PDE Control of String-Actuated Motion 9780691233505

Table of contents :
Contents
List of Figures
List of Tables
Preface
1. Introduction
Part I. Applications
2. Single-Cable Mining Elevators
3. Dual-Cable Elevators
4. Elevators with Disturbances
5. Elevators with Flexible Guides
6. Deep-Sea Construction
7. Deep-Sea Construction with Event-Triggered Delay Compensation
8. Offshore Rotary Oil Drilling
Part II. Generalizations
9. Basic Control of Sandwich Hyperbolic PDEs
10. Delay-Compensated Control of Sandwich Hyperbolic Systems
11. Event-Triggered Control of Sandwich Hyperbolic PDEs
12. Sandwich Hyperbolic PDEs with Nonlinearities
Part III. Adaptive Control of Hyperbolic PDE-ODE Systems
13. Adaptive Event-Triggered Control of Hyperbolic PDEs
14. Adaptive Control with Regulation-Triggered Parameter Estimation of Hyperbolic PDEs
15. Adaptive Control of Hyperbolic PDEs with Piecewise-Constant Inputs and Identification
Bibliography
Index

Citation preview

PDE Control of String-Actuated Motion

PRINCETON SERIES IN APPLIED MATHEMATICS Ingrid Daubechies (Duke University); Weinan E (Princeton University); Jan Karel Lenstra (Centrum Wiskunde & Informatica, Amsterdam); Endre Siili (University of Oxford), Series Editors

The Princeton Series in Applied Mathematics features high-quality advanced texts and monographs in all areas of applied mathematics. The series includes books of a theoretical and general nature as well as those that deal with the mathematics of specific applications and real-world scenarios. For a full list of titles in the series, go to https://press.princeton.edu/series /princeton-series-in-applied-mathematics PDE Control of String-Actuated Motion, Ji Wang and Miroslav Krstic Delay-Adaptive Linear Control, Yang Zhu and Miroslav Krstic Statistical Inference via Convex Optimization, Anatoli Juditsky and Arkadi Nemirovski A Dynamical Systems Theory of Thermodynamics, Wassim M. Haddad Formal Verification of Control System Software, Pierre-Lo¨ıc Garoche Rays, Waves, and Scattering: Topics in Classical Mathematical Physics, John A. Adam Mathematical Methods in Elasticity Imaging, Habib Ammari, Elie Bretin, Josselin Garnier, Hyeonbae Kang, Hyundae Lee, and Abdul Wahab Hidden Markov Processes: Theory and Applications to Biology, M. Vidyasagar Topics in Quaternion Linear Algebra, Leiba Rodman Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos Stability and Control of Large-Scale Dynamical Systems: A Vector Dissipative Systems Approach, Wassim M. Haddad and Sergey G. Nersesov Matrix Completions, Moments, and Sums of Hermitian Squares, Mih´ aly Bakonyi and Hugo J. Woerdeman Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation, Luca Zaccarian and Andrew R. Teel Totally Nonnegative Matrices, Shaun M. Fallat and Charles R. Johnson Graph Theoretic Methods in Multiagent Networks, Mehran Mesbahi and Magnus Egerstedt Matrices, Moments and Quadrature with Applications, Gene H. Golub and G´erard Meuran

PDE Control of String-Actuated Motion

Ji Wang Miroslav Krstic

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

c 2022 by Princeton University Press Copyright  Princeton University Press is committed to the protection of copyright and the intellectual property our authors entrust to us. Copyright promotes the progress and integrity of knowledge. Thank you for supporting free speech and the global exchange of ideas by purchasing an authorized edition of this book. If you wish to reproduce or distribute any part of it in any form, please obtain permission. Requests for permission to reproduce material from this work should be sent to [email protected] Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540 99 Banbury Road, Oxford OX2 6JX press.princeton.edu All Rights Reserved ISBN 9780691233482 ISBN (pbk.) 9780691233499 ISBN (e-book) 9780691233505 British Library Cataloging-in-Publication Data is available Editorial: Diana Gillooly, Kristen Hop, Kiran Pandey Jacket/Cover Design: Heather Hansen Production: Lauren Reese Publicity: Matthew Taylor, Charlotte Coyne Copyeditor: Wendy Lawrence ˇ Jacket/Cover Credit: Marko Celebonovi´ c (1902–1986), Coastal Motif, 1956, oil on canvas, exhibited at the 1957 Biennial in S˜ao Paulo. After graduating with a law ˇ degree from the Sorbonne, Celebonovi´ c studied sculpture with Antoine Bourdelle and then devoted the rest of his life to painting. He was a member of the Serbian Academy of Sciences and Arts. This book has been composed by using LATEX Printed on acid-free paper. ∞ Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents

List of Figures

ix

List of Tables

xv

Preface

xvii

1 Introduction 1.1 String-Actuated Mechanisms . . . . . . . . . . . 1.2 Hyperbolic PDE-ODE Systems . . . . . . . . . . 1.3 “Sandwich” PDEs . . . . . . . . . . . . . . . . . 1.4 Advanced Boundary Control of Hyperbolic PDEs 1.5 Notes . . . . . . . . . . . . . . . . . . . . . . . .

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I Applications

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2 Single-Cable Mining Elevators 2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . 2.2 State-Feedback for Vibration Suppression . . . . . 2.3 Observer and Output-Feedback Controller Using Cage Sensing . . . . . . . . . . . . . . . . . . . . . 2.4 Stability Analysis . . . . . . . . . . . . . . . . . . . 2.5 Simulation Test in a Single-Cable Mining Elevator 2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . 2.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Dual-Cable Elevators 3.1 Dual-Cable Mining Elevator Dynamics and Reformulation 3.2 Observer for Cable Tension . . . . . . . . . . . . . . . . . 3.3 Controller for Cable Tension Oscillation Suppression and Cage Balance . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.5 Simulation Test for a Dual-Cable Mining Elevator . . . . 3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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34 34 38

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43 46 49 54 63

4 Elevators with Disturbances 4.1 Problem Formulation . . . . . . . . . . . . . . . 4.2 Disturbance Estimator . . . . . . . . . . . . . . 4.3 Observer of Cable-and-Cage State . . . . . . . 4.4 Control Design for Rejection of Disturbances at 4.5 Simulation for a Disturbed Elevator . . . . . .

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64 64 67 71 73 80

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CONTENTS

vi 4.6 4.7

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Elevators with Flexible Guides 5.1 Description of Flexible Guides and Generalized Model 5.2 Observer for Distributed States of the Cable . . . . . . 5.3 Adaptive Disturbance Cancellation and Stabilization . 5.4 Adaptive Update Laws . . . . . . . . . . . . . . . . . . 5.5 Control Law and Stability Analysis . . . . . . . . . . . 5.6 Simulation for a Flexible-Guide Elevator . . . . . . . . 5.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 93

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95 95 99 102 108 110 117 121 129

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130 131 141 149 157 158 164 168

7 Deep-Sea Construction with Event-Triggered Delay Compensation 7.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . 7.2 Observer Design Using Delayed Measurement . . . . . . 7.3 Delay-Compensated Output-Feedback Controller . . . . 7.4 Event-Triggering Mechanism . . . . . . . . . . . . . . . 7.5 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . 7.6 Simulation for Deep-Sea Construction with Sensor Delay 7.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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170 170 175 181 183 190 193 198 201

8 Offshore Rotary Oil Drilling 8.1 Description of Oil-Drilling Models . . . . . . . . 8.2 Adaptive Update Laws for Unknown Coefficients 8.3 Output-Feedback Control Design . . . . . . . . . 8.4 Stability Analysis . . . . . . . . . . . . . . . . . . 8.5 Simulation for Offshore Oil Drilling . . . . . . . . 8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . .

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202 203 206 208 215 222 229

6 Deep-Sea Construction 6.1 Modeling Process and Linearization . . . . . . . 6.2 Basic Control Design Using Full States . . . . . . 6.3 Observer for Two-Dimensional Oscillations of the 6.4 Controller with Collocated Boundary Sensing . . 6.5 Simulation for a Deep-Sea Construction System . 6.6 Appendix . . . . . . . . . . . . . . . . . . . . . . 6.7 Notes . . . . . . . . . . . . . . . . . . . . . . . .

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II Generalizations

231

9 Basic Control of Sandwich Hyperbolic PDEs 9.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 9.2 Backstepping for the PDE-ODE Cascade . . . . . . . . . . . 9.3 Backstepping for the Input ODE . . . . . . . . . . . . . . . . 9.4 Controller and Stability Analysis . . . . . . . . . . . . . . . . 9.5 Boundedness and Exponential Convergence of the Controller 9.6 Extension to ODEs of Arbitrary Order . . . . . . . . . . . . .

233 233 234 238 239 244 255

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CONTENTS

9.7 9.8 9.9

vii

Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Delay-Compensated Control of Sandwich Hyperbolic Systems 10.1 Problem Formulation . . . . . . . . . . . . . . 10.2 Observer Design . . . . . . . . . . . . . . . . 10.3 Output-Feedback Control Design . . . . . . . 10.4 Stability Analysis of the Closed-Loop System 10.5 Application in Deep-Sea Construction . . . . 10.6 Appendix . . . . . . . . . . . . . . . . . . . . 10.7 Notes . . . . . . . . . . . . . . . . . . . . . .

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11 Event-Triggered Control of Sandwich Hyperbolic 11.1 Problem Formulation . . . . . . . . . . . . . . . . . 11.2 Observer . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Continuous-in-Time Control Law . . . . . . . . . . 11.4 Event-Triggering Mechanism . . . . . . . . . . . . 11.5 Stability Analysis of the Event-Based Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Application in the Mining Cable Elevator . . . . . 11.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . 11.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

265 268 272

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273 273 276 287 295 297 305 309

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310 310 312 315 320

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326 332 338 340

12 Sandwich Hyperbolic PDEs with Nonlinearities 12.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 12.2 State-Feedback Control Design . . . . . . . . . . . . . 12.3 Observer Design and Stability Analysis . . . . . . . . . 12.4 Stability of the Output-Feedback Closed-Loop System 12.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .

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341 341 342 349 356 359 362 373

III Adaptive Control of Hyperbolic PDE-ODE Systems 13 Adaptive Event-Triggered Control of Hyperbolic PDEs 13.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . 13.2 Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Adaptive Continuous-in-Time Control Design . . . . . . . 13.4 Event-Triggering Mechanism . . . . . . . . . . . . . . . . 13.5 Stability Analysis of the Closed-Loop System . . . . . . . 13.6 Application in the Flexible-Guide Mining Cable Elevator . 13.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

375 . . . . . . . .

377 377 379 381 385 388 393 399 403

14 Adaptive Control with Regulation-Triggered Parameter Estimation of Hyperbolic PDEs 14.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Nominal Control Design . . . . . . . . . . . . . . . . . . . . . . . .

405 405 408

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CONTENTS

viii 14.3 14.4 14.5 14.6 14.7

Regulation-Triggered Main Result . . . . . Simulation . . . . . . Appendix . . . . . . Notes . . . . . . . .

Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 Adaptive Control of Hyperbolic PDEs with Piecewise-Constant Inputs and Identification 15.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . 15.2 Nominal Control Design . . . . . . . . . . . . . . . . . . . 15.3 Event-Triggered Control Design with Piecewise-Constant Parameter Identification . . . . . . . . . . . . . . . . . . . 15.4 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Application in the Mining Cable Elevator . . . . . . . . . 15.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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410 418 425 428 438

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442 449 459 464 471

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Bibliography

473

Index

487

List of Figures

2.1 2.2 2.3 2.4

The mining cable elevator . . . . . . . . . . . . . . . . . . . . . . The hoisting velocity z˙ ∗ (t) . . . . . . . . . . . . . . . . . . . . . The open-loop responses of the plant (2.18)–(2.21) . . . . . . . . The closed-loop responses of the plant (2.18)–(2.21) with the PD controller (2.148) and the proposed state-feedback controller (2.52) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The responses of the closed-loop system (2.18)–(2.21) under the observer (2.66)–(2.69) and the output-feedback control law (2.109) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The hoisting acceleration . . . . . . . . . . . . . . . . . . . . . . 2.7 The closed-loop responses of the accurate plant (2.2)–(2.4) with (2.9) under the PD controller (2.148) and the proposed output-feedback controller (2.109) . . . . . . . . . . . . . . . . . 3.1 Diagram and prototype of a dual-cable mining elevator with flexible guide rails . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Diagram of the plant dynamics (3.16)–(3.20) . . . . . . . . . . . ˙ 3.3 The hoisting velocity l(t) . . . . . . . . . . . . . . . . . . . . . . 3.4 Tension oscillations EA × ux (l(t)/2, t) at the midpoint of cable 1 3.5 Tension oscillations EA × vx (l(t)/2, t) at the midpoint of cable 2 3.6 Errors EA × (ux (l(t)/2, t) − vx (l(t)/2, t)) of tension oscillations between cable 1 and cable 2 . . . . . . . . . . . . . . . . . . . . . 3.7 Axial vibration displacements y(t) of the cage . . . . . . . . . . 3.8 Cage roll angles θ(t) . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Observer errors z2 (l(t)/2, t) − zˆ2 (l(t)/2, t) and w2 (l(t)/2, t)− w ˆ2 (l(t)/2, t) between the plant (3.16)–(3.20) and the observer (3.27)–(3.32) . . . . . . . . . . . . . . . . . . . . . . . . 3.10 The proposed observer-based output-feedback control forces EA U1 (t) = EA (Ue2 + Ue1 ), EA U2 (t) = EA (Ue2 − Ue1 ) and the boundary dampers (3.102), (3.103) at two head sheaves . . . . . 4.1 The mining cable elevator with airflow disturbances . . . . . . . 4.2 Block diagram of the disturbance estimator . . . . . . . . . . . . 4.3 Block diagram of the state observer . . . . . . . . . . . . . . . . 4.4 The target hoisting trajectory l(t) . . . . . . . . . . . . . . . . . ˆ = −rd¯x (0, t) and the actual 4.5 The disturbance estimate d(t) disturbance d(t) (4.110) . . . . . . . . . . . . . . . . . . . . . . . ˜ = d(t) − d(t) ˆ between the actual 4.6 The estimation error d(t) disturbance d(t) (4.110) and the disturbance estimate ˆ = −rd¯x (0, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . d(t) 4.7 The observer error u ˜(l(t)/2, t) at the midpoint of the cable . . .

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x 4.8 4.9

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5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14

LIST OF FIGURES

The open-loop response u(0, t) of the plant (4.1)–(4.3) under the disturbance (4.110) at x = 0 . . . . . . . . . . . . . . . . . . . . . The output responses u(0, t) of the closed-loop system (4.94)–(4.108) under the disturbance (4.110) at x = 0 with the proposed output-feedback controller (4.91) and PD controller (4.111) . . . . . . The responses u(l(t)/2, t) of the closed-loop system (4.94)–(4.108) under the disturbance (4.110) at x = 0 with the proposed output-feedback controller (4.91) and PD controller (4.111) . . . . . . The output-feedback controller . . . . . . . . . . . . . . . . . . . . . . The output responses u(0, t) of the closed-loop system (4.94)–(4.108) under the disturbance (4.110) at x = 0 with the proposed output-feedback controller (4.91) under the model error and without the model error, as well as the PD controller (4.111) . . . . . . . . . . . . . . . . . . . . . . . . . . . . The output responses u(l(t)/2, t) of the closed-loop system (4.94)–(4.108) under the disturbance (4.110) at x = 0 with the proposed output-feedback controller (4.91) under the model error and without the model error, as well as the PD controller (4.111) . . . Lateral vibration control of a mining cable elevator with viscoelastic guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block of the closed-loop system . . . . . . . . . . . . . . . . . . . . . . Responses of x1 (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Responses of x2 (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Responses of the PDE states w(l(t)/2, t), z(l(t)/2, t) under the proposed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-adjustment of the control parameter kˆ1 to approach the ideal value k1 = −9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-adjustment of the control parameter kˆ2 to approach the ideal value k2 = −30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation a ˆ1 of the disturbance amplitude a1 = 5 . . . . . . . . . . . Estimation ˆb1 of the disturbance amplitude b1 = 2 . . . . . . . . . . . Responses of observer errors . . . . . . . . . . . . . . . . . . . . . . . . Control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram of a deep-sea construction vessel . . . . . . . . . . . . . . . . Diagram of the plant (6.32)–(6.37) . . . . . . . . . . . . . . . . . . . . Diagram of the system (6.44)–(6.53) . . . . . . . . . . . . . . . . . . . Descending trajectory and velocity—that is, the time-varying cable ˙ . . . . . . . . . . . . . . . . . . . length l(t) and the changing rate l(t) Responses of lateral vibrations w(x, t) without control . . . . . . . . . Responses of longitudinal vibrations u(x, t) without control . . . . . . Closed-loop responses of lateral vibrations w(x, t) . . . . . . . . . . . Closed-loop responses of longitudinal vibrations u(x, t) . . . . . . . . Control forces U1 (t) and U2 (t) . . . . . . . . . . . . . . . . . . . . . . Observer error of lateral vibrations w(x, ˜ t) . . . . . . . . . . . . . . . . Observer error of longitudinal vibrations u ˜(x, t) . . . . . . . . . . . . . Lateral oscillation drag forces from ocean current disturbances . . . . Closed-loop responses of longitudinal vibrations u(x, t) in the actual nonlinear model with unmodeled disturbances . . . . . . Closed-loop responses of lateral vibrations w(x, t) in the actual nonlinear model with unmodeled disturbances . . . . . . . . . . . . . .

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84 96 111 118 119 119 119 120 120 120 121 121 131 137 139 159 160 161 161 161 162 162 162 163 163 164

LIST OF FIGURES

˜(·, t) + w(·, ˜ t) in the actual nonlinear model 6.15 Observer errors u with unmodeled disturbances . . . . . . . . . . . . . . . . . . . . 6.16 Control forces in the actual nonlinear model . . . . . . . . . . . 7.1 Diagram of a deep-sea construction vessel . . . . . . . . . . . . . 7.2 Event-based closed-loop system . . . . . . . . . . . . . . . . . . . 7.3 Control forces (continuous-in-time U and event-based Ud ) . . . . 7.4 The evolution of z¯(¯ x, t) under the proposed event-based control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 The evolution of w(¯ ¯ x, t) under the proposed event-based control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The evolution of the observer error z˜(¯ x, t) . . . . . . . . . . . . . 7.7 The evolution of the observer error w(¯ ˜ x, t) . . . . . . . . . . . . . 7.8 The evolution of the internal dynamic variable m(t) in the ETM 7.9 The oscillation velocity and displacement of the payload . . . . 7.10 The cable oscillation energy . . . . . . . . . . . . . . . . . . . . . 8.1 A drill string used in offshore oil drilling . . . . . . . . . . . . . 8.2 Block diagram of the closed-loop system . . . . . . . . . . . . . 8.3 Open-loop responses of u(x, t) . . . . . . . . . . . . . . . . . . . . 8.4 Open-loop responses of u(0, t), ut (0, t) . . . . . . . . . . . . . . . 8.5 Open-loop responses of z(x, t), w(x, t) . . . . . . . . . . . . . . . 8.6 The proposed control input and the PD control input . . . . . . 8.7 Closed-loop responses of z(x, t), w(x, t) . . . . . . . . . . . . . . 8.8 Closed-loop responses of X(t) = [u(0, t), ut (0, t)]T under the proposed adaptive controller and the PD controller . . . . . . . . 8.9 Closed-loop response of u(x, t), which physically represents the torsional vibrations of the oil-drilling pipe under the proposed controller . . . . . . . . . . . . . . . . . . . . . . . . 1 8.10 Closed-loop response of the norm (ut (·, t)2 +ux (·, t)2 ) 2 under the proposed adaptive controller and the PD controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Adaptive estimation errors of the anti-damping coefficient c and the disturbance amplitudes a1 , b1 . . . . . . . . . . . . . . . 9.1 Response of u(x, t) in the plant (9.1)–(9.6) without control . . . 9.2 Response of u(x, t) in the plant (9.1)–(9.6) with the controller (9.52) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Response of v(x, t) in the plant (9.1)–(9.6) without control . . . 9.4 Response of v(x, t) in the plant (9.1)–(9.6) with the controller (9.52) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Response of input ODE states z(t), z(t) ˙ in the open-loop and closed-loop cases . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Response of ODE state X(t) in the open-loop and closed-loop cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Controller U (t) (9.52) in the closed-loop system . . . . . . . . . 10.1 Block diagram of the plant (10.1)–(10.7) . . . . . . . . . . . . . . 10.2 Diagram of the closed-loop system . . . . . . . . . . . . . . . . . 10.3 Schematic of a DCV used in seafloor installation . . . . . . . . . 10.4 Responses of w(x, t), z(x, t) without control . . . . . . . . . . . 10.5 Responses of w(x, t), z(x, t) with control . . . . . . . . . . . . . 10.6 Responses of X(t), Y (t) with control . . . . . . . . . . . . . . . 10.7 Observer errors w(x, ˜ t), z˜(x, t) . . . . . . . . . . . . . . . . . . . .

xi

. . . . .

. . . . .

. . . . .

164 164 172 192 195

. . .

195

. . . . . . . . . . . . .

. . . . . . . . . . . . .

196 196 196 197 197 197 203 215 223 224 224 226 226

. . .

227

. . .

228

. . .

228

. . . . . .

229 265

. . . . . .

266 266

. . .

266

. . .

267

. . . . . . . . .

267 267 274 296 298 302 302 303 303

. . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

xii 10.8 10.9 10.10 11.1

11.2 11.3 11.4

11.5 11.6 11.7 12.1

12.2 12.3 12.4 12.5 12.6 12.7 12.8 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

LIST OF FIGURES

Cable transverse oscillation energy ρ2 ut (·, t)2 + T20 ux (·, t)2 . . . Transverse displacement bL (t) of the payload . . . . . . . . . . . . . Control force of the onboard crane . . . . . . . . . . . . . . . . . . The relationship between the sandwich ODE-PDE-ODE hyperbolic system and the mining cable elevators consisting of a hydraulic-driven head sheave, mining cable, and cage . . . . . . Output-feedback event-based control input Ud (t) . . . . . . . . . . Axial vibration velocity of the cage X(t) and moving velocity of the hydraulic rod in the hydraulic cylinder . . . . . . . . . . . . Axial vibration displacement of the cage (initial elastic displacement 0.005 m) and movement of the hydraulic rod in the hydraulic cylinder (initial position 0.001 m) . . . . . . . . . . . . . Response of z(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . Response of w(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial vibration energy VE of the cable . . . . . . . . . . . . . . . . Block diagram of the output-feedback closed-loop system consisting of the plant (12.1)–(12.8), the observer (12.58)–(12.65), and the controller (12.127) . . . . . . . . . . . . . . . . . . . . . . . . . Moving boundary and its velocity . . . . . . . . . . . . . . . . . . . Open-loop responses of u(·, t) and v(·, t) . . . . . . . . . . . . . Responses of u(·, t) and v(·, t) under the proposed output-feedback controller . . . . . . . . . . . . . . . . . . . . . . . Responses of ODE states z(t), s1 (t), s2 (t), X(t) under the proposed output-feedback controller . . . . . . . . . . . . . . . . . . Observer errors of ˜ u(·, t), ˜ v (·, t) . . . . . . . . . . . . . . . . . . ˜ Observer errors of s˜2 (t), X(t) . . . . . . . . . . . . . . . . . . . . . Output-feedback control input . . . . . . . . . . . . . . . . . . . . . Mining cable elevator with viscoelastic guideways . . . . . . . . . . ˙ Time-varying domain l(t) and the according velocity l(t) . . . . . . ˆ ˆ Self-tuned control gains k1 , k2 , whose target values are k1 = −31.3, k2 = −78.2 . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive event-based control input and the continuous-in-time adaptive control input . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic internal variable md (t) in the ETM . . . . . . . . . . . . Observer errors at the midpoint of the time-varying spatial domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Responses of w l(t) ,t . . . . . . . . . . . . . . . . . . . . . . . .  2  l(t) Responses of z 2 , t . . . . . . . . . . . . . . . . . . . . . . . . .

13.9 Responses of x1 (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.10 Responses of x2 (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 13.11 Time evolution of the norm (ut (·, t)2 + ux (·, t)2 ) 2 , which physically reflects the vibration energy of the cable modeled by (13.104)–(13.106) . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 The adaptive certainty-equivalence control scheme with regulation-triggered batch least-squares identification . . . . . . . 14.2 Parameter estimates qˆ1 (t), qˆ2 (t). . . . . . . . . . . . . . . . . . . 14.3 The evolution of |ζ(t)| under the nominal control (14.23) and the proposed adaptive regulation-triggered control (14.36) . . . . . .

. . .

304 304 304

. .

333 336

.

337

. . . .

337 337 338 338

. . .

356 359 360

.

360

. . . . . .

361 361 361 362 394 395

.

396

. .

397 397

. .

397 397

.

398

. . . .

398 398

. .

399

. . . .

415 426

. .

426

LIST OF FIGURES

14.4 14.5 14.6 14.7 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9

xiii 1

The evolution of Ω(t) 2 under the nominal control (14.23) and the proposed adaptive regulation-triggered control (14.36) . . . . . . The evolution of w(x, t) under the proposed adaptive regulation-triggered control (14.36) . . . . . . . . . . . . . . . . . . . The evolution of z(x, t) under the proposed adaptive regulation-triggered control (14.36) . . . . . . . . . . . . . . . . . . . The control signals of the nominal control (14.23) and the proposed adaptive regulation-triggered control (14.36) . . . . . . . . Implementing the triggering mechanism (15.37)–(15.41) . . . . . . . . The continuous-in-state control signal Uc (t) in (15.18) and the piecewise-constant control signal Ud (t) in (15.17). . . . . . . . . . . . Parameter estimates dˆ3 and dˆ2 . . . . . . . . . . . . . . . . . . . . . . Parameter estimate a ˆ . . . . . . . . . . . . . . . . . . . . . . . . . . The evolution of d(t)2 and ϑV (t) − m(t) in (15.38) . . . . . . . . . . The evolution of ζ(t), m(t) under the control input Ud (t) in (15.17) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The evolution of w(x, t) under the control input Ud (t) in (15.17) . . . The evolution of z(x, t) under the control input Ud (t) in (15.17) . . . Density of the inter-execution times computed for eight different 2 ¯ , initial conditions given by z(x, 0) = 0.5 sin(2¯ nπx/L + π/6) + k(x/L) 2 ¯ for n ¯ = 1, 2, m ¯ = 1, 2, k¯ = 0, 1 w(x, 0) = 0.5 sin(3 mπx/L) ¯ + k(x/L)

426 427 427 427 447 461 462 462 462 463 463 463

464

List of Tables

1 2 3 2.1 3.1 4.1 4.2 5.1 6.1 6.2 7.1 8.1 10.1 11.1 12.1 13.1 13.2 15.1 15.2

The motion types . . . . . . . . . . . . . . . . . . . . . . . . . . . Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unmeasured and unknown effects . . . . . . . . . . . . . . . . . . Physical parameters of the mining cable elevator . . . . . . . . . Physical parameters of the dual-cable mining elevator . . . . . . Physical parameters of the mining elevator . . . . . . . . . . . . Simulation parameters of the mining elevator . . . . . . . . . . . Physical parameters of the mining cable elevator . . . . . . . . . Physical parameters of the DCV . . . . . . . . . . . . . . . . . . Comparison of previous results on the boundary vibration control of cables . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical parameters of the DCV . . . . . . . . . . . . . . . . . . Physical parameters of the oil-drilling system . . . . . . . . . . . Physical parameters of the DCV . . . . . . . . . . . . . . . . . . Physical parameters of the mining cable elevator . . . . . . . . . Comparison of results on the boundary control of sandwich systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical parameters of the descending mining cable elevator . . Parameters of the proposed adaptive event-based control system Physical parameters of the mining cable elevator . . . . . . . . . Recent results of triggered-type adaptive boundary control of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

xix xx xxi 27 50 65 80 117 132

. . . . .

. . . . .

. . . . .

168 171 223 300 334

. . . .

. . . .

. . . .

372 394 396 460

. . .

470

Preface

This book deals with the boundary control of wave partial differential equations (PDEs) in one dimension, with a moving actuated boundary, or with finite-dimensional dynamics (modeled as ordinary differential equations; ODEs) at either the unactuated boundary or in the actuation path. These structures are inspired by applications involving cables and strings that move mechanical loads. Cables moving mechanical loads are most conspicuously employed in elevators (both in buildings, where they may be hundreds of meters long, and in deep mining, where they may be kilometers long), but also in undersea construction, such as laying telecommunication cables along uneven seafloor or building artificial reefs that have an environmentally beneficial purpose for the sea life, and in other applications, including many yet to be revealed in other domains. In deep-sea drilling, the so-called drill string is not a cable but a kilometerslong thin cylinder, with a drill bit on its unactuated end, and its dynamics of rotation (torsional dynamics) are governed by the wave equation; that is, they are mathematically equivalent to string or cable vibration. Hence, our book’s overarching title for this multitude of physical configurations, load types, and operating purposes — PDE Control of String-Actuated Motion. The actuation of mechanical loads by means of strings (cables) has its advantages over rigid connections. It allows for significant improvements in energy efficiency, weight, size of the operation workspace, operation speed, and maximum payload, compared with rigid-body mechanisms, due to the string’s properties of lower weight, resisting relatively large axial loads, and low bending and torsional stiffness. However, the distributed parameter nature of a string or cable makes the control design of the cable-actuated mechanisms more challenging than the traditional ODE-based control designs for lumped parameter rigid-body mechanisms, giving rise to many new problems in boundary control of PDEs. The theoretical challenges and practical significance have led us to carry out research on many topics involving cables and strings of time-varying length, with moving loads, and with actuator dynamics. This book presents this collection of methodologies, whose meaning is predominantly mathematical, but whose inspiration comes entirely from applications and technology. Cable elevators and other lifting and depositing tasks, as well as drilling at a high penetration rate, introduce a heretofore unstudied problem of boundary control of wave PDEs with moving boundaries—that is, on time-varying domains. This is the central issue of our book—wave PDE control on one-dimensional domains of timevarying lengths—that is, vibration suppression in strings of time-varying lengths. One should note that, as the cable length varies with time, possibly fast, even if one could be talking about eigenvalues and eigenfunctions (that rapidly change),

xviii

PREFACE

the spectral approaches to control design certainly would not be applicable. We approach this challenge using time-domain and Lyapunov-based approaches. The second key challenge is that, at the moving end of a time-varying string, a load is present, and the motion of this load needs to be controlled with an actuator on the opposite end of the string. The objective is to suppress the vibration in both the string of varying length and in the load at the distal end from the actuator. Using feedback control to add damping artificially, where physical viscous damping might be absent or insufficient, would be easy at the location of (i.e., proximal to) the actuator. However, the actuator being at a boundary makes emulating viscous damping along the entire string, and at the load on the string’s distal end, challenging. This challenge is met using the method of PDE backstepping. Backstepping employs two tools, a Volterra transformation of the infinite-dimensional state, and feedback, to add damping at locations other than the actuator. But, prior to employing the backstepping transformation, we usually employ first a transformation of the state into the Riemann variables, which is a canonical representation for coupled hyperbolic PDEs, to which backstepping is readily applicable. However, the actuator in most boundary-controlled cable and string systems does not act directly and instantly. The actuator, be it hydraulic or electrical, has its own considerable inertia—namely, its own lumped-parameter dynamics modeled by an ODE. These dynamics themselves have to be overcome using finite-dimensional backstepping (the classical integrator backstepping). Hence, the overall system that arises in string-actuated motion is often a sandwiched ODE-PDE-ODE configuration, with an input acting on only one of the two ODEs at the end of the PDE, and not directly on the PDE. The reality of applications gives rise to additional effects: nonlinearities, disturbances, unknown parameters, input delays, sampled (or event-triggered) sensing, as well as many more which we deem beyond the page and time limits of this book. A vast literature exists on control of overhead cranes and gantry cables. A good entry point into this literature, in terms of both the theory and applications, is the tutorial article [23]. A large portion of the research on this topic justifiably focuses on transverse motion and employs the techniques of differential flatness, finite-time motion planning, and finite-horizon optimal control. Many, if not most of these approaches, are concerned with designing open-loop control signals. While some of our work in this book is applicable to the transversal motion of overhead cranes and gantry cables, these problems are not our focus here. Instead, we focus on vibration suppression, in axial and other directions, cables of varying length, Lyapunov stabilization techniques, and handling of uncertainties. What Does the Book Cover? The book comprises three parts. The first part is devoted to various control applications, as drivers for control design and theoretical study. Control problems for mining cable elevators are introduced in chapters 2–5, focusing on single-cable elevators in chapter 2, dual-cable elevators in chapter 3, and airflow disturbances and the influence of flexible guideways in chapter 4 and chapter 5, respectively. In addition to the mining cable elevator, the deep-sea construction vessel for undersea moving is also a cable-actuated manipulator to move mechanical loads. Its basic control design is introduced in chapter 6, and additional real-world effects—that is,

PREFACE

xix Table 1. The motion types Chapter 2 3 4 5 6

Axial √ √

√ √

13 15

√ √ √

8

11

Torsional

√

7

10

Transversal

√ √ √ √

sensor signal delays occurring in large-distance signal transmission through acoustic devices and the requirement of reducing changes in the actuator signal considering the massive ship-mounted crane—are dealt with in chapter 7. Apart from the cableactuated mining elevator and the deep-sea construction vessel, another distinct but kindred application, deep-sea drilling, is tackled in chapter 8. Inspired by but going beyond the applications in part I, generalized control problems are dealt with in part II—that is, boundary control of sandwich hyperbolic PDEs in particular. Control of the sandwich systems is covered in chapters 9–12, with the basic control design presented in chapter 9 and then extended to a variety of more challenging problems, including control of the sandwich systems with sensor delay in chapter 10, with event-triggered design in chapter 11, and with nonlinearities in chapter 12. The general results in part II are justified by applications in part I. The last of the three parts presents triggered-type adaptive control of hyperbolic PDEs. Three triggered adaptive control schemes (event-triggered control, regulation-triggered parameter estimation, and a combination of both) for hyperbolic PDE-ODE systems are developed in chapters 13–15, respectively. The triggeredtype adaptive control results in part III are also verified in the applications in part I. The book deals with all of the three possible motions of strings or cables: longitudinal/axial/stretching, lateral/transversal/bending, and angular/rotational/torsional. However, it is only in one chapter that we deal with more than one of these three motions. In chapter 6 we deal with coupled longitudinal-lateral vibrations. Table 1 gives an overview of the motion types that each chapter covers. The suppression of axial vibrations dominates our exposition, with transversal vibrations a close second.

PREFACE

xx Table 2. Configurations Chapter 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ODE at distal end of PDE √

ODE at proximal end of PDE

√ √ √ √ √ √ √

√

√

√

√

√

√

√

√ √ √

All of the problems considered in the book incorporate at least one PDE and one ODE. Some consider a second ODE as well. The configurations considered are given in table 2. Those configurations that include ODEs at both distal and proximal ends in chapters 9–12 (namely, about a third of the book) are sandwich systems. In addition, chapter 10 contains a sensor delay at the distal end. We consider both PDE-ODE systems that are fully known and those that contain unknown or unmeasured quantities—such as unmeasured states, unmeasured disturbances, and unknown parameters. In table 3 we overview the contents of the book based on the unmeasured and unknown effects. Virtually all of our exposition is for cables and (drill) strings that are not instrumented with distributed sensing, as is consistent with reality. Disturbances and unknown parameters occupy a large share of the book and create some of the most significant challenges for design and analysis. When it comes to adaptive control, a topic dealt with comprehensively for coupled hyperbolic PDEs in [9] (and for parabolic PDEs a decade earlier in [166]) is tackled in chapters 5, 8, 13, 14, and 15 of this book, as indicated in table 3. Chapters 5 and 8 employ a classical continuous-in-time Lyapunov-based approach. On the other hand, chapters 13–15 employ novel event-triggered approaches. Chapters 13 and 14 are very different in what event triggering is employed for. An adaptive controller consists of two components: the control law and the parameter estimator (update law). Both of these components can employ piecewise-constant values— the control input and the parameter estimate. And the changes in the piecewiseconstant values in both of these components can be triggered in various ways. In

PREFACE

xxi Table 3. Unmeasured and unknown effects

Chapter 2 3 4 5 6 7 8 10 11 12 13 14 15

Unmeasured

Unmeasured

Unmeasured

Unknown

PDE state √

ODE state

disturbance

parameters

√ √

√

√

√

√

√

√

√

√

√

√

√ √

√

√

√

√

√

√

√ √ √

chapter 13 we employ continuous parameter updates and event-triggered control inputs. Conversely, in chapter 14 we employ continuous control inputs (except for finite time instants) and event-triggered parameter updates. Because the parameter estimator is the more delicate of the two components of an adaptive control since it is generally not endowed with convergence guarantees, it is chapter 14 that is considerably more challenging of the two chapters. In chapter 15, simultaneous triggering is employed for the parameter update law and the control law, the result of which is that both the parameter estimates and the control input employ piecewise-constant values. For the researcher in coupled hyperbolic PDE systems who is interested in going beyond the basic 2 × 2 case, which is superbly covered in [16] and [9], there are interesting designs for 4 × 4 cases in this book—for example, in chapter 3 and chapter 6. Chapter 3 deals with axial oscillations in a pair of cables connected by a payload at the distal boundary. So the 4 × 4 system in chapter 3 is a set of two 2 × 2 pairs that are coupled not along the domain but at the boundary. In contrast, chapter 6 deals with a single cable but with axial and lateral oscillations that bring domain-wide coupling into the plant and, therefore, a fully coupled 4 × 4 hyperbolic system. While in multiphase flows, in both oil drilling and production, as well as in congested multiclass traffic flow, a larger number of first-order hyperbolic PDEs arise in the direction toward the actuated end than away from it, this interesting occurrence of underactuated heterodirectional hyperbolic PDEs does not arise with cables.

xxii

PREFACE

What Niche in the Literature Does This Book Fill? The main inspiration for this book comes from the cascade PDE-ODE configurations in [116]. The book [111] develops the cascade ideas from [116] in the parabolic PDE realm for applications in additive manufacturing. The present book expands the reach of [116] into cable-operated systems and on time-varying domains. But this book’s closest cousin may be [77], a major volume spawned from the classic [132]. While [77, 132] employ collocated static feedbacks for wave PDEs on static domains, and in the absence of ODEs, our focus is broader in terms of plant structure (varying domain, ODE included), methodology (backstepping controller and observer designs), and the emphasis on applications. We do not, however, deal with beam systems. For Whom Is This Book This book should be valuable to researchers working on control and dynamic systems—engineers, graduate students, and PDE system specialists in academia. Mathematicians with interest in control of distributed parameter systems will find the book stimulating, because it tackles and opens a door for control of sandwich PDE systems, which present many stimulating challenges and opportunities for further research on the stabilization of ever-expanding classes of unstable sandwich PDE systems. Engineers in mechanical, aerospace, and civil/structural engineering, focusing on vibration or motion control, especially for flexible structures or manipulators, will learn some new and useful methodologies for designing controller/observer algorithms, and addressing some problems they have no doubt faced in practice: time delay, disturbances, uncertainties, and so on. The background required to read this book includes little beyond the basics of function spaces and Lyapunov theory for ODEs. We hope that the reader will regard the book not as a collection of problems that have been solved but as a collection of tools and techniques that are applicable to open problems, particularly in interconnected systems of ODEs and PDEs, and to physical problems modeled by PDE-ODE coupled systems.

ACKNOWLEDGMENTS We would like to thank Shumon Koga for his contribution to chapter 2, Yangjun Pi for his contributions to chapters 2, 3, and 4, Shuxia Tang for her contributions to chapters 4, 5, and 8, and Iasson Karafyllis for his contribution to chapter 14. We are grateful to Rafael Vazquez, Jean-Michel Coron, Ole Morten Aamo, Florent Di Meglio, Federico Bribiesca Argomedo, Long Hu, Joachim Deutscher, Henrik Anfinsen, Delphine Bresch-Pietri, and their coauthors for their inspiring early work on boundary control of the 2 × 2 linear hyperbolic system. The pioneering work of Wei Guo, Hongyinping Feng, and Bao-Zhu Guo on extending the active disturbance rejection control (ADRC), introduced by Jingqing Han for finite dimension systems, to infinite dimension systems inspired chapters 4 and 8. We also would like to thank Nicolas Espitia, Christophe Prieur, and their coauthors for the original work on event-triggered designs in hyperbolic PDEs, from which we benefited in developing chapters 7, 11, and 13. We also thank CITIC Heavy Industries Co. Ltd.

PREFACE

xxiii

for providing practical motivations and expertise on the mining cable elevator in chapters 2–5. Ji Wang would like to give a special thanks to his family for their unwavering support and love. Miroslav Krstic thanks the National Science Foundation, the Air Force Office of Scientific Research, the Los Alamos National Laboratory, and Dr. Dan Alspach for research support, as well as his colleagues at General Atomics for enriching discussions on cable-actuated control systems. La Jolla, California July 2021

Ji Wang Miroslav Krstic

Chapter One Introduction

This chapter presents the motives, as well as a limited literature summary, for both the applied and the theoretical control design problems that we pursue in this book.

1.1

STRING-ACTUATED MECHANISMS

In this section, we introduce four string-actuated mechanisms: mining cable elevators, deep-sea construction vessels (DCVs), unmanned aerial vehicles (UAVs), and oil-drilling systems. Mining Cable Elevators In mining exploitation, a cable elevator, which is used to transport the cargo and miners between the ground and the working platform underground, is an indispensable piece of equipment [99]. The mining cable elevator is a cable-actuated moving load system. A common arrangement is a single-drum system [178]: a single-cable mining elevator comprising a driving winding drum, a steel wire cable, a head sheave, and a cage. The cable plays a vital role in mining elevators because its advantages of low bending and torsional stiffness, resisting relatively large axial loads, are helpful to heavy-load and large-depth transportation. However, the compliance property, or stretching and contracting abilities of cables, tends to cause mechanical vibrations, especially when the elevator is running at high speed, which leads to tension oscillations and premature fatigue fracture [87, 88, 91]. Therefore, the importance of suppressing the vibrations and tension oscillations cannot be overestimated, for the safety of both personnel and profitability. The economical and convenient way to suppress the vibrations is by designing an appropriate control input without modifying the original structure of the mining cable elevator. In chapter 2, the control design for axial vibration suppression of a high-speed, single-cable mining cable elevator is presented. For operation at a greater depth, such as over 2000 m, and carrying a heavier load, the single-cable elevator is not suitable. Because a very thick cable is required to bear the heavy load, such a thick cable, at high bending, suffers from problems in the winding on the winder drum. A dual-cable mining elevator [37], shown in figure 3.1 in chapter 3, is proposed to solve this problem, removing the requirement of a very thick cable because two cables tow the cage. However, an imbalance problem, such as cage roll, frequently appears in the dual-cable elevator [184], as shown in figure 3.1, for which, taut cables are used as flexible guide rails because traditional steel rails come with a high cost of manufacture and installation in deep mines. Cage roll will increase the differences in oscillation tension between two cables and

2

CHAPTER ONE

enlarge the oscillation amplitude of the tension, which accelerates premature fatigue and requires inspections and costly repairs. One feasible arrangement to balance the cage roll and suppress the vibrations and tension oscillations in cables of the dual-cable mining elevator is to design and apply additional control forces through actuators at floating head sheaves, as shown in figure 3.1. The control design for suppressing of the axial vibrations and tension oscillations and balancing the cage roll in a dual-cable mining elevator is presented in chapter 3. In an actual operating environment, the cage is usually subject to uncertain airflow disturbances. Additionally, the flexible guides, with their uncertain properties, may affect the smooth and steady running of the mining cable elevator. These factors inspired the control designs in chapters 4 and 5. Deep-Sea Construction Vessels In deep-sea oil exploration, some equipment, such as a subsea manifold, a subsea pump station, and a subsea distribution unit along with associated foundations, flow lines and umbilicals, is installed at designated locations [167, 168] around the drill center on the seafloor. The installation of the equipment is completed by a DCV [168, 182] because the installation sites are located outside a radius of 45 m from the floating drilling platform (see figure 2 in [168]) and cannot be accessed by the huge floating drilling platform, which has limited access and mobility [168], and because some of the equipment, such as flow lines and umbilicals, is installed in advance to prepare to hook up the floating drilling platform when it arrives. A DCV is shown in figure 6.1 in chapter 6, where the top of the cable is attached to a crane on a vessel at the ocean surface, and the cable’s bottom is attached to equipment to be installed at the sea floor. The traditional method in underwater installation by DCVs is to regulate the vessel dynamics position and manipulate the crane to obtain the desired heading for the payload [94]. Such a method is not suitable for the deeper water construction in offshore oil drilling (more than 1000 m) because the cable is very long when the payload is near the seabed, which would introduce large cable oscillations [94, 182], causing a large offset between the payload and the desired heading position of the crane—namely, the designated installation location. In chapter 6, the control forces at the onboard crane are designed to reduce the cable oscillations and then place the equipment in the target area on the sea floor. In chapter 7, we employ a piecewise-constant control input that is more suitable for the massive ship-mounted crane and compensate for delays in the transmission of sensing signals from the seabed to the vessel on the ocean surface through acoustic devices. Unmanned Aerial Vehicles In addition to DCVs, the control design in chapter 6 can also be applied to unmanned aerial vehicles (UAVs) used to aid delivery to dangerous and inaccessible areas, such as to flood, earthquake, fire, and industrial disaster victims [70, 140]. Food and first-aid kits are tied to the bottom of a cable, whose other end is attached to the UAV. The swing/oscillation of the cable-payload may appear during the transportation motion due to the properties of the cable and external disturbances, such as wind, which may cause damage to the suspended object, the environment, and the people nearby [70]. At the end of the transport motion, when the UAV arrives at the location directly over the rescue site and is ready to land the aid supplies,

INTRODUCTION

3

the suspended object naturally continues to swing [140], which makes precisely placing the aid supplies at the target position difficult. Therefore, rapid suppression of the oscillations of the cable and suspended object through a control force provided by the rotor wings of the UAV is required. In addition to aid delivery in disaster relief, UAV delivery is also used in some commercial cases to reduce labor cost. For example, some companies use UAVs to transport cargo in storehouses or lift and position building elements in architectural construction [191]. Some logistics companies have also begun to use UAVs to deliver packages in small areas [70]. Oil Drilling Oil-drilling systems used to drill deep boreholes for hydrocarbon exploration and production often suffer severe vibrations, which can cause the premature failure of drilling string components, damage to the borehole wall, and problems with precise control [98]. The vibrations also cause significant wastage of drilling energy [53]. The suppression of vibrations in the oil-drilling system is thus required for the economic interest and improvement of drilling performance [156]. There are three main types of vibration in oil-drilling systems [154]: vertical vibration, also called the bit-bouncing phenomenon, lateral vibration due to an out-of-balance drill string, which is called whirl motion, and torsional vibration, which appears due to friction between the bit and the rock. This nonlinear torsional interaction between the drill bit and the rock will cause the bit to slow down and even stall while the rotary table is still in motion. Once enough energy is accumulated, the bit will suddenly be released and start rotating at very high speed before slowing down again [24], settling into a limit cycling motion. This is called the stick-slip phenomenon. Several physical laws of bit-rock friction [156] are used to roughly describe stick-slip behavior in the oil-drilling system, such as the velocity-weakening law [31], the stiction plus Coulomb friction model [160], a class of Karnopp model [38, 107, 136], and so on. The stick-slip oscillations lead to instability from the lower end to travel up the drill string to the rotary table, which results in distributed instabilities and is the primary cause of fatigue to the drill collar connections as well as damage to the drill bit [154]. Therefore, suppressing torsional vibrations (stick-slip oscillations) in the oil-drilling system is important. In addition waves at the sea surface causing a heaving motion of the drilling rig [1] in an offshore rotary oil-drilling system [186] introduce an external disturbance at the bit, which is another instability source. As will be seen in chapter 8, the designed control input at the rotary table goes down from the rotating table, through the drill string, to the drill bit, to eliminate the stick-slip instability and, as a result, reduce the oscillations of the angular displacement and velocity of the drill bit. 1.2

HYPERBOLIC PDE-ODE SYSTEMS

The dynamics of the aforementioned string-actuated mechanisms are governed by hyperbolic partial differential equation-ordinary differential equation (PDE-ODE) systems. The design of controllers for such hyperbolic PDE-ODE systems requires boundary control approaches because the control input can only be applied at one end of the string in such mechanisms. In this section, we review boundary control

4

CHAPTER ONE

of elementary wave PDE-ODE systems, as well as of a class of coupled first-order hyperbolic PDE models, with the possible inclusion of an ODE in cascade with the hyperbolic PDEs. Wave PDE-ODE Systems A wave PDE-ODE system serves as a basic model of a string-actuated mechanism in which the wave PDE describes a vibrating string (without in-domain damping), and the attached payload is modeled as an ODE. Classical results on backstepping boundary control for wave PDEs with anti-damping terms in domain or on the uncontrolled boundary are found in [121, 162, 165]. In the past decade, many results on boundary control of wave PDE-ODE systems have been reported. The very first result on boundary control of a wave PDE-ODE plant was presented in [114], where the interconnection is of the Dirichlet type. Boundary control design for a wave PDE-ODE cascade system with a Neumann-type interconnection was also addressed in [170]. The boundary control problem was also tackled in [18, 28, 29] for a wave PDE-nonlinear ODE system. Coupled First-Order Hyperbolic PDEs For the sake of greater clarity in control design and analysis, wave PDEs can be converted to a class of heterodirectional coupled first-order hyperbolic PDE systems via the Riemann transformations [26]. Especially when considering the in-domain viscous damping terms describing the string material damping, there would exist in-domain coupling in the resulting coupled first-order hyperbolic PDE systems [147], which makes the control design more challenging. Some theoretical results on boundary control of coupled first-order hyperbolic PDEs have emerged over the last decade. The basic boundary stabilization problem of 2 × 2 coupled linear transport PDEs was addressed in [32, 177] by backstepping, based on which the extended results on boundary control of these 2 × 2 systems were presented in [4, 39]. The sliding mode approach and the proportional integral (PI) controller design applied to the control of such a 2 × 2 system was also considered in [127] and [173], respectively. Boundary control of the 2 × 2 transport PDE system was further extended to that of an n + 1 system in [50]. For a more general coupled transport PDE system where the number of PDEs in either direction is arbitrary, a boundary stabilization law was first designed by backstepping in [96], which is a systematic framework for control design for this class of systems, and other extended results were proposed in [6, 14, 40, 41, 97]. In addition to the applications to string/cable models, the boundary control design for coupled first-order hyperbolic PDEs has also been applied to water-level dynamics [45, 47, 51, 143, 142] and traffic flow [101, 194, 195, 197]. In the past five years, some results on the control of linear coupled hyperbolic PDEs cascaded with ODEs have also been reported. The first is [48], which addressed the state-feedback stabilization of a general linear hyperbolic PDE-ODE coupled system. The state-feedback boundary control design of a 2 × 2 linear hyperbolic PDE-ODE coupled system with nonlocal terms was also dealt with in [169]. Based on the observer design, in [44] an output-feedback controller with anticollocated measurements was proposed to stabilize general linear heterodirectional hyperbolic PDE-ODE systems with spatially varying coefficients.

INTRODUCTION

1.3

5

“SANDWICH” PDEs

In the results discussed in section 1.2, the control input enters the PDE boundary directly, neglecting the inertia and dynamics of the actuator. However, in some applications, actuator dynamics cannot be neglected, especially when the actuator has its own considerable inertia. Incorporating the actuator dynamics into the control design of the string-actuated mechanisms modeled by hyperbolic PDE-ODE systems gives rise to a more challenging problem: control of what we call sandwich ODE-PDE-ODE systems. The first backstepping state-feedback control design for sandwich hyperbolic systems was proposed in [113], which considered a transport PDE-ODE system with an integration (first-order ODE) at the input of the transport PDE. Also, the control problem of an ODE with input delay and unmodeled bandwidth-limiting actuator dynamics, which is represented by an ODE-transport PDE-ODE system where the input ODE is first order, was addressed in [118]. The boundary control design of a transport PDE sandwiched between two ODEs describing actuator and sensor dynamics was also proposed in [8]. Regarding coupled transport PDEs, state-feedback control of heterodirectional coupled transport PDEs sandwiched between two ODEs was proposed in [151, 152, 183]. Adding observer designs, output-feedback control of this type of sandwich systems was developed in [43, 49, 180, 181]. Parameter identification of a drill string, modeled as a wave PDE sandwich system from experimental boundary data, was studied in [150]. Regarding other types of PDEs, boundary control of viscous Burgers PDE, heat PDE, and n coupled parabolic PDE sandwich systems was also addressed in [130, 179] and [42], respectively. A fairly complete theory for boundary control of sandwich hyperbolic PDEs is derived in chapters 9–12, including basic design, delay compensation, event-triggered design, and design with nonlinearities.

1.4

ADVANCED BOUNDARY CONTROL OF HYPERBOLIC PDEs

Apart from the basic boundary control designs mentioned in sections 1.3 and 1.2, in this section we review some extended results on disturbance attenuation, adaptive control, delay compensation, and event-triggered control for hyperbolic PDEs. Disturbance Rejection/Cancellation Most research on disturbance rejection and adaptive cancellation for PDE systems focuses on disturbances collocated with control. Sliding mode control (SMC) schemes designed for heat, Euler-Bernoulli beam, and Schr¨odinger equations with boundary input disturbances were presented in [73, 76, 189]. The internal model principle [63] on the basis of the estimation/cancellation strategy was utilized in the beam equation [145]. For wave PDEs, adaptive disturbance cancellation was used in the output-feedback asymptotic stabilization of one-dimensional wave equations subject to harmonic disturbances at the controlled end and at the measured output in [82, 83] and [81], respectively. By applying the active disturbance rejection control method introduced by Han [86] for ODEs, state-feedback

6

CHAPTER ONE

or output-feedback control designs for wave PDEs with matched disturbances were presented in [60, 74, 75, 80, 172, 198]. The task of rejection or adaptive cancellation of unmatched disturbances—that is, the disturbances anti-collocated with the control input—is more difficult. While several results for this task have been developed for ODE systems, such as those found in [78, 129, 192] and so on, the literature is less ample in PDE systems, where the disturbance is on the far (distal) end from the control input. A state-feedback controller that practically stabilizes a Schr¨odinger equation-ODE cascade system in the presence of an unmatched disturbance assumed to be small and measurable was presented in [100]. An output-feedback controller was designed for output reference tracking in a wave equation with an anti-collocated harmonic disturbance at a stable damping boundary in [84]. The output regulation problem for a wave equation with a harmonic anti-collocated disturbance at a free boundary was further dealt with in [85]. In chapters 4, 5, and 8, the asymptotic rejection and adaptive cancellation of unmatched disturbances in hyperbolic PDEs are proposed, respectively, along with applications in the control of disturbed mining cable elevators and oil-drilling systems. Adaptive Control Three traditional adaptive schemes for PDEs with uncertain parameters are the Lyapunov design, the passivity-based design, and the swapping design [112, 124]. Using the three design methods initially developed for ODEs in [122], the same three adaptive control approaches were proposed for parabolic PDEs in [123, 163, 164]. For adaptive control of hyperbolic PDEs, many results based on the three conventional methods have also been achieved, as follows. In [24, 25, 26, 117], adaptive control laws were presented for a one-dimensional wave PDE that had an actuator on one boundary and an anti-damping instability with an unknown coefficient on the other boundary. The first result on adaptive control of general first-order hyperbolic partial integro-differential equations was proposed in [20]. An adaptive boundary control design of coupled first-order hyperbolic PDEs with uncertain boundary and spatially varying in-domain coefficients was developed in [5]. In [7], two adaptive boundary controllers of coupled hyperbolic PDEs with unknown in-domain and boundary parameters were proposed using identifier and swapping designs, respectively. More adaptive control results of coupled first-order hyperbolic PDEs have been collected in [9]. Adaptive control design methods for hyperbolic PDEs are employed in chapters 5 and 8 to deal with parameter uncertainties in mining cable elevator and oil-drilling systems. Also, an event-triggered adaptive controller is proposed in chapter 13. Recently, a new adaptive scheme using a regulation-triggered batch least-squares identifier was introduced in [102, 104], which has some advantages over the traditional adaptive approaches, such as guaranteeing exponential regulation of the states to zero as well as finite-time convergence of the estimates to the true values. This method has been successfully applied in the adaptive control of a parabolic PDE where the unknown parameters are the reaction coefficient and the highfrequency gain [106]. Regarding hyperbolic PDEs, using a scalar least-squares identifier updated at a sequence of times with fixed intervals, backstepping adaptive boundary control of a first-order hyperbolic PDE with an unknown transport speed was proposed in [11]. In chapters 14 and 15, adaptive controllers based on batch least-squares identifiers are designed for 2 × 2 hyperbolic PDE-ODE systems where

INTRODUCTION

7

the transport speeds of both transport PDEs and the coefficients of the in-domain couplings are unknown, respectively. Delay Compensation Time delays often exist in practical control systems and may destroy system stability [79]. For example, in a subsea installation by a DCV, sensor delays [94] exist due to the fact that the sensor signal is transmitted over a large distance from the seafloor to the vessel on the ocean surface, through a set of acoustic devices. Such sensor signal transmission may result in information distortion or even make the control system lose stability. Therefore, the time delay is a vital issue that should be considered in the control design. Recently, boundary control designs for hyperbolic PDEs have been proposed that take time delays into consideration. For example, delay-robust stabilizing feedback control designs for coupled first-order hyperbolic PDEs were introduced in [12, 13], achieving robustness to small delays in actuation. In order to compensate arbitrarily long delays, a delay compensation technique was developed in [116, 125], where the delay was captured as a transport PDE, and the original ODE plant with a sensor delay was treated as an ODE-transport PDE cascaded system in the controller and observer designs. The observer was built as a “full-order” type, which estimates both the plant states and the sensor states, compared with some classical results on delay-compensated observer designs [3, 27, 67], which only estimate plant states—namely, observers of the “reduced-order” type. While compensation for arbitrarily long delays by this technique are commonly available for finite-dimensional systems, very few examples exist where such compensation has been achieved for PDEs, including parabolic (reaction-diffusion) PDEs [108, 115]. Delay compensation for the wave PDEs with arbitrarily long delays is more complex than that for reaction-diffusion PDEs. The primary reason is the second-order-intime character of the wave equation. In [119], by treating the delay as a transport PDE and applying a backstepping design, a boundary controller was developed for a wave PDE with compensation for an arbitrarily long input delay and with a guarantee of exponential stability for the closed-loop system. In chapter 10, we design a delay-compensated control scheme for a sandwich hyperbolic PDE in the presence of a sensor delay of arbitrary length by treating the delay as a transport PDE. Event-Triggered Control When implementing the designed PDE control laws in an actual mining cable elevator, two challenges caused by high-frequency elements in the control law appear: 1) the massive actuator, comprising a hydraulic cylinder and head sheave, shown in figure 11.1 in chapter 11, is incapable of supporting the high-frequency control signal, and 2) the high-frequency components in the control input may in turn become vibration sources for the cable. It is thus necessary to reduce the actuation frequency and ensure the effective suppression of the vibrations in the mining cable elevator. Designing sampling schemes to apply to the control input is a potential solution. Designs of sampled-data control laws of parabolic and hyperbolic PDEs were presented in [64, 105] and [35, 103], respectively. Compared with the periodic sampled-data control, event-triggered control, where the input to the massive actuator is changed only at the necessary times determined by an event-triggering mechanism that acts by evaluating the operation of the elevator, is more feasible for

CHAPTER ONE

8

the mining cable elevator from the point of view of energy saving. This motivates us to design event-triggered PDE backstepping control laws. Most of the current designs on event-triggering mechanisms are for ODE systems, such as those in [69, 93, 133, 159, 171]. Designs of event-based controls for PDE systems are still rare in the existing literature. There exist results on the distributed (in-domain) control of PDEs, such as [158, 193]. For boundary control, an event-triggered feedback law was proposed for a reaction-diffusion PDE in [58]. For first-order linear hyperbolic PDEs with dissipativity boundary conditions, an event-triggered boundary control law was originally proposed in [55, 56]. Furthermore, a state-feedback event-based boundary controller for a class of 2 × 2 coupled linear hyperbolic PDEs was designed in [57]. Based on the observer design, the output-feedback event-triggered boundary control of 2 × 2 coupled linear hyperbolic PDEs was developed in [54]. In chapter 11, an event-triggered backstepping boundary control law is derived for a sandwich hyperbolic PDE, and an adaptive event-triggered boundary controller is further developed in chapter 13.

1.5

NOTES

Frequently used notations in this book are given next, with more specialized notational conventions introduced in the coming chapters. The partial derivatives and total derivatives are denoted as fx (x, t) = f  (x) =

∂f (x, t), ∂x

ft (x, t) =

∂f (x, t), ∂t

df (t) f˙(t) = . dt

df (x) , dx

By C k (A), where k ≥ 1, we denote the class of continuous functions on A. The single bars | · | denote the Euclidean norm for a finite-dimensional vector X(t). In contrast, norms of functions (of x) are denoted by double bars. For u(x, t), x ∈ [0, D], by default,  ·  denotes the L2 -norm of a function of x, namely,   D u(·, t) = u(x, t)2 dx, 0

and Sobolev norms such as H 1 [0, D] or even H 2 [0, D] are defined by   D  D u(·, t)H 1 = u(x, t)2 dx + ux (x, t)2 dx, 0

 u(·, t)H 2 =



0

D 0

 u(x, t)2 dx +

D 0

 ux (x, t)2 dx +

and the ∞-norm is denoted by u(·, t)∞ = sup {|u(x, t)|}. x∈[0,D]

D 0

uxx (x, t)2 dx,

Part I

Applications

Chapter Two Single-Cable Mining Elevators

In a high-speed mining cable elevator, the cable’s property of compliance, or its ability to stretch and contract, results in mechanical vibrations, leading to imprecise positioning and premature fatigue fracture [87]. For safe operation, vibration suppression is essential. Hence, an effective and feasible control design for suppressing the vibration of a mining cable elevator is needed. In section 2.1, the axial vibration dynamics of a single-cable mining elevator— that is, a varying-length string lifting a cage, shown in figure 2.1—is modeled as a coupled wave partial differential equation-ordinary differential equation (PDEODE) system with a Neumann interconnection, on a time-varying spatial domain. The control force can only be applied at the head sheave instead of at the cage— that is, a possibly unstable ODE on the far end from the control input of the wave PDE is not directly accessible for control. This is a more challenging task than the setting with the classical collocated boundary damper feedback control. Using the backstepping method, a state-feedback controller with explicit gains is designed to stabilize the coupled wave PDE-ODE system in section 2.2. To build an observerbased output-feedback controller, in section 2.3 an observer with explicit gains is designed to estimate the full distributed states of the varying-length string using only measurable boundary states in an anti-collocated setup. The exponential stability of the observer-based output-feedback control system is proved via Lyapunov analysis in section 2.4. The result is verified in the axial vibration suppression of a single-cable mining elevator via numerical simulations in section 2.5.

2.1

MODELING

Moving Boundary Wave PDE Model A schematic of a mining cable elevator is given in figure 2.1. Because the catenary cable, the part between the drum and the floating head sheave, in figure 2.1(a) is much shorter than the vertical cable (70 m compared to 2000 m), we suppose that the vibrations on the catenary part are negligible, which gives the simplified model of the varying-length cable with a cage shown in figure 2.1(b). Due to the help of the steel guides, the transverse vibrations in the vertical cable can be neglected since they are much smaller than the axial vibrations. Two external forces act on the system. One is the motion control force Ua (t) driven by a motor at the drum, and the other is the vibration control force Uv (t) manipulated by a hydraulic actuator at the floating head sheave. The axial transport motion z ∗ (t) is the rigid-body motion neglecting the compliant property of the cable in the fixed coordinate system O’, whereas z˙ ∗ (t) and z¨∗ (t) are the resulting

CHAPTER TWO

12

Vibration control force Uv(t)

Floating sheave Ua(t)

Motor Ua(t)

Hydraulic actuator Uv(t)

ż* (t)

hoisting velocity ż* (t)

x’

x y’

Acceleration sensor utt(0, t)

O

O’

y

Guide (b) Simplified representation.

(a) Original model.

Figure 2.1. The mining cable elevator.

velocity and acceleration. The dynamics of axial elastic deformations (vibration displacements) u(x, t), with ut (x, t) being the vibration velocity, are in reference to the moving coordinate system O associated with the motion z ∗ (t). Here we assume that the motion z ∗ (t) is controlled perfectly by the control force Ua (t) and plays the role of the known target hoisting trajectory. By using the extended Hamilton’s principle [134, 178], the axial vibration dynamics u(x, t) on a prescribed time-varying domain [0, l(t)], where l(t) = L − z ∗ (t),

(2.1)

−ρ(utt (x, t) + z¨∗ (t)) + EA uxx (x, t) = 0,

(2.2)

is derived as

∗

M (utt (0, t) + z¨ (t)) + EA ux (0, t) + c1 ut (0, t) = 0,

(2.3)

∗ ˙ Uv (t) − EA ux (l(t), t) − (M + ρl(t))g − l(t)ρ(u t (l(t), t) + z˙ (t))

−

JD u(l(t), t) + z¨∗ (t)) − c2 (u(l(t), ˙ t) + z˙ ∗ (t)) = 0, 2 (¨ RD

where the physical parameters are given as M mass of the load, L maximal length of the cable, l(t) time-varying length of the cable, E Young’s modulus of the cable, Aa cross-sectional area of the cable, JD moment of inertia of the drum, RD radius of the drum, ρ linear density of the cable,

(2.4)

SINGLE-CABLE MINING ELEVATORS

13

g gravitational acceleration, c1 cage-guide damping coefficient, c2 cable-head sheave damping coefficient. We neglect the target reference acceleration z¨∗ (t) in the model (2.2)–(2.4) for the sake of control design simplicity. The simplified model is obtained as −ρutt (x, t) + EA uxx (x, t) = 0,

(2.5)

M utt (0, t) + EA ux (0, t) + c1 ut (0, t) = 0,

(2.6)

∗ ˙ Uv (t) − EA ux (l(t), t) − (M + ρl(t))g − l(t)ρ(u t (l(t), t) + z˙ (t))

−

JD ¨(l(t), t) − c2 (u(l(t), ˙ t) + z˙ ∗ (t)) = 0. 2 u RD

(2.7)

Remark 2.1. The simplification z¨∗ (t) = 0 is reasonable because the velocity z˙ ∗ (t) of the reference motion z ∗ (t) can be set to be constant except for starting and stopping time intervals in the practical operation of the elevator. In the simulation, the controller designed based on the simplified model (2.5)–(2.7) will be tested in the accurate model (2.2)–(2.4) to show that our control design is effective in vibration suppression of the mining cable elevator. It should be noted that z¨∗ (t) = 0 is only for simplified presentation and is not required in our design and theory results. Toward bringing the model into the more readily recognizable form for control of wave equations, let us define the vibration control force to consist of two components, Uv (t) = Uv1 (t) + Uv2 (t), (2.8) and let us choose the component Uv2 (t) as ∗ ˙ Uv2 (t) = (M + ρl(t))g + l(t)ρ(u t (l(t), t) + z˙ (t)) JD + 2 u ¨(l(t), t) + c2 (u(l(t), ˙ t) + z˙ ∗ (t)), RD

(2.9)

which simplifies (2.5)–(2.7) to ρutt (x, t) = EA uxx (x, t), ∀(x, t) ∈ [0, l(t)] × [0, ∞),

(2.10)

M utt (0, t) + EA ux (0, t) + c1 ut (0, t) = 0,

(2.11)

Uv1 (t) = EA ux (l(t), t),

(2.12)

where the control input Uv1 (t) is yet to be designed. Before proceeding, we point out that feedback (2.9) employs the measurement of the floating head sheave acceleration u ¨(l(t), t). Standard accelerometers make such a measurement readily available. Coupled PDE-ODE Model on Time-Varying Domain The axial vibration dynamic system (2.10)–(2.12) is a wave PDE with the boundary conditions (2.12) and (2.11) which is a second-order ODE in time. To convert this boundary condition into the standard state-space form, we introduce new variables

CHAPTER TWO

14 x1 (t) and x2 (t) defined by x1 (t) = u(0, t),

(2.13)

x2 (t) = ut (0, t)

(2.14)

as the vibration displacement and the vibration velocity of the payload. Then the following relation is obtained: x˙ 1 (t) = x2 (t), c1 EA ux (0, t). x˙ 2 (t) = − x2 (t) − M M

(2.15) (2.16)

Let X(t) ∈ R2×1 be a state vector defined by X(t) = [x1 (t), x2 (t)]T .

(2.17)

Through the definition (2.17), we rewrite (2.10)–(2.12) as the interconnected (at x = 0) PDE-ODE system ˙ X(t) = AX(t) + Bux (0, t), u(0, t) = CX(t), utt (x, t) =

(2.19)

EA uxx (x, t), ρ

(2.20)

EA ux (l(t), t) = Uv1 (t), where

 A=

0 0

1



−c1 M

, B=

EA M



(2.18)

(2.21)

0 −1

 , C = [1, 0],

(2.22)

which is indeed the case for a rigid payload model. The Neumann interconnection in ODE (2.18) physically amounts to the force acting on the cage. The cage-guide boundary is damped when the damping coefficients c1 > 0 in (2.16). We develop our control design based on a more general model where c1 is arbitrary. In other words, we allow the uncontrolled boundary in the wave equation to be damped (c1 > 0), undamped (c1 = 0), or even anti-damped (c1 < 0). The general wave PDE-ODE model for which we pursue our control design is ˙ X(t) = AX(t) + Bux (0, t), utt (x, t) = quxx (x, t), u(0, t) = CX(t), ux (l(t), t) = U (t),

(2.23) (2.24) (2.25) (2.26)

∀(x, t) ∈ [0, l(t)] × [0, ∞), where q is an arbitrary positive constant. Matrices A ∈ R2×2 , B ∈ R2×1 , C ∈ R1×2 are assumed to be such that the pair (A, B) is controllable, the pair (A, C) is observable, and CB = 0. The function X(t) ∈ R2 is the ODE state, and u(x, t) ∈ R is the state of the wave PDE. The control input U (t) = E1A Uv1 (t) is to be designed.

SINGLE-CABLE MINING ELEVATORS

15

In the control design in this chapter, the time-varying spatial domain [0, l(t)] is assumed to have the following properties, which are reasonable for the string length of the ascending mining cable elevator. Assumption 2.1. Only the ascent of the cable elevator is considered, and the ˙ ≤ 0, and 0 < l(t) ≤ l(0) = varying length l(t) is decreasing and bounded—that is, l(t) L, ∀t ≥ 0. We only consider the ascending process—that is, l(t) being decreasing—in this chapter, because the vibratory energy of the cable is increasing as the cable length is being shortened [199], making ascent a harder control problem than descent.   ˙  √ Assumption 2.2. The hoisting speed l(t)  < q. Assumption 2.2—that is, the speed of the moving boundary is smaller than √ the wave speed q—is only for ensuring the well-posedness of the initial boundary value problem (2.23)–(2.26) according to [71, 72], and is not needed in the control design and stability proof for the ascending elevator in this chapter. Assumption √ 2.2 indeed holds in the mining cable elevator because the value of q in the cable is much larger than the speed of the cage. In remark 2.1 we indicate that we let z¨∗ (t) = 0 and explain why. While this would formally imply, from (2.1), that ¨l(t) = 0, we do not impose ¨l(t) = 0; that is, we do not restrict l(t) to be (piecewise) linear in time. We pursue control design, and provide theory, for general l(t), subject only to assumptions 2.1 and 2.2.

2.2

STATE-FEEDBACK FOR VIBRATION SUPPRESSION

In this section, we design the state-feedback controller that stabilizes the system (2.23)–(2.26), suppressing the axial vibration amplitudes of the cable and cage with the full-state measurements u(x, t) for ∀x ∈ [0, l(t)] and X(t). We seek an invertible transformation that converts the (X, u)-system into the following stable target system, described as ˙ X(t) = (A + BK)X(t) + Bwx (0, t), wtt (x, t) = qwxx (x, t), w(0, t) = 0, wx (l(t), t) = −dwt (l(t), t),

(2.27) (2.28) (2.29) (2.30)

where d > 0 is a positive arbitrary damping gain. The control parameter K is chosen to make A + BK Hurwitz. Based on [114] and [116], the backstepping transformation is formulated as  x w(x, t) = u(x, t) − γ(x, y)u(y, t)dy 0  x h(x, y)ut (y, t)dy − β(x)X(t), (2.31) − 0

where the kernel functions γ(x, y) ∈ R, h(x, y) ∈ R, and β(x) ∈ R1×2 are to be determined. Taking the second derivatives of (2.31) with respect to x and t, respectively,

CHAPTER TWO

16 along the solution of (2.23)–(2.26), we get

wtt (x, t) − qwxx (x, t)    x d γ(x, x) u(x, t) + q = 2q (hxx (x, y) − hyy (x, y))ut (y, t)dy dx 0    x d h(x, x) ut (x, t) +q (γxx (x, y) − γyy (x, y))u(y, t)dy + 2q dx 0  − β(x)AB − qγ(x, 0) + qhy (x, 0)CB ux (0, t)   + qh(x, 0) − β(x)B uxt (0, t) + qβ  (x) − β(x)A2 − qγy (x, 0)C − qhy (x, 0)CA X(t) = 0.

(2.32)

For (2.32) to hold, the following conditions must be satisfied: d γ(x, x) = 0, dx γxx (x, y) = γyy (x, y), d h(x, x) = 0, dx hxx (x, y) = hyy (x, y),

(2.33) (2.34) (2.35) (2.36)

qh(x, 0) = β(x)B,

(2.37)

β(x)AB = qγ(x, 0) − qhy (x, 0)CB,

(2.38)



2

qβ (x) = β(x)A + qγy (x, 0)C + qhy (x, 0)CA.

(2.39)

Substituting the transformation (2.31) into (2.27) and (2.29) and comparing them with (2.23) and (2.25), we can choose β(x) to satisfy β  (0) = K − γ(0, 0)C − h(0, 0)CA,

(2.40)

β(0) = C.

(2.41)

By conditions (2.33)–(2.36), γ(x, y) and h(x, y) can be written as γ(x, y) = m(x − y),

(2.42)

h(x, y) = n(x − y).

(2.43)

Let D ∈ R4×4 and Λ ∈ R1×2 be defined as   1 2 0 qA D= , I − 1q (BCA + ABC) 1 Λ = CABC. q

(2.44) (2.45)

Solving (2.37)–(2.41) with the help of (2.42) and (2.43), the explicit solutions of β(x), γ(x, y), and h(x, y) are obtained as  

 I β(x) = C, K − Λ eDx , (2.46) 0

SINGLE-CABLE MINING ELEVATORS

17

1 γ(x, y) = β(x − y)AB, q 1 h(x, y) = β(x − y)B, q

(2.47) (2.48)

where I ∈ R2×2 is an identity matrix. For the mining elevator modeled in section 2, the solutions of gain kernels (2.46)–(2.48) are written as

 ρ M ρ  ρx k1 − k 1 + e M , k2 − k2 e M x , ρ M

ρ  ρ (x−y) eM , γ(x, y) = k1 − k1 + M ρ h(x, y) = k2 − k2 e M (x−y) , β(x) = −

(2.49) (2.50) (2.51)

where k1 > 0, k2 > 0 are controller gains such that K = [k1 , k2 ] makes (A + BK) Hurwitz. For the boundary equation (2.30) to hold, the statefeedback controller is given by  1 U (t) = N2 ut (l(t), t) + N3 u(l(t), t) N1 + N4 ux (0, t) + N5 u(0, t) + N6 X(t)   l(t)  l(t) N7 u(x, t)dx + N8 ut (x, t)dx , + 0

(2.52)

0

where N1 = 1 − dKB,

(2.53)

N2 = −d, N3 (l(t)) = γ(l(t), l(t)) − qhxy (l(t), l(t)),

(2.54) (2.55)

N4 (l(t)) = dqhx (l(t), 0) − dβ(l(t))B, N5 (l(t)) = qdhxy (l(t), 0), N6 (l(t)) = βx (l(t)) + dβ(l(t))A,

(2.56) (2.57) (2.58)

N7 (l(t), x) = γx (l(t), x) + qhxyy (l(t), x), N8 (l(t), x) = hx (l(t), x) + dγ(l(t), x).

(2.59) (2.60)

In the same manner to obtain the direct transformation, we also obtain the inverse transformation  x u(x, t) = w(x, t) − ϕ(x, y)w(y, t)dy 0  x − λ(x, y)wt (y, t)dy − α(x)X(t), (2.61) 0

with α(x) =



−C

−K



 eZx

I 0

 ,

(2.62)

CHAPTER TWO

18 1 ϕ(x, y) = α(x − y)(A + BK)B, q

(2.63)

1 λ(x, y) = α(x − y)B, q

(2.64)

where

 Z=

0 I

1 2 q (A + BK)

0

 .

(2.65)

The detailed procedure to derive the inverse transformation (2.61) is shown in appendix 2.6.

2.3

OBSERVER AND OUTPUT-FEEDBACK CONTROLLER USING CAGE SENSING

In section 2.2, a state-feedback controller was designed to stabilize the system. However, the designed state-feedback control law requires an infinite number of sensors to obtain the distributed states in the whole domain, which is not feasible. In this section, we propose an observer-based output-feedback control law that requires only a few boundary values as measurements. An exponentially convergent observer is designed to reconstruct the distributed states using a finite number of available boundary measurements, and the output-feedback control law based on the observer is proposed. We assume that the full ODE state X is available for measurement, which does not make the problem a lot easier, as X is not collocated with the actuator. In the mining cable elevator, we usually place the acceleration sensor at the cage and use integration to obtain X(t) = [u(0, t), ut (0, t)], where the initial value of the vibration displacement at the cage can be obtained by the static equilibrium equation, and the initial cage velocity is zero. This paragraph only explains an acquisition method for X in practical mining cable elevators. No restrictions are imposed on any initial conditions in the design and theory. Observer Design The observer structure consists of a copy of the plant (2.23)–(2.26) plus the boundary state error injection, described as ˆ˙ ˆ + Bu ¯ ˆ X(t) = AX(t) ˆx (0, t) + LC(X(t) − X(t)), ˆ u ˆtt (x, t) = q u ˆxx (x, t) − D1 (X(t) − X(t)), ˆ u ˆ(0, t) = CX(t) − D2 (X(t) − X(t)), u ˆx (l(t), t) = U (t).

(2.66) (2.67) (2.68) (2.69)

We feed the full ODE state X into the observer component (2.66). This is not to ˆ but to estimate the unmeasured PDE state u(x, t) estimate the measured X by X by u ˆ(x, t). In other words, while one could pursue the design of a reduced-order observer, as for ODEs, we pursue a full-order observer here. The observer gains D1 , D2 and

SINGLE-CABLE MINING ELEVATORS

19 ¯ = [¯l1 , ¯l2 ]T L

are to be determined. Define the observer errors as u ˜(x, t) = u(x, t) − u ˆ(x, t), ˜ = X(t) − X(t). ˆ X(t)

(2.70) (2.71)

Then, subtracting (2.66)–(2.69) from (2.23)–(2.26) provides the observer error system, written as ˜˙ ¯ X(t) ˜ + Bu X(t) = (A − LC) ˜x (0, t), ˜ ˜xx (x, t) + D1 X(t), u ˜tt (x, t) = q u ˜ u ˜(0, t) = D2 X(t), u ˜x (l(t), t) = 0.

(2.72) (2.73) (2.74) (2.75)

To convert the system (2.72)–(2.75) into the following exponentially stable target system described as ˜˙ ¯ X(t) ˜ + Bw X(t) = (A − LC) ˜x (0, t), w ˜tt (x, t) = q w ˜xx (x, t), w(0, ˜ t) = 0, w ˜x (l(t), t) = −d¯w ˜t (l(t), t),

(2.76) (2.77) (2.78) (2.79)

¯ is chosen to make A − LC ¯ Hurwitz and d¯ is an arbitrary positive design where L parameter, the following direct and inverse transformations are formulated:  x u ˜(x, t) = w(x, ˜ t) − d0 (x, y)w(y, ˜ t)dy 0  x ˜ d1 (x, y)w ˜t (y, t)dy − Γ(x)X(t), (2.80) − 0  x d2 (x, y)˜ u(y, t)dy w(x, ˜ t) = u ˜(x, t) − 0  x ˜ − d3 (x, y)˜ ut (y, t)dy − ψ(x)X(t). (2.81) 0

By matching (2.72)–(2.75) and (2.76)–(2.79), the following conditions are obtained: Γ(x)AB = qd0 (x, 0), qd1 (x, 0) = Γ(x)B, ¯ 2 + D1 , qΓ (x) = Γ(x)(A − LC) D2 = −Γ(0),

(2.82) (2.83) (2.84) (2.85)

Γ (0) = 0, ¯ d1 (l(t), l(t)) = −d,

(2.86) (2.87)

d0 (l(t), l(t)) = 0, d0x (l(t), y) = 0, d1 x (l(t), y) = 0.

(2.88) (2.89) (2.90)

CHAPTER TWO

20 The solutions of the kernel needed in (2.80) are obtained as ¯ Γ(x) = −[0, q d][AB, B]−1 , d0 (x, y) = 0, ¯ d1 (x, y) = −d.

(2.91) (2.92) (2.93)

The observer gains are obtained as ¯ ¯ 2, D1 = [0, q d][AB, B]−1 (A − LC) ¯ B]−1 . D2 = [0, q d][AB, Here the matrix [AB, B] is invertible since the pair (A, B) is controllable. For the mining elevator modeled in section 2, the solutions of the gain (2.91)–(2.93) are written as d¯ Γ = [c1 , M ], ρ d0 = 0, ¯ d1 = −d, and the observer gains (2.94) and (2.95) are obtained as    d¯ c 1 l2 D1 = − c1 (l12 − l2 ) + M l1 l2 + , ρ M  2 

c  c1 1 + l1 + M − l , − c1 2 M M2 d¯ D2 = − [c1 , M ]. ρ

(2.94) (2.95) kernels

(2.96) (2.97) (2.98)

(2.99) (2.100)

¯ can be Hurwitz by choosing the positive parameters ¯l1 > 0 and The matrix A − LC ¯l2 > 0. Output-Feedback Control Design ˆ w)-subsystem, To design the output-feedback controller, we consider the target (X, ˆ which is constructed by the direct and inverse transformations with the same gain kernels as the state-feedback (2.31) and (2.61). Hence, we introduce the following ˆ u ˆ w), transformations from (X, ˆ) to (X, ˆ described as  w(x, ˆ t) = u ˆ(x, t) −  −

x 0

−

 x

0

γ(x, y)ˆ u(y, t)dy

0

ˆ h(x, y)ˆ ut (y, t)dy − β(x)X(t),

u ˆ(x, t) = w(x, ˆ t) − 

x

(2.101)

x 0

ϕ(x, y)w(y, ˆ t)dy

ˆ λ(x, y)wˆt (y, t)dy − α(x)X(t).

(2.102)

SINGLE-CABLE MINING ELEVATORS

21

Taking the time and spatial derivatives of (2.101) with the help of gain kernels ˆ u (2.46)–(2.48) and (X, ˆ)-system (2.66)–(2.69), we derive the following coupled PDEˆ w)-system: ODE (X, ˆ ˆ˙ ˆ + Bw X(t) = (A + BK)X(t) ˆx (0, t)   ¯ ˜ + LC + Bγ(0, 0)(C − D2 ) X(t),

(2.103)

˜ − f2 (x)w w ˆtt (x, t) = q w ˆxx (x, t) − f1 (x)X(t) ˜x (0, t),

(2.104)

˜ w(0, ˆ t) = (C − D2 )X(t),

(2.105)

w ˆx (l(t), t) = −dw ˆt (l(t), t),

(2.106)

where ¯ + β(x)LC(A ¯ ¯ f1 (x) = β(x)ALC − LC)   x ¯ h(x, y)D1 (A − LC)dy − − f2 (x) = −

0  x 0

x 0

γ(x, y)D1 dy,

¯ h(x, y)D1 Bdy + β(x)LCB.

(2.107) (2.108)

By (2.106), the output-feedback controller is designed as U (t) =

 1 ˆt (l(t), t) + N3 u ˆ(l(t), t) N2 u N1 + N4 (l(t))ˆ ux (0, t) + N5 (l(t))ˆ u(0, t)  l(t) ˆ + + N6 (l(t))X(t) N7 (l(t), x)ˆ u(x, t)dx 0



l(t)

+ 0



N8 (l(t), x)ˆ ut (x, t)dx ,

(2.109)

where N1 , . . . , N8 are defined in (2.53)–(2.60).

2.4

STABILITY ANALYSIS

In this section, we establish the stability proof of the target system via Lyapunov analysis for PDEs. The equivalent stability property between the target system and the original system is ensured due to the invertibility of the backstepping transformation. The main theorem of this chapter is stated next. ˆ Theorem 2.1. For all initial estimates (ˆ u(x, 0), u ˆt (x, 0), X(0)) compatible with the control law (2.109) and the initial values (u(x, 0), ut (x, 0), X(0)) which belong to H 1 (0, L) × L2 (0, L) × R2 , the closed-loop system consisting of the plant (2.23)– (2.26) and the observer design (2.66)–(2.69) with the output-feedback control law (2.109) is exponentially stable in the sense of the norm

CHAPTER TWO

22 

l(t) 0

ut 2 (x, t)dx +



l(t)

+ 0



l(t) 0

ux 2 (x, t)dx +

2 u ˆ2x (x, t)dx + |X(t)|



l(t) 0

u ˆ2t (x, t)dx

 1/2   ˆ 2 + X(t) .

(2.110)

˜ w)-subsystem. Proof. First, we show the stability of the (X, ˜ Define    ˜ 2 2 2 Ω1 (t) = ˜ ut (t) + ˜ ux (t) + X(t)  ,    ˜ 2 2 2 ˜t (t) + w ˜x (t) + X(t) Ξ1 (t) = w  ,

(2.111) (2.112)

 l(t) 2 2 ˜(x, t) dx. In addition, we employ the where ˜ u(t) is a compact notation for 0 u Lyapunov function ˜ + φ1 E1 (t), ˜ T (t)P1 X(t) (2.113) V1 = X where the matrix P1 = P1T > 0 is the solution to the Lyapunov equation ¯ + (A − LC) ¯ T P1 = −Q1 , P1 (A − LC)

(2.114)

for some Q1 = Q1 T > 0. The positive parameter φ1 is to be chosen later. The function E1 (t) is defined as 1 q 2 2 ˜t (t) + w ˜x (t) + δ1 E1 (t) = w 2 2



l(t) 0

(1 + x)w ˜x (x, t)w ˜t (x, t)dx,

(2.115)

where the parameter δ1 should satisfy 0 < δ1
0, 2 2   φ1 φ1 θ12 = max λmax (P1 ), (1 + δ1 (1 + L)), (q + δ1 (1 + L)) > 0. 2 2

The time derivative of V1 along (2.76)–(2.79) is obtained as 1  ˙  ¯ 1w V˙ 1 = −dqφ ˜t2 (l(t), t) − l(t) ˜t2 (l(t), t)  φ1 w 2 q  ˙  ˜ ˜ T (t)Q1 X(t) ˜x2 (l(t), t) − X − l(t)  φ1 w 2 ˜ + δ2 (1 + l(t))φ1 w + 2BP1 w ˜x (0, t)X(t) ˜t2 (l(t), t) 2 δ1 + q d¯2 (1 + l(t))φ1 w ˜t2 (l(t), t) 2

(2.118) (2.119)

SINGLE-CABLE MINING ELEVATORS

23

δ1 δ1 δ1 2 2 − q φ1 w ˜x2 (0, t) − φ1 w ˜t  − qφ1 w ˜x  2 2 2   ˙  ¯ + l(t) ˜t2 (l(t), t),  dδ1 (1 + l(t))φ1 w

(2.120)

  ˙  has been used. ˙ = − l(t) where assumption 2.1 yielding l(t) Applying Young’s inequality to (2.120), we obtain the following inequality:   1 δ1  ˜ 2 δ 1 2 2 V˙ 1 ≤ − λmin (Q1 )X(t) ˜t  − qφ1 w ˜x   − φ 1 w 2 2 2   ¯ − δ1 (1 + L) (1 + q d¯2 ) φ1 w − dq ˜t2 (l(t), t) 2     1 d¯2 q ˙  ¯ 1 (1 + L) φ1 w + − dδ − l(t) ˜t2 (l(t), t)  2 2  2  2|P1 B| δ1 − q φ1 − w ˜x2 (0, t). 2 λmin (Q1 )

(2.121)

Therefore, together with (2.116), the parameters δ1 and φ1 are chosen to satisfy the following:   ¯ 2dq 1 1 + q d¯2 min 1, q, 0 < δ1 < , (2.122) , 1+L 1 + q d¯2 2d¯ 2

φ1 >

4|P1 B| + , qδ1 λmin (Q1 )

(2.123)

with a positive parameter . Then we arrive at V˙1 ≤ −σ1 V1 − w ˜x2 (0, t) ≤ −σ1 V1 , where σ1 =

1 min θ12



 δ1 δ1 1 φ1 , qφ1 , λmin (Q1 ) . 2 2 2

(2.124)

(2.125)

ˆ w)-subsystem. Next, we show the stability of the (X, ˆ Define    ˆ 2 2 2 Ω2 (t) = ˆ ut (t) + ˆ ux (t) + X(t)  ,    ˆ 2 2 2 ˆt (t) + w ˆx (t) + X(t) Ξ2 (t) = w  .

(2.126) (2.127)

Let V2 be a Lyapunov function written as ˆ + φ2 E2 (t), ˆ T (t)P2 X(t) V2 = X

(2.128)

where the matrix P2 = P2T > 0 is the solution to the Lyapunov equation P2 (A + BK) + (A + BK)T P2 = −Q2 ,

(2.129)

CHAPTER TWO

24

for some Q2 = Q2 T > 0. The positive parameter φ2 is to be chosen later. Define E2 (t) as 1 q 2 2 ˆt (t) + w ˆx (t) + δ2 E2 (t) = w 2 2



l(t)

(1 + x)w ˆx (x, t)w ˆt (x, t)dx,

0

(2.130)

where the parameter δ2 must be chosen to satisfy 0 < δ2
0, 2 2   φ2 φ2 θ22 = max λmax (P2 ), (1 + δ2 (1 + L)), (q + δ2 (1 + L)) > 0. 2 2

(2.133) (2.134)

Taking the time derivative of V2 along (2.103)–(2.106), we get  V˙ 2 = φ2 q

w ˆt (x, t)w ˆxx (x, t)dx + φ2 q

0

 − φ2



l(t)

l(t) 0



l(t) 0

w ˆx (x, t)w ˆxt (x, t)dx

 ˜ w ˆt (x, t) f1 (x)X(t) + f2 (x)w ˜x (0, t) dx

 1  1 ˙  ˙  ˆt2 (l(t), t) − qφ2 l(t) ˆ 2 (l(t), t) − φ2 l(t)  w  w 2 2 x 1 1 + φ2 qδ2 (1 + l(t))w ˆx2 (l(t), t) − φ2 qδ2 w ˆx2 (0, t) 2 2 1 1 2 − φ2 qδ2 w ˆx  + φ2 δ2 (1 + l(t))w ˆt2 (l(t), t) 2 2 1 1 2 ˙ − φ 2 δ2 w ˆt2 (0, t) − φ2 δ2 w ˆt  + φ2 l(t)δ ˆt (l(t), t)w ˆx (l(t), t) 2 (1 + l(t))w 2 2 ˆ +X ˆ T (t)P2 X(t). ˆ˙ ˆ˙ T (t)P2 X(t) (2.135) +X Applying Young’s inequality to (2.135), as in (2.121), we obtain the following inequality,     1 1 1  ˆ 2 2 2 V˙ 2 ≤ − φ2 δ2 − (B1 + B2 )L w ˆt  − qφ2 δ2 w ˆx  − λmin (Q2 )X(t)  2 2 2   2 1 1 2 4|P2 B| qφ2 δ2 − q − w ˆx2 (0, t) − 2 2 λmin (Q2 )   δ2 2 ˆt2 (l(t), t) − φ2 qd − (1 + L)(1 + qd ) w 2     1 qd2 ˙  + − dδ2 (1 + L) w − φ2 l(t) ˆt2 (l(t), t) 2 2

SINGLE-CABLE MINING ELEVATORS

25



1 2 1 2 ¯ 2 φ2 + φ2 (C − D2 )2 (A − LC) 4 2   2 ¯ + Bγ(0, 0)(C − D2 ))2  4P2 (LC ˜  + 1 φ2 2 w + ˜x2 (0, t), X(t) λmin (Q2 ) 4 +

where Bi for i = 1, 2 is defined as



Bi = max

x∈[0,L]

|fi (x)|

2

(2.136)

 .

Therefore, by choosing the parameters δ2 and φ2 as   2dq 1 1 + qd2 min 1, q, 0 < δ2 < , , 1+L 1 + qd2 2d   2 4 |P2 B| 2 q , φ2 = max 2(B1 + B2 )L, + δ2 2 qλmin (Q2 )

(2.137) (2.138)

we arrive at    ˜ 2 V˙ 2 ≤ −σ2 V2 + ξ1 X(t) ˜x2 (0, t),  + ξ2 w

(2.139)

where σ2 = μ2 /θ22 > 0, and

 μ2 = min

 1 1 1 φ2 δ2 , qφ2 δ2 , λmin (Q2 ) , 4 2 2

1 1 ¯ 2 ξ1 = φ2 2 + φ2 2 (C − D2 )2 (A − LC) 4 2   ¯ + Bγ(0, 0)(C − D2 ))2 4P2 (LC , + λmin (Q2 ) 1 ξ2 = φ 2 2 . 4

(2.140)

(2.141) (2.142)

˜ w, ˆ w)-system Let V be the Lyapunov function of the overall (X, ˜ X, ˆ defined as V = RV1 + V2 .

(2.143)

Taking the time derivative of (2.143) and using (2.117), (2.124), and (2.139), we get Rσ1 V˙ ≤ − V1 − σ 2 V2 − 2



 Rσ1 θ12 2 ˜ − ξ1 |X(t)| 2

− (R − ξ2 ) w ˜x2 (0, t).

(2.144)

Therefore, choosing R sufficiently large, finally we arrive at V˙ ≤ −σV

(2.145)

CHAPTER TWO

26

for some positive σ. The differential inequality (2.145) implies that there exists a positive parameter η1 > 0 such that      ˜ 2  ˆ 2 2 2 2 2 w ˜ t  + w ˜x  + X(t) ˆ t  + w ˆx  + X(t)  + w   2  ˜  2 2 ≤ η1 w ˜x (0) + X(0) ˜t (0) + w      ˆ 2 −σt 2 2 + w ˆt (0) + w ˆx (0) + X(0) (2.146)  e . ˜ w, ˆ Therefore, the overall target (w, ˜ X, ˆ X)-system is exponentially stable. Due to the invertibility of the transformations (2.81) and (2.101) as explicitly written in (2.80) and (2.102), applying Poincare’s, Young’s, and the Cauchy-Schwarz inequalities to (2.146) in a similar manner as theorem 16.1 in [116] yields      ˜ 2  ˆ 2 2 2 2 2 ux  + X(t) ut  + ˆ ux  + X(t) ˜ ut  + ˜  + ˆ   2  ˜  2 2 ≤ η2 ˜ ux (0) + X(0) ut (0) + ˜      ˆ 2 −σt 2 2 + ˆ ut (0) + ˆ ux (0) + X(0) e

(2.147)

for some positive η2 . Therefore, the exponential stability of the overall original ˜ u ˆ ˆ, u ˆt , X)-system in the sense of (2.111) and (2.126) is proved, which con(˜ u, u ˜t , X, cludes theorem 2.1 with the help of (2.70) and (2.71).

