′ 6= 0

(5.50)

it follows that 1. If ulin = 0, then there exists a P > 0 such that v(x2 ) = x2′ Px2 is a Lyapunov function for the system; hence the system is globally stable. 2. Integrating J from 0 to T and taking the limit as T → ∞ kyd k22 < γ 2 kulin k22 + (x2 (0)′ Px2 (0) − x2 (∞)′ Px2 (∞))

(5.51)

which implies L2 gain less than γ from ulin to yd (and thus Item 3). The remainder of this part of the proof involves simplifying the expression for J. First, note that as the deadzone function is such that Dz ∈ Sector[0, I] we have that u˜i ui ≥ u˜2i ,

∀i ∈ {1, . . . , m}

(5.52)

From this, it follows that for some matrix W = diag(w1 , . . . , wm ) > 0 u˜′W (u − u) ˜ ≥0

(5.53)

Thus it follows that a sufficient condition for J < 0 is if d ˜ < 0 ∀[x2′ J˜ = x2 (t)′ Px2 (t) + kyd k2 − γ 2 kulin k2 + 2u˜′W (u − u) dt

u˜′

u′lin ]′ 6= 0 (5.54)

where u = ulin − ud . Hence using the state-space realisation (5.48), this becomes  (A p + B p F)′ P + P(A p + B p F) + (C p + D p F)′ (C p + D p F)  ⋆ ⋆  PBP + (C p + D p F)′ D p − F ′W 0 ′ −2W + D p D p W  0 means that A is positive definite and A 2 = (A 2 )T is the square root of a positive semi-definite symmetric matrix A. For a square matrix A, trace(A) is the trace of A. If P and K are Linear Time-Invariant (LTI) systems, the notation F (P, K) denotes the closed-loop system resulting from the interconnection of systems P and K. kG(s)k ∞ is the  A B H∞ norm of the LTI system G(s). kG(s)k 2 is the H2 norm of G(s). The notation CD represents a state space realization of G.

6.2 Preliminaries 6.2.1 LMI Formulation of Performance Specifications This section introduces the well-known H∞ and H2 performance measures for LTI systems. Algorithms for computing the H∞ and H2 norms of an LTI system are given in terms of LMIs. Throughout this section, we consider the LTI system G(s) with state space representation x˙ = Ax + Bw

(6.1)

z = Cx + Dw

(6.2)

where x(t) ∈ Rn is the state vector, w(t) ∈ Rnw is the system input, z(t) ∈ Rnz is the measurement output.

H∞ Performance Definition 6.1. Consider the LTI system G(s) with state space equations (6.1)-(6.2). Suppose G(s) is stable. The H∞ norm of G(s) can be defined as the largest singular value of its fre quency response across frequency, that is kG(s)k ∞ = sup σmax (G( jω )). w

The H∞ norm is useful to enforce the stability robustness of the closed loop with respect to model uncertainties . In addition , the H∞ norm facilitates frequency domain design

Output Feedback H ∞ Loop-Shaping Controller Synthesis

177

specifications such as bandwidth and high frequency roll-off in an H∞ controller synthesis.

The H∞ norm of an LTI system can be computed as the solution to a particular linear matrix inequality optimization problem. Lemma 6.1. (Bounded Real Lemma, see e.g., [6], [5]). Suppose the system G(s) given in (6.1)-(6.2) is stable. The H∞ norm of G(s) is smaller than γ if and only if there exists a symmetric matrix P such that  AT P + PA PB CT T T  B P −γ I D  < 0, C D −γ I 

P > 0.

(6.3)

H2 Performance Definition 6.2. Consider the LTI system (6.1)-(6.2) with D = 0. Assume w(t) ∈ Rnw is a zero mean white noise, with variance E {w(t)w(τ )T } = δ (t − τ )Inw , where δ (t) is the Dirac Delta function. The H2 norm of G is defined as   ZT 1 z(t)T z(t)dt (6.4) kGk22 = lim E T →∞ T 0

where the supremum is taken over all nonzero trajectories starting from x(0) = 0. The H2 norm is useful when disturbances/noises acting on the system are well represented by stochastic signals. The H2 objective can be computed as the solution of the LMI optimization problem given in Lemma 6.2. Lemma 6.2. (H2 norm computation, see e.g., [15]). kGk 22 is the global minimum of the optimization problem mintrace(Q)

(6.5)

AP + PAT + BB < 0   Q CP 0 and γ > 0 such that  T  T Acl Xcl + Xcl Acl Xcl Bcl Ccl  (6.13) BTcl Xcl −γ DTcl  < 0 Ccl Dcl −γ I then kTzw k∞ < γ . Now define

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ΩK :=



AK BK DK DK



.

(6.14)

It is easy to show that inequality (6.13) can be rewritten as Z + PT ΩK Q + QT ΩKT P < 0

(6.15)

where the matrices Z, P and Q depend only on the system matrices and on the matrix variable Xcl . Using the Elimination Lemma [2], the existence conditions for an internally γ -suboptimal H∞ controller reduce to solving two simpler LMIs NPT ZNP < 0,

NQT ZNQ < 0

(6.16)

where NP and NQ are bases of the null spaces of P and Q respectively. The next theorem, (Theorem 6.2), gives explicit expressions of inequalities (6.16) in terms of the system plant matrices and two Lyapunov matrices S and R which are related to Xcl in the following way. The matrix S is the matrix formed by the first n rows and n columns of Xcl while R is the matrix formed by the first n rows and n columns of Xcl−1 . The reader is referred to [6], [5] and [9] for more details. Theorem 6.2. (Existence conditions for a γ -suboptimal H∞ controller [6]). Consider the system given in (6.8)-(6.10) and let Nz and Nw be bases of the null spaces of (BTu , DTzu ) and (Cy , Dyw ) respectively. There exists an internally stabilizing output feedback controller K(s) such that the H∞ norm of the closed-loop transfer function from w to z is less than γ if and only if there exists two symmetric matrices R ∈ Rn×n and S ∈ Rn×n satisfying the following system of LMIs:   T AR + RAT RCzT Bw    Nz 0  Nz 0 Cz R −γ I Dzw  0.

⊔ ⊓

The optimization variables R, S and γ can be computed with efficient LMI solvers [7], [18]. Then it can be shown that matrix Xcl is uniquely determined as the solution of the following linear equation     I S R I = (6.20) Xcl 0 NT MT 0 where the matrices M and N are both in Rn×n and satisfy MN T = I − RS.

(6.21)

Once the matrix Xcl is determined, the state space matrices ΩK , of a γ -suboptimal H∞ controller, can be numerically obtained by solving the linear matrix inequality (6.15).

Output Feedback H∞ Loop-Shaping Controller Synthesis

181

It is worth mentioning that a reduced-order controller (i.e of order k < n) can be synthesized if the matrices R and S satisfy the additional rank constraint rank(I − RS) ≤ k.

(6.22)

A static output feedback controller (e.g., a controller with no states) can be obtained if rank(I − RS) = 0, that is, when R = S−1 . The literature is rich in algorithms that try to incorporate rank constraints into an optimization problem. The cone complementary algorithm [4] is one of them. However, because (6.22) is highly non convex in the variables R and S, its incorporation into an optimization problem may lead to formidable numerical difficulties. For this reason, there is still no efficient algorithm for fixed-order control law synthesis.

6.4 H∞ Loop-Shaping One H∞ synthesis procedure that meets stability and performance requirements together is the approach of McFarlane and Glover [10], [11]. The approach is based on the normalized coprime factors of a certain open-loop plant, known as the shaped plant. The method focuses on coprime factor robustness. The Glover–McFarlane design procedure is probably the H∞ design method preferred by most practical engineers since it combines classical open loop-shaping ideas. The method has many advantages over other H∞ control structures (see e.g., [16]) and tends to produce naturally robust controllers. This is essentially because the implicit H∞ minimization objective includes four closed loop sensitivity functions each of which is representative of a certain type of stability robustness. The design procedure is outlined below: 1) Pre and post filters are used to shape the plant singular values to give a desired open loop shape. The augmentation of the plant to be controlled with the shaping filters forms the shaped plant. 2) An H∞ feedback controller which robustly stabilizes with respect to perturbations of the the normalized left coprime factors is synthesized. 3) The feedback controller and shaping filters are combined to form the final controller. The design method was originally proposed by McFarlane and Glover [10] in the Riccati framework. Here we present the robust controller synthesis in terms of LMI optimization. With the LMI framework, we will see that it is relatively easy to extend the H∞ loop-shaping design procedure to low-order controller designs for LTI and polytopic systems.

6.4.1 LMI Formulation of the H∞ Loop-Shaping Controller Synthesis Let Gs be a strictly proper plant of order n having a stabilizable and detectable state-space realization:   AB (6.23) Gs := C0

with A ∈ Rn×n ,B ∈ Rn×nu , C ∈ Rny ×n . Gs represents the shaped plant in the Glover–McFarlane H∞ loop-shaping design procedure.

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˜ Such Suppose Gs has a minimal normalized left coprime factorization, that is Gs = M˜ −1 N. coprime factors can be computed with the formulae of Theorem 6.1 given in Section 6.2. With the normalized left coprime factorization of Gs we form the generalized plant depicted in Figure 6.1. A state space representation of the interconnection structure of Figure 6.1 is 

A 0 P :=  C C

−L 0 I I

 B I   0 0

(6.24)

where L = −ZCT and the matrix Z is the unique symmetric positive semi-definite solution to the algebraic Riccati equation AZ + ZAT − ZCT CZ + BBT = 0

(6.25)

w z 2 y

M˜ −1 

?  l

z1 6 N˜



u

Fig. 6.1. Open loop Glover–McFarlane H∞ loop-shaping interconnection Theorem 6.3. (Existence conditions for a full-order γ -suboptimal H∞ loop-shaping controller). Let L = −ZCT where Z ≥ 0 is the stabilizing solution to (6.25). There exists an output feedback controller K such that

 

K −1 ˜ −1

I (I + Gs K) M < γ ∞

(6.26)

if γ > 1 and if there exist positive definite matrices R and S solving the inequalities R(A + LC) + (A + LC)T R − γ CT C < 0   AR + RAT − γ BBT RCT −L  CR −γ I I  < 0 −LT I −γ I   RI > 0. I S

(6.27) (6.28)

(6.29)

Proof. The proof follows from the application of Theorem 6.2 to the augmented system (6.24). ⊔⊓ ⊓ ⊔

Corollary 6.1. (Existence conditions for a static γ -suboptimal H∞ loop-shaping controller). Let L = −ZCT where Z ≥ 0 is the stabilizing solution to (6.25). There exists a static output feedback controller K such that

Output Feedback H∞ Loop-Shaping Controller Synthesis

 

K −1 ˜ −1

(I + G K) M s

1 and if there exists a positive definite matrix R solving the system of inequalities (A + LC)R + R(A + LC)T < 0  AR + RAT − γ BBT RCT −L  CR −γ I I  < 0. T −L I −γ I 

(6.31) (6.32)

Proof. The proof follows from Theorem 6.3 if we set S = R−1 and eliminate the quadratic ⊔⊓ ⊓ ⊔ term −γ CT C in (6.27). Remark 6.2. The LMI conditions of Theorem 6.3 are necessary and sufficient for the existence of a full-order loop-shaping controller while those of Corollary 6.1 are only sufficient for the existence of a static H∞ loop-shaping controller. Therefore, Corollary 6.1 may lead to conservative results. ⊔ ⊓

6.4.2 Controller Reconstruction Full-order H∞ loop-shaping controllers can be computed numerically by solving the inequality (6.15). But it is preferable to use the reconstruction algorithms proposed in [9] and [5], which take advantage of the particular structure of (6.15). For the full-order controller syntheR hin f lmi of the LMI Control toolbox [7] can be directly used sis, the MATLAB command with the generalized plant Gs given in (6.24). For the static controller construction we need to proceed as follows. Since u = −Ky, a statespace realization of the closed-loop is given by

Tzw :=



Acl Bcl Ccl Dcl



 A − BKC BK − L =  −KC −K  . I (C − KC) 

(6.33)

From the Bounded Real Lemma the closed-loop system is stable and the H∞ -norm of Tzw is smaller than γ if and only if there exists a symmetric matrix R > 0 such that   T B Acl R + RATcl RCcl cl Ccl R −γ I Dcl  < 0. (6.34) Ω :=  BTcl DTcl −γ I

In this case,



 where A˜ =  

AR + RAT 0 CR −LT

0 −γ I 0 0

˜ C˜ − C˜ T K T B˜ T < 0 Ω = A˜ − BK    B RCT −L I
0 0  , B˜ =   and C˜ = CR 0 0 I .    D −γ I I 0 I −γ I

(6.35)

For given R and γ , one can solve the LMI in K (6.35) with any LMI solver or, more simply, K can be calculated using the explicit algebraic formulas given in [9], [7]. Static loop-shaping LMI conditions and controller reconstruction formulas for a non strictly proper plant (i.e., D 6= 0) are given in [14].

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6.4.3 Design Procedure for a Static H∞ Loop-Shaping Controller • •

• • •

The nominal plant G and the shaping functions W1 and W2 are combined to form the shaped plant Gs = W2 GW1 . Let (A, B,C) be a realization of Gs . Classical loop-shaping: Select W1 and W2 to get a desired open loop shape and compute the full-order McFarlane–Glover solution. Ensure that the closed-loop H∞ attenuation level γopt is small enough. Next, these weights are used as initial weights in the static output version of the design problem. Compute the matrix R > 0 and the scalar variable γ solution to the LMI system given in Corollary 6.1. If the LMI system is feasible, then make use of the Lyapunov matrix R > 0 and γ to solve the LMI problem (6.35) in K. The final feedback controller KST is then constructed using the static output feedback controller K with the shaping functions W1 and W2 such that KST = W1 KW2 . The order of KST is equal to sum of the orders of the weighting functions.

6.4.4 Static H∞ Flight Control System Design for the Bell 205 Helicopter This section describes the design of a static H∞ regulator which has been implemented on board a Bell 205 helicopter and tested in flight.

Plant Description The controller design was based on a 32-state nonlinear flight mechanic model of the Bell 205, provided by QinetiQ, Bedford. The linearized model corresponds to the hover situation. To simplify the synthesis of the resulting controller this model was first truncated and then residualised to 12 states. The main states which were removed were those associated with the rotor, which could be replaced with their steady state values. For control purposes, the helicopter can be thought of, roughly, as a three axis vehicle, the axes to be controlled being longitudinal, lateral and directional. Theses axes are associated with the three actuator positions producing the blade angle deflections as ordered and described in table 6.1. Generally speaking the longitudinal and lateral cyclic give lateral and longitudinal motion and the tail rotor generates a torque to counteract the main rotor torque and provides yaw motion. A fourth actuator, the collective, produces lift but, in this study, the torque axis is left under direct control of the pilot. An Attitude-Command Attitude-Hold strategy combined with Rate-Command Position-Hold is considered. The primary variables to be controlled are the pitch attitude (θ ), the roll attitude (φ ) and the yaw rate (r). The plant to be controlled is non square since we have 3 control inputs, 3 outputs to be controlled along with two extra rate gyro measurements p and q (table 6.1) used by the controller. In the sequel, G denotes the 12-state linearised model of the Bell 205 about the hover condition.

Static H∞ Helicopter Controller Design For the hover operating point the weights were chosen as W1 = diag(2

s + 0.6 s + 0.4 s + 0.6 ) , 1.4 ,2 s s s

(6.36)

Output Feedback H∞ Loop-Shaping Controller Synthesis

185

Table 6.1. Plant inputs and measured outputs

Description Measured outputs (y) Description Input (u) lateral cyclic actuator θ1c roll attitude φ longitudinal cyclic actuator θ1s pitch attitude θ tail rotor actuator δr yaw rate r roll rate p pitch rate q

W2 = diag(1, 1.1, 0.9, 0.7, 0.9)

(6.37)

Each input channel was augmented with a simple PI pre-compensator. The integral part was used to boost low frequency gain and improve performance. The proportional term was introduced to rectify the phase-lag added by the integrators at cross-over. W2 emphasizes the outputs to be controlled over the other measurements. The reader is referred to [12], [17], for a description of weighting function selection for helicopter control. The synthesis procedure described in section 2 was implemented using the solver in the LMI toolbox [7]. In this case, we get the following static controller   0.8897 −0.1374 0.2178 1.0567 −0.3132 K = −0.1915 −0.7287 −0.0967 −0.2070 −1.5119 (6.38) 0.1433 −0.1182 −1.0402 0.2405 −0.1470

with γ = 2.9877 which represents an upper bound on the H∞ closed-loop performance. The actual real closed-loop gain attenuation achieved with this static controller is equal to 2.69. The controller which is tested and implemented is KST = W1 KW2 , a 3rd-order controller called the static-based H∞ loop-shaping controller.

re f

 - KST  + 6−

u

-

G

y -

Fig. 6.2. Feedback configuration Figure 6.2 shows the feedback interconnection. The controller is the simple one-degree-ofT  freedom regulator KST . The reference vector, re f = φd θd rd 0 0 , includes the three pilot inputs corresponding to the roll, the pitch and the yaw rate demands. Its two last components are identically equal to zero as rate commands are not required here. Time domain responses of the static and full-order loop-shaping controllers are given in Figures 6.3 and 6.4 respectively. Clearly the static and full-order loop-shaping controllers present similar primary responses in pitch and roll. However, we can see, from Figure 6.4, that the full-order controller is less active in the lateral axis (roll and yaw responses are slower). Also, the full-order controller presents important roll to yaw and yaw to roll static cross couplings. The full-order controller time responses are not as good because the weighting functions were finely tuned for the static

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controller and these weighting functions do not correspond to the best choice for the full-order synthesis. It is relatively easy to select the weighting functions to obtain a full-order controller with good performance. However, in this application, it is worth noting that model reduction techniques do not lead to satisfactory low-order controllers. Residualization and Hankel norm model reduction techniques were applied on a set of good performance full-order loop-shaping controllers. The best reduced controller was obtained by Hankel norm model reduction and it was of order 7. Setting the order to 3 in the model reduction algorithm produces controllers with unacceptable robustness and performance properties. More details on this application, including flight test results, can be found in [14].

roll

rad & rad/s

1 0.5 0 −0.5

0

0.5

1

1.5

2

2.5 pitch

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 yaw

3

3.5

4

4.5

5

rad & rad/s

1 0.5 0 −0.5

rad & rad/s

1 0.5

Roll attidude pitch attitude yaw rate roll rate pitch rate

0 −0.5

0

0.5

1

1.5

2

2.5 sec

3

3.5

4

4.5

5

Fig. 6.3. Closed-loop time responses to unity step-demands in roll, pitch and yaw. Static-based output feedback controller

6.5 H∞ Loop-Shaping for Polytopic Systems This section generalizes the loop-shaping controller synthesis results to polytopic systems. Polytopic systems are parameter dependent systems whose state space representations depend affinely on some parameters. In this context, we will suppose that the plant parameters are available for measurement.

Output Feedback H∞ Loop-Shaping Controller Synthesis

187

roll

rad & rad/s

1 0.5 0 −0.5

0

0.5

1

1.5

2

2.5 pitch

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 yaw

3

3.5

4

4.5

5

rad & rad/s

1 0.5 0 −0.5

rad & rad/s

1 Roll attidude pitch attitude yaw rate roll rate pitch rate

0.5 0 −0.5

0

0.5

1

1.5

2

2.5 sec

3

3.5

4

4.5

5

Fig. 6.4. Closed-loop time responses to unity step-demands in roll, pitch and yaw. Full-order controller.

6.5.1 Left Coprime Factors for Polytopic Systems Let a realization of the parameter dependent system G(θ ) be given by   A(θ ) B C D

(6.39)

It is assumed that the parameter trajectories θ (t) = (θ1 , . . . , θm )T are bounded, that is, each component of the time-varying parameter θ (t) satisfies, for all t ≥ 0,

θ i ≤ θi (t) ≤ θ¯i . This assumption means that the parameter vector θ belongs to the hypercube Θ characterized by its r = 2m vertices Πi . More precisely, the hypercube Θ is defined as

Θ := [θ 1 , θ¯1 ] × . . . × [θ m , θ¯m ]. and, in this case, the parameter vector can be written as r

θ (t) =

∑ αi (t)Πi ,

i=1

r

αi (t) ≥ 0,

∑ αi (t) = 1.

i=1

The above expression is called a convex decomposition and the αi are called polytopic coordinates. Similarly, the parameter dependent evolution matrix A(θ ) can be rewritten as

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E. Prempain and I. Postlethwaite r

A(θ ) =

i=1

where Ai = A(Πi ),

r

∑ αi (t)Ai ,

αi (t) ≥ 0,

∑ αi (t) = 1

(6.40)

i=1

i = 1, . . . , r.

Throughout this section, it is assumed that the pairs (A(θ ), B) and (C, A(θ )) are quadratically stabilizable and detectable. Quadratic stabilizability is equivalent to the existence of a state feedback matrix F and a positive definite matrix Q such that (A(θ ) + BF)Q + Q(A(θ ) + BF)T < 0,

θ ∈ Θ.

Similarly, quadratic detectability is equivalent to the existence of an observer matrix L and a positive definite matrix P such that P(A(θ ) + LC) + (A(θ ) + LC)T P < 0,

θ ∈ Θ.

Theorem 6.4. (Left coprime factorization for polytopic systems, [13]). Define R˜ = I + DDT ,

R = I + DT D.

˜ θ) ˜ θ )−1 N( a) There is a left coprime factorization G(θ ) = M( # "   A(θ ) + LC L B + LD ˜ ˜ M(θ ) N(θ ) = 1 1 1 R˜ − 2 C R˜ − 2 R˜ − 2 D

with L = −(BDT + P−1CT )R˜ −1

(6.41)

if there exist matrices P > 0 and Z > 0 which are solutions to the optimization problem 

min trace(Z)  PAi + ATi P −CT C PB −CT D < 0, −R BT P − DT C i = 1, . . . , r   Z I > 0. I P

  ˜ θ )(s) is contractive, that is, at frozen values of parameter θ ˜ θ )(s) N( b) M( ˜ θ )(s)M˜ T (θ )(−s) + N( ˜ θ )(s)N˜ T (θ )(−s) ≤ I, M( s = jω ,

ω ∈R

and the equality holds (e.g., the factorization is normalized) when the polytope reduces to a single point. This result shows that the determination of a left coprime factorization for an LPV system reduces to solving a special H2 filtering problem. Moreover, it can be shown that if the size of the polytope vanishes to zero, the factorization given in (6.41) approaches the normalized left coprime factorization of the nominal plant; see [13] for more details.

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6.6 LMI Conditions The H∞ loop-shaping design procedure of McFarlane and Glover for LTI systems can now be extended to the class of linear parameter varying LPV systems. Two results are proposed. Theorem 6.5 gives existence conditions for a full-order gain-scheduled LPV controller, while Theorem 6.2 gives existence conditions for a static LPV controller. Theorem 6.5. Let us consider the LPV plant G(θ ) given in (6.39)-(6.40) and let L = −(BDT + P−1CT )R˜ −1 where P > 0 is the solution to the optimization problem of Theorem 6.4. There exists a dynamic output feedback LPV controller K(θ ) such that

 

K(θ ) −1 ˜ −1

M( θ )K( θ )) θ ) (6.42) (I + G(

1 and if there exist positive definite matrices Q and S solving the inequalities: S(Ai + LC) + (Ai + LC)T S − γ CT R˜ −1C < 0,

i = 1, . . . , r   1 T T T Ai Q + QAi − γ BB QC − γ BDT −LR˜ 2 1   CQ − γ DBT R˜ 2  < 0, −γ R˜  1 1 −γ Iny R˜ 2 −R˜ 2 LT

i = 1, . . . , r   QI ≥ 0. I S

(6.43)

(6.44) (6.45)

Proof. The proof follows from the application of Theorem 6.4 and makes use of some results given in [1] and [14]. ⊔⊓ ⊓ ⊔ Corollary 6.2. Let us consider the LPV plant G(θ ) given in (6.39)-(6.40) and let L = −(BDT + P−1CT )R˜ −1 where P > 0 is the solution to the optimization problem of Theorem 6.4. There exists a static output feedback LPV controller K(θ ) such that

 

K(θ ) −1 ˜ −1

(I + G( M( θ )K( θ )) θ )

1 and if there exists a positive definite matrix Q solution to the inequality system (Ai + LC)Q + Q(Ai + LC)T < 0, i = 1, . . . , r   1 T T T Ai Q + QAi − γ BB QC − γ BDT −LR˜ 2 1   R˜ 2  < 0, CQ − γ DBT −γ R˜  1 1 −γ Iny R˜ 2 −R˜ 2 LT

i = 1, . . . , r.

Proof. The proof is similar to that of Corollary 6.1.

(6.46)

(6.47) ⊔⊓ ⊓ ⊔

Remark 6.3. In Theorems 6.5 and 6.2, the plant matrices B, C, and D are required to be parameter independent. As is suggested in [1], low pass filters can be appended at the plant input (resp. output) to move the parameter dependence of B (resp. C) to the evolution matrix A. ⊓ ⊔

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6.6.1 Illustrative Example This example illustrates the design of an H∞ loop-shaping controller in the robust gain scheduled framework. We consider the LPV design example given in the LMI control toolbox [7]. We will see that it is possible to obtain good responses in terms of speed, damping and settling time with a low-order LPV regulator. The linear parameter dependent plant G0 represents a simplified model of the dynamics of a missile. G0 is a second-order system with two outputs and one input governed by the following equations  α˙ = −Zα (t)α + q    q˙ = −Mα (t)α + δm G0 : a = −α    zv q=q

(6.48)

The outputs are (azv ) the normalized vertical acceleration and (q) the pitch rate. The control input is (δm ) the fin deflection. The coefficients Zα (t) and Mα (t) are assumed to vary in Zα (t) ∈ [0.5, 4] Mα (t) ∈ [0, 106]

(6.49)

and are available in real-time. G0 is an affine 2-parameter plant, thus its corresponding polytopic representation has four vertices. More precisely, G0 (t) admits the following state space representation (A(t), B,C, D), where A(t) = ∑4k=1 ck (t)Ak ,     −0.5 1 −4 1 A1 = , A2 = , 0 0 0 0     −0.5 1 −4 1 , A4 = , A3 = −106 0 −106 0       0 −1 0 0 B= ,C = ,D = (6.50) 1 0 1 0 with polytopic coordinates

Mα (t) Zα (t) − 0.5 ), )(1 − 106 3.5 Mα (t) Zα (t) − 0.5 ), (1 − c2 (t) = 106 3.5 Zα (t) − 0.5 Mα (t) , ) c3 (t) = (1 − 106 3.5 Zα (t) − 0.5 Mα (t) ) c4 (t) = ( 106 3.5

c1 (t) = (1 −

(6.51)

In this problem, the vertical acceleration azv must be controlled. For extra realism, the actuator is modelled as a second-order system with a bandwidth of 188rad/s and a damping of 0.7. Similarly, each sensor has a bandwidth of 628 rad/s and a damping of 0.7.

Output Feedback H∞ Loop-Shaping Controller Synthesis

191

Therefore, the plant to be controlled Gaug has 8 states, 1 input, 2 outputs and four vertices. Values for the weighting matrices were chosen as W1 =

W2 =

80 s + 80

 1000s+4100 s

0

(6.52)

0 35



(6.53)

W1 is a low pass filter introduced to limit the control action of the polytopic controller. The first entry of W2 behaves like an integrator at DC. It is used to boost the gain of Gaug at low frequency. Now, define the loop-shaping plant as Gs = W2 GaugW1 .

(6.54)

In this case, the open loop shaped plant Gs is a polytopic system with 4 vertices, 10 states, 1 input and 2 outputs. The optimization problem of Theorem 6.4 is feasible and returns two matrices P > 0 and Z > 0 with Z such that mintraceZ = 0.633. Substituting the matrix P into (6.41) gives the following observer matrix   −6.2497e − 002 1.8757e − 003  −3.0161e − 005 1.1060e − 004     −2.7245e − 004 1.5691e − 004     1.2874e − 006 −1.8917e − 003     4.5150e − 003 −7.5989e − 003   (6.55) L=  2.7078e − 002 −1.6726e − 002  .    4.3663e − 001 −7.5981e − 001     −1.0303e − 001 5.9623e − 001     4.1795e − 002 −3.1349e − 001  7.9450e − 002 −6.0183e − 001

Substituting the values of the plant matrices of Gs and the matrix L into the LMI system of Theorem 6.2, leads to a feasible solution Q > 0 with the objective γ = 6.9. With the values of Q and γ a static polytopic controller K can be constructed using a procedure similar to the one given for LTI systems. The final loop-shaping controller KST = W1 KW2 , is a polytopic controller of order 2 with 4 vertices. In this case, it worth noting that the standard polytopic H∞ synthesis procedure of the LMI control toolbox [7] produces a polytopic controller of order 12. The closed-loop time-responses of the controlled system for a unit step demand in azv along the following spiral trajectories of the parameters are given in Figure 6.5. Zα = 2.25 + 1.7e−4t cos(100t) Mα = 50 + 49e−4t sin(100t) Figure 6.5 shows that the time response of the acceleration with the static based LPV controller KST (solid line) is very close to the specified reference. Note that the response with the static LPV controller is better damped than the response provided by the polytopic regulator designed with the procedure of the LMI toolbox [7] (dashed line). Figure 6.6 shows the corresponding control effort. We can see that the low-order LPV controller is a bit more active than the LPV controller of [7]. This is not surprising since KST provides a faster acceleration response.

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step response of the gain−scheduled autopilot 1.4

1.2

1

azv

0.8

0.6

0.4

0.2

0

0

0.1

0.2

0.3

0.4

0.5

time (s)

Fig. 6.5. Unit step closed-loop responses of the gain scheduled autopilots: Static based polyR toolbox [7] (dashed line) topic regulator KST (solid line); controller from the MATLAB LMI

6.7 Conclusions In this chapter, we have addressed the design of LTI and gain-scheduled H∞ loop-shaping controllers. The determination of a left coprime factorization for an LPV system is given as a special robust H2 filtering problem. This can be viewed as a natural extension of the wellknown normalized left coprime factors for LTI systems to LPV systems. Existence conditions for full-order and static LPV loop-shaping controllers are given in terms of linear matrix inequalities. The effectiveness of the proposed approaches are illustrated on a realistic helicopter model example and a polytopic missile model example.

References 1. P. Akparian and P. Gahinet. A Convex Characterzation of Gain-Scheduled H∞ Controllers. IEEE Transactions on Autmatic Control, 40(9):853–864, May 1995. 2. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM. Studies in Applied Mathematics, 1994. 3. J Doyle, K. Glover, P. Khargonekar, and B. Francis. State-space solutions to standard H2 and H∞ control problems. IEEE Transactions on Automatic Control, 34(8):831–847, August 1989. 4. L. El Ghaoui, F. Oustry, and M. AitRami. A cone complementary linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control, 42:1171–1176, August 1997.

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control action (fin deflection in degrees) 10

5

u

0

−5

−10

−15

0

0.1

0.2

0.3

0.4

0.5

time (s)

Fig. 6.6. Unit step closed-loop responses of the gain scheduled autopilots: Static based polyR toolbox [7] (dashed topic regulator KST (solid line); and controller from the MATLAB LMI line)

5. P. Gahinet. Explicit controller formulas for LMI-based H∞ synthesis. Automatica, 32(7):1007–1014, 1996. 6. P. Gahinet and P. Apkarian. A linear matrix inequality approach to H∞ control. International Journal of Robust and Nonlinear Control, 4:421–448, 1994. 7. P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali. LMI control toolbox. The Math Works, May 1995. 8. K. Glover and J. C. Doyle. State-space formulae for alll stabilizing controllers that satisfy an H∞ norm bound and relations to risk sensitivity. Systems and Control Letters, 11:167– 172, 1988. 9. T. Iwasaki and R. E. Skelton. All controllers for the general H∞ control problem: LMI existence conditions and state space formulas. Automatica, 30(8):1307–1317, 1994. 10. D. McFarlane and K. Glover. Robust controller design using normalized coprime factor plant description, Lecture notes in control and information Science. Springer-Verlag, Berlin, 1989. 11. D. McFarlane and K. Glover. A loop-shaping design procedure using H∞ synthesis. IEEE Transactions on Automatic Control, 37(6):759–769, 1992. 12. I. Postlethwaite, A. Smerlas, D. J. Walker, A. W. Gubbels, S. W. Baillie, M. E. Strange, and J. Howitt. H∞ control of the NRC Bell 205 fly-by-wire helicopter. Journal of American Helicopter Society, 44(4):276–284, 1999. 13. E. Prempain. Coprime Factorizations for Parameter-Dependent Systems. ROCOND’06, Toulouse, July 2006. 14. E. Prempain and I. Postlethwaite. Static H∞ loop-shaping control of a fly-by-wire helicopter. Automatica, 41:1517–1528, September 2005. 15. C. Scherer, P. Gahinet, and M. Chilali. Multi-objective output-feedback control via LMI optimization. IEEE Trans. Aut. Cont., 30(8):1307–1317, July 1997.

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16. S. Skogestad and I. Postlethwaite. Multivariable Feedback Control; Analysis and Design. Wiley, 2nd edition, 2005. 17. A. Smerlas, I. Postlethwaite, D. J. Walker, M. E. Strange, J. Howitt, R. I. Horton, A. W. Gubbels, and S. W. Baillie. Design and flight testing of an H∞ controller for the ncr bell 205 experimental fly-by-wire helicopter. AIAA-98-4300. Proc AIAA Guidance Navig. Contr. Conf. Boston, pages 1023–1033, August 1998. R ∗ toolbox for optimization over symmetric cones. 18. J. F. Sturm. SeDuMi-a MATLAB Department of Econometrics, Tilburg University, The Netherlands, October 2001. 19. M. Vidyasagar. Control system synthesis. The MIT Press, 1985. 20. K. Zhou, J. Doyle, and K. Glover. Robust and Optimal Control. Prentice-Hall International, Inc, 1995.

7 Stability and Asymptotic Behaviour of Nonlinear Systems: An Introduction Hartmut Logemann and Eugene P. Ryan

7.1

Introduction

To motivate a study of asymptotic behaviour of nonlinear systems modelled by ordinary differential equations and differential inclusions, we indicate how such equations/inclusions arise naturally in control of dynamical process by feedback. The concept of control pertains to modifying the behaviour of the process, by manipulation of inputs to the process, in order to achieve some prescribed goal. Fundamental to this is the notion of feedback: a strategy in which the inputs to the process are determined on the basis of concurrent observations on (or outputs from) the process. Consider first a finite-dimensional, continuous-time dynamical process, the state of which

inputs

6



-

process strategy

outputs -



?

Fig. 7.1. Control loop evolves in RN and is governed by a controlled ordinary differential equation, with initial data (t0 , x0 ), of the general form x(t) ˙ = g(t, x(t), u(t)),

x(t0 ) = x0 ,

(7.1)

where the function u is the input or control and the output or observation y is generated via an output map c: y(t) = c(t, x(t)). (7.2) Under feedback, the input u(t) at time t is determined by the output y(t) via a feedback map h: u(t) = h(t, y(t)) = h(t, c(t, x(t))). (7.3)

M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 195-220, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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Introducing the function f given by f (t, ξ ) = g(t, ξ , h(t, c(t, ξ ))), we see that the conjunction of (7.1) and (7.3) gives rise to an initial-value problem of the following type. x(t) ˙ = f (t, x(t)),

x(t0 ) = x0 .

(7.4)

Clearly, in order to ensure that this problem is well posed, the function f is required to be sufficiently regular. In due course, we will make precise the requisite regularity conditions which imply continuity – in their second arguments – of g, c and h. Are these continuity conditions reasonable? Perhaps not in the case of the feedback map h – since many feedback strategies are inherently discontinuous e.g., “bang-bang” or “on-off” control actions. A prototype example is the signum function sgn, which can be embedded in the following set-valued map defined on R:  x>0  {+1}, [−1, +1], x = 0. x 7→  {−1}, x 0, there exists δ > 0 such that kh(ξ0 ) − h(ξ )k ≤ ε for all ξ in A with kξ0 − ξ k ≤ δ . If h is continuous at ξ for all ξ in a subset B of A, then h is said to be continuous on B; if B = A, then we simply say that h is continuous. The function h is uniformly continuous on a subset B of A if, for every ε > 0, there exists δ > 0 such that kh(ξ1 ) − h(ξ2 )k ≤ ε for all points ξ1 and ξ2 of B with kξ1 − ξ2 k ≤ δ ; if B = A, then we say that h is uniformly continuous. It is convenient to adopt the convention that h is uniformly continuous on the empty set 0. / If h is continuous and B ⊂ A is compact, then h is uniformly continuous on B. If h is scalar-valued (that is, if P = 1), then h is lower semicontinuous if lim infξ ′ →ξ h(ξ ′ ) ≥ h(ξ ) for all ξ in A, while h is upper semicontinuous if −h is lower semicontinuous; we remark that h is continuous if, and only if, it is both upper and lower semicontinuous. The Euclidean distance function for a nonempty subset A ⊂ RN is the function dA : RN → R + given by dA (v) = inf{kv − ak : a ∈ A}. The function dA is globally Lipschitz with Lipschitz constant 1, that is, kdA (v) − dA (w)k ≤ kv − wk for all v, w ∈ RN . A function x : R + → RN is said to approach the set A if dA (x(t)) → 0 as t → ∞. For ε > 0, Bε (A) := {ξ ∈ RN : dA (ξ ) < ε } (the ε -neighbourhood of A); for a in RN , we write Bε (a) in place of Bε ({a}). It is convenient to set Bε (0) / = 0. / The closure of A is denoted by cl(A).

7.3

Background Concepts in Analysis

In the context of the real numbers R, Lebesgue measure µ is a map from a set (the Lebesgue measurable sets) of subsets of R to the extended non-negative reals [0, ∞]. It is an extension of the classical notation of length of an interval in R to more complicated sets. Whilst there

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exist subsets of R which are not measurable, the Lebesgue measurable sets include, for example, all open and closed sets and all sets obtained from these by taking countable unions and intersections. Lebesgue measure has a number of intuitively appealing properties. For example, (i) if A, B ⊂ R are measurable sets with A ⊂ B, then µ (A) ≤ µ (B), (ii) if A = ∪n∈N Bn is a disjoint union of countably many measurable sets, then µ (A) = ∑n∈N µ (Bn ), (iii) µ is translation invariant, that is, if A ⊂ R is measurable, then its translation by b ∈ R, given by B := {a + B : a ∈ A} is measurable and µ (A) = µ (B). The notion of a set of zero (Lebesgue) measure has a simple characterization: µ (A) = 0 if, for each ε > 0, there exists a countable collection of intervals In of length |In | such that

∑ |In | < ε

and

n∈N

A ⊂ ∪n∈N In .

If A is a closed and bounded interval [a, b], then its Lebesgue measure is the length of the interval µ (A) = b − a and, since each of the sets {a},{b} and {a, b} has measure zero, each of the intervals (a, b], [a, b) and (a, b) also has measure b − a.

