Resilient Control of Uncertain Dynamical Systems (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari 303 Springer Berlin Heidelberg NewYo...

Author: Magdi S. Mahmoud

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

303

Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo

Magdi S. Mahmoud

Resilient Control of Uncertain Dynamical Systems With 39 Figures and 29 Tables

13

Series Advisory Board A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Author Prof. Magdi S. Mahmoud UAE University College of Engineering Department of Electrical Engineering P.O. Box 17555-Al Ain Adu-Dhabi United Arab Emirates [email protected]

ISSN 0170-8643 ISBN 3-540-21351-1

Springer-Verlag Berlin Heidelberg New York

Library of Congress Control Number: 2004104464 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 other ways, 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-Verlag. Violations are liable to prosecution under German Copyright Law. Springer-Verlag is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Printed in Germany 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: Data conversion by the author. Final processing by PTP-Berlin Protago-TeX-Production GmbH, Berlin Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 62/3020Yu - 5 4 3 2 1 0

Dedicated to my Wife Salwa and my Children Medhat, Monda, and Mohamed

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Fragility Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Robust Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Non-fragile (Resilience) Controllers . . . . . . . . . . . . . . . . . . 1.2.4 Parameter Space Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Motivating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Inverted Pendulum on a Cart . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Flexible Disk Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Simple Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Stream Water Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Approaches to Resilient Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Guaranteed-Cost Approach . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 The Proposed Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Controller Gain Perturbations . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Multiplicative Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 A Glossary of Terminology and Notations . . . . . . . . . . . . . . . . . . 1.5.1 General Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Functional Differential Equations . . . . . . . . . . . . . . . . . . . 1.6 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 3 4 4 6 7 9 13 16 17 18 19 19 20 21 21 22 22 23 23 25

2

Resilient Control-Continuous Case . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model with Norm-Bounded Uncertainties . . . . . . . . . . . . . . . . . . . 2.3 Guaranteed Cost Control I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Additive Gain Perturbations . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 29 30 31

XII

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2.3.3 Multiplicative Gain Perturbations . . . . . . . . . . . . . . . . . . . 2.3.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Example 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H∞ Control I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Additive Gain Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Multiplicative Gain Perturbations . . . . . . . . . . . . . . . . . . . 2.4.3 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Example 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Feedback Control I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Example 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Example 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model with Convex-Polytopic Uncertainties . . . . . . . . . . . . . . . . . Guaranteed Cost Control II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Additive Gain Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Multiplicative Gain Perturbations . . . . . . . . . . . . . . . . . . . 2.7.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.5 Example 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H∞ Control II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Additive Gain Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Multiplicative Gain Perturbation . . . . . . . . . . . . . . . . . . . . 2.8.3 Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Results for Delayless Systems . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 Example 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Feedback Control II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Guaranteed Cost Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 H∞ Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.4 Example 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertain Model with State-Delays . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 Resilient Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.2 Error Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.3 Robust Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.5 Example 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.6 Example 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 36 37 37 39 39 41 41 43 44 46 47 47 49 49 51 52 53 55 55 57 58 58 59 60 61 63 65 67 69 71 71 73 74 77 81 81 82

Resilient Control-Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model of Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Gain Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 84 84 85

2.4

2.5

2.6 2.7

2.8

2.9

2.10

2.11 3

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XIII

3.3 Guaranteed Cost Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Additive Gain Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Multiplicative Gain Perturbations . . . . . . . . . . . . . . . . . . . 3.3.3 Example 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Example 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 H∞ Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 89 90 91 91 95 96

4

Resilient Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Continuous-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2.1 Nominal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2.2 Design Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3 Adaptive Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3.1 Known Perturbation Bound . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.2 Example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.3 Unknown Gain Perturbation Bound . . . . . . . . . . . . . . . . . 104 4.3.4 Example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3.5 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4 Polytopic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5 Design Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.5.1 Known Perturbation Bound . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5.2 Example 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5.3 Unknown Gain Perturbation Bound . . . . . . . . . . . . . . . . . 112 4.5.4 Example 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5.5 Model Reference State Regulation . . . . . . . . . . . . . . . . . . . 114 4.5.6 Example 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.5.7 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.6 Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.6.1 Stability and Stabilization Results . . . . . . . . . . . . . . . . . . . 119 4.7 Adaptive Stabilization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.7.1 Known Perturbation Bound . . . . . . . . . . . . . . . . . . . . . . . . 122 4.7.2 Unknown Perturbation Bound . . . . . . . . . . . . . . . . . . . . . . 123 4.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.8.1 Example 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.8.2 Example 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.9 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5

Resilient Linear Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2 A Class of Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.3 The Resilient Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.3.1 A Class of Linear Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.3.2 LMI-Based Design Conditions . . . . . . . . . . . . . . . . . . . . . . . 132

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5.4 5.5

5.6

5.7

5.8 6

5.3.3 Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.3.4 A Limiting Approach to Kalman Filtering . . . . . . . . . . . . 136 5.3.5 Multiplicative Gain Perturbations . . . . . . . . . . . . . . . . . . . 137 5.3.6 Example 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.3.7 Example 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Continuous Polytopic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 The Resilient Filtering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.5.1 Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.5.2 Delay-Independent Filter Synthesis . . . . . . . . . . . . . . . . . . 143 5.5.3 Multiplicative Gain Perturbations . . . . . . . . . . . . . . . . . . . 146 Resilient Delay-Dependent Filtering . . . . . . . . . . . . . . . . . . . . . . . . 149 5.6.1 A Descriptor Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.6.2 Extended Newton-Leibniz Approach . . . . . . . . . . . . . . . . . 154 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.7.1 Example 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.7.2 Example 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.7.3 Example 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.7.4 Example 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.7.5 Example 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Resilient Delay-Dependent Control . . . . . . . . . . . . . . . . . . . . . . . . 171 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.2.1 Descriptor Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.3 Feedback Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.3.1 Nominal H2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.3.2 Resilient H2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.3.3 Nominal H∞ Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.4 Resilient H∞ Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.3.5 Simultaneous Nominal H2 /H∞ Design . . . . . . . . . . . . . . . 180 6.3.6 Simultaneous Resilient H2 /H∞ Design . . . . . . . . . . . . . . . 181 6.3.7 Example 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.3.8 Example 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.3.9 Example 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.4 Continuous Polytopic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.4.1 Polytopic H2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.4.2 Polytopic H∞ Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.4.3 Simultaneous Polytopic H2 /H∞ Design . . . . . . . . . . . . . . 187 6.4.4 Example 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.5 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.5.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.5.2 Descriptor Model Transformation . . . . . . . . . . . . . . . . . . . . 189 6.5.3 Robust Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.5.4 Nominal Feedback Stabilization . . . . . . . . . . . . . . . . . . . . . 195

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XV

6.5.5 Resilient Feedback Stabilization . . . . . . . . . . . . . . . . . . . . . 199 6.5.6 Example 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.5.7 Example 6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.5.8 Example 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.5.9 Time-Varying Delay-Dependent Stability . . . . . . . . . . . . . 202 6.5.10 Example 6.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.5.11 Example 6.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7

Resilient Control-Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . 207 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7.2 Nonlinear Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . 207 7.3 Robust Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.3.1 Robust Delay-Independent Stability . . . . . . . . . . . . . . . . . 208 7.3.2 Example 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.3.3 Robust Delay-Dependent Stability . . . . . . . . . . . . . . . . . . . 211 7.3.4 Example 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.4 Robust Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.4.1 Nominal Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.4.2 Example 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.4.3 Example 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.4.4 Resilient Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.4.5 Example 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.5 Nonlinear Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.6 Robust Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.6.1 Example 7.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.6.2 Example 7.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.7 Robust Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.7.1 Example 7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.7.2 Example 7.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.7.3 Example 7.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.7.4 Resilient Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.7.5 Example 7.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.7.6 Example 7.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.8 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

8

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.1 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.1.1 Inequality 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.1.2 Inequality 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.1.3 Inequality 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 8.1.4 Inequality 4 (Schur Complements) . . . . . . . . . . . . . . . . . . . 239 8.1.5 Inequality 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 8.2 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.3 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

XVI

Contents

8.3.1 8.3.2 8.3.3 8.4 Some 8.5 Some 8.5.1 8.5.2

Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Some Standard Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 The S-Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Lyapunov-Krasovskii Functionals . . . . . . . . . . . . . . . . . . . . . 247 Formulas on Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . 247 Inverse of Block Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Matrix Inversion Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

1 Introduction

1.1 Overview Perhaps the problem of designing of optimal and robust controllers has been one of the most active research areas in linear control systems engineering. Over the past two decades or so, several methods have been developed that enable the controller to cope with parametric uncertainties in the plant dynamics. This is true for both types of uncertainties : structured and unstructured. A common denominator of these methods is that they rely on the celebrated YJBK parameterization [1, 7] of all stabilizing controllers for a fixed linear time-invariant plant, which provides a free parameter over which an appropriate function of a closed-loop transfer function may be minimized. Using this framework of parameterization, elegant techniques for minimizing different norms of the closed-loop transfer functions have been formulated. These include H∞ [111, 113], H2 [2] and L1 [13]. Moreover, efficient numerical approaches have been subsequently developed. Strictly speaking, the majority of the foregoing formulations represents a remarkable direction of systems research reflecting robust stability and/or robust performance. The controllers obtained using most robust design approaches are optimal if implemented exactly. It has been observed [43] that modern robust control design methods have been derived with one underlying assumption, that is, the resulting controller from off-line design will be implemented exactly in practice. It can be argued that this is generally a valid point when the main focus is on the plant uncertainty and considering that the controller will be implemented using high-precision hard-ware. Unfortunately, this is not the case in practice. Rather, it is expected that any controller being a part of a closed-loop system must be able to tolerate some uncertainty in its coefficients. Sources of this uncertainties include finite word length, imprecision inherent in analog-digital and digital-analog conversions , finite resolution measuring equipments and round-off errors in numerical computations, to name a few. More importantly, controller redesign becomes a basic operation in fine-tuning of the developed controllers to meet several practical considerations that might

M.S. Mahmoud: Resilient Control of Uncertain Dynamical Systems, LNCIS 303, pp. 1-25, 2004. © Springer-Verlag Berlin Heidelberg 2004

2

1 Introduction

have been overlooked at the beginning. This raises an important issue, that is a robust controller can be very sensitive, or fragile , with respect to errors or perturbations in the controller coefficients. In turn, that brings about a fundamental problem in robust control system design which has been recently termed the fragility problem and hence the design of non-fragile controller opens up as an important research topic that deserve further investigations. Throughout this monograph, we use the terms non-fragile and resilient interchangeably as equivalent terms. We anticipate that resilient controllers to minimize the cost of implementation and allow for on-line tuning of controller parameters .

1.2 The Fragility Problem 1.2.1 Introduction It is generally known that feedback control systems that are designed based on dynamic optimization of single performance criteria or to guarantee robustness with respect to parameter variations of the plant, by and large, may require precise or highly accurate controllers. In connection with the design of statefeedback gain K [1], it is reported that one needs a safety margin for K away from the boundaries of Kγ for the case of implementation inaccuracies. Additionally, in connection with the choice of structure for two-degree-offreedom controllers [124], it is stated that it may be necessary to single one (a structure) out on the basis of system sensitivity to controller uncertainty. To clarify the foregoing points further, consider the classical example [16] of designing an amplifier with open loop gain Ao to be insensitive to variation in Ao . For this purpose, the amplifier is embedded in a feedback configuration with a controller of fractional gain α in the feedback path and the goal is to ensure that the loop gain A β much greater than one. With reference to Figure 1.1, the closed-loop transfer function is

T =

Ao 1 + α Ao

(1.1)

and the sensitivity of T with respect to Ao is given by T = SA o

1 1 + α Ao

(1.2)

T It is readily seen that SA → 0 as Ao becomes very large. On the other hand, o the sensitivity of T with respect to the controller fractional gain α is given by

SαT =

α Ao 1 + α Ao

(1.3)

1.2 The Fragility Problem

3

Fig. 1.1. Amplifier with Feedback

In this case, we see that SαT → 1 as Ao becomes very large. The immediate conclusion [16] is that while a robust design guarantees zero sensitivity against changes in the open-loop gain Ao , it is 100% sensitive to changes in controller fractional gain α. As pointed out in [43], when one attempts to optimize a single performance index it is expected that small changes in controller parameters may lead to performance deterioration of other indeces. Typically a controller designed based on gain margin optimization is in effect very fragile with respect to phase margin. 1.2.2 Robust Control Problem In an engineering setting, the fragility problem discloses the issue of accuracy of controller implementation to the extent that it brings about a trade-off between implementation accuracy and performance deterioration. It is therefore crucial to address and understand all the effects of controller uncertainties in the implementation of robust controllers which optimize a prescribed performance measure in linear dynamical systems. Consider the classical control system of Figure 1.2. The robust design problem is: Given a linear plant P with additive uncertainties ∆P , it is required to find a controller K which internally stabilizes the family of plants P + ∆P and satisfies a given performance measure. There are different algorithms that provide an answer to this problem [1, 7]. Most of the time the focus is on structured uncertainties in the plant to represent the effect of general time-varying parameters whose exact values are unknown but which are known to belong to a given set. By and large, the available algorithms do not incorporate the problems associated with the implementation of uncertain controllers.

4

1 Introduction

Fig. 1.2. Plant with uncertain elements

1.2.3 Non-fragile (Resilience) Controllers At this stage, it is quite reasonable for various practical purposes to restrict attention to structured uncertainties in the controllers. Allowing some tuning procedures for some of the controller parameters will eventually lend the designer additional degrees of freedom to achieve a safety margin with respect to sampling methods, roundoff errors, to name a few. Therefore a more realistic robustness problem would be the one incorporating both plant uncertainties and controller uncertainties as depicted in Figure 1.3. In case of unstructured uncertainties in the plant and using well-known synthesis techniques (like linear quadratic optimization), the resulting controllers exhibit a poor stability margin if not implemented exactly. The term fragile controllers has been introduced in [43] to designate the class of highly accurate controllers. We remark that the fragility is recorded regardless of whether the designed controllers are optimal when implemented using their nominal parameters. It has been further suggested in [43] to overcome the fragility problem to: 1) Develop synthesis methods which incorporates some structured uncertainties in the controller and then search for the ”best” solution guaranteeing a compromise between optimality and fragility. 2) Employ a useful parameterization of the controller. 1.2.4 Parameter Space Design To illustrate the use of controller parameterization, consider Figure 1.4. Let ∆

p = [p1 , p2 , ...., ps ]

1.2 The Fragility Problem

5

Fig. 1.3. Uncertain plant with gain perturbations

denote the vector of real parameters representing the plant with p = po being the nominal value of the plant parameters and let x be the vector of real parameters representing the controller. We limit discussions to a class of parametric uncertainties represented by p ∈ P = {p : p− ≤ pj p+ j j , j = 1, 2, ..., s} ∆

Using a controller with fixed dynamic order, then ∆

x = [x1 , x2 , ...., xr ] denote the vector of real parameters representing the controller Additionally, the performance of the control system is characterized by a vector ∆

a = [a1 , a2 , ...., aq ] of real numbers which are functions of the plant and controller parameters: aj = aj (x, p),

j = 1, 2, ....q

The vector a(x, p) is referred to as the closed-loop performance vector. Suppose that the desired value of the closed-loop performance vector is ∆

d = [d1 , d2 , ...., dq ] ∈ IRn the components of which could represent the coefficient of a desired closed-loop transfer function. In this case, the goal of control system would be choosing the controller parameter vector x such that the set of inequalities aj (x, po ) = dj ,

j = 1, 2, ....q

6

1 Introduction

Fig. 1.4. General plant with uncertain elements

This represents a set of equations the solution of which yields the solution to model matching under nominal conditions. For practical convenience, we relax the desired closed-loop performance to a box-like set + D = {d : d− j ≤ dj ≤ dj , j = 1, 2, ....q ∆

which means that the robust design problem is to choose x if possible so that the following set of inequalities + d− j ≤ aj (x, p) ≤ dj , j = 1, 2, ....q

are satisfied for all p ∈ P. Indeed, the size of the desired box D would have to be adjusted to ensure satisfactory closed-loop performance. It should be observed that the foregoing formulation is carried out in the frequency-domain and its solution using linear programming techniques has been demonstrated [44].

1.3 Motivating Examples In this section, we provide some real-life examples to illustrate the fragility issue and related concepts. The first example deals with an open-loop unstable system (an inverted pendulum). The second example treats a continuoustime system with time-varying parameters (flexible disk drive with read/write head). In the remaining two examples, we shift attention to dynamical models with time-delays. Thus in the third example, we examine a simplified model of heat exchanger with output time-lag arising from finite measurement of

1.3 Motivating Examples

7

terminal temperature. Finally, in the fourth example we discuss a model of water quality in which time-delay phenomena is an intrinsic property due to mass conservation. Throughout the examples, we draw heavily on the use of MATLAB software in simulation. 1.3.1 Inverted Pendulum on a Cart In the course of parameter space designs of robust controllers , it has been shown [1] in case of car steering, cranes and flight control that small gain requirement (due to actuator constraints) frequently leads into a cusp of the stable region. This brings about a version of the fragility trap. In particular, it is not trivial to escape from there because almost all controller gain perturbations eventually lead to instability. To shed more light on this issue and related ideas, we consider the state-feedback design problem of an inverted pendulum shown schematically in Figure 1.5.

Fig. 1.5. An Inverted Pendulum

A linearized state-space model is given by     01 0 0 0  0 0 a23 0   b2     x˙ =  0 0 0 1 x +  0  u b4 0 0 a43 0 where −m g (m + mc ) g , a43 = mc mc −1 1 = , a23 = mc mc

a23 = b2

It can easily verified that the open-loop characteristic polynomial

(1.4)

8

1 Introduction

∆o (s) = mc s4 − (m + mc ) g s2 has double poles s1,2 at origin and two other real poles at s3,4 = ± (m + mc ) g/(mc ) Using the state-feedback u = −[k1 ,

k2 , k3 ,

k4 ] x

the closed-loop characteristic polynomial can be expressed as ∆c (s) = a4 s4 + a3 s3 + a2 s2 + a2 s + a0 where the coefficients are given by a4 = mc , a3 = k4 − k2 , a2 = k3 − k1 − g (m + mc ) a1 = g k2 , a0 = g k1 Since > 0, g > 0, mc > 0, m > 0mc > 0, it follows from Hurwitz theory that the necessary and sufficient conditions for stability are the following algebraic inequalities: kj > 0 j = 0, ...., 4 , k4 > k2 , k3 > k1 + (m + mc ) g g k2 [k3 − k1 − g(m + mc )][k4 − k2 ] > g 2 k22 mc + 2 m2c [k4 − k2 ]2 It is readily seen that the foregoing inequalities are difficult to cope with. Instead, we will explore some relevant special cases. Initially, we proceed to stabilize the system by shifting the positive real root to the stability boundary Re s = 0 using the feedback vector with smallest norm ||k||22 = k12 + k22 + k23 + k42 . The resulting gain vector turns out to be ko t = [0, 0, (m + mc ) g, 0] but the closed-loop characteristic polynomial becomes ∆co (s) = s4 implying a pole at origin with order 4. In attempting to use a marginal stabilization with a feedback vector kt = [ε1 ,

ε2 , (m + mc ) g, ε4 ]

it follows that the problem is not trivial. An alternative method would be to attempt a step-wise shifting by first placing the poles at to double imaginary poles (through manipulation of ε1 , ε2 ) then move the double roots into the left half plane (through manipulation of ε3 , ε4 ). This shows that the problem of minimum norm state-feedback stabilization for an unstable plant leads to a very fragile controller . Moreover, this controller is located in a cusp of the stability region of the four-dimensional


9

Table 1.1. Computational Results of Parameter Variations Case k % Mp yss ||k|| y1 [2 2 36 4] 12.37 0.3334 36.3318 y2 [3 3 54 6] 8.82 0.2498 54.4977 y3 [4 4 72 8] 6.90 0.1998 72.6636

gain space. A possible design route is to construct a singular path out of this cusp in several steps of pole shifting. In the sequel, we perform computer simulation with the following data g = 10, = 1/3, mc = 4/5, m = 1/2 The results of some technical properties are summarized in Table 1.1 The initial unity feedback response of the system with nominal state feedback vector k = [1, 1, 18, 2] is plotted in Figure 1.6. The step response of with different gain vectors are displayed in Figure 1.7

Fig. 1.6. Initial closed-loop response

1.3.2 Flexible Disk Drive Many dynamical systems including disk-drive read/write head have some flexibility between the sensor and actuator locations, which causes problems in the design of a control system [23]. Therefore, it is important to reflect this in the dynamical model. To elaborate on this point, consider the schematic of Figure 1.9 for a typical disk-drive read/write head system . Let Ih , Im stand for the head and motor inertias, respectively. Assume that the flexible

10

1 Introduction

Fig. 1.7. Different closed-loop responses

shaft has stiffness and damping coefficients k and b, respectively. Ignoring the disturbance torque and letting the angular displacements of the motor and the read head be θm , θr , then a state-space model of the system takes the form:

Fig. 1.8. Disk drive with flexible read head

z = [θm θr  0  0 z˙ =   −k/I1 k/I2

θ˙m θ˙r ]t , u = Tc    0 0 1 0   0 0 1   z +  0  u  1/I1  −b/I1 k/I1 b/I1  b/I2 −k/I2 −b/I2 0


y = [0 1 0 0]z

11

(1.5)

It is readily evident that system (1.5) represents an affine uncertain system [26] with four uncertain parameters in the system matrix. In the sequel, we consider the relative coefficients k(t)/I1 , k(t)/I2 , b(t)/I1 and b(t)/I2 are time-varying parameters over the intervals [0.5 → 2], [0.5 → 2], [0.2 → 1], [0.2 → 1], respectively. The nominal values are taken as k(t)/I1 |nominal = 1.25 , k(t)/I2 |nominal = 1.25 b(t)/I1 |nominal = 0.6 , b(t)/I2 |nominal = 0.6 Our objective hereafter is twofold: First is to study the sensitivities of the time-varying parameters on the dynamical behavior of the controlled system. Second is to examine the impact of the drift in the nominal controller gain factor. For the first objective, we proceed to design a nominal linear-quadratic static state-feedback controller Ko . We do this by using the MATLAB software system with state and controller weighting matrices Q = 10 × I4 , R = 1. The resulting controller gain is given by Ko = [26.6016, 16.8363, 6.3469, 17.8069] , ||k|| = 36.7216 By considering the nominal, maximum and minimum values of the parameters, the ensuing output responses are displayed for the following cases: 1) When all the parameters k(t)/I1 , k(t)/I2 , b(t)/I1 and b(t)/I2 vary simultaneously. The results are depicted in Figure 1.9 2) When the parameter k(t)/I1 varies and others are kept at their nominal values, he results are depicted in Figure 1.10 3) When the parameter k(t)/I2 varies and others are kept at their nominal values, he results are depicted in Figure 1.11 4) When the parameter b(t)/I1 varies and others are kept at their nominal values, he results are depicted in Figure 1.12 and finally 5) When the parameter b(t)/I2 varies and others are kept at their nominal values, he results are depicted in Figure 1.13 A summary of the effects on the controller gain (measured by ||k|| ) and the associated cost (measured by the trace of the Riccati matrix) is presented in Table 1.2 from which it is observed that the factor b/I2 has the most significant effect on the controller gain. To examine the fragility of the controller, we take the parameters at their maximum values and consider an affine family of perturbed controllers described by K = Ko (1 + δk ) where δk is the gain perturbation in the nominal coefficients. Observe that the form of K implies that each gain component has the same relative uncertainty range. By employing MATLAB facilities, we studied the impact of varying δk

12

1 Introduction

Fig. 1.9. Output Response with Variations . max value, .... nominal, − − −min value

in

All

Parameters:

−. −

Fig. 1.10. Output Response with Variations in k/I1 : −. − . max value, − − −min value

on the closed-loop quadratic stability. It is found that if δk is greater than 0.18, the quadratic stability is lost. Now by changing the gain vector over the domain −0.18 ≤ δk ≤ 0.8 and evaluating the associated cost, it is found that the ’critical’ gain is given by Kc = [29.1234, 13.4092, 11.0102, 25.3117] and the associated cost is 1.2478 × 103 implying that there is 13.07% degradation in the design cost as a price paid to guarantee non fragility of the state-feedback controller. Therefore, this example illuminates an important


13

Fig. 1.11. Output Response with Variations in k/I2 : −. − . max value, − − −min value

Fig. 1.12. Output Response with Variations in b/I1 : −. − . max value, − − −min value

issue in controller synthesis, that is, a trade-off exists between system performance and fragility of the controller. 1.3.3 Simple Heat Exchanger In a simple heat exchanger , steam enters the chamber at the top and cooler steam leaves at the bottom. There is a constant flow of water into a series of pipes to pick up the heat from the steam. The relevant variables here are the outflow steam temperature Ts and the out flowing water temperature Tw .

14

1 Introduction

Fig. 1.13. Output Response with Variations in b/I2 : −. − . max value, − − −min value Table 1.2. Computational Results of Parameter Variations Case 1) N ominal 1) M ax 1) M in 2) M ax 2) M in 3) M ax 3) M in 4) M ax 4) M in 5) M ax 5) M in

||k|| 36.7216 53.7593 27.3143 36.9879 37.1318 45.3366 42.6883 36.7623 36.9250 56.3917 29.6593

T r[P ] 1.1036 × 103 2.0343 × 103 0.7811 × 103 1.1545 × 103 1.1107 × 103 1.4413 × 103 1.6835 × 103 1.1039 × 103 1.1166 × 103 2.4018 × 103 0.7542 × 103

The heat process can be controlled by the steam inlet valve openingAs and monitored by the downstream water temperature Tm . Following [23], a simple transfer function of the heat exchanger has the form Ts (s) Ks e−d s = As (s) (1 + τ1 s) (1 + τ2 s)

(1.6)

where Ks is a flow coefficient of the inlet valve, d is the time delay in output measurement and the time constants τ1 , τ2 are functions of the heat exchanger parameters. Alternatively, a state-space model would be T˙s −α1 α1 Ts α4 = + (1.7) As α2 α3 Tw 0 T˙w Tm (t) = Tw (t − d)

(1.8)


15

where α1 , ..., α4 are scalars related to the time constants of the exchanger. To study the behavior, we select d = 5 , τ1 = 10 , τ2 = 60 The control objective is to design controller for the heat exchanger using the Smith predictor and pole placement [23]. Thus we choose to place the control poles at −0.05 ± j 0.087 and the estimator poles at −0.15 ± j 0.26. We have used Simulink to simulate the response of the closed-loop system (see Figure 1.14).

Fig. 1.14. Simulink diagram of closed-loop system

Standard computations showed that the state-feedback and estimator gains are given by Kc = [K11 K12 ] = [5.2

− 0.17 , Lte = [L11 L12 ] = [0.18 4.2]

We have considered small variations in Kc and the ensuing results are presented in Tables 1.3 and 1.4. The corresponding output responses are displayed in Figures 1.15 and 1.16. It is quite clear that small change in the controller coefficients has some effects on the output response. Similar simulations were performed with the Le coefficients but the results was almost negligible.

16

1 Introduction Table 1.3. Computational Results of Gain Variations-k11 Case y1 y2 y3 y4

Gain 6.5 -0.17 7.8 -0.17 9.1 -0.17 10.4 -0.17

Mp 0.2311 0.2835 0.3329 0.3777

Outputss 1.0224 1.0458 1.0631 1.0764

tp 35 32 30 28

Table 1.4. Computational Results of Gain Variations-k12 Case y1 y2 y3 y4

Gain 5.2 -0.2125 5.2 -0.2250 5.2 -0.2975 5.2 -0.3401

Mp 0.1920 0.1951 0.2286 0.2491

Outputss 0.9892 0.9894 0.9899 0.9905

tp 40 40 40 40

Fig. 1.15. Output response due to different gain coefficients-Case I

1.3.4 Stream Water Quality For all practical purposes, it is crucial to preserve the standards of water quality in streams. This can be measured by the concentrations of some water biochemical constituents. Let z(t), q(t) be the concentrations per unit volume of biological oxygen demand (BOD) and dissolved oxygen (DO), respectively, at time t. Under the simplifying assumptions that the stream has a constant flow rate and the water is well mixed. Further, we assume that there exists a τ > 0 such that the (BOD,DO) concentrations entering at time t are equal to the corresponding concentrations τ time units ago. Using mass balance concentration and employing a linearization about an equilibrium operating point, the growth of (BOD,DO) can be expressed as:

1.4 Approaches to Resilient Control

17

Fig. 1.16. Output response due to different gain coefficients-Case II

z(t) ˙ q(t) ˙

z(t) kcd kod z(t − τ ) kce 0 + = q(t) q(t − τ ) −kde −kre −kdd −krd mse 0 uz (t) + 0 mre uq (t)

(1.9)

where kce , ... , mre are composite rates some of which are unknown and uz (t), uq (t) are appropriate effluent controls. Using representative data on a single reach of the River Nile, system (1.9) has the form z(t) ˙ 1 0 z(t) −0.45 −0.50 z(t − τ ) = + q(t) ˙ −3 −2 q(t) −0.15 −0.10 q(t − τ ) 1.2 0 uz (t) + (1.10) 0 1.4 uq (t) With τ + = 0.65, the nominal state feedback control uz (t) 1.9237 −0.0307 z(t) = uq (t) −0.1145 0.8415 q(t) renders the water quality system asymptotically stable. Introducing 15% variations in the gain matrix coefficients can be easily shown to destabilize the system for 0 < τ ≤ 0.57.

1.4 Approaches to Resilient Control In this section, we outline some of the available approaches to resilient control design. Initially, from scanning the literature it is easy to observe that the

18

1 Introduction

majority of practical methods for control system design include the following ingredients: 1) A fixed structure controller with controller design parameters. It might be argued that smaller number of parameters would be an advantage in adjustment and tuning procedure. However with the growing software capabilities, versatile simulation packages are readily available so that large number of control parameters could be easily accommodated. 2) Performance specification in terms of multiple indices. In this way, several design objectives could be included for proper assessment of the desired behavior. 3) Selection of control system components including sensors and actuators followed by controller implementation. Let the controller parameters and the plant parameters be represented by q and p, respectively. Suppose that the performance specifications are formulated in terms of a set of N algebraic inequalities gj (r, p, q), j = 1, ..., N, where r is a generic variable stands for t in the time domain or ω in the frequency domain. Then, broadly speaking, the general objective is find the set Q of admissible parameter vector values that meet all of the performance specifications, that is gj (r, p, q) > 0,

r ∈ R , p ∈ P , j = 1, ..., N

(1.11)

Indeed, the foregoing family of inequalities encompass the procedure outlined in section (2.4). There are some techniques for the computations of the set Q in the frequency domain including A) Symbolic Quantifier-Elimination Techniques [15] These techniques are based on the development of quantifier-free logic formula involving polynomial inequalities and logic connectives. B) Interval Arithmetic Techniques [50] Interval arithmetic methods aim at exploring the range of functionals gj (ω, p, q) in the frequency domain and hence test for positivity. It further use brach-and-bound algorithms for the determination of the design parameter set Q. C) Statistical Learning Techniques [114] These probabilistic methods can be used to determine the design parameter set Q in an approximate manner with probabilistic guarantees. 1.4.1 Guaranteed-Cost Approach The main rationale behind the guaranteed-cost (GC) approach is to design feedback controllers that guarantee prescribed levels of performance for all plant parameter vectors p ∈ P [10]. In this case, the linear model of the plant has the form x(t) ˙ = A(p) x(t) + B(p) u(t)

(1.12)

1.4 Approaches to Resilient Control

19

where x(t) ∈ IRn is the state vector; u(t) ∈ IRm is the control input. A fixed state-feedback controller u(t) = K x(t) such that a prescribed level of integral-quadratic performance measure

∞ t x (t) Q x(t) + ut (t) R u(t) , Q > 0 , R > 0 (1.13) J = 0

is guaranteed for all possible plant vectors p ∈ P, where K is the unknown gain matrix. To employ this approach in the design of non-fragile controllers, one has to assume that the unknown feedback gain matrix is a function of the controller-uncertainty parameter vector, that is K = K(q), q ∈ Q. In this case, GC methods can be used to guarantee a level of performance with uncertainties in both p and q [18, 38, 41]. 1.4.2 The Proposed Framework In this monograph, we take a more general perspective by focusing on a widerclass of linear dynamical systems described by x(t) ˙ = A(∆p ) x(t) + Ad (∆p )x(t − τ ) + B(∆p )u(t) + Γ w(t) y(t) = C(∆p )x(t) + Cd (∆p )x(t − τ ) + Ψ w(t) z(t) = D(∆p )x(t) + F (∆p )u(t) + Φw(t)

(1.14)

where x(t) ∈ IRn is the state vector; u(t) ∈ IRm is the control input; w(t) ∈ q is the disturbance input which belongs to L2 [0, ∞); y(t) ∈ p is the measured output; z(t) ∈ r is the controlled output and τ ∈ [0, τ ∗ ] is an unknown time-varying delay factor satisfying 0 ≤ τ ≤ τ ∗ , 0 ≤ τ˙ ≤ τ + where the bounds τ ∗ , τ + < 1 are known constants and ∆p stands for the plant uncertainties. In terms of controller uncertainties ∆c , the feedback gain is represented by K(∆c ). A block diagram representation is shown in Figure 1.15. System (1.14) contains state and output time-varying delays. Throughout the book we will closely examine the development of control methods that generate various forms of the feedback gain K(∆c ) including state-feedback (instantaneous and delayed), dynamic output feedback , observers , adaptive controllers and filters . Interestingly enough, the proposed framework will encompass all previously published work [18, 38, 41, 118, 119, 120, 121, 122] and extend them in various technical ways 1.4.3 Controller Gain Perturbations In the subsequent chapters of the book we have to characterize the perturbed gain K(∆c ). Below, we describe some of the models that we are going to deal with. These are essentially two types of models:

20

1 Introduction

Fig. 1.17. Uncertain plant with time-delay and gain perturbations

Additive Type: K(∆c ) = Ko + ∆K(t) , ∆K(t) = Ha ∆c (t) Ea

(1.15)

where Ko is the nominal gain matrix, Ha and Ea are known constant matrices of appropriate dimensions and ∆c (t) is the uncertain parameter matrix which ∆ belongs to a compact set such that ∆c ∈ ∆ = {∆c : ∆tc ∆c ≤ I}. In the course of design, the matrices Ha and Ea could be manipulated to examine the robustness of the proposed method. A special case of (1.15) is that ∆K is a constant perturbation matrix to be determined in the part of controller synthesis [18] using appropriate numerical optimization. 1.4.4 Multiplicative Type K(∆c ) = Ko + ∆K(t) , ∆K(t) = Hm ∆c Em Ko

(1.16)

where Hm and Em are known constant matrices of appropriate dimensions. With Ko = {kij }, i = 1, ..., m, j = 1, ..., n, an alternative form of (1.16) is [18]:   k11 (1 + δ11 ) · · · k1n (1 + δ1n )   .. .. .. Ko + ∆K =   . . . km1 (1 + δm1 ) · · · kmn (1 + δmn )    k11 δ11 · · · k1n δ1n k11 · · · k1n     .. .. .. =  ... . . . ...  +   . . . km1 · · · kmn km1 δm1 · · · kmn δmn 

1.5 A Glossary of Terminology and Notations

−1 < δ¯ij ≤ δij ≤ δîj < 1

21

(1.17)

where δij represents the relative percentage drift from the nominal entries of the gain matrix Ko .

1.5 A Glossary of Terminology and Notations In this section, we assemble the terminologies and notations to be adopted throughout the book with the objective of paving the way to the technical development of subsequent chapters. These terminologies and notations are quite standard in the scientific media and only vary in form or character. 1.5.1 General Terms As a start, matrices as n × m dimensional arrays of elements with n-rows and m-columns are represented by capital letters while vectors as n-tuples or columns (unless otherwise specified) and scalars (single elements) are represented by lower case letters. We use IR, IR+ , IRn and IRn×m to denote the set of reals, positive real numbers, real n-tuples (vectors) and real n×m matrices, respectively. Alternatively, IRn is called the Euclidean space and is equipped ∆ with the vector-norm as ||x|| = [x21 + · · · x2n ]. The terms f (t), g(s) denote, respectively, scalar-valued functions of the real variables t and s. The quantities x, ˙ x ¨ are the first and the second derivative of x with respect to time, respectively. The symbols [., .], (., .], (., .) denote, respectively, closed, semiclosed, and open intervals; that is t ∈ (a, b] ⇒ a < t ≤ b. The open left-half (≡ {s : Re(s) ≤ 0}, the open proper left-half (≡ {s : Re(s) < 0} and the open proper right-half (≡ {s : Re(s) > 0} of the complex plane are represented by C I †, C I − and C I + , respectively. We use Up ∈ n×n and Uk ∈ m× to denote, respectively, the set of uncertain plant perturbations ∆A of the nominal dynamical system A and the set of uncertain controller perturbations ∆K of the nominal controller gain K. The Lebsegue space L2 [0, ∞) consists of square integrable functions on the interval [0, ∞) and equipped with the norm ∆

||p||2 =

∞

1/2 pt (τ ) p(τ ) dτ

(1.18)

0

For any square matrix W of arbitrary dimension n×n, let W t , W −1 , λ(W ), r(W ), tr(W ), det(W ), ρ(W ) and ||W || denote, respectively, the transpose, the inverse, the spectrum, the rank, the trace, the determinant, the spectral radius and the induced norm defined by 1/2

∆ ||W || = λ W W t (1.19) We use W > 0 (≥, 0 is termed the delay factor. In the sequel, if α ∈ IR, d ≥ 0 and x ∈ C([α I − τ, α + d], n ) then for any t ∈ [α, α + d], we let xt ∈ C I be defined ∆ by xt (θ) = x(t + θ), −τ ≤ θ ≤ 0 . In addition, if ID ⊂ IR × C, I f : ID → IRn is given function, then the relation x(t) ˙ = f (t, xt )

(1.22)

is a retarded functional differential equation (RFDE) on ID where xt , t ≥ t0 denotes the restriction of x(.) on the interval [t − τ, t] translated to [−τ, 0]. Here τ > 0 is termed the state-delay factor. A function x is said to be a solution of (1.22) on [α − τ, α + d] if there α ∈ IR and d > 0 such that x ∈ C([α I − τ, α + d], IRn ), (t, xt ) ∈ ID, t ∈ [α, α + d]

(1.23)

and x(t) satisfies (1.22) for t ∈ [α, α+d]. For a given α ∈ IR, φ ∈ C, I x(α, φ, f ) is said to be a solution of (1.22) with initial value φ at α.

1.6 Outline of the Book The chief objective of this monograph is to provide a complete description of resilient control and filtering theory. In our viewpoint, this theory virtually

1.6 Outline of the Book

23

stands as robust redesign approach and based thereon, it should be distinguished from robust control theory since the latter focuses mainly on parametric uncertainties. Thus this monograph brings together advanced methods for resilient control and filtering of dynamical systems. For generality in exposition, the main focus is on time-delay models as they capture additional features over conventional models described by ordinary differential equations. In writing up the different topics, emphasis is primarily placed on major developments attained thus far and then reference is made to other related work. Overall, the monograph is a suitable reference for graduate-students and researchers from wide-spectrum of engineering disciplines, science and mathematics. 1.6.1 Methodology Throughout the monograph, our methodology is composed of five-steps: • • • • •

Mathematical Modeling in which we discuss the main ingredients of the state-space model under consideration. Definitions and/or Assumptions here we state the definitions and/or constraints on the model variables to pave the way for subsequent analysis. Analysis and Examples this signifies the core of the respective sections and subsections which contains some solved examples for illustration. Results which are provided most of the time in the form of theorems, lemmas and corollaries. Remarks which are given to shed some light of the relevance of the developed results vis-a-vis published work.

In the sequel, theorems(lemmas, corollaries) are keyed to chapters and stated in italic font with bold titles, for example, Theorem 3.4 means Theorem 4 in Chapter 3 and so on. For convenience, we have grouped the reference in one major bibliography cited towards the end of the book. Relevant notes and research issues are offered at the end of each Chapter for the purpose of stimulating the reader. We hope that this way of articulating the information will attract the attention of a wide-spectrum of readership. 1.6.2 Chapter Organization The material covered is divided into 7 chapters whereby continuous-time results go in parallel with discrete-time results. All the developed results are conveniently expressed in LMI feasibility problem. It is fair to mention that

24

1 Introduction

almost all the analytical developments hereafter are new and original contributions to the control of time-delay systems. In Chapter 2 we start our guided tour by considering the design of resilient feedback controllers for a class of linear continuous-time systems with parametric uncertainties and state-delay. The main vehicle in the discussions to follow is the use of state-space modeling and design tools. Our effort treats both cases of norm-bounded and convex-bounded. The controller gain perturbations are either of additive and multiplicative forms. The control objective is to design resilient feedback stabilization schemes using guaranteed cost control and H∞ control approaches. Necessary and sufficient conditions for closed-loop robust quadratic stability are derived as opposed to the available sufficient conditions in the literature . The solution of all design problems under consideration is cast as convex optimization over linear matrix inequalities. The case of state-variable measurements is initially considered leading to resilient static feedback control. Then we move to examine the case of outputvariable measurements where the gain matrices of a dynamic output feedback control are constructed using an appropriate design procedure. The objective of Chapter 3 is to develop resilient controllers for a class of discrete-time systems with norm-bounded uncertainties and controller gain variations. The focus is on the development of resilient controllers for uncertain discrete-time systems with both types of gain variations. Necessary and sufficient conditions are established such that the resulting closed-loop feedback control system is quadratically stable for all admissible perturbations and uncertainties. These conditions are conveniently expressed in the form of linear matrix inequalities (LMIs). The feedback stabilization schemes are based on guaranteed cost control and H∞ control approaches. Chapter 4 addresses the problem of resilient adaptive control problem for classes of uncertain continuous-time and discrete-time systems with statedelays against controller gain variations. In the continuous case, design results on both norm-bounded and convex-bounded parametric uncertainties are derived. In the discrete-case, design results on norm-bounded parametric uncertainties are developed. Adaptive control schemes are constructed when gain perturbation bounds are known and then extended to accommodate unknown norm-bounded perturbations. To complete the design profile, special cases of delayless continuous and discrete systems are provided. The problem of robust nonfragile filtering for a class of linear uncertain continuous-time systems is investigated in Chapter 5. Initially we focus on norm-bounded uncertainties. We consider additive filter gain variations to reflect the imprecision in filter implementation. As a limiting procedure, the case of nonfragile Kalman filter is derived. All the developed results are conveniently extended to the case of multiplicative filter gain variations. Later on, we investigate the problem of designing a resilient L2 − L∞ filter for a class of linear uncertain state delay systems with uncertainties which belong to a convex bounded polytopic domain. The objective is to derive tractable synthesis conditions for the resilient design of full-order and reduced-order

1.7 Notes and References

25

filters such that a prescribed energy-to-peak disturbance-attenuation level is attained for all admissible uncertainties and gain perturbations. Both additive and multiplicative gain perturbations are considered. It is established that the filter design problems can be obtained from the solution of convex optimization problem over linear matrix inequalities. The design results are developed for both delay-independent and delay-dependent cases. In the latter case, two approaches are used: descriptor and extended Newton-Leibniz. In Chapter 6, we direct attention to delay-dependent methodologies for continuous-time and discrete-time systems by exhibiting the delay-dependence dynamics in the design procedure. For the continuous-time case, we develop a descriptor approach to simultaneous H2 /H∞ control design for a class of time-delay systems with additive and multiplicative controller gain perturbations. The delay factor is considered unknown-but-bounded constant or timevarying. The cases of nominal and convex polytopic uncertainties are treated. In the discrete-time case, delay-dependent stability and feedback stabilization methods are developed for the nominal, robust and resilience cases. We end the monograph by Chapter 7 in which we move further to examine classes of nonlinear continuous-time and discrete-time systems with statedelay. We develop an LMI-based analysis and design procedures to check primarily into the robust stability of both continuous-time and discrete-time systems. Then we address the robust stabilization using nominal and resilient feedback designs. In both cases the trade-off between the size of the controller gains and the bounding factors is illuminated and incorporated into the design formalism. Seeking computational convenience, all the developed results are cast in the format of family of LMIs. An appendix containing some relevant mathematical tools and basic results is provided.

1.7 Notes and References In addition to the numerous papers and articles on time-delay systems, there are a number of reference books which might have some connection to the topics to be discussed in this book. This includes [8, 81]. The discussions presented thus far follow the work of [7, 12, 15, 16]. It is hoped that this introductory chapter has succeeded in motivating the readers to the upcoming topics. We have made every effort to produce the book as a self-contained examination of the background and progress of resilient control and filtering of uncertain dynamical systems.

2 Resilient Control-Continuous Case

2.1 Introduction An integral part of robust control system design methods has been based on using a fixed quadratic Lyapunov function in order to guarantee robust stability [17, 73]. An implicit assumption inherent in these design methods is that the controller will be implemented exactly. In the presence of uncertain parameters, it is often desirable to perform the control system design not only to ensure stability but also to guarantee an adequate level of system performance. This brings about the notion of guaranteed cost control (GCC) [103] and H∞ -control [17]. When attending to the controller implementation based on different control design methods, it turns out that the controllers are very sensitive ”fragile” with respect to errors in the controller coefficients. This is due to imprecision in analog-digital conversion, fixed word length, finite resolution instrumentation and numerical roundoff errors. It is considered beneficial that the designed (nominal) controllers should be capable of tolerating some level of controller gain variations. This illuminates the controller fragility problem as discussed in Chapter 1 and in the discussions to follows we address methods to overcome this problem. This chapter considers the design of resilient feedback controllers for a class of linear continuous-time systems with parametric uncertainties and statedelay . The main vehicle in the discussions to follow is the use of state-space modeling and design tools. Our effort will treat both cases of norm-bounded and convex-polytopic. The controller gain perturbations are either of additive and multiplicative forms. The control objective is to design resilient (nonfragile) feedback stabilization schemes using guaranteed cost control and H∞ control approaches. Throughout this chapter, necessary and sufficient conditions for closed-loop robust quadratic stability are derived as opposed to the available sufficient conditions in the literature . The solution of all design problems under consideration is cast as convex optimization over linear matrix inequalities (LMIs) . This ensures the full use of convenient computational environment based on available commercial software. In the sequel, the case of


28


state-variable measurements is initially considered leading to state-feedback control. Then we move to examine the case of output-variable measurements where the gain matrices of a dynamic output feedback control are constructed using an appropriate design procedure. Several system examples are provided to illustrate the theoretical developments.