2.5

SIMULATION TEST IN A SINGLE-CABLE MINING ELEVATOR

The simulation is performed based on the simplified model and the accurate model. In the first case, the simulation based on the simplified model (2.18)–(2.21) under the designed state-feedback control law (2.52) and the output-feedback control law (2.109) is conducted to verify the theoretical result in theorem 2.1. In the second case, the simulation based on the accurate model (2.2)–(2.4) with (2.9), under the designed output-feedback control law (2.109), is conducted to test the controller performance on vibration suppression. The control input Uv1 applied in both cases is Uv1 = EA U (t), where U (t) is the designed control law, and the constant EA = E × Aa . The physical parameters of the mining cable elevator used in the simulation are shown in table 2.1. To highlight the controller performance on vibration suppression, we make the damping coefficient c1 in the elevator zero. The designed ˙ is plotted in figure 2.2. The initial proreference of the hoisting velocity z˙ ∗ (t) = l(t) file of the vibration displacement is u(x, 0) = −(ρxg + M g)/EA , which is obtained by the force balance equation at the static state, and the initial velocity is defined as ut (x, 0) = 0 because the initial velocity of each point in the cable is zero. The initial conditions u(x, 0) and ut (x, 0) defined in the simulations obviously impose no restrictions on the initial conditions in theorem 2.1. The closed-loop responses with the proposed control law (2.109) and the proportional-derivative (PD) control law,

SINGLE-CABLE MINING ELEVATORS

27

Table 2.1. Physical parameters of the mining cable elevator. Parameters (units)

Values

Initial length L (m) Final length (m) Cable effective steel area Aa (m2 ) Cable effective Young’s modulus E (N/m2 ) Cable linear density ρ (kg/m) Total hoisted mass M (kg) Gravitational acceleration g (m/s2 ) Maximum hoisting velocities Vmax (m/s) Total hoisting time tf (s)

2000 200 0.47×10−3 1.03×1010 8.1 15000 9.8 15 150

20 Vmax ż* (t) (m/s)

15 10 5 0

0

50

100

150

Time (s)

Figure 2.2. The hoisting velocity z˙ ∗ (t). which is traditionally used in industry, are examined to compare their performance in suppressing the axial vibrations of the mining cable. The PD control law is Upd (t) = kp u(l(t), t) + kd u(l(t), ˙ t),

(2.148)

where kp , kd are gain parameters. The values of kp and kd are tuned to attain efficient control performance. We have tested different values of kp and kd , and the best regulating performance is achieved with kp = 2000, kd = 7000. The gains of the ¯ and [k1 , k2 ] = [0.0035, ¯ and K = [k1 , k2 ] are chosen as d=d=1 proposed controller d, d, 0.03] in the simulation. The numerical simulation is performed by the finite-difference method for the discretization in time and space after converting the timevarying domain PDE to the PDE on a fixed domain [0, 1] but with time-varying x [184]. The time step and space step are chosen coefficients by introducing ηˆ = l(t) as 0.001 and 0.01, respectively. Comparison with the PD Control Figure 2.3 shows the open-loop responses of the plant (2.18)–(2.21). It illustrates that the large vibration occurs, at both the cage and the midpoint of the string during the total hoisting time. To suppress the vibration, the closed-loop responses

CHAPTER TWO

28 –0.03

u(0, t) (m)

Open loop –0.04 –0.05 –0.06 0

50

100 Time (s) (a) The axial vibration at the moving cage.

u(l(t)/2, t) (m)

–0.04

150

Open loop

–0.045

–0.05

0

50

100

150

Time (s) (b) The axial vibration at the midpoint of the cable.

Figure 2.3. The open-loop responses of the plant (2.18)–(2.21). The large vibration is caused both at the moving cage and at the midpoint of the cable.

with the PD control law (2.148) and the proposed control law are investigated and shown in figure 2.4. It shows that the vibration is suppressed and converges to zero on both the proposed control law and the PD control. Moreover, it can be observed that the responses with the proposed control law have faster convergence and less overshoot than the responses with the PD control law. Thus, the proposed control law shows better performance than the classical PD control. The Observer-Based Output-Feedback Responses With the available boundary measurements of the displacement and the velocity of the axial vibration at the cage u(0, t) and ut (0, t), the estimated variables of the distributed states required in the control law are obtained by the proposed observer (2.66)–(2.69). The closed-loop responses with an observer-based outputfeedback controller are simulated with the initial observer error u ˜(x, 0) = 0.002 (m) uniformly. Then the initial conditions of the observer used in the simulation are ˆ u ˆ(x, 0) = u(x, 0) + 0.002 and X(0) = X(0) + [0.002, 0]T , which satisfy the conditions in theorem 2.1. The dynamics of the observer error and the vibration displacement at the midpoint of the string are shown in figure 2.5(a) and (b), respectively. Because the locations of the actuator and the sensor are at opposite boundaries, the stabilization and the estimation of the vibration at the midpoint x = l(t)/2 is most challenging due to its accessibility. Figure 2.5(a) shows that the observer

SINGLE-CABLE MINING ELEVATORS

29

u(0, t) (m)

0.02

PD controller Proposed state-feedback controller

0 –0.02 –0.04 –0.06

0

50

100 Time (s) (a) The axial vibration at the moving cage.

u(l(t)/2, t) (m)

0.02

150

PD controller Proposed state-feedback controller

0 –0.02 –0.04 –0.06

0

50

100 Time (s) (b) The axial vibration at the midpoint of the cable.

150

Figure 2.4. The closed-loop responses of the plant (2.18)–(2.21) with the PD controller (2.148) (dashed line) and the proposed state-feedback controller (2.52) (solid line).

error converges to zero quickly, which implies that the estimates of the vibration displacements reconstruct their actual distributed states. Figure 2.5(b) shows that the convergence to zero of the vibration state at the midpoint of the cable is achieved with the output-feedback control law as well, although the initial observer error affects the controller performance in the initial stage compared with the statefeedback response in figure 2.4. Tests on the Accurate Model We process the controller design and stability analysis based on the simplified model. In this subsection, we test the performance of our controller based on the accurate model (2.2)–(2.4) with (2.9), which includes the boundary and distributed force disturbances from the motion acceleration z¨∗ (t) shown in figure 2.6. After applying the proposed output-feedback controller used in section 2.5, and the PD controller with the coefficients kp = 3500, kd = 9000, which are adjusted to obtain efficient performance, the results under the two controllers are compared in figure 2.7. We can see that our controller also performs better at vibration suppression than the PD controller, even though the boundary and distributed force disturbances from the motion acceleration z¨∗ (t) are incorporated, which introduce the possibility of saltation at 30 s and 120 s.

CHAPTER TWO

30

2.5

× 10–3 ∼ Observer error u(l(t)/2, t) (m)

∼ u(l(t)/2, t) (m)

2 1.5 1 0.5 0 –0.5

0

50

100 150 Time (s) (a) The observer error of the axial vibration at the midpoint of the cable. 0.02 u(l(t)/2, t) (m)

Vibration displacement u(l(t)/2, t) (m) 0 –0.02 –0.04 –0.06

0

50

100

150

Time (s) (b) The axial vibration at the midpoint of the cable.

Figure 2.5. The responses of the closed-loop system (2.18)–(2.21) under the observer (2.66)–(2.69) and the output-feedback control law (2.109). The observer achieves convergence to the actual distributed state, and the associated output-feedback controller performs similarly to the state-feedback.

0

..*

z (t) (m/s2)

0.5

–0.5

0

50

100 Time (s)

Figure 2.6. The hoisting acceleration.

150

SINGLE-CABLE MINING ELEVATORS

31

4 PD controller Proposed output-feedback controller

u(0, t) (m)

3 2 1 0 –1 –2

0

50

100 Time (s) (a) The axial vibration at the moving cage.

u(l(t)/2, t) (m)

3

150

PD controller Proposed output-feedback controller

2 1 0 –1

0

50

100 Time (s) (b) The axial vibration at the midpoint of the cable.

150

Figure 2.7. The closed-loop responses of the accurate plant (2.2)–(2.4) with (2.9) under the PD controller (2.148) (dashed line) and the proposed output-feedback controller (2.109) (solid line).

2.6

APPENDIX

The inverse transformation is defined as  x ϕ(x, y)w(y, t)dy, u(x, t) = w(x, t) − 0

 −

x 0

λ(x, y)wt (y, t)dy − α(x)X(t),

(2.149)

where kernel functions ϕ(x, y) ∈ R, λ(x, y) ∈ R, and α(x) ∈ R1×2 are to be determined. Taking the second derivatives of (2.149) with respect to x and t, respectively, and substituting them into (2.24), recalling (2.27)–(2.30), we get utt (x, t) − quxx (x, t)  x   = 2q ϕy (x, x) + ϕx (x, x) w(x, t) + q λxx (x, y) − λyy (x, y) wt (y, t)dy 0  x  ϕxx (x, y) − ϕyy (x, y) w(y, t)dy + 2q(λx (x, x) + λy (x, x))wt (x, t) +q 0

CHAPTER TWO

32 ˜ − qϕ(x, 0))wx (0, t) + (qλ(x, 0) − α(x)B)wxt (0, t) − (α(x)AB 

+ qα (x) − α(x)A˜2 X(t) = 0,

(2.150)

where A˜ = A + BK. Recalling the ODE (2.23) in the system (2.23)–(2.26) and the ODE (2.27) in the target system, we have ˙ X(t) = AX(t) + Bux (0, t) = AX(t) + Bwx (0, t) − Bα (0)X(t) = (A + BK)X(t) + Bwx (0, t).

(2.151)

By virtue of (2.25) and (2.29), we get u(0, t) = w(0, t) − α(0)X(t) = CX(t).

(2.152)

According to (2.150)–(2.152), we get the following conditions on the kernel functions in the inverse transformation: ϕy (x, x) + ϕx (x, x) = 0,

(2.153)

λy (x, x) + λx (x, x) = 0,

(2.154)

ϕyy (x, y) − ϕxx (x, y) = 0,

(2.155)

λyy (x, y) − λxx (x, y) = 0,

(2.156)

1 α (x) − α(x)A˜2 = 0, q

(2.157)

qλ(x, 0) − α(x)B = 0,

(2.158)

˜ = 0, qϕ(x, 0) − α(x)AB

(2.159)



−α (0) = K,

(2.160)

−α(0) = C.

(2.161)

According to (2.157), (2.160), and (2.161), the solution of α(x) can be obtained as  

 I , (2.162) α(x) = −C −K eZx 0 where  Z=

0 I

1 ˜2 qA

0

 .

According to (2.158), (2.159), and (2.162), we get ϕ(x, y) =

1

−C, q

1

−C, λ(x, y) = q

−K −K

 

 eZ(x−y)  e

Z(x−y)

I 0 I 0

 ˜ AB,

(2.163)

B.

(2.164)



SINGLE-CABLE MINING ELEVATORS

2.7

33

NOTES

Most of the existing studies on the vibration control of compliant strings, modeled by a wave PDE, focus on the fixed length [65, 88, 89, 137, 138, 139]. The timevarying length has a significant effect on the vibration dynamic characteristics of compliant string systems [157, 199, 201] and makes the design of the controller more challenging. Relatively few studies deal with the boundary vibration control problems of varying-length cables. One control design is presented in [90], where a boundary control law was developed to stabilize the transverse vibrations of a moving string system with varying length. In the above literature, however, either the actuators must be placed at both boundaries of the cable or the uncontrolled boundaries are assumed to be the fixed/damped types. For the control design in this chapter, we apply a control input in only one boundary of the cable with time-varying length and allow the uncontrolled boundary to be anti-stable. We deal with a wave PDE with a time-dependent moving boundary in this chapter, and the results regarding the boundary control of wave PDEs with a statedependent moving boundary can be seen in [28, 29]. Boundary control of other types of PDEs with moving boundaries can be found in [19, 46, 109, 110, 195]. While in this chapter we neglect the cable material damping and derive the controller based on an undamped wave PDE, control designs for a more realistic cable model that includes in-domain viscous damping can be found in chapters 3, 5, and 6.

Chapter Three Dual-Cable Elevators

Unlike chapter 2 in which we solved the problem of axial vibration suppression in a single-cable ascending mining elevator, in this chapter we advance to not only addressing the suppression of the axial vibrations and tension oscillations but also balancing the cage roll in an ascending/descending dual-cable mining elevator. In this chapter, we also employ a more realistic cable model that includes internal material damping—that is, a wave partial differential equation (PDE) with an in-domain viscous damping term. From a mathematical point of view, this introduction of damping leads to a more challenging control problem because of the PDE in-domain couplings that result from the in-domain viscous damping term [148]. The additional mathematical challenge is the requirement of stronger exponential stability estimates in the sense of the H 2 norm of the wave PDE regarding the suppression of tension oscillations. In this chapter, by representing the wave PDE-modeled system in the Riemann coordinates in section 3.1, the vibration dynamics of the dual-cable mining elevator are modeled by two pairs of 2 × 2 heterodirectional coupled hyperbolic PDEs on a time-varying domain, and all four PDE uncontrolled boundaries are coupled at one ordinary differential equation (ODE). The control task of suppressing axial vibrations and tension oscillations, as well as of balancing the cage roll in the dual-cable mining elevator, can be mathematically described as output-feedback boundary exponential stabilization of the aforementioned coupled hyperbolic PDE-ODE system in the sense of the H 1 norm. To this end, a state observer is designed and proved exponentially convergent to the plant in section 3.2, and then an observerbased output-feedback boundary controller is designed via backstepping in section 3.3. The required exponential stability of the closed-loop system and the exponential convergence of the control inputs are proved in section 3.4. A simulation test on a dual-cable mining elevator is provided in section 3.5. Throughout this chapter the cable numbers (1, 2) in the dual-cable mining elevator are denoted as the subscripts i = 1, 2 or j = 2, 1, j = i.

3.1

DUAL-CABLE MINING ELEVATOR DYNAMICS AND REFORMULATION

Dynamics of Dual-Cable Mining Elevators Following the modeling process in chapter 2, and with the inclusion of the material damping of the cables, the vibration dynamics of a dual-cable mining elevator are modeled as ˙ X(t) = AX(t) + B[ux (0, t), vx (0, t)]T , (3.1)

DUAL-CABLE ELEVATORS

35 utt (x, t) = quxx (x, t) − cut (x, t),

(3.2)

vtt (x, t) = qvxx (x, t) − cvt (x, t),

(3.3)

C3 X(t) + C4 X(t)l1 = ut (0, t),

(3.4)

C3 X(t) − C4 X(t)l1 = vt (0, t),

(3.5)

ux (l(t), t) = U1 (t),

(3.6)

vx (l(t), t) = U2 (t),

(3.7)

x ∈ [0, l(t)], t ∈ [0, ∞), where u(x, t), v(x, t) are the axial vibrations in two cables in a moving coordinate system associated with the elevator axial motion l(t) where ¯1 , B ¯2 ] are the origin is located at the cage. Matrices A and B = [B ⎛

0 ⎜ 0 A=⎜ ⎝ 0 0

0 0 0 0

1 0

−cd M

0

0 1 0

−ca Jc

⎞

⎛

⎟ ⎜ ⎟ , B = EA ⎜ ⎠ ⎝

0 0

−1 M −l1 Jc

⎞ 0 0 ⎟ −1 ⎟ , M ⎠

(3.8)

l1 Jc

where M denotes the mass of the load. The matrices C3 , C4 are C3 = [0, 0, 1, 0], C4 = [0, 0, 0, 1]. The function l(t) describes the time-varying length of the cables. The parameter Jc a is the moment of inertia of the cage, and q = E×A , where E, Aa , and ρ are Young’s ρ modulus, the cross-sectional area, and the linear density of the cables, respectively. The parameters cd and ca denote the damping coefficients of cage axial and roll motion, respectively, and c = ρc¯ , where c¯ is the material damping coefficient of the steel cables. The parameter l1 is the cage dimension shown in figure 3.1. The PDE states u(x, t), v(x, t) in (3.2), (3.3) describe the axial vibration displacements of the distributed points in cable 1 and cable 2, respectively. The functions ux (x, t) and vx (x, t) denote the distributed strain in the cables, and the tension oscillations are represented as EA ux (x, t) and EA vx (x, t), where the constant EA = E × Aa . The functions EA ux (0, t) and EA vx (0, t) that denote forces acting on the cage drive the ODE dynamics (3.1), where the ODE state X(t) = ˙ T describes the cage dynamics. The functions y(t), y(t) ˙ are the [y(t), θ(t), y(t), ˙ θ(t)] ˙ axial vibration displacement and velocity of the centroid of the cage, and θ(t), θ(t) are the cage roll angle and roll rate around the axis which is vertical to the door and through the centroid of the cage. Equations (3.4), (3.5) describe the velocity relationship between the cage and the bottom boundaries of the two cables. Equations (3.6), (3.7) come from EA ux (l(t), t) = U1v (t) and EA vx (l(t), t) = U2v (t) with the definition of U1v (t) = EA U1 (t) and U2v (t) = EA U2 (t), from which the two actual control forces at the two floating sheaves U1v (t), U2v (t) can be obtained by U1 (t), U2 (t) to be designed in this chapter. We pursue the control design for a general model where the damping coefficients ca , cd , c in (3.1)–(3.7) are arbitrary. In other words, these damping coefficients can be damped (> 0), undamped (= 0), or even anti-damped (< 0). Axial motion dynamics l(t) are regulated by a separate controller Ua (t) at the drum. We neglect the effect of the vibration dynamics on the motion dynamics because the vibration displacements u(x, t), v(x, t) are much smaller than the hoisting motion l(t) between 2000 m underground and the surface platform. We can then

CHAPTER THREE

36

x = l(t) Floating head sheave

U2(t) Double drums

U1(t) Cable 2

l1

Cable 1

y(t) x=0 θ(t)

Flexible guide

Cage roll

Figure 3.1. Diagram and prototype of a dual-cable mining elevator with flexible guide rails. (The prototype was built by CITIC Heavy Industries Co. Ltd.) consider that the l(t) reference governed by an independent ODE (motion dynamics) driven by Ua (t) is the known hoisting trajectory. Hence, we focus on the control design Ui (t) at the floating head sheave for the PDE vibration dynamics (3.1)–(3.7), where [0, l(t)] acts as a known time-varying domain. Assumption 3.1. The varying length l(t) is bounded—that is, 0 < l(t) ≤ L, ∀t ≥ 0, where the positive constant L is the maximal length. ˙ of the moving boundary is bounded by Assumption 3.2. The velocity l(t)   √ ˙  l(t) ≤ v¯max < q,

(3.9)

where v¯max is the maximum velocity of the mining cable elevator. As mentioned in chapter 2, in the mining cable elevator the wave speed, which √ may assume a value like q = r/ρ = 7.5 × 103 , is much larger than the value of the √ maximum hoisting velocity, which may be v¯max = 16.25 m/s, that is, v¯max q. ˙ According to [71, 72], the fact that the speed of the moving boundary |l(t)| is smaller √ than the wave speed q ensures the well-posedness of the initial boundary value √ problem (3.1)–(3.7). Since the value of q in the string is much larger than the speed of the moving payload, the assumption that leads to well-posedness indeed usually holds in almost all string-actuated motion mechanisms. In addition to the suppression of oscillations of tension in each cable, the suppression of tension oscillation discrepancy between two cables is required. Therefore,

DUAL-CABLE ELEVATORS

37

the error between tension oscillations in two cables should be a part of the model used to design a controller. Toward that end we introduce the new error and mean variables e(x, t) = v(x, t) − u(x, t), s(x, t) = v(x, t) + u(x, t),

(3.10) (3.11)

where EA ex (x, t) = EA vx (x, t) − EA ux (x, t) is the tension oscillation discrepancy between two cables, and EA sx (x, t) is the total of the tension oscillations in the two cables. The control objectives are formulated in terms of the following physical behavior and theoretical results: • Physically, suppress the oscillations of tension in each cable as fast as possible. Theoretically, make the (u, v)-system exponentially stable in the sense of terms including uxx (·, t) + vxx (·, t) , where the exponential decay rate can be chosen. • Physically, reduce the error of oscillations of tension between two cables as fast as possible. Theoretically, make the (e, s)-system exponentially stable in the sense of terms including exx (·, t) + sxx (·, t) , where the exponential decay rate can be chosen. • Physically, suppress the axial vibration displacement y(t) and the roll angle θ(t) of the cage as fast as possible. Theoretically, make |X(t)| exponentially convergent to zero when t → ∞, where the exponential decay rate can be chosen. This may be a convenient place to remind the reader of the preceding paragraphs l(t) 2 u(x, t) dx. that u(·, t) is a compact notation for 0 Reformulation in Riemann Coordinates In order to convert the wave models to first-order systems, for the sake of enabling the applicability of backstepping designs for first-order hyperbolic PDE-ODE systems, we introduce the following Riemann coordinates: √ z1 (x, t) = st (x, t) − qsx (x, t), √ w1 (x, t) = st (x, t) + qsx (x, t), √ z2 (x, t) = et (x, t) − qex (x, t), √ w2 (x, t) = et (x, t) + qex (x, t).

(3.12) (3.13) (3.14) (3.15)

With (3.10), (3.11), the system (3.1)–(3.7) is rewritten as 2  Bi √ wi (0, t), q i=1 c √ zit (x, t) = − qzix (x, t) − (zi (x, t) + wi (x, t)), 2 c √ wit (x, t) = qwix (x, t) − (zi (x, t) + wi (x, t)), 2 zi (0, t) = Di X(t) − wi (0, t), √ wi (l(t), t) = zi (l(t), t) + 2 qUei (t),

˙ ¯ X(t) = AX(t) +

(3.16) (3.17) (3.18) (3.19) (3.20)

CHAPTER THREE

38 Ue2(t)

Ue1(t)

w1(x, t) z1(x, t) w2(x, t)

l(t)

z2(t) X(t)

ODE

Figure 3.2. Diagram of the plant dynamics (3.16)–(3.20). for x ∈ [0, l(t)], t ∈ [0, ∞), where i = 1, 2, and

T ¯1 + B ¯2 )/2 = 0, 0, EA , 0 , B 1 = (B M

T EA l1 ¯ ¯ B2 = (B2 − B1 )/2 = 0, 0, 0, Jc ⎡ 0 ⎢ 2B1 2B2 ⎢ 0 A¯ = A − √ C3 + √ C4 l1 = ⎢ 0 q q ⎣ 0

(3.21) , 0 0 0 0

(3.22)

−cd M

1 0 2E√A −M q 0

⎤

0 1 0 −ca Jc

D1 = 4C3 = [0, 0, 4, 0]T , D2 = −4l1 C4 = [0, 0, 0, −4l1 ]T ,

2

A √l1 + 2E Jc q

⎥ ⎥ ⎥, ⎦

(3.23)

(3.24)

and Ue1 (t) = U2 (t) + U1 (t),

(3.25)

Ue2 (t) = U2 (t) − U1 (t).

(3.26)

The PDEs (3.17)–(3.20) are two pairs of 2 × 2 hyperbolic systems with states wi (x, t), zi (x, t), i = 1, 2 and with source terms that result from in-domain damping. The PDEs governing wi (x, t), zi (x, t) are coupled with an ODE (3.16) at the distal boundary. The first transport PDE pair wi is actuated at one boundary (3.20), and is driving the ODE (3.16) through its other boundary. The second transport PDE pair zi convects backward and is driven by the state of the ODE X(t) and the boundary states wi (0, t) in (3.19). The diagram describing these plant dynamics is shown in figure 3.2. Unlike [147, 148, 149], which achieve results robust to small positive in-domain viscous damping in wave PDEs, in our case the in-domain damping coefficient c can be an arbitrary constant.

3.2

OBSERVER FOR CABLE TENSION

In this section we pursue observer design to estimate the distributed states zi (x, t), wi (x, t) associated with tension oscillations ux (x, t), vx (x, t) in the two cables according to (3.10), (3.11), (3.12)–(3.15). Given that distributed tension usually cannot

DUAL-CABLE ELEVATORS

39

be measured in practice, though it is needed in the controller, we design an observer of distributed tension using only the available boundary measurements. Observer Structure The available measurements in the mining cable elevator are ¨ in the cage • axial vibration acceleration y¨(t) and roll angular acceleration θ(t) where accelerometers are placed. Measuring acceleration is the prevalent method in the control of vibrating mechanical systems because accelerations are easier to measure with accelerometers than displacements or velocities [17]; • force EA ux (l(t), t), EA vx (l(t), t) and velocity ut (l(t), t), vt (l(t), t) feedback signals of the actuators at two floating sheaves. The observer is constructed as a copy of the plant (3.16)–(3.20) with output error injections: 2

 Bi ˆ˙ ˆ + ˆi (0, t) X(t) =A¯X(t) √ w q i=1   ˆ ¯ (y(t) + θ(t)) − (C1 + C2 )X(t) , +L √

c zi (x, t) + w qˆ zix (x, t) − (ˆ ˆi (x, t)) 2 ¯ i (x, t)(zi (l(t), t) − zˆi (l(t), t)), +Γ c √ zi (x, t) + w ˆix (x, t) − (ˆ w ˆit (x, t) = q w ˆi (x, t)) 2 + Γi (x, t)(zi (l(t), t) − zˆi (l(t), t)), zˆit (x, t) = −

(3.27)

(3.28)

(3.29)

zˆ1 (0, t) =D1 X(t) − w ˆ1 (0, t) = 4y(t) ˙ −w ˆ1 (0, t),

(3.30)

˙ −w zˆ2 (0, t) =D2 X(t) − w ˆ2 (0, t) = −4l1 θ(t) ˆ2 (0, t), √ w ˆi (l(t), t) =zi (l(t), t) + 2 qUei (t),

(3.31) (3.32)

where zi (l(t), t) can be obtained by using sx (l(t), t), st (l(t), t), ex (l(t), t), et (l(t), t), which are computed by the measurements ux (l(t), t), vx (l(t), t), ut (l(t), t), vt (l(t), t). Because it is the acceleration (rather than displacement) sensor that is more con˙ in X(t) are calculated by integratvenient to place at the cage, y(t), θ(t), y(t), ˙ θ(t) ¨ ing the measured accelerations y¨(t), θ(t) with the known initial conditions y(0), ˙ θ(0), y(0), ˙ θ(0) (this is only the explanation of an implementable signal acquisition method in practice, and no restrictions are imposed on any initial conditions in the design and theory). The ODE measurements y(t), θ(t) and the matrix ¯ C1 + C2 ), where C1 = [1, 0, 0, 0], A¯ in (3.27) form an observable matrix pair (A, C2 = [0, 1, 0, 0]. Observer Error Dynamics and Backstepping Defining the observer errors ˜ (˜ zi (x, t), w ˜i (x, t), X(t)) =(zi (x, t), wi (x, t), X(t)) ˆ ˆi (x, t), X(t)), − (ˆ zi (x, t), w

(3.33)

CHAPTER THREE

40

we have the observer error dynamics between the plant and the observer as 2

 Bi ˜˙ ˜ + X(t) = AˆX(t) ˜i (0, t), √ w q i=1 c √ ¯ i (x, t)˜ zi (x, t) + w z˜it (x, t) = − q˜ zix (x, t) − (˜ ˜i (x, t)) − Γ zi (l(t), t), 2 c √ zi (x, t) + w ˜ix (x, t) − (˜ ˜i (x, t)) − Γi (x, t)˜ zi (l(t), t), w ˜it (x, t) = q w 2 z˜i (0, t) = −w ˜i (0, t), w ˜i (l(t), t) = 0,

(3.34) (3.35) (3.36) (3.37) (3.38)

where the system matrix ¯ 1 + C2 ) Aˆ = A¯ − L(C ¯ We would like to design the observer in (3.34) is made Hurwitz by choosing L. ¯ ¯ gains Γ1 (x, t), Γ2 (x, t), Γ1 (x, t), Γ2 (x, t) to make sure the observer error dynamics (3.34)–(3.38) are exponentially stable. Using the backstepping transformation  ˜ i (x, t) − z˜i (x, t) = α w ˜i (x, t) = β˜i (x, t) −

l(t)

x  l(t)

αi (y, t)dy, φ¯i (x, y)˜

(3.39)

αi (y, t)dy, ψ¯i (x, y)˜

(3.40)

x

we would like to convert the observer error dynamics (3.34)–(3.38) into the target observer error system as    l(t) 2  Bi ˜˙ ˜ + X(t) =AˆX(t) αi (y, t)dy , ψ¯i (0, y)˜ √ β˜i (0, t) + q 0 i=1 α ˜ it (x, t) = −

√



l(t)

qα ˜ ix (x, t) +

(3.41)

¯ i (x, y)β˜i (y, t)dy M

x

c c ˜ i (x, t) − β˜i (x, t), − α 2 2  l(t) c √ ˜ ˜ ˜ ¯i (x, y)β˜i (y, t)dy, N βit (x, t) = q βix (x, t) − βi (x, t) + 2 x α ˜ i (0, t) = − β˜i (0, t), β˜i (l(t), t) =0.

(3.42) (3.43) (3.44) (3.45)

The target system (3.41)–(3.45) is a PDE-ODE cascaded system where the PDE (3.42)–(3.45) has the same structure as the target system (17)–(20) in [21], and the ˆ PDE states flow into the ODE (3.41) with a Hurwitz system matrix A. Remark that the heterodirectional coupled hyperbolic PDEs (3.35), (3.36), which include in-domain unstable sources, especially the coupling terms z˜i (x, t), z˜i (l(t), t) in (3.36) whose propagate direction is from the right boundary to the left boundary and the ODE—that is, the “forward” direction, are intentionally converted into the target system where there is no coupling term (the states form the transport PDE in the reverse direction) in the “forward” transport PDE (3.43). This motivation is

DUAL-CABLE ELEVATORS

41

also used to construct the target system for (3.27)–(3.32), which will be shown in section 3.3. ¯ i (x, y), The kernels φ¯i (x, y), ψ¯i (x, y) in the transformation (3.39), (3.40), M ¯i (x, y) in (3.42), (3.43), and the observer gains Γ ¯ i (x, t), Γi (x, t) are determined N next. Calculation of the Kernels and Observer Gains Substituting the transformations (3.39), (3.40) into (3.35), and inserting (3.42), through a lengthy calculation, we get c √ ¯ i (x, t)˜ zi (x, t) + w z˜it (x, t) + q˜ zix (x, t) + Γ zi (l(t), t) + (˜ ˜i (x, t)) 2   l(t)  c √ √ ˜ i (y, t)dy = − q φ¯ix (x, y) − q φ¯iy (x, y) − ψ¯i (x, y) α 2 x   l(t)   y c ˜ ¯ ¯ ¯ ¯ φi (x, z)Mi (z, y)dz − Mi (x, y) − φi (x, y) βi (y, t)dy − 2 x  x ˙ φ¯i (x, l(t)) + √q φ¯i (x, l(t)) α ¯ i (x, t) − l(t) ˜ i (l(t), t). (3.46) + Γ Substituting the transformation (3.39), (3.40) into (3.36), and inserting (3.42), (3.43) with further lengthy calculation, we arrive at c √ ˜i (x, t) + z˜i (x, t)) ˜ix (x, t) + Γi (x, t)˜ zi (l(t), t) + (w w ˜it (x, t) − q w 2  c √ ¯ − 2 q ψi (x, x) α ˜ i (x, t) = 2    l(t) c¯ √ ¯ √ ¯ ˜ i (y, t)dy + − q ψiy (x, y) + q ψix (x, y) − φi (x, y) α 2 x   y  l(t)  c¯ ¯i (x, y) − ¯ i (z, y)dz β˜i (y, t)dy ψi (x, y) + N ψ¯i (x, z)M + 2 x  x √ ˙ ψ¯i (x, l(t)) + q ψ¯i (x, l(t)) α ˜ i (l(t), t). (3.47) + Γi (x, t) − l(t) To make sure the right-hand sides of the equal signs in (3.46), (3.47) are equal to zero, and matching (3.44) and (3.37), the kernels φ¯i (x, y), ψ¯i (x, y) in the backstepping transformation (3.39), (3.40) should satisfy c √ √ − q φ¯ix (x, y) − q φ¯iy (x, y) = ψ¯i (x, y), 2 c √ √ − q ψ¯iy (x, y) + q ψ¯ix (x, y) = φ¯i (x, y), 2 φ¯i (0, y) = −ψ¯i (0, y), c ψ¯i (x, x) = √ , 4 q ¯ i (x, y), N ¯i (x, y) in (3.42) and (3.43) should satisfy and M  y c¯ ¯ ¯ i (z, y)dz, Mi (x, y) = − φi (x, y) + φ¯i (x, z)M 2 x

(3.48) (3.49) (3.50) (3.51)

(3.52)

CHAPTER THREE

42 ¯i (x, y) = − c ψ¯i (x, y) + N 2



y

¯ i (z, y)dz. ψ¯i (x, z)M

(3.53)

x

¯ i (x, t) are thus obtained as The observer gains Γi (x, t), Γ ˙ ψ¯i (x, l(t)) − √q ψ¯i (x, l(t)), Γi (x, t) = l(t) ˙ φ¯i (x, l(t)) − √q φ¯i (x, l(t)). ¯ i (x, t) = l(t) Γ

(3.54) (3.55)

Lemma 3.1. The kernel equations (3.48)–(3.51) have a unique continuous solution (ψ¯i , φ¯i ) on D1 = {(x, y)|0 ≤ x ≤ y ≤ l(t)}. The proof of lemma 3.1 is shown in appendix 3.6. Exponential Convergence of Observer Errors ¯ i (x, t), Γi (x, t), we prove the exponential staAfter obtaining the observer gains Γ ¯ i (x, t), bility of the observer error dynamics (3.34)–(3.38) with the designed gains Γ Γi (x, t) in the following lemma, whose proof is shown in appendix 3.6. ˜i (x, t0 )) ∈ L2 (0, L0 ), the observer Lemma 3.2. For all initial values (˜ zi (x, t0 ), w error system (3.34)–(3.38) is uniformly exponentially stable in the sense of the norm  2 2  12    ˜ 

˜ zi (·, t) 2 + w ˜i (·, t) 2 + X(t) , (3.56) i=1

where L2 (0, L0 ) is the usual Hilbert space, with L0 = l(t0 ) the initial length of the cable that is being retracted. The word “uniformly” in lemma 3.2 refers to uniformity in the initial time t0 . Using lemma 3.2 and (3.33), it is straightforward to prove the following theorem. Theorem 3.1. For all initial values (zi (x, t0 ), wi (x, t0 )) ∈ L2 (0, L0 ) and (ˆ zi (x, t0 ), w ˆi (x, t0 )) ∈ L2 (0, L0 ), the observer (3.27)–(3.32) can track the system (3.16)–(3.20) with uniformly exponentially convergent errors in the sense of 2  

2   ˆ  .

zi (·, t) − zˆi (·, t) 2 + wi (·, t) − w ˆi (·, t) 2 + X(t) − X(t)

(3.57)

i=1

Theorem 3.1 shows that the proposed observer recovers the distributed states of the plant (3.16)–(3.20) using only the available boundary measurements. Moreover, the following lemma, whose proof is shown in appendix 3.6 holds as well. Lemma 3.3. For all initial data (˜ zi (x, t0 ), w ˜i (x, t0 )) ∈ H 1 (0, L0 ), the uniform exponential stability estimate of the observer error system (3.34)–(3.38) is obtained in the sense of 2  

˜ zix (·, t) 2 + w ˜ix (·, t) 2

 12

,

i=1

where H 1 (0, L0 ) = {u|u(·, t) ∈ L2 (0, L0 ), ux (·, t) ∈ L2 (0, L0 )}.

(3.58)

DUAL-CABLE ELEVATORS

3.3

43

CONTROLLER FOR CABLE TENSION OSCILLATION SUPPRESSION AND CAGE BALANCE

In section 3.2, we obtained an observer that can exponentially track the distributed states of the system (3.16)–(3.20). In this section, we design output-feedback control laws U1 (t), U2 (t) by using the states recovered from the observer via the backstepping method [116],[126]. Backstepping Transformation and Target System The design of the observer-based output-feedback controller is based on the observer (3.27)–(3.32). Using the backstepping transformation αi (x, t) ≡ zˆi (x, t),



(3.59) x

βi (x, t) = w ˆi (x, t) − ψi (x, y)ˆ zi (y, t)dy 0  x ˆ φi (x, y)w ˆi (y, t)dy − γi (x)X(t), −

(3.60)

0

we would like to convert the observer system (3.27)–(3.32) to the following target system:   2 2   Bi ˙ ¯ ˆ + ˆ X(t) = A + Bi κi X(t) √ βi (0, t) q i=1 i=1 ˜ ¯ 1 + C2 )X(t), + L(C c c √ αit (x, t) = − qαix (x, t) − βi (x, t) − αi (x, t) 2 2   c x ˆ c x ˆ Mi (x, y)αi (y, t)dy − Ni (x, y)βi (y, t)dy − 2 0 2 0 c ˆ +Γ ¯ i (x, t)˜ zi (l(t), t), − ϑˆi (x)X(t) 2 c √ βit (x, t) = qβix (x, t) − βi (x, t) 2 ˜ − Ni (x, t)z˜i (l(t), t) − N1i (x)X(t), ˆ − βi (0, t) + Di X(t), ˜ ¯ i X(t) αi (0, t) =D βi (l(t), t) =0, where



(3.62)

(3.63) (3.64) (3.65)

x

¯ i (y, t))dy − Γi (x, t), (φi (x, y)Γi (y, t) + ψi (x, y)Γ ¯ 1 + C2 ) + √qψi (x, 0)Di , N1i (x) = γi (x)L(C

Ni (x, t) =

(3.61)

(3.66)

0

and where



x

ˆi (x, y) = N y

ˆi (x, δ)φi (δ, y)dδ + φi (x, y), N

(3.67)

(3.68)

CHAPTER THREE

44 

x

ˆ i (x, y) = M  ϑˆi (x) =

ˆi (x, δ)ψi (δ, y)dδ + ψi (x, y), N

(3.69)

ˆi (x, y)γi (y)dy + γi (x). N

(3.70)

y x 0

Recalling (3.21), (3.22), (3.23), the matrix A´ = A¯ + ⎡ ⎢ ⎢ =⎢ ⎣

2 

B i κi

i=1

0 0

EA κ ˆ 11 M

0

0 0 0 E A l1 κ ˆ 22 Jc

2E√A −M q

1 0 13 −cd + EA κˆM 0

0 1 0 2

2EA√l1 Jc q

+ EA l1 κJˆc24 −ca

⎤ ⎥ ⎥ ⎥ ⎦

(3.71)

is made Hurwitz by choosing the row vectors κ1 = [kˆ11 , 0, kˆ13 , 0], κ2 = [0, kˆ22 , 0, kˆ24 ].

(3.72) (3.73)

¯ i are D ¯ i = Di − γi (0). In the following section, the kernels ψi (x, y), The matrices D φi (x, y), γi (x) are determined by mapping the observer system (3.27)–(3.32) and the target system (3.61)–(3.65) via the transformations (3.59), (3.60). Calculation of Kernels Substituting (3.59), (3.60) into (3.63), we get c √ ˜ βit (x, t) − qβix (x, t) + βi (x, t) + Ni (x, t)z˜i (l(t), t) + N1i (x)X(t) 2   c √ = − + 2 qψi (x, x) zˆi (x, t)  2x  c √ √ ψi (x, y) + qφix (x, y) + qφiy (x, y) w ˆi (y, t)dy + 2 0   x c √ √ φi (x, y) + qψix (x, y) − qψiy (x, y) zˆi (y, t)dy + 2  0  c  √  ˆ ¯ qγi (x) − γi (x) A + I4 − ψi (x, 0)Di X(t) + 2   1 √ √ + qφi (x, 0) − √ γi (x)Bi + qψi (x, 0) w ˆi (0, t) q 1 ˆj (0, t), − √ γi (x)Bj w q

(3.74)

where I4 is an identity matrix with dimension 4. To guarantee that the right-hand side of the equal sign in (3.74) is equal to zero, which ensures (3.63), and that the ODE (3.27) is mapped into (3.61), we get the following kernel conditions: √

qφix (x, y) +

√

c qφiy (x, y) = − ψi (x, y), 2

(3.75)

DUAL-CABLE ELEVATORS

45

c √ √ qψix (x, y) − qψiy (x, y) = − φi (x, y), 2 1 φi (x, 0) = γi (x)Bi − ψi (x, 0), q c ψi (x, x) = √ , 4 q  1 1 c  γi  (x) − √ γi (x) A¯ + I4 = √ ψi (x, 0)Di , q 2 q √ γi (0) = qκi .

(3.76) (3.77) (3.78) (3.79) (3.80)

According to (3.23), (3.24), (3.72), (3.73), it follows from (3.79), (3.80) that γ1 (x) and γ2 (x) are in the forms γ1 (x) = [ˆ γ11 (x), 0, γˆ13 (x), 0], γ2 (x) = [0, γˆ22 (x), 0, γˆ24 (x)].

(3.81) (3.82)

Recalling (3.21), (3.22), we have γi (x)Bj = 0,

i = j

(3.83)

in (3.74). The following lemma shows that there exists a unique continuous solution (ψi , φi , γi ) of (3.75)–(3.80). The proof is shown in appendix 3.6. Lemma 3.4. For a given κ1 , κ2 , the kernel equations (3.75)–(3.80) have a unique continuous solution (ψi , φi , γi ) on D = {(x, y)|0 ≤ y ≤ x ≤ l(t)}. Equations (3.62), (3.64) are obtained straightforwardly, recalling (3.28), (3.30), (3.31) and the transformations (3.59), (3.60). The boundary condition (3.65) will be achieved by choosing the control inputs in the next subsection. The inverse transformations are given by zˆi (x, t) ≡ αi (x, t),



(3.84) x

w ˆi (x, t) = βi (x, t) − ψiI (x, y)αi (y, t)dy 0  x I ˆ φi (x, y)βi (y, t)dy − γiI (x)X(t), −

(3.85)

0

and the existence of kernels ψiI (x, y), φIi (x, y), γiI (x) refers to section 2.4 in [183]. Control Law Taking into account the boundary condition (3.65) in the target system, the boundary condition (3.32) in the observer, and the transformation (3.60), we derive the controller as   l(t) −1 Ue1 (t) = √ z1 (l(t), t) − ψ1 (l(t), y)ˆ z1 (y, t)dy 2 q 0   l(t) ˆ − φ1 (l(t), y)w ˆ1 (y, t)dy − γ1 (l(t))X(t) , (3.86) 0

CHAPTER THREE

46

  l(t) −1 Ue2 (t) = √ z2 (l(t), t) − ψ2 (l(t), y)ˆ z2 (y, t)dy 2 q 0   l(t) ˆ − φ2 (l(t), y)w ˆ2 (y, t)dy − γ2 (l(t))X(t) .

(3.87)

0

ˆ The signals zˆi (x, t), w ˆi (x, t), X(t) are obtained from the observer (3.27)–(3.32) ¨ constructed by the measurements y¨(t), θ(t) and ux (l(t), t), vx (l(t), t), ut (l(t), t), vt (l(t), t) which are used to calculate zi (l(t), t). The gains (ψi (x, y), φi (x, y), γi (x)) are the solution of (3.75)–(3.80). Using (3.86), (3.87), the two control inputs U1 (t) and U2 (t) of (3.1)–(3.7) are derived as U1 (t) = Ue1 (t) − Ue2 (t), U2 (t) = Ue1 (t) + Ue2 (t).

(3.88)

The proposed controller requires only readily available measurements of the mining cable elevator. In particular, the highest time derivative signals used in the con˙ troller are the first-order derivatives ut (l(t), t), vt (l(t), t), y(t), ˙ θ(t). Physically, they are velocities and are measurable or easily deducible from acceleration measurements.

3.4

STABILITY ANALYSIS

The following lemma establishes the exponential stability of the system (3.27)– (3.32) under the control (3.86), (3.87). The proof is shown in appendix 3.6. Lemma 3.5. For all initial values (ˆ zi (x, t0 ), w ˆi (x, t0 )) ∈ L2 (0, L0 ), the system (3.27)–(3.32) under the control law (3.86), (3.87) is uniformly exponentially stable in the sense of the norm  2

2 1/2   ˆ 

ˆ zi (·, t) 2 + w ˆi (·, t) 2 + X(t) .



(3.89)

i=1

ˆ Based on the exponential stability result of the (ˆ zi , w ˆi , X)-system in the sense of 2 ˆ ˆi (·, t) 2 + |X(t)| , we can obtain the exponential stability estimate in

ˆ zi (·, t) 2 + w ˆix (·, t) 2 in the following lemma, whose proof is shown the sense of ˆ zix (·, t) 2 + w in appendix 3.6. Lemma 3.6. For all initial values (ˆ zi (x, t0 ), w ˆi (x, t0 )) ∈ H 1 (0, L0 ), the uniform exponential stability estimate of the system (3.27)–(3.32) under the control law (3.86), (3.87) is obtained in the sense of 

2  

2

2

ˆ zix (·, t) + w ˆix (·, t)



   ˆ 2 + X(t) 

 12 .

(3.90)

i=1

The following theorems are used to show the achievement of the control objectives formulated in section 3.1 under the proposed output-feedback controller, which is bounded and exponentially convergent to zero.

DUAL-CABLE ELEVATORS

47

Theorem 3.2. For all initial values (ˆ zi (x, t0 ), w ˆi (x, t0 ), zi (x, t0 ), wi (x, t0 )) ∈ H 1 (0, L0 ), the closed-loop system including the plant (3.16)–(3.20), the controller (3.86), (3.87), and the observer (3.27)–(3.32) has the following properties. 1) The closed-loop system is uniformly exponentially stable in the sense of the norm  2 

ˆ zi (·, t) 2 + w ˆi (·, t) 2 + zi (·, t) 2 + wi (·, t) 2 + ˆ zix (·, t) 2 i=1

1/2 2   ˆ  + |X(t)|2 + w ˆix (·, t) 2 + zix (·, t) 2 + wix (·, t) 2 + X(t)

(3.91)

with a decay rate σall that can be adjusted by the choices of the control parameters ¯ κi , L. 2) In the closed-loop system, there exist the positive constants σU i and Υ0i , making Ue1 (t), Ue2 (t) bounded and exponentially convergent to zero in the sense of |Ue1 (t)| ≤ Υ01 e−σU 1 t , |Ue2 (t)| ≤ Υ02 e−σU 2 t .

(3.92)

Proof. 1) We now prove the first of the two portions of the theorem. Recalling the exponential stability result in the sense of ˆ z1 (·, t) 2 + w ˆ1 (·, t) 2 + ˆ z2 (·, t) 2 + 2 2 ˆ

w ˆ2 (·, t) + |X(t)| proved in lemma 3.5 with the decay rate σ, and the exponen˜1 (·, t) 2 + ˜ z2 (·, t) 2 + w ˜2 (·, t) 2 + tial stability result in the sense of ˜ z1 (·, t) 2 + w 2 ˜ |X(t)| proved in lemma 3.2 with the decay rate σe , we get the exponential stability result in the sense of z1 (·, t) 2 + w1 (·, t) 2 + z2 (·, t) 2 + w2 (·, t) 2 + |X(t)|2 via (3.33) with a decay rate that can be adjusted by the control parameters κi ¯ and L. Similarly, recalling the exponential stability estimate in the sense of ˆ z1x (·, t) 2 2 2 2 + w ˆ1x (·, t) + ˆ z2x (·, t) + w ˆ2x (·, t) proved in lemma 3.6 with the decay rate z1x (·, t) 2 + w ˜1x (·, t) 2 + σH and the exponential stability estimate in the sense of ˜ 2 2 ˜2x (·, t) proved in lemma 3.3 with the decay rate σeH , we obtain

˜ z2x (·, t) + w the exponential stability estimate in the sense of z1x (·, t) 2 + w1x (·, t) 2 +

z2x (·, t) 2 + w2x (·, t) 2 with a decay rate that can be adjusted by the control ¯ as well. parameters κi and L The proof of property (1) in theorem 3.2 is complete. 2) We now prove the second portion of the theorem. Applying the CauchySchwarz inequality to (3.86), (3.87), we get 2

2

|Ue1 (t)| + |Ue2 (t)| ≤

where

2   1

1 zi (l(t), t)2 − M10i L ˆ zi (·, t) 2 q i=1    1 1  ˆ 2 − M11i L w ˆi (·, t) 2 − M12i X(t)  , q q q

  M10i = max ψi (l(t), y)2 ,   M11i = max φi (l(t), y)2 ,   M12i = max γi (l(t))2

for 0 ≤ y ≤ l(t) ≤ L, with L being the total length of the cables.

(3.93)

CHAPTER THREE

48 Using the Cauchy-Schwarz inequality and (3.65), we get

 2  l(t)    |βi (0, t)|2 ≤ |βi (l(t), t)|2 +  βix (x, t)dx ≤ L βix (·, t) 2 .  0 

(3.94)

According to the exponential stability estimate in the sense of βix (·, t) 2 in the proof of lemma 3.6, we establish that βi (0, t) is exponentially convergent to zero. ˆ Recalling (3.60) at x = 0 and the exponential convergence of |X(t)| proved in lemma 3.5, we find that w ˆi (0, t) is exponentially convergent to zero. Recalling the relationships (3.30), (3.31) and the exponential convergence of |X(t)| proved in property (1), we obtain the exponential convergence of zˆi (0, t). Similarly, using the Cauchy-Schwarz inequality and (3.45), recalling the exponential stability estimate in the sense of β˜ix (·, t) 2 in the proof of lemma 3.3, we find that β˜i (0, t) is exponentially convergent to zero and then obtain the exponential convergence of w ˜i (0, t) via (3.40) at x = 0. Together with the exponential convergence of w ˆi (0, t) proved above, we have that wi (0, t) is exponentially convergent to zero. Through (3.19), together with the exponential convergence of |X(t)|, we find that zi (0, t) is exponentially convergent to zero as well. Using the Cauchy-Schwarz inequality, we also have  2  l(t)    2 2 |zi (l(t), t)| ≤ |zi (0, t)| +  zix (x, t)dx  0  ≤ |zi (0, t)|2 + L zix (·, t) 2 .

(3.95)

We then find that zi (l(t), t) is exponentially convergent to zero when t → ∞ by recalling the exponential stability estimate in the sense of zix (·, t) 2 proved in property (1) and the exponential convergence of zi (0, t) proved above. Together ˆ1 (·, t) 2 + ˆ z2 (·, t) 2 + with the exponential stability in the sense of ˆ z1 (·, t) 2 + w 2 2 ˆ proved in lemma 3.5, we ascertain that the control inputs

w ˆ2 (·, t) + |X(t)| Ue1 (t), Ue2 (t) are bounded and exponentially convergent to zero according to (3.93). The proof of property (2) in theorem 3.2 is complete. According to theorem 3.2, we can obtain the stability properties of the (u, v) and (e, s) systems in the following theorem. Theorem 3.3. For all initial values (u(x, t0 ), v(x, t0 )) ∈ H 2 (0, L0 ), (e(x, t0 ), s(x, t0 )) ∈ H 2 (0, L0 ), the original closed-loop (u, v)-system including the plant (3.1)– (3.7) with the controllers (3.88) is uniformly exponentially stable in the sense of

ut (·, t) + ux (·, t) + vt (·, t) + vx (·, t) + uxt (·, t)

+ uxx (·, t) + vxt (·, t) + vxx (·, t) ,

(3.96)

where H 2 (0, L0 ) = {u|u(·, t) ∈ L2 (0, L0 ), ux (·, t) ∈ L2 (0, L0 ), uxx (·, t) ∈ L2 (0, L0 )}. The (e, s)-system obtained from (3.10), (3.11) is also uniformly exponentially stable in the sense of

et (·, t) + ex (·, t) + st (·, t) + sx (·, t) + ext (·, t)

+ exx (·, t) + sxt (·, t) + sxx (·, t) .

(3.97)

DUAL-CABLE ELEVATORS

49

¯ MoreThe exponential decay rates can be adjusted by the control parameters κi , L. over, the controllers U1 (t), U2 (t) realized by hydraulic actuators at the floating sheaves are bounded and exponentially convergent to zero. Proof. Recalling (3.12)–(3.15) and applying the Cauchy-Schwarz inequality, we get 1 1

et (·, t) 2 ≤ zi (·, t) 2 + wi (·, t) 2 , 2 2 1 1

ex (·, t) 2 ≤ wi (·, t) 2 + zi (·, t) 2 , 2q 2q 1 1

ext (·, t) 2 ≤ zix (·, t) 2 + wix (·, t) 2 , 2 2 1 1

exx (·, t) 2 ≤ wix (·, t) 2 + zix (·, t) 2 2q 2q

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

for i = 2, and st (·, t) 2 , sx (·, t) 2 , sxt (·, t) 2 , sxx (·, t) 2 have the same inequality relationships with (3.98)–(3.101) for i = 1. Therefore, recalling the exponential stability with the decay rate σall in the sense of zi (·, t) 2 + wi (·, t) 2 + zix (·, t) 2 +

wix (·, t) 2 proved in property (1) in theorem 3.2, we see that the (e, s)-system obtained from (3.10), (3.11) is exponentially stable in the sense of (3.97) with a decay rate of at least σall . According to the definition (3.10), (3.11), it is then straightforward to ascertain that the system (3.1)–(3.7) is exponentially stable in the sense of (3.96), with a ¯ decay rate of at least σall that can be adjusted by the control parameters κi , L. Recalling (3.88) and property (2) in theorem 3.2, we show that the controllers U1 (t), U2 (t) at the floating sheaves are bounded and exponentially convergent to zero. The proof of theorem 3.3 is complete. According to the exponential convergence of |X(t)| proved in property (1) of theorem 3.2 and the exponential stability estimate in the sense of the norms including uxx (·, t) + vxx (·, t) and exx (·, t) + sxx (·, t) proved in theorem 3.3, with adjustable exponential decay rates, we can state that the control objectives proposed in section 3.1 are achieved.

3.5

SIMULATION TEST FOR A DUAL-CABLE MINING ELEVATOR

The physical parameters of the mining cable elevator (3.1)–(3.7) used in the sim˙ ulation are shown in table 3.1. The design reference of the hoisting velocity l(t) ¯ = [1, 1, 1, 1]T , is plotted in figure 3.3. The control parameters chosen here are L κ1 = [0.0016, 0, 0.03, 0]T , and κ2 = [0, −0.0018, 0, −0.03]T . We also apply the boundary damper, which is traditionally utilized in industry at the head sheaves, to compare with the proposed output-feedback controller. The boundary damper feedback laws are given as ˙ t) Ud1 (t) = kd1 u(l(t), ˙ = kd1 (ut (l(t), t) + l(t)u x (l(t), t))

CHAPTER THREE

50  = kd1

1 (z1 (l(t), t) + w1 (l(t), t) − z2 (l(t), t) − w2 (l(t), t)) 4

 ˙ l(t) + √ ((w1 (l(t), t) − z1 (l(t), t)) − (w2 (l(t), t) − z2 (l(t), t))) , 4 q

(3.102)

Ud2 (t) = kd2 v(l(t), ˙ t) ˙ = kd2 (vt (l(t), t) + l(t)v x (l(t), t))  1 = kd2 (z1 (l(t), t) + w1 (l(t), t) + z2 (l(t), t) + w2 (l(t), t)) 4  ˙ l(t) + √ ((w1 (l(t), t) − z1 (l(t), t)) + (w2 (l(t), t) − z2 (l(t), t))) , 4 q

(3.103)

where kd1 , kd2 are tuned to attain efficient control performance. We have tested different values of kd1 , kd2 , and the best regulating performance is achieved with kd1 = −0.33, kd2 = −0.31. Computational Method The simulation is performed for the model (3.16)–(3.20) with the control law (3.86), (3.87) and the observer (3.27)–(3.32). The responses of tension oscillations EA ux (x, t), EA vx (x, t) in the cables can be calculated by those of z1 , w1 , z2 , w2 through (3.12)–(3.15) and (3.10), (3.11). The actual control forces EA U1 (t), EA U2 (t) applied at the two floating head sheaves are obtained via (3.88). The simulation is performed by the finite-difference method for the discretization in time and space after converting the time-varying domain PDE to a fixed-domain x [184], and the time step and space step are chosen as PDE via introducing ξˇ = l(t) Table 3.1. Physical parameters of the dual-cable mining elevator Parameters (units)

Values

Initial length L (m) Final length (m) Cable effective steel area Aa (m2 ) Cable effective Young’s modulus E (N/m2 ) Cable linear density ρ (kg/m) Total hoisted mass M (kg) Moment of inertia of the cage Jc (kg·m2 ) Gravitational acceleration g (m/s2 ) Maximum hoisting velocities v¯max (m/s) Total hoisting time tf (s) Cable material damping coefficient c¯ (N·s/m) Cage axial damping coefficient cd (N·s/m) Cage roll damping coefficient ca (N·m·s/rad) Cage dimension l1 (m)

2000 50 0.47×10−3 2.1×1010 8.1 15000 17500 9.8 16.25 150 0.6 0.4 0.4 2.5

Note: cage dimensions referred to figure 3.1.

DUAL-CABLE ELEVATORS

51

|i(t)| (m/s)

15 10 5 0

50

0

100

150

Time (s)

˙ Figure 3.3. The hoisting velocity l(t). 0.001 and 0.05, respectively. The kernel equations (3.48)–(3.51) and (3.75)–(3.80), which are coupled linear hyperbolic PDEs, are also solved by the finite-difference method. Initial Conditions The initial conditions of the plant (3.1)–(3.7) are obtained from the physical conditions of the dual-cable mining elevator with an initial unbalance. In detail, the initial profiles of the axial strain ux (x, 0), vx (x, 0) are obtained by the force balance equations at the static state, which are written as ux (x, 0) =

ρxg + Me1 g ρxg + Me2 g , vx (x, 0) = , EA EA

(3.104)

where Me1 and Me2 are loads at the bottoms of the two cables. Define the loads Me1 , Me2 as 9250 kg and 5750 kg. The difference between the loads supported by the two cables might come from the imprecise manufacturing and installation of the two cables, or eccentricity of the cage, which always happens in loading, resulting in the initial strain error of the two cables according to (3.104). This initial strain error of the plant is unknown in the control system design. The initial vibration velocities of the two cables are defined as ut (x, 0) = 0, vt (x, 0) = 0 because the initial vibration velocity of each point in the cable is zero. The initial condition of X(t) is defined as T T ˙ X(0) = [y(0), θ(0), y(0), ˙ θ(0)] = [0, 0, 0, 0] .

From the initial conditions of ut (x, 0), vt (x, 0), ux (x, 0), vx (x, 0) and (3.10), (3.11), (3.12)–(3.15), the initial conditions (zi (x, 0), wi (x, 0)) of (3.16)–(3.20) tested ˆi (x, 0) of in simulation can be determined. Similarly, the initial conditions zˆi (x, 0), w the observer (3.27)–(3.32) are defined. We use Me1 = Me2 = M/2 in (3.104) to define the observer initial conditions because the initial error between loads Me1 , Me2 is unknown in the observer design. Thus, parts of the observer initial conditions ˆ1 (x, 0) = are equal to those of the plant initial conditions as zˆ1 (x, 0) = z1 (x, 0), w w1 (x, 0), and the others are different from the plant initial conditions as zˆ2 (x, 0) = ˆ2 (x, 0) = w2 (x, 0). z2 (x, 0), w

CHAPTER THREE

52 Simulation Results

Tension oscillations in cables: Tension is a crucial physical variable when investigating the strength of a cable. The suppression of tension oscillations is beneficial for easing fatigue damage and prolonging the service life of the hoisting cables in the mining elevator. The responses of tension oscillations at the midpoint of cable 1 and cable 2 are shown in figures 3.4 and 3.5, respectively, with the open loop, the boundary damper, and the proposed control law. The dot-and-dash line, depicting the open-loop response in figures 3.4 and 3.5, shows that large tension oscillations would be caused in the accelerated ascending process for the hoist˙ ing velocity curve l(t) in figure 3.3. To suppress the oscillations of tension, the proposed output-feedback controller and the boundary damper are applied at the head sheaves, respectively, and the responses are shown in solid and dashed lines in figures 3.4 and 3.5. It can be seen that tension oscillations are suppressed by both the proposed control law and the boundary damper. However, we observe that the responses with the proposed control law have faster convergence and less overshoot than the responses with the boundary damper. Error of tension oscillations between cables: The error of tension oscillations between cables is also an important physical index to investigate fatigue damage and to prolong the service life of the cables in dual-cable mining elevators. Because the constant mass of the cage is carried by two cables, a larger tension oscillation

Tension oscillations (N)

× 105

6

Proposed controller Boundary damper Without control

4 2 0 –2 –4 –6 0

50

100

150

Time (s)

Figure 3.4. Tension oscillations EA × ux (l(t)/2, t) at the midpoint of cable 1.

Tension oscillations (N)

5 6 × 10

Proposed controller Boundary damper Without control

4 2 0 –2 –4 –6

0

100

50

150

Time (s)

Figure 3.5. Tension oscillations EA × vx (l(t)/2, t) at the midpoint of cable 2.

Errors of tension oscillations (N)

DUAL-CABLE ELEVATORS

53

× 105

Proposed controller Boundary damper Without control

1 0.5 0 –0.5 –1 0

100

50

150

Time (s)

Axial vibrations of the cage (m)

Figure 3.6. Errors EA × (ux (l(t)/2, t) − vx (l(t)/2, t)) of tension oscillations between cable 1 and cable 2. 5 0.4 × 10

Proposed controller Boundary damper Without control

0.2 0 –0.2 –0.4

0

50

100

150

Time (s)

Figure 3.7. Axial vibration displacements y(t) of the cage. error between cables means that a larger maximum load would be supported by one of the cables, whose service life would be shortened due to more serious fatigue damage. The errors between cables in the open loop, under the boundary damper, and under the proposed control law are shown in figure 3.6. It shows that the tension oscillation error increases in 0–30 s, which is the accelerated ascending process, in the open-loop case, while it is reduced and convergent to zero under the proposed output-feedback controller and, moreover, has a faster convergence than the boundary damper. Axial and roll vibrations of the cage: Axial and roll vibrations of the cage not only bring discomfort to passengers but also increase the tension error between two cables, which would cause overburdening to one of the cables and result in fatigue fracture. The responses of the axial and roll vibrations of the cage are shown in figures 3.7 and 3.8 in the cases without control, with the boundary damper, and with the proposed output-feedback controller. It can be observed that the large axial vibrations and roll of the cage in the open loop are suppressed to zero more efficiently under the proposed output-feedback controller than the traditional boundary damper, with faster convergence and less overshoot. Observer errors and output-feedback control forces: The states used in the output-feedback control forces (3.86), (3.87) are the states recovered from the observer (3.27)–(3.32). Figure 3.9 shows the convergent observer errors between zˆ2 ,

CHAPTER THREE

54

Cage roll angle (rad)

0.1 Proposed controller Boundary damper Without control

0.05 0 –0.05 –0.1

0

100

50

150

Time (s)

Figure 3.8. Cage roll angles θ(t).

Observer error for w2(l(t)/2, t) Observer error for z2(l(t)/2, t)

Observer error

5

0

–5 0

100

50

150

Time (s)

Figure 3.9. Observer errors z2 (l(t)/2, t)− zˆ2 (l(t)/2, t) and w2 (l(t)/2, t)− w ˆ2 (l(t)/2, t) between the plant (3.16)–(3.20) and the observer (3.27)–(3.32). w ˆ2 and the plant states z2 , w2 at the midpoint of the domain [0, l(t)], which indicates that the observer (3.27)–(3.32) can reconstruct the actual distributed states in (3.16)–(3.20). Because the locations of the actuator and the sensor are at the boundaries, the estimation of the states at the midpoint x = l(t)/2 is most challengˆ1 (x, 0) ing due to its accessibility. The initial conditions of the observer zˆ1 (x, 0), w are the same as the plant z1 (x, 0), w1 (x, 0), and the observer errors of z1 , w1 are thus at a very small magnitude 10−14 and convergent to zero as well. We only show the observer errors of z2 , w2 here to avoid repetition and excessive use of space. The control forces EA U1 (t), EA U2 (t) obtained from (3.86), (3.87) with (3.88) at the two head sheaves in the closed-loop system are given in figure 3.10, which shows that the control forces EA U1 (t), EA U2 (t) are bounded and convergent to zero.

3.6

APPENDIX

A. Proof of lemma 3.1 Extending the domain D1 = {(x, y)|0 ≤ x ≤ y ≤ l(t)} to a fixed triangular domain {(x, y)|0 ≤ x ≤ y ≤ L} (the boundary conditions (3.50), (3.51) are given along the lines y = x and x = 0 rather than on y = l(t)), the kernel equations (3.48)–(3.51) have the same form as the kernel equations (29)–(33) in [96]. In detail, (3.48) and (3.49) are two coupled transport PDEs with boundary conditions (3.50) and (3.51), which

DUAL-CABLE ELEVATORS

55

Control inputs (N)

4 6 × 10

Proposed controller EAU1(t) Proposed controller EAU2(t)

4 2 0 –2

Boundary damper EAUd1(t) Boundary damper EAUd 2(t)

–4 –6

0

50

100

150

Time (s)

Figure 3.10. The proposed observer-based output-feedback control forces EA U1 (t) = EA (Ue2 + Ue1 ), EA U2 (t) = EA (Ue2 − Ue1 ) and the boundary dampers (3.102), (3.103) at the two head sheaves. are the same form as the two coupled transport PDEs (33), (32) with boundary conditions (31), (29) in [96] through setting m = n = 1 and replacing K, L with φ¯i , ψ¯i . The kernel equations of (29)–(33) have been proved well-posed in [96]. Lemma 3.1 is then proved. B. Proof of lemma 3.2 ˜ The Lyapunov function Ve for the (˜ α1 , β˜1 , α ˜ 2 , β˜2 , X)-system is defined as Ve (t) =

2 

Vei (t),

(3.105)

i=1

where  a ¯i l(t) δ¯2i x ˜ 2 T ˜ ˜ Vei (t) =X (t)P2 X(t) + e βi (x, t) dx 2 0 ¯bi  l(t) ¯ 2 + e−δ1i x α ˜ i (x, t) dx, 2 0

(3.106)

where the matrix P2 = P2T > 0 is the solution to the Lyapunov equation P2 Aˆ + AˆT P2 = −Q2 , for some Q2 = Q2 T > 0. The positive parameters a ¯i , ¯bi , δ¯1i , δ¯2i are to be chosen later. Defining 2     ˜ 2 Ωe (t) = ( ˜ αi 2 + β˜i 2 ) + X(t)  i=1

yields the two positive constants θe1 , θe2 holding θe1 Ωe (t) ≤ Ve (t) ≤ θe2 Ωe (t). Taking the derivative of Ve along (3.41)–(3.45), using Young’s inequality and the Cauchy-Schwarz inequality, through a lengthy but straightforward calculation, recalling assumption 3.2, and choosing

CHAPTER THREE

56 √ q 2 c(1 + ξ0i ) ¯ , ξ0i + δ1i > √ max , q 2 2  ¯bi ξ0i cξ0i¯bi  2 c ¯ + ξ0i + + , − δ2i > √ q 2¯ ai a ¯i 2¯ ai 64|P2 Bi |2 , a ¯i > 4r0¯bi + √ q qλmin (Q2 )

where ξ0i is a positive constant only depending on the plant parameters, and large enough ¯bi and r0 , we get V˙ e (t) ≤ −σe Ve (t) − −

√  2 q a ¯i β˜i (0, t)2 4 i=1

2   ¯b  √ i −δ¯1i L ˙ e q − l(t) α ˜ i (l(t), t)2 , 2 i=1

(3.107)

¯ which affects where the decay rate σe can be adjusted by the control parameter L λmin (Q2 ). Therefore, we obtain the exponential stability result of the target observer error system (3.41)–(3.45) in the sense of Ωe (t). Due to the invertibility of the transformations (3.39), (3.40), the exponential stability with a decay rate at least σe of the ˜ ˜i , X)-system in the sense of the norm (3.56) is proved. (˜ zi , w C. Proof of lemma 3.3 Differentiating (3.42) and (3.43) with respect to x, and differentiating (3.44) and (3.45) with respect to t, we get c c √ ˜ ix (x, t) − β˜ix (x, t) + η¯1i (x, t), α ˜ ixt (x, t) = − q α ˜ ixx (x, t) − α 2 2 c˜ √ ˜ ˜ βixt (x, t) = q βixx (x, t) − βix (x, t) + η¯2i (x, t), 2 c 1 c α ˜ ix (0, t) = β˜ix (0, t) − √ β˜i (0, t) − √ η¯3i (t) − √ α ˜ i (0, t) q q 2 q β˜ix (l(t), t) = 0,

(3.108) (3.109)

(3.110)

where 

l(t)

η¯1i (x, t) = x  l(t)

η¯2i (x, t) =

¯ ix (x, y)β˜i (y, t)dy − M ¯ i (x, x)β˜i (x, t), M

(3.111)

¯ix (x, y)β˜i (y, t)dy − N ¯i (x, x)β˜i (x, t), N

(3.112)

x



η¯3i (t) = −

l(t) 0

¯ i (0, x) + N ¯i (0, x))β˜i (x, t)dx. (M

Taking the derivative of (3.45), we have ˙ β˜ix (l(t), t) + β˜it (l(t), t) = 0. l(t)

(3.113)

DUAL-CABLE ELEVATORS

57

Inserting (3.43), we get ˙ + (l(t)

√

q)β˜ix (l(t), t) = 0,

which gives (3.110), recalling assumption 3.2. Applying the Cauchy-Schwarz inequality into (3.111)–(3.113) yields positive constants M7i , M8i , M9i such that

¯ η1i (·, t) 2 ≤ M8i β˜i (·, t) 2 ,

(3.114)

¯ η2i (·, t) 2 ≤ M9i β˜i (·, t) 2 ,

(3.115)

|¯ η3i (t)|2 ≤ M7i β˜i (·, t) 2 ,

(3.116)

for i = 1, 2. Define a Lyapunov function a ˇi VieH (t) = 2



l(t) 0

ˇ

2

eδ2i x β˜ix (x, t) dx + R4 Ve (t)

ˇbi  l(t) ˇ 2 + e−δ1i x α ˜ ix (x, t) dx, 2 0

(3.117)

where the positive constants a ˇi , ˇbi , δˇ1i , δˇ2i , R4 are to be chosen later. Taking the derivative of (3.117) along (3.108)–(3.110), substituting the results of V˙ e (t) in (3.107), using Young’s inequality and the Cauchy-Schwarz inequality, and substituting (3.114)–(3.116), through a lengthy calculation we obtain   1√ R4 σ e √ V˙ ieH (t) ≤ − Ve (t) − qˇ ai − 3 qˇbi β˜ix (0, t)2 2 2     l(t) 1√ ˇ L |c| Lˇbi |c| ˇ 2 − − q δ2i − eδ2i x β˜ix (x, t) dx a ˇi − 2 2 r10i 4r11i 0    1√ ˇ |c| L ˇ l(t) −δˇ1i x 2 − |c|Lr11i − q δ1i − e α ˜ ix (x, t) dx bi − 2 2 r12i 0 √ ˙ ˇbi e−δˇ1i L ( q − l(t)) α ˜ ix (l(t), t)2 − 2   ˇ R4 ˇi r10i eδ2i L L ˜ 3ˇbi M7i M8i r12iˇbi L M9i a σe θe1 − √ − − −

βi (·, t) 2 2 q 4 4   R4 √ 6ˇbi c2 ˜ − q¯ ai − √ βi (0, t)2 4 q  ¯bi e−δ¯1i L  √ ˙ − R4 ( q − l(t)) (3.118) α ˜ i (l(t), t)2 , 2 where (3.44) has been used. The positive constants r10i , r11i , r12i are from using Young’s inequality. ˇi to satisfy Recalling assumption 3.2 and choosing δˇ1i , δˇ2i , a     L 2 L |c| 2 |c| + + δˇ1i > √ cLr11i + , δˇ2i > √ , aˇi > 6ˇbi , q r12i 2 q r10i 2 with large enough r11i , R4 and arbitrary ˇbi , r10i , r12i , we thus arrive at

CHAPTER THREE

58 V˙ ieH (t) ≤ −σieH VieH (t), where σieH > 0. Defining a Lyapunov function VeH (t) = V1eH (t) + V2eH (t) and taking the derivative of VeH (t), we get V˙ eH (t) ≤ −σeH VeH (t),

(3.119)

¯ which where the decay rate σeH > 0 can be adjusted by the control parameter L affects σe . We thus obtain the exponential stability estimate in the sense of

˜ α1x (·, t) 2 + β˜1x (·, t) 2 + ˜ α2x (·, t) 2 + β˜2x (·, t) 2 . Due to the invertibility of the transformation (3.39), (3.40), we can obtain the exponential stability estimate in the sense of (3.58) with a decay rate σeH . The proof of lemma 3.3 is complete. D. Proof of lemma 3.4 Extending the domain D = {(x, y)|0 ≤ y ≤ x ≤ l(t)} to a fixed triangular domain {(x, y)|0 ≤ y ≤ x ≤ L} (the boundary conditions (3.77), (3.78) are given along the lines y = x and y = 0 rather than on x = l(t)), the kernel equations (3.75)–(3.80) have the same form as the kernel equations (17)–(23) in [48]. In detail, (3.75), (3.76) are two coupled linear hyperbolic PDEs that correspond to (17), (18) in [48] by setting m = n = 1 and replacing K, L with ψi , φi . The boundary conditions (3.78), (3.77) correspond to (19), (21) in [48]. The ODE (3.79) with the initial condition (3.80) corresponds to (22) and (23) in [48]. The well-posedness of (17)–(23) in [48] has been proved. Lemma 3.4 can then be proved. E. Proof of lemma 3.5 ˆ We establish the stability proof of the target (α1 , β1 , α2 , β2 , X)-system via Lyapunov analysis. The equivalent stability properties between the target system and ˆ ˆ1 , zˆ2 , w ˆ2 , X)-system are ensured due to the invertibility of the the original (ˆ z1 , w backstepping transformation (3.59), (3.60). ˆ as Step 1. Consider a Lyapunov function for the (α1 , β1 , α2 , β2 , X)-system, follows:  l(t) 2 ˆ + ai ˆ T (t)P1 X(t) Vi (t) = X eδi2 x βi (x, t) dx 2 0  bi l(t) −δi1 x 2 + e αi (x, t) dx, (3.120) 2 0 where the matrix P1 = P1T > 0 is the solution to the Lyapunov equation P1 A´ + A´T P1 = −Q1 for some

Q1 = Q1 T > 0.

DUAL-CABLE ELEVATORS

59

The positive parameters ai , bi , δi1 , δi2 are to be chosen later. Defining 2 ˆ Ω1i (t) = βi (·, t) 2 + αi (·, t) 2 + |X(t)| ,

we get θ11i Ω1i (t) ≤ Vi (t) ≤ θ12i Ω1i (t), where  ai bi e−δi1 L θ11i = min λmin (P1 ), , > 0, 2 2  ai eδi2 L bi , θ12i = max λmax (P1 ), > 0. 2 2 Taking the derivative of Vi (t) along (3.61)–(3.65), applying Young’s inequality, and choosing the parameters bi , δi1 , δi2 to satisfy λmin (Q1 ) 8ξi bi 2|c| 0 < bi < √  2 , δi2 > √ + √ , a q ¯   i q 2 q Di  1 4ξi2 bi √ δi1 > √ max + 2|c| , q, 12ξi + q λmin (Q1 )

(3.121) (3.122)

where ξi are positive constants only depending on plant parameters and the design parameters κi , together with small enough positive constants r4i , r3i obtained from applying Young’s inequality, we get √  √ q 3 q 4 |P1 Bi | V˙ i (t) ≤ − σi Vi (t) − ai − bi − βi (0, t)2 2 2 qλmin (Q1 ) √  q −δi1 l(t) ˙ bi −δi1 l(t) bi e − − l(t) e αi (l(t), t)2 2 2  2 δi2 L  ai e L 4 |P1 Bj | b2 L βj (0, t)2 + + Hi + i Gi z˜i (l(t), t)2 qλmin (Q1 ) r4i 2r3i  √    2 δ L 3 qbi a e i2 L  ˜ 2 2 |Di | + i + Yi X(t) (3.123)  , 2 r4i for some positive σi that can be adjusted by the control parameters κi , where ai are chosen later. The constants Hi , Yi , Gi are defined as Hi = max {|Ni (x, t)|} , Yi = max {|N1i (x)|} ,   ¯ i (x, t)| Gi = max |Γ

(3.124) (3.125)

for x ∈ [0, L], t ∈ [0, ∞). Step 2. Recalling the exponential stability result in the sense of ˜ zi (·, t) , ˜ and ˜ zix (·, t) , w ˜ix (·, t) proved in lemma 3.2 and lemma 3.3, by

w ˜i (·, t) , |X(t)| virtue of w ˜i (l(t), t) = 0 in (3.38) and using the Cauchy-Schwarz inequality, similarly to (3.94), we can arrive at the fact that w ˜i (0, t) is exponentially convergent to zero. Recalling w ˜i (0, t) + z˜i (0, t) = 0 in (3.38) and using the Cauchy-Schwarz inequality again, similarly to (3.95), we get that z˜i (l(t), t) is exponentially convergent. Then ˜ we get that signals X(t) and z˜i (l(t), t) are exponentially convergent to zero in the sense of

CHAPTER THREE

60 "  ! ˜  max |˜ zi (l(t), t)| , X(t)  ≤ λ0 e−ηt := ηˇm (t)

(3.126)

for i = 1, 2, where the decay rate η > 0, which can be adjusted by the control param¯ and λ0 is a positive constant which depends on initial conditions only. eters L, Now we consider a Lyapunov function candidate V (t) =

2 

Vi (t) + Rˇ ηm (t)2 ,

(3.127)

i=1

where R > 0, to be determined later. Defining 2 ˆ Ωa (t) = β1 (·, t) 2 + α1 (·, t) 2 + β2 (·, t) 2 + α2 (·, t) 2 + |X(t)| + ηˇm (t)2 ,

we obtain θa1 Ωa (t) ≤ V (t) ≤ θa2 Ωa (t) with two positive constants θa1 , θa2 . Taking the derivative of (3.127) and recalling (3.123), we get 2   √

b i −δi1 l(t) ˙ e q − l(t) αi (l(t), t)2 2 i=1 √  √ q 3 q 8 |P1 Bi | ai − bi − + βi (0, t)2 2 2 qλmin (Q1 )   R a2 eδi2 L L b2 L η− i + Hi − i Gi ηˇm (t)2 + σi Vi (t) 2 r4i 2r3i  

√ 3 qbi R a2i eδi2 L L 2 2 + η− |Di | − Yi ηˇm (t) , 2 2 r4i

V˙ (t) ≤ −

(3.128)

˜ 2 in the last two lines of (3.123) where (3.126) is used to replace z˜i (l(t), t)2 and X(t) 2 as ηˇm (t) , respectively. We choose 16 |P1 Bi | ai > 3bi + √ q qλmin (Q1 )

(3.129)

and choose large enough R to make sure the coefficients before ηˇm (t)2 are positive. ˙ < √q—the coefficients of αi (l(t), t)2 in (3.128) Recalling assumption 3.2—that is, l(t) ˙ < 0), descending (l(t) ˙ > 0), and stop (l(t) ˙ = 0) cases. are positive in all ascending (l(t) We thus arrive at V˙ (t) ≤ −σV (t) −

2 

s¯i βi (0, t)2 ,

(3.130)

i=1

where  √ 2  3 qbi 1  a2 eδi2 L L 2 σ = min σ1 , σ2 , |Di | − i Yi Rη − R i=1 2 r4i  a2i eδi2 L L b2i L − Hi − Gi > 0, r4i 2r3i

(3.131)

DUAL-CABLE ELEVATORS

61

¯ and where which can be adjusted by the control parameters κi and L √ √ q 3 q 8 |P1 Bi | ai − bi − > 0. s¯i = 2 2 qλmin (Q1 ) Therefore, we obtain the exponential stability result in the sense of αi (·, t) 2 + 2 ˆ . Due to the invertibility of the transformations (3.59), (3.60),

βi (·, t) 2 + |X(t)| ˆ ˆi , X)-system in the exponential stability with a decay rate at least σ of the (ˆ zi , w the sense of the norm (3.89) in lemma 3.5 is proved. F. Proof of lemma 3.6 Differentiating (3.62) and (3.63) with respect to x, and differentiating (3.64) and (3.65) with respect to t, we get c c √ (3.132) αixt (x, t) = − qαixx (x, t) − βix (x, t) − αix (x, t) + ηi1 (x, t), 2 2 c √ βixt (x, t) = qβixx (x, t) − βix (x, t) + ηi2 (x, t), (3.133) 2 c αix (0, t) =βix (0, t) − √ βi (0, t) 2 q − Di

2  Bi i=1

βix (l(t), t) = where

q

1 (β˜i (0, t) + βi (0, t)) − √ ηi3 (t), q

Ni (l(t), t) N1i (l(t)) ˜ z˜ (l(t), t) + X(t), ˙ + √q i ˙ + √q l(t) l(t)

  c x ˆ c x ˆ Mix (x, y)αi (y, t)dy − Nix (x, y)βi (y, t)dy 2 0 2 0 c c ˆ ˆ +Γ ¯ i (x, t)˜ − ϑˆ (x)X(t) zi (l(t), t) − M i (x, x)αi (x, t) 2 2 c ˆ − N i (x, x)βi (x, t), 2  ˜ − Ni (x, t)z˜i (l(t), t), ηi2 (x, t) = − N1i (x)X(t) # $ ¯ 1 + C2 ) + Di Aˆ + c Di + N1i (0) X(t) ¯ i L(C ˜ ηi3 (t) = D 2 %   & 2  c¯ ˆ c ¯ ¯ Bi κi + ϑ(0) + D + Di A + i X(t) 2 2 i=1

(3.134) (3.135)

ηi1 (x, t) = −

¯ i (0, t) + Γi (0, t))˜ − (Γ zi (l(t), t)  2 l(t)  Bi αi (y, t)dy. ψ¯i (0, y)˜ + Di √ q 0 i=1

(3.136) (3.137) (3.138) (3.139) (3.140) (3.141)

Applying the Cauchy-Schwarz inequality into (3.136), (3.137) yields the positive constants N1i , N2i , N3i , N4i , N5i , N6i such that    ˆ 2

ηi1 (·, t) 2 ≤N1i αi (·, t) 2 + N2i X(t)  + N3i βi (·, t) 2 + N4i z˜i (l(t), t)2 ,

(3.142)

CHAPTER THREE

62    ˜ 2

ηi2 (·, t) 2 ≤N5i z˜i (l(t), t)2 + N6i X(t)  .

(3.143)

According to (3.126) and the exponential stability results proved in lemma 3.2 and lemma 3.5, we see that the signals ηi3 (t) are exponentially convergent to zero in the sense of |ηi3 (t)| ≤ λ03i e−η0i t := ηima (t),

(3.144)

where λ03i > 0 only depends on the initial values, and η0i are positive constants. Define a Lyapunov function VH (t) =

 2  a ˆi i=1

2

l(t) 0

ˆ

2

eδ2i x βix (x, t) dx +

ˆbi  l(t) ˆ 2 e−δ1i x αix (x, t) dx 2 0

1 + R3 ηima (t)2 + R2 ηˇm (t)2 + R1 V (t) + R0 Ve (t), 2

(3.145)

where the positive constants a ˆi , ˆbi , δˆ1i , δˆ2i , R0 , R1 , R2 , R3 are determined later, and ηm (t) is defined by (3.126). Taking the derivative of (3.145) along (3.132)–(3.135); using the Young and Cauchy-Schwarz inequalities; substituting the results of V˙ (t) (3.130), V˙ e (t) (3.107) and (3.142), (3.143); and recalling assumption 3.2, through a lengthy calculation we obtain √  2  qˆ ai R1 √ V˙ H (t) ≤ − σV (t) + − 2 qˆbi βix (0, t)2 − 2 2 i=1     l(t) 1√ ˆ 1 |c| r7iˆbi |c| ˆ 2 − − q δ2i − eδ2i x βix (x, t) dx a ˆi − 2 2 r6i 4 0    1√ ˆ |c| |c| 1 ˆ l(t) −δˆ1i x 2 − − q δ1i − − e αix (x, t) dx bi 2 2 r7i r8i 0     ˆ ˆbi N3i r σθ R R1 σθa1 r8i bi N1i 1 a1 8i − −

αi (·, t) 2 −

βi (·, t) 2 − 4 4 4 4     R1 σθa1 r8iˆbi N2i  ˆ 2 2ˆbi − ηima (t)2 − X(t) − R3 η0i − √ 4 4 q ˆ L   δ √ 2i ˆ a i ee Hi2 ai r6i eδ2i L N5i r8iˆbi N4i ( q + v¯max )ˆ − − − R2 η − ηˇm (t)2 √ 4 4 ( q − v¯max )2  ¯ 2 }Bi 2 c2ˆbi  R1 8 max{r9iˆbi D s¯i − − √ βi (0, t)2 − √ i 4 q q 2 q   √ ˆ ˆ ai eδ2i L Yi2  ˜ 2 R1 σθa1 ai r6i eδ2i L N6i ( q + v¯max )ˆ − − − X(t) √ 4 4 ( q − v¯max )2  √

 q 8 max{r9iˆbi Di2 }Bi 2 ˜ a ¯i − − R0 (3.146) βi (0, t)2 − R0 σe Ve (t) √ 4 q q

for i = 1, 2, and the expressions Hi , Yi are shown in (3.124). We have used (3.126) and (3.144) to replace z˜i (l(t), t)2 and ηi3 (t)2 with positive signs by ηˇm (t)2 and

DUAL-CABLE ELEVATORS

63

ηima (t)2 , respectively. From using Young’s inequality, r6i , r7i , r8i are arbitrary positive constants. Choose the positive constants a ˆi , δˆ1i , δˆ2i satisfying   |c| |c| 2 1 ˆ ˆ + + , (3.147) a ˆi > 4bi , δ1i > √ q 2 r7i r8i |c| 2 r7iˆbi |c| δˆ2i > , (3.148) √ +√ +√ 2ˆ ai q q qr6i with large enough R0 , R1 , R2 , R3 and arbitrary ˆbi , and then we arrive at V˙ H (t) ≤ −σH VH (t),

(3.149)

¯ where σH > 0 can be adjusted by the design parameters κi and L. We thus obtain the exponential stability estimate in the sense of αix (·, t) 2 +

βix (·, t) 2 . Differentiating (3.84), (3.85) with respect to x, using the Young and Cauchy-Schwarz inequalities, we get the exponential stability estimate in the sense ˆ ˆix (·, t) 2 . Together with the exponential convergence of |X(t)| of ˆ zix (·, t) 2 + w proved in lemma 3.5, we can obtain the exponential stability estimate in the sense of (3.90) with the decay rate σH .

3.7

NOTES

The in-domain viscous damping of cables in the plant has been considered here. It results in in-domain couplings between transport PDEs after applying Riemann transformations into the original wave PDEs, and the control design comes down to the boundary stabilization of heterodirectional coupled hyperbolic PDEs on a time-varying domain. The basic results and systematic framework of the boundary control of this kind of system on a fixed domain are provided in [96], [177]. In this chapter, we only considered an ideal mining cable elevator model without uncertainties and disturbances. These are further dealt with in chapters 4 and 5.

Chapter Four Elevators with Disturbances

In chapters 2 and 3 we conducted control designs based on the nominal models of mining cable elevators—that is, in the absence of disturbances or uncertainties. In an actual operating environment, the cage is usually subject to uncertain airflow disturbances, which would affect the smooth and steady running of the mining cable elevator. For this reason, in this chapter we pursue disturbance rejection control design for the mining cable elevator. When the disturbance is as far from the control input as being at the opposite boundary, and the control action has to be propagated through the wave partial differential equation (PDE) dynamics to achieve disturbance rejection, a new technical challenge arises: how to achieve the rejection of the uncertain external disturbance anti-collocated with the control input in a wave PDE. We start by presenting the wave PDE-modeled vibration dynamics of a cable elevator with an airflow disturbance at the cage in section 4.1, for which a disturbance estimator and a state observer are designed in section 4.2 and section 4.3, respectively, which allow exponential tracking of the unknown disturbance and unaccessible actual states. In section 4.4, based on the disturbance estimator and state observer, we present a two-step control design, where the first step is to move the anti-collocated disturbance terms to the controlled boundary, and the second step is to cancel the collocated disturbance terms and stabilize the overall system using the backstepping control design introduced in chapter 2. This is followed by proof of the exponential convergence of the state at the uncontrolled boundary and the uniform boundedness of all the states in the closed-loop system. The simulation results in section 4.5 show the achievement of rejection of the disturbance and of suppressing the vibrations at the moving cage via the control force at the head sheave.

4.1

PROBLEM FORMULATION

A mining cable elevator with an axial disturbance at the cage is depicted in figure 4.1. The drum drives the cable through the floating sheave to lift the cage which is subject to the disturbance. Input Uv (t) generated by the hydraulic actuator at the floating sheave is a control force for disturbance rejection and vibration suppression. Input Ua (t) at the drum is a separate control force to regulate motion dynamics. Following the modeling process in chapter 2, with the inclusion of an external disturbance force at the cage, the axial vibration dynamics of the mining cable elevator are obtained as utt (x, t) = quxx (x, t),

(4.1)

ELEVATORS WITH DISTURBANCES

65

ux (0, t) = −

m 1 utt (0, t) − d(t), r r

ux (l(t), t) = U (t),

(4.2) (4.3)

where x ∈ [0, l(t)] denotes the position coordinate along the cable in a moving coordinate system associated with the elevator axial motion l(t) and whose origin is located at the cage at the initial moment, and t ∈ [0, ∞) presents time. The physical parameters are shown in table 4.1 and r = E × Aa , q = E × Aa /ρ. The function u(x, t) denotes the axial vibration displacement distributed along the cable in the moving coordinate system. Equation (4.2) describes the cage dynamics. Equation (4.3) comes from rux (l(t), t) = Uv (t) with the definition of Uv (t) = rU (t), where U (t) is the control input to be designed in this chapter. Input Uv (t) is obtained by multiplying U (t) designed here by the constant gain r. The disturbance force d(t) caused by airflow at the cage x = 0 is anti-collocated with the control input U (t). We only consider the airflow disturbance acting on the cage and neglect that acting at the cable. Since the axial vibration displacement is parallel to the direction of

Floating sheave

Ua(t) Drum

Hydraulic actuator Uv(t)

Airflow disturbance

cage Guide rails

Figure 4.1. The mining cable elevator with airflow disturbances.

Table 4.1. Physical parameters of the mining elevator Parameters (units) Time-varying length of the cable (m) Maximal length (m) Cable effective steel area (m2 ) Cable effective Young’s modulus (N/m2 ) Cable linear density (kg/m) Total hoisted mass (kg) Maximum hoisting velocity (m/s)

Values l (t) L Aa E ρ m v¯

CHAPTER FOUR

66

the airflow disturbance and the cross section of the cable is much smaller than that of the cage, the airflow disturbance essentially affects only the cage. In practice, the measurements available in this system are u(l(t), ˙ t), utt (0, t), and u(0, t). The measurement u(l(t), ˙ t) is at the controlled boundary, which can be directly obtained from the velocity feedback signals of the actuator acting at the floating sheave. The cage vibration acceleration utt (0, t) is measured directly by an accelerometer placed at the cage, and then the vibration displacement u(0, t) is obtained by integrating twice the measured acceleration with the known initial conditions ut (0, 0), u(0, 0). This paragraph is only to explain the signal acquisition for the implementation of the feedback law, imposing no restrictions on any initial conditions in the design and theory. Assumption 4.1. Only the ascent of the cable elevator is considered, and the vary˙ ≤ 0 and 0 < l(t) ≤ l(0) = L, ing length l(t) is decreasing and bounded—that is, l(t) ∀t ≥ 0. ˙ is bounded: Assumption 4.2. The hoisting velocity l(t) ˙ ≤ 0, −¯ v ≤ l(t) where v¯, which is the maximum hoisting velocity, satisfies v¯
0 depends on a1 , and μd˜ is a positive constant that depends on the initial values only. Then we present two lemmas that are useful to complete the proof of theorem ¯ t) is exponentially convergent to u 4.1. The first lemma tells us that d(x, ´(x, t). The proof is shown in appendix 4.6. Lemma 4.2. For any initial values (˜ v (x, 0), v˜t (x, 0)) that belong to H 1 (0, L) × 2 ¯ t) is exponentially stable ´(x, t) − d(x, L (0, L), the error system defined by v˜(x, t) = u in the sense of the norm 1/2  vx (·, t)2 , (4.18) ˜ vt (·, t)2 + ˜ where  ·  denotes the L2 (0, l(t)) norm. The decay rate depends on a1 . Based on lemma 4.2, the subsequent lemma states that d¯t (x, t) is exponentially ¯t (x, t), from which d¯x (0, t) governed by (4.14)– convergent to u ´t (x, t) = ut (x, t) − u (4.16) can be proved exponentially convergent to u ´x (0, t) by using the CauchySchwarz inequality. The proof is shown in appendix 4.6. Lemma 4.3. For any a1 that satisfies (4.8), the system defined by e(x, t) = v˜t (x, t) with initial values (e(x, 0), et (x, 0)) which belong to H 1 (0, L) × L2 (0, L), is exponentially stable in the sense of the norm 

2

2

1/2

et (·, t) + ex (·, t)

.

(4.19)

Furthermore, |˜ vx (0, t)| ≤ μv˜ e−σd˜t , ∀t ≥ 0, where σd˜ > 0, and μv˜ is a positive constant that only depends on the initial values. With lemma 4.3, we prove theorem 4.1 as follows. Proof. According to (4.17) and (4.11), the estimation error of the proposed disturbance estimator is obtained as ˜ = d(t) − d(t) ˆ = −r´ d(t) ux (0, t) + rd¯x (0, t) = −r˜ vx (0, t).

(4.20)

With lemma 4.3, we conclude theorem 4.1. Theorem 4.1 can be regarded as an independent contribution about exponentially tracking a general disturbance d(t) in time-varying interval wave PDEs. ˜ proved in theorem In addition to the exponential convergence result of d(t) ˙ ¨ in the following 4.1, we obtain the exponential convergence estimates of d(t), d(t) lemma. The proof is shown in appendix 4.6. Lemma 4.4. For all initial values (˜ vt (x, 0), v˜xx (x, 0)) ∈ H 3 (0, L) × H 2 (0, L), the deri˙˜ ¨ ˜ of the estimation error d(t) ˜ are exponentially convergent to zero. vatives d(t), d(t) Furthermore, we can estimate each sinusoidal component in the harmonic disturbance (4.4) by using the disturbance estimate (4.17) in the following steps. Define  T Z(t)2N ×1 = a ¯1 cos(α1 t), ¯b1 sin(α1 t), . . . , a ¯N cos(αN t), ¯bN sin(αN t) . (4.21)

CHAPTER FOUR

70 The disturbance (4.4) can be written as ¨ = −Az Z(t), d(t) = Cz Z(t), Z(t) where Az = diag



α12 0



0 α12

 ,...,

2 αN 0

0 2 αN

(4.22)

 , Cz = [1, . . . , 1]1×2N .

(4.23)

ˆ = −rd¯x (0, t) According to the exponentially convergent disturbance estimate d(t) in theorem 4.1, the matrix Az consisting of the disturbance frequencies αj can ˆ be regarded as known because the frequencies of the disturbance estimation d(t) are considered to be equal to those of the actual periodic disturbance d(t) after eliminating the high-frequency noise with an appropriate cutoff frequency, which can be seen in figure 4.5 in section 4.5. In practice, we can use the real-time spectrum analyzer to obtain the frequencies αj , j = 1, 2, . . . , N by analyzing the signal −rd¯x (0, t). ˙ T , Z(t)T ]T , system (4.22) can be written as Denoting Y (t) = [Z(t) Y˙ (t) = Aˆz Y (t), d(t) = Cˆz Y (t), with

 Aˆz =

0 I

−Az 0

(4.24)

 4N ×4N

, Cˆz = [0, Cz ]1×4N ,

(4.25)

where I is an identity matrix with the appropriate dimension, and (Aˆz , Cˆz ) is observable. ˆ defined in (4.17), we can design an observer Using the disturbance estimate d(t) to estimate the state Y (t) that includes sinusoidal components of the harmonic disturbance. The observer is proposed in the form ˙ ˆ − Cˆz Yˆ (t)), Yˆ (t) = Aˆz Yˆ (t) + Lz (d(t)

(4.26)

where Lz is designed to make Aˆz − Lz Cˆz Hurwitz. Subtracting (4.26) from (4.24), we obtain the error system Y˜ (t) = Y (t) − Yˆ (t) as ˜ Y˜˙ (t) = (Aˆz − Lz Cˆz )Y˜ (t) + Lz d(t).

(4.27)

The following lemma holds. Lemma 4.5. The state Y˜ (t) of the system (4.27) is exponentially convergent to zero. ˜ proved in theorem 4.1, Proof. According to the exponential convergence of d(t) ˆ ˆ recalling that Az − Lz Cz is Hurwitz, it is straightforward to obtain lemma 4.5. Defining ˆ = Cˆ Yˆ (t), Z(t)

(4.28)

ELEVATORS WITH DISTURBANCES

71

with Cˆ = [0, diag(Cz )]2N ×4N , we obtain ˜ = Z(t) − Z(t) ˆ = CY ˆ (t) − Cˆ Yˆ (t) = Cˆ Y˜ (t). Z(t) ˆ According to lemma 4.5, we obtain that |Z(t)| is exponentially convergent to zero, which yields   ˜  (4.29) Z(t) ≤ ηZ˜ (t) := ΥZ˜ e−σZ˜ t for some positive ΥZ˜ , where σZ˜ > 0 depends on the exponential decay rate of the Y˜ -system.