Let I ⊂ R be an interval and X ⊂ RN an non-empty set. Two functions x, y : I → X are said to be equal almost everywhere (a.e.) if the subset (of I) on which they differ has zero measure, precisely, if {t ∈ I : x(t) 6= y(t)} is a set of zero measure. Let x : I → X and consider a sequence (xn ) of functions I → X. The sequence (xn ) is said to converge almost everywhere (a.e.) to x if the subset of points t ∈ I at which (xn (t)) fails to converge to x(t) has measure zero. A function x : I → RN is said to be a measurable function if there exists a sequence (xn ) of piecewise constant functions I → X converging almost everywhere to x. We remark that the composition f ◦x of a semicontinuous (upper or lower) function f and a measurable function x is a measurable function. A measurable function x : I → X is said to be essentially bounded if there exists K such that kx(t)k ≤ K for almost every (a.e.) t ∈ I (equivalently, the set of points t ∈ I at which kx(t)k > K has measure zero): the set of such functions x is denoted by L∞ (I; X). A measurable function x : I → X is said to be locally essentially bounded if the restriction of x to every compact (that is, closed and bounded) subset ∞ (I; X). The (Lebesgue) of I is essentially bounded: the set of such functions is denoted by Lloc integral Z x(t)dt

I

of a measurable function x : I → X may be defined via the limit of integrals of a suitably chosen sequence of piecewise constant approximants of x. The function x is said to be integrable if Z

I

kx(t)kdt < ∞.

The set of such integrable functions x is denoted1 by L1 (I; X). The function x is said to be locally integrable if the restriction x|J of x to every compact subinterval J of I is integrable R 1 (I; RN ). A function y (that is, J kx(t)kdt < ∞): the set of such functions x is denoted by Lloc that is the indefinite integral of a locally integrable function x is said to be locally absolutely continuous, that is, a function of the form 1

More generally, for 1 ≤ p < ∞, the set of measurable functions x : I → X with the property that Z kx(t)k p dt < ∞ I

is denoted by L p (I; RN ).

Stability and Asymptotic Behaviour of Nonlinear Systems

t 7→ y(t) =

Z t

199

x(s)ds

c

1 (I; X); moreover, y is differentiable almost everywhere (a.e.), that is, for some c ∈ I and x ∈ Lloc the set of points t ∈ I at which the derivative y(t) ˙ fails to exist has measure zero; furthermore y(t) ˙ = x(t) for almost all t ∈ I. Thus, one may identify locally absolutely continuous functions as those functions for which the Fundamental Theorem of Calculus holds in the context of Lebesgue integration.

7.4

Initial-Value Problems: Existence of Solutions

We proceed to develop a theory of existence of solutions of the initial-value problems (7.4) and (7.6): for clarity of exposition, we will assume t0 = 0 in each and, in the context of the latter problem, we restrict attention to the case of an autonomous differential inclusion x˙ ∈ F(x).

7.4.1

Ordinary Differential Equations

Consider the initial-value problem (7.4) (with t0 = 0), viz. x(0) = x0 ∈ RN .

x(t) ˙ = f (t, x(t)),

(7.7)

The ensuing results pertaining to problem (7.7) can be found in standard texts (see, for example, [1, 8, 28, 30]). In order to make sense of the notion solution of (7.7), it is necessary to impose some regularity on the function f : R+ × RN → RN . A classical result is: if f is continuous, then (7.7) has at least one solution, that is, a continuously differentiable function x : I → RN , on some non-trivial interval I containing 0, with x(0) = x0 and satisfying the differential equation in (7.7) for all t ∈ I. However, to insist on continuity of f in its t dependence is difficult to justify (for example, the t-dependence of the function f may arise from modelling extraneous disturbances impinging on a dynamical system - there is no reason to suppose that such disturbances are continuous). What can we say about existence of solutions in such cases: indeed, how do we even define the concept of solution? Given that we have decided against imposing, on f , continuity with respect to its first argument, we have to contend with the possibility of “solutions” of (7.7) which fail to be continuously differentiable. As a first attempt at arriving at a sensible notion of solution, consider the integrated version of (7.7): Z x(t) = x0 +

t

0

f (s, x(s))ds.

(7.8)

We might now consider a solution of (7.7) to be a function x : [0, ω ) → RN such that (7.8)) holds for all t ∈ [0, ω ). For this definition to have substance, the integral on the righthand side must make sense. As outlined in the previous section, the integral does indeed make sense (as a Lebesgue integral) if the integrand s 7→ f (s, x(s)) is a locally integrable function which implies, in particular, that its indefinite integral is locally absolutely continuous. Therefore, we define a (forward) solution of (7.7) to be a locally absolutely continuous function x : [0, ω ) → RN , 0 < ω ≤ ∞, such that (7.8) holds (or, equivalently, such that x(0) = x0 and the differential equation in (7.7) is satisfied for almost all t ∈ [0, ω )). Consequently, a basic requirement on f is sufficient regularity to ensure that, if x(·) is locally absolutely continuous,

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then s 7→ f (s, x(s)) is locally integrable. The following hypotheses (usually referred to as the Carath´eodory conditions are sufficient for this to hold:  for each fixed ξ , f (·, ξ ) is a measurable function;    for each fixed t, f (t, ·) is a continuous function; (H): for each compact K ⊂ RN , there exists a locally integrable function    m such that k f (t, ξ )k ≤ m(t) for all (t, ξ ) ∈ R × K.

A solution x : [0, ω ) → RN of (7.7) is said to be maximal, and [0, ω ) is said to be a maximal interval of existence, if x does not have a right extension that is also a solution. We are now in a position to state the fundamental existence result for the initial-value problem (7.7).

Theorem 7.1. Let f satisfy hypothesis (H). Then, for each x0 ∈ RN , (7.7) has a solution and every solution can be extended to a maximal solution. If x : [0, ω ) → RN is a maximal solution and ω < ∞, then, for every τ ∈ [0, ω ) and every compact set K ⊂ RN , there exists σ ∈ [τ , ω ) such that x(σ ) 6∈ K. This theorem asserts that, under hypothesis (H) and for each x0 , the initial-value problem (7.7) has at least one solution (there may be multiple solutions) and every solution can be maximally extended. By imposing further regularity on f , we can infer the existence of precisely one maximal solution (the uniqueness property). Additional regularity sufficient for uniqueness is the following local Lipschitz condition: (L) :



1 (R ; R ) such that for each compact K ⊂ RN , there exists λ ∈ Lloc + + k f (t, ξ ) − f (t, ζ )k ≤ λ (t)kξ − ζ k for all t ∈ R+ and all ξ , ζ ∈ K.

Theorem 7.2. Let f satisfy hypotheses (H) and (L). Then, for each x0 ∈ RN , (7.7) has a unique maximal solution. Next, we consider the autonomous counterpart of (7.7), viz. x(t) ˙ = f (x(t)),

x(0) = x0 ∈ RN ,

(7.9)

where f : RN → RN is locally Lipschitz: (La) :



for each compact K ⊂ RN , there exists λ > 0 such that k f (ξ ) − f (ζ )k ≤ λ kξ − ζ k for all ξ , ζ ∈ K.

In this autonomous setting, we are interested in solutions of (7.9) in both forwards and backwards time: thus, we deem a continuously differentiable function x : (α , ω ) → Rn to be a solution if 0 ∈ (α , ω ), x(0) = x0 and the differential equation in (7.9) holds for all t ∈ (α , ω ). A solution is maximal if it has no proper left or right extension that is also a solution. Theorem 7.3. Let f satisfy hypothesis (La). Then, for each x0 ∈ RN , the autonomous system (7.9) has unique maximal solution x : (α , ω ) → RN . This theorem implies the existence of a map (t, x0 ) 7→ ϕ (t, x0 ) defined by the property that, for each x0 , ϕ (·, x0 ) is the unique maximal solution x of (7.9). The domain of ϕ is given by D = dom(ϕ ) = {(t, x0 ) ∈ R × RN : t ∈ I(x0 )}

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where I(x0 ) = (α , ω ) denotes the maximal interval of existence of the maximal solution of (7.9). We refer to ϕ as the flow generated by f .

Proposition 7.1. D is an open set and ϕ is continuous. A set S ⊂ RN is said to be ϕ -invariant or invariant under the flow ϕ if, for every x0 ∈ S, the unique maximal solution of (7.9) has trajectory in S.

7.4.2

Autonomous Differential Inclusions

We now turn attention to the autonomous counterpart of the initial-value problem (7.6), viz. x(t) ˙ ∈ F(x(t)),

x(0) = x0 ∈ RN ,

(7.10)

where ξ 7→ F(ξ ) ⊂ RN is a set-valued map defined on RN . There is growing literature (see, for example, [2], [6], [7], [10], [12], [27]) pertaining to the study of differential inclusions. By a (forward) solution of (7.10), we mean a locally absolutely continuous function x : [0, ω ) → RN , 0 < ω ≤ ∞, with x(0) = x0 , such that the differential inclusion in (7.10) is satisfied for almost all t ∈ [0, ω ). A solution x : [0, ω ) → RN is maximal, and [0, ω ) is a maximal interval of existence, if x has no right extension that is also a solution. We proceed to address the issue of identifying regularity conditions on F sufficient to guarantee the existence of at least one solution of (7.10). To this end, let U denote the class of set-valued maps ξ 7→ F(ξ ) ⊂ RN , defined on RN , that (a) take nonempty convex compact values (that is, for each ξ ∈ RN , F(ξ ) is a non-empty, convex and compact subset of RN ) and (b) are upper semicontinuous at each ξ ∈ RN . A set-valued map F is upper semicontinuous at ξ ∈ RN if, for each ε > 0, there exists δ > 0 such that F(ξ ′ ) ⊂ Bε (F(ξ )) for all ξ ′ in Bδ (ξ ), as illustrated below.

ξ′

b

ξ

b

F

F(ξ ′ ) F(ξ )

Bδ (ξ ) Bε (F(ξ )) Fig. 7.3. Upper semi-continuity of set-valued map Theorem 7.4. Let F ∈ U . For each x0 ∈ RN , (7.10) has a solution and every solution can be extended to a maximal solution. If x : [0, ω ) → RN is a maximal solution with ω < ∞, then x is unbounded. A set S ⊂ RN is said to be weakly invariant with respect to the differential inclusion in (7.10) if, for each x0 ∈ S, there exists at least one maximal solution of (7.10) with trajectory in S.

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7.4.3 ω -L imit Sets In his well-known book [4, pp. 197], Birkhoff introduced the notion of an ω -limit point in the context of trajectories of dynamical systems. For the purposes of this paper, it is useful to define the concept of an ω -limit point for arbitrary RN -valued functions defined on R+ . Let x : R + → RN . A point ξ ∈ RN is an ω -limit point of x if there exists an unbounded sequence (tn ) ⊂ R + such that x(tn ) → ξ as n → ∞; the (possibly empty) ω -limit set of x, denoted by Ω (x), is the set of all ω -limit points of x. The following two lemmas highlight well-known properties of ω -limit sets (see, for example, [1], [12], [14], and [30]). Lemma 7.1. The following hold for any function x : R + → RN : (a) Ω (x) is closed. (b) Ω (x) = 0/ if and only if kx(t)k → ∞ as t → ∞. (c) If x is continuous and bounded, then Ω (x) is nonempty, compact, and connected, is approached by x, and is the smallest closed set approached by x. (d) If x is continuous and Ω (x) is nonempty and bounded, then x is bounded and x approaches Ω (x). If x happens to be a maximal solution of (7.9) or of (7.10), then we can say more. Lemma 7.2 (a) Let x : R+ → RN be a bounded solution of (7.9). Then Ω (x) is nonempty, compact, connected, is approached by x, is the smallest closed set approached by x, and is invariant under the flow ϕ generated by f . (b) Let x : R+ → RN be a bounded solution of (7.10). Then Ω (x) is nonempty, compact, connected, is approached by x, is the smallest closed set approached by x, and is weakly invariant with respect to the differential inclusion in (7.10).

7.5

Barbalat’s Lemma, LaSalle’s Invariance Principle, and ˘ Lyapunov Stability

A function y : R+ → R is Riemann integrable (on R+ ) if the improper Riemann integral exists, that is, y is Riemann integrable on [0,t] for each t ≥ 0 and the limit 0 y(s)ds R limt→∞ 0t y(s)ds exists and is finite. If y belongs to L1 and is Riemann integrable on [0,t] for each t ≥ 0, then y is Riemann integrable on R+ . First, we highlight the following simple observation, due to Barb˘alat [3]. R∞

Lemma 7.3 (Barb˘alat’s lemma). If y : R + → R is uniformly continuous and Riemann integrable, then y(t) → 0 as t → ∞. Proof. Suppose to the contrary that y(t) 6→ 0 as t → ∞. Then there exist ε > 0 and a sequence (tn ) in R + such that tn+1 − tn > 1 and |y(tn )| ≥ ε for all n in N. By the uniform continuity of y, there exists δ in (0, 1) such that, for all n in N and all t in R + , |tn − t| ≤ δ

=⇒

|y(tn ) − y(t)| ≤ ε /2 .

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Therefore, for all t in [tn ,tn + δ ] and all n in N, |y(t)| ≥ |y(tn )| − |y(tn ) − y(t)| ≥ ε /2, from which it follows that Z Z tn +δ tn +δ εδ = y(t)dt |y(t)|dt ≥ t 2 t n

n

for each n in N, contradicting the existence of the improper Riemann integral

R∞ 0

y(t)dt.

⊓ ⊔

Lemma 7.3 was originally derived in [3] to facilitate the analysis of the asymptotic behaviour of a class of systems of nonlinear second-order equations with forcing. Subsequently, Barb˘alat’s lemma has been widely used in mathematical control theory (see, for example, [9, p. 89], [23, p. 211], and [26, p. 205]). The following corollary is an immediate consequence of statement (c) of Lemma 7.1 and Lemma 7.3. Corollary 7.1. Let G be a nonempty closed subset of RN , and let g : G → R be continuous. Assume that x : R + → RN is bounded and uniformly continuous with x(R+ ) ⊂ G. If g ◦ x is Riemann integrable, then Ω (x) ⊂ g−1 (0) and x approaches g−1 (0). We will use Corollary 7.1 to derive LaSalle’s invariance principle. Let the vector field f : RN → RN be locally Lipschitz and consider the initial-value problem x˙ = f (x) ,

x(0) = x0 ∈ RN .

(7.11)

Let ϕ denote the flow generated by f , and so t 7→ ϕ (t, x0 ) is the unique solution of (7.11) defined on its maximal interval of existence I(x0 ). If R+ ⊂ I(x0 ) and ϕ (· , x0 ) is bounded on R+ , then, by assertion (a) of Lemma 7.2, Ω (ϕ (· , x0 )) is invariant with respect to the flow ϕ . The following integral-invariance principle provides an intermediate step towards LaSalle’s principle and is a consequence of Corollary 7.1. Theorem 7.5 (Integral-invariance principle). Let G be a nonempty closed subset of RN , let g : G → R be continuous, and let x0 be a point of G. Assume that R+ ⊂ I(x0 ), ϕ (· , x0 ) is bounded on R+ , and ϕ (R+ , x0 ) ⊂ G. If the function t 7→ g(ϕ (t, x0 )) is Riemann integrable on R+ , then ϕ (· , x0 ) approaches the largest invariant subset contained in g−1 (0). Proof. Since ϕ (· , x0 ) is bounded on R+ and satisfies the differential equation, it follows that the derivative of ϕ (· , x0 ) is bounded on R+ . Consequently, ϕ (· , x0 ) is uniformly continuous on R+ . An application of Corollary 7.1 together with the invariance property of Ω (ϕ (· , x0 )) establishes the claim. ⊓ ⊔. Before proceeding to derive LaSalle’s principle, we briefly digress to an example which illustrates that the above integral-invariance principle is of independent interest. Example 7.1. Theorem 7.5 is particularly useful in the context of observed systems. In applications, it is frequently impossible to observe or measure the complete state x(t) of (7.11) at time t. To illustrate the latter comment, consider the observed system given by (7.11) and the observation z = c(x) , (7.12) where c : RN → RP is continuous with c(0) = 0. The observation z (also called output or measurement) depends on the state and should be thought of as a quantity that can be observed

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x˙ = f (x), x(0) = x0

x

c

z = c(x)

Fig. 7.4. Observed system

or measured: an important special case occurring when z is given by one component of the state. Observability concepts relate to the issue of precluding the possibility that different initial states generate the same observation: the initial state of an observable system can, in principle, be recovered from the observation. The system given by (7.11) and (7.12) is said to be zero-state observable if the following holds for each x0 in RN : z(·) = c(ϕ (· , x0 )) = 0

=⇒

ϕ (· , x0 ) = 0 ,

that is, the system is zero-state observable if x(·) = 0 is the only solution generating the zero observation z(·) = 0. The following corollary of Theorem 7.5 is contained in [5, Theorem 1.3] and essentially states that, for a zero-state observable system, every bounded trajectory with observation in L p necessarily converges to zero. Corollary 7.2. Assume that the observed system given by (7.11) and (7.12) is zero-state observable. For given x0 in RN assume that R+ ⊂ I(x0 ) and that ϕ (· , x0 ) is bounded on R+ . If R∞ 0 p 0 0 kc(ϕ (t, x ))k dt < ∞ for some p in (0, ∞), then limt→∞ ϕ (t, x ) = 0.

Proof. By the continuity and boundedness of ϕ (· , x0 ), it follows from Lemma 7.1 that ϕ (· , x0 ) approaches its ω -limit set Ω := Ω (ϕ (· , x0 )) and that Ω is the smallest closed set approached by ϕ (· , x0 ). An application of Proposition 7.5 with G = RN and g(·) = kc(·)k p shows that Ω ⊂ g−1 (0) = c−1 (0). Let ξ be a point of Ω . By the invariance property of Ω , ϕ (t, ξ ) lies in Ω for all t in R. Consequently, c(ϕ (· , ξ )) = 0. Zero-state observability ensures that ϕ (· , ξ ) = 0, showing that ξ = 0. Hence Ω = {0}, so limt→∞ ϕ (t, x0 ) = 0. ⊓ ⊔ Theorem 7.5 is essentially contained in [5, Theorem 1.2]: the proof given therein is not based on Barb˘alat’s lemma. The above proof of Theorem 7.5 is from [11]. LaSalle’s invariance principle (announced in [15], with proof in [16]) is now a straightforward consequence of Theorem 7.5. For a continuously differentiable function V : D ⊂ RN → R (where D is open), it is convenient to define the directional derivative V f : D → R of V in the direction of the vector field f by V f (ξ ) = h∇V (ξ ), f (ξ )i. Corollary 7.3 (LaSalle’s invariance principle). Let D be a nonempty open subset of RN , let V : D → R be continuously differentiable, and let x0 be a point of D. Assume that R+ ⊂ I(x0 ) and that there exists a compact subset G of RN such that ϕ (R+ , x0 ) ⊂ G ⊂ D. If V f (ξ ) ≤ 0 for all ξ in G, then ϕ (· , x0 ) approaches the largest invariant subset contained in V f−1 (0) ∩ G. Proof. By the compactness of G and the continuity of V on G, the function V is bounded on G. Combining this with Z t 0

V f (ϕ (s, x0 ))ds =

Z t 0

R

(d/ds)V (ϕ (s, x0 ))ds = V (ϕ (t, x0 )) −V (x0 ) ,

we conclude that the function t 7→ 0t V f (ϕ (s, x0 ))ds is bounded from below: but this function is also nonincreasing (because V f ≤ 0 on G) and hence must converge to a finite limit as

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t → ∞. Therefore, the function t 7→ V f (ϕ (t, x0 )) is Riemann integrable on R+ . An application of Theorem 7.5 (with g = V f |G ) completes the proof. ⊓ ⊔ Assume that f (0) = 0, that is, 0 is an equilibrium of (7.11). The equilibrium 0 is said to be stable if for every ε > 0 there exists δ > 0 such that if kx0 k ≤ δ , then R+ ⊂ I(x0 ) and kϕ (t, x0 )k ≤ ε for all t in R+ . The equilibrium 0 is said to be asymptotically stable if it is stable and there exists δ > 0 such that kϕ (t, x0 )k → 0 as t → ∞ for every x0 satisfying kx0 k ≤ δ . Theorem 7.6 (Lyapunov’s stability theorem). Let D be a nonempty open subset of RN such that 0 ∈ D, and let V : D → R be continuously differentiable with V (0) = 0. If V (ξ ) > 0 for all ξ in D \ {0} and V f (ξ ) ≤ 0 for all ξ in D, then 0 is a stable equilibrium. Proof. Let ε > 0 be arbitrary. Without loss of generality, we may assume that the closed ball ¯ ε (0) is contained in D. Since the sphere Sε := {ξ ∈ RN : kξ k = ε } is compact and V is B continuous and positive-valued on Sε , we see that V achieves a minimum value m > 0 on Sε , that is, V (ξ ) ≥ m for all ξ ∈ Sε and V (ξ ) = m for some ξ ∈ Sε .

By continuity of the non-negative valued function V and since V (0) = 0, there exists δ ∈ (0, ε ), such that kξ k < δ =⇒ V (ξ ) < m. Let x0 be such that kx0 k < δ . Let x(·) = ϕ (·, x0 ) be the maximal solution of (7.11) with maximal interval of existence I(x0 ) = (α , ω ). Seeking a contradiction, suppose x(τ ) ∈ Sε for some τ ∈ (0, ω ) and assume τ is the first such time (and so kx(t)k < ε for all t ∈ [0, τ )). Then, d V (x(t)) = V f (x(t)) ≤ 0 dt

∀ t ∈ [0, τ ].

Therefore, t 7→ V (x(t)) is non-increasing on [0, τ ], whence the contradiction m ≤ V (x(τ )) ≤ V (x(0)) = V (x0 ) < m. Therefore, the positive trajectory x([0, ω )) is contained in the closed ball of radius ε centred ⊓ ⊔ at the origin in RN (and so ω = ∞). This completes the proof. Combining Corollary 7.3 and Theorem 7.6, we immediately obtain the following asymptotic stability theorem. Theorem 7.7 (Asymptotic stability theorem). Let D be a nonempty open subset of RN such that 0 ∈ D, and let V : D → R be continuously differentiable with V (0) = 0. If V (ξ ) > 0 for all ξ in D \ {0}, V f (ξ ) ≤ 0 for all ξ in D, and {0} is the largest invariant subset of V f−1 (0), then 0 is an asymptotically stable equilibrium. Example 7.2. In this example, which can also be found in [30], we describe a typical application of Theorem 7.7 in the context of a general class of nonlinear second-order systems. Consider the system y(t) ¨ + r(y(t), y(t)) ˙ = 0,

(y(0), y(0)) ˙ = (p0 , v0 ) ∈ R2 ,

(7.13)

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where r : R2 → R is locally Lipschitz and differentiable with respect to its second argument. Furthermore, we assume that r(0, 0) = 0. Setting x(t) = (x1 (t), x2 (t)) = (y(t), y(t)), ˙ the second-order system (7.13) can be expressed in the equivalent form (7.11), where f : R + × R2 → R2 and x0 ∈ R2 are given by f (p, v) = (v, −r(p, v)) ,

x0 = (p0 , v0 ) .

(7.14)

Let ε > 0, set D = (−ε , ε ) × (−ε , ε ), and define V : D → R,

(p, v) 7→

Z p 0

r(s, 0)ds + v2 /2.

It follows from the mean-value theorem that, for each (p, v) in D, there exists a number θ = θ (p, v) in the interval (0, 1) such that V f (p, v) = −v(r(p, v) − r(p, 0)) = −v2

∂r (p, θ v) . ∂v

(7.15)

C LAIM . Consider (7.11) with f and x0 given by (7.14). If pr(p, 0) > 0 for all p in (−ε , ε ) \ {0} and (∂ r/∂ v)(p, v) > 0 for all (p, v) in D satisfying pv 6= 0, then the equilibrium 0 is asymptotically stable. We proceed to establish this claim. Using the hypotheses and (7.15), we infer that V (p, v)>0 for all (p, v) in D \ {0} and V f (p, v) ≤ 0 for all (p, v) in D. Writing ϕ (t, x0 ) = (x1 (t), x2 (t)), we see that for x0 = (p0 , 0) in D with p0 6= 0, x˙2 (0) = −r(p0 , 0) 6= 0. Similarly, for x0 = (0, v0 ) in D with v0 6= 0, x˙1 (0) = v0 6= 0. We conclude that solutions with these initial conditions do not remain in V f−1 (0), showing that {0} is the largest invariant subset of V f−1 (0). The claim now follows from Theorem 7.7. As a special case of (7.13), consider the nonlinear oscillator usually referred to as the Li´enard equation

y(t) ¨ + d(y(t))y(t) ˙ + k(y(t)) = 0 ,

(y(0), y(0)) ˙ = (p0 , v0 ) ∈ R2 ,

where d(y)y˙ represents a friction term that is linear in the velocity and k(y) models a restoring force. We assume that the functions d : R → R and k : R → R are locally Lipschitz and k(0) = 0. It follows from the foregoing discussion on the stability of (7.13) (with r now given by r(p, v) = d(p)v + k(p)) that 0 is an asymptotically stable equilibrium state of the Li´enard equation, provided that there exists ε > 0 such that pk(p) > 0 and d(p) > 0 for all p in (−ε , ε ) with p 6= 0.

7.6

Generalizations of Barbalat’s Lemma ˘

In Theorems 7.8 and 7.9 below, we present generalizations of Barb˘alat’s lemma and of Corollary 7.1, which will be exploited in subsequent analyses of the behaviour of solutions of non-autonomous differential equations and autonomous differential inclusions. To this end, we introduce the notion of (weak) meagreness that will replace the assumption of Riemann integrability in Barb˘alat’s lemma. The concept of meagreness is defined via the Lebesgue measure µ . However, for the purposes of this tutorial paper, we do not wish to assume familiarity with measure theoretic concepts. As an alternative, we also introduce the notion of weak meagreness (which does not require measure theory).

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Definition 7.1 (a) A function y : R + → R is said to be meagre if y is Lebesgue measurable and µ ({t ∈ R + : |y(t)| ≥ λ }) < ∞ for all λ > 0. (b) A function y : R + → R is said to be weakly meagre if

lim ( inf |y(t)|) = 0

n→∞ t∈In

for every family {In : n ∈ N} of nonempty and pairwise disjoint closed intervals In in R + with infn∈N |In | > 0, where |In | denotes the length of the interval In . From Definition 7.1 it follows immediately that a meagre function is weakly meagre. The converse is not true, even in the restricted context of continuous functions. We remark that, if a function y : R+ → R is weakly meagre, then 0 belongs to Ω (y). The property of (weak) meagreness of a function implies that the function is “close to zero” in some sense: however, it is not the case that (weakly) meagre functions converge to zero as t → ∞. Indeed, there exist continuous and unbounded functions that are (weakly) meagre: the following is an example of one such function. Example 7.3. A continuous unbounded meagre function Consider the continuous unbounded function R + → R + , t 7→ x(t) =

∑ xn (t) n∈N

where, for each n ∈ N, xn is continuous and supported on In := [n, n + 1/n2 ], with graph as shown below. For every λ > 0, the total “length” (measure) of the set {t ∈ R + : |x(t)| ≥ λ } xn (t) n 0

b

b

n

n + 1/n2

t

Fig. 7.5. A continuous unbounded meagre function cannot exceed the sum of the lengths |In | = 1/n2 of the intervals In = [n , n + (1/n2 )], whence,
µ {t ∈ R + : |x(t)| ≥ λ } ≤

1 < ∞ ∀ λ > 0. 2 n∈N n

∑

and so x is meagre (and, a fortiori, weakly meagre). The above definitions of (weak) meagreness are somewhat obscure: the following result gives more tangible sufficient conditions for meagreness and weak meagreness, respectively. Proposition 7.2. Let y : R+ → R be measurable. Then the following statements hold:

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(a) If there exists a lower semicontinuous function α : R+ → R such that α −1 (0) = {0}, infs≥σ α (s) > 0 for all σ > 0, and α (|y(·)|) belongs to L1 , then y is meagre. R (b) If there exists τ > 0 such that limt→∞ tt+τ |y(s)|ds = 0, then y is weakly meagre. R (c) If y is continuous and for every δ > 0 there exists τ in (0, δ ) such that tt+τ y(s)ds converges to 0 as t → ∞, then y is weakly meagre. Proof. We prove only part (c) (the proofs of parts (a) and (b) are even more straightforward). Let y : R+ → R be continuous. We show that ifR y is not weakly meagre, then there exists δ > 0 such that for every τ in (0, δ ) the integral tt+τ y(s)ds does not converge to 0 as t → ∞. The claim follows then from contraposition. So assume that y is not weakly meagre. Then there exists a family {In : n ∈ N} of nonempty, pairwise disjoint closed intervals with δ = infn∈N µ (In ) > 0 and a number ε > 0 such that inft∈In |y(t)| ≥ ε for each n. Since y is continuous, the function y has the same sign on In for each n. Without loss of generality, we may assume that there are infinitely many intervals In on which y is positive. Then there exists a sequence (nk ) in N such that y has positive sign on Ink for all k. Denoting the left endpoint of Ink by tk , we obtain Z tk +τ tk

y(s)ds ≥ ετ > 0,

for each k in N and τ in (0, δ ), showing that the integral t → ∞.

R t+τ t

y(s)ds does not converge to 0 as ⊔ ⊓

It follows immediately from Proposition 7.2(a) that, if y belongs to L p for some p ∈ [1, ∞), then y is meagre (and, a fortiori weakly meagre). The following result will play a role in the subsequent derivation of generalized versions of Barb˘alat’s lemma. Lemma 7.4. Let A and B be nonempty subsets of RN such that cl(Bλ (B)) ⊂ A for some λ > 0. If x : R + → RN is uniformly continuous on x−1 (A), then there exists τ > 0 such that t ∈ R + , x(t) ∈ B

=⇒

x(s) ∈ Bλ (B) ∀ s ∈ [t − τ , t + τ ] ∩ R + .

(7.16)

Proof. Seeking a contradiction, suppose that property (7.16) does not hold. Then there exist sequences (sn ) and (tn ) in R + such that x(tn ) ∈ B and x(sn ) 6∈ Bλ (B) for all n, and sn −tn → 0 as n → ∞. Evidently, sn 6= tn for all n. Define In to be the closed interval with left endpoint min{sn ,tn } and right endpoint max{sn ,tn }, and write Tn = {s ∈ In : s 6∈ x−1 (Bλ (B))}. For each n, let τn in Tn (a compact set) be such that |τn − tn | = min |s − tn |. s∈Tn

Clearly, dB (x(τn )) = λ and dB (x(tn )) = 0 for each n. Combining this information with the facts that τn belongs to In and limn→∞ (sn − tn ) = 0, we conclude that (i) kx(tn ) − x(τn )k ≥ |dB (x(tn )) − dB (x(τn ))| = λ > 0,

(ii) tn , τn ∈ x−1 (A),

(iii) |tn − τn | → 0 as n → ∞, contradicting the hypothesis of the uniform continuity of x on x−1 (A). Therefore, property (7.16) holds. ⊔ ⊓

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The following two theorems, the main results of this section, provide our generalizations of Barb˘alat’s lemma. Theorem 7.8. Let G be a nonempty closed subset of RN , let g : G → R be a function, and let x : R + → RN be continuous with x(R+ ) ⊂ G. Assume that each ξ in G for which g(ξ ) 6= 0 has a neighbourhood U such that inf{|g(w)| : w ∈ G ∩U} > 0

(7.17)

and x is uniformly continuous on x−1 (U). If g ◦ x is weakly meagre, then the following statements hold: (a) (b) (c) (d)

Ω (x) is contained in g−1 (0). If g−1 (0) is bounded and Ω (x) 6= 0, / then x is bounded and x approaches g−1 (0). If x is bounded, then g−1 (0) 6= 0/ and x approaches g−1 (0). If x is bounded and g−1 (0) is totally disconnected, then Ω (x) consists of a single point x∞ which lies in g−1 (0) (in particular, limt→∞ x(t) = x∞ ).

Proof. If Ω (x) = 0, / then statement (a) holds trivially. Now assume that Ω (x) 6= 0. / Let ξ be a point of Ω (x). Since G is closed and x(R+ ) ⊂ G, Ω (x) ⊂ G and thus ξ belongs to G. We show that g(ξ ) = 0. Seeking a contradiction, suppose that g(ξ ) 6= 0. By the hypotheses, there exists a neighbourhood U of ξ such that (7.17) holds and x is uniformly continuous on x−1 (U). Choose δ > 0 such that the closure of Bδ (ξ ) lies in U. Then

ε = inf{|g(w)| : w ∈ G ∩ Bδ (ξ )} > 0 .

(7.18)

Choose δ1 in (0, δ ). Since ξ is an element of Ω (x), there exists a sequence (tn ) in R+ with tn+1 − tn > 1 and x(tn ) in Bδ1 (ξ ) for all n. An application of Lemma 7.4 (with A = U, B = Bδ1 (ξ ) and λ = δ − δ1 ) shows that there exists τ in (0, 1) such that x(t) is in Bδ2 (ξ ) for all t in ∪n∈N [tn ,tn + τ ]. Therefore, by (7.18), |(g ◦ x)(t)| ≥ ε

(t ∈ [tn ,tn + τ ], n ∈ N) .

(7.19)

Finally, since tn+1 −tn > 1 for all n and τ belongs to (0, 1), the intervals [tn ,tn + τ ] are pairwise disjoint. Combined with (7.19) this contradicts the weak meagreness of g ◦ x and establishes (a). A combination of statement (a) and Lemma 7.1 yields statements (b)-(d). ⊔ ⊓ We remark that lower semicontinuity of the function ξ 7→ |g(ξ )| is sufficient to ensure that (7.17) holds for some neighbourhood U of any ξ in G with g(ξ ) 6= 0. Barb˘alat’s lemma follows immediately from an application of Theorem7.8(b) tothesituation wherein N = 1, G = R, g = idR , and x = y, in conjunction with the observation that a uniformly continuous and Riemann integrable function y : R+ → R is weakly meagre, implying that 0 is a member of Ω (y) and thus ensuring that Ω (y) 6= 0. / Corollary 7.1 is a simple consequence of statements (a) and (c) of Theorem 7.8. When compared with Theorem 7.8, the next result (Theorem 7.9) posits that x be uniformly continuous on x−1 (Bε (g−1 (0))) for some ε > 0. We remark that, in certain situations (for

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example, if g−1 (0) is finite), this assumption is weaker than the uniform continuity assumption imposed on x in Theorem 7.8. On the other hand, the assumption imposed on g in Theorem 7.9 is stronger than that in its counterpart in Theorem 7.8. However, under these modified hypotheses, Theorem 7.9 guarantees that x approaches g−1 (0) 6= 0/ without assuming the nonemptiness of Ω (x) or the boundedness of x. Theorem 7.9. Let G be a nonempty closed subset of RN , and let g : G → R be such that g−1 (0) is closed and, for every nonempty closed subset K of G, K ∩ g−1 (0) = 0/

=⇒

inf |g(ξ )| > 0 .

ξ ∈K

(7.20)

Furthermore, let x : R + → RN be continuous with x(R+ ) ⊂ G. If (i) x is uniformly continuous on x−1 (Bε (g−1 (0))) for some ε > 0 and (ii) g ◦ x is weakly meagre, then the following statements hold: (a) g−1 (0) 6= 0, / x approaches g−1 (0), and Ω (x) is contained in g−1 (0). (b) If g−1 (0) is bounded, then x is bounded, x approaches g−1 (0), and Ω (x) is a nonempty subset of g−1 (0). (c) If g−1 (0) is bounded and totally disconnected, then Ω (x) is a singleton {x∞ }, where x∞ is a point of g−1 (0) (hence, limt→∞ x(t) = x∞ ). Proof. For convenience, we set Z = g−1 (0). It is clear that Z 6= 0/ (otherwise, by (7.20) and the closedness of G, γ = infξ ∈G |g(ξ )| > 0 and so |g(x(t))| ≥ γ for all t in R + , which contradicts the weak meagreness of g ◦ x). To prove statements (a) and (b), it now suffices to show that x approaches Z. From the closedness of Z it then follows immediately that Ω (x) ⊂ Z; moreover, if Z is bounded, then we can conclude that x is bounded and so Ω (x) 6= 0. / Since, by assumption, the trajectory of x is contained in G, it is immediate that, if G = Z, then x approaches Z. Consider the remaining case, wherein Z is a proper subset of G. By the closedness of Z, there / For θ in (0, δ ), define exists δ in (0, ε /3) such that G \ Bδ (Z) 6= 0.

ι (θ ) = inf{|g(ξ )| : ξ ∈ G \ Bθ (Z)} > 0 , wherein positivity is a consequence of (7.20) and the closedness of G \ Bθ (Z). Seeking a contradiction, we suppose that limt→∞ dZ (x(t)) 6= 0. Then there exist λ in (0, δ ) and a sequence (tn ) in R+ with tn → ∞ as n → ∞ and dZ (x(tn )) ≥ 3λ for all n. By the weak meagreness of g ◦ x, there exists a sequence (sn ) in R+ with sn → ∞ as n → ∞ and |g(x(sn ))| < ι (λ ) for all n, so dZ (x(sn )) ≤ λ for all n. Extracting subsequences of (tn ) and (sn ) (which we do not relabel), we may assume that sn is in (tn ,tn+1 ) for all n. We now have dZ (x(tn )) ≥ 3λ ,

dZ (x(sn )) ≤ λ ,

sn ∈ (tn ,tn+1 )

for all n. By the continuity of dZ ◦ x, there exists for each n a number σn in (tn , sn ) such that x(σn ) belongs to B := {ξ ∈ G : dZ (ξ ) = 2λ }. Extracting a subsequence (which, again,
we do not relabel), we may assume that σn+1 − σn > 1 for all n. Noting that cl Bλ (B) ⊂ Bε (Z) and invoking Lemma 7.4 (with A = Bε (Z)), we conclude the existence of τ in (0, 1) such that dZ (x(t)) ≥ λ for all t in [σn , σn + τ ] and all n. Therefore, {t ∈ R + : |g(x(t))| ≥ ι (λ )} ⊃ ∪n∈N [σn , σn + τ ],

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which (on noting that the intervals [σn , σn + τ ], are each of length τ > 0 and form a pairwise disjoint family) contradicts the weak meagreness of g ◦ x. Therefore, x approaches Z, implying that statements (a) and (b) hold. Finally, invoking the fact that the ω -limit set of a bounded continuous function is connected, we infer statement (c) from statement (b). ⊔ ⊓

7.7

Nonautonomous Ordinary Differential Equations

Consider again the initial-value problem (7.7) for a nonautonomous ordinary differential equation x(t) ˙ = f (t, x(t)),

x(0) = x0 ∈ RN ,

(7.21)

where f : R + ×RN → RN is a Carath´eodory function, that is, f satisfies hypothesis (H). Recall that, by Theorem 7.1, for each x0 ∈ RN , (7.21) has at least one solution and every solution can be extended to a maximal solution. With a view to highlighting a particular subclass of Carath´eodory functions f , we introduce the notion of uniform local integrability. 1 and Definition 7.2. A function m : R + → R is uniformly locally integrable if m belongs to Lloc if for each ε > 0 there exists τ > 0 such that

Z t+τ t

for all t in R + .

|m(s)|ds ≤ ε

Clearly, a locally integrable function m : R + → R is uniformly locally integrable if, and only R if, the function t 7→ 0t |m(s)|ds is uniformly continuous. It is readily verified that, if m belongs to L p for some p (1 ≤ p ≤ ∞), then m is uniformly locally integrable. We now introduce a particular subclass of Carath´eodory functions. Definition 7.3. For a nonempty subset A of RN , F(A) denotes the class of Carath´eodory functions f : R + × RN → RN with the property that there exists a uniformly locally integrable function m such that k f (t, ξ )k ≤ m(t) for all (t, ξ ) in R + × A. The next proposition shows that under suitable uniform local integrability assumptions relating to f , solutions of (7.21) satisfy the uniform continuity assumptions required for an application of Theorems 7.8 and 7.9. Proposition 7.3. Let A and B be nonempty subsets of RN with the property that Bε (A) ∩ B 6= 0/ for some ε > 0, and let f belong to F(Bε (A) ∩ B). If x : R + → RN is a global solution of (7.21) such that x(R+ ) ⊂ B, then x is uniformly continuous on x−1 (A). Proof. If x−1 (A) = 0, / then the claim holds trivially. Assume that x−1 (A) 6= 0. / Since f belongs to F(Bε (A) ∩ B), there exists a uniformly locally integrable function m such that k f (t, w)k ≤ m(t) for all (t, w) in R + × (Bε (A) ∩ B). Let δ in (0, ε ) be arbitrary. Choose τ > 0 such that R t+τ m ≤ δ for all t in R + . Let t1 and t2 be points of x−1 (A) with 0 ≤ t2 − t1 ≤ τ . We will t complete the proof by showing that kx(t2 ) − x(t1 )k ≤ δ . If we define J = {t > t1 : x(s) ∈ Bε (A) for all s ∈ [t1 ,t]}, it follows that kx(t) − x(t1 )k ≤

Z t t1

m(s)ds ≤

Z t1 +τ t1

m(s)ds ≤ δ

for all t in J with t ≤ t1 + τ . Since δ < ε , t1 + τ belongs to J, whence kx(t2 ) − x(t1 )k ≤ δ .