2.2 Model with Norm-Bounded Uncertainties A schematic of the problem setup is displayed in Figure 2.1 which shows a plant P subjected to uncertainties ∆p and a controller K having gain perturbations ∆c . Let the plant be represented by the class of linear continuous-time systems with state-delay : (Σ∆ ) :

x(t) ˙ = [Ao + ∆Ao (t)]x(t) + [Ad + ∆Ad (t)]x(t − τ ) + Bo u(t) + Γ w(t) = A∆o (t)x(t) + A∆d (t)x(t − τ ) + Bo u(t) + Γ w(t) y(t) = Co x(t) + Cd x(t − τ ) + Ψ w(t) z(t) = Do x(t) + Fo u(t) + Φw(t)

(2.1)

where x(t) ∈ IRn is the state vector; u(t) ∈ IRm is the control input; w(t) ∈ IRq is the disturbance input which belongs to L2 [0, ∞); y(t) ∈ IRp is the measured output; z(t) ∈ IRr is the controlled output and τ ∈ [0, τ ∗ ] is an unknown time-varying delay factor satisfying 0 ≤ τ ≤ τ ∗ , 0 ≤ τ˙ ≤ τ + where the bounds τ ∗ , τ + < 1 are known constants. The matrices Ao , Bo , Ad , Co , Do , Cd , Fo , Γ, Φ, Ψ are known real constant matrices of appropriate dimensions which describe the nominal system of (Σ∆ ). In this part, the matrices ∆Ao (t), ∆Ad (t) are real, time-varying matrix functions representing the norm-bounded parameter uncertainties and represented by: [∆Ao (t) ∆Ad (t)] = Ma ∆p (t)[Na Nd ] ∆

∆p (t) ∈ ∆p (t) = {∆p (t) : ∆tp (t)∆p (t) ≤ I ∀ t}

(2.2)

where Ma ∈ IRn×α , Na ∈ IRβ×n and Nd ∈ IRβ×n are known real constant matrices and the elements of ∆p (t) are being Lebesgue measurable. The initial condition is specified as x(0), x(s) = xo , φ(s) , where φ(.) ∈ L2 [−τ, 0]. We remark here that the norm-bounded uncertainty structure considered here has been widely adopted in robust control [73, 80, 82]. It covers the usual matching condition as a special case by setting N ≡ I. Observe that the unit upperbound condition on ∆(t) does not cause any loss of generality. Indeed ∆p (t) can always be normalized by appropriately selecting the matrices M and N .

2.3 Guaranteed Cost Control I

29

Fig. 2.1. Uncertain Plant with Controller Perturbations

2.3 Guaranteed Cost Control I Associated with system (Σ∆ ) is the cost function :

∞ [xt (t)Qx(t) + ut (t)Ru(t)] dt J=

(2.3)

0

where 0 < Q = Qt ∈ IRn×n and 0 < R = Rt IRm×m are given weighting matrices. Consider for the time being that we have a state-feedback controller u(t) = Ko x(t) with Ko being the static nominal (designed) gain matrix but the actual implemented controller is assumed to be u(t) = [Ko + ∆K(t)] x(t) = K∆ x(t)

(2.4)

where ∆K(t) represents the gain perturbation . Definition 2.1. Consider system (Σ∆ ) with cost function (2.3) and w(t) ≡ 0. The state-feedback control law (2.4) is said to be a guaranteed cost control (GCC) with quadratic cost matrix P > 0 given matrices 0 < Q = Qt , 0 < W = W t and 0 < R = Rt , if the following LMI t P A∆ + At∆ P + K∆ RK∆ + Q + W P A∆d < 0 ˆ • −W ˆ = (1 − τ + )W (2.5) W has a feasible solution with respect to P for all admissible uncertainties ∆A(t) satisfying (2.2) and gain perturbations ∆K(t) We note that the foregoing definition holds for arbitrary gain variations. In the absence of controller perturbations ∆K(t) ≡ 0, then Definition 2.1 reduces to the well known definition of guaranteed cost control for uncertain continuous-time systems [81].

30


2.3.1 Quadratic Stability Considering the GCC approach, the following theorem provides a link between the notion of quadratic cost matrix and an upper bound on the cost function J. Theorem 2.2. Consider system (Σ∆ ) with cost function (2.3). Suppose that the state-feedback control law (2.4) with controller gain perturbations ∆K(t) is a quadratic GCC with matrix P > 0. Then the resulting closed-loop system with w(t) ≡ 0 x(t) ˙ = A∆ (t)x(t) + A∆d (t)x(t − τ )

(2.6)

is quadratically stable for all admissible uncertainties satisfying (2.3) with the cost function bounded by

0 J ≤ xto P xo + xt (s) W x(s) ds (2.7) −τ

Proof (⇒) : If P > 0 is a quadratic cost matrix for system (Σ∆ ) and cost function (2.3), it follows from (2.5) and the Schur complements operations that t t t −1 t ˆ x P A∆ + A∆ P + K∆ RK∆ + Q + W + P A∆d W A∆d P x < 0 ∀ x = 0 , ∀ ∆(t) ∈ ∆p (t)

(2.8)

Hence, system (Σ∆ ) is quadratically stable . Next, introduce the Lyapunov-Krasovskii functional

0 xt (s)W x(s) ds V (x(t)) = xt (t)P x(t) + −τ

Standard matrix manipulations along the solutions of (2.6) yield: V˙ (x) = xt (t) At∆ P + P A∆ + W x(t) + 2xt P A∆d x(t − τ ) − xt (t − τ )(1 − τ˙ )W x(t − τ ) ≤ xt (t) At∆ P + P A∆ + W −1 t t t ˆ + P A∆d W A∆d P x(t) < − x (t) Q + K∆ RK∆ x(t) From which we conclude that t t x (t) Q + K∆ RK∆ x(t) < V˙ (x)

(2.9)

(2.10)

Integrating over the period t ∈ [0, ∞) and using 2.3, we get J ≤ V (xo ) − V (x(∞))

(2.11)


31

By (2.8), system (Σ∆ ) is quadratically stable thereby leading to V (x(t)) → 0 as t → ∞ and therefore (2.11) reduces to

J ≤ V (xo ) = xto P xo +

0

xt (s)W x(s) ds

(2.12)

−τ

(⇐) : Let system (Σ∆ ) be quadratically stable. It then follows on using the Schur complement operations that there exist 0 0 such that the following inequality t ˆ −1 At∆d P + Q + W ] = µ−1 [P A∆ + At∆ P + K∆ RK∆ + P A∆d W ¯ˆ −1 t ¯ t ¯ ¯+W ¯ < 0 A∆d P + Q RK∆ + P¯ A∆d W P¯ A∆ + At∆ P¯ + K∆

∀ ∆(t) ∈ ∆p (t)

(2.14)

This inequality implies that, with the help of Schur complements, there exists ¯ = µ−1 W, Q ¯ = µ−1 Q, R ¯ = µ−1 R, such that the matrix matrices W −1 P¯ = µ P is a quadratic cost matrix for system (Σ∆ ). This completes the proof. ∇∇∇ 2.3.2 Additive Gain Perturbations It is assumed that the gain perturbation matrix ∆K(t) has the form: ∆K(t) = Ha ∆c Ea

(2.15) ∆

where Ha and Ea are known constant matrices and ∆c (t) ∈ ∆c (t) = {∆c (t) : ∆tc (t)∆c (t) ≤ I ∀ t} is the uncertainty parameter matrix. In the sequel, we establish LMI-based results for GCC synthesis such that the closed-loop controlled system achieve quadratic stability for all admissible uncertainties ∆p (t) ∈ ∆p (t). For simplicity in exposition, we define the matrices: Af = Ao + Bo Ko , ∆Af (t) = ∆A(t) + Bo ∆K(t) A∆ = Af + ∆Af (t)

(2.16)

The following theorem provides a necessary and sufficient condition for GCC under additive gain perturbations. Theorem 2.3. Consider system (Σ∆ ) with cost function (2.3). There exist state feedback gain Ko such that the control law (2.4) with additive gain perturbations (2.15) is a GCC with a quadratic cost matrix P > 0 given matrices 0 < Q = Qt , 0 < W = W t and 0 < R = Rt if and only if there exist matrices

32


0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, µ > 0, such that the following LMIs   Y t R+ Ad + t t Ξ M B H ZN LE XQ XW a a  a a o a ZNat Nd µBo Ha Hat Bot      • −εI 0 0 0 0 0 0 0     • • −I 0 0 0 0 0 0     • • • −εI 0 0 0 0 0     • • • • −I 0 0 0 0  0, > 0. By the congruent transformation diag[P −1 , I, I], P −1 = X, Ko P −1 = Y, −1 = µ, εP −1 = Z, P −1 = L and followed by Schur complement operations we obtain the equivalent LMI conditions of (2.17). ∇∇∇


33

2.3.3 Multiplicative Gain Perturbations Here, it is assumed that the gain perturbation matrix ∆K(t) has the form: ∆K(t) = Hm ∆c Em Ko

(2.19) ∆

where Hm and Em are known constant matrices and ∆c (t) ∈ ∆c (t) = {∆c (t) : ∆tc )∆c (t) ≤ I ∀ t} is the uncertainty parameter matrix. The following theorem builds on Theorem 2.3 and provides necessary and sufficient conditions for GCC under multiplicative gain perturbations. Theorem 2.4. Consider system (Σ∆ ) with cost function (2.3). There exist state feedback gain Ko such that the control law (2.4) with multiplicative perturbations (2.19) is a GCC with a quadratic cost matrix P > 0 given matrices 0 < Q = Qt , 0 < W = W t and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, µ > 0, such that the following LMIs   Y t R+ Ad + t t Ξ M B H ZN GE XQ XW t a m  a a o m Bot  ZNat Nd µBo Hm Hm    • −εI  0 0 0 0 0 0 0    • • −I  0 0 0 0 0 0    • •  • −εI 0 0 0 0 0    • •  • • −I 0 0 0 0   0, > 0. By the congruent transformation diag[P −1 , I, I], P −1 = X , Ko P −1 = Y, −1 = µ εP −1 = Z, P −1 Kot = G and followed by Schur complement operations we obtain the LMI conditions of (2.20). ∇∇∇ 2.3.4 Special Cases We provide hereafter, resilient control results on some systems of particular interests. In the absence of system uncertainties (M ≡ 0, Na ≡ 0, Nd ≡ 0) we obtain the nominal system (Σo ) : x(t) ˙ = Ao x(t) + Ad x(t − τ ) + Bo u(t)

(2.22)

for which implementation of Theorem 2.3 reduces to: Theorem 2.5. Consider system (Σ∆ ) with cost function (2.3). There exist state feedback gain Ko such that the control law (2.4) with additive gain perturbations (2.15) is a GCC with a quadratic cost matrix P > 0 given matrices 0 < Q = Qt , 0 < W = W t and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y, Z, L and parameters ε > 0, > 0, µ > 0 such that the following LMIs   Y t R+ t  Ξa Bo Ha LEa XQ XW Ad µBo Ha Hat Bot     • −I 0  0 0 0 0    •  • −εI 0 0 0 0    •  0 given matrices 0 < Q = Qt , 0 < W = W t and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y, Z, L and parameters ε > 0, > 0, µ > 0 such that the following LMIs   Y t R+ t Ξ B H GE XQ XW A d t m  a o m µBo Hm Hm Bot      • −I 0 0 0 0 0     • • −εI 0 0 0 0    0 given matrices 0 < Q = Qt and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, µ > 0, such that the following LMIs   Y t R+ t t  Ξa Ma Bo Ha ZNa LEa XQ µBo Ha Hat Bot      • −εI 0 0 0 0 0     • • −I 0 0 0 0    0 given matrices 0 < Q = Qt and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, µ > 0, such that the following LMIs   Y t R+ t t Ξ XQ M B H ZN GE t a m  a a o m Bot  µBo Hm Hm    • −εI  0 0 0 0 0    • • −I  0 0 0 0    • •  0 if and only if there exists a matrix 0 < P t = P such that the following LMI   P A∆ + At∆ P + W P A∆d P Γ + Dkt Φ   < 0 ˆ (2.28) • −W 0 • • −γ 2 I + Φt Φ has a feasible solution for all admissible uncertainties ∆A(t) and ∆K(t) satisfying (2.2) and (2.15) or (2.19), where Dk = Do + Fo Ko . By a parallel development, the corresponding results for controller gain variations of both types are established by the following theorems. 2.4.1 Additive Gain Perturbations For the additive type of gain perturbations, we have Theorem 2.11. Consider system (Σ∆ ) with cost function (2.3). There exist a state feedback gain Ko such that the control law (2.4) with additive gain perturbations (2.15) is a H∞ control with disturbance attenuation γ > 0 if and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, such that the following LMIs

38


  Ξa   •   •   •   •   •    •    •

 Ad + Γ + XDot Φ ZNat Nd +Y t Fot Φ    −εI 0 0 0 0 0 0   • −I 0 0 0 0 0   • • −εI 0 0 0 0   0, > 0. By the congruent transformation diag[P −1 , I, I], P −1 = X and using

Ko P −1 = Y, −1 = µ, εP −1 = Z, P −1 = L

followed by Schur complement operations we obtain the equivalent LMI conditions of (2.29). ∇∇∇

2.4 H∞ Control I

39

2.4.2 Multiplicative Gain Perturbations For the multiplicative perturbation type, we have the following result Theorem 2.12. Consider system (Σ∆ ) with cost function (2.3). There exist state-feedback gain Ko such that the control law (2.4) with multiplicative gain perturbations (2.19) is a H∞ control with disturbance attenuation γ > 0 if and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, such that the following LMIs   Ad + Γ + XDot Φ t t  Ξa Ma Bo Hm ZNa GEm XW ZNat Nd +Y t Fot Φ     • −εI  0 0 0 0 0 0    • • −I  0 0 0 0 0    • •  • −εI 0 0 0 0    • •  0 if and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, such that the following LMIs   Γ + XDot Φ t  Ξa Bo Ha LEa XW Ad +Y t Fot Φ     • −I 0  0 0 0    •  • −εI 0 0 0   0 if

40


and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, such that the following LMIs   Ad + Γ + XDot Φ t Ξ B H GE XW m  a o m ZNat Nd +Y t Fot Φ     • −I  0 0 0 0    •  • −I 0 0 0   0 if and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, such that the following LMIs   t t t Γ + XDo Φ Ξ M B H ZN LE a a o a a a +Y t F t Φ   o    • −εI 0  0 0 0    • • −I  0 0 0   0 if and only if there exist matrices 0 < X = X t , Y, Z, L and some parameters ε > 0, > 0, such that the following LMIs   t t t Γ + XDo Φ  Ξa Ma Bo Hm ZNa GEm +Y t Fot Φ     • −εI  0 0 0 0    • • −I  0 0 0   0. 2.5.2 Design Procedure Our purpose hereafter is to derive tractable synthesis conditions for the resilient dynamic output feedback controller. To simplify the discussions to follow, we start by defining the following matrices Qx 0 Xx Xd ∈ IR2n×2n , X = ∈ IR2n×2n Q= • Qc • Xc Vx Vd Wx 0 ∈ IR2n×2n , W = ∈ IR2n×2n V= • Wc Vg Vc Zx Zd ∈ IR2n×2n Z= (2.51) Zg Zc It should be observed that (2.50) implies that V > 0 and thus we can assume that Vg > 0 and Vc > 0. Next we introduce the matrices Ω1 = Vgt Vc−t Vg , Ω2 = Vgt Vc−t Vdt

(2.52)

With (2.49) and (2.51), the general matrix inequality (2.50) becomes:   −Ξa Ξb Ξc Ξd Ξe Ξf V t  • −X 0 0 0 0 0     •  • −R 0 0 0 0    •  0, j ∈ IN if given matrices 0 < Qj = Qtj , 0 < Rj = Rjt , , 0 < Wj = Wjt , the following matrix inequalities t Pj A∆j + At∆j Pj + K∆ Rj K∆ + Qj + Wj Pj Adj ˆj < 0 • −W ˆ j = (1 − τ + )Wj , j ∈ IN W (2.66) has a feasible solution with respect to Pj , j ∈ IN, perturbations satisfying (2.15) or (2.19)

for all admissible gain

In the discussions to follow, the problem of interest in this section is as follows: Given the uncertain system (ΣJ ) with cost function (2.64) along with the state-feedback control law (2.65) with controller gain perturbations (2.15) or (2.19), it is required to determine the feedback gain matrix Ko such that the closed-loop controlled system achieve quadratic stability over the vertex set IN. Proceeding in parallel to the previous sections, we will derive expressions for gain matrix Ko , using guaranteed cost and H∞ approaches.

2.7 Guaranteed Cost Control II In this section, we establish results for GCC synthesis with both types of controller perturbations. Subsequently, we derive some important cases. We start by establishing the stability result 2.7.1 Quadratic Stability Considering the GCC approach, the following theorem provides a link between the notion of quadratic cost matrix and an upper bound on the cost function Jj , j ∈ IN. Theorem 2.21. Consider system (ΣJ ) with cost function (2.64). Suppose that the state-feedback control law (2.65) with controller gain perturbations (2.15) or (2.19) is a quadratic GCC with matrix P > 0. Then the resulting closed-loop system with w(t) ≡ 0 x(t) ˙ = A∆j (t)x(t) + Adj (t)x(t − τ )

(2.67)

is quadratically stable over the vertex set IN with the cost function bounded by Jj ≤

N j=1

λj xto Pj xo +

0

−τ

xt (s) Wj x(s) ds

(2.68)

50


Proof (⇒) : If Pj > 0, j ∈ IN is a quadratic cost matrix for system (ΣJ ) and cost function (2.64), it follows from (2.65) and the Schur complements that t ˆ −1 Atdj Pj x < 0 xt Pj A∆j + At∆j Pj + K∆ Rj K∆ + Qj + Wj + Pj Adj W j ∀ x = 0 , j ∈ IN

(2.69)

Hence, system (ΣJ ) is quadratically stable. Next, introduce the Lyapunov-Krasovskii functional N

V (x) =

λj xt (t)Pj x(t) +

0

−τ

j=1

xt (s)Wj x(s) ds .

Evaluation of V˙ (x) along the solutions of (2.67) yields: V˙ (x) =

N

t λj x (t) A∆j Pj + Pj A∆j + Wj x(t) + 2xt Pj Adj x(t − τ ) t

j=1

− x (t − τ )(1 − τ˙ )Wj x(t − τ ) ≤ x (t) At∆j Pj + Pj A∆j + Wj −1 t ˆ + Pj Adj Wj Adj Pj x(t) < t

−

N

t

t

λj x (t)

Qj +

t K∆ Rj K∆

x(t)

(2.70)

j=1

From which we conclude that N t λj xt (t) Qj + K∆ Rj K∆ x(t) < V˙ (x)

(2.71)

j=1

Integrating over the period t ∈ [0, ∞) and using 2.64, we get Jj ≤ V (xo ) − V (x(∞))

(2.72)

By (2.69), system (Σ∆ ) is quadratically stable thereby leading to V (x(t)) → 0 as t → ∞ and therefore (2.72) reduces to ∆

Jj ≤ V (xo ) =

N j=1

t λj xo Pj xo +

0

−τ

t

x (s)Wj x(s) ds

, j ∈ IN (2.73)

(⇐) : Let system (ΣJ ) be quadratically stable. It then follows on using the Schur complements operations that there exist 0 < Pj = Pjt such that t ˆ −1 Atdj Pj < 0 Pj A∆j + At∆j Pj + K∆ Rj K∆ + Qj + Wj + Pj Adj W j ∀ j ∈ IN (2.74)

2.7 Guaranteed Cost Control II

51

Hence, one can find some µj > 0, j ∈ IN such that the following inequality t t ˆ −1 t µ−1 j [Pj A∆j + A∆j Pj + K∆ Rj K∆ + Qj + Wj + Pj Adj Wj Adj Pj ] = t ¯ ¯j + W ¯ j + P¯j Adj W ˆ −1 Atdj P¯j < 0 P¯j A∆j + At∆j P¯j + K∆ Rj K∆ + Q j ∀ j ∈ IN (2.75)

This inequality implies that, with the help of Schur complements, there exists ¯ j = µ−1 Wj , Q ¯ j = µ−1 Rj , such that the matrix ¯ j = µ−1 Qj , R matrices W j j j −1 P¯j = µj Pj is a quadratic cost matrix for system (ΣJ ). This completes the proof. ∇∇∇ Next, we establish results for GCC synthesis with both types of controller perturbations. 2.7.2 Additive Gain Perturbations The following theorem provides a necessary and sufficient condition for GCC under additive gain perturbations. Theorem 2.22. Consider system (ΣJ ) with cost function (2.64). There exists a state feedback gain Ko such that the control law (2.65) with additive gain perturbations (2.15) is a GCC with a quadratic cost matrix Pj > 0, j ∈ IN given matrices 0 < Qj = Qtj , 0 < Wj = Wjt , 0 < Rj = Rjt , j ∈ IN if and only if there exist matrices 0 < Xj = Xjt , Lj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs   Xj − V+ W t+ t Ξ X Q X W L E B H A aj j j j j j oj a dj a   Aoj W t µj Boj Ha Hat    • −W − W t  0 0 0 0 0 0   t  •  • −R + µ H H 0 0 0 0 0 j j a a    •  0. By the congruent transformation diag[Pj−1 , I, I], Pj−1 = Xj with

−1 = Lj −1 j = µj , j Pj

and applying Lemma (8.5) of the Appendix along with Schur complement operations we obtain the equivalent LMI conditions of (2.76). ∇∇∇ 2.7.3 Multiplicative Gain Perturbations The following theorem builds on Theorem 2.22 and provides necessary and sufficient conditions for GCC under multiplicative gain perturbations. Theorem 2.23. Consider system (Σ∆ ) with cost function (2.64). There exists a state feedback gain Ko such that the control law (2.65) with multiplicative gain perturbations (2.19) is a GCC with a quadratic cost matrix Pj > 0, j ∈ IN given matrices 0 < Qj = Qtj , 0 < Wj = Wjt , 0 < Rj = Rjt , j ∈ IN if and only if there exist matrices 0 < Xj = Xjt , Gj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs   Xj − V W t+ t Xj Qj Xj Wj Gj Ea Boj Hm Adj  t  Ξaj +Aoj W t µj Boj Hm Hm   t  • −W − W 0 0 0 0 0 0    t  • • −Rj + µj Hm Hm 0 0 0 0 0     • 0 0 0 0  • • −Qj  0, j ∈ IN given matrices 0 < Qj = Qtj , 0 < Wj = Wjt , 0 < Rj = Rjt , j ∈ IN if and only if there exist matrices 0 < Xj = Xjt , j ∈ IN, V, W such that the following LMIs   Ξaj Xj − V + Aoj W t W t Xj Qj Xj Wj Adj −W − W t 0 0 0 0   •   0 0  • −Rj 0  • 0, j ∈ IN given matrices 0 < Qj = Qtj , 0 < Rj = Rjt , j ∈ IN if and only if there exist matrices 0 < Xj = Xjt , Lj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs   Ξaj Xj − V + Aoj W t W t + µj Boj Ha Hat Xj Qj Lj Eat Boj Ha  • −W − W t 0 0 0 0    t  • • −Rj + µj Ha Ha 0 0 0    0, j ∈ IN given matrices 0 < Qj = Qtj , 0 < Rj = Rjt , j ∈ IN if and only if there exist matrices 0 < Xj = Xjt , Gj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs   t t Ξaj Xj − V + Aoj W t W t + µj Boj Hm Hm Xj Qj Gj Em Boj Hm  • −W − W t 0 0 0 0    t  • + µ H H 0 0 0  • −R j j m m   0, j ∈ IN given matrices 0 < Qj = Qtj , 0 < Rj = Rjt , j ∈ IN if and only if there exist matrices 0 < Xj = Xjt , j ∈ IN, V, W such that the following LMIs 

 Ξaj Xj − V + Aoj W t W t Xj Qj  • −W − W t 0 0    0 if given a matrix 0 < Wjt = Wj , j ∈ IN there exists a matrix 0 < Pjt = Pj , j ∈ IN such that the following LMI 

 ˆ tΦ Pj A∆j + At∆j Pj + Wj Pj Adj Pj Γ + D j   ˆj 0 • −W 2 t • • −γ I + Φ Φ

< 0

(2.85)

has a feasible solution for all admissible uncertainties over the vertex IN and ˆ j = Doj + Foj Ko . gain perturbations ∆K(t) (2.15) or (2.19), where D

2.8 H∞ Control II

57

The following theorems summarize the corresponding results for controller gain variations of both types. We provides them in two consecutive subsections beginning with the additive type. 2.8.1 Additive Gain Perturbations Theorem 2.30. Consider system (ΣJ ) with additive gain perturbations (2.15). Given matrices 0 < Wj = Wjt , there exists an H∞ controller of the type (2.65) with disturbance attenuation γ > 0 if and only if there exist matrices 0 < Xj = Xjt , Lj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs   t Γ + VDoj Φ Xj − V+ W t+ t Ξ X W L E B H A t  aj Aoj W t µj Boj Ha Hat j j j a oj a dj +W t Foj Φ     • −W − W t  0 0 0 0 0 0     −Rj +  •  • 0 0 0 0 0 t   µj Ha Ha    •  0. Proceeding like Theorem 2.3, we use the = congruent transformation diag[Pj−1 , I, I] with Pj−1 = Xj and −1 j −1 µj , j Pj = Lj and applying Lemma (8.2) followed by Schur complement operations we obtain the equivalent LMI conditions of (2.86). ∇∇∇ 2.8.2 Multiplicative Gain Perturbation we have the following result Theorem 2.31. Consider system (ΣJ ) with multiplicative gain perturbations (2.19). Given matrices 0 < Wj = Wjt , there exists an H∞ controller of the type (2.65) with disturbance attenuation γ > 0 if and only if there exist matrices 0 < Xj = Xjt , Gj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs   Γ+ Xj − t W + t t  Ξaj VDoj Φ  V+ t Xj Wj Gj Em Boj Hm Adj   µj Boj Hm Hm t t t  A +W W Foj Φ  oj    • −W − W t  0 0 0 0 0 0     −Rj +  •  • 0 0 0 0 0 t   µj Hm Hm   0 if and only if there exist matrices 0 < Xj = Xjt , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs 

 t Φ Γ + VDoj Xj − V+ t Ξ W X W A aj j j dj t  +W t Foj Φ  Aoj W t    • −W − W t 0  0 0 0    •  0 if and only if there exist matrices 0 < Xj = Xjt , Lj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs   t Γ + VDoj Φ W t+ t t  Ξaj Xj − V + Aoj W µj Boj Ha Hat Lj Ea Boj Ha +W t F t Φ  oj    •  −W − W t 0 0 0 0     −Rj +  •  • 0 0 0 t   0 if and only if there exist matrices 0 < Xj = Xjt , Lj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs

60




 t Γ + VDoj Φ W t+ t t Ξ H X − V + A W G E B oj m aj j oj j t t m  +W t Foj Φ  µj Boj Hm Hm   t  •  −W − W 0 0 0 0     −Rj +  •  • 0 0 0 t   0 if and only if there exist matrices 0 < Xj = Xjt , Lj , j ∈ IN, V, W and some parameters j > 0, µj > 0, j ∈ IN such that the following LMIs   t t Φ Ξaj Xj − V + Aoj W t W t Γ + VDoj Φ + W t Foj  •  −W − W t 0 0   0. Now, it follows from [79] and Fact 1 that matrix inequality (2.97) with (2.98) holds if and only if the following inequality  ¯ ct Rj Pj Adξj  Pj Aξj + Atξj Pj + Qj + Wj K  • −Rj 0 + ˆj • • −W   t   t t   t ¯oj ¯ ¯oj ¯ B E B E c c −1        Rj Hc Rj Hc + εj 0 0  + εj 0 0 0 0   t   t t   t ¯ ¯ ¯ cj ¯ cj N N M M bj bj        0 0 −1 + 0 0  + j j 0 0 0 0   t   t t   t ¯ cj ¯ ¯ cj ¯ M N M N c c −1        0 0 0  + σj + σj 0 0 0 0 0     t   t ¯ cj ¯ cj M 0 0 M 0 , Qs = Qts > 0 , Qc = Qtc > 0

(2.132)

The following theorem establishes the first main stability result. Theorem 2.46. Consider system (ΣA ) with additive gain perturbations and given matrices 0 < Qs = Qts ∈ n×n , 0 < Ws = Wst ∈ IRn×n , 0 < Qc = Qtc ∈ n×n , 0 < Wc = Wct ∈ IRn×n . There exist matrices 0 < Xs = Xst ∈ n×n , 0 < Xc = Xct ∈ IRn×n , Ys , Yc , Za , Yd , Yh , Zb , Zc and scalars ε1 > 0, ..., ε4 > 0, such that the LMIs   Ao Xs + Xs Ato Π Π X Q A a s s s d   +Bo Ys + Yst Bot     • −Λs 0 0 0  < 0 (2.133)    0 • • −Λs 0     • • • −Qs 0 t ˆ • • • • −Qs + ε4 Nd Nd   t Ao Xc + Xc Ao t Πc Zc Xs Qs Ad  +Yc + Yct      • −Λ 0 0 0 c   < 0 (2.134)   • • −ε I 0 0 3     • • • −Qc 0 ˆ s + ε4 N t N d • • • • −Q d   Bo Yd + Ydt Bot + 2Ma 2Ma 2Ad   Yh + Yht     ≥ 0 • 2ε I 0 0 (2.135) 1     • • 2ε4 I 0 ˆ s − ε4 N t N d • • • 2Q d have a feasible solution. Moreover, the gain matrices are given by Kc = Ys Xs−1 , Ko = −Yc Xc−1 Co†

1

Co† = Cot [Co Cot ]−1 is the pseudo-inverse of Co .

1

.

2.10 Uncertain Model with State-Delays

75

Proof: Differentiating (2.131) with respect to t and using (2.127) and arranging terms, we obtain t PA∆ξ + At∆ξ P + Q PA∆dξ ξ(t) ξ(t) (2.136) V˙ (ξ) = ξ(t − τ ) ξ(t − τ ) • −(1 − τ˙ )Q By applying Inequalities I-II of the Appendix and using (2.128), it follows that: −1 ¯ ¯t ¯ ¯t ¯t ¯ P∆Aξ + ∆Atξ P ≤ ε−1 1 P M a Ma P + ε 1 N a N a + ε 2 P M b Mb P ¯f M ¯ht N ¯h + ε−1 P M ¯ ft P + ε3 N ¯gt N ¯g +ε2 N 3 −1 t ¯ −1 −1 t ˆ ¯ Atξdo P PA∆dξ (1 − τ˙ ) Q A∆dξ P ≤ PAξdo Q − ε4 Nd Nd

¯ ¯t +ε−1 4 P M a Ma P

(2.137)

ˆ = (1 − τ ∗ )Q . By invoking the Schur complements to (2.136) and where Q considering (2.128) and (2.137), we obtain t ˙ ¯ ¯t ¯t ¯ V (ξ) = ξ (t) PAξo + Atξo P + Q + ε−1 1 P M a Ma P + ε 1 N a N a −1 ¯ ¯t ¯ ¯t ¯t ¯ ¯t ¯ + ε−1 2 P M b Mb P + ε 2 N h N h + ε 3 P M f Mf P + ε 3 N g N g −1 ˆ − ε4 N ¯ aM ¯dt N ¯d ¯ at P ξ(t) + PAξdo Q Atξdo P + ε−1 P M 4

x(t) = ξ (t) Ξ ξ(t) = e(t) ∆

t

t

Ξs Ξd • Ξc

x(t) e(t)

(2.138)

From Lyapunov theory, it is known that V˙ (ξ) < 0 is a sufficient condition for stability of system (ΣA ), which corresponds from (2.138) to the inequality Ξ < 0. Expanding the composite matrix Ξ and incorporating the matrices (2.132) with lengthy but standard algebraic manipulations, we get: t Ξs = Ps Ao + Ato Ps + Ps Bo Kc + Kct Bot Ps + Qs + ε−1 1 Ps Ma Ma Ps

−1 t t t t + ε1 Nat Na + ε−1 2 Ps Bo Ms Ms Bo Ps + ε2 Nh Nh + ε4 Ps Ma Ma Ps ˆ s − ε4 N t Nd −1 At Ps + Ps Ad Q (2.139)

Ξc = + + Ξd = +

d d t t Pc Ao + Ao Pc − Pc Ko Co − Cot Kot Pc + Qc + ε−1 1 Pc Ma Ma Pc −1 −1 t t t t ε2 Nh Nh + ε3 Pc Mf Mf Pc + ε3 Co Ng Ng Co + ε4 Pc Ma Mat Pc ˆ s − ε4 N t Nd −1 At Pc Pc Ad Q (2.140) d d −1 t t −Ps Bo Kc + ε1 Ps Ma Ma Pc − ε2 Nh Nh ˆ s − ε4 N t Nd −1 At Pc + ε−1 Ps Ma M t Pc Ps Ad Q (2.141) d d a 4

Introducing the linearization matrices for some ε1 > 0, ε2 > 0, ε3 > 0

76


Xs = Ps−1 , Xc = Pc−1 , Ys = Kc Ps−1 , Yc = −Ko Co Pc−1 Yd = Kc Xc , Yh = ε2 Ps−1 Nht Nh Pc−1 , Zb = ε2 Nh Pc−1 Zc = ε3 Ng Co Pc−1 , Za = ε1 Na Ps−1

(2.142)

and define the block matrices for some scalars ε1 > 0, ..., ε4 > 0: Λs = [ε1 I, ε2 I] , Πa = [Ma , Bo Ms ] , Πs = [Zat , Zbt ] Λc = [ε1 I, ε3 I, ε4 I] , Πc = [Ma , Mf , Ma ] (2.143) Now, we premultiply (2.139) by Ps−1 and postmultiply the result by Ps−1 and use (2.142) to yield t Ξs = Ao Xs + Xs Ato + Bo Ys + Yst Bot + Xs Qs Xs + ε−1 1 M a Ma −1 −1 t t t t + ε−1 1 Za Za + ε2 Bo Ms Ms Bo + ε2 Zb Zb ˆ s − ε4 N t Nd −1 At + Ad Q (2.144) d d

Then, we premultiply (2.140) by Pc−1 and postmultiply the result by Pc−1 and use (2.142) again to yield t Ξc = Ao Xc + Xc Ato − Yc − Yct + Xc Qc Xc + ε−1 1 Ma Ma −1 t −1 t t + ε−1 3 Mf Mf + ε3 Zc Zc + ε4 Ma Ma ˆ s − ε4 N t Nd −1 At c + Ad Q d d

(2.145)

Next, we premultiply (2.141) by Ps−1 and postmultiply the result by Pc−1 and use (2.142) once more to yield t Ξd = −[Bo Kc + ε2 Ps−1 Nht Nh ]Xc + ε−1 1 Ma Ma ˆ s − ε4 N t Nd −1 At ε−1 Ma M t + Ad Q d d 4 a

(2.146)

Since it is sufficient for the stability requirement Ξ < 0 for system (2.127) that Ξs < 0, Ξd = 0, Ξc < 0, we enforce the latter conditions on (2.144)(2.146) and use the Schur complements with some matrix manipulations, we arrive at the LMIs (2.133)-(2.135). ∇∇∇ Had we considered the multiplicative gain perturbations, we would have obtained from (2.118)-(2.119) and (2.124), the error system (Σem ) : e(t) ˙ = [Ao − Ko Co − ∆Km (t)Ko Co ]e(t) + ∆Ao (t)x(t) + [Ad + ∆Ad (t)]x(t − τ ) (2.147) By combining systems (Σ∆ ) and (Σem ), we get the augmented system (ΣA ) along with (2.125) and (2.130). The following theorem establishes the second main stability result. Theorem 2.47. Consider system (ΣA ) with multiplicative gain perturbations and given matrices 0 < Qs = Qts ∈ n×n , 0 < Ws = Wst ∈ IRn×n , 0
0, ..., ε4 > 0, such that the LMIs   Ao Xs + Xs Ato Π Π X Q A a s s s d   +Bo Ys + Yst Bot     • −Λs 0 0 0    < 0  0 • • −Λs 0 (2.148)     0 • • • −Q s    ˆs+  −Q • • • • ε4 Ndt Nd   Ao Xc + Xc Ato ¯ t ¯ Ad  Πc Zc Xs Qs  +Yc + Yct     • −Λc 0 0 0     < 0 • • −ε I 0 0 (2.149) 3     0 • • • −Q c    ˆs+  −Q • • • • ε4 Ndt Nd   Bo Yd + Ydt Bot  +Yh + Yht 2Ma 2Ma 2Ad      • 2ε1 I 0 0   ≥ 0 (2.150)   I 0 • • 2ε 4    ˆs−  2Q • • • ε4 Ndt Nd have a feasible solution. Moreover, the gain matrices are given by Kc = Ys Xs−1 , Ko = −Yc Xc−1 Co† . ¯c = Proof: Follows by parallel development to Theorem 2.46 using Π [Ma , Md , Ma ] instead of Πc and the linearizations Z¯c = ε3 Nm Ko Co Pc−1 instead of Zc . ∇∇∇ 2.10.4 Special Cases Consider the nominal augmented system (ΣAO ) :

˙ = Aξo ξ(t) + Aξdo ξ(t − τ ) ξ(t)

(2.151)

where Aξo , Aξdo are given by (2.128). The following results hold for additive or multiplicative gain perturbations Corollary 2.48. Consider system (ΣAO ) with additive gain perturbations and given matrices 0 < Qs = Qts ∈ n×n , 0 < Ws = Wst ∈ IRn×n , 0 < Qc = Qtc ∈ n×n , 0 < Wc = Wct ∈ IRn×n . There exist matrices 0 < Xs = Xst ∈ n×n , 0 < Xc = Xct ∈ IRn×n , Ys , Yc , Zs , Yd , Yh , Zb , Zc and scalars ε2 > 0, ε3 > 0, such that the LMIs

78




 Ao Xs + Xs Ato + Bo Ys + Yst Bot Bo Ms Zb Xs Qs Ad  • −ε2 I 0 0 0      < 0 I 0 0 • • −ε 2    • • • −Qs 0  ˆs • • • • −Q   Ao Xc + Xc Ato + Yc + Yct Mf Zct Xs Qs Ad  0 0  • −ε3 0    • • −ε3 I 0 0    < 0  • • • −Qc 0  ˆs • • • • −Q Bo Yd + Ydt Bot + Yh + Yht 2Ad ˆs ≥ 0 • 2Q

(2.152)

(2.153)

(2.154)

have a feasible solution. Moreover, the gain matrices are given by Kc = Ys Xs−1 , Ko = −Yc Xc−1 Co† 2 . Proof: Follows from Theorem 2.46 by setting Ma ≡ 0, Na ≡ 0, Nd ≡ 0. ∇∇∇ Corollary 2.49. Consider system (ΣAO ) with multiplicative gain perturbations and given matrices 0 < Qs = Qts ∈ n×n , 0 < Ws = Wst ∈ IRn×n , 0 < Qc = Qtc ∈ n×n , 0 < Wc = Wct ∈ IRn×n . There exist matrices 0 < Xs = Xst ∈ n×n , 0 < Xc = Xct ∈ IRn×n , Ys , Yc , Za , Yd , Yh , Zb , Zc and scalars ε2 > 0, ε3 > 0, such that the LMIs   Ao Xs + Xs Ato + Bo Ys + Yst Bot Bo Ms Zb Xs Qs Ad  • −ε2 I 0 0 0      < 0 I 0 0 • • −ε (2.155) 2    • • • −Qs 0  ˆs • • • • −Q   t t t Ao Xc + Xc Ao + Yc + Yc Md Z¯c Xs Qs Ad  • −Λc 0 0 0      < 0 I 0 0 • • −ε (2.156) 3    • • • −Qc 0  ˆs • • • • −Q Bo Yd + Ydt Bot + Yh + Yht 2Ad (2.157) ˆs ≥ 0 • 2Q have a feasible solution. Moreover, the gain matrices are given by Kc = Ys Xs−1 , Ko = −Yc Xc−1 Co† . Proof: Follows from Theorem 2.46 by setting Ma ≡ 0, Na ≡ 0, Nd ≡ 0. ∇∇∇ 2

Co† = Cot [Co Cot ]−1 is the pseudo-inverse of Co .


79

In the absence of gain perturbation, we have the following result Corollary 2.50. Consider system (ΣAO ) and given matrices 0 < Qs = Qts ∈ n×n , 0 < Ws = Wst ∈ IRn×n , 0 < Qc = Qtc ∈ n×n , 0 < Wc = Wct ∈ IRn×n . There exist matrices 0 < Xs = Xst ∈ n×n , 0 < Xc = Xct ∈ IRn×n , Ys , Yc , Zs , Yd such that the LMIs   Ao Xs + Xs Ato + Bo Ys + Yst Bot Xs Qs Ad  • −Qs 0  < 0 (2.158) ˆs • • −Q   Ao Xc + Xc Ato + Yc + Yct Xs Qs Ad  • −Qc 0  < 0 (2.159) ˆs • • −Q Bo Yd + Ydt Bot 2Ad (2.160) ˆs ≥ 0 • 2Q have a feasible solution. Moreover, the gain matrices are given by Kc = Ys Xs−1 , Ko = −Yc Xc−1 Co† . Proof: Follows from Corollary 2.48 or 2.49 by setting Mf ≡ 0, Ng ≡ 0, Nh ≡ 0, Ms ≡ 0, Md ≡ 0. ∇∇∇ Next, consider the state-delay system (Σ0 ) : x(t) ˙ = Ao x(t) + Ad x(t − τ ) + Bo u(t) y(t) = Co x(t) + Go u(t)

(2.161) (2.162)

where the corresponding state-space matrices contain uncertainties represented by a real convex bounded polytopic model of the type Ao Ad Bo A(oλ) A(dλ) B(oλ) ∆ ∈ Sλ = = Co Go 0 C(oλ) G(oλ) 0 N Aoj Adj Boj ,λ ∈ Λ (2.163) λj Coj Goj 0 j=1

where Λ is the unit simplex ∆

Λ=

(λ1 , · · · , λN ) :

N

λ j = 1 , λj ≥ 0

(2.164)

j=1

Define the vertex set IN = {1, ..., N }. In the sequel, we use {Ao , · · · , Go } to imply generic system matrices and {Aoj , · · · , Goj , j ∈ IN} to represent the respective values at the vertices. The following two theorem are readily established

80


Corollary 2.51. Consider system (ΣO ) with additive gain perturbations and given matrices 0 < Qs = Qts ∈ n×n , 0 < Ws = Wst ∈ IRn×n , 0 < Qc = Qtc ∈ n×n , 0 < Wc = Wct ∈ IRn×n . There exist matrices 0 < Xs = Xst ∈ n×n , 0 < Xc = Xct ∈ IRn×n , Ys , Yc , Zs , Yd , Yh , Zb , Zc and scalars ε2 > 0, ε3 > 0, such that the LMIs   t Aoj Xs + Xs Atoj + Boj Ys + Yst Boj Boj Ms Zb Xs Qs Adj  • −ε2 I 0 0 0      < 0 (2.165) • • −ε I 0 0 2    • • • −Qs 0  ˆs • • • • −Q   t t t Aoj Xc + Xc Aoj + Yc + Yc Mf Zc Xs Qs Adj  • −ε3 0 0 0      < 0 I 0 0 • • −ε (2.166) 3    • • • −Qc 0  ˆs • • • • −Q t Boj Yd + Ydt Boj + Yh + Yht 2Adj (2.167) ˆs ≥ 0 • 2Q have a feasible solution over the vertex set IN. Moreover, the gain matrices are given by Kc = Ys Xs−1 , Ko = −Yc Xc−1 Co† . Theorem 2.52. Consider system (ΣO ) with multiplicative gain perturbations and given matrices 0 < Qs = Qts ∈ n×n , 0 < Ws = Wst ∈ IRn×n , 0 < Qc = Qtc ∈ n×n , 0 < Wc = Wct ∈ IRn×n . There exist matrices 0 < Xs = Xst ∈ n×n , 0 < Xc = Xct ∈ IRn×n , Ys , Yc , Za , Yd , Yh , Zb , Zc and scalars ε2 > 0, ε3 > 0, such that the LMIs   t Aoj Xs + Xs Atoj + Boj Ys + Yst Boj Boj Ms Zb Xs Qs Adj  0 0 0  • −ε2 I    • • −ε2 I 0 0    < 0 (2.168)  • • • −Qs 0  ˆs • • • • −Q   t t t Aoj Xc + Xc Aoj + Yc + Yc Md Z¯c Xs Qs Adj  • −Λc 0 0 0      < 0 • • −ε I 0 0 (2.169) 3    • • • −Qc 0  ˆs • • • • −Q t Boj Yd + Ydt Boj + Yh + Yht 2Aoj ≥ 0 (2.170) ˆs • 2Q have a feasible solution over the vertex set IN. Moreover, the gain matrices are given by Kc = Ys Xs−1 , Ko = −Yc Xc−1 Co† .