4.3

OBSERVER OF CABLE-AND-CAGE STATE

With the disturbance estimator in section 4.2, we can design a state observer to reconstruct the states in the system (4.1)–(4.3)—that is, axial vibration displacements in the cable and cage. Recalling that the control objective is to ensure the exponential convergence of u(0, t), in addition to the challenges from the anticollocated disturbance, the second-order boundary condition (4.2), which is also anti-collocated with the control input in the system (4.1)–(4.3), poses difficulties to the control problem as well. A new variable X(t) = [u(0, t), ut (0, t)]T is introduced to rewrite the system (4.1)–(4.3) as a PDE-ordinary differential equation (ODE) coupled system, as follows: 1 ˙ X(t) = AX(t) + Bux (0, t) + Bd(t), r utt (x, t) = quxx (x, t), u(0, t) = CX(t), ux (l(t), t) = U (t), with

 A=

0 1 0 0

,B =

r m



0 −1

(4.30) (4.31) (4.32) (4.33)

, C = [1, 0],

(4.34)

where the boundary order is reduced and CB = 0. By virtue of (4.34), the pair (A, B) is stabilizable, and the pair (A, C) is observable. In order to facilitate the design of observer-based output-feedback control, which depends on the construction of the observer, a copy of the plant (4.30)–(4.33) is used to build the observer by using the available measurements u(0, t), u(l(t), ˙ t). Consider the observer to be ¯ ˆ ˆ˙ ˆ + Bu t) − C X(t)) − B d¯x (0, t), X(t) = AX(t) ˆx (0, t) + L(u(0, u ˆtt (x, t) = q u ˆxx (x, t), u ˆ(0, t) = u(0, t), ˙ u ˆx (l(t), t) = (1 − a2 l(t))U ˙ t) − a2 u ˆt (l(t), t), (t) + a2 u(l(t),

(4.35) (4.36) (4.37) (4.38)

where d¯x (0, t) is governed by (4.14)–(4.16). The parameter a2 is a positive damping ¯ is chosen to make A − LC ¯ Hurwitz. gain, and L

CHAPTER FOUR

72 PDE-ODE system (u, X ) . u(l(t), t) u(0, t)

U(t)

PDE-ODE system copy (û, X ) – –rdx(0, t) Disturbance estimator – (d, –u)

. u(l(t), t) u(0, t) utt(0, t)

Figure 4.3. Block diagram of the state observer. Substituting the relation ˙ u(l(t), ˙ t) = ut (l(t), t) + l(t)u x (l(t), t)

(4.39)

into (4.38) and recalling (4.3), the equation (4.38) is equal to u ˆx (l(t), t) = U (t) + a2 (ut (l(t), t) − u ˆt (l(t), t)).

(4.40)

A block diagram of the state observer is shown in figure 4.3. Define the observer errors as ˜ = X(t) − X(t). ˆ u ˜(x, t) = u(x, t) − u ˆ(x, t), X(t)

(4.41)

Then the observer error system is 1 ˜ ˜˙ ¯ X(t) ˜ + Bu X(t) = (A − LC) ˜x (0, t) + B d(t), r u ˜tt (x, t) = q u ˜xx (x, t), u ˜(0, t) = 0, u ˜x (l(t), t) = −a2 u ˜t (l(t), t). Define

(4.42) (4.43) (4.44) (4.45)

H = H 2 (0, L) × H 1 (0, L).

The following lemma holds, indicating that the errors of the observer (4.35)–(4.38) exponentially converge to zero. The proof is shown in appendix 4.6. ˜ Lemma 4.6. For all initial values (˜ u(x, 0), u ˜t (x, 0)) ∈ H, X(0) ∈ R2 , there exists a T 2 ˜ ∈ R to the system (4.43)–(4.45) and its unique solution (˜ u, u ˜t ) ∈ C([0, ∞); H), X states are exponentially convergent to zero in the sense of 

 1   ˜ 2 2 ux (·, t) + X(t) . ˜ ut (·, t) + ˜ 2

2

(4.46)

According to lemma 4.6, we can state that the observer (4.35)–(4.38) exponentially converges to the original system (4.1)–(4.3). We then straightforwardly obtain the following theorem. Theorem 4.2. For all initial values (ˆ u(x, 0), u ˆt (x, 0)) ∈ H, (u(x, 0), ut (x, 0)) ∈ H, the observer (4.35)–(4.38) exponentially converges to the original system (4.1)–(4.3)

ELEVATORS WITH DISTURBANCES

in the sense of



73

2

2

ˆt (·, t) + ux (·, t) − u ˆx (·, t) ut (·, t) − u + |u(0, t) − x ˆ1 (t)|2 + |ut (0, t) − x ˆ2 (t)|2

 12 ,

(4.47)

ˆ ˆ2 (t)]T = X(t). where [ˆ x1 (t), x

4.4

CONTROL DESIGN FOR REJECTION OF DISTURBANCES AT THE CAGE

We obtained the disturbance estimator in section 4.2 and the state observer in section 4.3 with the measurements u(0, t), utt (0, t), and u(l(t), ˙ t). In this section, we design an output-feedback control law U (t) using the state and disturbance information recovered from the state observer and the disturbance estimator. The objective is to achieve rejection of the disturbance at the end x = 0 (cage) and regulate exponentially u(0, t) (cage axial vibrations) by using the control signal U (t) at the other end, x = l(t) (head sheave). Because of (4.37), the objective can be achieved by guaranteeing the exponential convergence of u ˆ(0, t) in the state observer via designing a control input at the actuated boundary, x = l(t). In the process of the controller design, we convert the system (4.35)–(4.38) to an exponentially stable target PDE-ODE system without the disturbance terms. This conversion is achieved in two stages. The anti-collocated disturbance terms are first moved to the actuated boundary, and u ˆ(0, t) = zˆ(0, t) is ensured via the first invertible transformation. Then the collocated disturbance terms are canceled, and the target PDE-ODE coupled system is made exponentially stable via the second backstepping transformation. Conversion to an Intermediate System We transform the system (4.35)–(4.38) into the following intermediate system: ¯ ˆ ˆ˙ ˆ + B zˆx (0, t) + L(u(0, t) − C X(t)) X(t) = AX(t) 1 1 ˜ ˜ + BCz Z(t), − B d(t) r r zˆtt (x, t) = qˆ zxx (x, t) + η(x, t), zˆ(0, t) = u ˆ(0, t),

(4.48) (4.49) (4.50)



ˆ ˜t (l(t), t) + ϑ (l(t))Z(t), zˆx (l(t), t) = U (t) + a2 u

(4.51)

where η(x, t) and ϑ(x) will be given later. Our goal is to make the control and the anti-collocated disturbance appear as if they are collocated, with zˆ(0, t) set equal to u ˆ(0, t) in the intermediate system (4.48)–(4.51). In the later derivation, the collocated disturbance can be easily canceled through control input design. Moreover, we can get the exponential stability of the intermediate system (4.48)–(4.51) via designing a control law through backstepping in the next stage so that u ˆ(0, t) is exponentially convergent to zero by virtue of (4.50).

CHAPTER FOUR

74 The transformation is defined as ˆ zˆ(x, t) = u ˆ(x, t) + ϑ(x)Z(t).

(4.52)

Also, (4.52) can be written as ˜ zˆ(x, t) = u ˆ(x, t) + ϑ(x)Z(t) − ϑ(x)Z(t).

(4.53)

Taking the second partial derivative of (4.53) with respect to x and t, respectively, we get ¨˜ ˜ − ϑ(x)Z(t). zˆtt (x, t) − qˆ zxx (x, t) = − (ϑ(x)Az + qϑ (x)) Z(t) + qϑ (x)Z(t) (4.54) Set η(x, t) in (4.49) as ¨ ˜ − ϑ(x)Z(t) ˜ η(x, t) = qϑ (x)Z(t) 

ˆ Aˆz − Lz Cˆz )2 Y˜ (t) = qϑ (x)Cˆ − ϑ(x)C( ˜ − ϑ(x)CL ˜˙ ˆ Aˆz − Lz Cˆz )Lz d(t) ˆ z d(t), − ϑ(x)C(

(4.55)

˜ = Cˆ Y˜ (t) and (4.27) have been used. where Z(t) Then we obtain that ϑ(x) satisfies the ODE qϑ (x) + ϑ(x)Az = 0, 1 ϑ (0) = Cz , r ϑ(0) = 0,

(4.56) (4.57) (4.58)

where (4.56) is obtained by comparing (4.54) with (4.49), and (4.57) is obtained by comparing (4.35) with the result of the substitution of (4.52) into (4.48) with (4.17), (4.22). For the boundary condition (4.50) to hold, the condition (4.58) is obtained. From (4.56)–(4.58), the solution of ϑ(x) is   Cz q Az ϑ(x) = x . (4.59) sin r Az q Conversion from the Intermediate System to the Target System Having completed the conversion from the state observer (4.35)–(4.38) to the intermediate system (4.48)–(4.51) where the collocated disturbance can be easily canceled and zˆ(0, t) = u ˆ(0, t), we next convert the intermediate system (4.48)–(4.51) into a target system via the PDE backstepping approach [114]. The backstepping transformation is formulated as  x w(x, ˆ t) = zˆ(x, t) + γ(x, y)ˆ z (y, t)dy 0  x ˆ h(x, y)ˆ zt (y, t)dy + β(x)X(t), (4.60) + 0

where the kernel functions γ(x, y), h(x, y), and β(x) in (4.60) are to be determined.

ELEVATORS WITH DISTURBANCES

75

The target system is 1 ˜ ˆ˙ ˆ + Bw X(t) = (A + BK)X(t) ˆx (0, t) − B d(t) r 1 ˜ ¯ ˜ + (LC − Bγ(0, 0)C)X(t) + BCz Z(t), r ˜ + η¯(x, t), w ˆtt (x, t) = q w ˆxx (x, t) − f¯1 (x)X(t) ˜ w(0, ˆ t) = C X(t), ˆt (l(t), t), w ˆx (l(t), t) = − a3 w

(4.61) (4.62) (4.63) (4.64)

where a3 in (4.64) is a positive damping gain, and K in (4.61) is chosen to make A + BK Hurwitz. The function f¯1 (x) in (4.62) is ¯ ¯ + β(x)ALC ¯ + qγy (x, 0)C f¯1 (x) = β(x)LC(A − LC) ¯ + qhy (x, 0)CA, + qhy (x, 0)C LC and η¯(x, t) in (4.62) is  η¯(x, t) = η(x, t) +



x 0

γ(x, y)η(y, t)dy +

x 0

h(x, y)ηt (y, t)dy.

(4.65)

Applying the Cauchy-Schwarz inequality to (4.65), recalling (4.55), with theorem 4.1, lemma 4.4, and lemma 4.5, we get max |¯ η (x, t)| ≤ max {|C2 (x)|, |C3 (x)|, |C4 (x)|, |C5 (x)|} 0≤x≤L           ˜   ˜˙   ¨˜   × Y˜ (t) + d(t)  + d(t) + d(t)

0≤x≤L

≤ Cmax Υη¯e−ση¯ t ,

(4.66)

where C2 (x), C3 (x), C4 (x), C5 (x) are some bounded gains, and Cmax , Υη¯, ση¯ are positive constants. Defining η¯m (t) = Υη¯e−ση¯ t ,

(4.67)

d¯ ηm (t)2 = −2ση¯η¯m (t)2 . dt

(4.68)

it follows that

The following lemma shows the exponential regulation of the target system (4.61)– (4.64). The proof is shown in appendix 4.6. ˆ ∈ R2 , there exists a Lemma 4.7. For all initial values (w(x, ˆ 0), w ˆt (x, 0)) ∈ H, X(0) T 2 ˆ ∈ R to the target system (4.61)–(4.64), unique solution (w, ˆ w ˆt ) ∈ C([0, ∞); H), X and its states are exponentially convergent to zero in the sense of 

 1   ˆ 2 2 ˆx (·, t) + X(t) . w ˆt (·, t) + w 2

2

(4.69)

By matching the system (4.48)–(4.51) with the system (4.61)–(4.64) through (4.60), we obtain the conditions for the kernels to be determined in the transformation (4.60) as

CHAPTER FOUR

76 γxx (x, y) = γyy (x, y), d γ(x, x) = 0, dx 1 γ(x, 0) = β(x)AB + hy (x, 0)CB, q hxx (x, y) = hyy (x, y), 1 h(x, 0) = β(x)B, q d h(x, x) = 0, dx 1 β  (x) = β(x)A2 + γy (x, 0)C + hy (x, 0)CA, q β  (0) = K − γ(0, 0)C − h(0, 0)CA, β(0) = −C,

(4.70) (4.71) (4.72) (4.73) (4.74) (4.75) (4.76) (4.77) (4.78)

which is a coupled system of an ODE and two PDEs. According to (4.70)–(4.78), the kernel functions γ(x, y), h(x, y), and β(x) in (4.60) are calculated as   1 I −C Λ − K eD(x−y) γ(x, y) = AB, (4.79) 0 q   1 I −C Λ − K eD(x−y) B, (4.80) h(x, y) = 0 q    I , (4.81) β(x) = −C Λ − K eDx 0 where I denotes the identity matrix of the appropriate dimension, and D, Λ are defined as  1 2 0 1 qA , Λ = CABC. D= I − 1q (BCA + ABC) q Similarly, the inverse transformation of (4.60) can be obtained. For the boundary condition (4.64) to hold, the controller U (t) can be obtained as  1 U (t) = z (l(t), t) c2 zˆt (l(t), t) + f3 (l(t))ˆ c1 ˆ zx (0, t) + f5 (l(t))ˆ z (0, t) + f6 (l(t))X(t) + f4 (l(t))ˆ   l(t)  l(t) f7 (l(t), x)ˆ z (x, t)dx + f8 (l(t), x)ˆ zt (x, t)dx + 0

0



ˆ ˜t (l(t), t) − ϑ (l(t))Z(t), − a2 u

(4.82)

where c1 = 1 − a3 KB, c2 = −a3 , f3 (l(t)) = γ(l(t), l(t)) − qhxy (l(t), l(t)),

(4.83) (4.84) (4.85)

ELEVATORS WITH DISTURBANCES

77

f4 (l(t)) = a3 qhx (l(t), 0) − a3 β(l(t))B, f5 (l(t)) = qa3 hxy (l(t), 0),

(4.86) (4.87)

f6 (l(t)) = βx (l(t)) + a3 β(l(t))A, f7 (l(t), x) = γx (l(t), x) + qhxyy (l(t), x),

(4.88) (4.89)

f8 (l(t), x) = hx (l(t), x) + a3 γ(l(t), x).

(4.90)

ˆ The following lemma shows the exponential regulation of the closed-loop (ˆ z , X)system, which will be used in the proof of the exponential convergence of u(0, t) in ˆ the original system because the (ˆ z , X)-system and the original system are connected at x = 0. Lemma 4.8. For all initial values (ˆ z (x, 0), zˆt (x, 0)) ∈ H, the states of the system consisting of the plant (4.48)–(4.51) and the control law (4.82) are exponentially convergent to zero in the sense of 

 1/2   ˆ 2 zx (·, t) + X(t) . ˆ zt (·, t) + ˆ 2

2

Proof. Based on lemma 4.7 and the invertibility and continuity of the transformation (4.60), the proof is straightforward. Stability of the Closed-Loop System Controller (4.82) can be written by u ˆ and the available measurements as   1 1 U (t) = ˆt (l(t), t) + f3 (l(t))ˆ u(l(t), t) c2 u ˙ c (1 + a2 |l(t)|) 1 + f4 (l(t))ˆ ux (0, t) + f5 (l(t))ˆ u(0, t)  l(t) ˆ + f7 (l(t), x)ˆ u(x, t)dx + f6 (l(t))X(t) 0



l(t)

+ 0



f8 (l(t), x)ˆ ut (x, t)dx − a2 u(l(t), ˙ t)

ˆ z d¯x (0, t) ˆt (l(t), t) + rL(l(t))CL + a2 u

ˆ ˆ ˆ ˆ − [P(l(t)) + L(l(t))(Az − Lz Cz )C]Z(t) , where

(4.91)



f3 (l(t)) f4 (l(t)) ϑ(l(t)) + c1 c1  l(t) f7 (l(t), x) + ϑ(x)dx + ϑ (l(t)) , c1 0   l(t) c2 f8 (l(t), x) ϑ(l(t)) + ϑ(x)dx , L(l(t)) = c1 c1 0

P(l(t)) =

(4.92) (4.93)

and ϑ(x) is defined in (4.59). All signals required in the control law (4.91) are obtained from the measurable boundary quantities u(l(t), ˙ t), u(0, t), and utt (0, t).

CHAPTER FOUR

78

The measurements u(l(t), ˙ t), u(0, t) are used to construct the observer (4.35)–(4.38) ˆ to estimate the distributed states u(x, t) and X(t)—that is, to obtain u ˆ(x, t), X(t). The measurements u(l(t), ˙ t), u(0, t), and, utt (0, t) are also used to implement the disturbance estimator (4.5)–(4.7) and (4.14)–(4.16) to obtain the disturbance estimate ˆ = −rd¯x (0, t), which is used to get Z(t) ˆ based on (4.26)–(4.28). In practice, the d(t) estimator and the observer can be calculated by using the finite-difference method, where different spatial steps are chosen by considering the trade-off between the model accuracy and the computational speed in different cases, whereas the time step depends on the sample period of the data acquisition. With the controller (4.91), which uses the information from the disturbance estimator in section 4.2 and the state observer in section 4.3, the complete closedloop system is utt (x, t) = quxx (x, t),

(4.94)

m 1 utt (0, t) − d(t), r r ux (l(t), t) = U (t), ux (0, t) = −

(4.95) (4.96)

ˆ˙ ˆ + Bu ¯ ˆ X(t) = AX(t) ˆx (0, t) + L(u(0, t) − C X(t)) − B d¯x (0, t),

(4.97)

ˆxx (x, t), u ˆtt (x, t) = q u

(4.98)

u ˆ(0, t) = u(0, t),

(4.99)

˙ ˙ t) − a2 u ˆt (l(t), t), (t) + a2 u(l(t), u ˆx (l(t), t) = (1 − a2 l(t))U ¯ ¯ dtt (x, t) = q dxx (x, t),

(4.100) (4.101)

¯ t) = u(0, t) − u d(0, ¯(0, t), ¯ ¯ dx (l(t), t) = − a1 dt (l(t), t),

(4.102)

¯xx (x, t), u ¯tt (x, t) = q u m u ¯x (0, t) = − utt (0, t), r ˙ u ¯x (l(t), t) = (1 − a1 l(t))U ˙ t) − a1 u ¯t (l(t), t), (t) + a1 u(l(t),

(4.104)

(4.103)

(4.105) (4.106)

˙ Yˆ (t) = (Aˆz − Lz Cˆz )Yˆ (t) − Lz rd¯x (0, t),

(4.107)

ˆ = Cˆ Yˆ (t), Z(t)

(4.108)

where U (t) is shown in (4.91)–(4.93). We next present the main theorem of this chapter. Theorem 4.3. The closed-loop system consisting of the plant (4.94)–(4.96) with the unmatched disturbance d(t); the disturbance estimator (4.101)–(4.106) and (4.107), (4.108); the state observer (4.97)–(4.100); and the controller (4.91) has the following properties. 1. There exist positive constants μ1 and σ such that the output state u(0, t) of the closed-loop system is exponentially convergent to zero in the sense of |u(0, t)| ≤ μ1 e−σt , ∀t ≥ 0. 2. All states in the closed-loop system are uniformly bounded in the sense of

ELEVATORS WITH DISTURBANCES



l(t)

sup t≥0

0



79

ˆ2t (x, t) + u ˆ2x (x, t) + d¯2t (x, t) + d¯2x (x, t) u2t (x, t) + u2x (x, t) + u

        ˆ 2  ˆ  2  ˆ  2 +u ¯2t (x, t) + u ¯2x (x, t) dx + X(t)  + Y (t) + Z(t) < ∞.

(4.109)

3. The control input U (t) in (4.91) is bounded—that is, sup |U (t)| < ∞. t≥0

Proof. 1) We now prove the first of the three portions of the theorem. According to (4.99) and (4.50), we get u(0, t) = u ˆ(0, t) = zˆ(0, t). This, together with lemma 4.8, from which we can infer that zˆ(0, t) is exponentially convergent to zero with the decay rate σ, gives property 1. 2) We now prove the second of the three portions of the theorem. According to assumption 4.4 and theorem 4.1, we know that dˆ= −rd¯x (0, t) is bounded. Then ˆ Z(t) obtained from (4.108) is bounded because Aˆz − Lz Cˆz in (4.107) is Hurwitz, which shows the boundedness of Yˆ (t) thanks to the boundedness of dˆ= −rd¯x (0, t). Together with lemma 4.8 and the invertible transformation (4.52), we obtain that 2 ˆ ux (x, t)2 , |X(t)| are uniformly bounded. For the sake of brevity, when ˆ ut (x, t)2 , ˆ we mention boundedness, we refer to the corresponding state norms in (4.109). Then, from (4.41), with lemma 4.6, we conclude that u(x, t) is uniformly bounded. Based on lemma 4.1, which proves the uniform boundedness of the system u ´(x, t), and lemma 4.2, which means the exponential stability of the v˜(x, t) system, we ¯ t) uniformly bounded thanks to v˜(x, t) = u ¯ t). Then we find get d(x, ´(x, t) − d(x, ¯ t). that u ¯(x, t) is also uniformly bounded because of v˜(x, t) = u(x, t) − u ¯(x, t) − d(x, Therefore, all subsystems in the closed-loop system (4.94)–(4.108) are uniformly bounded as (4.109). Thus, we get property 2. 3) We now prove the third and last of the three portions of the theorem. In the proof of property 2, we have proved the boundedness of all states in the closed-loop system in the sense of (4.109). Now we prove the boundedness of the control input U (t) in (4.91). Due to (4.91) and property 2, we know that the ˆx (0, t), ux (l(t), t), ut (l(t), t) in boundedness analysis of the four signals u ˆt (l(t), t), u (4.91) needs to be conducted, for which the L2 estimates need to be produced for ˆxx (x, t), u ˆxt (x, t). That is, estimates of ˜ uxx (·, t), ˜ uxt (·, t) uxx (x, t), uxt (x, t), u uxt (·, t) need to be found. and ˆ uxx (·, t), ˆ Toward that end, we now present two lemmas. The first one shows the bounded uxt (·, t)2 . The proof is shown in appendix 4.6. estimates in terms of ˆ uxx (·, t)2 + ˆ uxt (·, t)2 . The second one gives the bounded estimates in terms of ˜ uxx (·, t)2 + ˜ Lemma 4.9. For all initial values (ˆ u(x, 0), u ˆt (x, 0)) ∈ H, the states of the u ˆ(x, t)2 2 uxt (·, t) . system are bounded in the sense of ˆ uxx (·, t) + ˆ Through a similar process as in the proof of lemma 4.9, it is straightforward to prove the following lemma. Lemma 4.10. For all initial values (˜ u(x, 0), u ˜t (x, 0)) ∈ H, the states of the u ˜(x, t)2 2 uxt (·, t) . system are bounded in the sense of ˜ uxx (·, t) + ˜ Recalling the bounded estimate for the norm ˆ uxx (·, t) + ˆ uxt (·, t) proved in lemma 4.9 and using the Sobolev inequality, we show that u ˆx (l(t), t), u ˆx (0, t), and u ˆt (l(t), t) are bounded.

CHAPTER FOUR

80

Similarly, using lemma 4.10, we obtain the boundedness of u ˜x (l(t), t), u ˜t (l(t), t). ˆx (l(t), t), u ˆt (l(t), t), u ˜x (l(t), t), According to the boundedness of u ˆx (0, t), u ˆt (l(t), t), u ˆx (0, t), u ˜t (l(t), t), we can obtain the boundedness of the four signals u ux (l(t), t), ut (l(t), t) required for proving that U (t) is bounded. Thus, we get property 3. This completes the proof of theorem 4.3. 4.5

SIMULATION FOR A DISTURBED ELEVATOR

The system parameters of the mining cable elevator used in the simulation are shown in table 4.2. Consider the cage subject to an airflow disturbance of the form d(t) = 150 sin(0.3t) + 100 sin(0.4t) + 200 cos(0.2t) + 140 cos(0.25t).

(4.110)

The simulation is performed based on a priori-known l(t) in figure 4.4, which is considered to be a monotonically decreasing curve from 2000 m to 200 m with a maximum velocity v¯ = 15 s during the total hoisting time of 150 s. The control force is (4.91) multiplied by the constant r = E × Aa mentioned in section 4.1, with the gain functions (4.83)–(4.90) and (4.92), (4.93), where kernels γ(x, y), h(x, y), β(x), ϑ(x) are defined in (4.79)–(4.81) and (4.59), respecˆ tively. In the controller, the states u ˆ(x, t), X(t) are defined by the state observer ¯ (4.97)–(4.100). The function dx (0, t) is defined by the disturbance estimator (4.101)– ˆ (4.106), and Z(t) is defined by (4.107), (4.108). The constant control parameters required in the controller are shown next. The parameters K = [k1 , k2 ] are ¯ = [l1 , l2 ] are chosen as [1.5, 1], and chosen as [0.0012, 0.011]. The parameters L Lz = [1, . . . , 1]1×16 . Other design parameters are a1 = 0.022, a2 = 0.07, a3 = 0.01. The PDE on the time-varying domain [0, l(t)] is converted to a PDE on the fixed x , and then the domain [0, 1] with time-varying coefficients by introducing ξˇ = l(t) Table 4.2. Simulation parameters of the mining elevator Parameters Values

L

r

ρ

q

m

2000

0.48×107

8.1

5.9×105

15000

2000

l(t) (m)

1500 1000 500 0

0

50

100 Time (s)

Figure 4.4. The target hoisting trajectory l(t).

150

ELEVATORS WITH DISTURBANCES

81

500

(N)

0

–500 Estimated disturbance d(t) Actual disturbance d(t) –1000

0

50

100

150

Time (s)

ˆ = −rd¯x (0, t) and the actual disturbance Figure 4.5. The disturbance estimate d(t) d(t) (4.110) (dashed line). 600 Error of disturbance estimator d(t) (N)

400 200 0 –200

0

50

100

150

Time (s)

˜ = d(t) − d(t) ˆ between the actual disturbance Figure 4.6. The estimation error d(t) ˆ d(t) (4.110) and the disturbance estimate d(t) = −rd¯x (0, t). simulation is conducted based on the finite-difference method with a time step of 0.001 and a space step of 0.05. ˙ t), By using the available boundary measurements u(0, t), utt (0, t), and u(l(t), the disturbance estimator (4.101)–(4.106) is implemented with the initial conditions ¯ 0) = 0 and u d(x, ¯(x, 0) = 0. Figure 4.5 shows that the estimate from the disturbance estimator (4.101)–(4.106) can quickly track the actual unknown disturbance (4.110). In practice, the high-frequency noise at the beginning of the estimation process can be eliminated with a low-pass filter by setting an appropriate cutoff frequency. The error between the estimated and actual values of the disturbance (4.110) is shown in figure 4.6. The estimated variables of the distributed states are obtained by the proposed observer (4.35)–(4.38) with available boundary measurements u(0, t) and u(l(t), ˙ t). The initial errors of the observer are defined as 0.005 m. Because the locations of the actuator and the sensor are at opposite boundaries, the estimation of the state at the midpoint x = l(t)/2 is most challenging due to its accessibility. Figure 4.7 shows that the observer error at the midpoint of the cable converges to zero quickly, which implies that the estimates from the state observer (4.35)–(4.38) can reconstruct the distributed states. The closed-loop responses under the proposed control law (4.91) and the proportional-derivative (PD) control law, which is traditional in industry, are examined, to compare their performance at suppressing the axial vibrations at the cage. Consider the PD control law ˙ t), (4.111) Upd (t) = kp u(l(t), t) + kd u(l(t),

CHAPTER FOUR

82 × 10–3

6

Error of state observer

ũ(l(t))/2, t) (m)

5 4 3 2 1 0 –1

0

100

50

150

Time (s)

Figure 4.7. The observer error u ˜(l(t)/2, t) at the midpoint of the cable. 0 Open loop u(0, t) (m)

–1 –2 –3 –4 0

50

100

150

Time (s)

Figure 4.8. The open-loop response u(0, t) of the plant (4.1)–(4.3) under the disturbance (4.110) at x = 0. where kp , kd are gain parameters. The values of kp , kd are tuned to attain efficient control performance. Here we choose kp = 700, kd = 14000. From figure 4.8, we observe that the oscillation appearing at the cage is becoming larger and larger because of the disturbance. Figure 4.9 shows that both the proposed output-feedback control law and the PD control law suppress the enlargement of the vibration displacement. Moreover, the proposed control law can regulate the vibration displacement u(0, t) of the cage to zero with faster convergence and less overshoot despite the disturbance at the cage. In addition, according to figure 4.10, we can see that the proposed control law also has better control performance for the internal state, such as the midpoint u(l(t)/2, t) of the cable. This illustrates that the states in the domain’s interior are uniformly bounded. Figure 4.11 shows that the outputfeedback control input in the closed-loop system is uniformly bounded. The control input is not zero at the final moment t = 150 s because the disturbance in figure 4.5 is not zero at that time, so the action of disturbance rejection is continuing in the controller to ensure the convergence of the controlled states. The model parameter error between the actual plant and the nominal plant often appears in practice. In order to test the robustness of the proposed controller to the model parameter error, we change some plant parameters with respect to their nominal values in table 4.2, such as r = 0.56 × 107 , ρ = 8.5, q = r/ρ = 6.6 × 105 , and M = 15500. These plant parameters are considered as actual plant parameters and those in table 4.2 are nominal plant parameters, and the difference between them is the model error. For the actual plant, a comparison of the proposed controller

ELEVATORS WITH DISTURBANCES

83

0.5

u(0, t) (m)

Proposed output-feedback controller PD controller

0

–0.5

0

50

100

150

Time (s)

Figure 4.9. The output responses u(0, t) of the closed-loop system (4.94)–(4.108) under the disturbance (4.110) at x = 0 with the proposed output-feedback controller (4.91) (solid line) and PD controller (4.111) (dashed line).

u(l(t)/2, t) (m)

0.2 0.1 0 –0.1 –0.2

Proposed output-feedback controller PD controller

–0.3 –0.4

0

50

100

150

Time (s)

Figure 4.10. The responses u(l(t)/2, t) of the closed-loop system (4.94)–(4.108) under the disturbance (4.110) at x = 0 with the proposed output-feedback controller (4.91) (solid line) and the PD controller (4.111) (dashed line).

1500 Control input

U(t)(N)

1000 500 0 –500 –1000 –1500

0

50

100

150

Time (s)

Figure 4.11. The output-feedback controller. based on the nominal plant parameters (under the model error), the proposed controller based on the actual plant parameters (without the model error), and the PD controller with new control gains kp = 630, kd = 15000, which are tuned to attain efficient control performance, is presented in figure 4.12 and figure 4.13, which show the vibration responses of the cage and the midpoint of the cable, respectively. We observe that, although the vibration amplitudes under the proposed controller with

CHAPTER FOUR

84 0.4

u(0, t) (m)

0.2 0 –0.2

PD controller Proposed controller without model error Proposed controller under model error

–0.4 50

0

100

150

Time (s)

Figure 4.12. The output responses u(0, t) of the closed-loop system (4.94)–(4.108) under the disturbance (4.110) at x = 0 with the proposed output-feedback controller (4.91) under the model error (solid line) and without the model error (dot-and-dash line), as well as the PD controller (4.111) (dashed line).

u(l(t) /2, t) (m)

0.2 0.1 0 –0.1 –0.2

PD controller Proposed controller without model error Proposed controller under model error

–0.3 –0.4

0

50

100

150

Time (s)

Figure 4.13. The output responses u(l(t)/2, t) of the closed-loop system (4.94)– (4.108) under the disturbance (4.110) at x = 0 with the proposed output-feedback controller (4.91) under the model error (solid line) and without the model error (dot-and-dash line), as well as the PD controller (4.111) (dashed line).

the model error are slightly larger than the vibration amplitudes under the proposed controller without the model error before 65 s, both exhibit similar good results as time goes on and perform better than the standard PD controller.

4.6

APPENDIX

A. Proof of lemma 4.1 As in [121], we employ a Lyapunov function 1 q 2 2 ut (·, t) + ´ ux (·, t) Vu´ (t) = ´ 2 2  l(t) + δu´ (1 + x)´ ux (x, t)´ ut (x, t)dx, 0

where the parameter δu´ is to be determined and needs to at least satisfy

(4.112)

ELEVATORS WITH DISTURBANCES

85

0 < δu´ < 1/(1 + L) min{1, q} to guarantee the positive definiteness of Vu´ (t). Defining Ωu´ (t) as the square of the norm (4.13), we get the inequality θu´1 Ωu´ (t) ≤ Vu´ (t) ≤ θu´2 Ωu´ (t),

(4.113)

where 1 θu´1 = [min{1, q} − δu´ (1 + L)] > 0, 2 1 θu´2 = [max{1, q} + δu´ (1 + L)] > 0. 2 Taking the derivative of Vu´ (t) along the system (4.10)–(4.12), applying Young’s inequality and 0 < l(t) ≤ L, and choosing   2a1 q 1 1 + qa21 min 1, q, 0 < δu´ < , , (4.114) 1+L 1 + qa1 2 2a1 we obtain V˙ u´ ≤ −λu´ Vu´ + M ,

(4.115)

where λu´ = δu´ /(2θu´2 ), M = 1/r2 [1/(2r1 ) − δu´ /2]D

2

with 0 < r1 < δu´ /q 2 . Multiplying both sides of (4.115) by eλu´ t , we obtain d(Vu´ eλu´ t ) ≤ M eλu´ t . dt

(4.116)

Integration of (4.116) yields Ωu´ (t) ≤

1 1 Vu´ (t) ≤ θu´1 θu´1

 Vu´ (0) −

M λu´



e−λu´ t +

M , θu´1 λu´

which implies that Ωu´ (t) is uniformly bounded by Vu´ (0)/θu´1 . Moreover, it is uniformly ultimately bounded with the ultimate bound M /(θu´1 λu´ ). The proof is complete. B. Proof of lemma 4.2 According to (4.10)–(4.12) and (4.14)–(4.16), the system v˜ is governed by v˜tt (x, t) = q˜ vxx (x, t), v˜(0, t) = 0, v˜x (l(t), t) = − a1 v˜t (l(t), t).

(4.117) (4.118) (4.119)

CHAPTER FOUR

86 We employ a Lyapunov function 1 q 2 2 vt (·, t) + ˜ vx (·, t) Vv˜ (t) = ˜ 2 2  l(t) + δv˜ (1 + x)˜ vx (x, t)˜ vt (x, t)dx,

(4.120)

0

where the parameter δv˜ is to be determined and needs to at least satisfy 0 < δv˜ < 1/(1 + L) min{1, q} to guarantee the positive definiteness of Vv˜ (t). Defining Ωv˜ (t) as the square of the norm (4.18), we can then get the inequality θv˜1 Ωv˜ (t) ≤ Vv˜ (t) ≤ θv˜2 Ωv˜ (t),

(4.121)

where 1 θv˜1 = [min{1, q} − δv˜ (1 + L)] > 0, 2 1 θv˜2 = [max{1, q} + δv˜ (1 + L)] > 0. 2 Taking the derivative of Vv˜ (t) along the trajectory of the system (4.117)–(4.119), we obtain δv˜ δv˜ δv˜ 2 2 vt  − q˜ vx  − v˜x2 (0, t) V˙ v˜ ≤ − ˜ 2 2 2   δv˜ (1 + L) 2 (1 + qa1 ) v˜t2 (l(t), t) − a1 q − 2     1 a 2q ˙  1 + − a1 δv˜ (1 + L) v˜t2 (l(t), t), − l(t) 2 2 ˙ ≤ 0 in assumption 4.1 have been used. where 0 < l(t) ≤ L and l(t) Choosing δv˜ to satisfy   2a1 q 1 + qa21 1 0 < δv˜ < min 1, q, , , 1+L 1 + qa21 2a1 we obtain V˙ v˜ ≤ −λv˜ Vv˜ , where λv˜ = δv˜ /(2θv˜2 ) > 0. With (4.121), we get Ωv˜ (t) ≤ The proof is complete.

θv˜2 Ωv˜ (0)e−λv˜ t . θv˜1

(4.122)

ELEVATORS WITH DISTURBANCES

87

C. Proof of lemma 4.3 According to the system (4.117)–(4.119), the e-system can be written as ett (x, t) = qexx (x, t), e(0, t) = 0, ex (l(t), t) = −b1 et (l(t), t), where b1 =

˙ qa1 − |l(t)| . ˙ q − qa1 |l(t)|

(4.123) (4.124) (4.125)

(4.126)

With the choice v¯/q < a1 < 1/¯ v,

(4.127)

we ensure b1 > 0, given that v¯/q < 1/¯ v in assumption 4.2. Consider a Lyapunov function for the system (4.123)–(4.125): 1 q 2 2 Ve (t) = et (·, t) + ex (·, t) 2 2  l(t) + δe (1 + x)ex (x, t)et (x, t)dx,

(4.128)

0

where the parameter δe is to be determined and needs to at least satisfy 0 < δe < 1/(1 + L) min{1, q} to guarantee the positive definiteness of Ve (t). Defining Ωe (t) as the square of the norm (4.19), we get the inequality θe1 Ωe (t) ≤ Ve (t) ≤ θe2 Ωe (t),

(4.129)

where 1 θe1 = [min{1, q} − δe (1 + L)] > 0, 2 1 θe2 = [max{1, q} + δe (1 + L)] > 0. 2 Taking the derivative of Ve (t) along the trajectory of the system (4.123)–(4.125), through a similar computation as (4.122), and using (4.129), we get the exponential stability of the system e(x, t): V˙ e ≤ −σd˜Ve ,

(4.130)

where σd˜ = δe /(2θe2 ) and δe satisfy   2b1 q 1 1 + qb1 2 min 1, q, 0 < δe < , . 1+L 2b1 1 + qb1 2

(4.131)

CHAPTER FOUR

88 We can then get σd˜ =

δe . max{1, q} + δe (1 + L)

(4.132)

From (4.131) and (4.132), it can be seen that the decay rate σd˜ depends on b1 . By combining with (4.126), we observe that σd˜ depends on a1 . From (4.129) and (4.130), we obtain Ωe (t) ≤

θe2 Ωe (0)e−σd˜t . θe1

(4.133)

According to the v˜-system (4.117)–(4.119) and using (4.133), from the CauchySchwarz inequality, we obtain   l(t)     |˜ vx (0, t)| ≤ |˜ vx (l(t), t)| +  v˜xx (x, t)dx 0

√   l(t)  12 L 2 ≤ |a1 v˜t (l(t), t)| + |˜ vtt (x, t)| dx q 0 √ L et (·, t) ≤ μv˜ e−σd˜t , = |a1 e(l(t), t)| + q

(4.134)

where the positive constant μv˜ depends only on the initial data. The proof is complete. D. Proof of lemma 4.4 Defining y¯ = et and taking the derivative of (4.123)–(4.125), recalling assumption 4.3, we obtain y¯tt (x, t) = q y¯xx (x, t), y¯(0, t) = 0, y¯x (l(t), t) = −b2 y¯t (l(t), t), where b2 =

˙ qb1 − |l(t)| . ˙ q − qb1 |l(t)|

(4.135) (4.136) (4.137)

(4.138)

Substituting (4.126) into (4.138), we get b2 =

˙ ˙ q(q + |l(t)|)a 1 − 2q|l(t)| . 2 2 2 ˙ ˙ q + q|l(t)| − 2q a1 |l(t)|

There exists a positive constant a1 to make b2 > 0. This is because there is a mapping between the dependent variable interval b1 ∈ (0, ∞) and the independent variable v /q, 1/¯ v ) according to the function (4.126). Therefore, we can choose interval a1 ∈ (¯ some a1 in the range (¯ v /q, 1/¯ v) to make b1 stay in the range 0 < v¯/q < b1 < 1/¯ v < ∞, which yields b2 > 0 according to (4.138).

ELEVATORS WITH DISTURBANCES

89

A calculation similar to (4.128)–(4.134), leads to    ˜˙  vxt (0, t)| ≤ μd˜t e−σd˜t t d(t) = r|˜ for some positive constants σd˜t and μd˜t which depends on the system initial values only.    ¨˜  vxtt (0, t)|. Similarly, we can prove the exponential convergence to zero of d(t)  = r|˜ The proof is complete. E. Proof of lemma 4.6 First, we illustrate the well-posedness of the observer error system (4.42)–(4.45). Define an operator A: D(A) → H by A(z, v)T = (v, qz  )T , ∀(z, v) ∈ D(A), D(A) = {(z, v) ∈ H|z(0) = v(0) = 0, z  (l(t)) = −a2 v(l(t))}. The system (4.43)–(4.45) can be written as     d u ˜(·, t) u ˜(·, t) =A . u ˜t (·, t) u ˜t (·, t) dt Under the boundedness and regularity assumptions (4.1–4.3) on l(t), according to [121], A generates an exponential stable C0 semigroup, which also can be obtained through the Lyapunov analysis in the sequel of this proof. Then there exist K, μ2 > 0 such that eAt  ≤ Ke−μ2 t . By [190], one concludes that for all initial values u, u ˜t )T ∈ C([0, ∞); H) (˜ u(x, 0), u ˜t (x, 0))T ∈ H, and there exists a unique solution (˜ to the system (4.43)–(4.45) as     u ˜(·, 0) u ˜(·, t) = eAt . u ˜t (·, 0) u ˜t (·, t) With the initial values X(0) ∈ R2 , it is straightforward to demonstrate that there exists a unique solution X ∈ R2 to the ODE (4.42), which is cascaded with the u ˜-PDE proved to be well-posed above. The signal ˜ = d(t) − d(t) ˆ = d(t) − (−rd¯x (0, t)) d(t)

(4.139)

in the ODE (4.42) is well-defined, because d¯x (0, t) is defined by three well-posed ¯ systems where the well-posedness of the d-system in (4.14)–(4.16) and the u ¯-system in (4.5)–(4.7) are proved in [121], whereas the well-posedness of the u-system in (4.1)–(4.3) is proved in section 5 of [190]. The well-posedness of (4.42)–(4.45) can then be obtained. Next, we prove the exponential stability of the observer error system (4.42)– (4.45) via Lyapunov analysis. Recalling theorem 4.1, define ηd˜(t) = μd˜e−σd˜t .

(4.140)

˜ + φu˜ Eu˜ (t) + η1 η ˜(t)2 , ˜ T (t)P1 X(t) Vu˜ (t) = X d

(4.141)

We employ a Lyapunov function

CHAPTER FOUR

90

where the positive parameters φu˜ and η1 are to be chosen later. The matrix P1 = P1T > 0 is the unique solution to the Lyapunov equation ¯ + (A − LC) ¯ T P1 = −Q1 P1 (A − LC) for some Q1 = Q1 T > 0, and Eu˜ (t) is defined as 1 q 2 2 ut (·, t) + ˜ ux (·, t) Eu˜ (t) = ˜ 2 2  l(t) + δu˜ (1 + x)˜ ux (x, t)˜ ut (x, t)dx,

(4.142)

0

where δu˜ should at least satisfy 0 < δu˜ < 1/(1 + L) min{1, q} to guarantee the positive definiteness of Eu˜ (t). Taking the derivative of Vu˜ along (4.42)–(4.45), (4.140); applying Young’s inequality and 0 < l(t) ≤ L; and choosing the parameters δu˜ , η1 , and φu˜ to satisfy the inequalities   2a2 q 1 + qa22 1 min 1, q, 0 < δu˜ < , , 1+L 1 + qa22 2a2  2 8 1r P1 B  , η1 > σd˜λmin (Q1 ) 2

φu˜ >

4|P1 B| , qδu˜ λmin (Q1 )

we arrive at V˙ u˜ ≤ −σu˜ Vu˜ ,

(4.143)

where  2   4 1r P1 B  δu˜ 1 δu˜ 1 σu˜ = φu˜ , qφu˜ , λmin (Q1 ), η1 σd˜ − min > 0. θu˜2 2 2 2 λmin (Q1 )

(4.144)

The proof of lemma 4.6 is complete. F. Proof of lemma 4.7 ˆ First, we establish well-posedness of the target (w, ˆ X)-system in (4.61)–(4.64). Define an operator A1 : D(A1 ) → H by A1 (z, v)T = (v, qz  )T , ∀(z, v) ∈ D(A1 ), D(A1 ) = {(z, v) ∈ H|z(0) = v(0) = 0, z  (l(t)) = −a3 v(l(t))}. The system (4.62)–(4.64) can be written as       d 0 w(·, ˆ t) w(·, ˆ t) ˜ + BC X(t), = A1 + f (·, t) w ˆt (·, t) w ˆt (·, t) dt

ELEVATORS WITH DISTURBANCES

91

where ˜ + η¯(x, t) f (x, t) = −f¯1 (x)X(t) and B = (0, δ(x))T . Similar to A in lemma 4.6, A1 generates an exponentially stable C0 -semigroup, which also can be obtained through the following Lyapunov analysis. Then there exist K1 , μ3 > 0 such that eA1 t  ≤ K1 e−μ3 t . It is a routine exercise that B is admissible for A1 . By [190], recalling lemma 4.6 and (4.65), one concludes that for all initial values (w(x, ˆ 0), w ˆt (x, 0))T ∈ H there T exists a unique solution (w, ˆ w ˆt ) ∈ C([0, ∞); H) to the system (4.62)–(4.64) as     t    w(·, ˆ t) w(·, ˆ 0) 0 A1 t A1 (t−s) e =e + ds w ˆt (·, t) w ˆt (·, 0) f (·, s) 0  t ˜ eA1 (t−s) BC X(s)ds. + 0

ˆ With the initial value X(0) ∈ R2 , it is straightforward to show that there exists a 2 ˆ proved unique solution X ∈ R to the ODE (4.61) cascaded with the w-subsystem ˜ as well-posed above. It should be noted that the signal Z(t) in the ODE (4.61) depends on the ODE Yˆ in (4.26), which is obviously a well-posed system. The well-posedness of (4.61)–(4.64) is thus obtained. Next, we prove the exponential regulation of the target system (4.61)–(4.64) in the sense of (4.69) via Lyapunov analysis. Let Vwˆ (t) be a Lyapunov function defined as ˆ + φwˆ Ewˆ (t) + ξ3 η¯m (t)2 + ξ4 η ˜(t)2 + ξ5 η ˜ (t)2 , ˆ T (t)P2 X(t) Vwˆ (t) = X Z d

(4.145)

where η¯m (t)2 , ηZ˜ (t), ηd˜(t) are defined in (4.67), (4.29), (4.140), and the matrix P2 = P2T > 0 is the unique solution to the Lyapunov equation P2 (A + BK) + (A + BK)T P2 = −Q2 for some matrix Q2 = Q2 T > 0. The function Ewˆ (t) in (4.145) is defined as 1 q 2 2 ˆt (·, t) + w ˆx (·, t) Ewˆ (t) = w 2 2  l(t) + δwˆ (1 + x)w ˆx (x, t)w ˆt (x, t)dx,

(4.146)

0

where the parameter δwˆ is to be determined and needs to at least satisfy 0 < δwˆ < 1/(1 + L) min {1, q} to guarantee the positive definiteness of Ewˆ (t). The positive parameters φwˆ and ξ3 , ξ4 , ξ5 are to be chosen later. Taking the derivative of Vwˆ along (4.61)–(4.64), recalling (4.29), (4.140) and (4.66)–(4.68), and applying Young’s inequality through the similar computation of (4.141)–(4.144), we arrive at    ˜ 2 V˙ wˆ ≤ −σwˆ Vwˆ + ξ2 X(t) (4.147) 

CHAPTER FOUR

92 for some positive σwˆ and

  ¯ + Bγ(0, 0)C)2 4P2 (LC 1 2 1 2 2 2 ¯ ξ2 = φwˆ + φwˆ C (A − LC) + . 4 2 λmin (Q2 ) The parameters ξ3 , ξ4 , ξ5 , δwˆ , φwˆ should satisfy 2 L Cmax , 4r0 ση¯ 8|P2 B| , ξ4 > σd˜r2 λmin (Q2 ) 8|P2 BCz | ξ5 > , σZ˜ r2 λmin (Q2 )

ξ3 >

  2a3 q 1 + qa23 1 min 1, q, , , 1+L 1 + qa23 2a3   2 4 |P2 B| 2 q , max 2C1 L, + φwˆ > δwˆ 2 qλmin (Q2 ) 0 < δwˆ
0.

(4.150)

(4.151)

The proof of lemma 4.7 is complete. G. Proof of lemma 4.9 Differentiating (4.62) with respect to x, and differentiating (4.63), (4.64) with respect to t, we obtain ˜ + η¯x (x, t), w ˆttx (x, t) = q w ˆxxx (x, t) − f¯1 (x)X(t)

(4.152)

ELEVATORS WITH DISTURBANCES

93

¯ X(t), ˜ w ˆt (0, t) = C(A − LC) ˆxt (l(t), t) − w ˆtt (l(t), t) = −b3 w +

(4.153) ˙ f¯1 (l(t)) l(t) ˜ X(t) ˙ a3 q + l(t)

˙ l(t) η¯(l(t), t), ˙ a3 q + l(t)

(4.154)

where (4.42) and CB = 0 are recalled, and where b3 > 0 by choosing v¯/q < a3 < 1/¯ v. We know that η¯(l(t), t) is exponentially convergent to zero according to (4.66). Similarly, we find that η¯x (x, t) is also exponentially convergent to zero by using (4.55), (4.65), (4.66) and theorem 4.1, lemma 4.4, and lemma 4.5. According to ˜ lemma 4.6, we know that X(t) is exponentially convergent to zero. Through a similar calculation with the proof of lemma 4.7, we obtain the exponential stability ˆxx (·, t)2 . of the system (4.152)–(4.154) in the sense of w ˆxt (·, t)2 + w Through the invertible transformations (4.60), we get an exponential stability zxx (·, t)2 )1/2 . Recalling the transformation estimate for the norm (ˆ zxt (·, t)2 + ˆ (4.52), we get ˆ 2, ˆ uxx (·, t)2 ≤ 2ˆ zxx (·, t)2 + 2L|ϑm Z(t)| 2 ˆ˙ zxt (·, t)2 + 2L|ϑm Z(t)| , ˆ uxt (·, t)2 ≤ 2ˆ

where ϑm = max {|ϑ (x)|}, x∈[0,L]

ϑm = max {|ϑ (x)|}. x∈[0,L]

ˆ and The terms |ϑm Z(t)|       ˆ˙  ˆ ˆ  Az − Lz Cˆz )Yˆ (t) − Lz rd¯x (0, t)] ϑm Z(t) = |ϑm | C[( are bounded. We thus obtain the bounded estimates for the norm (ˆ uxt (·, t)2 + ˆ uxx (·, t)2 )1/2 . The proof of lemma 4.9 is complete.

4.7

NOTES

The first results on dealing with anti-collocated disturbances in wave PDEs is given in [85] using an adaptive cancellation scheme, where the PDE is on a fixed domain and the asymptotic convergence of the output state is achieved. In this chapter, we proposed a disturbance rejection scheme for a wave PDE-ODE system on a time-varying domain, where a disturbance estimator that generates estimates

94

CHAPTER FOUR

exponentially convergent to external uncertain disturbances was designed, and the exponential convergence of the state at the uncontrolled and disturbed boundary was achieved. Differing from this chapter’s topic of disturbance rejection in wave PDEs, chapter 5 will present adaptive cancellation of unmatched disturbances in more complex coupled hyperbolic PDEs with spatially varying coefficients.

Chapter Five Elevators with Flexible Guides

In chapters 2–4 we developed control designs for suppression of the axial vibrations in mining cable elevators with steel guideways. In this chapter, we address suppression of the lateral vibrations in mining cable elevators moving along flexible guideways. The elastic support of flexible guides is usually approximated as a spring-damper system [174, 200], where the stiffness and damping coefficients are not known exactly. This uncertainty leads to unknown parameters existing in the system matrix of the ordinary differential equation (ODE) (cage dynamics) at the uncontrolled boundary of the partial differential equation (PDE) with a timevarying domain. Moreover, as in chapter 4, the cage is always subject to an airflow disturbance, which increases the unmatched uncertainties in the PDE system and makes the control design more challenging. The objective in this chapter is to design a control law at the top of a vibrating cable with a time-varying length to regulate the cage at the cable’s bottom, where information about the viscoelastic guides (parameters in the system matrix of the ODE) are unknown, and the cage is subject to uncertain external disturbances. The content of this chapter is organized as follows. In section 5.1, the lateral vibration dynamics of the mining cable elevator with flexible guides are shown, and the general mathematical problem is introduced. With the ODE state fully measured, an observer is designed to estimate the PDE states in section 5.2. The design of the output-feedback controller via the backstepping method is proposed in section 5.3. Adaptive update laws for the unknown parameters are given in section 5.4. In section 5.5, the adaptive output-feedback control law is presented, and the stability result of the closed-loop control system is proved, followed by simulation tests in a mining cable elevator in section 5.6.

5.1

DESCRIPTION OF FLEXIBLE GUIDES AND GENERALIZED MODEL

Model of the Mining Cable Elevator with Flexible Guides For lateral vibrations, an important factor of influence is the interaction between the cage and the flexible guides. The elastic support of flexible guides is approximated as a spring-damper system—that is, as a viscoelastic guide [174, 200] where the stiffness and damping coefficients kc , cd are not exactly known (see figure 5.1). The wave PDE-modeled lateral vibration dynamics of the mining cable elevator are given by ρutt = T (x)uxx (x, t) + T  (x)ux (x, t) − c¯ut (x, t),

(5.1)

CHAPTER FIVE

96

Control input U(t) Hydraulic cylinder driving head sheaves Cable lateral vibration

Modeled by a wave PDE with an in-domain damping term Riemann transformation Coupled transport PDEs z(x,t), w(x,t) Flexible guide

Cage lateral vibration described by the ODE X(t)

kc

cd

Cage

Disturbance d(t)

Uncertainties

Figure 5.1. Lateral vibration control of a mining cable elevator with viscoelastic guides. Mc utt (0, t) = −kc u(0, t) − cd ut (0, t) + T (0)ux (0, t) + d(t), −T (l(t))ux (l(t), t) = U (t),

(5.2) (5.3)

where u(x, t) denotes the lateral vibration displacements along the cable shown in figure 5.1, and x ∈ [0, l(t)] are the positions along the cable in a moving coordinate system associated with the motion l(t), with the origin located at the cage. The function T (x) = Mc g + xρg is the static tension along the cable, and ρ is the linear density of the cable. The coefficient c¯ is the material damping of the cable. The signal d(t) is the uncertain airflow disturbance [188] acting at the cage. The constants kc , cd are the unknown equivalent stiffness and damping coefficients of the viscoelastic guide. The modeling process of the lateral vibration dynamics of the mining cable elevator (5.1)–(5.3) is based on [30]. By applying the Riemann transformations  T (x) ux (x, t), (5.4) z(x, t) = ut (x, t) − ρ  T (x) w(x, t) = ut (x, t) + ux (x, t) (5.5) ρ

ELEVATORS WITH FLEXIBLE GUIDES

97

and defining X(t) = [x1 (t), x2 (t)]T = [u(0, t), ut (0, t)]T , which physically means the lateral displacement and velocity of the cage, we convert (5.1)–(5.3) into a 2 × 2 hyperbolic system coupled with an ODE, given by ˙ X(t) = AX(t) + Bw(0, t) + B1 d(t), z(0, t) = CX(t) − p1 w(0, t),

(5.6) (5.7)

zt (x, t) = −q1 (x)zx (x, t) + c1 (x)z(x, t) + c2 (x)w(x, t),

(5.8)

wt (x, t) = q2 (x)wx (x, t) + c3 (x)z(x, t) + c4 (x)w(x, t),

(5.9)

w(l(t), t) = U (t) + p2 z(l(t), t),

(5.10)

with x ∈ [0, l(t)], t ∈ [0, ∞), and the coefficients defined as  q1 (x) = q2 (x) =

T  (x) T (x) −¯ c , c1 (x) = c3 (x) = −  , ρ 2ρ 4 ρT (x)

T  (x) −¯ c +  , p1 = p2 = 1, 2ρ 4 ρT (x)     0 1 0 M  √c A= , ,B = ρg Mc −kc −cd − Mc ρg Mc   0 B1 = , C = [0, 2]. 1 c2 (x) = c4 (x) =

(5.11) (5.12)

(5.13)

(5.14)

Mc

It should be noted that the control input designed based on (5.6)–(5.10) with the √ ρT (l(t)) in order to convert the input above coefficients should be multiplied by − 2 U (t) in (5.10) into a control force in the practical mining cable elevator—that is, into the control input U (t) in the boundary condition (5.3) in the wave PDE model (5.1)–(5.3). In the practical mining cable elevator, l(t) is obtained by the product of the radius and the angular displacement of the rotating drum driving the cable, where the angular displacement is measured by the angular displacement sensor at the drum. Generalization In this chapter, we conduct the control design based on (5.6)–(5.10) in a general form with the following conditions. The vector X(t) ∈ Rn is an ODE state, whereas z(x, t), w(x, t) are PDE states. The spatially varying transport speeds q1 , q2 are positive-valued C 1 ([0, L]) functions, and c1 , c2 , c3 , c4 are C 0 ([0, L]) functions where the positive constant L is the upper bound of l(t), as will be seen in assumption 5.3. The constant p1 is nonzero, and the constant p2 is arbitrary. The matrix C ∈ R1×n is arbitrary. The matrix A ∈ Rn×n is the system matrix, and B ∈ Rn×1 is the input matrix and B1 = Bbd , where bd is an arbitrary constant. The matrices A, B and the signal d(t) are expected to satisfy the following assumptions:

CHAPTER FIVE

98 Assumption 5.1. The matrices A, B are in the form of ⎛ ⎞ ⎛ 0 1 0 0 ··· 0 ⎜ 0 0 1 ⎟ 0 ··· 0 ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ . . A=⎜ ⎟,B =⎜ . ⎜ ⎜ ⎟ ⎝ ⎝ 0 0 0 0 ··· 1 ⎠ g1 g2 g3 · · · gn−1 gn

0 0 0 0 hn

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

(5.15)

where the constants g1 , g2 , g3 , . . . , gn−1 , gn are unknown and arbitrary, and their lower and upper bounds are known and arbitrary. The constant hn is nonzero and known. Assumption 5.1 indicates that the ODE is in the controllable form, which covers many practical models, including the cage dynamics modeled in (5.6), (5.13). Choose a target Hurwitz matrix ⎞ ⎛ 0 1 0 0 ··· 0 ⎜ 0 0 1 0 ··· 0 ⎟ ⎟ ⎜ ⎟ ⎜ . .. (5.16) Am = ⎜ ⎟, ⎟ ⎜ ⎠ ⎝ 0 0 0 0 ··· 1 g¯1 g¯2 g¯3 · · · g¯n−1 g¯n where g¯1 , g¯2 , g¯3 , . . . , g¯n−1 , g¯n are determined by the user according to the desired performance for the specific application, such as the required stiffness coefficient and damping coefficient of the cage in figure 5.1. According to assumption 5.1 and (5.16), we know that there exists a unique, though unknown, row vector K1×n = [k1 , . . . , kn ]

(5.17)

Am = A + BK,

(5.18)

g¯i = gi + hn ki , i = 1, 2, . . . , n.

(5.19)

such that

and

By virtue of (5.18), while the ki ’s are unknown, the lower and upper bounds on the ki ’s—that is, [k i , k¯i ], i = 1, 2, . . . , n, are thus known, given that the lower and upper bounds of the gi ’s are known in assumption 5.1, and the g¯i ’s are chosen by the user. Assumption 5.2. The disturbance d(t) is of the general harmonic form as d(t) =

N 

[aj cos(θj t) + bj sin(θj t)],

(5.20)

j=1

where the integer N is arbitrary. The frequencies θj , j ∈ {1, 2, . . . , N } are known and arbitrary constants. The amplitudes aj , bj are unknown constants bounded by ¯j ], bj ∈ [0, ¯bj ]. the known and arbitrary positive constants a ¯j , ¯bj , that is, aj ∈ [0, a

ELEVATORS WITH FLEXIBLE GUIDES

99

Assumption 5.2 can model all periodic disturbance signals to an arbitrarily high degree of accuracy by choosing N sufficiently large. The time-varying domain [0, l(t)] associated with the moving boundary l(t), which is a known time-varying function, is under the following two assumptions. Assumption 5.3. The function l(t) is bounded—that is, 0 < l(t) ≤ L, ∀t ≥ 0, where L is a positive constant. The constant L is the maximal length of the cable in the application of vibration suppression of mining cable elevators. ˙ is bounded as Assumption 5.4. The function l(t)   ˙  l(t) < min {q1 (x), q2 (x)}, ∀t ≥ 0. 0≤x≤L

(5.21)

As in chapters 2–4, the limit of the speed of the moving boundary in assumption 5.4 ensures the well-posedness of the initial boundary value problem (5.6)–(5.10) according to [71, 72]. This assumption holds in the applications of the mining cable elevator, as we shall see in the simulation section.

5.2

OBSERVER FOR DISTRIBUTED STATES OF THE CABLE

Observer Structure To estimate the PDE states (z(x, t), w(x, t))T —that is, the distributed state in the cable—which usually cannot be fully measured in practice but are required in the controller, an observer using the measurements X(t), z(l(t), t) is introduced as ˙ X(t) = AX(t) + B w(0, ˆ t) + B w(0, ˜ t) + B1 d(t), zˆ(0, t) = CX(t) − p1 w(0, ˆ t), zˆt (x, t) = −q1 (x)ˆ zx (x, t) + c1 (x)ˆ z (x, t) + c2 (x)w(x, ˆ t) + Φ2 (x, t)(z(l(t), t) − zˆ(l(t), t)), w ˆt (x, t) = q2 (x)w ˆx (x, t) + c3 (x)ˆ z (x, t) + c4 (x)w(x, ˆ t) + Φ3 (x, t)(z(l(t), t) − zˆ(l(t), t)), w(l(t), ˆ t) = U (t) + p2 z(l(t), t),

(5.22) (5.23) (5.24) (5.25) (5.26)

where (5.22) is exactly the ODE (5.6) with w(0, t) = w(0, ˆ t) + w(0, ˜ t). Equation (5.22) is a part of the plant dynamics instead of representing a part of the observer. The functions Φ2 (x, t), Φ3 (x, t) are observer gains to be determined. The ODE state X(t) physically means the vibration displacement and velocity of the cage in the mining cable elevator, which can be obtained by an acceleration sensor placed at the cage plus an integral algorithm [178]. Because the ODE state is available, the task for the observer (5.23)–(5.26) is only to estimate the PDE states. Define the observer error state as (˜ z (x, t), w(x, ˜ t)) = (z(x, t), w(x, t)) − (ˆ z (x, t), w(x, ˆ t)).

(5.27)

According to (5.7)–(5.10) and (5.23)–(5.26), the observer error system is z˜(0, t) = −p1 w(0, ˜ t),

(5.28)

CHAPTER FIVE

100 z˜t (x, t) = −q1 (x)˜ zx (x, t) + c1 (x)˜ z (x, t) + c2 (x)w(x, ˜ t) − Φ2 (x, t)˜ z (l(t), t),

(5.29)

w ˜t (x, t) = q2 (x)w ˜x (x, t) + c3 (x)˜ z (x, t) + c4 (x)w(x, ˜ t) − Φ3 (x, t)˜ z (l(t), t),

(5.30)

w(l(t), ˜ t) = 0.

(5.31)

Observer gains Φ2 (x, t), Φ3 (x, t) are to be designed to ensure convergence to zero of the observer errors. Determining Observer Gains via Backstepping Postulate the invertible backstepping transformation 

l(t)

z˜(x, t) = α ˜ (x, t) −  −

l(t)

ˇ y)β(y, ˜ t)dy, φ(x,

x



˜ t) − w(x, ˜ t) = β(x,  −

¯ y)˜ φ(x, α(y, t)dy

x

l(t)

(5.32)

¯ y)˜ ψ(x, α(y, t)dy

x l(t)

ˇ y)β(y, ˜ t)dy ψ(x,

(5.33)

x

to convert the original observer error system (5.28)–(5.31) to the following target observer error system: ˜ t), α ˜ (0, t) = −p1 β(0,

(5.34)

α ˜ t (x, t) = −q1 (x)˜ αx (x, t) + c1 (x)˜ α(x, t),

(5.35)

˜ t), β˜t (x, t) = q2 (x)β˜x (x, t) + c4 (x)β(x,

(5.36)

˜ β(l(t), t) = 0.

(5.37)

The form of the backstepping transformation (5.32), (5.33) for coupled hyperbolic PDEs is taken from [96]. The integration interval chosen in the transformation (5.32), (5.33) is [x, l(t)] because the PDE boundary measurement used in the observer is at the boundary x = l(t). Even though the integration interval is time varying, the kernels in (5.32), (5.33) need not include the time argument because the extra terms introduced by the time-varying integration interval during the calculation of the kernel conditions will be “absorbed” by the time-dependent observer gains Φ2 (x, t), Φ3 (x, t), which will be seen clearly later. By matching (5.28)–(5.31) and (5.34)–(5.37) using (5.32), (5.33) (the details are shown in appendix 5.7A), ¯ y), ψ(x, ¯ y), φ(x, ˇ y), ψ(x, ˇ y) in (5.32), (5.33) are the conditions on the kernels φ(x, obtained as the following two well-posed hyperbolic systems: ¯ x) = ψ(x,

−c3 (x) , q1 (x) + q2 (x)

¯ y) = −p1 ψ(0, ¯ y), φ(0,

(5.38) (5.39)

ELEVATORS WITH FLEXIBLE GUIDES

101

q2 (x)ψ¯x (x, y) − q1 (y)ψ¯y (x, y) ¯ y) + c3 (x)φ(x, ¯ y) = 0, +(c4 (x) − c1 (y) − q1  (y))ψ(x,

(5.40)

−q1 (x)φ¯x (x, y) − q1 (y)φ¯y (x, y) ¯ y) + c2 (x)ψ(x, ¯ y) = 0, +(c1 (x) − c1 (y) − q1  (y))φ(x,

(5.41)

and c2 (x) , q1 (x) + q2 (x)

(5.42)

ˇ y) = − 1 φ(0, ˇ y), ψ(0, p1

(5.43)

ˇ x) = φ(x,

q2 (x)ψˇx (x, y) + q2 (y)ψˇy (x, y) ˇ y) + c3 (x)φ(x, ˇ y) = 0, +(c4 (x) − c4 (y) + q2  (y))ψ(x,

(5.44)

−q1 (x)φˇx (x, y) + q2 (y)φˇy (x, y) ˇ y) + c2 (x)ψ(x, ˇ y) = 0, +(c1 (x) − c4 (y) + q2  (y))φ(x,

(5.45)

on D = {0 ≤ x ≤ y ≤ l(t)}. The observer gains are thus determined as ˙ φ(x, ¯ l(t)) − q1 (l(t))φ(x, ¯ l(t)), Φ2 (x, t) = l(t)

(5.46)

˙ ψ(x, ¯ l(t)) − q1 (l(t))ψ(x, ¯ l(t)). Φ3 (x, t) = l(t)

(5.47)

Remark 5.1. The equations (5.38)–(5.41) and (5.42)–(5.45) are in the same form as the kernel equations (24)–(31) in [177] if we extend the domain D to a fixed triangular domain. The boundary conditions in (5.38)–(5.41) and (5.42)–(5.45) on the triangular domain D = {0 ≤ x ≤ y ≤ l(t)} are given along the lines y = x and x = 0 rather than on y = l(t). Therefore, it is feasible to extend the domain D to a fixed triangular domain D1 = {0 ≤ x ≤ y ≤ L} (L is defined in assumption 5.3) and obtain the solutions of (5.38)–(5.41) and (5.42)–(5.45) on D by solving (5.38)–(5.41) and (5.42)–(5.45) on D1 , whose well-posedness is proved in [177]. This ensures the existence of the observer gains Φ2 (x, t) and Φ3 (x, t) in (5.46), (5.47) consisting of ¯ l(t)), ψ(x, ¯ l(t)) obtained by extracting the results along y = l(t) in the solution φ(x, of (5.38)–(5.41) on D1 . Stability of the Observer Error System Lemma 5.1. For all initial data (˜ z (·, 0), w(·, ˜ 0)) ∈ H 1 (0, L), the states z˜(·, t), w(·, ˜ t) of the observer error system (5.28)–(5.31) with the observer gains (5.46), (5.47) become and remain zero no later than the time t = ta , where ta =

L L + . min0≤x≤L {q1 (x)} min0≤x≤L {q2 (x)}

Proof. According to the target observer error system (5.34)–(5.37) and the result ˜ t) reach zero by the time ta , at the latest. Applyin [96], we know that α ˜ (x, t), β(x, ing the Cauchy-Schwarz inequality into (5.32), (5.33), the proof of this lemma is complete.

CHAPTER FIVE

102

Lemma 5.1 physically means that the designed observer (5.23)–(5.26), which uses only boundary measurements, can effectively recover the actual distributed states of the vibrating string.

5.3

ADAPTIVE DISTURBANCE CANCELLATION AND STABILIZATION

In this section, we design an observer-based output-feedback controller. We conduct the state-feedback control design based on the observer using the backstepping method, which makes the resulting control law employ only the observer states. Three transformations are used to convert the observer (5.22)–(5.26) to a target system, with the intention of adaptively canceling the unmatched disturbance (the disturbance at the cage), removing the coupling in the PDE domain, and making the system matrix of the ODE Hurwitz (stabilizing the vibrating cable and cage). The output injection signals z˜(l(t), t), w(0, ˜ t) in the observer are regarded as zero in the control design, following which the separation principle, which is verified by the fact that the stability of the observer error system is independent of the control design according to lemma 5.1, which shows that the observer errors vanish in finite time only depending on the plant parameters, is applied in the stability analysis of the resulting closed-loop system. The First Transformation for Adaptively Canceling the Unmatched Disturbance We introduce the transformation (w, ˆ zˆ) → (ˆ v , sˆ): vˆ(x, t) = w(x, ˆ t) + Γ(x, t)Z(t),

(5.48)

sˆ(x, t) = zˆ(x, t) + Γ1 (x, t)Z(t),

(5.49)

where Γ(x, t), Γ1 (x, t) are to be determined, and Z(t) = [cos(θ1 t), sin(θ1 t), . . . , cos(θN t), sin(θN t)]T .

(5.50)

˙ = Az Z(t), Z(t)

(5.51)

We then have

where

 Az = diag

0 θ1

−θ1 0



 ,...,

0 θN

−θN 0

 .

(5.52)

According to assumption 5.2, the disturbance can be written as d(t) = [a1 , b1 , . . . , aN , bN ]Z(t), ˆ as and we define the disturbance estimate d(t) ˆ = [ˆ d(t) a1 (t), ˆb1 (t), . . . , a ˆN (t), ˆbN (t)]Z(t),

(5.53)

ELEVATORS WITH FLEXIBLE GUIDES

103

where a ˆ1 (t), ˆb1 (t), . . . , a ˆN (t), ˆbN (t) are estimates of a1 , b1 , . . . , aN , bN , which will be shown in section 5.4. Through (5.48), (5.49), we convert the system (5.22)–(5.26) into the following system: ˜ ˙ X(t) = AX(t) + Bˆ v (0, t) + B1 d(t), sˆ(0, t) + p1 vˆ(0, t) = CX(t), sˆt (x, t) = −q1 (x)ˆ sx (x, t) + c1 (x)ˆ s(x, t) + c2 (x)ˆ v (x, t) + Γ1t (x, t)Z(t), vˆt (x, t) = q2 (x)ˆ vx (x, t) + c3 (x)ˆ s(x, t) + c4 (x)ˆ v (x, t) + Γt (x, t)Z(t), vˆ(l(t), t) = U (t) + p2 sˆ(l(t), t) + (Γ(l(t), t) − p2 Γ1 (l(t), t))Z(t),

(5.54) (5.55) (5.56) (5.57) (5.58)

˜ is given as where d(t) ˜ = d(t) − d(t) ˆ d(t) =

N 

[(aj − a ˆj (t)) cos(θj t) + (bj − ˆbj (t)) sin(θj t)]

j=1

=

N 

[˜ aj (t) cos(θj t) + ˜bj (t) sin(θj t)].

(5.59)

j=1

The functions Γ1 (x, t), Γ(x, t) in (5.48), (5.49) are determined as follows. Taking the time and spatial derivatives of (5.48), (5.49), substituting the result into (5.56), (5.57), and recalling (5.24), (5.25), and (5.51), we get sx (x, t) − c1 (x)ˆ s(x, t) sˆt (x, t) + q1 (x)ˆ − c2 (x)ˆ v (x, t) − Γ1t (x, t)Z(t) = zˆt (x, t) + q1 (x)ˆ zx (x, t) + Γ1t (x, t)Z(t) + Γ1 (x, t)Az Z(t) − c2 (x)w(x, ˆ t) − c1 (x)ˆ z (x, t) + q1 (x)Γ1x (x, t)Z(t) − c2 (x)Γ(x, t)Z(t) − c1 (x)Γ1 (x, t)Z(t) − Γ1t (x, t)Z(t) = (Γ1 (x, t)Az + q1 (x)Γ1x (x, t) − c2 (x)Γ(x, t) − c1 (x)Γ1 (x, t))Z(t) = 0,

(5.60)

and vˆt (x, t) − q2 (x)ˆ vx (x, t) − c3 (x)ˆ s(x, t) − c4 (x)ˆ v (x, t) − Γt (x, t)Z(t) =w ˆt (x, t) − q2 (x)w ˆx (x, t) + Γt (x, t)Z(t) + Γ(x, t)Az Z(t) − c4 (x)w(x, ˆ t) − c3 (x)ˆ z (x, t) − q2 (x)Γx (x, t)Z(t) − c4 (x)Γ(x, t)Z(t) − c3 (x)Γ1 (x, t)Z(t) − Γt (x, t)Z(t) = (Γ(x, t)Az − q2 (x)Γx (x, t) − c4 (x)Γ(x, t) − c3 (x)Γ1 (x, t))Z(t) = 0.

(5.61)

CHAPTER FIVE

104 For (5.60), (5.61) to hold, we obtain the conditions

−q2 (x)Γx (x, t) + Γ(x, t)(Az − c4 (x)I2N ) − c3 (x)Γ1 (x, t) = 0,

(5.62)

q1 (x)Γ1x (x, t) + Γ1 (x, t)(Az − c1 (x)I2N ) − c2 (x)Γ(x, t) = 0,

(5.63)

where I2N is an identity matrix with dimension 2N . Defining ζ(x, t) = [Γ(x, t), Γ1 (x, t)], we rewrite (5.62), (5.63) as ¯ ζx (x, t) = −ζ(x, t)A(x), where

 ¯ = A(x)

Az − c4 (x)I2N −c2 (x)I2N −c3 (x)I2N Az − c1 (x)I2N  −1 −q2 (x)I2N 02N × . 02N q1 (x)I2N

(5.64) 

By mapping (5.22), (5.23) and (5.54), (5.55) through the transformation (5.48), (5.49), recalling (5.53) and B1 = Bbd , we obtain the condition   ζ(0, t) = Γ(0, t), Γ1 (0, t) = bd [ˆ a1 (t), ˆb1 (t), . . . , a ˆN (t), ˆbN (t), − p1 a ˆ1 (t), −p1ˆb1 (t), . . . , −p1 a ˆN (t), −p1ˆbN (t)].

(5.65)

The solution to (5.64), (5.65) is ¯ ζ(x, t) = ζ(0, t)H(x),

(5.66)

¯ where H(x) is the unique solution of the following initial value problem: ¯ x (x) = −H(x) ¯ ¯ ¯ H A(x), H(0) = I4N

(5.67)

for x ∈ [0, L]. The Second Transformation for Decoupling PDEs We postulate the backstepping transformation  x ¯ y)ˆ α ˆ (x, t) = sˆ(x, t) − λ(x, s(y, t)dy 0  x ˇ y)ˆ λ(x, v (y, t)dy, − 0  x ˆ t) = vˆ(x, t) − ¯ Υ(x, y)ˆ s(y, t)dy β(x, 0  x ˇ Υ(x, y)ˆ v (y, t)dy −

(5.68)

(5.69)

0

to convert sˆ, vˆ, X (5.54)–(5.58) into the following system: ˆ t) + B1 d(t), ˜ ˙ X(t) =AX(t) + B β(0,

(5.70)

ELEVATORS WITH FLEXIBLE GUIDES

105

ˆ t), α ˆ (0, t) = CX(t) − p1 β(0, αx (x, t) + c1 (x)ˆ α(x, t) α ˆ t (x, t) = − q1 (x)ˆ ¯ − λ(x, 0)q1 (0)CX(t)   x ¯ y)Γ1t (y, t)dy λ(x, + Γ1t (x, t) − 0   x ˇ y)Γt (y, t)dy Z(t), λ(x, −

(5.71)

(5.72)

0

ˆ t) βˆt (x, t) = q2 (x)βˆx (x, t) + c4 (x)β(x, ¯ − Υ(x, 0)q1 (0)CX(t)  x ˇ Υ(x, y)Γt (y, t)dy − 0   x ¯ Υ(x, y)Γ1t (y, t)dy − Γt (x, t) Z(t), +

(5.73)

0

ˆ β(l(t), t) =U (t) + p2 sˆ(l(t), t) + (Γ(l(t), t) − p2 Γ1 (l(t), t))Z(t)  l(t) ¯ Υ(l(t), y)ˆ s(y, t)dy −  −

0

l(t) 0

ˇ Υ(l(t), y)ˆ v (y, t)dy.

(5.74)

By matching (5.70)–(5.74) and (5.54)–(5.58) via (5.68), (5.69) (the details are shown ¯ y), λ(x, ˇ y), Υ(x, ¯ ˇ in appendix 5.7B), the conditions of kernels λ(x, y), Υ(x, y) are obtained as the following two well-posed hyperbolic systems: c2 (x) , q1 (x) + q2 (x) ¯ 0) = − q2 (0) λ(x, ˇ 0), λ(x, q1 (0)p1 ˇ x (x, y) + q2 (y)λ ˇ y (x, y) −q1 (x)λ ˇ x) = λ(x,

ˇ y) − c2 (y)λ(x, ¯ y) = 0, +(q2  (y) + c1 (x) − c4 (y))λ(x, ¯ ¯ q1 (x)λx (x, y) + q1 (y)λy (x, y)  ¯ y) + c3 (y)λ(x, ˇ y) = 0, +(q1 (y) + c1 (y) − c1 (x))λ(x,

(5.75) (5.76)

(5.77) (5.78)

and c3 (x) , q1 (x) + q2 (x) q1 (0)p1 ¯ ˇ Υ(x, 0) = − Υ(x, 0), q2 (0) ˇ x (x, y) + q2 (y)Υ ˇ y (x, y) q2 (x)Υ ¯ Υ(x, x) = −

ˇ ¯ +(q2  (y) + c4 (x) − c4 (y))Υ(x, y) − c2 (y)Υ(x, y) = 0, ¯ x (x, y) + q1 (y)Υ ¯ y (x, y) −q2 (x)Υ  ¯ ˇ +(q1 (y) + c1 (y) − c4 (x))Υ(x, y) + c3 (y)Υ(x, y) = 0

(5.79) (5.80)

(5.81) (5.82)

CHAPTER FIVE

106

on {0 ≤ y ≤ x ≤ l(t)}. The equation sets (5.75)–(5.78) and (5.79)–(5.82) are in the same form as (5.38)–(5.41) and (5.42)–(5.45). Please refer to remark 5.1 and appendix B of [6] for the well-posedness of (5.75)–(5.78) and (5.79)–(5.82). The Third Transformation for a Stable ODE We postulate the backstepping transformation  x ˆ t) − ˆ t)dy − D(x; K(t))X(t), ˆ (x, y; K(t)) ˆ ˆ ηˆ(x, t) =β(x, N β(y,

(5.83)

0

ˆ ∈ R1×n is developed in the next secwhere the update law for the control gain K(t) ˆ (x, y; K(t)), ˆ ˆ tion. The conditions for the kernels N D(x; K(t)) are to be determined later. The inverse transformation is defined as  x ˆ ˆI (x, y; K(t))ˆ ˆ ˆ N η (y, t)dy − DI (x; K(t))X(t), (5.84) β(x, t) =ˆ η (x, t) − 0

ˆI , DI are the kernels whose existence and continuity will be shown later. where N Through the transformation (5.83), we convert (5.70)–(5.74) into the following target system: ˜ − B K(t)X(t), ˙ ˜ X(t) = Am X(t) + B ηˆ(0, t) + B1 d(t) ˆ α ˆ (0, t) = (C − p1 D(0; K(t)))X(t) − p1 ηˆ(0, t), ¯ 0)q1 (0)CX(t) αx (x, t) + c1 (x)ˆ α(x, t) − λ(x, α ˆ t (x, t) = −q1 (x)ˆ   x ¯ y)Γ1t (y, t)dy + Γ1t (x, t) − λ(x,

(5.86)

0

 −

(5.85)

x 0

 ˇ λ(x, y)Γt (y, t)dy Z(t),

(5.87)

ηˆt (x, t) = q2 (x)ˆ ηx (x, t) + c4 (x)ˆ η (x, t)   x ˇ Υ(x, y)Γt (y, t)dy + Γt (x, t) − 0

 −  −  −

x 0 x 0 y 0

¯ Υ(x, y)Γ1t (y, t)dy   ˆ ˆ N (x, y; K(t)) −

y 0

ˇ z)Γt (z, t)dz Υ(y,

  ¯ z)Γ1t (z, t)dz + Γt (y, t) dy Z(t) Υ(y,



 ˆ ˜ − K(t)D ˆ˙ ˆ + D(x; K(t))B K(t) (x; K(t)) X(t) ˆ K(t) ˜ ˆ˙ ˆ t) − D(x; K(t))B − K(t)R(x, 1 d(t), ηˆ(l(t), t) = 0,

(5.88) (5.89)

where ˜ = K − K(t), ˆ K(t)

(5.90)

ELEVATORS WITH FLEXIBLE GUIDES

107

and where  R(x, t) = 

x 0 x

= 0

ˆ t)dy ˆ ˆ (x, y; K(t)) ˆ N β(y, K(t)  ˆ ˆ (x, y; K(t)) ˆ N ηˆ(y, t) K(t)



−

y 0

 ˆ ˆ ˆ NI (y, σ; K(t))ˆ η (σ, t)dσ − DI (y; K(t))X(t) dy

(5.91)

and ˆ (x; K(t)) = DK(t) ˆ

ˆ ∂D(x; K(t)) , ˆ ∂ K(t)

ˆ (x, y; K(t)) ˆ ∂N ˆ ˆ (x, y; K(t)) ˆ N = . K(t) ˆ ∂ K(t)

(5.92) (5.93)

By matching (5.70)–(5.74) and (5.85)–(5.89) with the aid of (5.83) (the details are ˆ ˆ shown in appendix 5.7C), the conditions on the kernels N (x, y; K(t)), D(x; K(t)) in (5.83) are determined as follows: ˆ ˆ D(0; K(t)) = K(t), ˆ ˆ ˆ + D(x; K(t))(A −q2 (x)D (x; K(t)) m − c4 (x)In − B K(t))  x ¯ ˆ (x, y; K(t)) ˆ ¯ 0)q1 (0)Cdy = 0, +Υ(x, 0)q1 (0)C − N Υ(y,

(5.94)

(5.95)

0

ˆy (x, y; K(t)) ˆ ˆx (x, y; K(t)) ˆ + q2 (x)N q2 (y)N ˆ (x, y; K(t)) ˆ + q2  (y)N = 0,

(5.96)

ˆ (x, 0; K(t)) ˆ ˆ − D(x; K(t))B = 0. q2 (0)N

(5.97)

The equation set (5.94)–(5.97) is a transport PDE-ODE coupled system consisting of the transport PDE (5.96) with the boundary condition (5.97) on {(x, y)|0 ≤ y ≤ x ≤ l(t)} and the ODE (5.95) with the initial value (5.94) on {0 ≤ x ≤ l(t)}. It ˆ should be noted that K(t) is a parameter rather than a variable in the transport PDE (5.96), (5.97) with respect to the independent variables x, y and in the ODE (5.94), (5.95) with respect to the independent variable x when solving (5.94)–(5.97). To establish the well-posedness of (5.94)–(5.97), the transport PDE state ˆ (x, y; K(t)) ˆ ˆ N can be represented by its boundary value D(x; K(t))B. By substiˆ ˆ tuting the result into ODE (5.95) to replace N (x, y; K(t)), the solution of the ODE ˆ ˆ ˆ (x, y; K(t)) ˆ D(x; K(t)) can be obtained. The well-posedness of the transport PDE N (5.96), (5.97) is thus obtained because of the well-defined boundary condition (5.97). ˆI , DI in the inverse transformation The existence and continuity of the kernels N (5.84) are shown as follows. Rewrite (5.83) as  ˆ t) − ˆ ηˆ(x, t) + D(x; K(t))X(t) = β(x,

x 0

ˆ t)dy. ˆ (x, y; K(t)) ˆ N β(y,

(5.98)

CHAPTER FIVE

108

ˆ (x, y; K(t)) ˆ Because N is continuous, according to [169], there exists a unique conˆ tinuous (x, y; K(t)) on {(x, y)|0 ≤ x ≤ y ≤ l(t)} such that ˆ t) = ηˆ(x, t) + D(x; K(t))X(t) ˆ β(x,  x ˆ ˆ

(x, y; K(t))(ˆ η (y, t) + D(y; K(t))X(t))dy + 0



= ηˆ(x, t) + 

x

+ 0

x 0

ˆ

(x, y; K(t))ˆ η (y, t)dy

 ˆ ˆ ˆ

(x, y; K(t))D(y; K(t))dy + D(x; K(t)) X(t).

(5.99)

Comparing with (5.99) and the inverse transformation (5.84), we obtain the exisˆ ˆ ˆI (x, y; K(t)), DI (x; K(t)). tence and continuity of the kernels N Finally, for (5.89) to hold, recalling (5.83) and (5.74), we derive the boundary control input U (t), the expression for which is shown in section 5.5.

5.4

ADAPTIVE UPDATE LAWS

Using normalization and projection operators to guarantee boundedness, as is typical in adaptive control designs, the adaptive update laws for the self-tuned control gains ˆ = [kˆ1 (t), . . . , kˆn (t)] K(t) (5.100) and for the unknown parameters a ˆj (t), ˆbj (t), j ∈ {1, . . . , N } are built as   ˙ kˆi (t) = Proj[ki ,k¯i ] τi (t), kˆi (t) , a ˆ˙ j (t) = Proj[0,¯aj ] (τ1j (t), a ˆj (t)) ,   ˆb˙ j (t) = Proj ¯ τ2j (t), ˆbj (t) , [0,bj ]

(5.101) (5.102) (5.103)

i ∈ {1, . . . , n}, j ∈ {1, . . . , N }. For any m ≤ M and any r, p, Proj[m,M ] is the standard projection operator given by ⎧ ⎪ ⎨ 0, if p = m and r < 0, 0, if p = M and r > 0, Proj[m,M ] (r, p) = ⎪ ⎩ r, else. The role of the projection operator is to keep the parameter estimates bounded. The bounds k i , k¯i and a ¯j , ¯bj are defined in section 5.1. The functions τi , τ1j , τ2j in (5.101)–(5.103) are defined as [τ1 (t), . . . , τn (t)]T = γc

−2XB T P X + ra

 l(t) 0

T ˆ eδx ηˆ(x, t)XB T D(x; K(t)) dx , 1 + Ω(t)

(5.104)

ELEVATORS WITH FLEXIBLE GUIDES

τ1j (t)



= γaj τ2j (t) = γbj

2X T P B1 − ra

 l(t) 0

109

 ˆ eδx ηˆ(x, t)D(x; K(t))B 1 dx cos(θj t) 1 + Ω(t)



2X T P B1 − ra

 l(t) 0

 ˆ eδx ηˆ(x, t)D(x; K(t))B 1 dx sin(θj t) 1 + Ω(t)

,

(5.105)

,

(5.106)

where γc = diag{γc1 , . . . , γcn }. The positive update gains γci , γaj , γbj are to be chosen by the user, and   2c4 + q2 2c1 + 1 + q1 δ > max , q2 q1

(5.107)

(5.108)

with q1 = min {q1 (x)},

q1 = max {|q1 (x)|},

(5.109)

q2 = min {q2 (x)},

q2 = max {|q2 (x)|},

(5.110)

c1 = max {|c1 (x)|}, c4 = max {|c4 (x)|}.

(5.111)

0≤x≤L 0≤x≤L 0≤x≤L

0≤x≤L 0≤x≤L

0≤x≤L

The scalar Ω(t) is defined as  l(t) 1 eδx ηˆ(x, t)2 dx Ω(t) =X T P X(t) + ra 2 0  l(t) 1 + rb e−δx α ˆ (x, t)2 dx. 2 0

(5.112)

The determination of the positive constants ra , rb will be shown in the next section. The matrix P = P T > 0 is the unique solution to the following Lyapunov equation: P Am + ATm P = −Q

(5.113)

for some Q = QT > 0. Because Am in (5.16) is known (chosen by the user, in spite of A and K being unknown), the matrix P is known. The normalization Ω(t) + 1 is introduced in the denominator in (5.104)–(5.106) ˙ ˙ ˆ˙ j (t), ˆbj (t). to limit the rates of change of the parameter estimates—that is, kˆi (t) and a The functions ηˆ(·, t) and α ˆ (·, t) in (5.104)–(5.106), (5.112) can be represented by the observer states through (5.48), (5.49), (5.68), (5.69), (5.83). The idea of conˆj (t), ˆbj (t) i ∈ {1, . . . , n}, j ∈ {1, . . . , N } structing the adaptive update laws kˆi (t), a in (5.101)–(5.106) will be clear from the Lyapunov analysis in the next section.