⊔ ⊓

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In the following, we combine Proposition 7.3 with Theorems 7.8 and 7.9 to derive results on the asymptotic behaviour of solutions of (7.21).

Theorem 7.10. Let G be a nonempty closed subset of RN , and let g : G → R be a function. Assume that each ξ in G for which g(ξ ) 6= 0 has a neighbourhood U such that (7.17) holds and f belongs to F(U ∩ G). If x : R + → RN is a global solution of (7.21) with x(R+ ) ⊂ G and g ◦ x is weakly meagre, then statements (a)-(d) of Theorem 7.8 hold. Proof. Let ξ in G be such that g(ξ ) 6= 0. By the hypotheses, there exists a neighbourhood U of ξ such that (7.17) holds and f belongs to F(U ∩ G). Let ε > 0 be sufficiently small so that B2ε (ξ ) lies in U. Then, setting A = Bε (ξ ), we see that f is in the class F(Bε (A) ∩ G). By Proposition 7.3, it follows that x is uniformly continuous on x−1 (A). An application of Theorem 7.8 completes the proof. ⊓ ⊔ We remark that Theorem 7.10 contains a recent result by Teel [29, Theorem 1] as a special case. In the next theorem, it is assumed that f is a member of F(Bε (g−1 (0)) ∩ G) for some ε > 0. Under the additional assumption that g satisfies (7.20), it is then guaranteed that x approaches g−1 (0) (without positing the boundedness of x). Theorem 7.11. Let G be a nonempty closed subset of RN , and let g : G → R be such that g−1 (0) is closed and (7.20) holds for every nonempty closed subset K of G. Assume that f belongs to F(Bε (g−1 (0)) ∩ G)) for some ε > 0. If x : R + → RN is a global solution of (7.21) with x(R+ ) ⊂ G and g ◦ x is weakly meagre, then statements (a)-(c) of Theorem 7.9 hold. Proof. Fix δ in (0, ε ). By Proposition 7.3, x is uniformly continuous on the set x−1 (Bδ (g−1 (0))). An application of Theorem 7.9 completes the proof. ⊓ ⊔ In the following we use Theorem 7.10 to obtain a version of a well-known result on ω limit sets of solutions of nonautonomous ordinary differential equations. For a nonempty open subset D of RN and a continuously differentiable function V : R+ × D → R, we define V f : R+ ×D → R (the derivative of V with respect to (7.21) in the sense that (d/dt)V (t, x(t)) = V f (t, x(t)) along a solution x of (7.21)) by V f (t, ξ ) =

N ∂V ∂V (t, ξ ) + ∑ (t, ξ ) fi (t, ξ ) ∂t ∂ i=1 ξi

for all (t, ξ ) ∈ R+ × D, where f1 , . . . , fN denote the components of f . Corollary 7.4. Let D be a nonempty open subset of RN , and let V : R+ × D → R be continuously differentiable. Assume that V satisfies the following two conditions: (a) for each ξ in cl(D) there exists a neighbourhood U of ξ such that V is bounded from below on the set R+ × (U ∩ D); (b) there exists a lower semicontinuous continuous function W : cl(D) → R+ such that V f (t, ξ ) ≤ −W (ξ ) for all (t, ξ ) in R+ × D. Furthermore, assume that for every ξ in cl(D) there exists a neighbourhood U ′ of ξ such that f belongs to F(U ′ ∩ D). Under these assumptions, if x : R+ → RN is a global solution of (7.21) with x(R+ ) ⊂ D, then Ω (x) ⊂ W −1 (0).

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Proof. If Ω (x) = 0/ there is nothing to prove, so we assume that Ω (x) 6= 0. / Since (d/dt)V (t x(t)) , = V f (t, x(t)) for all t in R+ , it follows from assumption (b) that the function t 7→ V (t, x(t)) is nonincreasing, showing that the limit l of V (t, x(t)) as t → ∞ exists, where possibly l = −∞. Let ξ ∈ Ω (x) ⊂ cl(D). Then there exists a nondecreasing unbounded sequence (tn ) in R+ such that limn→∞ x(tn ) = ξ . By assumption (a) there exists a neighbourhood U of ξ such that V is bounded from below on R+ × (U ∩ D). Now x(R+ ) ⊂ D, so there exists n0 such that x(tn ) ∈ U ∩ D whenever n ≥ n0 . Consequently, the nonincreasing sequence (V (tn , x(tn ))) is bounded from below, showing that l > −∞. Therefore 0≤

Z ∞ 0

(W ◦ x)(t)dt ≤ −

Z ∞ 0

V f (t, x(t))dt =−

Z ∞ 0

(d/dt)V (t, x(t))dt = V (0, x0 ) − l < ∞ ,

verifying that W ◦ x is in L1 , hence is weakly meagre. By assumption, for each ξ in cl(D) there exists an open neighbourhood U ′ of ξ such that f belongs to F(U ′ ∩ D), implying that f is in F(U ′ ∩ cl(D)). Therefore, an application of Theorem 7.10 with G = cl(D) and g = W establishes the claim. ⊓ ⊔ Corollary 7.4 is essentially due to LaSalle [17] (see also [14, Satz 6.2, p. 140]). However, we point out that the assumption imposed on f in Corollary 7.4 is weaker then that in [14] and [17], wherein it is required that, for every ξ in cl(D), there exists a neighbourhood U ′ of ξ such that f is bounded on the set R+ × (U ′ ∩ D). Furthermore, we impose only lower semicontinuity on the function W (in contrast to [14] and [17], wherein continuity of W is assumed). The next result is a consequence of Theorem 7.11. It shows, in particular, that under a mild assumption on f every global L p -solution of (7.21) converges to zero. Corollary 7.5. Assume that there exists ε > 0 such that f belongs to F(Bε (0)), and let x : R+ → RN be a global solution of (7.21). Then the following statements hold: (a) If kx(·)k is weakly meagre, then limt→∞ x(t) = 0. (b) If x belongs to L p for some p in (0, ∞), then limt→∞ x(t) = 0. Proof. If kx(·)k is weakly meagre, then an application of Theorem 7.11 with G = RN and g = k · k shows that limt→∞ x(t) = 0. This establishes statement (a). To prove statement (b), let x be in L p for some p in (0, ∞). Then, by Proposition 7.2(a), the function kx(·)k is meagre and hence is weakly meagre. By part (a) of the present result, limt→∞ x(t) = 0. ⊓ ⊔ Obviously, if (7.21) is autonomous (i.e., the differential equation in (7.21) has the form x(t) ˙ = f (x(t))), then the assumption that f belongs to F(Bε (0)) for some ε > 0 is trivially satisfied. Thus we may conclude that every weakly meagre global solution t 7→ x(t) of an autonomous ordinary differential equation converges to 0 as t → ∞.

7.8

Autonomous Differential Inclusions

In Section 5, we investigated the behaviour of systems within the framework of ordinary differential equations with Carath´eodory righthand sides. However, as already alluded to in the

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Introduction, there are many meaningful situations wherein this framework is inadequate for purposes of analysis of dynamic behaviour. A prototypical example is that of a mechanical system with Coulomb friction which, formally, yields a differential equation with discontinuous right-hand side. Other examples permeate control theory and applications: a canonical case is a discontinuous feedback strategy associated with an on-off or switching device. Reiterating earlier comments, such discontinuous phenomena can be handled mathematically by embedding the discontinuities in set-valued maps, giving rise to the study of differential inclusions. The next goal is to extend our investigations on ordinary differential equations in this direction. Recall that U denotes the class of set-valued maps ξ 7→ F(ξ ) ⊂ RN , defined on RN , that are upper semicontinuous at each ξ in RN and take nonempty, convex, and compact values. The object of our study is the initial-value problem (7.10) for an autonomous differential inclusion, viz. x(t) ˙ ∈ F(x(t)), x(0) = x0 ∈ RN , F ∈ U . (7.22)

Recall that, by Theorem 7.4, for each x0 ∈ RN , (7.22) has at least one solution and every solution can be extended to a maximal solution. The following proposition shows that, under suitable local boundedness assumptions on F, the solutions of (7.22) satisfy the uniform continuity assumptions required for an application of Theorems 7.8 and 7.9. For a subset A of RN and for a member F of U we denote (in a slight abuse of notation) the set ∪a∈A F(a) by F(A). Proposition 7.4. Let A and B be subsets of RN . Assume that F(Bε (A) ∩ B) is bounded for some ε > 0 and that x : R + → RN is a solution of (7.22) with x(R+ ) ⊂ B. Then x is uniformly continuous on x−1 (A). / then the assertion holds trivially. Assume that x−1 (A) 6= 0, / and let δ Proof. If x−1 (A) = 0, in (0, ε ) be arbitrary. Define θ = sup{kvk : v ∈ F(Bε (A) ∩ B)}, and let τ > 0 be sufficiently small so that τθ ≤ δ . Adopting an argument similar to that used in the proof of Proposition 7.3, it can be shown that kx(t2 ) − x(t1 )k ≤ δ for all t1 and t2 in x−1 (A) with 0 ≤ t2 − t1 ≤ τ , proving that x is uniformly continuous on x−1 (A). ⊔ ⊓ We now invoke Theorems 7.8 and 7.9 to derive counterparts of Theorems 7.10 and 7.11 for differential inclusions. Theorem 7.12. Let G be a nonempty closed subset of RN , let g : G → R have the property that each ξ in G for which g(ξ ) 6= 0 has a neighbourhood U such that (7.17) holds. If x : R + → RN is a solution of (7.22) with x(R+ ) ⊂ G and g ◦ x is weakly meagre, then statements (a) and (d) of Theorem 7.8 hold. Moreover, the following statements are true: (b′ ) If g−1 (0) is bounded and Ω (x) 6= 0, / then x is bounded and x approaches the largest subset of g−1 (0) that is weakly invariant with respect to (7.22). (c′ ) If x is bounded, then g−1 (0) 6= 0/ and x approaches the largest subset of g−1 (0) that is weakly invariant with respect to (7.22). Proof. Let ξ in G be such that g(ξ ) 6= 0. By hypothesis, there exists ε > 0 such that (7.17) holds with U = Bε (ξ ). By the upper semicontinuity of F, together with the compactness of its values, F(Bε (U) ∩ G) is bounded (see [2, Proposition 3, p. 42]. By Proposition 7.4, x is uniformly continuous on x−1 (U). Therefore, the hypotheses of Theorem 7.8 are satisfied, so statements (a)-(d) thereof hold. Combining statements (b) and (c) of Theorem 7.8 with the ⊔ ⊓ weak invariance of Ω (x) yields statements (b′ ) and (c′ ).

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215

Theorem 7.13. Let G be a nonempty closed subset of RN , let g : G → R be such that g−1 (0) is closed and (7.20) holds for every nonempty closed subset K of G. Assume that F(Bε (g−1 (0))∩ G) is bounded for some ε > 0. If x : R + → RN is a global solution of (7.21) with x(R+ ) ⊂ G and g ◦ x is weakly meagre, then statements (a) and (c) of Theorem 7.9 hold. Moreover, the following also holds: (b′ ) If g−1 (0) is bounded, then x is bounded and x approaches the largest subset of g−1 (0) that is weakly invariant with respect to (7.22). Proof. Fix δ in (0, ε ). By Proposition 7.4, x is uniformly continuous on the set x−1 (Bδ (g−1 (0))). It follows immediately from Theorem 7.9 that statements (a)-(c) thereof hold. Assuming that g−1 (0) is bounded, a combination of statements (b) of Theorem 7.9 with the weak invariance of Ω (x) yields statement (b′ ). ⊓ ⊔ If there exists a locally Lipschitz function f : RN → RN such that F(x) = { f (x)} (in this case, the differential inclusion (7.22) “collapses” to an autonomous differential equation which, for every x0 ∈ RN , has a unique solution satisfying x(0) = x0 ), then the conclusions of Theorems 7.12 and 7.13 remain true when every occurence of “weakly invariant” is replaced with “invariant”. We mention that precursors of Theorems 7.12 and 7.13 have appeared in [11] and [25]. Next, we exploit Theorem 7.13 to generalize LaSalle’s invariance principle(see Corollary 7.3) to differential inclusions. Corollary 7.6. Let D be a nonempty open subset of RN , let V : D → R be continuously differentiable, and set VF (ξ ) = maxy∈F(ξ ) h∇V (ξ ), yi for all ξ in D. Let x : R+ → RN be a solution of (7.22) and assume that there exists a compact subset G of RN such that x(R+ ) ⊂ G ⊂ D. If VF (ξ ) ≤ 0 for all ξ in G, then x approaches the largest subset of VF−1 (0) ∩ G that is weakly invariant with respect to (7.22). Proof. For later convenience, we first show that the function VF : D → R is upper semicontinuous. Let (ξn ) be a convergent sequence in D with
limit ξ in D. Define l = lim supn→∞ VF (ξn ). From (VF (ξn )) extract a subsequence VF (ξnk ) with VF (ξnk ) → l as k → ∞. For each k, let yk be a maximizer of the continuous function y 7→ h∇V (ξnk ), yi over the compact set F(ξnk ), so VF (ξnk ) = h∇V (ξnk ), yk i. Let ε > 0 be arbitrary. By upper semicontinuity of F, F(ξnk ) ⊂ Bε (F(ξ )) for all sufficiently large k. Since yk lies in F(ξnk ), F(ξ ) is compact and ε > 0 is arbitrary, we infer that (yk ) has a subsequence (which we do not relabel) converging to a point y∗ in F(ξ ). Therefore, lim sup VF (ξn ) = l = lim VF (ξnk ) = lim h∇V (ξnk ), yk i n→∞

k→∞

k→∞

= h∇V (ξ ), y∗ i ≤ VF (ξ ) , confirming that VF is upper semicontinuous. Evidently, d V (x(t)) = h∇V (x(t)), x(t)i ˙ ≤ VF (x(t)) ≤ 0 dt for almost every t in R + , which leads to

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V (x(t)) −V (x(0)) ≤

Z t 0

VF (x(s))ds ≤ 0

(7.23) R

for all t in R+ . Since x is bounded, we conclude that the function t 7→ 0t VF (x(s))ds is bounded from below. But this function is also nonincreasing (because VF ≤ 0 on G), which ensures that R limt→∞ 0t VF (x(s))ds exists and is finite. Consequently, VF ◦ x is an L1 -function, showing that VF ◦ x is weakly meagre. Since VF is upper semicontinuous and VF ≤ 0 on G, the function G → R given by ξ 7→ |VF (ξ )| is lower semicontinuous. Therefore, each ξ in G with VF (ξ ) 6= 0 has a neighbourhood U such that inf{|VF (w)| : w ∈ G ∩U} > 0. By statement (c′ ) of Theorem 7.12 (with g = VF |G ) it follows that x approaches the largest subset of VF−1 (0) ∩ G that is weakly invariant with respect to (7.22). ⊓ ⊔ In Corollary 7.6, it is assumed that the solution x is global (that is, defined on R+ ) and has trajectory in some compact subset G of D. These assumptions may be removed at the expense of strengthening the conditions on V by assuming that its sublevel sets are bounded and that VF (ξ ) ≤ 0 for all ξ in D. Corollary 7.7. Let D, V , F, and VF be as in Corollary 7.6. Assume that the sublevel sets of V are bounded and that VF (ξ ) ≤ 0 for all ξ in D. If x : [0, ωx ) → RN is a maximal solution of (7.22) such that cl(x([0, ωx ))) ⊂ D, then x is bounded, ωx = ∞, and x approaches the largest subset of VF−1 (0) that is weakly invariant with respect to (7.22). Proof. Since (d/dt)V (x(t)) = VF (x(t)) ≤ 0 for almost all t in [0, ωx ), we have the counterpart of (7.23): V (x(t)) ≤ V (x(0)) for all t in [0, ωx ). Since the sublevel sets of V are bounded, it follows that x is bounded. By assertion (b) of Lemma 7.2, ωx = ∞. An application of Corollary 7.6, with G = cl(x(R + )), completes the proof. ⊓ ⊔ Example 7.4. In this example we describe a typical application of Corollary 7.7. In part (a) of the example we analyze a general class of second-order differential inclusions; in part (b) we discuss a special case, a mechanical system subject to friction of Coulomb type. (a) Let k : R → R be continuous with the property lim

Z p

|p|→∞ 0

k = ∞.

(7.24)

Let (p, v) 7→ C(p, v) ⊂ R be upper semicontinuous with nonempty, convex, compact values and with the property that, for all (p, v) in R2 , C∗ (p, v) := max{vw : w ∈ C(p, v)} ≤ 0 .

(7.25)

Consider the system y(t) ¨ + k(y(t)) ∈ C(y(t), y(t)), ˙

(y(0), y(0)) ˙ = (p0 , v0 ) ∈ R2 .

(7.26)

Setting x(t) = (y(t), y(t)), ˙ the second-order initial-value problem (7.26) can be expressed in the equivalent form x(t) ˙ ∈ F(x(t)),

x(0) = x0 = (p0 , v0 ) ∈ R2 ,

where the set-valued map F ∈ U is given by

(7.27)

Stability and Asymptotic Behaviour of Nonlinear Systems

F(p, v) = {v} × {−k(p) + w : w ∈ C(p, v)}.

217

(7.28)

By Theorem 7.4, (7.27) has a solution and every solution can be extended to a maximal solution; moreover, every bounded maximal solution has interval of existence R + . C LAIM A. For each x0 = (p0 , v0 ) in R2 , every maximal solution x = (y, y) ˙ of (7.27) is bounded (hence, has interval of existence R + ) and approaches the largest subset E of C∗−1 (0) that is weakly invariant with respect to (7.27). To establish this claim, we define V : R2 → R by V (p, v) =

Z p 0

k(s)ds + v2 /2 .

Observe that, by property (7.24) of k, V is such that, for every sequence (ξn ) in R2 , V (ξn ) → ∞ as n → ∞ and, as a result, every sublevel set of V is bounded. Moreover, VF (p, v) = max h∇V (p, v), θ i = C∗ (p, v) ≤ 0 θ ∈F(p,v)

for all (p, v) ∈ R2 .

˙ be a maximal solution of (7.27). An Let x0 = (p0 , v0 ) be a point in R2 and let x = (y, y) application of Corollary 7.7, with D = R2 , completes the proof of Claim A. (b) As a particular example, consider a mechanical system wherein a mass is subject to a restoring force k friction force of Coulomb type: the system can be written formally as y(t) ¨ + sgn(y(t)) ˙ + k(y(t)) = 0. Again, we assume that k is continuous with property (7.24). This system may be embedded in the differential inclusion (7.26) with the set-valued map C given by   {−1}, v > 0 C(p, v) := [−1, 1], v = 0  {+1}, v < 0.

(7.29)

C LAIM B. For each x0 = (p0 , v0 ) in R2 , every maximal solution x = (y, y) ˙ of (7.27) (with F and C given by (7.28) and (7.29)) is bounded and approaches the set k−1 ([−1, 1]) × {0}. To prove this claim, we first note that in this case the function C∗ (defined in (7.25)) is given by C∗ (p, v) = −|v| ≤ 0. Therefore, C∗−1 (0) = R × {0}. By Claim A, for each x0 = (p0 , v0 ) in R2 , every maximal solution x = (y, y) ˙ of (7.27) is bounded, is defined on R + , and approaches the largest subset E of R×{0} that is weakly invariant with respect to (7.27) (equivalently, (7.26)). Clearly (0, 0) ∈ E and so E is non-empty. To conclude Claim B, it suffices to show that E ⊂ k−1 ([−1, 1])×{0} . Let (p1 , 0) ∈ E be arbitrary. By weak invariance of E, there exists a solution (z, z˙) : R + → R2 of z¨ + k(z(t)) ∈ C(z(t), z˙(t)), with (z(0), z˙(0)) = (p1 , 0), such that (z(t), z˙(t)) ∈ E ⊂ R × {0} for all t ∈ R + . Therefore, for all t ∈ R + , z(t) = p1 and z˙(t) = 0 = z¨(t). By the differential inclusion, it follows that k(p1 ) ∈ C(p1 , 0) = [−1, 1] and so p1 ∈ k−1 ([−1, 1]). This completes the proof of Claim B.

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References 1. H. Amann, Ordinary Differential Equations: An Introduction to Nonlinear Analysis, de Gruyter, Berlin, 1990 2. J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. 3. I. Barb˘alat, Syst`emes d’´equations diff´erentielles d’oscillations non lin´eaires, Revue de Math´ematiques Pures et Appliqu´ees IV (1959) 267-270. 4. G. D. Birkhoff, Dynamical Systems, American Mathematical Society, Colloquium Publications Vol 9, Providence, 1927. 5. C. I. Byrnes and C. F. Martin, An integral-invariance principle for nonlinear systems, IEEE Trans. Automatic Control AC-40 (1995) 983-994. 6. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. 7. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998. 8. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGrawHill, New York, 1955. 9. C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973. 10. K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992. 11. W. Desch, H. Logemann, E. P. Ryan, and E.D. Sontag, Meagre functions and asymptotic behaviour of dynamical systems, Nonlinear Analysis: Theory, Methods & Applications 44 (2001) 1087-1109. 12. A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer, Dordrecht, 1988. 13. A. T. Fuller, The General Problem of the Stability of Motion (A.M. Lyapunov) (translation), Int. J. Control, bf 55 (1992), 531-773. 14. H. W. Knobloch and F. Kappel, Gew¨ohnliche Differentialgleichungen, B.G. Teubner, Stuttgart, 1974. 15. J. P. LaSalle, The extent of asymptotic stability, Proc. Nat. Acad. Sci. USA 46 (1960) 363-365. 16. , Some extensions of Liapunov’s Second Method, IRE Trans. Circuit Theory CT-7 (1960) 520-527. , Stability theory for ordinary differential equations, J. Differential Equations 4 17. (1968) 57-65. 18. , The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. 19. H. Logemann and E. P. Ryan, Non-autonomous systems: asymptotic behaviour and weak invariance principles, J. Differential Equations 189 (2003) 440-460. 20. H. Logemann and E. P. Ryan, Asymptotic Behaviour of Nonlinear Systems, American Mathematical Monthly, 111 (2004), 864-889. 21. A. M. Lyapunov, Probl`eme g´en´eral de la stabilit´e du mouvement, Ann. Fac. Sci. Toulouse 9 (1907), 203-474. Reprinted in Ann. Math. Study No. 17, 1949, Princeton University Press. 22. A. M. Lyapunov, The general problem of the stability of motion (translator: A. T. Fuller), Int. J. Control 55 (1992) 531-773. 23. V. M. Popov, Hyperstability of Control Systems, Springer-Verlag, Berlin, 1973. 24. E. P. Ryan, Discontinuous feedback and universal adaptive stabilization, in Control of Uncertain Systems, (D. Hinrichsen and B. M˚artensson, eds.), Birkh¨auser, Boston, 1990, pp. 245-258.

Stability and Asymptotic Behaviour of Nonlinear Systems

25. 26. 27. 28. 29. 30.

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, An integral invariance principle for differential inclusions with application in adaptive control, SIAM J. Control & Optim. 36 (1998) 960-980. S. Sastry, Nonlinear Systems: Analysis, Stability and Control, Springer-Verlag, New York, 1999. G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, 2002. E. D. Sontag, Mathematical Control Theory, 2nd Edition, Springer, New York, 1998. A. R. Teel, Asymptotic convergence from L p stability, IEEE Trans. Auto. Control AC-44 (1999) 2169-2170. W. Walter, Ordinary Differential Equations, Springer-Verlag, New York, 1998.

8 Sliding-Mode Observers † Christopher Edwards, Sarah K. Spurgeon, Chee P. Tan, and Nitin Patel

8.1

Introduction

The focus of much of the research in the area of control systems theory during the seventies and eighties addressed the issue of robustness – i.e., designing controllers with the ability to maintain stability and performance in the presence of discrepancies between the plant and model. One nonlinear approach to robust controller design which emerged during this period is the Variable Structure Control Systems methodology. Variable Structure Control Systems evolved from the pioneering work in Russia of Emel’yanov and Barbashin in the early 1960’s. The ideas did not appear outside the Soviet Union until the mid 1970’s when a book by Itkis [20] and a survey paper by Utkin [29] were published in English. Variable structure systems concepts have subsequently been utilized in the design of robust regulators, model-reference systems, adaptive schemes, tracking systems and state observers. The ideas have successfully been applied to problems as diverse as automatic flight control, control of electrical motors, chemical processes, helicopter stability augmentation, space systems and robotics [12, 23, 28, 30]. Variable structure control systems comprise a collection of different, usually quite simple, feedback control laws and a decision rule. Depending on the status of the system, a decision rule, often termed the switching function, determines which of the control laws is ‘on-line’ at any one time. Unlike, for example, a gain-scheduling methodology, the decision rule is designed to force the system states to reach, and subsequently remain on, a pre-defined surface within the state-space. The dynamical behaviour of the system when confined to the surface is described as the ideal sliding motion. The advantages of obtaining such a motion are twofold: firstly there is a reduction in order and secondly the sliding motion is insensitive to parameter variations implicit in the input channels. The latter property of invariance towards so-called matched uncertainty makes the methodology an attractive one for designing robust controllers for uncertain systems [12, 23, 28, 30]. This chapter considers the use of these ideas for robust estimation of states (and parameters). It will provide a perspective on the development of sliding-mode observers for continuous † This

is based in part on C. Edwards, S.K. Spurgeon and C.P. Tan, ‘On the Development and Application of Sliding-mode Observers’, in ‘Variable Structure Systems: Towards the 21st Century’, X.Yu and J.X. Xu, Springer-Verlag, 2002.

M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 221-242, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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time systems – primarily those which can be well represented as linear systems subject to bounded nonlinearities/uncertainties. This encompasses a wide range of real engineering systems. Sliding-mode observers for systems modelled by nonlinear systems, and whose design methodology explicitly exploits the nonlinear system structure, have recently been considered in [2]; sliding-mode observers for discrete time systems is addressed elsewhere in this volume. Consider initially a nominal linear system x(t) ˙ = Ax(t) + Bu(t)

(8.1)

y(t) = Cx(t)

(8.2)

where A ∈ Rn×n , B ∈ Rn×m and C ∈ R p×n . Without loss of generality assume that C has full row rank; i.e., there is no redundancy amongst the measurements. The objective here is to obtain an estimate of the state x(t) by measuring the quantities y(t) and u(t). The design of a linear time invariant dynamical system to estimate x(t) from the inputs and outputs was extensively studied in the 1960’s by Luenberger. An algebraic condition on the matrix pair (A,C) – the notion of observability – was proposed as a necessary and sufficient condition for state reconstruction [22]. (For simplicity, throughout this chapter, the observability condition will be assumed to hold, although technically some of the developments only require the weaker restriction of detectability on the pair (A,C). See for example [4].) One way of viewing the approach of Luenberger is to regard the observer as a model of the system in which the discrepancy between its output, and the output of the system, is fedback through a designer specified gain. Generally speaking, in sliding-mode observers, instead of feeding back the output error between the observer and the system linearly, the output error is fedback via a discontinuous switched signal.

8.2

A Discontinuous Observer

To be specific, consider the dynamical system z˙(t) = Az(t) + Bu(t) + Gn ν

(8.3)

where Gn ∈ Rn×p is to be specified, and ν is a discontinuous injection signal depending on the output error ey (t) = Cz(t) − y(t). As is common in sliding-mode approaches, the analysis is greatly simplified if an appropriate change of coordinates is introduced. In the case of observer design, the coordinate transformation associated with the invertible matrix  T Nc Tc = (8.4) C where Nc ∈ Rn×(n−p) spans the null-space of C, is appropriate. As a result of this transformation, the triple (A, B,C) has the form       A11 A12 B1 A= B= (8.5) C = 0 Ip A21 A22 B2

where the partitions of A and B are conformable with respect to C. Define the state estimation error e(t) = z(t) − x(t) and suppose (in the new coordinate system) that the output error injection gain

Sliding-Mode Observers

Gn =



L Ip



223

(8.6)

where L ∈ R(n−p)×p is a gain matrix to be specified.

Suppose the state estimation error e(t) in the coordinates of (8.5) is partitioned as (e1 , ey ) then e˙1 (t) = A11 e1 (t) + A12 ey (t) + Lν

(8.7)

e˙y (t) = A21 e1 (t) + A22 ey (t) + ν

(8.8)

As argued in [30], if the components of the discontinuous term

ν = −ρ

ey key k

if ey 6= 0

(8.9)

then for a large enough scalar ρ an ideal sliding motion is induced in finite time on the surface S = {e ∈ Rn : ey = Ce = 0}

(8.10)

During the sliding motion ey = e˙y = 0 and equation (8.8) can be written as 0 = A21 e1 + νeq

(8.11)

where νeq represents the equivalent output error injection term necessary to maintain a sliding motion on S . This is the natural analogue of the so-called equivalent control occurring in the design of sliding-mode controllers [30]. Rearranging (8.11) and substituting back into equation (8.7) it follows that the reduced order sliding motion is governed by e˙1 (t) = (A11 − LA21 )e1 (t)

(8.12)

In order to be able to sustain a sliding motion, and for (asymptotic) state estimation error decay, the eigenvalues of (A11 − LA21 ) must be stable. However if (A,C) is observable then so is (A11 , A21 ); see [30] for example. Therefore there exists L for which (A11 − LA21 ) is stable. Consider the example       0 1 0 A= B= C= 1 1 (8.13) −1 0 1

An appropriate coordinate change to generate the canonical form in (8.5) is given by x 7→ Tc x where   1 −1 Tc = (8.14) 1 1 This results in (with abuse of notation)   0 1 A= −1 0



−1 B= 1



  C= 0 1

(8.15)

In the following the gain Gn,1 = −1 which gives (A11 − Gn,1 A21 ) = −1 and the nonlinear gain ρ = 2. The sliding-mode observer from (8.3) can then be written as z˙1 (t) = z2 (t) − u(t) + 2sign ey (t)

z˙2 (t) = −z1 (t) + u(t) − 2sign ey (t)

(8.16) (8.17)

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Output Estimation Error

0.4

.

0.2

0

−0.2

.

.

.

.

.

.

−0.4

−0.6

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5 Time

6

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Fig. 8.1. Switching function

For simplicity assume that u(t) ≡ 0. In the simulations which follow the initial condition for the observer is [ 0 0 ]T and the initial condition of the system is [ 1 0 ]T . Figure 8.1 is a plot of the output estimation error ey against time. This may also be viewed as the switching function, and as expected, it is forced to zero in finite time. Figure 8.2 shows a comparison of the states of the system and the states of the plant. It is apparent that although ey = 0 in finite time (after approximately 0.5 seconds) the state estimation error e(t) → 0 exponentially and is different from zero for several seconds. 1.5 Plant Observer

1st State

1

0.5

0

−0.5

−1

−1.5

0

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0

1

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3

4

5 Time

6

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9

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1.5

2nd State

1

0.5

0

−0.5

−1

−1.5

Fig. 8.2. A comparison of the observer and system states

Sliding-Mode Observers

225

The description thus far describes the first published1 work on sliding-mode observers by Utkin [30]. An important consequence of inducing a sliding motion is that robust state estimation can be obtained. In the same way that sliding-mode controllers exhibit complete rejection of a class of uncertainty, so will the observer just described. Suppose that the real system is governed by x(t) ¨ = − sin x(t)

(8.18)

and that the measured output corresponds to y(t) = x(t) + x(t) ˙

(8.19)

Physically this may be viewed as a pendulum system where x(t) represents angular displacement from the vertical. The system in (8.13) may be viewed as a linearization of the system above at the equilibrium point (0, 0). Using the observer given in (8.16)–(8.17), the error system is governed by e˙1 (t) = ey (t) − sin x1 (t) − ν

e˙y (t) = −e1 (t) + sin x1 (t) + ν

(8.20) (8.21)

where ν = −2 sign ey . During the sliding motion ey = e˙y = 0 and

νeq = e1 (t) − sin x1 (t)

(8.22)

Substituting into equation (8.20) the sine terms cancel, leaving e˙1 (t) = −e1 (t)

(8.23)

The sliding motion is therefore independent of the nonlinearity/uncertainty resulting from the linearization, and the observer states track the real states asymptotically. Once again, in the simulations which follow the initial condition for the observer is given by [ 0 1 ]T and the initial condition of the (nonlinear) system is [ 1 0 ]T . Figure 8.3 is a plot of the output estimation error ey against time. Again the switching function is forced to zero in finite time. Figure 8.4 shows a comparison of the states of the system and the states of the plant. Once again perfect asymptotic tracking is obtained despite the mismatch between the system and the linear model about which the observer was designed.

8.3

Observers with Linear and Discontinuous Injection

The formulation just described generally requires large values of ρ in order to ensure sliding for a broad range of initial state estimation errors – particularly if the underlying system is unstable. As discussed in [7], from a practical viewpoint, this may cause difficulties. A tradeoff is usually necessary between the requirement of a large ρ to ensure a sliding-mode occurs and its subsequent reduction to prevent excessive chattering (whilst still ensuring sliding). For 1

It is more accurate to say the first published work in English on sliding observers is [30]. Earlier published work on this theme appears in Russian; see for example the references in [8].

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Switching function

1

0.5

0

0

0.5

1

1.5

2

2.5 Time, sec

3

3.5

4

4.5

5

Fig. 8.3. Switching function 1.5

1

1st State

0.5

0

−0.5

−1 0

0.5

1

1.5

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2.5

3

3.5

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2.5 Time, sec

3

3.5

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1

2nd state

0.5

0

−0.5

−1

−1.5 0

Fig. 8.4. A comparison of the observer and system states

this reason it may be preferable to include a linear output error injection term. Rather than the formulation described in (8.3) consider the dynamical system z˙(t) = Az(t) + Bu(t) − Gl ey (t) + Gn ν

(8.24)

where Gl , Gn ∈ Rn×p and ey (t) = Cz(t) − y(t). This is effectively the observer structure of Slotine et.al. [25], where it is argued that the linear gain should be chosen to enhance the size of the so-called sliding patch i.e., the domain in the state estimation error space in which sliding occurs. With a well designed linear gain Gl this observer enjoys the same robustness

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227

prop erties. Indeed in certain situations (which will be discussed in more detail later) global state estimation error convergence properties can be proven. To demonstrate this, consider once more the nonlinear system (8.18)–(8.19) and its linearization (8.13). Again change coordinates according to (8.14) to obtain the realization in (8.15). The observer design and analysis will be performed in this coordinate system. Choose as the output error injection feedback gains     0 −1 Gl = Gn = (8.25) 2 1

It can be shown that λ (A − Gl C) = {−1, −1}. Furthermore the state estimation error system is governed by e˙1 (t) = ey (t) − sin x1 (t) − ν

e˙y (t) = −e1 (t) − 2ey (t) + sin x1 (t) + ν

(8.26) (8.27)

Consider now the quadratic form V (e1 , ey ) = e21 + 2e1 ey + 2e2y = (e1 + ey )2 + e2y

(8.28)

as a candidate Lyapunov function for the error system above. Clearly the expression in (8.28) is positive definite. Furthermore, after some algebra, it can be shown that V˙ = −2e21 − 6ey e1 − 6e2y + 2ey sin x1 + 2ey ν

≤ −2e21 − 6ey e1 − 6e2y + 2|ey || sin x1 | − 2ρ |ey |

≤ −2e21 − 6ey e1 − 6e2y − 2|ey |

≤ −2(e1 + 23 ey )2 − 23 e2y − 2|ey | ≤ 0 and therefore (global) asymptotic stability of the error system has been proven.

8.4

The Walcott and˙ Zak Observer

The problem of robust state estimation for systems with bounded matched uncertainty was ˙ [32]. They sought to design a sliding-mode observer for the first explored by Walcott & Zak system x(t) ˙ = Ax(t) + Bu(t) + B f (t, y, u)

(8.29)

y(t) = Cx(t)

(8.30)

where f : R+ × R p × Rm 7→ Rm represents lumped uncertainty or nonlinearities. The function is assumed to be unknown but bounded so that k f (t, y, u)k ≤ ρ (t, y, u)

(8.31)

where ρ (·) is known. The observer proposed in [32] has the form z˙(t) = Az(t) + Bu(t) − GCe(t) − P−1CT F T ν where the discontinuous scaled unit vector term

(8.32)

C. Edwards et al.

228

FCe(t) ν = ρ (t, y, u) kFCe(t)k

(8.33)

and e(t) = z(t) − x(t). The symmetric positive definite matrix P ∈ Rn×n and the gain matrix G are assumed to satisfy (8.34) PA0 + AT0 P < 0 where A0 := A − GC, and the structural constraint PB = (FC)T

(8.35)

for some F ∈ Rm×p . Under these circumstances the quadratic form given by V (e) = eT Pe can be shown to guarantee quadratic stability [32]. Furthermore an ideal sliding motion takes place on SF = {e ∈ Rn : FCe = 0}

in finite time. Remarks • •

If p > m then sliding on SF is not the same as sliding on Ce(t) = 0 and so the observer structure in (8.32) is different from the observer in (8.24). A system theoretic interpretation of (8.34)–(8.35) by Steinberg & Corless [27] is that the transfer function matrix G(s) = FC(sI − A0 )−1 B is strictly positive real.