81

Remark 2.53. It is significant to observe that Corollaries 2.48-2.49 for the nominal system and Theorems 2.51-2.52 for the polytopic system new contributions to the area of resilient control. In addition, Corollary 2.50 provides an improved LMI-based conditions for the observer-based control over the results of [79, 87, 94, 90]. 2.10.5 Example 2.8 Consider a time-delay system of the type (2.118)-(2.119) with 1 0.7 0.5 0.5 0.32 Ao = , Bo = , Ad = , Ms = 1.2 0 −2 0.75 0 −0.5 Co = [−1 0] , Go = 1 , Mat = [1 1.4] , Na = [0.35 0.45] Mft = [0.7 0.6] , Mp = 0.3 Ng = 0.5 , Nh = [0.25 0.15] , Nf = 0.8 , Mdt = [0.5 0.5] , Nm = [0.6 0.7] By resorting to MATLAB, the ensuing results are summarized in Table 1 for both types of gain perturbations Table 2.10. Results of Dynamic Controllers: Example 2.8 P erturbatiion Kc Ko Additive -0.0411 -1.6345 -4.4615 -3.7004 Multiplicative -0.1209 -1.4786 -5.3155 -2.9874

2.10.6 Example 2.9 Consider a third-order time-delay system with the following data       −0.1 0.2 0 00 −2 0.95 −0.05 Ao =  0.1 −2 0.03  , Bo =  0 1  , Ad =  0 −0.4 0.1  0 0 −0.6 10 0.1 0 4   01 10 Co =  0 0  , Go = 01 10 Mat = [1 1.4 1.3] , Na = [0.35 0.45 0.55] , Mft = [0.07 0.06 0.05] Mpt = [1 2] , Nf = [0.7 0.3] , Nm = [0.6 0.7 0.4] , Mdt = [0.5 0.5 0.5] Ng = [−2

− 1] , Nh = [−1

−2

− 1] , Mst = [0.7 0.3]

The results are summarized in Table 2.11 for both types of gain perturbations

82

2 Resilient Control-Continuous Case Table 2.11. Results of Dynamic Controllers- Example 2.9 Kc Ko P erturbatiion 0.0155 13.9865 -0.1874 -0.2453 10.1027 Additive -3.0308 45.1042 11.8975 7.0135 -1.0025 12.1253 -2.0145 Multiplicative 0.1035 14.0015 -0.2038 -0.3055 10.2105 -4.0126 46.8865 12.9084 7.1004 -1.0845 11.8954 -1.9035

2.11 Notes and References This Chapter contains the basic methods of resilient feedback control theory applied to classes of time-delay systems. There are a lot of other methods/approaches to be examined or developed to complete the resilient control theory. Some of these methods will be exposed in the coming Chapters. Others are left to the interested readers and researchers. One of the challenging problems is to address the gain perturbation representation. In our view, we felt that the norm-bounded representation is general enough to encompass other related form. However, other forms would deserve further investigations.

3 Resilient Control-Discrete Case

3.1 Introduction A basic ingredient of robust control system design is the employment of a fixed quadratic Lyapunov function in order to guarantee robust stability [74, 105, 126]. In the presence of uncertain parameters, it is often desirable to perform the control system design not only to ensure stability but also to guarantee an adequate level of system performance. This brings about the notion of guaranteed cost control (GCC) [103]. In the course of controller implementation based on different control design methods it has been reported turns that the controllers are very sensitive with respect to errors in the controller coefficients [43]. Broadly speaking, the sources for this include, but not limited to, imprecision in analog-digital conversion, fixed word length, finite resolution instrumentation and numerical roundoff errors and it has been demonstrated that relatively small perturbations in controller parameters could even destabilize the closed loop system. This calls for fine tuning or, loosely speaking, additional redesign. Thus it is considered beneficial, and almost certainly essential, that the designed (nominal) controllers should be capable of tolerating some level of controller gain variations. This has been the heart of the controller fragility problem (see Chapter 1). The objective of this Chapter is to develop resilient controllers for a class of discrete-time systems with norm-bounded uncertainties and controller gain variations. Previous results to the analysis, robustness and design of discretetime systems can be found in [28, 29, 101, 115, 116] and their references. In the sequel, we focus on the development of resilient controllers and extend the results of [118, 121] to uncertain discrete-time systems with both types of gain variations. Necessary and sufficient conditions are established such that the resulting closed-loop feedback control system is quadratically stable for all admissible perturbations and uncertainties. These conditions are conveniently expressed in the form of linear matrix inequalities (LMIs). The feedback stabilization schemes are based on guaranteed cost control and H∞


84


control approaches. System examples are provided to illustrate the theoretical developments.

3.2 Model of Discrete-Time Systems The class of linear discrete-time systems under consideration is represented by: (Σ∆ ) : xk+1 = [Ao + ∆Ak ]xk + Bo uk + Γ wk = [Ao + M ∆sk N ]xk + Bo uk + Γ wk zk = Co xk + Do uk

(3.1)

where xk ∈ n is the state, uk ∈ p is the control input, wk ∈ r is the external disturbance, zk ∈ q is the controlled output and matrices Ao ∈ n×n , Bo ∈ n×p , Co ∈ q×n , Do ∈ q×p , M ∈ n×α , and N ∈ β×n are real and known constant matrices with ∆sk ∈ α×β being an unknown matrix representing the norm-bounded uncertainties and belong to a bounded set, that is ∆ ∆sk ∈ ∆s = {∆tsk : ∆tsk ∆sk ≤ I , ∀k} The cost function associated with system (Σ∆ ) is given by: J=

∞

xtk Qxk + utk Ruk

(3.2)

k=0

where 0 < Q = Qt and 0 < R = Rt are given weighting matrices. 3.2.1 Gain Perturbations For a given state-feedback controller uk = Ko xk with Ko being the nominal controller gain, the actual implemented controller is assumed to be uk = [Ko + ∆Kk ] xk

(3.3)

where ∆Kk represents the gain perturbation, which is assumed to be either of two forms: 1) Additive Type ∆Kk = Ha ∆ck Ea

(3.4)

where Ha and Ea are known constant matrices and ∆ck is the uncertain parameter matrix represented by ∆

∆tck ∈ ∆c = {∆tck : ∆tck ∆ck ≤ I , ∀k

3.2 Model of Discrete-Time Systems

85

2) Multiplicative Type ∆Kk = Hm ∆ck Em Ko

(3.5)

where Hm and Em are known constant matrices. For simplicity in exposition, we define the following matrices: Ao = Ao + Bo Ko , ∆Ak = ∆Ak + Bo ∆Kk , A∆ = Ao + ∆Ak K∆ = Ko + ∆Kk

(3.6)

3.2.2 Quadratic Stability We start by the following definition Definition 3.1. Consider system (Σ∆ ) with cost function (3.2) and wk ≡ 0. The state-feedback control law (3.3) with controller gain perturbations (3.4) or (3.5) is said to be a guaranteed cost control (GCC) with quadratic cost matrix P > 0 if given matrices 0 < Q = Qt and 0 < R = Rt , the following LMI   t −P + Q At∆ P K∆ R  • −P 0  < 0 (3.7) • • −R has a feasible solution with respect to P for all admissible uncertainties ∆Ak and ∆Kk satisfying (3.1) and (3.4) or (3.5) Remark 3.2. The norm-bounded uncertainty structure considered here has been widely adopted in robust control [64, 74, 80, 82, 105, 64]. We note that it covers the usual matching condition as a special case by setting N ≡ I. Observe that the unit upperbound condition on ∆k does not cause any loss of generality. Indeed ∆k can always be normalized by appropriately selecting the matrices M and N . In the absence of controller perturbations ∆Kk ≡ 0, then Definition 3.1 reduces to the well known definition of guaranteed cost control for uncertain discrete-time systems [74]. In this chapter, we closely examine the following problem: Given the uncertain system (Σ∆ ) with cost function (3.2) along with the state-feedback control law (3.3) with controller gain perturbations (3.4) or (3.5), it is required to determine the feedback gain matrix Ko such that the closed-loop controlled system achieve quadratic stability for all admissible uncertainties satisfying ∆tk ∆k ≤ I , ∀k. We will derive expressions for gain matrix Ko using guaranteed cost and H∞ approaches. Considering the guaranteed cost approach, the following theorem provides a link between the notion of quadratic cost matrix, addressed in Definition 3.1, and an upper bound on the cost function J.

86


Theorem 3.3. Consider system (Σ∆ ) with cost function (3.2). Suppose that the state-feedback control law (3.3) with controller gain perturbations (3.4) or (3.5) is a quadratic GCC with quadratic cost matrix (QCM) P > 0. Then the resulting closed-loop system with wk ≡ 0 xk+1 = A∆ xk

(3.8)

is quadratically stable with the cost function satisfies the bound J ≤ xto P xo

(3.9)

Conversely, if system (Σ∆ ) is quadratically stable then there will be a QCM for this system and cost function (3.2). Proof (=⇒) : Let 0 < P = P t be a quadratic cost matrix for system (Σ∆ ) and cost function (3.2), it follows from (3.7) and the Schur complements that t t (3.10) x A∆ P A∆ − P x < 0 ∀ x = 0 , ∀ ∆tk ∈ ∆s Hence, system (Σ∆ ) is quadratically stable. Next, introduce the Lyapunov functional V (xk ) = xtk P xk . Along the state sequences of (3.8), we have V (xk+1 ) − V (xk ) = xtk+1 P xk+1 − xtk P xk = xtk [Ao + ∆Ak ]t P [Ao + ∆Ak ] − P xk t < − xtk Q + K∆ RK∆ xk

(3.11)

Summing up k over [0, ∞) and using the quadratic stability of (Σ∆ ), we obtain ∞ t RK∆ xk ≤ V (xo ) = xto P xo xtk Q + K∆ (3.12) J = k=0

since limk→∞ x(k) = 0. (⇐=) : Let system (Σ∆ ) be quadratically stable . If follows that there exists 0 0 such that the following inequality holds: t RK∆ ] = −1 [Ao + ∆Ak ]t P [Ao + ∆Ak ] − −1 [P + Q + K∆ t¯ t ¯ ¯ ¯ [Ao + ∆Ak ] P [Ao + ∆Ak ] − P + Q + K∆ RK∆ ] < 0 ∀∆sk ∈ ∆s , ∀∆ck ∈ ∆c

(3.14)

3.3 Guaranteed Cost Control Synthesis

87

¯ = −1 Q, R ¯ = −1 R The above inequality implies that there exists matrices Q −1 ¯ such that the matrix P = P is a QCM for system (Σ∆ ) and the proof is completed ∇∇∇ Having developed the basic stability theory, we proceed to the synthesis problem.

3.3 Guaranteed Cost Control Synthesis In this section, we establish results for GCC synthesis with both types of controller perturbations. We formulate the core of the work in the form of theorems such that their derived conditions will eventually lead to the design procedure. 3.3.1 Additive Gain Perturbations The following theorem provides a necessary and sufficient condition for GCC under additive gain perturbations. Theorem 3.4. Consider system (Σ∆ ) with cost function (3.2). There exist state feedback gain Ko such that the control law (3.3) with additive gain perturbations (3.4) is a GCC with a quadratic cost matrix P > 0 given matrices 0 < Q = Qt and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y and some parameters ε1 > 0, ε2 > 0 such that the following LMIs   −X XQ XN t XEat XAto + Y t Bot Yt  • −Q 0  0 0 0    • • −ε1 I 0  0 0    • •  0 0 • −ε2 I   < 0 t   −X + ε1 M M  • •  • • 0 t t   +ε2 Bo Ha Ha Bo   −1  −R +  • • • • • ε2 Ha Hat   −X ε1 M ε2 Bo Ha −R−1 ε2 Ha  • −ε1 I  0 < 0, < 0 (3.15) • −ε2 I • • −ε2 I are feasible with respect to X and Y . The feedback gain is Ko = Y X −1 . Proof: By Definition 3.1 and [74], it follows that inequality (3.7) holds if and only if the LMI   −P + Q (Ao + Bo Ko )t P Kot R  • −P 0  + • • −R

88




 t  t   t t 0 0 N N    0 0  ε1  P M   P M  + ε−1 1 0 0 0 0 t  t t  t t   Ea Ea 0 0    0 0  + ε2  P Bo Ha   P Bo Ha  + ε−1 2 0 0 RHa RHa   −P + Q+ t t Ko R   ε−1 N t N + ε−1 E t Ea [Ao + Bo Ko ] P a 2   1   −P +   t t  = ε P M M P + • R P B H H ε 1 o a a  2    ε2 P Bo Ha Hat Bot P     −R+ • • ε2 RHa Hat R

< 0(3.16)

holds some parameters ε1 > 0, ε2 > 0. Grouping like terms and multiplying every block entry of (3.16) on the left and on the right by P −1 and setting X = P −1 and Y = Ko P −1 with some Schur complement operations we obtain the equivalent conditions of (3.15). ∇∇∇ Remark 3.5. It should be observed that matrix R > 0 is a part of the data and hence its inverse is readily available. However the presence of R−1 render the LMIs formalism ill-posed. To overcome this problem, we introduce a relaxation variable S = R−1 thereby in implementation we seek a feasible solution of the following system of LMIs:   −X XQ XN t XEat XAto + Y t Bot Yt  • −Q 0  0 0 0    • • −ε1 I 0  0 0    • •  • −ε2 I 0 0   < 0 t   −X + ε1 M M  • •  • • 0 t t   +ε2 Bo Ha Bo Ha    −S+  • • • • • ε2 Ha Hat   −X ε1 M ε2 Bo Ha S I −S ε2 Ha  • −ε1 I  0 > 0, ≥ 0 > 0, • −ε2 I •R • • −ε2 I with respect to X, Y and S. In the absence of system uncertainties (M ≡ 0, N ≡ 0) we obtain the nominal system (Σo ) : x(k + 1) = Ao x(k) + Bo u(k) for which implementation of Theorem 3.4 reduces to:

(3.17)

3.3 Guaranteed Cost Control Synthesis

89

Theorem 3.6. Consider system (Σo ) with cost function (3.2). There exist state feedback gain Ko such that the control law (3.3) with additive gain perturbations (3.4) is a GCC with a quadratic cost matrix P > 0 given matrices 0 < Q = Qt and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y, 0 < S = S t and some parameter ε2 > 0 such that the following LMIs   −X XQ XEat XAto + Y t Bot Yt  • −Q 0  0 0    • • −ε2 I  < 0 0 0   t t  • •  0 • −X + ε2 Bo Ha Bo Ha • • • • −S + ε2 Ha Hat −X ε2 Bo Ha −S ε2 Ha S I < 0, < 0, ≥ 0 (3.18) •R • −ε2 I • −ε2 I are feasible with respect to X, Y and S. The feedback gain is Ko = Y X −1 . 3.3.2 Multiplicative Gain Perturbations The following theorem provides a necessary and sufficient condition for GCC under multiplicative gain perturbations. Theorem 3.7. Consider system (Σ∆ ) with cost function (3.2). There exist state-feedback gain Ko such that the control law (3.3) with multiplicative gain perturbations (3.5) is a GCC with a quadratic cost matrix P > 0 given matrices 0 < Q = Qt and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y and some parameters ε1 > 0, ε2 > 0 such that the following LMIs   t −X XQ XN t Y t Em XAto + Y t Bot Yt  • −Q 0  0 0 0    • • −ε1 I 0  0 0    • •  • −I 0 0   < 0 t   −X + ε1 M M  • •  • • 0 t t   +ε2 Bo Hm Bo Hm   −1  −R +  • • • • • t ε 2 Hm Hm   −X ε1 M ε2 Bo Hm −R−1 ε2 Hm  • −ε1 I  0 < 0 (3.19) < 0, • −ε2 I • • −ε2 I are feasible with respect to X and Y . The feedback gain is Ko = Y X −1 . Proof: Following Theorem 3.4, it follows that inequality (3.7) under multiplicative perturbations (3.5) holds if and only if the LMI

90




 −P + Q+ −1 t   [Ao + Bo Ko ]t P Kot R  −1ε1 N N +   ε K t E t Em K o  o m  2    −P +   t t  ε1 P M M P + ε2 P Bo Hm Hm R  •   t   P ε2 P Bo Hm Bot Hm     −R+ • • t ε2 RHm Hm R

< 0 (3.20)

holds some parameters ε1 > 0, ε2 > 0. Multiplying every block entry of (3.20) on the left and on the right by P −1 and setting X = P −1 and Y = Ko P −1 with Schur complement operations we obtain the equivalent conditions of (3.19). ∇∇∇ In the absence of system uncertainties (M ≡ 0, N ≡ 0) we obtain the nominal system (3.17) for which Theorem 3.4 reduces to: Theorem 3.8. Consider system (Σo ) with cost function (3.2). There exist state-feedback gain Ko such that the control law (3.3) with additive gain perturbations (3.5) is a GCC with a quadratic cost matrix P > 0 given matrices 0 < Q = Qt and 0 < R = Rt if and only if there exist matrices 0 < X = X t , Y, 0 < S = S t and some parameter ε2 > 0 such that the following LMIs   t −X XQ Y t Em XAto + Y t Bot Yt  • −Q 0  0 0    • •  < 0 −I 0 0   t t  • •  0 • −X + ε2 Bo Hm Bo Hm −1 t • • • • −R + ε2 Hm Hm −X ε2 Bo Hm −R−1 ε2 Hm < 0, (3.21) • −ε2 I • −ε2 I are feasible with respect to X and Y . The feedback gain is Ko = Y X −1 . Remark 3.9. Needless to stress that implementation of Theorems 3.7 and 3.8 is performed in the mannar of Remark 3.5. 3.3.3 Example 3.1 Consider a discrete-time system of the type (3.1) with 0.99 0.01 1 0.4 0.1 0 , Bo = , Γ = , M= , Hm = 0.2 Ao = 0.01 0.51 0 0.5 0 0.1 0.1 0.05 0.2 N = , Co = [0.5 0.5] , Do = 1 , Ha = −0.02 0.1 0.1 Ea = [0.3 0.6] , Em = 0.4 A summary of the numerical computations is given in Table 1, from which it is clear that the case of multiplicative gain perturbations yields less cost bound than additive gain perturbations.

3.4 H∞ Control Synthesis

91

Table 3.1. Computational Results of Example 3.1 P erturbation Theorem Q R J o ε1 ε2 Ko Additive 3.4 1 0 0.75 0.0823 0.01 0.3 -0.9457 -0.1697 0 1 Multiplicative 3.7 1.2 0 0.5 0.0376 0.05 0.7 -1.0612 -0.1925 0 1.2

3.3.4 Example 3.2 Consider the following discrete-time system of the type (3.1) with       0.2113 0.0087 0.4524 0.6135 0.6538 100 Ao =  0.0824 0.8096 0.8075  , Bo =  0.2749 0.4899  , Γ =  0 1 0  0.7599 0.8474 0.4832 0.8807 0.7741 001     0.1 100 1 0 N = [0.05 − 0.02 0.1] , Co =  0 1 0  , Do = , M =  0.2  01 0.3 001 0.15 0 0.6 0 Em = , Hm = , Ea = [0.3 0.6 − 0.1] 0 0.44 0 −0.38 0.2 −0.3 Ha = 0.1 0.2 Observe that the system is open-loop unstable as λ(Ao ) = {0.3827, −0.4919, 1.6133} A summary of the numerical computations is given in Table 2 which shows comparable results in both cases of gain perturbations. Table 3.2. Computational Results of Example 3.2 P erturbation Theorem Additive 3.4 1 0 0 Multiplicative 3.7 1.2 0 0

Q 0 1 0 0 1.2 0

0 0 1 0 0 1.4

R Jo Ko 1 0 5.6532 0.3024 0.3758 0.3114 01 0.2299 0.4852 0.5001 1 0 5.4899 0.3016 0.3809 0.3632 01 0.2304 0.5014 0.4988

3.4 H∞ Control Synthesis In the sequel, we extend the results attained in the forgoing section to the case of H∞ control. For a given nominal state-feedback controller uk = Ko xk , the

92


closed loop transfer function from wk to zk is given by: Hwz (ζ) = C∆ [ζ I − A∆ ]−1 Γ where

∆

C∆ = Co + Do ∆K ,

(3.22)

Co = Co + Do Ko

This means that given a prespecified H∞ -norm level γ, it is required to develop conditions for the state-feedback controller (3.3) with controller gain perturbations (3.4) or (3.5) to render the closed-loop system (3.6) quadratically stable and ||Hwz ||∞ ≤ γ for all admissible uncertainties ∆tk ∆k ≤ I. Following [110], we have the following definition: Definition 3.10. Consider system (Σ∆ ). The state-feedback control law (3.3) with controller gain perturbations (3.4) or (3.5) is said to be a H∞ with disturbance attenuation γ > 0 if there exists a matrix 0 < P t = P such that the following LMI   t −P At∆ P C∆ At∆ P Γ  • −P 0  0   < 0 (3.23)  •  • −I 0 2 t • • • −γ I + Γ P Γ has a feasible solution for all admissible uncertainties ∆Ak and ∆Kk satisfying (3.1) For simplicity in exposition, we introduce the following vector quantities:     t   t 0 N 0 Ea PM   0   P Bo Ha   0            , σ4 =  0     0 σ1 =     0  , σ2 =  0  , σ3 =     0   0    0  0 0 0 0 0       0 0 0   0   0   0             0 σ5 =  Do Ha  , σ6 =  0  , σ7 =  (3.24)   0   Γ tP M   Γ t P Bo Ha  0 0 0 The following theorems four summarize the corresponding results for controller gain variations of both types. For the additive type, we have Theorem 3.11. Consider system (Σ∆ ) with cost function (3.2). There exist state-feedback gain Ko such that the control law (3.3) with additive gain perturbations (3.4) is a H∞ control with disturbance attenuation γ > 0 if and only if there exist matrices 0 < X = X t , Y and some parameters ε1 > 0, ........., ε7 > 0 such that the following LMIs

3.4 H∞ Control Synthesis



93



¯ t XE ¯ t XAt + Y t B t XC t + Y t Dt XAt + Y t B t 0 −X X N o o o o o a o • −Φ 0 0 0 −XΓ 0   • • −Ψ 0 0 0 0   • • • −Υ1 0 0 0   < 0 0 0  • • • • −Υ2  • • • • • −γ 2 I Γt  • • • • • • −Υ3     −X ε1 M ε2 Bo Ha −X ε4 M ε5 Ea  • −ε1 I  < 0,  • −ε4 I 0  < 0, −I ε3 Do Ha < 0(3.25) 0 • −ε3 I • • −ε2 I • • −ε5 I

        

are feasible with respect to X, Y where   N ¯ = N  , E ¯a = Ea , Υ1 = X − ε1 M M t − ε2 Bo Ha Bot Hat N Ea N Ψ = diag[ε2 I ε3 I] , Υ3 = X − ε4 M M t − ε5 Ea Eat Φ = diag[ε1 I ε4 I ε5 I] , Υ2 = I − ε3 Do Ha Hat Dot

(3.26)

The feedback gain is Ko = Y X −1 . Proof: By Definition 3.1 and Theorem 3.4 with simple Schur operations, it follows that inequality (3.23) holds if and only if the LMI   −P (Ao + Bo Ko )t P (Co + Do Ko )t (Ao + Bo Ko )t P 0  • −P 0 Γ 0     • • −I 0 0     • • • −γ 2 I Γ tP  • • • • −P −1 −1 t t t t t + ε1 σ1 σ1t + ε−1 1 σ 2 σ 2 + ε2 σ 3 σ 3 + ε2 σ 4 σ 4 + ε3 σ 5 σ 5 + ε3 σ 4 σ 4

−1 t t t + ε4 σ6 σ6t + ε−1 4 σ 2 σ 2 + ε5 σ 7 σ 7 + ε5 σ 2 σ 2 < 0

(3.27)

holds for some parameters ε1 > 0, ....., ε7 > 0. Grouping like terms and by the congruent transformation [X X I I X] with X = P −1 , Y = Ko P −1 and some Schur complement operations we obtain the equivalent conditions of (3.25). ∇∇∇ Theorem 3.12. Consider system (Σo ) with cost function (3.2). There exist state-feedback gain Ko such that the control law (3.3) with additive gain perturbations (3.4) is a H∞ control with disturbance attenuation γ > 0 if and only if there exist matrices 0 < X = X t , Y and some parameters ε2 > 0, ε3 > 0 such that the following LMIs

94




 ¯ t XAt + Y t B t XC t + Y t Dt XAt + Y t B t 0 −X X E o o o o o a o  • −Ψ 0 0 0 0     •  • −Υ 0 0 0 1o   < 0  •  • • −Υ 0 0 2    • Γt  • • • −γ 2 I • • • • • −Υ3o −X ε2 Bo Ha −X ε5 Ea −I ε3 Do Ha > 0, > 0, > 0 (3.28) • −ε2 I • −ε5 I • −ε3 I are feasible with respect to X, Y where Υ1o = X + ε2 Bo Ha Hat Bot , Υ3o = X − ε5 Ea Eat

(3.29)

The feedback gain is Ko = Y X −1 . For the multiplicative type, we have the following results Theorem 3.13. Consider system (Σ∆ ) with cost function (3.2). There exist state feedback gain Ko such that the control law (3.3) with multiplicative gain perturbations (3.5) is a H∞ control with disturbance attenuation γ > 0 if and only if there exist matrices 0 < X = X t , Y , Z and some parameters ε1 > 0, ....., ε7 > 0 such that the following LMIs   ¯ t Y tE ¯ t XAt + Y t Bot XC t + Y t Dot XAto + Y t Bot 0 −X X N a o o  • −Φ 0 0 0 −XΓ 0     •  • −Ψ 0 0 0 0    •  < 0 • • −Υ 0 0 0 1    •  • • • −Υ 0 0 2    • Γt  • • • • −γ 2 I • • • • • • −Υ33     −X ε1 M ε2 Bo Hm −X ε4 M ε5 Em Y  • −ε1 I  0, ε3 > 0, ε5 > 0, ε6 > 0, ε7 > 0 such that the following LMIs

3.5 Design Example



95



¯ t XAt + Y t B t XC t + Y t Dt XAt + Y t Bot 0 −X Y t E a o o o o o • −Ψ 0 0 0 0   • • −Υ1 0 0 0   < 0 • • • −Υ2 0 0   Γt  • • • • −γ 2 I • • • • • −Υ33 −X ε2 Bo Hm −I ε3 Do Hm −X ε5 Em Y < 0, < 0, 0, > 0 such that the LMIs   Ao X + XAto + Z + Z t M M XQ L Y A d  +βBo X + βXBot      • −εI 0 0 0 0 0     • • −I 0 0 0 0     < 0 • • • −Q 0 0 0 (4.10)     • • • • −εI 0 0     • • • • • −I 0    −Q+  • • • • • • Ndt Nd have a feasible solution. Moreover, the feedback gain is Ko = Y X −1 . Proof: Evaluation of the derivative of Vn (x, µ) along the solutions of system (4.1)-(4.2) using adaptive controller (4.8) with some algebraic manipulations yields: t t ˙ ˜) = x (t) P A∆c + A∆c P + Q x(t) + 2xt P A∆d x(t − τ ) Vn (x, µ ˜µ ˜˙ − xt (t − τ ) Q x(t − τ ) + µ t t −1 t t = x (t) P A∆c + A∆c P + Q + P A∆d Q A∆d P + Ko Ko x(t) t t −1 t − x (t − τ ) − x (t)P A∆d Q x(t − τ ) − A∆d P x(t) − g µ ˜2

4.3 Adaptive Schemes

103

t t −1 t t ≤ x (t) P A∆c + A∆c P + Q + P A∆d Q A∆d P + Ko Ko x(t) ∆

= xt (t) Ξn x(t) A∆c = A∆ + µ ˜ Bo Ko + Bo ∆K(t) = [Ao + µ ˜ Bo Ko ] + ∆A + Bo ∆K(t) = Ac + ∆A + Bo ∆K(t)

(4.11)

(4.12)

From Lyapunov theory, it follows that V˙ n (x, µ ˜) < 0 is guaranteed if Ξn < 0. By [81] with Inequalities I1-I-2 of the Appendix, it is a straightforward task to show that P A∆c + At∆c P ≤ P Ao + Ato P + ε−1 P M M t P + εNat Na P A∆d Q−1 At∆d P ≤ −1 P M M t P + P Ad [Q − Ndt Nd ]−1 Atd P for some scalars ε > 0, > 0. Using these inequalities and the Schur complements the stability condition holds if the inequality:   P Ao + Ato P +  Q + Kot Ko + εNat Na    P Ad  +βP Bo + βBot P + P M P M    t  µ  0 guarantees the asymptotic stability of system (4.8) and different values will only affect the speed of convergence. This is illustrated by the following example.

104

4 Resilient Adaptive Control

4.3.2 Example 4.1 This example is motivated by the dynamics of bio-strata in water-quality studies on the river Nile. A pilot-scale model of the form (4.1) is described by:       −0.2 0 0 0 0 0 0.8 0 Ao =  0 −0.9 −0.3  , Ad =  −0.7 0 0  , Bo =  0.2 0.3  0.8 0 −1 0 −0.8 0 0 0.4         −0.1 −0.1 −0.2 10 M =  0.1  , Nat =  0  , Ndt =  0  , Cot =  0 0  0.3 0.1 0.2 01 The feasible solution of LMIs (4.10) is given by   −2.146 0.01 0  K =  0.157 −3.804 −1.626 In Figures 4.2-4.3, the behavior of the output variables and the gain factor µ ˜ are displayed for different values of g, from which it is clear that relatively-high values of g tend to yield effective stabilization.

Fig. 4.2. Plot of Output Response for different values of g

4.3.3 Unknown Gain Perturbation Bound Now we consider the application of controller (4.6) subject to bound (4.7) where β is unknown. The following adaptive scheme is then proposed u(t) = µ ¯ Kx(t) µ ¯˙ = −g µ ¯ + xt K t µ ¯−1 Kx + β ||x||2 , µ ¯(0) = µ∗ β˙ = −h β − µ ¯ ||x||2 + β −1 xt Rx , β(0) = β ∗

(4.14)

4.3 Adaptive Schemes

105

Fig. 4.3. Plot of Gain Factor µ ˜ for different values of g

where h > 0 represents a growth factor. Note that scheme (4.14) is constructed in the same way as scheme (4.8). A convenient Lyapunov functional V (.) is given by

t Vb (x, µ ¯, β) = xt (t)P x(t) + xt (s)Qx(s)ds + 1/2 µ ¯2 + 1/2 β 2 (4.15) t−τ

The following theorem summarizes the desired result: Theorem 4.3. System (4.1) under the adaptive controller (4.14) is asymptotically stable if for given matrices 0 < Q = Qt , 0 < R = Rt there exist matrices 0 < X = X t , Y, L, Z and scalars g > 0, ε > 0, > 0 such that the LMIs   Ao X + XAto + Z + Z t t M M Na Y XQ XR Ad   +L + Lt     • −εI 0 0 0 0 0 0     • • −I 0 0 0 0 0     • • • −σI 0 0 0 0   < 0 (4.16)   • • • • −I 0 0 0     • • • • • −Q 0 0     • • • • • • −R 0    −Q+  • • • • • • • Ndt Nd have a feasible solution. Moreover, the feedback gain is K = Y X −1 and the adjustable factor µ ¯−1 = Bo Y Z −1 . ¯, β) along the solutions of Proof: An evaluation of the derivative of Vb (x, µ system (4.1)-(4.2) using (4.12) and adaptive controller (4.8), yields:

106


t t ˙ ¯, β) = x (t) P A∆c + A∆c P + Q x(t) + 2xt P A∆d x(t − τ ) Vb (x, µ ¯µ ¯˙ + β β˙ − xt (t − τ ) Q x(t − τ ) + µ t t −1 t t = x (t) P A∆f + A∆f P + Q + P A∆d Q A∆d P + K K + R x(t) t t −1 t − x (t − τ ) − x (t)P A∆d Q x(t − τ ) − A∆d P x(t) − gµ ¯2 − hβ 2 ≤ xt (t) P A∆f + At∆f P + Q + P A∆d Q−1 At∆f P + K t K + R x(t) ∆

A∆f

= xt (t) Ξf x(t) ¯ Bo K] + ∆A = Af + ∆A = [Ao + µ

(4.17) (4.18)

Following parallel development to Theorem (4.1), it is readily evident that ˜) < 0 holds if there exist scalers ε > 0, > 0 the stability condition V˙ n (x, µ P A∆f + At∆f P + Q + P A∆d Q−1 At∆d P + K t K + R + S < 0 =⇒ ¯ P Bo K + µ ¯ K t Bot P P Ao + Ato P + Q + K t K + εNat Na + µ + ε−1 P M M t P + −1 P M M t P + P Ad [Q − Ndt Nd ]−1 Atd P < 0 ⇐⇒   P Ao + Ato P + R+  Q + K t K + εNat Na P M P M  P Ad   t t  +¯  ¯ K Bo P  µ P Bo K + µ  0, there exist matrices 0 < X = X t , Y, Z and a scalar ε > 0 such that the LMIs   Ao X + XAto + Z + Z t M L Y   +βBo X + βXBot    • −εI 0 0  (4.21)   < 0  • • −εI 0  • • • −I have a feasible solution. Moreover, the feedback gain is Ko = Y X −1 . Corollary 4.5. System (4.20) under the adaptive controller (4.14) is asymptotically stable if for given matrices 0 < Q = Qt , 0 < R = Rt there exist matrices 0 < X = X t , Y, L, Z and scalars g > 0, ε > 0 such that the LMIs   Ao X + XAto + Z + Z t t M Na Y XR   +L + Lt    • −εI 0 0 0    < 0 (4.22)  • • −σI 0 0     • • • −I 0  • • • • −R have a feasible solution. Moreover, the feedback gain is K = Y X −1 . Thus far, we have developed LMI-based sufficient conditions for checking the feasibility of resilient feedback controller to stabilize a class of dynamical systems with norm-bounded parametric uncertainties. In the next section, we direction attention to the case of convex-bounded parametric uncertainties.

4.4 Polytopic Model Consider a plant represented by the class of linear continuous-time systems with state-delay: (ΣJ ) :

x(t) ˙ = Ao x(t) + Ad x(t − τ ) + Bo u(t) , y(t) = Co x(t) (4.23)

4.5 Design Results

109

where x ∈ IRn is the system state, u(t) ∈ IRm is the control input, y(t) ∈ IRp is the measured output and τ is a time-delay factor such that 0 < τ ≤ τ ∗ , 0 < τ˙ ≤ τ + < 1 The corresponding state-space matrices contain uncertainties represented by a real convex bounded polytopic model of the type [Ao ,

Ad ,

∆

Co ] ∈ Sλ =

Bo ,

=

N

[A(oλ) , A(dλ) , B(oλ) ,

C(oλ) ]

λj [Aoj , Adj , Boj , Coj ], λ ∈ Λ

(4.24)

j=1

where Λ is the unit simplex ∆

Λ=

(λ1 , · · · , λN ) :

N

λ j = 1 , λj ≥ 0

(4.25)

j=1

Define the vertex set IN = {1, ..., N }. Throughout this section, we use {Ao , · · · , Fo } to imply generic system matrices and {Aoj , · · · , Cdj , j ∈ IN} to represent the respective values at the vertices. The initial condition is specified as x(0), x(s) = xo , φ(s) , where φ(.) ∈ L2 [−τ, 0]. In line of the analytical developments of the foregoing section, we focus hereafter on the design of adaptive feedback control schemes that ensures that the closed-loop system of (4.23) is asymptotically stable. The actual implemented controller is assumed to have two-terms: u(t) = [µ Ko + ∆K(t)] u(t)

(4.26)

where µ is an adjustable gain factor, Ko is the gain matrix to be determined and ∆K(t) represents the gain perturbation, which is assumed to be normbounded of the form: ||∆K(t)|| ≤ β

(4.27)

where β > 0 is an upper bound to be dealt with in the subsequent analysis.

4.5 Design Results The control design proceeds in two stages. In the first stage, we attempt to construct an adaptive schemes for the uncertain time-delay system (4.23) assuming that the gain perturbation bound is known. Then in the second stage, we extend the results to accommodate bounded-but-known gain perturbations.

110


4.5.1 Known Perturbation Bound When the gain perturbation bound is known, then the purpose of adaptation is to accommodate the uncertainties of system (4.1). The following adaptive scheme is proposed u(t) = [˜ µ Ko + ∆K(t)]x(t) µ ˜˙ = −g µ ˜ +

N

λj xt Kot µ ˜−1 Ko x , µ ˜(0) = µ+ , g > 0

(4.28)

j=1

where Ko , g > 0 represent, respectively, a control gain matrix and a growth factor, both will be determined in the sequel. Combining system (ΣJ ) and controller (4.28), we get the closed-loop dynamics ˜ Bo Ko + Bo ∆K(t) x˙ = A∆c x(t) + Ad x(t − τ ) , A∆c = Ao + µ µ ˜˙ = −g µ ˜ +

N

λj xt Kot µ ˜−1 Ko x , µ ˜(0) = µ+ , g > 0

(4.29)

j=1

A parameter-dependent Lyapunov functional V (.) is given by Vn (x, µ) =

N

t λj x (t)Pj x(t) +

t t

x (s)Qj x(s)ds

+ 1/2 µ ˜2 (4.30)

t−τ

j=1

where 0 < Pj = Pjt ∈ IRn×n and 0 < Qj = Qtj ∈ IRn×n . The following theorem summarizes the main result: Theorem 4.6. System (4.23) under the adaptive controller (4.28) is asymptotically stable if for a given matrix 0 < Qj = Qtj , j ∈ IN and a scalar β > 0, there exist matrices 0 < Xj = Xjt , Yj , Zj , j ∈ IN such that the LMIs   Aoj Xj + Xj Atoj + Zj + Zjt Xj Qj Yj Adj  t  +βBoj Xj + βXj Boj    • −Qj 0 0  (4.31)  < 0 , j ∈ IN   • • −I 0  ˆj • • • −Q have a feasible solution. Moreover, the feedback gain is Ko = Yj Xj−1 . Proof: Evaluation of the derivative of Vn (x, µ) along the solutions of system (4.29) with some algebraic manipulations yields: V˙ n (x, µ ˜) =

N

λj xt (t){Pj A∆cj + At∆cj Pj + Qj }x(t) + 2xt Pj Adj x(t − τ )

j=1

N 2 − (1 − τ˙ )x (t − τ ) Qj x(t − τ ) − g µ ˜ + λj xt Kot Ko x t

j=1

(4.32)

4.5 Design Results

111

where ˜Boj Ko + Boj ∆K = Acj + Boj ∆K A∆cj = Aoj + µ

(4.33)

ˆ j = (1 − τ + )Qj , completing the squares and using (4.27) it follows Letting Q that N V˙ n (x, µ ˜) ≤ λj xt (t) Pj [Aoj + µ ˜Boj Ko ] + [Aoj + µ ˜Boj Ko ]t Pj + Qj j=1

t ˆ −1 Atdj Pj + Kot Ko + βPj Boj + βBoj P + Pj Adj Q j x(t) j ∆

=

N

λj xt (t) Ξnj x(t)

(4.34)

j=1

From Lyapunov theory, it follows that V˙ n (x, µ ˜) < 0 is guaranteed if Ξnj < 0. By the Schur complements, it follows that Ξnj < 0 holds if and only if the matrix inequality :   t Pj Aoj + Atoj Pj + µ ˜Pj Boj Ko + µ ˜Kot Boj Pj A P j dj  t  Pj +Qj + Kot Ko + βPj Boj + βBoj 0 guarantees the asymptotic stability of system (4.28) and different values will only affect the speed of convergence. This is illustrated by the following example. 4.5.2 Example 4.3 This example is motivated by the dynamics of bio-strata in water-quality studies on the river Nile. A pilot-scale model of the form (4.23) with two modes of operation is described by:       −0.2 0 0 0 0 0 0.8 0 Ao1 =  0 −0.9 −0.3  , Ad1 =  −0.7 0 0  , Bo1 =  0.2 0.3  0.8 0 −1 0 −0.8 0 0 0.4       −0.6 0 0.1 −0.1 0 0 0.3 0 Ao2 =  0 −0.8 0.1  , Ad2 =  0.5 0 −0.2  , Bo2 =  0.4 0.4  0.6 0 −1.2 0 0.4 −0.5 0 0.9

112


The feasible solution of LMIs (4.31) is given by   −2.3015 0.0104  0 K =  0.1682 −2.9704 −1.8246 In Figure 4.6, the behavior of the output variables is displayed for different values of g, from which it is clear that relatively-high values of g tend to yield effective stabilization.

Fig. 4.6. Plot of Output Response for different values of g

4.5.3 Unknown Gain Perturbation Bound Now we consider the application of controller (4.26) subject to bound (4.27) where β is unknown. The following adaptive scheme is then proposed u(t) = µ ¯ Kx(t) µ ¯˙ = −g µ ¯ +

N

λj xt K t µ ¯−1 Kx + β ||x||2 , µ ¯(0) = µ∗

j=1

β˙ = −d β − µ ¯ ||x||2 +

N

λj β −1 xt Rx , β(0) = β ∗

(4.36)

j=1

where d > 0 represents a growth factor. Note that scheme (4.36) is constructed in the same way as scheme (4.28). A parameter-dependent Lyapunov functional V (.) is given by

4.5 Design Results

Vb (x, µ ¯, β) =

N

λj xt (t)Pj x(t) +

j=1

113

t t

x (s)Qj x(s)ds

t−τ

+ 1/2 µ ¯2 + 1/2 β 2

(4.37)

The following theorem summarizes the desired result: Theorem 4.8. System (4.23) under the adaptive controller (4.36) is asymptotically stable if for given matrices 0 < Q = Qt , 0 < R = Rt there exist matrices 0 < X = X t , Y, L, Z and scalars g > 0 such that the LMIs   Aoj Xj + Xj Atoj Yj Xj Qj Xj Rj Adj   +Lj + Ltj    • −I 0 0 0    < 0 (4.38)  0 0  • • −Qj    • • • −Rj 0  ˆ • • • • −Q have a feasible solution. Moreover, the feedback gain is K = Yj Xj−1 . Proof: An evaluation of the derivative of Vb (x, µ ¯, β) along the solutions of system (4.23) using (4.32) and adaptive controller (4.28), yields: ¯, β) = V˙ b (x, µ

N

λj xt (t){Pj [Aoj + µ ¯Boj K] + [Aoj + µ ¯Boj K]t Pj + Qj }x(t)

j=1

+ 2xt Pj Adj x(t − τ ) − (1 − τ˙ )xt (t − τ ) Qj x(t − τ ) +µ ˜µ ˜˙ + β β˙

(4.39)

Proceeding like Theorem 4.6, it follows that ¯, β) = xt (t) Pj [Aoj + µ ¯Boj K] + [Aoj + µ ¯Boj K]t Pj + Qj V˙ b (x, µ t x(t) + K t K + Rj Pj Adj (1 − τ˙ )Q−1 A P j dj j t t −1 t − x (t − τ ) − x (t)P A∆d Q x(t − τ ) − A∆d P x(t) − g µ ¯2 − dβ 2 t ≤ x (t) Pj [Aoj + µ ¯Boj K] + [Aoj + µ ¯Boj K]t Pj + Qj ∆ −1 t t ˆ + K K + Rj Pj Adj Qj Adj Pj x(t) = xt (t) Ξf x(t) (4.40) It is readily evident that the stability condition V˙ n (x, µ ˜) < 0 holds if Ξf < 0 and by the Schur complements this corresponds to

114




Pj Aoj + +Atoj Pj t µ ¯Pj KBoj  ¯Pj Boj K + µ  •   •   • •

Kt Q −I • • •

0 −Q • •

 R P Adj   0 0   0

(4.44)

Figure 4.7 shows the structure of the controlled system. Over the vertex set IN, the matrices Do , Fo , Ho have the values Doj , Foj , Hoj , j ∈ IN. It follows on combining (4.23) and (4.42)-(4.44) with K re placing Ko that the closed-loop dynamics takes the form −1 x(t) ˙ = Doj x(t) + µm Foj Hoj Γj x(t) + Adj x(t − τ )

µ˙m (t) = −σ µm +

N

−1 λj xt (t)Γjt Hoj Γj x(t)

(4.45)

j=1

In this respect, a convenient Lyapunov functional Va (.) is given by

t N λj xt (t)Pj x(t) + xt (s)Qj x(s)ds + µ2m (4.46) Va (x, µm ) = j=1

t−τ

where 0 < Pj = Pjt ∈ IRn×n and 0 < Qj = Qtj ∈ IRn×n . In the sequel we t select the gain matrix Γj = Foj Pj . The following theorem summarizes the third main result: Theorem 4.9. System (4.1) with the reference model (4.42) under the adaptive controller (4.43)-(4.44) is asymptotically stable if for a given matrix 0 < Qj = Qtj , j ∈ IN there exist matrices 0 < Xj = Xjt , Yj , j ∈ IN such that the LMIs   t Aoj Xj + Xj Atoj + Foj Yj + Yjt Foj Xj Qj Adj  • −Qj 0  < 0 , j ∈ IN (4.47) ˆj • • −Q have a feasible solution. Moreover, the feedback gain is K = −Yj Xj−1 .