CHAPTER FIVE

110 5.5

CONTROL LAW AND STABILITY ANALYSIS

Control Law According to (5.89), (5.83), and (5.74), U (t) is obtained as U (t) = −p2 sˆ(l(t), t) − (Γ(l(t), t) − p2 Γ1 (l(t), t))Z(t)  l(t)  l(t) ¯ ˇ Υ(l(t), y)ˆ s(y, t)dy + Υ(l(t), y)ˆ v (y, t)dy + 

0

0

l(t)

+ 0

ˆ t)dy + D(l(t); K(t))X(t). ˆ (l(t), y; K(t)) ˆ ˆ N β(y,

(5.114)

Recalling (5.83), (5.69), (5.48), (5.49), (5.26), the control law (5.114) is rewritten as U (t) = − p2 z(l(t), t) − Γ(l(t), t)Z(t)  l(t) ¯ Υ(l(t), y)(ˆ z (y, t) + Γ1 (y, t)Z(t))dy + 

0

l(t)

+ 

0 l(t)

+ 0 y

ˇ Υ(l(t), y)(w(y, ˆ t) + Γ(y, t)Z(t))dy  ˆ ˆ N (l(t), y; K(t)) w(y, ˆ t) + Γ(y, t)Z(t)

¯ σ)(ˆ Υ(y, z (σ, t) + Γ1 (σ, t)Z(t))dσ   y ˇ ˆ Υ(y, σ)(w(σ, ˆ t) + Γ(σ, t)Z(t))dσ dy + D(l(t); K(t))X(t), − −

0

(5.115)

0

which is the output-feedback adaptive controller sought in this chapter. The signals z(l(t), t), X(t) are measurements, and Z(t) is defined in (5.50). The states zˆ(x, t), w(x, ˆ t) are obtained from the observer (5.23)–(5.26). The functions Γ1 (y, t) and Γ(y, t) are solutions of (5.66), (5.67) where the adaptive estimates a ˆj (t), ˆbj (t) ¯ ˇ σ) are defined in (5.102), (5.103) and (5.105), (5.106). The functions Υ(y, σ), Υ(y, ¯ ˇ ˆ are solutions of (5.79)–(5.82). The functions Γ(l(t), t), Υ(l(t), y), Υ(l(t), y), N (l(t), y; ˆ ˆ K(t)), D(l(t); K(t)) are the solutions of (5.66), (5.67), (5.79)–(5.82), and (5.94)– ˆ (5.97) on x = l(t), respectively. The adaptive estimate K(t) is defined in (5.101), (5.104). The block diagram of the closed-loop system is shown in figure 5.2, whose stability result is given in the next subsection. Stability Analysis ˆ Lemma 5.2. For all initial data (ˆ α(·, 0), ηˆ(·, 0)) ∈ H 1 (0, L), X(0) ∈ Rn , K(0) ∈ Rn , a ˆj (0) ∈ R, ˆbj (0) ∈ R, j = 1, . . . , N , the target system (5.85)–(5.89) is asymptotically regulated in the sense of α(·, t) + ˆ η (·, t) + |X(t)|) = 0. lim ( ˆ

(5.116)

Θ(t) = ˆ η (·, t) 2 + ˆ α(·, t) 2 + |X(t)|2 ,

(5.117)

t→∞

Proof. Define

ELEVATORS WITH FLEXIBLE GUIDES

111 Disturbance

Control law

U(t)

Plant

Kˆ (t)

X(t), z(l(t), t)

g–i

Adaptive adjustment for ideal control parameters K

zˆ(x, t), wˆ (x, t)

X(t)

PDE state observer

aˆj(t), bˆj(t)

X(t), z(l(t), t)

Adaptive estimation of disturbance amplitudes aj, bj

X(t)

Figure 5.2. Block of the closed-loop system.

where · denotes the L2 norm. Recalling (5.112), we get μ1 Θ(t) ≤ Ω(t) ≤ μ2 Θ(t),

(5.118)

with positive μ1 , μ2 determined as 1 min{ra , rb e−δL , λmin (P )}, 2 1 μ2 = min{ra eδL , rb , λmax (P )}, 2

μ1 =

(5.119) (5.120)

where λmin and λmax denote the minimum and maximum eigenvalues of the corresponding matrix. Let us choose a Lyapunov function as V (t) = ln (1 + Ω(t)) +

N N   1 1 ˜ a ˜j (t)2 + bj (t)2 2γ 2γ aj bj j=1 j=1

1˜ T −1 ˜ K(t) , + K(t)γ c 2

(5.121)

˜ are given in (5.59), (5.90). Recalling (5.112), we rewrite the Lyawhere a ˜j , ˜bj , K punov function as 

 l(t) 1 eδx ηˆ(x, t)2 dx V (t) = ln X P X(t) + ra 2 0    l(t) N 1 1 −δx 2 e α ˆ (x, t) dx + 1 + a ˜j (t)2 + rb 2 2γ aj 0 j=1 T

N  1 ˜ 1˜ T −1 ˜ K(t) . + bj (t)2 + K(t)γ c 2γ 2 bj j=1

(5.122)

CHAPTER FIVE

112

Taking the derivative of V (t), through a lengthy calculation in appendix 5.7D we obtain    1 3 q1 (0)2 2 2 2 2 ˙ ¯ ¯ V (t) ≤ λmin (Q) − q1 (0)rb D − Lrb J |C| |X(t)| − 1 + Ω(t) 4 2   1 8 2 2 q2 (0)ra − q1 (0)rb p1 − |P B| ηˆ(0, t)2 − 2 λmin (Q)    l(t) 1 1 δq2 ra − ra c4 − ra q2 − eδx ηˆ(x, t)2 dx 2 2 0  1 −δL ˙ − q1 (l(t)) − l(t) rb e α ˆ (l(t), t)2 2     l(t) 1 1 1  −δx 2 q1 δrb − rb c1 − rb − rb q1 − e α(x, ˆ t) dx 2 2 2 0  ˆ˙ T ˜ − K(t) γc −1 K(t) − −

N 

 a ˜j (t)

j=1

−

N  j=1



 ˜bj (t) 

+ ra

0

 + rb

l(t)

l(t) 0

e

1 ˙ a ˆj (t) − γaj

 l(t)

 T ˆ eδx ηˆ(x, t)X(t)B T D(x; K(t)) dx 1 + Ω(t)  l(t) δx  ˆ (2X T P B1 − ra e ηˆ(x, t)D(x; K(t))B 1 dx) cos(θj t)

−2X(t)B T P X(t) + ra

0

0

1 + Ω(t)

(2X T P B1 − ra 1 ˆ˙ bj (t) − γbj

 l(t) 0

 ˆ eδx ηˆ(x, t)D(x; K(t))B 1 dx) sin(θj t) 1 + Ω(t)

  ˆ˙ ˆ X(t) − KR(x, ˆ˙ eδx ηˆ(x, t) Hb Z(t) + KD t) dx K(t) −δx

 α ˆ (x, t)Ha Z(t)dx

1 . 1 + Ω(t)

(5.123)

Regarding (5.101), (5.102), (5.103), we know that there exist positive constants m2 , m3 such that  2  2  ˙  ˙ 2 2 ˆj (t) , ˆbj (t) ≤ m2 max {γaj , γbj }(|X(t)|2 + ˆ η 2 ), (5.124) max a j∈{1,...,N }

   ˆ˙ 2 K(t) ≤ m3

j∈{1,...,N }

2 max {γci },

i∈{1,...,n}

(5.125)

where m2 , m3 only depend on the parameters of the plant and the coefficients introduced in the Lyapunov function. With (5.66), we obtain ! " |Γt (x, t)|2 , |Γ1t (x, t)|2 max x∈[0,L],t∈[0,∞)

≤ 2N

max

j∈{1,...,N }

≤ 2N m2

 2  2  ˙  ¯2 ˙ ˆj (t) , ˆbj (t) h a m

max

j∈{1,...,N }

2 2 ¯2 , {γaj , γbj }(|X(t)|2 + ˆ η 2 )h m

(5.126)

ELEVATORS WITH FLEXIBLE GUIDES

113

¯ m is defined as where h ¯ m = max {¯ ¯ h σ (h(x))}, 0≤x≤L

(5.127)

¯ and H(x) is the solution of (5.67), and σ ¯ stands for the largest singular value at x. According to the definitions of Ha , Hb in (5.154), (5.155), recalling (5.125), (5.126), there exist positive constants ξ1 , ξ2 such that max{|Ha Z(t)|2 , |Hb Z(t)|2 } ≤ ξ1 2    ˆ˙ X(t)  ≤ ξ2 KDK(t) ˆ

max

j∈{1,...,N }

2 2 {γaj , γbj }(|X(t)|2 + ˆ η 2 ),

2 max {γci }|X(t)|2 ,

i∈{1,...,n}

(5.128) (5.129)

with ξ1 , ξ2 depending only on the kernels, the parameters of the plant, and the coefficients used in the Lyapunov analysis but not on γci , γaj , γbj . Applying the Young and Cauchy-Schwarz inequalities, we obtain the inequality  ra

l(t) 0

 l(t) 1 eδx ηˆ(x, t)Hb Z(t)dx ≤ ra eδx ηˆ(x, t)2 dx 2 0 1 2 2 + ra eδL Lξ1 max {γaj , γbj }(|X(t)|2 + ˆ η 2 ), 2 j∈{1,...,N } (5.130)

where we have used (5.128); the inequality  −ra

l(t) 0

ˆ˙ ˆ X(t)dx eδx ηˆ(x, t)KD K(t)

1 ≤ ra 2



l(t) 0

1 2 eδx ηˆ(x, t)2 dx + ra eδL L max {γci }ξ2 |X(t)|2 , 2 i∈{1,...,n}

(5.131)

where we have used (5.129); and the inequality  −ra

l(t) 0

ˆ˙ eδx ηˆ(x, t)KR(x, t)dx

 l(t) 1 ≤ ra eδx ηˆ(x, t)2 dx 2 0 1 2 + ra eδL m3 max {γci }ξ3 (|X(t)|2 + ˆ η 2 ) 2 i∈{1,...,n}

(5.132)

for which we have employed (5.125), (5.91). The positive constant ξ3 in (5.132) only ˆI, N ˆ ˆ . Finally, we also obtain the inequality ˆI , D depends on kernels N K(t)  rb

l(t) 0

≤

rb 2

e−δx α ˆ (x, t)Ha Z(t)dx 

l(t) 0

e−δx α ˆ (x, t)2 dx

1 2 2 + rb Lξ1 max {γaj , γbj }(|X(t)|2 + ˆ η 2 ) 2 j∈{1,...,N }

(5.133)

CHAPTER FIVE

114 by applying (5.128). Applying (5.130)–(5.133), we obtain  ra

l(t) 0

  ˆ˙ ˆ X(t) − KR(x, ˆ˙ eδx ηˆ(x, t) Hb Z(t) − KD t) dx K(t)

 l(t) + rb e−δx α(x, ˆ t)Ha Z(t)dx 0  1 1 2 2 2 ≤ ra eδL L max {γci }ξ2 + ra eδL Lξ1 max {γaj , γbj } 2 2 i∈{1,...,n} j∈{1,...,N } 1 2 + ra eδL m3 max {γci }ξ3 L 2 i∈{1,...,n}  1 2 2 + rb Lξ1 max {γaj , γbj } |X(t)|2 2 j∈{1,...,N }  3 1 2 2 + ra eδL + ra eδL Lξ1 max {γaj , γbj } 2 2 j∈{1,...,N } 1 2 + ra eδL m3 max {γci }ξ3 L 2 i∈{1,...,n}  1 rb 2 2 α 2 + rb Lξ1 max {γaj , γbj } ˆ η 2 + ˆ 2 2 j∈{1,...,N }   2 2 2 2 ≤ max {γci , γaj , γbj }λb |X(t)| + ˆ η 2 + ˆ α 2 , i∈{1,...,n},j∈{1,...,N }

(5.134)

where λb > 0 depends only on the kernels, the parameters of the plant, and the coefficients used in the Lyapunov analysis. Choosing 3 λmin (Q) ,  24 q1 (0)2 2 ¯ + ¯2 q1 (0)D 2 LJ |C|   2 8 2 |P B| , ra > q1 (0)rb p21 − q2 (0) λmin (Q)

rb
0, is exponentially convergent to zero, where the decay rate of the vibrational energy is adjustable with the control parameters. Theorem 6.1. For all initial values (w(x, 0), wt (x, 0)) ∈ H 2 (0, L) × H 1 (0, L), (u(x, 0), ut (x, 0)) ∈ H 2 (0, L) × H 1 (0, L), the closed-loop system consisting of the plant (6.32)–(6.37) and the state-feedback control law (6.104), (6.105) is exponentially stable in the sense that there exist positive constants Υ1 , σ1 such that 

wt (·, t) 2 + wx (·, t) 2 + ut (·, t) 2 + ux (·, t) 2 1/2 + w(0, t)2 + wt (0, t)2 + u(0, t)2 + ut (0, t)2  ≤Υ1 wt (·, 0) 2 + wx (·, 0) 2 + ut (·, 0) 2 + ux (·, 0) 2 + w(0, 0)2 + wt (0, 0)2 + u(0, 0)2 + ut (0, 0)2 where u(·, t) 2 is a compact notation for is adjustable by the control parameters.

 l(t) 0

1/2

e−σ1 t ,

(6.106)

u(x, t)2 dx. The convergence rate σ1

DEEP-SEA CONSTRUCTION

145

Proof. We start by studying the stability of the target system (6.77)–(6.81). The equivalent stability property between the target system (6.77)–(6.81) and the original system (6.32)–(6.37) is ensured via the definitions (6.39)–(6.42), (6.58)–(6.60) and the backstepping transformations (6.72), (6.73) and (6.101), (6.102). Consider the following Lyapunov function for the target system (6.77)–(6.81):  1 l(t) δ2 x T −1 e β(x, t) Ra Q(x) β(x, t)dx V1 = W (t)P1 W (t) + 2 0  1 l(t) −δ1 x T −1 + e α(x, t) Rb Q(x) α(x, t)dx, 2 0 T

(6.107)

where the positive definite matrix P1 = P1T is the solution to the Lyapunov equaˆ 1 , for some Q ˆ1 = Q ˆ T > 0, and Ra , Rb are diagonal matrices tion P1 Aˆ + AˆT P1 = −Q 1 given as Ra = diag{ra1 , ra2 }, Rb = diag{rb1 , rb2 }. (6.108) The positive parameters ra1 , ra2 , rb1 , rb2 , δ1 , δ2 are to be chosen later. According to (6.107), we get μ1 Ω(t) ≤ V1 (t) ≤ μ2 Ω(t),

(6.109)

Ω(t) = |W (t)|2 + β(x, t) 2 + α(x, t) 2

(6.110)

where

and μ1 , μ2 are some positive constants. The time derivative of V1 (t) along (6.77)– (6.81) is obtained as ˙ (t) ˙ T (t)P1 W (t) + W T (t)P1 W V˙ 1 = W  l(t) T −1 + eδ2 x β(x, t) Ra Q(x) βt (x, t)dx 

0

l(t)

+ 0

e−δ1 x α(x, t) Rb Q(x) T

−1

αt (x, t)dx

˙ l(t) e−δ1 l(t) α(l(t), t)T Rb Q(l(t))−1 α(l(t), t) 2 ˆ (t) + 4W T (t)P1 Bβ(0, ¯ t) = W (t)T (AˆT P1 + P1 A)W +

˙ l(t) e−δ1 l(t) α(l(t), t)T Rb Q(l(t))−1 α(l(t), t) 2  l(t) T −1 eδ2 x β(x, t) Ra Q(x) T¯a (x)β(x, t)dx +

+



0

l(t)

+ 0

T

eδ2 x β(x, t) Ra Q(x)

−1

g(x)β(0, t)dx

 1 δ2 l(t) δ2 x T − β(0, t)T Ra β(0, t) − e β(x, t) Ra β(x, t)dx 2 2 0 1 1 − e−δ1 l(t) α(l(t), t)T Rb α(l(t), t) + α(0, t)T Rb α(0, t) 2 2

CHAPTER SIX

146  δ1 l(t) −δ1 x T e α(x, t) Rb α(x, t)dx 2 0  l(t) T −1 + e−δ1 x α(x, t) Rb Q(x) T¯b (x)α(x, t)dx

−



0

l(t)

+ 0

e−δ1 x α(x, t) Rb Q(x) T

−1

g1 (x)β(0, t)dx.

(6.111)

Applying Young’s inequality, by virtue of the boundedness of the elements √ √

1 , d6 (x)

1 , d1 (x)

ga (x), gb (x) in the matrices Q(x)−1 , g(x), g1 (x) yields ξ > 0 such that the

following inequalities hold: 

l(t) 0



≤ξ

l(t)

0 l(t)

 0



≤ξ

T

eδ2 x β(x, t) Ra Q(x)

g(x)β(0, t)dx 

l(t)

T

eδ2 x β(x, t) Ra β(x, t)dx + ξ

e−δ1 x α(x, t) Rb Q(x) T

l(t) 0

−1

e

−δ1 x

−1

0

 α(x, t) Rb α(x, t)dx + ξ  Λa =

ra2 0

(6.112)

β(0, t) Λb β(0, t)dx,

(6.113)

g1 (x)β(0, t)dx l(t)

T

where

T

eδ2 L β(0, t) Λa β(0, t)dx,

0 0

0



, Λb =

T

rb2 0

0 0

.

(6.114)

Inserting (6.112), (6.113) and applying Young’s inequality into (6.111), one obtains  Rb 1 2 T Ra ˙ − V1 (t) ≤ − λmin (Q2 )|W (t)| − β(0, t) 2 2 2  2   l(t) ¯1  8P1 B T δ2 L I2 − e ξΛa − ξΛb β(0, t) − eδ2 x β(x, t) − λmin (Q2 ) 0    l(t) δ2 −1 ¯ × Ra ( − ξ)I2 − Q(x) Ta (x) β(x, t)dx − e−δ1 x 2 0   δ1 T −1 ¯ × α(x, t) Rb ( − ξ)I2 − Q(x) Tb (x) α(x, t)dx 2   1 −δ1 l(t) ˙ α(l(t), t)T Rb I2 − Q(l(t))−1 l(t) − e α(l(t), t), (6.115) 2 where I2 is a 2 × 2 identity matrix. The parameters ra1 , ra2 , rb1 , rb2 , δ1 , δ2 are chosen to satisfy   ¯ 1 2 16P1 B ra1 > rb1 + + 2ra2 eδ2 L ξ + 2rb2 ξ, λmin (Q2 )   ¯ 1 2 16P1 B ra2 > rb2 + λmin (Q2 )

(6.116)

(6.117)

DEEP-SEA CONSTRUCTION

147

with sufficiently large δ1 , δ2 . The positive constants rb1 , rb2 can be arbitrary. We ˙ are smaller than unity know that the elements in the diagonal matrix Q(l(t))−1 l(t) by recalling assumption 6.4. Additionally, the boundedness of all elements in the diagonal matrix Q(x)−1 , T¯a (x), T¯b (x) is assured by recalling assumption 6.1. We thus arrive at V˙ 1 (t) ≤ −η1 V1 (t) (6.118) for some positive η1 . It then follows that V1 (t) ≤ V1 (0)e−η1 t ,

(6.119)

μ2 Ω(0)e−η1 t μ1

(6.120)

and hence, Ω(t) ≤

by recalling (6.109). In sum, we have obtained exponential stability in Ω(t). Establishing the relationship between the Ω(t) and the appropriate norm of the u(x, t), w(x, t)-system is the key to establishing exponential stability in the original variables. Defining Ξ(t) = ux (·, t) 2 + ut (·, t) 2 + |u(0, t)|2 + |ut (0, t)|2 + wx (·, t) 2 + wt (·, t) 2 + |w(0, t)|2 + |wt (0, t)|2 ,

(6.121)

recalling (6.39)–(6.42), (6.43)–(6.60), (6.101), (6.102), and applying the CauchySchwarz inequality, the following inequality holds θ¯1a Ξ(t) ≤ Ω(t) ≤ θ¯1b Ξ(t)

(6.122)

for some positive θ¯1a and θ¯1b . Therefore, we get Ξ(t) ≤

μ2 θ¯1b Ξ(0)e−η1 t . μ1 θ¯1a

(6.123)

μ2 θ¯1b , μ1 θ¯1a

(6.124)

Thus, (6.106) is achieved with  Υ1 =

σ1 =

η1 , 2

where the convergence rate σ1 can be adjusted by the control parameter κ through ˆ 1 ). With this, the proof of theorem 6.1 is complete. affecting λmin (Q

Next, we prove the exponential convergence of the control input. We state first a lemma that shows the exponential stability result for the closed-loop system in the sense of the H 2 norm. Lemma 6.2. With arbitrary initial data (w(x, 0), wt (x, 0)) ∈ H 2 (0, L) × H 1 (0, L), (u(x, 0), ut (x, 0)) ∈ H 2 (0, L) × H 1 (0, L), the exponential stability of the closed-loop system consisting of the plant (6.32)–(6.37) and the state-feedback control law (6.104),

CHAPTER SIX

148

(6.105) holds in the sense that there exist positive constants Υ1a and σ1a such that

1

uxx (·, t) 2 + wxx (·, t) 2 + utx (·, t) 2 + wtx (·, t) 2 2  ≤ Υ1a ux (·, 0) 2 + wx (·, 0) 2 + ut (·, 0) 2 + wt (·, 0) 2 + uxx (·, 0) 2 + wxx (·, 0) 2 + utx (·, 0) 2 + wtx (·, 0) 2  +|w(0, 0)|2 + |wt (0, 0)|2 + |u(0, 0)|2 + |ut (0, 0)|2 e−σ1a t .

(6.125)

Proof. Taking the spatial and time derivatives of (6.78)–(6.79) and (6.126), (6.129), (6.130), respectively, we get ¯¯ ¨ (t) = AˆW ˙ (t) + 2BQ(0)β ¯ W x (0, t) + 2B Ta (0)β(0, t),

(6.126)

αxt (x, t) = − Q(x)αxx (x, t) + (T¯b (x) − Q (x))αx (x, t) + T¯b (x)α(x, t),

(6.127)

βxt (x, t) = Q(x)βxx (x, t) + (T¯a (x) + Q (x))βx (x, t) + T¯a (x)β(x, t),

(6.128)

Q(0)αx (0, t) = Q(0)βx (0, t) + T¯a (0)β(0, t) + T¯b (0)α(0, t),

(6.129)

βx (l(t), t) = 0,

(6.130)

where (6.79), (6.81) have been used. Equation (6.130) results from ˙ + Q(l(t)))βx (l(t), t) = 0, (l(t)

(6.131)

˙ + Q(l(t)) are nonzero for all t by where the elements in the diagonal matrix l(t) recalling assumption 6.4. Define new variables ˙ (t). (x, t) = αx (x, t), ζ(x, t) = βx (x, t), Z(t) = W

(6.132)

Consider a Lyapunov function as V2 (t) = R1 V1 (t) + Z(t)T P1 Z(t)  1 l(t) δ¯1 x T ¯ −1 + e ζ(x, t) R ζ(x, t)dx a Q(x) 2 0  1 l(t) −δ¯2 x ¯ b Q(x)−1 (x, t)dx, e (x, t)T R + 2 0

(6.133)

¯ b are diagonal matrices given as ¯a, R where R ¯ a = diag{¯ ¯ b = diag{¯ R ra1 , r¯a2 }, R rb1 , r¯b2 }. The parameters r¯a1 , r¯a2 , r¯b1 , r¯b2 , δ¯1 , δ¯2 , and R1 are positive. Taking the derivative of (6.133) along (6.126)–(6.130), recalling (6.115), determining r¯a1 , r¯a2 , r¯b1 , r¯b2 , δ¯1 , δ¯2 through a process similar to that in (6.111)–(6.119), and choosing large enough positive constant R1 , we arrive at V˙ 2 (t) ≤ −η2 V2 (t)

(6.134)

DEEP-SEA CONSTRUCTION

149

for some positive η2 . Recalling the backstepping transformations (6.101), (6.102), we obtain

px (·, t) 2 + rx (·, t) 2 ≤Υ1b W (0)2 + p(·, 0) 2 + r(·, 0) 2  + px (·, 0) 2 + rx (·, 0) 2 e−η2 t (6.135) for some positive Υ1b . Applying (6.39)–(6.42), (6.58), (6.59), the proof of lemma 6.2 is complete. Theorem 6.2. In the closed-loop system (6.32)–(6.37), (6.104), (6.105), the statefeedback control signals U1 (t), U2 (t) in (6.104), (6.105) are bounded and exponentially convergent to zero in the sense that there exist positive constants σ2 and Υ2 such that  2 2 |U1 (t)| + |U2 (t)| ≤Υ2 ux (·, 0) 2 + wx (·, 0) 2 + ut (·, 0) 2 + wt (·, 0) 2 + uxx (·, 0) 2 + wxx (·, 0) 2 + utx (·, 0) 2 + wtx (·, 0) 2 + |w(0, 0)|2 + |wt (0, 0)|2  + |u(0, 0)|2 + |ut (0, 0)|2 e−σ2 t . (6.136) Proof. According to (6.103) and the exponential stability result in theorem 6.1, we know that once   p(l(t), t) = [ut (l(t), t) − d6 (l(t))ux (l(t), t), wt (l(t), t) − d1 (l(t))wx (l(t), t)]T is established to be exponentially convergent to zero in the sense of |p(l(t), t)|2 , the exponential convergence of the control input is obtained. Applying the Cauchy-Schwarz inequality and recalling (6.35), (6.36), we get √ |p(l(t), t)| ≤ 2|p(0, t)| + 2 L px (·, t)

√ ≤ 4|r(0, t)| + 4|C3 W (t)| + 2 L px (·, t)

√ √ ≤ 8|r(l(t), t)| + 8 L rx (·, t) + 4|C3 W (t)| + 2 L px (·, t) . (6.137) Recalling (6.102), (6.81) and the exponential convergence of α(·, t) 2 , β(·, t) 2 , |W (t)|2 proved in theorem 6.1, we have |r(l(t), t)| as exponentially convergent to zero. Recalling lemma 6.2, we thus have |p(l(t), t)| as exponentially convergent to zero. The proof of theorem 6.2 is complete.

6.3

OBSERVER FOR TWO-DIMENSIONAL OSCILLATIONS OF THE CABLE

Observer Structure We consider only sensors placed at the actuated boundary (ship-mounted crane). An observer should then be designed to estimate the states w, u in the domain and at the uncontrolled boundary—that is, the lateral-longitudinal coupled vibration

CHAPTER SIX

150

states of the cable and the attached payload. These estimates are required in the state-feedback control laws (6.104), (6.105). The available measurements are ut (l(t), t), wt (l(t), t), which makes p(l(t), t) known through the conversion  p(l(t), t) = [ut (l(t), t) − d6 (l(t))d19 (l(t))U1 (t),  (6.138) wt (l(t), t) − d1 (l(t))d20 (l(t))U2 (t)] by recalling (6.36), (6.37), (6.58), (6.39), and (6.42). Using the known signal p(l(t), t), the observer for the coupled wave PDE plant (6.32)–(6.37) is constructed as 1 w ˆt (x, t) = (ˆ z (x, t) + vˆ(x, t)), 2 1 w ˆx (x, t) =  (ˆ z (x, t) − vˆ(x, t)), 2 d1 (x) 1 ˆ u ˆt (x, t) = (k(x, t) + yˆ(x, t)), 2 1 ˆ t) − yˆ(x, t)), u ˆx (x, t) =  (k(x, 2 d6 (x)

(6.139) (6.140) (6.141) (6.142)

px (x, t) = Ta (x)ˆ r(x, t) + Tb (x)ˆ p(x, t) pˆt (x, t) + Q(x)ˆ + Γ1 (x, t)(p(l(t), t) − pˆ(l(t), t)),

(6.143)

rˆt (x, t) − Q(x)ˆ rx (x, t) = Ta (x)ˆ r(x, t) + Tb (x)ˆ p(x, t) + Γ2 (x, t)(p(l(t), t) − pˆ(l(t), t)), ˆ (t) − rˆ(0, t), pˆ(0, t) = C3 W

(6.144)

ˆ˙ (t) = (A¯ − BC ¯ 3 )W ˆ (t) + 2Bˆ ¯ r(0, t) W + Γ3 (t)(p(l(t), t) − pˆ(l(t), t)), rˆ(l(t), t) = R(l(t))U (t) + p(l(t), t),

(6.145)

(6.146) (6.147)

where pˆ = [ˆ y (x, t), vˆ(x, t)]T , ˆ t), zˆ(x, t)]T , rˆ = [k(x, ˆ (t) = [X(t), ˆ W Yˆ (t)]T = [w(0, ˆ t), w ˆt (0, t), u ˆ(0, t), u ˆt (0, t)]T .

(6.148) (6.149) (6.150)

The observer consists of two parts: 1. The system (6.143)–(6.147), as a copy of plant (6.61)–(6.65) with the addition of output injection, is to estimate p(x, t), r(x, t). 2. Once p(x, t), r(x, t) are estimated successfully by (6.143)–(6.147), the estimates of the original plant are simply obtained as (6.139)–(6.142) by virtue of the Riemann transformations (6.39)–(6.42). Next, the observer gains Γ1 (x, t), Γ2 (x, t), and Γ3 (t) are determined, in order to achieve exponential stability of the observer error system, which is done in the next subsection. A difference from the traditional observer gain functions should be noted. The gains Γ1 , Γ2 depend not only on the spatial variable x but also on time t because of the time-varying domain.

DEEP-SEA CONSTRUCTION

151

Observer Error System The observer’s task is to ensure that the observer errors (differences between the estimated and real states) are driven to zero, which is to be achieved by designing adequate observer gains. Denote the observer errors as w ˜t (x, t) = wt (x, t) − w ˆt (x, t),

(6.151)

w ˜x (x, t) = wx (x, t) − w ˆx (x, t), u ˜t (x, t) = ut (x, t) − u ˆt (x, t),

(6.152) (6.153)

u ˜x (x, t) = ux (x, t) − u ˆx (x, t), ˜ ˆ (t) W (x, t) = W (x, t) − W

(6.154)

ˆ = [X(t), Y (t)] − [X(t), Yˆ (t)] = [w(0, t), wt (0, t), u(0, t), ut (0, t)]T − [w(0, ˆ t), w ˆt (0, t), u ˆ(0, t), u ˆt (0, t)]T ˜ = [X(t), Y˜ (t)] ˜(0, t), u ˜t (0, t)]T , = [w(0, ˜ t), w ˜t (0, t), u

(6.155)

p˜(x, t) = p(x, t) − pˆ(x, t) = [˜ y (x, t), v˜(x, t)], ˜ t), z˜(x, t)]. r˜(x, t) = r(x, t) − rˆ(x, t) = [k(x,

(6.156) (6.157)

Recalling (6.61)–(6.65), (6.39)–(6.42), and (6.139)–(6.147), the resulting observer error dynamics are given by 1 w ˜t (x, t) = (˜ z (x, t) + v˜(x, t)), 2 1 w ˜x (x, t) =  (˜ z (x, t) − v˜(x, t)), 2 d1 (x) 1 ˜ t) + y˜(x, t)), u ˜t (x, t) = (k(x, 2 1 ˜ t) − y˜(x, t)), u ˜x (x, t) =  (k(x, 2 d6 (x)

(6.158) (6.159) (6.160) (6.161)

p˜t (x, t) + Q(x)˜ px (x, t) = Ta (x)˜ r(x, t) p(x, t) + Γ1 (x, t)˜ p(l(t), t), + Tb (x)˜

(6.162)

rx (x, t) = Ta (x)˜ r(x, t) r˜t (x, t) − Q(x)˜ p(x, t) + Γ2 (x, t)˜ p(l(t), t), + Tb (x)˜ ˜ (t) − r˜(0, t), p˜(0, t) = C3 W

(6.163) (6.164)

˜˙ (t) = (A¯ − BC ¯ 3 )W ˜ (t) + 2B˜ ¯ r(0, t) W p(l(t), t), + Γ3 (t)˜ r˜(l(t), t) = 0,

(6.165) (6.166)

where the subsystem (6.162)–(6.166) describing the dynamics of the observer error of the system (6.61)–(6.65) determines the observer error of the plant (6.32)–(6.37) via (6.158)–(6.161). Therefore, the exponential stability of (6.162)–(6.166) is the key to making sure that the proposed observer is exponentially convergent to the actual states of the original plant (6.32)–(6.37).

CHAPTER SIX

152 Observer Backstepping Design

To find the observer gains Γ1 (x, t), Γ2 (x, t), Γ3 (t) that guarantee that (6.162)–(6.166) is exponentially stable, we use a transformation to map (6.162)–(6.166) to a target observer error system whose exponential stability result is straightforward to obtain. The transformation is introduced as  l(t) ϕ(x, ¯ y)˜ α(y, t)dy, (6.167) p˜(x, t) = α(x, ˜ t) −  ˜ t) − r˜(x, t) = β(x,

x l(t)

 ˜ (t) = S(t) ˜ + W

x l(t)

0

¯ y)˜ ψ(x, α(y, t)dy,

¯ α(y, t)dy, K(y)˜

(6.168) (6.169)

where the kernels ¯ y) = {ψ¯ij (x, y)}1≤i,j≤2 ϕ(x, ¯ y) = {ϕ¯ij (x, y)}1≤i,j≤2 , ψ(x, on a triangular domain D1 = {0 ≤ x ≤ y ≤ l(t)} and ¯ ¯ ij (y)}1≤i≤4,1≤j≤2 , K(y) = {K are to be determined. The target observer error system is set up as ˜ t) + T¯b (x)˜ α ˜ t (x, t) + Q(x)˜ αx (x, t) = Ta (x)β(x, α(x, t)  l(t) ˜ t)dy, ¯ (x, y)β(y, M + x l(t)

 β˜t (x, t) − Q(x)β˜x (x, t) =

˜ t)dy + Ta (x)β(x, ˜ t), ¯ (x, y)β(y, N

x



˜ t) + ˜ − β(0, α ˜ (0, t) = C3 S(t) ˜ β(l(t), t) = 0, ˜ t) + ˜˙ = AˇS(t) ˜ +E ˇ β(0, S(t)

(6.170) (6.171)

l(t)

H(y)˜ α(y, t)dy,

0

(6.172) (6.173)



l(t) 0

˜ t)dy, G(y)β(y,

(6.174)

where the matrix ¯ 3 − L0 C 3 Aˇ = A¯ − BC is made Hurwitz by choosing L0 = {L0ij }1≤i≤4,1≤j≤2 ¯ (x, y), N ¯ (x, y) satisfy and recalling assumption 6.5, and where M  y ¯ ¯ (z, y)dz + ϕ(x, M (x, y) = ϕ(x, ¯ z)M ¯ y)Ta (y), x  y ¯ z)M ¯ y)Ta (y). ¯ (x, y) = ¯ (z, y)dz + ψ(x, N ψ(x, x

(6.175) (6.176)

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153

The function H(y) = {hij (y)}1≤i,j≤2 in (6.172) is a strictly lower triangular matrix given by   0 0 (6.177) H(y) = ¯ 21 (y) 0 , ψ¯2,1 (0, y) + ϕ¯2,1 (0, y) + K ˇ in (6.174) are and G(y), E  ¯ y)T¯a (y) − G(y) = {Gij (y)}1≤i≤4,1≤j≤2 = −K(0,

y 0

¯ z)M ¯ (z, y)dz, K(0,

ˇ = {E ˇij }1≤i≤4,1≤j≤2 = L0 + 2B. ¯ E The exponential stability of the target system (6.170)–(6.174) will be given in lemma 6.4. By matching (6.162)–(6.166) and (6.170)–(6.174) through the transformation (6.167)–(6.169), the conditions on the kernels in (6.167)–(6.169) and the observer ¯ y), gains in (6.143), (6.144), (6.146) are obtained as follows. Kernels ϕ(x, ¯ y), ψ(x, ¯ K(y) should satisfy the matrix equations − ϕ¯y (x, y)Q(y) − Q(x)ϕ¯x (x, y) − ϕ(x, ¯ y)Q (y) ¯ y) + Tb (x)ϕ(x, ¯ y) − ϕ(x, ¯ y)T¯b (y) = 0, + Ta (x)ψ(x,

(6.178)

¯ y)Q (y) − ψ¯y (x, y)Q(y) + Q(x)ψ¯x (x, y) − ψ(x, ¯ y) − ψ(x, ¯ y)T¯b (y) + Tb (x)ϕ(x, + Ta (x)ψ(x, ¯ y) = 0,

(6.179)

Q(x)ϕ(x, ¯ x) − ϕ(x, ¯ x)Q(x) − Tb (x) + T¯b (x) = 0, ¯ x) + ψ(x, ¯ x)Q(x) + Tb (x) = 0, Q(x)ψ(x,

(6.180)

¯ y) + ϕ(0, ¯ − H(y) = 0, ψ(0, ¯ y) + C3 K(y)  ¯ (y)Q(y) + (A¯ − BC ¯ 3 − L0 C3 )K(y) ¯ −K

(6.182)

(6.181)

 ¯ − K(y)[Q (y) + T¯b (y)] − L0 ϕ(0, ¯ y) ¯ y) = 0, ¯ + L0 )ψ(0, − (2B

¯ K(0) = L0 Q(0)

(6.183) −1

,

(6.184)

and the observer gains are obtained as ˙ ϕ(x, Γ1 (x, t) = l(t) ¯ l(t)) − ϕ(x, ¯ l(t))Q(l(t)), ˙ ψ(x, ¯ l(t)) − ψ(x, ¯ l(t))Q(l(t)), Γ2 (x, t) = l(t)

(6.186)

˙ K(l(t)) ¯ ¯ Γ3 (t) = l(t) − K(l(t))Q(l(t)).

(6.187)

(6.185)

Lemma 6.3. After adding an additional artificial boundary condition ϕ¯21 (x, L) = 0 ¯ the matrix equations (6.178)–(6.184) have a for the element ϕ¯21 in the matrix ϕ, ¯ ∈ L∞ ([0, l(t)]). unique solution ϕ, ¯ ψ¯ ∈ L∞ (D1 ), K Proof. After swapping the positions of the arguments in B.9–B.10 in [6]—that is, by changing the domain D1 to D—(6.178)–(6.184) has the analogous form with kernels F (x, y), N (x, y), λ(y) in (6.87)–(6.93). Following the steps in the proof of lemma 6.1, including the introduction of the extended domain D0 and the addition of the additional artificial boundary condition, lemma 6.3 is obtained.

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154

Following similar steps as above, the inverse transformation of (6.167)–(6.169) is determined as  l(t) ϕ(x, ˇ y)˜ p(y, t)dy, (6.188) α(x, ˜ t) = p˜(x, t) − 

x l(t)

˜ t) = r˜(x, t) − β(x,  ˜ =W ˜ (t) + S(t)

ˇ y)˜ ψ(x, p(y, t)dy,

(6.189)

x l(t) 0

ˇ r(y, t)dy, K(y)˜

(6.190)

ˇ y) ∈ R2×2 , and K(y) ˇ where ϕ(x, ˇ y) ∈ R2×2 , ψ(x, ∈ R4×2 are kernels on D1 and 0 ≤ y ≤ l(t), respectively. Stability Analysis of Observer Error System Before showing the performance of the proposed observer on tracking the actual states in the original plant (6.32)–(6.37) in the next theorem, the stability result of the observer error subsystem (6.162)–(6.166), which dominates the observer errors of the original plant (6.32)–(6.37), is given in the following lemma.

Lemma 6.4. For the observer error subsystem (6.162)–(6.166), there exist positive constants Υ3 , σ3 such that   1   ˜ 2 2 (t) r(·, t) 2 + W

˜ p(·, t) 2 + ˜  2  12   ˜ (0) ≤ Υ3 ˜ r(·, 0) 2 + W e−σ3 t . p(·, 0) 2 + ˜

(6.191)

Proof. Expanding (6.170)–(6.174) as α ˜ = [˜ α1 , α ˜ 2 ]T , β˜ = [β˜1 , β˜2 ]T , one obtains αix (x, t) = α ˜ it (x, t) + Qi (x)˜

2 

Taij (x)β˜j (x, t) + T¯bi (x)˜ αi (x, t)

j=1



l(t)

+ x



l(t)

β˜it (x, t) − Qi (x)β˜ix (x, t) =

x

+

2 

2 

¯ ij (x, y)β˜j (y, t)dy, M

(6.192)

j=1 2 

¯ij (x, y)β˜j (y, t)dy N

j=1

Taij (x)β˜j (x, t),

(6.193)

j=1

˜ − β˜i (0, t) α ˜ i (0, t) = C3 S(t)  l(t) h21 (y)˜ α1 (y, t)dy, + (i − 1)

(6.194)

0

β˜i (l(t), t) = 0

(6.195)

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155

˜ is governed by for i = 1, 2, and S(t) ˜˙ = AˇS(t) ˜ + E[ ˇ β˜1 (0, t), β˜2 (0, t)]T + S(t)



l(t) 0

G(y)[β˜1 (y, t), β˜2 (y, t)]T dy.

(6.196)

In (6.192)–(6.196), β˜i (·, t) are independent, and β˜i (·, t) ≡ 0 after a finite time because ˜ is exponentially convergent to zero because Aˇ is Hurwitz. The of (6.195). Thus, S(t) signals α ˜ 1 (·, t) are exponentially convergent to zero because of the exponential con˜ 1 (·, t) flow into α ˜ 2 (0, t) vergence of α ˜ 1 (0, t) due to (6.194) for i = 1. The signals α through the boundary (6.194), where the exponential convergence of α ˜ 2 (0, t) is also obtained for i = 2 because all signals on the right-hand side of the equal sign are exponentially convergent to zero. It follows that α ˜ 2 (·, t) are exponentially convergent to zero as well. The exponential stability result is seen more clearly by using the Lyapunov function  rˇb1 l(t) −δˇ1 x T Ve (t) = e α ˜ 1 (x, t) Q1 (x)−1 α ˜ 1 (x, t)dx 2 0  rˇa1 l(t) δˇ2 x ˜ T + e β1 (x, t) Q1 (x)−1 β˜1 (x, t)dx 2 0  rˇa2 l(t) δˇ2 x ˜ T ˜ ˜ T P2 S(t) + e β2 (x, t) Q2 (x)−1 β˜2 (x, t)dx + S(t) 2 0  rˇb2 l(t) −δˇ1 x T + e α ˜ 2 (x, t) Q2 (x)−1 α ˜ 2 (x, t)dx, (6.197) 2 0 where a positive definite matrix P2 = P2T is the solution to the Lyapunov equation ˆ2 P2 Aˇ + AˇT P2 = −Q ˆ2 = Q ˆ T > 0, and rˇa1 , rˇa2 , rˇb1 , rˇb2 , δˇ1 , δˇ2 are positive constants. The followfor some Q 2 ing inequality holds μe1 Ωe (t) ≤ Ve (t) ≤ μe2 Ωe (t)

(6.198)

for some positive μe1 , μe2 , where 2  ˜ t) 2 + S(t) ˜  , Ωe (t) = ˜ α(·, t) 2 + β(·,

(6.199)

and

˜ α(·, t) 2 =



l(t) 0

α ˜ 1 (·, t)2 dx +



l(t) 0

α ˜ 2 (·, t)2 dx.

(6.200)

Taking the derivative of (6.197) along (6.192)–(6.196) and choosing rˇa1 , rˇa2 , rˇb1 , rˇb2 , δˇ1 , δˇ2 in a process similar to that in (6.111)–(6.119), we obtain V˙ e (t) ≤ −ηe Ve (t)

(6.201)

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156

for some positive ηe , which is associated with the choice of L0 . The exponential stability result follows in the sense of 1 ˜ t) 2 + |S(t)| ˜ 2 2

˜ α(x, t) 2 + β(x,  1 2 2 −ηe t ˜ 0) 2 + |S(0)| ˜ α(x, 0) 2 + β(x, ≤ ξe ˜ e 

(6.202)

for some positive ξe and ηe . Recalling the direct and inverse backstepping transformations (6.167)–(6.169), (6.188)–(6.190), and applying the Cauchy-Schwarz inequality, the proof of lemma 6.4 is complete. Applying the exponential stability result of the observer error subsystem (6.162)– (6.166) in lemma 6.4 and recalling the relationships (6.158)–(6.161), we acquire the following theorem about the performance of the observer on tracking the actual states in the original plant (6.32)–(6.37). Theorem 6.3. For the observer error system (6.158)–(6.166) with the observer gains Γ1 (x, t) in (6.185), Γ2 (x, t) in (6.186), and Γ3 (t) in (6.187), with arbitrary initial data (w(x, 0), wt (x, 0)) ∈ H 2 (0, L) × H 1 (0, L), (u(x, 0), ut (x, 0)) ∈ H 2 (0, L) × H 1 (0, L), there exist positive constants Υ4 , σ4 such that  ux (·, t) 2 + w ˜t (·, t) 2 + w ˜x (·, t) 2

˜ ut (·, t) 2 + ˜  12 + w(0, ˜ t)2 + w ˜t (0, t)2 + u ˜(0, t)2 + u ˜t (0, t)2  ≤ Υ4 ˜ ux (·, 0) 2 + w ˜t (·, 0) 2 + w ˜x (·, 0) 2 ut (·, 0) 2 + ˜ + w(0, ˜ 0)2 + w ˜t (0, 0)2 + u ˜(0, 0)2 + u ˜t (0, 0)2

 12

e−σ4 t ,

(6.203)

which means that the observer states in (6.139)–(6.147) are exponentially convergent to the actual values in (6.32)–(6.37) according to (6.151)–(6.154). Proof. Recalling lemma 6.4 and (6.155)–(6.157), the following inequality holds   1    ˜  2  ˜ 2 2 ˜ t) 2 + ˜ v (·, t) 2 + k(·, z (·, t) 2 + X(t)

˜ y (·, t) 2 + ˜  + Y (t)  ≤ Υ4a ˜ v (·, 0) 2 y (·, 0) 2 + ˜ 

  1    ˜ 2  ˜ 2 2 −σ4a t 2 2 ˜ + k(·, 0) + ˜ z (·, 0) + X(0) + Y (0) e for some positive constants Υ4a , σ4a . ˜x (·, t), w ˜t (·, t), w ˜x (·, t) are repAccording to (6.158)–(6.161), with which u ˜t (·, t), u ˜ t), y˜(·, t), the proof of theorem 6.3 is complete, recalling resented by z˜(·, t), v˜(·, t), k(·, (6.155).

DEEP-SEA CONSTRUCTION

6.4

157

CONTROLLER WITH COLLOCATED BOUNDARY SENSING

The output-feedback control law, employing axial and lateral control forces at the ship-mounted crane with sensors placed only at that location, is obtained by combining the state-feedback controller from section 6.2 with the observer from section 6.3. After inserting the observer states into the state-feedback controller (6.104), (6.105) to replace the unmeasurable states, the control laws U1 (t), U2 (t) in (6.36), (6.37) assume the output-feedback form denoted as Uo1 (t), Uo2 (t) and given by Uo1 (t) =

  l(t)  −1  (F11 (l(t), y) + N11 (l(t), y))ˆ ut (l(t), t) − ut (y, t) d19 (l(t)) d6 (l(t)) 0 ˆt (y, t) + (F12 (l(t), y) + N12 (l(t), y))w  + (N11 (l(t), y) − F11 (l(t), y)) d6 (y)ˆ ux (y, t)   + (N12 (l(t), y) − F12 (l(t), y)) d1 (y)w ˆx (y, t) dy − λ11 (l(t))w(0, ˆ t) − λ12 (l(t))w ˆt (0, t) u(0, t) − λ14 (l(t))ˆ ut (0, t) , − λ13 (l(t))ˆ

(6.204)

  l(t)  −1  (F21 (l(t), y) + N21 (l(t), y))ˆ wt (l(t), t) − ut (y, t) Uo2 (t) = d20 (l(t)) d1 (l(t) 0 ˆt (y, t) + (F22 (l(t), y) + N22 (l(t), y))w  + (N21 (l(t), y) − F21 (l(t), y)) d6 (y)ˆ ux (y, t)   + (N22 (l(t), y) − F22 (l(t), y)) d1 (y)w ˆx (y, t) dy − λ21 (l(t))w(0, ˆ t) − λ22 (l(t))w ˆt (0, t) u(0, t) − λ24 (l(t))ˆ ut (0, t) . − λ23 (l(t))ˆ

(6.205)

The control inputs Uo1 (t), Uo2 (t) are implemented based on the boundary measurements ut (l(t), t), wt (l(t), t) mentioned in section 6.1. To be exact, ut (l(t), t), wt (l(t), t) directly act as the first terms in the expressions (6.204), (6.205) and are also used to obtain the solutions of the observer system (6.139)–(6.147) that are required in the remaining terms in (6.204), (6.205), through the conversion (6.138). Theorem 6.4. For the closed-loop system consisting of the plant (6.32)–(6.37), the observer (6.139)–(6.147), and the output-feedback controller (6.204), (6.205), with arbitrary initial values (w(x, 0), wt (x, 0)) ∈ H 2 (0, L) × H 1 (0, L), (u(x, 0), ut (x, 0)) ∈ H 2 (0, L) × H 1 (0, L), one obtains 1) there exist positive constants Υ5 and σ5 such that  1/2  1/2 ˆ ˆ Ξ(t) + Ξ(t) ≤ Υ5 Ξ(0) + Ξ(0) e−σ5 t ,

(6.206)

ˆ where Ξ(t) is given in (6.121), and Ξ(t) is defined as 2 2 ˆ = ˆ Ξ(t) ux (·, t) 2 + ˆ ut (·, t) 2 + |ˆ u(0, t)| + |ˆ ut (0, t)| 2

2

+ w ˆx (·, t) 2 + w ˆt (·, t) 2 + |w(0, ˆ t)| + |w ˆt (0, t)| ;

(6.207)

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158

2) the output-feedback control signals (6.204), (6.205) are bounded and exponentially convergent to zero. Proof. The output-feedback controller (6.204), (6.205) can be written as ˜ [Uo1 (t), Uo2 (t)]T = [Usf 1 (t), Usf 2 (t)]T + δ(t)

(6.208)

by virtue of (6.151)–(6.155), where Usf 1 (t), Usf 2 (t) are the state-feedback laws given ˜ ∈ R2×1 is by (6.104), (6.105), and δ(t)   l(t)  ˜ = 2R(l(t))−1 F (l(t), y)[˜ ut (y, t) − d6 (y)˜ ux (y, t), δ(t) 0  w ˜t (y, t) − d1 (y)w ˜x (y, t)]T dy  l(t)  N (l(t), y)[˜ ut (y, t) + d6 (y)˜ ux (y, t), + 0  ˜x (y, t)]T dy w ˜t (y, t) + d1 (y)w ˜(0, t), u ˜t (0, t)]T . + λ(l(t))[w(0, ˜ t), w ˜t (0, t), u

(6.209)

Applying the output-feedback controller (6.208) to the plant (6.32)–(6.37), that is, U1 (t) = U1o (t), U2 (t) = U2o (t), and recalling theorems 6.1 and 6.3, together with (6.151)–(6.155), we arrive at (6.206), which is property (1) in theorem 6.4. Moreover, by applying theorem 6.2, which shows that the state-feedback controllers Usf 1 (t), Usf 2 (t) are exponentially convergent to zero, and theorem 6.3, which guarantees ˜ (6.209), we obtain the results of boundthe exponential convergence to zero of δ(t) edness and exponential convergence of the output-feedback control input in light of (6.208)—that is, property (2) of theorem 6.4 is established. Therefore, the proof of theorem 6.4 is complete.

6.5

SIMULATION FOR A DEEP-SEA CONSTRUCTION SYSTEM

The simulation is conducted based both on the linear model (6.21)–(6.26) and on the actual nonlinear model (6.13)–(6.18) with unmodeled disturbances, where the simulation on the former model is performed to verify the theoretical results, and the second simulation is done to illustrate the effectiveness in the application of vibration control to a reasonably realistic model of a DCV. The plant on the time-varying domain with pre-determined time-varying func˙ shown in figure 6.4 is converted to a plant on the fixed domain tions l(t) and l(t) ˙ ¨l(t) by introducing ι ∈ [0, 1], with time-varying coefficients related to l(t), l(t), ι=

x l(t)

(6.210)

that is, representing u(x, t) by u(ι, t), as ux (x, t) =

1 uι (ι, t), l(t)

(6.211)

DEEP-SEA CONSTRUCTION

159 12

1250

l(t) (m)

8 6

750

4 Descending trajectory Descending velocity 250

0

40

80

i(t) (m/s)

10

2 0 120

Time (s)

Figure 6.4. Descending trajectory and velocity—that is, the time-varying cable ˙ length l(t) and the changing rate l(t). uxx (x, t) =

1 uιι (ι, t), l(t)2

ut (x, t) = ut (ι, t) −

˙ l(t)ι uι (ι, t), l(t)

utt (x, t) = utt (ι, t) − −

(6.212) (6.213)

˙ ˙ 2 ι2 2l(t)ι l(t) uιt (ι, t) − uιι (ι, t) l(t) l(t)2

(l(t)¨l(t) − 2¨l(t)2 )ι uι (ι, t). l(t)2

(6.214)

Then the simulation is conducted based on the finite-difference method with the time and space steps of 0.001 and 0.05, respectively. The observer (6.139)–(6.147) is solved in the same way, and the following equations are used to obtain u ˆ, w ˆ from ˆ yˆ, zˆ, vˆ: k,  ι 1 ˆ ι, t) − yˆ(¯ι, t))d¯ι + C¯1 W ˆ (t),  (k(¯ u ˆ(ι, t) = ι) 0 2 d6 (¯  ι 1 ˆ (t)  w(ι, ˆ t) = (ˆ z (¯ι, t) − vˆ(¯ι, t))d¯ι + C¯2 W ι) 0 2 d1 (¯ according to (6.139)–(6.142) and (6.150), where C¯1 = [0, 0, 1, 0], and C¯2 = [1, 0, 0, 0]. The initial conditions are defined according to the steady state, as ux (·, 0) = ε¯(·), ¯ wt (·, 0) = 0. By defining u(0, 0) = 0 and ut (·, 0) = 0, and wx (·, 0) = −φ(·), w(l(0), 0) = 0, the initial conditions of (6.21)–(6.26) are thus defined completely ˆ 0), yˆ(·, 0), in the finite-difference numerical calculation. All initial conditions k(·, ˆ zˆ(·, 0), vˆ(·, 0), W (0) of the observer (6.139)–(6.147) are set as zero. Test on the Linear Model Matching (6.32)–(6.37) with (6.21)–(6.26), we obtain the specific expressions of the coefficients in (6.32)–(6.37) as d1 (x) =

3 ¯ 2 2 EAa φ(x)

mc

+ T (x)

,

(6.215)

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160

ω(m)

8 4 0

–4 –8 1 0.5 l

0 0

40

80

120

t(s)

Figure 6.5. Responses of lateral vibrations w(ι, t) without control.

d2 (x) =

EAa ε¯ (x) + ρg −EAa φ¯ (x) , d3 = , mc mc

−cv EAa −EAa φ¯ (x) , d5 = 0, d6 = , d7 (x) = , mc mc mc ¯ 2 −cu −cw −EAa φ(0) d8 = d9 = 0, d10 = , d11 = , d12 = , mc ML 2ML ¯ −EAa φ(0) −ch −EAa d13 = 0, d14 = , d15 = , d16 = , ML ML ML ¯ EAa φ(0) 1 d17 = 0, d18 = , d19 = , 2ML EAa 1 d20 (l(t)) = , EAa ¯ EAa ε¯(l(t)) + 2 φ(l(t))2 + T (l(t)) d4 =

(6.216) (6.217) (6.218) (6.219) (6.220) (6.221)

¯ where T (x), ε¯(x), and φ(x) are given in (6.6), (6.19), (6.20), and the values of the physical parameters are shown in table 6.1. The variable x in (6.215)–(6.221) can be represented by ι via (6.210). We apply the proposed controllers (6.204), (6.205) into (6.21)–(6.26), where the approximate solution of the kernel equations (6.87)– (6.93) is also solved by the finite-difference method on a fixed triangular domain D0 = {0 ≤ y ≤ x ≤ L}. Then we extract F (l(t), y), N (l(t), y), which are used in the controller. The control parameters κ are chosen as   0.8 1.2 4.5 6 κ11 κ12 κ13 κ14 (6.222) = × 103 , κ21 κ22 κ23 κ24 2.5 3 1.5 2 which determines the kernel λ(x) used in the controllers. The same process is used ¯ l(t)), which are used in the observer gains (6.185), (6.186). All to get ϕ(x, ¯ l(t)), ψ(x, elements in L0 are defined as 1. Figure 6.5 shows that the large lateral vibrations whose oscillation range is up to 10 m persist in the whole operation time 120 s in the case without control. Even though the longitudinal vibrations in figure 6.6 are decaying because of the material damping coefficient cu of the cable in table 6.1, the longitudinal vibration at the top of the cable (ι = 1 in figure 6.5) is excessive and decays slowly because this point bears the whole mass of the cable and payload, resulting in large elastic deflections. Applying the proposed output-feedback lateral and longitudinal vibration control forces at the ship-mounted crane, the lateral vibrations decay with a satisfied decay

DEEP-SEA CONSTRUCTION

161

0.3 u(m)

0.15 0

–0.15 –0.3 1 0.5 0 0

l

40 t(s)

80

120

Figure 6.6. Responses of longitudinal vibrations u(ι, t) without control.

ω(m)

6 3 0 –3 –6 1 0.5 0 0

l

40 t(s)

80

120

Figure 6.7. Closed-loop responses of lateral vibrations w(ι, t).

u(m)

0.3 0.15 0

–0.15 –0.3 1 0.5 l

0 0

40 t(s)

80

120

Figure 6.8. Closed-loop responses of longitudinal vibrations u(ι, t).

rate according to figure 6.7, and figure 6.8 shows that the longitudinal vibrations are suppressed very quickly. The output-feedback control forces at the ship-mounted crane are shown in figure 6.9, where the states of the proposed observer are used. The performance of the observer on tracking the actual states can be seen in figures 6.10 and 6.11, which show that the observer errors of both the lateral and the longitudinal vibrations are convergent to zero. Test on the Actual Nonlinear Model with Ocean Current Disturbances The nonlinear model (6.13)–(6.18) is converted to the one on the fixed domain ι ∈ [0, 1] through the process mentioned at the beginning of this section. The time

CHAPTER SIX

162

(N)

1.5

× 106

1

U2(t)

0.5

U1(t)

0 –0.5 –1 –1.5

0

20

40

60

80

100

120

Time (s)

Figure 6.9. Control forces U1 (t) and U2 (t).

0.3 w (m)

0.15 0

–0.15 –0.3 1 0.5 l

0 0

40 t(s)

80

120

Figure 6.10. Observer error of lateral vibrations w(ι, ˜ t).

0.4 u (m)

0.2 0

–0.2 1 0.5 l

0 0

40 t(s)

80

120

Figure 6.11. Observer error of longitudinal vibrations u ˜(ι, t).

and space steps are changed to 0.0005 and 0.1, respectively, in the finite-difference method, to ensure numerical stability. In practice, ocean current disturbances act as external lateral oscillating drag forces f (ι, t) on the cable. In the simulation, f (ι, t), which is added in (6.14) and (6.16), converted to a fixed domain, as above, is defined as follows. Consider the time-varying ocean surface current velocity P (t) modeled by a first-order GaussMarkov process [61], as follows: P˙ (t) + μP (t) = G(t), Pmin ≤ P (t) ≤ Pmax ,

(6.223)

Disturbances f (N)

DEEP-SEA CONSTRUCTION

163

500 250 0 –250 –500 1 0.5 l

80

40 t(s)

0 0

120

Figure 6.12. Lateral oscillation drag forces from ocean current disturbances.

u (m)

0.3 0.15 0 –0.15 –0.3 1 0.5 l

0

0

40 t(s)

80

120

Figure 6.13. Closed-loop responses of longitudinal vibrations u(ι, t) in the actual nonlinear model with unmodeled disturbances. where G(t) is Gaussian white noise. The constants Pmin , Pmax , and μ are chosen, respectively, as 1.6 ms−1 , 2.4 ms−1 , and 0 [95]. The function f (ι, t) is then given according to [95] as   St P (t) 1 t+ς , (6.224) f (ι, t) = (0.9ι + 0.1) ρs Cd P (t)2 RD AD cos 4π 2 RD where 0.9ι + 0.1 means that the full disturbance load is applied at the top of the cable, at the ocean surface, and linearly declines to 0.1 at the bottom of the cable, at the payload. The constant Cd = 1 denotes the drag coefficient, and ς = π is the phase angle. The constant AD = 400 denotes the amplitude of the oscillating drag force, and St = 0.2 is the Strouhal number [59]. The ocean disturbances f used in the simulation are shown in figure 6.12. The control parameters κ11 , κ12 , κ13 , κ14 and κ21 , κ22 , κ23 , κ24 are increased to twice and ten times those in (6.222), respectively, due to the unmodeled disturbances. The observer parameters are kept the same as those in section 6.5. We apply the proposed output-feedback controller to the actual nonlinear model with the ocean current disturbances. Figures 6.13 and 6.14 show that the longitudinal vibrations and lateral vibrations are reduced as time goes on. It is particularly remarkable in figure 6.14 that while the control law is designed only for suppressing oscillations in response to initial lateral displacements and not for attenuating a lateral disturbance, like that in figure 6.12, the controller is effective in disturbance attenuation after an initial transient of about 40 seconds. From figure 6.15, the observer errors converge to a small range around zero. Simulation results in

CHAPTER SIX

164

w (m)

10 0 –10 1 0.5 0 0

l

120

80

40 t(s)

Observer error (m)

Figure 6.14. Closed-loop responses of lateral vibrations w(ι, t) in the actual nonlinear model with unmodeled disturbances.

5 0 –5 1 0.5 0 0

l

40

120

80

t(s)

Figure 6.15. Observer errors u ˜(·, t) + w(·, ˜ t) in the actual nonlinear model with unmodeled disturbances.

1

× 107

Control inputs (N)

0 –1 –2 –3 –4

Ū2(t)

–5

Ū1(t)

–6

0

20

40

60 Time (s)

80

100

120

Figure 6.16. Control forces in the actual nonlinear model. this section illustrate the effectiveness of the proposed control design when applied to vibration suppression of the DCV. The output-feedback control inputs in this actual model are shown in figure 6.16.

6.6

APPENDIX

Proof of lemma 6.1 The boundary conditions in (6.87)–(6.93) are along the lines y = x and y = 0, and there are no conditions at the boundary x = l(t) on the triangular domain D. Therefore, we can extend the boundary x = l(t) in D to x = L

DEEP-SEA CONSTRUCTION

165

due to 0 < l(t) ≤ L in assumption 6.3 to solve F, N on a fixed triangular domain D0 = {0 < y < x < L} and λ on a constant interval 0 ≤ x ≤ L in (6.87)–(6.93). The lines y = x and y = 0 are overlapped boundaries of the triangular domains D0 and D and D ⊆ D0 . Once the solutions of (6.87)–(6.93) on D0 are obtained, those on the subset D are the required kernels F, N and λ. To ensure the well-posedness of the kernel equations (6.87)–(6.93) on D0 , we add an additional artificial condition at x = L for N21 , as done in [96]. Next, we will prove that there exists a unique solution of (6.87)–(6.93) on D0 by showing the system is in the form of a class of wellposed equations in [48], which ensures there exists a unique solution F, N ∈ L∞ (D), λ ∈ L∞ ([0, l(t)]). Expanding (6.87)–(6.93) on D0 , we know that Fij (x, y)1≤i,j≤2 , Nij (x, y)1≤i,j≤2 should satisfy the following coupled hyperbolic PDEs:   d6 (x)F11x (x, y) − d6 (y)F11y (x, y) =[

d3 (y) d5 (y) d10 (y) − s1 (y)]N11 (x, y) + [ −  ]N12 (x, y) 2 2 2 d6 (y)

  d10 (x) d10 (y) − s1 (y) + d6 (y) − + s1 (x)]F11 (x, y) 2 2 d3 (y) d5 (y) −  +[ ]F12 (x, y), 2 2 d6 (y)   d6 (x)F12x (x, y) − d1 (y)F12y (x, y) +[

=[

(6.225)

d7 (y) d9 (y) d4 (y) −  − s2 (y)]N12 (x, y) ]N11 (x, y) + [ 2 2 2 d1 (y)

  d10 (x) d4 (y) − s2 (y) + d1 (y) − + s1 (x)]F12 (x, y) 2 2 d7 (y) d9 (y) −  +[ ]F11 (x, y), 2 2 d1 (y)   d1 (x)F21x (x, y) − d6 (y)F21y (x, y) +[

=[

(6.226)

d3 (y) d5 (y) d10 (y) − s1 (y)]N21 (x, y) + [ −  ]N22 (x, y) 2 2 2 d6 (y)

  d4 (x) d10 (y) − s1 (y) + d6 (y) − + s2 (x)]F21 (x, y) 2 2 d3 (y) d5 (y) −  ]F22 (x, y), +[ 2 2 d6 (y)   d1 (x)F22x (x, y) − d1 (y)F22y (x, y) +[

=[

(6.227)

d7 (y) d4 (y) d9 (y) −  − s2 (y)]N22 (x, y) ]N21 (x, y) + [ 2 2 2 d1 (y)

  d4 (x) d4 (y) − s2 (y) + d1 (y) − + s2 (x)]F22 (x, y) 2 2 d7 (y) d9 (y) −  ]F21 (x, y), +[ 2 2 d1 (y) +[

(6.228)

CHAPTER SIX

166 

d6 (x)N11x (x, y) +

= [s1 (y) +



d6 (y)N11y (x, y)

d5 (y) d10 (y) d3 (y) ]F11 (x, y) + [  ]F12 (x, y) + 2 2 2 d6 (y)

 d10 (y)  d10 (x) − d6 (y) − s1 (x) − ]N11 (x, y) 2 2 d3 (y) d5 (y) +[  ]N12 (x, y), + 2 2 d6 (y)   d6 (x)N12x (x, y) + d1 (y)N12y (x, y)

+ [s1 (y) +

d7 (y) d9 (y) d4 (y) =[  ]F11 (x, y) + [s2 (y) + ]F12 (x, y) + 2 2 2 d1 (y)  d4 (y)  d10 (x) − d1 (y) − s1 (x) − ]N12 (x, y) + [s2 (y) + 2 2 d7 (y) d9 (y) +[  ]N11 (x, y), + 2 2 d1 (y)   d1 (x)N21x (x, y) + d6 (y)N21y (x, y) = [s1 (y) +

(6.229)

(6.230)

d5 (y) d10 (y) d3 (y) ]F21 (x, y) + [  ]F22 (x, y) + 2 2 2 d6 (y)

 d10 (y)  d4 (x) − d6 (y) − s2 (x) − ]N21 (x, y) 2 2 d3 (y) d5 (y) +[  ]N22 (x, y), + 2 2 d6 (y)   d1 (x)N22x (x, y) + d1 (y)N22y (x, y)

+ [s1 (y) +

d7 (y) d9 (y) d4 (y) ]F21 (x, y) + [s2 (y) + ]F22 (x, y) =[  + 2 2 2 d1 (y)  d4 (y)  d4 (x) − d1 (y) − s2 (x) − ]N22 (x, y) + [s2 (y) + 2 2 d7 (y) d9 (y) +[  ]N21 (x, y) + 2 2 d1 (x)

(6.231)

(6.232)

along with the following set of boundary conditions: −d10 (x) + 2s1 (x)  , 4 d6 (x)  −d9 (x) d1 (x) + d7 (x)   F12 (x, x) =  , 2( d6 (x) + d1 (x)) d1 (x)  −d5 (x) d6 (x) + d3 (x)   F21 (x, x) =  , 2( d6 (x) + d1 (x)) d6 (x) F11 (x, x) =

F22 (x, x) =

−d4 (x) + 2s2 (x)  , 4 d1 (x)

(6.233)

(6.234)

(6.235) (6.236)

DEEP-SEA CONSTRUCTION

167 −d7 (x)

− d92(x)  N12 (x, x) =  , d6 (x) − d1 (x) 2

√

d1 (x)

− d52(x)  , N21 (x, x) =  d1 (x) − d6 (x)

(6.237)

−d3 (x)

2

√

d6 (x)

(6.238)

N11 (x, 0) = −F11 (x, 0) +

d14 λ12 (x) d16 λ14 (x) + , d6 (0) d6 (0)

(6.239)

N12 (x, 0) = −F12 (x, 0) +

d12 λ12 (x) d18 λ14 (x) + , d1 (0) d1 (0)

(6.240)

N21 (x, 0) = −F21 (x, 0)  g0 (x) d6 (0) + d14 λ22 (x) + d16 λ24 (x) , + d6 (0) N22 (x, 0) = −F22 (x, 0) +

d12 λ22 (x) d18 λ24 (x) + , d1 (0) d1 (0)

N21 (L, y) = 0,

(6.241) (6.242) (6.243)

where (6.243) is the artificial boundary condition, and assumption 6.2 ensures that the denominators of (6.237), (6.238) are nonzero. Additionally, λij (x)1≤i≤2,1≤j≤4 should satisfy the following ODEs: 

d10 (x) ]λ11 (x) = 0, 2  d10 (x) d12 − d11 +  ]λ12 (x) d6 (x)λ12  (x) + [s1 (x) + 2 d1 (0)  d18 )λ14 (x) − 2 d1 (0)F12 (x, 0) = 0, − λ11 (x) − (d17 −  d1 (0)  d10 (x) ]λ13 (x) = 0, d6 (x)λ13  (x) + [s1 (x) + 2  d10 (x) d16 − d15 +  ]λ14 (x) d6 (x)λ14  (x) + [s1 (x) + 2 d6 (0) d6 (x)λ11  (x) + [s1 (x) +

 d14 )λ12 (x) − 2 d6 (0)F11 (x, 0) = 0, − λ13 (x) − (d13 −  d6 (0)  d4 (x) ]λ21 (x) + g0 (x)λ11 (0) = 0, d1 (x)λ21  (x) + [s2 (x) + 2  d4 (x) d12 − d11 +  ]λ22 (x) d1 (x)λ22  (x) + [s2 (x) + 2 d1 (0) d18 )λ24 (x) − λ21 (x) − (d17 −  d1 (0)  − 2 d1 (0)F22 (x, 0) + g0 (x)λ12 (0) = 0, 

d1 (x)λ23  (x) + [s2 (x) +

d4 (x) ]λ23 (x) + g0 (x)λ13 (0) = 0, 2

(6.244)

(6.245) (6.246)

(6.247) (6.248)

(6.249) (6.250)

CHAPTER SIX

168 

d1 (x)λ24  (x) + [s2 (x) +

d4 (x) d16 − d15 +  ]λ24 (x) 2 d6 (0)

d14 )λ22 (x) − λ23 (x) − (d13 −  d6 (0)  − 2 d6 (0)F21 (x, 0) + g0 (x)λ14 (0) = 0, with initial conditions

  =

λ11 (0) λ21 (0) κ11 κ21

λ12 (0) λ22 (0)

κ12 κ22

κ13 κ23

λ13 (0) λ23 (0) κ14 . κ24

λ14 (0) λ24 (0)

(6.251)



(6.252)

The equation set (6.225)–(6.252) has the same structure as the kernel equations (17)–(24) in [48] with setting m = n = 2. More precisely, (6.225)–(6.228) corresponds to (17); (6.229)–(6.232) corresponds to (18); (6.233)–(6.236) corresponds to (19); (6.238) corresponds to (20); (6.239)–(6.242) corresponds to (21); the additional artificial boundary condition (6.243) corresponds to (24); the initial value problem (6.244)–(6.252) corresponds to (22), (23). Even though (17)–(24) in [48] are with constant coefficients, the well-posedness result still holds in (6.225)–(6.252) with spatially varying coefficients because (6.225)–(6.252) are in the form of a general class of hyperbolic PDEs (30), (31), whose well-posedness has been proved in [48]. Therefore, (6.225)–(6.252)—that is, (6.87)–(6.93)—are well-posed on the domain D0 , which straightforwardly yields to a unique solution F, N ∈ L∞ (D), λ ∈ L∞ ([0, l(t)]) of (6.87)–(6.93) because of D ⊆ D0 , l(t) ≤ L. Kernel equations (6.94)– (6.100) of J(x, y), K(x, y), γ(x) have the same structure as (6.87)–(6.93). Through a similar process as above, adding an additional artificial boundary condition K21(L,y) = 0, one obtains a unique solution J, K ∈ L∞ (D), γ ∈ L∞ ([0, l(t)]) of (6.94)–(6.100). The proof of lemma 6.1 is complete.

6.7

NOTES

Chapters 2–5 presented control designs for one-dimensional cable vibrations, either axial or transversal. In this chapter, we dealt with axial-transversal coupled Table 6.2. Comparison of previous results on the boundary vibration control of cables Multi-dimensional coupled vibrations in cables [90, 131] [92] [22] [178], [185] This chapter

× √ √ × √

Time-varying cable lengths √ √ × √ √

Number of controlled/fixed/ damped boundaries 2 2 1 1 1

DEEP-SEA CONSTRUCTION

169

vibrations of the cable, whose dynamics are linear wave PDEs with in-domain couplings obtained from linearizing the original nonlinear model around a steady state. After applying Riemann transformations, the plant becomes a 4 × 4 hyperbolic system. Compared to the 4 × 4 system in the dual-cable mining elevator in chapter 3, which is a set of two 2 × 2 pairs that are coupled not along the domain but at the boundary, the plant is a fully coupled 4 × 4 hyperbolic system in this chapter. Comparisons with some previous results on the boundary vibration control of cables are summarized in table 6.2.

Chapter Seven Deep-Sea Construction with Event-Triggered Delay Compensation

A backstepping control law for the deep-sea construction vessel (DCV) was presented in chapter 6. One challenge to its practical implementation is that the massive actuator, the ship-mounted crane, is incapable of supporting the fast-changing control signal due to its low natural frequency. In chapter 6, we placed the sensors at the ship-mounted crane—that is, at the top of the cable. When the sensor is placed at the object, referred to as the payload, at the bottom of the cable, there exists another challenge: a sensor delay, due to the fact that the sensor signal is transmitted over a large distance from the seabed to the vessel on the ocean surface, through a set of acoustic devices. Such a sensor signal delay may result in information distortion or even make the control system lose stability. In order to solve these problems, in this chapter we pursue a delay-compensated event-triggered control scheme, which compensates for the sensor delay of arbitrary length and reduces the changes in the actuator signal—that is, the control input employs piecewise-constant values. The DCV model and the general model used for control are presented in section 7.1. The observer design is proposed in section 7.2, where the observer gains are determined in two transformations that convert the observer error system to a target observer error system whose exponential stability is straightforward to obtain. An observer-based output-feedback controller with delay compensation is designed in section 7.3 using the backstepping method. The dynamic event-triggering mechanism (ETM) is designed in section 7.4. The existence of a minimal dwell time between two successive triggering times and exponential convergence in the eventbased closed-loop system are proved in section 7.5. The proposed control design is verified in the application of DCV control to seabed installation via a simulation in section 7.6. 7.1

PROBLEM FORMULATION

Model of DCV We follow the modeling process in section 6.1 in chapter 6 but make the following three simplifications: 1) We take the cable length as constant, setting l(t) as L; 2) The spatially dependent static tension of the cable, which considers the effect of the cable mass on the tension in the cable, is assumed to be constant as a result of the payload mass and buoyancy—that is, we set T (x) as T0 ; 3) We only focus on lateral vibrations instead of longitudinal-lateral coupled vibrations and model the

EVENT-TRIGGERED DELAY COMPENSATION

171

Table 7.1. Physical parameters of the DCV Parameters (units)

Values

Cable length L (m) Cable diameter Rd (m) Cable linear density ρ (kg/m) Payload mass ML (kg) Gravitational acceleration g (m/s2 ) Cable material damping coefficient dc (N·s/m) Height of payload modeled as a cylinder hc (m) Diameter of payload modeled as a cylinder Dc (m) Damping coefficient at payload dL (N·s/m) Seawater density ρs (kgm−3 )

1000 0.2 8.02 4.0 × 105 9.8 0.5 10 5 2.0×105 1024

dynamics of a DCV as T0 ux¯ (0, t) = U (t), ρutt (¯ x, t) = T0 ux¯x¯ (¯ x, t) − dc ut (¯ x, t),

(7.1) (7.2)

u(L, t) = bL (t), ML¨bL (t) = −dL b˙ L (t) + T0 ux¯ (L, t),

(7.3) (7.4)

∀(¯ x, t) ∈ [0, L] × [0, ∞). The function u(¯ x, t) denotes the distributed transverse displacement along the cable. The function bL (t) represents the transverse displacement of the payload. Input U (t) is the control force at the ship-mounted crane. The static tension T0 is defined as T0 = ML g − Fbuoyant ,

(7.5)

where the buoyancy Fbuoyant is 1 Fbuoyant = πDc2 hc ρs g. (7.6) 4 The physical parameters of the DCV are given in table 7.1, whose values are taken from [95]. We simplify the DCV model derived in chapter 6 with the purpose of presenting the delay-compensated event-triggered control design in this chapter more clearly. This design can be applied to the model in chapter 6 with a modest extension. In chapter 6, we placed the sensors at the ship-mounted crane, at the top of the cable, whereas the design in which the sensor is placed at the object, referred to as the payload, at the bottom of the cable, is considered here. As shown in figure 7.1, an accelerometer is placed at the payload to measure the lateral acceleration ¨bL (t). However, there exists a sensor delay τ because the sensor signal is transmitted over a large distance from the seabed to the vessel on the ocean surface through a set of acoustic devices. Therefore, the acquisition of the wireless receiver at the vessel is the delayed measurement of the lateral acceleration of the payload, denoted as ˙ − τ ), yout (t) = ζ(t

t ∈ [τ, ∞),

(7.7)

CHAPTER SEVEN

172 u(0,t)

Wireless receiver

Piecewise-constant control force Ud(t)

L

Sensor delay τ

u(x,t) Acceleration sensor with acoustics emitter bL(t) Mass ML

hc

Dc Target position on seabed

Figure 7.1. Diagram of a deep-sea construction vessel.

where a new variable is introduced as ζ(t) = b˙ L (t).

(7.8)

In practice, the lateral velocity ζ of the payload is obtained by integrating the delayed acceleration sensing yout (t)—that is, 

t

ζ(t − τ ) =

yout (h)dh + ζ(0), t ∈ [τ, ∞),

(7.9)

τ

which is a delayed signal of the lateral velocity where the initial value ζ(0) of the lateral velocity at the payload is used. Equation (7.9) only explains an acquisition method for ζ in practical DCVs via the acceleration sensor, and no restrictions are imposed on any initial conditions in the design and theory in this chapter. In the practical DCV there always exists, caused by ocean currents, a distributed oscillating drag force f (¯ x, t) along the cable and a drag force fL (t) at the payload; that is, disturbances f (¯ x, t), fL (t) would exist in (7.2), (7.4). Although we do not deal with the disturbances f (¯ x, t), fL (t) in the control design in this chapter, they are considered in the simulation model for testing the robustness of the proposed controller.

EVENT-TRIGGERED DELAY COMPENSATION

173

Representing the Model in Riemann Coordinates Applying the Riemann transformations  z¯(¯ x, t) = ut (¯ x, t) −  x, t) + w(¯ ¯ x, t) = ut (¯

T0 ux¯ (¯ x, t), ρ

(7.10)

T0 ux¯ (¯ x, t), ρ

(7.11)

introducing a space normalization variable x=

x ¯ ∈ [0, 1], L

(7.12)

and defining z¯(¯ x, t) = z(x, t),

w(¯ ¯ x, t) = w(x, t),

(7.13)

we can rewrite the equations (7.1)–(7.4) as the plant (7.16)–(7.20), to be shown shortly, with the coefficients   1 1 T0 dc c¯ = 2 , q1 = q 2 = , c1 = c2 = c3 = c4 = , (7.14) T0 ρ L ρ 2ρ √ √ T0 ρ T0 ρ −dL q = −1, p = 1, c = 2, a1 = + , b1 = , (7.15) ML ML ML x, t) = L1 zx (x, t), w ¯x¯ (¯ x, t) = L1 wx (x, t) have been used. where z¯x¯ (¯ General Model The considered plant is in a general form, given by z(0, t) = pw(0, t) + c¯U (t),

(7.16)

zt (x, t) = −q1 zx (x, t) − c1 w(x, t) − c2 z(x, t), wt (x, t) = q2 wx (x, t) − c3 w(x, t) − c4 z(x, t), w(1, t) = qz(1, t) + cζ(t), ˙ = a1 ζ(t) − b1 z(1, t), ζ(t)

(7.17) (7.18) (7.19) (7.20)

∀(x, t) ∈ [0, 1] × [0, ∞), and the output measurement is ˙ − τ )]T , Yout (t) = [ζ(t − τ ), ζ(t

t ∈ [τ, ∞)

(7.21)

according to the obtained signals shown in (7.7)–(7.9). The scalar ζ(t) is an ODE state. The scalars z(x, t), w(x, t) are states of the 2 × 2 coupled hyperbolic PDEs. The positive parameter τ is an arbitrary constant denoting the time delay in the measurement. The control input U (t) in (7.16) is to be designed. The parameters c1 , c2 , c3 , c4 , c ∈ R are arbitrary. The positive constants q1 and q2 are transport speeds. The constants a1 and b1 = 0 are arbitrary. The parameters q, p ∈ R (q = 0) satisfy assumption 7.1.

CHAPTER SEVEN

174 Assumption 7.1. The plant parameters p, q satisfy c3

c2

|pq| < e q2 + q1 . One can readily check that assumption 7.1, which will be used in section 7.4, is satisfied in the DCV model with the parameters (7.14), (7.15), whose values are given in table 7.1. Rewrite Delay as a Transport PDE By defining v(x, t) = ζ(t − τ (x − 1)),

(7.22)

we obtain from (7.21) a transport PDE v(1, t) = ζ(t), 1 vt (x, t) = − vx (x, t), τ

(7.23) x ∈ [1, 2],

Yout (t) = [v(2, t), vt (2, t)]T

(7.24) (7.25)

for all (x, t) ∈ [1, 2] × [0, ∞), to describe the time delay in the measurement (7.21). Replacing (7.21) with (7.23)–(7.25), we obtain a hyperbolic PDE-ODE system connecting with another transport PDE, the coupled hyperbolic PDE-ODE-transport PDE system (7.16)–(7.20), (7.23)–(7.25). The time delay is “removed” at a cost of adding a transport PDE into the plant (7.16)–(7.20). The continuous-in-time boundary control design for a PDE-ODE-PDE configuration was also studied in [2]. Now the task is equivalent to exponentially stabilizing the overall system (7.16)– (7.20), (7.23)–(7.25)—that is, the (z, w, v)-PDEs and ζ-ODE, by employing an event-triggered output-feedback control input U (t) at the left boundary of the z PDE using only the right boundary state of the v-PDE. We adopt the following notation: • The symbol R− denotes the set of negative real numbers, whose complement of the real axis is R+ := [0, +∞). The symbol Z+ denotes the set of all nonnegative integers. • Let U ⊆ Rn be a set with a nonempty interior, and let Ω ⊆ R be a set. By C 0 (U ; Ω), we denote the class of continuous mappings on U , which take values in Ω. By C k (U ; Ω), where k ≥ 1, we denote the class of continuous functions on U , which have continuous derivatives of order k on U and take values in Ω. • We use the notation L2 (0, 1) for the standard space of the equivalence class of square-integrable, measurable functions defined on (0, 1), and f =   12 1 2 f (x) dx < +∞ for f ∈ L2 (0, 1). 0

• For an I ⊆ R+ , the space C 0 (I; L2 (0, 1)) is the space of continuous mappings I t → u[t] ∈ L2 (0, 1). • Let u : R+ × [0, 1] → R be given. We use the notation u[t] to denote the profile of u at certain t ≥ 0—that is, (u[t])(x) = u(x, t) for all x ∈ [0, 1].

EVENT-TRIGGERED DELAY COMPENSATION

7.2

175

OBSERVER DESIGN USING DELAYED MEASUREMENT

In order to build the observer-based event-triggered output-feedback controller of the plant (7.16)–(7.21), in this section we design a state observer to track the overall system (7.16)–(7.21) using only the delayed measurement yout (t). Through the reformulation in section 7.1, the estimation task is equivalent to designing a state observer to recover the overall system (7.16)–(7.20), (7.23)–(7.25) using only the measurements at the right boundary x = 2 of the v PDE. The observer is built as a copy of the plant (7.16)–(7.20), (7.23)–(7.25) plus some output error injection terms, as follows: zˆ(0, t) = pw(0, ˆ t) + c¯U (t),

(7.26)

ˆ t) − c2 zˆ(x, t) zˆt (x, t) = − q1 zˆx (x, t) − c1 w(x, + H2 (x)(Yout (t) − Yˆout (t)), x ∈ [0, 1],

(7.27)

ˆx (x, t) − c3 w(x, ˆ t) − c4 zˆ(x, t) w ˆt (x, t) = q2 w + H3 (x)(Yout (t) − Yˆout (t)), x ∈ [0, 1],

(7.28)

ˆ + H4 (Yout (t) − Yˆout (t)), w(1, ˆ t) = qˆ z (1, t) + cζ(t)

(7.29)

ˆ − b1 zˆ(1, t) + H1 (Yout (t) − Yˆout (t)), ˆ˙ = a1 ζ(t) ζ(t)

(7.30)

ˆ vˆ(1, t) = ζ(t),

(7.31)

1 vˆt (x, t) = − vˆx (x, t) + H5 (x)(Yout (t) − Yˆout (t)), x ∈ [1, 2], τ

(7.32)

where Yˆout (t) = [ˆ v (2, t), vˆt (2, t)]T ,

(7.33)

and the row vectors H1 = [H1a , H1b ] ∈ R2 , H2 (x) = [H2a (x), H2b (x)] ∈ R2 , H3 (x) = [H3a (x), H3b (x)] ∈ R2 , H4 = [H4a , H4b ] ∈ R2 , H5 (x) = [H5a (x), H5b (x)] ∈ R2 are observer gains to be determined. Defining the observer error states as ˜ (˜ z (x, t), w(x, ˜ t), ζ(t), v˜(x, t)) ˆ = (z(x, t), w(x, t), ζ(t), v(x, t)) − (ˆ z (x, t), w(x, ˆ t), ζ(t), vˆ(x, t)),

(7.34)

according to (7.16)–(7.20), (7.23)–(7.25), and (7.26)–(7.32), yields the observer error system z˜(0, t) = pw(0, ˜ t),

(7.35)

˜ t) − c2 z˜(x, t) − H2 (x)[˜ v (2, t), v˜t (2, t)]T , z˜t (x, t) = −q1 z˜x (x, t) − c1 w(x, x ∈ [0, 1],

(7.36)

˜x (x, t) − c3 w(x, ˜ t) − c4 z˜(x, t) − H3 (x)[˜ v (2, t), v˜t (2, t)] , w ˜t (x, t) = q2 w T

x ∈ [0, 1],

(7.37)

˜ − H4 [˜ w(1, ˜ t) = q˜ z (1, t) + cζ(t) v (2, t), v˜t (2, t)] , ˜˙ = a1 ζ(t) ˜ − b1 z˜(1, t) − H1 [˜ ζ(t) v (2, t), v˜t (2, t)]T , ˜ v˜(1, t) = ζ(t), T

1 v˜t (x, t) = − v˜x (x, t) − H5 [˜ v (2, t), v˜t (2, t)]T , τ

x ∈ [1, 2].

(7.38) (7.39) (7.40) (7.41)

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176

The determination of the observer gains H1 , H2 (x), H3 (x), H4 , H5 (x) in (7.26)– (7.32) will be completed through two transformations presented next, which convert the observer error system (7.35)–(7.41) to a target observer error system whose exponential stability is straightforward to obtain. First Transformation We apply the transformation ˜ v˜(x, t) = η˜(x, t) + ϕ(x)ζ(t),

(7.42)

ϕ(x) = e−τ a1 (x−1) , x ∈ [1, 2],

(7.43)

where ϕ(x) satisfies

with the purpose of converting (7.39)–(7.41) to ˜˙ = −¯ ˜ − b1 z˜(1, t) − eτ a1 L1 η˜(2, t), ζ(t) a1 ζ(t) η˜(1, t) = 0,

(7.44) (7.45)

1 η˜t (x, t) = − η˜x (x, t), x ∈ [1, 2], τ

(7.46)

where a ¯1 = L1 − a1 > 0,

(7.47)

by choosing the design parameter L1 . Matching (7.39) and (7.44) via (7.42), by virtue of (7.47), we obtain H1 as H1 = [eτ a1 L1 , 0].

(7.48)

Taking the time and spatial derivatives of (7.42), submitting the result into (7.41), and applying (7.39), (7.46), and (7.48), we obtain 1 v˜t (x, t) + v˜x (x, t) + H5 (x)[˜ v (2, t), v˜t (2, t)]T τ ˜ − ϕ(x)b1 z˜(1, t) − ϕ(x)eτ a1 L1 v˜(2, t) = η˜t (x, t) + ϕ(x)a1 ζ(t) 1 1 ˜ + H5 (x)[˜ v (2, t), v˜t (2, t)]T + η˜x (x, t) + ϕ (x)ζ(t) τ τ = −ϕ(x)b1 z˜(1, t) − ϕ(x)eτ a1 L1 v˜(2, t) + H5 (x)[˜ v (2, t), v˜t (2, t)]T  1 ˜ = 0, + ϕ(x)a1 + ϕ (x) ζ(t) τ

(7.49)

where ϕ (x) + τ a1 ϕ(x) = 0 obtained from (7.43) has been used. The row vector H5 (x) is determined to ensure −ϕ(x)b1 z˜(1, t) − ϕ(x)eτ a1 L1 v˜(2, t) + H5 (x)[˜ v (2, t), v˜t (2, t)]T = 0.

(7.50)

We solve the observer gains by using the Laplace transforms, considering the convenience of the algebraic relationships between the output estimation error injections and other observer error states in the frequency domain.

EVENT-TRIGGERED DELAY COMPENSATION

177

First, we derive the algebraic relationships between z˜(1, s) and v˜(2, s). Taking the Laplace transform of (7.44), we get ˜ = −b1 z˜(1, s) − eτ a1 L1 e−τ s η˜(1, s). (s + a ¯1 )ζ(s)

(7.51)

For brevity, we consider all initial conditions to be zero when we take the Laplace transform. (Arbitrary initial conditions could be incorporated into the stability statement through an expanded analysis, which is routine but heavy on additional ¯1 ) does not notation.) Recalling (7.45) as well as a ¯1 > 0, which ensures that (s + a have any zeros in the closed right-half plane, we use (7.51) to get ˜ = −(s + a ζ(s) ¯1 )−1 b1 z˜(1, s).

(7.52)

According to (7.42), (7.45), (7.46), applying (7.43), (7.52), we have ˜ = −r(s)˜ v˜(2, s) = e−τ s η˜(1, s) + ϕ(2)ζ(s) z (1, s),

(7.53)

r(s) = e−τ a1 (s + a ¯1 )−1 b1 .

(7.54)

where

The function r(s) is an asymptotically stable and strictly proper transfer function because a ¯1 > 0. After obtaining the algebraic relationship (7.53), we determine H5 to make (7.50) hold. Taking the Laplace transform of (7.50) and inserting (7.43) and (7.53), we get − ϕ(x)b1 z˜(1, s) − ϕ(x)eτ a1 L1 v˜(2, s) + H5 (x)[˜ v (2, s), s˜ v (2, s)]T = −ϕ(x)b1 z˜(1, s) + ϕ(x)eτ a1 L1 r(s)˜ z (1, s) − H5a (s; x)r(s)˜ z (1, s) − H5b (s; x)sr(s)˜ z (1, s) = −ϕ(x)b1 z˜(1, s) + ϕ(x)eτ a1 L1 r(s)˜ z (1, s) − H5a (s; x)r(s)˜ z (1, s) − H5b (s; x)(e−τ a1 b1 − a ¯1 r(s))˜ z (1, s) = −(ϕ(x)b1 + H5b (s; x)e−τ a1 b1 )˜ z (1, s) + (ϕ(x)eτ a1 L1 + H5b (s; x)¯ a1 − H5a (s; x))r(s)˜ z (1, s) = 0,

(7.55)

where sr(s) = (s + a ¯1 − a ¯1 )e−τ a1 (s + a ¯1 )−1 b1 = e−τ a1 b1 − a ¯1 r(s)

(7.56)

by recalling (7.54) has been used. For (7.55) to hold, the transfer function H5 (x) is then defined as H5 (x) = [e−τ a1 (x−2) L1 − e−τ a1 (x−2) a ¯1 , −e−τ a1 (x−2) ].

(7.57)

In the above derivation, we have completed the conversion between (7.39)–(7.41) and (7.44)–(7.46) through (7.42), under the designed H1 , H5 (x). ˜ t) in the boundary condition In what follows, H4 is determined to make w(1, (7.38) be zero—that is, to render ˜ − H4 [˜ w(1, ˜ t) = q˜ z (1, t) + cζ(t) v (2, t), v˜t (2, t)]T = 0.

(7.58)

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178

Taking the Laplace transform of (7.58) and inserting (7.52) and (7.53), we obtain ˜ − H4a (x)˜ w(1, ˜ s) = q˜ z (1, s) + cζ(s) v (2, s) − H4b s˜ v (2, s) z (1, s) + H4b sr(s)˜ z (1, s) = q˜ z (1, s) − c(s + a ¯1 )−1 b1 z˜(1, s) + H4a r(s)˜ τ a1 = q˜ z (1, s) − ce r(s)˜ z (1, s) + H4a r(s)˜ z (1, s) + H4b (e−τ a1 b1 − a ¯1 r(s))˜ z (1, s) z (1, s) + (H4a − H4b a ¯1 − ceτ a1 )r(s)˜ z (1, s) = (q + H4b e−τ a1 b1 )˜ = 0,

(7.59)

where (7.54), (7.56) have been used. For (7.59) to hold, the row vector H4 is chosen as

 q¯ a1 q τ a1 . H4 = − −τ a1 + ce , − −τ a1 e b1 e b1

(7.60)

Through applying the first transformation (7.42), with H4 , H5 (x), and H1 , (7.35)– (7.41) is converted to the intermediate system as z˜(0, t) = pw(0, ˜ t),

(7.61) T

˜ t) − c2 z˜(x, t) − H2 (x)[˜ v (2, t), v˜t (2, t)] , z˜t (x, t) = −q1 z˜x (x, t) − c1 w(x,

(7.62)

˜x (x, t) − c3 w(x, ˜ t) − c4 z˜(x, t) − H3 (x)[˜ v (2, t), v˜t (2, t)]T , w ˜t (x, t) = q2 w w(1, ˜ t) = 0, ˜˙ = −¯ ˜ − b1 z˜(1, t) − eτ a1 L1 η˜(2, t), ζ(t) a1 ζ(t) η˜(1, t) = 0, 1 η˜t (x, t) = − η˜x (x, t). τ

(7.63) (7.64) (7.65) (7.66) (7.67)

Next, we introduce the second transformation to decouple the couplings in (7.62), (7.63). Second Transformation We now apply the second transformation [21] 

1

˜ t) − w(x, ˜ t) = β(x,

x

 z˜(x, t) = α(x, ˜ t) −

ψ(x, y)˜ α(y, t)dy,

(7.68)

φ(x, y)˜ α(y, t)dy,

(7.69)

1

x

with the kernels ψ(x, y), φ(x, y) satisfying c4 , q1 + q2

(7.70)

φ(0, y) = pψ(0, y),

(7.71)

ψ(x, x) =

−q1 ψy (x, y) + q2 ψx (x, y) − c4 φ(x, y) + (c2 − c3 )ψ(x, y) = 0,

(7.72)

−q1 φx (x, y) − q1 φy (x, y) − c1 ψ(x, y) = 0.

(7.73)

EVENT-TRIGGERED DELAY COMPENSATION

179

The purpose of the transformations (7.68), (7.69) is to convert the intermediate system (7.61)–(7.67) to the target system (the subsystem η˜(·, t), given in (7.66), (7.67), is removed for brevity because η˜(·, t) ≡ 0, t ≥ τ ), as follows: ˜ t), α ˜ (0, t) = pβ(0,



α ˜ t (x, t) = − q1 α ˜ x (x, t) +  β˜t (x, t) = q2 β˜x (x, t) +

1

(7.74) 1

˜ t)dy − c2 α ˜ t), ¯ (x, y)β(y, M ˜ (x, t) − c1 β(x,

(7.75)

x

˜ t)dy − c3 β(x, ˜ t), ¯ (x, y)β(y, N

(7.76)

x

˜ t) = 0, β(1, ˜˙ = − a ˜ − b1 α ˜ (1, t) ζ(t) ¯1 ζ(t) ¯ and N ¯ are defined as for t ≥ τ , where the integral operator kernels M  y ¯ (x, y) = ¯ (δ, y)dδ − c1 φ(x, y), M φ(x, δ)M x  y ¯ ¯ (δ, y)dδ − c1 ψ(x, y). N (x, y) = ψ(x, δ)M

(7.77) (7.78)

(7.79) (7.80)

x

In what follows, H2 (x), H3 (x) are determined by matching the intermediate system (7.61)–(7.65) and the target system observer error system (7.74)–(7.78) via (7.68), (7.69). Inserting (7.68), (7.69) into (7.63) along (7.75), (7.76) and applying (7.70)– (7.72), (7.80), we get ˜x (x, t) + c4 z˜(x, t) + c3 w(x, ˜ t) + H3 (x)[˜ v (2, t), v˜t (2, t)]T w ˜t (x, t) − q2 w = q1 ψ(x, 1)˜ α(1, t) + H3 (x)[˜ v (2, t), v˜t (2, t)]T = 0,

(7.81)

where the detailed calculation is shown in (7.193) in appendix 7.7A. We thus know that the following equation needs to be satisfied: q1 ψ(x, 1)˜ z (1, t) + H3 (x)[˜ v (2, t), v˜t (2, t)]T = 0,

(7.82)

where α ˜ (1, t) = z˜(1, t) according to (7.69) has been used. Rewriting (7.82) in the frequency domain and applying (7.53), (7.56), we obtain q1 ψ(x, 1)˜ z (1, s) + H3 (x)[˜ v (2, s), s˜ v (2, s)]T = q1 ψ(x, 1)˜ z (1, s) − H3a (x)r(s)˜ z (1, s) − H3b (x)sr(s)˜ z (1, s) = q1 ψ(x, 1)˜ z (1, s) − H3a (x)r(s)˜ z (1, s) − H3b (x)(e−τ a1 b1 − a ¯1 r(s))˜ z (1, s) = (q1 ψ(x, 1) − H3b (x)e−τ a1 b1 )˜ z (1, s) + (H3b (x)¯ a1 − H3a (x))r(s)˜ z (1, s) = 0. The transfer function H3 (x) is chosen as

 a ¯1 q1 ψ(x, 1) q1 ψ(x, 1) H3 (x) = , −τ a1 . e−τ a1 b1 e b1

(7.83)

(7.84)

Inserting (7.68), (7.69) into (7.62) along (7.75), (7.76) and applying (7.73), (7.79), we get

CHAPTER SEVEN

180

z˜t (x, t) + q1 z˜x (x, t) + c1 w(x, ˜ t) + c2 z˜(x, t) + H2 (x)[˜ v (2, t), v˜t (2, t)]T = q1 φ(x, 1)˜ α(1, t) + H2 (x)[˜ v (2, t), v˜t (2, t)]T = 0,

(7.85)

where the detailed calculation is shown in (7.194) in appendix 7.7A. Therefore, H2 (x)[˜ v (2, t), v˜t (2, t)]T should satisfy q1 φ(x, 1)˜ z (1, t) + H2 (x)[˜ v (2, t), v˜t (2, t)]T = 0,

(7.86)

where α ˜ (1, t) = z˜(1, t) according to (7.69) has been used. Taking the Laplace transform of (7.86) and recalling (7.53), (7.56), we obtain q1 φ(x, 1)˜ z (1, s) + H2 (x)[˜ v (2, s), s˜ v (2, s)]T = q1 φ(x, 1)˜ z (1, s) − H2a (x)r(s)˜ z (1, s) − H2b (x)sr(s)˜ z (1, s) = q1 φ(x, 1)˜ z (1, s) − H2a (x)r(s)˜ z (1, s) − H2b (x)(e−τ a1 b1 − a ¯1 r(s))˜ z (1, s) = (q1 φ(x, 1) − H2b (x)e−τ a1 b1 )˜ z (1, s) + (H2b (x)¯ a1 − H2a (x))r(s)˜ z (1, s) = 0.

(7.87)

The transfer function H2 (x) is determined to be

 a ¯1 q1 φ(x, 1) q1 φ(x, 1) , . H2 (x) = e−τ a1 b1 e−τ a1 b1

(7.88)

The boundary condition (7.74) follows directly from inserting x = 0 into (7.68), (7.69) and applying (7.61), (7.71). The second conversion is thus completed, and the two PDEs (7.62), (7.63) are decoupled, which can be seen in (7.75), (7.76). After performing the above two transformations, we have converted the original observer error system (7.35)–(7.41) to the target observer error system (7.74)– (7.78) (for t ∈ [τ, ∞), η˜(x, t) ≡ 0 according to (7.66), (7.67) is removed for brevity). The exponential convergence of the observer error system is shown in the next subsection, which guarantees that the output estimation error injections in the observer (7.26)–(7.32) are exponentially convergent to zero. Stability Analysis of the Observer Error System ˜ ∈ C 0 ([0, 1]; R2 ) × Theorem 7.1. For all initial data ((˜ z [0], w[0]) ˜ T , v˜[0], ζ(0)) 1 − C ([1, 2]; R) × R and m(0) ∈ R , the exponential convergence of the observer error system (7.35)–(7.41) holds in the sense of the norm ˜ ˜˙ (7.89) ˜ z (·, t) ∞ + w(·, ˜ t) ∞ + ˜ v (·, t) ∞ + ˜ vt (·, t) ∞ + ζ(t) + ζ(t) , where the decay rate is adjustable by L1 . Proof. The stability of the original observer error system is obtained by analyzing the stability of the target observer error system (7.74)–(7.78) and using the invertibility of the transformations. With the method of characteristics, it is easy to show ˜ ≡ 0 after t = τ + 1 , considering (7.76), (7.77). According to (7.74), (7.75), that β[t] q2 ˜ exponenwe have α ˜ [t] ≡ 0 after t0 = τ + 1 + 1 . Because a ¯1 > 0, we then have ζ(t) q1

q2

tially convergent to zero after t0 . Recalling η˜[t] ≡ 0 for t ≥ τ , we obtain η˜t (x, t) ≡ 0 ˜ is exponentially convergent for t ≥ τ . According to α ˜ (1, t) ≡ 0 and the fact that ζ(t)

EVENT-TRIGGERED DELAY COMPENSATION

181

˜˙ is exponentially convergent to zero after t0 via to zero, after t0 , we find that ζ(t) (7.78). Therefore, we know that ˜ ˜˙ ˜ t) ∞ + ˜ ¯ = ˜ Ω(t) α(·, t) ∞ + β(·, η (·, t) ∞ + ˜ ηt (·, t) ∞ + ζ(t) + ζ(t) is bounded by an exponential decay with the decay rate λe for t ≥ t0 . The decay ˜ In other words, the decay rate rate λe depends on the decay rate of the ODE ζ(t). λe depends on the choice of L1 according to (7.47). It should be noted that the transient in the finite time [0, t0 ) can be bounded by an arbitrarily fast decay rate considering the trade-off between the decay rate and the overshoot coefficient— that is, the higher the decay rate, the higher the overshoot coefficient. Therefore, ¯ we conclude that there is exponential stability in the sense that Ω(t) is bounded by an exponential decay rate λe with some overshoot coefficients for t ≥ 0. Applying the transformations (7.42) and (7.68), (7.69), we respectively have   ˜ ˜ v (·, t) ∞ ≤ Υ1a ˜ η (·, t) ∞ + ζ(t) ,   ˜˙ ˜ vt (·, t) ∞ ≤ Υ1b ˜ ηt (·, t) ∞ + ζ(t) ,   ˜ t) ∞ , ˜ z (·, t) ∞ + w(·, α(·, t) ∞ + β(·, ˜ t) ∞ ≤ Υ1c ˜ for some positive Υ1a , Υ1b , Υ1c . The proof is complete.