8.4.1

Synthesizing the Gains

The problem of synthesizing P, G and F (and incorporating some sort of design element) is non-trivial. The original work in [32] postulated the use of symbolic manipulation tools to solve a sequence of constraints arising from ensuring that the principal minors of both P and the right hand-side of (8.34) are positive and negative respectively. In the original work, the class of systems for which the design problem has a solution is not identified. For low order systems the synthesis problem is quite tractable: the observer designed for the pendulum ˙ observer. The structural requirements (8.34)–(8.35) described earlier is in fact a Walcott & Zak ˙ were shown in [14] to be solvable if and only if of Walcott & Zak • •

rank(CB) = m any invariant zeros of (A, B,C) ∈ C−

Details of the constructive design algorithms are described in [14] and follow on from earlier work described in [9]. Under these circumstances a (semi) analytic expression for the solution to the observer design problem is given in terms of a gain matrix L ∈ R(n−p)×(p−m) and a stable matrix As22 ∈ R p×p .

8.5

A Convex Parameterization

˙ and proposes a This section considers a more general problem than that of Walcott & Zak numerically based solution methodology which exploits all the degrees of freedom available in the design.

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229

Consider the dynamical system x(t) ˙ = Ax(t) + Bu(t) + Dξ (t, x, u)

(8.36)

y(t) = Cx(t)

(8.37)

where A ∈ Rn×n , B ∈ Rn×m ,C ∈ R p×n and D ∈ Rn×q where p ≥ q. Assume C and D are full rank and the function ξ : R+ × Rn × Rm → Rq is unknown but bounded so that kξ (t, x, u)k ≤ r1 kuk + α (t, y)

(8.38)

where r1 is a known scalar and α : R+ × R p → R+ is a known function. An observer of the form (as described earlier in §8.3) z˙(t) = Az(t) + Bu(t) − Gl ey (t) + Gn ν

(8.39)

will be considered where Gl ∈ Rn×p and Gn ∈ Rn×p and ey (t) = Cz(t) − y(t). The discontinuous vector ν is defined by ( e −ρ (t, y, u)kPo D2 k key k if ey 6= 0 y ν= (8.40) 0 otherwise where Po ∈ R p×p is symmetric positive definite. The matrices Po and D2 will be defined formally later. The function ρ : R+ × R p × Rm → R+ satisfies

ρ (t, y, u) ≥ r1 kuk + α (t, y) + γ0

(8.41)

where γ0 is a positive scalar. If the state estimation error e(t) := z(t) − x(t), then it is straightforward to show from equations (8.36)–(8.37) and (8.39) that e(t) ˙ = A0 e(t) + Gn ν − Dξ (t, x, u)

(8.42)

where A0 := A − Gl C. In [14] it is argued that necessary and sufficient conditions for the existence of a stable sliding motion on S = {e ∈ Rn : ey = 0} that is independent of ξ are 1. rank (CD)=q 2. any invariant zeros of (A, D,C) lie in the left half plane ˙ observer described in §8.4.1. These of course are analogous to those for the Walcott & Zak Again a (semi) analytic expression for the gains Gl and Gn in terms of a matrix L ∈ R(n−p)×(p−q) and a stable matrix As22 ∈ R p×p can be demonstrated [14]. This parameterization however only represents a specific subclass of possible solutions. Additional degrees of freedom are available which are not exploited because the simplicity of solution is lost. The next section considers the design of the matrices Gl , Gn and Po so that a sliding motion takes place on S . Assuming that conditions 1 and 2 are satisfied, a new parameterization is given which seeks to exploit all of the design freedom which is available. A canonical form from [9] will constitute a useful starting point. If conditions 1 and 2 above are satisfied there exists a coordinate transformation T0 in which the system (A, D,C) can be written as

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

 A¯ 11 A¯ 12 A¯ =  A¯ 211 ¯  A A¯ 212 22

D¯ =



0 D¯ 2



  C¯ = 0 T

(8.43)

where the sub-matrices A¯ 11 ∈ R(n−p)×(n−p) , A¯ 211 ∈ R(p−q)×(p−n) represent a detectable pair, D¯ 2 ∈ Rq×q is nonsingular, and T ∈ R p×p is orthogonal. Further the unobservable modes of (A¯ 11 , A¯ 211 ) are the invariant zeros of (A, D,C) [12]. In order to make the partitions in (8.43) conformable it is convenient to introduce the term   0 D2 := (p−q)×q (8.44) D¯ 2 Let G¯ l and G¯ n represent the observer gain matrices in the new coordinate system and define ¯ The gain matrix G¯ l is to be determined but assume A¯ 0 = A¯ − G¯ l C.   ¯ T −1 −LT (8.45) Po G¯ n = TT   where L¯ ∈ R(n−p)×p and L¯ = L 0 with L ∈ R(n−p)×(p−q) , and the orthogonal matrix T is part of the output distribution matrix C¯ from (8.43). ¯ that satisfies P¯ A¯ 0 + Proposition 8.1. If there exists a positive definite Lyapunov matrix P, A¯ T0 P¯ < 0, with the structure   P¯ P¯ L¯ P¯ = ¯ T1¯ ¯ 1¯ T ¯ ¯ > 0 (8.46) L P1 P2 + L P1 L where P¯1 ∈ R(n−p)×(n−p) and P¯2 ∈ R p×p , then the error system in equation (8.42) is quadratically stable. Proof. Consider the quadratic form given by V (e) ¯ = e¯T P¯ e¯

(8.47)

as a candidate Lyapunov function where e¯ := T0 e. Notice that if P¯1 , P¯2 > 0 then P¯ > 0 from the Schur expansion. From (8.42) the derivative along the system trajectory V˙ = e¯T (A¯ T0 P¯ + P¯ A¯ 0 )e¯ + 2e¯T P¯ G¯ n ν − 2e¯T P¯ D¯ ξ From the definitions in (8.43), (8.45) and (8.46)   0 ¯ ¯ PGn = ¯ T Po−1 = C¯ T P2 T

(8.48)

(8.49)

if the symmetric positive definite matrix Po := T P¯2 T T ¯ 2 = 0 and therefore Using the special structures of L¯ and D2 , LD   0 P¯ D¯ = ¯ = C¯ T Po D2 P2 D2 if the matrix

(8.50)

(8.51)

Sliding-Mode Observers D2 := T D2

231 (8.52)

Consequently, (8.48) becomes V˙ = e¯T (A¯ T0 P¯ + P¯ A¯ 0 )e¯ + 2ey T ν − 2ey T Po D2 ξ ≤ e¯T (A¯ T0 P¯ + P¯ A¯ 0 )e¯ − 2ρ kPo D2 kkey k − 2ey T Po D2 ξ Using the uncertainty bounds for ξ from equations (8.38) and (8.41) V˙ ≤ e¯T (A¯ T0 P¯ + P¯ A¯ 0 )e¯ − 2ρ kPo D2 kkey k + 2kPo D2 k [r1 kuk + α (y)] key k ≤ e¯T (A¯ T0 P¯ + P¯ A¯ 0 )e¯ − 2γ0 kPo D2 kkey k Since (A¯ T0 P¯ + P¯ A¯ 0 ) < 0 it follows that V˙ < 0 for all e¯ 6= 0.

⊓ ⊔

Corollary 8.1. An ideal sliding motion takes place on S in finite time. Furthermore the sliding dynamics are given by the system matrix A¯ 11 + LA¯ 211 . Proof. Using Proposition 8.1, a modification to Corollary 6.1 in [12] shows that sliding takes place on S in finite time. Using the concept of equivalent output error injection, the sliding motion is governed by   ¯ ¯ ¯ ¯¯ ¯ A¯ 0 = A11 + LA211 A12 + LA22 (I − G¯ n (C¯ G¯ n )−1C) 0 0 Hence the sliding motion is governed by A¯ 11 + LA¯ 211 as claimed.

⊓ ⊔

Remarks: Since (A¯ 11 , A¯ 211 ) is detectable by construction, there exists a family of matrices L ∈ R(n−p)×(p−q) such that A¯ 11 + LA¯ 211 is stable. If a further linear change of co-ordinates   In−p L¯ TL = (8.53) 0 T ¯ D, ¯ and its Lyapunov matrix P, ¯ C) ¯ the system matrix, disturbance is applied to the triple (A, distribution matrix and the output distribution matrix will be in the form       A11 A12 0 A = C = 0 Ip (8.54) D= D2 A21 A22

where A11 = A¯ 11 + LA¯ 211 . In the new co-ordinate system, the Lyapunov matrix will be   P¯1 0 −1 T ¯ −1 P = (TL ) P(TL ) = (8.55) 0 Po where Po is defined in (8.50). The nonlinear output error injection gain matrix   0 Gn = Po−1

(8.56)

As argued in [9], the fact that P is block diagonal Lyapunov matrix for A0 = A − Gl C implies that A11 is stable and hence the sliding motion is stable. From a design and synthesis perspective the problem of determining Gl , Gn and P can be posed in such a way that Linear Matrix Inequalities (LMIs) [3] can be used to numerically synthesize the required matrices.

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A Case Study: Road Tyre Friction Estimation

Increasingly, commercial vehicles are being fitted with micro-processor based systems to enhance the safety, improve driving comfort, increase traffic circulation, and reduce environmental pollution associated with vehicles. Examples of such products are anti-lock brake systems (ABS), traction control systems (TCS), adaptive cruise control (ACC), active yaw control, active suspension systems, and engine management systems (EMS). Many of these systems rely on the physical parameters of the vehicle and the conditions in which it is required to operate. Some of these vehicle related parameters are fixed (or are at least subject to negligible variation); some depend on specific scenarios relating to the way in which the vehicle is being used (e.g., loading) and may be thought of as constant unknown parameters (which may be estimated or accounted for by the robustness of the control system); but others – particularly the tyre/road friction coefficient – are subject to severe and short term variations. Various methods have been developed to predict tyre/road friction [1, 6, 15, 21, 24, 33]. Most are passive, in the sense that they make use of available data from existing sensors to compute an estimate of the tyre/road friction and assume the parameters related to the vehicle and the tyres are constant. The methods from [1, 6, 15, 21, 24, 33] use observers designed around mathematical models of friction and simple vehicle models. Ray [21, 24] primarily uses a Kalman filter but [21] also examines a least squares approach. Extensive work by Canudas de Wit et. al [6] and Horowitz et. al [33] has used a LuGre friction model [5] and investigates nonlinear adaptive observers based on measurements of wheel speed. The work described here uses the same friction model as in [1, 6, 33] but employs a slidingmode observer to estimate the road/tyre friction coefficient. During sliding, an equivalent output estimation error is used to estimate the road surface parameter rather than the adaptive scheme in [33].

8.6.1

Tyre/Road Friction and Vehicle Modelling

The tyre/road coefficient of friction is defined as

µ=

Friction force Fx = Fn Normal force

(8.57)

The quantity µ is a nonlinear function of many physical variables including the velocity of the vehicle, the road surface conditions and so-called slip. Longitudinal slip, s, can be defined for a braking scenario as v − rω where v > rω and v 6= 0 (8.58) s= v where r is the effective rolling radius of the tyre, v is the linear speed of the tyre centre and ω is the angular speed of the tyre. Slip is an indirect measure of the fraction of the contact patch on the road surface and it creates braking and accelerating forces. The plot in Figure 8.5 is a typical µ -slip curve associated with a dry asphalt road surface. The figure has been obtained from using an expression known as the pseudo-steady state LuGre friction model2 [6]. When s = 0, free rolling of the wheel takes place, whilst s = 1 represents 2

Specifically equation (8.40) from [6] has been used together with the parameter values in Table 8.1 from the appendix. In [6] slip during a braking manoeuvre is defined as s = (rω − v)/v, i.e., with opposite polarity to the definition used here – accordingly s has been replaced by −s to obtain the plots in Figures 8.5-8.6.

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233

Relationship between µ and s

1

0.8

µ

0.6

0.4

0.2

0

0

0.1

0.2

0.3

0.4 0.5 0.6 longitudinal slip s

0.7

0.8

0.9

1

Fig. 8.5. A typical µ -slip curve

a locked wheel condition. It can be seen from Figure 8.5 that there is an unique value, s, ˆ associated with the maximum value of µ . During a braking manoeuvre, maintaining the slip at s = sˆ provides an optimal stopping distance. Less favourable (and potentially more dangerous) road surface conditions tend to decrease the peak value of the mu/slip curve (by up to 25% in icy conditions) and alter the value of s. ˆ In many friction models this effect is taken into account by means of a ‘road surface condition’ parameter. Figure 8.6 captures this effect by means of a parameter θ which forms part of the pseudo-steady state LuGre friction model [34]. Figure 8.6 shows curves representing the relationship between µ and s for different values of θ . This has been obtained keeping the vehicle velocity v at 13.41 m/s (30 mph) and using different values for θ . For the purpose of developing a friction estimator, consider the dynamic LuGre friction model from [6] together with the vehicle dynamics: z˙ = vr − θ σ0 |vr |z/h(vr )

(8.59)

J ω˙ = −rFx − σω ω − kb Pb

(8.61)

mv˙ = 4Fx − Fav

(8.60)

where vr = rω − v is the relative velocity and z is an internal frictional state. The friction force Fx produced by the tyre/road contact is given by Fx = Fn (σ0 z + σ1 z˙ + σ2 vr )

(8.62)

where σ0 is the stiffness coefficient, σ1 is the damping coefficient, and σ2 is the viscous relative damping coefficient. The scalar function

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0.8

θ = 1.5

µ

0.6

θ=2

θ = 2.5 0.4

θ=4 0.2

0

0

0.1

0.2

0.3

0.4 0.5 0.6 longitudinal slip s

0.7

0.8

0.9

1

Fig. 8.6. Pseudo static model at v = 13.41 m/sec and θ between 1 and 4 vr 1/2

h(vr ) := µc + (µs − µc )e−| vs |

(8.63)

where vs is the Stribeck relative velocity, µs is the normalized static friction coefficient, µc is the normalized Coulomb friction. The parameter θ in (8.59) captures the changes in the road characteristics: typically, θ =1 represents dry, θ = 2.5 represents wet and θ = 4 represents icy road conditions. In (8.61) and (8.60), which represent a simple model of the vehicle, J is the moment of inertia of the wheel, σω represents the viscous rotational friction coefficient, m is the total mass of the vehicle, Fav represents the aerodynamic force, kb the brake system gain and Pb is the actual applied braking pressure (the control variable). The values of these quantities used in this paper are given in Table 8.1. In what follows the state ω has been replaced by vr = rω − v. This clearly just represents a linear change of coordinates. Assuming the vehicle is on a flat road and the load is equally distributed on all four tyres then Fn = mg 4 . This removes the appearance of the normal force Fn from the state-space model that will be developed.

8.6.2

Observer Design

In this section a sliding-mode observer will be designed based on the assumption that only the angular velocity ω is available (which can be measured easily). The states of the system have been chosen as x = (z, v, vr ). From equations (8.59)– (8.62) it can be verified that the state-space equations can be written as x(t) ˙ = Ax(t) + Bu(t) + Dθ x1 (t) f (x3 )

(8.64)

Sliding-Mode Observers where the control signal u(t) = Pb (t) and   0 0 1 g(σ1 + σ2 )  A =  gσ0 −gσv qσ0 gσv − σJw q(σ1 + σ2 ) − σJω

where the static nonlinearity

 0 B= 0  − rkJb





 −1 D =  −gσ1  −qσ1

235

(8.65)

σ0 |x3 | (8.66) h(x3 ) Since it is assumed in this section that only angular wheel speed ω is measured, the output distribution matrix   C = 0 1r 1r (8.67) f (x3 ) :=

The following nonlinear observer is proposed

˙ˆ = Ax(t) x(t) ˆ + Bu(t) + Gl ey + Dν

(8.68)

where Gl = col(g1 , g2 , g3 ) and g1 , g2 and g3 are scalar gains; and the nonlinear injection ν = −ksgn(Fey ) where F and k are scalar gains. The main objective is to synthesize an observer to generate an estimated angular velocity ωˆ = Cxˆ such that ey = ω − ωˆ ≡ 0 in finite time despite the nonlinear friction terms in (8.59) – or equivalently (8.64) – which have been ignored. Define e = x − xˆ as the state estimation error. The dynamics for the error system can be obtained from (8.64) and (8.68) as
e˙ = (A − Gl C)e + D θ f (x3 )x1 − k sgn(Fey ) (8.69)

Suppose e = col(e1 , e2 , e3 ), then in the coordinates e1 , e2 and ey it can be shown that ¯ D, ¯ where ¯ C) (A, D,C) 7→ (A,    0 −1 r ¯ 11 A¯ 12 A  g(σ1 + σ2 )r A¯ = ¯ ¯ =  gσ0 −g(σv + σ1 + σ2 ) A21 A22 σω Fn r Fn r Fn r2 − J σ0 − J (σ1 + σ2 ) − J J (σ1 + σ2 ) 

and

   −1 ¯1 D D¯ = ¯ =  −gσ1  D2 Fn r J σ1 

(8.70)

(8.71)

The output distribution matrix is now

  C¯ = 0 0 1

A useful choice for the linear gain of the observer is   A¯ L¯ + A¯ 12 − L¯ α G¯ l = ¯11 A22 + A¯ 21 L¯ − α

(8.72)

(8.73)

¯ where L¯ := D¯ −1 2 D1 and α is a negative scalar. After some algebra it can be shown that     r + (g + σα1 ) FJn r g¯1   G¯ l =  g¯2  =  g(α + g(σv + σ1 + σ2 ) − σσ10 ) FJn r + gr(σ1 + σ2 )  (8.74) σ σ w 0 g¯3 (σ1 + σ2 )q − J + σ1 − α

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In the original coordinates e1 , e2 and e3 the linear output error injection gain g1 = g¯1 , g2 = g¯2 and g3 = rg¯3 − g¯2 . Proposition 8.2. The error system (8.69) is quadratically stable. Proof. Consider as a potential Lyapunov candidate V (e) = eT Pe where      1 0 0 10 0  ¯ ¯1 L¯ P − P 1 1 0 1 0  P = 0 1 r  −L¯ T P¯1 P¯2 + L¯ T P¯1 L¯ 0 1r 1r 0 0 1r

(8.75)

where P¯1 ∈ R2×2 is an s.p.d. matrix which is a Lyapunov matrix for  σ0 σ2  − σ1 σ1 A¯ 11 − L¯ A¯ 21 = 0 −σv g

(8.76)

and P¯2 is a sufficiently large positive scalar such that P(A − Gl C) + (A − Gl C)T P =: −Q < 0

(8.77)

It is clear from (8.76) that A¯ 11 − L¯ A¯ 21 is stable with eigenvalues at {− σσ10 , −σv g} and so the existence of P1 is guaranteed. Also the existence of a sufficiently large P2 to ensure (8.77) is also guaranteed [9]. It can be verified by direct substitution that PD = FCT for some scalar F. Then
V˙ = −eT Qe − 2eT PD k sgn(Fey ) − θ f (x3 )x1 ≤ −eT Qe − 2|Fey |(k − |θ f (x3 )x1 |)

≤0

for large enough k and so the state estimation error e is quadratically stable as claimed. As argued in [9], in a domain of the origin, a sliding motion takes place on S = {e : Ce = 0}. ♯ Now, from first principles, the reduced order motion whilst sliding will be investigated. The output error derivative satisfies re˙y = (g + q)(σ0 e1 + σ1 (e3 − θ f (x3 )x1 ) + σ2 e3 + σ1 ν ) − (k2 + k3 +

σω )ey J

and so if a sliding-mode is enforced in finite time, ey = e˙y = 0 [30] and the above equation becomes 0 = (g + q)(σ0 e1 + σ1 (e3 − θ f (x3 )x1 ) + σ2 e3 + σ1 νeq ) (8.78) where νeq represents the equivalent output error injection signal necessary to maintain a sliding motion in the state estimation error space. Therefore from (8.78) an expression for νeq is given by 1 (8.79) νeq = − (σ0 e1 + σ1 (e3 − θ f (x3 )x1 ) + σ2 e3 ) σ1

2 During the sliding motion ey := e3 +e = 0 which implies e3 = −e2 . Substituting this value r of νeq into the first two equations of the components of (8.69), and using the fact that the

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237

e3 = −e2 , it can be shown that the sliding motion is governed by the reduced order linear system    σ0 σ2    − σ1 σ1 e˙1 e1 = (8.80) e˙2 0 −σv g e2 and so the sliding motion is stable and e1 → 0 and e2 → 0. When e1 , e2 = 0 the expression for the equivalent output error injection signal (8.79) becomes

νeq = θ f (x3 )x1

(8.81)

Equation (8.81) indicates that the expression

θˆ =

νeq f (xˆ3 )xˆ1

(8.82)

may be used as an estimate for the parameter θ during the sliding motion. Since xˆ1 → x1 and xˆ3 → x3 it follows f (xˆ3 )xˆ1 → f (x3 )x1 and so θˆ → θ . Notice (8.81) is only valid when xˆ1 6= 0 and xˆ3 6= 0 since f (xˆ3 ) > 0 if x3 6= 0. The following tables give the friction model parameters and the vehicle parameters used in the subsequent simulations. Table 8.1. Vehicle and LuGre friction model parameters parameter vs σ0 σ1 σ2 µc µs

value 6.57 181.54 0.70 0.0018 0.8 1.55

unit m/s 1/m s/m s/m

parameter r J m σv σw kb

value unit 0.323 m 2.603 kg/m2 1701 kg 0.005 1 0.9

In the simulations which follow the initial conditions for the plant represented by z, v and vr are 0, 40 and 0, whilst 0, 40.1 and 0.1 are used for the observer. This represents a deliberate mis-match for the purpose of demonstration. The discontinuous term in the observer has been replaced by a sigmoidal approximation. In the following simulation a brake pressure signal u(t) has been selected to generate an ‘emergency braking’ (Figure 8.7). In the following simulation the gain on the nonlinear injection term k = 50 whilst g1 = 0.3392, g2 = −2.57864 and g3 = 46.4196. These have been calculated according to the formula in (8.73) and guarantee a sliding motion will take place in finite time. Figure 8.8 shows the evolution of the plant states (as dotted lines) and the estimated states (as solid lines). Figure 8.9 shows the state estimation errors and the normalized estimation error for θ . As expected from Figure 8.8, the error in the friction state estimate given in Figure 8.9 is small and quickly becomes zero. The error in the velocity estimate and the relative velocity estimate show an asymptotic decay but at a slow rate. This is to be expected since one of the poles associated with the system matrix in (8.80) is −gσv = −0.049 and is therefore slow. The fact that the state estimation errors have not decayed, results in a small error for the estimate of θ . Figure 8.10 shows the switching function associated with the observer i.e., the output estimation error. It shows sliding occurs almost instantly in the state estimation error space and so the estimated angular velocity ωˆ from the observer exactly tracks the plant output ω throughout the simulation.

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Brake input P (Pa) b

2500

2000

P

b

1500

1000

500

0

0

1

2

3 time(sec.)

4

5

6

Fig. 8.7. Brake input signal −3

0

x 10

Internal friction state z

Velocity v m/s 40

−2 30

−4

20

−6

−8

10

0

2

4

6

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time(sec.)

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time(sec.)

Relative velocity v m/s

θ

r

10

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4

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6

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4

2.5

2

2

1.5

0

−2

1

0

2

4

time(sec.)

6

0.5

0

2

time(sec.)

Fig. 8.8. Evaluation of plant and estimated states, road condition parameter θ

8.7

Summary

This chapter has discussed the development of sliding-mode observers for a class of uncertain systems. Recent work has been reported which formulates the design problem in convex terms so that efficient numerical methods can be employed to synthesize the observer gains. A case

239

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Error in friction state z

−4

5

Error in velocity v m/s

x 10

−0.08

−0.085

0

−0.09

−0.095

−5 −0.1

−0.105

−10

−0.11

−15

0

2

4

6

−0.115

0

2

time(sec.)

4

6

time(sec.)

Error in relative velocity v m/s

Normalised error in estimated θ

r

0.15

0.1

0.1 0.05

0.05

0 0

−0.05

−0.1

−0.05

−0.15

−0.2

0

2

4

−0.1

6

0

2

time(sec.)

4

6

time(sec.)

Fig. 8.9. Estimation errors ey

0.1

0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

time(sec.)

Fig. 8.10. Output estimation error study has also been presented which demonstrates how sliding-mode observers can be used for parameter estimation as well as state estimation.

8.8

Notes and References

The pendulum example is effectively the case study considered in [31]. Interestingly in this paper a sliding-mode observer was compared to other direct nonlinear observer design

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methodologies for the pendulum example. It was argued that the sliding-mode observer had the best performance and was the least involved in terms of design. ˙ observer problem is given in [11] An almost completely analytic solution to the Walcott & Zak for square systems. In [10] an analysis is given of the closed loop system arising from using a state feedback sliding-mode controller implemented using the state estimates generated from a sliding-mode observer. For a class of systems with matched uncertainty the closed loop retains the invariance properties of its full state feedback counterpart. An overview of much of this work appears in [12]. In Herrmann et. al [19] the minimum phase condition is alleviated to some extent for systems with many outputs. In essence, rather than seeking a closed-loop observer based controller which is robust to matched uncertainty, robustness with respect to a strict subset of matched uncertainty is achieved by effectively generating an uncertainty distribution matrix of lower dimension which is contained within the range space of the input distribution matrix. For details see [19]. The use of sliding-mode observers for disturbance estimation is discussed in [17]. In fact [17] describes a design framework for a broader class of systems than the one described in this chapter – namely linear time varying systems. Sliding-mode ideas have previously been used in the literature for fault detection. Sreedhar et. al [26] consider a model-based sliding-mode observer approach although in their design procedure it is assumed that the states of the system are available; a different approach is adopted by Hermans & Zarrop [18] who attempt to design an observer in such a way that in the presence of a fault the sliding motion is destroyed. The approach in [14] seeks to design the sliding-mode observer to maintain sliding even in the presence of faults. More detailed simulations and analysis of a (slightly different) sliding-mode based FDI scheme is described in [13]. An interesting use of the concept of equivalent output error injection is described in [16]. A recursive design procedure based on successive evaluations of appropriate output error injection signals is shown to provide estimation of all unknown states of observable linear systems in finite time.

Acknowledgements The first author would like to thank the Royal Academy of Engineering for the provision of a Luverhulme Senior Research Fellowship.

References 1. L. Alvarez, R. Horowitz J. Yi, and L. Olmos. Dynamic friction model-based tyre/road friction estimation and emergency braking control. Journal of Dynamic Systems, Measurement and Control, 2005. 2. J.P. Barbot. Sliding-mode observers. In W. Perruquetti and J.P. Barbot, editors, Slidingmodes in Automatic Control. Marcel Dekker, 2001. 3. S.P. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in Systems and Control Theory. SIAM: Philadelphia, 1994.

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4. W.L. Brogan. Modern Control Theory. Prentice Hall, Englewood Cliffs NJ., 1991. ´ om, and P. Lischinsky. A new model for control 5. C. Canudas de Wit, H. Olsson, K.J. Astr¨ of systems with friction. IEEE Transactions on Automatic Control, pages 419–425, 1995. 6. C. Canudas de Wit, P. Tsiotras, X. Claeys, J. Yi, and R.Horowitz. Friction tyre/road modeling, estimation and optimal braking control. In R. Johansson and A.Rantzer, editors, Nonlinear and Hybrid Systems in Automotive Control, pages 147–210. Springer-Verlag, London, UK, 2003. 7. C.M. Dorling and A.S.I. Zinober. A comparative study of the sensitivity of observers. In Proceedings of the IASTED Symposium on Applied Control and Identification– Copenhagen, pages 6.32–6.38, 1983. 8. S. Drakunov and V.I. Utkin. Sliding-mode observers: tutorial. In Proceedings of the 34th IEEE Conference of Decision and Control, pages 3376–3378, 1995. 9. C. Edwards and S.K. Spurgeon. On the development of discontinuous observers. International Journal of Control, 59:1211–1229, 1994. 10. C. Edwards and S.K. Spurgeon. Robust output tracking using a sliding-mode controller/observer scheme. International Journal of Control, 64:967–983, 1996. 11. C. Edwards and S.K. Spurgeon. Sliding-mode output tracking with application to a multivariable high temperature furnace problem. International Journal of Robust and Nonlinear Control, 7:337–351, 1997. 12. C. Edwards and S.K. Spurgeon. Sliding Mode Control: Theory and Applications. Taylor & Francis, 1998. 13. C. Edwards and S.K. Spurgeon. A sliding-mode control observer based FDI scheme for the ship benchmark. European Journal of Control, 6:341–356, 2000. 14. C. Edwards, S.K. Spurgeon, and R.J. Patton. Sliding-mode observers for fault detection. Automatica, 36:541–553, 2000. 15. F. Gustafsson. Slip-based tyre-road friction estimation. Automatica, pages 1087–1099, 1997. ¨ uner, and V.I. Utkin. On sliding-mode observers via equivalent control 16. I. Hasakara, U. Ozg¨ approach. International Journal of Control, 71:1051–1067, 1998. 17. H. Hashimoto, V.I. Utkin, J.X. Xu, H.Suzuki, and F. Harashima. Vss observers for linear time varying systems. In Proceedings of the 16th Annual Conference of the IEEE Industrial Electronic Society, pages 34–39, 1990. 18. F.J.J. Hermans and M.B. Zarrop. Sliding-mode observers for robust sensor monitoring. In Proceedings of the 13th IFAC World Congress, pages 211–216, 1996. 19. G. Herrmann, S.K. Spurgeon, and C. Edwards. A robust sliding-mode output tracking control for a class of relative degree zero and non-minimum phase plants: a chemical process application. International Journal of Control, 72:1194–1209, 2001. 20. U. Itkis. Control Systems of Variable Structure. Wiley, New York, 1976. 21. U. Kiencke. Real-time estimation of adhesion characteristic between tyre and road. In Proceedings of the IFAC World Congress, 1993. 22. D.G. Luenberger. An introduction to observers. IEEE Transactions on Automatic Control, 16:596–602, 1971. 23. W. Perruquetti and J.P. Barbot (Eds). Sliding Modes in Automatic Control. Marcel Dekker, 2001. 24. L.R. Ray. Nonlinear tyre force estimation and road friction identification - simulation and experiments. Automatica, 33:1819–1833, 1997. 25. J.J.E. Slotine, J.K. Hedrick, and E.A. Misawa. On sliding observers for nonlinear systems. Transactions of the ASME: Journal of Dynamic Systems, Measurement and Control, 109:245–252, September 1987.

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26. R. Sreedhar, B. Fern´andez, and G.Y. Masada. Robust fault detection in nonlinear systems using sliding-mode observers. In Proceedings of the IEEE Conference on Control Applications, pages 715–721, 1993. 27. A. Steinberg and M.J. Corless. Output feedback stabilisation of uncertain dynamical systems. IEEE Transactions on Automatic Control, 30:1025–1027, 1985. 28. V. Utkin, J. Guldner, and J. Shi. Sliding Mode Control in Electromechanical Systems. Taylor & Francis, 1999. 29. V.I. Utkin. Variable structure systems with sliding-modes. IEEE Transactions on Automatic Control, 22:212–222, 1977. 30. V.I. Utkin. Sliding Modes in Control Optimization. Springer-Verlag, Berlin, 1992. ˙ 31. B.L. Walcott, M.J. Corless, and S.H. Zak. Comparative study of nonlinear state observation techniques. International Journal of Control, 45:2109–2132, 1987. ˙ 32. B.L. Walcott and S.H. Zak. State observation of nonlinear uncertain dynamical systems. IEEE Transaction on Automatic Control, 32:166–170, 1987. 33. J. Yi, L. Alvarez, X. Claeys, and R. Horowitz. Emergency braking control with an observer-based dynamic tyre/road friction model and wheel angular velocity measurement. Vehicle System Dynamics, 39:81–97, 2003. 34. J. Yi, L. Alvarez, R. Horowitz, and C. Canudas de Wit. Adaptive emergency braking control using a dynamic tyre/road friction model. In Proc. of 39th IEEE CDC, Sydney, Australia, pages 456–461, 2000.

9 Sliding-M ode Control in Systems with Output Time Delay Alan S.I. Zinober, G. Liu , and Yuri B. Shtessel

9.1

Introduction

The main features of sliding-mode control (SMC) and the associated feedback control law of Variable Structure Control (VSC) systems will be summarized in this chapter. SMC is a well-known solution to the problem of the deterministic control of uncertain systems, since it yields invariance to a class of parameter variations [4,7,33–36,43]. The characterizing feature is sliding motion, which occurs when the system state repeatedly crosses certain subspaces, or sliding hyperplanes, in the state space. A sliding controller may comprise nonlinear and linear parts, and has been well studied in the literature. Numerous practical applications of SMC have been reported in the literature. These include aircraft flight control [32] ,helicopter flight control, spacecraft flight control, ship steering, turbogenerators, electrical drives, overhead cranes, industrial furnace control, electrical power systems, robot manipulators, automobile fuel injection and magnetic levitation. There has been research on determining sliding hyperplanes by various approaches including complete and partial eigenstructure assignment, and reduction of the sensitivity to unmatched parameter variations [5, 6]. After presenting the underlying theory of the sliding-mode, we shall describe some of the techniques relating to the design of the sliding hyperplanes. For completeness we also present a suitable control law to ensure the attainment of the slidingmode. A straightforward scalar illustrative example is presented. Sliding-mode observers have also been designed and there have been important extensions of SMC to higher order sliding-modes (see, for example, [9, 10, 20, 21]. We then consider an interesting application of SMC (ully reported in [22] to the tracking of a nonminimum phase control system which is difficult to control [28, 29]. Output tracking in SISO fully linearizable nonlinear systems with an output time delay is considered using sliding-mode control and the system centre approach. Using Pad´e approximations for the delay, the problem is reduced to the tracking of a nonminimum phase control system. This system is transformed into a corresponding state tracking problem, where the state tracking profiles are generated by the equations of the stable system centre. The sliding-mode control approach is developed and numerical examples are presented. Also the describing function is used to evaluate the limit cycle generated by the feedback of the actual delayed output and these results agree very closely with the system simulations. M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 243-264, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

244

A. S. I. Zinober, G. Liu, and Y.B. Shtessel

The paper is organized as follows: Section 7.7 is dedicated to the output tracking problem formulation. Section 7.10 uses Pad´e approximation to approximate the delayed system. Section 4 presents the system centre and sliding-mode control to solve the output tracking problem. A numerical example demonstrating various aspects of different Pad´e approximations is presented. We use the describing function to evaluate the limit cycle generated by the actual delayed output. Conclusions are summarized in Section 9.4.

9.2 9.2.1

Sliding-Mode Control Regulator System

As our basic control system we consider the uncertain linear regulator x(t) ˙ = [A + ∆ A(t)]x(t) + [B + ∆ B(t)]u(t) + f (x, u,t)

(9.1)

where x is an n-vector of states and u is an m-vector of controls. It is assumed that n > m, that B is of full rank m and that the pair (A, B) is controllable. The matrices ∆ A and ∆ B represent the variations and uncertainties in the plant parameters and the control interface respectively. f represents uncertain time-varying additive terms. It is assumed further that the parameter uncertainties and disturbances are matched, occurring only on the control channels, i.e., R(B) = R([B, ∆ B]) (where R(·) denotes the range space); and that rank [B + ∆ B(t)] = m for all t ≥ 0. This implies that for suitable choice of limiting values of the control, one can achieve total invariance to parameter variations and disturbances [7]. The overall aim is to drive the system state from an initial condition x(0) to the state space origin as t → ∞. The jth component u j ( j = 1, . . . , m) of the state feedback control vector u(x) has a discontinuity on the jth switching surface which is a hyperplane M j passing through the state origin. Defining the hyperplanes by M j = {x : c j x = 0}, ( j = 1, 2, . . . , m)

(9.2)

(where c j is a row n-vector), the sliding-mode occurs when the state lies in M j for all j, i.e., in the sliding subspace M=

m \

Mj

(9.3)

j=1

In practice the control discontinuity may be replaced by a soft nonlinearity to reduce chattering [3].

9.2.2

Model-Following Control System

Model-following control systems are very widely used in practice and VSC can be designed in a manner very similar to the regulator system [18, 32, 44]. Consider the system x(t) ˙ = [A + ∆ A(t)]x(t) + [B + ∆ B(t)]u(t) + f (x, u) x˙m (t) = Am xm (t) + Bm r(t)

(9.4)

Sliding-Mode Control in Systems with Output Time Delay

245

where the first equation (as in (9.1)) describes the actual plant, and the second equation is the model plant with xm an n-vector of model states and r a vector of reference inputs. It is desired that the actual plant states follow the model states. The error e(t) = xm (t) − x(t)

(9.5)

should be forced to zero as time t → ∞ by suitable choice of the control u. Subject to the matrices A, B, ∆ A, ∆ B, Am and Bm satisfying certain structural and matching properties [18], we can achieve the desired objective with suitable control. The error model satisfies e(t) ˙ = Am e(t) + [(Am − A)x(t) − f + Bm r(t)] − Bu(t) (9.6)

and, subject to certain matching conditions [32], the model equations with suitable linear control components, reduce to e(t) ˙ = Am e(t) − Bu(t) (9.7)

The VSC of this error system may be readily designed, using the techniques previously described, by associating e with x in earlier sections. The sliding hyperplanes are now in the error state space. Further details and examples of practical time-varying and nonlinear avionics systems are given in [32]. Tracking problems can also be controlled and in the second half of this chapter a tracking problem is studied.

9.2.3

Sliding-Mode

When considering the synthesis of the sliding hyperplanes, it is sufficient to study the ideal regulator system, without uncertainties and disturbances, given by x(t) ˙ = Ax(t) + Bu(t)

(9.8)

Matched uncertainties are handled by suitable choice of the control function. From (9.2) the sliding-mode satisfies s = Cx(t) = 0 , t ≥ ts (9.9) where ts is the time when the sliding subspace is reached, and C is an m × n matrix. Differentiating equation (9.9) with respect to time, and substituting for x(t) ˙ from (9.8) gives Cx(t) ˙ = CAx(t) +CBu(t) = 0 ,

t ≥ ts

(9.10)

Equation (9.10) may be rearranged to give CBu(t) = −CAx(t)

(9.11)

The hyperplane matrix C is selected so that |CB| 6= 0, and therefore the product CB is invertible. Hence (9.11) may be rearranged to give the following expression for the equivalent control [33] ueq (t) = −(CB)−1CAx(t) = −Kx(t) (9.12) where ueq (t) is the linear open-loop control which would force the trajectory to remain in the null space of C, during sliding. Substituting for ueq (t) from equation (9.12) into (9.8) gives x(t) ˙ = {I − B(CB)−1C }Ax(t) , = (A − BK)x(t)

t ≥ ts

which is the system equation for the closed-loop system dynamics during sliding.