116


Proof: Evaluation of the derivative of Vn (x, µ) along the solutions of system (4.45) with some algebraic manipulations in the manner of Theorem 4.6, it yields: ˜) = V˙ a (x, µ

N

−1 t t λj xt (t){Pj Doj + Doj Pj − 2µm Pj Foj Hoj Foj Pj + Qj }x(t)

j=1

+ 2x Pj Adj x(t − τ ) − (1 − τ˙ )x (t − τ ) Qj x(t − τ ) t

t

− 2σ µ2m + 2

N

−1 t Foj Pj x(t) λj xt (t)µm Pj Foj Hoj

j=1

≤

N

t Pj λj x (t) Pj Aoj + Atoj Pj − Pj Foj Kj + Qj + Kjt Foj t

j=1

t ˆ + Pj Adj Qj Aoj x(t) ∆

=

N

λj xt (t)Ξaj x(t)

(4.48)

j=1

By the Schur complements, the stability condition Ξaj < 0 using Xj = Pj−1 and introducing the linearization variables Yj = −KXj can be readily written into the LMI (4.47) and thus the proof is completed. ∇∇∇ In the single-input single-output case, we use the following change of variables Hoj −→ hj , Goj −→ gj , Loj −→ rj Foj −→ fj = hj boj , rj gjt −→ ctj , Γj −→ γj = hj Pj boj (4.49) A little algebra yields the following result Corollary 4.10. Consider system (4.23) with single input and single output (m = 1, p = 1). The reference model (4.42) under the adaptive controller (4.43)-(4.44) renders the closed-loop system asymptotically stable if for a given matrix 0 < Qj = Qtj , j ∈ IN there exist matrices 0 < Xj = Xjt ∈ n×n , yj ∈ n×1 , j ∈ IN such that the LMIs   Aoj Xj + Xj Atoj + hj boj yjt + hj yj btoj Xj Qj Adj  • −Qj 0  < 0 , j ∈ IN (4.50) ˆj • • −Q have a feasible solution. Moreover, the feedback gain is k = −Xj−1 yj .

4.5 Design Results

117

Fig. 4.7. Structure of the controlled system

4.5.6 Example 4.5 In order to demonstrate the model-reference state regulation, we consider a system of the type (4.23) with the following data       11 0.01 0 0 −1 0.5 −0.2 Ao1 =  0 −0.5 −2  , Ad1 =  0 −0.02 0  , Bo1 =  0 1  10 0 0 0.03 0.15 0 −2       0 −0.1 0 10 −0.8 0 0.2 Ao2 =  −0.5 2 0.1  , Ad2 =  0.2 0 0  , Bo2 =  1 1  0 −0.1 0 01 0.2 0 −2.5     10 01 t t Co1 =  1 0  , Co2 = 1 0 01 01 Let the invertible feedforward matrix 0 < Ho be selected as 1.3 0.1 0.7 −0.2 20 40 Ho1 = , Ho2 = , ; Lo1 = , Lo2 = −0.1 0.8 0.2 1.1 02 04 Using Q1 = 0.3 × I3 , Q2 = 1.2 × I3 , the feasible solution of LMI (4.47) is given by 1.0102 0.9945 −0.0015 K= −0.0013 0.0004 2.9675

118


4.5.7 Special Cases Once again, in case of the delay-less system (ΣF ) :

x(t) ˙ = Ao x(t) + Bo u(t) , y(t) = Co x(t)

(4.51)

where the matrices Ao , Bo , Co are represented by (4.24). The following corollaries can be easily established. Corollary 4.11. System (4.51) under the adaptive controller (4.28) is asymptotically stable if for a given scalar β > 0, there exist matrices 0 < Xj = Xjt , Yj , Zj , j ∈ IN such that the LMIs 

 Aoj Xj + Xj Atoj + Zj + Zjt Y j  t  +βBoj Xj + βXj Boj < 0 , • −I

j ∈ IN

(4.52)

have a feasible solution. Moreover, the feedback gain is Ko = Yj Xj−1 . Corollary 4.12. System (4.51) under the adaptive controller (4.36) is asymptotically stable if for a given matrix 0 < R = Rt there exist matrices 0 < X = X t , Y, L, Z and a scalar g > 0 such that the LMIs   Aoj Xj + Xj Atoj Y X R j j j  +Lj + Ltj   < 0 (4.53)  • −I 0  • • −Rj have a feasible solution. Moreover, the feedback gain is K = Yj Xj−1 . Corollary 4.13. System (4.51) with the reference model (4.42) under the adaptive controller (4.43)-(4.44) is asymptotically stable if there exist matrices 0 < Xj = Xjt , Yj , j ∈ IN such that the LMIs t < 0 , Aoj Xj + Xj Atoj + Foj Yj + Yjt Foj

j ∈ IN

(4.54)

have a feasible solution. Moreover, the feedback gain is K = −Yj Xj−1 . We now proceed to consider uncertain discrete-time systems

4.6 Discrete-Time Model In this section, we focus on the development of resilient adaptive controllers for a class of linear discrete-time systems with norm-bounded parametric uncertainties and bounded controller gain variations. In the absence of controller gain variations, adaptive control schemes for uncertain time-delay systems are developed in [96, 21, 63, 72, 81]. We extend these results to a class of

4.6 Discrete-Time Model

119

uncertain discrete-time systems with state-delay. In this regard, the results compiled hereafter are new contributions to the research investigations on resilient control of uncertain time-delay systems. Towards our goal, we consider the following class of time-delay systems with parametric uncertainties: xk+1 = A∆ xk + Bo uk + A∆d xk−d yk = Co xk

(4.55)

where xk ∈ n is the state vector; uk ∈ p is the control input, y(t) ∈ q is the controlled output, d is a constant time-delay and the uncertain matrices A∆ ∈ n×n and A∆d ∈ n×n , are represented by [A∆ A∆d ] = [Ao Ad ] + M ∆k [Na Nd ]

(4.56)

where Ao ∈ n×n , Bo ∈ n×p , Co ∈ q×n , Ad ∈ n×n , M ∈ n×α , Na ∈ β×n and Nd ∈ β×n , are real and known constant matrices with ∆k is a bounded matrix of uncertainties such that ∆

∆k ∈ ∆k = {∆tk : ∆k ∆k < I , ∀ k} Remark 4.14. The class of systems (4.55) emerges in many areas dealing with functional difference equations or delay-difference equations [35]. On the application side, these systems appear in cold rolling mills [51] and decision-making of manufacturing systems [112]. Our purpose now is to proceed to design a resilient feedback controller for system (4.55) in the sense that it should be capable of accommodating the norm-bounded parametric uncertainties and guaranteeing desirable behavior against controller gain perturbations. The following definition is recalled [81]. Definition 4.15. Given matrix 0 < Q = Qt ∈ n×n . System (4.55) is said to robustly quadratically stable (RQS) if and only if the following inequality −1 t At∆ P A∆ − P + At∆ P A∆d Q − At∆d P A∆d A∆d P A∆ + Q < 0 ∆

∆k ∈ ∆k = {∆tk : ∆k ∆k < I , ∀ k}

(4.57)

has a feasible solution 0 < P = P t ∈ n×n . 4.6.1 Stability and Stabilization Results To proceed further, we introduce the Lyapunov function k−1

Vk = xtk P xk +

j=k−d

A preliminary stability result is provided

xtj Qxj

(4.58)

120


Theorem 4.16. Given a matrix 0 < Q = Qt ∈ n×n , system (4.55) with uk ≡ 0 is RQS if and only if the linear matrix inequality (LMI)   −P + Q + Nat Na Nat Nd Ato P 0  • −Q + Ndt Nd Atd P 0    < 0 (4.59)  • • −P P M  • • • −I has a feasible solution 0 < P = P t ∈ n×n . Proof (Sufficiency): Suppose that 0 < P = P t ∈ n×n is a feasible solution of LMI (4.59) for all admissible uncertainties ∆k ∈ ∆k . Given Vk of (4.58), the corresponding Lyapunov difference ∆Vk = Vk+1 − Vk along the solutions of (4.55) has the form: ∆Vk = xtk [At∆ P A∆ − P + Q]xk + xtk At∆ P A∆d xk−d + xtk−d At∆d P A∆ xk − xtk−d Qxk−d t t A∆ P A∆ − P + Q At∆ P A∆d xk xk = xk−d • −Q xk−d ∆

= ηkt Ξk ηk , ηk = [xtk

xtk−d ]t

(4.60)

By Schur complements operation and in view of Definition 4.15, it follows that ∆Vk < 0 for ηk = 0 and hence system (4.55) with uk ≡ 0 is RQS. (Necessity): Suppose that system (4.55) with uk ≡ 0 is RQS and given a matrix 0 < W = W t . It follows from Definition 4.15 using (4.56) and Schur complements that there exists a matrix 0 < S = S t such that     −S + W 0 At∆ S −S + W 0 Ato S  • −W At∆d S  =  • −W Atd S  + • • −S • • −S    t 0 Na  0  ∆k [Na Nd 0] +  Ndt  ∆tk [0 0 S t M t ] < 0 SM 0 ∀ ∆k ∈ ∆k Therefore, given any ∆k ∈ ∆k we get   −S + Q 0 Ato S ∆ • −W Atd S  < Υ =  • • −S    t 0 Na  0  ∆k [Na Nd 0] +  Ndt  ∆tk [0 0 S t M t ] SM 0 Letting σ = [0 0 S t M t ]t , ψ = [Na Nd 0], we have the inequality

(4.61)

(4.62)

4.6 Discrete-Time Model

ξ t Υ ξ < −2ξ t σ ∆k ψξ ∀ ∆k ∈ ∆k , ξ = 0

121

(4.63)

From which it follows that given ξ = 0 ξ t Υ ξ < −2 max {ξ t σ ∆k ψξ : ∆k ∈ ∆k } ≤ 0

(4.64)

Hence, given ξ = 0 [ξ t Υ ξ]2 > 4 max {[ξ t σ ∆k ψξ]2 : ∆k ∈ ∆k }

(4.65)

By Lemma 8.3 of the Appendix, we get [ξ t Υ ξ]2 > 4 ξ t σσ t ξ ξ t ψ t ψξ , ξ = 0

(4.66)

Application of Lemma 8.4 of the Appendix implies that there exists a α > 0 such that α2 σσ t + α Υ + ψ t ψ < 0

(4.67)

∆

With P = αS, W = αQ, inequality (4.67) reduces to the LMI   −P + Q + Nat Na Nat Nd Ato P   < 0 • −Q + Ndt Nd Atd P • • −P + P M M t P

(4.68)

Finally, a simple Schur complements operation turns (4.68) to (4.59) and therefore the proof is completed. ∇∇∇ Theorem 4.17. Given a matrix 0 < Q = Qt ∈ n×n , system (4.55) is robustly quadratically stabilizable by the feedback control uk = Ko xk if and only if the LMI   −X XNat Nd XAto + Y t Bot XQ XNat  • −Q + N t Nd Atd P 0 0  d   t  • 0 0  • −X + M M (4.69)   < 0  • • • −Q 0  • • • • −I has a feasible solution 0 < X = X t ∈ n×n , Y ∈ m×n . Moreover the gain matrix is given by Ko = Y X −1 . Proof: Follows from Theorem 4.16 on re placing Ao by Ao + Bo Ko and employing the congruent transformation diag[X I I] with X = P −1 and ∇∇∇ Y = Ko P −1 . As we learned before in the continuous-time model, there are at least two sources of inaccuracies when implementing the state-feedback controller uk = Ko xk beyond the availability of state measurements. The first source is obviously due to the presence of uncertainties in the system matrices and the

122


second source arises from gain perturbations due to various reasons [16, 38]. Therefore, it is naturally to consider uk = Ko xk as a nominal feedback controller but the actual implemented controller is assumed to have two-terms: uk = [µ Ko + ∆Ko ] xk

(4.70)

where µ is an adjustable gain factor, Ko is the gain matrix to be determined and ∆Ko represents the gain perturbation, which is assumed to be belong to a compact set of the form: ∆

∆Ko ∈ ∆k = {∆Ko : ∆Ko = β Imn }

(4.71)

where Imn is the unit matrix of order m × n and β > 0 is an upper bound to be dealt with in the subsequent analysis. The problem of interest in the sequel is to develop a feedback control scheme that ensures that the closed-loop system of (4.55)-(4.56) and (4.70)is robustly quadratically stable. Among the various possible approaches, we aim at constructing an adaptive scheme to achieve the cited design objective.

4.7 Adaptive Stabilization Methods We will proceed along two consecutive stages. In the first stage, we attempt to construct an adaptive stabilization method for the uncertain time-delay system (4.55) assuming that the gain perturbation bound β is known. Then in the second stage, we extend the results to accommodate bounded-butknown gain perturbations thereby establishing another . adaptive stabilization method 4.7.1 Known Perturbation Bound When the gain perturbation bound is known, then the purpose of adaptation is to accommodate the uncertainties of system (4.1). The following adaptive scheme is proposed uk = [Ko + ∆Ko ]xk µ ˜k+1 = g µ ˜k + αt Ko xk , µ ˜0 = µ+ , |g| < 1

(4.72)

where Ko ∈ m×n , α ∈ m×1 represent, respectively, a control gain matrix and an adjustable factor, both will be determined in the sequel. A convenient Lyapunov functional Vk is given by Vk = xtk P xk +

k−1

xtj Qxj + µ ˜2k

(4.73)

j=k−d

where 0 < P = P t ∈ IRn×n and 0 < Q = Qt ∈ IRn×n . The following theorem summarizes the first main result:

4.7 Adaptive Stabilization Methods

123

Theorem 4.18. Consider system (4.55) under the adaptive controller (4.72) and given matrices 0 < Q = Qt , α and a scalar β . The closed-loop system is RQS if and only if the fo1lowing LMI   XAto + Y t Bot t t t Z  −X XNa Nd +βXInm Bot XQ XNa Y α      −Q+ t  •  XAd 0 0 0 0 t   N N d d     −X+  •  < 0, • 0 0 0 0 t   MM    •  • • −Q 0 0 0    •  • • • −I 0 0    •  • • • • −I 0 • • • • • • −1 + g 2 |g| < 1

(4.74)

have a feasible solution 0 < X = X t , Y, Z, g. Moreover the gain matrix is given by Ko = Y X −1 . Proof: By similarity to Theorem 4.17, we replace Ao by Ao +Bo Ko +Bo ∆Ko and apply the congruent transformation diag[X I I] to (4.68) with X = P −1 , Y = Ko P −1 and the linearization Z = gP −1 Kot α. Then performing Schur complements, we obtain LMI (4.74). ∇∇∇ 4.7.2 Unknown Perturbation Bound When the gain perturbation bound is unknown, then the adaptive scheme should be constructed to accommodate the uncertainties of system (4.55) and robustify the closed-loop system. For this purpose, the following adaptive scheme is provided uk = [Ko + βk Imn Ko ]xk µ ¯k+1 = g µ ¯k + αt Ko xk , |g| < 1, µ ¯0 = µ+ βk+1 = d βk + ψ t Ko xk , |d| < 1, β0 = β +

(4.75)

where α ∈ m×1 , ψ ∈ m×1 represent adjustable factors, both will be determined in the sequel. In this case, a convenient Lyapunov functional Vk is given by Vk = xtk P xk +

k−1

xtj Qxj + µ ¯2k + βk2

(4.76)

j=k−d

where 0 < P = P t ∈ IRn×n and 0 < Q = Qt ∈ IRn×n . The following theorem summarizes the first main result:

124


Theorem 4.19. Consider system (4.55) under the adaptive controller (4.75) and given matrices 0 < Q = Qt , α, ψ and a scalar β. The closed-loop system is RQS if and only if the fo1lowing LMI   XAto + Y t Bot t t t t XQ XNa Y α W ψ Z R  −X XNa Nd  +W t Bot     −Q+ t  •  0 0 0 0 0 0 XAd t   N N d d   t  •  • −X + M M 0 0 0 0 0 0    •  < 0 • • −Q 0 0 0 0 0    •  • • • −I 0 0 0 0    •  • • • • −I 0 0 0    •  • • • • • −I 0 0   2  •  • • • • • • −1 + g 0 • • • • • • • • −1 + d2 |g| < 1

,

|d| < 1

(4.77)

have a feasible solution 0 < X = X t , Y, Z, W, R, g, d. Moreover the gain matrix is given by Ko = Y X −1 . Proof: By similarity to Theorem 4.16, we replace Ao by Ao + Bo Ko + Bo βImn Ko and apply the congruent transformation diag[X I I] to (4.75) with X = P −1 , Y = Ko P −1 and the linearizations Z = gP −1 Kot α, W = βImn P −1 , R = dP −1 Kot ψ. Performing Schur complements lead to the LMIs (4.77). ∇∇∇ Remark 4.20. In implementing the LMI (4.74) or LMIs (4.77), we start by selecting Q as diagonal matrix of weighting factors and α (same for ψ) as row vector of equal adjustable elements and update the selection till reaching a feasible solution. The availability of convenient software based on interiorpoint minimization [26] helps much in performing computer simulation.

4.8 Examples Some examples will be solved to demonstrate the theoretical developments 4.8.1 Example 4.6 In terms of system (4.55)-(4.56), the data are: −2 0 0 0.1 1 0.7 Ao = , Ad = , Bo = , M= 0 1 0.1 0.5 1 0.8 0.4 0 0.7 Nat = , Γ = , Ndt = , Cot = [0.5 0.5] 0.6 1 0.8

4.8 Examples

125

Observe that the system is open-loop unstable as λ(Ao ) = {−2, 1}. It took several computations to reach at appropriate weighting matrices. A summary of the feasible solutions is given in Table 4.1. It is quite clear that the selection of input data Q, β, α, ψ plays a crucial role in the computational burden. However, the resulting gain Ko changes slightly. A typical state response of the closed-loop system is depicted in Figures 4.8 and 4.9.

Fig. 4.8. Closed-loop State Response-x1: Example 4.6

Fig. 4.9. Closed-loop State Response-x2: Example 4.6

126


4.8.2 Example 4.7 Consider the following discrete-time system of the type (4.55)-(4.56) with 0.5 0.2 0.1 0.2 0.1 1 −0.6 Ao = , Bo = , M= , Ad = 0.6 0.4 0 0.1 0.1 0.4 0.5 0.02 2 0.02 , Γ = , Ndt = , Cot = [1 0] Nat = 0.01 1.5 0.03 A summary of the numerical computations is given in Table 4.2 which shows comparable results in both cases of gain perturbations. Table 4.1. Computational Results of Example 4.6 Theorem Q 4.17 2 0 0 2 4.18 5 0 0 5 4.19 7 0 0 3.5 4.17 3 0 0 6 4.18 8 0 0 8 4.19 10 0 0 5 4.17 4 0 0 8 4.18 10 0 0 10 12 0 4.19 0 15

β

α

1.4 2.3

ψ

Ko -0.7514 0.5496 -0.7386 0.4068

1.9 3.25 1.75 -0.8025 0.3576 -0.8223 0.6755 2.7 3.7

-0.7669 0.6407

3.1 5.4 2.66 -0.8787 0.5617 -0.8345 0.9116 3.6 6.1

-0.8785 0.8237

4.9 7.3 7.5 -0.8066 0.7446

Table 4.2. Computational Results of Example 4.7 Theorem Q β α ψ 10 2 4.17 2 14 4.18 15 1 3.5 6.15 1.45 1 5 4.19 8 3 2.7 7.44 2.37 6.75 1.56 3 16

Ko 4.6818 -0.4981 -6.2863 0.6743 -3.9785 0.4068 7.1235 -1.0578 -5.2775 1.1616 6.1765 -0.6452


127

4.9 Notes and References Most of the results reported in this Chapter are new. As such, it paves the way to more extension by exploiting other ideas and methods of adaptive control theory. This is equally true for continuous and discrete-time systems. Of particular interest is the sliding-mode and variable-structure control methods.

5 Resilient Linear Filtering

5.1 Introduction An integral part of deterministic and stochastic dynamical models describing physical and man-made systems can be conveniently cast into the framework of time-delay systems TDs, that is a wide class of retarded or neutral differential equations containing state and/or input delays [81, 93]. It becomes increasingly apparent that delays occur due to various reasons including finite capabilities of information processing among different parts of the system, inherent phenomena like mass transport flow and recycling and/or by product of computational delays [81]. Considerable discussions on delays and their stabilization/destabilization effects in control systems have commanded the interests of numerous investigators in recent years, see [22, 52, 78, 79, 86, 87, 89, 90, 95] and the references cited therein. On another direction of research, robust filtering arose out of the desire to determine estimates of non-measurable state variables for dynamical systems with uncertain parameters. From this perspective, robust filtering can be viewed as an extension of the celebrated Kalman filter to uncertain dynamical systems. The past decade has witnessed major developments in robust filtering problem using various approaches [6, 24, 33, 97, 104]. In the course of filter implementation based on different design algorithms, it turns out that the filters are very sensitive with respect to errors in the filter coefficients [16, 122]. The sources for this include, but not limited to, imprecision in analog-digital conversion, fixed word length, finite resolution instrumentation and numerical roundoff errors. By means of several examples, it is demonstrated in the control design formalism [43] that relatively small perturbations in controller parameters could even destabilize the closed loop system. Such controllers are often termed ”fragile”. Hence, it is considered beneficial that the designed (nominal) controllers should be capable of tolerating some level of controller gain variations. This illuminates the controller fragility problem for which some relevant results are available in [16, 38]. Extension of this effort to the


130


problem of robust nonfragile Kalman filtering with respect to estimator gain perturbations was considered in [122]. The objective of this Chapter is to contribute to the further development of resilient linear filtering for a class of uncertain systems. Initially, we consider the robust linear filtering problem and extend the results of [122] to a class of continuous-time systems with norm-bounded uncertainties. We start by describing a class of nonfragile linear filters with additive filter gain perturbations then we formulate the design problem of robust nonfragile filtering as a convex optimization over linear matrix inequalities (LMIs). Further, we establish an important special case which recovers, in the nominal case, previously known results. By a limiting approach, we derive an LMI result on robust nonfragile Kalman filter and finally we show that the case of multiplicative filter gain perturbations can be developed conveniently as an extension of the forgoing results. Later on, we look into the class of time-delay systems with parametric uncertainties belong to a convex bounded polytopic domain. In particular, we focus attention to the resilient L2 − L∞ filter and establish complete results under additive and multiplicative gain perturbations. Specifically, we tackle both the filter design problem from a general viewpoint that conveniently yield both full-order and reduced-order filters. We formulate the design problem of linear resilient filtering as a convex optimization over linear matrix inequalities (LMIs). In the absence of gain perturbations, we develop tractable synthesis conditions. Initially, the delay-independent case is considered. Then we proceed to develop delay-dependent case using the descriptor and extended Newton-Leibniz approaches. Simulation results show that the descriptor approach yields less conservative results for both full-order and reduced-order filters.

5.2 A Class of Uncertain Systems We consider a class of uncertain systems described by: Σ∆ :

x(t) ˙ = [A + ∆A(t)]x(t) + Bw(t) y(t) = [C + ∆C(t)]x(t) + Dw(t)

(5.1) (5.2)

z(t) = Lx(t)

(5.3)

where x(t) ∈ IRn is the system state, y(t) ∈ IRm is the measurement output, z(t) ∈ IRp is the vector of state combination to be estimated, w(t) ∈ IRm is a zero-mean white noise input with unity power spectrum density matrix. The initial condition x(0) is a zero-mean random variable uncorrelated with the input noise w(t) for all t ≥ 0. The matrices A, B, C, D, L are known real constants of appropriate dimensions which describe the nominal system whereas ∆A(t) and ∆C(t) are real, time-varying matrix functions representing the norm-bounded parameter uncertainties where

5.3 The Resilient Filter

∆A(t) ∆C(t)

=

Ma Na

131

∆a (t) Ha , ∆ta (t) ∆a (t) ≤ I , ∀ t

(5.4)

where Ma ∈ IRn×α , Na ∈ IRp×α and Ha ∈ IRβ×n are known real constant matrices and ∆a (t) ∈ IRα×β is an unknown matrix with Lebesgue measurable elements. The problem of robust linear filtering considered hereafter is to design an estimate zf of z given by zˆ = F · y, where F is a linear operator, such that as t → ∞ it minimizes an upper bound σ(F) of the estimation error variance for all admissible parameter uncertainty. Define the estimation error ∆ e(t) = z(t) − zf (t), then the problem of interest becomes min σ(F) ,

F ∈Cf

sup ||∆||2 ≤I

IE[et (t)e(t)] ≤ σ(F)

(5.5)

where the feasible set Cf represents the set of all linear operators with minimum state-space realization of the form: ˆ(0) = 0 x˙ f (t) = Af xf (t) + Bf w(t) , x zf (t) = Lf xf (t) nf ×nf

nf ×m

(5.6) (5.7)

, Bf ∈ IR and Lf ∈ IR and the where the matrices Af ∈ IR scalar nf > 0 are the design parameters. We will focus on the stationary case where all matrices of the uncertain system as well as the filter are timeinvariant. The solution to the robust linear filtering cited here before is known [24, 33, 81, 97, 104] using different algorithms. One of our prime interests in this Chapter is the issue of filter realization or implementation. It turns out that implementing the designed filter (5.6)(5.7), much in line with the controller design counterpart, one might encounter inaccuracies or parameter perturbations which eventually leads to the fragility problem [16]. In this regard, this problem has been overlooked in the literature. The only exception to this is the problem of robust nonfragile Kalman filtering with respect to estimator gain perturbations that was considered in [122]. In this part, we generalize this result in different ways by considering the robust nonfragile linear filtering with respect to perturbations in both the estimator gain and dynamics matrices. p×nf

5.3 The Resilient Filter This section contains the analytical developments and main results. Initially, we describe a class of linear filters with additive gain perturbations then we formulate the design problem of robust nonfragile filter as a convex optimization over LMIs. Further, we establish an important special case which recovers, in the nominal case, previously known results. By a limiting approach, we derive an LMI result on robust nonfragile Kalman filter and finally we show that the case of multiplicative filter gain perturbations can be developed conveniently as an extension of the forgoing results.

132


5.3.1 A Class of Linear Filters In the sequel, to cope with the filtering fragility problem we consider that the feasible set Cf now represents the set of all linear operators with minimum state-space realization of the form: Σf :

x˙ f (t) = [Af + ∆Af (t)]xf (t) + Bf w(t) , zf (t) = [Lf + ∆Lf (t)]xf (t)

xf (0) = 0

where the additive gain perturbations are represented by ∆Af (t) Mf ∆f (t) Hf , ∆tf (t) ∆f (t) ≤ I , ∀ t = Nf ∆Lf (t)

(5.8) (5.9)

(5.10)

Consider the case of full-order filtering, n = nf , and connecting the filter (5.8)-(5.9) to the system (5.1)-(5.3), we can write: Σ∆f :

where

∆

x ˆ(t) =

x(t) xf (t)

ˆx(t) + Bw(t) ˆ x ˆ˙ (t) = Aˆ ˆ x(t) e(t) = Lˆ

ˆ= ∈ 2n , B

B Bf D

(5.11)

ˆ = [L − [Lf + ∆Lf (t)] = [L − Lf ] − Nf ∆f (t)Hf L A 0 A + ∆A(t) 0 = Aˆ = Bf C Af Bf [C + ∆C(t)] Af + ∆Af (t) Ma 0 ∆a (t) 0 Ha 0 + Bf Na Mf 0 ∆f (t) 0 Hf

(5.12)

Therefore, the estimation error variance as t → ∞ satisfies ˆ L ˆt] IE[et (t) e(t)] = T r[LIP

(5.13)

for all admissible uncertainties satisfying (5.4) where 0 < IP = IPt ∈ 2n×2n is the solution of the Lyapunov equation ˆ +B ˆB ˆt = 0 IPAˆt + AIP

(5.14)

5.3.2 LMI-Based Design Conditions It follows from [24, 97], using monotonicity property of the Lyapunov equation and invoking Schur complements, that the design problem of robust linear filtering can be cast as:


min T r[W ] ˆ + IPAˆt B ˆ AIP < 0 ˆt −I B ˆt IP IPL > 0 ˆ LIP W

133

(5.15) (5.16)

for all admissible uncertainties satisfying (5.4). Our immediate goal is to transform inequalities (5.15) and (5.16) into LMIs which implies that the foregoing design problem is converted into a convex optimization problem. To achieve our goal, we start by expressing matrix IP in the form Y R Px Uc −1 , IP = (5.17) IP = Uct Pf Rt Qf where Px , Pf , Y, Qf ∈ n×n are all symmetric and positive definite matrices. Among several relations one can derive from (5.17) and the fact IPIP−1 = I are Px Y +Uc Rt = I and Uct Y +Pf Rt = 0. It then follows from these conditions that Px = Y −1 + Uc Pf Uct from which we observe that for any given symmetric and positive definite matrices Px > Y −1 > 0 and Uc nonsingular, then R must nonsingular and by the Schur complements it is always possible to find Pf > 0. In turn, this assures that IP > 0. This guarantees that the matrix   −1 Px Y 0 (5.18) T =  0 Rt 0  0 0 I is nonsingular. On multiplying (5.15) to the right by T and to the left by T t then overbounding the uncertainties using Fact 1, we get for some , µ > 0.   X A + At X Υa XB X Ma 0  • Υb YB + GD 0 YMa + GNa      < 0 • • −I 0 0     • • • −µI 0 • • • • −µI 1 ≥ 0 (5.19) •µ where Υa = X A + At Y + C t G t + Γ t + (X Ma Mat Y + X Ma Nat G t ) + µHat Ha Υb = YA + At Y + GC + C t G t + µHat Ha (5.20) and X = Px−1 , G = RBf , Γ = RAf Uct X are relaxation variables introduced to insure the linearity of inequality (5.19). The second inequality in (5.19) is incorporated to guarantee well-posedness of the matrix inequalities [25]. In a

134


similar way, performing same congruent transformation to inequality (5.16), we obtain for some ε > 0 the following linear matrix inequality:   X X −Z + Lt Rt • Y 0  Lt   (5.21)  • • W − εNf Nft 0  > 0 • • • −εI where Z = Lf Uct X , R = Hf Uc X are relaxation variables introduced again to insure the linearity of inequality (5.21). Therefore, the foregoing analysis leads to the following general result: the robust nonfragile linear filtering design problem of (5.15)-(5.16) with additive gain perturbations is readily equivalent to the following convex programming problem over LMIs: min

W, X , Y, G, Z, R, µ, ε,

T r[W ]

subject to LM Is (5.19) − (5.21) In view of the invertibility of the linear transformation, we immediately obtain the nonfragile filter gain matrices Af = Rc−1 Γ [Uct X ]−1 , Bf = Rc−1 G , Lf = Z[Uct X ]−1

(5.22)

Remark 5.1. One should observe that a necessary condition for the feasibility of inequality (5.19) is that X A + At X < 0 which means that A is a stable matrix. This is an implicit essential requirement of Kalman filtering. Remark 5.2. It is readily seen that in the absence of system uncertainties ∆a ≡ 0, the nonfragile linear filtering design problem reduces to the following convex programming problem over LMIs: min

W,X ,Y,G,Z,R,ε

T r[W ]

   X X −Z t + Lt Rt XB X A + At X Υ¯a • Y Lt 0    • Υ¯b YB + GD  < 0 ,   • • W − εNf Nft 0  > 0 • • −I • • • −εI t t t t t ¯ ¯ Υa = X A + A Y + C G + Γ , Υb = YA + A Y + GC + C t G t 

for which the filter gain matrices are given by (5.22).

∇∇∇


135

5.3.3 Special Case The next theorem summarizes an important special case: Theorem 5.3. Consider system (5.2)-(5.3) with nonfragile linear filter (5.8)(5.9) subject to uncertainties (5.4) and additive gain perturbations (5.10). The gain matrices of the minimum error variance filter F ∈ Cf defined by Lf = L , Af = −Y

−1

−1 −1 Γ I −Y X , Bf = −Y −1 G

(5.23)

where 0 < W = W t , 0 < X = X t , 0 < Y = Y t , Γ, G, R, ε, µ, are the optimal feasible solution of the following convex programming problem over LMIs: min T r[W ]  X X Rt t • Y 0  > 0 , Y L > 0 • W • • −εI   X A + At X Υ a XB  • Υb YB + GD  < 0 • • −I 

(5.24)

Proof: We start by inequality (5.21) and performing some Schur operations, it follows that:     X X −Z t + Lt Rt X −Z t + Lt X Rt t • Y   Lt 0    > 0 ⇐⇒  • W − εNf Nf L 0  > 0 t  • • W − εNf Nf 0  • • Y 0  • • • −εI • • • −εI −1 −1 t −1 t t t X − X Y X + ε R R −X Y L − Z + L > 0 ⇐⇒ • W − εNf Nft − LY −1 Lt   X X Rt t • Y 0  > 0 , Y L > 0 , Z = L (I − Y −1 X ) (5.25) • W • • −εI On making the selection Uc = Uct = X −1 − Y −1 and using Lf from (5.22) along with Z from (5.25), we get −1 Lf = Z[Uct X ]−1 = L I − Y −1 X X −1 X −1 − Y −1 = L

(5.26)

136


which proves the first part of (5.23). From (5.17), we have Px Y + Uc Rt = I =⇒ R = Rt = −Y which eventually leads to the remaining two parts of (5.23).

∇∇∇

Remark 5.4. It is interesting to observe that Theorem 5.3 has a fewer variables to handle and a simpler LMIs to manipulate than the case of general result derived earlier. This took place on the expense of constraining the choice of the optimal output filter matrix Lf to be Lf = L. 5.3.4 A Limiting Approach to Kalman Filtering Following Remark 5.1, we introduce a matrix 0 < Xˆ = Xˆ t such that Xˆ = ω X , ω > 0 is an arbitrary small scalar and note that inequality (5.19) remains feasible as long as ω → 0+ . Bearing in mind that in Kalman filtering, the gain matrix satisfies [6]: Af = A − Bf C In the present analysis and recalling the results of Theorem 5.3, this requires in the limiting case that Γ = − YA + GC + +µHat Ha holds for some µ > 0. This leads to the following result: the robust nonfragile Kalman filtering design problem of (5.15)-(5.16) is readily equivalent to the following convex programming problem with respect to Y, G, µ over LMIs: min T r[W ] YA + At Y + GC + C t G t + µHat Ha YB + GD < 0 • −I Y Lt > 0 • W

(5.27)

The nonfragile filter gain matrices are given by Af = A − Bf C , Bf = Y −1 G , Lf = L

(5.28)

Remark 5.5. The significance of the foregoing result stems from the fact that by enforcing the regularity conditions B Dt = 0, D Dt = I and further ∆ substituting S = Y −1 and G = −C t , the robust nonfragile Kalman filtering problem after some Schur complements operations becomes min



S>0,µ>0

T r[L S Lt ]

 SAt + AS + µHat Ha SC t B  • I 0  < 0 • • −I

(5.29)


137

The nonfragile filter gain matrices are now given by Af = A − Bf C , Bf = S C t , Lf = L

(5.30)

5.3.5 Multiplicative Gain Perturbations Here we consider the feasible set Cf to represent the set of all linear operators with minimum state-space realization of the form with multiplicative gain perturbations: x˙ f (t) = Af [I + ∆Af (t)]xf (t) + Bf w(t) , zf (t) = Lf [I + ∆Lf (t)]xf (t)

xf (0) = 0

(5.31) (5.32)

where the matrices Af ∈ IRnf ×nf , Bf ∈ IRnf ×m and Lf ∈ IRp×nf and the scalar nf > 0 are the design parameters and ∆Af (t), ∆Lf (t) are given by (5.10). The results pertaining to this case can be derived from the foregoing ¯f = Lf Nf ¯ f = Af Mf , N design results by simply defining two matrices M to replace Mf and Nf , respectively. Therefore, we summarize the following results: the robust nonfragile linear filtering design problem of (5.15)-(5.16) with multiplicative gain perturbations corresponds to the following convex programming problem over LMIs: min



W, X , Y, G, Z, R, K, µ, ε,

X A + At X Υ a  • Υb   • •   • • • •  X X −Z + Lt • Y Lt   • • W − εK • • •

T r[W ]

 XB X Ma 0 YB + GD 0 YMa + GNa    < 0 , 1 −I 0 0 ≥ 0  •µ  • −µI 0 • • −µI  t R 0   > 0 0  −εI

¯f N ¯ t is a linearizing matrix variable. The filter gain matrices where K = N f are given by (5.22). In a similar way, in the absence of system uncertainties ∆a ≡ 0, the nonfragile filtering design problem with multiplicative gain perturbations reduces to: min    X X A + At X Υ¯a XB •  • Υ¯b YB + GD  < 0 ,  • • • −I •

W,X ,Y,G,Z,R,K,ε

T r[W ]

 X −Z t + Lt Rt Y Lt 0   > 0 • W − εK 0  • • −εI

138


5.3.6 Example 5.1 In order to illustrate the theoretical results of this paper, we consider a pilotscale model of a stirred-tank heater which can fall into the type (5.1)-(5.3) with the following data at a particular operating point: −0.4 0.3 10 0.15 0 A= , C= , B= , Ha = [0.3 0.3] 3 −4.5 01 0 0.25 0.2 0.3 0.6 0 Ma = , Na = , L= 0.1 −0.2 0 0.4 In the implementation of the linear filter, we consider that the gain perturbations have 0.6 −0.3 Mf = , Nf = , Hf = [0.2 0.4] 0.4 0.5 Numerical computations of the robust nonfragile linear filter show that 0 1 0.201 0 0.598 0.01 Af = , Bf = , Lf = −0.524 −5.132 −0.02 0.245 −0.2 0.4 T r[W ] = 3.1246 , ε = 1.2097 , µ = 2.3416 , = 0.8412 On the other hand, application of Theorem 5.3 yields 0 1 0.223 −0.03 Af = , Bf = −0.511 −5.723 0.01 0.264 T r[W ] = 5.0634 , ε = 3.4066 , µ = 3.4615 , = 0.4436 The higher value of the minimum upper bound to the error variance is due to the constrained choice of Lf = L. 5.3.7 Example 5.2 As another example, we consider a single-reach model for stream water-quality which can fall into the type (5.1)-(5.3) with the following parameters:     0 1.01 0.49 1 A =  −5.01 −0.02 0  , C t =  1  1.51 0 −0.1 −2   00 B =  1 0  , Ha = [0.3 0.3 0.5] 10   0.2 Ma =  0.1  , Na = 0.6 0.7   1.2 Lt =  0.8  −2

5.4 Continuous Polytopic Systems

139

In the implementation of the linear filter, we consider that the gain perturbations have     0.6 −0.3 Mf =  0.4  , Nf =  0.5  , Hf = [0.2 0.4 0.4] 0.2 −0.2 Numerical computations of the robust nonfragile linear filter show that     0 0.1 1 0.201 0 0 0.65  , Bf =  −0.02 0.245  Af =  −0.1 −1.337 −1.196 −6.025 0 0.311   0.911 Ltf =  −0.04  0.63 T r[W ] = 2.9835 , ε = 3.7625 , µ = 4.6115 , = 1.0245 On the other hand, application of Theorem 5.3 yields     0 1 −0.05 0.311 0 0 −0.2 0.75  , Bf =  −0.03 0.264  Af =  −1.4011 −1.201 −5.986 0.01 0.401 T r[W ] = 4.9711 , ε = 2.6576 , µ = 3.6588 , = 1.4016 Once again, the higher value of the minimum upper bound to the error variance is due to the constrained choice of Lf = L. In the remaining part of the Chapter, we shift attention to the resilient linear filtering problem for a class of uncertain time-delay systems with parametric uncertainties belong to a convex bounded polytopic domain. In particular, we focus attention to the resilient L2 − L∞ filter and establish complete results under additive and multiplicative gain perturbations.

5.4 Continuous Polytopic Systems Consider a class of uncertain continuous time-delay systems described by (ΣJ ) :

x(t) ˙ = Ao x(t) + Ad x(t − τ ) + Γ w(t) y(t) = Co x(t) + Φw(t) z(t) = Do x(t)

(5.33) (5.34) (5.35)

where x ∈ IRn is the system state, y(t) ∈ IRp is the measured output, z(t) ∈ IRr is the controlled output and τ is a delay factor which is an unknown and bounded time-varying function such that 0 < τ ≤ β , 0 < τ˙ ≤ α < 1. The corresponding state-space matrices contain uncertainties represented by a real convex bounded polytopic model of the type

140


[Ao , =

N

Ad ,

∆

Co , Do ] ∈ Sλ = [A(oλ) ,

λj [Aoj ,

A(dλ) , C(oλ) ,

Adj , Coj , Doj ], λ ∈ Λ

D(oλ) ] (5.36)

j=1

where Λ is the unit simplex N ∆ Λ = (λ1 , · · · , λN ) : λ j = 1 , λj ≥ 0

(5.37)

j=1

Define the vertex set IN = {1, ..., N }. Throughout the paper, we use {Ao ,· · ·, Fo } to imply generic system matrices and {Aoj , · · · , Foj , j ∈ IN} to represent the respective values at the vertices. The initial condition is specified as x(0), x(s) = xo , φ(s) , where φ(.) ∈ L2 [−τ, 0]. Consider the problem of designing a filter of order k x˙ f (t) = Af xf (t) + Bf y(t) zf = Cf xf (t)

(Σf o ) :

(5.38)

where the matrices Af ∈ k×k , Bf ∈ k×p and Cf ∈ r×k are the unknown (nominal) filter parameters to be determined subject to some prescribed criteria. Note that model (5.38) is quite general in the sense that we consider k = n for full-order and 1 ≤ k ≤ n for reduced-order filter. In the literature, there have been several methods to cope with this problem [78, 79, 86, 87, 88, 89, 90]. In practice it turns out however that these gains cannot be implemented exactly or can be tolerated for fine tuning purposes thereby leading to the so-called fragility (resilience) problem [16, 43]. It is well known that this problem is closely related to the issue of performance deterioration. In the remainder of this paper, we closely investigate the resilience problem in filtering design of time-delay systems

5.5 The Resilient Filtering Problem To address the resilience problem in filtering design and related issues, we consider the filter (5.38) to be expressed as (Σf a ) :

x˙ f a (t) = [Af + ∆Af a (t)] xf a (t) + [Bf + ∆Bf a (t)] y(t) zf a = [Cf + ∆Cf a (t)] xf a (t) (5.39)

where ∆Af a (t), ∆Bf a (t), ∆Cf a (t) are additive gain perturbations represented by ∆Af a (t) = Ma ∆f a (t)Na , ∆Bf a (t) = Ma ∆f a (t)Nb ∆Cf a (t) = Mc ∆f a (t)Nc ∆

(5.40)

∆f a (t) ∈ ∆FA (t) = {∆f a (t) : ∆tf a (t)∆f a (t) ≤ I} ⊂ α×β (5.41)

5.5 The Resilient Filtering Problem

141

Fig. 5.1. The resilience filtering system

A schematic diagram of the resilience filtering problem is depicted in Figure 5.1. Now by combining systems (Σ∆ ) and (Σf a ), we get the augmented system ˙ = Aξ∆ ξ(t) + Aξd ξ(t − τ ) + Γξ w(t) (ΣAa ) : ξ(t) ∆

¯ ξ(t) zξ (t) = ze (t) = L where

ξ(t) =

x(t) xf a (t)

(5.42)

∈ n+k×n+k , Aξ∆ = Aξo + ∆Aξ

Ao 0 Bf Co Af Ad Ad 0 [I 0] = A¯d E = Aξd = 0 0 0 0 0 ¯ = [Do − Cf − ∆Cf a ] = L ¯ o + ∆L ¯ ∆Aξ = , L ∆Bf Co ∆Af ¯ = [0 − ∆Cf a ] ¯ o = [Do − Cf ] , ∆L L Γ Γ Γξ = = Bf Φ (Bf + ∆Bf a )Φ 0 + = Γo + ∆Γ (5.43) ∆Bf a Φ Aξo =

The design objective is to develop a tractable synthesis procedure for selecting the gain matrices Af , Bf and Cf such that for all admissible uncertainties over the vertex IN and gain perurbations ∆Af (t), ∆Bf (t), ∆Cf (t) system (ΣA ) is robustly exponentially stable and ||zξ ||22 < γ 2 ||w||22 .