7.3

DELAY-COMPENSATED OUTPUT-FEEDBACK CONTROLLER

In the last section, we designed an observer that compensates for the time delay in the output measurements of the distal ODE to track the states of the overall system (7.16)–(7.21). In this section, we design an output-feedback control law U (t) based on the observer (7.26)–(7.32) with output estimation error injection terms assumed absent, in accordance with the result of their convergence to zero in theorem 7.1. The separation principle is then verified and applied in the stability analysis of the resulting closed-loop system. With the purpose of removing the coupling terms in (7.27), (7.28) and making the system parameter in the distal ODE (7.30) negative, a PDE backstepping transformation in the form [48]  1 K3 (x, y)ˆ z (y, t)dy α(x, t) = zˆ(x, t) −  −

1

x

ˆ J3 (x, y)w(y, ˆ t)dy − γ(x)ζ(t),

x



β(x, t) = w(x, ˆ t) −  −

1

(7.90)

1

K2 (x, y)ˆ z (y, t)dy x

ˆ J2 (x, y)w(y, ˆ t)dy − λ(x)ζ(t)

(7.91)

x

is introduced, where the kernels K3 (x, y), J3 (x, y), γ(x), K2 (x, y), J2 (x, y), λ(x) satisfy

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182

q1 K3 (x, 1) = q2 J3 (x, 1)q − γ(x)b1 , c1 , J3 (x, x) = q2 + q1 − q1 J3x (x, y) + q2 J3y (x, y) + c1 K3 (x, y) − (c2 − c3 )J3 (x, y) = 0, − q1 K3x (x, y) − q1 K3y (x, y) + c4 J3 (x, y) = 0, γ(1) = F1 ,

(7.92) (7.93)

(7.94) (7.95) (7.96)

− q1 γ  (x) − γ(x)(a1 + c2 ) − q2 J3 (x, 1)c = 0, q2 qJ2 (x, 1) = q1 K2 (x, 1) + λ(x)b1 , −c4 K2 (x, x) = , q1 + q2 q2 J2x (x, y) + q2 J2y (x, y) + c1 K2 (x, y) = 0,

(7.100)

q2 K2x (x, y) − q1 K2y (x, y) + c4 J2 (x, y) + c2 K2 (x, y) − c3 K2 (x, y) = 0, q2 λ (x) − λ(x)(a1 + c3 ) − q2 J2 (x, 1)c = 0,

(7.101) (7.102)

λ(1) = qγ(1) + c

(7.97) (7.98) (7.99)

(7.103)

in order to convert (7.26)–(7.32) into the target system: α(0, t) = pβ(0, t), αt (x, t) = − q1 αx (x, t) − c2 α(x, t), βt (x, t) = q2 βx (x, t) − c3 β(x, t), β(1, t) = qα(1, t), ˆ˙ = − am ζ(t) ˆ − b1 α(1, t), ζ(t) ˆ vˆ(1, t) = cζ(t), 1 vˆt (x, t) = − vˆx (x, t), τ

(7.104) (7.105) (7.106) (7.107) (7.108) (7.109) (7.110)

where a m = b1 F 1 − a 1 > 0

(7.111)

is imposed by choosing the design parameter F1 , which appears within the gain functions of the feedback law   1 1 ¯ 1¯ ˆ 1 1 ¯ K1 (x)ˆ K2 (x)w(x, z (x, t)dx + ˆ t)dx + K (7.112) U (t) = 3 ζ(t), c¯ 0 c¯ 0 c¯ where ¯ 1 (x) = K3 (0, x) − pK2 (0, x), K ¯ 2 (x) = J3 (0, x) − pJ2 (0, x), K ¯ 3 = γ(0) − pλ(0). K

(7.113) (7.114) (7.115)

The conditions of the kernels (7.92)–(7.103) are obtained by matching (7.105)– (7.108) and (7.27)–(7.30), whose calculation details are given in steps 1–3 in

EVENT-TRIGGERED DELAY COMPENSATION

183

appendix 7.7B. The equation sets (7.92)–(7.97) and (7.98)–(7.103) have an analogous structure with (19)–(24) in [183]. Following the proof of lemma 1 in [183], we obtain the well-posedness of (7.92)–(7.103). Following section 2.4 in [183], we have that the inverse transformation of (7.90), (7.91) exists as  1 M(x, y)α(y, t)dy zˆ(x, t) = α(x, t) −  −

x 1

ˆ N (x, y)β(y, t)dy − G(x)ζ(t),

x



w(x, ˆ t) = β(x, t) −  −

1

(7.116)

D(x, y)α(y, t)dy

x 1

ˆ T (x, y)β(y, t)dy − P(x)ζ(t),

(7.117)

x

where the well-posedness of the kernels M(x, y), N (x, y), G(x), D(x, y), T (x, y), P(x) can be obtained from the well-posedness of (7.92)–(7.103), according to chapter 9.9 of [175].

7.4

EVENT-TRIGGERING MECHANISM

In this section we introduce an event-triggered control scheme for the stabilization of the overall PDE-ODE-PDE system (7.16)–(7.20), (7.23)–(7.25). This scheme relies on the delay-compensated observer-based output-feedback continuous-in-time control U (t) designed in the last section, and on a dynamic event-triggering mechanism (ETM) that is realized using the states from the delay-compensated observer. The event-triggered control signal Ud (t) is the value of the continuous-in-time U (t) at the time instants tk but applied until time tk+1 —that is, Ud (t) = U (tk ) =

1 c¯



1 0

1 + c¯

¯ 1 (x)ˆ K z (x, tk )dx



1 0

1¯ ˆ ¯ 2 (x)w(x, K ˆ tk )dx + K 3 ζ(tk ), c¯

t ∈ [tk , tk+1 )

(7.118)

for k ∈ Z+ . A deviation d(t) between the continuous-in-time control signal and the event-based one is given as d(t) = U (t) − Ud (t). (7.119) Inserting (7.118), the right boundary of the target system (7.104)–(7.110) becomes α(0, t) = pβ(0, t) − c¯d(t).

(7.120)

Taking the Laplace transform of (7.105)–(7.110), according to section 3.2 in [49], we obtain the following algebraic relationships between α(0, s) and β(0, s) as β(0, s) = qe

−(c3 +s) (c +s) − 2q q2 1

α(0, s).

(7.121)

CHAPTER SEVEN

184 Inserting (7.121) into (7.120) in the frequency domain, we have α(0, s) =

−¯ c d(s), (1 − h(s))

(7.122)

where c3

c2

1

1

h(s) = pqe−( q2 + q1 ) e−( q2 + q1 )s .

(7.123)

From (7.122), (7.123), assumption 7.1 guarantees |α(0, t)| ≤ |h(0)| sup |α(0, ξ)| + |¯ c| sup |d(ξ)|, 0≤ξ≤t

0≤ξ≤t

where the constant |h(0)| is strictly smaller than 1. By the small gain theorem, we obtain (7.124) α(0, t)2 ≤ λd sup d(ξ)2 , 0≤ξ≤t

where λd =

c¯2 . (1 − |h(0)|)2

Please note that we obtain (7.124), which will be used in the Lyapunov analysis, based on (7.105)–(7.110) (state-feedback loop)—that is, assuming the observer error injections absent—with the purpose of avoiding heavy additional notation. Incorporating these observer error injections will not change the stability result obtained in the Lyapunov analysis, which will be shown later, by virtue of exponential convergence of the observer errors in theorem 7.1. Similarly to [57], the ETM to determine the triggering times is designed to be governed by the following dynamic triggering condition: tk+1 = inf{t ∈ R+ |t > tk |d(t)2 ≥ θV (t) − μm(t)}.

(7.125)

The internal dynamic variable m(t) satisfies the ordinary differential equation: 2 m(t) ˙ = − ηm(t) − κ1 α(1, t)2 − κ2 α(0, t)2 − κ3 vˆ(2, t)2 − κ4 Y˜ (t) ,

(7.126)

where Y˜ (t) = Yout (t) − Yˆout (t), and where the initial condition of m is m(0) < 0, which guarantees that m(t) < 0.

(7.127)

Inequality (7.127) follows from the ODE in (7.126), the nonpositivity of the nonhomogeneous terms on its right-hand side, the strict negativity of m(0), the variationof-constants formula, and the comparison principle. Thus, the fact that m(t) < 0 would not be affected by (7.125). Introducing the internal dynamic variable m(t) defined by (7.126) is used in proving the existence of a minimal dwell time, which will be seen clearly in the proof of lemma 7.2. The Lyapunov function V (t) in (7.125) is defined as

EVENT-TRIGGERED DELAY COMPENSATION

 1 ˆ 2 1 1 δ1 x V (t) = ζ(t) + e β(x, t)2 dx 2 2 0  1  2 1 1 + rb e−δ2 x α(x, t)2 dx + rc e−x vˆ(x, t)2 dx. 2 2 0 1

185

(7.128)

The positive design parameters θ, μ, η, κ1 , κ2 , κ3 , δ1 , δ2 , rc , rb satisfy κ1 > 2λ1 ,

(7.129)

κ2 > 2λ1 ,

(7.130)

δ1 >

2|c3 | , q2

(7.131)

2|c2 | , q1 am eτ , rc < 2 1 b2 2 rb > q 2 q2 eδ1 +δ2 + 1 eδ2 + κ1 eδ2 , q1 q1 a m q1 1 κ3 < rc e−2 , 2τ σa μ< 1 , 2 q1 rb λd + κ2 λd

 σa μτ μ μ μ 2 θ < min 1 , κ3 e , κ1 , κ2 , κ4 , rc 2λα1 2λα0 2 2 q1 rb λd + κ2 λd δ2 >

(7.132) (7.133) (7.134) (7.135) (7.136) (7.137)

where η is free and associated with the dwell time, which can be seen later, and λ1 is a positive constant that will be given in lemma 7.1 and only depends on the parameters of the plant and the design parameters L1 , F1 in the continuous-in-time control law, and where

 2κ4 > 0, (7.138) σa = min λa , η, Re

 1 1 1 1 λa = ¯ min am − rc e−1 , δ1 q2 − |c3 |, 2 τ 2 ξ2  −2 1 −δ2 −δ2 e δ2 q1 rb e rc , − rb |c2 |e , (7.139) 2 2τ 2κ4 , λe 1 1 λα1 = |b1 | + q1 rb e−δ2 , 2 2 1 λα0 = 2 q2 max{1, c¯2 }, q Re >

(7.140) (7.141) (7.142)

where λe is a positive constant depending on the design parameter L1 , as shown in proof of theorem 7.1. The conditions of all the parameters L1 , F1 , κ1 , κ2 , δ1 , δ2 , rc , rb , κ3 , μ, θ are given in (7.47), (7.111), (7.129)–(7.137), which are cascaded rather than coupled.

CHAPTER SEVEN

186

The event-triggering condition, including (7.125), (7.126), (7.128), only uses the observer states, recalling the transformations (7.90), (7.91). Lemma 7.1. For d(t) defined in (7.119), there exists a positive constant λ1 such that  2 ˙ 2 ≤ λ1 Ω(t) + α(1, t)2 + α(0, t)2 + d(t)2 + Y˜ (t) d(t) (7.143) for t ∈ (tk , tk+1 ), where ˆ 2 Ω(t) = α(·, t) 2 + β(·, t) 2 + ζ(t)

(7.144)

and where λ1 only depends on the parameters of the plant and the design parameters L1 , F1 in the continuous-in-time control law. Proof. Taking the time derivative of (7.119), we have ˙ 2 = U˙ (t)2 d(t)

(7.145)

because U˙ d (t) = 0 for t ∈ (tk , tk+1 ). Recalling (7.112), we have ˙ 2 = U˙ (t)2 d(t)

1 ¯ 1 (0)ˆ ¯ 2 (0)w(0, = 2 q1 K z (0, t) − q2 K ˆ t) c¯  ˆ ¯ ¯ ¯ ¯ 2 (1)q2 )ζ(t) ¯ 3 a1 + c K − K1 (1)q1 − q K2 (1)q2 + K3 b1 zˆ(1, t) + (K 

L

 ¯ 1 (x)q1 − K ¯ 1 (x)c2 − K ¯ 2 (x)c4 zˆ(x, t)dx K

L

  ¯ ¯ ¯ K2 (x)c3 − K2 (x)q2 + K1 (x)c1 w(x, ˆ t)dx

+  −

0

0



1

+ 0

 ¯ 1 (x)H2 (x)dx + K

1 0

¯ 2 (x)H3 (x)dx K

2 ¯ 3 H1 + K ¯ 2 (1)q2 H4 Y˜ (t) +K

≤ λ0 zˆ(1, t)2 + w(0, ˆ t)2 + zˆ(0, t)2 + ˆ z (·, t) 2 + w(·, ˆ t) 2  ˜ 2 2 ˆ + ζ(t) + Y (t)

(7.146)

for some positive λ0 depending only on the parameters of the plant and the design parameters L1 , F1 in the continuous-in-time control law. Applying the inverse transformation and (7.120), we obtain (7.143). The proof of lemma 7.1 is complete. Remark 7.1. The target system corresponding to the output-feedback loop is obtained by applying the backstepping transformation (7.90), (7.91) into the observer (7.26)–(7.32) and replacing U in (7.26) by Ud defined in (7.118). The resulting system is (7.120), (7.105)–(7.110) with output estimation error injection terms

EVENT-TRIGGERED DELAY COMPENSATION

187

G2 (x)Y˜ (t), G3 (x)Y˜ (t), H4 Y˜ (t), H1 Y˜ (t), H5 (x)Y˜ (t) in (7.105), (7.106), (7.107), (7.108), (7.110), respectively, where the bounded functions G2 , G3 depend on the observer gains obtained in section 7.2 and the kernels in the backstepping transformation (7.90), (7.91). In the following analysis in this chapter, we denote this system as the S-system. The following lemma proves the existence of a minimal dwell time independent of initial conditions. Lemma 7.2. For some κ1 , κ2 , θ, there exists a minimal dwell time independent of initial conditions between any two successive triggering times—that is, tk+1 − tk ≥ T for all k ≥ 0 where T is a positive constant. Proof. Let us introduce the function 2

ψ(t) =

d(t) + μ2 m(t) , θV (t) − μ2 m(t)

(7.147)

which is proposed in [57]. We see that ψ(tk+1 ) = 1 because the event is triggered, and ψ(tk ) < 0 because of m(t) < 0 and d(tk ) = 0. The function ψ(t) is continuz (·, 0), w(·, ˆ 0))T ∈ C 0 ([0, 1]; R2 ), ous on [tk , tk+1 ] for the given initial conditions (ˆ 1 − ˆ ∈ R, m(0) ∈ R , following the proof of proposition 14.1 vˆ(·, 0) ∈ C ([1, 2]; R), ζ(0) in chapter 14 and recalling the backstepping transformations (7.90), (7.91). By the intermediate value theorem, there exists t∗ > tk such that ψ(t) ∈ [0, 1] when t ∈ [t∗ , tk+1 ]. The lower bound of the minimal dwell time T can be defined as the minimal time it takes for ψ(t) from 0 to 1. For all t ∈ [t∗ , ti+1 ], taking the time derivative of V (t) (7.128) along the system defined in remark 7.1, applying integration by parts, one gets ˆ 1 α(1, t) + 1 q2 eδ1 β(1, t)2 − 1 q2 β(0, t)2 ˆ 2 − ζ(t)b V˙ (t) = − am ζ(t) 2 2  1  1 1 − δ 1 q2 eδ1 x β(x, t)2 dx − c3 eδ1 x β(x, t)2 dx 2 0 0 1 1 −δ2 2 − rb q1 e α(1, t) + q1 rb α(0, t)2 2 2  1  1 1 −δ2 x − δ2 q1 rb e α(x, t)2 dx − c2 rb e−δ2 x α(x, t)2 dx 2 0 0  2 1 1 1 e−x vˆ(x, t)2 dx − rc e−2 vˆ(2, t)2 + rc e−1 vˆ(1, t)2 − rc 2τ 2τ 2τ 1  1  1 δ1 x + e β(x, t)H3 (x)dx + rb e−δ2 x α(x, t)H2 (x)dx 0

 + rc

1

0

2

e

−x

ˆ vˆ(x, t)H5 (x)dx + ζ(t)H1 Y˜ (t).

(7.148)

We then have 2 1 V˙ (t) ≥ − μ0 Ω1 (t) − λα1 α(1, t)2 − λα0 (α(0, t)2 + d(t)2 ) − rc e−2 vˆ(2, t)2 − Y˜ (t) 2τ (7.149)

CHAPTER SEVEN

188 where ˆ 2 Ω1 (t) = α(·, t) 2 + β(·, t) 2 + ˆ v (·, t) 2 + ζ(t) and

μ0 = max

1 δ1 q2 eδ1 + |c3 |eδ1 + e2δ1 max {G3 (x)2 }, 2 x∈[0,1] 1 δ2 q1 rb + |c2 |rb + rb 2 max {G2 (x)2 }, 2 x∈[0,1]  1 1 rc e−1 + rc 2 max {H5 (x)2 }, am + |b1 | + H12 , 2τ 2 x∈[1,2]

(7.150)

(7.151)

and where the positive constants λα1 , λα0 are shown in (7.141), (7.142). Taking the derivative of (7.147), applying Young’s inequality, using (7.143) in lemma 7.1, and inserting (7.149), we have ˙ + μ m(t)) ˙ (θV˙ (t) − μ2 m(t)) (2d(t)d(t) 2 ˙ − ψ ψ˙ = μ μ θV (t) − 2 m(t) θV (t) − 2 m(t) 

1 1 2 2 2 2 ≤ λ Ω(t) + r λ α(1, t) + r λ α(0, t) + + r λ r 1 1 1 1 1 1 1 1 d(t) θV (t) − μ2 m(t) r1 

 2 μ 1 ˙ − + r1 λ1 Y˜ (t) + m(t) θ − μ0 Ω1 (t) − λα1 α(1, t)2 2 θV (t) − μ2 m(t)  2 1 μ ˙ ψ, (7.152) − λα0 (α(0, t)2 + d(t)2 ) − Y˜ (t) − rc e−2 vˆ(2, t)2 − m(t) 2τ 2 where the positive constant r1 comes from applying Young’s inequality. Inserting (7.126) to replace m(t), ˙ one obtains 

1 1 2 2 2 λ Ω(t) + r λ α(1, t) + r λ α(0, t) + + r λ ψ˙ ≤ r 1 1 1 1 1 1 1 1 d(t) θV (t) − μ2 m(t) r1 2 μ 2  μ μ μ + r1 λ1 Y˜ (t) − ηm(t) − κ1 α(1, t)2 − κ2 α(0, t)2 − κ4 Y˜ (t) 2 2 2 2

1 − − θμ0 Ω1 (t) − θλα1 α(1, t)2 θV (t) − μ2 m(t) 2 θ − θλα0 α(0, t)2 − θλα0 d(t)2 − θ Y˜ (t) − rc e−2 vˆ(2, t)2 2τ  μ μ μ μ μ ˜ 2 2 2 2 + ηm(t) + κ1 α(1, t) + κ2 α(0, t) + κ3 vˆ(2, t) + κ4 Y (t) ψ. 2 2 2 2 2 (7.153) Recalling (7.128), (7.144), (7.150) and the fact that ξ1 Ω1 (t) ≤ V (t) ≤ ξ2 Ω1 (t), where the positive constants ξ1 , ξ2 are

 1 1 −δ2 1 −2 ξ1 = min , rb e , rc e , 2 2 2

(7.154)

(7.155)

EVENT-TRIGGERED DELAY COMPENSATION

ξ2 = max

189

 1 1 δ1 1 1 −1 , e , rb , rc e , 2 2 2 2

(7.156)

we have ψ˙ ≤



  1 μ  1 2 2 + r λ 1 1 d(t) + r1 λ1 − κ1 α(1, t) μ θV (t) − 2 m(t) r1 2  2   λ1 μ  μ  μ + r1 λ1 − κ2 α(0, t)2 + r1 λ1 − κ4 Y˜ (t) + r1 V (t) − ηm(t) 2 2 ξ1 2

  1 μ μ0 − − θ V (t) − θλα1 − κ1 α(1, t)2 μ θV (t) − 2 m(t) ξ1 2   μ − θλα0 − κ2 α(0, t)2 − θλα0 d(t)2 2  2  θ  μ  μ μ rc e−2 − κ3 vˆ(2, t)2 + ηm(t) ψ, (7.157) − θ − κ4 Y˜ (t) − 2 2τ 2 2

where Ω1 (t) ≥ Ω(t) has been used. Choosing r1 = μ and applying (7.129), (7.130), (7.137) (the last four terms) and the inequalities −

θV

μ 2 ηm(t) (t) − μ2 m(t)

≤−

μ 2 ηm(t) − μ2 m(t)

= η,

V (t) V (t) 1 ≤ = , θV (t) − μ2 m(t) θV (t) θ d(t)2 + μ2 m(t) − μ2 m(t) d(t)2 = ≤ ψ(t) + 1, θV (t) − μ2 m(t) θV (t) − μ2 m(t) which hold because m(t) < 0, we see that (7.157) becomes ( μ1 + μλ1 )d(t)2 + μ λξ11 V (t) − μ2 ηm(t)

μ0 + ηψ + ψ + θλα0 ψ 2 + θλα0 ψ θV (t) − μ2 m(t) ξ1  1 μλ1 1 + μλ1 + η + μ0 + θλα0 ψ + + μλ1 + ≤ θλα0 ψ 2 + + η. (7.158) μ μ θξ1

˙ ≤ ψ(t)

The differential inequality (7.158) has the form ψ˙ ≤ n1 ψ 2 + n2 ψ + n3 ,

(7.159)

where n1 = θλα0 ,

are positive constants.

(7.160)

n2 =

1 + μλ1 + η + μ0 + θλα0 , μ

(7.161)

n3 =

1 μλ1 + μλ1 + +η μ θξ1

(7.162)

CHAPTER SEVEN

190

It follows that the time needed by ψ to go from 0 to 1 is at least  1 1 d¯ s > 0, T= n + n ¯ + n3 s¯2 1 2s 0

(7.163)

which is independent of initial conditions. 7.5

STABILITY ANALYSIS

In this section we state the main result of the chapter after we first establish an intermediate result. Lemma 7.3. With the arbitrary initial data (α(x, 0), β(x, 0))T ∈ C 0 ([0, 1]; R2 ), ˆ ∈ R, m(0) ∈ R− , for (7.105)–(7.110), (7.120), (7.126), vˆ(x, 0) ∈ C 1 ([1, 2]; R), ζ(0) ˆ the exponential convergence is achieved in the sense of the norm |ζ(t)| + α(·, t) + β(·, t) + ˆ v (·, t) + |m(t)|. Proof. There exists a function e(t) = Υd e−λe t , for some positive Υd , λe such that

˜ Y (t) ≤ e(t),

(7.164)

recalling the exponential convergence result proved in theorem 7.1. Let us consider the Lyapunov function 1 Va (t) = V (t) − m(t) + Re e(t)2 (7.165) 2 where m(t) is defined in (7.126) (m(t) < 0), and V (t) is given in (7.128). By virtue of (7.150), we have d1 (Ω1 (t) + |m(t)| + e(t)2 ) ≤ Va (t) ≤ d2 (Ω1 (t) + |m(t)| + e(t)2 )

(7.166)

for some positive d1 , d2 . Taking the derivative of (7.165) along (7.105)–(7.110), (7.120) and recalling (7.126), we obtain V˙ a (t) = V˙ − m(t) ˙ + Re e(t)e(t) ˙ ˆ 1 α(1, t) + 1 q2 eδ1 β(1, t)2 − 1 q2 β(0, t)2 ˆ 2 − ζ(t)b = − am ζ(t) 2 2  1  1 1 − δ 1 q2 eδ1 x β(x, t)2 dx − c3 eδ1 x β(x, t)2 dx 2 0 0 1 1 − rb q1 e−δ2 α(1, t)2 + q1 rb α(0, t)2 2 2  1  1 1 − δ2 q1 rb e−δ2 x α(x, t)2 dx − c2 rb e−δ2 x α(x, t)2 dx 2 0 0  2 1 1 1 e−x vˆ(x, t)2 dx − rc e−2 vˆ(2, t)2 + rc e−1 vˆ(1, t)2 − rc 2τ 2τ 2τ 1

EVENT-TRIGGERED DELAY COMPENSATION

191

2 − Re λe e(t)2 + ηm(t) + κ1 α(1, t)2 + κ2 α(0, t)2 + κ3 vˆ(2, t)2 + κ4 Y˜ (t) . (7.167) Recalling (7.124), (7.31) and applying Young’s inequality and the Cauchy-Schwarz inequality, we have   1 1 1 −1 ˆ 2 q2 b21 −δ2 δ1 ˙ Va (t) ≤ − q1 rb e − q2 e − − κ1 α(1, t)2 ζ(t) − am − rc e 2 τ 2 2 2am    1 1 1 q1 rb λd + κ2 λd sup d(ξ)2 − δ1 q2 − |c3 | + eδ1 x β(x, t)2 dx 2 2 0≤ξ≤t 0  1   2 1 1 −δ2 x 2 δ2 q1 rb − rb |c2 | e α(x, t) dx − rc e−x vˆ(x, t)2 dx − 2 2τ 0 1  1 −2 rc e − κ3 vˆ(2, t)2 − (Re λe − κ4 )e(t)2 + ηm(t). (7.168) − 2τ Applying (7.131), (7.132), (7.133), (7.134), (7.135), we thus arrive at  1 2 ˙ Va (t) ≤ − λa V (t) + ηm(t) − κ4 e(t) + q1 rb λd + κ2 λd sup d(ξ)2 , 2 0≤ξ≤t where λa is given in (7.139). Recalling (7.165), we have  1 ˙ Va (t) ≤ − σa Va (t) + q1 rb λd + κ2 λd sup d(ξ)2 , 2 0≤ξ≤t where σa is given in (7.138). Multiplying both sides of (7.170) by eσa t , we have  1 q1 rb λd + κ2 λd sup d(ξ)2 . eσa t V˙ a (t) + eσa t σa Va (t) ≤ eσa t 2 0≤ξ≤t

(7.169)

(7.170)

(7.171)

σa t

The left-hand side of (7.171) is d(e dtV (t)) . The integration of (7.171) from 0 to t yields  1 1 −σa t −σa t q1 rb λd + κ2 λd sup d(ξ)2 Va (t) ≤ Va (0)e + (1 − e ) σa 2 0≤ξ≤t  1 1 q1 rb λd + κ2 λd sup d(ξ)2 . ≤ Va (0)e−σa t + (7.172) σa 2 0≤ξ≤t The triggering condition (7.125) guarantees sup d(ξ)2 ≤ θ sup V (ξ) + μ sup |m(ξ)|.

0≤ξ≤t

0≤ξ≤t

0≤ξ≤t

(7.173)

Inserting (7.173) into (7.172) and then recalling (7.165) yields  1 1 q1 rb λd + κ2 λd sup (θV (ξ) + μ|m(ξ)|) Va (t) ≤ Va (0)e−σa t + σa 2 0≤ξ≤t −σa t ¯ ≤ Va (0)e + Φ sup Va (ξ), (7.174) 0≤ξ≤t

CHAPTER SEVEN

192 w(x, t)

z(x, t) Ud (t)

v(x, t)

ζ(t)

Sensor delay

Transport PDEs z,w

Transport PDE v

ODE ζ ZOH –

+

ETM

Continuous-in-time controller U(t)

Observer

Figure 7.2. Event-based closed-loop system. where ¯= 1 Φ σa



1 q1 rb λd + κ2 λd max{θ, μ}. 2

(7.175)

Recalling (7.136), (7.137) (the first term), we find that ¯ < 1. Φ

(7.176)

The following estimate then holds     ¯ sup Va (ξ)eσa ξ sup Va (ξ)eσa ξ ≤ Va (0) + Φ 0≤ξ≤t

0≤ξ≤t

(7.177)

as a consequence of (7.174). It follows that sup Va (ξ) ≤ ΥV Va (0)e−σa t ,

0≤ξ≤t

(7.178)

where the constant ΥV =

1 ¯ >0 1−Φ

(7.179)

by recalling (7.176). The choices of ETM parameters affect the overshoot coefficient in the exponential result according to (7.175), (7.179). Recalling (7.166), lemma 7.3 is obtained. The block diagram of the event-based output-feedback closed-loop system is shown in figure 7.2, where an observer-based continuous-in-time control input U (t) in (7.112) is updated at time instants tk determined by ETM (7.125), (7.126) realized based on the observer (7.26)–(7.32). Between updates the input is held constant in a zero-order-hold (ZOH) fashion. The properties of the output-feedback closed-loop system are shown in the following theorem. Theorem 7.2. For all initial data (z(x, 0), w(x, 0))T ∈ C 0 ([0, 1], R2 ), v(x, 0) ∈ z (x, 0), w(x, ˆ 0))T ∈ C 0 ([0, 1], R2 ), vˆ(x, 0) ∈ C 1 ([1, 2], R), C 1 ([1, 2], R), ζ(0) ∈ R, (ˆ − ˆ ζ(0) ∈ R, m(0) ∈ R , choosing the design parameters to satisfy (7.47), (7.111), (7.129)–(7.137), the output-feedback closed-loop system—that is, the plant (7.16)– (7.21) under the event-based control input Ud (t) in (7.118), which is realized using

EVENT-TRIGGERED DELAY COMPENSATION

193

the observer (7.26)–(7.32), and the ETM (7.125), (7.126), has the following properties: 1) No Zeno phenomenon occurs—that is, lim ti = +∞.

i→∞

(7.180)

2) The exponential convergence in the event-based output-feedback closed-loop system is achieved in the sense of the norm |ζ(t)| + z(·, t) + w(·, t) + v(·, t) + ˆ |ζ(t)| + ˆ z (·, t) + w(·, ˆ t) + ˆ v (·, t) + |m(t)|. 3) The event-triggered control input is convergent to zero in the sense of lim Ud (t) = 0.

t→∞

(7.181)

Proof. 1) Recalling lemma 7.2, we have ti ≥ iT,

i ∈ Z+ .

(7.182)

Property (1) is thus obtained. 2) Through an additional analysis that is routine but heavy on additional notation, we know that the stability result in lemma 7.3 still holds (by choosing a sufficiently large Re ) for the target system corresponding to the output-feedback loop, that is, the S-system defined in remark 7.1. Thus, the separation principle holds. Recalling the invertibility of the backstepping transformation (7.90), (7.91) and applying theorem 7.1 as well as (7.34), property (2) is thus obtained. 3) Recalling (7.112) and the stability result proved in property (2), we have that the continuous-in-time control input U (t) is convergent to zero. According to the definition (7.118), property (3) is obtained.

7.6

SIMULATION FOR DEEP-SEA CONSTRUCTION WITH SENSOR DELAY

Here we consider this simulation’s applicability to the DCV when placing equipment to be installed on the seabed for offshore oil drilling. The equipment must be installed accurately at a predetermined location with a tight tolerance. The permissible maximum tolerance for a typical subsea installation was 2.5 m, according to [95]. Applying the design presented above, we obtain an output-feedback control force employing piecewise-constant values at the crane to reduce oscillations of the long cable and position the equipment in the target area while compensating for the sensor delay. Initial Conditions and Design Parameters The DCV model in the simulation is (7.16)–(7.21), with coefficients in (7.14), (7.15) whose values are shown in table 7.1. The obtained simulation results are represented in the (¯ z , w) ¯ and u models that are on the spatial domain [0, L] via (7.10), (7.11), (7.12), (7.13). The disturbances f , fL , which will be formulated in the next subsection, are included (with multiplying by ρ1 considering the conversion from the wave PDE (7.1)–(7.4)) in the model (7.16)–(7.21). The initial conditions are chosen as

CHAPTER SEVEN

194

  π π , w(x, 0) = 6 cos 5πx + , z(x, 0) = 6 sin 5πx + 4 3 which gives the ODE initial conditions   21π 16π ζ(0) = 3 sin + 3 cos 4 3

(7.183)

(7.184)

by recalling (7.19), and the quantity is, physically, the initial oscillation velocity of the payload. According to the Riemann transformation (7.10), (7.11) and (7.12), (7.13), the initial conditions of z¯, w ¯ and the initial oscillation velocity of the cable are hence   5π¯ x π 5π¯ x π + + , w(¯ ¯ x, 0) = 6 cos , (7.185) z¯(¯ x, 0) = 6 sin L 4 L 3 1 z (¯ x, 0) + w(¯ ¯ x, 0)). (7.186) x, 0) = (¯ ut (¯ 2 The initial distributed oscillation displacement of the cable is defined as u(¯ x, 0) = 0, which, recalling (7.1), (7.3), implies that the initial displacement of the payload is bL (0) = 0. The initial values of the observer are defined as zero—that is, ˆ = 0, vˆ[0] = 0. The initial value of the internal dynamic varizˆ[0] = 0, w[0] ˆ = 0, ζ(0) able m(t) in ETM is set as m(0) = −1.5 × 105 . The design parameters are chosen as L1 = 1, F1 = 50, κ1 = 3000, κ2 = 2000, δ1 = 0.017, δ2 = 0.01, rc = 0.2, rb = 40, κ3 = 0.025, μ = 0.2, and θ = 0.08 according to (7.47), (7.111), (7.129)–(7.137), and the free design parameter η is picked as 8. Ocean Current Disturbances The time-varying ocean surface current velocity is modeled by a first-order GaussMarkov process [61]: ¯ P˙ (t) + μ ¯P (t) = G(t), Pmin ≤ P (t) ≤ Pmax ,

(7.187)

¯ is Gaussian white noise. The constants Pmin , Pmax and μ where G(t) ¯ are chosen as 1.6 ms−1 , 2.4 ms−1 , and 0 [95]. The full current load P (t) is applied at the cable from x ¯ = 0 to x ¯ = 300 m and thereafter linearly declines to 0.1P (t) at the bottom of the cable, which is at x ¯ = 1000 m [95]. The depth-dependent ocean current profile P (¯ x, t) is thus obtained as ⎧ ¯ ≤ 300 ⎨P (t), 0 ≤ x P (¯ x, t) = 970 − 0.9¯ (7.188) x ⎩ P (t), 300 ≤ x ¯ ≤ L, 700 which determines the ocean current disturbances f¯(¯ x, t) and fL (t) next. The disturbance f¯(¯ x, t) is the distributed oscillating drag force [95] modeled as  x, t) St P (¯ 1 x, t)2 RD AD cos 4π t+ς , (7.189) f¯(¯ x, t) = ρs Cd P (¯ 2 RD where Cd = 1 denotes the drag coefficient, ς = π is the phase angle, AD = 400 denotes the amplitude of the oscillating drag force, and St = 0.2 is the Strouhal number [59].

EVENT-TRIGGERED DELAY COMPENSATION

6

195

× 108 U Ud

Control force (N)

4 2 0 –2 0

5

15

10 Time (s)

20

Figure 7.3. Control forces (continuous-in-time U and event-based Ud ).

– – t) z(x,

10 0 –10 –20 1000 500 Position (m)

0 0

5

15 10 Time (s)

20

Figure 7.4. The evolution of z¯(¯ x, t) under the proposed event-based control input.

Then the distributed disturbance applied in the model (7.16)–(7.21) is f (x, t) = f¯(xL, t). The drag force fL (t) at the payload, which is considered to be a cylinder, is derived from Morison’s equation [95] as 1 fL (t) = Cd ρs hc Dc |P (L, t)| P (L, t). 2

(7.190)

Simulation Results We concentrate on the end phase (20 s) of the descending process, when the payload is near the seabed, and the cable is at its fully extended total length L. Our task is to reduce the oscillations of the cable and place the payload at the bottom of the cable in the target area on the seabed—namely, within the permissible tolerance of 2.5 m around the predetermined location [95] by applying a piecewise-constant control input at the onboard crane driving the top of the cable, where the measurements are transmitted from the seabed and subject to a 0.5 s delay. The event-based control input defined in (7.118) and the continuous-in-time control input defined in (7.112) are shown in figure 7.3, where the number of triggering times is 15, and the minimal dwell time is 0.097 s. The control input is zero at the beginning due to the sensor delay and the zero initial conditions of the observer based on which the control law is realized. With the event-based control input, we know from figures 7.4 and 7.5 that the PDE states z¯(¯ x, t) and w(¯ ¯ x, t) are regulated to a small range around zero, under the unknown ocean current disturbances (7.189), (7.190) and the sensor delay τ = 0.5 s. Similar results from the

CHAPTER SEVEN

196

– – t) w(x,

20 0

–20 1000 500 Position (m)

5

0 0

10 Time (s)

15

20

Figure 7.5. The evolution of w(¯ ¯ x, t) under the proposed event-based control input.

~ – t) z(x,

30 20 10 0 –10 –20 1000 500 0 0

Position (m)

5

10

15

20

Time (s)

~ – t) w(x,

Figure 7.6. The evolution of the observer error z˜(¯ x, t).

30 15

0 –15 –30 1000 500 0 0

Position (m)

5

10

15

20

Time (s)

Figure 7.7. The evolution of the observer error w(¯ ˜ x, t). x ¯ x ¯ observer errors z˜(¯ x, t) = z¯(¯ x, t) − zˆ( L , t), w(¯ ˜ x, t) = w(¯ ¯ x, t) − w( ˆ L , t) are observed in figures 7.6 and 7.7. Figure 7.8 shows that the internal dynamic variable m(t) in ETM is less than zero all the time. By virtue of (7.3), (7.8), the lateral displacement of the payload bL (t) is

 bL (t) =

t 0

ζ(δ)dδ + u(L, 0).

(7.191)

From figure 7.9, we know that the oscillation velocity of the payload, b˙ L (t) = ζ(t), is convergent to zero, and the position error of the payload is −0.98 m from the desired location on the seabed, which satisfies the requirement of being within the permissible tolerance of 2.5 m given in [95]. Through (7.10), (7.11), the cable x, t) 2 and lateral oscillation energy, including the oscillation kinetic energy ρ2 ut (¯

EVENT-TRIGGERED DELAY COMPENSATION

0

197

× 105

–1 m(t)

–2 –3 –4 0

5

15

10 Time (s)

20

0.5 0 –0.5 –1 –1.5 –2 –2.5 –3 –3.5 –4

bL bL 0

5

10 Time (s)

15

0.5 0 –0.5 –1 –1.5 –2 –2.5 –3 –3.5 –4 20

bL (m)

bL (m/s)

Figure 7.8. The evolution of the internal dynamic variable m(t) in the ETM.

T ρ –2 || ut (.,t) ||2 + –20 || ux– (.,t) ||2

Figure 7.9. The oscillation velocity and displacement of the payload.

5

× 106 Disturbance×1 Disturbance×2 Disturbance×3

4 3 2 1 0

0

5

10 Time (s)

15

20

Figure 7.10. The cable oscillation energy. potential energy 7.4 and 7.5, as

T0 x, t) 2 , ¯ (¯ 2 ux

is represented by z¯(¯ x, t), w(¯ ¯ x, t), shown in figures

ρ T0 ut (·, t) 2 + ux¯ (·, t) 2 2 2 ρ ρ ¯ t) + z¯(·, t) 2 + w(·, ¯ t) − z¯(·, t) 2 . = w(·, 8 8

(7.192)

As shown in figure 7.10, the oscillation energy of the cable with the proposed control law in (7.118) is reduced to a low level after t = 10 s under the external ocean current disturbances (7.189), (7.190). This result, the solid line in figure 7.10, shows

CHAPTER SEVEN

198

the robustness of the proposed control to small disturbances. However, as we continue to increase the amplitude of the disturbance (7.189), from the baseline value to its twice and to its triple, the results on cable oscillation energy are shown as the dashed line and dot-and-dash line in figure 7.10, where performance deterioration of the proposed controller is observed with the increase of disturbance amplitudes, and relatively large vibrations appear at the triple disturbance.

7.7

APPENDIX

A. Calculations of (7.81) and (7.85) Details of calculating (7.81) are shown as follows: w ˜t (x, t) − q2 w ˜x (x, t) + c4 z˜(x, t) + c3 w(x, ˜ t) + H3 (x)[˜ v (2, t), v˜t (2, t)]T  1 = β˜t (x, t) − ψ(x, y)˜ αt (y, t)dy x



− q2 β˜x (x, t) + q2

x 1

 + c4 α ˜ (x, t) − c4 ˜ t) − c3 + c3 β(x, 1

=

ψ(x, y)˜ α(y, t)dy 

˜ t)dy + c2 ¯ (x, y)β(y, N 

1

ψ(x, y)˜ α(y, t)dy x

1

+ q1 

φ(x, y)˜ α(y, t)dy + H3 (x)[˜ v (2, t), v˜t (2, t)]T

x

x

−

ψx (x, y)˜ α(y, t)dy − q2 ψ(x, x)˜ α(x, t)

x 1

 

1



1

ψ(x, y)˜ αx (y, t)dy + 1

x

x



˜ t)dy c1 ψ(x, y)β(y,

x y



1

˜ t)dy + q2 ¯ (δ, y)dδ β(y, ψ(x, δ)M

x



− q2 ψ(x, x)˜ α(x, t) − c3  + c4 α ˜ (x, t) − c4

1

ψx (x, y)˜ α(y, t)dy x

1

ψ(x, y)˜ α(y, t)dy x

φ(x, y)˜ α(y, t)dy + H3 (x)[˜ v (2, t), v˜t (2, t)]T

x

= (c4 − (q1 + q2 )ψ(x, x)) α ˜ (x, t)  1 (−q1 ψy (x, y) + q2 ψx (x, y) − c4 φ(x, y) + (c2 − c3 )ψ(x, y)) α ˜ (y, t)dy + x  1  y ˜ t)dy ¯ (x, y) − ¯ (δ, y)dδ β(y, + ψ(x, δ)M c1 ψ(x, y) + N x

x

α(1, t) + H3 (x)[˜ v (2, t), v˜t (2, t)]T + q1 ψ(x, 1)˜ = q1 ψ(x, 1)˜ α(1, t) + H3 (x)[˜ v (2, t), v˜t (2, t)]T = 0. Details of calculating (7.85) are shown as follows: z˜t (x, t) + q1 z˜x (x, t) + c1 w(x, ˜ t) + c2 z˜(x, t) + H2 (x)[˜ v (2, t), v˜t (2, t)]T

(7.193)

EVENT-TRIGGERED DELAY COMPENSATION



1

=α ˜ t (x, t) −

199 

φ(x, y)˜ αt (y, t)dy + q1 α ˜ x (x, t) − q1

x



1

˜ t) − c1 + q1 φ(x, x)˜ α(x, t) + c1 β(x, 

φx (x, y)˜ α(y, t)dy x

ψ(x, y)˜ α(y, t)dy + H2 (x)[˜ v (2, t), v˜t (2, t)]T

x

1

+ c2 α ˜ (x, t) − c2

1

φ(x, y)˜ α(y, t)dy x  1

 1 ˜ t) + q1 = − c1 β(x, φ(x, y)˜ αx (y, t)dy + (−q1 φx (x, y) − c1 ψ(x, y)) α ˜ (y, t)dy x x  1  y ˜ t)dy ¯ (y, δ)dδ + φ(x, y)c1 β(y, + φ(x, δ)M − x

x



1

˜ t) + α(x, t) + c1 β(x, + q1 φ(x, x)˜  =

˜ t)dy + H2 (x)[˜ ¯ (x, y)β(y, M v (2, t), v˜t (2, t)]T

x

1

(−q1 φx (x, y) − c1 ψ(x, y) − q1 φy (x, y)) α ˜ (y, t)dy  y  1 ˜ t)dy ¯ (x, y) − ¯ (δ, y)dδ + c1 φ(x, y) β(y, M + φ(x, δ)M x

x

x

α(1, t) + H2 (x)[˜ v (2, t), v˜t (2, t)]T + q1 φ(x, 1)˜ = q1 φ(x, 1)˜ α(1, t) + H2 (x)[˜ v (2, t), v˜t (2, t)]T = 0.

(7.194)

B. Matching (7.27)–(7.30) and (7.105)–(7.108) Step 1: Taking the time and spatial derivative of (7.91) along (7.26)–(7.30) and substituting the results into (7.106), we get βt (x, t) − q2 βx (x, t) + c3 β(x, t)  1  K2 (x, y)ˆ zt (y, t)dy − =w ˆt (x, t) − x



ˆ˙ J2 (x, y)w ˆt (y, t)dy − λ(x)ζ(t)

x

1

− q2 w ˆx (x, t) + q2

1



1

K2x (x, y)ˆ z (y, t)dy + q2

J2x (x, y)w(y, ˆ t)dy

x

x

ˆ − q2 K2 (x, x)ˆ z (x, t) − q2 J2 (x, x)w(x, ˆ t) + q2 λ (x)ζ(t)  1  1 ˆ ˆ t) − c3 K2 (x, y)ˆ z (y, t)dy − c3 J2 (x, y)w(y, ˆ t)dy − c3 λ(x)ζ(t) + c3 w(x,  = − c4 zˆ(x, t) + q1 

1

+

x 1

x

K2 (x, y)ˆ zx (y, t)dy x



K2 (x, y) (c2 zˆ(y, t) + c1 w(y, ˆ t)) dy − q2

x  1

+

1

J2 (x, y)w ˆx (y, t)dy x

J2 (x, y) (c4 zˆ(y, t) + c3 w(y, ˆ t)) dy x





1

+ q2

1

K2x (x, y)ˆ z (y, t)dy+q2 x

J2x (x, y)w(y, ˆ t)dy x

z (x, t) − q2 J2 (x, x)w(x, ˆ t) − q2 K2 (x, x)ˆ

CHAPTER SEVEN

200 ˆ ˆ + λ(x)b1 zˆ(1, t) + q2 λ (x)ζ(t) − λ(x)a1 ζ(t)  1  1 ˆ K2 (x, y)ˆ z (y, t)dy − c3 J2 (x, y)w(y, ˆ t)dy − c3 λ(x)ζ(t) − c3 x

x

= − (c4 + (q1 + q2 )K2 (x, x)) zˆ(x, t)  1 (c1 K2 (x, y) + q2 J2x (x, y) + q2 J2y (x, y)) w(y, ˆ t)dy + 

x 1

(c4 J2 (x, y) + c2 K2 (x, y) − c3 K2 (x, y) + q2 K2x (x, y) − q1 K2y (x, y)) zˆ(y, t)dy

+ x

+ (q1 K2 (x, 1) − q2 J2 (x, 1)q + λ(x)b1 ) zˆ(1, t) ˆ = 0. + (q2 λ (x) − λ(x)a1 − c3 λ(x) − q2 J2 (x, 1)c) ζ(t)

(7.195)

For (7.195) to hold, conditions (7.98)–(7.102) should be satisfied. Step 2: Taking the time and spatial derivative of (7.90) along (7.26)–(7.30) and substituting the results into (7.105), we obtain αt (x, t) + q1 αx (x, t) + c2 α(x, t)  1  K3 (x, y)ˆ zt (y, t)dy − = zˆt (x, t) − x

ˆ˙ + q1 zˆx (x, t) − q1 − γ(x)ζ(t)  − q1

1



1 x

 = − c1 w(x, ˆ t) + q1 1

1

K3x (x, y)ˆ z (y, t)dy

ˆ + q1 K3 (x, x)ˆ J3x (x, y)w(y, ˆ t)dy − q1 γ  (x)ζ(t) z (x, t) + q1 J3 (x, x)w(x, ˆ t)

+ c2 zˆ(x, t) − c2





− q1

x

1

K3 (x, y)ˆ z (y, t)dy − c2 

1

x



ˆ J3 (x, y)w(y, ˆ t)dy − c2 γ(x)ζ(t)

x 1

K3 (x, y)ˆ zx (y, t)dy +

K3 (x, y)(c2 zˆ(y, t) + c1 w(y, ˆ t))dy x

1

J3 (x, y)w ˆx (y, t)dy + x  1

J3 (x, y)w ˆt (y, t)dy x

x

x

− q2



1

J3 (x, y)(c4 zˆ(y, t) + c3 w(y, ˆ t))dy x

K3x (x, y)ˆ z (y, t)dy − q1



1

J3x (x, y)w(y, ˆ t)dy + q1 K3 (x, x)ˆ z (x, t) x

ˆ ˆ + γ(x)b1 zˆ(1, t) − q1 γ  (x)ζ(t) ˆ t) − γ(x)a1 ζ(t) + q1 J3 (x, x)w(x,  1  1 ˆ K3 (x, y)ˆ z (y, t)dy − c2 J3 (x, y)w(y, ˆ t)dy − c2 γ(x)ζ(t) − c2 x x  1 = ((q2 + q1 )J3 (x, x) − c1 ) w(x, ˆ t) + c1 K3 (x, y) − q1 J3x (x, y) x ˆ t)dy + q2 J3y (x, y) − (c2 − c3 )J3 (x, y) w(y, 

1

+

(c4 J3 (x, y) − q1 K3x (x, y) − q1 K3y (x, y)) zˆ(y, t)dy

x

+ (q1 K3 (x, 1) − q2 J3 (x, 1)q + γ(x)b1 ) zˆ(1, t) ˆ = 0. + (−q1 γ  (x) − γ(x)a1 − c2 γ(x) − q2 J3 (x, 1)c) ζ(t) For (7.196) to hold, conditions (7.92)–(7.95), (7.97) should be satisfied.

(7.196)

EVENT-TRIGGERED DELAY COMPENSATION

201

Step 3: Inserting (7.90), (7.91) into (7.108), (7.107), respectively, and applying (7.29), (7.30), we obtain ˆ˙ + am ζ(t) ˆ + b1 α(1, t) ζ(t) ˆ ˆ + b1 F1 ζ(t) ˆ + b1 zˆ(1, t) − b1 γ(1)ζ(t) ˆ˙ − a1 ζ(t) = ζ(t) ˆ = 0, = b1 (F1 − γ(1))ζ(t)

(7.197)

β(1, t) − qα(1, t) ˆ = w(1, ˆ t) − qˆ z (1, t) + (qγ(1) − λ(1))ζ(t) ˆ = 0. = (qγ(1) − λ(1) + c)ζ(t)

(7.198)

For (7.197), (7.198) to hold, conditions (7.96) and (7.103) should be satisfied.

7.8

NOTES

Most existing results [108, 115, 119, 180] on the delay-compensated control of PDEs are in a continuous-in-time form, while the control input in this chapter is in a piecewise-constant fashion. In the control design in this chapter, we only considered the dynamics of the cable and payload and neglected the ship-mounted crane dynamics in the DCV. Incorporating the crane dynamics, the plant becomes an ODE-PDE-ODE sandwich structure, which is dealt with in chapter 10. Even though chapter 10 only provides a continuous-in-time control law, readers can refer to the result in chapter 11 where an event-triggered control design for sandwich systems is presented to develop an event-triggered-type controller.

Chapter Eight Offshore Rotary Oil Drilling

Besides the control designs for the mining cable elevators and deep-sea construction vessels in chapters 2–6, here we present a torsional vibration control scheme for offshore rotary oil drilling. We adopt the one-dimensional wave partial differential equation (PDE) [24, 25, 153] to describe the torsional vibration dynamics of the drill string, avoiding the spillover phenomenon [15] caused by a control design based on lumped parameter models, which neglects the distributed nature of the system. A spillover instability could potentially occur as a result of applying, to a long drill string, a feedback law designed using traditional ordinary differential equation (ODE) control strategies [68, 146] for an approximated mass-spring-damper system [156]. According to [24, 25, 26, 120, 153], the stick-slip instability between the bit and rock is characterized by a linear anti-damped term with a highly uncertain friction parameter (depending on the nature of the rock or soil that the drill is passing through) on the the wave PDE boundary opposite to the control input. The external disturbance at the bit, which results from a wave-induced heaving motion of the drill rig [1], is usually described by a harmonic form with known frequencies and unknown amplitudes [128], because the amplitudes of the disturbance caused by the heaving motion may be affected by wind or ocean currents and are difficult to define in advance, while the dominant frequency components of the heaving motion in a specific sea area are usually accessible. In this chapter, the disturbance at the bit is modeled as the harmonic function with known frequencies and unknown amplitudes. The control problem in this chapter is to design an adaptive output-feedback controller for the wave PDE with both high uncertainty and instability at the anticollocated boundary. We begin this chapter by introducing the torsional vibration dynamics of oil drilling with stick-slip instability and a disturbance at the bit in section 8.1. The adaptive update laws for the unknown coefficients are presented in section 8.2, followed by the design of the output-feedback controller to adaptively cancel the disturbance and eliminate the destabilizing terms at the anti-collocated boundary in section 8.3. The asymptotic convergence to zero of the uncontrolled boundary states—that is, the oscillations of the angular displacement and velocity at the bit—and the boundedness of all states in the closed-loop system, are proved via Lyapunov analysis in section 8.4. Simulation tests in offshore rotary oil drilling are provided in section 8.5.

OFFSHORE ROTARY OIL DRILLING

203 U(t) x=L

Rotary table

Drill pipe u(x, t)

Drill collar Drill bit x=0

Uncertain stick-slip instability and disturbance

Figure 8.1. A drill string used in offshore oil drilling. The function u(x, t) denotes the distributed elastic angular displacement of the drill string. The drill bit is subject to the uncertain stick-slip instability and disturbance, which are anti-collocated with the torque U (t) at the rotary table. 8.1

DESCRIPTION OF OIL-DRILLING MODELS

A Wave PDE Model The offshore oil-drilling system rotates around its vertical axis, penetrating through the rock on the seafloor (see figure 8.1). It consists of the assembly of a drill pipe, a drill collar, and a rock-cutting tool referred to as a drill bit. At the top of the drill string, the rotary table provides the necessary torque to push the system into a rotary motion [153]. The torsional vibration dynamic model of the rotary offshore oil drill string system [24] is given by utt (x, t) = quxx (x, t), ux (L, t) = U (t), Ib utt (0, t) = cut (0, t) − kux (0, t) − d(t),

(8.1) (8.2) (8.3)

where the state u(x, t) denotes the distributed elastic angular displacement of the drill pipe, x ∈ [0, L], with L the length of the drill pipe and t ∈ [0, ∞) representing the time. The coefficients q = GJ/Id , k = GJ,

(8.4)

with G, J, Id representing the shear modulus of the drill pipe, drill pipe second moment of area, and drill pipe moment of inertia per unit of length, respectively, and Ib as the moment of inertia of the bottom-hole assembly (BHA). Input U (t) is scalar, and kU (t) represents the control torque. The function cut (0, t) describes the stick-slip instability-introducing force between the bit and the rock. Moreover, d(t) denotes the pressure oscillations on the drill bit caused by the wave-induced heaving motion of the drilling rig. Two assumptions are made for the unknown coefficient c and the uncertain disturbance d(t).

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204

Assumption 8.1. The anti-damping coefficient c in (8.3) is unknown but bounded by a known and arbitrary constant c¯—that is, c ∈ [0, c¯]. The coefficient c is treated as nonnegative without loss of generality. If, instead of anti-damping, damping were actually introduced by the rock cutting process, the control problem would be easier, the design developed here would still apply, and the resulting stability result would still hold. Assumption 8.2. The disturbance d(t) is of the general harmonic form as d(t) =

N 

[aj cos(θj t) + bj sin(θj t)],

(8.5)

j=1

where N is an arbitrary integer. The frequencies θj are known and arbitrary constants. The amplitudes aj , bj are unknown constants bounded by the known and ¯j ], bj ∈ [0, ¯bj ]. arbitrary constants a ¯j , ¯bj —that is, aj ∈ [0, a Equation (8.5) can model all periodic disturbance signals to an arbitrarily high degree of accuracy by choosing N sufficiently large. The frequency information requirement is reasonable in the wave-introduced disturbance considered in this chapter because the dominant frequency of the ocean waves is usually known for a particular area of the ocean [62]. The available boundary measurements are the torsional vibration acceleration utt (0, t) measured by the acceleration sensor placed at the bit, and the torsional vibration velocity ut (L, t) obtained by the feedback signal from the actuator. We obtain the torsional vibration displacement and velocity at the bit u(0, t), ut (0, t) by twice integrating the measurement utt (0, t) with known initial conditions u(0, 0), ut (0, 0) because the installation of the acceleration sensor is more economical, reliable, and accurate. This paragraph only explains a signal acquisition method, imposing no restrictions on any initial conditions in the design and theory. With the coefficient c of the anti-damping term unknown and the amplitudes aj , bj of the harmonic disturbance d(t) unknown at the uncontrolled boundary x = 0, the control objective in this chapter is to design a control input U (t) using the available boundary measurements to guarantee the asymptotic convergence to the origin of the torsional vibrational angular displacement u(0, t) and velocity ut (0, t) of the drill bit at the downhole boundary. The uniform boundedness of all the states in the closed-loop system should be ensured as well. A Wave PDE-ODE Model We define X(t) = [u(0, t), ut (0, t)]T ,

(8.6)

and then the system (8.1)–(8.3) is written as a PDE-ODE coupled system 1 ˙ X(t) = AX(t) + Bux (0, t) + Bd(t), k u(0, t) = CX(t),

(8.7) (8.8)

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205

utt (x, t) = quxx (x, t),

(8.9)

ux (L, t) = U (t),

(8.10)

where  A=

0 0

1



c Ib

k ,B = Ib



0 −1

 , C = [1, 0].

(8.11)

The unknown anti-damping coefficient c is in the matrix A. The control objective is thus to ensure the asymptotic convergence of X(t). A 2 × 2 Transport PDEs-ODE Model In order to reduce the plant (8.7)–(8.10) from second order to the conventional first order, we use the following Riemann coordinates: z(x, t) = ut (x, t) − w(x, t) = ut (x, t) +

√ √

qux (x, t),

(8.12)

qux (x, t)

(8.13)

to reversibly rewrite the system (8.7)–(8.10) as   BC1 1 B ˙ X(t) = A − √ X(t) + √ w(0, t) + Bd(t), q q k z(0, t) = 2C1 X(t) − w(0, t), √ zt (x, t) = − qzx (x, t), √ wt (x, t) = qwx (x, t), √ w(L, t) = 2 qU (t) + z(L, t).

(8.14) (8.15) (8.16) (8.17) (8.18)

The constant q is positive according to the definition in (8.4), and the system matrix in ODE (8.14) is rewritten as 1 1 A − √ BC1 = AE + Ac − √ BC1 , q q

(8.19)

C1 = [0, 1]

(8.20)

where

by defining  AE =

0 1 0 0



,  0 Ac = A − AE = 0

(8.21) 0 c Ib

 ,

(8.22)

which will be used in the following control design. In what follows we design the adaptive controller U (t) based on the plant (8.14)– (8.19) with the condition that AE − √1q BC1 is controllable. This condition is satisfied in the oil-drilling model according to (8.11), (8.20), and (8.21).

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206 8.2

ADAPTIVE UPDATE LAWS FOR UNKNOWN COEFFICIENTS

The objective in this section is to build adaptive update laws for the unknown coefficients c and aj , bj , respectively, where normalization and projection operators are used to guarantee boundedness, as is typical in adaptive control designs. We propose adaptive update laws for the unknown coefficient c in the matrix A in (8.11) and the unknown coefficients aj , bj in d(t) in (8.5) as follows: cˆ˙(t) = γc Proj[0,¯c] (τ (t), cˆ(t)) , a ˆ˙ j (t) = γaj Proj[0,¯aj ] (τ1j (t), a ˆj (t)) ,   ˆb˙ j (t) = γbj Proj ¯ τ2j (t), ˆbj (t) , [0,bj ] where the positive update gains γc , γaj , γbj For all m ≤ M and all r, p, Proj[m,M ] is given by ⎧ ⎨ 0, 0, Proj[m,M ] (r, p) = ⎩ r,

(8.23) (8.24) (8.25)

are tuning parameters to be determined. the standard projection operator [116] if p = m and r < 0, if p = M and r > 0, else.

The role of the projection operator is to keep the parameter estimates bounded. The bounds c¯ and a ¯j , ¯bj are defined in assumption 8.1 and assumption 8.2, respectively. The functions τ, τ1j , τ2j in (8.23)–(8.25) are defined as  L 1 ex β(x, t) X(t)T P Am − λa 1 + Ω(t) 0   √   1 qˆ c(t) √ c(t)Am − √1q BC1 )x q (AE +ˆ × κ ¯ + 0, Am dx X(t), e k  L 1 T τ1j (t) = ex β(x, t) X(t) P B cos(θj t) − λa k(1 + Ω(t)) 0   √   1 qˆ c(t) ˆ (t)− √1 BC1 )x √ (A +A q × κ ¯ + 0, B cos(θj t)dx , e q E c k  L 1 τ2j (t) = ex β(x, t) X(t)T P B sin(θj t) − λa k(1 + Ω(t)) 0   √   1 qˆ c(t) ˆ (t)− √1 BC1 )x √ (A +A q × κ ¯ + 0, B sin(θj t)dx . e q E c k τ (t) =

(8.26)

(8.27)

(8.28)

The choices of τ (t), τ1j (t), τ2j (t) will be clear from Lyapunov analysis, which will be shown in section 8.4. The definition of the parameters and states used in (8.26)– (8.28) are shown as follows—that is, (8.29)–(8.41):   0 0 Am = , (8.29) 0 I1b   0 0 ˆ Ac (t) = , (8.30) 0 cˆI(t) b

OFFSHORE ROTARY OIL DRILLING

1 Ω(t) = λa 2



L 0

207

1 ex β(x, t)2 dx + λb 2



L 0

1 e−x α(x, t)2 dx + X(t)T P X(t). 2

(8.31)

The normalization Ω(t) is introduced in the denominator in (8.26)–(8.28) to limit ˙ the rates of changes of the parameter estimates—that is, in cˆ˙(t) and a ˆ˙ j (t), ˆbj (t). The normalization constants λa > 0, λb > 0 are tuning parameters to be determined. The matrix P = P T > 0, where the superscript T means transposition, is the unique solution to the Lyapunov equation P A¯ + A¯T P = −Q

(8.32)

for some Q = QT > 0, and the known matrix 1 1 κ A¯ = AE − √ BC1 + √ B¯ q q

(8.33)

is made Hurwitz by appropriately choosing the control parameters κ ¯ = [k¯1 , k¯2 ] later. Next, disturbance-shifted state variables α(x, t), β(x, t) are introduced as √ q √z x −A [ˆ a1 (t), ˆb1 (t), . . . , a ˆN (t), ˆbN (t)]e q Z(t), (8.34) α(x, t) = z(x, t) − k √ Az q √ x [ˆ a1 (t), ˆb1 (t), . . . , a β(x, t) = w(x, t) + ˆN (t), ˆbN (t)]e q Z(t)   k√  1 qˆ c(t) ˆ (t)− √1 BC1 )x √ (A +A q X(t) − κ ¯ + 0, e q E c k  √  x  1 qˆ c(t) ˆ (t)− √1 BC1 )(x−y) 1 √ (A +A q B − κ ¯ + 0, e q E c k 0 q   √ Az q √ y [ˆ a1 (t), ˆb1 (t), . . . , a ˆN (t), ˆbN (t)]e q Z(t) dy, (8.35) × w(y, t) + k where

 Az = diag

0 θ1

−θ1 0



 ,...,

0 θN

−θN 0

 ,

(8.36)

and Z(t) = [cos(θ1 t), sin(θ1 t), . . . , cos(θN t), sin(θN t)]T .

(8.37)

The signal Z(t) in (8.37) will be used in constructing an important transformation in section 8.3. The functions α(x, t), β(x, t) in (8.34), (8.35) are related to transformations (8.42), (8.43) and (8.62), (8.63), which will be shown later. The states z(x, t), w(x, t) in (8.34) and (8.35) are calculated by means of the available boundary states through the transport PDEs (8.14)–(8.18), as follows:   1 (8.38) w(x, t) = w L, t − √ (L − x) , q     1 1 z(x, t) = 2C1 X t − √ x − w 0, t − √ x q q     L 1 1 = 2C1 X t − √ x − w L, t − √ − √ x . (8.39) q q q

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208

Applying (8.6), (8.13), (8.20), we write the states w(x, t), z(x, t) as     1 1 √ w(x, t) = ut L, t − √ (L − x) + qux L, t − √ (L − x) , q q     1 1 L z(x, t) = 2ut 0, t − √ x − ut L, t − √ − √ x q q q   1 L √ − qux L, t − √ − √ x . q q

(8.40)

(8.41)

The functions ut (L, t − √Lq − √1q x), ut (L, t − √1q (L − x)) are the measurement

ut (L, t) at previous time moments, and recalling (8.2), ux (L, t − √Lq − √1q x) and

ux (L, t − √1q (L − x)) can be replaced by U (t − √Lq − √1q x) and U (t − √1q (L − x)) which are control inputs at previous time moments, and are, therefore, known quantities. Hence, according to (8.34)–(8.37), (8.40), (8.41), and (8.6), we have that α(x, t), β(x, t), and X(t) can be obtained by the available boundary measurements proposed in section 8.1. Therefore, the adaptive update laws can be expressed using the available boundary states proposed in section 8.1.

8.3

OUTPUT-FEEDBACK CONTROL DESIGN

In this section, we design an output-feedback controller to compensate for the uncertain stick-slip instability and cancel the external disturbance at the ODE, which is anti-collocated with the controller. The controller employs the adaptive laws presented in section 8.2, using the measurements mentioned in section 8.1. To develop our output-feedback design, we first propose a transformation to make the unmatched external disturbance collocated with the control input, and then the disturbance is easily canceled via control design. The Transformation Making Unmatched Disturbances Collocated with Control We now transform the system (8.14)–(8.18) into an intermediate system so that the control input at x = L and the anti-collocated disturbance at the ODE become collocated. We introduce the invertible transformations (w, z) → (v, s) as v(x, t) = w(x, t) + Γ(x, t)Z(t),

(8.42)

s(x, t) = z(x, t) + Γ1 (x, t)Z(t),

(8.43)

where Γ(x, t), Γ1 (x, t) are to be determined. Through (8.42), (8.43), we convert the system (8.14)–(8.18) into the following system:   1 ˜ BC1 B ˙ (8.44) X(t) = A− √ X(t) + √ v(0, t) + B d(t), q q k s(0, t) + v(0, t) = 2C1 X(t), √ st (x, t) = − qsx (x, t) + Γ1t (x, t)Z(t),

(8.45) (8.46)

OFFSHORE ROTARY OIL DRILLING

209

√ qvx (x, t) + Γt (x, t)Z(t), √ v(L, t) = 2 qU (t) + s(L, t) + (Γ(L, t) − Γ1 (L, t))Z(t),

vt (x, t) =

(8.47) (8.48)

˜ is defined as where d(t) ˜ = d(t)

N 

 a ˜j (t) cos(θj t) + ˜bj (t) sin(θj t) ,

(8.49)

j=1

with a ˜j (t), ˜bj (t), j = 1, . . . , N defined as a ˜j (t) = aj − a ˆj (t), ˜bj (t) = bj − ˆbj (t).

(8.50) (8.51)

Recalling (8.36), (8.37), we immediately have ˙ = Az Z(t). Z(t)

(8.52)

Taking the time and spatial derivatives of (8.42) and substituting the result into (8.47), we get √ vt (x, t) − qvx (x, t) − Γt (x, t)Z(t) √ = wt (x, t) − qwx (x, t) + Γt (x, t)Z(t) + Γ(x, t)Az Z(t) √ − qΓx (x, t)Z(t) − Γt (x, t)Z(t) √ = (Γ(x, t)Az − qΓx (x, t))Z(t) = 0.

(8.53)

For (8.53) to hold, we obtain the sufficient condition Γ(x, t)Az −

√

qΓx (x, t) = 0.

(8.54)

By mapping (8.14) and (8.44) through the transformation (8.42), we get the condition √ q Γ(0, t) = [ˆ a1 (t), ˆb1 (t), . . . , a ˆN (t), ˆbN (t)]. (8.55) k According to (8.54), (8.55), we obtain the solution √ Az q √ x [ˆ a1 (t), ˆb1 (t), . . . , a ˆN (t), ˆbN (t)]e q , Γ(x, t) = k where e

Az √ q

x

is written as e

Az √ q



x

⎛

= diag ⎝

θ1 cos( √ q x),

θ1 − sin( √ q x)

θ1 sin( √ q x),

θ1 cos( √ q x)

⎛ ⎝

according to (8.36).

(8.56)

⎞ ⎠,··· ,

θN cos( √ q x),

θN − sin( √ q x)

θN sin( √ q x),

θN cos( √ q x)

⎞

 ⎠ ,

(8.57)

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210

Similarly, mapping (8.15), (8.16) and (8.45), (8.46) through (8.42), (8.43), we obtain √ q √z x −A [ˆ a1 (t), ˆb1 (t), . . . , a ˆN (t), ˆbN (t)]e q . (8.58) Γ1 (x, t) = − k The functions |Γ(x, t)| and |Γ1 (x, t)| are bounded as √ 2N q {|Γ(x, t)|, |Γ1 (x, t)|} ≤ max k x∈[0,L],t∈[0,∞)

max {¯ aj , ¯bj },

j=1,...,N

(8.59)

where | · | is a Euclidean norm, recalling (8.57) being a rotation matrix and the adaptive estimates a ˆj (t), ˆbj (t) bounded by [0, a ¯j ] and [0, ¯bj ] via the projection operator in section 8.2. Thus, through the transformation (8.42), (8.43) with (8.56)–(8.58), we complete the conversion from the plant (8.14)–(8.18) to the intermediate system (8.44)– (8.48) where the control input and the unmatched disturbance information are collocated. Backstepping Control Design In addition to making the cancellation of the disturbance term Z(t) at (8.48) straightforward, the objective in this section is to compensate for the uncertain anti-damped term cut (0, t) included in (8.44) by designing the control input U (t) at (8.48). The PDE backstepping method [114] is used to design the controller. Recalling (8.44) and (8.19), through a backstepping transformation, we assign the poles of AE − √1q BC1 to form a Hurwitz matrix and compensate for the anti-damping term Ac X(t) by Aˆc (t)X(t). Recalling (8.22) and (8.30), we obtain  A˜c (t) = Ac − Aˆc (t) =

0 0

0 c˜(t) Ib

 ,

(8.60)

where c˜(t) is the estimation error of c, given by c˜(t) = c − cˆ(t).

(8.61)

The backstepping transformation is defined as α(x, t) = s(x, t),

(8.62)



β(x, t) = v(x, t) −

x 0

φ(x, y, t)v(y, t)dy − γ(x, t)X(t),

(8.63)

where the time-varying kernel functions φ(x, y, t), γ(x, t) are to be determined later. From the spatially causal structure, the inverse of the transformation (8.62), (8.63) is written as s(x, t) = α(x, t), v(x, t) = β(x, t) −

(8.64)



x 0

ψ(x, y, t)β(y, t)dy − χ(x, t)X(t),

where ψ(x, y, t) and χ(x, t) are kernel functions to be determined.

(8.65)

OFFSHORE ROTARY OIL DRILLING

211

Remark 8.1. The fact that the kernel functions are time-varying in the backstepping transformation (8.63) is due to the adaptive estimate cˆ(t) included. Because cˆ(t) and cˆ˙(t) are bounded according to the designed update laws, cˆ(t) is continuously differentiable. Through the backstepping transformation (8.62), (8.63) and (8.64), (8.65), we convert the intermediate system (8.44)–(8.48) into the following target system: 1˜ 1 ˙ ¯ X(t) = AX(t) + A˜c (t)X(t) + B √ β(0, t) + B d(t), q k x √ βt (x, t) = qβx (x, t) + Γt (x, t)Z(t) − φ(x, y, t)Γt (y, t)dyZ(t) 0 x φt (x, y, t)β(y, t)dy − (γt (x, t) + γ(x, t)A˜c )X(t) − 0 y x φt (x, y, t) ψ(y, ω, t)β(ω, t)dωdy + 0 0 x B˜ φt (x, y, t)χ(y, t)X(t)dy − γ(x, t) d(t), + k 0 √ αt (x, t) = − qαx (x, t) + Γ1t (x, t)Z(t),

(8.66)

(8.67) (8.68)

β(0, t) = −α(0, t) + (γ(0, t) + 2C1 )X(t),

(8.69)

β(L, t) = 0,

(8.70)

where k 1 1 A¯ = AE − √ BC1 + √ B¯ κ= √ q q qIb



√

0 −¯ κ1

qIb k



(1 − κ ¯2 )

(8.71)

is Hurwitz by choosing the control parameters κ ¯1 , κ ¯ 2 to satisfy κ ¯ 2 > 1, 0 < κ ¯1
0, {γaj , γbj , γc }

min

1 > 0. {γaj , γbj , γc }

j∈{1···N }

j∈{1···N }

(8.94)

Taking the derivative of (8.92) and inserting (8.66), we obtain  1 1 ˙ V (t) = − X(t)T QX(t) + √ X T P Bβ(0, t) 1 + Ω(t) q N B [˜ aj (t) cos(θj t) + ˜bj (t) sin(θj t)] k j=1 L T + X(t) P c˜(t)Am X(t) + λa ex β(x, t)βt (x, t)dx

+ X(t)T P

0

+ λb −

N  j=1

L 0

  N 1 ˙ e−x α(x, t)αt (x, t)dx − aj (t) a ˆj (t)˜ γ j=1 aj

1 ˆ˙ 1 c(t), bj (t)˜bj (t) − cˆ˙(t)˜ γbj γc

(8.95)

˜ = c˜(t)Am implied by (8.29) and (8.60) is used. where A(t) Inserting (8.67)–(8.70) into (8.95) and applying the Cauchy-Schwarz inequality, we obtain    3 1 √ 2 λmin (Q) − qλb (γ(0, t) + 2C1 )2 |X(t)| V˙ (t) ≤ − 1 + Ω(t) 4 L L 1√ x 2 − e β(x, t) dx − λa ex β(x, t)γt (x, t)dxX(t) qλa 2 0 0  x L x e β(x, t) φt (x, y, t)β(y, t)dy − λa x0 y 0 − φt (x, y, t) ψ(y, ω, t)β(ω, t)dωdy 0  0 x 1√ − φt (x, y, t)χ(y, t)X(t)dy dx − qλb e−L α(L, t)2 2 0

OFFSHORE ROTARY OIL DRILLING

217

 1√ |P B|2 √ − qλb β(0, t)2 qλa − − 2 qλmin (Q) L L 1√ − qλb e−x α(x, t)2 dx + λb e−x α(x, t)Γ1t (x, t)Z(t)dx 2 0 0  x L L + λa ex β(x, t)Γt (x, t)Z(t)dx − λa ex β(x, t) φ(x, y, t)Γt (y, t)dyZ(t)dx 

+

0  N  j=1

+

N   X(t)T P B sin(θj t) j=1

 +

X(t)T P B cos(θj t) λa − k(1 + Ω(t))

k(1 + Ω(t))

X(t)T P Am X(t) λa − 1 + Ω(t)

−

λa

L 0

L 0

L 0

0

0

 e β(x, t)γ(x, t)B cos(θj t)dx 1 ˙ − a ˜j (t) ˆj (t) a k(1 + Ω(t)) γa x

 ex β(x, t)γ(x, t)B sin(θj t)dx 1 ˙ − ˆbj (t) ˜bj (t) k(1 + Ω(t)) γb

 ex β(x, t)γ(x, t)Am dxX(t) 1 − cˆ˙(t) c˜(t). 1 + Ω(t) γc

(8.96)

Step 1. Consider the third and fourth terms in the square bracket in (8.96). Because cˆ(t) ∈ [0, c¯], one can see that γ(x, t), φ(x, y, t) are bounded according to (8.80), (8.81). Here we define γ¯ = max{|γ(x, t)|; x ∈ [0, L], t ∈ [0, ∞)}, φ¯ = max{|φ(x, y, t)|; x ∈ [0, L], y ∈ [0, L], t ∈ [0, ∞)}.

(8.97) (8.98)

Similarly, the boundedness of χ(x, t), ψ(x, y, t) can also be obtained, defining χ ¯ = max{|χ(x, t)|; x ∈ [0, L], t ∈ [0, ∞)}, ψ¯ = max{|ψ(x, y, t)|; x ∈ [0, L], y ∈ [0, L], t ∈ [0, ∞)}.

(8.99) (8.100)

Applying the Cauchy-Schwarz and Young’s inequalities to the numerator of (8.26), we have that the absolute value of the numerator is less than or equal to m1 (|X(t)|2 + β(·, t)2 ) for some positive m1 . Recalling the form of Ω(t), given by (8.31), which appears in the denominator of (8.26), we also have 1 + Ω(t) >

1 min{λa , λb e−L , λmin (P )}(|X(t)|2 + β(·, t)2 ). 2

It follows that   ˙  cˆ(t) ≤

2γc m1 (|X(t)|2 + β(·, t)2 ) min{λa , λb e−L , λmin (P )}(|X(t)|2 + β(·, t)2 ) 2γc m1 = min{λa , λb e−L , λmin (P )}

(8.101)

via (8.23). Recalling (8.80), (8.81), we obtain the bounds on γt (x, t) and φt (x, t) as   √ ˙    √  ˙   q cˆ(t) qˆ c(t) cˆ(t)  |γt (x, t)| =  0, + κ ¯ + 0, √ Am me k k q ≤

γ c mc me , min{λa , λb e−L , λmin (P )}

(8.102)

CHAPTER EIGHT

218 |φt (x, t)| ≤

γc mc me |B| , q min{λa , λb e−L , λmin (P )}

(8.103)

where the constants mc , me are  √ 2 qm1 2 q¯ c2 + √ |¯ κ | 2 + 2 m1 , mc = k Ib q k    1 √ c(t)Am − √1q BC1 )x q (AE +ˆ me = max σ ¯ e . cˆ(t)∈[0,¯ c],x∈[0,L]

(8.104) (8.105)

 1  √ (A +ˆ c(t)Am − √1q BC1 )x The symbol σ ¯ e q E in (8.105) stands for the largest singular 1 √

(A +ˆ c(t)A − √1 BC )x

m 1 q at c(t) and x. value of e q E Applying the Young and Cauchy-Schwarz inequalities into the third and fourth terms in the square bracket in (8.96), using (8.99)–(8.103), we get L ex β(x, t)γt (x, t)dxX(t) λa

0

¯0   γc M β(·, t)2 + |X(t)|2 , −L min{λa , λb e , λmin (P )}  x L λa ex β(x, t) φt (x, y, t)β(y, t)dy 0 0 x y − φt (x, y, t) ψ(y, ω, t)β(ω, t)dωdy 0 0  x − φt (x, y, t)χ(y, t)X(t)dy dx ≤

(8.106)

0

≤

¯1   γc M β(·, t)2 + |X(t)|2 , −L min{λa , λb e , λmin (P )}

(8.107)

where ¯ 0 = 2λa eL mc me max{L, 1}, M ¯ 1 = λa eL L 1 mc me |B| max{1 + Lψ¯ + 2χ, M ¯ 2χL}. ¯ q Step 2. Consider the eighth to tenth terms in the square bracket in (8.96). By virtue of the numerator of (8.27), (8.28), together with 1 + Ω(t) ≥ 1 in the denominator, we straightforwardly show that   √ max {|τ1j (t)|, |τ2j (t)|} ≤ m2 |X(t)| + Lβ(·, t) (8.108) j=1,...,N

for all t, where m2 =

√ 2 max{|P B|, eL λa L¯ γ |B|} > 0. k

(8.109)

Therefore, we obtain       √  ˙  ˙ ˆj (t) , ˆbj (t) ≤ m2 max {γaj , γbj } |X(t)| + Lβ(·, t) , max a j=1,...,N

j=1,...,N

(8.110) establishing the boundedness of (8.24), (8.25).

OFFSHORE ROTARY OIL DRILLING

219

According to (8.56)–(8.58), we obtain max

x∈[0,L],t∈[0,∞)

{|Γt (x, t)|, |Γ1t (x, t)|}

√   q ˙  ˙ ˙ ˆ1 (t), ˆb1 (t), . . . , a ˆ˙ N (t), ˆbN (t) ≤ a k √   √ 2qN ≤ m2 max {γaj , γbj } |X(t)| + Lβ(·, t) . j=1,...,N k

(8.111)

Recalling (8.111) and applying Young’s inequality to the eighth, ninth, and tenth terms in the square bracket in (8.96), we obtain λb

L

e−x α(x, t)Γ1t (x, t)Z(t)dx

0

  ¯ 2 α(·, t)2 + β(·, t)2 + |X(t)|2 , ≤ max {γaj , γbj }M j=1,...,N



λa

L 0

ex β(x, t)Γt (x, t)Z(t)dx



− λa

(8.112)



L 0

x

x

e β(x, t)

0

φ(x, y, t)Γt (y, t)dyZ(t)dx

  ¯ 3 β(·, t)2 + |X(t)|2 , ≤ max {γaj , γbj }M j=1,...,N

(8.113)

√ ¯ 2, M ¯ 3 , where |Z(t)| = N and (8.98) are used. for some positive constants M Step 3. Substituting (8.106), (8.107), (8.112), (8.113), (8.23)–(8.25), and (8.80) into (8.96) and again applying Young’s inequality, we obtain  1 2 V˙ (t) ≤ − h1 |X(t)| − h2 β(·, t)2 1 + Ω(t)  − h3 α(·, t)2 − h4 α(L, t)2 − h5 β(0, t)2 , where 3 √ h1 = λmin (Q) − qλb (γ(0, t) + 2C1 )2 4 ¯ 0 + γc M ¯1 γc M ¯2 + M ¯ 3 ), − max {γaj , γbj }(M − −L min{λa , λb e , λmin (P )} j=1,...,N ¯ 0 + γc M ¯1 1√ γc M h2 = qλa − −L 2 min{λa , λb e , λmin (P )} ¯2 + M ¯ 3 ), − max {γaj , γbj }(M j=1,...,N

1√ ¯ 2, qλb e−L − max {γaj , γbj }M j=1,...,N 2 1√ qλb e−L > 0, h4 = 2 1√ |P B|2 √ − qλb . qλa − h5 = 2 qλmin (Q) h3 =

CHAPTER EIGHT

220 Choosing

2|P B|2 + 2λb λa > √ q qλmin (Q)

to guarantee h5 > 0 and using sufficiently small positive constants λb , γaj , γbj , γc to make h1 > 0, h2 > 0, and h3 > 0, we get   −ξ V˙ (t) ≤ (8.114) |X(t)|2 + β(·, t)2 + α(·, t)2 + α(L, t)2 + β(0, t)2 1 + Ω(t) with a positive ξ = min{h1 , h2 , h3 , h4 , h5 }, and hence V (t) ≤ V (0), ∀t ≥ 0.

(8.115)

Step 4. Recalling (8.93), one easily gets that c˜(t), a ˜j (t), ˜bj (t), j = 1, . . . , N , and Θ(t) are uniformly bounded. Therefore, together with (8.88), we find that β(·, t), α(·, t), |X(t)| are uniformly bounded. It follows that v(·, t), s(·, t) are uniformly bounded via (8.64), (8.65). By recalling (8.49) and (8.60), we also ˜ and |A˜c (t)| are uniformly bounded. According to (8.111), Γt (x, t) know that d(t) and Γ1t (x, t) are bounded as well. According to (8.66)–(8.70), we further get   d 1˜ 1 2 T ¯ ˜ |X(t)| = 2X (t) AX(t) + Ac (t)X(t) + B √ β(0, t) + B d(t) , (8.116) dt q k  L d √ β(·, t)2 = − qβ(0, t)2 + 2 β(x, t) Γt (x, t)Z(t) dt 0 x − φ(x, y, t)Γt (y, t)dyZ(t) − (γt (x, t) + γ(x, t)A˜c (t))X(t) −

0



x

φt (x, y, t)β(y, t)dy +

0



x 0

φt (x, y, t)

y 0

ψ(y, ω, t)β(ω, t)dωdy

 B˜ φt (x, y, t)χ(y, t)X(t)dy − γ(x, t) d(t) dx, + k 0 L d √ √ α(·, t)2 = − qα(L, t)2 + qα(0, t)2 + 2 α(x, t)Γ1t (x, t)Z(t)dx. dt 0

x

(8.117) (8.118)

Recalling (8.63) and (8.70), we have v(L, t) as uniformly bounded. According to (8.42) and the boundedness of Γ(x, t), we find that w(L, t) is uniformly bounded. Because w(0, t) = w(L, t − √Lq ), w(0, t) is uniformly bounded for t > √Lq . Therefore, z(0, t) is uniformly bounded via (8.15). Because z(L, t) = z(0, t − √Lq ), z(L, t) is uniformly bounded for t > √Lq . According to (8.42), (8.43) together with the boundedness of Γ(x, t), Γ1 (x, t) and (8.62), (8.63), we have β(0, t), β(L, t), α(L, t), α(0, t) as uniformly bounded. Applying the Cauchy-Schwarz inequality, we get   d ˜ 2 , |X(t)|2 ≤ μ3 |X(t)|2 + β(0, t)2 + d(t) dt   d ˜ 2 , β(·, t)2 ≤ μ4 β(·, t)2 + |X(t)|2 + β(0, t)2 + d(t) dt

OFFSHORE ROTARY OIL DRILLING

221

  d 2 2 2 2 2 2 α(·, t) ≤ μ5 α(·, t) + β(·, t) + |X(t)| + α(0, t) + α(L, t) , dt d d d with some positive constants μ3 , μ4 , μ5 . Thus, dt |X(t)|2 , dt β(·, t)2 , and dt α(·, t)2 are uniformly bounded thanks to the boundedness results established above. Finally, integrating (8.114) from 0 to ∞, it follows that |X(t)|, α(·, t), β(·, t) are square integrable. Following Barbalat’s lemma, we conclude that |X(t)|, α(·, t), β(·, t) tend to zero as t → ∞. Due to the invertibility and continuity of the backstepping transformations (8.62), (8.63) and (8.64), (8.65), the proof of lemma 8.1 is complete.

The closed-loop system is utt (x, t) = quxx (x, t), ux (L, t) = U (t), Ib utt (0, t) = cut (0, t) − kux (0, t) − d(t), d(t) =

N 

[aj cos(θj t) + bj sin(θj t)],

(8.119) (8.120) (8.121) (8.122)

j=1

cˆ˙(t) = γc Proj[0,¯c] {τ (t), cˆ(t)},

(8.123)

ˆj (t)}, a ˆ˙ j (t) = γaj Proj[0,¯aj ] {τ1j (t), a

(8.124)

ˆb˙ j (t) = γbj Proj ¯ {τ2j (t), ˆbj (t)}, [0,bj ]

(8.125)

where the control input U (t) is defined in (8.86). The functions τ (t), τ1j (t), τ2j (t), which are defined in (8.26)–(8.28), can be represented as the original state u by applying (8.34)–(8.37), (8.40), (8.41), and (8.6). Define H = H 2 (0, L) × H 1 (0, L),

(8.126)

where H 1 (0, L) = {u|u(·, t) ∈ L2 (0, L), ux (·, t) ∈ L2 (0, L)}, H 2 (0, L) = {u|u(·, t) ∈ L2 (0, L), ux (·, t) ∈ L2 (0, L), uxx (·, t) ∈ L2 (0, L)} and let u(·, t) ∈ L2 (0, L) mean that u(·, t) is square integrable in x. The main result is presented in the following theorem. Theorem 8.1. For all initial values (u(·, 0), ut (·, 0)) ∈ H, the closed-loop system consisting of the plant (8.119)–(8.121) and the controller (8.86) with the adaptive update laws (8.123)–(8.125) has the following properties: 1. The outputs u(0, t), ut (0, t) of the closed-loop system are asymptotically convergent to zero—that is, lim u(0, t) = 0,

t→∞

lim ut (0, t) = 0.

t→∞

2. Distributed states in the closed-loop system are uniformly ultimately bounded in the sense of the norm  1 ux (·, t)2 + ut (·, t)2 2 .

CHAPTER EIGHT

222

Proof. According to the asymptotic stability result in lemma 8.1, we know that X(t) = [u(0, t), ut (0, t)]T is asymptotically convergent to zero, and thus property 1 of theorem 8.1 is proved. Recalling (8.42), (8.43) and applying the Cauchy-Schwarz inequality, we obtain w(·, t)2 ≤ 2v(·, t)2 + 2N Γ(·, t)2 ,

(8.127)

z(x, t)2 ≤ 2s(·, t)2 + 2N Γ1 (·, t)2 ,

(8.128)

where Z(t)2 = N is used. The functions Γ(·, t)2 , Γ1 (·, t)2 are bounded by a positive constant according to (8.59). Together with the convergence to zero and the uniform boundedness of v(·, t)2 + s(·, t)2 proved in lemma 8.1, we obtain the uniform ultimate boundedness of w(·, t)2 , z(·, t)2 . Recalling (8.12) and (8.13), we get 1 (8.129) ut (x, t) = (z(x, t) + w(x, t)), 2 1 ux (x, t) = √ (w(x, t) − z(x, t)). (8.130) 2 q Applying the Cauchy-Schwarz inequality, we obtain ut (·, t)2 + ux (·, t)2 L L ≤ (z(·, t)2 + w(·, t)2 ) + (z(·, t)2 + w(·, t)2 ) 2 2q q+1 2 L(z(·, t) + w(·, t)2 ). ≤ 2q Property 2 of theorem 8.1 is thus proved. Summary of the implementation of the control algorithm supported by the above theorem: According to theorem 8.1, by applying the control torque GJU (t) to the rotary table of the oil-drilling system with the uncertain stick-slip instability and external disturbance at the drill bit as shown in figure 8.1, the torsional vibration displacement u(0, t) and velocity ut (0, t) of the drill bit are driven toward zero as time goes on. The constants G, J are physical parameters given in section 8.1. Input U (t) in (8.86) includes the adaptive estimates cˆ(t), a ˆj (t), ˆbj (t) defined in section 8.2 and is constructed by measuring the signals u(0, t), ut (0, t), ut (L, t), which are obtained from the acceleration sensor placed at the bit and the feedback signal of the actuator at the rotary table, as mentioned in section 8.1. All signals in the controller are obtained from the direct measurements or integrals without using derivatives, which avoids the measurement noise amplification.

8.5

SIMULATION FOR OFFSHORE OIL DRILLING

The Oil-Drilling Model The offshore oil-drilling model tested in the simulation is (8.1)–(8.3), with the physical parameters shown in table 8.1, which are borrowed from [24, 153]. The disturbance at the drill bit is given as a harmonic form of d(t) = 2 cos(2t) + sin(2t).

(8.131)

OFFSHORE ROTARY OIL DRILLING

223

Table 8.1. Physical parameters of the oil-drilling system Parameters (units)

Values

Length of the drill pipe L (m) Shear modulus of the drill pipe G (N/m2 ) Drill pipe moment of inertia per unit of length Id (kg·m) Moment of inertia of the BHA Ib (kg·m2 ) Drill pipe second moment of area J (m4 ) Anti-damping parameter c (N·m·s/rad)

2000 7.96×1011

× 108

u (x, t)

4 2 0 –2 –4 –6 –8 2000

0.095 311 1.19×10−5 1

1000 x(m)

10 0 0

5 t(s)

Figure 8.3. Open-loop responses of u(x, t). Therefore, the unknown amplitudes a1 , b1 in the disturbance are 2, 1, respectively. We only know that their upper bounds a ¯1 , ¯b1 are 4, 2. The unknown anti-damping coefficient c is 1, and the upper bound is known as c¯ = 2. According to section 8.1, the system (8.1)–(8.3) is written as (8.14)–(8.18), where X(t) = [u(0, t), ut (0, t)]T . The main simulation is conducted based on (8.14)–(8.18), and then the responses z, w are converted to the responses u of the system (8.1)– (8.3) through x 1 (w(y, t) − z(y, t))dy + u(0, t) (8.132) u(x, t) = √ 2 q 0 by recalling (8.12), (8.13). The finite-difference method is adopted to conduct the simulation with a time step and space step of 0.0005 and 0.05, respectively. According to the physical parameters in table 8.1, the coefficients in (8.14)–(8.18) are obtained as q = 9.971 × 107 , k = 9.472 × 106 through the definitions in section 8.1. Consider the initial conditions in (8.1)–(8.3) to be u(x, 0) = 0.15, ut (x, 0) = sin( 2π L x). Then the according initial conditions in (8.14)–(8.18) are w(x, 0) = z(x, 0) = sin( 2π L x) by virtue of (8.12), (8.13), (8.132). Open-Loop Responses In the open-loop case, it is shown that the plant (8.1)–(8.3) is unstable in figures 8.3 and 8.4 because of the effect of the anti-damping term at the bit x = 0. The expected diverging results for z(x, t), w(x, t) (8.14)–(8.18) are seen in figure 8.5. To be exact, the diverging phenomenon starts at X(t) = [u(0, t), ut (0, t)]T in (8.14) (shown in figure 8.4), flowing into z(0, t) via (8.15) and giving rise to the increase

CHAPTER EIGHT

224 × 108 2

u(0, t)

0 –2 –4 –6 –8

0

2

4

6

8

10

6

8

10

Time (s) (a) u(0, t). × 109 0 ut(0, t)

–1 –2 –3 –4 0

2

4 Time (s)

(b) ut (0, t).

Figure 8.4. Open-loop responses of u(0, t), ut (0, t).

z(x, t)

5

× 109

0 –5

–10 2000 1000 x(m)

0 0

2

4

6

8

10

t(s)

(a) z(x, t). × 109

w(x, t)

2 0 –2

–4 2000

1000 x(m)

0 0

5 t(s)

(b) w(x, t).

Figure 8.5. Open-loop responses of z(x, t), w(x, t).

10

OFFSHORE ROTARY OIL DRILLING

225

of |z(0, t)|, following which the diverging response develops for z(x, t) via (8.16), with instability traveling up to the boundary x = L, which is shown in figure 8.5(a). According to (8.18), diverging performance then appears at w(L, t), which leads to the instability of w(x, t) via (8.17) and causes the increase of |w(0, t)|, which is shown in figure 8.5(b). Closed-Loop Responses We apply the control law (8.86) into (8.14)–(8.18), with the control parameters chosen as κ ¯ = [0.1, 1.5], γa = 0.005, γb = 0.008, and γc = 0.006. The constant N in (8.86) is 1. The function ut in (8.86) is represented by the states in (8.14)–(8.18) ˆ1 (t), ˆb1 (t) in (8.86) are calculated via ut = 12 (w + z). The adaptive estimates cˆ(t), a from (8.23) and (8.26), (8.24) and (8.27), and (8.25) and (8.28). Recalling (8.62), L L (8.63), (8.42), (8.43), (8.38), (8.39), the integrals 0 e−x α(x, t)2 dx, 0 ex β(x, t)2 dx in the weight norm Ω(t) defined in (8.31) and appearing in the adaptive update laws for the estimates are also represented by the states in (8.14)–(8.18) as

L 0

e−x α(x, t)2 dx



L

e−x (z(x, t) + Γ1 (x, t)Z(t))2 dx  t √ L − q(t−δ1 ) e =− 2C1 X(δ1 ) − w(L, δ1 − √ ) L q t− √q 2 √ q [ˆ a1 (t), ˆb1 (t)]e−Az (t−δ1 ) Z(t) dδ1 , − k

=

0

and



L

0 L

= 0

ex β(x, t)2 dx  x e w(x, t) + Γ(x, t)Z(t)

2 φ(x, y, t)(w(y, t) + Γ(y, t)Z(t))dy − γ(x, t)X(t) dx 0  t √ √ √ Az q √ (L− q(t−δ1 )) L− q(t−δ1 ) [ˆ a1 (t), ˆb1 (t)]e q = e Z(t) w(L, δ1 ) + k t− √Lq   √  δ1 qˆ c(t) ˆ (t)− √1 BC1 )(δ1 −δ2 ) 1 (A +A q B − κ ¯ + 0, e E c L k t− √q q   √ √ Az q √ (L− q(t−δ2 )) q ˆ × w(L, δ2 ) + [ˆ a1 (t), b1 (t)]e Z(t) dδ2 k   √ 2  1 qˆ c(t) ˆ (t)− √1 BC1 )(L−√q(t−δ1 )) √ (A +A q − κ ¯ + 0, X(t) dδ1 . e q E c k −



(8.133)

x

(8.134)

L The same process is adopted to calculate 0 β(x, t)dx used in (8.26)–(8.28). 2L L √ The controller is activated at t = √ q = 0.4 s, ensuring that δ1 − q with δ1 ∈ [t − √Lq , t] in w(L, δ1 − √Lq ) in (8.133) is nonnegative.

CHAPTER EIGHT

226

0.2

U(t)

0 –0.2 Proposed controller PD controller

–0.4 0

2

4

6

8

10

Time (s)

Figure 8.6. The proposed control input and the PD control input.

z(x, t)

1 0 –1 2000

10

1000 x(m)

5 0 0

t(s)

(a) z(x, t).

w(x, t)

1 0

–1 2000

10

1000 x(m)

5 0 0

t(s)

(b) w(x, t).