(9.13) (9.14)

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This motion is independent of the actual nonlinear control and depends only on the choice of C, which determines the matrix K. The purpose of the control u is to drive the state into the sliding subspace M , and thereafter to maintain it within the subspace M . The convergence of the state vector to the origin is ensured by suitable choice of the feedback matrix K. The determination of the matrix K or alternatively, the determination of the matrix C defining the subspace M , may be achieved without prior knowledge of the form of the control vector u. (The reverse is not true). The null space of C, N (C), and the range space of B, R(B), are, under the hypotheses given earlier, complementary subspaces, so N (C) ∩ R(B) = {0}. / Since motion lies entirely within N (C) during the ideal sliding-mode, the dynamic behaviour of the system during sliding is unaffected by the controls, as they act only within R(B). The development of the theory and design principles is simplified by using a particular canonical form for the system, which is closely related to the Kalman canonical form for a multivariable linear system. By assumption the matrix B has full rank m; so there exists an orthogonal n × n transformation matrix T such that   0 T B= (9.15) B2 where B2 is m × m and nonsingular [5]. The orthogonality restriction is imposed on T for reasons of numerical stability, and to remove the problem of inverting T when transforming back to the original system. The transformed state is y = T x, and the state equation becomes y(t) ˙ = T AT T y(t) + T Bu(t)

(9.16)

The sliding condition is CT T y(t) = 0, ∀ t ≥ ts . If the transformed state y is now partitioned as yT = (yT1 yT2 ); y1 ∈ R n−m , y2 ∈ R m (9.17)

and the matrices T AT T , T B and CT T are partitioned accordingly, then equation (9.16) may be written as the following pair of equations y˙1 (t) = A11 y1 (t) + A12 y2 (t) y˙2 (t) = A21 y1 (t) + A22 y2 (t) + B2 u(t)

(9.18)

The sliding condition becomes C1 y1 (t) +C2 y2 (t) = 0 , where T AT T =



A11 A12 A21 A22



,

t ≥ ts

CT T = (C1

(9.19)

C2 )

(9.20)

and C2 is nonsingular because CB is nonsingular. This canonical form is central to hyperplane design methods and it plays a significant role in the solution of the reachability problem i.e., the determination of the control form ensuring the attainment of the sliding-mode in M [5, 6, 36, 38–42]. Equation (9.19) defining the sliding-mode is equivalent to y2 (t) = −Fy1 (t) where the m × (n − m) matrix F is defined by

(9.21)

Sliding-Mode Control in Systems with Output Time Delay

F = C2−1C1

247 (9.22)

so that in the sliding-mode y2 is related linearly to y1 . The sliding-mode satisfies equation (9.21) and y˙1 = A11 y1 (t) + A12 y2 (t) (9.23) This represents an (n − m) th order system in which y2 plays the role of a state feedback control. So we get y˙1 (t) = (A11 − A12 F)y1 (t) (9.24)

which is known as the reduced-order equivalent system. The design of a stable sliding-mode such that y → 0 as t → ∞, requires the determination of the gain matrix F such that A11 − A12 F has n − m left-hand half-plane eigenvalues.

9.2.4

Feedback Control

Once the sliding hyperplanes have been selected, attention must be turned to solving the reachability problem. This involves the selection of a state feedback control function u : R n → R m which will drive the state x into N (C) and thereafter maintain it within this subspace. There is a virtually unlimited number of possible forms for this control function, the only essential features of the form chosen being discountinuity on one or more subspaces containing N (C). In general the variable structure control law consists of two additive parts; a linear control law uℓ and a nonlinear part un , which are added to form u. The linear control is merely a state feedback controller uℓ (x) = Lx (9.25) while the nonlinear feedback controller un incorporates the discountinuous elements of the control law. Consider here the unit vector control un (x) = ρ

s Cx =ρ , ksk kCxk

ρ >0

(9.26)

in the form [23] u(x) = Lx + ρ

Nx k Mx k

(9.27)

where the null spaces of N, M and C are coincident : N (N) = N (M) = N (C). Starting from the transformed state y, we form a second transformation T2 : R n → R n such that z = T2 y where T2 =



In−m 0 F Im

(9.28) 

The matrix T2 is clearly nonsingular, with inverse   I 0 T2 −1 = n−m −F Im Partitioning zT = (zT1

(9.29)

(9.30)

zT2 ) with z1 ∈ R n−m and z2 ∈ R m z1 = y1 ;

z2 = Fy1 + y2

(9.31)

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from which it is clear that the conditions s ≡ 0 and z2 ≡ 0 are equivalent (in the sense that the points of the state space at which s = 0 are precisely the points at which z2 = 0). The transformed system equations become z˙1 = Σ z1 + A12 z2

(9.32)

z˙2 = Θ z1 + Φ z2 + B2 u

(9.33)

where

Σ = A11 − A12 F

Θ = F Σ − A22 F + A21 Φ = FA12 + A22

(9.34)

In order to attain the sliding-mode it is necessary to force z2 and z˙2 to become identically zero. Define the linear part of the control to be uℓ (z) = −B−1 2 {Θ z1 + (Φ − Φ∗ )z2 }

(9.35)

where Φ∗ is any m × m matrix with left-hand half-plane eigenvalues. Transforming back into the original state space (x-space) gives L = −B−1 2 (Θ

Φ − Φ∗ )T2 T

(9.36)

The linear control law uℓ drives the state component z2 to zero asymptotically; to attain N (C) in finite time, the nonlinear control component un is required. This nonlinear control must be discontinuous whenever z2 = 0, and continuous elsewhere. Letting P2 denote the positive definite unique solution of the Lyapunov equation P2 Φ∗ + Φ∗ T P2 = −Im

(9.37)

then P2 z2 = 0 if and only if z2 = 0, and we may take un (z) = −ρ

B−1 2 P2 z2 , k P2 z2 k

z2 6= 0

(9.38)

where ρ > 0 is a scalar parameter to be selected by the designer. When z2 = 0, un may be arbitrarily defined as any function satistying k un k ≤ ρ . Expressing the control in x-space, we have N = −B−1 2 (0 P2 )T2 T M = (0 P2 )T2 T

(9.39) (9.40)

For the more general system (9.1) in which disturbances and uncertainties are present, a similar control structure may be employed. However, in this case the scalar ρ of (9.38) is replaced by a time-varying state-dependent function incorporating two design parameters γ1 , γ2 , upon which the time to reach N (C) also depends [23]. Discontinuous control produces chatter motion in the neighbourhood of the sliding surface. In may practical applications this cannot be tolerated. There are numerous techniques to “smooth” the control function. Perhaps the most straightforward smoothed continuous nonlinear control, which eliminates the chatter motion, is (see, for example, [3, 43]

Sliding-Mode Control in Systems with Output Time Delay

u(x) = Lx + ρ

9.2.5

Nx , kMxk + δ

δ > 0+

249

(9.41)

Second-Order Example

To illustrate some of the main ideas consider the simple scalar double-integrator plant x(t) ¨ = bu(t) + f (t) with the positive parameter b uncertain but taking known bounded maximum and minimum values, and f (t) a disturbance. Here the sliding subspace will be a one-dimensional space, a straight line through the state origin s = gx + x˙ = 0 During sliding for t > ts we require lim s˙ > 0

and

s=0

and

s→0−

lim s˙ < 0

s→0+

and then s˙ = 0

i.e., the state remains on the sliding surface. So x˙ = −gx with eigenvalue −g i.e., the dynamics of a reduced first-order system (with order n − 1). x(t) = x(ts )e−gt One obtains exactly the closed-loop eigenvalue −g by specifying the sliding line (9.2.5). The dynamics in the sliding-mode are independent of the parameter b. The discontinuous control s u=ρ |s| can maintain sliding motion on s = 0 within a bounded region of the state origin, for a range of values of b with the precise value of b not required to be known [33]. The equivalent control which theoretically can maintain the state on the sliding line, is the linear control ueq = −

gx b

To achieve the sliding-mode with this linear control would require exact knowledge of b; unlike for the case of nonlinear control. Smooth nonlinear control has the form s u=ρ |s| + δ Simulation results are presented in Figures. 9.2.5 and 9.2.5 for discontinuous (δ = 0) and smooth control (δ = 0.01) with ρ = 1. The state trajectories are very similar. During the sliding-mode the smooth control is equal to the equivalent control ueq , which is included in the control graph of Figure 9.2.5. Note the elimination of chatter when using the smoothing control. The system is invariant to a matched disturbance function f (t).

A.S.I. Zinober, G. Liu, and Y. B. Shtessel

250

states; relay)

control

0.5

0.5

0

0

u

1

x1 x2

1

−0.5

−1

−0.5

0

2

4

−1

6

0

2

4

t

6

t

s(t)

phase plane 0

1.2

1

−0.2

0.8

s

x2

0.6

−0.4

0.4

0.2

−0.6

0

−0.2

0

2

4

t

6

−0.8

0

0.2

0.4

0.6

0.8

1

x1

Fig. 9.1. Double integrator with discontinuous relay control

9.3

Application Problem

We now consider an interesting application of SMC to the tracking of a nonminimum phase control system which is difficult to control [28, 29]. This was fully reported in [22]. Output tracking in SISO fully linearizable nonlinear systems with an output time delay is considered using sliding-mode control and the system centre approach. Using Pad´e approximations for the delay, the problem is reduced to the tracking of a nonminimum phase control system. This system is transformed into a corresponding state tracking problem, where the state tracking profiles are generated by the equations of the stable system centre. The sliding-mode control approach is developed and numerical examples are presented. Also the describing function is used to evaluate the limit cycle generated by the feedback of the actual delayed output and these results agree very closely with the system simulations. Output time delay is a common feature in many systems and must be taken into account when designing a controller. The output tracking of a real-time reference profile in nonlinear systems with output delay by sliding-mode control was addressed in the paper [SHT 03a]. In addition to the first-order Pad´e approximation, the more precise second and third-order Pad´e approximations have been used to replace the output delay element so that we can transfer the problem into the output tracking of a nonminimum phase system. For the output reference profile given by a linear exogenous system, the nonminimum phase output tracking problem is transformed to the corresponding state tracking problem. Bounded state tracking profiles are generated by equations of the stable system centre [27]. The sliding-mode control algorithm is designed for the approximate nonminimum phase model of the system. In simulations many comparisons are made between different order Pad´e approximations and different delay time.

Sliding-Mode Control in Systems with Output Time Delay

251

Also the describing function is used to evaluate the limit cycle generated by the feedback of the actual delayed output and these results agree very closely with the system simulations. control 1

0.5

0.5

0

0

u

x1 x2

states; smoothed (d=0.01) 1

−0.5

−1

−0.5

0

2

4

−1

6

0

2

4

t

6

t

s(t)

phase plane 0

1.2

1

−0.2

0.8

s

x2

0.6

−0.4

0.4

0.2

−0.6

0

−0.2

0

2

4

6

−0.8

0

0.2

0.4

t

0.6

0.8

1

x1

Fig. 9.2. Double integrator with smoothed control

9.3.1

Problem Formulation

Consider a controllable fully feedback linearizable nonlinear SISO dynamic system without time delay x˙ = f (x,t) + g(x,t)u, y = h(x) (9.42) where x(t) ∈ Rn is a state vector, y(t) ∈ R1 a controlled output and u(t) ∈ R1 is a control input. As a fully linearizable relative degree n system, it can be transformed [ISI 95] to y(n) = ϕ (ξ ,t) + b(ξ ,t)u

(9.43)

where ξ = [y, y, ˙ · · · , y(n−1) ]T ∈ Rn . Following the approach developed in [12], we define a coordinate transformation   1 0 ··· ··· 0    0 1 ··· ··· 0  z  =  ··· ··· ··· ··· 0 ξ q a0 a1 · · · an−2 1

with ai > 0, where

q = yn−1 + an−2 yn−2 + · · · + a1 y˙ + a0 y

is a new output to get a relative degree 1 system,

z(t) ∈ Rn−1

and

q(t) ∈ R1 .

(9.44)

(9.45)

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A.S. I. Zinober, G. Liu, and Y. B. Shtessel

The system (9.42) is rewritten in the new basis (9.44) as  z˙1 = z2 ,      z˙2 = z3 , ··· ···   z˙ = q − an−2 zn−1 − · · · − a1 z2 − a0 z1    n−1 ˆ q,t)u q˙ = ϕˆ (z, q,t) + b(z, where

and

(9.46)

ϕˆ (z, q,t) = −a0 an−2 z1 − (a1 an−2 − a0 )z2 − · · · − (a2n−2 − an−3 )zn−1 +an−2 q + ϕ (z, q − a0 z1 − · · · − an−2 zn−1 ,t) bˆ = b(q − a0 z1 − · · · − an−2 zn−1 ,t).

Note that the internal dynamics of the system (9.46) are stable and can be disregarded when solving the output tracking problem as time increases. So output tracking in system (9.42) can be transformed to output tracking in the scalar system ˆ q,t)u q˙ = ϕˆ (z, q,t) + b(z,

(9.47)

using the transformation (9.44). Assume here the desired command output profile is given in the form yc = A¯ + B¯ sin ωn t

(9.48)

Then we can assume the output tracking ( command ) profile qc (corresponding to q) as follows: qc = A + B sin ωn t +C cos ωn t (9.49) where A, B,C and ωn are piecewise constants and A, B,C are unknown. The signal (9.49) can be described by a linear system of exogenous differential equations with the following characteristic equation: (9.50) P3 (λ ) = λ 3 + 0λ 2 + ωn2 λ + 0 ˆ = bˆ 0 (·)(1 + δ (·)), |δ (·)| ≤ β1 < 1. The function Now we restrict ϕˆ (·) < α1 , α1 > 0 and b(·) bˆ 0 (·) is assumed known. Also assume the output of (9.47) is accessible with time delay y(t) ˆ = q(t − τ ).

(9.51)

The problem is to design sliding-mode control u(t) that forces the output variable y(t) ˆ to track asymptotically the command profile.

9.3.2

Pade´ Approximations and Time Delay Systems

In this section, we consider the Pad´e approximation to approximate the time delay. The Pad´e approximation [2] uses the quotient of two polynomials to estimate a power series. We will use the direct solutions of the Pad´e equations [L/M] = PL (x)/QM (x), where PL (x) is a polynomial of degree L and QM (x) is a polynomial of degree M. When we approximate a formal power series f (x), the explicit equation is

Sliding-Mode Control in Systems with Output Time Delay

lim

x→∞

253

QM (x) f (x) − PL (x) =0 xL+M+1

Letting f (x) = ex , we can get the first, second and third-order Pad´e approximations x 2 x 1− 2 x x2 1+ + 2 12 ex ≈ x x2 1− + 2 12 x x2 x3 1+ + + 2 10 120 ex ≈ x x2 x3 1− + − 2 10 120 ex ≈

1+

y(s) ˆ

The corresponding Laplace transform of equation (9.51) q(s) = e−sτ can always be approximated by first-order, second-order and third-order Pad´e approximation as e−sτ

e−sτ

e−sτ

sτ 2 ≈ sτ 1+ 2 sτ 1− 2 ≈ sτ 1+ 2 sτ 1− 2 ≈ sτ 1+ 2 1−

s2 τ 2 12 s2 τ 2 + 12 s2 τ 2 s3 τ 3 + − 10 120 2 2 s τ s3 τ 3 + + 10 120 +

(9.52)

where s is the Laplace variable. We introduce a new output variable y˜ to make the approximation exact and the system (9.51) can be always approximated by

For the first-order

for the second-order

and for the third-order



ζ˙ = Q˜1 ζ + Q˜2 y˜ ˜ ζ ,t) + ψ (y, ˜ ζ )u y˜˙ = ϕ¯ (y,

  Q˜1 = τ2 , Q˜2 = − τ4 ϕ¯ (y, ˜ ζ ,t) = −ϕˆ (z, q,t) + τ2 ζ − τ4 y˜  ˆ q,t) ψ (y, ˜ ζ ) = −b(z,

     0 1  12 1 ˜   Q˜1 = 12 6 , Q2 = − τ 6 − τ2 τ τ ˜1 ζ + Q˜2 y˜ ¯ ˆ ϕ ( y, ˜ ζ ,t) = ϕ (z, q,t) + Q    ˆ q,t) ψ (y, ˜ ζ ) = b(z,

(9.53)

(9.54)

(9.55)

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A.S.I. Zinober, G. Liu, and Y.B. Shtessel

  0      Q˜1 =  0

  24  1 0 τ  0 1  , Q˜2 = −  288 τ2 60 12 120 2256 − τ3 τ2 τ τ3    ϕ¯ (y, ˜ ζ ,t) = −ϕˆ (z, q,t) + Q˜1 ζ + Q˜2 y˜   ˆ q,t) ψ (y, ˜ ζ ) = −b(z,

(9.56)

The system (9.53) is nonminimum phase because Q˜ 1 has at least one non-Hurwitz eigenvalue. Note that the output delay system tracking problem has been transformed into a nonminimum phase system output tracking problem (with no delay) by the Pad´e approximation.

9.3.3

System Centre Method and Sliding-Mode Control

The nonminimum phase system tracking problem is solved in this section by the system centre method and sliding mode-control. The equation of the system centre (ideal internal dynamics) [27] that defines a command (tracking) profile ζc (t) for the internal state vector for the system with delay approximated by the nonminimum phase system (9.53) is

ζ˙c = Q˜1 ζc + Q˜2 qc

(9.57)

which is unstable. According to the theorem of system centre in [27], the stable system centre ζ˜c (t) that converges asymptotically to the bounded particular solution ζc (t) of the unstable equation of system centre [16], i.e. ζ˜c (t) → ζc (t), is given by (3) (2) (1) (2) (1) ζ˜c + c2 ζ˜c + c1 ζ˜c + c0 ζ˜c = −(P2 θc + P1 θc + P0 θc )

where θc = Q˜2 qc , and the coefficients of P0 , P1 , P2 are computed as follows:  −1 P = c0 Q˜1    0 −1 −2  (c −ω 2 )Q˜ +(c −c ω 2 )Q˜ P1 = 1 n 1 2 ˜0 −2 2 n 1 Q I+ ω n 1  −1 −2 −3   c Q˜ +(c1 −ωn2 )Q˜1 +c0 Q˜1  P2 = 2 1 2 ˜ −2

(9.58)

(9.59)

I+ωn Q1

The coefficients c0 , c1 , c2 are chosen to provide the specified eigenvalues of the homogeneous differential equation. (3) (2) (1) (9.60) ζ˜c + c2 ζ˜c + c1 ζ˜c + c0 ζ˜c = 0. The sliding-mode control is introduced with the sliding function defined as

σ = eq +C1 e˜ζ

(9.61)

where eq = qc − y, ˜ and e˜ζ = ζ˜c − ζ .

Assume that the sliding-mode exists on the sliding surface σ = 0, then eq = −C1 e˜ζ . Therefore e˙˜ζ = (Q˜1 − Q˜2C1 )e˜ζ + (ζ˙˜c − Q˜1 ζ˜ + Q˜2 qc ), eq = −C1 e˜ζ

(9.62)

Sliding-Mode Control in Systems with Output Time Delay

255

Since ζ˜c (t) → ζc (t) with increasing time, then e˜ζ (t) → eζ (t) and ζ˙˜c − Q˜1 ζ˜ + Q˜2 qc → 0. So the sliding-mode dynamics (9.62) asymptotically approaches the homogeneous differential equation e˙˜ζ = (Q˜1 − Q˜2C1 )e˜ζ (9.63) The coefficient C1 is selected to provide asymptotic tracking error dynamics (9.63) to zero. The sliding-mode control is designed as follows: u = bˆ −1 [q˙c − ϕˆ (·) +C1 (Q1 eζ + Q2 eq ) − ρ sign(σ )]

(9.64)

where ρ is a sufficiently large positive gain. Note that the standard sliding-mode controller design can be found in the classical book [8].

9.3.4

Numerical Example and Simulations

Consider the relative degree 2 second-order system: x˙1 = x2 x˙2 = −x2 + u y = x1

(9.65)

Transforming the plant to the form (9.46) with relative degree equal to one, introduce a new state vector in accordance with (9.44)       z1 1 0 x = · 1 (9.66) q a0 1 x2 The system (9.65) is rewritten in the new basis (9.66) as z˙1 = −a0 z1 + q q˙ = −(a0 − 1)a0 z1 + (a0 − 1)q + u

(9.67)

The desired command output profile has the characteristic equation (9.50). Now assume that the system output (9.67) is accessible with a time delay yˆ = q(t − τ ). The problem is to design the SMC that provides asymptotic tracking yˆ → qc as time increases. Replacing the time-delay function by the respective first-order, second-order and third-order Pad´e approximation to the system output y, ˆ the system (9.67) is approximately represented by the nonminimum phase systems without delay z˙1 = −a0 z1 − y˜ + ζ ζ˙ = τ2 ζ − τ4 y˜ y˙˜ = ( τ2 − a0 + 1)ζ + (a0 − 1 − τ4 )y˜ + (a0 − 1)a0 z1 − u

(9.68)

z˙1 = −a0 z1 + y˜ − ζ1 ζ˙1 = ζ2 − 12 τ y˜ ζ˙2 = τ6 ζ2 − 12 − 72 y˜ τ2 τ2 ˙y˜ = (− 12 + a0 − 1)y˜ + ζ2 − (a0 − 1)ζ1 − a0 (a0 − 1)z1 + u τ

(9.69)

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A.S.I. Zinober, G. Liu, and Y. B. Shtessel

z˙1 = −a0 z1 − y˜ + ζ1 ζ˙1 = ζ2 − 24 τ y˜ ζ˙2 = ζ3 − 288 y˜ τ2 60 − ζ˙3 = 12 ζ ζ + 120 y˜ ζ − 2256 3 τ τ2 2 τ3 1 τ3 ˙y˜ = (− 24 + a0 − 1)y˜ + ζ2 − (a0 − 1)ζ1 + a0 (a0 − 1)z1 − u τ

(9.70)

where the output y˜ is an approximation to the original output y. ˆ Now we select parameters as follows: c0 = 1000, c1 = 300, c2 = 30, ωn = 2, a0 = 20 and A¯ = 1, B¯ = 2. Also we choose C1 = −0.75 for the first-order Pad´e approximation, [−0.75, 0] for the secondorder Pad´e approximation and [−0.75, 0, 0] for the third-order Pad´e approximation. Notice that the sign of control in the second-order Pad´e approximation is positive and those in the first and third-order Pad´e approximations are negative. The control is u = 25sign(σ ) for the second-order Pad´e approximation and u = −25sign(σ ) for the first-order and the third-order Pad´e approximations. R R We have used MATLAB Simulink

and Scilab to get the tracking results for different τ (see Figures. 1–9 ). Note that all the errors mentioned are the maximum magnitude of absolute errors after reaching steady state. Comparing Figures 1–3 we see that for τ = 0.2 the first-order Pad´e approximation has good tracking results (0.0068) although the errors for the second-order (0.0005) and the third-order (0.0045) Pad´e are smaller. However when τ increases to 0.5 (see Figures 4–6), the first-order Pad´e results is bad since the output error is about 0.068. So the advantages of second-order (0.00078) and third-order Pad´e (0.0048) become obvious. For τ = 1.0 the first-order Pad´e approximation is quite clearly bad (see Figure 7), so we should not use the first-order Pad´e approximation to estimate the delayed output. Also the second-order Pad´e error (0.035) is bigger than the third-order Pad´e error (0.004). In conclusion higher-order Pad´e approximations will generally have better results for larger τ and the first-order Pad´e can only be used in smaller time delays.

pade1 0.06 0.05 0.04 0.03

error

0.02 0.01 0.00 −0.01 −0.02 −0.03 −0.04 1

2

3

4

5

6

7

8

9

time

Fig. 9.3. 1st-order Pad´e τ = 0.2: error

10

11

Sliding-Mode Control in Systems with Output Time Delay

257

pade2 2.0e−003

1.5e−003

1.0e−003

error

5.0e−004

0.0e+000

−5.0e−004

−1.0e−003

−1.5e−003

−2.0e−003 3

11

10

9

8

7

6

5

4

time

Fig. 9.4. 2nd-order Pad´e τ = 0.2: error pade3 0.006

0.004

0.002

error

0.000

−0.002

−0.004

−0.006

−0.008

−0.010 0

1

2

3

4

5

6

7

8

9

10

time

Fig. 9.5. 3rd-order Pad´e τ = 0.2: error Also the effects of variations in the parameters (C1 , a0 , ρ and the system centre formulation) have been tested. Detailed results will appear in a future paper.

9.3.5

Feedback by yˆ and Describing Function

In a real-life situation the feedback should be the actual delayed output yˆ and not y, ˜ i.e., yˆ should be included in the sliding-mode surface design. Let us consider the first-order case to establish the effect of feedback by yˆ by both simulation and also using the Describing Function approach. Using the Scilab program for the whole system with the actual delayed output, we get a limit cycle as in Figure 10. In fact, it can be shown that the frequency and the magnitude of the limit cycle are not altered by A and B, so we can assume both of them zero, i.e., qc = 0. The Describing Function method (DF) is briefly introduced and used to analyse the limit cycle above. DF is a well-known method that can be used to detect oscillations (limit cycles). The basic assumptions for our system are as follows:

A.S.I. Zinober, G. Liu, and Y. B. Shtessel

258

pade1 0.15

0.10

error

0.05

0.00

−0.05

−0.10

−0.15 1

9

8

7

6

5

4

3

2

10

time

Fig. 9.6. 1st-order Pad´e τ = 0.5: error pade2 2.0e−003

1.5e−003

1.0e−003

error

5.0e−004

0.0e+000

−5.0e−004

−1.0e−003

−1.5e−003

−2.0e−003 2

4

3

5

6

7

8

9

10

time

Fig. 9.7. 2nd-order Pad´e τ = 0.5: error No input, i.e. qc = 0. Nonlinearity is symmetric. Nonlinearity does not depend on frequency. Assume at the input point of the nonlinearity, e(t) = E sin(ω t)), i.e., first-order sinusoidal approximation. 5. Linear part (Laplace transform for the whole system) acts as a low-pass filter, i.e., higherorder harmonic components are damped.

1. 2. 3. 4.

In general, the describing function at the output point of the nonlinearity can be defined as n(t) = N(e(t)) = N(E sin(ω t)) = n(ω t) where n(ω t) is defined by the Fourier series n(ω t) = From the assumptions above

∞ ∞ A0 + ∑ Ak cos(kω t) + ∑ Bk sin(kω t) 2 k=1 k=1

Sliding-Mode Control in Systems with Output Time Delay

pade3 0.020

0.015

0.010

error

0.005

0.000

−0.005

−0.010

−0.015

−0.020

−0.025 1

7

6

5

4

3

2

8

9

10

time

Fig. 9.8. 3rd-order Pad´e τ = 0.5: error pade1 3

2.5

2

1

c

y and y

1.5

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

time t

Fig. 9.9. 1st-order Pad´e τ = 1.0: yc and y n(ω t) = A1 cos(ω t) + B1 sin(ω t) where A1 =

2 T

Z T /2

n(ω t) cos(ω t)dt =

1 π

Z π

n(α ) cos(α )d α

B1 =

2 T

Z T /2

n(ω t) sin(ω t)dt =

1 π

Z π

n(α ) sin(α )d α

and

−T /2

−T /2

The nonlinearity is the sign function

−π

−π

  M t >0 0 t =0 N(t) =  −M t < 0

So N(t) is an odd function and therefore A1 = 0, B1 =

1 π

Z π

−π

n(α ) sin(α )d α =

4M π

25

259

260

A.S.I. Zinober, G. Liu, and Y.B. Shtessel pade2 0.06

0.04

0.02

error

0.00

−0.02

−0.04

−0.06

−0.08 2

5

4

3

6

9

8

7

10

time

Fig. 9.10. 2nd-order Pad´e τ = 1.0: error pade3 0.020 0.015 0.010 0.005

error

0.000 −0.005 −0.010 −0.015 −0.020 −0.025 −0.030 4

6

5

8

7

9

10

time

Fig. 9.11. 3rd-order Pad´e τ = 1.0: error

The describing function is 4M πE It remains to calculate the total Laplace transform function for the whole system.The Laplace transform ( from u to z1 ) for the system with feedback yˆ is N( jω ) =

1 Z1 (s) = U(s) s(s + 1)

(9.71)

4 (s + a0 ) τ τ −s −s τ ˆ For qc = 0, σ = −yˆ + 0.75ζ . Also Y (s) = Q(s)e = (s + a0 )Z(s)e and ζ (s) = 2 s+ τ

σ (s) = (s + a0 )Z1 (s)( Therefore the total transfer function

3 τ

s + τ2

− e−sτ )

(9.72)

Sliding-Mode Control in Systems with Output Time Delay

261

Solution 0.10

0.05

y

0.00

−0.05

−0.10

−0.15 0

2

1

3

4

5

6

7

8

9

10

time

Fig. 9.12. Feedback by delayed output yˆ for 1st-order Pad´e τ = 0.2: y

G(s) = −

3 4M s + a0 ( τ 2 − e−sτ ) π E s(s + 1) s + τ

(9.73)

For a limit cycle to exist G( jω )N( jω ) = −1 + 0 j.

(9.74)

Many ‘stable’ solutions exist and the values of E and T (corresponding to ω ) are listed for largest E value in the tables. From the Tables 1–3, we can see that the period T and the Table 9.1. τ = 0.1: magnitudes of E and yˆ and period T Program for τ = 0.1 E T yˆ Simulink DF 1.40 0.148 0.949 Simulink for whole system 1.405 0.148 0.945

Table 9.2. τ = 0.2: magnitudes of E and yˆ and period T Program for τ = 0.2 E T yˆ Simulink DF 4.10 0.36 2.4815 Simulink for whole system 4.035 0.361 2.47

Table 9.3. τ = 0.3: magnitudes of E and yˆ and period T Program for τ = 0.3 E T yˆ Simulink DF 8.9 0.569 5.452 Simulink for whole system 8.896 0.57 5.427 magnitudes of E and yˆ are consistent. This indicates that the describing function method can be used to evaluate the limit cycle.

262

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A.S.I. Zinober, G. Liu, and Y. B. Shtessel

Conclusions

Sliding-mode control has been described in detail. Output tracking in causal nonlinear systems with an output delay via sliding-mode control is considered. The problem is reduced to the tracking of a real-time output reference profile given by an exogenous system in a nonminimum phase system without delay via the first, second and third-order Pad´e approximations. The nonminimum phase output tracking problem is transformed to a corresponding state tracking problem and the system centre method is used to deal with the unstable internal dynamics. We use sliding-mode control to solve the problem and present many numerical results. The describing function is used to evaluate the limit cycle generated when the actual delayed output feedback is used in the feedback and these results agree very closely with system simulations.

References 1. D.P. Atherton (1982) Nonlinear Control Engineering, Van Nostrand Reinhold Company Ltd 2. G.A. Baker Jr. (1973) Recursive Calculation of Pad´e Approximation, Pad´e Approximants and Their Applications. ed Graves-Morris PR 83–91 3. J.A. Burton and A.S.I. Zinober (1986) Continuous approximation of variable structure control. Int J Systems Science 17: 876–885 ˙ and G.P. Matthews (1988) Variable structure control of nonlinear 4. R.A. DeCarlo, S.H. Zak multivariable systems: a tutorial. Proc IEEE 26: 1139–1144 5. C.M. Dorling and A.S.I. Zinober (1986) Two Approaches to Hyperplane Design in Multivariable Variable Structure Control Systems. Int J Control44: 65–82 6. C.M. Dorling and A.S.I. Zinober (1988) Robust hyperplane design in multivariable variable structure control systems. Int J Control48: 2043–2054 7. B. Dra˘zenovi´c (1969) The invariance conditions in variable structure systems. Automatica 5: 287–295 8. C. Edwards C and S.K. Spurgeon (1998) Sliding Control. Taylor and Francis, Bristol 9. L. Fridman and A. Levant (1996) Higher order sliding modes as the natural phenomena of control theory. in: Robust Control Variable Structure and Lyapunov Techniques, eds Garafalo F, Glielmo G, Lecture Notes in Control and Information Science, 217, Springer Verlag, London 107–133 10. L. Fridman and A. Levant (2002) Higher order sliding modes in Sliding Mode Control in Engineering, eds Barbot JP, Perruguetti W, Marcel Dekker, New York 3–101 11. K. Furuta and S.B. Kim (1987) Pole Assignment in a Specified Disc. IEEE Trans Aut ContrAC-32: 423–427 12. S. Gopalswamy and J.K. Hedrick (1993) Tracking Nonlinear Nonminimum Phase Systems Using Sliding Control. International Journal of Control 57: 1141–4458 13. A. Isidori (1995) Nonlinear control systems. Springer-Verlag, London 14. Y.-T. Juang Z.-C. Hong and Y.-T. Wang (1989) Robustness of Pole Assignment in a Specified Region. IEEE Trans Aut ContrAC-34: 758–760 15. J. Kautsky, N.K. Nichols and P. Van Dooren (1985) Robust Pole Assignment in Linear State Feedback. Int J Control41: 1129–1155 16. G. Klein and B.C. Moore (1977) Eigenvalue generalised eigenvector assignment with state feedback. IEEE Trans Aut ContrAC–22: 140–141

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17. A. Koshkouei, K.J. Burnham KJ and A.S.I. Zinober A (2004) Dynamic Sliding Mode Control Design. Proceeding of IEE Conference, Bath 18. I.D. Landau (1979) Adaptive control: The model reference approach, M. Dekker, New York 19. A. Levant (2003) Higher-order sliding modes, differentiation and output-feedback control. Int J Control 76: 924-=941, Special Issue on Sliding-Mode Control 20. A. Levant (2004) Finite Differences in Homogeneous Discontinuous Control.Proc. of the 44th IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas 14–17 21. A. Levant (2005) Homogeneity Approach to High-order Sliding Mode Design. Automatica41: 823–830 22. G. Liu, A.S.I. Zinober and Y.B. Shtessel (2006) Output Delay Systems Tracking Using System Centre Approach and Sliding Mode Control. Proc CTS-HYCON Workshop, Paris, July 2006 23. E.P. Ryan and M. Corless (1984) Ultimate boundedness and asymptotic stability of a class of uncertain dynamical systems via continuous and discontinuous feedback control. IMA J Math Control Information 1: 223–242 24. M.G. Safonov and R.Y. Chiang RY (1988) CACSD using the state-space L∞ theory – a design example. IEEE Trans Aut ContrAC-33: 477–479 25. M.G. Safonov, A.J. Laub and C.L. Hartmann (1981) Feedback properties of multivariable systems: the role and use of the return difference matrix. IEEE Trans Aut ContrAC-26: 47–65 26. S.L. Shah, D.G. Fisher and D.E. Seborg (1975) Eigenvalue/eigenvector assignment for multivariable systems and further results output feedback control. Electron Letters 11: 388–389 27. I. Shkolnikov and Y.B. Shtessel (2002) Tracking in a Class of Nonmimumum Phase Systems with Nonlinear Internal Dynamics via Sliding Mode Control using Method of System Centre. Automatica: 38: 837–842 28. Y.B. Shtessel, A.S.I. Zinober and I. Shkolnikov (2003) Sliding Mode Control for Nonlinear Systems with Output delay via Method of Stable System Centre. Journal of Dynamic Systems, Measurement, and Control 125: 29. Y.B. Shtessel, A.S.I. Zinober and I. Shkolnikov (2003) Sliding Mode Control of Boost and Buck-boost Power Converters using Method of Stable System Centre. Automatica 39: 1061–1067 30. Y.B. Shtessel and I. Shkolnikov (2003) Aeronautical and Space Vehicle Control in Dynamic Sliding Manifolds. International Journal of Control 25: 253–257 31. V. Sinswat and F. Fallside (1977) Eigenvalue/eigenvector assignment by state-feedback. Int J Control26: 389–403 32. S.K. Spurgeon, M.K. Yew, A.S.I. Zinober and R.J. Patton (1990) Model-following control of time-varying and nonlinear avionics systems, in Deterministic control of uncertain systems, ed. Zinober ASI, Peter Peregrinus Press, 96–114 33. V.I. Utkin (1977) Variable structure systems with sliding mode. IEEE Trans Aut ContrAC-22: 212–222 34. V.I. Utkin (1978) Sliding Modes and Their Application in Variable Structure Systems, MIR, Moscow 35. V.I. Utkin (1992) Sliding Modes in Control Optimization, Springer-Verlag, Berlin 36. V.I. Utkin and K.D. Yang (1978) Methods for Constructing Discontinuity Planes in Multidimensional Variable Structure Systems. Autom. Remote Control 39: 1466–1470

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37. V. Utkin, J. Guldner, Shi J (1999) Sliding Modes in Electromechanical Systems, Taylor and Francis, London, 38. C.A. Woodham (1991) Eigenvalue Placement for Variable Structure Control Systems, PhD Thesis, University of Sheffield 39. C.A. Woodham, A.S.I. Zinober (1990) New Design Techniques for the Sliding Mode. Proc IEEE International Workshop on VSS and their Applications, Sarajevo, 220–231 40. C.A. Woodham, A.S.I. Zinober (1991) Eigenvalue Assignment for the Sliding Hyperplanes. Proc IEE Control Conference, Edinburgh, 982–988 41. C.A. Woodham, A.S.I. Zinober (1991) Robust Eigenvalue Assignment Techniques for the Sliding Mode. IFAC Symposium on Control System Design, Zurich, 529–533 42. C.A. Woodham, A.S.I. Zinober(1993) Eigenvalue placement in a specified sector for variable structure control systems. Int J Control57: 1021–1037 43. A.S.I. Zinober (1990), Deterministic control of uncertain systems ed, Peter Peregrinus Press, London 44. A.S.I. Zinober, O.M.E. El-Ghezawi and S.A. Billings (1982) Multivariable variablestructure adaptive model-following control systems. Proc IEE129D: 6–12

10 Control Engineering and Systems Biology Burton W. Andrews and Pablo A. Iglesias

Summary. Engineers use feedback, both positive and negative, to perform a wide array of signaling functions. Biological systems are also faced with many of the same requirements In this tutorial we examine examples from different cellular signaling systems to show how biology also uses feedback paths to perform many of the same tasks.

10.1 Introduction Though the components differ greatly, there are striking similarities between cellular organisms and engineered, man-made objects. In both classes of systems, most functions are performed by operational modules consisting of a wide range of interacting subsystems [1]. In biology, these modules consist of many classes of molecules. In fact, as argued in [1], much of twentieth-century biology attempted to reduce biological phenomena to the behavior of individual molecules. Much effort has been spent in determining the identity and specific function of molecules responsible for different phenomelogical processes. This is a necessary first step to reverse engineering biology, and is one that has produced great results [2]. However, just as the individual components of an electronic circuit do not uniquely identify the system’s function, neither do the molecules of a biological entity solely determine the cellular function. Understanding how these biological modules achieve their function will require both knowledge of the identities of the molecules as well as how these components are arranged. Recently, the study of biology at this system level has received considerable attention, giving rise to the field of “systems biology” [3, 4]. Interest in understanding biology in terms of a system-theoretic viewpoint is not new — Wiener’s cybenetics provides an early viewpoint [5] — yet the interest in the field from biologists is. Inherent in all systems biology approaches is the notion that biological function does not arise out of individual genes or proteins, but that it is the concerted interaction of different components that gives rise to the observed phenomena [3]. Thus, to understand a biological system, it is necessary not only to grasp the role played by each of the individual components, but also how these individual biochemical species interact to form the signaling network. M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 267-288, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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To elucidate these networks, systems biology relies on several complementary approaches. One is the so-called high-throughput model of biological investigation. High-throughput techniques allow the automated perturbation of large numbers of components (eg. genes) thus probing the cellular response to different changes in a systematic way. A second arrow in the systems biology quiver is the use of computational methods which allow biologists to test conceptual models that involve more than a few variables. More importantly, they provide predictive power by generating hypotheses that can be tested experimentally. The new focus on biology at the systems level affords an immense opportunity for control and systems engineers. One common aspect of systems and control analysis is that it works with systems at an abstract mathematical level that does not rely on specific knowledge of the components. As such, subsystems can represent electronic, mechanical, chemical or biochemical components — or combinations thereof. In this paper we consider mathematical models of several well-known signaling pathways. We show how feedback connections are crucial in achieving the desired function. In doing so we highlight how control engineering tools may prove to be fruitful tools for reverse engineering biology.