142


5.5.1 Stability Results To study the stability of the foregoing augmented system, we introduce the Lyapunov-Krasovskii functional

t t V (ξ) = ξ (t)Pξ(t) + ξ t (s)E t QEξ(s) ds (5.44) t−τ

where 0 < P = P t , 0 < Q = Qt are weighting matrices. Evaluation of V˙ (ξ, j) along the solutions of (5.42) with (w ≡ 0) and completing the squares, it yields: V˙ (ξ) = ξ t (t) Atξ∆ P + PAξ∆ + E t QE ξ(t) + 2ξ t P A¯d Eξ(t − τ ) ¯ − ξ t (t − τ )(1 − τ˙ )E t QEξ(t − τ ) ≤ ξ t (t) Atξ∆ P + PAξ∆ + Q ∆ ˆ −1 A¯td P ξ(t) = + P A¯d Q ξ t (t)Ξξ(t) (5.45) ˆ = (1 − α)Q, Q ¯ = E t QE. where Q The following theorem establishes an LMI-based sufficient condition for the robust exponential stability of the closed-loop system (ΣAa ). Theorem 5.6. Consider system (ΣAa ) and given matrices 0 < Q = Qt ∈ n×n . If there exists a matrix 0 0 (5.47) • γ2I has a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FA (t), then the filtering-error system (5.42) is exponentially stable with an L2 − L∞ disturbance attenuation level γ>0 Proof: Given the Lyapunov functional V (ξ) and its derivative V˙ (ξ), it follows that from (5.46) and the fact that PΓξt Γξ P > 0, it follows that Ξ < 0 is guaranteed and we conclude that V˙ (ξ) < 0 for all ξ = 0 and V˙ (ξ) ≤ 0 for all ξ. Since ||ξ(t+β)|| ≤ ϕ||ξ(t)|, ∀β ∈ [−τ, 0] and some ϕ > 0 [55], it follows from (5.44) that V (ξ) ≤ ξ t (t)Pξ(t) + µ||ξ||2 where µ = ϕ τ λM [P] + λM [E t QE] . Therefore, for all ξ = 0, we have ξ t Ξξ V˙ (ξ) λm [−Ξ] ∆ ≤ t (5.48) ≤ − σ = − min V (ξ) ξ Pξ + µ||ξ||2 j∈IN λM [P] + µ


143

It is readily seen from (5.48) that σ > 0 and hence we get V˙ (ξ) ≤ −σ V (ξ), which implies that V (ξ) ≤ e−σ t V (ξo )

(5.49)

Therefore system (5.39) is exponentially stable at a rate σ > 0. To establish the L2 −L∞ performance we consider that ξo ≡ ξ(t = 0) = 0 thereby leading to V (ξo ) = 0 . Now, consider the performance index

t ∆ J = V (ξ) − wt (φ) w(φ) d φ 0

For any 0 = w(t) ∈ L2 [0, ∞), t ≥ 0, it follows that

t wt (φ) w(φ) d φ J = V (ξ) − V (ξo ) − 0

t

t t ˙ V (ξ, j) − w (φ) w(φ) d φ ≤ ϑt (φ) Υ ϑ(φ) d φ = 0

where

(5.50)

0



 ξ t (φ) ϑ(φ) =  ξ t (φ − τ )  w(φ)   PA∆ξ + At∆ξ P + E t QE P A¯d PΓξ Υ = • −Q 0  • • −I

(5.51)

Obviously (5.46) guarantees that J < 0 over the entire domain Sλ . Furthermore, we have

t ξ t (t) P ξ(t) ≤ V (ξ) < wt (φ) w(φ) d φ (5.52) 0

¯tL ¯ < γ 2 P which in turn, for all t ≥ 0 In view of (5.47), it is readily seen that L leads to ¯tL ¯ ξ(t) < γ 2 ξ t (t)P ξ(t) zet (t) ze (t) = ξ t (t) L

t

wt (φ) w(φ) d φ ≤ γ 2 < γ2 0

∞

wt (φ) w(φ) d φ (5.53)

0

||ze (t)||2∞

< γ 2 ||w(t)||2∞ ∀ 0 = By taking supremum over t ≥ 0, we have ∇∇∇ w(t) ∈ L2 [0, ∞) and therefore the proof is completed. 5.5.2 Delay-Independent Filter Synthesis On considering (5.46)-(5.47) in view of (5.43) and applying Fact 1, it follows that

144


¯ P A¯d PΓξ   PAξo + At P + Q ¯ P A¯d PΓo  PAξ∆ + Atξ∆ P + Q ξo  ˆ 0 = ˆ 0 + • −Q • −Q • • −I • • −I     t ¯ Π P Ad PΓo P∆Aξ + ∆Aξ P 0 P∆Γ   < 0 (5.54) ˆ • 0 0  ≤  • −Q 0 t t • • 0 • • −I + ε2 Φ Nb Nb Φ 

and in a similar way ¯t ¯t ¯ ¯ to P L L P − ε−1 3 Nc Nc ≥ ¯ cM ¯t > 0 • γ2I • γ 2 I − ε3 M c

(5.55)

where ¯ + ε−1 P M¯a M ¯bt N ¯b ¯ at P + ε−1 P M¯a M ¯ at P + ε1 N Π = PAξo + Atξo P + Q 2 1 ¯b = [Nb Co Na ] ¯ at = [0 Mat ] , N M ¯ ¯ Nc = [0 Nc ] , Mct = [0 Mct ] (5.56) For simplicity in exposition, we introduce the following matrices W RΛ I 0 n×n k×k P= , W∈ , V =∈ , T= • V 0 V −1 Λt Ga = RΛV −1 Λt , Gc = ΛMa Gb = Na VΛt , Λ ∈ k×k Rt = [Ik 0k×(n−k) ] ∈ IRk×n , Θa = [RGc , RGc , RGc , RGc ] Θb = [ε1 I, ε2 I, ε1 I, ε2 I] (5.57) The existence of resilient L2 −L∞ filter is established by the following theorem. Theorem 5.7. Consider system (ΣAa ) and given a matrix 0 < Q = Qt ∈ n×n and a scalar γ > 0. A kth−order resilient L2 − L∞ filter exists if there ˆf , Cˆf , Ga , Gb , Gc , exist matrices 0 < W = W t , 0 < V = V t , 0 < Λ, Aˆf , B Θa , Θb , and scalars ε1 > 0, ε2 > 0, ε3 > 0, > 0, µ > 0 such that the LMIs   WAoj + Atoj W t ˆ RAf + Aoj Ga   ˆf Coj + WΓ + +RB   t ˆ WAdj Θ +Coj Bf + a   t ˆt t ˆf Φ RB Coj Bf R   t t ε C N G   1 b t oj b Nbt Nb Coj + Q   +ε1 Coj     Aˆtf + Aˆf t   G Γ + a t t  < 0(5.58)  • G A 0 G G + +ε dj 1 b a b ˆf Φ   B t t   Gc Gc + µGc Gc   ˆ   • • − Q 0 0     −I+  0  • • •   ε2 Nb Nbt • • • • −Θb




µ 1 • ε1

≥ 0 ,

1 • ε2

145

≥ 0

 t W − ε−1 Dot 3 Nc Nc R Λ W Nct   > 0 , • V −Cft > 0 • ε3 I • • γ 2 I − ε3 Mct Mc

(5.59)

(5.60)

have a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FA (t). Moreover, the filter matrices are given ˆf , Cf = Cˆf Af = Λ−1 Aˆf Λ−t V , Bf = Λ−1 B

(5.61)

Proof: It follows from (5.60) and the Schur complements for some ε3 > 0 that 0 < W = W t ∈ IRn×n , 0 < V = V t ∈ IRk×k and more importantly t −1 t t W − ε−1 Λ R > 0. Observe that P as constructed in (5.57) is 3 Nc Nc − RΛV positive-definite since W > 0, V > 0 and W − RΛV −1 Λt Rt > 0. Introducing the following change of variables: −1 −t ˆf Aˆf B Af Bf Λ 0 Λ V0 (5.62) = Cf 0 0 I 0 I Cˆf 0 On performing the congruent transformation C I a = [T, I, I], C I b = [T, I] to the last inequalities of (5.54) and (5.55), respectively, and using (5.56), we obtain   Πs Π d WAd WΓ + RΛBf Φ  • Πf ΛV −1 Λt Rt Ad ΛV −1 Λt Rt Γ + ΛBf Φ    < 0 (5.63)  • •  ˆ −Q 0 • • • −I + ε2 Nbt Nb   t W − ε−1 Dot 3 Nc Nc R Λ   < 0 • −V −Cft (5.64) 2 t • • γ I − ε 3 Mc M c where for j ∈ IN t Πs = WAoj + Atoj W + RΛBf Coj + Coj Bft Λt Rt + Q −1 t t t t t t t + ε1 Coj Nbt Nb Coj + ε−1 1 RΛMa Ma Λ R + ε2 RΛMa Ma Λ R −1 t t t t Πf = ΛAf V −1 Λt + ΛV −1 Atf Λt + ε−1 1 ΛMa Ma Λ + ε2 ΛMa Ma Λ

+ ε1 ΛV −1 Nat Na V −1 Λt t t Πd = RΛAf V −1 Λt + Atoj RΛV −1 Λt + Coj Bft Λt + ε1 Coj Nbt Na V −1 Λt −1 t t t t + ε−1 1 RΛMa Ma Λ + ε2 RΛMa Ma Λ

(5.65)

By using the linearizations (5.57), invoking Schur complements and performing standard matrix manipulations, we cast the matrix inequalities (5.63)(5.64) with (5.65) into the LMIs (5.58) and (5.60) subject to the constraints

146


ε2 = 1, µ ε1 = 1. This is readily expressed by (5.59). Therefore, in view of Theorem 5.6, we conclude that the filter (5.39)-(5.41) with realization defined by (5.61) guarantees that the filtering error system (5.42) has an L2 − L∞ disturbance attenuation level bounded by γ. This completes the proof of the theorem. ∇∇∇ Interestingly enough, in the absence of gain perturbations (∆Af a (t) ≡ 0, ∆Bf a (t) ≡ 0, ∆Cf a (t) ≡ 0), we have the nominal augmented system (ΣAao ) :

˙ = Aξ0 ξ(t) + A¯d Eξ(t − τ ) + Γo w(t) ξ(t) ∆ ¯ o ξ(t) zξ (t) = ze (t) = L

(5.66)

for which the following result holds: Corollary 5.8. Consider the nominal augmented system (ΣAao ) and given a matrix 0 < Q = Qt ∈ n×n and a scalar γ > 0. A kth−order robust L2 − L∞ ¯f , C¯f , , Ga filter exists if there exist matrices 0 < W = W t , 0 < V, A¯f , B such that the LMIs   WAoj + Atoj W ¯f + At Ga R A WΓ +  oj ¯f Coj + C t B ¯t t  +RB WAdj oj f R t ˆ ¯f Φ   R B +C B f oj   +Q   t   < 0 G Γ + (5.67) t t a ¯ ¯   Ga Adj ¯ • A f + Af   B Φ f    ˆ • • −Q 0  • • • −I   t W R Λ Do W Nct  • V −Cft  > 0 , > 0 (5.68) • ε3 I • • γ2I have a feasible solution for all admissible uncertainties over the vertex IN. Moreover, the filter matrices are given by ¯f , Cf = C¯f Af = Λ−1 A¯f Λ−t V , Bf = Λ−1 B

(5.69)

5.5.3 Multiplicative Gain Perturbations Had we considered the filter perturbation matrices to be gain dependent, an alternative representation of the resilient filter would then be (Σf m ) :

x˙ f m (t) = [I + ∆Af m (t)]Af xf m (t) + [I + ∆Bf m (t)]Bf y(t) zf m = [I + ∆Cf m (t)]Cf xf m (t) (5.70)

where ∆Af m (t), ∆Bf m (t), ∆Cf m (t) are multiplicative gain perturbations represented by


147

∆Af m (t) = Md ∆f m (t)Ne , ∆Bf m (t) = Md ∆f m (t)Nf ∆Cf m (t) = Mh ∆f m (t)Ng ∆

∆f m (t) ∈ ∆FM (t) = {∆f m (t) : ∆tf m (t)∆f m (t) ≤ I} ⊂ α×β (5.71) Augmenting systems (Σ∆ ) and (Σf m ) yields: (ΣAm ) :

˙ = Aζ∆ ζ(t) + Aζd ζ(t − τ ) + Γζ w(t) ζ(t) ∆ ¯ ζ(t) zζ (t) = ze (t) = H

where

x(t) xf m (t)

(5.72)

∈ 2n×2n , Aζ∆ = Aζo + ∆Aζ Ad Ao 0 Ad 0 , Aζd = = [I 0] = A¯d E Aζo = Bf Co Af 0 0 0 0 0 Γ ∆Aζ = , Γζ = = Γζo + ∆Γ ∆Bf m Bf Co ∆Af m Af (I + ∆Bf m )Bf Φ 0 Γ Γζo = , ∆Γ = ∆Bf m Bf Φ Bf Φ ¯ o + ∆H ¯ ¯ = [Do − [I + ∆Cf m ]Cf ] = H H ¯ = [0 − ∆Cf m Cf ] ¯ o = [Do − Cf ] , ∆H (5.73) H ζ(t) =

By similarity to the case of system (ΣAa ), the following results are easily established for system (ΣAm ): Theorem 5.9. Consider system (ΣAm ) and given matrices 0 < Q = Qt ∈ n×n . If there exists a matrix 0 0 (5.75) • γ2I has a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FM (t), then the filtering-error system (5.72) is exponentially stable with an L2 − L∞ disturbance attenuation level γ>0 Likewise, the existence of resilient L2 − L∞ filter in case of multiplicative gain perturbations is established by the following theorem. Theorem 5.10. Consider system (ΣAm ) and given a matrix 0 < Q = Qt ∈ n×n and a scalar γ > 0. A kth− order resilient L2 − L∞ filter exists if

148


there exist matrices Θa , Θb , Ψa , Ψb , Ψc , such that the LMIs  Σm Σa WAdj   • Σb Gta Adj    • • −Q ˆ    • • • 

0 < W = W t , 0 < V = V t , Aˆf , , Cˆf , Λ, Ga , Ψe and scalars ε1 > 0, ε2 > 0, ε3 > 0, > 0, µ > 0

 Θa Σc  Gta Γ + 0  ˆ  Bf Φ  0 0   < 0  −I+ 0  t  ε2 Ψb Ψb • • • • −Θb µ 1 1 ≥ 0 , ≥ 0 • ε1 • ε2   t W − ε−1 Dot 3 Ψc Ψc R Λ W Ψct   > 0 , • V −Cft > 0 • ε3 I • • γ 2 I − ε3 Mht Mh

(5.76)

(5.77)

(5.78)

have a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FM (t). Moreover, the filter matrices are given ˆf , Cˆf = Cf Af = Λ−1 Aˆf Λ−t V , Bf = Λ−1 B

(5.79)

where ˆf Coj + C t B ˆt t Σm = WAoj + Atoj W + RB oj f R t + ε1 Coj Ψbt Ψb Coj + Q t ˆ t Σa = RAˆf + Atoj Ga + Coj Ψet Bf + ε1 Coj ˆf Φ Σc = WΓ + RB

Σb = Aˆtf + Aˆf + ε1 Gtb Gb + Gd Gtd + µGd Gtd

(5.80)

Proof: Follows from a parallel development of Theorem 5.10 using the linearizations Gd = Λ Md , Ψa = Bft Nft Ne Af VΛt , Ψb = Nf Bf , Ψc = Ng Cf , Ψe = ∇∇∇ ΛVNat Nf Bf . Remark 5.11. It is crucial to note that the approach developed here before provides a tractable procedure of resilient filter design for polytopic state-delay systems since the case of resilient full-order filter can be readily derived by setting R ≡ In . Interestingly enough, this has been enabled by the significant employment of linearization procedures in Theorem 5.7 to Theorem 5.10 and carried out in the same way in all subsequent theorems. By adopting the approach of [90], the ensuing resilient results can be extended to multiple statedelays. In the absence of gain perturbations, it yields a filter structure, which is basically quite different from the results of [27].

5.6 Resilient Delay-Dependent Filtering

149

Remark 5.12. When solving for the resilient filter matrices in the case of additive gain perturbations, it is possible to cast the scalar γ as one of the LMI variables in which case we have to solve the following convex optimization problem: M inimize π = γ 2 ˆf , Cˆf , Ga , Gb , Gc , Θa , Θb over W, V, Λ, Aˆf , B subject to(5.58) and (5.60) √ The minimum attainable value will be γ ∗ = π ∗ . A similar convex optimization problem can be also formulated for multiplicative gain perturbations case.

5.6 Resilient Delay-Dependent Filtering In order to exhibit the delay-dependence dynamics in the filter design procedure, one can utilize different methods [81], each of which has merits and demerits. In the sequel, we are going to develop two methods to resilient delaydependent filtering. The first method is based on the descriptor transformation approach [22] and the resulting design conditions belong to the class of strong delay-dependent [93] since they require the availability of the time-delay factor at every time instant. The second uses the conventional Newton-Leibniz method in addition to some appropriate relaxation variables [95] and since the results require bounds on the time-delay factor and its derivative, the filter design is weak delay-dependent [93]. 5.6.1 A Descriptor Approach Unlike several transformations [93] for time-delay systems, it has been reported that the descriptor approach leads to a system which is equivalent to the original time-delay system, it does not essentially depend on additional assumptions on stability of the transformed system and requires bounding fewer cross-terms. Thus it is expected that it yields least conservative results [22]. In the discussions to follow, we initially focus on the additive gain perturbations case. Thus, we rewrite system (ΣAa ) into the form: ˙ = σ(t) (ΣD ) : ξ(t) 0 = −σ(t) + [Aξ∆ + Aξd ]ξ(t) + Γξ w(t)

t σ(s)ds − Aξd

(5.81)

t−τ (t)

∆ ¯ ξ(t) zξ (t) = ze (t) = L

(5.82)

where the variable σ(t) is treated as fast variable, that is σ˙ = 0. Proceeding further, we introduce the matrices

150


0 0 I0 IPξ 0 ˜ ¯ , Q= P = U IP , U = ; IP = 0 Qσ IPd IPσ 00 I 0 ∆ ¯ ˜= E1 = , E2 = , L [L 0] 0 I

(5.83)

Let the Lyapunov-Krasovskii functional for system (ΣDa ) be selected as: ∆

Vt (η) = Vξ (ξ) + Vσ (ξ)

Vξ (ξ) = ξ t (t)IPξ ξ(t) , Vσ (ξ) =

0

−τ (t)

t

σ t (s) Qσ σ(s) ds d θ (5.84) t+θ

Using (5.82) and (5.83), we get: ˙ ˙ = 2ξ t (t)IPt ξ(t) ˙ = 2η t IPt ξ(t) ∂Vξ /∂ξ ξ(t) ξ 0 σ(t) = 2[ξ t (t) σ t (t)] IPt −σ(t) + [Aξ∆ + Aξd ]ξ(t) 0 t t t + 2[ξ (t) σ (t)] IP t −Aξd t−τ σ(s)ds Simple manipulations using (5.82) yield: σ(t) 2[ξ t (t) σ t (t)] IPt = −σ(t) + [Aξ∆ + Aξd ]ξ(t) ξ(t) [ξ t (t) σ t (t)] IPt [Aξ∆ + Aξd ] + [Aξ∆ + Aξd ]t IP σ(t) Using Fact 1, it follows that t t t 2[ξ (t) σ (t)] IP

t−τ

(5.86)

σ(s)ds

t

[ξ t (t) σ t (t)] IPt A¯d σ(s)ds

=2 t−τ

≤ τ

−Aξd

0 t

(5.85)

ξ(t) σ(t)

t

t ξ(t) −1 ¯t ¯ IP Ad Qσ Ad IP + σ t (s)Qσ σ(s) ds (5.87) σ(t) t−τ t

Therefore from (5.85)-(5.87) we get: t ξ(t) ξ(t) t t ¯ t −1 ¯t ˙ IP [Aξ∆ + Aξd ] + [Aξ∆ + Aξd ] IP + τ IP Ad Qσ Ad IP Vξ ≤ σ(t) σ(t)

t σ t (s)Qσ σ(s) ds (5.88) + t−τ

It is straightforward to show that,


V˙ σ = τ σ t (t)Qσ σ(t) −

0

−τ

σ t (t + θ)Qσ σ(t + θ) dθ

151

(5.89)

By using (5.82)-(5.84) and (5.88), it follows that V˙ t = V˙ ξ + V˙ σ t x(t) IPt (Aξ∆ + Aξd ) + (Aξ∆ + Aξd )t IP = σ(t) x(t) t ¯ −1 ¯t + τ IP Ad Qσ Ad IP + τ Q σ(t)

(5.90)

Letting Aξ∆d = Aξ∆ + Aξd 0 0 Ao + Ad 0 + = ∆Bf Co ∆Af Bf Co Af

(5.91)

Then in line of Theorem 5.6, we have the following stability result Theorem 5.13. Consider system (ΣD ) subject to additive gain perturbations (5.41) and given a matrix 0 < Qσ = Qtσ ∈ n×n . If there exists a matrix 0 < IP = IPt ∈ n+k×n+k and scalars ε1 > 0, ε2 > 0, ε3 > 0 such that the inequality   ˜ τ IPt A¯d ΠD + Q IPt Γξ   < 0 (5.92) • −τ Qσ 0 • • −I + ε2 Φt Nbt Nb Φ t ¯t ¯ ˜t IP − ε−1 L 3 Nc Nc (5.93) ¯ cM ¯ ct > 0 • γ 2 I − ε3 M t ¯ ¯t ΠD = IPt [Ao + Ad ] + [Ao + Ad ]t IP + ε−1 1 IP Ma Ma IP + t ¯ ¯t ¯t ¯ ε−1 2 IP Ma Ma IP + ε1 Nb Nb

(5.94)

has a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FA (t), then the filtering-error system (5.81)-(5.82) is exponentially stable with an L2 − L∞ disturbance attenuation level γ > 0 Proof: Inequality (5.92) is obtained from applying the Schur complements to the sufficient stability condition V˙ t < 0 followed by bounding the result via inequality I.1 of the Appendix. Inequality (5.94) results from the L2 − L∞ performance consideration followed by bounding the result via inequality I.1 as well. ∇∇∇ The existence of resilient delay-dependent L2 − L∞ filter with additive gain perturbations is established by the following theorem.

152


Theorem 5.14. Consider system (ΣD ) subject to additive gain perturbations (5.41) and given a matrix 0 < Qσ = Qtσ ∈ n×n and a scalar γ > 0. A kth−order resilient L2 − L∞ filter exists if there exist matrices 0 < IPξ = ˜f , C˜f , Z, Y and scalars IPtξ ∈ n×n , 0 < IPσ = IPtσ ∈ k×k , IPd , A˜f , B ε1 > 0, ε2 > 0, ε3 > 0 such that the LMIs   Ωmj Ωaj IPtd Ma IPtd Ma IPtξ Adj IPtξ Γ + YΦ   A˜tf + A˜f  •  ˜t Φ 0 0 0 B f t   +ε1 Na Na + Qσ    •  0 0 0 • −ε1 I   < 0 (5.95)  •  0 0 • • −ε2 I    •  0 • • • −Qσ    −I+  • • • 0 0 ε2 Nb Nbt   t IPξ − ε−1 Dot 3 Nc Nc IPd IPξ Nct t   • IPσ −Cf > 0 , > 0 (5.96) • ε3 I • • γ 2 I − ε3 Mct Mc have a feasible solution for all admissible uncertainties over the vertex IN. Moreover, the filter matrices are given by −t ˜ ˜ ˜ (5.97) Af = IP−t σ Af , Bf = IPσ Bf , Cf = Cf where Ωmj = IPtξ (Aoj + Adj ) + (Atoj + Atdj )IPξ + Cot Y t Ωaj

t + YCo + ε1 Coj Nbt Nb Coj t ˜ t = Coj Nbt Na Bf + Z + ε1 Coj

(5.98)

Proof: Follows directly from Theorem 5.13 using (5.83) and applying the ∇∇∇ linearizations Y = IPtd Bf , Z = IPtd Af . Had we considered the case of multiplicative gain perturbations, we would had arrived at the following results Theorem 5.15. Consider system (ΣD ) subject to multiplicative gain perturbations (5.71) and given a matrix 0 < Qσ = Qtσ ∈ n×n . If there exists a matrix 0 < IP = IPt ∈ n+k×n+k and scalars ε1 > 0, ε2 > 0, ε3 > 0 such that the inequality   ˜ τ IPt A¯d IPt Γξ ΠE + Q  < 0  (5.99) • −τ Qσ 0 t t t • • −I + ε2 Φ Bf Nf Nf Bf Φ ˜t ˜ ˜t IPξ − ε−1 L 3 Nc Nc (5.100) ˜ hM ˜t > 0 • γ 2 I − ε3 M h

t ¯ ¯t ΠE = IPt [Ao + Ad ] + [Ao + Ad ]t IP + ε−1 1 IP Md Md IP t ¯ ¯t ˜t ˜ +ε−1 2 IP Md Md IP + ε1 Nb Nb ˜c = [0 Ng Cf ] , M ˜ t = [0 M t ] (5.101) ˜b = [Nf Bf Co Ne Af ] , N N h h


153

has a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FM (t), then the filtering-error system (5.81)-(5.82) is exponentially stable with an L2 − L∞ disturbance attenuation level γ > 0 Proof: Follows from parallel development to Theorem 5.13. ∇∇∇ The existence of resilient delay-dependent L2 − L∞ filter is established by the following theorem. Theorem 5.16. Consider system (ΣD ) subject to multiplicative gain perturbations (5.71) and given a matrix 0 < Qσ = Qtσ ∈ n×n and a scalar γ > 0. A kth−order resilient L2 − L∞ filter exists if there exist matrices 0 < IPξ = ˜f , C˜f , Y, Z, Bj , D, K, N IPtξ ∈ n×n , 0 < IPσ = IPtσ ∈ k×k , IPd , A˜f , B and scalars ε1 > 0, ε2 > 0, ε3 > 0 such that the LMIs   Θaj IPtd Md IPtd Md IPtξ Adj IPtξ Γ + YΦ Θmj   A˜tf + A˜f   • ˜t Φ IPtσ Md 0 0 B f t   +ε D D + Q 1 σ     • I 0 0 0 • −ε 1  < 0  (5.102)   • • • −ε I 0 0 2     • • • • −Qσ 0    −I+  • • • 0 0 ε2 KKt   t IPξ − ε−1 Dot 3 N N IPd IPξ N t   > 0 , • IPσ −Cft > 0 (5.103) • ε3 I 2 t • • γ I − ε 3 Mh M h have a feasible solution for all admissible uncertainties over the vertex IN. Moreover, the filter matrices are given by −t ˜ ˜ ˜ Af = IP−t σ Af , Bf = IPσ Bf , Cf = Cf

(5.104)

where Θmj = IPtξ (Aoj + Adj ) + (Atoj + Atdj )IPξ + Cot Y t + YCo Θaj

+ ε1 Bjt Bj t ˜ = Coj Bf + Z + ε1 Bjt Na

(5.105)

Proof: Follows directly from Theorem 5.15 using (5.83) and applying the linearizations Y = IPtd Bf , Z = IPtd Af , Bj = Nb Bf Coj , D = Ne Af , K = Nf Bf , N = Ng Cf ∇∇∇ As a final closure to this part of our work, in the absence of gain perturbations (∆Af a (t) ≡ 0, ∆Bf a (t) ≡ 0, ∆Cf a (t) ≡ 0), we have the nominal augmented system (ΣAao ) for which the following result holds:

154


Corollary 5.17. Consider the nominal augmented system (ΣAao ) and given a matrix 0 < Q = Qt ∈ n×n and a scalar γ > 0. A kth−order robust L2 − L∞ filter exist if there exist matrices 0 < IPξ = IPtξ ∈ n×n , 0 < IPσ = IPtσ ∈ ˜f , C˜f such that the LMIs k×k , IPd , A˜f , B 

 IPtξ (Aoj + Adj )+ t ˜  (Atoj + Atdj )IPξ + Coj Bf + Z IPtξ Adj IPtξ Γ + YΦ    t t  Co Y + YCo      < 0 ˜t + A˜f A t f   ˜ • 0 Bf Φ   +Q σ     0 • • −Qσ • • • −I   IPξ IPd Dot  • IPσ −Cft  > 0 • • γ2I

(5.106)

(5.107)

have a feasible solution for all admissible uncertainties over the vertex IN. Moreover, the filter matrices are given by −t ˜ ˜ ˜ Af = IP−t σ Af , Bf = IPσ Bf , Cf = Cf

(5.108)

Remark 5.18. Similarly, when solving for the resilient filter matrices, it is possible to cast the scalar γ as one of the LMI variables in which case we have to solve the following convex optimization problem: M inimize π = γ 2 ˆf , Cˆf , Ga , Gb , Gc , Θa , Θb , over W, V, Λ, Aˆf , B Y, Z, Bj , D, K, N subject to(5.102) and (5.103) √ The minimum attainable value will be γ ∗ = π ∗ . 5.6.2 Extended Newton-Leibniz Approach All Newton-Leibniz methods have the starting point of using the classical rule

t ˙ ξ(t − τ ) = ξ(t) − ξ(s) ds t−τ

which eventually transforms (5.42) into the form:

(ΣN ) :

t

˙ = (Aξ∆ + Aξd ) ξ(t) − A¯d E ξ(t)

˙ ξ(s) ds + Γξ w(t) t−τ

∆

¯ ξ(t) zξ (t) = ze (t) = L

(5.109)


155

In preparation for the stability analysis based on Lyapunov theory, there are several Lyapunov-Krasovskii functionals that have been used [81, 93]. Hereafter, we introduce the following Lyapunov-Krasovskii functional

t t Vs (ξ) = ξ (t)Pξ(t) + ξ t (s)E t QEξ(s) ds + (1 − α)−1

0

−τ

t−τ

t

˙ dsdφ ξ˙t (s)E t RE ξ(s)

(5.110)

t+φ

where 0 < P = P t , 0 < Q = Qt , 0 < R = R are weighting matrices. The following theorem establishes the main result: Theorem 5.19. Consider system (ΣN ) and given matrices 0 < Q = Qt ∈ n×n . If there exists a matrix 0 < P = P t ∈ n+k×n+k such that the inequality  t  Aξ∆ P + PAξ∆ + E t QE+ ¯ t t P A − Y PΓ A E R d ξ α ξ∆   βX + E t Yt + YE   t t  ˆ ¯ • −Q 0 Ad E Rα    (5.111)  • • −I Γξt E t Rα  • • • −Rα t ¯ P L > 0 (5.112) • γ2I has a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FA (t) with Rα = β/(1 − α) R such that XY ≥ 0 (5.113) •R then the filtering-error system (5.42) is exponentially stable with an L2 − L∞ disturbance attenuation level γ > 0 Proof: Evaluation of V˙ s (ξ) along the solutions of (5.109) and bounding some terms yields: V˙ s (ξ) ≤ ξ t (t) (Aξ∆ + Aξd )t P + P(Aξ∆ + Aξd ) ξ(t) + 2ξ t (t)PΓξ w(t)

t

˙ ξ t (s)P A¯d E ξ(s) ds + ξ(t)E t QEξ(t)

−2 t−τ

− (1 − α)ξ t (t − τ )E t QEξ(t − τ )

t β t t ˙ ˙ ˙ ˙ RE ξ(s) ds ξ(s)E + ξ(t)E RE ξ(t) − (1 − α) t−τ ≤ ξ t (t) (Aξ∆ + Aξd )t P + P(Aξ∆ + Aξd ) ξ(t) + 2ξ t (t)PΓξ w(t)

156


t

˙ ξ t (s)P A¯d E ξ(s) ds + ξ(t)E t QEξ(t)

−2 t−τ

− (1 − α)ξ t (t − τ )E t QEξ(t − τ ) ¯ + Aξ∆ ξ(t) + Ad Eξ(t − τ ) + Γξ w(t) t ¯ E Rα E Aξ∆ ξ(t) + Ad Eξ(t − τ ) + Γξ w(t)

t t ˙ ˙ RE ξ(s) ds ξ(s)E

−

(5.114)

t−τ ∆ ∆ ∆ ∆ ˙ Using Lemma 8.5 with h = ξ(t), m = E ξ(t), IM = P A¯d , Z = R, we get:

t ˙ ξ t (s)P A¯d E ξ(s) ds −2 t−τ t t t t ¯ ¯ ≤ ξ (t) βX + E Y − E Ad P + YE − P Ad E ξ(t)

t t ˙ ˙ RE ξ(s) ds (5.115) ξ(s)E − 2ξ t (t) Y − P A¯d Eξ(t − τ ) + t−τ

such that X, Y, R satisfy (5.113). Setting w ≡ 0, it follows from (5.114) and (5.115) that t t ˙ Vs (ξ) ≤ ξ (t) (Aξ∆ + Aξd ) P + P(Aξ∆ + Aξd ) ξ(t) + 2ξ t (t)PΓξ w(t) t ¯ − 2ξ (t) N − P Ad Eξ(t − τ ) + ξ(t)E t QEξ(t) − (1 − α)ξ t (t − τ )E t QEξ(t − τ ) t + Aξ∆ ξ(t) + Aξd ξ(t − τ ) + Γξ w(t) E t Rα E Aξ∆ ξ(t) + Aξd ξ(t − τ ) + Γξ w(t) t t t t ¯ ¯ + ξ (t) βX + E Y − E Ad P + YE − P Ad E ξ(t) = where

ξ(t) ξ(t − τ )E t

t

Ξ

ξ(t) ξ(t − τ )E t

(5.116)

 Atξ∆ P + PAξ∆ + E t QE+  βX + E t Yt + YE P A¯d − Y + Atξ∆ E t Rα EAξd   (5.117) Ξ= t t   +Aξ∆ E Rα EAξ∆ t t ˆ • −Q + Aξd E Rα EAξd 


157

By a parallel development to Theorem 5.6, the internal exponential stability of the filtering-error system (5.42) with w ≡ 0 is guaranteed by (5.111) and the Schur complements which in turn ensure that Ξ < 0 over the vertex set IN. t ∆ By considering the performance measure J = Vs (ξ) − 0 wt (s)w(s)ds for 0 = w(t) ∈ L2 [0, ∞) at zero initial condition and noting that internal stability implies Vs (ξ) → 0 as t → 0, it follows that  t  

t

t ξ(t) ξ(t)  ξ(t − τ )E t  Ξ¯  ξ(t − τ )E t  V˙ s (ξ) − wt (s)w(s) ds ≤ J = 0 0 w(t) w(t) where

 ¯ Ξ¯c Ξ¯d Ξa ˆ + A¯t E t Rα E A¯d Ξ¯ =  • −Q A¯td E t Rα EΓξ  d • • −I + Γξt E t Rα EΓξ Ξ¯a = At P + PAξ∆ + E t QE + βX ξ∆

+ E t Yt + YE + Atξ∆ E t Rα EAξ∆ Ξ¯c = P A¯d − Y + Atξ∆ E t Rα E A¯d Ξ¯d = PΓξ + Atξ∆ E t Rα EΓξ

(5.118)

It is readily evident once again on using the Schur complements that (5.111) guarantees Ξ¯ < 0 over the vertex set IN and therefore J < 0, which in turn t leads to ξ t (t)Pξ(t) ≤ Vs (ξ) < 0 wt (s)w(s)ds. Moreover, by Theorem 5.6, inequality (5.53) can be easily derived and hence the proof is completed. ∇∇∇ Finally, on considering (5.118) in view of (5.43) and applying Inequalities 1-2 of the Appendix, it follows that  t  Aξo P + PAξo + E t QE+ ¯ P A − Y PΓ d o   βX + E t Yt + YE  Ξ¯ =   ˆ • −Q 0  • • −I    t   t t t Aξ∆ E t Aξ∆ E t ∆Aξ P + P∆Aξ 0 P∆Γ + • 0 0  +  A¯td E t  Rα  A¯td E t  Γξt E t Γξt E t • • 0   Atξo P + PAξo + E t QE+ ¯ tN ¯   βX + E t Yt + YE + ε1 N P A¯d − Y b b+  −1 PΓo + Atξo R−1 Γo  ρ ¯ ¯ aM ¯ aM ¯ at P + ε−1 P M ¯ at P +Atξo R−1   ε1 P M ρ Ad 2   −1 t t t −1   ≤  + E Na Na E + Aξo Rρ Aξo  t −1 ¯ t −1   ˆ ¯ ¯ • −Q + Ad Rρ Ad Ad Rρ Γo   t −1  −I + Γo Rρ Γo  • • +ε2 Φt Nbt Nb Φ

158


= Ξ+ < 0 ∆

t , Rρ = R−1 α − ψMa Ma ∆

With reference to (5.113), we let X1 0 X= 0 X2

(5.119)

,

Y1 Y= Y2

(5.120)

then the existence of resilient L2 − L∞ filter is established by the following theorem. Theorem 5.20. Consider system (ΣN ) subject to additive gain perturbations (5.41) and given a matrix 0 < Q = Qt and a scalar γ > 0. A kth−order resilient L2 − L∞ filter exists if there exist matrices 0 < W = W t , 0 < V = ¯ 2 , Y2 and scalars ¯a , Θ ¯ b , X1 , Y1 , X ˆf , Cˆf , Ga , Gb , Gc , Θ V t , 0 < Λ, Aˆf , B ε1 > 0, ε2 > 0, ε3 > 0, > 0, µ > 0, ψ > 0 such that the LMIs 

 WΓ + t ¯a Υ Υ WA A Θ m d dj oj ˆf Φ   RB   t   t ˆt  • Υn Gta Adj Ga Γ + Coj  0 B f   ˆ Bf Φ    • • −Q t ˆ (5.121) 0 Adj 0    < 0   −I+  • • • Γot 0    ε2 Nb Nbt    • • • • −Rρ 0  ¯b • • • • • −Θ µ 1 1 ≥ 0 , ≥ 0 (5.122) • ε1 • ε2   t W − ε−1 Dot 3 Nc Nc R Λ W Nct   > 0 , • V −Cft > 0 (5.123) • ε3 I • • γ 2 I − ε3 Mct Mc have a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FA (t) where t t ˆf Coj + C t B ˆt t Υm = WAoj + Atoj W + RB oj f R + ε1 Coj Nb Nb Coj

+ Q + αX1 + Y1 + Y1t t ˆ t Υd = RAˆf + Atoj Ga + Coj Nbt Gb + Y2t Bf + ε1 Coj Υn = Aˆtf + Aˆf + ε1 Gtb Gb + αV −1 Λt X2 Λ + Gc Gtc + µGc Gtc ¯ b = [ε1 I, ε2 I, ε1 I, ε2 I, ψI] ¯ a = [RGc , RGc , RGc , RGc , Nat ] , Θ Θ −1 t ¯ 2 = V Λ X2 Λ (5.124) X Moreover, the filter matrices are given ˆf , Cf = Cˆf Af = Λ−1 Aˆf Λ−t V , Bf = Λ−1 B

(5.125)


159

Proof: Follows parallel developments to Theorem 5.7 applied to Ξ¯ of (5.118). ∇∇∇ The case of multiplicative gain perturbations can be readily stated Theorem 5.21. Consider system (ΣN ) subject to multiplicative gain perturbations (5.41) and given a matrix 0 < Q = Qt ∈ n×n and a scalar γ > 0. A kth−order resilient L2 − L∞ filter exists if there exist matriˆf , Cˆf , Ga , Gb , Gc , ces 0 < W = W t , 0 < V = V t , 0 < Λ, Aˆf , B ¯ 2 , Y2 Ψb and scalars ε1 > 0, ε2 > 0, ε3 > 0, > 0, µ > ¯a , Θ ¯ b , X1 , Y1 , X Θ 0, ψ > 0 such that the LMIs 

¯ ¯  Υm Υd WAdj    • Υn Gta Adj    • • −Q ˆ    • • •    • • • • • • µ 1 ≥ 0 , • ε1  t W − ε−1 3 Ψc Ψc  • •

 WΓ + t ¯a A Θ oj ˆf Φ  RB  t  Ga Γ + t ˆt  ˆf Φ Coj Bf 0  B  0 Atdj 0   < 0  −I+ Γot 0   t ε2 N b N b  • −Rρ 0  ¯b • • −Θ 1 ≥ 0 • ε2  RΛ Dot W Ψct t  V −Cf > 0 , > 0 • ε3 I • γ 2 I − ε3 Mht Mh

(5.126)

(5.127)

(5.128)

have a feasible solution for all admissible uncertainties over the vertex IN and for all gain perturbations ∆f a (t) ∈ ∆FM (t) where t t ˆf Coj + C t B ˆt t Υ¯m = WAoj + Atoj W + RB oj f R + ε1 Coj Ψb Ψb Coj

+ Q + αX1 + Y1 + Y1t t ˆ t Ψet + Y2t Υ¯d = RAˆf + Atoj Ga + Coj Bf + ε1 Coj

(5.129)

Moreover, the filter matrices are given ˆf , Cf = Cˆf Af = Λ−1 Aˆf Λ−t V , Bf = Λ−1 B

(5.130)

We end this part of our work by considering the nominal augmented system (ΣAao ), we have for which the following result holds: Corollary 5.22. Consider the nominal augmented system (ΣAao ) and given a matrix 0 < Q = Qt ∈ n×n and a scalar γ > 0. A kth−order robust L2 − L∞ filter exists if there exist matrices 0 < W = W t , 0 < V = V t , 0 < ¯ 2 , Y2 such that the LMIs ˆf , Cˆf , Ga , Gb , Gc , Θ ¯a , Θ ¯ b , X1 , Y1 , X Λ, Aˆf , B

160




 ˆm Υˆd WAdj WΓ + At Υ oj  ˆf Φ  RB   t   t ˆt   • Υn Gta Adj Ga Γ + Coj B f < 0  ˆ (5.131) Bf Φ    • • −Q  t ˆ 0 A  dj   • • • −I Γot  • • • • −Rρ µ 1 1 ≥ 0 , ≥ 0 (5.132) • ε1 • ε2   t W − ε−1 Dot 3 Nc Nc R Λ W Nct   > 0 , • V −Cft > 0 (5.133) • ε3 I • • γ 2 I − ε3 Mct Mc have a feasible solution for all admissible uncertainties over the vertex IN where t ˆt t ˆf Coj + Coj Bf R + Q + αX1 + Y1 + Y1t Υˆm = WAoj + Atoj W + RB t ˆ Bf + Y2t Υˆd = RAˆf + Atoj Ga + Coj (5.134) Υˆn = Aˆt + Aˆf + αV −1 Λt X2 Λ f

Moreover, the filter matrices are given by −t ˜ ˜ ˜ Af = IP−t σ Af , Bf = IPσ Bf , Cf = Cf

(5.135)

Remark 5.23. Likewise, when solving for the resilient filter matrices, it is possible to cast the scalar γ as one of the LMI variables in which case we have to solve the following convex optimization problem: M inimize π = γ 2 ˆf , Cˆf , Ga , Gb , Gc , over W, V, Λ, Aˆf , B ¯ 2 , Y2 Ψb ¯a , Θ ¯ b , X1 , Y1 , X Θ subject to(5.126) and (5.128) √ The minimum attainable value will be γ ∗ = π ∗ . Remark 5.24. A simple comparison between the resilient delay-dependent filtering approaches shows that the descriptor approach depends on a larger information set than the extended Newton-Leibniz approach since the former approach demands instantaneous value of the delay τ, whereas the latter approach needs only the bounds α, β . This suggests that the descriptor approach belongs to the family of strong delay-dependent methods and extended NewtonLeibniz approach is a member of the weak delay-dependent methods.

5.7 Examples

161

5.7 Examples In the sequel, we provide some examples to shed some light on the theoretical development of this part of the Chapter. 5.7.1 Example 5.3 We consider a pilot-scale single-reach water quality system which can fall into the type (5.33)-(5.35) with the following associated date :      0.9 δd 0 1 + δa 2δa −0.2 1 + δd  , Γ =  −0.7  0 0 Ao =  0 1.5δa 3δa  , Ad =  0.1 0.5 −3 −7 −5 −0.5 −0.2 −0.4 + 2δd 

Co = [2, 0, 1] , Do = [1, 3, 1] , Φ = 0.65 τ = 0.6 , −0.4 ≤ δa ≤ 0.4 , −0.2 ≤ δd ≤ 0.2 , α = 0.6 , β = 0.5 Invoking the MATLAB software environment, we solve the convex optimization problem outlined in Remark 5.12 using   200 Qσ =  0 5 0  008 Considering the delay-independent resilient filter synthesis, the feasible solutions for both full-order and reduced-order resilient filters with additive and multiplicative gain perturbations are summarized in Table 5.1. It is quite Table 5.1. Delay-Independent Filter Design Results: Example 5.3 T ype k Additive 2 0.5174 0.9175 -7.1156 Additive 3 0.7236 0.8535 -11.6015 Multipl- 2 0.1764 icative 2.7885 -12.1586 Multipl- 3 1.1764 icative 3.4175 -14.1673

Af 11.0175 2.4567 -12.6457 12.2011 3.0517 -17.5337 10.1856 3.7464 -17.6557 1.7775 1.4557 -12.4587

-2.5466 -0.4176 -8.6953 -3.1633 -0.1616 -12.9045 -2.6236 -2.1796 -38.5553 -8.5626 -3.4146 -28.6924

Bf 0.7846 -0.9464 1.0314 0.8655 -1.1244 1.2254 0.4995 -2.6333 1.0544 1.4356 -1.4694 5.0304

Cf γ∗ -1.0482 -23.7784 6.3714 1.8765

-1.2015 -19.5466 7.9805 1.6224

-1.2357 -22.4788 5.7815 2.0134

-1.4882 -17.7894 11.3144 1.7988

evident that minimum disturbance-attenuation level bound γ ∗ = 1.6624 corresponding to full-order with additive gain perturbations. Next we examine the variation of the resultant filtering-error system over the vertex set

162

5 Resilient Linear Filtering ∆

IN = {(δa , δd ) : −0.4 ≤ δa ≤ 0.4 , −0.2 ≤ δd ≤ 0.2} through computer simulation of Theorem 5.6. This is displayed in Figures 2 and 3 from which it is clear that the maximum L2 − L∞ gain bound over the set IN is γ ∗ = 1.4932 for full-order filter and is attained at the point {δa = 0.4 , δd = −0.2} and is γ ∗ = 1.5158 for reduced-order filter and is attained at the point {δa = 0.4 , δd = 0.2}. In turn, this illuminates the effectiveness of the resilient filter design.

Fig. 5.2. Resilient L2 − L∞ Gain Bound: full-order delay-independent filter

Turning to the delay-dependent approaches, the results related to the feasible solutions are included in Table 5.2. As expected, we record two important observations: 1) the descriptor approach yields less conservative results that the extended Newton-Leibniz approach and 2) the delay-dependent resilient filter design outperforms the delay-independent design. For the purpose of completeness, the variations of the resultant filtering-error system over the vertex set IN are is displayed in Figures 4 and 5 for full-order resilient filter design with additive gain perturbations using both delay-dependent approaches.