Figure 8.7. Closed-loop responses of z(x, t), w(x, t).

We compare the proposed controller with the classical proportional-derivative (PD) controller, which uses the signal X(t) = [u(0, t), ut (0, t)]T and is given by UPD (t) = kp u(0, t) + kd ut (0, t).

(8.135)

The best regulating PD performance is achieved with kp = 0.13 and kd = 1.2 in (8.14)–(8.18). The evolution of the proposed controller, activated at t = 0.4 s, and the PD controller, activated at t = 0, is shown in figure 8.6. According to figure 8.7, we know that the responses of z(x, t), w(x, t) under the proposed controller are convergent to a small neighborhood of zero. From figure

OFFSHORE ROTARY OIL DRILLING

0.15

0.01

Proposed controller PD controller

0.1 u(0, t)

227

0 –0.01 7

0.05

8

9

10

0 –0.05

0

2

6

4

8

10

Time (s) (a) u(0, t). time = 0.2 s

time = 5.5 s Proposed controller PD controller

ut (0, t)

0.1 0

0.02

–0.1

0

–0.2

–0.02 –0.3

0 2 time = 0.6 s

4

6

7

8

9 8

10 10

Time (s) (b) ut (0, t).

Figure 8.8. Closed-loop responses of X(t) = [u(0, t), ut (0, t)]T under the proposed adaptive controller and the PD controller, which physically represents the torsional vibration angular displacement and velocity at the bit (the reasons for the phenomenon of the PD controller’s better transient performance before about t = 5.5 s are discussed in remark 8.2). 8.8, even though the transient performance of u(0, t), ut (0, t)—that is, of the vector state X(t)—under the proposed controller is worse than that under the PD controller before about t = 5.5 s (the reasons for this phenomenon are discussed in remark 8.2), the responses of u(0, t), ut (0, t) under the proposed controller are convergent to a smaller neighborhood of zero than those under the PD controller as time goes on. It physically means that the proposed controller achieves better performance in the suppression of the torsional vibration displacement and velocity at the bit. Recalling (8.132) and the closed-loop responses of z(x, t), w(x, t), u(0, t), the response of u(x, t) under the proposed controller is obtained and shown in figure 8.9, which indicates that the torsional vibrations of the oil-drilling pipe have been sup1 pressed. The norm (ut (·, t)2 + ux (·, t)2 ) 2 obtained from (8.129), (8.130) denotes torsional vibration energy consisting of kinetic energy and potential energy. The 1 responses of (ut (·, t)2 + ux (·, t)2 ) 2 under the proposed controller and the PD controller are shown in figure 8.10. We see that, even though the PD controller has a better transient performance before about t = 5.5 s (the two reasons for this phenomenon are explained in remark 8.2), the proposed controller reduces the vibration to a smaller range around zero as time goes on, which verifies that the proposed adaptive controller performs better at suppressing vibrations in the offshore oildrilling platform.

CHAPTER EIGHT

228

u(x, t)

0.2 0.1 0 –0.1 2000 1000 x(m)

10 5 t(s)

0 0

Figure 8.9. Closed-loop response of u(x, t), which physically represents the torsional vibrations of the oil-drilling pipe under the proposed controller.

1

– (|| ut (.,t) ||2 + || ux (.,t) ||2) 2

35 Proposed controller PD controller

30 25 1

20 15

0.5

10

0

6

8

10

5 0

0

2

6

4

8

10

Time (s) 1

Figure 8.10. Closed-loop response of the norm (ut (·, t)2 +ux (·, t)2 ) 2 under the proposed adaptive controller and the PD controller, which physically represents torsional vibration energy, including kinetic energy and potential energy of the oildrilling pipe (the reasons for the phenomenon of the PD controller’s better transient performance before about t = 5.5 s are discussed in remark 8.2). The adaptive estimation action is activated at t = 0.4 s. From figure 8.11, which shows the adaptive estimation errors of the constants c, a1 , b1 in the above regulation process, we know that the estimates cˆ(t), a ˆ1 (t), ˆb1 (t) converge to values that are close to the actual c, a1 , b1 as time goes on. Even though the estimates do not exactly arrive at their actual values, the state convergence is achieved, which is typical in adaptive control in the absence of persistence of excitation. Remark 8.2. There are two reasons for the phenomenon that the simpler PD controller has a better transient performance than the proposed adaptive controller before t = 5.5 s. First, the proposed model-based adaptive controller is activated later than the PD controller with a 0.4 s delay. The proposed adaptive controller is activated at t = 0.4 s and the regulation action reaches the ODE at x = 0 until t = 0.6 s because the propagation time from x = L to x = 0 is √Lq = 0.2 s in (8.14)– (8.18), while the PD controller is activated at t = 0 and the regulation action reaches the ODE at t = 0.2 s. This is shown in figure 8.8(b), where the PD controller starts regulating the ODE states toward zero after t = 0.2 s, and the response under the proposed adaptive controller continues to deteriorate until about t = 0.6 s because of

OFFSHORE ROTARY OIL DRILLING

229

c˜(t)

1 0.5 0 0

2

4

6

8

10

6

8

10

6

8

10

Time (s) (a) c˜(t).

a˜1(t)

2 1 0 0

2

4 Time (s) (b) a˜1(t).

b˜1(t)

1 0 –1

0

2

4 Time (s) (c) b˜1(t).

Figure 8.11. Adaptive estimation errors of the anti-damping coefficient c and the disturbance amplitudes a1 , b1 .

no regulation action to stabilize the anti-stable ODE. Second, the PD controller is running under the very best parameters that we have chosen over many simulation tests (which is equivalent to knowing the model parameters), to be more than fair to the model-free PD controller and to give it a maximal advantage. In contrast, the proposed adaptive controller is operating with a poor knowledge of the anti-damping parameter—that is, with very bad initial gains, and a poor knowledge of the disturbance parameters, where the adaptive estimation of these parameters introduces an adaptive learning transient until as late as about t = 5.5 s, which is shown in figure 8.11. In summary, as soon as our model-based adaptive outputfeedback disturbance-canceling controller completes, or nearly completes, the learning process, this controller vastly outperforms the PD controller in subsequent performance, as expected.

8.6

NOTES

Chapters 2–6 dealt with the longitudinal or lateral vibrations of cables, while the torsional vibrations in the “drill string,” which is not a cable but a kilometerslong thin cylinder, were dealt with in this chapter, along with stick-slip instability and external disturbances resulting from the wave-induced heaving motion of the drilling rig, at the bit. The plant is a one-dimensional wave PDE system with

230

CHAPTER EIGHT

an uncontrolled dynamic boundary, which includes an anti-damping term with an unknown coefficient and a harmonic disturbance with unknown amplitudes. The control design incorporates adaptive control designs in [24, 25, 26, 117, 124], which were developed for a one-dimensional wave PDE with an actuator on one boundary and an anti-damping instability with an unknown coefficient on the other boundary, as well as the idea of adaptive cancellation of anti-collocated uncertain disturbances for wave PDEs in [85].

Part II

Generalizations

Chapter Nine Basic Control of Sandwich Hyperbolic PDEs

In part I of this book we neglected the actuator dynamics, assuming that the control input directly flows into the partial differential equation (PDE) boundary. In many situations, however, the actuator dynamics must be considered in the control design, especially when the dominant time constant of the actuator is close to that of the plant. When the model incorporates the actuator dynamics in the input channel of the string-payload model considered in part I, such as the hydraulic cylinder and head sheaves in chapters 2–5, the ship-mounted crane in chapter 6, or the rotary table in chapter 8, the plant becomes a hyperbolic PDE sandwiched between two ordinary differential equations (ODEs). We address a new theoretical problem: boundary control of sandwich PDEs, starting from a basic design presented in this chapter, moving on to delaycompensated control in chapter 10, developing an event-triggered controller in chapter 11, and dealing with nonlinearities in chapter 12. In this chapter, we solve the problem of stabilization of 2 × 2 coupled linear first-order hyperbolic PDEs sandwiched between two ODEs by combining PDE backstepping and ODE backstepping. In section 9.2, we seek a PDE backstepping transformation that maps the plant into a stable system where the in-domain couplings between the hyperbolic PDEs are removed, and the system matrix of the ODE at the left boundary is Hurwitz. The resulting right boundary condition is an ODE with a number of perturbation given in terms of the PDE states. We then deal with this ODE in the input channel via the ODE backstepping method in section 9.3. In section 9.4, a controller is built, and the exponential stability of the overall closed-loop system is proved by Lyapunov analysis, where some control parameters, which exist in the ODE backstepping design procedure, are determined, with the goal of tolerating the PDE perturbations. The boundedness and exponential convergence of the controller in the closed-loop system are proved in section 9.5. In section 9.6, we extend the proposed method and the associated proofs to a more general case where the input ODE is of arbitrary order. The simulation results are provided in section 9.7.

9.1

PROBLEM FORMULATION

The sandwich hyperbolic PDE system considered in this chapter is ˙ X(t) = AX(t) + Bv(0, t),

(9.1)

ut (x, t) = −pux (x, t) + c1 v(x, t),

(9.2)

vt (x, t) = pvx (x, t) + c2 u(x, t),

(9.3)

CHAPTER NINE

234 u(0, t) = qv(0, t) + CX(t),

(9.4)

v(1, t) = z(t),

(9.5)

z¨(t) = c0 z(t) ˙ + ru(1, t) + U (t),

(9.6)

∀(x, t) ∈ [0, 1] × [0, ∞), where X(t) ∈ Rn×1 , and Z T (t) = [z(t), z(t)] ˙ = [z1 (t), z2 (t)] ∈ R2×1 are ODE states. The scalars u(x, t) ∈ R, v(x, t) ∈ R are states of the PDEs. The matrices A ∈ Rn×n , B ∈ Rn×1 satisfy that the pair [A; B] is controllable. The matrix C ∈ R1×n and constants c0 , c1 , c2 , r, q ∈ R are arbitrary. The positive constant p denotes the arbitrary transport speed. We consider the transport speed of (9.2), (9.3) to be equal in this chapter but with a possible extension to the case where the transport speeds of the two transport PDEs are different. The control input U (t) is to be designed. The full relative degree in (9.5), (9.6) is assumed for the design. The objective here is to exponentially stabilize all ODE states Z(t), X(t) and PDE states u(x, t), v(x, t) by the control input U (t). Moreover, the result is extended to a more general system with arbitrary-order ODEs sandwiching a PDE.

9.2

BACKSTEPPING FOR THE PDE-ODE CASCADE

Backstepping Transformations and the Target System We consider the infinite-dimensional backstepping transformation of the PDE state u(x, t), v(x, t) as follows: α(x, t) ≡ u(x, t),



β(x, t) = v(x, t) −

x 0

(9.7)

 ψ(x, y)u(y, t)dy −

x 0

φ(x, y)v(y, t)dy − γ(x)X(t).

(9.8)

The kernel equations for ψ(x, y), φ(x, y), γ(x) are determined later. The reason why we only apply the backstepping transformation on v is that given the difference in the propagation directions in the u and v subsystems, only the u-term in the v-subsystem in (9.3) acts as a potentially destabilizing feedback term, whereas the v-term in the u-subsystem in (9.2) acts as a feedforward term, which disturbs the u system but cannot destabilize it, once the presence of the u-term in (9.3) is eliminated. Using the partial backstepping transformation (9.7), (9.8) results in a requirement of the calculation of fewer kernels and a simpler structure of the controller. The inverse of (9.7), (9.8) is postulated as u(x, t) ≡ α(x, t), v(x, t) = β(x, t) −



x 0

(9.9)

 ψ I (x, y)α(y, t)dy −

x 0

φI (x, y)β(y, t)dy − γ I (x)X(t), (9.10)

where φI (x, y), ψ I (x, y), γ I (x) are the kernels of the inverse transformation (9.10). The well-posedness of these kernels is shown later. Our aim is to convert the original system (9.1)–(9.5) to the following target system: ˙ X(t) = (A + Bκ)X(t) + Bβ(0, t),

(9.11)

SANDWICH HYPERBOLIC PDE CONTROL

235

 x ψ I (x, y)α(y, t)dy αt (x, t) = −pαx (x, t) + c1 β(x, t) − c1 0  x I φ (x, y)β(y, t)dy − c1 γ I (x)X(t), − c1

(9.12)

βt (x, t) = pβx (x, t), α(0, t) = qβ(0, t) + C0 X(t),

(9.13) (9.14)

0

where C0 = C + qγ(0).

(9.15)

Since the pair [A; B] is controllable, there exists indeed κ such that A + Bκ is Hurwitz. The system (9.13) contains no feedback connection, as we had indicated a few lines above, whereas the system (9.12) contains only feedforward connections from β in (9.13) and X in (9.11). While less obvious, the integral of α(y, t) from y = 0 to y = x is also a feedforward connection, given the transport direction in the α-PDE from y = 0 toward y = 1. Let us now consider the boundary state β(1, t). It is easily seen that βtt (1, t) = vtt (1, t) + pψ(1, 1)ut (1, t) − pψ(1, 0)ut (0, t)   + p pψy (1, 1) + c2 φ(1, 1) u(1, t)   − p pψy (1, 0) + c2 φ(1, 0) u(0, t) − pφ(1, 1)vt (1, t)   + pφ(1, 0) − γ(1)B vt (0, t)   2 − γ(1)A X(t) − p c1 ψ(1, 1) − pφy (1, 1) v(1, t)   2 + pc1 ψ(1, 0) − p φy (1, 0) − γ(1)AB v(0, t)  −  −

1 0

0





2

p ψyy (1, y) + c1 c2 ψ(1, y) u(y, t)dy 1



 p2 φyy (1, y) + c1 c2 φ(1, y) v(y, t)dy.

(9.16)

By virtue of (9.5), (9.6), we obtain vtt (1, t) = c0 vt (1, t) + ru(1, t) + U (t).

(9.17)

Plugging (9.13) and the inverse transformations (9.9), (9.10) into (9.16), after a lengthy calculation, which involves a change of the order of integration in a double integral, we get βtt (1, t) = h1 βt (1, t) + h5 β(1, t) + U (t) + h2 αt (1, t) + h3 βt (0, t) + h4 αt (0, t) + (h6 + r)α(1, t) + h7 β(0, t) + h8 α(0, t)  1  1 h9 (y)β(y, t)dy + h10 (y)α(y, t)dy + H11 X(t), + 0

0

(9.18)

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236

which also belongs to the target system, where h1 , h2 , h3 , h4 , h5 , h6 , h7 , h8 , h9 (y), h10 (y), H11 are shown in appendix 9.8. It should be noted that (9.18) is a secondorder ODE system (β(1, t), βt (1, t)) with a number of PDE state perturbation terms. We can obtain well-posedness of the closed-loop system through analyzing the well-posedness of the target system (9.11)–(9.14), (9.18). It can be expected that well-posedness of the target system depends on that of the 2 × 2 coupled linear hyperbolic PDE-ODE (9.11)–(9.14), which can be obtained similarly to the proof of lemma 9.1 which will be shown later. Kernel Equations Taking the derivative of (9.8) with respect to x and t, respectively, along the solution of (9.1)–(9.4) and substituting the results to (9.13), we get βt (x, t) − pβx (x, t)  x  ψ(x, y)ut (y, t)dy − = vt (x, t) − 0



− pvx (x, t) + p

x 0

φ(x, y)vt (y, t)dy 

x 0

ψx (x, y)u(y, t)dy + p

x 0

φx (x, y)v(y, t)dy

˙ + pψ(x, x)u(x, t) + pφ(x, x)v(x, t) − γ(x)X(t) + pγ  (x)X(t)  x  x pψ(x, y)ux (y, t)dy − c1 ψ(x, y)v(y, t)dy = c2 u(x, t) + 0

 − 

x 0

pφ(x, y)vx (y, t)dy − pψx (x, y)u(y, t)dy +

0

x

c2 φ(x, y)u(y, t)dy

0



x

+

0



x 0

pφx (x, y)v(y, t)dy

˙ + pψ(x, x)u(x, t) + pφ(x, x)v(x, t) − γ(x)X(t) + pγ  (x)X(t)     = c2 + 2pψ(x, x) u(x, t) + pφ(x, 0) − γ(x)B − pψ(x, 0)q v(0, t) 

x



x



+  −

0

 c2 φ(x, y) − pψx (x, y) + pψy (x, y) u(y, t)dy

0



 − c1 ψ(x, y) + pφx (x, y) + pφy (x, y) v(y, t)dy





+ pγ (x) − γ(x)A − pψ(x, 0)C X(t) = 0.

(9.19)

For (9.19) to hold and matching (9.11), (9.14) with (9.1), (9.4) via the transformations (9.8), we obtain the following kernel equations: c2 + 2pψ(x, x) = 0, pφ(x, 0) = γ(x)B + pψ(x, 0)q,

(9.20) (9.21)

pφx (x, y) + pφy (x, y) − c1 ψ(x, y) = 0,

(9.22)

−pψx (x, y) + pψy (x, y) + c2 φ(x, y) = 0,

(9.23)

SANDWICH HYPERBOLIC PDE CONTROL

237

pγ  (x) − γ(x)A − pψ(x, 0)C = 0,

(9.24)

γ(0) = κ,

(9.25)

for 0 ≤ y ≤ x ≤ 1. Well-Posedness of the Kernel Equations We show the well-posedness of the kernel equations (9.20)–(9.25) by using the methods of characteristics and successive approximations [50]. Lemma 9.1. The kernel equations (9.20)–(9.25) have a unique solution (ψ, φ) ∈ C 1 (D) × C 1 (D), where D = {(x, y)|0 ≤ y ≤ x ≤ 1}. Proof. The proof of this lemma is presented in appendix 9.8. Inverse Transformation In order to ensure the invertibility of the transformation (9.8), we search for the inverse transformation of (9.8), which can convert the target system (9.11)–(9.14) into the original system (9.1)–(9.4). Recalling the transformation (9.8) and rewriting it as  x  x v(x, t) − φ(x, y)v(y, t)dy = β(x, t) + ψ(x, y)u(y, t)dy + γ(x)X(t). (9.26) 0

0

According to lemma 9.1, φ(x, y) is continuous, and we obtain a unique continuous χ(x, y) existing on D = {(x, y)|0 ≤ y ≤ x ≤ 1} such that (see, e.g., [169])  x v(x, t) = β(x, t) + ψ(x, y)u(y, t)dy + γ(x)X(t) 0    x  y + χ(x, y) β(y, t) + ψ(y, z)u(z, t)dz + γ(y)X(t) dy, (9.27) 0

0

whose proof can be seen in chapter 9.9 of [175]. Equation (9.27) is rewritten in the form of (9.10), as follows:  x v(x, t) = β(x, t) + χ(x, y)β(y, t)dy   x   0x + χ(x, z)ψ(z, y)dz + ψ(x, y) α(y, t)dy 0 y    x + γ(x) + χ(x, y)γ(y)dy X(t).

(9.28)

Comparing (9.28) with (9.10), we obtain  x I ψ (x, y) = − χ(x, z)ψ(z, y)dz − ψ(x, y),

(9.29)

0

y

φI (x, y) = −χ(x, y),  I γ (x) = −γ(x) −

(9.30) x 0

χ(x, y)γ(y)dy.

(9.31)

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238

According to the well-posedness of ψ(x, y), χ(x, y), γ(y) in D = {(x, y)|0 ≤ y ≤ x ≤ 1}, we obtain the well-posedness of the kernels ψ I (x, y), φI (x, y), γ I (x) on D in (9.10), which shows the invertibility between the target system (α(x, t), β(x, t)) and the original one (u(x, t), v(x, t)).

9.3

BACKSTEPPING FOR THE INPUT ODE

The following backstepping transformation for the (β(1, t), βt (1, t)) system (9.18) is made: y1 (t) = β(1, t), y2 (t) = βt (1, t) + τ1 [β(1, t)],

(9.32) (9.33)

where the function τ1 , which is to be defined in the following steps, is the virtual control law in the ODE backstepping method. Step 1. We consider a Lyapunov function candidate as 1 Vy1 = y1 (t)2 . 2

(9.34)

Taking the derivative of (9.34), we obtain V˙ y1 = y1 (t)y˙ 1 (t) = y1 (t)(y2 (t) − τ1 ).

(9.35)

τ1 (y1 ) = c¯1 y1 ,

(9.36)

Define

where c¯1 is a positive constant to be determined later. Substituting (9.36) into (9.35) yields V˙ y1 = −¯ c1 y1 (t)2 + y1 (t)y2 (t).

(9.37)

Step 2. Similarly, a Lyapunov function candidate is considered as 1 1 1 Vy = Vy1 + y2 (t)2 = y1 (t)2 + y2 (t)2 . 2 2 2

(9.38)

Taking the derivative of (9.38), we get V˙ y = −¯ c1 y1 (t)2 + y1 (t)y2 (t) + y2 (t)(βtt (1, t) + τ˙1 ).

(9.39)

Recalling (9.18), we obtain  2 ˙ Vy = −¯ c1 y1 (t) + y1 (t)y2 (t) + y2 (t) U (t) + h1 βt (1, t) + h5 β(1, t) + h2 αt (1, t) + h3 βt (0, t) + h4 αt (0, t) + (h6 + r)α(1, t)  1 + h7 β(0, t) + h8 α(0, t) + h9 (y)β(y, t)dy 0   1 + h10 (y)α(y, t)dy + H11 X(t) + τ˙1 , 0

(9.40)

SANDWICH HYPERBOLIC PDE CONTROL

239

where the gains h1 , . . . , h10 , H11 shown in appendix 9.8 are related to the kernel functions (ψ(x, y), φ(x, y)) ∈ W 2,1 (D). Choosing U (t) = −¯ c2 y2 (t) − y1 (t) − τ˙1 − h1 βt (1, t) − h5 β(1, t) − h2 αt (1, t) − h3 βt (0, t) − h4 αt (0, t) = −(¯ c2 + c¯1 + h1 )βt (1, t) − (¯ c1 c¯2 + 1 + h5 )β(1, t) − h2 αt (1, t) − h3 βt (0, t) − h4 αt (0, t),

(9.41)

where c¯2 is a positive constant to be determined later, we get  V˙ y = −¯ c1 y1 (t)2 − c¯2 y2 (t)2 + y2 (t) (h6 + r)α(1, t) + h7 β(0, t) + h8 α(0, t)  + 0

9.4



1

h9 (y)β(y, t)dy +



1 0

h10 (y)α(y, t)dy + H11 X(t) .

(9.42)

CONTROLLER AND STABILITY ANALYSIS

Control Law Substituting the PDE transformation (9.7), (9.8) into (9.41), we get the controller expressed by the original states, as follows: U (t) = −n1 vt (1, t) + n2 v(1, t) − h2 ut (1, t) − n3 u(1, t) − h3 vt (0, t) − n4 v(0, t) − h4 ut (0, t) + n5 u(0, t) + N8 X(t)  1  1 n6 (y)u(y, t)dy + n7 (y)v(y, t)dy, + 0

(9.43)

0

where n1 = c¯2 + c¯1 + h1 ,

(9.44)

n2 = (¯ c2 + c¯1 + h1 )φ(1, 1) − (¯ c1 c¯2 + 1 + h5 ), n3 = (¯ c2 + c¯1 + h1 )ψ(1, 1), n4 = (¯ c2 + c¯1 + h1 )(φ(1, 0) − γ(1)B) − h3 γ(0)B,

(9.45) (9.46) (9.47)

n5 = (¯ c2 + c¯1 + h1 )ψ(1, 0), n6 (y) = (¯ c2 + c¯1 + h1 )(ψy (1, y) + c2 φ(1, y))

(9.48)

+ (¯ c1 c¯2 + 1 + h5 )ψ(1, y), n7 (y) = (¯ c2 + c¯1 + h1 )(c1 ψ(1, y) − φy (1, y)) + (¯ c1 c¯2 + 1 + h5 )φ(1, y),

(9.49) (9.50)

N8 = h3 γ(0)A + (¯ c2 + c¯1 + h1 )γ(1)A + (¯ c1 c¯2 + 1 + h5 )γ(1).

(9.51)

The pending control parameters c¯1 , c¯2 will be determined in the following stability analysis. By substituting (9.2), (9.3) at x = 0 and x = 1 into (9.43), the controller is rewritten as U (t) = −n1 pvx (1, t) + (n2 − h2 c1 )v(1, t) + h2 pux (1, t) − (n3 + n1 c2 )u(1, t) − h3 pvx (0, t)

CHAPTER NINE

240

− (n4 + h4 c1 )v(0, t) + h4 pux (0, t) + (n5 − h3 c2 )u(0, t)  1  1 n6 (y)u(y, t)dy + n7 (y)v(y, t)dy + N8 X(t). + 0

(9.52)

0

Stability Analysis of States Theorem 9.1. For all initial values (u(x, 0), v(x, 0)) ∈ W 2,2 (0, 1), with some c¯1 , c¯2 , the closed-loop system consisting of the plant (9.1)–(9.6) and the control law (9.52) is exponentially stable at the origin in the sense of the norm 

1 0

2



u (x, t)dx +

1 0

2

2

2

v (x, t)dx + |X(t)| + z1 (t) + z2 (t)

2

1/2 ,

(9.53)

where | · | denotes the Euclidean norm. Proof. We start from studying the stability of the target system. The equivalent stability property between the target system and the original system is ensured due to the invertibility of the PDE backstepping transformation (9.7), (9.8) and the ODE backstepping transformation (9.32), (9.33). First, we study the stability proof of the target system via Lyapunov analysis of the PDE-ODE system. Second, with the Lyapunov analysis of the input ODE in section 9.3, Lyapunov analysis of the whole ODE-PDE-ODE system is provided, where the control parameters c¯1 , c¯2 in the control law (9.52) are determined. 1) Lyapunov analysis for the PDE-ODE system Define 2

2

2

Ω1 (t) = β(·, t) + α(·, t) + |X(t)| , 1 2 2 where β(·, t) is a compact notation for 0 β(x, t) dx. Now consider a Lyapunov function   a1 1 δ1 x b1 1 −δ1 x 2 2 e β(x, t) dx + e α(x, t) dx, V1 (t) = X T (t)P1 X(t) + 2 0 2 0

(9.54)

(9.55)

where there exists a matrix P1 = P1T > 0, which is the unique solution to the Lyapunov equation (9.56) P1 (A + Bκ) + (A + Bκ)T P1 = −Q1 , for some Q1 = Q1 T > 0, by recalling that A + Bκ is Hurwitz. The positive parameters a1 , b1 , δ1 are to be chosen later. From (9.54), we obtain θ11 Ω1 (t) ≤ V1 (t) ≤ θ12 Ω1 (t),

(9.57)

  a1 b1 e−δ1 θ11 = min λmin (P1 ), , > 0, 2 2

(9.58)

where

SANDWICH HYPERBOLIC PDE CONTROL

241

  a1 eδ1 b1 , θ12 = max λmax (P1 ), > 0. 2 2

(9.59)

The time derivative of V1 (t) along (9.11)–(9.14) is obtained as p p 2 V˙ 1 (t) ≤ −λmin (Q1 )|X(t)| + 2X T P1 Bβ(0, t) + a1 eδ1 β(1, t)2 − a1 β(0, t)2 2 2  1  1 p p 2 2 δ1 x −δ1 x − δ1 a1 e β(x, t) dx − δ1 b1 e α(x, t) dx 2 2 0 0 p p − b1 e−δ1 α(1, t)2 + b1 α(0, t)2 2 2   1  x + b1 e−δ1 x α(x, t)c1 β(x, t) − ψ I (x, y)α(y, t)dy 0 0   x φI (x, y)β(y, t)dy − γ I (x)X(t) dx. (9.60) − 0

Let us consider the final part in (9.60) first. Using Young’s inequality and the Cauchy-Schwarz inequality for the the final part in (9.60) yields the existence of ξ > 0 such that  1  1 2 e−δ1 x α(x, t)c1 β(x, t)dx < ξ e−δ1 x α(x, t) dx 0

0



+ξ 

1 0



0

e−δ1 x α(x, t)c1

1

e

−δ1 x



x 0

 α(x, t)c1

x 0

ψ I (x, y)α(y, t)dydx
0, 2 p 4 |P B| > 0, η2 = a1 − pb1 q 2 − 2 λmin (Q1 ) ξ p η3 = δ1 a1 − b1 ξ − b1 > 0, 2 δ1   p 2b1 ξ ξ 2 b21 δ 1 b1 − − b1 ξ e−δ1 > 0. − η4 = 2 δ1 λmin (Q1 )

(9.69) (9.70) (9.71) (9.72)

Defining p η5 = b1 e−δ1 > 0, 2

p η6 = a1 eδ1 > 0, 2

(9.73)

we arrive at 2 V˙ 1 (t) ≤ − η1 |X(t)| − η2 β(0, t)2 − η3

 − η4

1 0



1 0

2

β(x, t) dx

2

α(x, t) dx − η5 α(1, t)2 + η6 β(1, t)2 .

(9.74)

2) Lyapunov analysis for the whole ODE-PDE-ODE system Recalling (9.38), we define a Lyapunov function V (t) = V1 (t) + Vy (t).

(9.75)

Denoting the norm 2

2

2

Ω2 (t) = β(·, t) + α(·, t) + |X(t)| + y1 (t)2 + y2 (t)2 ,

(9.76)

θ21 Ω2 (t) ≤ V (t) ≤ θ22 Ω2 (t),

(9.77)

  a1 b1 e−δ1 1 θ21 = min λmin (P1 ), , , > 0, 2 2 2

(9.78)

we get

where

SANDWICH HYPERBOLIC PDE CONTROL

243

  a1 eδ1 b1 1 , , θ22 = max λmax (P1 ), > 0. 2 2 2

(9.79)

Taking the derivative of (9.75) and using (9.74) and (9.42), we get  1 2 2 β(x, t) dx − η4 α(x, t) dx − η5 α(1, t)2 0 0  2 2 2 + η6 β(1, t) − c¯1 y1 (t) − c¯2 y2 (t) + y2 (t) (h6 + r)α(1, t) + h7 β(0, t)

2 V˙ ≤ −η1 |X(t)| − η2 β(0, t)2 − η3



 + h8 (qβ(0, t) + C0 X(t)) +

1

1

h9 (y)β(y, t)dy   1 + h10 (y)α(y, t)dy + H11 X(t) , 0

(9.80)

0

where we have used (9.14). Applying Young’s inequality, the Cauchy-Schwarz inequality, and (9.32) to (9.80), we obtain  2 2 2 V˙ ≤ − η1 − r1 |H11 | − r7 h8 2 |C0 | |X(t)| − (η2 − h27 r3 − r6 h8 2 q 2 )β(0, t)2  1  1 2 2 − (η3 − r5 h29 max ) β(x, t) dx − (η4 − r2 h210 max ) α(x, t) dx 0

2

2

2

0

− (η5 − (h6 + r) r4 )α(1, t) − (¯ c1 − η6 )y1 (t)    1 1 1 1 1 1 1 − c¯2 − + + + + + + y2 (t)2 . 4r1 4r2 4r3 4r4 4r5 4r6 4r7

(9.81)

We choose the positive constants r1 , r2 , r3 , r4 , r5 , r6 , r7 as r1 < r5
0.

From (9.77) and (9.86), we conclude that the target system (β(x, t), α(x, t), X(t), y1 (t), y2 (t)) is exponentially stable in the sense of the norm 

1



2

α (x, t)dx +

0

1 0

2

2

2

β (x, t)dx + |X(t)| + y1 (t) + y2 (t)

2

1/2 .

(9.87)

Using the invertibility between the target (α(x, t), β(x, t)) system and the original (u(x, t), v(x, t)) system via the transformation (9.8) and its inverse (9.10) and the invertibility between (y1 (t), y2 (t)) and (β(1, t), βt (1, t)) via the invertible transformations (9.32), (9.33), together with (9.5), we obtain that the (v(x, t), u(x, t), X(t), z1 (t), βt (1, t)) system is exponentially stable in the sense of the norm 

1 0

u2 (x, t)dx +



1 0

2

v 2 (x, t)dx + |X(t)| + z1 (t)2 + βt (1, t)2

1/2 .

(9.88)

Taking the derivative of the inverse transformation (9.10) and setting x = 1, together with (9.5), we get z(t) ˙ = vt (1, t) = βt (1, t) + pψ I (1, 1)α(1, t) − pφI (1, 1)β(1, t)   − pψ I (1, 0)α(0, t) + pφI (1, 0) − γ I (1)B β(0, t) 

1



1

+ 

0 1



c1 ψ (1, σ)ψ (σ, y)dσ − pψy (1, y) α(y, t)dy I

I

y 1

+ 0

 I

 c1 ψ (1, σ)φ (σ, y)dσ − c1 ψ (1, y) + pφy (1, y) β(y, t)dy I

I

y



1

+ 0

I

I



c1 ψ (1, y)γ (y)dy − γ (1)(A + Bκ) X(t). I

I

I

(9.89)

Applying the Cauchy-Schwarz inequality into (9.89), with the exponential stability results in terms of α(·, t)2 + β(·, t)2 + |X(t)|2 + |z(t)|2 + |βt (1, t)|2 , shown in (9.87) and (9.88), we obtain the exponential convergence results in terms of |z(t)| ˙ 2, 2 namely z2 (t) . With this, the proof of theorem 9.1 is complete.

9.5

BOUNDEDNESS AND EXPONENTIAL CONVERGENCE OF THE CONTROLLER

In the last section, we proved that all states of PDEs and ODEs are exponentially stable in the closed-loop system, which consists of the plant (9.1)–(9.6) and the

SANDWICH HYPERBOLIC PDE CONTROL

245

controller (9.52). In this section, we prove the exponential convergence and boundedness of the controller U (t) (9.52) in the closed-loop system. Theorem 9.2. In the closed-loop system, which consists of the plant (9.1)–(9.6) and the controller U (t) (9.52), there exist positive constants λ2 and Υ0 such that |U (t)| ≤ Υ0 e−

λ2 2

t

,

∀t ≥ 0,

(9.90)

namely, ensuring that U (t) is bounded and exponentially convergent to zero. According to (9.52) and theorem 9.1, we know that if we want to show the exponential convergence of the controller (9.52), the exponential convergence of eight signals u(1, t), ux (1, t), v(0, t), vx (0, t), v(1, t), vx (1, t), u(0, t), ux (0, t) in (9.52) needs to be proved, which can be obtained through producing L2 estimates of ux (x, t), vx (x, t), uxx (x, t), vxx (x, t). Before the proof of theorem 9.2, we present two lemmas. The first shows the exponential stability estimates in terms of ux (x, t)2 + vx (x, t)2 . The second gives the exponential stability estimates in terms of uxx (x, t)2 + vxx (x, t)2 . Lemma 9.2. For all initial data (u(x, 0), v(x, 0)) ∈ H 1 (0, 1), the closed-loop system (u(x, t), v(x, t)), comprising (9.1)–(9.6) with the controller (9.52), is exponentially stable in the sense of ux (·, t)2 + vx (·, t)2 .

(9.91)

Proof. Differentiating (9.12) and (9.13) with respect to x and differentiating (9.14) with respect to t, we obtain 

αxt (x, t) = −pαxx (x, t) + c1 βx (x, t) − c1 γ I (x)X(t) − c1 ψ I (x, x)α(x, t) − c1 φI (x, x)β(x, t)  x  x − c1 ψx I (x, y)α(y, t)dy − c1 φx I (x, y)β(y, t)dy, 0

(9.92)

0

βxt (x, t) = pβxx (x, t),

(9.93)

 1

C0 (A + Bκ) + c1 γ I (0) X(t) −αx (0, t) = qβx (0, t) + p 1 + (C0 B − c1 )β(0, t). p Let us define

1 A1 = 2 A2 =

1 2

 

1 0

0

1

(9.94)

2

b2 e−δ2 x αx (x, t) dx,

(9.95)

2

a2 eδ2 x βx (x, t) dx,

(9.96)

where b2 is an arbitrary positive constant, which can adjust the convergence rate, and the positive constants δ2 , a2 shall be chosen later. Taking the derivative of (9.95) along (9.92), (9.93), we obtain p p p A˙ 1 = − b2 e−δ2 αx (1, t)2 + b2 αx (0, t)2 − b2 δ2 2 2 2



1 0

2

e−δ2 x αx (x, t) dx

CHAPTER NINE

246  −  −  −  − 

1 0 1 0 1 0 1 0 1

+  −

0 1 0

b2 e−δ2 x αx (x, t)c1 ψ I (x, x)α(x, t)dx b2 e−δ2 x αx (x, t)c1 φI (x, x)β(x, t)dx 

b2 e−δ2 x αx (x, t)c1



b2 e−δ2 x αx (x, t)c1

x 0 x 0

ψx I (x, y)α(y, t)dydx φx I (x, y)β(y, t)dydx

b2 e−δ2 x αx (x, t)c1 βx (x, t)dx 

b2 e−δ2 x αx (x, t)c1 γ I (x)X(t)dx.

(9.97)

Let us consider the last six terms in (9.97) first. Using Young’s inequality and the Cauchy-Schwarz inequality yields the existence of ξ2 > 0 such that  1 e−δ2 x αx (x, t)c1 ψ I (x, x)α(x, t)dx 0



< ξ2 

1 0

1 0

−δ2 x



2

αx (x, t) dx + ξ2

1 0

2

e−δ2 x α(x, t) dx,

2

e−δ2 x αx (x, t) dx + ξ2

e−δ2 x αx (x, t)c1



x 0



1 0

2

eδ2 x β(x, t) dx,


0, 2 2

(9.111)

where

CHAPTER NINE

248 

a2 eδ2 b2 , θ32 = max R1 θ22 , 2 2

 > 0.

(9.112)

Taking the derivative of (9.108), recalling (9.107), (9.86), and applying the CauchySchwarz inequality into (9.94) to rewrite αx (0, t)2 in (9.107) as αx (0, t)2 ≤ 3q 2 βx (0, t)2 + +

2 3 C0 (A + Bκ) + c1 γ I (0) |X(t)|2 2 p

3 (C0 B − c1 )2 β(0, t)2 , p2

(9.113)

we get V˙ 2 (t) = A¯˙ + R1 V˙ p ≤ − b2 e−δ2 αx (1, t)2 2   p 3pb2 q 2 a2 − − βx (0, t)2 2 2   1 p 2ξ2 b2 2 b2 δ2 − 4ξ2 b2 − − αx (x, t) dx 2 δ2 0   1 p 2 − a 2 δ 2 − ξ 2 b2 eδ2 x βx (x, t) dx 2 0   3b2 (c1 + C0 B)2 β(0, t)2 − R1 ηˆ0 − 2p     a2 eδ2 a2 eδ2 2 2 c¯ β(1, t)2 − R1 θ22 λ − y2 (t) − R1 θ22 λ − p p 1   1 ξ 2 b2 2 − R1 θ22 λ − ξ2 b2 − α(x, t) dx δ2 0   1 ξ2 b2 δ2 2 − R1 θ22 λ − ξ2 b2 eδ2 − e β(x, t) dx δ2 0   3b2 2 I 2 |c1 γ (0) + C0 (A + Bκ)| |X(t)| − R1 ηˆ1 α(1, t)2 − R1 θ22 λ − ξ2 b2 − 2p ≤ − λ1 θ32 Ω3 (t) − R1 ηˆ1 α(1, t)2 −ˆ η2 β(0, t)2 − ηˆ3 βx (0, t)2 ≤ − λ1 V2 (t) − R1 ηˆ1 α(1, t)2 −ˆ η2 β(0, t)2 − ηˆ3 βx (0, t)2 for some positive λ1 , where we have chosen     2ξ2 12ξ2 2 , 3b2 q δ2 > max 1, , a2 > max p δ2

(9.114)

(9.115)

and sufficiently large R1 . The coefficients ηˆ1 , ηˆ2 , ηˆ3 are 3pb2 (c1 + C0 B)2 > 0, 2p2 p 3pb2 q 2 > 0. ηˆ3 = a2 − 2 2 ηˆ2 = R1 ηˆ0 −

(9.116) (9.117)

SANDWICH HYPERBOLIC PDE CONTROL

249

Recalling (9.110), (9.114), we obtain the exponential stability estimates in the sense of αx (·, t)2 + βx (·, t)2 .

(9.118)

Differentiating (9.9), (9.10) with respect to x, we get ux (x, t) = αx (x, t),



vx (x, t) = βx (x, t) −

x 0

(9.119)

 ψxI (x, y)α(y, t)dy −

x 0

φIx (x, y)β(y, t)dy

− γ˙ I (x)X(t) − ψ I (x, x)α(x, t) − φI (x, x)β(x, t).

(9.120)

Using the Young and Cauchy-Schwarz inequalities, we get the inequalities  vx (x, t)2 ≤ 6 βx (x, t)2 + K∞ α(x, t)2   + L∞ β(x, t)2 + max {|γ I (x)|2 }|X(t)|2 , (9.121) x∈[0,1]

where K∞ = max {|ψx (x, y)|2 } + max {|ψ I (x, x)|2 },

(9.122)

L∞ = max {|φx (x, y)|2 } + max {|φI (x, x)|2 }.

(9.123)

(x,y)∈D

x∈[0,1]

(x,y)∈D

x∈[0,1]

Based on the exponential stability estimates in terms of αx (·, t)2 + βx (·, t)2 proved above, together with the exponential stability results in terms of the norm that includes α(·, t)2 + β(·, t)2 + |X(t)|2 , provided in theorem 9.1, we obtain the exponential stability estimates in terms of ux (·, t)2 + vx (·, t)2 . The proof of lemma 9.2 is complete. Lemma 9.3. For all initial data (u(x, 0), v(x, 0)) ∈ H 2 (0, 1), the closed-loop system (u(x, t), v(x, t)) (9.1)–(9.6) with the controller (9.52) is exponentially stable in the sense of 2

2

vxx (·, t) + uxx (·, t) .

(9.124)

Proof. Differentiating (9.12) and (9.13) twice with respect to x and differentiating (9.14) twice with respect to t, we obtain 

αxxt (x, t) = −pαxxx (x, t) + c1 βxx (x, t) − c1 γ I (x)X(t)  x − c1 ψxx I (x, y)α(y, t)dy − c1 ψ I (x, x)αx (x, t) 0  x φxx I (x, y)β(y, t)dy − c1 φI (x, x)βx (x, t) − c1  0 − 2c1 ψx I (x, x) + c1 ψy I (x, x) α(x, t)  − 2c1 φx I (x, x) + c1 φy I (x, x) β(x, t),

(9.125)

βxxt (x, t) = pβxxx (x, t),

(9.126)

CHAPTER NINE

250 and

1 1 αxx (0, t) = qβxx (0, t) + C0 Bβx (0, t) − c1 ψ I (0, 0)α(0, t) p p 1   − 2 pc1 γ I (0) − C0 (A + Bκ)2 − c1 γ(0)(A + Bκ) X(t) p  1

− 2 C0 (A + Bκ)B + pc1 φI (0, 0) − c1 γ(0)B β(0, t). p We denote B1 =

1 2

B2 =

1 2

 

1 0

0

1

2

b3 e−δ3 x αxx (x, t) dx,

(9.127)

(9.128)

2

a3 eδ3 x βxx (x, t) dx,

(9.129)

where the positive constant b3 can be chosen arbitrarily to adjust the convergence rate, and the the positive constants δ3 , a3 shall be defined later. Taking the derivative of (9.128) along (9.125), (9.126), we get p p p B˙ 1 (t) = − b3 e−δ3 αxx (1, t)2 + b3 αxx (0, t)2 − b3 δ3 2 2 2  1 + b3 e−δ3 x αxx (x, t)c1 βxx (x, t)dx 

0



0

− − 

0



0



0



0



0

− − − − −

1

1 0



b3 e

−δ3 x

 αxx (x, t)c1

b3 e−δ3 x αxx (x, t)c1



x 0 x 0

ψxx I (x, y)α(y, t)dydx φxx I (x, y)β(y, t)dydx

1

 b3 e−δ3 x αxx (x, t) 2c1 ψx I (x, x) + c1 ψy I (x, x) α(x, t)dx

1

 b3 e−δ3 x αxx (x, t) 2c1 φx I (x, x) + c1 φy I (x, x) β(x, t)dx

1

1

0

2

e−δ3 x αxx (x, t) dx

b3 e−δ3 x αxx (x, t)c1 γ I (x)X(t)dx

1

1



b3 e−δ3 x αxx (x, t)c1 ψ I (x, x)αx (x, t)dx b3 e−δ3 x αxx (x, t)c1 φI (x, x)βx (x, t)dx.

(9.130)

Now let us deal with the last eight terms in (9.130) by using Young’s inequality and the Cauchy-Schwarz inequality. Similarly to (9.98)–(9.103), there exists a ξ3 > 0 such that  1 e−δ3 x αxx (x, t)c1 βxx (x, t)dx 0



≤ ξ3

1 0

2

e−δ3 x αxx (x, t) dx + ξ3



1 0

2

eδ3 x βxx (x, t) dx,

(9.131)

SANDWICH HYPERBOLIC PDE CONTROL



1 0

e−δ3 x αxx (x, t)c1 γ¨ I (x)X(t)dx



≤ ξ3 1 0

≤

1 0



251

2

e−δ3 x αxx (x, t)c1



x 0

(9.132)

ψxx I (x, y)α(y, t)dydx

 ξ3 1 −δ3 x ξ3 2 2 e−δ3 x αxx (x, t) dx + e α(x, t) dx, δ3 0 δ3 0  1  x e−δ3 x αxx (x, t)c1 φxx I (x, y)β(y, t)dydx 0



2

e−δ3 x αxx (x, t) dx + ξ3 |X(t)| ,



1

0

 ξ3 ξ3 1 −δ3 x 2 2 −δ3 x ≤ e αxx (x, t) dx + e β(x, t) dx, δ3 0 δ3 0  1  e−δ3 x αxx (x, t) 2c1 ψx I (x, x) + c1 ψy I (x, x) α(x, t)dx 1

0



≤ ξ3

1 0



1 0

1 0



1 0

1 0



1 0

1 0

2

eδ3 x α(x, t) dx,

(9.135)

2

e−δ3 x αxx (x, t) dx + ξ3



1 0

2

eδ3 x β(x, t) dx,

(9.136)

2

e−δ3 x αxx (x, t) dx + ξ3



1 0

2

e−δ3 x αx (x, t) dx,

(9.137)

e−δ3 x αxx (x, t)c1 φI (x, x)βx (x, t)dx



≤ ξ3



e−δ3 x αxx (x, t)c1 ψ I (x, x)αx (x, t)dx



≤ ξ3

2

e−δ3 x αxx (x, t) dx + ξ3

(9.134)

 e−δ3 x αxx (x, t) 2c1 φx I (x, x) + c1 φy I (x, x) β(x, t)dx



≤ ξ3

(9.133)

1 0

2

e−δ3 x αxx (x, t) dx + ξ3



1 0

2

eδ3 x βx (x, t) dx.

(9.138)

Substituting (9.131)–(9.138) into (9.130), we obtain p p B˙ 1 (t) ≤ − b3 e−δ3 αxx (1, t)2 + b3 αxx (0, t)2 2 2   1 p ξ 3 b3 2 b3 δ3 − 6ξ3 b3 − 2 − e−δ3 x αxx (x, t) dx 2 δ3 0  1  1 2 2 eδ3 x βxx (x, t) dx + ξ3 b3 eδ3 x βx (x, t) dx + ξ 3 b3 0 0   1   1 ξ3 ξ3 2 2 + b3 ξ 3 + eδ3 x β(x, t) dx + b3 ξ3 + e−δ3 x α(x, t) dx δ3 δ3 0 0  1 2 2 + ξ3 b3 e−δ3 x αx (x, t) dx + ξ3 b3 |X(t)| . (9.139) 0

CHAPTER NINE

252 Taking the derivative of (9.129) along (9.125), (9.126), we get  B˙ 2 (t) =

1 0

a3 eδ3 x βxx (x, t)βxxt (x, t)dx



=p

1 0

a3 eδ3 x βxx (x, t)βxxx (x, t)dx

p p p = a3 eδ3 βxx (1, t)2 − a3 βxx (0, t)2 − a3 δ3 2 2 2



1 0

2

eδ3 x βxx (x, t) dx.

(9.140)

Let us define ¯ = B1 + B2 . B

(9.141)

Applying the Cauchy-Schwarz inequality into (9.127) yields 5 5 |C0 B|2 βx (0, t)2 + 2 c21 ψ I (0, 0)2 α(0, t)2 p2 p 2 5   + 4 pc1 γ I (0) − C0 (A + Bκ)2 − c1 γ(0)(A + Bκ) |X(t)|2 p 2 5

+ 4 C0 (A + Bκ)B + pc1 φI (0, 0) − c1 γ(0)B β(0, t)2 , (9.142) p

αxx (0, t)2 ≤ 5q 2 βxx (0, t)2 +

which is used to replace αxx (0, t)2 in (9.139). Then by recalling (9.139), (9.140), the inequality of the derivative of (9.141) is obtained as ¯˙ = B˙ 1 + B˙ 2 B   1 p ξ 3 b3 2 ≤ − b3 δ3 − 6ξ3 b3 − 2 e−δ3 x αxx (x, t) dx 2 δ3 0    1  p p 5pb3 q 2 2 δ3 x a 3 δ 3 − ξ 3 b3 a3 − − e βxx (x, t) dx − βxx (0, t)2 2 2 2 0  1 p −δ3 2 2 − b3 e αxx (1, t) + ξ3 b3 eδ3 x βx (x, t) dx 2 0   1 ξ3 2 eδ3 x β(x, t) dx + b3 ξ 3 + δ3 0   1 ξ3 2 + b3 ξ 3 + e−δ3 x α(x, t) dx δ3 0  1 p 2 + ξ 3 b3 e−δ3 x αx (x, t) dx + a3 eδ3 βtt (1, t)2 2 0    2 5b3 2 I 2 pc1 γ (0) − C0 (A + Bκ) − c1 γ(0)(A + Bκ) + ξ3 b3 |X(t)| + 2p3 2 5b3

+ 3 C0 (A + Bκ)B + pc1 φI (0, 0) − c1 γ(0)B β(0, t)2 2p +

5b3 2 I 5|C0 B|2 b3 c1 ψ (0, 0)2 α(0, t)2 + βx (0, t)2 . 2p 2p

(9.143)

SANDWICH HYPERBOLIC PDE CONTROL

253

Equation (9.143) includes a positive term βtt (1, t)2 (“positive term” here means a quadratic term with a plus sign) which is dealt with next. Substituting (9.41) into (9.18) yields βtt (1, t) = −(¯ c2 + c¯1 )βt (1, t) − (¯ c1 c¯2 + 1)β(1, t) + (h6 + r)α(1, t) + h7 β(0, t) + h8 α(0, t)  1  1 h9 (y)β(y, t)dy + h10 (y)α(y, t)dy + H11 X(t). + 0

(9.144)

0

Applying the Cauchy-Schwarz inequality in (9.144), we get βtt (1, t)2 ≤ 8(¯ c2 + c¯1 )2 βt (1, t)2 + 8(¯ c1 c¯2 + 1)2 β(1, t)2 + 8(h6 + r)2 α(1, t)2 + 8h7 2 β(0, t)2 + 8h8 2 α(0, t)2 2

+ 8h9 max β(·, t)2 + 8h10 max α(·, t)2 + 8H11 2 |X(t)| .

(9.145)

Replacing βtt (1, t)2 in (9.143) with (9.145) will be used in the following Lyapunov analysis. Defining a Lypunov function ¯ Vu = R 2 V2 + B

(9.146)

and 2

2

2

Ω4 (t) = βxx (·, t) + αxx (·, t) + βx (·, t) 2

2

2

+ αx (·, t) + β(·, t) + α(·, t) 2

+ |X(t)| + β(1, t)2 + y2 (t)2 ,

(9.147)

θ41 Ω4 (t) ≤ Vu (t) ≤ θ42 Ω4 (t),

(9.148)

we obtain

where   a3 b3 e−δ3 θ41 = min R2 θ31 , , > 0, 2 2   a3 eδ3 b3 , θ42 = max R2 θ32 , > 0. 2 2

(9.149) (9.150)

By virtue of (9.32), (9.33), (9.36), (9.112), (9.114), (9.143), (9.145), taking the derivative of Vu , we get ¯˙ V˙ u = R2 V˙ 2 + B

  1 1 p ξ 3 b3 2 ≤ − R 2 λ1 V2 − b3 δ3 − 6ξ3 b3 − 2 αxx (x, t) dx 2 2 δ3 0    1  p p 5pb3 q 2 2 − a 3 δ 3 − ξ 3 b3 a3 − eδ3 x βxx (x, t) dx − βxx (0, t)2 2 2 2 0   1 1 p 2 R2 λ1 θ32 − ξ3 b3 eδ3 − b3 e−δ3 αxx (1, t)2 − βx (x, t) dx 2 2 0

CHAPTER NINE

254 

 1 1 ξ3 b3 δ3 2 δ3 δ3 R2 λ1 θ32 − ξ3 b3 e − − e − 4pa3 e h9 max β(x, t) dx 2 δ3 0   1 1 ξ 3 b3 2 δ3 − R2 λ1 θ32 − ξ3 b3 − − 4pa3 e h10 max α(x, t) dx 2 δ3 0   1 1 2 − R2 λ1 θ32 − ξ3 b3 αx (x, t) dx 2 0

 2  1 5b3 I 2 − R2 λ1 θ32 − γ (0) − C (A + Bκ) − c γ(0)(A + Bκ) + ξ 3 b3 pc 1 0 1 2 2p3     5b3 2 2 2 2 δ3 2 δ3 2 c ψ(0, 0) + 4pa3 e h8 |C0 | |X(t)| + 4pa3 e H11 − 2 2p 1

  2 5b3

2 I δ3 C − R2 ηˆ2 − (A + Bκ)B + pc φ (0, 0) − c γ(0)B + 4pa e h 0 1 1 3 7 2p3   5b3 2 c ψ(0, 0)2 + 4pa3 eδ3 h8 2 β(0, t)2 − 2q 2 2p 1   5|C0 B|2 b3 − R2 ηˆ3 − βx (0, t)2 2p   1 R2 λ1 θ32 − 4pa3 eδ3 (¯ − c2 + c¯1 )2 βt (1, t)2 2   1 δ3 2 R2 λ1 θ32 − 4pa3 e (¯ − c1 c¯2 + 1) β(1, t)2 2   − R2 R1 ηˆ1 − 4pa3 eδ3 (h6 + r)2 α(1, t)2 . (9.151) Choosing     2ξ3 b3 16ξ3 , 5b3 q 2 δ3 > max 1, , a3 > max p pδ3 and sufficiently large R2 , we obtain 1 ¯ V˙ u (t) ≤ − R2 λ1 V2 − σ2 B, 2

(9.152)

with the positive constant σ2 =

2 min{ p2 b3 δ3 − 6ξ3 b3 − 2 ξ3δ3b3 , p2 a3 δ3 − ξ3 b3 } max{a3 , b3 }eδ3

.

(9.153)

Then we arrive at V˙ u (t) ≤ −λ2 Vu (t), where

 λ2 = min

 1 λ1 , σ 2 . 2

(9.154)

(9.155)

SANDWICH HYPERBOLIC PDE CONTROL

255

Hence, we obtain Vu (t) ≤ e−λ2 t Vu (0), ∀t ≥ 0.

(9.156)

Therefore, we obtain the exponential stability estimates in terms of αxx (·, t)2 + βxx (·, t))2 . Differentiating (9.9), (9.10) twice with respect to x, we get uxx (x, t) = αxx (x, t), 

vxx (x, t) = βxx (x, t) − γ I (x)X(t)     I I I I − 2ψ x (x, x) + ψ y (x, x) α(x, t) − 2φx (x, x) + φy (x, x) β(x, t)  x  x I − ψ xx (x, y)α(y, t)dy − φIxx (x, y)β(y, t)dy 0

0

− ψ I (x, x)αx (x, t) − φI (x, x)βx (x, t).

(9.157)

Through a similar calculation as in (9.121), with the exponential stability estimates in terms of αxx (·, t)2 + βxx (·, t)2 proved above, and recalling the exponential stability estimates in terms of αx (·, t)2 + βx (·, t)2 shown in lemma 9.2, together with the exponential stability results in terms of the norm including α(·, t)2 + β(·, t)2 + |X(t)|2 provided in theorem 9.1, we obtain the exponential stability estimates in terms of uxx (·, t)2 + vxx (·, t)2 . The proof of lemma 9.3 is thus complete. Using lemmas 9.2 and 9.3, we prove theorem 9.2 as follows. Proof. Recalling (9.52) and using the Cauchy-Schwarz inequality, we obtain |U (t)|2  ≤ ξ¯1 vx (1, t)2 + ξ¯2 v(1, t)2 + ξ¯3 ux (1, t)2 + ξ¯4 u(1, t)2 + ξ¯5 vx (0, t)2 + ξ¯6 v(0, t)2  2 2 2 2 2 ¯ ¯ ¯ ¯ ¯ + ξ7 ux (0, t) + ξ8 u(0, t) + ξ9 |X(t)| + ξ10 u(·, t) + ξ11 v(·, t) (9.158) for some positive constants ξ¯1 , ξ¯2 , ξ¯3 , ξ¯4 , ξ¯5 , ξ¯6 , ξ¯7 , ξ¯8 , ξ¯9 , ξ¯10 , ξ¯11 . Recalling the exponential estimates in terms of the norms u(·, t)H2 + v(·, t)H2 proved in lemmas 9.2 and 9.3, using the Sobolev inequality, we obtain the exponential estimate in terms of the norm u(·, t)C 1 + v(·, t)C 1 , which gives the exponential convergence of U (t) by recalling (9.158) and theorem 9.1. The upper boundedness Υ0 of |U (t)| depends on the initial values of the terms in (9.158). The proof of theorem 9.2 is complete.

9.6

EXTENSION TO ODES OF ARBITRARY ORDER

In this section, we allow the possibility that the input ODE is not of second-order but of arbitrary order m and provide a sketch of the design and analysis for this general case. To avoid repetition, we omit some detailed calculations that can be developed by relying on calculations in sections 9.2–9.5.

CHAPTER NINE

256 Replace (9.5), (9.6) with v(1, t) = Cz Z(t), ˙ = Az Z(t) + Bz U (t) + φz , Z(t)

(9.159) (9.160)

where Z(t) ∈ Rm×1 , and Az = [0, 1, 0, . . . , 0; 0, 0, 1, 0, . . . , 0; · · · ; 0, . . . , 0, 1; az1 , . . . , azm ] ∈ Rm×m ,

(9.161)

with az1 , . . . , azm as arbitrary constants. The matrices Bz and Cz are Bz = [0, 0, . . . , 1]T ∈ Rm×1 , Cz = [1, 0, . . . , 0] ∈ R1×m , and φz is

 φz = Rz1 v(0, t) + Rz2 u(1, t) + 

1

+ 0

1 0

Rz3 (x)u(x, t)dx

Rz4 (x)v(x, t)dx + Rz5 X(t),

(9.162)

where the constants Rz1 , Rz2 , the bounded functions Rz3 (x), Rz4 (x), and the constant matrix Rz5 are arbitrary. It is assumed that the system (9.159), (9.160) has the full relative degree. Control Design The equation (9.160), with the state variable choices Z(t) = [z1 (t), . . . , zm (t)]T , is in the form of a chain of m integrators given by z˙1 (t) = z2 (t), z˙2 (t) = z3 (t), .. . z˙m−1 (t) = zm (t), z˙m (t) = az1 z1 (t) + · · · + azm zm (t) + U (t) + φz .

(9.163) (9.164) (9.165) (9.166) (9.167)

The form of system (9.18) extended to order m is obtained as ∂tm β(1, t) = U (t) + [qm ∂tm−1 β(1, t) + qm−1 ∂tm−2 β(1, t) + · · · + q2 βt (1, t) + q1 β(1, t)] + [pm ∂tm−1 α(1, t) + pm−1 ∂tm−2 α(1, t) + · · · + p2 αt (1, t) + p1 α(1, t)] + [¯ qm ∂tm−1 β(0, t) + q¯m−1 ∂tm−2 β(0, t) + · · · + q¯2 βt (0, t) + q¯1 β(0, t)] + [¯ pm ∂tm−1 α(0, t) + p¯m−1 ∂tm−2 α(0, t) + · · · + p¯2 αt (0, t) + p¯1 α(0, t)]  1  1 ¯ Q(y)β(y, t)dy + P¯ (y)α(y, t)dy + Hz X(t), (9.168) + 0

0

¯ P¯ (y), Hz are coefficients where qm , . . . , q1 , pm , . . . , p1 , q¯m , . . . , q¯1 , p¯m , . . . , p¯1 , Q(y), consisting of the kernel functions in the backstepping transformation (9.8), (9.10) and the plant parameters in (9.1)–(9.4), (9.159)–(9.162).

SANDWICH HYPERBOLIC PDE CONTROL

257

The following backstepping transformation for the (β(1, t), βt (1, t), . . . , ∂tm−1 β(1, t)) system (9.168) is postulated as y1 (t) = β(1, t), y2 (t) = βt (1, t) + τ1 [β(1, t)], .. . ym (t) = ∂tm−1 β(1, t) + τm−1 [β(1, t), . . . , ∂tm−2 β(1, t)],

(9.169) (9.170) (9.171) (9.172)

where τ1 , . . . , τm−1 are determined in the following steps as the virtual control laws in the ODE backstepping method. Step 1. We consider a Lyapunov function candidate 1 Vy1 = y1 (t)2 . 2

(9.173)

Taking the derivative of (9.173), we obtain V˙ y1 = −ˆ c1 y1 (t)2 + y1 (t)y2 (t),

(9.174)

τ1 = cˆ1 y1 (t),

(9.175)

with the choice of

where cˆ1 is a positive constant to be determined later. Step 2. A Lyapunov function candidate is considered as 1 1 1 Vy2 = Vy1 + y2 (t)2 = y1 (t)2 + y2 (t)2 . 2 2 2

(9.176)

Taking the derivative of (9.176), we obtain V˙ y2 = −ˆ c1 y1 (t)2 + y1 (t)y2 (t) + y2 (t)(y3 (t) − τ2 + τ˙1 ).

(9.177)

τ2 = τ˙1 + y1 (t) + cˆ2 y2 (t),

(9.178)

τ˙1 = cˆ1 y˙ 1 = cˆ1 βt (1, t)

(9.179)

Choosing

where

according to (9.175), (9.169), we get V˙ y2 = −ˆ c1 y1 (t)2 − cˆ2 y2 (t)2 + y2 (t)y3 (t).

(9.180)

Step 3. through step m-1. We make an induction hypothesis that for a Lyapunov function candidate 1 Vyi = Vyi−1 + yi (t)2 2 1 1 1 1 = y1 (t)2 + y2 (t)2 + · · · + yi−1 (t)2 + yi (t)2 , ∀ 2 ≤ i ≤ m − 2, 2 2 2 2

(9.181)

CHAPTER NINE

258 we have

V˙ yi = − cˆ1 y1 (t)2 − cˆ2 y2 (t)2 − · · · − cˆi−1 yi−1 (t)2 − cˆi yi (t)2 + yi (t)yi+1 (t), ∀i ≥ 2

(9.182)

by choosing τi = τ˙i−1 + yi−1 (t) + cˆi yi (t).

(9.183)

Consider a Lyapunov function candidate 1 Vyi+1 = Vyi + yi+1 (t)2 2 1 1 1 1 = y1 (t)2 + y2 (t)2 + . . . + yi (t)2 + yi+1 (t)2 . 2 2 2 2

(9.184)

Taking the derivative of (9.184), using (9.182) and (9.169)–(9.172), we obtain V˙ yi+1 = V˙ yi + yi+1 (t)y˙ i+1 (t) = −ˆ c1 y1 (t)2 − cˆ2 y2 (t)2 − . . . − cˆi−1 yi−1 (t)2 − cˆi yi (t)2 + yi (t)yi+1 (t) + yi+1 (t)(τ˙i + yi+2 − τi+1 ) = −ˆ c1 y1 (t)2 − cˆ2 y2 (t)2 − . . . − cˆi−1 yi−1 (t)2 − cˆi yi (t)2 − cˆi+1 yi+1 (t)2 + yi+1 (t)yi+2 (t)

(9.185)

by choosing τi+1 = τ˙i + yi (t) + cˆi+1 yi+1 (t).

(9.186)

Therefore, (9.181)–(9.183) holds for i + 1. Recalling steps 1 and 2, it follows that for a Lyapunov function candidate 1 1 1 Vym−1 = y1 (t)2 + y2 (t)2 + · · · + ym−1 (t)2 , 2 2 2

(9.187)

we have V˙ ym−1 = − cˆ1 y1 (t)2 − cˆ2 y2 (t)2 − . . . − cˆm−2 ym−2 (t)2 − cˆm−1 ym−1 (t)2 + ym−1 (t)ym (t)

(9.188)

τm−1 = τ˙m−2 + ym−2 (t) + cˆm−1 ym−1 (t).

(9.189)

by choosing

Step m. A Lyapunov function candidate is considered as 1 Vym = Vym−1 + ym (t)2 2 1 1 1 1 2 = y1 (t) + y2 (t)2 + · · · + ym−1 (t)2 + ym (t)2 . 2 2 2 2 Taking the derivative of (9.190), recalling (9.188), we obtain

(9.190)

SANDWICH HYPERBOLIC PDE CONTROL

259

V˙ ym = − cˆ1 y1 (t)2 − cˆ2 y2 (t)2 − · · · − cˆm−1 ym−1 (t)2 + ym−1 (t)ym (t) + ym (t)y˙ m (t).

(9.191)

According to (9.168) and (9.172), (9.191) is rewritten as V˙ ym 2

2

2

= −ˆ c1 y1 (t) − cˆ2 y2 (t) − · · · − cˆm−1 ym−1 (t) + ym−1 (t)ym (t) + ym (t) U (t)   + qm ∂tm−1 β(1, t) + qm−1 ∂tm−2 β(1, t) + · · · + q2 βt (1, t) + q1 β(1, t)   + pm ∂tm−1 α(1, t) + pm−1 ∂tm−2 α(1, t) + · · · + p2 αt (1, t) + p1 α(1, t)   + q¯m ∂tm−1 β(0, t) + q¯m−1 ∂tm−2 β(0, t) + · · · + q¯2 βt (0, t) + q¯1 β(0, t)   + p¯m ∂tm−1 α(0, t) + p¯m−1 ∂tm−2 α(0, t) + · · · + p¯2 αt (0, t) + p¯1 α(0, t)   1  1 ¯ Q(y)β(y, t)dy + P¯ (y)α(y, t)dy + Hz X(t) + τ˙m−1 , + (9.192) 0

0

where τ˙m−1 = cˆ1 y1m−1 (t) +

i=m−2 



m−1−i yim−1−i (t) + cˆi+1 yi+1 (t)



(9.193)

i=1

for m − 1 ≥ 2, by virtue of (9.189), and yin (t) denotes the n order derivative of yi (t), ∀i = 1, . . . , m. For m − 1 = 1, τ˙1 is given in (9.179). Design the controller as   U (t) = − qm ∂tm−1 β(1, t) + qm−1 ∂tm−2 β(1, t) + · · · + q2 βt (1, t) + q1 β(1, t)   − pm ∂tm−1 α(1, t) + pm−1 ∂tm−2 α(1, t) + · · · + p2 αt (1, t)   − q¯m ∂tm−1 β(0, t) + q¯m−1 ∂tm−2 β(0, t) + · · · + q¯2 βt (0, t)   − p¯m ∂tm−1 α(0, t) + p¯m−1 ∂tm−2 α(0, t) + · · · + p¯2 αt (0, t) − ym−1 (t) − τ˙m−1 − cˆm ym (t) 

 m−2  cˆi ∂tm−1 β(1, t) = − qm + 

i=1

+ qm−1 + m − 2 +

m−2 

 cˆi cˆi−1

 ∂tm−2 β(1, t) + · · · + q1 β(1, t)

i=2

− [pm ∂tm−1 α(1, t) + pm−1 ∂tm−2 α(1, t) + · · · + p2 αt (1, t)] − [¯ qm ∂tm−1 β(0, t) + q¯m−1 ∂tm−2 β(0, t) + · · · + q¯2 βt (0, t)] − [¯ pm ∂tm−1 α(0, t) + p¯m−1 ∂tm−2 α(0, t) + · · · + p¯2 αt (0, t)] − ym−1 (t) − cˆm−1 y˙ m−1 (t) − cˆm ym (t).

(9.194)

Using the transformations (9.169)–(9.172), (9.7), (9.8), the controller (9.194) is expressed as a function of the original state u(x, t), v(x, t) as U (t) = n ˆ m−1 ∂xm−1 v(1, t) + n ˆ m−2 ∂xm−2 v(1, t), · · · + n ˆ 0 v(1, t) m−1 m−2 ˆ ˆ + km−1 ∂ u(1, t) + km−2 ∂ u(1, t) + · · · + kˆ0 u(1, t) x

x

CHAPTER NINE

260

ˆ m−1 ∂ m−1 v(0, t) + h ˆ m−2 ∂ m−2 v(0, t) + · · · + h ˆ 0 v(0, t) +h x x + ˆlm−1 ∂xm−1 u(0, t) + ˆlm−2 ∂xm−2 u(0, t) + · · · + ˆl0 u(0, t)  1  1 ˆ ˆ ˆ N (y)u(y, t)dy + L(y)v(y, t)dy, + DX(t) + 0

(9.195)

0

where the equations (9.2), (9.3) have been used to replace the time derivatives arising in (9.194) with the spatial derivatives in (9.195). ˆ m−1 , . . . , h ˆ 0 , ˆlm−1 , . . . , ˆl0 , and The gains n ˆ m−1 , . . . , n ˆ 0 , kˆm−1 , . . . , kˆ0 , h ˆ ˆ ˆ D, N (y), L(y) consist of the kernels in the backstepping transformations (9.8), (9.10), the plant parameters in (9.1)–(9.4), (9.159), (9.160), and the control parameters cˆ1 , . . . , cˆm , κ. Now we get V˙ ym = − cˆ1 y1 (t)2 − cˆ2 y2 (t)2 − · · · − cˆm ym (t)2  + ym (t) q¯1 β(0, t) + p¯1 α(0, t) + p1 α(1, t) 

1

+ 0

 ¯ Q(y)β(y, t)dy +

1 0

 ¯ P (y)α(y, t)dy + Hz X(t) ,

(9.196)

where cˆ1 , . . . , cˆm are positive constants to be determined later. Stability Analysis Theorem 9.3. For all initial values (u(x, 0), v(x, 0)) ∈ W m,2 (0, 1), the closed-loop system consisting of the plant (9.1)–(9.4), (9.159), (9.160) and the control law (9.195) is exponentially stable at the origin in the sense of the norm 

1 0

u2 (x, t)dx +



1 0

2

v 2 (x, t)dx + |X(t)| + z1 (t)2 + · · · + zm (t)2

1/2 .

(9.197)

Proof. Recalling (9.190) and (9.55), we define a Lyapunov function as Vm (t) = V1 (t) + Vym (t).

(9.198)

Denoting 2

2

2

Ω2m (t) = β(·, t) + α(·, t) + |X(t)| + y1 (t)2 + · · · + ym (t)2 ,

(9.199)

we get θ21 Ω2m (t) ≤ Vm (t) ≤ θ22 Ω2m (t),

(9.200)

where   a1 b1 e−δ1 1 θ21 = min λmin (P1 ), , , > 0, 2 2 2   a1 eδ1 b1 1 , , θ22 = max λmax (P1 ), > 0. 2 2 2 Taking the derivative of (9.198) and using (9.74) and (9.196), we get

(9.201) (9.202)

SANDWICH HYPERBOLIC PDE CONTROL

2 V˙ m ≤ −η1 |X(t)| − η2 β(0, t)2 − η3

261



1 0



2

β(x, t) dx − η4

1

0

2

α(x, t) dx − η5 α(1, t)2

+ η6 y1 (t)2 − cˆ1 y1 (t)2 − cˆ2 y2 (t)2 − · · · − cˆm−1 ym−1 (t)2 − cˆm ym (t)2   1 + ym (t) q¯1 β(0, t) + p¯1 α(0, t) + p1 α(1, t) + Q(y)β(y, t)dy  + 0

0



1

P (y)α(y, t)dy + Hz X(t) .

(9.203)

Recalling (9.14) and applying Young’s inequality, the Cauchy-Schwarz inequality, and (9.169) into (9.203), we obtain  2 2 2 V˙ m ≤ − η1 − rˆ1 |Hz | − rˆ7 p¯21 |C0 | |X(t)| − (η2 − q¯12 rˆ3 − rˆ6 p¯21 q 2 )β(0, t)2 ¯ 2max ) − (η3 − rˆ5 Q



1 0

2

2 β(x, t) dx − (η4 − rˆ2 P¯max )

− (η5 − p21 rˆ4 )α(1, t)2

2



1 0

2

α(x, t) dx

2

− (ˆ c1 − η6 )y1 (t) − cˆ2 y2 (t) − · · · − cˆm−1 ym−1 (t)2    1 1 1 1 1 1 1 − cˆm − + + + + + + ym (t)2 . (9.204) 4ˆ r1 4ˆ r2 4ˆ r3 4ˆ r4 4ˆ r5 4ˆ r6 4ˆ r7 We choose the positive constants rˆ1 , rˆ2 , rˆ3 , rˆ4 , rˆ5 , rˆ6 , rˆ7 as rˆ1
η6 ,   1 1 1 1 1 1 1 1 + + + + + + . cˆm > 4 rˆ1 rˆ2 rˆ3 rˆ4 rˆ5 rˆ6 rˆ7

(9.207) (9.208)

The positive control parameters cˆ2 , . . . , cˆm−1 can be chosen arbitrarily to adjust the exponential decay rate of the closed-loop system. Finally, we arrive at ˆ m (t) − gˆ0 β(0, t)2 − gˆ1 α(1, t)2 V˙ m (t) ≤ −λV ˆ and for some positive λ, gˆ0 = η2 − q¯12 rˆ3 − rˆ6 p¯21 q 2 > 0, gˆ1 = η5 − p21 rˆ4 > 0. Through a process similar to (9.87)–(9.89), we arrive at theorem 9.3.

(9.209)

CHAPTER NINE

262

Boundedness and Exponential Convergence of the Controller In this section, we prove the exponential convergence and boundedness of the controller U (t) (9.195) in the closed-loop system including the input ODE of order m. Theorem 9.4. In the closed-loop system including the plant (9.1)–(9.4), (9.159), (9.160) and the controller U (t) (9.195), there exist positive constants λm and Υ0m such that |U (t)| ≤ Υ0m e−

λm 2

t

,

∀t ≥ 0,

(9.210)

namely, ensuring that U (t) is bounded and exponentially convergent to zero. Before showing the proof of theorem 9.4, we present two lemmas where we produce and analyze L2 estimates of ux (x, t), vx (x, t), · · · , ∂xm−1 u(x, t), ∂xm−1 v(x, t), ∂xm u(x, t), ∂xm v(x, t) in order to prove the exponential convergence of the bounds of signals ∂xm−1 v(1, t), ∂xm−1 u(1, t), ∂xm−1 v(0, t), ∂xm−1 u(0, t), ∂xm−2 u(1, t), ∂xm−2 v(0, t), ∂xm−2 u(0, t),· · · , v(1, t), u(1, t), v(0, t), u(0, t) in the controller (9.195). Lemma 9.4. For all initial data (u(x, 0), v(x, 0)) ∈ H m−1 (0, 1), the closed-loop (u(x, t), v(x, t)) system (9.1)–(9.4), (9.159), (9.160) with the controller (9.195) is exponentially stable in the sense of ∂xm−1 u(·, t)2 + ∂xm−1 v(·, t)2 .

(9.211)

Proof. We consider the Lyapunov function  1 1 2 ¯ Bm−1 (t) = bm−1 e−δm−1 x ∂xm−1 α(x, t) dx 2 0  1 1 2 + am−1 eδm−1 x ∂xm−1 β(x, t) dx, 2 0

(9.212)

where the positive constant bm−1 can be chosen arbitrarily to adjust the convergence rate, and the positive constants am−1 , δm−1 shall be defined later. Taking the derivative of (9.212) along the system obtained from differentiating (9.12), (9.13) m − 1 times with respect to x and differentiating (9.14) m − 1 times with respect to t, using the Cauchy-Schwarz inequality, we can choose positive am−1 and δm−1 (see remark 9.1) such that ¯˙ m−1 (t) B  ¯1 ≤ −M

1 0

2

¯2 e−δm−1 x ∂xm−1 α(x, t) dx − M



1 0

2

eδm−1 x ∂xm−1 β(x, t) dx

¯ 3 ∂ m−1 β(0, t)2 − 1 bm−1 e−δm−1 ∂ m−1 α(1, t)2 + 1 am−1 eδm−1 ∂ m−1 β(1, t)2 −M x x x 2 2  1   2 2 2 ¯4 +M eδm−1 x ∂xm−2 β(x, t) + · · · + βx (x, t) + β(x, t) dx 

0

  2 2 2 e−δm−1 x ∂xm−2 α(x, t) + · · · + αx (x, t) + α(x, t) dx 0   ¯ 6 |X(t)|2 + M ¯ 7 ∂xm−2 β(0, t)2 + · · · + βx (0, t)2 + β(0, t)2 , +M

¯5 +M

1

¯ 1, M ¯ 2, M ¯ 3, M ¯ 4, M ¯ 5, M ¯ 6, M ¯ 7 are positive constants. where M

(9.213)

SANDWICH HYPERBOLIC PDE CONTROL

263

Remark 9.1. As in (9.143), we can choose am−1 , δm−1 to make sure that the terms 2 of order m − 1, namely the terms ∂xm−1 α(·, t)2 , ∂xm−1 β(·, t)2 , ∂xm−1 β(0, t) , are ¯˙ m−1 , except for ∂xm−1 β(1, t)2 . The positive term ∂xm−1 β(1, t)2 with negative signs in B (“positive term” here means a quadratic term with a plus sign) bounded by p12 ∂tm−1 β(1, t)2 can be accommodated by the exponential results in terms of y1 (t)2 , . . . , 2 ym (t)2 provided in theorem 9.3. As in (9.142), the positive term ∂xm−1 α(0, t) has 2 2 2 m−1 m−2 been written as the positive terms ∂x β(0, t) , ∂x β(0, t) , · · · , β(0, t) , and |X(t)|2 by using the Cauchy-Schwarz inequality in the time derivative of (9.14) 2 of m − 1 order and using (9.12). As in (9.143), the positive term ∂xm−1 β(0, t) is ¯ 6, M ¯7 “canceled” by choosing am−1 , and other positive terms with coefficients M are kept in (9.213). The remaining positive terms will be dealt with in the following steps. All positive terms of order m − 2, . . . , 1 can be accommodated by the exponential estimates in the sense of ∂xm−2 α(·, t)2 + ∂xm−2 β(·, t)2 , . . ., αx (·, t)2 + βx (·, t)2 , which can be obtained according to lemma 9.2 and lemma 9.3. Together with the exponential results in the sense of α(·, t)2 + β(·, t)2 + |X(t)|2 + y1 (t)2 + · · · + ym (t)2 provided in theorem 9.3, we define a Lyapunov function Vu(m−1) (t) = Rm−1

m−2 

Ri Vm (t) +

i=1

+

m−2 

m−2 

¯ + Ri A(t)

i=2



m−2 

¯ Ri B(t)

i=3

¯3 (t) + · · · + B ¯m−2 (t) + B ¯m−1 (t), Ri B

(9.214)

i=4

¯i (t), ∀i = 3, . . . , m − 2 are Lyapunov functions similar to (9.212) by replacwhere B ing m − 1 with i. Taking the derivative of (9.214) and choosing sufficiently large Ri > 0, we get V˙ u(m−1) (t) = −λm−1 Vu(m−1) (t) − gˆ2 α(1, t)2 − gˆ3 [∂xm−1 β(0, t)2 + ∂xm−2 β(0, t)2 + · · · + βx (0, t) + β(0, t)2 ] (9.215) for some positive λm−1 , gˆ2 , gˆ3 . Thus, we obtain the exponential stability estimates in terms of ∂tm−1 α(·, t)2 + m−1 β(·, t)2 . ∂t Through a similar process with (9.119)–(9.121), we obtain lemma 9.4. Lemma 9.5. For all initial data (u(x, 0), v(x, 0)) ∈ H m (0, 1), the closed-loop system (u(x, t), v(x, t)) (9.1)–(9.4), (9.159), (9.160) with the controller (9.195) is exponentially stable in the sense of ∂xm u(·, t)2 + ∂xm v(·, t)2 . Proof. Define a Lyapunov function  1  1 1 2 2 ¯m (t) = 1 B bm e−δm x ∂xm α(x, t) dx + am eδm x ∂xm β(x, t) dx, 2 0 2 0

(9.216)

(9.217)

where the positive constant bm can be chosen arbitrarily to adjust the convergence rate, and the positive constants am , δm will be defined later.

CHAPTER NINE

264

Taking the derivative of (9.217) along the system obtained from differentiating (9.12), (9.13) m times with respect to x and differentiating (9.14) m times with respect to t, using the Cauchy-Schwarz inequality, we can choose positive am and δm (see remark 9.2) such that ¯˙ m (t) ≤ −M1 B



1 0

2 e−δm x ∂xm α(x, t) dx − M2



1 0

2

eδ3 x ∂xm β(x, t) dx

1 1 − bm e−δm ∂xm α(1, t)2 + am eδm ∂xm β(1, t)2 2 2  1   2 2 2 + M4 eδm x ∂xm−1 β(x, t) + · · · + βx (x, t) + β(x, t) dx − M3 ∂xm β(0, t)2



0

  2 2 2 e−δ3 x ∂xm−1 α(x, t) + · · · + αx (x, t) + α(x, t) dx 0   2 + M6 |X(t)| + M7 ∂xm−1 β(0, t)2 + · · · + βx (0, t)2 + β(0, t)2 , 1

+ M5

(9.218)

where M1 , M2 , M3 , M4 , M5 , M6 , M7 , M8 are positive constants. Remark 9.2. As in (9.143), we can choose am , δm to make sure that all terms of 2 order m, namely, ∂xm α(·, t)2 , ∂xm β(·, t)2 , ∂xm β(0, t) , are with negative signs in ¯˙ m , except for ∂ m β(1, t)2 . As in (9.142), the positive term ∂ m α(0, t)2 (“positive B x x term” here means a quadratic term with a plus sign) has been written as positive 2 2 2 terms ∂xm β(0, t) , ∂xm−1 β(0, t) , · · · , β(0, t) , and |X(t)|2 via applying the CauchySchwarz inequality in the time derivative of (9.14) of m order and using (9.12). As 2 in (9.143), the positive term ∂xm β(0, t) is “canceled” by choosing am , and other positive terms are kept in (9.218), with coefficients M6 , M7 . In (9.218), the function ∂xm β(1, t)2 can be written as the positive terms

  M9 ∂tm−1 β(1, t)2 + ∂tm−2 β(1, t)2 + · · · + βt (1, t) + β(1, t)2 + β(0, t)2 + α(1, t)2 +



1 0

β(y, t)2 dy +



1 0

 α(y, t)2 dy + |X(t)|2 ,

(9.219)

where M9 is a positive constant, obtained by using (9.13), substituting (9.194) into (9.168), and applying the Cauchy-Schwarz inequality, as well as by recalling that α(0, t)2 is bounded by β(0, t)2 and |X(t)|2 via applying the Cauchy-Schwarz inequality in (9.14). According to lemma 9.2, lemma 9.3, and lemma 9.4, we obtain the exponential estimates in the sense of ∂xm−1 α(·, t)2 + ∂xm−1 β(·, t)2 + · · · + αx (·, t)2 + ¯˙ m βx (·, t)2 , which accommodate the positive terms of order m − 1, . . . , 1 in B 2 2 (9.218). Together with the exponential results in terms of α(·, t) + β(·, t) + |X(t)|2 + y1 (t)2 + · · · + ym (t)2 provided in theorem 9.3, we define a Lyapunov function ¯m . Vum = Rm Vu(m−1) + B

(9.220)

Taking the derivative of Vum and choosing sufficiently large Rm , we arrive at V˙ um (t) ≤ −λm Vum (t)

(9.221)

SANDWICH HYPERBOLIC PDE CONTROL

265

2

u(x, t)

1 0 –1 –2 1

0.5

0 0

2

1

x

3

4

5

t

Figure 9.1. Response of u(x, t) in the plant (9.1)–(9.6) without control.

for some positive λm . Through a similar process with (9.156), (9.157), we arrive at lemma 9.5. Now we give the proof of theorem 9.4. Proof. Using the exponential estimates in terms of u(·, t)2 + v(·, t)2 + ux (·, t)2 + vx (·, t)2 + · · · + ∂xm u(·, t)2 + ∂xm v(·, t)2 + z1 (t)2 + z2 (t)2 + · · · + zm (t)2 + |X(t)|2 provided in lemma 9.2, lemma 9.3, lemma 9.4, lemma 9.5, and theorem 9.3, through a similar process with proof of theorem 9.2, we obtain theorem 9.4.

9.7

SIMULATION

We use the finite-difference method to conduct the simulation with a time step of 0.00025 and a spatial step of 0.005. The solutions of the kernel equations (9.20)– (9.25), which are coupled linear hyperbolic PDEs on D = {(x, y)|0 ≤ y ≤ x ≤ 1}, are also solved by the finite-difference method. We define the plant parameters in (9.1)– (9.6) as [A, B, c1 , c2 , q, p, C, c0 , r] = [0.5, 1, 0.5, 0.5, 1, 1, 1, 1, 1], and the control parameters are chosen as [κ, c¯1 , c¯2 ] = [−2, 5, 13]. The initial conditions of v(x, t) and u(x, t) are defined as v(x, 0) = u(x, 0) = sin(2πx), and the initial conditions of X(t) and z(t) are X(0) = u(0, 0) − v(0, 0) = 0, z(0) = v(1, 0) = 0 according to (9.4) and (9.5). Comparing figure 9.1, which shows the open-loop response of u(x, t), and figure 9.2, which gives the closed-loop response of u(x, t), one can observe that in the latter case the convergence to zero is achieved, whereas the states grow unbounded

CHAPTER NINE

266

u(x, t)

2 1 0 –1 1 0.5 x

1

0 0

3

2

4

5

t

Figure 9.2. Response of u(x, t) in the plant (9.1)–(9.6) with the controller (9.52).

1 v(x, t)

0 –1 –2 –3 1 0.5 x

0 0

1

3

2

4

5

t

Figure 9.3. Response of v(x, t) in the plant (9.1)–(9.6) without control.

v(x, t)

1 0 –1 1 0.5 x

0 0

1

2

3

4

5

t

Figure 9.4. Response of v(x, t) in the plant (9.1)–(9.6) with the controller (9.52).

in the former case. Similar observations are made by comparing figure 9.3 and figure 9.4, which show the open-loop and closed-loop responses of v(x, t), respectively. In figure 9.5, we show the responses of the input ODE states z(t), z(t) ˙ in both open-loop (left side) and closed-loop (right side) cases. We observe that the states z(t), z(t) ˙ grow unbounded in the open-loop case and converge to zero under control. Similar results are observed in figure 9.6, which shows both the open-loop and closed-loop responses of the ODE state X(t). In figure 9.7, we show the response of the control input U (t) (9.52) in the closed-loop system. As one can observe, U (t) converges to zero.

SANDWICH HYPERBOLIC PDE CONTROL

267

1

0.02 0 z(t)

z(t)

0 –1

–0.02

–2 –3

–0.04 0

1

2

3

4

–0.06

5

0

1

2

3 4 t (b) z(t), closed-loop case

5

0

1

2

5

0.5 0 –0.5 –1 –1.5 –2 –2.5 –3

0.2 0.1 ż(t)

ż(t)

t (a) z(t), open-loop case

0 –0.1

0

1

4 3 t (c) ˙z(t), open-loop case

–0.2

5

2

3 4 t (d) ˙z(t), closed-loop case

Figure 9.5. Response of input ODE states z(t), z(t) ˙ in the open-loop and closed-loop cases.

0.5 0.3 X(t)

X(t)

0 –0.5 –1

0.2 0.1 0

0

1

2

3

4

5

0

t (a) X(t), open-loop case

1

2

3 4 t (b) X(t), closed-loop case

5

Figure 9.6. Response of ODE state X(t) in the open-loop and closed-loop cases.

3 2

U(t)

1 0 –1 –2 –3

0

1

2

3

4

5

t

Figure 9.7. Control input U (t) (9.52) in the closed-loop system.

CHAPTER NINE

268 9.8

APPENDIX

A. The quantities h1 , h2 , h3 , h4 , h5 , h6 , h7 , h8 , h9 (y), h10 (y), H11 The quantities h1 , h2 , h3 , h4 , h5 , h6 , h7 , h8 , h9 (y), h10 (y), H11 are given by h1 = c0 − pφ(1, 1), h2 = pψ(1, 1), h3 = pφ(1, 0) − γ(1)B, h4 = − pψ(1, 0), h5 = − pc1 ψ(1, 1) + p2 φy (1, 1) + p2 φ(1, 1)φI (1, 1) − c0 φI (1, 1)p, h6 = p2 ψy (1, 1) + pc2 φ(1, 1) + c0 ψ I (1, 1)p − p2 φ(1, 1)ψ I (1, 1), h7 = pc1 ψ(1, 0) − p2 φy (1, 0) − γ(1)AB − p2 φ(1, 1)φI (1, 0) − c0 γ I (1)B + pφ(1, 1)γ I (1)B + c0 φI (1, 0)p − (pφ(1, 0) − γ(1)B) γ I (0)B, h8 = p2 φ(1, 1)ψ I (1, 0) − c0 ψ I (1, 0)p − p (pψy (1, 0) + c2 φ(1, 0)) ,  1 pφ(1, 1)ψ I (1, δ)c1 φI (δ, y)dδ h9 (y) = pφ(1, 1)ψ I (1, y)c1 − y



1

+



 p2 φyy (1, δ) + c1 c2 φ(1, δ) φI (δ, y)dδ

y

− p2 φyy (1, y) + c1 c2 φ(1, y)



+ p (c1 ψ(1, 1) − pφy (1, 1)) φI (1, y) + c0 φy I (1, y)p  1 c0 ψ I (1, δ)c1 φI (δ, y)dδ − c0 ψ I (1, y)c1 − p2 φ(1, 1)φI y (1, y), + y



1

h10 (y) = y

c0 ψ I (1, δ)c1 ψ I (δ, y)dδ − c0 pψ I y (1, y)

 − p2 ψyy (1, y) + c2 c1 ψ(1, y) + p (c1 ψ(1, 1) − pφy (1, 1)) ψ I (1, y)  1

2  p φyy (1, δ) + c1 c2 φ(1, δ) ψ I (δ, y)dδ + y

+ p2 φ(1, 1)ψy I (1, y) −

1

pφ(1, 1)ψ I (1, δ)c1 ψ I (δ, y)dδ,

y



1

I

H11 = pφ(1, 1)γ (1)A +



0



 p2 φyy (1, δ) + c1 c2 φ(1, δ) γ I (δ)dδ

+ p (c1 ψ(1, 1) − pφy (1, 1)) γ I (1)

 − pc1 ψ(1, 0) − p2 φy (1, 0) − γ(1)AB γ I (0) + c0 γ I (1)Bγ I (0) − pφ(1, 1)γ I (1)Bγ I (0)

 − γ(1)A2 + (pφ(1, 0) − γ(1)B) γ I (0)2 B − pφ(1, 0) − γ(1)B γ I (0)A  1  1 I I c0 ψ (1, y)c1 γ (y)dy − pφ(1, 1)ψ I (1, y)c1 γ I (y)dy − c0 γ I (1)A. + 0

0

SANDWICH HYPERBOLIC PDE CONTROL

269

B. Proof of lemma 9.1 First, we transform the kernel equations (9.20)–(9.25) into integral equations using the method of characteristics. By virtue of (9.24), (9.25), we express the solution of γ(x) in terms of ψ, as follows:  x 1 1 Ax p γ(x) = κe +C e p A(x−τ ) ψ(τ, 0)dτ. (9.222) 0

Then we use the method of characteristic lines, as in [50], to give the successive approximations of ψ(x, y), φ(x, y). Along the line x = −y + a ¯1 , according to (9.23), (9.20), we obtain dψ(s) = −c2 φ(¯ a1 − s, s), ds c2 a ¯1 ψ( ) = − , 2 2p

(9.223) (9.224)

with the characteristic (−s + a ¯1 , s) that reaches to (x, y). According to the ODE (9.223), (9.224), we obtain  y c2 c2 φ(x + y − τ, τ )dτ . (9.225) ψ(x, y) = − − x+y 2p p 2 Then (9.225) is rewritten as the integral form ψ(x, y) = G0 (x, y) + G[φ](x, y),

(9.226)

where, G0 (x, y) = − G[φ](x, y) = −

c2 , 2p  y

(9.227)

x+y 2

c2 φ(x + y − τ, τ )dτ . p

(9.228)

Substituting (9.222) into (9.21), with (9.21), (9.22) along the line x = y + a ¯2 , we get dφ(s) c1 = ψ(¯ a2 + s, s), s p  a¯2 1 1 1 A¯ a2 p φ(0) = κe B+ C e p A(¯a2 −τ ) ψ(τ, 0)dτ B + ψ(¯ a2 , 0)q, p 0

(9.229) (9.230)

with the characteristic (s + a ¯2 , s) that reaches to (x, y). According to the ODE (9.229), (9.230), we obtain 1 1 φ(x, y) = κe p A(x−y) B + C p

+ ψ(x − y, 0)q +

1 p



x−y 0



y 0

1

e p A(x−y−τ ) ψ(τ, 0)dτ B

c1 ψ(x − y + τ, τ )dτ.

(9.231)

CHAPTER NINE

270 Substituting (9.225) into (9.231) yields

 x−y 1 1 φ(x, y) = κe B+ C e p A(x−y−τ ) ψ(τ, 0)dτ B p 0  x−y 2 c1 c2 c2 qc2 +q φ(x − y − τ, τ )dτ − 2 y − 2p p 2p  y τ 0 c1 c2 φ(x − y + 2τ − μ, μ)dμdτ, − x−y+2τ p2 0 2 1 p A(x−y)

(9.232)

which is rewritten as the integral form φ(x, y) = F0 (x, y) + F [ψ, φ](x, y),

(9.233)

where 1

F0 (x, y) = κe p A(x−y) B − 1 F [ψ, φ](x, y) = C p  −



x−y

0 y τ 0

qc2 c1 c2 − 2 y, 2p 2p

(9.234)

1

e p A(x−y−τ ) ψ(τ, 0)dτ B + q

x−y+2τ 2



x−y 2

0

c2 φ(x − y − τ, τ )dτ p

c1 c2 φ(x − y + 2τ − μ, μ)dμdτ. p2

(9.235)

Second, we use the method of successive approximations to construct a solution to the integral equations (9.226), (9.233) in the form of a converging series. Setting ψ 0 (x, y) = 0,

(9.236)

0

φ (x, y) = 0, ψ

n+1

φ

n+1

(9.237) n

(x, y) = G0 (x, y) + G[φ ](x, y), n

n

(x, y) = F0 (x, y) + F [ψ , φ ](x, y),

(9.238) (9.239)

for n = 0, 1..., with the definition of increments Δψ n+1 = ψ n+1 − ψ n and Δφn+1 = φn+1 − φn , where

Δψ 0 = G0 (x, y), Δφ0 = F0 (x, y),

it is easy to see that Δψ n+1 = G[Δφn ](x, y), Δφ

n+1

n

= F [Δψ , Δφ ](x, y).

Define

 a ¯ = max

n

x∈[0,1]

 1 1 Ce p Ax B, κe p Ax B ,

(9.240) (9.241)

(9.242)

SANDWICH HYPERBOLIC PDE CONTROL

271

  ¯b = 1 max c2 , qc2 , 1 c1 c2 , p p ¯ η = 2¯ a + 4b.

(9.243) (9.244)

According to the definition of Δψ 0 , Δφ0 , we observe Δψ 0 < η and

Δφ0 < η.

Assume now that |Δψ n | ≤ η n+1

n

(x − y) , n!

|Δφn | ≤ η n+1

(x − y) n!

(9.245)

n

(9.246)

are true for some n ∈ N∗ . Substituting (9.245), (9.246) into (9.240), (9.241) expressed in (9.228) and (9.235), through a straightforward calculation we obtain n+1 Δψ n+1 ≤ η n+2 (x − y) , (n + 1)!

(9.247)

n+1 n+1 Δφ ≤ η n+2 (x − y) . (n + 1)!