10.2 Negative Feedback Negative feedback loops are used in engineering systems to provide stability, to regulate systems and provide a level of robustness. As we demonstrate in this section, these same requirements give rise to negative feedback loops throughout biology.

10.2.1 Negative Feedback: Regulation Many problems in control can be framed as regulation problems, where the goal is to design a controller that can drive the output of a system that has been perturbed by possibly nonmeasurable disturbances to a constant set point [6]. Output regulation is also crucial for the functioning of all biological systems, where it is known as homeostasis, from the simplest bacteria to humans. Not surprisingly, feedback control is essential in achieving homeostasis. For example, the production of tryptophan, an essential amino acid, is regulated in bacteria by a series of three feedback loops [7]. Mammals rely on negative feedback loops to maintain blood calcium and glucose levels within tight bounds for their survival [8, 9].

Feedback Inhibition In biology, the term feedback inhibition is used when the catalytic function of an enzyme acting early in a reaction pathway is inhibited by a downstream product of that pathway [10]. This is illustrated by the system in Figure 10.1. This functional cascade appears in many cellular systems. One of the better known is the MAPK (Mitogen-Activated Protein Kinase) class of pathways that are found in a diverse array of signaling sytems1 . In this “module,” the 1

A kinase is an enzyme that transfers a phosphate group to substrates. This phosphorylation process is one of the fundamental switches in biology. The enzyme that removes the phosphate group is known as a phosphatase.

Control Engineering and Systems Biology

269

Fig. 10.1. Feedback inhibition in a cascade of biochemical reactions. In this system, several proteins stimulate downstream signals in a linear fashion (S stimulates X, which stimulates Y , which stimulates Z.) Feedback inhibition is present when one of the downstream molecules inhibits an earlier process. In this case, Z reduces the production rate of X. Examples in biology include the large class of mitogen-activated protein kinase (MAPK) systems that are found throughout biology.

MAPK (pronounced MAP kinase, and corresponding to element Z in Figure 10.1) is activated by the MAPKK (MAP kinase kinase, element Y ) which is activated by the MAPKKK (MAP kinase kinase kinase, element X). Comprehensive analyses of MAPK systems can be found elsewhere [11, 12]. Here, we assume that the MAPK pathway can be described by the system of differential equations2 d[X] = −k−1 [X] + k1 [S], dt d[Y ] = −k−2 [Y ] + k2 [X], dt d[Z] = −k−3 [Z] + k3 [Y ]. dt

(10.1) (10.2) (10.3)

This is, of course, the cascade of three linear first order systems and its transfer function, from signal S to output Z is: k1 k2 k3 . (10.4) (s + k−1 )(s + k−2 )(s + k−3 ) If we assume that a negative feedback exists between Z and the signaling molecule S, we replace (10.1) by d[X] = −k−1 [X] + k1 [S] − k[Z], (10.5) dt and obtain the new closed-loop transfer function: k1 k2 k3 . (s + k−1 )(s + k−2 )(s + k−3 ) + kk1 k2 k3

(10.6)

What difference can this feedback make? Clearly, at steady-state the gain of the system k1 k2 k3 kk1 k2 k3 + k−1 k−2 k−3 is a decreasing function of k. However, note that for small t (corresponding to |s| ↑ ∞ in (10.4) and (10.6)), the gain of the system is not affected. In effect, negative feedback leaves the 2

In computational biology, it is customary that square brackets around the name of a biochemical species denote its concentration.

270

B. W. Andrews and P.A. Iglesias

no feedback

160

Response

120

k = 0.005 80

k = 0.01 k = 0.02

40

0 0

10

20

30

40

50

60

Time (sec)

Fig. 10.2. Step response for the cascade system in Figure 10.1 and described by the equations (10.1–10.5). Parameters used are k1 = k2 = k3 = 1, k−1 = 0.1, k−1 = 0.2, k−1 = 0.3 and variable feedback gain k.

initial response unchanged, but decreases the steady-state. This inhibition has the effect of qualitatively changing the response of the system; see Figure 10.2. Without feedback, the system response rises to a persistent signal; with feedback, the response of the system is more transient.

Growth Factor Signaling We have just demonstrated how negative feedback can give rise to a transient signal. It is interesting to consider a specific biological example where this observation is of importance, in particular, in resolving a perplexing observation in the response in a cell line that is used extensively for studying nerve growth factor3 (NGF) signaling [13]. In these cells, two different growth factors, nerve growth factor (NGF) and epidermal growth factor (EGF), stimulate the same MAPK cascade. However, whereas EGF stimulation induces a rapid transient activation of the MAPK cascade, NGF stimulus results in sustained activation. Though these two behaviors had been observed experimentally, the reason behind these contrasting responses was unknown. Using a computer simulation of growth factor-induced MAPK cascade activation, Brightman and Fell determined that it is differences in the feedback regulation of the signals initiated by EGF and NGF that determines the duration of cascade activation [13]. Their model, of course, is considerably more complicated than that described above, as the system includes many more elements than the system of Figure 10.1 and the model uses more realistic dynamics than those used here. Nevertheless, the basic systems analysis is the same: negative feedback can lead to transient responses and attenuation of signal responses at steady-state.

3

Growth factors are proteins that bind to transmembrane receptors and are used to signal cells to activate differentiation or proliferation.

Control Engineering and Systems Biology

L + R GGDPαβγ

271

RL GGTPα + Gβγ

downstream effectors

RGS GGDPα

Fig. 10.3. Reactions describing the GPCR response. Ligand (L) binds to the GPCR (R) leading to an occupied receptor (RL). This induces the dissociation of the G-protein into its Gα - and Gβ γ subunits. The latter stimulates the synthesis of the RGS proteins. RGS proteins increase the rate of GGTP hydrolysis. Once GGDP is formed, the subunits reunite into a trimer. α α

G-protein Signaling We consider another ubiquitous cellular signaling mechanism that relies on negative feedback to generate transient responses to external stimuli. G-protein4 coupled receptors (GPCRs) are found in the signaling networks of eukaryotic organisms from single-celled amoebae and yeast to humans, where it is used in the eye (rhodopsin) as well as in the nose. GPCRs form the most common type of receptors in mammalian cells — it estimated that about 3–5% of genes in the genome are used to encode such receptors. Incredibly, about half of all known drugs target GPCRs [10]. GPCRs consist of a transmembrane component that traverses the cell membrane seven times. The intracellular component is coupled to G-proteins consisting of three subunits: Gα , Gβ and Gγ . The Gα subunit is bound to either GTP or GDP. In its inactive state, GGDP αβ γ , all three subunits are coupled, and Gα is bound to GDP. Binding of a stimulus to the transmembrane receptor causes Gα to release GDP and bind to GTP, and the Gα and Gβ γ subunits to separate. The Gβ γ subunit then signals downstream processes. The dissociation of the G-protein subunits is reversible. Hydrolysis5 of GTP leads to the formation of GGDP which reassociates α with free Gβ γ . This process can be accelerated by proteins known as regulators of G-protein signaling (RGS). Hao and coworkers developed a computational model for the effect of negative feedback on GPCR signaling that highlights the effect of negative feedback [14]. In this model, described in Figure 10.3, the dynamics of Gβ γ can be described by the ordinary differential equation
GDP d[Gβ γ ] GTP GTP = k1 + k1′ [RL] [Gαβ γ ] − k2 [Gα ] − k3 [RGS][Gα ] dt

= k1 + k1′ [RL] [Gβ γ ]T − [Gβ γ ] − k2 [GαGTP ] − k3 [RGS][GGTP α ], 4

5

(10.7)

G-protein is short for guanine nucleotide binding proteins. These are signaling proteins that are activated through the exchange of guanosine diphosphate (GDP) for guanosine triphosphate (GTP). In this case, hydrolysis refers to the exchange of GTP with GDP.

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where we have assumed that GTP hydrolysis is the rate limiting step in reformation of GGDP αβ γ and the total concentration of Gβ γ is constant: GDP [Gβ γ ]T = [Gβ γ ] + [Gαβ γ ]. GDP in the absence of stimulus, We have also assumed that some Gβ γ is produced from Gαβ γ indicated by the rate constant k1 . We assume the total Gα concentration is constant and given by GTP GDP [Gα ]T := [GGDP αβ γ ] + [Gα ] + [Gα ].

Moreover, by the assumption that GTP hydrolysis is the rate limiting step in reformation of GDP rapidly combines with G , we can assume that [GGDP ] ≈ 0 so that GGDP βγ α αβ γ , so that Gα GDP GDP GDP [GαGTP ] = [Gα ]T − [Gαβ γ ] − [Gα ] ≈ [Gα ]T − [Gαβ γ ].

Thus, [GGTP α ] ≈ [Gα ]T − [Gβ γ ]T + [Gβ γ ].

Substituting this into (10.7) leads to:



d[Gβ γ ] = k1 + k1′ [RL] [Gβ γ ]T − [Gβ γ ] − (k2 + k3 [RGS]) [Gβ γ ], dt

where, in the last line, we assumed that [Gβ γ ]T ≈ [Gα ]T . We rewrite this last equation as:

d[Gβ γ ] = k1 + k1′ [RL] [Gβ γ ]T − k7 + k1′ [RL] + k3 [RGS] [Gβ γ ], dt

(10.8)

where k7 := k1 + k2 . We now focus on the feedback path. Assume that

d[RGS] = k4 + k5 [Gβ γ ] − k6 [RGS], dt

(10.9)

where k4 is a constant rate of RGS production in the absence of stimulus, k5 is the stimulusinduced rate of RGS production, and k6 is the degradation rate. The complete system of equations can be rewritten as:

d[Gβ γ ] = k1 + k1′ [RL] [Gβ γ ]T − k7 + k1′ [RL] + k3 [RGS] [Gβ γ ], dt d[RGS] = k4 + k5 [Gβ γ ] − k6 [RGS]. dt Making the substitutions x :=

[Gβ γ ] , [Gβ γ ]T

the system becomes

τ := k7 t,

u :=

k1′ [RL], k7

and

y :=

k7 [RGS], k5 [Gβ γ ]T

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1.0

A

273

B

1.0

g(x,y,u)=0 0.8

0.6

0.6

f(x,y,u)=0, k=1

x

no feedback

0.8

f(x,y,u)=0, k=0

x

k=1 0.4

0.4

0.2

0.2

k = 10

f(x,y,u)=0, k=10 0

1.0

1.5

2.0

0

0

1

2

3

4

5

Time (sec)

y

Fig. 10.4. Nullclines (A) and time response (B) for the GPCR signaling system, described by (10.10) and (10.11). Parameters chosen are u = 1, α = 0.5, and β = γ = 1. dx = f (x, y, u) := (α + u) − (1 + u + ky) x, dτ dy = g(x, y, u) := β + x − γ y, dτ where

α :=

k1 , k7

β :=

k4 , k5 [Gβ γ ]T

γ :=

k6 k7

and

k :=

(10.10) (10.11)

k3 k5 [Gβ γ ]T k72

.

The equilibrium is determined by the intersection of the two nullclines 0 = f (x, ¯ y, ¯ u) ¯ and 0 = g(x, ¯ y, ¯ u); ¯ or α + u¯ x¯ = (10.12) = γ y¯ − β , 1 + u¯ + ky¯ as shown in Figure 10.4, along with the time response, for varying k. As the feedback strength increases, the equilibrium value decreases demonstrating the regulating role of RGS on the system response.

10.2.2 Negative Feedback: Sensitivity and Robustness A common use of negative feedback in engineered systems is to reduce the effect of parameter variation on the system response.

Gene regulation We consider a simple example, due to Vinnicombe [15], of feedback reduction of sensitivity to parameter variation during gene regulation. The system can be described by the reactions: c

α

0/ −→ mRNA −−→ 0/ where [mRNA] represents the concentration of a particular messenger RNA (mRNA) being translated by DNA at a constant average rate c. Degradation is assumed at constant rate α . The mean rate of change of [mRNA] can be described by the differential equation

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d[mRNA] = c − α [mRNA]. dt

(10.13)

At steady-state, the production and degradation rates are equal, and the average number of molecules of mRNA is [mRNA]0 = c/α . Small changes, δα , to the parameter α lead to a change δ[mRNA] = [mRNA] − [mRNA]0 in the number of mRNA molecules. Linearizing (10.13) about α and [mRNA]0 yields dδ[mRNA] dt

= −αδ[mRNA] − [mRNA]0 δα ,

which has a steady-state value of

δ[mRNA] δα ≈− . [mRNA]0 α Thus, a small percentage change in the degradation parameter α leads to approximately the same percentage change in the steady-state number of mRNA molecules. If a consistent response is needed by the cell, this effect is undesirable because variations in parameters within an individual cell or across a population lead to response variations of the same order. A solution to the above sensitivity problem is to decrease these fluctuations through negative feedback. If we assume that the mRNA codes for a DNA-binding protein that promotes further transcription of the mRNA, then the system becomes f ([mRNA])

mRNA α

∅

(10.14)

where the transcription rate f ([mRNA]) is a function of the number of mRNA molecules. In this case, small perturbations in α yield dδ[mRNA] dt

= ( f ′ ([mRNA]0 ) − α )δ[mRNA] − [mRNA]0 δα ,

where the linearization is about [mRNA]0 and α . With feedback, the relative steady-state number of mRNA molecules is   δ[mRNA] 1 δα ≈− . [mRNA]0 1 − f ′ ([mRNA]0 )/α α If f ′ ([mRNA]0 ) < 0, then decreasing the slope (more negative) reduces the sensitivity of the system output to changes in the parameter α . While perhaps overly simplistic, this example provides a useful insight into sensitivity reduction in biological networks through negative feedback. Other, more realistic examples are described in [16–19].

Bacterial Chemotaxis Robust systems maintain consistent, stable behavior in the face of uncertainties; both internal (eg. component variability) and external (eg. disturbances). Robustness has been found to exist in many biological systems [20–24].

Control Engineering and Systems Biology

I

k1

D+L

R+L k-1

k-r

E

kr

k-d

I

RL

275

k2 k-2

kd

DL E

Fig. 10.5. General four-state receptor model of receptor-mediated adaptation in E. coli. The receptors can be found bound (RL and DL) or not (R and D). Methylation by CheR, labeled I, inactivates the receptors. Similarly, demethylation, by CheB, labeled E, activates the system.

To illustrate the connection between biological robustness and control engineering, we consider the chemotactic response of Escherichia coli. E. coli is a rod-shaped bacteria, aproximately one micron in diameter and two microns long, that is propelled by rotation of 6–10 flagella. These bacteria move towards high concentrations of chemical attractants (a process known as chemotaxis) by monitoring and responding to temporal changes in attractant concentration. The signaling network governing this response is one of the best-understood signaling networks in all of biology. The response to a step change in chemoattractant consists of an initial transient followed by a return to the prestimulus level, a property known to biologists as adaptation. Adaptation allows cells to detect changes in the level of attractant and hence assess the values of the direction the cell is moving. This adaptation has been shown to be robust, in the sense that the bacteria will return to their prestimulus state even when internal conditions are varied greatly [20, 25]. To illustrate the control-theoretic aspects of robust adaptation, we consider a general framework for adaptation that is representative of the signaling network found in E. coli [26]. Chemotactic receptors can exist in one of two configurations, which we term either modified or unmodified. Biologically, these correspond to whether receptors are methylated or not6 . Methylation and demethylation are mediated by two enzymes: CheR and CheB, respectively. Because methylation “inactivates” the receptor, we will refer to CheR and CheB as the inactivator (I) and excitation (E) processes, respectively. Additionally, these receptors can be occupied by the chemoattractant molecules (so-called, ligand-bound) or not. Thus, as illustrated in Figure 10.5, receptors can be found in one of four states: R, RL, D and DL. To develop a numerical model of this system, we makes some assumptions about how the enzymes act on their substrates. An enzymatic reaction takes a substrate, S, and an enzyme, E, and creates a product, P, without consuming the enzyme. A classical model of enzyme catalysis, due to Michaelis–Menten, states that the product of an enzymatic reaction appears at the rate: [S][E] d[P] ∝ . dt kM + [S] 6

Methylation refers to the replacement of a hydrogen atom (H) with a methyl group (CH3 ).

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The constant, kM , is known as the Michaelis–Menten constant [27]. Note that there are two distinct regimes in this equation. If the amount of substrate is small relative to kM then: d[P] ∝ [S][E]. dt This is known as the linear regime, as the rate of production is linear in both substrate and enzyme concentrations. If, however, the amount of substrate is large relative to kM , then d[P] ∝ [E]. dt This is known as the saturated regime, in that extra substrate will not increase the rate of production. In our model of receptor-mediated adapation in bacterial chemotaxis, we make contrasting assumptions on the enzymatic production of CheB and CheR: that excitation (CheB) acts at saturation but that inhibition (CheR) acts in the linear regime. We make two more assumptions. We assume that fractions of the unmodified receptors, α1 R and α2 RL are active, and that the total activity of the system is given by the sum of these two fractions. Moreover, the inactivator CheB acts only on this fraction of receptors. Finally, we assume that the rate at which CheB acts on modified and unmodified receptors is equal: k−1 = k−2 in Figure 10.5. Based on these assumptions, we can describe the system dynamics with the following set of differential equations: d[R] = k−1 [E] − k1 [I]α1 [R] − kr [R][L] + k−r [RL], dt d[RL] = k−2 [E] − k1 [I]α2 [RL] + kr [R][L] − k−r [RL], dt d[D] = −k−1 [E] + k1 [I]α1 [R] − kd [D][L] + k−d [DL], dt d[DL] = −k−2 [E] + k2 [I]α2 [RL] + kd [D][L] − k−d [DL]. dt Note that the first two equations are decoupled from the second two. Making the variable transformation: [A] := α1 [R] + α2 [RL], which corresponds to the system’s activity, and [B] := leads to:

where

d dt



[R] + [RL] , k1 [I]

      [A] −a1 ([L], [I]) a0 ([L], [I]) [A] b([I]) = + [A]0 , [B] −1 0 [B] 1 a0 ([L], [I]) := (α1 k−r + α2 kr [L] + α1 α2 k1 [I])k1 [I], a1 ([L], [I]) := kr [L] + k−r + (α1 + α2 )k1 [I], b1 ([I]) :=

(α1 k−1 + α2 k−2 )k1 [I] , k−1 + k−2

[A]0 = [E](k−1 + k−2 )/k1 [I].

(10.15)

Control Engineering and Systems Biology

x

277

y

Fig. 10.6. A. Autocrine loop of the epidermal growth factor receptor (EGFR) system. Epidermal growth factor (EGF) binding to EGFRs trigger the MAPK response. This, in turn, leads to the secretion of EGF completing a positive feedback loop. B. Linear system incorporating a positive feedback loop.

The system has one stable steady-state at [A] = [A]0 , which is independent of [L]. Thus, adaptation is a robust feature of the signaling network, and the steady-state response is robust to changes in ligand. It is worth highlighting how the system achieves adapation. If we consider the second row of (10.15): Z d[B] = −([A] − [A]0 ) ⇐⇒ [B](t) = − e(t)dt, dt | {z } e(t), error

we see that an integral feedback loop is being implemented where the activity [A](t) is being compared to the “set-point” [A]0 . The error [A] − [A]0 is then integrated. This is an example of the internal model principle: step changes in the external “disturbance” — which in this case is the chemoattractant concentration — can be rejected in a robust manner because the system has an integrator inside the loop [23]. The experimental verification of this robustness has been confirmed [20].

10.3 Positive Feedback 10.3.1 Positive Feedback: Amplification In engineering systems, the use of positive feedback for signal amplification dates back to the early 1900’s when Edwin Armstrong discovered the gain benefits of positive feedback for a triode amplifier [28]. Amplification through positive feedback is also common in biological systems. We consider one example: the epidermal growth factor receptor (EGFR) system where positive feedback from an autocrine loop — a loop where a cell secretes some of the very signal that stimulates its signaling network — leads to an amplified MAPK response [29]. Here we present an illustrative model that provides insight into the means by which positive feedback can induce gain. A detailed model of the EGFR system can be found in [29].

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Response

0.4

0.3

Increasing k 0.2

0.1

0

0

5

10

15

20

25

Time (sec)

Fig. 10.7. Response of the linear system (10.16) for varying values of the positive feedback gain k to a one-second pulse of unit amplitude. Parameter values used are α = 1 and k = 0.2, 0.4, 0.6 and 0.8.

Consider the system shown in Figure 10.6A. A primary stimulus (eg., an exogenous growth factor or some intracellular stimulant) and membrane receptor-bound ligands (eg. transforming growth factor [TGF]) serve as inputs to an intracellular signaling pathway (the extracellular signal-regulated MAPK cascade in the EGFR system). The output of the signaling pathway activates ligand-releasing proteases7 . The released ligand can then rebind to surface receptors creating a positive feedback loop. To illustrate the effect of such a feedback, we describe the signaling pathway by the transfer function 1 H(s) = , (s + α )2 which is a reasonable linear model of a two-level MAPK cascade [30]. The autocrine loop is modeled as a simple positive feedback gain k as in Figure 10.6B. The transfer function from stimulus x to output y is Y (s) 1 . (10.16) = X(s) (s + α )2 − k Figure 10.7 shows the response of this system to a pulse of duration one second, a perturbation similar to that studied in [29], for increasing values of k. In the EGFR system, increases in the effect of positive feedback may arise from factors such as an increase in protease activity (increased ligand release rate) or an increase in the number of receptors [29]. Figure 10.7 shows that positive feedback yields amplification and an increased duration of the response of the system. This effect is representative of observations both experimentally and in theoretical models of the EGFR system [29, 31] and demonstrates the use of positive feedback as an amplifier in biological systems.

10.3.2 Positive Feedback: Switching and Memory Positive feedback is often used in engineering applications to induce switching behavior, usually characterized by an all-or-nothing response to an input. The response remains relatively constant for some range of input values but changes abruptly once the input reaches some threshold value; see Figure 10.8. Monostable switches have a single stable (input-dependent) 7

A protease is an enzyme that breaks bonds between the amino acids of proteins.

Control Engineering and Systems Biology

B

Output

Output

A

279

stable “high” equilibrium

unstable stable “low” equilibrium

Input

Input

Fig. 10.8. Monostable (A) and bistable (B) switches

steady-state response at all input values (Figure 10.8A) while bistable switches exhibit hysteresis (Figure 10.8B) and are often said to have memory due to the delayed response to input transitions. An example is the Schmitt trigger, which uses an operational amplifier and positive feedback to implement a bistable circuit [32]. cooperative feedback

S

A

A*

Fig. 10.9. Model representative of the positive feedback found in the oocyte maturation signaling network

Many cells rely on switch-like behavior for response regulation [33–38], enabling the discretization of responses to graded inputs and, in some cases, making the response irreversible. For example, positive feedback loops have been shown to yield irreversible maturation8 of Xenopus oocytes (frog egg cells) when stimulated with sufficient amounts of progesterone. In particular, a MAPK, known as p42, and the protein kinase Cdc2, both of which are activated during oocyte maturation, are further activated through positive feedback. Active p42 MAPK increases the activity of both p42 MAPK and Cdc2 via intermediate proteins, and active Cdc2 increases the activity of both kinases as well [35]. A simplified model of this system, taken from [35], is shown in Figure 10.9. A signaling protein A is activated both by an external stimulus S (progesterone in the Xenopus oocyte system) and by positive feedback from the active form A∗ of A. If the feedback is cooperative9 with Hill coefficient n, the system response can be described by the differential equation 8

9

Maturation refers to the process in which egg cells reduce the number of chromosomes in half. Thus leaving them with a single copy of each chromosome. This enables them to be fertilized. An enzymatic reaction is said to be cooperative if the rate of production is proportional to n + [S]n ). The exponent n is known as the Hill coefficient. the term [S]n [E]/(kM

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0.02

A

s3

B

s3

Synthesis rate Degradation rate 0.01

s2

s2

s1

s1

0 0.02

C

s3

D

s3

0.01

s2

s2

s1

s1

0 0

0.5

1

0

Active response (A*)

0.5

1

Active response (A*)

Fig. 10.10. The strength of the positive feedback loop gain can change the number of equilibria. These panels plot the degradation (dotted line) and synthesis rates (solid lines) for (10.17) as a function of the active response [A∗ ]. Equilibria for (10.17) are characterized by points of intersection. The synthesis rate is plotted for three different constant signaling strengths: s1 = 0.001, s2 = 0.005 and s3 = 0.02. Each panel represents a different feedback strength: k1 = 0, 0.05, 0.15 and 0.25 in panels A–D respectively. For no or low feedback strengths, each of the external signal values leads to a single equilibrium (panels A and B). For higher values of the feedback gain, multiple equilibria can exist — note that for s2 in panel C, three equilibria appear. The two black circles correspond to stable equilibria, the open circle is an unstable equilibrium. Smaller input signals lead to a low response; higher input signals to a high output. Finally, for even greater feedback gains (panel D), these multiple equilibria exist only for small input values. The location of the equilibria can be plotted as a function of the signaling strength s as in Figure 10.11. [A∗ ]n d[A∗ ] = s([Atot ] − [A∗ ]) + k1 n ([Atot ] − [A∗ ]) − k−1 [A∗ ] , dt K + [A∗ ]n | {z } | {z } degradation synthesis

(10.17)

where k−1 is the degradation rate of A∗ , K is the Michaelis–Menten constant of the feedback reaction, k1 is the strength of the feedback and [Atot ] := [A] + [A∗ ]. As shown in Figure 10.10, as the strength of the feedback gain is increased, multiple equilibria can arise. These are shown in Figure 10.11 as a function of stimulus concentration for varying feedback gains. As the feedback strength increases, the system progresses from a monostable (panels a–c) to a bistable, but reversible switch (panels d–e). Systems in this regime will show hysteretic responses to the external stimulus s. For greater feedback strengths (panel f), the

Control Engineering and Systems Biology

281

1

Response (A*)

0.5

A

k1=0.04

B

k1=0.07

C

k1=0.11

D

k1=0.15

E

k1=0.18

F

0.02

0.04

k1=0

0 1

0.5

0 0

0

0.02

0.04

0

0.02

0.04

Stimulus (s) Fig. 10.11. Bistability and irreversibility of the system of Figure 10.9, representative of the behavior of Xenopus oocyte maturation. Parameters used are [Atot ] = 1, K = 1, k−1 = 0.01, and n = 5 with s ranging from 0 to 0.04.

switch becomes irreversible. Once the input signal increases beyond the low-to-high transition point, the system is in the high state. Because the high-to-low transition is not admissible (it is negative, which is not biologically plausible) the system can not return to the low state. Thus, this is appropriate for cell fate decisions that cannot be reversed, such as differentiation and maturation.

10.3.3 Positive Feedback: Oscillations One of the uses of positive feedback in engineering is the generation of oscillatory behavior. Originally discovered around 1915 [28], positive feedback-induced oscillators are found in numerous engineering applications. Many biological systems also rely on oscillatory behavior for various cellular functions [39]. A system which has drawn considerable interest is the oscillatory signaling network controlling cAMP in the amoebae Dictyostelium discoideum [40–46]. When nutrients are abundant, Dictyostelium are single-celled amoebae that live in soil and feed on bacteria. However, when the food supply is exhausted, they begin a developmental process that lasts approximately 24 hours. In this time they develop a mechanism that enables them to secrete cAMP10 periodically. These oscillatory signals synchronize and generate propagating cAMP waves that can be observed under a microscope [47]. Cells react to these waves by chemotaxing towards the source of cAMP. There, up to 100 000 cells aggregate and form a multicellular organism. This aggregate forms a slug with the ability to migrate towards light (phototaxis). Eventually, a fruiting body consisting of a stalk and spore are formed. The approximately 5 000 cells in the 10

Cyclic Adenosine MonoPhosphate; pronounced “cyclic AMP,” is a common second messenger in biology.

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Fig. 10.12. Oscillation-inducing autocrine loop found in Dictyostelium

spore are inactive but remain viable and can revert to their single-cell existence in the presence of new nutrients [48]. The circuit underlying oscillatory cAMP signaling in Dictyostelium is based on competing positive and negative feedback loops, as shown in Figure 10.12, which is adapted from [46,49]. Cells sense cAMP from the external using GPCRs. This sensed cAMP is used by cells to chemotax toward the sensed signal. Though the exact chemotaxis pathway is different, relying on a spatial, rather than temporal signal, the cell uses an adaptation mechanism similar to that in the bacterial chemotaxis pathway to process the sensed signal [50–53]. Signals downstream of this receptor are used to stimulate synthesis of intracellular cAMP which is then secreted by the cell, thereby completing an autocrine loop. We now show how this interplay of positive and negative feedback in the chemotactic signaling network of Dictyostelium leads to these oscillations [46]. As with the bacterial chemotaxis pathway, the signaling network of Dictyostelium can be modeled as a response regulator R that is excited and inhibited by processes E and I, respectively [51, 54, 55]. The dynamics of the active form of R are described by the differential equation d[R∗ ] = −k−r [I][R∗ ] + kr [E][R] dt = −(k−r [I] + kr [E])[R∗ ] + kr [E][R]T ,

(10.18)

where [R]T := [R] + [R∗ ]. The excitation and inhibition processes are each regulated by the external stimulus S (cAMP in the case of Dictyostelium): d[E] = −k−e [E] + ke [S], dt d[I] = −k−i [I] + ki [S]. dt

(10.19) (10.20)

As shown in [46], the system described by (10.18), (10.19) and (10.20) forms an integral control mechanism with negative feedback that yields a steady-state [R∗ ] that is independent of [S]. The concentration of cAMP within the cell, [C], is governed by

Control Engineering and Systems Biology

283

4

3

y g(x,y)=0 2

limit cycle 1

f(x,y)=0 0

0

1

2

x

3

4

Fig. 10.13. Phase-plane analysis for the oscillator of (10.25) and (10.26) following [46]. The parameters a = 1/2, b = 1/6 and ε = 0.01 were chosen so as to make the equilibrium point at x = y = 4/9 unstable. The nullcline for f (x, y) increases sharply as x ↓ 0 allowing for the application of the Poincar´e–Bendixson theorem. d[C] = −k1 [C] − k2 [C] + k3 [R∗ ]2 + k4 , dt

(10.21)

where the first term represents degradation, the second secretion, the third stimulus-induced synthesis, and the last constitutive synthesis. The quadratic dependence on [R∗ ] accounts for amplification in the signaling pathway [52, 56–58]. The cAMP concentration outside the cell, S (also the stimulus in (10.19) and (10.20)) is governed by d[S] = −k5 [S] + k6 [C]. dt

(10.22)

For simplicity, we assume fast dynamics of [R∗ ], [E] and [C] and replace these variables with their corresponding steady-states values to obtain d[I] = −a1 [I] + a2 [S], dt d[S] a4 [S]2 + a6 , = −a3 [S] + dt (a5 [S] + [I])2 where a1 := k−i , a2 := ki , a3 := k5 , a4 :=

k3 k6 ke2 kr2 [R]2T , 2 k2 ) (k1 + k2 )(k−e −r

a5 := kr ke /(k−r k−e ),

and a6 := k4 k6 /(k1 + k2 ). The change of variables x := and τ := a3 t, yields:

a22 a3 [S], a21 a4

y :=

a2 a3 [I], a1 a4

(10.23) (10.24)

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dx x2 = f (x, y) := −x + + ε, dt (ax + y)2 dy = g(x, y) := −b(y − x), dt

(10.25) (10.26)

where a = a1 a5 /a2 , b = a1 /a3 , and ε = a22 a6 /(a21 a4 ). It can be shown that the steady-state x = y ≈ 1/(1+a)2 is unstable iff b ≤ (1−a)/(1+a), and by the Poincar´e–Bendixson theorem, a stable limit cycle exists under this condition [46].

10.4 Discussion Feedback is used throughout engineering to accomplish a wide range of objectives. In this paper we have tried to demonstrate how biology, faced by many of the same functional requirements, has also relied on feedback regulation. Our goal has been to highlight how understanding how engineering systems are synthesized can help in understanding biological systems. We believe that this is one way in which control engineering can make a significant contribution to systems biology. As expected, we are starting to see this effect [52, 59–63]. Control engineers will also have an impact through the development of new theoretical and computational tools to study models. For example, by applying computational algorithms from robust control theory, it has been possible to evaluate and select between proposed models [24, 64–68]. Finally, it is our expectation that by having control engineers actively working in biology, the field of control systems will be enriched. In particular, these new systems will give rise to interesting control theory, thus closing the intellectual loop between biology and control systems. Again, we are beginning to see this effect [69, 70].

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11 Robust Control of a Distillation Column Da-Wei Gu

Summary. In this Chapter we present the design of a robust control system for a high-purity distillation column. The original nonlinear model of the column is of high order and it includes parametric gain and time delay uncertainty. A low order linearized distillation column model is used in design of a two-degree-of freedom (2DOF) H∞ loop shaping controller and a µ controller. Both controllers ensure robust stability of the closed loop system and fulfillment of a mixture of time domain and frequency domain specifications. A reduced order µ -controller is then found which preserves the robust stability and robust performance of the closed loop system. The simulation of the closed loop system with the nonlinear distillation column model shows very good performance for different reference and disturbance signals as well as for different values of the uncertain parameters.

11.1 Introduction Distillation is an important process in the separation and purification of chemicals. The process exploits the difference at boiling points of multi-component liquids. The control of distillation columns is difficult, because the distillation process is highly nonlinear and the corresponding linearized models are often ill-conditioned around the operating point. The aim of the design, presented in this chapter, is to find a controller which achieves robust stability and robust performance of the closed loop control system of a high-purity distillation column. The original nonlinear model of the column is of 82nd order and it includes uncertainties in the form of parametric gains and time delay. The uncertainty model is considered in the form of an input multiplicative complex uncertainty. In our design exercises, we found that it is difficult to achieve the desired performance of the closed loop system using one-degree-of-freedom controllers. Hence we turned to H∞ two-degree-of-freedom loop shaping design procedure and µ -synthesis/analysis method. The designs are based on a 6th order linearized distillation column model. Both designed controllers ensure robust stability of the closed loop system and achieve a mixed set of time domain and frequency domain † This

is based in part on Chapter 11 of Robust Control Design with Matlab by D.W. Gu, M. Konstatinov and P. Petkov, Springer, 2005.

M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 289-328, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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specifications. We present several time domain and frequency domain characteristics of the corresponding closed loop systems which makes possible the comparison of controllers efficiency. An 11th order reduced order µ -controller is found which preserves the stability and performance of the closed loop system in the presence of uncertainties. The simulation of the closed loop system with this µ controller and with the nonlinear distillation column model is R conducted in Simulink and shows very good performance for different reference and disturbance signals as well as for different values of the uncertain parameters.

11.2 Dynamic Model of the Distillation Column

Overhead vapour VT

Condenser

MD

P

Condenser holdup

Reflux N N-1

Distillate D, yD

L

Feed F, zF

3 2 1 Reboiler holdup

Boilup V Reboiler

Bottom product B, xB

Fig. 11.1. The distillation column system A typical two-product distillation column is shown in Figure 11.1. The objective of the distillation column is to split the feed F, which is a mixture of a light and a heavy component with composition zF , into a distillate product D with composition yD , which contains most of the light component, and a bottom product B with composition zB , which contains most of the heavy component. For this aim the column contains a series of trays that are located along its height. The liquid in the columns flows through the trays from top to bottom, while the vapor in the column rises from bottom to top. The constant contact between the vapor and liquid leads to increasing the concentration of the more volatile component in the vapor, while

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simultaneously increasing the concentration of the less volatile component in the liquid. The operation of the column requires that some of the bottom product is reboiled at a rate of V to ensure the continuity of the vapor flow and some of the distillate is refluxed to the top tray at a rate of L to ensure the continuity of the liquid flow. The notations used in the derivation of the column model are summarized in Table 11.1 and the column data are given in Table 11.2. Table 11.1. Column nomenclature Symbol F zF qF D and B yD and xB

Description Feed rate [kmol/min] feed composition [mole fraction] fraction of liquid in feed distillate (top) and bottom product flowrate [kmol/min] distillate and bottom product composition (usually of light component) [mole fraction] L reflux flow [kmol/min] V boilup flow [kmol/min ] N number of stages (including reboiler) Ntot = N + 1 total number of stages (including condenser) i stage number (1 - bottom, NF - feed stage, NT - total condenser) Li and Vi liquid and vapor flow from stage i [kmol/min] xi and yi liquid and vapor composition of light component on stage i Mi liquid holdup on stage i [kmol] (MB - reboiler, MD -condenser holdup) α relative volatility between light and heavy component τL time constant for liquid flow dynamics on each stage [min]

Table 11.2. Column Data N Ntot NF 40 41 21 B L V 0.5 2.70629 3.20629

F 1 yD 0.99

zF 0.5 xB 0.01

qF D 1 0.5 Mi τL 0.5 0.063

The index i denotes the stages numbered from the bottom (i = 1) to the top (i = Ntot ) of the column. Index B denotes the bottom product and D the distillate product. A particular highpurity distillation column with 40 stages (39 trays and a reboiler) plus a total condenser is considered. The nonlinear model equations are: 1. Total material balance on stage i

292

D.-W. Gu dMi /dt = Li+1 − Li +Vi−1 −Vi .

2. Material balance for light component on each stage i d(Mi xi )/dt = Li+1 xi+1 +Vi−1 yi−1 − Li xi −Vi yi . This equation leads to the following expression for the derivative of the liquid mole fraction dxi /dt = (d(Mi xi )/dt − xi (dMi /dt))/Mi .

3. Algebraic equations The vapor composition yi is related to the liquid composition xi on the same stage through the algebraic vapor-liquid equilibrium yi = α xi /(1 + (α − 1)xi ).

From the assumption of constant molar flows and no vapor dynamics, one obtains the following expression for the vapor flows Vi = Vi−1 . The liquid flows depend on the liquid holdup on the stage above and the vapor flow as follows Li = L0i + (Mi − M0i )/τL + λ (Vi−1 −V 0i−1 )

where L0i [kmol/min] and M0i [kmol] are the nominal values for the liquid flow and holdup on stage i and V 0i is the nominal boilup flow. If the vapor flow into the stage effects the holdup then the parameter λ is different from zero. For the column under investigation λ = 0.

The above equations apply at all stages except in the top (condenser), feed stage and bottom (reboiler). 1. For the feed stage, i = NF (it is assumed that the feed is mixed directly into the liquid at this stage) dMi /dt = Li+1 − Li +Vi−1 −Vi + F, d(Mi xi )/dt = Li+1 xi+1 +Vi−1 yi−1 − Li xi −Vi yi + FzF .

2. For the total condenser, i = Ntot (MNtot = MD , LNtot = LT ) dMi /dt = Vi−1 − Li − D,

d(Mi xi )/dt = Vi−1 − Li xi − Dxi .