5.7 Examples

163

Fig. 5.3. Resilient L2 − L∞ Gain Bound: reduced-order delay-independent filter

Fig. 5.4. Resilient L2 − L∞ Gain Bound: descriptor delay-dependent filter

164

5 Resilient Linear Filtering Table 5.2. Delay-Dependent Filter Design Results: Example 5.3 Approach Descriptor

T ype Additive Multiplicative Descriptor Additive Multiplicative Newton-Leibniz Additive Multiplicative Newton-Leibniz Additive Multiplicative

k 2 2 3 3 2 2 3 3

γ∗ 1.4985 1.5026 1.4213 1.4355 1.5546 1.5787 1.5136 1.5248

Fig. 5.5. Resilient L2 − L∞ Gain Bound: Newton-Leibniz delay-dependent filter

5.7.2 Example 5.4 Consider a linear dynamical system with polytopic uncertainties and described by     0 −3 1 + δa −0.2 0.1 + δd 0.5 + δd  , Ad =  0.5 1 −1 −0.7 + δd  Ao =  0.3 −2.5 + 0.5δa 0 1 + δd −2.5 + δd −0.1 0.3 + δa −4 + 0.6δa   1 Γ =  0  , Cot = [0.8, 0.3, 0.1] 1

5.7 Examples

165

Do = [0.2, −0.3, −0.6] , Φ = 0.5 , α = 0.6 , β = 0.5 τ = 0.7 , −0.3 ≤ δa ≤ 0.3 , −0.1 ≤ δd ≤ 0.1 By resorting the MATLAB software environment, we solve the convex optimization problem outlined in Remark 5.12 using   4 0 0 Qσ =  0 12 0  0 0 7 Considering the delay-independent resilient filter synthesis, the feasible solutions for both full-order and reduced-order resilient filters with additive and multiplicative gain perturbations are summarized in Table 5.3. It is quite Table 5.3. Delay-Independent Filter Design Results: Example 5.4 T ype k Additive 2 -6.1654 1.9105 0.7256 Additive 3 -5.12567 1.6255 0.5045 Multipl- 2 -7.1764 icative 1.7895 0.8156 Multipl- 3 -5.5006 icative 1.6715 0.7113

Af 2.0195 -2.5967 -8.1247 1.8767 -3.7907 -7.3007 2.1856 -3.4694 -7.6597 1.7687 -3.8037 -6.8237

2.4646 -2.4336 -3.6773 2.3722 -1.5305 -3.4895 2.6446 -2.2796 -3.8853 2.4572 -1.5605 -4.2774

Bf 0.4776 2.7484 3.9314 -1.8995 3.4224 4.9854 0.4785 2.6773 4.0344 -1.9685 3.6105 5.0314

Cf γ∗ -1.0204 -2.4101 2.7144 2.1135

-1.0105 -1.6026 2.4305 1.7014

-1.0317 -2.4258 2.7801 2.3224

-1.1123 -1.5928 2.5105 1.8108

evident that minimum disturbance-attenuation level bound γ ∗ = 1.7014 corresponding to full-order with additive gain perturbations. 5.7.3 Example 5.5 Consider another linear dynamical system with polytopic uncertainties and described by     −0.1 −2.5 0.5 + δa 0 1 + δd 0.6 + δd  , Ad =  0.1 0.5 −0.1 + δd  0.3 Ao =  0.1 −3.5 + 0.1δa −0.1 0.8 + δa −1.6 + 0.4δa −0.6 1 + δd −0.8 + δd   −0.6 Γ =  0.5  , Cot = [−0.5, 0.2, 0.3] 0 Do = [0, −0.6, 0.2] , Φ = 0.8 , α = 0.8 , β = 0.6 τ = 0.7 , −0.2 ≤ δa ≤ 0.2 , −0.6 ≤ δd ≤ 0.6

166


Observe that the variation in the delay matrix takes on wider range. By resorting the MATLAB software environment, we solve the convex optimization problem outlined in Remark 5.12 using   14 0 0 Qσ =  0 4 0  0 0 10 Considering the delay-independent resilient filter synthesis, the feasible solutions for both full-order and reduced-order resilient filters with additive and multiplicative gain perturbations are summarized in Table 5.4. It is quite Table 5.4. Delay-Independent Filter Design Results: Example 5.5 T ype k Additive 2 -6.1654 1.9105 0.7256 Additive 3 -8.4813 5.2715 -4.5005 Multipl- 2 -7.1764 icative 1.7895 0.8156 Multipl- 3 -8.9863 icative 5.4705 -4.4945

Af 2.0195 -2.5967 -8.1247 1.8007 -4.4107 2.1601 2.1856 -3.4694 -7.6597 1.8101 -4.4231 2.1691

2.4646 -2.4336 -3.6773 -1.6322 1.5905 -1.9235 2.6446 -2.2796 -3.8853 -1.6412 1.6005 -1.9246

Bf 0.4776 2.7484 3.9314 -2.4469 1.6644 -2.3140 0.4785 2.6773 4.0344 -2.4519 1.6714 -2.3240

Cf γ∗ -1.0204 -2.4101 2.7144 3.9345

-1.1340 -1.0927 2.3105 3.0184

-1.0317 -2.4258 2.7801 4.1246

-1.2340 -1.1325 2.5555 3.2209

evident that minimum disturbance-attenuation level bound γ ∗ = 3.0184 corresponding to full-order with additive gain perturbations. 5.7.4 Example 5.6 Consider a second-order linear dynamical system described by 0 0 0 11 −0.5 0.7 + δd , Ad = , Γ = Ao = 1 + δc 0 −11 −2.2 + δa −0.4 + δd −0.6 Co = [0, 1] , Do = [1, 0] , Φ = [0, 1 + δc ] τ = 0.8 , −1 ≤ δa ≤ 1 , −0.7 ≤ δd ≤ 0.7 α = 0.8 , β = 0.6 , 0 ≤ δc ≤ 1 Observe that the variation in the input and delay matrices take on wider range and the amplitude of the bounded energy input can vary. By resorting the MATLAB software environment, we solve the convex optimization problem outlined in Remark 5.12 using

5.7 Examples

Qσ =

10 0 0 6

167

Considering the delay-independent resilient filter synthesis, the feasible solution of full-order resilient filter with additive and multiplicative gain perturbations are summarized in Table 5.5. Table 5.5. Filter Design Results: Example 5.6 T ype k Af Additive 2 -0.2675 10.9903 -11.0013 -1.3055 Multipl- 2 -0.4604 11.1053 icative -11.1155 -2.1525

Bf Cf γ∗ 0.0091 0.5705 -0.0084 1.2154 0.6525 0.0105 0.8205 -0.0205 1.5325 0.4275

5.7.5 Example 5.7 Consider a fourth-order uncertain system of the form   1 −1.5 −0.7 δa  0 δa 1 1.3   , Co = [1, δo , 0, 0] Ao =   −0.1 0 0  δa −0.4 −2.5 −3.5 −4.7   0 0 −0.1 + δd 0.05 + δd  −0.1 0 −0.5 −0.1 + δd   , Φ=1 Ad =   0 0 −0.6 0.1 + δd  0 −0.9 0 −0.5 + δd Γ t = [0, 0, 0, δc ] , Do = [1, 1, 0, 0] τ = 1 , 0.9 ≤ δa ≤ 1 , 0.6 ≤ δd ≤ 0.7 , α = 0.8 β = 0.6 , 0 ≤ δo ≤ 1 , 0.08 ≤ δc ≤ 0.09 Observe that there are four parameters in the model matrices. By resorting the MATLAB software environment, we solve the convex optimization problem outlined in Remark 5.12 using   14 0 0 0 0 9 0 0  Qσ =   0 0 12 0  0 0 0 20 Considering the delay-independent resilient filter synthesis, the feasible solutions for full-order k = 4 resilient filters with additive and multiplicative gain perturbations are summarized in Table 5.6. With focus on the resilient

168

5 Resilient Linear Filtering Table 5.6. Filter Design Results: Example 5.7

T ype Additive -1.1585 1.0845 0 -0.0445 Multipl- -1.2565 icative 2.0510 0.0155 -0.4455

Af 1.3928 -6.9155 -1.8718 3.1555 -0.0728 -4.5775 1.1775 2.4675 1.8115 -8.5775 -1.2718 3.0055 -1.7208 -1.7575 3.0075 2.7465

4.5705 2.6683 0.1185 -2.7783 11.0544 4.8663 0.0186 -2.0183

Bf Cf γ∗ 0.8764 1.1678 -0.8577 -10.4456 -2.4582 24.5445 0.2757 0 0.2015 1.6054 1.7698 -1.5787 -10.7756 -5.5812 30.4872 0.4757 0.0025 0.4115

Table 5.7. Computational Results: Example 5.7

γ

γ

γ

γ

V ertix1 δa = 0.9 δd = 0.6 δo = 0 δc = 0.08 2.3456 V ertix5 δa = 0.9 δd = 0.7 δo = 0 δc = 0.08 3.6986 V ertix9 δa = 1.0 δd = 0.6 δo = 0 δc = 0.08 5.9855 V ertix13 δa = 1.0 δd = 0.7 δo = 0 δc = 0.08 7.2345

V ertix2 δa = 0.9 δa = 0.6 δa = 0 δa = 0.09 2.5789 V ertix6 δa = 0.9 δa = 0.7 δa = 0 δa = 0.09 3.5789 V ertix10 δa = 1.0 δa = 0.6 δa = 0 δa = 0.09 5.8779 V ertix14 δa = 1.0 δa = 0.7 δa = 0 δa = 0.09 7.7895



filter with full-order subject to additive gain perturbations, a summary of the variations of the resultant filtering-error system over the vertex set IN is summarized in Table 5.7. It is quite evident that minimum disturbanceattenuation level bound γ ∗ = 9.4155 and occurs at the utmost corner of the vertex set.


169

5.8 Notes and References This Chapter has established the basic foundations for resilient filtering of uncertain time-delay systems. It should open the door for possible build up and extensions along the results of [33, 34, 97, 100, 101, 102]. The topic is exciting and more results should clarify questions like what is the best (least conservative) delay-dependent approach to resilient filtering? Is possible to integrate the delay-factor as one the perturbation parameters. Indeed, the topic of resilient filtering of uncertain discrete-time delay systems is still fresh.

6 Resilient Delay-Dependent Control

6.1 Introduction It becomes increasingly apparent that delays occur in physical and man-made systems due to various reasons including finite capabilities of information processing among different parts of the system, inherent phenomena like mass transport flow and recycling and/or by product of computational delays [81]. Considerable discussions on delays and their stabilization/destabilization effects in control systems have commanded the interests of numerous investigators in recent years, see [81, 8, 22] and their references. In control design, it turns out that the design objectives have to incorporate the impact of parameter shifting, component and interconnection failures which are frequently occurring in practical situations. Robust control theory provides tractable design tools using the time domain and the frequency domain when the plant modeling uncertainty or external disturbance uncertainty is of major concern in control systems. In this Chapter, we direct attention to delay-dependent methodologies for continuous-time and discrete-time systems by exhibiting the delay-dependence dynamics in the design procedure. For the continuous-time case, we develop a descriptor approach to simultaneous H2 /H∞ control design for a class of time-delay systems with additive and multiplicative controller gain perturbations. The delay factor is considered unknown-but-bounded constant or timevarying. The cases of norm-bounded and convex polytopic uncertainties are treated. In the discrete-time case, delay-dependent stability and feedback stabilization methods are developed for the nominal, robust and resilience cases. Seeking computational convenience, all the developed results are cast in the format of linear matrix inequalities (LMIs) and several numerical examples are presented.


172


6.2 Continuous-Time Systems We consider a class of time-delay systems by: (ΣJ ) :

x(t) ˙ = Ao x(t) + Ad x(t − τ (t)) + Bo u(t) + Γ w(t) x(s) = φ(s), t ∈ [−τ, 0] y(t) = Co x(t) + Do u(t) z(t) = Go x(t) + Fo u(t) + Φw(t)

(6.1) (6.2) (6.3)

where x(t) ∈ n is the state vector; u(t) ∈ m is the control input; w(t) ∈ q is the disturbance input which belongs to L2 [0, T ]; y(t) ∈ p , z(t) ∈ p is the controlled output, τ ∈ [τ0 , τ ∗ ] is a constant delay factor and Ao , Bo , Ad , Co , Do , Go , Fo , Γ, Φ are known real constant matrices of appropriate dimensions which describe the nominal system of (ΣJ ). The initial condition is specified as x(0), x(s) = xo , φ(s) , where φ(.) ∈ L2 [−τ, 0]. Our purpose in the sequel is to develop methods for determining delaydependent stabilizing controllers which minimizes the upper bound of an H2 performance measure while guaranteeing that a prescribed upper bound on an H∞ performance is attained for all possible w(t) ∈ L2 [0, ∞]. It will be shown that this problem is to the existence of a feasible solution of linear matrix inequalities (LMIs) thereby providing a clear key to designing the state feedback controller. Note that system ΣJ is continuous-time system with point-delay. In the sequel, the main thrust is to transform this system to an appropriate form in order to exhibit its delay dependence behavior. There are several methods to achieve this [81] each of which has merits and demerits. We will accomplish this by the following method which is known to be less conservative than other methods. 6.2.1 Descriptor Transformation We employ the descriptor system approach [22] to exhibit the delay-dependence dynamics in the design procedure. Thus, we rewrite system (ΣJ ) into the form: (ΣD ) : x(t) ˙ = σ(t) 0 = −σ(t) + [Ao + Ad ]x(t) + Bo u(t)

t σ(s)ds + Γ w(t) − Ad

(6.4)

t−τ (t)

y(t) = Co x(t) + Do u(t),

(6.5)

z(t) = Go x(t) + Fo u(t) + Φw(t),

(6.6)

Under the state-feedback control law u(t) = Ko x(t)

(6.7)

6.3 Feedback Control Design

173

into system (ΣDo ), it becomes: (ΣDK ) :

x(t) ˙ = σ(t)

0 = −σ(t) + AKd x(t) − Ad

t

σ(s)ds + Γ w(t) (6.8) t−τ (t)

y(t) = CK x(t)

(6.9)

z(t) = GK x(t) + Φw(t)

(6.10)

where CK = Co + Do Ko , GK = Go + Fo Ko , Aod = Ao + Ad AKd = Ao + Bo Ko + Ad = Aod + Bo Ko

(6.11)

In the discussions to follow, we develop tractable synthesis procedures to compute the feedback gain Ko .

6.3 Feedback Control Design We focus attention on three design methodologies based on H2 , H∞ and simultaneous H2 /H∞ approaches, respectively. For each approach, we divide our effort into two schemes: a nominal scheme where the gain Ko is assumed to be exactly implemented and a perturbed scheme that takes into consideration various sources of inaccuracies. 6.3.1 Nominal H2 Design In this regard, we introduce the H2 performance measure ∞ ∆ t J2 = y (s)y(s) ds

(6.12)

0

The problem of H2 state-feedback control could be phrased as follows: Given system ΣJ determine the control law (6.7) which achieves the minimal value of H2 performance measure (6.12) Since our objective is to develop tractable synthesis to determine the feedback gain Ko , we introduce, for convenience, the following matrices I0 Px 0 P¯ = U IP ; U = ; IP = 00 P d Pσ I 0 Yx 0 E1 = (6.13) , E2 = , Y = IP−1 = Yd Yσ 0 I Yx = Px−1 , Yσ = Pσ−1 , Yd = −Yx Pd Yσ 0 I 0 ¯ , C¯K = [CK 0] , Ad = AoKd = AKd −I Ad and provide the following stability result.

(6.14)

174


Theorem 6.1. In the absence of input disturbance w(t) ≡ 0, controller (6.7) is an H2 -optimal controller for system (ΣDK ) minimizing the H2 -performance measure (6.12) if, given matrix sequence Qσ = Qtσ > 0, i ∈ S there exist matrices Px = Pxt > 0, Pσ = Pσt > 0, Pd = Pdt > 0, i ∈ S, satisfying the system of LMIs   t CK Ω2 τ Pdt Ad Ω1 + CK ∆ • −Ω3 τ Pσt Ad  < 0 (6.15) Πt =  • • −Qσ where Ω1 = Pdt AKd + AtKd Pd , Ω2 = Pxt − Pdt + AtKd Pσ Ω3 = Pσt + Pσ − τ Qσ

(6.16)

An upper bound on the H2 performance measure is given by

∆ J2 ≤ J+ = xt (0)Px x(0) + τo

1/2

0

−τo

σ t (s) Qσ σ(s) ds

(6.17)

Proof: Let the Lyapunov-Krasovskii functional V (·) of the transformed system (ΣDKo ) be selected as: ∆

Vt (x) = Vx (x) + Vσ (x)

Vx (x) = xt (t)Px x(t) , Vσ (x, i) =

0

−τ (t)

t

σ t (s) Qσ σ(s) ds d θ (6.18) t+θ

Using (6.8) and (6.18), we get: ∂Vx /∂x x(t) ˙ = 2xt (t)Pxt x(t) ˙

x(t) ˙ = 2[x (t) σ (t)]IP 0 σ(t) t t t = 2[x (t) σ (t)] IP −σ(t) + AoKd x(t) 0 t + 2[xt (t) σ t (t)] IPt −Ad t−τ σ(s)ds t

t

t

Simple manipulations using (6.8) yield: σ(t) 2[xt (t) σ t (t)] IPt = −σ(t) + AoKd x(t) x(t) [xt (t) σ t (t)] IPt AoKd + AtoKd IP σ(t) Using Inequality I.1 of the Appendix, it follows that

(6.19)

(6.20)


2[xt (t) σ t (t)] IPt

−Ad

t

0

t−τ

175

σ(s)ds

t

[xt (t) σ t (t)] IPt A¯d σ(s)ds

=2 t−τ

≤ τ

x(t) σ(t)

t

¯t IPt A¯d Q−1 σ Ad IP

x(t) σ(t)

t

σ t (s)Qσ σ(s) ds

+

(6.21)

t−τ

Therefore from (6.19)-(6.21) we get: t x(t) x(t) t ¯ V˙ x ≤ IPt AoKd + AtoKd IP + τ IPt A¯d Q−1 IP A d σ σ(t) σ(t)

t σ t (s)Qσ σ(s) ds (6.22) + t−τ

It is straightforward to show that, V˙ σ = τ σ t (t)Qσ σ(t) −

0

−τ

σ t (t + θ)Qσ σ(t + θ) dθ

(6.23)

By using (6.13)-(6.14) and (6.22), it follows that V˙ t = V˙ x + V˙ σ t x(t) x(t) t t ¯ IPt AoKd + AtoKd IP + τ IPt A¯d Q−1 = IP + τ E Q E A 2 σ σ d 2 σ(t) σ(t) t x(t) x(t) ∆ (6.24) Πo = σ(t) σ(t) Let χ(t) = [xt (t) σ t (t)]t . Considering Πo and using (6.13)-(6.14), some matrix t manipulations convert it to LMIs (6.15). In view of the fact that CK CK > 0, ˙ it follows that Πo < 0 is guaranteed and we conclude that Vt < 0 for all χ(t) = 0 and V˙ t ≤ 0 for all χ(t). Since ||χ(t + β)|| ≤ ϕ||χ(t)|, ∀β ∈ [−τ, 0] t 2 ¯ and some ϕ > 0 [53], it follows from (6.18) that Vt (.) ≤ χ (t)P χ(t) + µ||χ|| t ¯ where µ = ϕ τ maxi λM [P ] + τ λM [E2 Qσ E2 ] . Therefore, for all χ = 0, we have χt Πo χ V˙ t (.) λm [−Πo ] ∆ (6.25) ≤ t¯ ≤ −ξ = − Vt (.) λM [P¯ ] + µ χ P χ + µ||χ||2 ∆

It is readily seen from (6.25) that ξ > 0 and hence, on letting Vt (0) = Vt (x = xo ) we get V˙ t ≤ −ξ Vt , which implies Vt (.) ≤ e−ξ t Vt (0)

(6.26)

Therefore system (ΣDK ) is exponentially stable at a rate ξ > 0 which completes the proof. ∇∇∇

176


Remark 6.2. It should be noted that Theorem 6.1 establishes an LMI-based sufficient condition for the existence of state-feedback controller (6.7) and hence it depends on Ko . The following theorem provides a method for computing the feedback gains Theorem 6.3. The feedback gain associated with the H2 -optimal controller for system (ΣDK ) is given by Ko = Z Yx−1 where the matrices Yx = Yxt > 0, Yσ = Yσt > 0, Yd = Ydt > 0, satisfy the system of LMIs   Λ1 Λ2 τ Ydt Yxt Cot + Z t Dot  • −Λ3 τ Yσt  0   < 0 (6.27)  • • −τ Qσ  0 • • • −I where Λ1 = Ydt + Yd , Λ2 = Yσ − Ydt + Yxt Atod + Z t Bot Λ3 = Yσt + Yσ − τ Ad Q−1 σ Ad

(6.28)

Proof: By the Schur complements, condition (6.24) is equivalent to t ¯ t ¯t CK + τ IPt A¯d Q−1 AtoKd IP + IPt AoKd + C¯K σ Ad IP + τ E2 Qσ E2 < 0 (6.29)

Using (6.13)-(6.14), premultiplying (6.29) by Y t = IP−t and postmultiplying the result by Y , we obtain: t ¯ t t ¯t CK Y + τ A¯d Q−1 Y t AtoKd + AoKd Y + Y t C¯K σ Ad + τ Y E2 Qσ E2 Y < 0 (6.30)

Expanding inequality (6.30) with the help of (6.11) and using Ko Yx = Z, LMIs (6.27) subject to (6.28) follow immediately. ∇∇∇ 6.3.2 Resilient H2 Design In the course of controller implementation, it turns out that the controllers are very sensitive with respect to errors in the controller coefficients [43]. The sources for this include, but not limited to, imprecision in analog-digital conversion, fixed word length, finite resolution instrumentation and numerical roundoff errors. By means of several examples, it is demonstrated [43] that relatively small perturbations in controller parameters could even destabilize the closed loop system. Such controllers are often termed ”fragile”. Hence, it is considered beneficial that the designed (nominal) controllers should be capable of tolerating some level of controller gain variations [16]. This illuminates the controller fragility problem for which some relevant results are available in [38] and further effort to alleviate this problem can also be found in [118, 121]. Looked at in this light, we consider the actual (implemented) control law be u(t) = [Ko + ∆K] x(t) , ∆K = H∆c (t)E

(6.31)


177

where ∆c (t) is uncertainty matrix represented by ∆

∆c (t) ∈ ∆C (t) = {∆c (t) : ∆tc (t)∆c (t) ≤ I}

(6.32)

The closed-loop system becomes: (ΣDK ) : x(t) ˙ = σ(t) 0 = −σ(t) + A∆Kd x(t) − Ad

t

σ(s)ds + Γ w(t) (6.33) t−τ (t)

y(t) = C¯K x(t)

(6.34) (6.35)

where A∆Kd = Ao + Bo Ko + Ad + Bo H∆c E = Aod + Bo Ko + Bo H∆c E CˆK = Co + Do Ko + Do H∆c E (6.36) Extending on Theorem 6.3 for system (ΣDK ) and by the Schur complements, it follows that a sufficient condition for exponential stability can be expressed t ˆ t ¯t At∆Kd IP + IPt A∆Kd + CˆK CK + τ IPt A¯d Q−1 σ Ad IP + τ E2 Qσ E2 < 0 (6.37)

Inequality (6.37) holds if and only if the following inequality t ¯t At IP + IPt AtoKd + τ IPt A¯d Q−1 σ Ad IP + τ E2 Qσ E2   oKd −1 −1 t (ε + )E E+ 0 t  0, > 0. Using (6.13)-(6.14), premultiplying (6.29) by Y t = IP−t and postmultiplying the result by Y , we obtain: t t ¯t Y t AtoKd + AoKd Y + τ A¯d Q−1 σ Ad + τ Y E2 Qσ E2 Y   (ε−1 + −1 )E t E+ 0 t t Y < 0 CK [I − Do HH t Dot ]CK +Y t t • εBo HH Bo

(6.39)

The following summarizes the main result. Theorem 6.4. The perturbed feedback gain (6.31) associated with the H2 optimal controller for system (ΣDK ) is given by Ko = Z Yx−1 for some scalars ε > 0, > 0. where the matrices Yx = Yxt > 0, Yσ = Yσt > 0, Yd = Ydt > 0, satisfy the system of LMIs

178




Yxt Cot t t t εYd Bo H Y  Λ1 Λ2 Do +Z   • −Λ3 τ Yσt 0 εYσt Bo H 0   • • −τ Qσ 0 0 0   • • • −I 0 0   • • • • −εI 0   • • • • • −Λ e   • • • • • • τ Ydt

Yxt Cot +Z t Dot 0 0 0 0 0 −I+ Do HH t Dot

        < 0      

(6.40)

for all admissible gain perturbations, where Λe = [εI,

I] , Y = [Yxt E t ,

Yxt E t ]

Proof: Upon expanding inequality (6.39) with the help of (6.11) and applying Inequality I.3 of the Appendix, then we use Ko Yx = Z. Thus LMIs (6.40) follows immediately. ∇∇∇ 6.3.3 Nominal H∞ Design In the sequel, for a prescribed γ > 0, we introduce the performance measure ∞ ∆ J∞ = {z t (s)z(s) − γ 2 wt (s)w(s)} ds (6.41) 0

The problem of H∞ state-feedback control could be phrased as follows: Given system ΣJ determine the control law (6.7) which guarantees that H∞ performance measure (6.41) is bounded by γ for all w(t) ∈ L2 [0, ∞] The following two theorems summarize the main results pertaining to the H∞ performance: Theorem 6.5. Given a prescribed scalar γ > 0. State feedback controller (6.7) renders system (ΣDK ) exponentially stable with a disturbance attenuation level γ for all w(t) ∈ L2 [0, ∞) if, given matrix sequence Qx = Qtx > 0 there exist matrices Px = Pxt > 0, Pσ = Pσt > 0, Pd = Pdt > 0 satisfying the system of LMIs   ¯t Φ Ω1 + GtK GK Ω2 τ Pdt Ad IPt Γ¯ + G K  • −Ω3 τ Pσt Ad 0 ∆   < 0 (6.42) Πh =    0 • • −Qσ • • • −γ 2 I + Φt Φ Moreover

||z(t)|| ≤ γ 2 ||w(t)||2 + xt (0)Px (ηo )x(0)

+ τo

0

−τo

1/2 σ t (s)Qσ (ηo )σ(s)ds

(6.43)


179

Proof: The exponential stability follows as a result of Theorem 6.1. To show that system (ΣDKo ) has a disturbance attenuation γ, we let the Lyapunov functional V (t) be given by (6.18). Using (6.13)-(6.14) and (6.24), we get t t ¯ t t −1 ¯t t ˙ Vh = χ (t) IP AoKd + AoKd IP + τ IP Ad Qσ Ad IP + τ E2 Qσ E2 χ(t) + χt (t)IPΓ¯ w(t) + wt (t)Γ¯ t IPχ(t)

(6.44)

Γ ¯ ¯ K = [GK 0] Γ = , G 0

where

It follows from (6.44) that: z t (t)z(t) − γ 2 wt (t)w(t) + V˙ h   t IPt AoKd + AtoKd IP t ¯ ¯ χ(t) χ(t)  IP Γ + G Φ t ¯ −1 t t K ¯  = +τ IP Ad Qσ Ad IP + τ E2 Qσ E2 w(t) w(t) • −γ 2 I + Φt Φ t χ(t) χ(t) ∆ (6.45) = Πe w(t) w(t) By the Schur complements, it is easy to show from (6.42) that Πe < 0. Therefore from (6.41), we have: J∞ ≤ Vh (x0 ) =⇒ 2

2

2

||z(t)|| − γ ||w(t)|| ≤

xt0 Px x0

0

+τ −τ

φt (β)Qx φ(β)dβ =⇒

||z(t)||2 ≤ γ 2 ||w(t)||2 + xt0 Px x0

0 +τ φt (β)Qx φ(β)dβ

(6.46)

−τ

which completes the proof.

∇∇∇

Theorem 6.6. Given a prescribed constant γ > 0. The feedback gain associated with the H∞ -controller for system (ΣDK ) is given by Ko = Z Yx−1 where the matrices Yx = Yxt > 0, Yσ = Yσt > 0, Yd = Ydt > 0 , satisfy the system of LMIs   Λ1 Λ2 τ Ydt Yxt Gto + Z t Fot Γ + Y t Gto Φ  • −Λ3 τ Yσt  0 0    • • −τ Qσ  < 0 0 0 (6.47)    • •  • −I 0 • • • • −γ 2 I + Φt Φ Proof: Follows from parallel development to Theorem 6.3.

∇∇∇

180


6.3.4 Resilient H∞ Design Following the procedure of resilient H2 design, we develop hereafter a tractable method for resilient H∞ design. The following summarizes the main result. Theorem 6.7. Given a prescribed scalar γ > 0, the perturbed feedback gain (6.31) associated with the H∞ -optimal controller for system (ΣDK ) is given by Ko = Z Yx−1 for some scalars ε > 0, > 0. where the matrices Yx = Yxt > 0, Yσ = Yσt > 0, Yd = Ydt > 0, satisfy the system of LMIs   t t Γ+ Yxt Cot + t Yx Go + t εYd Bo H Y   Λ1 Λ2 τ Yd Z t Dot Z t Fot Y t Gto Φ   t t   • −Λ3 τ Yσ B H 0 0 0 0 εY σ o     • • −τ Qσ 0 0 0 0 0     • • • −I 0 0 0 0   2  < 0 (6.48)  −γ I+   • • • • 0 0 0 t   Φ Φ     • • • • • −εI 0 0     • • • • • • −Λ 0 e     −I+ • • • • • • • Do HH t Dot for all admissible gain perturbations Proof: Upon expanding inequality (6.47) with the help of (6.11) and (6.32)(6.34) in the manner of Theorem 6.4 and using Ko Yx = Z, LMI (6.48) follows immediately. ∇∇∇ 6.3.5 Simultaneous Nominal H2 /H∞ Design Having developed two separate nominal design methods, one is obviously tempted to combine both methods in one way or another. This can successfully be achieved by a simultaneous H2 /H∞ design approach. Adopting one approach, the problem of simultaneous H2 /H∞ state-feedback control could be phrased as follows: Given system ΣJ determine a control law of the type (6.7) which achieves the minimal value of H2 performance measure while guaranteeing that H∞ performance measure is bounded by γ for all w(t) ∈ L2 [0, ∞]. Remark 6.8. It should be observed that the objective of the simultaneous H2 /H∞ feedback control under consideration is to minimize the energy of the output y(t) while simultaneously satisfying the prescribed H∞ -norm bound of the controlled system w(t) → z(t). By exploiting the benefits of the foregoing results, the solution to the simultaneous nominal H2 /H∞ control problem is established by the following theorem:


181

Theorem 6.9. Given a prescribed scalar γ > 0. The feedback controller (6.7) with gain Ko = Z Yx−1 is a simultaneous H2 /H∞ controller satisfying the performance measure (6.41) for system (ΣDK ) if there exist matrices Y = Y t > 0, Z, L = Lt > 0, W = W t > 0, such the system of generalized eigenvalue problems min λ + T r(W )

−L I I −Qx

subject to (6.27), (6.47) and −λ φt (0) −W X t 0, W = W t > 0 such that the system of generalized eigenvalue problems min λ + T r(W )

−L I I −Qx

subject to (6.40), (6.48) and −λ φt (0) −W X t 0, Pd = Pdt > 0, satisfying the system of LMIs   Ω1j + CPt K CP K Ω2j τ Pdt Adj ∆ • −Ω3j τ Pσt Adj  < 0 , j ∈ IN (6.62) Πt =  • • −Qσ where Ω1j = Pdt AP Kd + AtP Kd Pd , Ω2j = Pxt − Pdt + AtP Kd Pσ Ω3j = Pσt + Pσ − τ Qσ

(6.63)

An upper bound on the H2 performance measure is given by

∆ J2 ≤ J+ = xt (0)Px x(0) + τo

0

−τo

1/2 σ t (s) Qσ σ(s) ds

(6.64)

Theorem 6.12. The feedback gain associated with the H2 -optimal controller for the polytopic system (ΣP K ) is given by Ko = Z Yx−1 where the matrices Yx = Yxt > 0, Yσ = Yσt > 0, Yd = Ydt > 0, satisfy the system of LMIs   t t Λ1 Λ2j τ Ydt Yxt Coj + Z t Doj  • −Λ3j τ Y t  0 σ   < 0 , j ∈ IN (6.65)  •  0 • −τ Qσ • • • −I where t Λ2j = Yσ − Ydt + Yxt Atodj + Z t Boj , Λ3j = Yσt + Yσ − τ Adj Q−1 σ Adj (6.66)

186


Theorem 6.13. The perturbed feedback gain (6.31) associated with the H2 optimal controller for the polytopic system (ΣP K ) is given by Ko = Z Yx−1 for some scalars ε > 0, > 0. where the matrices Yx = Yxt > 0, Yσ = Yσt > 0, Yd = Ydt > 0, satisfy the system of LMIs   t t Yxt Coj Yxt Coj t t t εYd Boj H Y t  Λ1 Λ2j τ Yd +Z t Doj  +Z t Doj   t  • −Λ3j τ Y t  0 εY B H 0 0 σ σ oj    •  • −τ Q 0 0 0 0 σ    •  < 0 (6.67) • • −I 0 0 0    •  • • • −εI 0 0    •  • • • • −Λ 0 e     −I+ • • • • • • t Doj HH t Doj for all admissible gain perturbations 6.4.2 Polytopic H∞ Design The main design results are summarized by the following theorems: Theorem 6.14. Given a prescribed scalar γ > 0. State feedback controller (6.7) renders the polytopic system (ΣP K ) exponentially stable with a disturbance attenuation level γ for all w(t) ∈ L2 [0, ∞) if, given matrix sequence Qx = Qtx > 0 there exist matrices Px = Pxt > 0, Pσ = Pσt > 0, Pd = Pdt > 0 satisfying the system of LMIs   ¯t Φ Ω1j + GtP K GP K Ω2j τ Pdt Adj IPt Γ¯ + G PK  • −Ω3j τ Pσt Adj 0 ∆   < 0 (6.68) Πh =    0 • • −Qσ • • • −γ 2 I + Φt Φ Moreover

||z(t)|| ≤ γ 2 ||w(t)||2 + xt (0)Px (ηo )x(0)

+ τo

0

−τo

1/2 t

σ (s)Qσ (ηo )σ(s)ds

(6.69)

Theorem 6.15. Given a prescribed constant γ > 0. The feedback gain associated with the H∞ -controller for system (ΣP K ) is given by Ko = Z Yx−1 where the matrices Yx = Yxt > 0, Yσ = Yσt > 0, Yd = Ydt > 0 , satisfy the system of LMIs   t Γ + Y t Gtoj Φ Λ1 Λ2j τ Ydt Yxt Gtoj + Z t Foj  • −Λ3j τ Y t  0 0 σ    •  < 0 (6.70) • −τ Qσ 0 0    •  • • −I 0 • • • • −γ 2 I + Φt Φ

6.4 Continuous Polytopic Systems

187

Theorem 6.16. Given a prescribed scalar γ > 0, the perturbed feedback gain (6.31) associated with the H∞ -optimal controller for system (ΣP K ) is given by Ko = Z Yx−1 for some scalars ε > 0, > 0. where the matrices Yx = Yxt > 0, Yσ = Yσt > 0, Yd = Ydt > 0, satisfy the system of LMIs   t Yxt Coj Yxt Gtoj Γ+ t t t   Λ1 Λ2j τ Yd +Z t F t oj Y t Gt ojΦ εYd Boj H Y +Z t Doj   t   • −Λ3j τ Y t 0 0 εY B H 0 0 σ σ oj     • • −τ Q 0 0 0 0 0 σ     • • • −I 0 0 0 0   2 t   • • • • −γ I + Φ Φ 0 0 0     • • • • • −εI 0 0     • 0 • • • • • −Λ e     −I+ • • • • • • • t Doj HH t Doj (6.71) ;< 0

(6.72)

for all admissible gain perturbations 6.4.3 Simultaneous Polytopic H2 /H∞ Design By similarity to the nominal design methods, we provide below the polytopic counterpart of the simultaneous H2 /H∞ design approach. Keeping the same pace, the main results are summarized by the following theorems: Theorem 6.17. Given a prescribed scalar γ > 0. The feedback controller (6.7) with gain Ko = Z Yx−1 is a simultaneous H2 /H∞ controller satisfying the performance measure (6.41) for system (ΣP K ) if there exist matrices Y = Y t > 0, Z, L = Lt > 0, W = W t > 0, such the system of generalized eigenvalue problems min λ + T r(W )

subject to (6.65), (6.70) and −λ φt (0) −W X t −L I 0, W = W t > 0 such that the system of generalized eigenvalue problems

188


min λ + T r(W )

subject to (6.67), (6.72) and −L I −λ φt (0) −W X t 0

(6.87)

j=k−d

We observe that Vk has a general form and is constructed from three terms: Vk,a signifies necessary and sufficient conditions for the stability of discrete descriptor system without delay [117], Vk,b corresponds to delay-dependent criteria [81] and Vk,c is common for delay-independent stability conditions (see Chapter 3). For simplicity in exposition, we introduce the following matrix expressions: Z Y W 0 Wq = , ≥ 0 , Z ∈ IR2n×2n , Y ∈ IR2n×n (6.88) • Q • d+ Q We now address Problem A. The following theorem establishes LMI-based sufficient conditions for asymptotic stability of system (6.77). Theorem 6.22. Consider system (6.77) without uncertainties ∆k ≡ 0 with the delay factor dk = d being an unknown constant satisfying 0 ≤ dk ≤ d+ . Given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , this system

192


is asymptotically stable if there exist matrices 0 < IP = IPt ∈ IR2n×2n , Y ∈ IR2n×n , and Z ∈ IR2n×2n satisfying (6.88) and the LMI   A¯tξ IPA¯ξ − E t IPE+ t ¯ ¯ Y − Aξ IPAd  ∆ < 0 (6.89) Υ (d+ ) =  Wq + d+ Z + YE + E t Y t • −W + A¯t IPA¯d d

Proof: Let ζkt = [xtk 0] , πk = xk − xk−dk

(6.90)

We consider Vk and evaluate the first difference of the functionals Vk,a , Vk,b and Vk,c . Thus using (6.80), we have t E t IPEζk+1 − ζkt E t IPEζk Vk+1,a − Vk,a = ζk+1 t = ζk+1 IPζk+1 − ξkt E t IPEξk t k−1 k−1 ¯ ¯ yj IP A¯ξ ξk − A¯d = Aξ ξk − Ad j=k−dk

yj

j=k−dk

− ξkt E t IPEξk t ¯t t ¯ = ξk Aξ IPAξ − E IPE ξk + πkt A¯td IPA¯d πk k−1

− 2

ξkt A¯tξ IPA¯d yj

(6.91)

j=k−d

Similarly k−1

Vk+1,b − Vk,b = d+ ykt Qyk − =

ξkt

yjt Qyj

j=k−d+

0 0 ξ − 0 d+ Q k

k−1

yjt Qyj

(6.92)

j=k−d+

Vk+1,c − Vk,c = xtk Wxk − xtk−d Wxk−d W0 ξ − xtk−d Wxk−d = ξk 0 0 k

(6.93)

It follows from (6.90) through (6.93) that Vk+1 − Vk = ξ t A¯t IPA¯ξ − E t IPE + Wq ξk + π t A¯t IPA¯d πk k

− 2

ξ

k−1

k

d

ξkt A¯tξ IPA¯d yj

j=k−d

− xtk−d Wxk−d −

k−1 j=k−d+

yjt Qyj

(6.94)

6.5 Discrete-Time Systems

193

Using Lemma 8.6 of the Appendix with IN = A¯tξ IPA¯d , β = yj , α = ξk we obtain the following −2

k−1

k−1

ξkt A¯tξ IPA¯d yj ≤

j=k−d

j=k−d k−1

=

ξk yj

t

yjt Qyj

Z Y − A¯tξ IPA¯d • Q

+d

+

2ξkt Zξk

ξk yj

t ¯ ¯ + 2 Y − Aξ IPAd xk

j=k−d

−2

ξkt

t ¯ ¯ Y − Aξ IPAd xk−d

(6.95)

Using (6.95) into (6.94), applying Schur complements and arranging terms we get t ξk ξk Υ (d+ ) Vk+1 − Vk = xk−d xk−d Based on LMI (6.89), we infer that Vk+1 − Vk < 0 and more importantly from (6.86)-(6.88), it follows that Vk ≥ 0 and Vk+1 − Vk < −κ|xk |2 , ∀k ≥ 0. By Theorem 6.21, the asymptotic stability is therefore guaranteed. ∇∇∇ Remark 6.23. Consider the nominal delayless system xk+1 = Aod xk , x0 = ψ0

(6.96)

obtained from (6.77) by setting ∆k ≡ 0 , dk ≡ 0 , uk ≡ 0. If |λ(Ao + Ad )| < 1 implying that system (6.96) is asymptotically stable and then, equivalently stated, its descriptor form ξk+1 = A¯ξ ξk would be asymptotically stable, that is, the LMI A¯tξ IPA¯ξ − E t IPE < 0 has a feasible solution IP > 0. This is equally true for all possible and sufficiently small d+ > 0. Since the results of Theorem 6.22 depend on the bound d+ , the stability condition is termed weakly-delay dependent. In the special case of Z = υ I , Q = υ I , Y = υ[0, I] , υ −→ 0 then LMI (6.89) reduces to t ¯ IPA¯ξ − E t (IP − W)E Atξ IPA¯d ∆ A ξ Υ = < 0 , IP > 0 • −W + A¯td IPA¯d which provides an LMI characterization of delay-independent stability.