(9.248)

Therefore, the series ψ(x, y) =

∞ 

Δψ n (x, y),

(9.249)

n=0

φ(x, y) =

∞ 

Δφn (x, y)

(9.250)

n=0

uniformly converges to the solution (ψ(x, y), φ(x, y)) of the kernel equations (9.20)– (9.25) on D = {(x, y)|0 ≤ y ≤ x ≤ 1}, and then the solution γ(x) is obtained via (9.222). Now we show the continuity of the sum (9.249), (9.250). First, it is straightforward to show that for n ∈ N∗ , Δψ 0 = G0 (x, y), Δφ0 = F0 (x, y) are continuous on D. Indeed, Δψ 0 , Δφ0 are continuous on D as a composition of continuous functions. Besides, if we assume that Δψ n and Δφn are continuous, then Δψ n+1 and Δφn+1 are continuous as the integral (with continuous limits of integration) of continuous functions times Δψ n and Δφn composed of continuous functions. Finally, the normal convergence ensures continuity of the solutions ψ(x, y), φ(x, y) on D [50].

CHAPTER NINE

272

The proof of uniqueness of the solutions, which directly relies on the linearity of the kernel equations, is identical to [32]. We do not get into detail here for the sake of brevity. The proof of lemma 9.1 is complete.

9.9

NOTES

In chapters 2–8, we presented control designs for PDE-ODE systems. In this chapter, we dealt with ODE-PDE-ODE sandwich configurations. In these configurations the control input is directly applied not at the PDE boundary but at an ODE in the input channel of the PDE-ODE system. As mentioned at the beginning of the chapter, the ODE-PDE-ODE sandwich configuration physically represents the incorporation of the dynamics of the actuators (the hydraulic cylinder and head sheaves, the ship-mounted crane, and the rotary table) whose inertias are considerable, into the mining cable elevator, deep-sea construction vessel, and oil-drilling systems considered in chapters 2–8. The control design in this chapter is also applicable in the unmanned aerial vehicle (UAV) for the rapid suppression of oscillations of the cable and suspended object through a control force provided by the UAV’s rotor wings, where the plant is structured as a UAV-cable-payload (ODE-PDEODE), and in control of an overhead crane [34] that consists of a motorized platform (ODE) driving a cable (PDE) connecting a payload (ODE) at the bottom. The basic control design of sandwich systems was presented in this chapter, and more additional effects: delay, sampled (or event-triggered) sensing, and nonlinearities are dealt with in chapters 10–12. A summary of theoretical results on the boundary control of sandwich systems will be given in the notes section of chapter 12.

Chapter Ten Delay-Compensated Control of Sandwich Hyperbolic Systems

Compared with the basic control of sandwich hyperbolic partial differential equations (PDEs) in chapter 9, here we develop the delay-compensated boundary control of sandwich hyperbolic PDEs and propose an observer-based output-feedback controller. We solve the problem by employing a framework where the sensor delay is represented as a transport PDE, and by estimating the delay value as the reciprocal of the convection speed in the transport PDE, generating an ODE (ordinary differential equation)-PDE-ODE-PDE plant. Moreover, the restriction on the proximal ODE structure that is a chain of integrators in chapter 9 is relaxed. The control design is applied in the deep-sea construction vessel (DCV), to build output-feedback control forces at the ship-mounted crane to reduce oscillations of the long cable and place the equipment at the predetermined location on the seafloor while compensating for the sensor delay that arises from the long-distance transmission (from the seabed to the ship at the ocean surface) of the sensing signal via acoustic devices. We start this chapter by introducing the model of the system and the control task in section 10.1. The observer design is proposed in section 10.2. Therein, three transformations are used to convert the observer error system to a target observer error system whose exponential stability is straightforward to obtain, where all the output injection terms required for constructing the observer are determined. An observer-based output-feedback control design is proposed in section 10.3, where two transformations are applied to transform the observer into a so-called target system in a stable form, except for the proximal ODE that is influenced by perturbations originating from PDEs and the distal ODE. After representing this target system in the frequency domain to obtain the relationships between the states of the proximal ODE and those perturbation states, the proximal ODE is reformulated as a new ODE without external perturbations in the frequency domain, and then the stabilizing control input is designed. The exponential stability of the closed-loop system and the boundedness and exponential convergence to zero of the control input are proved in section 10.4. In simulation, the obtained theoretical result is applied to the oscillation suppression and position control of a DCV in section 10.5. 10.1

PROBLEM FORMULATION

Model Description The plant considered in this chapter is

CHAPTER TEN

274 ˙ X(t) = A0 X(t) + E0 w(0, t) + B0 U (t),

(10.1)

z(0, t) = pw(0, t) + C0 X(t),

(10.2)

zt (x, t) = −q1 zx (x, t) − c1 w(x, t) − c1 z(x, t),

(10.3)

wt (x, t) = q2 wx (x, t) − c2 w(x, t) − c2 z(x, t),

(10.4)

w(1, t) = qz(1, t) + C1 Y (t),

(10.5)

Y˙ (t) = A1 Y (t) + B1 z(1, t),

(10.6)

yout (t) = C1 Y (t − τ ),

(10.7)

∀(x, t) ∈ [0, 1] × [0, ∞). The block diagram of (10.1)–(10.7) is shown in figure 10.1. The vectors X(t) ∈ Rn×1 , Y (t) ∈ Rm×1 are ODE states. The scalars z(x, t) ∈ R, w(x, t) ∈ R are states of the 2 × 2 coupled hyperbolic PDEs with initial conditions (z(x, 0), w(x, 0)) ∈ L2 (0, 1) × L2 (0, 1). The parameter τ is an arbitrary constant denoting the time delay in the measurement. Input U (t) is the control input to be designed. The parameters c1 , c2 ∈ R and E0 ∈ Rn×1 are arbitrary. The positive constants q1 and q2 are transport velocities. The parameters q, p ∈ R satisfy assumption 10.1. The matrices A0 ∈ Rn×n , B0 ∈ Rn×1 , C0 ∈ R1×n , A1 ∈ Rm×m , B1 ∈ Rm×1 , C1 ∈ R1×m satisfy assumptions 10.2 and 10.3. Assumption 10.1. The plant parameters p, q satisfy c2

c1

|pq| < e q2 + q1

(10.8)

and q = 0. This assumption will be used in the output-feedback control design in section 10.3. Assumption 10.2. The pairs (A0 , B0 ), (A1 , B1 ) are stabilizable, and (A0 , C0 ), (A1 , C1 ) are detectable. According to assumption 10.2, there exist constant matrices L0 , L1 , F0 , F1 to make the following matrices Hurwitz: A¯0 = A0 − L0 C0 , A¯1 = A1 − eτ A1 L1 C1 e−τ A1 ,

(10.9) (10.10)

Aˆ0 = A0 − B0 F0 , Aˆ1 = A1 − B1 F1 ,

(10.11) (10.12)

where A1 − eτ A1 L1 C1 e−τ A1 has the same eigenvalues as A1 − L1 C1 [125]. z(x, t)

w(x, t)

X(t)

Y(t)

Transport PDEs z, w U(t)

ODE X

ODE Y

C1Y(t – τ)

Figure 10.1. Block diagram of the plant (10.1)–(10.7).

DELAY-COMPENSATED CONTROL OF PDES

Assumption 10.3. The matrices C0 , A0 , B0 satisfy   sI − A0 B0 det = 0 C0 0

275

(10.13)

for all s ∈ C, (s) ≥ 0. Assumption 10.3 is about matrices of the proximal ODE X(t), namely actuator dynamics. Even though zeros in the closed right-half plane are excluded here while zeros are allowed in some previous results on the control of sandwich PDE systems, such as [43], this assumption relaxes some restrictions on the structure of the proximal ODE in the existing literature (such as A0 , B0 , C0 being scalar in [8, 49], B0 being invertible in [151], det(C0 B0 ) = 0 in [43], or a form of a chain of integrators in [179, 183]). This assumption, which is first used in sandwich systems in [152], is equal to the existence of a stable left inversion system [135] of (10.1) and is used in the control input design in section 10.3. Assumption 10.4. The matrices C1 , A1 , B1 satisfy   sI − A1 B1 = 0 det C1 e−τ A1 0

(10.14)

for all s ∈ C, (s) ≥ 0. Assumption 10.4 is about matrices of the distal ODE Y (t) with a sensor delay τ in the measurement output state. This assumption also prohibits the zeros of the ODE subsystem (C1 , A1 , B1 ) from being located in the closed right-half plane. It is not particularly restrictive and (C1 , A1 , B1 ) is still quite general, covering many application cases. This assumption is used in the observer design for the overall sandwich system with the delayed measurement in section 10.2. If the sensor delay is zero, this assumption has the same form as assumption 10.3. The design in this chapter is also suitable for collocated control, namely, when the measurement is the output state of the proximal ODE X(t) with a time delay τ , provided the matrix triple (C0 , A0 , E0 ) satisfies   sI − A0 E0 = 0 det C0 e−τ A0 0 for all s ∈ C, (s) ≥ 0. The control objective of this chapter is to exponentially stabilize the overall sandwich system—that is, the ODE states Y (t), X(t) and the PDE states u(x, t), v(x, t) by constructing an output-feedback control input U (t) applied at the proximal ODE X(t) using the delayed measurement yout (t). Rewrite Delay as a Transport PDE By defining v(x, t) = C1 Y (t − τ (x − 1)), ∀(x, t) ∈ [1, 2] × [τ, ∞),

(10.15)

we obtain a transport PDE v(1, t) = C1 Y (t),

(10.16)

CHAPTER TEN

276 1 vt (x, t) = − vx (x, t), τ yout (t) = v(2, t)

(10.17) (10.18)

for all (x, t) ∈ [1, 2] × [τ, ∞) to describe the time delay in the measurement (10.7). Replacing (10.7) by (10.16)–(10.18), we obtain a sandwich hyperbolic PDEODE system connecting with another transport PDE—that is, the following ODEcoupled hyperbolic PDEs-ODE-transport PDE system: ˙ X(t) = A0 X(t) + E0 w(0, t) + B0 U (t), z(0, t) = pw(0, t) + C0 X(t), zt (x, t) = −q1 zx (x, t) − c1 w(x, t) − c1 z(x, t), x ∈ [0, 1], wt (x, t) = q2 wx (x, t) − c2 w(x, t) − c2 z(x, t), x ∈ [0, 1], w(1, t) = qz(1, t) + C1 Y (t), Y˙ (t) = A1 Y (t) + B1 z(1, t), v(1, t) = C1 Y (t), 1 vt (x, t) = − vx (x, t), x ∈ [1, 2], τ yout (t) = v(2, t)

(10.19) (10.20) (10.21) (10.22) (10.23) (10.24) (10.25) (10.26) (10.27)

for t ≥ τ . The time delay is “removed” at a cost of adding a transport PDE into the plant (10.1)–(10.7). Now, the control task is equivalent to exponentially stabilizing the overall system (10.19)–(10.27)—that is, ODE(X)-PDE(z, w)-ODE(Y )-PDE(v)—by constructing an output-feedback control input U (t) at the first ODE (10.19) using the right boundary state of the last PDE (10.27).

10.2

OBSERVER DESIGN

In order to build the observer-based output-feedback controller of the plant (10.1)– (10.7), in this section we design a state observer to track the overall system (10.1)– (10.7) using only the delayed measurement yout (t). Through the reformulation in section 10.1, the estimation task is equivalent to designing a state observer to recover the overall system (10.19)–(10.27) using only measurements at the right boundary x = 2 of the last transport PDE v. The observer is built as a copy of the plant (10.19)–(10.27) plus some dynamic output error injection terms, as follows: ˆ + E0 w(0, ˆ˙ X(t) = A0 X(t) ˆ t) + B0 U (t) + h1 (yout (t) − vˆ(2, t)), ˆ zˆ(0, t) = pw(0, ˆ t) + C0 X(t),

(10.28) (10.29)

zˆt (x, t) = − q1 zˆx (x, t) − c1 w(x, ˆ t) − c1 zˆ(x, t) + h2 (yout (t) − vˆ(2, t); x),

(10.30)

ˆx (x, t) − c2 w(x, ˆ t) − c2 zˆ(x, t) + h3 (yout (t) − vˆ(2, t); x), w ˆt (x, t) = q2 w

(10.31)

w(1, ˆ t) = qˆ z (1, t) + C1 Yˆ (t) + h4 (yout (t) − vˆ(2, t)), ˙ Yˆ (t) = A1 Yˆ (t) + B1 zˆ(1, t) + Γ1 (yout (t) − vˆ(2, t)),

(10.32) (10.33)

DELAY-COMPENSATED CONTROL OF PDES

277

vˆ(1, t) = C1 Yˆ (t),

(10.34)

1 vˆt (x, t) = − vˆx (x, t) + h5 (yout (t) − vˆ(2, t); x), τ

(10.35)

where a constant matrix Γ1 and the dynamics h1 , h2 , h3 , h4 , h5 are to be determined. Initial conditions are taken as (ˆ z (x, 0), w(x, ˆ 0), vˆ(x, 0)) ∈L2 (0, 1)×L2 (0, 1)×L2 (1, 2). Defining the observer error states as ˜ (˜ z (x, t), w(x, ˜ t), X(t), Y˜ (t), v˜(x, t)) = (z(x, t), w(x, t), X(t), Y (t), v(x, t)) ˆ − (ˆ z (x, t), w(x, ˆ t), X(t), Yˆ (t), vˆ(x, t)),

(10.36)

according to (10.19)–(10.27) and (10.28)–(10.35), we obtain the observer error system ˜˙ ˜ + E0 w(0, X(t) = A0 X(t) ˜ t) − h1 (˜ v (2, t)), ˜ z˜(0, t) = pw(0, ˜ t) + C0 X(t),

(10.37) (10.38)

z˜t (x, t) = −q1 z˜x (x, t) − c1 w(x, ˜ t) − c1 z˜(x, t) − h2 (˜ v (2, t); x),

(10.39)

˜x (x, t) − c2 w(x, ˜ t) − c2 z˜(x, t) − h3 (˜ v (2, t); x), w ˜t (x, t) = q2 w

(10.40)

v (2, t)), w(1, ˜ t) = q˜ z (1, t) + C1 Y˜ (t) − h4 (˜

(10.41)

Y˜˙ (t) = A1 Y˜ (t) + B1 z˜(1, t) − Γ1 v˜(2, t),

(10.42)

v˜(1, t) = C1 Y˜ (t),

(10.43)

1 v˜t (x, t) = − v˜x (x, t) − h5 (˜ v (2, t); x), τ

(10.44)

v (2, t)), h2 (˜ v (2, t); x), h3 (˜ v (2, t); x), where Γ1 v˜(2, t) is an output injection, and h1 (˜ v (2, t)), h5 (˜ v (2, t); x) are dynamic output injections, which are defined as h4 (˜ h1 (˜ v (2, t)) = L−1 [H1 (s)˜ v (2, s)], h2 (˜ v (2, t); x) = L

−1

h3 (˜ v (2, t); x) = L

−1

[H2 (s; x)˜ v (2, s)],

(10.46)

[H3 (s; x)˜ v (2, s)],

(10.47)

h4 (˜ v (2, t)) = L−1 [H4 (s)˜ v (2, s)], h5 (˜ v (2, t); x) = L

−1

(10.45)

[H5 (s; x)˜ v (2, s)],

(10.48) (10.49)

where L−1 denotes the inverse Laplace transform, and the transfer functions H1 (s), H2 (s; x), H3 (s; x), H4 (s), H5 (s; x) are to be determined later. It should be noted that x in H2 , H3 , H5 is only a parameter. Introducing (10.45)–(10.49) is helpful in constructing the dynamics hi (·) in (10.28)–(10.35) because the algebraic relationships between v˜(2, s) and other states can be obtained by using the Laplace transform, and the transfer functions in (10.45)–(10.49) can be solved in algebraic equations after rewriting the conditions required to achieve an exponentially stable observer error system in the frequency domain. The determination of H1 (s), H2 (s; x), H3 (s; x), H4 (s), H5 (s; x), and Γ1 in the observer (10.28)–(10.35) will be completed through three transformations presented

CHAPTER TEN

278

next, which convert the observer error system (10.37)–(10.44) to a target observer error system whose exponential stability is straightforward to obtain. First Transformation Applying the transformation v˜(x, t) = η˜(x, t) + ϕ(x)Y˜ (t),

(10.50)

where ϕ(x) is to be determined, we convert (10.42)–(10.44) to a stable form as Y˜˙ (t) = A¯1 Y˜ (t) + B1 z˜(1, t) − Γ1 η˜(2, t), η˜(1, t) = 0,

(10.51) (10.52)

1 η˜t (x, t) = − η˜x (x, t), x ∈ [1, 2], τ

(10.53)

where A¯1 is a Hurwitz matrix defined in (10.10). In what follows, ϕ(x), Γ1 , H5 (s; x) are determined by matching (10.42)–(10.44) and (10.51)–(10.53) via (10.50). Inserting the transformation (10.50) into (10.42), we obtain Y˜˙ (t) = (A1 − Γ1 ϕ(2))Y˜ (t) + B1 z˜(1, t) − Γ1 η˜(2, t).

(10.54)

By virtue of (10.10), (10.51), Γ1 should satisfy Γ1 ϕ(2) = eτ A1 L1 C1 e−τ A1 .

(10.55)

Evaluating (10.50) at x = 1 and applying (10.43), (10.52), we get v˜(1, t) = η˜(1, t) + ϕ(1)Y˜ (t) = ϕ(1)Y˜ (t) = C1 Y˜ (t).

(10.56)

ϕ(1) = C1 .

(10.57)

Therefore,

Taking the time and spatial derivatives of (10.50) and submitting the result into (10.44), we obtain 1 v˜t (x, t) + v˜x (x, t) + h5 (˜ v (2, t); x) τ = η˜t (x, t) + ϕ(x)A1 Y˜ (t) + ϕ(x)B1 z˜(1, t) − ϕ(x)Γ1 v˜(2, t) 1 1 + η˜x (x, t) + ϕ (x)Y˜ (t) + h5 (˜ v (2, t); x) τ τ = ϕ(x)B1 z˜(1, t) − ϕ(x)Γ1 v˜(2, t) + h5 (˜ v (2, t); x)   1 + ϕ(x)A1 + ϕ (x) Y˜ (t) = 0, τ

(10.58)

where (10.42), (10.53) are used, and ϕ(x) should satisfy ϕ (x) = −τ A1 ϕ(x)

(10.59)

DELAY-COMPENSATED CONTROL OF PDES

279

to make [ϕ(x)A¯1 + τ1 ϕ (x)]Y˜ (t) zero. The transfer function H5 (s; x) that determines v (2, t); x) via (10.49) should be defined to ensure the remainder of the signal h5 (˜ (10.58) is zero—that is, ϕ(x)B1 z˜(1, t) − ϕ(x)Γ1 v˜(2, t) + h5 (˜ v (2, t); x) = 0.

(10.60)

Before determining H5 (s; x), we solve conditions (10.55), (10.57), (10.59) to obtain ϕ(x), Γ1 as ϕ(x) = C1 e−τ A1 (x−1) , x ∈ [1, 2], Γ1 = e

τ A1

L1 .

(10.61) (10.62)

With (10.52), (10.53), we know that η˜(2, t) = 0, t ≥ τ.

(10.63)

Y˜˙ (t) = A¯1 Y˜ (t) + B1 z˜(1, t)

(10.64)

Thus, (10.51) is written as

for t ≥ τ . Taking the Laplace transform of (10.64), we get (sI − A¯1 )Y˜ (s) = B1 z˜(1, s),

(10.65)

where I is an identity matrix with appropriate dimensions. For brevity, we consider all initial conditions to be zero when we take the Laplace transform. (Arbitrary initial conditions could be incorporated into the stability statement through an expanded analysis which is routine but heavy on additional notation.) Recalling that A¯1 is Hurwitz, det(sI − A¯1 ) does not have any zeros in the closed right-half plane. Then the matrix sI − A¯1 is invertible for any s ∈ C, (s) ≥ 0. Multiplying both sides of (10.65) by (sI − A¯1 )−1 , we obtain Y˜ (s) = (sI − A¯1 )−1 B1 z˜(1, s).

(10.66)

According to (10.50) and (10.63), we get v˜(2, t) = ϕ(2)Y˜ (t), t ≥ τ.

(10.67)

Writing (10.67) in the frequency domain and inserting (10.61), (10.66), we obtain v˜(2, s) = ϕ(2)Y˜ (s) = r(s)˜ z (1, s),

(10.68)

r(s) = C1 e−τ A1 (sI − A¯1 )−1 B1 .

(10.69)

where

Notice r(s) ∈ R due to C1 ∈ R1×m and B1 ∈ Rm×1 . Lemma 10.1. The function r(s) = C1 e−τ A1 (sI − A¯1 )−1 B1 is nonzero for any s ∈ C with (s) ≥ 0 under assumptions 10.2 and 10.4, or, in plain words, r(s) is a stable, strictly proper transfer function.

CHAPTER TEN

280

Proof. Using (10.10) in assumption 10.2, we get     sI − A1 B1 I −(sI − A¯1 )−1 B1 I eτ A 1 L1 0 I 0 I C1 e−τ A1 0   ¯ sI − A1 0 = . C1 e−τ A1 −C1 e−τ A1 (sI − A¯1 )−1 B1

(10.70)

Recalling assumption 10.4, we know that   0 sI − A¯1 = 0 det C1 e−τ A1 −C1 e−τ A1 (sI − A¯1 )−1 B1 for any s ∈ C, (s) ≥ 0. Therefore, C1 e−τ A1 (sI − A¯1 )−1 B1 = 0. The proof of the lemma is complete.

According to lemma 10.1, we have the existence of r(s)−1 =

1 , r(s)

which is a stable though improper transfer function. Let us now go back to (10.60) to determine H5 (s; x). Taking the Laplace transform of (10.60), recalling (10.49), and inserting (10.61) and (10.68), we obtain ϕ(x)B1 z˜(1, s) − ϕ(x)Γ1 v˜(2, s) + H5 (s; x)˜ v (2, s)     −τ A1 (x−1) −τ A1 (x−1) = C1 e B 1 − C1 e Γ1 − H5 (s; x) r(s) z˜(1, s) = 0.

(10.71)

The transfer function H5 (s; x) is then chosen as H5 (s; x) = C1 e−τ A1 (x−1) Γ1 − C1 e−τ A1 (x−1) B1 r(s)−1 = C1 e−τ A1 (x−1) Γ1 −

C1 e−τ A1 (x−1) B1 , C1 e−τ A1 (sI − A¯1 )−1 B1

(10.72)

where lemma 10.1 has been used to ascertain that this transfer function is stable. This transfer function, as well as all the other transfer functions Hi (s; x), yet to be determined, is improper. In remark 10.1, we discuss how to implement these transfer functions despite their improperness. Therefore, (10.71) holds. Then (10.60) holds by rewriting (10.71) in the time dov (2, t); x) main. Together with (10.59), then (10.58) holds for t ≥ τ . The function h5 (˜ can then be defined via (10.72) and (10.49). In the above derivation, we have completed the conversion between (10.42)– v (2, t); x) needed (10.44) and (10.51)–(10.53) through (10.50) and determined Γ1 , h5 (˜ in the observer. In what follows, H4 (s) is determined to make the boundary condition (10.41) to be zero—that is, to render w(1, ˜ t) = q˜ z (1, t) + C1 Y˜ (t) − h4 (˜ v (2, t)) = 0.

(10.73)

DELAY-COMPENSATED CONTROL OF PDES

281

Taking the Laplace transform of (10.73), recalling (10.48), and inserting (10.66) and (10.68), we obtain w(1, ˜ s) = q˜ z (1, s) + C1 Y˜ (s) − H4 (s)˜ v (2, s)  −1 ¯ = q + C1 (sI − A1 ) B1 − H4 (s)r(s) z˜(1, s) = 0.

(10.74)

The transfer function H4 (s) is chosen as H4 (s) = [q + C1 (sI − A¯1 )−1 B1 ]r(s)−1 q + C1 (sI − A¯1 )−1 B1 = C1 e−τ A1 (sI − A¯1 )−1 B1

(10.75)

to make (10.74) hold. This transfer function is stable, thanks to lemma 10.1, but it is improper. We discuss its implementation in remark 10.1. We then get w(1, ˜ t) = 0 v (2, t)) is in (10.73) by rewriting w(1, ˜ s) = 0 in the time domain. The function h4 (˜ thus determined by (10.48), (10.75). Therefore, through the first transformation (10.50), with determining the dynav (2, t)), h5 (˜ v (2, t); x), (10.37)–(10.44) is converted to mic output injection terms h4 (˜ the first intermediate system as ˜˙ ˜ + E0 w(0, X(t) = A0 X(t) ˜ t) − h1 (˜ v (2, t)), ˜ z˜(0, t) = pw(0, ˜ t) + C0 X(t),

(10.76) (10.77)

z˜t (x, t) = −q1 z˜x (x, t) − c1 w(x, ˜ t) − c1 z˜(x, t) − h2 (˜ v (2, t); x), w ˜t (x, t) = q2 w ˜x (x, t) − c2 w(x, ˜ t) − c2 z˜(x, t) − h3 (˜ v (2, t); x),

(10.78) (10.79)

w(1, ˜ t) = 0, Y˜˙ (t) = A¯1 Y˜ (t) + B1 z˜(1, t), η˜(1, t) = 0, 1 η˜t (x, t) = − η˜x (x, t) τ

(10.80) (10.81) (10.82) (10.83)

for t ≥ τ , where (10.80)–(10.83) are in a stable form while coupling terms exist in the domain x ∈ [0, 1]—that is, (10.78), (10.79). Next, we introduce the second transformation to decouple the couplings in (10.78), (10.79). Second Transformation We now apply the second transformation [21]

1

˜ t) − w(x, ˜ t) = β(x,

ψ(x, y)˜ α(y, t)dy, x

z˜(x, t) = α(x, ˜ t) −

(10.84)

1

φ(x, y)˜ α(y, t)dy

(10.85)

x

with the kernels ψ(x, y), φ(x, y) satisfying ψ(x, x) =

c2 , q1 + q2

(10.86)

CHAPTER TEN

282 φ(0, y) = pψ(0, y) − C0 K1 (y), −q1 ψy (x, y) + q2 ψx (x, y) = (c2 − c1 )ψ(x, y) + c2 φ(x, y),

(10.87) (10.88)

−q1 φx (x, y) − q1 φy (x, y) = c1 ψ(x, y),

(10.89)

where the function K1 (y) will be defined later and then the well-posedness of (10.86)–(10.89) will be shown. The purpose of the transformations (10.84), (10.85) is to convert the first intermediate system (10.76)–(10.81) to a second intermediate system, which is given by

1 ˜ t) − E0 ˜˙ ˜ + E0 β(0, X(t) = A0 X(t) ψ(0, y)˜ α(y, t)dy + h1 (˜ v (2, t)), (10.90) 0

˜ t) + C0 X(t) ˜ − α ˜ (0, t) = pβ(0,

˜ x (x, t) + α ˜ t (x, t) = − q1 α

1

1 0

C0 K1 (y)˜ α(y, t)dy,

(10.91)

˜ t)dy ¯ (x, y)β(y, M

x

˜ t), − c1 α ˜ (x, t) − c1 β(x,

1 ˜ t)dy − c2 β(x, ˜ t), ¯ (x, y)β(y, N β˜t (x, t) = q2 β˜x (x, t) +

(10.92) (10.93)

x

˜ t) = 0, β(1,

(10.94)

Y˜˙ (t) = A¯1 Y˜ (t) + B1 α ˜ (1, t) ¯ and N ¯ are defined as for t ≥ τ , where the integral operator kernels M

y ¯ ¯ (δ, y)dδ − c1 φ(x, y), M (x, y) = φ(x, δ)M x

y ¯ (x, y) = ¯ (δ, y)dδ − c1 ψ(x, y). N ψ(x, δ)M

(10.95)

(10.96) (10.97)

x

The subsystem η˜(·, t), given in (10.82), (10.83), is removed from the second intermediate system (10.90)–(10.95) for brevity because η˜(·, t) ≡ 0, t ≥ τ . In what follows, H2 (s; x), H3 (s; x) are determined in matching the first intermediate system (10.76)–(10.81) and the second intermediate system (10.90)–(10.95) via (10.84), (10.85). Inserting (10.84), (10.85) into (10.79) along (10.92), (10.93) and applying (10.86)– (10.88), (10.97), we get ˜x (x, t) + c2 z˜(x, t) + c2 w(x, ˜ t) + h3 (˜ v (2, t); x) w ˜t (x, t) − q2 w = q1 ψ(x, 1)˜ α(1, t) + h3 (˜ v (2, t); x) = 0,

(10.98)

where the detailed calculation is shown in (10.254) in appendix 10.6B. We thus know that the following equation needs to be satisfied: q1 ψ(x, 1)˜ z (1, t) + h3 (˜ v (2, t); x) = 0,

(10.99)

where α ˜ (1, t) = z˜(1, t) according to (10.85) is used. Writing (10.99) in the frequency domain and applying (10.47), (10.68), we obtain

DELAY-COMPENSATED CONTROL OF PDES

283

q1 ψ(x, 1)˜ z (1, s) + H3 (s; x)˜ v (2, s) = (q1 ψ(x, 1) + H3 (s; x)r(s)) z˜(1, s) = 0.

(10.100)

The transfer function H3 (s; x) is chosen as H3 (s; x) = −q1 ψ(x, 1)r(s)−1 =

−q1 ψ(x, 1) C1 e−τ A1 (sI − A¯1 )−1 B1

(10.101)

to make (10.100) hold. This transfer function is stable, thanks to lemma 10.1, but it is improper. We discuss its implementation in remark 10.1. We get that (10.98) v (2, t); x) can then holds by rewriting (10.100) in the time domain. The function h3 (˜ be obtained by (10.47), (10.101). Inserting (10.84), (10.85) into (10.78) along (10.92), (10.93) and applying (10.89), (10.96), we get ˜ t) + c1 z˜(x, t) + h2 (˜ v (2, t); x) z˜t (x, t) + q1 z˜x (x, t) + c1 w(x, = q1 φ(x, 1)˜ α(1, t) + h2 (˜ v (2, t); x) = 0,

(10.102)

where the detailed calculation is shown in (10.255) in appendix 10.6B. Therefore, h2 (˜ v (2, t); x) should satisfy q1 φ(x, 1)˜ z (1, t) + h2 (˜ v (2, t); x) = 0,

(10.103)

where α ˜ (1, t) = z˜(1, t) according to (10.85) is used. Taking the Laplace transform of (10.103) and recalling (10.46), (10.68), we obtain q1 φ(x, 1)˜ z (1, s) + H2 (s; x)˜ v (2, s) = (q1 φ(x, 1) + H2 (s; x)r(s)) z˜(1, s) = 0.

(10.104)

The transfer function H2 (s; x) is obtained as H2 (s; x) = −q1 φ(x, 1)r(s)−1 =

−q1 φ(x, 1) −τ A ¯1 )−1 B1 1 (sI − A C1 e

(10.105)

to ensure (10.102) holds. This transfer function is stable, thanks to lemma 10.1, but it is improper. We discuss its implementation in remark 10.1. The function v (2, t); x) can thus be defined by (10.105), (10.46). h2 (˜ The boundary conditions (10.77), (10.80) follow directly from inserting x = 0, x = 1 into (10.84), (10.85) and applying (10.87), (10.91), (10.94). The ODEs (10.76), (10.81) are obtained directly from (10.90), (10.95) via (10.84), (10.85), respectively. The second conversion is thus completed, and two PDEs (10.78), (10.79) are decoupled now, which can be seen in (10.92), (10.93). Third Transformation In order to decouple the ODE (10.90) with the PDEs and rebuild the ODE in a stable form, we intend to convert the second intermediate system (10.90)–(10.95) to the following target observer error system: ˜ ˜˙ = A¯0 Z(t), Z(t)

(10.106)

CHAPTER TEN

284 ˜ α ˜ (0, t) = C0 Z(t), ˜ x (x, t) − c1 α ˜ (x, t), α ˜ t (x, t) = −q1 α Y˜˙ (t) = A¯ Y˜ (t) + B α(1, ˜ t) 1

1

(10.107) (10.108) (10.109)

for t ≥ t0 = τ + q12 , where A¯0 is a Hurwitz matrix defined in (10.9). According to ˜ t) are removed for brevity. ˜ t) ≡ 0 after t0 = τ + 1 and β(x, (10.93), (10.94), β(x, q2 Equations (10.90)–(10.95) are thus rewritten as ˜ − E0 ˜˙ X(t) = A0 X(t)

˜ − α ˜ (0, t) = C0 X(t)



1

1 0

ψ(0, y)˜ α(y, t)dy + h1 (˜ v (2, t)),

C0 K1 (y)˜ α(y, t)dy,

0

α ˜ t (x, t) = −q1 α ˜ x (x, t) − c1 α ˜ (x, t), ˙ Y˜ (t) = A¯ Y˜ (t) + B α ˜ (1, t) 1

(10.110) (10.111) (10.112) (10.113)

1

for t ≥ t0 . Equations (10.112), (10.113) are the same as (10.108), (10.109). We thus only need to convert (10.110), (10.111) to (10.106), (10.107). The transformation

1 ˜ = X(t) ˜ − Z(t) K1 (y)˜ α(y, t)dy (10.114) 0

is applied to complete the conversion, where K1 (y) satisfies L0 − q1 K1 (0) = 0, (A¯0 + c1 )K1 (y) − q1 K1  (y) − E0 ψ(0, y) + L0 C0 K1 (y) = 0.

(10.115) (10.116)

The equation set (10.86)–(10.89) and (10.115), (10.116) is a 2 × 2 hyperbolic PDEODE system, which is a scalar case of the well-posed kernel equations (17)–(23) in [48] (setting dimensions in [48] to 1). Therefore, the conditions of the kernels ψ(x, y), φ(x, y) in (10.84), (10.85) and K1 (y) in (10.114)—that is, (10.86)–(10.89), (10.115), (10.116), are well-posed. In what follows, H1 (s) is determined by matching (10.110), (10.111) and (10.106), (10.107) via (10.114). Substituting (10.114) into (10.106) and applying (10.110)– (10.112), (10.115), (10.116), we obtain ˜˙ − A¯0 Z(t) ˜ Z(t) ˜ − E0 = A0 X(t)



1 0

ψ(0, y)˜ α(y, t)dy + h1 v˜(2, t)

α(1, t) − q1 K1 (0)˜ α(0, t) + q1 K1 (1)˜

1

1  − q1 K1 (y)˜ α(y, t)dy + c1 K1 (y)˜ α(y, t)dy 0

˜ + L0 C0 X(t) ˜ + A¯0 − A0 X(t)



0

1 0

K1 (y)˜ α(y, t)dy

α(1, t) + [L0 − q1 K1 (0)]˜ α(0, t) = h1 v˜(2, t) + q1 K1 (1)˜

DELAY-COMPENSATED CONTROL OF PDES



1

+ 0



285

A¯0 K1 (y) + c1 K1 (y) − q1 K1  (y) − E0 ψ(0, y)

 + L0 C0 K1 (y) α ˜ (y, t)dy = h1 (˜ v (2, t)) + q1 K1 (1)˜ α(1, t) = 0, t ≥ t0 .

(10.117)

The function H1 (s), which defines h1 (˜ v (2, t)) by (10.45), is solved from h1 (˜ v (2, t)) + q1 K1 (1)˜ z (1, t) = 0,

(10.118)

α ˜ (1, t) = z˜(1, t),

(10.119)

where

according to (10.85), has been used. Writing (10.118) in the frequency domain and applying (10.45), (10.68) yields H1 (s)˜ v (2, s) + q1 K1 (1)˜ z (1, s) = (H1 (s)r(s) + q1 K1 (1)) z˜(1, s) = 0.

(10.120)

Solving for H1 (s), we get it as H1 (s) = −q1 K1 (1)r(s)−1 =

−q1 K1 (1) −τ A ¯1 )−1 B1 . 1 (sI − A C1 e

(10.121)

This transfer function is stable, thanks to lemma 10.1, but it is improper. We discuss its implementation in remark 10.1. We get that (10.117) holds by rewritv (2, t)) can then be defined via ing (10.120) in the time domain. The function h1 (˜ (10.45), (10.121). Inserting (10.114) into (10.111), it is straightforward to obtain (10.107). Therefore, (10.106), (10.107) is converted from (10.110), (10.111) through (10.114) for t ≥ t0 . The third transformation is completed, and the ODE (10.106) is independent and exponentially stable now. After the above three transformations, we have converted the original observer error system (10.37)–(10.44) to the target observer error system (10.106)–(10.109) ˜ t) ≡ 0, according (for t ∈ [t0 , ∞), η˜(x, t) ≡ 0, according to (10.82), (10.83), and β(x, to (10.93), (10.94), are removed for brevity). Because the original observer error system (10.37)–(10.44) is bounded in the finite time t ∈ [0, t0 ), we prove the exponential stability of (10.37)–(10.44) for t ∈ [t0 , ∞) in the next subsection. The spatially dependent transfer functions Hi determined, whose outputs are dynamic output injections in the observer (10.28)–(10.35), are employed as follows: y1 (t) = h1 (˜ v (2, t)),

(10.122)

y2 (x, t) = h2 (˜ v (2, t); x), y3 (x, t) = h3 (˜ v (2, t); x),

(10.123) (10.124)

y4 (t) = h4 (˜ v (2, t)), y5 (x, t) = h5 (˜ v (2, t); x).

(10.125) (10.126)

The signals yi are proved exponentially convergent to zero in the next subsection.

CHAPTER TEN

286

Remark 10.1. While stable, thanks to lemma 10.1, the transfer functions Hi (s) v (2, s) contain time derivatives of are improper and, therefore, the signals Hi (s)˜ v˜(2, t). In practice, one way to avoid taking the time derivatives, which may lead to measurement noise amplification, is by measuring n-order time-derivative states ∂tn v(2, t) and calculating v˜(2, t) by n times integrations of ∂tn v˜(2, t), which amounts to multiplying Hi (s) by s1n to make Hi (s) proper. Measuring ∂tn v(2, t) is not viable in general, but in the case of the control application to the DCV in section 10.5 the payload oscillation acceleration is measured, and the velocity is calculated by integration starting from the known initial conditions, as mentioned in chapter 7. Measuring acceleration is a prevalent method in many mechanical systems because the acceleration sensor is cheaper and far easier to manufacture and install [17]. Stability Analysis of the Observer Error System ˜ Theorem 10.1. For all initial data (˜ z (x, 0), w(x, ˜ 0), v˜(x, 0), X(0), Y˜ (0)) ∈ L2 (0, 1) × 2 2 n m L (0, 1) × L (1, 2) × R × R , the internal exponential stability of the observer error system (10.37)–(10.44) holds in the sense of the norm      ˜  ˜  ˜ z (·, t) ∞ + w(·, ˜ t) ∞ + ˜ v (·, t) ∞ + X(t)  + Y (t) + |y1 (t)| + |y4 (t)| + y2 (·, t) ∞ + y3 (·, t) ∞ + y5 (·, t) ∞ ,

(10.127)

with the decay rate adjustable by L0 , L1 . Proof. The stability of the original observer error system is obtained by analyzing the stability of the target observer error system (10.106)–(10.109) and using the invertibility of the transformations. Equations (10.106)–(10.109) are a cascade of ˜ ˜ Z(t) into α ˜ (·, t) into Y˜ (t). From (10.106), Z(t) is exponentially convergent to zero because A¯0 is Hurwitz. With the method of characteristics, as in [43] it is easy to show that α ˜ (x, t) in the PDE subsystem (10.106), (10.107) is exponentially convergent to zero. Because A¯1 is Hurwitz, Y˜ (t) is exponentially convergent to zero. The decay rate λe of (10.106)–(10.109) depends on the decay rate of the ODEs ˜ Z(t), Y˜ (t). In other words, the decay rate λe is adjustable by L0 , L1 according to ˜ t) ≡ 0 after t0 = 1 + τ , we find that (10.9), (10.10). Recalling η˜(x, t) ≡ 0 and β(x, q2      ˜  ˜  ˜ t) ∞ + ˜ ¯ = ˜ Ω(t) α(·, t) ∞ + β(·, η (·, t) ∞ + Z(t)  + Y (t) is bounded by an exponential decay with the decay rate λe for t ≥ t0 . It should be noted that the transient in the finite time [0, t0 ) can be bounded by an arbitrarily fast decay rate considering the trade-off between the decay rate and the overshoot coefficient—that is, the higher the decay rate, the higher the overshoot coefficient. ¯ is bounded Therefore, we conclude that the exponential stability in the sense of Ω(t) by an exponential decay rate λe with some overshoot coefficients for t ≥ 0. Applying the transformation (10.50), (10.114) and (10.84), (10.85), we respectively have that      ˜ v (·, t) ∞ ≤ Υ1a ˜ η (·, t) ∞ + Y˜ (t) ,      ˜  ˜  α(·, t) ∞ + Z(t) X(t) ≤ Υ1b ˜  ,   ˜ t) ∞ ˜ z (·, t) ∞ + w(·, α(·, t) ∞ + β(·, ˜ t) ∞ ≤ Υ1c ˜ for some positive Υ1a , Υ1b , Υ1c .

DELAY-COMPENSATED CONTROL OF PDES

287

According to (10.45)–(10.49), (10.72), (10.75), (10.101), (10.105), (10.121), we know that the output injection states y1 (t), y2 (x, t), y3 (x, t), y4 (t), y5 (x, t) are the output states of the following dynamical systems given by their spatially dependent transfer functions: H1 (s) =

−q1 K1 (1) , C1 e−τ A1 (sI − A¯1 )−1 B1

(10.128)

H2 (s; x) =

−q1 φ(x, 1) −τ A ¯1 )−1 B1 , 1 (sI − A C1 e

(10.129)

−q1 ψ(x, 1) , C1 e−τ A1 (sI − A¯1 )−1 B1 q + C1 (sI − A¯1 )−1 B1 H4 (s) = , C1 e−τ A1 (sI − A¯1 )−1 B1

H3 (s; x) =

H5 (s; x) = C1 e−τ A1 (x−1) Γ1 −

C1 e−τ A1 (x−1) B1 , C1 e−τ A1 (sI − A¯1 )−1 B1

(10.130) (10.131) (10.132)

whose input signal is v˜(2, t), which is exponentially convergent to zero. Recalling lemma 10.1, we know that there is no pole in the closed right-half plane in the transfer function (10.128)–(10.132). The exponential convergence of |y1 (t)|, y2 (·, t) ∞ , y3 (·, t) ∞ , |y4 (t)|, y5 (·, t) ∞ is thus obtained. It should be noted that x ∈ [0, 1] is just a parameter in the numerators of the transfer functions (10.129), (10.130), (10.132), and the stability result is not affected.

10.3

OUTPUT-FEEDBACK CONTROL DESIGN

In the last section, we obtained the observer that compensates for the time delay in the output measurement yout (t) of the distal ODE, which is the only measurement used in the observer, to track the states of the overall sandwich PDE system (10.1)– (10.7). In this section, we design an output-feedback control law U (t) based on the observer (10.28)–(10.35) by using backstepping transformations and frequencydomain designs. First, two transformations are introduced to transform the observer (10.28)– (10.35) into a target system (10.173)–(10.180), which is in a stable form except for the proximal ODE influenced by perturbations originating from the PDEs and the distal ODE. Representing this “target system” in the frequency domain by using the Laplace transform, the algebraic relationships (10.196)–(10.202) between the states of the proximal ODE and the states of the PDEs and the distal ODE are obtained. Inserting these algebraic relationships to rewrite the perturbations in the proximal ODE, a new ODE (10.208) without external perturbations is obtained in the frequency domain, where the control input to exponentially stabilize this ODE is to be designed. First Transformation The aim of the first transformation is to remove the source terms in the PDE domain x ∈ [0, 1]—that is, the couplings in (10.30), (10.31)—and to make the system matrix of the distal ODE (10.33) Hurwitz. A PDE backstepping transformation in the form [48]

CHAPTER TEN

288

1

α(x, t) = zˆ(x, t) −

−

K3 (x, y)ˆ z (y, t)dy x

1

J3 (x, y)w(y, ˆ t)dy − γ(x)Yˆ (t),

x



β(x, t) = w(x, ˆ t) −

−

(10.133)

1

K2 (x, y)ˆ z (y, t)dy x

1

J2 (x, y)w(y, ˆ t)dy − λ(x)Yˆ (t)

(10.134)

x

is introduced, where the kernels K3 (x, y), J3 (x, y), γ(x), K2 (x, y), J2 (x, y), λ(x) are to be determined later, to convert (10.28)–(10.35) into the following intermediate system:

1 ˆ + E0 β(0, t) + ¯ 4 (x)α(x, t)dx ˆ˙ K X(t) = A0 X(t) 0



1

+ 0

¯ 5 (x)β(x, t)dx + K ¯ 6 Yˆ (t) + B0 U (t) + h1 (˜ K v (2, t)),

ˆ + α(0, t) = pβ(0, t) + C0 X(t)

¯ 3 Yˆ (t) + +K

1 0

1 0

(10.135)

¯ 1 (x)α(x, t)dx K

¯ 2 (x)β(x, t)dx, K

(10.136)

αt (x, t) = − q1 αx (x, t) − c1 α(x, t) − γ(x)Γ1 v˜(2, t)

1 − J2 (x, y)h3 (˜ v (2, t); y)dy

−

x 1

K3 (x, y)h2 (˜ v (2, t); y)dy x

+ h2 (˜ v (2, t); x) − q2 J3 (x, 1)h4 (˜ v (2, t)),

(10.137)

βt (x, t) = q2 βx (x, t) − c2 β(x, t)

1 − λ(x)Γ1 v˜(2, t) − J2 (x, y)h3 (˜ v (2, t); y)dy

−

x 1

K2 (x, y)h2 (˜ v (2, t); y)dy x

v (2, t); x) − q2 J2 (x, 1)h4 (˜ v (2, t)), + h3 (˜ β(1, t) = qα(1, t) + h4 (˜ v (2, t)), ˙ Yˆ (t) = Aˆ1 Yˆ (t) + B1 α(1, t) + Γ1 v˜(2, t), vˆ(1, t) = C1 Yˆ (t), 1 vˆt (x, t) = − vˆx (x, t) + h5 (˜ v (2, t); x), τ

(10.138) (10.139) (10.140) (10.141) (10.142)

where the matrix Aˆ1 is made Hurwitz by choosing the control parameter F1 accord¯ 2 (x), K ¯ 3, K ¯ 4 (x), K ¯ 5 (x), K ¯ 6 satisfy ¯ 1 (x), K ing to assumption 10.2. The functions K

DELAY-COMPENSATED CONTROL OF PDES

¯ 1 (x) = pK2 (0, x) − K3 (0, x) + K

x

+ 0

x 0



(10.143)

x 0

¯3 = K

1 0



¯ 2 (x)λ(x)dx + K

1 0

(10.144) ¯ 1 (x)γ(x)dx K

− E0 K2 (0, x),

x

¯ ¯ K5 (x) = K4 (y)J3 (y, x)dy + 0

1 0

¯ 1 (y)J3 (y, x)dy K

¯ 2 (y)J2 (y, x)dy, K

0



x 0

+ pλ(0) − γ(0),

x

¯ ¯ K4 (x) = K4 (y)K3 (y, x)dy +

¯6 = K

¯ 1 (y)K3 (y, x)dy K

¯ 2 (y)K2 (y, x)dy, K

¯ 2 (x) = − pJ2 (0, x) + J3 (0, x) + K +

289

¯ 5 (x)λ(x)dx + K

1 0

(10.145) x 0

¯ 5 (y)K2 (y, x)dy K (10.146)

x 0

¯ 5 (y)J2 (y, x)dy + E0 J2 (0, x), K

¯ 4 (x)γ(x)dx + E0 λ(0), K

(10.147) (10.148)

which are obtained by matching (10.135), (10.136) and (10.28), (10.29) via (10.133), (10.134) (the details are given in step 4 in appendix 10.6A). The following conditions of the kernels in the transformations (10.133), (10.134) are obtained by matching (10.137)–(10.140) and (10.30)–(10.33), with the details given in steps 1–3 in appendix 10.6A: q1 K3 (x, 1) = q2 J3 (x, 1)q + γ(x)B1 , c1 , J3 (x, x) = q2 + q1 −q1 J3x (x, y) + q2 J3y (x, y)

(10.149) (10.150)

+ (c2 − c1 )J3 (x, y) + c1 K3 (x, y) = 0, −q1 K3x (x, y) − q1 K3y (x, y) + c2 J3 (x, y) = 0,

(10.151) (10.152)

γ(1) = − F1 , −q1 γ  (x) − γ(x)(A1 + c1 ) − q2 J3 (x, 1)C1 = 0, q2 qJ2 (x, 1) = q1 K2 (x, 1) − λ(x)B1 , −c2 K2 (x, x) = , q1 + q2 q2 J2x (x, y) + q2 J2y (x, y) + c1 K2 (x, y) = 0,

(10.153) (10.154) (10.155)

q2 K2x (x, y) − q1 K2y (x, y) + (c1 − c2 )K2 (x, y) + c2 J2 (x, y) = 0, 

q2 λ (x) − λ(x)(A1 + c2 ) − q2 J2 (x, 1)C1 = 0, λ(1) = qγ(1) + C1 .

(10.156) (10.157) (10.158) (10.159) (10.160)

The well-posedness of (10.149)–(10.160) is established in the following lemma.

CHAPTER TEN

290

Lemma 10.2. The kernel equations (10.149)–(10.154) have a unique solution K3 , J3 ∈ C 1 (D), γ ∈ C 1 ([0, 1]), and the kernel equations (10.155)–(10.160) have a unique solution K2 , J2 ∈ C 1 (D), λ ∈ C 1 ([0, 1]) on D = {(x, y)|0 ≤ x ≤ y ≤ 1}. Proof. The equation sets (10.149)–(10.154) and (10.155)–(10.160) have a structure analogous to (9.20)–(9.25) in chapter 9. Following the proof of lemma 1 in chapter 9, we obtain this lemma. Similarly, the inverse of the transformation (10.133)–(10.134) is obtained as

1

zˆ(x, t) = α(x, t) −

1

−

N (x, y)β(y, t)dy − G(x)Yˆ (t),

x



w(x, ˆ t) = β(x, t) −

1

−

M(x, y)α(y, t)dy

x

1

(10.161)

D(x, y)α(y, t)dy

x

T (x, y)β(y, t)dy − P(x)Yˆ (t),

(10.162)

x

where M(x, y), N (x, y), G(x), D(x, y), T (x, y), P(x) are kernels that can be determined through a process similar to that in appendix 10.6A. The first transformation in the control design is completed. Second Transformation In order to remove the last three terms in the boundary condition (10.136) and render Hurwitz the system matrix of the proximal ODE (10.135), we introduce the second transformation

1 + ˆ ˆ ¯ 1 (x)α(x, t)dx Z(t) = X(t) + C0 K + C0 +



1 0

0

¯ 2 (x)β(x, t)dx + C0 + K ¯ 3 Yˆ (t), K

(10.163)

where C0 + denotes the Moore-Penrose right inverse of C0 . Because C0 is full-row rank (with rank equal to 1), a right inverse exists for C0 —that is, C0 C0 + = I. A choice of C0 + is C0 + = C0T (C0 C0T )−1 . Using (10.163), we convert (10.135), (10.136) to ˆ˙ = Aˆ0 Z(t) ˆ + q1 C 0 + K ¯ 1 (0)C0 Z(t) ˆ + B0 U ¯ (t) Z(t)

1

1 ˆ + MY Y (t) + Mα (x)α(x, t)dx + Mβ (x)β(x, t)dx 0

0

+ N1 α(1, t) + N2 β(0, t) + H[h1 (˜ v (2, t)), h2 (˜ v (2, t); x), h3 (˜ v (2, t); x), h4 (˜ v (2, t)), h5 (˜ v (2, t); x), v˜(2, t)], ˆ α(0, t) = pβ(0, t) + C0 Z(t),

(10.164) (10.165)

DELAY-COMPENSATED CONTROL OF PDES

291

where H[h1 (˜ v (2, t)), h2 (˜ v (2, t); x), h3 (˜ v (2, t); x), h4 (˜ v (2, t)), h5 (˜ v (2, t); x), v˜(2, t)] + ¯ + ¯ v (2, t)) + C0 K3 Γ1 v˜(2, t) + q2 C0 K2 (1)qh4 (˜ v (2, t)) = h1 (˜

1

1 ¯ 1 (x) K + C0 + J2 (x, y)h3 (˜ v (2, t); y)dydx − C0 + + C0 + − C0 + − C0 + − C0

+

− C0 + + C0 + − C0 +



0



0



0



0



0



0



0



0

0

1

1

1

1

1

1

1

1

¯ 1 (x) K

x 1

K3 (x, y)h2 (˜ v (2, t); y)dydx x

¯ 1 (x)h2 (˜ K v (2, t); x)dx ¯ 1 (x)q2 J3 (x, 1)dxh4 (˜ K v (2, t)) ¯ 2 (x)λ(x)dxΓ1 v˜(2, t) K

1



x 1

¯ 2 (x) K ¯ 2 (x) K

J2 (x, y)h3 (˜ v (2, t); y)dydx K2 (x, y)h2 (˜ v (2, t); y)dydx x

¯ 2 (x)h3 (˜ K v (2, t); x)dx ¯ 2 (x)q2 J2 (x, 1)dxh4 (˜ K v (2, t)),

(10.166)

and ˆ ¯ (t) = U (t) − F0 Z(t). U

(10.167)

The matrix Aˆ0 is made Hurwitz by choosing the control parameter F0 recalling assumption 10.2, and N1 , N2 , Mα (x), Mβ (x), MY in (10.164) are ¯ 3 B 1 − q1 C 0 + K ¯ 1 (1) + q2 C0 + K ¯ 2 (1)q, N1 = C 0 + K ¯ 2 (0) + q1 C0 + K ¯ 1 (0)p, N2 = E 0 − q 2 C 0 + K ¯  (x) − (Aˆ0 + c1 )C0 + K ¯ 1 (x), ¯ 4 (x) + q1 C0 + K Mα (x) = K 1 ¯ 2 (x) − (Aˆ0 + c2 )C0 + K ¯ 2 (x), ¯ 5 (x) − q2 C0 + K Mβ (x) = K +

+

¯ 3 Aˆ1 + K ¯ 3. ¯ 6 − Aˆ0 C0 K MY = C 0 K

(10.168) (10.169) (10.170) (10.171) (10.172)

We thus arrive at the target system consisting of (10.137)–(10.142), (10.164), (10.165), which includes dynamic output injections in (10.166). Considering theov (2, t)),h2 (˜ v (2, t); x),h3 (˜ v (2, t); x), rem 10.1 and (10.122)–(10.126), we know that h1 (˜ v (2, t)), h5 (˜ v (2, t); x), and Γ1 v˜(2, t) in the target system (10.137)–(10.142), h4 (˜ (10.164), (10.165) can be regarded as zero, at least after the time gets large—that is, H[h1 , h2 , h3 , h4 , h5 , v˜] = 0—for brevity. Therefore, the target system (10.137)– (10.142), (10.164), (10.165) is rewritten as ˆ˙ = Aˆ0 Z(t) ˆ + q1 C 0 + K ¯ 1 (0)C0 Z(t) ˆ Z(t)

CHAPTER TEN

292

+ MY Yˆ (t) +



1 0

Mα (x)α(x, t)dx +

1 0

Mβ (x)β(x, t)dx

¯ (t), + N1 α(1, t) + N2 β(0, t) + B0 U ˆ α(0, t) = pβ(0, t) + C0 Z(t),

(10.174)

αt (x, t) = − q1 αx (x, t) − c1 α(x, t), x ∈ [0, 1], βt (x, t) = q2 βx (x, t) − c2 β(x, t), x ∈ [0, 1],

(10.175) (10.176)

(10.173)

β(1, t) = qα(1, t), ˙ Yˆ (t) = Aˆ1 Yˆ (t) + B1 α(1, t), vˆ(1, t) = C1 Yˆ (t),

(10.177) (10.178)

1 vˆt (x, t) = − vˆx (x, t), x ∈ [1, 2]. τ

(10.180)

(10.179)

Control Design in the Frequency Domain In the last two subsections, the system (10.28)–(10.35) is converted to the target system (10.173)–(10.180) through the two transformations, (10.133), (10.134) ¯ (t) in (10.173) of the target system and (10.163). In this subsection, the control U (10.173)–(10.180) will be designed in the frequency domain by using the Laplace transform. Taking the Laplace transform of (10.173)–(10.180), we obtain ¯ 1 (0)C0 Z(s) ˆ + MY Yˆ (s) ˆ = q1 C 0 + K (sI − Aˆ0 )Z(s)

1

1 + Mα (x)α(x, s)dx + Mβ (x)β(x, s)dx 0

0

¯ (s), + N1 α(1, s) + N2 β(0, s) + B0 U ˆ α(0, s) = pβ(0, s) + C0 Z(s), sα(x, s) = − q1 αx (x, s) − c1 α(x, s), sβ(x, s) = q2 βx (x, s) − c2 β(x, s), β(1, s) = qα(1, s), ˆ (sI − A1 )Yˆ (s) = B1 α(1, s), vˆ(1, s) = C1 Yˆ (s), 1 sˆ v (x, s) = − vˆx (x, s). τ

(10.181) (10.182) (10.183) (10.184) (10.185) (10.186) (10.187) (10.188)

For brevity, we consider all initial conditions to be zero when we take the Laplace transform. (Arbitrary initial conditions could be incorporated into the stability statement through an expanded analysis which is routine but heavy on additional notation.) We use the definition of a (strictly) proper transfer function for infinitedimensional systems from [33]. Definition 10.1. The function G is said to be proper if, for sufficiently large ρ, sup 

Re(s)≥0

|s|>ρ

|G(s)| < ∞.

(10.189)

If the limit of G(s) at infinity exists and is 0, we say that G is strictly proper.

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According to [33], the definition of a stable transfer function for infinite-dimensional systems in this book is given next. Definition 10.2. An irrational transfer function G(s) appearing in this book is said to be stable if it satisfies sup |G(s)| < ∞.

(10.190)

Re(s)≥0

Definition 10.2 indicates there is no pole in the closed right-half plane. Defining h(s) = 1 − pqe

−



c2 q2

c

+ q1



e

1

−



1 q2

+ q1

1



s

,

(10.191)

according to (10.182)–(10.188) and section 3.2 in [49], we obtain the following algeˆ and other states in (10.182)–(10.188) as braic relationships between C0 Z(s) h(s)α(x, s) = e

−(c1 +s) x q1

h(s)β(x, s) = qe

ˆ C0 Z(s),

(10.192)

−(c2 +s) (c +s) (1−x)− 1q q2 1

h(s)ˆ v (x, s) = C1 (sI − Aˆ1 )−1 B1 e h(s)ˆ v (1, s) = C1 (sI − Aˆ1 )−1 B1 e

ˆ C0 Z(s),

−(c1 +s) −τ (x−1)s q1 −(c1 +s) q1

(10.193) ˆ C0 Z(s),

ˆ C0 Z(s),

ˆ h(s)α(0, s) = C0 Z(s), h(s)β(1, s) = qe h(s)β(0, s) = qe h(s)α(1, s) = e

−(c1 +s) q1

ˆ C0 Z(s),

(10.197)

ˆ C0 Z(s),

ˆ C0 Z(s),

ˆ h(s)Yˆ (s) = (sI − Aˆ1 )−1 B1 e C0 Z(s),

1

1 −(c2 +s) (c1 +s) ˆ h(s) Mβ (y)β(y, s)dy = Mβ (y)qe q2 (1−y)− q1 dyC0 Z(s), h(s)

Mα (y)α(y, s)dy =

0

(10.198) (10.199)

−(c1 +s) q1

0

1

(10.195) (10.196)

−(c2 +s) (c +s) − 1q q2 1

−(c1 +s) q1

(10.194)

(10.200) (10.201)

0

1 0

Mα (y)e

−(c1 +s) y q1

ˆ dyC0 Z(s).

(10.202)

Multiplying both sides of (10.181) by scalar h(s) and substituting (10.196)– (10.202) therein yields ˆ h(s)(sI − Aˆ0 )Z(s) ¯ 1 (0)C0 Z(s) ˆ = h(s)q1 C0 + K −(c +s)

1 ˆ + MY (sI − Aˆ1 )−1 B1 e q1 C0 Z(s)

1 −(c1 +s) ˆ + Mα (y)e q1 y dyC0 Z(s)



0

1

+ 0

Mβ (y)qe

+ N1 e

−(c1 +s) q1

−(c2 +s) (c +s) (1−y)− 1q q2 1

ˆ + N2 qe C0 Z(s)

ˆ dyC0 Z(s)

−(c2 +s) (c +s) − 1q q2 1

ˆ + h(s)B0 U ¯ (s). C0 Z(s)

(10.203)

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294

Recalling assumption 10.1, we know that h(s) is nonzero for any s ∈ C, (s) ≥ 0, and then h(s) has an inverse h(s)−1 that is a stable, proper transfer function according to definitions 10.1 and 10.2. Multiplying both sides of (10.203) by h(s)−1 and defining ˆ = C0 Z(t), ˆ ξ(t)

(10.204)

we rewrite (10.203) as ˆ (sI − Aˆ0 )Z(s) ˆ + h(s)−1 MY (sI − Aˆ1 )−1 B1 e ¯ 1 (0)ξ(s) = q1 C 0 + K

1 −(c1 +s) ˆ Mα (y)e q1 y dy ξ(s) + h(s)−1 + h(s)−1



−(c1 +s) q1

ˆ ξ(s)

0

1

Mβ (y)qe

0

+ h(s)−1 N1 e

−(c1 +s) q1

+ h(s)−1 N2 qe

−(c2 +s) (c +s) (1−y)− 1q q2 1

ˆ dy ξ(s)

ˆ ξ(s)

−(c2 +s) (c +s) − 1q q2 1

ˆ + B0 U ¯ (s) ξ(s)

for any s ∈ C, (s) ≥ 0. Defining G(s) = q1 C0

+



 −(c1 +s) −1 ¯ K1 (0) + h(s) MY (sI − Aˆ1 )−1 B1 e q1

1

+ 0

Mα (y)e

+ N1 e

−(c1 +s) q1

−(c1 +s) y q1

dy +

1

Mβ (y)qe 0  −(c2 +s) (c1 +s) + N2 qe q2 − q1 ,

−(c2 +s) (c +s) (1−y)− 1q q2 1

dy (10.205)

which is a stable, proper transfer matrix according to definitions 10.1 and 10.2, we get ˆ + B0 U ¯ (s). ˆ = G(s)ξ(s) (sI − Aˆ0 )Z(s)

(10.206)

Recalling Aˆ0 being Hurwitz, det(sI − Aˆ0 ) does not have any zeros in the closed right-half plane. Then the matrix (sI − Aˆ0 ) is invertible for any s ∈ C, (s) ≥ 0. Multiplying both sides of (10.206) with C0 (sI − Aˆ0 )−1 from the left, we obtain ˆ + C0 (sI − Aˆ0 )−1 B0 U ˆ = C0 (sI − Aˆ0 )−1 G(s)ξ(s) ¯ (s). C0 Z(s)

(10.207)

That is, ˆ = C0 (sI − Aˆ0 )−1 G(s)ξ(s) ˆ + W0 U ¯ (s), ξ(s) where

(10.208)

W0 (s) = C0 (sI − Aˆ0 )−1 B0 .

Recall assumption 10.3, which is equivalent to the existence of a right inverse for W0 . A possible choice is given by the Moore-Penrose right inverse W0+ (s) = W0T (s)(W0 (s)W0T (s))−1 [152].

DELAY-COMPENSATED CONTROL OF PDES

295

¯ (s) in (10.208) as Choose U ˆ = F (s)ξ(s), ˆ ¯ (s) = −W + (s)Ω(s)C0 (sI − Aˆ0 )−1 G(s)ξ(s) U 0

(10.209)

where a single-input single-output (SISO) low-pass filter Ω(s) satisfying |1 − Ω(jω)|
0 is associated with the L1 norm of the impulse response (chapter 2 in [52], appendix B of [116]) of the entries of D(s) in (11.107). It follows that d¯ depends only on the parameters of the plant and of the low-pass-filter-based backstepping continuous-in-time control law. Recalling (11.63)–(11.66), (11.80), and (11.106), we obtain the event-based target system in the time domain: ˆ (t) + D(d(t)), ˆ˙ (t) = Aˆ0 W W ˆ (t), α(0, t) = C0 W

(11.110) (11.111)

αt (x, t) = −q1 αx (x, t) − c1 α(x, t), βt (x, t) = q2 βx (x, t) − c2 β(x, t), β(1, t) = qα(1, t), ˆ˙ ˆ + Bα(1, t). X(t) = AˆX(t)

(11.112) (11.113) (11.114) (11.115)

The ETM to determine the triggering times is designed to be governed by the following dynamic triggering condition: ˆ (t) − μm(t)}, ˆ T (t)P0 W tk+1 = inf{t ∈ R+ |t > tk |d(t)2 ≥ θW

(11.116)

where the internal dynamic variable m(t) satisfies the ordinary differential equation ˆ (ζ)2 − μd sup d(ζ)2 m(t) ˙ =−ηm(t) − μW sup W 0≤ζ≤t

0≤ζ≤t

(11.117)

with initial condition m(0) < 0, which guarantees that m(t) < 0.

(11.118)

CHAPTER ELEVEN

322

Inequality (11.118) follows from the ODE (11.117), the nonpositivity of the nonhomogeneous terms on its right-hand side, the strict negativity of m(0), the variation-of-constants formula, and the comparison principle. The positive definite matrix P0 = P0T in (11.116) is the unique solution to the Lyapunov equation AˆT0 P0 + P0 Aˆ0 = −Q0

(11.119)

for some Q0 = Q0 T > 0. The positive constants θ, μ, η, μW , μd in the ETM are to be determined later. The observer-based event-triggering condition (11.116) uses the transformed ˆ (t) because it contains the estimated states of the overall ODE-PDEODE state W ODE system through the transformations (11.59), (11.60), (11.67), (11.78). ˙ 2 , on which the minimal dwell time relies, is As will be seen in lemma 11.2, d(t) 2 ˆ ˆ (t)|2 , d(t)2 in (11.116)). bounded by sup0≤ζ≤t |W (ζ)| , sup0≤ζ≤t d(ζ)2 (instead of |W In order to avoid the Zeno phenomenon, namely, to ensure that lim tk = +∞,

(11.120)

k→∞

the internal dynamic variable m(t) is introduced in (11.116) to offset sup0≤ζ≤t ˆ (ζ)|2 , sup0≤ζ≤t d(ζ)2 when proving the existence of a minimal dwell time, which |W will be seen clearly in the proof of lemma 11.3. Proposition 11.1. For the given (z(·, tk ), w(·, tk ))T ∈ L2 ((0, 1); R2 ), X(tk ) ∈ Rn , ˆ k ) ∈ Rn , Yˆ (tk ) ∈ Rn¯ , m(tk ) ∈ z (·, tk ), w(·, ˆ tk ))T ∈ L2 ((0, 1); R2 ), X(t Y (tk ) ∈ Rn¯ and (ˆ − T R , there exist unique (weak) solutions ((z, w) , X, Y ) ∈ C 0 ([tk , tk+1 ]; L2 (0, 1); R2 ) ˆ Yˆ ) ∈ C 0 ([tk , tk+1 ]; L2 (0, 1); z , w) ˆ T , X, × C 0 ([tk , tk+1 ]; Rn ) × C 0 ([tk , tk+1 ]; Rn¯ ) and ((ˆ 2 0 n 0 n ¯ 0 R ) × C ([tk , tk+1 ]; R ) × C ([tk , tk+1 ]; R ), m ∈ C ([tk , tk+1 ]; R− ) to the systems (11.1)–(11.6) and (11.13)–(11.18), (11.117) with the event-based control input Ud (t) applied in (11.1) and (11.13), respectively, between two time instants tk and tk+1 . Proof. Adopting the definition of the weak solution for linear hyperbolic PDEODE systems, which will be shown in chapter 14 as definition 14.1, for the given (z(·, tk ), w(·, tk ))T ∈ L2 ((0, 1); R2 ), X(tk ) ∈ Rn , and Y (tk ) ∈ Rn¯ , with the results in [35, 141], we can show that there exist unique (weak) solutions ((z, w)T , X, Y ) ∈ C 0 ([tk , tk+1 ]; L2 (0, 1); R2 ) × C 0 ([tk , tk+1 ]; Rn ) × C 0 ([tk , tk+1 ]; Rn¯ ) to the system (11.1)–(11.6) under the event-based control input Ud (t). Similarly, the unique solution of the observer error system also can be obtained by recalling the target observer error system (11.28)–(11.31), (11.54), (11.55), as well as the backstepping transformation (11.26), (11.27), and (11.49). According to (11.19), it can then ˆ Yˆ ) ∈ C 0 ([tk , tk+1 ]; be shown that there exist unique (weak) solutions ((ˆ z , w) ˆ T , X, 2 2 0 n 0 n ¯ L (0, 1); R ) × C ([tk , tk+1 ]; R ) × C ([tk , tk+1 ]; R ) and m ∈ C 0 ([tk , tk+1 ]; R− ) to the system (11.13)–(11.18), (11.117) under the event-based control input Ud (t). Proposition 11.1 is thus obtained. Lemma 11.2. For d(t) defined in (11.103), there exist the positive constants λW , λd such that ˙ 2 ≤λW sup W ˆ (ζ)2 + λd sup d(ζ)2 d(t) 0≤ζ≤t

0≤ζ≤t

(11.121)

EVENT-TRIGGERED CONTROL OF PDES

323

for t ∈ (tk , tk+1 ), where λW , λd depend only on the parameters of the plant and the low-pass-filter-based backstepping continuous-in-time control law. Proof. Taking the time derivative of (11.103), we obtain ˙ 2 = U˙ (t)2 d(t)

(11.122)

because U˙ d (t) = 0 for t ∈ (tk , tk+1 ). Recalling (11.78), (11.86), (11.101), (11.106), we get ˆ (s) + Rd (s)d(s), sU (s) = R(s)W where

        c2 −s c −s + 1q + q2 1 R(s) = K0 − K0 C0 pqe + F (s) C0 Aˆ0

is an n ¯ -dimensional row vector of stable, proper transfer functions, and         c2 −s c −s + 1q 1 + F (s) C0 D(s) Rd (s) = K0 − K0 C0+ pqe q2

(11.123)

(11.124)

(11.125)

is a stable, proper transfer function (F (s) and D(s) are stable and proper). We thus have BIBO with the estimate ˆ (ζ)2 + λd sup d(ζ)2 , U˙ (t)2 ≤ λW sup W 0≤ζ≤t

0≤ζ≤t

(11.126)

where λW is associated with the · 1 norm of the impulse response of the entries of the transfer function vector R(s), and λd is, likewise, associated with Rd 1 , which depend only on the plant and the design of the continuous-in-time control law. The proof is complete. The following lemma proves the existence of a minimal dwell time between two triggering times (independent of initial conditions). It ensures the Zeno phenomenon does not occur and a reduction of changes in the value in the actuator signal compared with the continuous-in-time control. Lemma 11.3. For some μW , μd , θ, there exists a τ > 0, independent of initial conditions, such that tk+1 − tk ≥ τ for all k ≥ 0. Proof. Let us introduce a function 2

ψ(t) =

d(t) + μ2 m(t) ˆ (t) − μ m(t) ˆ T (t)P0 W θW

(11.127)

2

according to [57]. We have ψ(tk+1 ) = 1

(11.128)

ψ(tk ) < 0

(11.129)

because the event is triggered, and

CHAPTER ELEVEN

324

because m(t) < 0 and d(tk ) = 0. The function ψ(t) is continuous on [tk , tk+1 ] due to proposition 11.1. By the intermediate value theorem, there exists t∗ > tk such that ψ(t) ∈ [0, 1] when t ∈ [t∗ , tk+1 ]. The lower bound τ of the minimal dwell time can be defined as the minimal time it takes for ψ(t) from 0 to 1. Recalling (11.109), (11.110), we obtain 2 ¯ ˆ T (t)P0 W ˆ (t) W 3 ˆ (t)2 − 2dλmax (P0 ) sup d(ζ)2 . ≥ − λmax (Q0 )W dt 2 λmax (Q0 ) 0≤ζ≤t

(11.130)

Taking the derivative of ψ(t) (11.127) and using (11.121), (11.130), we have ˆ

T

ˆ

0 W (t) ˙ + μ m(t) − μ2 m(t)) ˙ (θ W (t)P 2d(t)d(t) 2 ˙ dt − ψ(t) μ μ ˆ (t) − m(t) θW ˆ (t) − m(t) ˆ T (t)P0 W ˆ T (t)P0 W θW 2 2  1 ˆ (ζ)2 ≤ × r1 λW sup W μ T ˆ ˆ θW (t)P0 W (t) − 2 m(t) 0≤ζ≤t  1 μ 2 2 ˙ + r1 λd sup d(ζ) + d(t) + m(t) r1 2 0≤ζ≤t   1 3 ˆ (t)2 − θ − λmax (Q0 )W μ T ˆ ˆ 2 θW (t)P0 W (t) − 2 m(t)   ¯ max (P0 )2 2dλ μ sup d(ζ)2 − m(t) ˙ ψ(t), (11.131) − λmax (Q0 ) 0≤ζ≤t 2

˙ = ψ(t)

where r1 is a positive constant from Young’s inequality. Inserting (11.117), one obtains  1 ˙ ˆ (ζ)2 ψ(t) ≤ sup W r λ ˆ (t) − μ m(t) 1 W 0≤ζ≤t ˆ T (t)P0 W θW 2 + r1 λd sup d(ζ)2 + 0≤ζ≤t

1 μ 2 d(t) − ηm(t) r1 2

 μ μ 2 2 ˆ − μW sup W (ζ) − μd sup d(ζ) 2 2 0≤ζ≤t 0≤ζ≤t  1 3θ ˆ (t)2 − − λmax (Q0 )W μ T ˆ ˆ 2 θW (t)P0 W (t) − 2 m(t) −

¯ max (P0 )2 2θdλ μ sup d(ζ)2 + ηm(t) λmax (Q0 ) 0≤ζ≤t 2

 μ ˆ (ζ)2 + μ μd sup d(ζ)2 ψ(t) μW sup W 2 2 0≤ζ≤t 0≤ζ≤t 

 μ 1 ˆ (ζ)2 r sup W μ λ − ≤ 1 W W μ T ˆ ˆ 2 θW (t)P0 W (t) − 2 m(t) 0≤ζ≤t  

μ 1 μ 2 2 + r1 λd − μd sup d(ζ) + d(t) − ηm(t) 2 r1 2 0≤ζ≤t +

EVENT-TRIGGERED CONTROL OF PDES

325

 1 μ + − ηm(t) μ T ˆ ˆ 2 θW (t)P0 W (t) − 2 m(t)   3θ μ ˆ (ζ)2 λmax (Q0 ) − μW sup W + 2 2 0≤ζ≤t  ¯   2θdλmax (P0 )2 μ − μd sup d(ζ)2 ψ(t). + λmax (Q0 ) 2 0≤ζ≤t

(11.132)

Choosing positive constants μW , μd , θ in the ETM such that they satisfy μW ≥

2r1 λW , μ

(11.133)

2r1 λd , μ

μμW μd μλmax (Q0 ) θ ≤ min , ¯ , 2 3λmax (Q0 ) 4dλ max (P0 )

μd ≥

(11.134) (11.135)

we get ˙ ≤ ψ(t)

μ 1 2 r1 d(t) − 2 ηm(t) ˆ (t) − μ m(t) ˆ T (t)P0 W θW 2

+

− μ2 ηm(t) ψ(t). ˆ (t) − μ m(t) ˆ T (t)P0 W θW

(11.136)

2

Applying the inequalities − μ2 ηm(t) − μ2 ηm(t) = η, ≤ ˆ (t) − μ m(t) ˆ T (t)P0 W − μ2 m(t) θW 2

d(t)2 + μ2 m(t) − μ2 m(t) d(t) = ˆ (t) − μ m(t) θW ˆ (t) − μ m(t) ˆ T (t)P0 W ˆ T (t)P0 W θW 2 2 2

≤ ψ(t) + 1, which hold because m(t) < 0, inequality (11.136) becomes   ˙ ≤ 1 + η + 1 + η ψ(t). ψ(t) r1 r1

(11.137)

(11.138)

It follows that the time needed by ψ(t) to go from 0 to 1 is at least  τ=

1 0

1 d¯ s > 0, ( r11 + η)¯ s + r11 + η

(11.139)

which is independent of initial conditions, where η is a free design parameter appearing in (11.117), and the condition on the positive constant r1 from Young’s inequality will be determined later.

CHAPTER ELEVEN

326 11.5

STABILITY ANALYSIS OF THE EVENT-BASED CLOSED-LOOP SYSTEM

In the event-based output-feedback closed-loop system, a low-pass-filter-based backstepping control law U (t) in (11.101), using the states from the observer, is updated at time instants tk determined by the ETM (11.116), (11.117) implemented based on the observer, to regulate the PDE plant (11.1)–(11.6). ˆ Lemma 11.4. With arbitrary initial data (Yˆ (0), zˆ(x, 0), w(x, ˆ 0), X(0)) ∈ χ, in the event-based state-feedback loop, the exponential convergence is achieved in the sense that there exist positive constants Υf , λf such that −λf t ˆ ˆ ≤ Υf Ξ(0)e Ξ(t) ,

(11.140)

where 2   2 ˆ  + Yˆ (t) + ˆ ˆ = X(t) ¯ (t)2 . Ξ(t) z (·, t) 2 + w(·, ˆ t) 2 + |m(t)| + U

(11.141)

¯ (t) in (11.95). According to (11.92), we Proof. The state of the low-pass filter is U get ¯ (s) = −F (s)C0 W ˆ (s) U where F (s) (11.96) is a stable, proper transfer function guaranteeing the BIBO property with the estimate as ¯ (t)2 ≤ λlp |C0 |2 sup W ˆ (ζ)2 , U 0≤ζ≤t

(11.142)

where the positive constant λlp is associated with the L1 norm of the impulse response of F (s). Let us consider the Lyapunov function  1 ˆ T P X(t) ˆ (t) + 1 ra ˆ +W ˆ (t)T P0 W eδ1 x β(x, t)2 dx V (t) =rw X(t) 2 0  1 1 + rb e−δ2 x α(x, t)2 dx − m(t), (11.143) 2 0 where a positive definite matrix P = P T is the solution to the Lyapunov equation AˆT P + P Aˆ = −Q

(11.144)

for some Q = QT > 0. The positive constants ra , rb , δ1 , δ2 , rw are to be determined later. The Lyapunov function (11.143) is positive definite because m(t) < 0. Defining      ˆ 2  ˆ 2 (11.145) Ω0 (t) = α(·, t) 2 + β(·, t) 2 + X(t)  + W (t) + |m(t)|, recalling (11.143)–(11.145), the following inequality holds μ1 Ω0 (t) ≤ V (t) ≤ μ2 Ω0 (t)

(11.146)

EVENT-TRIGGERED CONTROL OF PDES

327

for positive constants

1 1 μ1 = min rw λmin (P ), λmin (P0 ), ra , rb e−δ2 , 1 , 2 2

1 1 μ2 = max rw λmax (P ), λmax (P0 ), ra eδ1 , rb , 1 . 2 2

(11.147) (11.148)

Taking the derivative of (11.143) along (11.110)–(11.115), recalling (11.117), one obtains ˆ T QX(t) ˆ T P Bα(1, t) − W ˆ (t) + 2W ˆ T P0 D(d(t)) ˆ + 2rw X ˆ (t)T Q0 W V˙ (t) = − rw X(t)  1  1 eδ1 x β(x, t)βx (x, t)dx − q1 rb e−δ2 x α(x, t)αx (x, t)dx + q2 ra  − ra c2

0

0

1

eδ1 x β(x, t)2 dx − rb c1



1 0

0

e−δ2 x α(x, t)2 dx

ˆ (ζ)2 + μd sup d(ζ)2 + ηm(t) + μW sup W 0≤ζ≤t

0≤ζ≤t

ˆT

ˆ ˆ (t) + 2W ˆ T P0 D(d(t)) ˆ + 2rw X P Bα(1, t) − W ˆ (t)T Q0 W = − rw X(t) QX(t)  1 1 1 1 eδ1 x β(x, t)2 dx + q2 ra eδ1 β(1, t)2 − q2 ra β(0, t)2 − δ1 q2 ra 2 2 2 0  1 1 1 ˆ (t))2 − 1 δ2 q1 rb − q1 rb e−δ2 α(1, t)2 + q1 rb (C0 W e−δ2 x α(x, t)2 dx 2 2 2 0  1  1 δ1 x 2 −δ2 x 2 − ra c2 e β(x, t) dx − rb c1 e α(x, t) dx T

0

0

ˆ (ζ)2 + μd sup d(ζ)2 . + ηm(t) + μW sup W 0≤ζ≤t

0≤ζ≤t

(11.149)

Applying Young’s inequality and the Cauchy-Schwarz inequality and recalling (11.109), we obtain   1 rw 1 2 ˆ ˆ (t)|2 λmin (Q)|X(t)| λmin (Q0 ) − q1 rb |C0 |2 |W V˙ (t) ≤ − − 2 2 2   1 2rw |P B|2 1 q1 rb e−δ2 − − q2 ra eδ1 q 2 α(1, t)2 − (11.150) 2 λmin (Q) 2 ¯ max (P0 )2 2dλ 1 sup d(ζ)2 − q2 ra β(0, t)2 + 2 λmin (Q0 ) 0≤ζ≤t   1 1 δ1 q2 ra − ra |c2 | − eδ1 x β(x, t)2 dx (11.151) 2 0  1  1 δ2 q1 rb − rb |c1 | e−δ2 x α(x, t)2 dx (11.152) − 2 0 ˆ (ζ)2 + μd sup d(ζ)2 . − η|m(t)| + μW sup W (11.153) 0≤ζ≤t

0≤ζ≤t

Choosing δ1 , δ2 , ra , rb , rw as δ1 >

2|c2 | , q2

(11.154)

CHAPTER ELEVEN

328 2|c1 | , q1 λmin (Q0 ) rb < , q1 |C0 |2 q1 rb e−δ2 λmin (Q) rw < , 8|P B|2 q1 rb −(δ2 +δ1 ) ra < e , 2q2 q 2 δ2 >

(11.155) (11.156) (11.157) (11.158)

we thus arrive at  ¯ max (P0 )2  2dλ 2 ˙ ˆ V (t) ≤ − σa V (t) + μW sup W (ζ) + μd + sup d(ζ)2 , (11.159) λmin (Q0 ) 0≤ζ≤t 0≤ζ≤t where σa =

rw 1 1 1 λmin (Q), λmin (Q0 ) − q1 rb |C0 |2 , min μ2 2 2 2  

1 1 δ1 q2 ra − ra |c2 |, δ2 q1 rb − rb |c1 | e−δ2 , η > 0. 2 2

(11.160)

Multiplying both sides of (11.159) by eσa t , we get eσa t V˙ (t) + eσa t σa V (t)

  2 ¯ ˆ (ζ)2 + eσa t μd + 2dλmax (P0 ) ≤ eσa t μW sup W sup d(ζ)2 . λmin (Q0 ) 0≤ζ≤t 0≤ζ≤t

The left side of (11.161) is

(11.161)

d(eσa t V (t)) . dt

Integrating (11.161) from 0 to t yields  1 −σa t −σa t ˆ (ζ)2 V (t) ≤ V (0)e + (1 − e ) μW sup W σa 0≤ζ≤t   ¯ max (P0 )2  2dλ + μd + sup d(ζ)2 λmin (Q0 ) 0≤ζ≤t  1 ˆ (ζ)2 ≤ V (0)e−σa t + μW sup W σa 0≤ζ≤t   ¯ max (P0 )2  2dλ 2 + μd + (11.162) sup d(ζ) . λmin (Q0 ) 0≤ζ≤t

The triggering condition (11.116) guarantees ˆ (ζ)2 + μ sup |m(ζ)|. sup d(ζ)2 ≤ θλmax (P0 ) sup W

0≤ζ≤t

0≤ζ≤t

0≤ζ≤t

(11.163)

Inserting (11.163) into (11.162) and then recalling (11.145), (11.146) yields    2 ¯ 1 ˆ (ζ)2 + μd + 2dλmax (P0 ) μW sup W V (t) ≤ V (0)e−σa t + σa λmin (Q0 ) 0≤ζ≤t

 ˆ (ζ)2 + μ|m(ζ)| × sup θλmax (P0 )W 0≤ζ≤t

EVENT-TRIGGERED CONTROL OF PDES

329

   ¯ max (P0 )2  1 2dλ ≤V (0)e + μW + μd + θλmax (P0 ) σa λmin (Q0 )   2 ¯ ˆ (ζ)2 + 1 μd + 2dλmax (P0 ) μ sup |m(ζ)| × sup W σa λmin (Q0 ) 0≤ζ≤t 0≤ζ≤t     ¯ 1 2dλmax (P0 )2 ≤V (0)e−σa t + max μW + μd + θλmax (P0 ) , σa λmin (Q0 )   2 ¯ 1 2dλmax (P0 ) μd + μ sup Ω0 (ζ) σa λmin (Q0 ) 0≤ζ≤t −σa t ¯ sup V (ζ), ≤V (0)e +Φ (11.164) −σa t

0≤ζ≤t

where  ¯ max (P0 )2  1 1 1 2dλ ¯ Φ = max μW + μd + θλmax (P0 ), μ1 σa σa λmin (Q0 )

¯ max (P0 )2 μd μ 2μdλ + . σa σa λmin (Q0 )

(11.165)

In order to ensure that ¯ < 1, Φ

(11.166)

along with combining the conditions (11.133)–(11.135) used to avoid the Zeno phenomenon, the design parameters μ, μd , μW , θ are chosen according to the following guidelines. 1) Choose μ as μ1 σa λmin (Q0 ) ¯ max (P0 )2 4dλ

(11.167)

¯ max (P0 )2 μ1 2μdλ < σa λmin (Q0 ) 2

(11.168)

μ< to ensure that

in (11.165). Before showing the guidelines for the other design parameters, we define the analysis parameter r1 , which is from Young’s inequality applied in (11.131), as

μ1 σa μμ1 σa , . (11.169) r1 < min 4λd 4λW The choice of r1 comes from the need to guarantee that both (11.166) and (11.133), (11.134) hold, which will be clearly seen later. 2) Choose μd to satisfy 2r1 λd μ1 σ a ≤ μd < μ 2μ

(11.170)

μd μ μ1 < σa 2

(11.171)

in order to ensure that

CHAPTER ELEVEN

330

in (11.165) (with the right inequality of (11.170)). Recalling (11.167), the final term in (11.165) is less than μ1 . The condition (11.134) is incorporated as the left inequality of (11.170). 3) Choose μW to satisfy 2r1 λW μ1 σ a ≤ μW < μ 2

(11.172)

1 μ1 μW < σa 2

(11.173)

in order to ensure that

in (11.165) (with the right inequality of (11.172)), where the condition (11.133) is incorporated as the left inequality of (11.172). The parameter r1 (11.169) is chosen to ensure that the far-left terms of (11.170), (11.172) are less than the far-right ones, where the far-right terms are from ensuring (11.166), and the far-left terms are from the conditions (11.133), (11.134) on avoiding the Zeno phenomenon. 4) Choose θ to satisfy ⎧ ⎫ ⎨ σ a μ1 μd μλmax (Q0 ) ⎬ μμW 

, ¯ , θ < min (11.174) ¯ max (P0 )2 2 ⎩ 2 μ + 2dλ λ (P ) 3λmax (Q0 ) 4dλmax (P0 ) ⎭ d

λmin (Q0 )

max

0

in order to ensure that 1 σa



¯ max (P0 )2 2dλ μd + λmin (Q0 )

 θλmax (P0 )
0 1−Φ

(11.177)

by recalling (11.166). The choice of the low-pass filter and the ETM parameters affect the overshoot coefficient in the exponential result according to (11.165). Since, according to definitions 10.1 and 10.2, the transfer function between ˆ (s) is stable and proper, (11.86) leads to β(0, s) and W

EVENT-TRIGGERED CONTROL OF PDES

331

ˆ (ζ)2 , β(0, t)2 ≤ γβ sup W 0≤ζ≤t

(11.178)

where the positive constant γβ only depends on the plant parameters. Recalling (11.78), the following inequality holds ˆ (ζ)2 + 2|C0 + |2 p2 sup β(0, ζ)2 ˆ 2 = 2 sup W Z(t) 0≤ζ≤t

0≤ζ≤t

ˆ (ζ)2 , ≤ sup γZ W 0≤ζ≤t

(11.179)

where the positive constant γZ = max{2, 2|C0 + |2 p2 γβ } depends only on the plant parameters. According to (11.179), (11.176), (11.142), we obtain       ˆ 2  ˆ  2 ¯ (t)2 X(t) + Z(t) + α(·, t) 2 + β(·, t) 2 + |m(t)| + U      ˆ 2  ˆ 2 X(0) + Z(0) + α(·, 0) 2 + β(·, 0) 2  ¯ (0)2 e−λ¯ f t + |m(0)| + U

¯f ≤Υ

(11.180)

¯ f , which are associated with ΥV (11.177) and σa (11.160), ¯f,λ for some positive Υ respectively. Applying the invertibility of the backstepping transformations (11.59), (11.60), (11.67), we arrive at (11.140), where Υf , λf are associated with ΥV (11.177) and σa (11.160), respectively. The guidelines for the choices of all the parameters are given by (11.7)–(11.10), (11.97), (11.98), (11.154)–(11.158), (11.167)–(11.174), which are cascaded rather than coupled. The optimal choices of these parameters are not studied in this book, but in future work, the trade-off between the convergence rate and the lower bound of the minimal dwell time is worth studying. Theorem 11.1. For all initial data (Y (0), z(x, 0), w(x, 0), X(0)) ∈ χ and (Yˆ (0), ˆ zˆ(x, 0), w(x, ˆ 0), X(0)) ∈ χ, m(0) ∈ R− , choosing the design parameters to satisfy (11.7)–(11.10), (11.97), (11.98), (11.167), (11.170)–(11.174), the output-feedback closed-loop system—that is, the plant (11.1)–(11.6) under the event-based control input Ud (t) in (11.102), which is realized using the observer (11.13)–(11.18), the low-pass filter Ω(s) and the event-triggering mechanism (11.116), (11.117), has the following properties: 1) No Zeno phenomenon occurs—that is, lim tk = +∞.

k→∞

(11.181)

2) The closed-loop system has unique (weak) solutions ((z, w)T , X, Y ) ∈ C 0 ([0, ∞); ˆ Yˆ ) ∈ C 0 ([0, ∞); L2 (0, 1); R2 ) × C 0 ([0, ∞); Rn ) × C 0 ([0, ∞); Rn¯ ), ((ˆ z , w) ˆ T , X, 2 2 0 n 0 n ¯ 0 L (0, 1); R ) × C ([0, ∞); R ) × C ([0, ∞); R ), and m ∈ C ([0, ∞); R− ).

CHAPTER ELEVEN

332

3) The exponential convergence in the closed-loop system is achieved in the sense that there exist the positive constants Υa , λa such that   ˆ ˆ ≤ Υa Ξ(0) + Ξ(0) (11.182) e−λa t , Ξ(t) + Ξ(t) where

Ξ(t) = |X(t)|2 + |Y (t)|2 + z(·, t) 2 + w(·, t) 2 ,

ˆ and Ξ(t) is defined in (11.141), which includes |m(t)|. 4) The event-triggered control input is convergent to zero in the sense of lim Ud (t) = 0.

t→∞

(11.183)

Proof. 1) Recalling lemma 11.3, property (1) is obtained. 2) By virtue of proposition 11.1 and lemma 11.3, through iterative constructions between successive triggering times, property (2) is obtained. 3) Rewriting the observer states in the output-feedback control input as a sum of the plant states and the observer errors according to (11.19) and inserting the result into the plant (11.1)–(11.6), through the same steps as in the above statefeedback control designs, it follows that the closed-loop dynamics are a cascade of the observer error dynamics feeding into the target system dynamics in the form of (11.110)–(11.115) (the state-feedback loop). Because the stability of the observer error dynamics (which depends on the choices of L, L0 ) and the stability of the state-feedback loop dynamics (which depends on the choices of K, K0 ) are independent, the separation principle holds. Equation (11.182) is thus obtained recalling lemma 11.1 and lemma 11.4, where the overshoot Υa is associated with Υe , Υf , and the decay rate σa is associated with λe , λf . Property (3) is obtained. 4) Recalling (11.101) and the stability result proved in property (3), we have that the continuous-in-time control input U (t) is convergent to zero. According to the definition (11.102), property (4) is obtained. 

11.6

APPLICATION IN THE MINING CABLE ELEVATOR

In this section, the proposed event-triggered backstepping boundary control design is applied to the axial vibration control of a mining cable elevator that is 2000 m deep and whose dynamics include a hydraulic actuator, mining cable, and cage. Model Figure 11.1 shows that the axial vibration dynamics of the mining cable elevator consisting of the hydraulic actuator, mining cable, and cage are described by a wave PDE sandwich system. The figure also shows that this system can be transformed, using a Riemann transformation, into a 2 × 2 coupled transport PDE sandwich system, considered in this chapter, and for which assumptions 11.1–11.3 are satisfied. The vibration model of the mining cable elevator, which includes the hydraulic actuator dynamics, is described (as in chapter 2) by a wave PDE that is sandwiched between two ODEs, as follows: 2

πRd Eux (0, t) + Ud (t), Mh¨b0 (t) = −ch b˙ 0 (t) + 4

(11.184)

EVENT-TRIGGERED CONTROL OF PDES

Transport PDE z(x,t) Transport PDE w(x,t) ODE X(t)

Ud(t)

Ud(t)

ODE

ODE

333

Hydraulic actuator

Riemann transformation

Low nature frequency

Wave PDE with Vibrating string with viscous material damping damping

ODE Y(t) Vibrating cage

ODE

ODE

General model (10.1)-(10.6)

Simplify as singlecable model

Head sheaves

Hydraulic cylinders

Electronically controlled values

Ud(t)

Hydraulic actuator

Figure 11.1. The relationship between the sandwich ODE-PDE-ODE hyperbolic system and the mining cable elevators consisting of a hydraulic-driven head sheave, mining cable, and cage.

u(0, t) = b0 (t), ρutt (x, t) =

(11.185)

πRd2 ¯ Euxx (x, t) − dc ut (x, t), x ∈ [0, L], 4

¯ t) = bL (t), u(L, Mc¨bL (t) = −cL b˙ L (t) +

(11.186) (11.187)

πRd2 ¯ t), Eux (L, 4

(11.188)

where Ud (t) is the event-triggered backstepping control input of the electronically controlled valves that regulates the hydraulic actuator to suppress vibration in the mining cable elevator. The PDE state u(x, t) denotes the distributed axial vibration dynamics along the cable. The ODE state b0 (t) represents the displacement of the hydraulic actuator, and bL (t) is the vibration displacement of the cage. The physical parameters in (11.184)–(11.188) of the mining cable elevator are shown in table 11.1. We apply the Riemann transformations  Eπ Rd z(x, t) = ut (x, t) − ux (x, t), (11.189) ρ 2  Eπ Rd ux (x, t) (11.190) w(x, t) = ut (x, t) + ρ 2 and define the new variables Y (t) = b˙ 0 (t), X(t) = b˙ L (t), which allows us to rewrite (11.184)–(11.188) as (11.1)–(11.6), with  Eπ Rd −dc q1 = q 2 = , c1 = c2 = , ρ 2 2ρ q = p = −1, C0 = C1 = 2, √ √ −ch Rd Eπρ Rd Eπρ 1 − , E0 = , B0 = , A0 = Mh 2Mh 2Mh Mh

(11.191)

(11.192) (11.193) (11.194)

CHAPTER ELEVEN

334

Table 11.1. Physical parameters of the mining cable elevator Parameters (units)

Values

¯ (m) Depth L Cable diameter Rd (m) Cable effective Young’s modulus E (N/m2 ) Cable linear density ρ (kg/m) Mass of hydraulic actuator Mh (kg) Mass of cage Mc (kg) Damping coefficient of hydraulic actuator ch Damping coefficient of cage cL Cable material damping coefficient dc Gravitational acceleration g (m/s2 )

2000 0.2 1.02×109 8.1 300 15000 0.4 0.4 0.5 9.8

√ √ −cL Rd Eπρ Rd Eπρ A= + , B =− , Mc 2Mc 2Mc

(11.195)

which satisfy assumptions 11.1–11.3. The initial conditions of z(x, t) and w(x, t) are defined as ¯ − x)/L ¯ + π/6), z(x, 0) = 0.01 sin(2π(L (11.196) ¯ ¯ w(x, 0) = 0.01 sin(3π(L − x)/L) (11.197) and 1 ¯ 0) − qz(L, ¯ 0)), X(0) = (w(L, 2 1 Y (0) = (z(0, 0) − pw(0, 0)) 2

(11.198) (11.199)

according to (11.5). The observer initial conditions are defined as zˆ(x, 0) = z(x, 0) + 0.2,

(11.200)

w(x, ˆ 0) = w(x, 0) + 0.2,

(11.201)

where 0.2 is an initial observer error, and 1 ˆ ¯ 0) − qˆ ¯ 0)), X(0) = (w( ˆ L, z (L, 2 1 Yˆ (0) = (ˆ z (0, 0) − pw(0, ˆ 0)) 2

(11.202) (11.203)

according to (11.17). We pick the initial value of m(t) as m(0) = −0.001. The simulation is conducted based on the finite-difference method with the time step of 0.0015 s and the space step of 0.5 m. Determining Design Parameters The free design parameter η in (11.117) is selected as η = 0.11. The parameters affecting the decay rate of the states in the closed-loop system are determined

EVENT-TRIGGERED CONTROL OF PDES

335

next. According to A0 , A, B0 , B, C0 , C in (11.193)–(11.195) and the parameter values in table 11.1, recalling (11.7)–(11.10), the control gains and observer gains are chosen as K0 = 1, K = 1.5 and L0 = 1, L = 2, respectively, yielding Aˆ0 = −106.7, Aˆ = −1.067, A¯ = −2.9, A¯0 = −55.4. Defining P = P0 = 12 , we then have λmin (Q0 ) = 106.7, λmin (Q) = 1.067 via (11.119), (11.144). According to (11.97), (11.98) and C0 (Is − Aˆ0 )−1 B0 =

2 300(s + 106.7)

in F (s) in (11.96), the low-pass filter is chosen as the first-order type Ω(s) =

1 , 1 + 0.0011s

which can be implemented with a resistor-capacitor (RC) circuit. Next, choosing δ1 = 0.5, δ2 = 0.5 according to (11.154), (11.155) and then determining rb = 0.013, rw = 7.3, ra = 0.0023 from (11.156)–(11.158) leads to μ2 = 3.65, μ1 = 0.0011 according to the formulae (11.147), (11.148). Therefore, the estimate of the decay rate σa obtained from (11.160) is 0.108. The parameters of the ETM are determined next. According to the transfer functions (11.107), (11.124), (11.125), the plant parameters, and the choices of K0 , ¯ λW , λd is K, a group of conservative estimates of d, d¯= 5, λW = 250, λd = 600. Recalling (11.167), (11.169), μ is defined as μ = 0.0024 and r1 as

r1 = 0.22 × 10−9 .