3. For the reboiler, i = 1(Mi = MB ,Vi = VB = V )

d(Mi xi )/dt = Li+1 xi+1 −Vi yi − Bxi . As a result we obtain a nonlinear model of the distillation column of 82nd order. There are two states per tray, one representing the liquid composition and the other one representing the liquid holdup. The model has four manipulated inputs (LT ,VB , D and B) and three disturbances (F, zF and qF ). In order to find a linear model of the distillation column it is necessary to have a steady state operating point around which the column dynamics is to be linearized. However the model

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contains two integrators, because the condenser and reboiler levels are not under control. To stabilize the column, we make use of the so called LV-configuration of the distillation column where we use D to control MD and B to control MB . This is done by two proportional controllers with both gains equal to 10. The nonlinear model is linearized at the operating point given in Table 11.2 (the values of F, L,V, D, B, yD , xB and zF ). These steady state values correspond to an initial state where all liquid compositions are equal to 0.5 and the tray holdups are also equal to 0.5 [kmol]. The steady state vector is obtained for t = 5000 min by numerical integration of the nonlinear model equations of the LV-configuration given in the M-file cola lv.m. The linearization is carried out by implementing the M-file cola lin which makes use of the equations given in the file cola lv lin.m. The 82nd order, linear model is stored in the variable G4 u and has four inputs (the latter two are actually disturbances) [LT VB F zF ] and two outputs [yD xB ]. Before reducing the model order, the model G4 u is scaled in order to make all inputs/disturbances and all outputs at about the same magnitude. This is done by dividing each variable by its maximum change, i.e., u = U/Umax ; y = Y /Ymax where U, Y are the input and output of the model G4 u in original units, Umax , Ymax are the corresponding maximum values allowed, and u, y are the scaled variables. The scaling is achieved by using the input scaling matrix   10 0 0 0 1 0 0   Si =   0 0 0.2 0  0 0 0 0.1 and output scaling matrix

So =



 100 0 . 0 100

The scaled model is then found as G4 = SoG4 uSi . The final stage in selecting the column model is the order reduction of the scaled model G4 . This is done by using the commands sysbal and hankmr. As a result we obtain a 6th order model saved in the variable G. All commands for finding the 6th order liner model of the distillation column are contained in the file mod col.m. The frequency responses of the singular values of G are compared with the singular values of the 82nd order linearized model G4 in Figure 11.2. It is seen that the behaviour of both models is close until the frequency 2 rad/min.

11.3 Uncertainty Modelling The uncertainties considered in the distillation column control systems are a gain uncertainty of ±20% and a time delay of up to 1 min in each input channel. Thus the uncertainty may be represented by the transfer matrix

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D.-W. Gu Frequency responses of the original plant and shaped plant

6

10

5

10

4

10

3

Magnitude

10

2

10

1

10

0

10

−1

solid line: singular values of G

10

dashed line: singular values of G4

−2

10

−3

10

−4

10

−3

10

−2

−1

10

10 Frequency (rad/min)

0

1

10

2

10

10

Fig. 11.2. Singular values of G and G4 Gd

W∆

∆1

1

u1

+

+ y1

∆2

W∆ 2 u2

G +

+

y2

Fig. 11.3. Distillation column with input multiplicative uncertainty

Wu =



0 k1 e−Θ1 s 0 k2 e−Θ2 s



where ki ∈ [0.8 1.2]; Θi ∈ [0.0 1.0]; i = 1, 2. It is convenient to represent this uncertainty by an input multiplicative uncertainty as shown in Figure 11.3 with   ∆1 0 ∆= 0 ∆2 where |∆1 | ≤ 1, |∆2 | ≤ 1. The uncertainty weighting function   W∆1 0 W∆ = 0 W∆2 is determined in the following way. Denote by W ui = 1 the nominal transfer function in the ith channel for ki = 1 and Θi = 0; i =1, 2.

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According to Figure 11.3 we have that Wui = (1 +W∆i ∆i )W ui , i = 1, 2. Taking into account that |∆i | ≤ 1 it follows that the relative uncertainty should satisfy Wu ( jω ) − W u ( jω ) i i ≤ W∆i ( jω ) , i = 1, 2 W u ( jω ) i

where Wui ( jω ) = ki e jωΘi = ki (cos(ωΘi ) + j sin(ωΘi )). In this way, to choose the uncertainty weight W∆i is equivalent to determine an upper bound of the frequency response of the relative uncertainty Wu ( jω ) − W u ( jω ) q i i = (ki cos(ωΘi ) − 1)2 + (ki sin(ωΘi ))2 . W u ( jω ) i

The frequency responses of of the relative uncertainty

Approximation of uncertain time delay by multiplicative perturbation 2.5

2

Magnitude

1.5

1

0.5

0 −2 10

−1

10

0

10 Frequency (rad/min)

1

10

2

10

Fig. 11.4. Approximation of the uncertain time delay

Wu ( jω ) − W u ( jω ) i i W u ( jω ) i

are computed by the file unc col.m and shown in Figure 11.4. These responses are then approximated by 3rd order transfer functions by using the file wfit.m. As a result one obtains W∆i =

2.2138s3 + 15.9537s2 + 27.6702s + 4.9050 , i = 1, 2. 1.0000s3 + 8.3412s2 + 21.2393s + 22.6705

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11.4 Closed-Loop System Performance Specifications The aim of the distillation column control system design is to determine a controller which meets robust stability and robust performance specifications for the LV configuration. Since these specifications are difficult to be satisfied with a one-degree-of-freedom controller, we present the design of two-degree-of-freedom controllers which ensure robust stability and robust performance of the closed loop system. In the given case, the robust stability means guaranteed closed loop stability for all 0.8 ≤ k1 , k2 ≤ 1.2 and 0 ≤ Θ1 , Θ2 ≤ 1 minute. The time domain specifications are given in terms of step response requirements, which must be met for all values of k1 , k2 , Θ1 and Θ2 . Specifically, for a unit step command to the first input channel at t = 0, the scaled plant outputs y1 (tracking) and y2 (interaction) should satisfy: • • • • •

y1 (t) ≥ 0.9 for all t ≥ 30 min; y1 (t) ≤ 1.1 for all t; 0.99 ≤ y1 (∞) ≤ 1.01; y2 (t) ≤ 0.5 for all t; −0.01 ≤ y2 (∞) ≤ 0.01.

Correspondingly, similar requirements should be met for a unit step command at the second input channel. In addition, the following frequency domain specification should be met: •

•

ˆ jω ) < 316, for each ω , where Kˆ y denotes the feedback part of the unscaled conσ (Kˆ y S)( troller. (Here and further on, a variable with hat refers to the case of unscaled plant.) This specification is included mainly to avoid saturation of the plant inputs. ˆ jω ) ≤ 1, for ω ≥ 150. σ (Gˆ Kˆ y )( jω ) < 1, for ω ≥ 150; or σ (Kˆ y S)(

In the above, σ denotes the largest singular value, and Sˆ = (I + Gˆ Kˆ y ) < 1 is the sensitivity ˆ function for G. Wu

W∆

r

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u

eu

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y+

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ey

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Fig. 11.5. Closed-loop interconnection structure of the Distillation Column system The block diagram of the closed-loop system incorporating the design requirements consideration represented by weights is shown in Figure 11.5. The plant enclosed by the dashed

Robust Control of a Distillation Column

297

rectangle consists of the nominal scaled model G plus the input multiplicative uncertainty. The controller K implements a feedback from outputs yD and xB and a feedforward from the reference signal r. The measurement of the distillate and bottom products composition is corrupted by the noise n. The desired dynamics of the closed loop system is sought by implementation of a suitably chosen model M. The model M represents the desired dynamic behavior of the closed-loop system from the reference signal to the outputs. The usage of a model of the desired dynamics allows to take easily into account the design specifications. The transfer function matrix of the model M is selected as # " 1 0 T s2 +2ξ T s+1 . M= 1 0 T s2 +2ξ T s+1

The coefficients of the transfer functions (T = 6, ξ = 0.8) in both channels of the model are chosen such that to ensure an overdamped response with the settling time of about 30 min. The off-diagonal elements of the transfer matrix are set as zeros in order to minimize the interaction between the channels. Model frequency response

0

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The frequency response of the model M is shown in Figure 11.6. Let the scaled, two-degree-of-freedom controller be partitioned as   K(s) = Ky (s) Kr (s) .

where Ky is the feedback part of the controller and Kr is the pre-filter part. It is easy to show that      ˜ r − M) −Wp TWn Wp (SGK r ep = ˜ −1 Kr −Wu Ky SWn n eu Wu (I + Ky G)

298

D.-W. Gu

˜ y )−1 is the sensitivity function for the scaled plant, T = (I + GK ˜ y )−1 GK ˜ y where S = (I + GK ˜ is the complementary sensitivity function and G = G(I +W∆ ∆ ) is the uncertain, scaled plant model. The performance objective is to satisfy

  ˜ r − M) −Wp TWn

Wp (SGK

1. The uncompensated and compensated open-loop responses are presented in Figure 15(a), with the latter emphasising the OL integral action at low frequencies. The difficulties in the early-type classical nulling strategy are the following: (i) the sensor exists within the control loop resulting in interactions between controller and suspension dynamics, and (ii) the system is also non-minimum phase and there is a stability threshold point set by the non-minimum phase zero, which result in major limitations in the controller design. Later on in the section we will see how we can improve the performance of nulling-type schemes with the use of modern control methods.

Command-driven with Precedence Control The problems with the early-type nulling schemes and also with heavy low pass filtering in attempting local command-driven strategies (to reduce the effect of bogie high frequency com-

Modelling and Control of Railway Vehicle Suspensions

Body roll gyroscope (abs roll rates) @ 58 m s

Body lateral acceleration @ 58 m s 2

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Fig. 13.14. Early-type nulling PI scheme for deterministic track

ponents in the feedback signal used), led to the currently used by industry command-driven with precedence scheme (the concept can be seen in Figure 13.8(b)). For illustrative purposes we consider the command-driven with precedence scheme in Figure 13.16. It employs an accelerometer on the leading bogie of the leading vehicle (a preview of 29m was assumed) provided the curving acceleration signal, passed via a 0.45Hz low pass filter. The signal is then processed to provide 60% compensation for the lateral acceleration (K is set to 0.6). The actuator is controlled using an additional feedback of tilt angle (i.e., the suspension roll). For appropriate comparison to the local tilt strategies, the filter delay was chosen to match the precedence time, however this can be changed to emphasise precedence information if necessary. The transfer function of the LP filter is given by HLP2 (s) =

w2c2 , wc2 = 2π × 0.45, ζ2 = 0.707 s2 + 2ζ2 wc2 + w2c2

(13.20)

392

A.C. Zolotas and R.M. Goodall Bode Diagrams T , S, KS

Nichols Chart 20

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(b) Complementary Sensitivity, Sensitivity and Control Sensitivity

Fig. 13.15. Early-type nulling PI system frequency responses

and the time delay introduced is tdLP =

2ζ2 wc2 w2c2

, which for the current case is 0.5s. Thus for

the precedence to match the filter delay, it takes l = 58( ms ) × 0.5s = 29m precedence, i.e., approximately 1.5 vehicle length. Note that the tilting response for the leading vehicle will unavoidably be too late.
1 The leading vehicle controller was chosen as K pi (s) = 1 + s0.5 . For the trailing vehicle, the controller designed to actively tilt the body is a PI compensator in series with a low-pass filter (LPF)⋆⋆ , in order to remove high frequencies from the secondary suspension roll (those are introduced due to the bogie roll contribution). The overall controller transfer function is   1.5 + s0.75 400 Ktotal (s) = × 2 s0.5 s + 28.28s + 400 The compensated and uncompensated open loop together with the overall compensator frequency response can be viewed in Figure 17(a). The corresponding sensitivity and complementary sensitivity of the closed loop system are presented in Figure 17(b), where it is evident that the control action influences the system over a wider range of frequencies compared to the classical nulling case. A set of time-domain results for the deterministic track case is shown in Figure 13.18, where it is obvious that the precedence scheme is superior to the classical nulling approach. The tilt controller performance is presented in Table 13.4, and it is closer to the ideal performance as expected in all cases. In the stochastic case there is an improvement in ride quality by the active system, because the precedence time matches the filter delay meaning that the reference and track input are uncorrelated, thus the tilt command will compensate for long wavelengths.

⋆⋆

The LP filter will be redundant in the case of actuator dynamics, i.e., actuator will have limited bandwidth by default.

Modelling and Control of Railway Vehicle Suspensions

393

Precedent Vehicle LPF

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direction of travel 1.5 Vehicle length preview (~29m) Leading

Trailing Tilt comm.

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for simulation

(b) Interpretation for simulation purposes Fig. 13.16. Command-driven with precedence approach

Emphasising more precedence information improves the deterministic performance, subject of course to the amount of precedence used, i.e., too much precedence (over-precedence) can be disastrous for the normal operation of the train (tilt action will apply on straight track segments much sooner than the intended start of the curve!). In addition, the amount of precedence used will influence the stochastic ride quality either in a positive or negative way depending upon the correlation of the signals (in the case where the precedence time differs from the filter delay, the reference and track input signals are no longer uncorrelated). It should be noted that, even in the precedence schemes, sensors located on each vehicle (i.e., local sensors) are used to ensure the correct operation of the overall tilting system (the sensors are always present for safety purposes).

A.C. Zolotas and R.M. Goodall

394

Bode Diagrams OL (δa ) to θ2sr

Bode Diagrams T , S

Mag. (dB)

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Fig. 13.17. Command-driven with precedence system frequency responses Table 13.4. Command-driven with precedence - assessment @ 58(m/s) D ETERMINISTIC - steady-state - R.M.S. deviation error - peak value Roll gyroscope - R.M.S. deviation - peak value (P-factor) - peak jerk level - standing - seated S TOCHASTIC Passenger comfort - R.M.S. passive (equiv.) - R.M.S. active - degradation Lateral accel. (actual vs ideal)

9.53 1.54 12.18 0.018 0.104 6.80 47.62 13.455

(%g) (%g) (%g) (rad/s) (rad/s) (%g/s) (% of passng) (% of passng)

3.78 (%g) 3.31 (%g) -12.12 (%)

13.2.5 Nulling-Type Tilt Via Robust Control Techniques In this section we will see how to improve the performance nulling-type schemes by employing robust control methods, in order to achieve comparable results to the command-driven with precedence scheme. In particular the case study involves an LQG/LTR approach [11] and a multiple objective H∞ /H2 scheme [12].

LQG/LTR Nulling-Type Tilt Control Linear Quadratic Gaussian control is well documented in [8], [10], defining the following state-space plant model

Modelling and Control of Railway Vehicle Suspensions

395

Body roll gyroscope (abs roll rates) @ 58 m s

Body lateral acceleration @ 58 m s 1.2

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actual "ideal"

active actual active "ideal" prof passive actual

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(d) Suspension Deflection

Fig. 13.18. Command-driven with precedence on deterministic track

x˙ = Ax + Bu + Γ w

(13.21)

y = Cx + v,

(13.22)

where w, v are (ideally) white uncorrelated process and measurement noises which excite the system, and are characterised by covariance matrices W,V respectively. The separation principle can be applied to first find the optimal control u = −Kr x which minimises (13.23) Z T  1 J = lim E [xT Qx + uT Ru]d τ , (13.23) T →∞ T 0 where Kr = R−1 BT X and X is the positive semi-definite solution of the following Algebraic Riccati Equation (ARE)      I A −BR−1 BT X −I =0 (13.24) T X −Q −A

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Next find the optimal state estimate xˆ of x where xˆ = Axˆ + Bu + K f (y −Cx) ˆ (13.25) n o to minimise E [x − x] ˆ T [x − x] ˆ . The optimal Kalman gain is given by K f = YCT V −1 and Y is the positive semi-definite solution of the following ARE      AT −CT V −1C I Y −I =0 (13.26) Y −Γ W Γ T −A

Weighting matrices Q (pos. semidefn.), R (pos. defn.) for control, and W (pos. semidefn.), V (pos. defn.) for estimation, can be tuned to provide the desired result. Note that it is also possible to follow the dual procedure, i.e., solve for the state estimate sub-problem and next for the optimal gain sub-problem (although this is not considered in our study). For synthesizing the tilt controller we consider a simple extension to the classical nulling ′ , with approach in an optimal control framework by deriving the SISO model, from δa to θdm all disturbance signals set to zero. Thus, on measurement is effectively use for the Kalman filter. We will synthesize the controller using the weighting matrices Q, R, W, V purely as tuning parameters until an appropriate design is obtained. In particular, the structure of the LQG tilt compensator is found by shaping the principal gains of the system, i.e., return ratios. First the LQR is synthesized via Q, R to obtain a satisfactory a satisfactory return ratio −Kr (sI − A)−1 B, with the Kalman Filter designed via W,V such that the return ratio at the input of the compensated plant converges sufficiently close to −Kr (sI − A)−1 B over the frequency range of interest (Loop Transfer Recovery to recover as much as possible of the robust properties of LQR). For disturbance rejection and/or reference tracking (which is zero in this case), the system ′ . should be augmented using an extra state, the integral of the effective cant deficiency θdm This approach will produce an optimal controller with integral action [8]. Hence, the system is becomes        x˙ A 0 x B = + u (13.27) x˙′ C′ 0 x′ 0 R

′ and C′ is the selector matrix for integral action and is found from θ ′ = C′ x. where x′ = θdm dm The control signal has the form  
x u = − K p Ki (13.28) x′

We start with the simplest possible choices for Q, R   0 0 Q = 9×9 9×1 , R = 1 01×9 qi

(13.29)

thus adjusting only the weight of the integral state, and imposing no constraints on the remaining states. Figure 13.19 illustrates the return ratio −Kr (sI − A)−1 B for various qi . We choose the return ratio for qi = 100 with a crossover of approx 20rad/s, to recover for in the next steps. However a simple calculation of the transmission zeros for the design plant reveals a non-minimum phase zeros at approximately 6.0rad/s (this

Modelling and Control of Railway Vehicle Suspensions

397

is characteristic for such a setup in tilting trains [12]), thus making full recovery cumbersome. For illustration, the usual LTR procedure is followed up to the limit of recovery allowed from the non-minimum phase zero (usually the achievable bandwidth of the system is less than half of the RHP zero frequency [10]).

−1

K (sI−A) B; q =[.1 1 5 10 100 500 1e3] r

i

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Fig. 13.19. Return ration −Kr (sI − A)−1 B for various qi Moreover, for the design of the Kalman Filter, the zero eigenvalue of the augmented A matrix needs to be placed just to the left of the origin for the solution to exist. For controller implementation this should move back to the origin for proper integration. equal to B, 1 respectively (still for the SISO model). Setting Γ = B refers to any (virtual) disturbances on the plant acting via the input, rather than the actual track disturbances from the track. The sensor noise covariance matrices is set to V = 1 (can be reduced to characterise better quality measurements), while the process noise covariance matrix is set to W = Wo + wI ˜ (Wo = 0). Figure 13.20 illustrates the amount of recovery at plant output for increasing values of w. ˜ It is seen that there is no point in recovering after w˜ = 500 as appropriate integral action is already recovered and the actual crossover limit is placed by the non-minimum phase zero. Figure 13.21 presents the principal gains of the designed closed loop system w.r.t. sensitivity S and complementary sensitivity T . The bandwidth of the system is rather low (approx. 1rad/s due to the NMP zero) however there is some compensation at higher frequencies which cater for few stochastic components (on straight track ride). The synthesized LQG controller realization is given by   s A − BKr − K f C K f Klqg = −Kr 0

(13.30)

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A.C. Zolotas and R.M. Goodall

LQG/LTR @ input for w=[.1 1 5 10 100 500 1e3] 200

150

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K (s)G(s) lqg

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Fig. 13.20. LTR at plant output for increasing w˜ and is 10th order and its frequency response can be seen in Figure 13.22. This can be further reduced either in an open loop or closed sense [10], however at the expense of performance quality.

Table 13.5. LQG nulling-type control assessment @ 58(m/s) D ETERMINISTIC - steady-state - R.M.S. deviation error - peak value Roll gyroscope - R.M.S. deviation - peak value (PCT-factor) - peak jerk level - standing - seated S TOCHASTIC Passenger comfort - R.M.S. passive (equiv.) - R.M.S. active - degradation

Lateral accel. (actual vs ideal)

9.53 4.09 17.3 0.033 0.101 8.98 65.76 19.95

(%g) (%g) (%g) (rad/s) (rad/s) (%g/s) (% of passng) (% of passng)

3.78 (%g) 3.97 (%g) 4.83 (%)

The time domain results for the lateral acceleration felt by the passengers and the related body tilt angle can be seen in Figure 13.23, while the performance assessment of the controller is presented in Table 13.5. It is seen that although the LQG-based is a simple straightforward

Modelling and Control of Railway Vehicle Suspensions

399

Principal Gains 10

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Fig. 13.21. Designed system Sensitivity and Compl. Sensitivity with LQG controller optimal extension of the classical nulling scheme, the performance is much improved (emphasizing robustness with the additional damping injected). It is worth mentioning that the performance of the controller can be further improved by using extra sensor information, i.e., passenger acceleration, body roll rate, vehicle yaw rate [13]. The design system in this case will be non-square with more outputs than inputs, as a result the LQG controller will be non-square (more inputs than outputs). However, we can still synthesize via the separation principle, but make the system square for LTR [8].

13.2.6 Multi-objective H∞ /H2 Nulling-Type Control Via LMIs It is well known that the H∞ norm of a system represents the worst-case energy transfer between (bounded energy) disturbances to (bounded-energy) regulated outputs, and as a result can be conservative when disturbances are naturally modelled as persistent or white noise signals. In cases where the interests falls upon minimising the RMS value of a regulated output, the H2 norm [10] of the corresponding closed-loop transfer function is a more appropriate measure of stochastic performance. This section considers a multiple-objective H∞ /H2 via LMIs [14] design method for the local nulling-type tilt control [15]. Details on some preliminary concepts related to the following design procedure can be found in Appendix B. The design objectives are formulated as an optimisation problem, defined in the generalisedregulator setting shown in Figure 13.24, where P(S) and K(s) are the generalised plant (inclusive of all weighting factors) and the controller to be designed. The vector of external disturbances was set to w = [w1 w2 ]T , where w1 denotes ( R1 ) the (deterministic) track-curvature (low-frequency) disturbance signal and w2 is the (stochastic) lateral track position (higherfrequency) signal yo . Scaling factors Wi1 ,Wi2 emphasise the relative weighting between the

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two disturbances for the design. Note that in this case we choose to employ two outputs y1 and y2 which are the measured body lateral acceleration and the secondary suspension roll angle, respectively (chosen for the output vector to replicate the sensors used in the basic classical nulling control). It is worth noting that the system transfer function with the two aforementioned measurements is not NMP. It is very important to meet both deterministic (curve track) and stochastic (straight track) requirements, thus the following multi-objective optimisation problem was formulated min α kW1 Tz1 w k2∞ + β kW2 Tz2 w k22

K∈S

(13.31)

in which S denotes the set of all internally stabilising controllers. The first regulated output ′ . For z1 for infinity-norm minimisation, was chosen as the effective cant deficiency z1 = θdm the minimisation of the 2-norm, z2 was chosen as the control input u denoting the actuator roll angle δa . Regulating z1 to zero corresponds to 60% tilt compensation and thus attains the desired (steady-state) level of acceleration on steady state curve. Tzi w (i = 1, 2) denotes the (closed-loop) transfer functions between signals w and z1 , z2 respectively. Multi-objective optimisation typically refers to the joint optimisation of a vector consisting of two or more functions, typically representing conflicting objectives. Common types of multi-objective optimisation problems include “Pareto-optimal” (non-inferior) optimality criteria, minimax optimality criteria, etc. In the context of this exercise, the term “multi-objective” refers simply to the fact that the cost function of the optimisation problem involves two different types of norms, capturing the deterministic and stochastic objectives of the design. The two different norms

Modelling and Control of Railway Vehicle Suspensions Body acceleration 2

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401

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z1 z2

W1

Wi1

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K Fig. 13.24. Generalised Regulator configuration for multi-objective control

1. Constrained w minimisation: w Minimise wW1 Txy w2 subject to kW2 Tzw k∞ < γ , 2. Unconstrained w minimisation: w Minimise β wW1 Txy w2 + α kW2 Tzw k∞ , and 3. Feasibility problem: Find a stabilising K(s) (if one exists) such that w w wW1 Txy w ≤ γ1 and kW2 Tzw k ≤ γ2 ∞ 2 Txy and Tzw represent two general closed-loop transfer functions, weighted via W1 and W2 . Scalars α and β , in (13.31), are positive definite design parameters which may be used to shift the emphasis of the optimisation problem between the minimisation of the kTz1 w k∞ term (deterministic objective) and the kTz2 w k2 term (stochastic objective). The frequency-domain weights W1 and W2 have been chosen as: W1 (s) = 104 W2 (s) = 0.5

s 200 + 1 s 0.0001 + 1 s3 + 1.59s2 + 0.58s + 0.06

s3 + 13.81s2 + 38.4s + 2.98

(13.32) (13.33)

W1 is essentially a low-pass filter with a very low pole cut-off frequency (10−4 rads s ) and high gain at low frequencies, Figure 25(a). Thus W1 emphasises minimisation of the kTz1 w k∞ term in the low frequency range and effectively enforces integral control on the regulated output rads (z1 ). W2 is a high-pass filter with pole (10 rads s ) and zero (0.2 s ) cut-off frequencies. A rads rads lead/lag network is also included in W2 , in the range of [.1 s , 6 s ], which found to have a positive effect on controller design (by enhancing the cross-over frequency of W1 ,W2 ), Figure 25(a). By limiting the high-frequency components of the control input (z2 ), effectively places a limit on the closed-loop bandwidth of the system, which in turn limits the RMS acceleration on straight track (stochastic case). Additional benefits include a smoother control signal and improved robustness properties of the controller when the effects of uncertainty in P(s) and

Modelling and Control of Railway Vehicle Suspensions

403

in the actuator dynamics are taken into account. Moreover, the relative weighting between w1 and w2 were simply set to unity, i.e.     W 0 10 Wi = i1 = (13.34) 0 Wi2 01 Thus the energy of either of the signals is equally incorporated in the cost function. Increasing either Wi1 or Wi2 with respect to the other will put more emphasis on the deterministic or the stochastic track respectively. However, the current choice of Wi provides the best results.

Bode diagram

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Fig. 13.25. Multi-objective H∞ /H2 LMI approach scheme R The minimisation problem in (13.31) was solved in MATLAB using the LMI toolbox [14], i.e., representing the problem in a set of Linear Matrix Inequalities and follow a convex optimisation approach. This technique has very attractive computational properties and is widely used in systems and control theory. For controller design, the generalised plant was formulated as follows

x˙ = Ax + B1 w + B2 u z∞ = C∞ x + D∞1 w + D∞2 u

(13.35)

z2 = C2 x + D21 w + D22 u

(13.36)

y = Cy x + Dy1 w

(13.37)

where all matrices can be formed based upon the state space model of the system and the specifications in the generalised plant discusses earlier in this section. The controller can be R then found by using MATLAB function hinfmix(). The optimisation problem was solved for a few combinations of the α and β parameters and the results can be seen in Table 13.6. The results shown in the table clearly illustrate the fundamental trade-off between the deterministic and the stochastic objectives of the design. As expected, increasing the value of β relative to α places more emphasis on the stochastic aspects of the design, and as a result the RMS acceleration on straight track is further

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A.C. Zolotas and R.M. Goodall Table 13.6. α -β combinations for the H∞ /H2 problem

α β Ride Quality-Degrad.(%) Deviations-Determ.(%g) 1 1 21.7 1.95 1 2.5 10 2.15 1 5 4.95 2.37 1 10 3.4 2.62 1 20 2.1 2.9 Ride Quality-Degrad.: ride-quality degradation @58m/s of active system compared to passive @58m/s (straight track) Deviations-Determ.: RMS acceleration deviation from the ideal response of an ideal tilting controller @58m/s (curved track)

reduced. This is at the expense of deterministic performance and, therefore, the curved track response becomes slower (larger deviations from the ideal tilt response). Since it is required that stochastic performance deteriorates by no more than 7.5% compared to the passive system, R the “best” design was obtained for α = 1 and β = 5. The result returned in MATLAB for the “best” configuration is shown below (summary)

Optimization of 1.000 * G^2 + 5.000 * H^2 : ---------------------------------------------Solver for linear objective minimization under LMI constraints Iterations : Best objective value so far 1 . . . 16 * switching to QR 17 . . . . 35 36 1.018621e+005 37 1.018621e+005 38 6.830550e+004 39 6.830550e+004 40 5.029532e+004 41 5.029532e+004 42 4.475980e+004 43 4.226245e+004 *** new lower bound: 3159.072336 44 4.067042e+004 45 3.932891e+004 .

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. . 62

3.493807e+004 new lower bound: 3.437031e+004 63 3.488358e+004 *** new lower bound: 3.441555e+004 64 3.483742e+004 *** new lower bound: 3.450646e+004 Result: feasible solution of required accuracy best objective value: 3.483742e+004 guaranteed relative accuracy: 9.50e-003 f-radius saturation: 88.219\% of R = 1.00e+008 Guaranteed Hinf performance: 1.54e+002 Guaranteed H2 performance: 4.68e+001 ***

Note that, in the first few iterations, the algorithm does not find any solutions, however the solution converges soon after. The resulting controller is of 2-input/1-output dimension due to the two measurements used in the formulation. The singular value plot is shown in Figure 25(b), by definition depicting the largest singular of the two output/one input system transfer function. Note that the controller order is equal to 13, i.e., 9 states from the train model + 4 states from the weights. However, it can be easily reduced down to a 7th order equivalent, e.g., via balanced truncation [10], with minimal degradation in performance. The performance of the designed system is assessed in Table 13.7, where it can be seen that it is significantly improved compared to the classical nulling and LQG nulling-type control schemes. This exercises illustrate the usability of employing two measurements (compared to only one in the LQG scheme) and the effectiveness of distinguishing the design objectives in the cost function. For completeness the associated time history analysis for the design track is presented in Figure 13.26.

Table 13.7. H∞ /H2 multi-objective LMI approach @ 58(m/s) D ETERMINISTIC - steady-state - R.M.S. deviation error - peak value Roll gyroscope - R.M.S. deviation - peak value (PCT-factor) - peak jerk level - standing - seated S TOCHASTIC Passenger comfort - R.M.S. passive (equiv.) - R.M.S. active - degradation

Lateral accel. (actual vs ideal)

9.53 2.37 13.66 0.023 0.101 7.07 51.7 14.93

(%g) (%g) (%g) (rad/s) (rad/s) (%g/s) (% of passng) (% of passng)

3.78 (%g) 3.96 (%g) 4.95 (%)

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(c) Tilt angles Fig. 13.26. Design track time history results for H∞ /H2 design

13.2.7 Case Study Remarks This exercise has considered the design of local tilt controllers (a form of secondary railway suspension control) based upon advanced control concepts. The problems with earlytype classical nulling approaches has been presented, and briefly discussed the currently-used precedence strategy. It has been shown that by using modern control methods, i.e., LQG and H∞ based schemes, the performance of nulling-type controllers can be significantly improved. Problems with the two model-based schemes include high-order controller size and the choice of weighting functions. Controller reduction can be employed for the former, while the latter requires a realistic setup of the design problem (to reduce the complexity of choosing the structure of the weights) and usually designer experience (this being the case for the majority of engineering applications).

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13.3 Appendix A- Tilting Train Parameter Values and Notation

yv , yb , yo θv , θb , θr δa θo , R v mv ivr mb ibr g kaz ksz krz crz ksy csy kvr kpz cpz kpy cpy d1 d2 h1 h2 h3 hg2 hg1

Lateral displacem. of body, bogie, track Roll displacement of body, bogie, airspring reservoir ARB applied tilt Track cant, curve radius Vehicle forward speed Half body mass, 19,000(kg) Half body roll inertia, 25,000(kgm2 ) Bogie mass, 2,500(kg) Bogie roll inertia, 1,500(kgm2 ) gravitational acceleration, 9.81(ms−2 ) — Values per bogie side — Airspring area stiffness, 210,000( N m) Airspring series stiffness, 620,000( N m) Airspring reservoir stiffness, 244,000( N m) Airspring reservoir damping, 33,000( Ns m) Secondary lateral stiffness, 260,000( N m) Secondary lateral damping, 33,000( Ns m) Nm Anti-roll bar stiffness/bogie, 2,000,000( rad ) N Primary vertical stiffness, 2,000,000( m ) Primary vertical damping, 20,000( Ns m) Primary lateral stiffness, 35,000,000( N m) Primary lateral damping, 16,000( Ns m) Airspring semi-spacing, 0.90(m) Primary vertical suspension semi-spacing, 1.00(m) 2ndary lateral susp. height(body cog), 0.9(m) 2ndary lateral susp. height(bogie cog), 0.25(m) Primary lateral susp. height(bogie cog), -0.09(m) Bogie cog height(rail level), 0.37(m) Body cog height(rail level), 1.52(m)

13.4 Appendix B- H∞ Based Controllers: Preliminaries 13.4.1 Basic Notation A continuous time, linear time invariant, state space system is given by x(t) ˙ = Ax(t) + Bu(t)

(13.38)

y(t) = Cx(t) + Du(t)

(13.39)

p×n

p×m

n×m , C ∈ R and D ∈ Re . The above state space system is charwhere A ∈ Rn×n e e , B ∈ Re acterised by the following transfer function with dimension p × n

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G(s) = C(sI − A)−1 B + D

(13.40)

This thesis adopts the following conventional state-space representation to represent G(s)   A B s G(s) = (13.41) CD Note also that the complex conjugate of G(s) is given by   T T s −A −B G∗ (s) = GT (−s) = CT DT

(13.42)

and if G(s)G∗ (s) = I = G∗ (s)G(s) for all s ∈ jRe , then G(s) is said to be all-pass. Finally if matrix D is invertible, then G−1 (s) is given by⋆⋆⋆   −1 −1 s A − BD C BD −1 G (s) = −D−1C D−1

(13.43)

• Frequency Domain Spaces and Norms This part introduces the meaning of frequency domain spaces and norms of real rational, matrix valued, transfer functions. For a more comprehensive study the reader is referred to Zhou and Doyle [9]. Let R denote the space of all real rational transfer function matrices. The L2 /H2 norm of G(s) is given by r Z 1 ∞ kGk2 , (13.44) tr (G∗ ( jω )G( jω )) d ω 2π −∞ which is used to define the following spaces (i). RL 2 refers to the space of all real rational transfer function matrices with no poles on the imaginary axis and is characterised by a finite L2 norm. (ii). RH 2 defines the space of all transfer function matrices in RL 2 with no poles in Re(s) > 0. The L∞ /H∞ norm of G(s) is given by kGk∞ , sup σ [G( jω )] ω ∈Re

(13.45)

and (i). RL ∞ refers to the space of all real rational transfer function matrices with no poles on the imaginary axis (with finite L∞ norm). (ii). RH ∞ defines the space of all transfer function matrices in RL ∞ with no poles in Re(s) > 0. ⋆⋆⋆

Using the matrix inversion lemma: −1 −1 −1 −1 −1 (A1 + A2 A3 A4 )−1 = A−1 1 − A1 A2 (A4 A1 A2 + A3 ) A4 A1 .

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Also, the H∞ norm of a stable transfer function G(s) is its largest input/output RMS gain kGk∞ , sup

u∈L2 u6=0

kykL2 kukL2

(13.46)

where L2 is the space of signals having finite energy and y is the output of the system G for a given input u. Thus, for any input u of unit energy, the output energy in y is bounded by the H∞ norm of G(s).

• Linear Fractional Transformations The basic concept of Linear Fractional Transformations is outlined in this section. Linear Fractional Transformations (LFT) are frequently used in the area of H∞ optimisation as well as in other areas of control theory. They do represent a means of standardising a wide variety of feedback arrangements [10]. Let P(s) define a transfer function matrix with the following state-space representation   A B1 B2 s  P(s) = C1 D11 D12  (13.47) C2 D21 D22

which can be also partitioned as

s

P(s) = where



P11 P12 P21 P22



(13.48)

Pi j (s) = Ci (sI − A)−1 B j + Di j

z(s) ¾

(13.49)

¾ w(s)

P (S) ¾

¾

y(s)

u(s) - K(S)

-

Fig. 13.27. The Generalised Regulator Configuration Referring to Figure 13.27, which presents the generalised regulator configuration, the (lower) linear fractional transformation† of P and K is given by †

There is also the concept of the upper LFT which is employed in representing uncertainties in a system [10].

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(13.50)

for det(I −P22 K) 6= 0. P(s) represents the “generalised plant”, consisting of the nominal model G(s) combined with all frequency weightings appropriately chosen to shift the emphasis with frequency between different design objectives. In addition, the signals are: u the control variables, w the exogenous inputs such as disturbances wd and commands r, y the measured variables and z the regulated outputs, i.e., the signals need to minimise to meet the design objectives. In fact FL (P, K) represents the transfer function between w and z in Figure 13.27, i.e. h i z(s) = P11 + P12 K(I − P22 K)−1 P21 w(s) (13.51) H∞ and H2 control implies the minimisation of the H∞ -norm and the H2 -norm of FL (P, K) respectively.

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References 1. S. Iwnicki (Ed.) (2006) Handbook of Railway Vehicle Dynamics. Taylor and Francis 2. A. Wickens (2003) Fundamentals of Rail Vehicle Dynamics. Swets and Zeitlinger 3. R.M. Goodall RM and S. Brown (2001) Tilt technology still evolving as the cost falls. Railway Gazette International. 521–525 4. R.M. Goodall (1999) Tilting Trains and Beyond – the Future for Active Railway Suspensions: Part 1 Improving Passenger Comfort. Computing and Control Engineering Journal. August 153–159 5. R.M. Goodall, S. Bruni S and T.X. Mei (2005) Concepts and prospects for activelycontrolled railway running gear. Procs 19th IAVSD Symposium. Milano 6. J.T. Pearson, R.M. Goodall and I. Pratt (1998) Control System Studies of an Active AntiRoll Bar Tilt System for Railway Vehicles. Proceedings of the Institution of Mechanical Engineers Part F. 212(F1):43–60 7. R.M. Goodall, A.C. Zolotas and J. Evans (2000) Assessment of the Performance of Tilt System Controllers. Proceedings of the Railway Technology Conference IMechE. C580/028/2000. Birmingham UK. November 231–239 8. J.M. Maciejowski (1989) Multivariable Feedback Design. Addison-Wesley 9. K. Zhou and J.C. Doyle (1998) Essentials of Robust Control. Prentice-Hall 10. S. Skogestad and I. Postlethwaite (2000) Multivariable Feedback Control: Analysis and Desgn. Wiley and Sons 11. A.C. Zolotas, G.D. Halikias, R.M. Goodall and J. Wang (2006) Model Reduction Studies in LQG Optimal Control Design for High-Speed Tilting Railway Carriages. Proceedings of the 2006 American Control Conference. Minneapolis MN USA. June 1796–1801 12. A.C. Zolotas (2002) Advanced Control Strategies for Tilting Trains. PhD Thesis, Loughborough University, UK 13. A.C. Zolotas and R.M. Goodall (2005) Improving the tilt control performance of high-speed railway vehicles: an LQG approach. Proceedings of the 16th IFAC World Congress. Prague 14. P. Gahinet, A. Nemirovski, A.J. Laub and M. Chilali (1994) LMI Control Toolbox. Natick, MA, The MathWorks. 15. A.C. Zolotas, G.D. Halikias and R.M. Goodall (2000) A Comparison of Tilt Control Approaches for High Speed Railway Vehicles. Proceedings of the 14th International Conference on Systems Engineering ICSE 2000, Coventry UK. 2:632–636

14 Case Study on Anti-w indup Compensation Micro-actuator Control in a Hard-D isk Drive Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite

Summary. This chapter demonstrates the use of anti-windup compensation in the control loop of a micro-actuator which is nominally controlled by a linear, discrete robust controller. The micro-actuator is part of a hard disk drive dual-stage servo-control system for positioning of the read/write head. The actuator inputs are constrained to retain the micro-actuator’s displacement range of less than 0.4 µ m for mechanical protection. In the first part of the chapter, the anti-windup compensation scheme exemplifies an approach suggested by Weston & Postlethwaite [29]. Here, the scheme is posed as a discrete full-order compensator and the closed loop analysis uses a generalized circle citerion approach. The design of the compensator is posed in LMI-form. In the second part of the chapter, it is shown how the linear micro-actuator control loop with anti-windup compensation is incorporated into the non-linear servo-control scheme for positioning of the read/write head in a hard disk drive.