(6.97)

194


6.5.3 Robust Stability Had we considered system (6.77) and applied the descriptor model transformation, we have obtained the descriptor system E ξk+1

= A¯∆ξ ξk − A¯∆d

k−1

yj

(6.98)

j=k−dk

where

I I 0 0 I = + Aod − I −I M ∆k [Na + Nd ] 0 A∆od − I −I ¯ ¯ ¯ ¯ = Aξ + M ∆k Nad , Nad = [Na + Nd 0] = [Nad 0] ¯ ∆k Nd , M ¯ t = [0 M t ] (6.99) = A¯d + M

A¯∆ξ = A¯∆d

I

Considering Theorem 6.22, it follows that the uncertain system (6.77) is asymptotically stable if given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , there exist matrices 0 < IP = IPt ∈ IR2n×2n , Y ∈ IR2n×n , and Z ∈ IR2n×2n satisfying (6.88) and the LMI   A¯t∆ξ IPA¯∆ξ − E t IPE+ ¯t IPA¯∆d Y − A ∆ ∆ξ  < 0 (6.100) Υ∆ =  Wq + d+ Z + YE + E t Y t t ¯ ¯ • −W + A∆d IPA∆d In terms of (6.99), we rewrite the LMI (6.100) in the form:   −E t IPE + Wq + Y  ∆ Υ∆ =  d+ Z + YE + E t Y t • −W t t ¯ ¯ A∆ξ N t ¯t ad IP + ∆k M IP IP−1 + Ndt −A¯t∆d t t t t ¯ A¯∆ξ ¯ ∆k Nad IP < 0 + IP M t ¯ Ndt −A∆d

(6.101)

The robust stability result is now established by the following theorem: Theorem 6.24. Consider system (6.77) with the delay factor dk = d being an unknown constant satisfying 0 ≤ dk ≤ d+ . Given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , this system is robustly asymptotically stable if there exist matrices 0 < IP = IPt ∈ IR2n×2n , Y ∈ IR2n×n , and Z ∈ IR2n×2n and scalar > 0, satisfying (6.88) and the LMI   −E t IPE + Wq + ¯ t Nd A¯t IP 0   d+ Z + YE + E t Y t Y + N ad ξ   t ¯ ¯   +Nad Nad   < 0 (6.102) t ¯  • −W + Nd Nd −Ad IP 0    ¯  • • −IP IPM • • • −I


195

Proof: Starting from (6.101) and using Inequality 2 of the Appendix with some matrix arrangement, we can obtain the matrix bound Υ∆ ≤   −E t IPE + Wq + ¯ t Nd  d+ Z + YE + E t Y t Y + N  A¯tξ IP ad   t ¯ ¯   < 0 (6.103) +Nad Nad   t ¯   −Ad IP • −W + Nd Nd t −1 ¯ ¯ • • −IP + IPM M IP for some > 0. The latter part corresponds to (6.102) by a simple Schur complement operation. ∇∇∇ Extending on Remark 6.23, we have the following result Theorem 6.25. Consider system (6.77) with the delay factor dk = d being an unknown constant satisfying 0 ≤ dk ≤ d+ . Given a matrix 0 < W = W t ∈ IRn×n , this system is robustly asymptotically stable independent of delay if there exist matrices 0 1, > 0 satisfying the LMI   −E t (IP − W)E t t ¯ ¯ N IP 0 N A ad d ξ ¯t N ¯  +N  ad ad   t t ¯  • −W + Nd Nd −Ad P 0  (6.104)   < 0 ¯  • • −IP IPM • • • −I 6.5.4 Nominal Feedback Stabilization In the sequel, we consider the state stabilization problem for system (6.77). Initially, we use the nominal feedback controller uk = Ko xk

(6.105)

Applying controller (6.105) to system (6.75) with ∆k ≡ 0 yields the nominal closed-loop system xk+1 = [Ao + Bo Ko ]xk + Ad xk−dk , x0 = ψ0 = AKo xk + Ad xk−dk

(6.106)

With reference to (6.80), it is readily evident that the corresponding descriptor system matrix takes the form I I ¯ AKξ = , AKod = AKo + Ad (6.107) AKod − I −I Now, given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , it follows from Theorem 6.22 that system (6.106) is asymptotically stable if

196


there exist matrices 0 < IP = IPt ∈ IR2n×2n , Y ∈ IR2n×n , and Z ∈ IR2n×2n satisfying (6.88) and the LMI   A¯tKξ IPA¯Kξ − E t IPE+ t ¯ ¯ Y − AKξ IPAd  ∆ ΥK (d+ ) =  Wq + d+ Z + YE + E t Y t < 0 (6.108) • −W + A¯td IPA¯d Performing a Schur complement operation, to  −E t IPE + Wq + +  d Z + YE + E t Y t Y   • −W • •

LMI (6.108) becomes equivalent  t ¯ AKξ IP   < 0 −A¯td IP  −IP

(6.109)

The feedback stabilization result is readily established by the following theorem: Theorem 6.26. Consider system (6.75) with ∆k ≡ 0 and the delay factor dk = d being an unknown constant satisfying 0 ≤ dk ≤ d+ . Given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , this system is asymptotically stable if there exist matrices 0 < ILx = ILtx ∈ IRn×n , , 0 < ILd = ILtd ∈ IRn×n , ILf ∈ IRn×n , ILa ∈ IRn×n , ILb ∈ IRn×n , ILg ∈ IRn×n , ILh ∈ IRn×n , IBx ∈ IRm×n , IBd ∈ IRm×n , IMx ∈ IRn×n , IMd ∈ IRn×n satisfying the LMI   ILx (Atod − I) −IL IL + IM IL IL IL x f x a x g   +IBtx Bot   t   ILf (Aod − I)  •  −IM IL IL + IL IL d b f d h  t t  −ILd + IBd Bo   < 0 (6.110)  • 0  • −W 0 −Atd    • • • −ILx 0 0     • −ILd 0  • • −ILf • • • • • −IHx Moreover, the nominal gain is given by Ko = IBx IL−1 x Proof: On observing that −1 IPx 0 ∆ ILx 0 IL = = , ILx = IP−1 x ILf ILd IPf IPd −1 ILd = IP−1 d , ILf = −IPd IPf IPx

We perform the congruent transformation C I = diag[IP−t , on the LMI (6.109) using

I, IP−t ]


Z=

Zx 0 0 Zd

, Y=

Yx Yd

197

and manipulating with the help of (6.107), we finally arrive at:   IL Ξ IL ILY ILA¯tKξ  • −W −A¯td  < 0 • • −IL

(6.111)

where Ξ = −E t IPE + Wq + d+ Z + YE + E t Y t Ydt −IPx + W + d+ Zx + Yx + Yxt = • d+ (Q + Zd )

(6.112)

Introducing the linearizations IMx = ILx Ydt ILd , IHx = W + d+ Zx + Yx + Yxt ILg = ILx IHx , ILh = ILf IHx t t t IMd = ILf IL−1 x ILf − ILf IHx ILf − ILd Yd ILf − ILf Ydt ILd − ILd d+ [Q + d+ Zd ]ILd

(6.113)

it is readily seen that  −ILx ILf + IMx ILg −IMd ILh  IL Ξ IL =  • • • −IHx 

and taking into consideration that ILx Yx ∆ ILa = IL Y = ILf Yx + ILd Yd ILb

(6.114)

(6.115)

then by performing some matrix manipulations of (6.111) using (6.112)(6.115), the LMI (6.110) is readily obtained. ∇∇∇ Had we considered the uncertain case, ∆k = 0, we would have obtained the closed-loop system xk+1 = A∆Ko xk + Ad xk−dk , x0 = ψ0 A∆Ko = [AKo + M ∆Nk ]

(6.116)

In a similar way, the descriptor system matrix takes the form I I 0 0 I I ¯ = + A∆Kξ = AKod − I −I M ∆[Na + Nd ] 0 A∆Kod − I −I ¯ ∆k N ¯ad , AKod = Aod + Bo Ko (6.117) = A¯Kξ + M

198


By Theorem 6.24, it follows that system (6.116) is asymptotically stable if given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , there exist matrices 0 < IP = IPt ∈ IR2n×2n , Y ∈ IR2n×n , and Z ∈ IR2n×2n and scalar > 0, satisfying (6.88) and the LMI   −E t IPE + Wq + ¯ t Nd A¯t IP 0   d+ Z + YE + E t Y t Y + N ad Kξ   t ¯ ¯   + N N ad ad   < 0 (6.118) t t ¯  • −W + Nd Nd −Ad IP 0    ¯  • • −IP IPM • • • −I Applying Schur complements convert LMI (6.118) into the form   −E t IPE + Wq + t t ¯ ¯ AKξ IP Nad   d+ Z + YE + E t Y t Y    • −W −A¯td IP Ndt    < 0 (6.119) ¯M ¯ t IP 0   • • −IP + −1 IPM • • • −I The robust feedback stabilization result is readily established by the following theorem: Theorem 6.27. Consider system (6.75) with the delay factor dk = d being an unknown constant satisfying 0 ≤ dk ≤ d+ . Given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , this system is robustly asymptotically stable if there exist matrices 0 < ILx = ILtx ∈ IRn×n , , 0 < ILd = ILtd ∈ IRn×n , ILf ∈ IRn×n , ILa ∈ IRn×n , ILb ∈ IRn×n , ILg ∈ IRn×n , ILh ∈ IRn×n , IBx ∈ IRm×n , IBd ∈ IRm×n , IMx ∈ IRn×n , IMd ∈ IRn×n and scalars µ > 0 satisfying the LMI   ILx (Atod − I) t ILx ILg ILx Nad   −ILx ILf + IMx ILa +IBtx Bot   t   ILf (Aod − I) t   • −IMd ILb ILf + ILd ILh ILf Nad  t t  + IB B −IL d d o    • 0 Ndt  • −W 0 −Atd   < 0 (6.120)  •  • • −IL 0 0 0 x   t  •  −IL + µM M 0 0 • • −IL f d    • 0  • • • • −IHx • • • • • • −µI Moreover, the nominal gain is given by Ko = IBx IL−1 x Proof: By a parallel development to Theorem 6.26 we apply the congruent transformation C I = diag[IP−t , I, IP−t , I] on the LMI (6.119) using the block partitions of Z, Y and IL = IP−1 and manipulating, we finally arrive at:


 t IL Ξ IL ILY ILA¯tKξ ILNad  • −W −A¯td Ndt   < 0   • • −IL 0  • • • −I

199



(6.121)

where (6.112)-(6.113) have been incorporated. The substitution of (6.114)(6.115) into LMI (6.121) subject to the constraint µ = 1 with some Schur complements operation yields LMIs (6.120) . ∇∇∇ 6.5.5 Resilient Feedback Stabilization In line of the goals of this monograph, we consider the practical case that the controller gain Ko is subject to perturbations due to various technical reasons. Thus the actual implemented control law would be uk = [Ko + ∆Ko ]xk = [Ko + H∆c Nc ]xk

(6.122)

which implies that the gain perturbations are of additive type. Extension to multiplicative type can be derived in the manner of the foregoing chapters. In (6.122), H ∈ IRm×αk and Nc ∈ IRβk ×n are real and known constant matrices ∆ with ∆c is a bounded matrix of uncertainties such that ∆c ∈ ∆c = {∆tc : ∆c ∆c < I , ∀ k}. Applying controller (6.122) to system (6.75) yields the perturbed closedloop system xk+1 = [Ao + M ∆k Na ]xk + Bo [Ko + H∆c Nc ]xk + [Ad + M ∆k Nd ]xk−dk = [AKo + M ∆k Na + Bo H∆c Nc ] = A¯∆Ko xk + A∆d xk−dk , x0 = ψ0 The perturbed descriptor system matrix now takes the form I I ˆ A∆Kξ = A¯∆Kod − I −I I I 0 0 0 = + + AKod − I −I M ∆[Na + Nd ] 0 Bo H∆Nc ¯ ∆k N ¯ cN ¯ad + H∆ ¯c = A¯Kξ + M ¯c = [Nc 0] ¯ t = [0 H t B t ] , N H o

(6.123)

0 0

(6.124)

Again, by Theorem 6.24, it follows that system (6.123) is asymptotically stable if given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , there exist matrices 0 < IP = IPt ∈ IR2n×2n , Y ∈ IR2n×n , and Z ∈ IR2n×2n and scalars > 0, ψ > 0 satisfying (6.88) and the LMI

200




 −E t IPE + Wq + ¯ t Nd A¯t IP 0 IPM ¯  d+ Z + YE + E t Y t Y + N ad Kξ   ¯t N ¯ tN ¯ad + ψ N ¯c  +N  c ad   t t ¯   < 0 N − A IP 0 0 • −W + N d d d   ¯ 0   • • −IP IP M    • • • −I 0  • • • • −ψI

(6.125)

Applying Schur complements convert LMI (6.125) into the form   −E t IPE + Wq + ¯ t IPH ¯  d+ Z + YE + E t Y t Y A¯tKξ IP N ad   t ¯ ¯   +ψ N N c c   t ¯t IP   < 0 (6.126) • −W − A N 0 d d   −1 t ¯ ¯  0  • • −IP + IPM M IP 0    • • • −I 0  • • • • −ψI The resilient feedback stabilization result is readily established by the following theorem: Theorem 6.28. Consider system (6.75) with the delay factor dk = d being an unknown constant satisfying 0 ≤ dk ≤ d+ and subject to the perturbed feedback control (6.122). Given matrices 0 < Q = Qt ∈ IRn×n , 0 < W = W t ∈ IRn×n , this system is robustly asymptotically stable if there exist matrices 0 < ILx = ILtx ∈ IRn×n , , 0 < ILd = ILtd ∈ IRn×n , ILf ∈ IRn×n , ILa ∈ IRn×n , ILb ∈ IRn×n , ILg ∈ IRn×n , ILh ∈ IRn×n , IBx ∈ IRm×n , IBd ∈ IRm×n , IMx ∈ IRn×n , IMd ∈ IRn×n and scalars µ > 0, ψ > 0 satisfying the LMI   ILx (Atod − I) t ILx ILg ILx Nad 0   −ILx ILf + IMx ILa +IBtx Bot   t   ILf (Aod − I) t  • ILb ILf + ILd ILh ILf Nad Bo H  −IMd t t   + IB B −IL d d o   t t  •  • −W 0 −A 0 N 0 d d   1 The following result holds Theorem 6.30. Consider system (6.77) without uncertainties ∆k ≡ 0. and the delay factor dk = d being an unknown time-varying satisfying 0 ≤ dk ≤ ∆ d+ and ∆k ∈ ∆k = {∆tk : ∆k ∆k 1 satisfying the LMI t Ato PAd ∆ Ao PAo − ϑP < 0 (6.128) Υˆ = • Atd PAd − (1 − ϑ)P/σ Proof: Evaluation of the first forward-difference of Vk along the solutions of system (6.77) without uncertainties yields: Vk+1 − Vk = [xtk Ato + xtk−dk Atd ]P[Ao xk + Ad xk−dk ] − xtk Pxk = xtk [Ato PAo − ϑP]xk + 2xtk−dk Atd PAo + xtk−dk Atd PAd xk−dk − (1 − ϑ)xtk Pxk t xk xk ˆ ≤ Υ xk−dk xk−dk

(6.129)


203

In view of (6.128) we conclude that Vk+1 − Vk < 0, which implies the asym∇∇∇ ptotic stability of system (6.77) with ∆k ≡ 0. Extending on Remark 6.23 in the mannar of Theorem 6.25, we have the following result Theorem 6.31. Consider system (6.77) with the delay factor dk = d being ∆ an unknown time-varying satisfying 0 ≤ dk ≤ d+ and ∆k ∈ ∆k = {∆tk : ∆k ∆k 1, > 0 satisfying the LMI   t t Nad Nad Nd Ato P 0 −ϑP + Nad  • −(1 − ϑ)P/σ + Ndt Nd −Atd P 0   < 0 (6.130)   • • −P PM  • • • −I On considering the feedback stabilization problems, we establish the following results Theorem 6.32. Consider system (6.75) without uncertainties ∆k ≡ 0. and the delay factor dk = d being an unknown time-varying satisfying 0 ≤ dk ≤ ∆ d+ and ∆k ∈ ∆k = {∆tk : ∆k ∆k 1 satisfying the LMI   −ϑX 0 X Ato + Λt Bot  < 0  • −(1 − ϑ)X /σ X Atd (6.131) • • −X Moreover the feedback gain is given by Ko = Λ X −1 . Proof: Consider the LMI (6.128) with Ao → Ao + Bo Ko . Performing Schur complements operation and applying the congruent transformation C I t = diag[X , X , X ], X = P −1 with Ko = Λ X −1 we obtain LMI (6.131) as desired.

∇∇∇

Theorem 6.33. Consider system (6.75) with the delay factor dk = d being ∆ an unknown time-varying satisfying 0 ≤ dk ≤ d+ and ∆k ∈ ∆k = {∆tk : ∆k ∆k 1, > 0 satisfying the LMI   −ϑX + X Ato + t Nad Nd 0   N t Nad Λt Bot ad     −(1 − ϑ)X /σ t  • X Ad 0  (6.132) t   < 0 +N N d d    • • −X M  • • • −I

204


Moreover the feedback gain is given by Ko = Λ X −1 . Proof: Follows from Theorem 6.32 and Theorem 6.33. ∇∇∇ Finally, we consider that the actual feedback controller is (6.122). The resilient feedback stabilization results are summarized by the following theorems. Theorem 6.34. Consider system (6.75) without uncertainties ∆k ≡ 0. subject to the perturbed control (6.122) and the delay factor dk = d being an ∆ unknown time-varying satisfying 0 ≤ dk ≤ d+ and ∆k ∈ ∆k = {∆tk : ∆k ∆k 1, ψ > 0 satisfying the LMI   −ϑX 0 X Ato + Λt Bot X Nct  • −(1 − ϑ)X /σ 0  X Atd   < 0 (6.133)  • • −X + ψBo HH t Bot 0  • • • −ψI Moreover the feedback gain is given by Ko = Λ X −1 . Proof: Follows from a development parallel to Theorem 6.32 taking into account the additive gain perturbations in (6.122) . ∇∇∇ Theorem 6.35. Consider system (6.75) subject to the perturbed control (6.122) with the delay factor dk = d being an unknown time-varying satisfying ∆ 0 ≤ dk ≤ d+ and ∆k ∈ ∆k = {∆tk : ∆k ∆k 1, ψ > 0, > 0 satisfying the LMI   −ϑX + X Ato + t t N N 0 X N c ad d  N t Nad Λt Bot ad     −(1 − ϑ)X /σ t   • X A 0 0 t d   +Nd Nd   < 0 (6.134)   −X +   • • M 0 t t   ψBo HH Bo    • • • −I 0  • • • • −ψI Moreover the feedback gain is given by Ko = Λ X −1 . Proof: Follows from Theorem 6.33 and Theorem 6.34. 6.5.10 Example 6.8 Consider the following nominal discrete-time system 0 0.5 −0.6 0 Ao = , Ad = 0.5 0.3 0 0.02

∇∇∇


205

In the case of constant delay factor, application of Remark 6.23 shows that this system is delay-independently stable. In the case of time-varying delay, application of Theorem 6.30 with ϑ = 0.55, σ = 1.05 shows that this system is asymptotically stable for 0 < dk ≤ 3. We take into account the gain perturbations with H = 1 , Nc = [0.2 0.8]. 6.5.11 Example 6.9 Consider the discrete-time system of the type (6.77) having the following parameters 0.88 0.77 0.07 0.11 0.45 , Ad = , Bo = Ao = 0.95 0.88 0.09 0.12 0.5 0.1 0.1 0.12 0.15 0.07 0.6 M = , Na = , Nd = 0.2 0.1 0.17 0.2 0.05 0.08 We take into account the gain perturbations with H = 1 , Nc = [0.2 0.8]. A summary of the implementation of the LMI-based feedback design is summarized in Table 6.1 Table 6.1. Computational Results of Example 6.9 Theorem 6.26 6.27 6.28 6.32 6.33 6.33 6.33

Ko -1.8335 -1.7714 -1.7988 -1.8115 -1.8009 -1.7268 -1.8124 -1.7645 -1.7869 -1.8017 -1.8375 -1.7517 -1.8045 -1.7315

Delay d+ = 5 d+ = 7 d+ = 12 0 < dk ≤ 6 0 < dk ≤ 8 0 < dk ≤ 10 0 < dk ≤ 14

6.6 Notes and References With focus on resilience control issues with delay-dependent methodologies for continuous-time and discrete-time systems, our experience shows that this topic is overlooked in the literature on time-delay systems. In this Chapter, we have established some new results in this direction. Certainly, there are several problems to be addressed including a recently developed method [47]. The topic of discrete-time systems deserves more attention since a lot of continuous-time results do not simply carry over. Different gain perturbation patterns should be attempted to reach a reasonable representation.

7 Resilient Control-Nonlinear Systems

7.1 Introduction In control design of dynamical systems, it turns out that the design objectives have to incorporate the impact of parameter shifting, component and interconnection failures which are frequently occurring in practical situations. Robust control theory provides tractable design tools using the time domain and the frequency domain when the plant modeling uncertainty or external disturbance uncertainty is of major concern in control systems. When implementing the designed controllers, issues and problems associated with the available precision and finite computational capabilities has called for consideration of redesign procedures. Such procedures have been addressed in the foregoing chapters for classes of linear time-delay systems using various techniques. In this Chapter, we direct attention to classes of nonlinear continuoustime and discrete-time systems with state-delay. We develop an LMI-based analysis and design procedures to check primarily into the robust stability of both continuous-time and discrete-time systems. Then we address the robust stabilization using nominal and resilient feedback designs. In both cases the trade-off between the size of the controller gains and the bounding factors is illuminated and incorporated into the design formalism. Seeking computational convenience, all the developed results are cast in the format of linear matrix inequalities (LMIs) and several numerical examples are presented throughout the chapter.

7.2 Nonlinear Continuous-Time Systems We consider a class of nonlinear time-delay systems described by: x(t) ˙ = Ao x(t) + Ad x(t − d) + Bo u(t) + h(t, x, x(tψ− d)) x(s) = φ(s),ψ t ψ∈ [−d, 0]

(7.1)


208


where x(t) ∈ IRn is the state vector; u(t) ∈ IRm is the control input; d ∈ [d0 , d∗ ] is an unknown constant delay factor with known bound d∗ and Ao , Bo , Ad are known real constant matrices of appropriate dimensions. In contrast to all system representation previously discussed in this book, system (7.1) has a nonlinear term h(., ., .) : IR2n+1 −→ IRn . In the sequel, we assume that h(t, x, x(t − d)) is a piecewise-continuous function in its arguments. We note that piecewise-continuity of h(t, ., .) implies the same property of the right hand side Ao x(t) + Ad x(t − d) + h(t, x, x(t − d)) of system (7.1) with u ≡ 0, and their domains of continuity D coincide. In order to explore the dynamical behavior and stability properties of system (7.1), we have to speak more about the nonlinear function h(t, x, x(t − d)). In the discussions to follow, we assume that this function is uncertain and all what we know is that, in the domains of continuity D, it satisfies the quadratic inequality ht (t, x, x(t − d)) h(t, x, x(t − d)) ≤ α2 xt Hot Ho x + σ 2 xt (t − d)Hdt Hd x(t − d) (7.2) where α > 0, σ > 0 are the bounding parameters and Ho ∈ IRr×n , Hd ∈ IRp×n are constant matrices. It is crucial to observe at this stage that for any given Ho , Hd , inequality (7.2) furnishes a class of piecewise-continuous functions H = {h : IR2n+1 −→ IRn | ht h ≤ α2 xt Hot Ho x + σ 2 xt (t − d)Hdt Hd x(t − d) ⊂ D} ∆

(7.3)

over the domain of continuity. Obviously, the class H include functions that satisfy h(t, 0, 0) = 0 the domains of continuity D and that the point x = 0 is an equilibrium of system (7.1) for d ≥ 0. It is readily evident that when d = 0, the origin is an equilibrium point. When d > 0, we recall [55] that ||x(t − d)|| ≤ π||x(t)||, t ∈ (−d, 0), π > 0.

7.3 Robust Stability In this section, our objective is to establish tractable conditions guaranteeing stability of the origin (x = 0). In view of the crucial role of the delay factor d, we divide our effort along two directions: the first deals with delay-independent stability and the second considers delay-dependent stability. 7.3.1 Robust Delay-Independent Stability Towards our goal, we recall the following definition : Definition 7.1. System (7.1) with u ≡ 0, is robustly delay-independent stable if the equilibrium x = 0 is globally asymptotically stable for all h(t, x, x(t − d)) ∈ H.

7.3 Robust Stability

209

It is important to note that Definition 7.1 is similar to the standard definition of absolute stability [45] applied to time-delay systems. It is important to note that we do not assume here any structure of dependence of h(t, x, x(t−d)) on the state x or the delayed state x(t−d). Rather the function h(t, x, x(t−d)) depends on both the state x and the delayed state x(t − d) and not the output of the system. To examine the robust delay-independent stability in the sense of Definition 7.1, we introduce a Lyapunov-Krasovskii function of the form

t xt (s) Q x(s) ds (7.4) V (x) = xt (t)P x(t) + t−d

where 0 < P = P t ∈ IRn×n , 0 < Q = Qt ∈ IRn×n . We assume that V (x) is a C1 function satisfying φ1 (||x||) ≤ V (x) ≤ φ2 (||x||) , ∀x ∈ IRn

(7.5)

where φ1 , φ2 ∈ K∞ . Using the well-known results of Lyapunov theory, we can establish robust delay-independent stability in the sense of Definition 7.1 by negative definiteness of the Lyapunov derivative V˙ (x) ≤ −φ3 ||x|| ∈ D when evaluated along the trajectories of system (7.1). Since our vehicle throughout the book is to base the solvability conditions on the LMI approach and hence exploit its main computational features, we start with the basic observation that the bounding constraint (7.2) is equivalent to the quadratic inequality     t  −α2 Hot Ho 0 0 x(t) x(t)  x(t − d)   • −σ 2 Hdt Hd 0   x(t − d)  ≤ 0 (7.6) • • I h h Next, we compute the derivative V˙ (x) along the trajectories of system (7.1) to get: V˙ (x) = xt (t)[P Ao + Ato P + Q]x(t) + 2xt (t)P h(., ., .) + 2xt (t)P Ad x(t − d) − xt (t − d)Qx(t − d) (7.7) and for stability, we require for a given Q > 0 P > 0 , xt (t)[P Ao + Ato P + Q]x(t) + 2xt (t)P h(., ., .) + 2xt (t)P Ad x(t − d) − xt (t − d)Qx(t − d) < 0 (7.8) Equivalently stated P > 0 ,     t  P Ao + Ato P + Q P Ad P x(t) x(t)  x(t − d)   • −Q 0   x(t − d)  < 0 h • • 0 h

(7.9)

210


On using the S-procedure [9] we conclude that, when (7.6) is satisfied, inequalities (7.9) are equivalent to the existence of a matrix 0 0 such that P > 0 ,   P Ao + Ato P + Q P A P d  +ωα2 Hot Ho    < 0 t 2  • −Q + ωσ Hd Hd 0  • • −ωI

(7.10)

which, in turn, with the linearization W = ω −1 Q, X = ωP −1 is equivalent to the existence of a matrix 0 < X = X t satisfying the inequality X > 0 ,   Ao X + XAto + XW X A I d   +α2 XHot Ho X   < 0 2 t  • −W + σ Hd Hd 0  • • −I

(7.11)

By resorting to the Schur complements, inequalities (7.11) with α−2 = γ, σ −2 = ψ can be cast into the form X > 0 ,   Ao X + XAto XW XHot I Ad   • −W 0 0 0     < 0 • • −γI 0 0     • • • −I 0 t • • • • −W + ψHd Hd

(7.12)

This is a pleasing robust stability result. In practice, we would like to establish robust delay-independent stability for as large a H as possible. One attractive way to achieve this is by first selecting the matrices Ha , Hd and then attempt to maximize the parameters α, σ by solving an LMI problem in X, γ, ψ. Based thereon, we are led to state the following result. Theorem 7.2. System (7.1) with u ≡ 0, is robustly delay-independently stable if the following problem minimize γ , ψ subject to Y > 0  Ao X + XAto XW XHot  • −W 0   • • −γI   • • • • • •

 I Ad  0 0   < 0 0 0   −I 0 • −W + ψHdt Hd

is feasible The following example illustrates a numerical application

(7.13)


211

7.3.2 Example 7.1 Consider the uncertain time-delay system 0 1 0.1 −0.1 , Ad = Ao = −2 −3 0.2 −0.2 1 0.3 h=ϑ x+ x(t − d) , d+ = 3 1 0.4 On solving the LMI problem (7.13) using MATLAB software with the input data 1.5 0.2 W = , Ho = I , Hd = 2I 0.2 0.8 we get the bounds α = 0.7645 , along with

X =

σ = 0.3825

1.3455 0.2143 0.2143 0.5477

which verifies Theorem 7.2. Next, we direct attention to the other type of robust stability 7.3.3 Robust Delay-Dependent Stability To deal with delay-dependent stability, we have to transform the system into an appropriate form that exhibits the necessary dynamics. Among the different transformation methods [81], the Newton-Leibniz method will be adopted. Towards our objective, we express the delayed state with u ≡ 0 as:

t x(t − d) = x(t) − x(t ˙ + θ)d θ −d t

= x(t) −

−

−d

Ao x(t + θ)d θ −

t

−d

Ad x(t + θ − d)d θ

t

−d

h(s, x, x(t − d))ds

(7.14)

Substituting (7.14) back into (7.1) u ≡ 0, we get: t

Ao x(t + θ)d θ − x(t) ˙ = (Ao + Ad )x(t) − Ad

−

t

−d

h(s, x, x(t − d))ds

−d

t

−d

Ad x(t + θ − d)d θ (7.15)

Needless to stress that λ((Ao + Ad )) < 0 is a necessary condition for robust stability. Now we introduce a Lyapunov-Krasovskii function of the form

212


t

Vd (x) = xt (t)P x(t) +

t

xt (s) Q x(s) ds + W1 + W2 + W3 t−d

t

r1 [xt (s) Ato Ao x(s)] dsd θ

W1 = t−d t

t+θ t

t−d

t

t+θ−d

t

r2 [xt (s) Atd Ad x(s)] dsd θ

W2 =

r3 ht (s, ., .) h(s, ., .) dsd θ

W3 = t−d

(7.16)

t+θ

where 0 0, r2 > 0, r3 > 0 . Note that Vd (x) builds on V (x) by adding the quadratic terms W1 , W2 and W3 . More on their role will be discussed later on. Now by differentiating Vd (x) along the trajectories of system (7.15) and manipulating, it yields: t t ˙ Vd (x) = x (t) P (Ao + Ad ) + (Ao + Ad ) P + Q x(t)

− 2xt P Ad − 2xt P Ad

t

Ao x(t + θ)d θ

−d

t −d

t

− 2xt P Ad + d

Ad x(t + θ − d)d θ h(s, x, x(t − d))ds + d r3 ht h

−d r1 xt (t)Ato Ao x(t)

+ d r2 xt (t)Atd Ad x(t)

− xt (t − d)Qx(t − d)

(7.17)

Applying Inequality 1 of the Appendix, it follows that

t

t −1 t − 2x P Ad Ao x(t + θ)d θ ≤ r1 [xt (t)P Ad Atd P x(t)]d θ

+ r1

−d

t

−d

t

+ r1

−d

− 2x P Ad

xt (t + θ)Ato Ao x(t + θ)d θ = d r1−1 xt (t)P Ad Atd P x(t) xt (t + θ)Ato Ao x(t + θ)d θ

t

t

+ r2

t

−d

t

+ r2

−d

− 2x P Ad

−d

−d

Ad x(t + θ − d)d θ ≤

(7.18) r2−1

t

−d

[xt (t)P Ad Atd P x(t)]d θ

xt (t + θ − d)Atd Ad x(t + θ − d)d θ = d r2−1 xt (t)P Ad Atd P x(t) xt (t + θ − d)Atd Ad x(t + θ − d)d θ

t

t

−d

dθ ≤

r3−1

t

−d

[xt (t)P Ad Atd P x(t)]d θ

(7.19)


+ r3 + r3

t

−d

t

213

ht (t, ., .)h(t, ., .)d θ = d r3−1 xt (t)P Ad Atd P x(t) ht (t, ., .)h(t, ., .)d θ

(7.20)

−d

Next, by combining (7.17)-(7.20), we arrive at t ˙ Vd (x) = x (t) P (Ao + Ad ) + (Ao + Ad )t P + Q + d r1 Ato Ao + d r2 Atd Ad

+ d [r1−1 + r2−1 + r3−1 ]P Ad Atd P x(t) + d r3 ht h − xt (t − d)Qx(t − d)   t  x(t) x(t) =  x(t − d)  Ω  x(t − d)  h h   P (Ao + Ad ) + (Ao + Ad )t P + Q  +d r1 Ato Ao + d r2 Atd Ad 0 0    −1 −1 −1 t   Ω =  +d [r1 + r2 + r3 ]P Ad Ad P   • −Q 0  • • d r3 I

(7.21)

If V˙ d (x) < 0 when x = 0 (implying x(t − d) = 0, h = 0), then x(t) → 0 as t → ∞ and therefore the time-delay system (7.1) is globally asymptotically stable with delay-dependent. For a given Q > 0, the stability requirement is expressed as P > 0 ,   P (Ao + Ad ) + (Ao + Ad )t P + Q  +d r1 Ato Ao + d r2 Atd Ad 0 0     < 0  +d [r−1 + r−1 + r−1 ]P Ad At P 1 2 3 d    • −Q 0  • • d r3 I

(7.22)

On using the S-procedure [9] we conclude that, when (7.6) is satisfied, inequalities (7.22) are equivalent to the existence of a matrix 0 0 such that

214


P > 0 ,   P (Ao + Ad ) + (Ao + Ad )t P + Q   +d r1 Ato Ao + d r2 Atd Ad  0 0   +d [r−1 + r−1 + r−1 ]P Ad At P  1 2 3 d   2 t   +ωα H H o o   < 0   −Q+  • 0  2 t   ωσ H H d d    −ωI+  • • d r3 I

(7.23)

Using the linearizations Y = P −1 , R = P −1 QP −1 , ε1 = (d r1 )−1 , ε2 = (d r2 )−1 ε3 = (d [r1−1 + r2−1 + r3−1 ])−1 , γ = ω −1 α−2 ψ = ω −1 σ −2 , ε4 = (ω − d r3 ) and resorting to the Schur complements operation, inequalities (7.23) are equivalent to the existence of matrices 0 < Y = Y t , 0 < R = Rt and scalars ε1 > 0, ε2 > 0, ε3 > 0, ε4 > 0, such that Y > 0 , R >0 ,  (Ao + Ad )Y + t t t  Y (Ao + Ad )t Y Ao Y Ad Ad Y Ho   • −ε1 I 0 0 0   • • −ε I 0 0 2   • • • −ε3 I 0   • • • • −γI   • • • • •   • • • • • • • • • •

 0

0

0

  0 0 0   0 0 0   0 0 0   < 0 0 0 0   −R Y Hdt 0   • −ψI 0  • • −ε4 I

(7.24)

Once again, we would like to establish robust delay-dependent stability for as large a H as possible. By similarity to the delay-independent case, one attractive way to achieve this is by selecting the matrices Ha , Hd and subsequently attempt to maximize the parameters α, σ by solving an LMI problem in Y, R, γ, ψ, ε1 , ..., ε4 . The result is summarized by the following theorem. Theorem 7.3. System (7.1) with u ≡ 0, is robustly delay-dependently stable for any d ∈ [0, d∗ ] if the following problem minimize subject to

γ, ψ Y > 0, R > 0, ε1 > 0, ε2 > 0, ε3 > 0, ε4 > 0




(Ao + Ad )Y + t t t  Y (Ao + Ad )t Y Ao Y Ad Ad Y Ho   • −ε1 I 0 0 0   • • −ε I 0 0 2   • • • −ε3 I 0   • • • • −γI   • • • • •   • • • • • • • • • •

215

 0

0

0

  0 0 0   0 0 0   0 0 0   < 0 0 0 0   −R Y Hdt 0   • −ψI 0  • • −ε4 I

(7.25)

is feasible Remark 7.4. The motivation behind the Lyapunov-Krasovskii functional (7.16) is purely technical in order to absorb some terms appearing in the Lyapunov derivative later on. This technique is quite standard [45] and the basic role is played by the scalars r1 , r2 and r3 . On the other hand, note that while system (7.1) has initial condition over [−d, 0], system (7.15) requires initial data on [−2d, 0]. Interestingly enough, Theorem 7.3 provides robust stability results for any constant time-delay d satisfying 0 ≤ d ≤ d∗ where d∗ is a delay bound determined by the onset of the in feasibility of LMIs (7.25). We note illustrate the numerical computation of Theorem 7.3. 7.3.4 Example 7.2 Consider the uncertain system −3 −2 0 0.3 1 0.6 Ao = , Ad = , h=ϑ x+ x(t − d) 1 0 −0.3 −0.2 1 0.4 Note that λ(Ao + Ad ) = {−0.7225, −24775}. On solving the LMI problem (7.25) using Ho = 1.5I , Hd = 2I, we get the feasible solution 0.3797 −0.3741 1.1324 0.1298 Y = , R= , d∗ = 2 −0.3741 0.8855 0.1298 0.3758 ε1 = 0.2314 , ε2 = 11.2435 , ε3 = 4.1166 , ε4 = 0.2745 γ = 0.6995 , ψ = 0.4315 which means that the time-delay system is robustly stable for constant delay satisfying d ∈ [0, 0.7105].

216


7.4 Robust Stabilization In this section, we extend the results of the foregoing stability section to state feedback stabilization. We achieve our objective in two-stages. In the first stage, we treat nominal feedback design in which the gain is implemented accurately. Later on, in the second stage, we direct attention on resilient feedback design with consideration to gain perturbations. 7.4.1 Nominal Feedback Design It is well-known that [45] when the linear part of system (7.1) with u ≡ 0 is not stable we can introduce feedback to stabilize the overall system and, at the same time, maximize its tolerance to uncertain nonlinear perturbations. W elaborate on this fact hereafter. Let the linear state feedback be given by u(t) = Ko x(t)

(7.26)

where Ko ∈ IRm×n is a constant gain matrix. We assume that the pair (Ao , Bo ) is stabilizable. The application of control law (7.26) to system (7.1) yields the closed-loop system x(t) ˙ = (Ao + Bo Ko )x(t) + Ad x(t − d) + h(t, x, x(t − d)) x(s) = φ(s), t ∈ [−d, 0]

(7.27)

for which the following definition holds Definition 7.5. System (7.27) is robustly delay-independent stabilized by control law (7.26) if the closed-loop system (7.27) is robustly stable. A direct application of Theorem 7.2 leads to the following optimization problem minimize γ , ψ subject to X > 0   (Ao + Bo Ko )X+ t Ad  X(Ao + Bo Ko )t XW XHo I      • −W 0 0 0   < 0   • • −γI 0 0     • • • −I 0 • • • • −W + ψHdt Hd

(7.28)

To standardize inequality (7.28), we introduce the slack variable Ko X = M or Ko = M X −1 to convert it into the format of convex optimization problems over LMIs:

7.4 Robust Stabilization

minimize γ , ψ subject to X > 0 , M  Ao X + XAto + t  Bo M + M t Bot XW XHo   • −W 0   • • −γI   • • • • • •

217

 I

Ad

   0 0  < 0  0 0   −I 0 • −W + ψHdt Hd

(7.29)

We have just established the following theorem Theorem 7.6. System (7.1) is robustly delay-independently stabilized by control law (7.26) if the LMI problem (7.29) has a feasible solution. A numerical application of LMI (7.29) is in order. 7.4.2 Example 7.3 Consider the uncertain time-delay system 0.1 −0.1 0 0 1 Ao = , Bo = , Ad = 0.2 −0.2 1 −2 −3 1 0.3 h=ϑ x+ x(t − d) , d+ = 3 1 0.4 which is the same like Example 7.1. On solving the LMI problem (7.29) using Ho = I , Hd = 2I, we get the bounds α = 0.9565 , with

σ = 0.6125

Ko = 106 [−11.2565,

−0.1346]

which, despite the fact that it verifies Theorem 7.6, the resulting gain is relatively of high magnitude. In an attempt to alleviate this issue, we use in another simulation Bot = [1, 1] while retaining all data as before. The results are summarized by α = 2.8914 , with

σ = 2.1745

Ko = 104 [−7.3715,

Continuing our effort further with Bo =

20 02

3.1772]

218


it turns out that the ensuing bounds are α = 0.91664 × 105 , σ = 3.2344 × 104 with 3

Ko = 10

−4.9817 0.1211 0.3122 −5.6724

The results of Example 7.3 illuminates an important issue concerning the size of the feedback gains viz-a-vis the bounding factors. One possible approach to overcome this problem is to change the underlying design procedure so as to include bounds the gain matrix Ko in the form Kot Ko < κ I This condition, from a technical standpoint, corresponds to constraints on the component matrices M and X −1 . A candidate choice to realize this would be by setting Mt M < µ I , µ > 0

,

X −1 < ϕ I , ϕ > 0

In turn, these are equivalent to the LMIs −µ I M t −ϕ I I < 0 , < 0 • −I • −X

(7.30)

Apparently, we have achieved our goal since Kot Ko = X −1 M t M X −1 < µ X −1 X −1 < µ ϕ2 I = κ I ∆

Finally, in order to guarantee desired values {¯ α, σ ¯ } of the bounding factors {α, σ}, we recall that α−2 = γ, σ −2 = ψ . Thus we require 1 1 < 0 , ψ − 2 < 0 α ¯2 σ ¯ To sum up, the foregoing modifications pave the way to constructing the following convex optimization problem over LMIs γ −

minimize γ + ψ + µ + ϕ subject to Y > 0 , M   Ao X + XAto + t XW XH I A d o  Bo M + M t Bot      • −W 0 0 0   < 0   • • −γI 0 0     • • • −I 0 t • • • • −W + ψHd Hd 1 1 γ − 2 < 0 , ψ − 2 < 0 α ¯ ¯ σ −µ I M t −ϕ I I < 0 , < 0 • −I • −X The following theorem summarizes the main result

(7.31)


219

Theorem 7.7. System (7.1) is robustly delay-independently stabilized by control law (7.26) with constrained feedback gains if the LMI problem (7.31) has a feasible solution. 7.4.3 Example 7.4 Consider the uncertain time-delay system 0 1 0.1 −0.1 20 , Ad = , Bo = Ao = −2 −3 0.2 −0.2 02 1 0.3 h=ϑ x+ x(t − d) , d+ = 2 , Ho = I , Hd = 2I 1 0.4 which is the same data like Example 7.3. On solving the LMI problem (7.31) using the bounds α ¯ = 6.75 , σ ¯ = 4.65 We obtain the gain Ko =

−12.3526 0.4277 0.8873 −8.9354

which verifies Theorem 7.7. 7.4.4 Resilient Feedback Design In the sequel, we address the performance deterioration issue by considering that the actual linear state-feedback controller has the form u(t) = [Ko + ∆Ko ] x(t)

(7.32)

where Ko ∈ IRm×n is a constant gain matrix and ∆Ko is an additive gain perturbation matrix represented by ∆

∆Ko = Mc ∆c Nc , ∆c ∈ ∆c = {∆c : ∆tc ∆c ≤ I}

(7.33)

As before, we assume that the pair (Ao , Bo ) is stabilizable. The application of control law (7.32) to system (7.1) yields the perturbed closed-loop system x(t) ˙ = (Ao + Bo Ko + Bo ∆Ko )x(t) + Ad x(t − d) + h(t, x, x(t − d)) x(s) = φ(s), t ∈ [−d, 0]

(7.34)

It follows by applying Theorem 7.2 to system (7.34), we obtain the following problem

220


minimize γ , ψ subject to Y > 0   (Ao + Bo Ko + Bo ∆Ko )X+ t Ad   X(Ao + Bo Ko + Bo ∆Ko )t XW XHo I     • −W 0 0 0  < 0 (7.35)    • • −γI 0 0     • • • −I 0 • • • • −W + ψHdt Hd over all possible perturbations ∆c ∈ ∆c . To standardize inequality (7.35) and bypass the exhaustive search over the perturbation set ∆c , we start by focusing on the main inequality of (7.35) and manipulating with the aid of Inequality 1 of the Appendix to reach   (Ao + Bo Ko )X + X(Ao + Bo Ko )t XW XHot I Ad   • −W 0 0 0     + • • −γI 0 0     • • • −I 0 t • • • • −W + ψHd Hd   Bo Mc  0     0  ∆c [Nc X, 0, 0, 0, 0] +    0  0   XNct  0     0  ∆tc [Mct Bot , 0, 0, 0, 0] ≤    0  0   (Ao + Bo Ko )X+   X(Ao + Bo Ko )t + XW XHot I Ad    ηXNct Nc X + η −1 Bo Mc Mct Bot      < 0 (7.36) • −W 0 0 0     • • −γI 0 0     • • • −I 0 t • • • • −W + ψHd Hd Then we introduce the slack variables Ko X = M, Z = ηXNct which will convert (7.36) into the LMI format:


221





Ao X + XAto + t Bo Mc Z  Ad  Bo M + M t Bot XW XHo I    • −W 0 0 0 0 0     • • −γI 0 0 0 0   < 0   • • • −I 0 0 0      • • • • −W + ψHdt Hd 0    • • • • • −ηI 0  • • • • • • −ηI

(7.37)

To summarize, we have just established the following theorem Theorem 7.8. System (7.1) is robustly delay-independently stabilized by control law (7.32) for all possible gain variations if the following convex optimization problems: minimize γ , ψ, η subject to X > 0 , M,  Ao X + XAto + t  Bo M + M t Bot XW XHo   • −W 0   • • −γI   • • •   • • •   • • • • • •

Z

 Bo Mc Z   0 0 0 0   0 0 0 0   < 0 (7.38) −I 0 0 0   0  • −W + ψHdt Hd 0  • • −ηI 0  • • • −ηI I

Ad

has a feasible solution. On the other hand, by taking the modifications made to constrain the feedback gains we reach the following convex optimization problem over LMIs minimize γ + ψ + µ + ϕ subject to X > 0 , M, Z   Ao X + XAto + t Ad Bo Mc Z   Bo M + M t Bot XW XHo I    • −W 0 0 0 0 0     • • −γI 0 0 0 0    < 0 (7.39)  • • • −I 0 0 0     • • • • −W + ψHdt Hd 0 0     • • • • • −ηI 0  • • • • • • −ηI 1 1 γ − 2 < 0 , ψ − 2 < 0 α ¯ σ ¯ −ϕ I I −µ I M t < 0 , < 0 • −I • −X In this regard, the following theorem summarizes the main result

222


Theorem 7.9. System (7.1) is robustly delay-independently stabilized by the actual control law (7.32) for possible perturbations ∆c ∈ ∆c with constrained feedback gains if the LMI problem (7.39) has a feasible solution. A numerical application of LMIs (7.38) and (7.39) is in order. 7.4.5 Example 7.5 Consider the uncertain time-delay system       0 1 0 0.1 −0.1 −0.5 200 Ao =  0 0 1  , Ad =  0.2 −0.2 −0.8  , Bo =  0 5 0  −3 −4 −5 0.6 −0.7 −1 003     1 0.3 h = ϑ  1  x +  0.4  x(t − d) , d+ = 2 , Ho = I , Hd = I 0 0.2     0.3 1 Mc =  0.8  , Nct =  0.4  0.5 1.2 On solving the convex optimization problem (7.38) using MATLAB software we obtain the gain   −7.2355 0.4277 −0.2985 Ko =  6.4387 −2.7411 0.3976  , ||Ko || = 9.9989 , α = 2.9855 1.0645 −0.4776 −1.2357 σ = 1.1477 , η = 0.7578 which verifies Theorem 7.8. On the other hand, by solving the convex optimization problem (7.39) we obtain the gain   −4.5571 0.3245 −0.3015 Ko =  4.4873 −0.9877 0.3976  , ||Ko || = 6.9407 , α = 3.0145 2.4533 −0.6468 −1.8352 σ = 1.1532 , η = 0.7729 which verifies Theorem 7.9. A simple comparison shows that the feasible results of the convex optimization problem (7.39) given a less magnitude of the feedback gain than those of the convex optimization problem (7.38) and the bounding factors are almost the same. Remark 7.10. It is significant to observe that all the results developed thus far can be carried over for the class of nonlinear continuous-time systems with convex polytopic uncertainties. In such case, the corresponding state-space matrices contain uncertainties represented by a real convex bounded polytopic model of the type

7.5 Nonlinear Discrete-Time Systems





Ao ∆  Ad  ∈ Sλ = Bo









N Aoj A(oλ)  A(dλ)  = λj  Adj  , λ ∈ Λ j=1 B(oλ) Boj

223

where Λ is the unit simplex N ∆ λ j = 1 , λj ≥ 0 Λ = (λ1 , · · · , λN ) :

(7.40)

(7.41)

j=1

Define the vertex set IN = {1, ..., N }. In short, we use {Ao , · · · , Bo } to imply generic system matrices and {Aoj , · · · , Boj , j ∈ IN} to represent the respective values at the vertices. Thus we simply make the substitutions Ao −→ Aoj , Bo −→ Boj , Ad , −→ Adj in all the developed LMIs and adds the condition ”over the set IN”. A typical example would be Theorem 7.2. It should read in this case Theorem 7.11. Consider system (7.1) with u ≡ 0 and the system matrices have the polytopic representation (7.40). This system is robustly delayindependently stable if the following problem minimize γ , ψ subject to X > 0  t Aoj X + XAtoj XW XHoj  • −W 0   • • −γI   • • • • • •

 I Adj  0 0   < 0 0 0   −I 0 t • −W + ψHdj Hdj

(7.42)

is feasible over the vertex set IN.

7.5 Nonlinear Discrete-Time Systems In this section, we focus attention on the development of design results for a class of nonlinear discrete-time systems. The ensuing results are the discrete counter-part of the foregoing sections. Consider the following class of discretetime state-delay systems: xk+1 = Ao xk + Bo uk + Ad xk−dk + g(k, xk , x(t − dk ) , x0 = ψ0 (7.43) ∆

where for k ∈ Z+ = {0, 1, ....}, xk ∈ IRn is the state vector; uk ∈ IRp is the control input; dk is a positive number representing the delay such that dk ≤ d+ and the matrices Ao ∈ IRn×n ; Bo ∈ IRn×p Ad ∈ IRn×n , are real and known constant matrices. The function g : Z+ × IR2n → IRn is a nonlinear perturbations. We assume that the matrix Ao is Schur stable, that is it has all of its eigenvalues inside the unit circle.

224


Remark 7.12. The class of systems (7.43) emerges in many areas dealing with the applications functional difference equations or delay-difference equations [35] while preserving the nonlinear character of the models. These applications appear in cold rolling mills [51] and decision-making of manufacturing systems [112]. It should be stressed that, in line with our rationale in this book, we consider only the case of single-delay since multiple-delay systems can be easily handled using the augmentation procedure in [90]. To examine the dynamical behavior and stability properties of system (7.43), we have to provide a characterization of the nonlinear function g(t, x, x(t − d)). In the sequel, we assume that this function is uncertain and the available information is that, in the domains of continuity D, it satisfies the quadratic inequality g t (k, xk , xt−dk ) g(k, xk , xt−dk ) ≤ δ 2 xtk Gto Go xk + ν 2 xtk−dk Gtd Gd xk−dk

(7.44)

which can be conveniently written as   t  2 t −δ Go Go xk xk 0 0  xk−d   • −ν 2 Gtd Gd 0   xk−d  < 0 • • I g g 

(7.45)

where δ > 0, ν > 0 are the bounding parameters and Go ∈ IRr×n , Gd ∈ IRp×n are constant matrices. At this stage it is crucial to observe that for any given Go , Gd , inequality (7.44) furnishes a class of piecewise-continuous functions G = {g : IR2n+1 −→ IRn | g t g ≤ δ 2 xtk Gto Go xk + ν 2 xtk−dk Gtd Gd xk−dk ⊂ D} ∆

(7.46)

over the domain of continuity. The class G include functions that satisfy g(t, 0, 0) = 0 the domains of continuity D and that the point x = 0 is an equilibrium of system (7.43) for dk ≥ 0. It is readily evident that when dk = 0, the origin is an equilibrium point. When dk > 0, we recall [55] that ||xk−dk || ≤ π||xk ||, t ∈ (−dk , 0), π > 0.

7.6 Robust Stability Our goal now is to establish tractable conditions guaranteeing stability of the origin (x = 0). In order to streamline ideas we concentrate our effort on delay-independent stability in which we consider dk = d as unknown constant satisfying 0 ≤ dk ≤ d+ . Towards our goal, we recall the following definition:


225

Definition 7.13. System (7.43) with uk ≡ 0, is robustly delay-independent stable if the equilibrium x = 0 is globally asymptotically stable for all g(k, xk , xk −dk ) ∈ G. To derive tractable conditions for stability, we introduce the LyapunovKrasovskii functional Vk = xtk P xk +

k−1

xtk Qxk

(7.47)

j=k−d

(7.48) where 0 < P = P t ∈ IRn×n , 0 < Q = Qt ∈ IRn×n are weighting matrices. We ∆ consider Vk and evaluate the first difference ∆Vk = Vk+1 − Vk . Thus using (7.43), we have ∆Vk = xtk+1 P xk+1 − xtk P xk + xtk Qxk − xtk−d Qxk−d = xtk Ato P Ao − P + Q xk + xtk Ato P g(k, xk , xk−d ) + g t (k, xk , xk−d )P Ao xk − xtk−d Qxk−d + g t (k, xk , xk−d )P g(k, xk , xk−d ) + xtk Ato P Ad xk−d + xtk−d Atd P Ao xk + g t (k, xk , xk−d )P Ad xk−d + xtk−d Atd P g(k, xk , xk−d )   t  t  Ao P Ao − P + Q Ato P Ad Ato P xk xk ∆ • −Q + Atd P Ad Atd P   xk−d  =  xk−d   • • P g g

(7.49)

Since a sufficient condition of stability is ∆Vk < 0, it follows that  t  t  Ao P Ao − P + Q xk Ato P Ad Ato P xk  xk−d   • −Q + Atd P Ad Atd P   xk−d  < 0 (7.50) • • P g g 

By resorting to the S-procedure [9], inequalities (7.45) and (7.50) can be rewritten together as P > 0 , ω ≥ 0  t  Ao P A o − P + Q t t A P A A P d o o  +ωδ 2 Gto Go    t   < 0 −Q + Ad P Ad t   • A P t 2 d   +ων Gd Gd • • P − ωI

(7.51)

226


which describes non-strict LMIs since ω ≥ 0. However, recalling the known result [9] that minimization under non-strict LMIs corresponds to the same result as minimization under strict LMIs when both strict and non-strict LMI constraints are feasible. Moreover, if there is a solution for (7.51) for ω = 0, there will be also a solution for some ω > 0 and sufficiently small δ, ν. Therefore, we safely replace ω ≥ 0 by ω > 0. Equivalently, we may further rewrite (7.51) in the form ¯ P  >t 0  ¯ Ao P¯ Ao − P¯ + Q t ¯ t ¯ Ao P A d Ao P   +δ 2 Gto Go   ¯ + At P¯ Ad   −Q t ¯  < 0 d  Ad P  •  +ν 2 Gtd Gd • • P¯ − I

(7.52)

¯ = ω −1 Q. Using the linearizations Y = P¯ −1 , W = where P¯ = ω −1 P, Q −1 ¯ ¯ −1 ¯ P QP with φ = δ −2 and π = ν −2 , transformation By the Schur complement operations we express (7.52) as Y  > −Y       

0  +W 0 0 Y Ato Y Ato Y Gto • −W Y Gtd Y Atd Y Atd 0   • • −πI 0 0 0   < 0 • • • I −Y 0 0   • • • • −Y 0  • • • • • −φI

(7.53)

which is an LMI in the variables Y, W, π, φ. Robust stability in the sense of Definition 7.13 of the nonlinear system (7.43) under the constraint(7.44) with maximal δ, ν is established by the following theorem Theorem 7.14. System (7.43) with u ≡ 0, is robustly delay-independently stable if the following convex optimization problem over LMIs minimize π , φ subject to Y > 0   −Y + W 0 0 Y Ato Y Ato Y Gto  • −W Y Gtd Y Atd Y Atd 0     • • −πI 0 0 0    < 0  • • • I −Y 0 0     • • • • −Y 0  • • • • • −φI is feasible

(7.54)


227

7.6.1 Example 7.6 Consider the uncertain time-delay system     0.01 −0.1 −0.05 0 1 0 Ao =  0 0 1  , Ad =  0.02 −0.2 −0.08  0.2 −0.07 −0.1 0.3 −0.4 0.6     1 0.3 h = ϑ  1  x +  0.4  x(t − d) , d+ = 2 , Ho = I , Hd = I 0 0.2 On solving the convex optimization problem (7.54) using MATLAB software we get the bounds α = 0.5245 , σ = 0.4345 along with



 0.9855 0.4322 0.0105 Y =  0.4322 1.7687 0.3305  0.0105 0.3305 1.6429

which verifies Theorem 7.14. 7.6.2 Example 7.7 Consider the uncertain time-delay system    0.1 0.2 0.3 0.4 0.01  0 0.2 0.3 0   −0.02   Ao =   0.05 0.01 0.25 0.4  , Ad =  0 0 0.15 0.2 0.3 0     1 0.3 1  0.4  +    h=ϑ   0  x +  0.2  x(t − d) , d = 1 0.5

 −0.1 0 0 0.2 0 −0.01   −0.07 −0.1 0  0 0.05 −0.05 3

which is a delayed-version of example 2 in [109]. On solving the convex optimization problem (7.54) using MATLAB software with Ho = I , Hd = I, we get the bounds α = 0.2927 , σ = 0.4566 along with



 1.1155 0.0937 0.0005 0.1136  0.0937 1.1647 0.1305 0.0976   Y =   0.0005 0.1305 1.3075 0.0428  0.1136 0.0976 0.0428 0.9865

which verifies Theorem 7.14. We observe that the bounding factor is far from the results of [109] and the difference is indeed due to the presence of delay.

228


7.7 Robust Stabilization We now consider the application of a linear feedback controller of the form uk = Ko xk

(7.55)

to system (7.43), where Ko ∈ IRm×n is a constant gain matrix. Given the pair (Ao , Bo ) is stabilizable with Bo being of full row-rank, we substitute (7.55) into (7.43) to yield xk+1 = (Ao + Bo Ko )xk + Ad xk−dk + g(k, xk , x(t − dk ) , x0 = ψ0

(7.56)

for which the following definition holds Definition 7.15. System (7.43) is robustly delay-independent stabilized by control law (7.55) if the closed-loop system (7.56) is robustly stable. A direct application of Theorem 7.14 leads to the following optimization problem mimimize π , φ subject to Y > 0   −Y + W 0 0 Y (Ao + Bo Ko )t Y (Ao + Bo Ko )t Y Gto  • −W Y Gtd Y Atd Y Atd 0     • • −πI 0 0 0    < 0 (7.57)  • • • I −Y 0 0     • • • • −Y 0  • • • • • −φI To standardize inequality (7.57), we introduce the slack variable Ko Y = M or Ko = Y −1 M to convert it into the format of convex optimization problems over LMIs: minimize

γ, ψ

subject to Y > 0 , M   −Y + W 0 0 Y Ato + M t Bot Y Ato + M t Bot Y Gto  • −W Y Gtd Y Atd Y Atd 0     • • −πI 0 0 0    < 0  • • • I −Y 0 0     • • • • −Y 0  • • • • • −φI

(7.58)

We have just established the following theorem Theorem 7.16. System (7.43) is robustly delay-independently stabilized by control law (7.55) if the convex optimization problem (7.58) has a feasible solution. Some numerical applications of LMI (7.58) are in order.


229

7.7.1 Example 7.8 Consider the uncertain time-delay system 1 0.5 0.01 0.1 0 Ao = , Ad = , Bo = 1 0.01 −0.5 0.01 0.1 0.5 1 h=ϑ x+ x(t − d) , d+ = 3 0.5 1 On solving the LMI problem (7.58) using Ho = 2I , Hd = I, we get the bounds α = 0.7455 , σ = 0.3725 with

Ko = 102 [−1.2565,

−0.1634]

which verifies Theorem 7.16. In another simulation run, we let Bo =

20 02

while retaining all data as before. The results are summarized by α = 1.9014 ,

σ = 1.0715

with Ko = [−2.3157,

1.1335]

which showed again a trade-off between the size of feedback gains and the bounding factor. 7.7.2 Example 7.9 Consider the a fourth-order uncertain time-delay system     0.8 −0.25 0 0 0.1 0 −0.1 0.1  1   0 0 0   , Ad =  0.01 0.1 0.02 0  Ao =   0   0 0.2 0.03 0.03 0 −0.2 0.1  0 0 1 0 0 0.1 0 −0.3       0.5 1 0  0.5  0 0 +      h=ϑ  x +   x(t − d) , d = 2 , Bo =   0  0 1 0.3 1 0 On solving the LMI problem (7.58) using Ho = 2I , Hd = 2I, we get the bounds α = 1.1250 , σ = 0.5614

230


with Ko = [−12.2565,

−7.1634]

which verifies Theorem 7.16. When we used Bot [1, 0, 1, 0], we obtained α = 0.6347 ,

σ = 0.6014 ,

Ko = [−4.1947,

−2.3164]

which reflects the fact that the second case the input matric has more accessibility to the system states. By a parallel development to the continuous-time case, we can include bounds the gain matrix Ko in the form Kot Ko < κ I This condition, from a technical standpoint, corresponds to constraints on the component matrices M and Y −1 . A candidate choice to realize this would be by setting Mt M < µ I , µ > 0

,

X −1 < ϕ I , ϕ > 0

In turn, these are equivalent to the LMIs −µ I M t −ϕ I I < 0 , < 0 • −I • −X

(7.59)

Since Kot Ko = X −1 M t M X −1 < µ X −1 X −1 < µ ϕ2 I = κ I ∆

our goal has been achieved. In a similar way, in order to guarantee desired values {¯ α, σ ¯ } of the bounding factors {α, σ}, we recall that α−2 = γ, σ −2 = ψ . Thus we require 1 1 < 0 , ψ − 2 < 0 α ¯2 σ ¯ The foregoing modifications have constructed the basic elements of establishing the following convex optimization problem over LMIs γ −

minimize γ + ψ + µ + ϕ subject to Y > 0 , M   −Y + W 0 0 Y Ato + M t Bot Y Ato + M t Bot Y Gto  • −W Y Gtd Y Atd Y Atd 0     • • −πI 0 0 0    < 0 (7.60)  • • • I −Y 0 0     • • • • −Y 0  • • • • • −φI 1 1 γ − 2 < 0 , ψ − 2 < 0 σ ¯ α ¯ −µ I M t −ϕ I I < 0 , < 0 • −I • −X


231

The following theorem summarizes the main result Theorem 7.17. System (7.43) is robustly delay-independently stabilized by control law (7.55) with constrained feedback gains if the LMI problem (7.60) has a feasible solution. 7.7.3 Example 7.10 Consider the uncertain time-delay system 0.1 −0.2 10 0.7 0.1 , Bo = Ao = , Ad = 0.1 −0.3 01 −0.1 −0.5 1 1 h=ϑ x+ x(t − d) , d+ = 4 1 1 31 W = , Ho = I , Hd = I 18 On solving the LMI problem (7.60) using the bounds α ¯ = 1.1147 , σ ¯ = 0.9824 We obtain the gain Ko =

1.9252 −0.2937 −0.0214 2.8878

which verifies Theorem 7.17. 7.7.4 Resilient Feedback Design In the sequel, we address the performance deterioration issue by considering that the actual linear state-feedback controller has the form u(t) = [Ko + ∆Ko ] x(t)

(7.61)

where Ko ∈ IRm×n is a constant gain matrix and ∆Ko is an additive gain perturbation matrix represented by ∆

∆Ko = Mc ∆c Nc , ∆c ∈ ∆c = {∆c : ∆tc ∆c ≤ I}

(7.62)

Given that the pair (Ao , Bo ) is stabilizable. the application of control law (7.61) to system (7.43) yields the perturbed closed-loop system x(t) ˙ = (Ao + Bo Ko + Bo ∆Ko )x(t) + Ad x(t − d) + h(t, x, x(t − d)) x(s) = φ(s), t ∈ [−d, 0]

(7.63)

232


It follows by applying Theorem 7.14 to system (7.63), we obtain the following problem mimimize π , φ subject to Y > 0   Y (Ao + Bo Ko )t + Y (Ao + Bo Ko )t + t −Y + W 0 0 Y G o  Y (Bo ∆Ko )t Y (Bo ∆Ko )t   t t t  Y Ad Y Ad 0  • −W Y Gd    • • −πI 0 0 0     • • • I −Y 0 0     • • • • −Y 0  • • • • • −φI < 0

(7.64)

over all possible perturbations ∆c ∈ ∆c . In order to inequality (7.64) into proper LMI format and bypass the exhaustive search over the perturbation set ∆c , we start by focusing on the main inequality of (7.64) and manipulating with the aid of Inequality 1 of the Appendix to reach   −Y + W 0 0 Y (Ao + Bo Ko )t Y (Ao + Bo Ko )t Y Gto  • −W Y Gtd Y Atd Y Atd 0     • • −πI 0 0 0    +  • • • I −Y 0 0     • • • • −Y 0  • • • • • −φI   0  0     0     Bo Mc  ∆c [Nc Y, 0, 0, 0, 0, 0] +    0  0   XNct  0     0  t t t    0  ∆c [0, 0, 0, Mc Bo , 0, 0] +    0  0   0  0     0     0  ∆c [Nc Y, 0, 0, 0, 0, 0] +    Bo Mc  0


        

233

 t

XNc 0   0   ∆tc [0, 0, 0, 0, Mct Bot , 0] ≤ 0   0  0

 −Y + W + t t Y Ao + Y Ao +  ηa Y Nct Nc Y + 0 Y Gto  0 t t t t   B B Y K Y K t o o o o  ηc Y Nc Nc Y    t t t   Y A Y A 0 • −W Y G d d d     • • −πI 0 0 0     I − Y +  • • • 0 0  −1 t t   η B M M B o c c o a     −Y +  • • • • 0    ηc−1 Bo Mc Mct Bot • • • • • −φI < 0

(7.65)

We now introduce the slack variables Ko Y = L, Za = ηa Y Nct , Zc = ηc Y Nct , Z¯ = [Za , which   −Y              

Zc ]

will eventually convert (7.65) into the LMI format: +W • • • • • • • •

0 −W • • • • • • •

Y Ato + Y Ato + 0 0 Y Gto t t L Bo Lt Bot 0 Y Atd 0 0 Y Gtd Y Atd −πI 0 0 0 0 0 • I − Y Bo Mc 0 0 0 • • −ηa I 0 0 0 • • • −Y Bo Mc 0 • • • • −ηc I 0 • • • • • −φI • • • • • • 0

 Z¯   0   0   0   < 0 0   0   0   0  −I

(7.66)

In effect, we have established the following theorem Theorem 7.18. System (7.43) is robustly delay-independently stabilized by the actual control law (7.61) for all possible gain variations if the following convex optimization problems: minimize subject to

γ , ψ, ηa , ηc Y > 0 , M, Za , Zb

234


  −Y              

+W • • • • • • • •

0 −W • • • • • • •

Y Ato + Y Ato + 0 0 0 Y Gto Lt Bot Lt Bot Y Gtd Y Atd 0 Y Atd 0 0 −πI 0 0 0 0 0 • I − Y Bo Mc 0 0 0 0 0 0 • • −ηa I • • • −Y Bo Mc 0 • • • • −ηc I 0 • • • • • −φI • • • • • •

 ¯ Z   0   0   0   < 0 (7.67) 0   0   0   0  −I

has a feasible solution. On the other hand, by taking the modifications made to constrain the feedback gains we reach the following convex optimization problem over LMIs minimize

γ + ψ + µ + ϕ

subject to Y > 0 , M, Za , Zb  Y Ato + Y Ato + −Y + W 0 0 0 0 Y Gto t t  L Bo Lt Bot   0 Y Atd 0 0 • −W Y Gtd Y Atd   • • −πI 0 0 0 0 0   • • • I − Y Bo Mc 0 0 0   I 0 0 0 • • • • −η a   • • • • • −Y B M 0 o c   • • • • • • −η I 0 c   • • • • • • • −φI • • • • • • • • 1 1 γ − 2 < 0 , ψ − 2 < 0 α ¯ σ ¯ −ϕ I I −µ I M t < 0 , < 0 • −I • −X

 ¯ Z   0   0   0   < 0 (7.68) 0   0   0   0  −I

In conclusion, the following theorem summarizes the main result Theorem 7.19. System (7.43) is robustly delay-independently stabilized by the actual control law (7.61) for possible perturbations ∆c ∈ ∆c with constrained feedback gains if the LMI problem (7.68) has a feasible solution. 7.7.5 Example 7.11 Consider the uncertain time-delay system       1 0 −1 0.1 −0.1 0.02 −1 0 Ao =  −1 2 −1  , Ad =  0.02 −0.2 0.05  , Bo =  1 −1  0 −2 0 0.01 0.04 −0.3 0 −1










1 0.3 h = ϑ  0.5  x +  0  x(t − d) , d+ = 3 0 0.6     1 0.3 Mc =  0.6  , Nct =  0.6  , Ho = I , Hd = 2I 1.4 0.6 On solving the LMI problem (7.60) using the bounds α ¯ = 2.1175 , σ ¯ = 1.7765 We obtain the gain 

 2.0136 0.7427 Ko =  0.8385 0.5934  0.2462 2.9815 which verifies Theorem 7.18. One final example is now presented. 7.7.6 Example 7.12 Consider the uncertain time-delay system 0 1 0.1 0 0 , Ad = , Bo = Ao = −2 −3 0 −0.2 1 1 0.3 h=ϑ x+ x(t − d) , d+ = 2 1 0.4 0.7 0.5 t Mc = , Nc = , Ho = 2I , Hd = 2I 1.4 0.8 On solving the LMI problem (7.60) using the bounds α ¯ = 1.1125 , σ ¯ = 1.2148 We obtain the following 2.5110 0.3205 Y = , α = 0.7055 , σ = 0.7845 0.3205 1.1444 M = [6.0125, 3.5578] , Ko = [2.1145, 3.0018] which verifies Theorem 7.19.

235

236


7.8 Notes and References With focus on robust and resilience control methodologies for classes of nonlinear continuous-time and discrete-time systems with state delay, our experience shows that this topic is overlooked in the literature on time-delay systems. In this Chapter, we have established some new results in this direction. Certainly, there are several problems to be addressed including a recently developed method [47]. The topic of discrete-time systems deserves more attention since a lot of continuous-time results do not simply carry over. Different gain perturbation patterns should be attempted to reach a reasonable representation. [106, 107, 109, 40, 123].

8 Appendix

In this appendix, we collect some useful mathematical inequalities and lemmas which have been extensively used throughout the book.

8.1 Inequalities All mathematical inequalities are proved for completeness. They are termed facts due to their high frequency of usage in the analytical developments. 8.1.1 Inequality 1 For any real matrices Σ1 , Σ2 and Σ3 with appropriate dimensions and Σ3t Σ3 ≤ I, it follows that Σ1 Σ3 Σ2 + Σ2t Σ3t Σ1t ≤ α Σ1 Σ1t + α−1 Σ2t Σ2 , ∀α > 0 Proof: This inequality can be proved as follows. Since Φt Φ ≥ 0 holds for any matrix Φ, then take Φ as Φ = [α1/2 Σ1 − α−1/2 Σ2 ] Expansion of Φt Φ ≥ 0 gives ∀α > 0 α Σ1 Σ1t + α−1 Σ2t Σ2 − Σ1t Σ2 − Σ2t Σ1 ≥ 0 which by simple arrangement yields the desired result.

∇∇∇

8.1.2 Inequality 2 Let Σ1 , Σ2 , Σ3 and 0 < R = Rt be real constant matrices of compatible dimensions and H(t) be a real matrix function satisfying H t (t)H(t) ≤ I. Then for any ρ > 0 satisfying ρΣ2t Σ2 0 with appropriate dimensions, it follows that −2ρt β ≤ ρt Q ρ + β t Q−1 β Proof: Starting from the fact that [ρ + Q−1 β]t Q [ρ + Q−1 β] ≥ 0 , Q > 0 which when expanded and arranged yields the desired result.

∇∇∇

8.1 Inequalities

239

8.1.4 Inequality 4 (Schur Complements) Given a matrix Ω composed of constant matrices Ω1 , Ω2 , Ω3 where Ω1 = Ω1t and 0 < Ω2 = Ω2t as follows Ω1 Ω3 Ω = Ω3t Ω2 We have the following results (A) Ω ≥ 0 if and only if either  Ω2 ≥ 0  Π = Υ Ω2  Ω1 − Υ Ω 2 Υ t ≥ 0 or

(8.1)

 

Ω1 ≥ 0 Π = Ω1 Λ  Ω2 − Λt Ω1 Λ ≥ 0

(8.2)

hold where Λ, Υ are some matrices of compatible dimensions. (B) Ω > 0 if and only if either Ω2 > 0 Ω1 − Ω3 Ω2−1 Ω3t > 0 or

Ω1 ≥ 0 Ω2 − Ω3t Ω1−1 Ω3 > 0

hold where Λ, Υ are some matrices of compatible dimensions. In this regard, matrix Ω3 Ω2−1 Ω3t is often called the Schur complement Ω1 (Ω2 ) in Ω. Proof: (A) To prove (8.1), we first note that Ω2 ≥ 0 is necessary. Let z t = [z1t z2t ] be a vector partitioned in accordance with Ω. Thus we have z t Ω z = z1t Ω1 z1 + 2z1t Ω3 z2 + z2t Ω2 z2

(8.3)

Select z2 such that Ω2 z2 = 0. If Ω3 z2 = 0, let z1 = −πΩ3 z2 , π > 0. Then it follows that z t Ω z = π 2 z2t Ω3t Ω1 Ω3 z2 − 2π z2t Ω3t Ω3 z2 which is negative for a sufficiently small π > 0. We thus conclude Ω1 z2 = 0 which then leads to Ω3 z2 = 0, ∀ z2 and consequently Ω3 = Υ Ω 2 for some Υ .

(8.4)

240

8 Appendix

Since Ω ≥ 0, the quadratic term z t Ω z possesses a minimum over z2 for any z1 . By differentiating z t Ω z from (8.3) wrt z2t , we get ∂(z t Ω z) = 2Ω3t z1 + 2Ω2 z2 = 2Ω2 Υ t z1 + 2Ω2 z2 ∂z2t Setting the derivative to zero yields Ω2 Υ z1 = −Ω2 z2

(8.5)

Using (8.4) and (8.5) in (8.3), it follows that the minimum of z t Ω z over z2 for any z1 is given by min z t Ω z = z1t [Ω1 − Υ Ω2 Υ t ]z1 z2

which prove the necessity of Ω1 − Υ Ω2 Υ t ≥ 0. On the other hand, we note that the conditions (8.1) are necessary for Ω ≥ 0 and since together they imply that the minimum of z t Ω z over z2 for any z1 is nonnegative, they are also sufficient. Using similar argument, conditions (8.2) can be derived as those of (8.1) by starting with Ω1 . The proof of (B) follows as direct corollary of (A). ∇∇∇ 8.1.5 Inequality 5 For any quantities u and v of equal dimensions and for all ηt = i ∈ S, it follows that the following inequality holds ||u + v||2 ≤ [1 + β −1 ] ||u||2 + [1 + β]||v||2 for any scalar β > 0, Proof: Since

(8.6)

i∈S [u + v]t [u + v] = ut u + v t v + 2 ut v

(8.7)

It follows by taking norm of both sides of (8.7) for all i ∈ S that ||u + v||2 ≤ ||u||2 + ||v||2 + 2 ||ut v||

(8.8)

We know from the triangle inequality that 2 ||ut v|| ≤ β −1 ||u||2 + β ||v||2 On substituting (8.9) into (8.8), it yields (8.6).

(8.9) ∇∇∇

8.2 Lemmas

241

8.2 Lemmas The basic tools and standard results that are utilized in robustness analysis and resilience design in the different chapters are collected hereafter. Lemma 8.1. The matrix inequality − Λ + S Ω −1 S t < 0 holds for some 0 < Ω = Ω t ∈ n×n , if and only if −Λ SX < 0 • −X − X t + Z

(8.10)

(8.11)

holds for some matrices X ∈ n×n and Z ∈ n×n . Proof: (=⇒) By Schur complements, inequality (8.10) is equivalent to −Λ SΩ −1 < 0 (8.12) • −Ω −1 Setting X = X t = Z = Ω −1 , we readily obtain inequality (8.11). (⇐=) Since the matrix [I S] is of full rank, we obtain

I St

t

−Λ SX • −X − X t + Z

I St

< 0 ⇐⇒

− Λ + S Z S t < 0 ⇐⇒ − Λ + S Ω −1 S t < 0 , Z = Ω −1 (8.13) which completes the proof.

∇∇∇

Lemma 8.2. The matrix inequality AIP + IPAt + Dt IR−1 D + IM < 0 holds for some 0 < IP = IPt ∈ n×n , if and only if   AV + V t At + IM IP + AW − V Dt IR  • −W − W t 0  < 0 • • −IR

(8.14)

(8.15)

holds for some V ∈ n×n and W ∈ n×n . Proof: (=⇒) By Schur complements, inequality (8.14) is equivalent to AIP + IPAt + IM Dt IR < 0 (8.16) • −IR Setting V = V t = IP, W = W t = IR, it follows from Lemma (8.1) with Schur complements that there exists IP > 0, V, W such that inequality (8.15) holds.

242

8 Appendix

(⇐=) In a similar way, Schur complements to inequality (8.15) imply that:   AV + V t At + IM IP + AW − V Dt IR  • −W − W t 0  < 0 • • −IR t I I AV + V t At + IM + Dt IR−1 D IP + AW − V < 0 ⇐⇒ t A A • −W − W ⇐⇒ AIP + IPAt + Dt IR−1 D + IM < 0 , V = V t

(8.17) ∇∇∇

which completes the proof. The following lemmas are found in [102] Lemma 8.3. Given any x ∈ n : max {[xt RH∆G x]2 : ∆ ∈ } = xt RHH t R x xt Gt G x

Lemma 8.4. Given matrices 0 ≤ X = X t ∈ p×p , Y = Y t < 0 ∈ p×p , 0 ≤ Z = Z t ∈ p×p , such that [ξ t Y ξ]2 − 4 [ξ t X ξ ξ t Z ξ]2 > 0 for all 0 = ξ ∈ p is satisfied. Then there exists a constant α > 0 such that α2 X + α Y + Z < 0 The following lemma can be found in [95] Lemma 8.5. For a given two vectors α ∈ IRn , β ∈ IRm and matrix IN ∈ IRn×m defined over a prescribed interval Ω, it follows for any matrices X ∈ IRn×n , Y ∈ IRn×m , and Z ∈ IRm×m , the following inequality holds t

X Y − IN α(s) α(s) −2 αt (s) IN β(s) ds ≤ ds β(s) β(s) Y t − INt Z Ω Ω where

X Y Yt Z

≥0

An algebraic version of Lemma 8.5 is stated below Lemma 8.6. For a given two vectors α ∈ IRn , β ∈ IRm and matrix IN ∈ IRn×m defined over a prescribed interval Ω, it follows for any matrices X ∈ IRn×n , Y ∈ IRn×m , and Z ∈ IRm×m , the following inequality holds t X Y − IN α α − 2 αt IN β ≤ β β Y t − INt Z = αt Xα + β t (Y t − INt )α + αt (Y − IN)β + β t Zβ

8.2 Lemmas

243

subject to

X Y Yt Z

≥0

The following lemma can be found in [81] Lemma 8.7. Let 0 < Y = Y t and M, N be given matrices with appropriate dimensions. Then it follows that Y + M ∆ N + N t ∆t M t

< 0 , ∀ ∆t ∆ ≤ I

holds if and only if there exists a scalar ε > 0 such that Y + ε M M t + ε−1 N t N

< 0

In the following lemma, we let X(z) ∈ IRn×p be a matrix function of the variable z. A matrix X∗ (z) is called the orthogonal complement of X(z) if X t (z) X∗ (z) = 0 and X(z) X∗ (z) is nonsingular (of maximum rank). Lemma 8.8. Let 0 < L = Lt and X, Y be given matrices with appropriate dimensions. Then it follows that the inequality L(z) + X(z) P Y (z) + Y t (z) P t X t (z)

> 0

(8.18)

holds for some P and z = zo if and only if the following inequalities X∗t (z) L(z) X∗ (z) > 0 , Y∗t (z) L(z) Y∗ (z) > 0

(8.19)

hold with z = zo . It is significant to observe that feasibility of matrix inequality (8.18) with variables P and z is equivalent to the feasibility of 8.19) with variable z and thus the matrix variable P has been eliminated from 8.18) to form 8.19). Using Finsler’s lemma [9], we can express (8.19) in the form L(z) − β X(z) X t (z) > 0 ,

L(z) − β Y (z) Y t (z) > 0 (8.20)

for some β ∈ IR. The following is a statement of the reciprocal projection Lemma [3] Lemma 8.9. Let P > 0 be a given matrix. The following statements are equivalent: i) IM + Z + Z t < 0 ii) the LMI problem IM + P − (V + V t ) V t + Z t < 0 V +Z −P is feasible with respect to the general matrix V .

244

8 Appendix

8.3 Linear Matrix Inequalities It has been shown in [9] that a wide variety of problem arising in system and control theory can conveniently reduced to a few standard convex or quasiconvex optimization problems involving linear matrix inequalities (LMIs). The resulting optimization problems can then be solved numerically very efficiently using commerically available interior-point methods. 8.3.1 Basics One of the earliest LMIs arises in Lyapunov theory. It is well-known that the differential equation x(t) ˙ = A x(t)

(8.21)

has all of its trajectories converge to zero (stable) id and only if there exists a matrix P > 0 such that At P + A P < 0

(8.22)

This leads to the LMI formulation of stability, that is, a linear time-invariant system is asymptotically stable if and only if there exists a matrix 0 0

Given a vector variable x ∈ IRn and a set of matrices 0 < Gj = Gtj ∈ IRn×n , j = 0, ..., p, then a basic compact formulation of a linear matrix inequality is ∆

G(x) = G0 +

p

xj Gj

> 0

(8.23)

j=1

Notice that (8.23) implies that v t G(x)v > 0 ∀0 = v ∈ IRn . More importantly, the set {x |G(x) > 0 is convex. Nonlinear (convex) inequalities are converted to LMI form using Schur complements in the sense that Q(x) S(x) > 0 (8.24) • R(x) where Q(x) = Qt (x), R(x) = Rt (x), S(x) depend affinely on x, is equivalent to R(x) > 0

, Q(x) − S(x)R−1 (x)S t (x) > 0

More generally, the constraint

(8.25)

8.3 Linear Matrix Inequalities

T r[S t (x) P −1 (x) S(x)] < 1 ,

245

P (x) > 0

where P (x) = P t (x) ∈ IRn×n , S(x) ∈ IRn×p depend affinely on x, is handled by introducing a new (slack) matrix variable Y (x) = Y t (x) ∈∈ IRp×p and the LMI (in x and Y ): Y S(x) T rY < 1 , > 0 (8.26) • P (x) Most of the time, our LMI variables are matrices. It should clear from the foregoing discussions that a quadratic matrix inequality (QMI) in the variable P can be readily expressed as linear matrix inequality (LMI)in the same variable. 8.3.2 Some Standard Problems Here we provide some common convex problems that we encountered throughout the monograph. Given an LMI G(x) > 0, the corresponding LMI problem (LMIP) is to find a feasible x ≡ xf such that G(xf ) > 0, or determine that the LMI is infeasible. It is obvious that this is a convex feasibility problem. The generalized eigenvalue problem (GEVP) is to minimize the maximum generalized eigenvalue of a pair of matrices that depend affinely on a variable, subject to an LMI constraint. GEVP has the general form minimize λ subject to λB(x) − A(x) > 0

,

B(x) > 0 , C(x) > 0

(8.27)

where A, B, C are symmetric matrices that are affine functions of x. Equivalently stated minimize λM [A(x), B(x)] subject to B(x) > 0 , C(x) > 0

(8.28)

where λM [X, Y ] denotes the largest generalized eigenvalue of the pencil λY − X with Y > 0. This is problem is quasiconvex optimization problem since the constraint is convex and the objective , λM [A(x), B(x)], is quasiconvex. The eigenvalue problem (EVP) is to minimize the maximum eigenvalue of a matrix that depend affinely on a variable, subject to an LMI constraint. EVP has the general form minimize λ subject to λ I − A(x) > 0

, B(x) > 0

(8.29)

where A, B are symmetric matrices that are affine functions of the optimization variable x. This is problem is convex optimization problem.

246

8 Appendix

EVPs can appear in the equivalent form of minimizing a linear function subject to an LMI, that is minimize ct x subject to G(x) > 0

(8.30)

where G(x) is an affine function of x. Examples of G(x) include P A + At P + C t C + γ −1 P BB t P < 0 , P > 0 It should be stressed that the standard problems (LMIPs, GEVPs, EVPs) are tractable, from both theoretical and practical viewpoints: They can be solved in polynomial-time. They can solved in practice very efficiently using commercial software [26]. 8.3.3 The S-Procedure In some design applications, we faced the constraint that some quadratic function be negative whenever some other quadratic function is negative. In such cases, this constraint can be expressed as an LMI in the data variables defining the quadratic functions. Let Go , ..., Gp be quadratic functions of the variable ξ ∈ IRn : ∆

Gj (ξ) = ξ t Rj ξ + 2utj ξ + vj , j = 0, ..., p,

Rj = Rjt

We consider the following condition on Go , ..., Gp : Go (ξ) ≤ 0 ∀ξ

such that

Gj (ξ) ≥ 0, j = 0, ..., p

(8.31)

It is readily evident that if there exist scalars ω1 ≥ 0, ...., ωp ≥ 0 such that ∀ξ,

Go (ξ) −

p

ωj Gj (ξ) ≥ 0

(8.32)

j=1

then inequality (8.31) holds. Observe that if the functions Go , ..., Gp are affine, then Farkas lemma [9] state that (8.31) and (8.32) are equivalent. Interestingly enough, inequality (8.32) can written as

Ro u o • vo

−

p

ωj

j=1

Rj u j • vj

≥ 0

(8.33)

The foregoing discussions were stated for non strict inequalities. In case of strict inequality, we let Ro , ..., Rp ∈ IRn×n be symmetric matrices with the following qualifications ξ t Ro ξ > 0 ∀ξ

such that

ξ t Gj ξ ≥ 0,

j = 0, ..., p

(8.34)

8.5 Some Formulas on Matrix Inverses

247

Once again, it is obvious that there exist scalars ω1 ≥ 0, ...., ωp ≥ 0 such that p

Go (ξ) −

∀ξ,

ωj Gj (ξ) > 0

(8.35)

j=1

then inequality (8.34) holds. Observe that (8.35) is an LMI in the variables Ro , ω1 , ..., ωp . It should be remarked that the S-procedure deals with non strict inequalities allows the inclusion of constant and linear terms. In the strict version, only quadratic functions can be used.

8.4 Some Lyapunov-Krasovskii Functionals In this section, we provide some Lyapunov-Krasovskii functionals and their time-derivatives which are of common use in stability studies throughout the text.

V1 (x) = xt P x +

V2 (x) = V3 (x) =

0

0

xt (t + θ)Qx(t + θ) dθ

(8.36)

−τ

xt (α)Rx(α) dα dθ

t

−τ

0

t+θ t

−τ

t+θ

(8.37)

t

x˙ (α)W x(α) ˙ dα dθ

(8.38)

where x is the state vector, τ is a constant delay factor and the matrices 0 < P t = P, 0 < Qt = Q, 0 < Rt = R, 0 < W t = W are appropriate weighting factors. Standard matrix manipulations lead to V˙ 1 (x) = x˙ t P x + xt P x˙ + xt (t)Qx(t) − xt (t − τ )Qx(t − τ )

0 t t ˙ V2 (x) = x (t)Rx(t) − x (t + α)Rx(t + α) d θ −τ

= τ xt (t)Rx(t) −

0

xt (t + θ)Rx(t + θ) d θ

(8.39)

(8.40)

−τ

t

V˙ 3 (x) = τ x˙ t (t)W x(t) −

x˙ t (α)W x(α) ˙ dα

(8.41)

t−τ

8.5 Some Formulas on Matrix Inverses This concerns some useful formulae for inverting of matrix expressions in terms of the inverses of its constituents.

248

8 Appendix

8.5.1 Inverse of Block Matrices Let A be a square matrix of appropriate dimension and partitioned in the form A1 A2 A = (8.42) A 3 A4 where both A1 and A4 are square matrices. If A1 is invertible, then ∆1 = A4 − A3 A−1 1 A2 is called the Schur complement of A1 . Alternatively, if A4 is invertible, then ∆4 = A1 − A2 A−1 4 A3 is called the Schur complement of A4 . It is well-known [42] that matrix A is invertible if and only if either A1 and

∆1 are invertible,

A4 and

∆4 are invertible.

or Specifically, we have the following equivalent expressions −1 A1 A2 −A−1 A2 ∆−1 Υ1 1 1 = −1 A 3 A4 −∆−1 ∆−1 1 A 3 A1 1 or

A1 A2 A 3 A4

−1

=

−1 −∆−1 ∆−1 4 4 A2 A4 −1 −1 −A4 A3 ∆4 Υ4

(8.43)

(8.44)

where −1 −1 + A−1 Υ1 = A−1 1 1 A2 ∆ 1 A3 A1

−1 −1 Υ4 = A−1 + A−1 4 4 A3 ∆ 4 A2 A4

(8.45)

Important special cases are −1 0 A−1 A1 0 1 = −1 −1 A3 A4 −A−1 4 A3 A1 A4 and

A1 A2 0 A4

−1

=

−1 −1 A−1 1 −A1 A2 A4 −1 0 A4

(8.46)

(8.47)

8.5.2 Matrix Inversion Lemma Let A ∈ IRn×n and C ∈ IRm×m be nonsingular matrices. By using the definition of matrix inverse, it can be easily verified that [A + B C D]−1 = A−1 − A−1 B [D A−1 B + C −1 ]−1 DA−1 (8.48)

8.5 Some Formulas on Matrix Inverses

249

About the Author MagdiSadek Mahmoud received the Ph.D. degree from Cairo University, Egypt in 1974. He has been a Professor of Systems Engineering since 1984. He served on the faculties of several universities world-wide including Cairo University; the American University in Cairo; the Egyptian Air Academy; MSA University; the Arab Academy of Sciences and Technology (Egypt), Kuwait University and KISR (Kuwait), UMIST (UK), Pittsburgh University and Case Western Reserve University (USA) and NTU (Singapore). He lectured in Europe (UK, Germany, Switzerland), (Australia) and (Venezuela). He has been actively engaged in teaching and research in the development of modern methodologies to computer control, systems engineering and information technology and has been a technical consultant on information, computer and systems engineering for numerous companies and agencies at all levels of government and the private sector. Dr. Mahmoud is the principal author of ten (10) books, nine (9) bookchapters and the author/co-author of more than 350-refereed papers. He is the recipient of 1978, 1986 Science State Incentive Prizes for outstanding research in engineering (Egypt), of the Abdul-Hameed Showman Prize for Young Arab Scientists in engineering sciences, 1986 (Jordan) and of the Prestigious Award for Best Researcher at Kuwait University, 1992 (Kuwait), the State Medal of Science and Arts-first class, 1979 (Egypt) and the State Distinguished Award-first class, 1995 (Egypt). He is listed in the 1979 edition of Who’s Who in Technology Today (USA). He was the vice-chairman of the IFAC-SECOM working group on large-scale systems methodology and applications (1981-1986), and an associate editor of LSS Journal (1985-1988) and editor-at-large of the EEE series, Marcel-Dekker, USA. He is an associate editor of the International Journal of Parallel and Distributed Systems of Networks, IASTED, since 1997. He is a member of the New York Academy of Sciences. He is a fellow of the IEE, a senior member of the IEEE, a member of Sigma Xi, the CEI (UK), the Egyptian Engineers society, the Kuwait Engineers society and a registered consultant engineer of information engineering and systems (Egypt).

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Index

H∞ Control 37, 40, 55, 91 H∞ state-feedback control 178 adaptive controllers 19 adaptive schemes 101 adaptive stabilization methods 122 additive perturbations 20, 31, 37, 51, 57, 84, 87, 130, 171 additive uncertainties 3 analog-digital 1, 27 closed-loop system 30 congruent transformation 32, 34, 38, 45, 93, 196 continuous polytopic systems 139, 184 continuous-time model 99 controller 1 controller parameters 2–4 controller redesign 1 controller uncertainties 3, 4 conversion 1, 27 convex optimization 27, 46 convex-polytopic 27, 47, 171 cost function 29, 30

disk drive

9

error dynamics 73 estimation error variance estimator 15

feedback controllers 18 feedback stabilization 195, 198 filters 19 fragile 2 fragile controller 4, 8 fragility problem 4, 27 functional differential equations 22 gain perturbations 7, 19, 27–30, 34, 49, 55, 84 guaranteed-cost 18 guaranteed-cost control 27, 29, 49, 87 guaranteed-cost matrix 30 heat exchanger

13, 95

inverted pendulum Kalman filter

delay-dependent methodologies 171 delay-dependent stability 171, 191 delay-dependent stabilizing controllers 172 descriptor approach 171 descriptor transformation 172, 189 digital-analog 1 discrete-time model 118 discrete-time systems 83

132

7

129, 136

Lebesgue space 28 Lebsegue space 21 linear filtering 132 linear matrix inequalities 27 LMIs 32, 34, 35, 39, 40, 80, 87 Lyapunov functional 86 Lyapunov-Krasovskii functional 50, 74, 174

30,

258

Index

model-reference state regulator 114 multiplicative perturbations 20, 33, 35, 39, 52, 58, 85, 137, 171 nominal H2 design 173 nominal H∞ design 178 nominal controller 84 nominal feedback design 216 nominal model 100 non-fragile controller 2 nonlinear continuous-time systems 207 nonlinear discrete-time systems 223 norm-bounded 27 norm-bounded uncertainties 28, 83, 84, 171 observers 19 output feedback

19, 28, 41, 44, 61

parameter space design 4 parametric uncertainties 1, 27 polytopic H2 design 185 polytopic H∞ design 186 quadratic cost matrix 29, 31, 34, 35, 49, 85 quadratic stability 30, 49, 85 quadratically stable 30, 31, 42, 50, 55, 83, 86, 92 resilient resilient resilient resilient resilient resilient resilient resilient

2 H∞ design 180 H2 design 176 adaptive control 99 control 17, 27, 34, 69 controllers 2, 4, 27, 83 delay-dependent control 171 delay-dependent filtering 149

resilient feedback design 219, 231 resilient feedback stabilization 199 resilient filter 131 resilient filtering 140 resilient linear filtering 129, 130 resilient nonlinear control 207 robust control 1, 27 robust controllers 1–3, 7 robust delay-dependent stability 211, 214 robust delay-independent stability 208 robust design 1 robust filtering 129 robust linear filtering 131 robust performance 1 robust stability 1, 27, 194 robustness 2 robustness problem 4 Schur complement 32, 34, 38, 43, 64 Schur complements 31, 50 simultaneous H2 /H∞ control 171 simultaneous H2 /H∞ design 180 simultaneous polytopic H2 /H∞ design 187 stabilizing controllers 1 state-delay 27, 28, 47, 71, 79 state-feedback 2, 7, 8, 15, 19, 28–30 state-feedback control 100, 172 structured uncertainties 3, 4 the fragility problem

2, 3

uncertain systems 130 uncertainties 1 unstructured uncertainties water quality

16, 17

4

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