14.1 Introduction Micro-actuators have been gaining in importance in practical systems during recent years. For instance, it is nowadays a target to integrate the actuator, sensor and associated electronics for powerful computations into one device of micro or nano-scale. These technologies can be useful in optical communication systems, in electromechanical signal processing systems or in healthcare systems, such as BIO-MEMS for microchip-lab diagnostics or micro-devices for therapeutic targeting and delivery. Mechanical micro-actuators for nano-positioning are of interest to the University of Leicester because of their importance in the servo-controller used in the positioning of the read-write head in hard disk data-storage systems [5, 8, 11]. The micro-actuator is being used in a dual-stage control system to achieve high-banwidth positioning control. Significant research effort on hard-disk drive (HDD)-servo techniques has been invested in the area of dual-stage servo control [14, 17, 18]. The reason for this is the continuous increase in the track density and in storage capacity of HDDs. Recently, a track density of more than 420 kTPI (TPI- track per inch) has been demonstrated (see [2, 23]) in a laboratory environment M.C. Turner et al. (Eds.): Mathe. Methods for Robust & Nonlin. Ctrl., LNCIS 367, pp. 413- 430, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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Fig. 14.1. Schematic of Hard Disk Drive with PZT-actuator

and 149 kTPI density HDDs are nowadays available in the consumer market. In addition to the significant increase in data density, the increased demand for higher data rates requires the improved performance of the head-positioning servo system. A promising way to meet these demands is to augment the conventional voice coil motor (VCM) actuator with a second-stage, high bandwidth micro-actuator. Dual-stage servo systems in HDDs are now a feasible alternative to the single stage VCM-servo system. PZT-based micro-actuators using PZT-elements embedded in the head suspension are popular, e.g., the ‘FUMA’-actuator in [19] (Figure 14.1). However, the displacement range of secondary actuators is very limited, typically less than 1-2 µ m, and the input signal for the actuator is limited to prevent damage. In dual-stage servosystems, the two actuators have to deal with the following servo-tasks: seek/settling and track following. Seek/settling control has to ensure a fast movement of the read/write head from one track to another. For track following, high bandwidth controllers are necessary to ensure good error rejection capabilities to counteract disturbances. Therefore, the primary VCM-actuator is required for large displacements and the secondary actuator provides large bandwidth. For servo-control of such a dual-stage actuator system, the method of [11, 12] is employed. It is based on the well known decoupled dual-stage controller structure of [13], where design and stability for the primary VCM-control loop and the secondary PZT-control loop are guaranteed independently. It will be shown [11, 12] that the primary and secondary loop remain stable independently regardless of the seek/settling/track-following control method used for the VCM-loop, providing the secondary control loop is stable in the presence of saturation limits. Hence, it seems logical to use in the secondary loop an AW compensator to guarantee overall large and small signal stability despite actuator saturation limits. AW compensators are most suitable, as they have been developed to retain the nominal performance of the original control system [7,9,20], e.g., high-bandwidth tracking in the servo-control of hard disk drives. This case study on AW-compensation will consider the following issues: 1. Controller design for the micro-actuator loop and wind-up problems due to actuator limits.

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2. Discrete anti-windup compensator design and anti-windup compensation for the microactuator control loop. 3. The micro-actuator control loop as part of the overall seek-settling scheme for the hard disk drive. We will consider, for reasons of simplicity, the micro-actuator loop first.

14.2 The Micro-actuator Control Loop and Windup Problems

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Fig. 14.2. Continuous model of a PZT-actuator The model of a micro-actuator is usually very simple. Ignoring high frequency resonances, the model of the micro-actuator could be considered to be just a constant gain (Figure 14.2). However, in practice these resonances (at 6.5 kHz and 9.6 kHz in Figure 14.2) are important. They are usually subject to phase and gain uncertainty and the centre frequency of a resonance might even shift due to the influence of temperature. This usually forces the control engineer to design the open and closed loop bandwidth lower than 50 % of the smallest significant resonance. Hence, for our micro-actuator, an open loop crossover frequency of no more than 3 kHz is deemed possible. It must be also noted that hard disk drives are usually controlled using digital technology. The sampling frequency is directly correlated to the rotational speed of the hard disk and the servo-information which is placed along the concentric data tracks. The amount of servoinformation will limit the available space for user data. Hence, it is necessary to limit the amount of servo-information which in turn limits the sampling frequency (Figure 14.3). In

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Fig. 14.3. Hard-disk with servo-information and data tracks

our case, we have used a sampling frequency of 27 kHz which is reasonable for modern hard disk drives. However, this sampling frequency is rather low, since micro-actuators often also have resonance frequencies at 20 kHz and above (not depicted here) i.e., above the Nyquist frequency of 13.5 kHz. Thus, a hard disk drive servo system is a sampled-data control system for which it is necessary to use discrete or sampled-data control methods to design the servocontroller. Notch filters 0

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4

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sys_notch_d1*sys_notch_d2

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R R Fig. 14.5. MATLAB Simulink

model of PZT-actuator with notch filter and protective saturation element

In the hard disk drive industry, it is common practice to counteract resonances by simple notch filters. In our case, this may be done by using two discrete notch filters which are digitally implemented (see Figure 14.4). Hence, these notch filters are easily augmented at the input of the PZT-actuator via a digital-to-analogue unit (DAU). In general, a saturation is placed between the filter and the DAU to protect the actuator from any damage due to high voltage R peaks (see Figure 14.5 for a MATLAB respresentation). The saturation limits are also tuned in such a way as to protect the micro-actuator from too large displacements. The micro-actuator augmented with the notch filters is easily modelled by a second order system PPZT (z) (see Figure 14.6): −0.07997z − 0.05363 z2 − 0.7805z + 0.3097

PPZT =

(14.1)

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Fig. 14.6. Zero-order hold discretization of PZT-model with notch (line); Second order model of PZT-actuator (dash) The first step in the closed loop controller design for this micro-actuator is to design a suitable linear controller K(z) based on robust µ -design methods. The design problem is standard.

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We will minimize the µ -value for the following control problem with two blocks [ WT T ]:

h

Ws S Wu SK

i

and

 where S =

1 1+KPPZT

and T =

KPPZT 1+KPPZT

 Ws S  Wu SK    WT T

(see Figure 14.7).

Fig. 14.7. µ -control problem The weights (as shown in Figure 14.8) are given by WT =

1.3637(z + 0.04294) −0.25664(z − 2.578) , Ws = , Wu = 0.000001. z + 0.6843 z − 0.9605

R HowThe design of the controller is easily achieved using the µ -toolbox of MATLAB . R ever, a small trick has to be used. MATLAB does not provide µ -tools for discrete control problems. Hence, the discrete control problem has to be converted into a continuous-time problem using a bilinear transformation [3, 5]. Note that the bilinear transformation creates an equivalent continuous-time µ -problem. The discrete controller can then be obtained from the solution to the continuous-time control problem by again using the bilinear transformation. The discrete controller will always be stabilizing and µ -(sub)-optimal with the same µ -value as that obtained from the continuous-time design. √ Thus, for our design problem, a controller is obtained using DK-iteration for a µ -value of 2. Balanced truncation methods then leads to a third order controller which perfectly matches the originally designed controller. The discrete controller of third order is:

K(z) =

−6.584z3 − 1.403z2 + 3.15z − 2.079 z3 + 0.4509z2 − 0.9169z − 0.4211

The control loop is easily simulated with a sine sweep in the frequency range from 10 Hz to 500 Hz for the position demand. For a small demand amplitude of 0.2 µ m, the controller is able

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Fig. 14.8. µ -control problem weights (WT (line), Ws (dashed)) to follow this demand without significant phase lag but with a small amplitude loss (Figure 14.9). However, for a large demand amplitude of 1 µ m, the actuator saturation limit of 1.54 V is activated, limiting the actuator displacement range. Thus, in this case, the controller is not able to track the controller demand as the actuator displacement is limited to a magnitude of less than 0.5 µ m. Hence, the actuator starts to oscillate and the controller exhibits a lag (Figure 14.10). These two issues of resonance and lag can be easily resolved by introducing an antiwindup compensator to recover performance quickly following saturation. The anti-windup compensator of interest to us will be discussed next.

14.3 Anti-windup Compensation for Discrete Linear Control Systems We consider the system in Figure 14.11 as introduced in [21, 29] for continuous-time antiwindup (AW) compensation and in [7] for discrete-time control systems. P(z) is the discrete plant and K(z) is the nominal linear discrete controller. The AW-compensator is given by the blocks, M(z) − I and PM(z). The anti-windup compensation is designed using the free parameter M(z). The control signal is constrained by the limits of a saturation function. The saturation function is defined as sat(u) := [sat1 (u1 ), . . . , satm (um ))]T (14.2) where sati (ui ) := sign(ui ) × min {|ui |, u¯i } and u¯i > 0 is the i’th saturation limit. The following identity holds

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0.2

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Fig. 14.10. Control result for 1 µ m demand amplitude (demand (dashed), actuator-position (line)) Dz(u) = u − sat(u)

(14.3)

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Fig. 14.11. Conditioning with M(z)

Fig. 14.12. Equivalent representation of conditioning with M(z)

where Dz(u) is the deadzone function. For the deadzone nonlinearity Dz(u), it follows that there exists a diagonal matrix W such that Dz(u)′W (u − Dz(u)) ≥ 0, u ∈ Rm

(14.4)

This inequality will be used later for stability analysis employing the S-procedure [2]. This is the discrete-time version of the continuous time configuration introduced in [21, 29]. As in [21, 29], it is straightforward to show that, using the identity (14.3), Figure 14.11 can be re-drawn as Figure 14.12. These diagrams essentially cast the anti-windup problem as one of choosing an appropriate M(z). However, notice that Figure 14.12 reveals an attractive decoupling into nominal linear system, nonlinear loop and disturbance filter which motivates a useful approach to synthesizing M(z). As noted in [29], most linear conditioning schemes can be interpreted in the framework of Figure 14.11. In [7], the mapping T : ulin 7→ yd was picked as a measure of the anti-windup

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compensator’s performance (Figure 14.12). We shall choose to minimise the l2 gain, kT ki,l2 , in our anti-windup synthesis. The detectable plant, P(z), has the following state-space description

P(z) ∼



x p (k + 1) = A p x p (k) + B p um (k) y(k) = C p x p (k) + D p um (k)

(14.5)

where x p ∈ Rn p is the plant state, um ∈ Rm is the actual control input to the plant and y ∈ Rq is the output which is fed back to the controller. We are making the assumption that the plant P(z) is asymptotically stable i.e., |λmax (A p )| < 1, for global stability of the non-linear control system with anti-windup compensation. As in [29] for continuous-time, M(z) can be chosen as a coprime factor of P(z). So if P(z) = N(z)M −1 (z), we can search for a coprime factor of P(z) such that the anti-windup closedloop has the best performance in terms of the gain of T . This approach is also related to that of [20] and, to a lesser extent, that of [16]. To achieve full-order stabilisation we would like to choose coprime factors, which share the same state space and are of order equal to that of P(z). Employing Figure 14.12, such coprime factorisations can be characterised by 

   Ap + BpF Bp M(z) − I ∼ F 0  N(z) Cp + D p F D p

(14.6)

where u(k) ˜ = Dz(ulin (k) − ud (k)). Note that these equations are parameterised by the free parameter F and therefore we attempt to choose F such that kT ki,l2 is minimised. Theorem 14.1. There exists a dynamic compensator Θ (z) = [M(z)′ − I (P(z)M(z))′ ]′ which solves strongly the anti-windup problem if there exist matrices Q > 0,U = diag(µ1 , . . . µm ) > 0, L ∈ R(m+q)×m and a scalar γ > 0 such that the following linear matrix inequality is satisfied  −Q −L′ 0 QC′p + L′ D′p QA′p + L′ B′p ′ ′   ⋆ −2U I UD p UB p    0. We define the Lyapunov difference as ∆ V (k) := V (k + 1) −V (k). Next we consider the function ∆ V˜ (k) which is defined as 1 ∆ V˜ (k) := ∆ V (k) + 2u(k) ˜ ′W [ulin (k) − ud (k) − u(k)] ˜ + kyd (k)k2 − γ kulin (k)k2 γ

(14.8)

This function is a combination of the Lyapunov difference (first term), the sector bounds from (14.4) associated with the deadzone nonlinearity (second term), and two final terms which ensure we have a certain level of l2 performance. If we can ensure that equation (14.8) is negative definite, we have

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1. Asymptotic stability When ulin (k) = 0 and

1 + kyd (k)k2 < 0 ∆ V˜ (k) = ∆ V (k) + 2u(k) ˜ ′W [ulin (k) − ud (k) − u(k)] ˜ } γ {z | | {z } ≥0

(14.9)

≥0

for [x(k) u(k) ˜ ulin (k)] 6= 0 asymptotic stability is implied from ∆ V (k) < 0. 2. l2 gain < γ Summing ∆ V˜ (k) from 0 to ∞ gives: ∞

∞

k=0

k=0

1

˜ ′W [ulin (k) − ud (k) − u(k)] ˜ + kyd k2l − γ kulin k2l ∑ ∆ V (k) + 2 ∑ u(k) γ 2

2

0 ≥0

for [x(k) u(k) ˜ ulin (k)] 6= 0, which implies kyd k2l2 < γ 2 kulin k2l2 + V (0) and therefore kT ki,l2 < γ . This procedure of extending the Lyapunov difference by some extra non-negative terms is the well established S-procedure [2]. Strictly speaking m + 1 terms are included. For the sector non-linearity (14.4), m terms are included, while the m + 1-th term is contributed by the l2 gain constraint. It is well known that the inclusion of one term only, creates an equivalent inequality for both conditions to be satisfied. However, once more than one term is considered only sufficiency is achieved and conservatism is introduced. Furthermore, the sector bound as given in (14.4) may also introduce conservatism as discussed earlier. These facts are wellknown and accepted trade-offs [2, 4] as they allow practically feasible solutions (e.g., [10]). Substituting for x(k), ud (k) and yd (k) in (14.8), we have that ∆ V˜ (k) < 0 if and only if  ′   VF11 VF12 0 x(k) x(k)  u(k)   ⋆ VF22 W   u(k)  < 0, ˜ ˜ ⋆ ⋆ −γ I ulin (k) ulin (k) | {z } 

[x(k) u(k) ˜ ulin (k)] 6= 0

(14.12)

VF

where

1 VF11 = (A p + B p F)′ P(A p + B p F) − P + (C p + D p F)′ (C p + D p F) γ 1 VF12 = (A p + B p F)′ PB p − F ′W + (C p + D p F)′ D p γ 1 ′ VF22 = −2W + D p D p + B′p PB p . γ

(14.13) (14.14) (14.15)

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424

The remainder of the proof has to show that VF < 0 is equivalent to (14.7). This follows by standard Schur complement and congruence transformation arguments. The Schur complement implies that (14.12) holds if and only if 

 −P −F ′W 0 C′p + F ′ D′p A′p + F ′ B′p ′ ′  ⋆ −2W W  Dp Bp    ⋆  0. Figure 15.1 shows the typical ⋆⋆ time evolution of the two populations for initial values g0 = 1000, w0 = 10 demonstrating its cyclic nature. Figure 15.2 depicts the phase plane for the same initial conditions. Notice how the phase plane divides naturally into four quadrants corresponding to periods: (A) when goats are increasing while wolves decrease; (B) both populations are growing; (C) predation takes over and the goats are severely reduced resulting in a large rise in wolves, and finally (D) the wolf population reduces owing to
a lack of food, slowing the rate of goat reduction. The point marked with a square at dc , ba = (4000, 80) is an equilibrium point of the system.

⋆ ⋆⋆

Don’t forget this is a model – we have not taken the finiteness of resources such as land area and food into account . Parameter values of a = 0.04, b = 0.0005, c = 0.2,d = 0.00005 are chosen in this abstract example.

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12000

10000

Population Size

8000

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0

10

20

30

40

50

60

70

80

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Fig. 15.1. Evolution of goat (solid) and wolf (dashed) populations from an initial state of g0 = 1000, w0 = 10. This depicts one cycle.

15.2.1

Analysis of Equilibria

One of the easiest tools for analyzing non-linear dynamics is known as Lyapunov’s First or Indirect Method. This tells us about dynamic behaviour in the locality of the (possibly many) equilibrium points defined by x˙ = f (x) = 0 – the overbar is used to denote a fixed-point. From this type of analysis it is sometimes possible to infer wider behaviour although global methods are more reliable. Lyapunov’s Second or Direct Method provides a means for doing this although, in general, it can be difficult to apply. A good source of information on this topic is [8]. To apply Lyapunov’s Indirect Method we must find the (static) equilibrium points⋆⋆⋆ of the system, linearize about them and examine the eigenvalues of the Jacobian matrices evaluated at the equilibrium points. Real Eigenvalues: If all eigenvalues are real and negative (positive) then all states asymptotically approach (leave) equilibrium, monotonically. These are known as stable (unstable) nodes. If the eigenvalues have a mixture of signs then some states will approach equilibrium and some will diverge, again monotonically, resulting in so-called saddle-points. Complex Eigenvalues: Here, the eigenvalues must occur in conjugate pairs and those with negative (positive) real parts will converge (diverge) with a cyclical form. These are known as stable (unstable) foci. Again, a mixture can occur. ⋆⋆⋆

The points where the state derivative is zero, i.e., the system can be said to be “at rest” .

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500

Wolves

400

300 D

C

A

B

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0

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6000

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Fig. 15.2. Phase-plane depiction of one cycle of the evolution of goat and wolf populations from an initial state of g0 = 1000, w0 = 10 Imaginary Eigenvalues: Often associated with purely oscillatory behaviour known as a centre but in fact, indeterminate. What this tells us is that the higher-order terms in the Taylor expansion of f may play an important role. So, for our system, defining x1 as the number of goats and x2 as the number of wolves, we can write its two-dimensional state-space form, x˙ = f (x), thus

x˙1 = ax1 − bx2 x1 ; x1 (0) = x10

(15.2)

x˙2 = −cx2 + dx1 x2 ; x2 (0) = x20 giving two fixed points, (x1 , x2 ) =

c a d, b






and the origin. The Jacobian matrix is given by

a − bx2 −bx1 dx2 −c + dx1



(15.3)

√ At the first equilibrium the two eigenvalues of the Jacobian are given by λ = ± j ac. For a linear system this corresponds to a centre but in fact we can conclude nothing about the stability of the dynamics although, in our case, they do happen to cycle. At the origin we clearly have λ = a and λ = −c defining a saddle-point. Obviously the origin corresponds to the extinction of both species but the saddle suggests that extinction is not easily arrived at naturally.

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The situation described in equation (15.2) is not very realistic but it does in troduce the ideas of how two species can interact. Obviously the ideas can be extended. For instance, if food for the goats becomes short there would be intra-species competition. This is easily accounted for by introducing a term proportional to x12 in the first equation in (15.2). Because this will have the effect of reducing goat numbers, it must be subtracted. Likewise, culling of wolves could be introduced as a control variable on the RHS of the second equation – a negative constant, for instance, to indicate the rate of removal. Introduction of any number of additional species is straightforward and for added realism, time delays can be introduced to account for gestation times – obviously the consumption of a goat does not instantaneously result in a new wolf. However, while simulation of time-delay systems is relatively straightforward, analysis and control is vastly more advanced. We won’t consider it here. Figure 15.3 shows what happens when there is competition between goats for food (a term −ex12 is added to the first equation in (15.2). We see that a new equilibrium is reached instead of the continuous cycling (a dynamic equilibrium). This occurs at (4000, 72) after about 2500 time units (e=0.000001) so, for the farmer’s investment of the original 1000 goats a steadystate population of 4000 arises even though some wolves have had to be fed.

500 450 400 350

Wolves

300 250 200 150 100 50 0

0

2000

4000

6000

8000

10000

12000

Goats

Fig. 15.3. Phase-plane depiction of the goats & wolves populations when goats compete for food

15.3 Optimal Control The topic of “optimal control” is enormous so I will restrict the problem to that of choosing a control policy to minimize a quadratic function of the system states and control effort subject

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to non-linear dynamics. While this is a restricted statement of the optimal control problem, it is widely applicable in practical circumstances. We will solve it by generalizing the ideas of linear quadratic control to this class of system to create a sequence of approximations to the control trajectory that can be shown to converge to the global optimum. These ideas are quite recent but provide, for the first time, an easily computed and implemented solution to this class of feedback control problem [4].

15.3.1

Linear, Time-Varying Quadratic Optimal Control

To keep things simple and because, for the purpose of these notes, it is all we shall need, I shall only look at the regulator problem under the assumption that all states are available for feedback. I’ll say a bit more about this rather strong assumption at the end. The problem of deriving the optimal linear feedback for a linear, time-varying (i.e., the A and B matrices of the state-space form are functions of time) system under quadratic cost is well understood (see, e.g., [9]). Consider the time-varying linear dynamics x˙ = A(t)x + B(t)u; x (t0 ) = x0

(15.4)

where x(t) is the n-dimensional state vector and u, the m-dimensional control vector. A(t) and B(t) are matrix-valued functions of time of appropriate dimension. The objective is to choose the control law that minimizes the quadratic cost functional over the interval t0 –t f

1 1 J (u) = xT t f Fx t f + 2 2

Z tf  t0

 xT (t) Q(t)x (t) + uT (t) R(t)u (t) dt

(15.5)

It is well-known that the solution to this problem is reached via a two-point boundary value problem which, owing to linearity, happens to lead to a full-state, linear feedback formulation given by uopt (t) = −R−1 (t)BT P(t)x(t)

(15.6)

where P(t) is the positive-definite solution of the matrix Riccati equation




˙ P = −Q(t) − P(t)A(t) − AT (t)P(t) + P(t)B(t)R−1 (t)BT (t)P(t)

(15.7)

with P t f = F.

Since only the terminal condition for equation (15.7) is given, this must must first be solved backwards in time and stored so that it can be used in the forward dynamics. This can be done by substituting τ = t f − t and solving an initial value problem. Clearly, optimal control of a time varying system and/or over a finite time horizon is not causal and questions arise as to its application in a real-time setting. It turns out that the state evolution of a wide class of non-linear dynamical system can be computed as the limit of a sequence of linear, time-varying approximations and that this can be exploited to find the optimal control policy using the above idea [4, 10].

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The class of dynamical system of interest is of the form x˙ = f (x, u) ; x (t0 ) = x0

(15.8)

with f (0, 0) = 0 and smooth in some sense† . It is necessary that the system (15.8) can be re-written in a form resembling the familiar, linear, state-space equations, thus x˙ = A (x) x + B (x, u) u; x(t0 ) = x0

(15.9)

where A is an appropriately dimensioned, non-linear, matrix-valued function of the state only and likewise B, except that it is allowed to be a function of the control as well. The fact that A may only be a function of the state and not the control imposes a further, small, restriction on the dynamics. The objective is to find the control law that minimizes the cost functional

1 1 J (u) = xT t f Fx t f + 2 2

Z tf  t0

 xT (t) Qx (t) + uT (t) Ru (t) dt

(15.10)

with F, Q positive semi-definite and R positive-definite matrices of appropriate order. Note that to simplify notation I have chosen these to be constant but the theory allows them to be functions of the state vector‡ . From here on I have also suppressed the dependence of quantities on t Analogously to the linear time-varying optimal feedback control problem, it can be shown that the sequence of approximating solutions, given by the following, converges globally to the optimal solution [4].   u[i] = R−1 BT x[i−1] , u[i−1] P[i] x[i]

(15.11)

for i ≥ 0, where P[i] is the solution of

    P˙ [i] = −Q − P[i] A x[i−1] − AT x[i−1] P[i]     +P[i] B x[i−1] , u[i−1] R−1 BT x[i−1] , u[i−1] P[i]

(15.12)


with P[i] t f = F. It follows that the sequence of state trajectories is given by

        x˙ [i] = A x[i−1] − B x[i−1] , u[i−1] R−1 BT x[i−1] , u[i−1] P[i] x

(15.13)

with x[i] (t0 ) = x0 . The crucial point here is that the non-linear problem has been reduced to a sequence of linear, time-varying problems that is guaranteed to converge to the correct solution. All that remains is to start the sequence off, which can be done by taking x[i−1] = x0 and u[i−1] = 0 for i = 0. The algorithmic steps for computing the control law are as follows. † ‡

Locally Lipschitz continuous. This means that, unlike many approaches, ASRE is not restricted to quadratic functions of the states and controls in the cost function – these can be very general indeed.

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Compute P[0] (t) from (15.12) with i = 0 and x[i−1] = x0 and u[i−1] = 0. Substitute P[0] (t) and compute x[0] (t) from (15.13) with A (x0 ), B (x0 , 0). Apply (15.12) followed by (15.13) repeatedly, until convergence.

15.4 Immune System Dynamics What, then, has the Lotka-Volterra approach to do with the immune system and how can we apply optimal control to assist it in responding to attack by microbes? The answer is that we can use the same intuitive arguments to come up with a plausible model for the dynamics of the immune system which then fits nicely into the formulation required for the ASRE or other optimal control strategies. Now we have different populations – typically concentrations of organisms and chemicals that interact by coming into contact with one another. We shall see how these ideas relate to modelling the immune system. What happens when the body is attacked by an external agent or pathogen ? Roughly speaking, plasma cells specialized to the particular antigens associated with these pathogens are produced. The plasma cells produce antibodies that bind to the antigens resulting in the destruction of the pathogen in some way. Much more detail is given in [11] and in the series of papers by Stengel and colleagues [1–3]. In the model we shall study we define x1 x2 x3 x4

– pathogen concentration – plasma cell concentration – antibody concentration – measure of organ health (0 – healthy, 1 – dead)

and for the controls u1 u2 u3 u4

– pathogen killer – plasma cell enhancer – antibody enhancer – organ health enhancer

The model proposed in [11] and modified through the addition of controls [1] is given by dx1 dt dx2 dt dx3 dt dx4 dt

= (a11 − a12 x3 ) x1 + b1 u1

(15.14)

= a21 (x4 ) a22 x1 x3 − a23 (x2 − x2 ) + b2 u2 = a31 x2 − (a32 + a33 x1 ) x3 + b3 u3 = a41 x1 − a42 x4 + b4 u4

which is clearly of the Lotka-Volterra type. All disease parameters, ai j > 0. For instance, taking the first equation governing the concentration of pathogens, in the absence of antibodies

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(x3 ) or pathogen killer (u1 ) the pathogen concentration is free to grow exponentially, however, the interaction with the antibodies has a moderating effect as in the goats and wolves example. Similar reasoning underpins, qualitatively, the processes behind the rest of the equations. One significant departure, though, is the introduction of the function a21 (x4 ). This has the effect of reducing the ability of the vital organs to produce plasma cells as they becomes more diseased and is chosen to be a21 (x4 ) =



cos (π x4 ) , 0 ≤ x4 < 1 0, 2 ≤ x4

1 2



which means that for a healthy organ there is little degradation while for a dead organ the excess plasma cell production decays exponentially to its normal level. Evidently the parameters have to be determined and this can be done from biochemical analysis or, in principle, through system identification. For this example we will use a plausible set as provided by [1]: a11 = a12 = a23 = a31 = a42 = b2 = b3 = 1, b1 = b4 = −1, a22 = 3, a32 = 1.5 and a33 = 0.5. The initial conditions for equation (15.8) determine the uncontrolled respo nse of the states. In all cases we assume an initially healthy organ (x4 (0) = 0),  a normal level ofplasma cells

31 (x2 (0) = x2 = 2), and an equilibrium level of antibody x3 (0) = x3 = aa32 x2 . The initial value of x1 determines the strength of the pathogen attack. In [1] four cases of interest are identified

Case 1 Sub-clinical – no medical intervention required, (x1 (0) = 1.50) Case 2 Clinical – needs medical assessment, (x1 (0) = 2.00) Case 3 Chronic – pathogen level goes into steady state, (x1 (0) = 2.57) Case 4 Lethal – organ health is damaged beyond repair and pathogen levels diverge, (x1 (0) = 3.00) These situations can be simulated by judicious choice of initial conditions for x1 (see above). Figure 15.4 demonstrates the natural response of the immune system when attacked by the different levels of pathogen. h i   It is easy to verify that uT = 0 0 0 0 , xT = 0 x2 aa3132x2 0 is an equilibrium point satisfying x˙ = 0. We can therefore define a new state vector, z = x − x and a new centred RHS, f (z, u), so that the equilibrium of the centred system lies at the origin as required by the ASRE theory. This gives dz1 dt dz2 dt dz3 dt dz4 dt

= ((a11 − a12 x3 ) − a12 z3 ) z1 + b1 u1

(15.15)

= a21 (z4 ) a22 z1 z3 + a21 (z4 ) a22 x3 z1 − a23 z2 + b2 u2 = a31 z2 − (a32 + a33 z1 ) z3 + a31 x2 − (a32 + a33 z1 ) x3 + b3 u3 = a41 z1 − a42 z4 + b4 u4

which has a non-unique decomposition into the required form of equation (15.9) where A (z) is given by

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R.F. Harrison 6 plasma conc.

pathogen conc.

6

4

2

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5

0

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antibody conc.

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Fig. 15.4. Uncontrolled response for the four cases, Sub-clinical (solid), Clinical (dashed), Chronic (chained) & Lethal (dotted)



 0 0 ((a11 − a12 x3 ) − a12 z3 ) 0  a21 (z4 ) a22 (z3 + x3 ) −a23 0 0   A (z) =   a33 x3 a31 − (a32 + a33 z1 ) 0  a41 0 0 −a42

(15.16)

and B (z) = B = diag (bi ). Clearly, A (z) and B satisfy the required Lipschitz condition.

15.4.1

Optimal Enhancement of the Immune Response

It is worth asking why anyone would want to apply control to a problem such as this. The over-riding reason is to make sure that as much external agent as needed is given, in the right dose and at the right time, but no more. Many drugs are no more than poisons that we take because the danger they pose is less than that of the illness. Antibiotics fall into this class as do many others. So the driver for this type of research is taking the guesswork out of drug treatment. To demonstrate the effectiveness of the proposed method, we must make some design choices. It is also important to realize that both the states and the controls must be constrained to remain non-negative. Being a new development, ASRE is not yet able explicitly to cope with such constraints§ , however, the dynamics under consideration here are sufficiently benign that the constraints are typically satisfied by the closed-loop system.

§

A version of the theory under control constraints is currently under development.

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To illustrate how individual therapies (each control applied separately) can be used, we examine the application of pathogen killer, u1 , alone, as this was found to be most effective in an open-loop strategy [1]. Of course, a variety of combinations can be used as well. Note that the method described here provides a feedback strategy directly, whereas in [2, 3] a separate method based on the idea of neighbouring optimal solutions is used. This involves linearizing about the optimal open-loop trajectory and obtaining the time-varying (approximate) optimal policy. Feedback is important because the open-loop optimal control is unable to account for different initial conditions or for any disturbances that might enter the system. We repeat the situation dealt with by open-loop control in [1] (Figure 2) with q11 = q44 = 1 f11 = f44 = 1 and b2 = b3 = b4 = 0 so that only the pathogen killer is used with R = r = 1. Figure 15.5 shows how the system recovers from all four conditions. The dotted curves correspond to the solid curves in Figure 2 of [1].

6 plasma conc.

pathogen conc.

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Fig. 15.5. Optimal administration of pathogen killer for the four cases, Sub-clinical (solid), Clinical (dashed), Chronic (chained) & Lethal (dotted). The lethal case using an open-loop optimal strategy is shown in Figure 2 of [1] (solid).

To emphasize the importance of feedback, we examine the lethal situation and compare the open-loop solution computed for an initial value of pathogen concentration of 3.0 but applied to a case when the initial concentration is, in fact, 4.5. Figure 4 of [3] demonstrates that the open-loop strategy would fail to accommodate the mismatch and would lead to organ death. In that paper, the approximate neighbouring optimal solution is applied and it can be seen that this rectifies the situation. I have used the ASRE method on the same problem and Figure 15.6 shows how it is able to cope with the perturbation. By eye it is clear that, in this case, the neighbouring optimal trajectory [3] and the ASRE solution are virtually indistinguishable.

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R.F. Harrison 6 plasma conc.

pathogen conc.

6

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antibody conc.

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Fig. 15.6. Comparison of open-loop response (solid) with the optimal feedback response (dashed) for a perturbed initial pathogen concentration. c.f. [3] (Figure 4) where the no longer optimal open-loop solution is seen to lead rapidly to organ failure.

What I have done here is to establish the principle that the ASRE method can deliver a viable feedback control policy in spite of it being unable explicitly to handle the positivity constraints on states and controls. To do this I have used the abstract example of [1] with arbitrary choices of F, Q and R. Clearly, even in this case, there is much experimentation to be done to discover an acceptable set of weighting matrices and manipulated variables.

15.4.2

Some Practical Considerations

There are some pretty strong assumptions in the development of the control strategy and I want to deal with two of them. The first, is relatively easy and is to do with the question of causality. The finite-time optimal control problem does not have a causal solution, i.e., information from the future is required at the current time to compute the optimal value of control input. This is compounded in the ASRE approach because the computations are iterative and cannot happen instantaneously. In general, therefore, a real-time optimal policy is not attainable. However, when the system dynamics are “slow ” relative to the speed of the computer used to perform the calculations which will usually be the case for immune systems, then all computations can be performed within the time required. This may be more problematic for “fast” dynamics. Handling unexpected disturbances in the interval requires a re-computation over the remaining time from the revised state – effectively a re-start. Another way of dealing with uncertainty, whether in the model or in external disturbances, is to use an explicit robust control strategy. The ASRE technique has been used in this way

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to provide robust rejection of unexpected disturbances and model error in a time-domain H ∞ setting [12]. Application of this idea to the enhancement of immune response is underway. The major difficulty associated with any control signal that makes use of th e system state is that exact measurements of the states are rarely, if ever, available. In the case of linear dynamics, the celebrated Kalman-Bucy filter provides a solution, whereby, given a good model of the dynamics and some knowledge of any measurement and process noise entering the system of interest¶ , an optimal estimate of the states can be computed causally, in real-time [13]. In the non-linear situation, it is usual to adopt the extended Kalman filter (EKF) which accommodates the non-linearity by a local linearization along the estimated state trajectory. While this no longer possesses optimality and may not even converge, the EKF has been found to work well in practice and its use is widespread [13]. More recently attention has been given to other types of state estimator, the most practical being the so-called unscented Kalman filter (UKF) [14]. This provides a more accurate estimate of the state error covariance matrix than the EKF and has been shown to perform better in a number of cases. More recently still, the ASRE technique has been applied to the problem of state estimation by appealing to the duality of the control and filtering problems [15]. While this goes some way toward solving the problem, it must be recognized that since the state estimates are the limit of a sequence of linear, time-varying approximations, such a method cannot properly represent the effects of non-linearity on the noise processes. In [2] a time-varying Kalman filter is established through linearization along the closed-loop trajectory and this is shown to work well in the current setting.

15.5 Conclusion I hope that I have given you a flavour of how relatively simple dynamic models can be derived that explain, with some degree of plausibility, quite complex behaviour and how they can be made to behave in a desirable way through the application of optimal control theory. As I stated earlier, it is my belief that the application of such methods has much to offer in the biomedical and life sciences and that we should expect to see applications leaving the computer laboratory and going live in the next decade or so.

References 1. R.F. Stengel, R. Ghigliazza, N. Kulkarni, and O. Laplace. Optimal control of innate immune response. Optimal Control Applications and Methods, 23:91–104, 2002. 2. R.F. Stengel and R. Ghigliazza. Stochastic optimal therapy for enhanced immune response. Mathematical Biosciences, 191:123–142, 2004. 3. R.F. Stengel, R. Ghigliazza, and N. Kulkarni. Optimal enhancement of immune response. Bioinformatics, 18:1227–1235, 2002. 4. T. C¸imen and S.P. Banks. Global optimal feedback control for general non-linear systems with non-quadratic parformance criteria. Systems & Control Letters, 53:327–346, 2004.

¶

Linear time-varying or invariant dynamics are admissible with uncorrelated, Gaussian distributed process and measurement noise of known covariance.

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5. V. Volterra. Variations and fluctuations of the number of individuals in animal species living together. In R.N. Chapman, editor, Animal Ecology. McGraw-Hill, New York, 1931. 6. A.J. Lotka. Elements of Physical Biology. Williams & Wilkins Co., Baltimore, 1925. 7. Anonymous. Uist Hedgehogs: Factsheet No. 3. World Wide Web, http://www.snh. org.uk/pdfs/news/nw-uwp03.pdf, 2002. 8. J.-J. E. Slotine and W. Li. Applied Nonlinear Control. Prentice Hall, New Jersey, 1991. 9. B. Friedland. Control System Design: An Introduction to State-Space Methods. McGrawHill, Singapore, 1987. 10. T. C ¸ imen and S.P. Banks. Nonlinear optimal trackingcontrol with application to supertankers for autopilot design. Automatica, 40:1845–1863, 2004. 11. A. Asachenkov, G. Marchuk, R. Mohler, and S. Zuev. Disease Dynamics. Birkh¨auser. 12. S.F. Fahmy and S.P. Banks. Robust H ∞ control of uncertain non-linear dynamical systems via linear time-varying approximations. Non-linear Analysis, 63:2315–2327, 2005. 13. A.H. Jazwinski. Stochastic Processes and Filtering Theory. Academic Press, New York, 1970. 14. S.J. Julier and J.K. Uhlmann. A new extension of the kalman filter to non-linear systems. In The Proceedings of Aerosense: The 11th International Symposium on Aerospace/Defense Sensing, Simulation and Controls, Orlando, 1997. 15. T. C¸imen and S.P. Banks. Stochastic optimal control of partially observable non-linear systems. In CDROM Preprints of the 16th IFAC World Congress.

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