CHAPTER ELEVEN

336 150 100 Ud (t)

50 0 –50 –100 –150

0

50

100

150

Time (s)

Figure 11.2. Output-feedback event-based control input Ud (t).

Then μW , μd are determined by (11.170), (11.172) as μd = 0.01, μW = 0.5 × 10−4 . Finally, pick

θ = 0.36 × 10−9

via (11.174). Recalling (11.139) for the highly conservative minimal dwell-time estimate τ , we get τ = 0.15 × 10−9 s. Substituting the above parameters into (11.165), we arrive at ¯ = 0.6754. Φ The approximate solutions of the kernels M (x, y), N (x, y), γ(x), H(x, y), J(x, y), λ(x) are obtained from the conditions (11.205)–(11.216), which are two groups of coupled linear hyperbolic PDE-ODE systems on the domain {(x, y)|0 ≤ x ≤ y ≤ ¯ The finite-difference method is employed with a step length of 1 m for y L}. ¯ The approximate solutions of K ¯ 1 (x), K ¯ 2 (x), K ¯ 3, K ¯ 4 (x), running from x to L. ¯ ¯ K5 (x), K6 are obtained from conditions (11.217)–(11.222) by the finite-difference ¯ with a step length of 1 m as well. Based on method with respect to x ∈ [0, L] the above approximate solutions, N1 , N2 , Mα (x), Mβ (x), MX are obtained using (11.223)–(11.227). Closed-Loop Responses Figure 11.2 shows the event-triggered control input, where the minimal dwell time is 0.297 s, which is much larger than the conservative estimate τ = 0.15 × 10−9 s. If the design parameter η is picked as a smaller one of 0.106 (other design parameters are not changed), compared with the first value η = 0.11 defined at the beginning of this subsection, the number of update times of the control input decreases from 373 to 361, further reducing the actuation frequency. However, the control performance is slightly degraded because η also affects the convergence rate of the closed-loop system. Figure 11.3 shows the convergence of the ODE states X(t), Y (t)—that is, the suppression of the axial vibration velocity of the cage and the regulation of the

EVENT-TRIGGERED CONTROL OF PDES

337

0.01 Hydraulic cylinder rod velocity Y(t) Cage vibration velocity X(t)

(m/s)

0.005 0 –0.005 –0.01

0

50

100

150

Time (s)

Figure 11.3. Axial vibration velocity of the cage X(t) and moving velocity of the hydraulic rod Y (t) in the hydraulic cylinder.

0.01 Hydraulic cylinder rod displacement Cage vibration displacement

0.005 (m)

0 –0.005 –0.01 –0.015

0

50

100

150

Time (s)

Figure 11.4. Axial vibration displacement of the cage (initial elastic displacement 0.005 m) and movement of the hydraulic rod in the hydraulic cylinder (initial position 0.001 m).

z(x, t)

0.02 0 –0.02 2000 1000 x(m)

0 0

50 t(s)

100

150

Figure 11.5. Response of z(x, t).

moving velocity of the hydraulic rod in the hydraulic cylinder at the head sheaves. Integrations of X(t), Y (t)—the axial vibration displacement of the cage and the movement of the hydraulic rod in the hydraulic cylinder—are shown in figure 11.4 under the initial elastic displacement of 0.005 m at the cable-cage connection point and the initial position at 0.001 m of the hydraulic rod. Figures 11.5 and 11.6 show the convergence of the PDE states z(x, t), w(x, t). The axial vibration energy of the

CHAPTER ELEVEN

w (x, t)

338

0.015 0.01 0 –0.01 –0.02 2000 1000 x(m)

0 0

100

50

150

t(s)

4 3

2

Rd π –12 ρ|| ut (., t)||2 + — E|| ux (., t) ||2 8

Figure 11.6. Response of w(x, t).

2 1 0

0

50

100

150

Time (s)

Figure 11.7. Axial vibration energy VE of the cable.

cable, 1 R2 πE VE = ρ ut (·, t) 2 + d

ux (·, t) 2 , 2 8

(11.204)

is converted into ρ ρ VE = z(·, t) − w(·, t) 2 + z(·, t) + w(·, t) 2 8 8 using (11.189), (11.190). Figure 11.7 shows that the axial vibration energy of the cable is reduced with the help of the proposed event-based vibration control system.

11.7

APPENDIX

A. The conditions on the kernels M (x, y), N (x, y), γ(x), H(x, y), J(x, y), λ(x) The conditions on the kernels M (x, y), N (x, y), γ(x), H(x, y), J(x, y), λ(x) in (11.59), (11.60) are given by q1 M (x, 1) − q2 N (x, 1)q − γ(x)B = 0, (q2 + q1 )N (x, x) − c1 = 0,

(11.205) (11.206)

EVENT-TRIGGERED CONTROL OF PDES

339

c1 M (x, y) − q1 Nx (x, y) + q2 Ny (x, y) + (c2 − c1 )N (x, y) = 0,

(11.207)

c2 N (x, y) − q1 Mx (x, y) − q1 My (x, y) = 0, γ(1) = −K,

(11.208) (11.209)

−q1 γ  (x) − γ(x)A − c1 γ(x) − q2 N (x, 1)C = 0, q1 H(x, 1) − q2 J(x, 1)q − λ(x)B = 0,

(11.210) (11.211)

−c2 − (q1 + q2 )H(x, x) = 0, c1 H(x, y) + q2 Jx (x, y) + q2 Jy (x, y) = 0, c2 J(x, y) + (c1 − c2 )H(x, y)

(11.212) (11.213)

+ q2 Hx (x, y) − q1 Hy (x, y) = 0, q2 λ (x) − λ(x)A − c2 λ(x) − q2 J(x, 1)C = 0,

(11.214) (11.215)

qγ(1) − λ(1) + C = 0.

(11.216)



¯ 1 (x), K ¯ 2 (x), K ¯ 3 (x), K ¯ 4 (x), K ¯ 5 (x), K ¯6 B. The conditions of K ¯ 1 (x), K ¯ 2 (x), K ¯ 3, K ¯ 4 (x), K ¯ 5 (x), K ¯ 6 are the solutions of the In (11.61), (11.62), K following linear Volterra integral equations of the second kind:  x ¯ 1 (y)M (y, x)dy ¯ 1 (x) = pH(0, x) − M (0, x) + K K 0  x ¯ 2 (y)H(y, x)dy, K (11.217) + 0  x ¯ 2 (x) = −pJ(0, x) + N (0, x) + ¯ 1 (y)N (y, x)dy K K 0  x ¯ 2 (y)J(y, x)dy, K (11.218) +  ¯3 = K

1

0



1

¯ 1 (x)γ(x)dx + pλ(0) − γ(0), K  x  x ¯ ¯ ¯ 5 (y)H(y, x)dy − E0 H(0, x), K4 (x) = K4 (y)M (y, x)dy + K 0 0  x  x ¯ 5 (x) = ¯ 4 (y)N (y, x)dy + ¯ 5 (y)J(y, x)dy + E0 J(0, x), K K K 0

 ¯6 = K

0

0

1

¯ 2 (x)λ(x)dx + K

 ¯ 5 (x)λ(x)dx + K

(11.219)

0

1 0

(11.220) (11.221)

0

¯ 4 (x)γ(x)dx + E0 λ(0). K

(11.222)

C. Expressions for N1 , N2 , Mα , Mβ , MX Expressions for N1 , N2 , Mα , Mβ , MX are ¯ 3 B − q1 C 0 + K ¯ 1 (1) + q2 C0 + K ¯ 2 (1)q, N1 = C 0 + K + ¯ + ¯ N2 = E0 − q2 C0 K2 (0) + q1 C0 K1 (0)p, +

¯ 1 (x) − (Aˆ0 ¯ 4 (x) + q1 C0 K Mα (x) = K ¯ 2 (x) − (Aˆ0 ¯ 5 (x) − q2 C0 + K Mβ (x) = K MX = C 0

+

+

¯ 1 (x), + c1 )C0 K ¯ 2 (x), + c2 )C0 + K

¯ 3 Aˆ + K ¯ 6 − Aˆ0 C0 + K ¯ 3. K

(11.223) (11.224) (11.225) (11.226) (11.227)

CHAPTER ELEVEN

340 11.8

NOTES

The sandwich system control designs in chapters 9 and 10 used continuous-in-time control input signals, whereas the control input in this chapter is piecewise-constant by designing an event-triggering mechanism. As a result, changes in the actuator signal are reduced, which physically facilitates the practical implementation of many industrial string-actuated mechanisms, where the control inputs are provided by massive actuators. The first result of the event-triggered backstepping control of a 2 × 2 hyperbolic system is given in [57]. In this chapter, besides two additional ODEs, the proximal reflection term was required to be compensated by the eventbased control input that goes through the input ODE which is stabilized meanwhile, making the control design more challenging.

Chapter Twelve Sandwich Hyperbolic PDEs with Nonlinearities

In this chapter, we extend the results in chapter 9 for linear sandwich partial differential equations (PDEs) to a more challenging case: the control of sandwich hyperbolic PDEs with nonlinearities, motivated by the brake control of cable mining elevators, where the dynamics consist of a brake, a shock absorber (described by a nonlinear ordinary differential equation (ODE)), a cable of time-varying length, and a cage. We begin by describing, in section 12.1, a particular class of coupled hyperbolic PDEs sandwiched between a nonlinear ODE on the actuated side and a linear ODE on the distal side, with a PDE domain that is time-varying. A state-feedback controller entering a single ODE state is designed to exponentially stabilize the overall system through several backstepping transformations in section 12.2. An observer that only uses the boundary values at the actuated side is constructed to recover all the states of the overall system in section 12.3. The global exponential stability of the observer-based output-feedback closed-loop control system, as well as the boundedness and exponential convergence of the control input, are proved via Lyapunov analysis in section 12.4. The performance is investigated via numerical simulation in section 12.5.

12.1

PROBLEM FORMULATION

The plant considered in this chapter is ˙ X(t) = AX(t) + Bv(0, t),

(12.1)

ut (x, t) = −p1 ux (x, t) + c1 v(x, t), vt (x, t) = p2 vx (x, t) + c2 u(x, t), u(0, t) = qv(0, t) + CX(t), v(l(t), t) = s1 (t),  s˙ 1 (t) = c3 s2 (t) + f1

s1 (t),



(12.2) (12.3)



l(t) 0

u(x, t)dx ,

s˙ 2 (t) = f2 (s1 (t), s2 (t), u(l(t), t)) + z(t), z(t) ˙ = c4 z(t) + ru(l(t), t) + U (t),

(12.4) (12.5) (12.6) (12.7) (12.8)

∀(x, t) ∈ [0, l(t)] × [0, ∞), where X(t) ∈ Rn×1 , z(t) ∈ R are ODE states, which describe the vibration dynamics of the cage and drum. The nonlinear ODE with the vector state S(t) = [s1 (t), s2 (t)]T ∈ R2×1 represents the shock absorber dynamics, where s1 (t) represents the displacement of the connection point with the cable,

CHAPTER TWELVE

342

and s2 (t) means the velocity of the connection point with the drum. The functions u(x, t) ∈ R, v(x, t) ∈ R are the states of the 2 × 2 coupled hyperbolic PDEs, which model the vibration states of the cable. The matrices A ∈ Rn×n , B ∈ Rn×1 , C ∈ R1×n are assumed to be such that the pair (A, B) is controllable, and (A, C) is observable. The constants c1 , c2 , c3 , c4 , r, q ∈ R are arbitrary, whereas p1 and p2 are arbitrary positive transport velocities. The signal U (t) is the control input to be designed. Functions f1 , f2 , and l(t) satisfy the following assumptions. Assumption 12.1. The functions f1 and f2 vanish at the origin—that is, f1 (0, 0) = 0 and f2 (0, 0, 0) = 0. Assumption 12.2. The functions f1 (x1 , x2 ) and f2 (x1 , x2 , x3 ) are continuously differentiable and globally Lipschitz in (x1 , x2 ) and (x1 , x2 , x3 ), respectively. Assumption 12.3. The function l : [0, ∞) → C 2 is uniformly bounded—that is, 0 < l(t) ≤ L, ∀t ≥ 0, where L is a positive constant. Assumption 12.4. The speed of the moving boundary is lower than the transport speed in the PDEs (12.2), (12.3)—that is,   ˙  ∀t ≥ 0. (12.9) l(t) < min{p1 , p2 }, Equations (12.1)–(12.5) can be regarded as reversibly converted from a wave PDE with in-domain damping through a Riemann coordinate transformation (as in chapter 3). Therefore, according to the conclusions in [71, 72], the fact that the speed of the moving boundary l(t) is smaller than the wave speed—that is, assumption 12.4—ensures the well-posedness of the initial boundary value problem (12.1)–(12.5). The signal flow of the plant (12.1)–(12.8) is configured as follows. The control input U (t) goes through a linear ODE (12.8), which acts as a filter and whose output signal z(t) drives a nonlinear ODE (12.6), (12.7), which includes the states s2 (t) and s1 (t), the latter of which, in turn, flows into the right boundary x = l(t) (12.5) of the transport PDE (12.3), which is coupled with another transport PDE (12.2), (12.4) and connected with a linear ODE (12.1) at the left boundary x = 0. The reflected signal u flows back to the ODEs (12.6)–(12.8) via the transport PDE (12.2). The control objective is to exponentially stabilize all the ODE states S(t), X(t), z(t) and the PDE states u(x, t), v(x, t) by designing a control input U (t) applied at the first ODE (12.8) using the measurements v(l(t), t), u(l(t), t), z(t).

12.2

STATE-FEEDBACK CONTROL DESIGN

In this section, a PDE backstepping transformation is used to convert the coupled hyperbolic PDE-ODE subsystem to a stable intermediate system where the indomain couplings between the hyperbolic PDEs are removed, and the system matrix of the ODE at the left boundary is Hurwitz. The right boundary condition of the intermediate system can be regarded as a cascade of a nonlinear ODE and a linear ODE under some perturbations from the PDE states in the time-varying domain and the left boundary. The right boundary condition will be dealt with by using an ODE backstepping procedure. The global exponential stability of the closed-loop system is then proved, where control parameters are determined that come from

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343

the ODE backstepping design procedure and whose purpose is to retain stability in the face of the PDE perturbations. Moreover, the boundedness and exponential convergence of the designed control input are proved as well. Backstepping for a PDE-ODE Subsystem We use the infinite-dimensional backstepping transformation in chapter 9 for the PDE states u(x, t), v(x, t), as follows: α(x, t) = u(x, t),



(12.10) x

β(x, t) = v(x, t) − ψ(x, y)u(y, t)dy 0  x − φ(x, y)v(y, t)dy − γ(x)X(t).

(12.11)

0

The kernels ψ(x, y), φ(x, y) on D = {0 ≤ y ≤ x ≤ l(t)} and the row vector {0 ≤ x ≤ l(t)} satisfy −c2 ψ(x, x) = , p1 + p2 1 p1 q φ(x, 0) = γ(x)B + ψ(x, 0), p2 p2 p2 φx (x, y) + p2 φy (x, y) − c1 ψ(x, y) = 0, −p2 ψx (x, y) + p1 ψy (x, y) + c2 φ(x, y) = 0, p2 γ  (x) − γ(x)A − p1 ψ(x, 0)C = 0, γ(0) = κ,

γ(x) on (12.12) (12.13) (12.14) (12.15) (12.16) (12.17)

where κ is a row vector such that A + Bκ is Hurwitz since the pair (A, B) is controllable. Please refer to lemma 9.1 of chapter 9 for the well-posedness of (12.12)– (12.17). As in chapter 9, the inverse of (12.10), (12.11) is u(x, t) = α(x, t),



(12.18) x

v(x, t) = β(x, t) − D(x, y)α(y, t)dy 0  x M(x, y)β(y, t)dy − J (x)X(t), −

(12.19)

0

where D(x, y), M(x, y) and the row vector J (x) are the kernels of the inverse transformation (12.19), whose well-posedness is shown in section 9.2 in chapter 9. Applying the above backstepping transformations, the original system (12.1)– (12.5) is converted to the following intermediate system (without the right boundary condition): ˙ X(t) = (A + Bκ)X(t) + Bβ(0, t),



(12.20) x

αt (x, t) = − p1 αx (x, t) + c1 β(x, t) − c1 D(x, y)α(y, t)dy 0  x M(x, y)β(y, t)dy − c1 J (x)X(t), − c1 0

(12.21)

CHAPTER TWELVE

344 βt (x, t) = p2 βx (x, t),

(12.22)

α(0, t) = qβ(0, t) + C0 X(t),

(12.23)

where the row vector C0 = C + qγ(0).

(12.24)

Let us now consider the right boundary condition. Inserting x = l(t) into (12.11) and taking the derivative with respect to t, we obtain ˙ ˙ β(l(t), t) = v(l(t), ˙ t) − l(t)ψ(l(t), l(t))u(l(t), t)  l(t) ˙ ˙ − l(t)φ(l(t), l(t))v(l(t), t) − l(t) ψx (l(t), y)u(y, t)dy 0

 ˙ − l(t)

l(t) 0

φx (l(t), y)v(y, t)dy

 ˙ (l(t))X(t) − − l(t)γ

 −

l(t) 0



l(t) 0

ψ(l(t), y)ut (y, t)dy

˙ φ(l(t), y)vt (y, t)dy − γ(l(t))X(t).

(12.25)

Using (12.5), (12.6) to replace v(l(t), ˙ t) in (12.25) and then plugging the inverse transformations (12.18), (12.19) into (12.25) to replace u, v with α, β, through a ˙ change of the order of integration in a double integral, we get β(l(t), t) as   ˙ β(l(t), t) = c3 s2 (t) + f1 β(l(t), t) −  −

l(t) 0

l(t) 0

D(l(t), y)α(y, t)dy

M(l(t), y)β(y, t)dy − J (l(t))X(t),





l(t) 0

α(y, t)dy

+ F(β(l(t), β(0, t), α(l(t), t), α(0, t), β(x, t), α(x, t), X(t)),

(12.26)

where F is a perturbation that includes β(l(t), t), β(0, t), α(l(t), t), α(0, t), β(x, t), α(x, t), and X(t). The complete expression of F is shown in appendix 12.6A. Recalling (12.5), (12.7), (12.8) and (12.18), (12.19) yields   s˙ 2 (t) = f2 β(l(t), t) −  −

l(t) 0

l(t) 0

D(l(t), y)α(y, t)dy 

M(l(t), y)β(y, t)dy − J (l(t))X(t), s2 (t), α(l(t), t) + z(t), (12.27)

z(t) ˙ = c4 z(t) + rα(l(t), t) + U (t).

(12.28)

The equation set (12.26)–(12.28) is the right boundary condition of the intermediate system in the form of several ODEs regulated by the control input U (t). The equations (12.26)–(12.28) governing the variables (β(l(t), t), s2 (t), z(t)) are a cascade of ODEs converted from the equations (12.5)–(12.8) for (s1 (t), s2 , z(t)) via transformation (12.10), (12.11). The system (12.26), (12.27) is a second-order nonlinear ODE for (β(l(t), t), s2 (t)) with perturbations F. Equation (12.28) is a first-order linear ODE z(t) with a perturbation α(l(t), t).

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345

Through the backstepping transformations (12.10), (12.11), the original (u(x, t), v(x, t), X(t), s1 (t), s2 (t), z(t))-system in (12.1)–(12.8) is converted to the intermediate (α(x, t), β(x, t), X(t), β(l(t), t), s2 (t), z(t))-system in (12.20)–(12.23), (12.26)– (12.28). Next, we propose a backstepping design for the ODEs for (β(l(t), t), s2 (t), z(t)) in (12.26)–(12.28) at the right boundary of the intermediate system. Backstepping for ODEs in the Input Channel The following backstepping transformation for the (β(l(t), t), s2 (t))-system (12.26), (12.27) is introduced: (12.29) y1 (t) = β(l(t), t), y2 (t) = s2 (t) + τ1 (t),

(12.30)

where τ1 (t) is to be chosen in the following steps as the virtual control law in the ODE backstepping method. Step 1. We consider the Lyapunov function candidate 1 Vy1 = y1 (t)2 . 2 Taking the derivative of Vy1 , recalling (12.22), (12.26), and (12.30), we obtain ˙ V˙ y1 = y1 (t)y˙ 1 (t) = y1 (t)β(l(t), t)  = y1 (t) c3 y2 (t) − c3 τ1 (t) + f1 + F .

(12.31)

The arguments of f1 and F are omitted in (12.31), which are the same as those in (12.26). We choose c¯1 (12.32) τ1 (t) = y1 (t), c3 where c¯1 is a positive constant to be determined later. Substituting (12.32) into (12.31) yields V˙ y1 = −¯ c1 y1 (t)2 + c3 y1 (t)y2 (t) + y1 (t)f1 + y1 (t)F.

(12.33)

Step 2. A Lyapunov function candidate for y1 (t), y2 (t) is considered to be 1 1 1 Vy = Vy1 + y2 (t)2 = y1 (t)2 + y2 (t)2 . 2 2 2

(12.34)

Taking the derivative of (12.34), we get V˙ y = − c¯1 y1 (t)2 + c3 y1 (t)y2 (t) + y1 (t)f1  + y1 (t)F + y2 (t) f2 + z(t) + τ˙1 ,

(12.35)

where (12.30) and (12.27) are used, and the argument omitted in f2 is the same as that in (12.27). Step 3. Define a new variable E(t) as E(t) = z(t) + c¯2 y2 (t) + c3 y1 (t), where the positive constant c¯2 is to be determined later.

(12.36)

CHAPTER TWELVE

346 Inserting (12.36) into (12.35) to replace z(t), we obtain V˙ y = − c¯1 y1 (t)2 − c¯2 y2 (t)2 + y1 (t)f1 + y1 (t)F c¯1 + y2 (t)E(t) + y2 (t)f2 + y2 (t)y˙ 1 (t). c3

(12.37)

Using (12.36), we write (12.28) as E˙ = c4 E(t) + rα(l(t), t) + c¯2 y˙ 2 (t) + c3 y˙ 1 (t) − c4 c¯2 y2 (t) − c4 c3 y1 (t) + U (t).

(12.38)

Choosing U (t) in (12.38) as U (t) = −¯ a0 E(t) − rα(l(t), t) + c4 c¯2 y2 (t) + c4 c3 y1 (t),

(12.39)

˙ = −kE E(t) + c¯2 y˙ 2 (t) + c3 y˙ 1 (t), E(t)

(12.40)

kE = a ¯ 0 − c4 > 0

(12.41)

we then have

where

by choosing the control gain a ¯0 . Through the transformations (12.10), (12.11), (12.29), (12.30), and (12.36), the original (u(x, t), v(x, t), X(t), s1 (t), s2 (t), z(t))-system is converted to the target (α(x, t), β(x, t), X(t), y1 (t), y2 (t), E(t))-system. The exponential stability of the target system will be ensured in the following Lyapunov analysis by appropriately ¯0 . choosing the control parameters c¯1 , c¯2 , a

Stability Analysis of the State-Feedback Closed-Loop System Substituting (12.36), (12.32), (12.29), (12.30), (12.10), (12.11) into (12.39), we express the controller in terms of the original states, as U (t) = − a ¯0 z(t) + (c4 − a ¯0 )¯ c2 s2 (t) − ru(l(t), t)    l(t) c¯1 c¯2 ¯0 ) + c3 ψ(l(t), y)u(y, t)dy + (c4 − a s1 (t) − c3 0   l(t) φ(l(t), y)v(y, t)dy − γ(l(t))X(t) . −

(12.42)

0

¯0 will be determined in the stability analysis. Because The control parameters c¯1 , c¯2 , a the control law (12.42) uses the signal u(l(t), t), in order to ensure that the control law is sufficiently regular, we will require the initial value u(x, 0) to be in H 1 (0, L), where the positive constant L, given in assumption 12.3, is the maximum value of the time-varying PDE domain. Theorem 12.1. For all initial values (u(x, 0), v(x, 0)) ∈ H 1 (0, L), with some c¯1 , c¯2 , a ¯0 , the closed-loop system consisting of the plant (12.1)–(12.8) and the control law (12.42) is exponentially stable in the sense that there exist the positive constants

PDES WITH NONLINEARITIES

347

Υ1 , λ1 such that Ωa (t) ≤ Υ1 Ωa (0)e−λ1 t ,

(12.43)

where 2

Ωa (t) = u(·, t)2 + v(·, t)2 + |X(t)| + s1 (t)2 + s2 (t)2 + z(t)2 .

(12.44)

Proof. We start by studying the stability of the target system. The equivalent stability property between the target system and the original system is ensured due to the invertibility of the transformations (12.10), (12.11), (12.29), (12.30), and (12.36). First, we produce the stability proof of the target system via Lyapunov analysis of the PDE-ODE subsystem. Second, combining the Lyapunov analysis for the ODEs under the input channel in the backstepping design, Lyapunov analysis of ¯0 in the overall system is provided, through which the control parameters c¯1 , c¯2 , a the control law (12.42) are determined. a) Lyapunov analysis for the PDE-ODE subsystem-(α(x, t), β(x, t), X(t)): Consider the Lyapunov function  a1 l(t) δ1 x 2 e β(x, t) dx V1 (t) = X (t)P1 X(t) + 2 0  b1 l(t) −δ1 x 2 + e α(x, t) dx, 2 0 T

(12.45)

where P1 = P1T > 0 is the solution to the Lyapunov equation P1 (A + Bκ) + (A + Bκ)T P1 = −Q1 for some Q1 = Q1 T > 0. The positive parameters a1 , b1 , δ1 shall be chosen later. Taking the derivative of V1 (t), we arrive at 2 V˙ 1 (t) ≤ − η1 |X(t)| − η2 β(0, t)2 − η3

 − η4

l(t) 0



l(t) 0

2

β(x, t) dx

2

α(x, t) dx − η5 α(l(t), t)2 + η6 β(l(t), t)2 ,

(12.46)

where the detailed process of calculating V˙ 1 (t) is shown in appendix 12.6B. In this appendix, the choices of a1 , b1 , δ1 and the expressions of positive constants η1 , η2 , η3 , η4 are also given. Defining

  ˙  vmax = max l(t)  , t∈[0,∞)

we know, by recalling assumption 12.4, that η5 = (p1 − vmax )

b1 −δ1 L e > 0, 2

as well as that η6 = (p2 + vmax )

a1 δ1 L e > 0. 2

CHAPTER TWELVE

348

b) Lyapunov analysis for the overall system: Consider the Lyapunov function 1 V (t) = V1 (t) + Vy (t) + E(t)2 , 2

(12.47)

where Vy was introduced in (12.34). Defining 2

2

Ω1 (t) = β(·, t) + α(·, t) + |X(t)|

2

+ y1 (t)2 + y2 (t)2 + E(t)2 ,

(12.48)

θ1a Ω1 (t) ≤ V (t) ≤ θ1b Ω1 (t)

(12.49)

we get

for some positive constants θ1a and θ1b . Taking the derivative of (12.47), using (12.46), (12.40) and (12.37) with (12.149)– (12.156) in appendix 12.6A, recalling assumptions 12.1, 12.2, 12.4, we show that V˙ (t) ≤ −λV (t) − ηˆ0 β(0, t)2 − ηˆ1 α(l(t), t)2

(12.50)

for some positive λ, where ηˆ0 , ηˆ1 are positive constants given as (12.181), (12.182) in appendix. The detailed process of calculating V˙ (t) is shown in appendix 12.6C, where the choices of the control parameters c¯1 , c¯2 , a0 in the ODE backstepping are chosen to dominate the PDE perturbations. We thus have V (t) ≤ V (0)e−λt . It then follows that Ω1 (t) ≤

(12.51)

θ1b Ω1 (0)e−λt θ1a

by recalling (12.49). By defining Ξ(t) = u(·, t)2 + v(·, t)2 + |X(t)|2 + s1 (t)2 + s2 (t)2 + z(t)2

(12.52)

and applying the Cauchy-Schwarz inequality and transformations (12.10), (12.11), (12.18), (12.19), (12.29), (12.30), and (12.36), it is straightforward to obtain θ¯1a Ξ(t) ≤ Ω1 (t) ≤ θ¯1b Ξ(t)

(12.53)

for some positive θ¯1a and θ¯1b . Therefore, we get Ξ(t) ≤

θ1b θ¯1b Ξ(0)e−λt . θ1a θ¯1a

(12.54)

θ1b θ¯1b , λ1 = λ. θ1a θ¯1a

(12.55)

Thus, (12.43) is achieved with Υ1 =

The proof of theorem 12.1 is complete.

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349

In theorem 12.1, we proved that all PDEs and ODEs are exponentially stable in the closed-loop system including the plant (12.1)–(12.8) and the controller (12.42). Moreover, next we prove that the controller U (t) (12.42) in the closed-loop system is also bounded and exponentially convergent to zero. According to (12.42) and the exponential stability result proved in theorem 12.1, the exponential convergence of the control input requires, additionally, the exponential convergence of the signal u(l(t), t), which can be obtained by proving the exponential stability estimate of ux (·, t) + vx (·, t). Before proving the exponential convergence of the control input, we state a lemma first. Lemma 12.1. With arbitrary initial data (u(x, 0), v(x, 0)) ∈ H 1 (0, L), the exponential stability estimate of the closed-loop system (u(x, t), v(x, t)) is obtained in the sense that there exist the positive constants Υ1a and λ1a such that ux (·, t)2 + vx (·, t)2  ≤ Υ1a Ξ(0) + ux (·, 0)2 + vx (·, 0)2 e−λ1a t ,

(12.56)

where Ξ(t) is given in (12.52). The proof of lemma 12.1 is shown in appendix 12.6D. Lemma 12.1 will be used in proving the exponential convergence and boundedness of the controller (12.42) in the following theorem. Theorem 12.2. In the closed-loop system, which consists of the plant (12.1)–(12.8) and the controller given by (12.42), there exist the positive constants λ2 and Υ2 , which ensure that U (t) is bounded and exponentially convergent to zero in the sense of  2 |U (t)| ≤ Υ2 u(·, 0)2 + v(·, 0)2 + |X(0)| + s1 (0)2 2

2

2

2

+ s2 (0) + z(0) + ux (·, 0) + vx (·, 0)

 12

e−λ2 t .

(12.57)

Proof. The proof is shown in appendix 12.6E.

12.3

OBSERVER DESIGN AND STABILITY ANALYSIS

In section 12.2, a state-feedback controller that requires distributed states is designed to stabilize the original system exponentially. However, it is difficult to measure distributed states in practice. We propose an output-feedback control law, which only requires measurements u(l(t), t), v(l(t), t), z(t) at the controlled boundary of the PDE—that is, a “collocated” output-feedback law—based on a state observer designed in this section. The observer gains are determined in two transformation steps from the observer error system to an intermediate observer error system, and then to a target observer error system. The exponential stability of the observer error system is inferred from the stability of the target system and the invertibility of the transformations.

CHAPTER TWELVE

350 Observer Design

Using the measurements u(l(t), t), v(l(t), t), z(t), the observer is designed as ˆ˙ ˆ + Bˆ X(t) = AX(t) v (0, t) + Γ0 (t)(u(l(t), t) − u ˆ(l(t), t)),

(12.58)

u ˆt (x, t) = −p1 u ˆx (x, t) + c1 vˆ(x, t) + Γ1 (x, t)(u(l(t), t) − u ˆ(l(t), t)),

(12.59)

vˆt (x, t) = p2 vˆx (x, t) + c2 u ˆ(x, t) + Γ2 (x, t)(u(l(t), t) − u ˆ(l(t), t)),

(12.60)

ˆ u ˆ(0, t) = qˆ v (0, t) + C X(t), vˆ(l(t), t) = v(l(t), t),



sˆ˙ 1 (t) = c3 sˆ2 (t) + f1

(12.61) 

sˆ1 (t),

l(t) 0

(12.62)

 u ˆ(y, t)dy + μ1 (v(l(t), t) − sˆ1 (t))),

s1 (t), sˆ2 (t), u ˆ(l(t), t)) + z(t) + μ2 (v(l(t), t) − sˆ1 (t))), sˆ˙ 2 (t) = f2 (ˆ zˆ˙ (t) = c4 zˆ(t) + ru(l(t), t) + μ3 (z(t) − zˆ(t)) + U (t),

(12.63) (12.64) (12.65)

where Γ0 (t), Γ1 (x, t), Γ2 (x, t), μ1 , μ2 , μ3 are observer gains to be determined later. The initial values u ˆ(x, 0), vˆ(x, 0) are required to be in H1 (0, L) to be consistent with section 12.2. Define observer errors as ˜ [X(t), u ˜(x, t), v˜(x, t), s˜1 (t), s˜2 (t), z˜(t)] = [X(t), u(x, t), v(x, t), s1 (t), s2 (t), z(t)] ˆ − [X(t), u ˆ(x, t), vˆ(x, t), sˆ1 (t), sˆ2 (t), zˆ(t)].

(12.66)

According to (12.58)–(12.65) and (12.1)–(12.8), the observer error dynamics can be obtained as ˜˙ ˜ + B˜ X(t) = AX(t) v (0, t) − Γ0 (t)˜ u(l(t), t),

(12.67)

u ˜t (x, t) = −p1 u ˜x (x, t) + c1 v˜(x, t) − Γ1 (x, t)˜ u(l(t), t),

(12.68)

v˜t (x, t) = p2 v˜x (x, t) + c2 u ˜(x, t) − Γ2 (x, t)˜ u(l(t), t),

(12.69)

˜ u ˜(0, t) = q˜ v (0, t) + C X(t),

(12.70)

v˜(l(t), t) = 0,

(12.71)

s˜˙ 1 (t) = c3 s˜2 (t) + f˜1 − μ1 s˜1 (t),

(12.72)

s˜˙ 2 (t) = f˜2 − μ2 s˜1 (t),

(12.73)

z˜˙ (t) = −kz z˜(t),

(12.74)

where kz = μ 3 − c 4 > 0 by choosing the control parameter μ3 , and where     l(t)  ˜ u(y, t)dy − f1 sˆ1 (t), f1 = f1 s1 (t), 0

l(t) 0

 u ˆ(y, t)dy ,

(12.75)

PDES WITH NONLINEARITIES

351

s1 (t), sˆ2 (t), u ˆ(l(t), t)) . f˜2 = f2 (s1 (t), s2 (t), u(l(t), t)) − f2 (ˆ

(12.76)

Defining ˜ = [˜ S(t) s1 (t), s˜2 (t)]T ,

(12.77)

we rewrite the system (12.72), (12.73) as

T ˜˙ = (As − BC2 )S(t) ˜ + f˜1 , f˜2 , S(t) where

 As =

(12.78)



0 c3 0 0

, C2 = [1, 0], B = [μ1 , μ2 ]T .

(12.79)

The matrix As − BC2 can be made Hurwitz by choosing B because (As , C2 ) is observable. In order to remove the potentially destabilizing feedback terms—that is, the u ˜-terms in v˜ in (12.69), as mentioned in chapter 9—we apply, as in [96], the invertible backstepping transformation for the PDE states (˜ u, v˜), 

l(t)

u ˜(x, t) = α(x, ˜ t) − ˜ t) − v˜(x, t) = β(x,

x  l(t)

¯ y)˜ φ(x, α(y, t)dy,

(12.80)

¯ y)˜ ψ(x, α(y, t)dy,

(12.81)

x

to convert (12.67)–(12.74) to the intermediate observer error system as ˜ t) − B ˜˙ ˜ + B β(0, X(t) = AX(t)  α ˜ t (x, t) = − p1 α ˜ x (x, t) +  β˜t (x, t) = p2 β˜x (x, t) +

l(t)

x l(t)



l(t) 0

¯ y)˜ α(l(t), t), ψ(0, α(y, t)dy − Γ0 (t)˜

˜ t)dy + c1 β(x, ˜ t), ¯ (x, y)β(y, M

˜ t)dy, ¯ (x, y)β(y, N

x

˜ t) + C X(t) ˜ + α ˜ (0, t) = q β(0,



l(t) 0

¯ y) − q ψ(0, ¯ y))˜ (φ(0, α(y, t)dy,

˜ β(l(t), t) = 0, ˙ ˜ = (As − BC2 )S(t) ˜ + [f˜1 , f˜2 ]T , S(t) z˜˙ (t) = − kz z˜(t).

(12.82) (12.83) (12.84) (12.85) (12.86) (12.87) (12.88)

¯ ψ¯ on By matching (12.67)–(12.71) and (12.82)–(12.86), the kernel functions φ, D1 = {0 ≤ x ≤ y ≤ l(t)} should satisfy ¯ y) = 0, −p1 φ¯x (x, y) − p1 φ¯y (x, y) − c1 ψ(x, ¯ x) = c2 , ψ(x, p1 + p2 ¯ ¯ ¯ −p1 ψy (x, y) + p2 ψx (x, y) − c2 φ(x, y) = 0.

(12.89) (12.90) (12.91)

CHAPTER TWELVE

352 The boundary condition of φ¯ is set as ¯ y) = q ψ(0, ¯ y) − CK0 (y), φ(0,

where K0 (x) is shown later. The choice of (12.92) will be clear later. ¯ (x, y), N ¯ (x, y) in (12.82)–(12.88) satisfy The functions M  y ¯ z)M ¯ y), ¯ (x, y)= ¯ (z, y)dz + c1 φ(x, M φ(x, x  y ¯ z)M ¯ y). ¯ (x, y)= ¯ (z, y)dz + c1 ψ(x, N ψ(x,

(12.92)

(12.93) (12.94)

x

The observer gains Γ1 (x, t) and Γ2 (x, t) are obtained as ˙ φ(x, ¯ l(t)) − p1 φ(x, ¯ l(t)), Γ1 (x, t) = l(t) ˙ ψ(x, ¯ l(t)) − p1 ψ(x, ¯ l(t)). Γ2 (x, t) = l(t)

(12.95) (12.96)

In order to decouple the ODE (12.82) from the PDE state α ˜ (β˜ reaches to zero after a finite time because of (12.84), (12.86)) and to make the system matrix in the ODE (12.82) Hurwitz, with the observer gain Γ0 (t) in (12.82) still to be defined, we apply another transformation as  ˜ − Y˜ (t) = X(t)

l(t) 0

 K0 (x)˜ α(x, t)dx −

l(t) 0

˜ t)dx K1 (x)β(x,

(12.97)

to convert (12.82) into Y˜˙ (t) = (A − L0 C)Y˜ (t) − 

x

+ 0



l(t)



0

x 0

¯ (y, x)dy K0 (y)M

 ˜ t)dx, ¯ K1 (y)N (y, x)dy β(x,

(12.98)

where the matrix A − L0 C is made Hurwitz by recalling that (A, C) is observable and suitably choosing L0 , and where K0 (x), K1 (x) are determined next. Substituting (12.97) into (12.98), with (12.82)–(12.86), using integration by parts and a change of the order of integration in a double integral, we obtain

˙ K0 (l(t))p1 − l(t)K ˜ (l(t), t) 0 (l(t)) − Γ0 (t) α  l(t)    ¯ x) α − K0 (x)p1 − AK0 (x) + B ψ(0, ˜ (x, t)dx 0

 l(t)  + (L0 − K0 (0)p1 )˜ − K0 (x)c1 + K1 (x)p2 α(0, t) + 0  ˜ t)dx + (A − L0 C)K1 (x) β(x, ˜ t) = 0. + [K1 (0)p2 − L0 q + B]β(0,

(12.99)

For (12.99) to hold, K0 (x), K1 (x) should satisfy ¯ x) = 0, K0 (x)p1 − AK0 (x) + B ψ(0,

(12.100)

PDES WITH NONLINEARITIES

353

L0 , p1 K1 (x)p2 + (A − L0 C)K1 (x) − K0 (x)c1 = 0, L0 q − B K1 (0) = . p2 K0 (0) =

(12.101) (12.102) (12.103)

¯ y), ψ(x, ¯ y), K0 (x), Equations (12.89)–(12.92), (12.100)–(12.103) for the kernels φ(x, K1 (x) are well-posed, which is established using the fact that, after swapping positions of arguments, like B.9–B.10 in [6]—that is, by changing the domain D1 to D— ¯ ψ, ¯ K0 have the same form the conditions (12.89)–(12.92), (12.100), (12.101) on φ, as the conditions (12.12)–(12.17) on the kernels φ, ψ, γ, which have been proved to be well-posed in chapters 6 and 9. The explicit solution of K1 (x) is then easy to obtain for the initial value problem (12.102), (12.103). The observer gain Γ0 (t) is obtained as ˙ Γ0 (t) = −l(t)K 0 (l(t)) + K0 (l(t))p1 . The target observer error system thus can be written as  l(t)   x ˙ ˜ ˜ ¯ (y, x)dy Y (t) = (A − L0 C)Y (t) − K0 (y)M 0 0   x ˜ t)dx, ¯ (y, x)dy β(x, + K1 (y)N 0



α ˜ t (x, t) = − p1 α ˜ x (x, t) +  β˜t (x, t) = p2 β˜x (x, t) +

l(t)

x l(t)

˜ t)dy, ¯ (x, y)β(y, N

x



˜ t) + C Y˜ (t) + α ˜ (0, t) = q β(0, ˜ β(l(t), t) = 0,

˜ t)dy + c1 β(x, ˜ t), ¯ (x, y)β(y, M



l(t) 0

˜ t)dy, CK1 (y)β(y,

˜ + f˜1 , f˜2 ˜˙ = (As − BC2 )S(t) S(t)

T

(12.104)

(12.105) (12.106) (12.107) (12.108) (12.109)

,

z˜˙ (t) = − kz z˜(t),

(12.110) (12.111)

where (12.92) has been used in forming (12.108). The following theorem shows the exponential stability of the observer error system (12.67)–(12.74), which is obtained through the stability analysis of the target observer error system (12.105)–(12.111) and by applying the invertibility of the transformations. The initial data (˜ u(x, 0), v˜(x, 0)) of the observer error system, defined by the initial conditions of the plant and the observer via (12.66), belong to H 1 (0, L). Stability Analysis of the Observer Error System Theorem 12.3. Consider the observer system (12.58)–(12.65) with observer gains Γ0 (t) (12.104), Γ1 (x, t) (12.95), Γ2 (x, t) (12.96). The observer error system (12.67)– (12.74) is exponentially stable in the sense that there exist the positive constants Υe , λe such that

CHAPTER TWELVE

354 Ωe (t) ≤ Υe Ωe (0)e−λe t , where

   ˜ 2 Ωe (t) = ˜ u(·, t)2 + ˜ v (·, t)2 + X(t)  + s˜1 (t)2 + s˜2 (t)2 + z˜(t)2 .

(12.112)

(12.113)

˜ Proof. 1) Analysis for the observer error subsystems of (˜ u(x, t), v˜(x, t), X(t), z˜(t)): The ODE (12.110) with the state z˜(t) is exponentially stable because kz > 0. From ˜ ˜ t) ≡ 0 after tf 0 = (12.107), (12.109), the β-dynamics are independent of α ˜ and β(x, L p2 —that is, when the boundary condition (12.109) has propagated through the whole domain. The subsystem (12.105)–(12.109) becomes Y˜˙ (t) = (A − L0 C)Y˜ (t), α ˜ t (x, t) = −p1 α ˜ x (x, t), α ˜ (0, t) = C Y˜ (t)

(12.114) (12.115) (12.116)

    for t ≥ tf 0 . The signal Y˜ (t) is exponentially convergent to zero because A − L0 C in the ODE (12.114) is Hurwitz. Define ba Va (t) = Y˜ (t)T Pa Y˜ (t) + 2



l(t) 0

e−x α ˜ (x, t)2 dx,

(12.117)

where ba is a positive constant, and Pa = PaT > 0 is the solution to the Lyapunov equation Pa (A − L0 C) + (A − L0 C)T Pa = −Qa for some Qa = Qa T > 0. Taking the derivative of Va (t) along (12.114)–(12.116), we get V˙ a (t) ≤ −λmin (Qa )Y˜ (t)2 − p1 ba



l(t) 0

e−x α ˜ (x, t)˜ αx (x, t)dx

ba ˙ l(t)e−l(t) α ˜ ((t), t)2 2   1 1 2 −l(t) ˙ ≤ − λmin (Qa ) − p1 ba |C| Y˜ (t)2 − ba (p1 − l(t))e α ˜ (l(t), t)2 2 2  l(t) 1 − p 1 ba e−x α ˜ (x, t)2 dx. (12.118) 2 0 +

Choosing ba
0 is the solution to the Lyapunov equation P2 (A + Bκ) + (A + Bκ)T P2 = −Q2 for some Q2 = Q2 T > 0. Through the same steps as in theorem 12.1, using (12.138), (12.125), (12.111), we obtain V˙ of ≤ −λof Vof (t) for some positive λof . We then obtain Ω4 (t) ≤ Υ4a Ω4 (0)e−λof t for some positive Υ4a , where 2  ˆ t)2 + X(t) ˆ  + yˆ1 (t)2 + yˆ2 (t)2 + E(t) ˆ 2 Ω4 (t) = ˆ α(·, t)2 + β(·,      ˜ 2  ˜ 2 ˜ t)2 + z˜(t)2 . + S(t) α(·, t)2 + β(·,  + Y (t) + ˜

(12.140)

Applying all transformations and their inverses, through the same steps as in (12.52)–(12.54), we get −λof t ¯ ≤ Υ4b Ω(0)e ¯ Ω(t) ,

(12.141)

where Υ4b is a positive constant, and    ˆ 2 ¯ = ˆ Ω(t) u(·, t)2 + ˆ v (·, t)2 + X(t)  + sˆ1 (t)2 + sˆ2 (t)2 + zˆ(t)2 + ˜ u(·, t)2 + ˜ v (·, t)2 2  ˜  + X(t)  + s˜1 (t)2 + s˜2 (t)2 + z˜(t)2 . Then recalling (12.66) and applying the Cauchy-Schwarz inequality, we obtain (12.128). 2) In order to prove the boundedness and exponential convergence of the outputfeedback controller (12.127), having proved the above exponential stability results in property (1), we need to additionally prove the exponential convergence of u ˆ(l(t), t) to zero. This result can be obtained with an exponential stability estimate in the vx (·, t), which can be obtained through the same steps as sense of ˆ ux (·, t) + ˆ lemma 12.1, with the aid of lemma 12.2 and theorem 12.3. Then, through the same steps as in theorem 12.2, we show that the output-feedback controller (12.127) is bounded and exponentially convergent to zero as well. The proof of theorem 12.4 is complete.

PDES WITH NONLINEARITIES

359 1 . |l (t)|

l(t)

1 0.5

0.05

0 0

2

4 6 Time (s)

8

10

0

0

2

(a) l(t).

4 6 Time (s) . (b) |l (t)|.

8

10

Figure 12.2. Moving boundary and its velocity. 12.5

SIMULATION

Consider the system given by ˙ X(t) = 0.4X(t) + v(0, t),

(12.142)

ut (x, t) = −ux (x, t) + 0.5v(x, t),

(12.143)

vt (x, t) = vx (x, t) + 0.5u(x, t),

(12.144)

u(0, t) = v(0, t) + X(t), v(l(t), t) = s1 (t),  l(t) s˙ 1 (t) = s2 (t) + s1 (t)2 + u(x, t)dx,

(12.145) (12.146)

0

s˙ 2 (t) = s1 (t)s2 (t) + u(l(t), t) + z(t),

(12.147)

z(t) ˙ = 0.5z(t) + u(l(t), t) + U (t),

(12.148)

for x ∈ [0, l(t)]. The time-varying function l(t) is known ahead of time and is decreasing from l(0) = 1 to 0.2 over a span of 10 s, as shown in figure 12.2. The initial values are given as u(x, 0) = 3 sin(4πx), v(x, 0) = 3 sin(4πx), X(0) = u(0, 0) − v(0, 0), s1 (0) = v(l(0), 0), s2 (0) = z(0) = 0. The initial values of the observer are given as u ˆ(x, 0) = u(x, 0) + 0.2 sin(2π(l(0) − x)), vˆ(x, 0) = v(x, 0) + 0.2 sin(2π(l(0) − x)), ˆ X(0) =u ˆ(0, 0) − vˆ(0, 0), sˆ1 (0) = vˆ(l(0), 0), sˆ2 (0) = s2 (0) + 0.5, ˆ = Z(0) + 0.5 Z(0) where the additional terms are the initial observer errors. The simulation is performed by the finite-difference method for the discretization in time and space after converting the time-varying PDE domain to a fixed PDE

CHAPTER TWELVE

360 7

||u(x, t)|| ||v(x, t)||

6 5 4 3 2 1 0

0

2

4

6

8

10

Time (s)

Figure 12.3. Open-loop responses of u(·, t) and v(·, t).

1.5 ||u(x, t)|| ||v(x, t)|| 1

0.5

0

0

2

4

6

8

10

Time (s)

Figure 12.4. Responses of u(·, t) and v(·, t) under the proposed output-feedback controller. x , and then the time step and space step are chosen as domain by introducing ξˇ = l(t) 0.001 and 0.02, respectively. The kernels (12.12)–(12.17), (12.89)–(12.92), (12.100), (12.101), used in the control input, are also solved by the finite-difference method. The control parameters are chosen as

c1 = 80, c2 = 150, a ¯0 = 350, κ = −10, L0 = 10, μ1 = μ2 = μ3 = 5. The simulation results are shown next. Comparing figure 12.3, which shows the open-loop responses of u(·, t), v(·, t), and figure 12.4, which gives the closed-loop responses of u(·, t), v(·, t), one can observe that in the latter case the convergence to zero is achieved, whereas the states grow unbounded in the former case. According to figure 12.5, we see that the responses of the z(t)-ODE, the nonlinear (s1 (t), s2 (t))-ODE, and the X(t)-ODE at the opposite boundary converge to zero under the proposed output-feedback controller. Moreover, figures 12.6 and 12.7 show that the proposed observer converges to the actual plant for both PDE and ODE states. Because v(l(t), t) and z(t) are the measurements, s˜1 (t) and z˜(t) are of small magnitude and converge to zero fast, so their plots are omitted here to avoid repetition. Figure 12.8 shows that the observer-based output-feedback control input is bounded and convergent to zero. We limited ourselves in this chapter to known nonlinearities, with no uncertainties. In future work, it might be of interest to extend the control design to a

PDES WITH NONLINEARITIES

361 0.2 s1(t)

z(t)

50 0

0 –0.2 –0.4

–50 0

2

4 6 Time (s)

8

0

10

2

4 6 Time (s)

2

8

10

0.2

0

X(t)

s2(t)

10

(b) s1(t).

(a) z(t).

–2 –4 –6

8

0.1 0 –0.1

0

2

4 6 Time (s)

8

10

0

2

4 6 Time (s) (d) X(t).

(c) s2(t).

Figure 12.5. Responses of ODE states z(t), s1 (t), s2 (t), X(t) under the proposed output-feedback controller.

0.2 ˜ t)|| ||u(x, ˜ t)|| ||v(x,

0.15 0.1 0.05 0 0

2

4

6

8

10

Time (s)

Figure 12.6. Observer errors of ˜ u(·, t), ˜ v (·, t).

0.1

0

X˜ (t)

s˜2(t)

0.2

–0.2

0.05 0

–0.4 0

2

4 6 Time (s) (a) ˜s2(t).

8

10

0

2

4 6 Time (s) (b) X˜ (t).

˜ Figure 12.7. Observer errors of s˜2 (t), X(t).

8

10

CHAPTER TWELVE

362 1000

U(t)

500 0 –500 –1000 0

2

4

6

8

10

Time (s)

Figure 12.8. Output-feedback control input.

more complicated and practical case with some system parameters unknown and an adaptive design developed. 12.6

APPENDIX

A. The expression of F F(β(l(t), β(0, t), α(l(t), t), α(0, t), β(x, t), α(x, t), X(t)) = h1 (l(t))β(l(t), t) + h2 (l(t))β(0, t) + h3 (l(t))α(l(t), t) + h4 (l(t))α(0, t)  l(t) h5 (l(t), y)β(y, t)dy + 

0

l(t)

+

h6 (l(t), y)α(y, t)dy + H7 (l(t))X(t),

0

(12.149)

where ˙ h1 (l(t)) = − p2 φ(l(t), l(t)) − l(t)φ(l(t), l(t)), h2 (l(t)) = p2 φ(l(t), 0) − γ(l(t))B, ˙ h3 (l(t)) = p1 ψ(l(t), l(t)) − l(t)ψ(l(t), l(t)), h4 (l(t)) = − p1 ψ(l(t), 0),  ˙ h5 (l(t), y) = p2 φ(l(t), l(t)) + l(t)φ(l(t), l(t)) M(l(t), y) ˙ + p2 φy (l(t), y) − c1 ψ(l(t), y) − l(t)φ x (l(t), y)  l(t)  p2 φy (l(t), z) − c1 ψ(l(t), z) −

(12.150) (12.151) (12.152) (12.153)

y

˙ − l(t)φ x (l(t), z) M(z, y)dz,

(12.154)

˙ h6 (l(t), y) = p1 ψy (l(t), y) + c2 φ(l(t), y) + l(t)ψ x (l(t), y)  ˙ − p2 φ(l(t), l(t)) + l(t)φ(l(t), l(t)) D(l(t), y)  l(t)  p2 φy (l(t), z) − c1 ψ(l(t), z) − y

˙ − l(t)φ x (l(t), z) D(z, y)dz,

(12.155)

PDES WITH NONLINEARITIES

363

 ˙ H7 (l(t)) = p2 φ(l(t), l(t)) + l(t)φ(l(t), l(t)) J (l(t))   ˙ (l(t)) − p2 φ(l(t), 0) − γ(l(t))B J (0) − γ(l(t))A − l(t)γ  l(t)  p2 φy (l(t), y) − c1 ψ(l(t), y) − 0

˙ − l(t)φ x (l(t), y) J (y)dy.

(12.156)

B. Calculation of V˙ 1 (t) Taking the time derivative of (12.45) along (12.20)–(12.23), we obtain 2 V˙ 1 (t) ≤ − λmin (Q1 )|X(t)| + 2X T P1 Bβ(0, t) p2 p2 + a1 eδ1 l(t) β(l(t), t)2 − a1 β(0, t)2 2 2 ˙ ˙ a1 l(t) δ1 l(t) b1 l(t) 2 2 e e−δ1 l(t) α(l(t), t) + β(l(t), t) + 2 2  l(t)  l(t) p2 p1 2 2 − δ1 a1 eδ1 x β(x, t) dx − δ1 b1 e−δ1 x α(x, t) dx 2 2 0 0 p1 p1 −δ1 l(t) 2 2 α(l(t), t) + b1 α(0, t) − b1 e 2 2   x  l(t) + b1 c 1 e−δ1 x α(x, t) β(x, t) − D(x, y)α(y, t)dy 0

 −

x 0

0



M(x, y)β(y, t)dy − J (x)X(t) dx.

(12.157)

Recalling (12.23), using Young’s inequality and the Cauchy-Schwarz inequality for the last part in (12.157) yields the existence of ξ > 0 such that   1 2 2 ˙ λmin (Q1 ) − p1 b1 |C0 | |X(t)| V1 (t) ≤ − 2   p2 4 |P1 B| a 1 − p 1 b1 q 2 − − β(0, t)2 2 λmin (Q1 )    l(t) p2 ξ 2 δ 1 a 1 − b1 ξ − b1 − β(x, t) dx 2 δ1 0    l(t) p 2 δ 1 b1 b1 ξ ξb21 2 − − b1 ξ e−δ1 L − − α(x, t) dx 2 δ1 λmin (Q1 ) 0 b1 −δ1 l(t) e α(l(t), t)2 2 a1 δ1 l(t) ˙ e + (p2 + l(t)) β(l(t), t)2 , 2 ˙ − (p1 − l(t))

where

 ξ = max

c1 ¯ ¯ LJ¯2 c2 , 1 c1 , c1 DL ¯ [D(1 + L) + M], 1 4 2 4

(12.158)  (12.159)

CHAPTER TWELVE

364 and ¯= D

max {|D(x, y)|},

(12.160)

0≤y≤x≤L

¯= M

max {|M(x, y)|},

(12.161)

0≤y≤x≤L

J¯ = max {|J (x)|}.

(12.162)

0≤x≤L

Choosing the parameters b1 , δ1 , a1 to satisfy 0 < b1
max 1, 2ξ + , p2 λmin (Q1 )   8 |P1 B| p1 2b1 ξ 2b1 ξ + 2q 2 b1 , + , a1 > max p2 λmin (Q1 ) p2 p2 δ1 p2 δ12

(12.163) (12.164) (12.165)

we arrive at (12.46), where 1 2 η1 = λmin (Q1 ) − p1 b1 |C0 | > 0, 2 p2 4 |P1 B| > 0, η 2 = a 1 − p 1 b1 q 2 − 2 λmin (Q1 ) ξ p2 η3 = δ1 a1 − b1 ξ − b1 > 0, 2 δ1   p 2 δ 1 b1 b1 ξ ξb21 − − b1 ξ e−δ1 L > 0. − η4 = 2 δ1 λmin (Q1 )

(12.166) (12.167) (12.168) (12.169)

C. Calculation of V˙ (t) Taking the time derivative of (12.47) and recalling (12.46), (12.40) and (12.37) with (12.149)–(12.156), we obtain 2 V˙ ≤ − η1 |X(t)| − η2 β(0, t)2 − η3

 − η4

l(t) 0



l(t) 0

2

β(x, t) dx

2

α(x, t) dx − η5 α(l(t), t)2 + η6 β(l(t), t)2

− c¯1 y1 (t)2 − c¯2 y2 (t)2 + y1 (t)f1  + y1 (t) h1 (l(t))β(l(t), t) + h2 (l(t))β(0, t) + h3 (l(t))α(l(t), t) + h4 (l(t))α(0, t)  l(t)  l(t) h5 (l(t), y)β(y, t)dy + h6 (l(t), y)α(y, t)dy + 0 0  c¯1 + H7 (l(t))X(t) + y2 (t)E(t) + y2 (t)f2 + y2 (t)y˙ 1 (t) c3 c2 y˙ 2 (t) + E(t)c3 y˙ 1 (t). − kE E(t)2 + E(t)¯

(12.170)

PDES WITH NONLINEARITIES

365

Recalling assumptions 12.1 and 12.2 and (12.29), (12.30), (12.32), we obtain  f12 ≤ γf 1 4y1 (t)2 + 4β(·, t)2 + 5α(·, t)2 + 4|X(t)|2 , (12.171)   2 2¯ c f22 ≤ γf 2 4 + 21 y1 (t)2 + 4β(·, t)2 + 4α(·, t)2 c3  + 4|X(t)|2 + 2y2 (t)2 + α(l(t), t)2 , (12.172) where γf 1 , γf 2 are positive constants associated with the kernels D, M, J and the Lipschitz constants. The omitted arguments of f1 , f2 are the same as those in (12.26), (12.27). Applying Young’s inequality and the Cauchy-Schwarz inequality into, respectively, the ninth and tenth terms, and y2 (t)E(t) + y2 (t)f2 + cc¯13 y2 (t)y˙ 1 (t) + E(t)¯ c2 y˙ 2 (t) + E(t)c3 y˙ 1 (t) in (12.170), where (12.30), (12.27), (12.32), (12.36) are used to rewrite y˙ 2 (t) as y˙ 2 (t) = f2 + E(t) − c¯2 y2 (t) − c3 y1 (t) +

c¯1 y˙ 1 (t), c3

using (12.171), (12.172) to replace the resulting f12 , f22 , and recalling (12.23), (12.29) to rewrite the resulting α(0, t)2 , β(l(t), t)2 , respectively, we get    2 ¯ 2 |C0 |2 − 4r8 γf 1 − 4r9 γf 2 − 4r11 γf 2 |X(t)|2 ¯ 7  − 2r4 h V˙ ≤ − η1 − r7 H 4  ¯ 2 r2 − 2r4 h ¯ 2 q 2 β(0, t)2 − η2 − h 2 4    l(t) 2 2 − η3 − r5 h5 max L − 4r8 γf 1 − 4r9 γf 2 − 4r11 γf 2 β(x, t) dx 



0 l(t)

2

α(x, t) dx − η4 − r6 h26 max L − 5r8 γf 1 − 4r9 γf 2 − 4r11 γf 2 0  ¯ 2 r3 − r9 γf 2 − r11 γf 2 α(l(t), t)2 − η5 − h 3  1 1 1 1 1 1 − c¯1 − 1 − − − − − − − η6 4r2 4r3 4r4 4r5 4r6 4r7      1 c21 2¯ c21 ¯ 1 − 4 + 2¯ − − 4r8 γf 1 − h γ − 4 + γ r r y1 (t)2 9 f 2 11 f 2 4r8 c23 c23   1 c¯2 3 − c¯2 − − 21 − − 2r11 γf 2 − 2r9 γf 2 y2 (t)2 4r9 4c3 r10 2   c¯2 c2 1 c¯4 c¯2 c¯2 c¯2 c2 − (kE − c¯2 ) − − 2 − 2 − 2 1 2 − 2 3 − 3 E(t)2 2 4 4r11 4r12 c3 4 4r13 + (r10 + r12 + r13 )y˙ 1 (t)2 ,

(12.173)

where r1 , . . . , r13 are positive constants from using Young’s inequality, h5 max = h6 max =

max

{|h5 (x, l(t))|} ,

(12.174)

max

{|h6 (x, l(t))|} ,

(12.175)

x∈[0,L],l(t)∈[0,L] x∈[0,L],l(t)∈[0,L]

¯1, h ¯2, h ¯ 3, h ¯ 4, H ¯ 7 are the maximum values of |h1 (l(t))|, |h2 (l(t))|, |h3 (l(t))|, and h |h4 (l(t))|, |H7 (l(t))| for l(t) ∈ [0, L] in (12.150)–(12.156).

CHAPTER TWELVE

366

According to (12.29), (12.30), (12.32) and (12.26) with (12.149)–(12.156), we get  y˙ 1 (t)2 ≤ ξ¯c c¯1 y1 (t)2 + y1 (t)2 + y2 (t)2 + f12 + β(0, t)2  + α(l(t), t)2 + α(0, t)2 + β(·, t)2 + α(·, t)2 + X(t)2 (12.176) for some positive constants ξ¯c depending on the kernels D, M, J and gains in (12.150)–(12.156). Inserting (12.176) into (12.173) to replace y˙ 1 (t)2 and using (12.171), we arrive at   2 ¯ 2 |C0 |2 − 4r8 γf 1 − 4r9 γf 2 ¯ 7  − 2r4 h ˙ V ≤ − η1 − r7 H 4 − 4r11 γf 2 − (r10 + r12 + r13 )ξ¯c − 2(r10 + r12 + r13 )ξ¯c |C0 |2   2 ¯ ¯ 2 r2 − 2r4 h ¯ 2 q2 − 4γf1 (r10 + r12 + r13 )ξc |X(t)| − η2 − h 2 4   − (1 + 2q 2 )(r10 + r12 + r13 )ξ¯c β(0, t)2 − η3 − r5 h25 max L − 4r8 γf 1 − 4r9 γf 2 − 4r11 γf 2 − (r10 + r12 + r13 )ξ¯c    l(t) 2 ¯ − 4γf1 (r10 + r12 + r13 )ξc β(x, t) dx − η4 − 5r8 γf 1 0

− r6 h26 max L − 4r9 γf 2 − 4r11 γf 2 − (r10 + r12 + r13 )ξ¯c  l(t)  2 ¯ 2 r3 − 5γf (r10 + r12 + r13 )ξ¯c α(x, t) dx − η5 − h 1

0



3

− r9 γf 2 − r11 γf 2 − (r10 + r12 + r13 )ξ¯c α(l(t), t)2  1 1 1 1 1 1 − c¯1 − 1 − − − − − − 4r2 4r3 4r4 4r5 4r6 4r7   1 2¯ c21 ¯ − η6 − − 4r8 γf 1 − h1 − 4 + 2 (r9 + r11 )γf 2 4r8 c3  − (1 + 4γf1 + c¯1 )(r10 + r12 + r13 )ξ¯c y1 (t)2  1 c¯2 3 − c¯2 − − 21 − − 2r11 γf 2 − 2r9 γf 2 4r9 4c3 r10 2   1 2 ¯ − (r10 + r12 + r13 )ξc y2 (t) − a ¯0 − c4 − c¯2 − 2  c¯22 c23 c¯21 c¯22 c¯22 c23 c¯42 − − − E(t)2 , − − 4 4r11 4r12 c23 4 4r13

(12.177)

where ¯ 0 − c4 kE = a is recalled from (12.41). Choosing small enough positive r2 , r3 , r4 , r5 , r6 , r7 , r8 , r9 , ¯0 large enough to satisfy r10 , r11 , r12 , r13 and making the control parameters c¯1 , c¯2 , a c¯1 >1 +

1 1 1 1 1 1 1 ¯1, + + + + + + η6 + + 4r8 γf 1 + h 4r2 4r3 4r4 4r5 4r6 4r7 4r8

(12.178)

PDES WITH NONLINEARITIES

367

1 c¯2 3 + 21 + + 2r11 γf 2 + 2r9 γf 2 + (r10 + r12 + r13 )ξ¯c , 4r9 4c3 r10 2 c¯2 c2 1 c¯4 c¯2 c¯2 c¯2 c2 a ¯0 >c4 + c¯2 + + 2 + 2 + 1 2 2 + 2 3 + 3 , 2 4 4r11 4r12 c3 4 4r13 c¯2 >

(12.179) (12.180)

we obtain (12.50) with ¯ 2 r2 − 2r4 h ¯ 2 q 2 − (1 + 2q 2 )(r10 + r12 + r13 )ξ¯c > 0, ηˆ0 = η2 − h 2 4 ¯ 2 r3 − r9 γf 2 − r11 γf 2 − (r10 + r12 + r13 )ξ¯c > 0. ηˆ1 = η5 − h 3

(12.181) (12.182)

D. Proof of lemma 12.1 Differentiating (12.21) and (12.22) with respect to x and differentiating (12.23) with respect to t, we obtain αxt (x, t) = −p1 αxx (x, t) + c1 βx (x, t) − c1 J  (x)X(t) − c1 D(x, x)α(x, t) − c1 M(x, x)β(x, t)  x  x − c1 Dx (x, y)α(y, t)dy − c1 Mx (x, y)β(y, t)dy, 0

(12.183)

0

βxt (x, t) = p2 βxx (x, t), p2 1 (C0 (A + Bκ) − c1 J (0)) X(t) αx (0, t) = −q βx (0, t) − p1 p1 1 − (C0 B − c1 )β(0, t). p1

(12.184)

(12.185)

Defining b2 A¯ = 2



l(t) 0

2

e−δ2 x αx (x, t) dx +

a2 2



l(t) 0

2

eδ2 x βx (x, t) dx,

(12.186)

where b2 is an arbitrary positive constant that can adjust the convergence rate. The positive constants δ2 , a2 will be chosen later. Taking the derivative of (12.186) along (12.183), (12.184), we obtain p1 p1 A¯˙ = − b2 e−δ2 l(t) αx (l(t), t)2 + b2 αx (0, t)2 2 2  l(t) ˙ b2 l(t) p1 2 e−δ2 l(t) αx (l(t), t)2 − b2 δ2 + e−δ2 x αx (x, t) dx 2 2 0 a2 δ2 l(t) p2 ˙ + (p2 + l(t)) e βx (l(t), t)2 − a2 βx (0, t)2 2 2  l(t)  l(t) p2 2 − a2 δ2 eδ2 x βx (x, t) dx − b2 e−δ2 x αx (x, t)c1 D(x, x)α(x, t)dx 2 0 0  l(t) − b2 e−δ2 x αx (x, t)c1 M(x, x)β(x, t)dx  −  −

0

l(t) 0

b2 e−δ2 x αx (x, t)c1

l(t) 0

b2 e

−δ2 x

 

αx (x, t)c1

x 0 x 0

Dx (x, y)α(y, t)dydx Mx (x, y)β(y, t)dydx

CHAPTER TWELVE

368 

l(t)

+  −

0 l(t) 0

b2 e−δ2 x αx (x, t)c1 βx (x, t)dx b2 e−δ2 x αx (x, t)c1 J  (x)X(t)dx.

(12.187)

Using Young’s inequality and the Cauchy-Schwarz inequality in the last six terms in (12.187) yields the existence of ξ2 > 0 such that  b p1 2 −δ2 l(t) ˙ ¯˙ ≤ − p1 − l(t) A(t) e αx (l(t), t)2 + b2 αx (0, t)2 2   a2 p2 2 δ2 l(t) ˙ e + p2 + l(t) βx (l(t), t)2 − a2 βx (0, t)2 2 2    l(t) p1 2ξ2 b2 2 b2 δ2 − 4ξ2 b2 − − e−δ2 x αx (x, t) dx 2 δ2 0    l(t) p2 2 a 2 δ 2 − ξ 2 b2 eδ2 x βx (x, t) dx − 2 0    l(t) ξ 2 b2 2 2 e−δ2 x α(x, t) dx + ξ2 b2 |X(t)| + ξ 2 b2 + δ2 0    l(t) ξ 2 b2 2 + ξ 2 b2 + eδ2 x β(x, t) dx. δ2 0

(12.188)

The function αx (0, t)2 in (12.188) can be replaced by p22 2 3 q βx (0, t)2 + 2 (C0 B − c1 )2 β(0, t)2 p21 p1 3 2 + 2 |C0 (A + Bκ) + c1 J (0)| |X(t)|2 p1

αx (0, t)2 ≤ 3

(12.189)

using the Cauchy-Schwarz inequality on (12.185). Recalling (12.22), (12.26) with (12.149)–(12.156), (12.29), (12.30), (12.32), (12.171), using the Cauchy-Schwarz inequality  (for the integrals appearing in (12.26), (12.149)), the positive term  ˙ p2 + l(t) a22 eδ2 l(t) βx (l(t), t)2 in (12.188) can be replaced as a2 δ2 l(t) e βx (l(t), t)2 2 ≤ ξ¯2 y2 (t)2 + ξ¯3 y1 (t)2 + ξ¯4 β(0, t) + ξ¯5 α(l(t), t)2 + ξ¯6 α(0, t)2

˙ (p2 + l(t))

+ ξ¯7 α(·, t)2 + ξ¯8 β(·, t)2 + ξ¯9 |X(t)|2

(12.190)

for some positive ξ¯i , i = 2, . . . , 9. We propose a Lyapunov function ¯ + R1 V (t) V2 (t) = A(t)

(12.191)

and define the norm 2

2

2

Ω2 (t) = βx (·, t) + αx (·, t) + β(·, t) 2

2

+ α(·, t) + |X(t)| + y1 (t)2 + y2 (t)2 + E(t)2

(12.192)

PDES WITH NONLINEARITIES

369

so that θ2a Ω2 (t) ≤ V2 (t) ≤ θ2b Ω2 (t)

(12.193)

for some positive θ2a and θ2b . Taking the derivative of (12.191) and recalling (12.188)–(12.190), (12.50), we then get V˙ 2 (t) = A¯˙ + R1 V˙  b 2 −δ2 l(t) ˙ e ≤ − p1 − l(t) αx (l(t), t)2 2   p2 3p2 q 2 a2 − 2 − βx (0, t)2 2 2b2 p1    l(t) p1 2ξ2 b2 2 b2 δ2 − 4ξ2 b2 − − e−δ2 x αx (x, t) dx 2 δ2 0    l(t) p2 R1 2 a 2 δ 2 − ξ 2 b2 λV (t) eδ2 x βx (x, t) dx − − 2 2 0   3b2 − R1 ηˆ0 − (C0 B − c1 )2 − ξ¯4 − 2ξ¯6 q 2 β(0, t)2 2p1     R1 R1 2 ¯ ¯ θ1a λ − ξ2 y2 (t) − θ1a λ − ξ3 y1 (t)2 − 2 2    l(t) R1 ξ 2 b2 ¯ 2 θ1a λ − ξ2 b2 − − − ξ7 α(x, t) dx 2 δ2 0    l(t) R1 b ξ 2 2 2 δ2 L δ2 L ¯ θ1a λ − ξ2 b2 e − e − ξ8 β(x, t) dx − 2 δ2 0  R1 3b2 θ1a λ − ξ2 b2 − |c1 J (0) + C0 (A + Bκ)|2 − 2 2p1  2 2 ¯ ¯ − ξ9 − 2ξ6 |C0 | |X(t)| − (R1 ηˆ1 − ξ¯5 )α(l(t), t)2  ≤ − σ1 V2 (t) − R1 ηˆ1 − ξ¯5 α(l(t), t)2  b 2 −δ2 l(t) ˙ e − p1 − l(t) αx (l(t), t)2 2 − ηˆ2 β(0, t)2 − ηˆ3 βx (0, t)2 (12.194) for some positive σ1 , and ηˆ2 = R1 ηˆ0 −

3b2 (C0 B − c1 )2 − ξ¯4 − 2ξ¯6 q 2 > 0, 2p1 p 3p2 q 2 ηˆ3 = a2 − 2 > 0, 2 2b2 p1 R1 ηˆ1 − ξ¯5 > 0

by choosing     2ξ2 b2 3p2 q 2 12ξ2 δ2 > max 1, , , a2 > max p1 p 2 δ 2 b2 p 1

(12.195)

CHAPTER TWELVE

370

˙ > 0 by recalling assumption 12.4, and sufficiently large R1 . According to p1 − l(t) we thus have V˙ 2 (t) ≤ −σ1 V2 (t). It then follows that

V2 (t) ≤ V2 (0)e−σ1 t .

Recalling (12.193), we obtain Ω2 (t) ≤

θ2b Ω2 (0)e−σ1 t . θ2a

(12.196)

Differentiating (12.18), (12.19) with respect to x, we get ux (x, t) = αx (x, t),



(12.197) x

vx (x, t) = βx (x, t) − Dx (x, y)α(y, t)dy 0  x Mx (x, y)β(y, t)dy − J  (x)X(t) − 0

− D(x, x)α(x, t) − M(x, x)β(x, t).

(12.198)

Similarly differentiating (12.10), (12.11) with respect to x, together with (12.197), (12.198), using (12.10), (12.11), (12.18), (12.19), (12.29), (12.30), and (12.36), we obtain θ¯2a Ξ1 (t) ≤ Ω2 (t) ≤ θ¯2b Ξ1 (t) for some positive θ¯2a , θ¯2b , where Ξ1 (t) is defined as Ξ1 (t) = Ξ(t) + ux (·, t)2 + vx (·, t)2 . Therefore, we get θ¯2b θ2b ux (·, t)2 + vx (·, t)2 ≤ Ξ1 (t) ≤ ¯ Ξ1 (0)e−σ1 t . θ2a θ2a Thus, (12.56) is obtained with θ¯2b θ2b , Υ1a = ¯ θ2a θ2a

λ1a = σ1 .

(12.199)

The proof of lemma 12.1 is complete. E. Proof of theorem 12.2 Applying the Cauchy-Schwarz inequality in (12.42), we get  |U (t)|2 ≤ ξd |s1 (t)|2 + |s2 (t)|2 + |z(t)|2 + |X(t)|2  + |u(l(t), t)|2 + u(·, t)2 + v(·, t)2 for some positive ξd .

(12.200)

PDES WITH NONLINEARITIES

371

Applying the Cauchy-Schwarz inequality and recalling (12.4), (12.5), we obtain √ |u(l(t), t)| ≤ |u(0, t)| + Lux (·, t) √ ≤ |qv(0, t)| + |CX(t)| + Lux (·, t) √ ≤ |q||s1 (t)| + |q| Lvx (·, t) √ (12.201) + |C||X(t)| + Lux (·, t). According to (12.200), (12.201), using theorem 12.1 and lemma 12.1, we know that the control input (12.42) is bounded by  1 |U (t)| ≤ Υ2 Ξ(0) + ux (·, 0)2 + vx (·, 0)2 2 e−λ2 t .

(12.202)

Then (12.57) is obtained by recalling (12.52). The proof of theorem 12.2 is complete. F. Calculation of V˙ s (t) Taking the derivative of (12.120) along (12.110), we obtain  ˜ T (As − BC2 )T P0 + P0 (As − BC2 ) S(t) ˜ V˙ s (t) ≤ S(t) ˜ T P0 f˜, + 2S(t) where

(12.203)

T f˜ = f˜1 , f˜2 .

Recalling (12.75), (12.76), according to assumption 12.2, we get        l(t) l(t)  2 2 f˜1 = f1 s1 , u(x, t)dx − f1 sˆ1 , u ˆ(x, t)dx  0

0

 2  l(t)   2 2 2 ≤ γ1 |s1 − sˆ1 | + γ1  (u(x, t) − u ˆ(x, t))dx  0  ≤ γ12 s˜21 + γˆ12 ˜ α(·, t)2 ,

(12.204)

f˜22 = |f2 (s1 , s2 , u(l(t), t)) − f2 (ˆ s1 , sˆ2 , u ˆ(l(t), t))|2 ≤ γ22 |(s1 , s2 , u(l(t), t)) − (ˆ s1 , sˆ2 , u ˆ(l(t), t))|2 ≤ γ22 s˜21 + γ22 s˜22 + γ22 α ˜ (l(t), t)2 ,

(12.205)

where γˆ1 is a positive constant and (12.80) is used. Then, f˜2 = f˜12 + f˜22 ≤ (γ12 + γ22 )˜ s1 (t)2 + γ22 s˜2 (t)2 + γˆ12 ˜ α(·, t)2 + γ22 α ˜ (l(t), t)2 2  ˜  ≤ (γ12 + 2γ22 ) S(t) α(·, t)2 + γ22 α ˜ (l(t), t)2 .  + γˆ12 ˜ Thus, we obtain 2  ˜ T P0 f˜(t) ≤ (γ12 + 2γ22 ) P0 S(t) ˜  + 2S(t)

1 |f˜(t)|2 γ12 + 2γ22

(12.206)

Note:

√

9 10 11 12

First-order and scalar First-order and scalar A chain of integrators det(C1 B) = 0 det(C0 B0 ) = 0 B0 being invertible Left invertible A chain of integrators Left invertible Left invertible Nonlinear

Transport PDE 2 × 2 coupled transport PDEs Heat PDE n coupled parabolic PDEs n coupled transport PDEs 2 × 2 coupled transport PDEs 2 × 2 coupled transport PDEs 2 × 2 coupled transport PDEs 2 × 2 coupled transport PDEs 2 × 2 coupled transport PDEs 2 × 2 coupled transport PDEs

Types of PDEs

denotes “included,” and × denotes “not included.”

[8] [49] [179] [42] [43] [151] [152] Chapter Chapter Chapter Chapter

Types of ODEs of actuation Output-feedback Output-feedback Output-feedback Output-feedback Output-feedback State-feedback State-feedback Output-feedback Output-feedback Output-feedback Output-feedback

Types of control systems

× ×

× × × × × × × × √

Delay compensation

Table 12.1. Comparison of results on the boundary control of sandwich systems

Continuous-in-time Continuous-in-time Continuous-in-time Continuous-in-time Continuous-in-time Continuous-in-time Continuous-in-time Continuous-in-time Continuous-in-time Piecewise-constant Continuous-in-time

Types of control signals

PDES WITH NONLINEARITIES

373

     ˜ 2  ˜ 2 ≤ (γ12 + 2γ22 ) P0 S(t)  + S(t) +

γˆ12 γ22 2 ˜ α (·, t) + α ˜ (l(t), t)2 γ12 + 2γ22 γ12 + 2γ22

˜ + S(t) ˜ T S(t) ˜ ˜ T P0T P0 S(t) = (γ12 + 2γ22 )S(t) +

γ12

γˆ12 γ2 ˜ α(·, t)2 + 2 2 2 α ˜ (l(t), t)2 , 2 + 2γ2 γ1 + 2γ2

(12.207)

where Young’s inequality is used. Substituting (12.207) into (12.203) yields  ˜ T (A¯ − BC2 )T P0 + P0 (A¯ − BC2 ) S(t) ˜ V˙ s (t) ≤ S(t) ˜ + S(t) ˜ T S(t) ˜ + (γ12 + 2γ22 )S˜T (t)P0T P0 S(t) γˆ12 γ2 ˜ α(·, t)2 + 2 2 2 α ˜ (l(t), t)2 2 + 2γ2 γ1 + 2γ2  T ˜ ≤ S(t) (A¯ − BC2 )T P0 + P0 (A¯ − BC2 ) +

γ12

+ (γ12 +

γ12

+ 2γ22 )P0T P0

 +I

T

˜ S(t)

γˆ12 γ2 ˜ α(·, t)2 + 2 2 2 α ˜ (l(t), t)2 . 2 + 2γ2 γ1 + 2γ2

(12.208)

Recalling (12.121), we arrive at (12.123).

12.7

NOTES

Comparisons of the results on the boundary control of sandwich systems are summarized in table 12.1.

Part III

Adaptive Control of Hyperbolic PDE-ODE Systems

Chapter Thirteen Adaptive Event-Triggered Control of Hyperbolic PDEs

The adaptive control designs in chapters 5 and 8 employed continuous-in-time approaches. In this part, however, we present triggered-type adaptive control designs for hyperbolic partial differential equation-ordinary differential equation (ODE) systems, with the purpose of alleviating the adaptive learning transient and making the adaptive control laws more user-friendly in string-actuated mechanisms with massive actuators. In this chapter, we extend the continuous-in-time adaptive controller in chapter 5 to an event-triggered form. Compared with the nonadaptive event-triggered control design in chapter 11, the adaptive update law in this chapter needs an appropriately modified event-triggering mechanism design to guarantee the absence of a Zeno phenomenon. The content of this chapter is organized as follows. The problem formulation is shown in section 13.1. An observer is designed to estimate the PDE states in section 13.2. Using the approach in chapter 5, the design of a continuous-in-time adaptive observer-based backstepping controller is presented in section 13.3, followed by an observer-based event-triggering mechanism in section 13.4, where the existence of a minimal dwell time is proved. The asymptotic stability of the overall adaptive event-based output-feedback closed-loop system is proved via Lyapunov analysis in section 13.5. In a numerical simulation in section 13.6, the proposed adaptive event-triggered controller is tested in the lateral vibration suppression of a mining cable elevator with a viscoelastic guideway whose stiffness and damping coefficients are unknown.

13.1

PROBLEM FORMULATION

The class of plants considered in this chapter is ˙ X(t) = AX(t) + Bw(0, t),

(13.1)

z(0, t) = CX(t) + p1 w(0, t),

(13.2)

zt (x, t) = −q1 (x)zx (x, t) + c1 (x)z(x, t) + c2 (x)w(x, t),

(13.3)

wt (x, t) = q2 (x)wx (x, t) + c3 (x)z(x, t) + c4 (x)w(x, t),

(13.4)

w(l(t), t) = U (t),

(13.5)

with x ∈ [0, l(t)], t ∈ [0, ∞). The vector X(t) ∈ Rn is an ODE state, and the scalars z(x, t), w(x, t) are PDE states. Equation (13.5) is the boundary condition with control. The spatially varying transport speeds q1 (x), q2 (x) ∈ C 1 are positive, and c1 (x), c2 (x), c3 (x), c4 (x) ∈ C 0 are arbitrary. The constant p1 is nonzero. The matrix

CHAPTER THIRTEEN

378

C ∈ R1×n is arbitrary. The input matrix B ∈ Rn×1 and the system matrix A ∈ Rn×n with unknown parameters satisfy the following assumptions. Assumption 13.1. The ⎛ 0 ⎜ 0 ⎜ ⎜ A=⎜ ⎜ ⎝ 0 g1

matrices A, B are in the form ⎞ ⎛ 1 0 0 ··· 0 0 1 0 ··· 0 ⎟ ⎜ ⎟ ⎜ ⎟ .. ⎜ , B = ⎟ . ⎜ ⎟ ⎝ 0 0 0 ··· 1 ⎠ g2 g3 · · · gn−1 gn

0 0 0 0 hn

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

(13.6)

where the constants g1 , g2 , g3 , . . . , gn−1 , gn are unknown and arbitrary, and their lower and upper bounds are known and arbitrary. The constant hn is nonzero and known. Assumption 13.1 indicates that the ODE is in the controllable form, which covers many practical models, including the payload dynamics in the string-actuated mechanisms. The known target matrix ⎞ ⎛ 0 1 0 0 ··· 0 ⎜ 0 0 1 0 ··· 0 ⎟ ⎟ ⎜ ⎟ ⎜ . .. (13.7) Am = ⎜ ⎟ ⎟ ⎜ ⎠ ⎝ 0 0 0 0 ··· 1 g¯1 g¯2 g¯3 · · · g¯n−1 g¯n indicates the coefficients g¯1 , g¯2 , g¯3 , . . . , g¯n−1 , g¯n , which are chosen by the user to make Am Hurwitz and to achieve a desired degree of performance for the specific application. According to assumption 13.1 and (13.7), we know that there exists a unique, though unknown, row vector K1×n = [k1 , . . . , kn ]

(13.8)

Am = A + BK

(13.9)

g¯i = gi + hn ki , i = 1, 2, . . . , n.

(13.10)

such that

and

By virtue of (13.10), while the ki ’s are unknown, the lower and upper bounds on the ki ’s—that is, [k i , k¯i ], i = 1, 2, . . . , n—are known because the lower and upper bounds of the system matrix coefficients gi ’s are known in assumption 13.1, and the target matrix coefficients g¯i ’s are chosen by the user. The time-varying domain—that is, the moving boundary l(t)—is under the following two assumptions. Assumption 13.2. The function l(t) is uniformly bounded—that is, 0 < l(t) ≤ L, ∀t ≥ 0, where L is a positive constant.

ADAPTIVE EVENT-TRIGGERED CONTROL

˙ is bounded as Assumption 13.3. The function l(t)   ˙  l(t) ≤ vm < min {q1 (x), q2 (x)}, 0≤x≤L

379

(13.11)

where vm is the maximum velocity of the moving boundary. The limit of the speed of the moving boundary in assumption 13.3 is to ensure the well-posedness of the plant (13.1)–(13.5) according to [71, 72]. The complete set of the plant parameters is given by ζp = {p1 , q1 , q2 , c1 , c2 , c3 , c4 , A, B, L, vm },

(13.12)

where x is omitted for conciseness.

13.2

OBSERVER

To estimate the PDE states z(x, t), w(x, t), which usually cannot be fully measured in practice but are employed in the controller, an observer using the measurements X(t), z(l(t), t) is formulated as ˙ X(t) = AX(t) + B w(0, ˆ t) + B w(0, ˜ t), ˆ t), zˆ(0, t) = CX(t) + p1 w(0,

(13.13) (13.14)

zx (x, t) + c1 (x)ˆ z (x, t) + c2 (x)w(x, ˆ t) zˆt (x, t) = − q1 (x)ˆ + Ψ2 (x, l(t))(z(l(t), t) − zˆ(l(t), t)),

(13.15)

ˆx (x, t) + c3 (x)ˆ z (x, t) + c4 (x)w(x, ˆ t) w ˆt (x, t) = q2 (x)w + Ψ3 (x, l(t))(z(l(t), t) − zˆ(l(t), t)), w(l(t), ˆ t) = U (t),

(13.16) (13.17)

where (13.13) is exactly the ODE (13.1) with w(0, t) = w(0, ˆ t) + w(0, ˜ t), providing the measured signal X(t) into (13.14). It should be noted that (13.13) is not computed online as a part of the observer. The observer gains Ψ2 (x, l(t)), Ψ3 (x, l(t)) are to be determined. As we indicate, because the ODE state X is available, the observer (13.14)– (13.17) is only to estimate the PDE states. Let us denote the observer error states as (˜ z (x, t), w(x, ˜ t)) = (z(x, t), w(x, t)) − (ˆ z (x, t), w(x, ˆ t)).

(13.18)

According to (13.2)–(13.5) and (13.14)–(13.17), the observer error system is obtained as z˜(0, t) = p1 w(0, ˜ t),

(13.19)

zx (x, t) + c1 (x)˜ z (x, t) + c2 (x)w(x, ˜ t) z˜t (x, t) = −q1 (x)˜ z (l(t), t), − Ψ2 (x, l(t))˜

(13.20)

CHAPTER THIRTEEN

380

w ˜t (x, t) = q2 (x)w ˜x (x, t) + c3 (x)˜ z (x, t) + c4 (x)w(x, ˜ t) − Ψ3 (x, l(t))˜ z (l(t), t),

(13.21)

w(l(t), ˜ t) = 0.

(13.22)

The observer gains Ψ2 (x, l(t)), Ψ3 (x, l(t)) are to be designed to ensure convergence to zero of the observer errors (13.18). Next, we postulate the backstepping transformation l(t) ¯ y)˜ z˜(x, t) = α ˜ (x, t) − φ(x, α(y, t)dy −

x l(t)

ˇ y)β(y, ˜ t)dy, φ(x,

x



l(t)

˜ t) − w(x, ˜ t) = β(x, −

(13.23)

¯ y)˜ ψ(x, α(y, t)dy

x l(t)

ˇ y)β(y, ˜ t)dy ψ(x,

(13.24)

x

to convert the original observer error system (13.19)–(13.22) to the following target observer error systems: ˜ t), α ˜ (0, t) = p1 β(0,

(13.25)

αx (x, t) + c1 (x)˜ α(x, t), α ˜ t (x, t) = −q1 (x)˜

(13.26)

˜ t), β˜t (x, t) = q2 (x)β˜x (x, t) + c4 (x)β(x,

(13.27)

˜ β(l(t), t) = 0.

(13.28)

Even though the integration interval [0, l(t)] is time-varying, the kernels in (13.23), ˙ (13.24) need not include the argument l(t) because the extra terms in which l(t), l(t) appear in the course of calculating the kernel conditions will be “absorbed” by the observer gains Ψ2 (x, l(t)), Ψ3 (x, l(t)). By matching (13.19)–(13.22) and (13.25)– ¯ y), (13.28) via (13.23), (13.24), we obtain PDE conditions on the kernels φ(x, ˇ y), ψ(x, ¯ y), ψ(x, ˇ y). These PDEs are well-posed because they belong to a genφ(x, eral class of kernel equations whose well-posedness is proved in theorem 3.2 of [48]. The observer gains are then deduced as ˙ φ(x, ¯ l(t)) − q1 (l(t))φ(x, ¯ l(t)), Ψ2 (x, l(t)) = l(t)

(13.29)

˙ ψ(x, ¯ l(t)) − q1 (l(t))ψ(x, ¯ l(t)). Ψ3 (x, l(t)) = l(t)

(13.30)

The detailed calculations of matching (13.19)–(13.22) and (13.25)–(13.28) via ¯ y), φ(x, ˇ y), ψ(x, ¯ y), (13.23), (13.24) and of deriving the conditions on the kernels φ(x, ˇ ψ(x, y) are shown in appendix 5.7A in chapter 5. Lemma 13.1. For the observer error system (13.19)–(13.22), the state estimation errors z˜(x, t), w(x, ˜ t) become zero after tf =

L min0≤x≤L {q1 (x)}

+

L min0≤x≤L {q2 (x)}

.

ADAPTIVE EVENT-TRIGGERED CONTROL

381

Proof. According to the target observer error system (13.25)–(13.28) and the result ˜ t) become zero after a finite time tf . Applying in [96], we know that α ˜ (x, t), β(x, the Cauchy-Schwarz inequality into (13.23), (13.24), the proof of this lemma is complete. The finite time in which lemma 13.1 establishes that the observer errors vanish depends only on the plant parameters and not on the controller parameters. In the next section, we first design a continuous-in-time adaptive control law to stabilize the coupled transport PDEs coupled with a highly uncertain ODE at the uncontrolled boundary. Then we design an event-triggering mechanism, which uses the signals from the observer and includes an internal dynamic variable and which produces triggering times based on evaluating the size of the deviation of the control input applied over the interval between the triggers from the continuousin-time control signal. The combined continuous-in-time adaptive controller and the event-triggering mechanism constitute the adaptive event-triggered boundary controller.

13.3

ADAPTIVE CONTINUOUS-IN-TIME CONTROL DESIGN

In this section, we conduct a state-feedback control backstepping design, with the intent of feeding into this full-state design the observer states from the observer in the previous section. In other words, we will design an observer-based outputfeedback controller that we then make adaptive. The output error injections z˜(l(t), t), w(0, ˜ t) in the observer are regarded as zero in the state-feedback design, and then the separation principle, which is verified by the fact that the stability of the observer error system is independent of the control design according to lemma 13.1, is applied in the stability analysis of the resulting closed-loop system. Backstepping Two transformations are employed to convert (13.13)–(13.17) to a target system, with the purposes of removing the couplings in the PDE domain and of making the ODE system matrix Hurwitz. a) The first transformation to decouple PDEs We postulate the backstepping transformation x x J(x, y)ˆ z (y, t)dy − G(x, y)w(y, ˆ t)dy, α ˆ (x, t) = zˆ(x, t) − 0 0 x x ˆ t) = w(x, F (x, y)ˆ z (y, t)dy − N (x, y)w(y, ˆ t)dy β(x, ˆ t) − 0

(13.31) (13.32)

0

to convert (13.13)–(13.17) to the following system: ˆ t), ˙ X(t) = AX(t) + B β(0, ˆ t), α ˆ (0, t) = CX(t) + p1 β(0, αx (x, t) + c1 (x)ˆ α(x, t) − J(x, 0)q1 (0)CX(t), α ˆ t (x, t) = −q1 (x)ˆ

(13.33) (13.34) (13.35)

CHAPTER THIRTEEN

382

ˆ t) − F (x, 0)q1 (0)CX(t), βˆt (x, t) = q2 (x)βˆx (x, t) + c4 (x)β(x, l(t) ˆ F (l(t), y)ˆ z (y, t)dy β(l(t), t) = U (t) − 0

−

(13.36)

l(t)

N (l(t), y)w(y, ˆ t)dy.

0

(13.37)

By matching (13.33)–(13.37) and (13.13)–(13.17) via (13.31), (13.32) through a process similar to that shown in appendix 5.7B in chapter 5, we obtain the conditions of the kernels J(x, y), G(x, y), F (x, y), N (x, y), which is a special case of (6.87)–(6.100) in chapter 6, where (6.88) and (6.95) hold here because the PDE states are scalar in this chapter. The well-posedness proof can be found in lemma 6.1 in chapter 6. b) The second transformation to form a stable ODE We postulate the backstepping transformation x ˆ t) − ˆ t)dy − D(x; K(t))X(t), ˆ (x, y; K(t)) ˆ ˆ ηˆ(x, t) = β(x, N β(y,

(13.38)

0

ˆ ∈ R1×n is the estimate of the ideal control gains and will be shown later. where K(t) ˆ (x, y; K(t)), ˆ ˆ The conditions on the kernels N D(x; K(t)) are to be determined next. The inverse transformation is postulated as x ˆ t) = ηˆ(x, t) − ˆI (x, y; K(t))ˆ ˆ ˆ N β(x, η (y, t)dy − DI (x; K(t))X(t), (13.39) 0

ˆ ˆ ˆI (x, y; K(t)), DI (x; K(t)) are kernels which can be determined after the where N ˆ ˆ ˆ determination of N (x, y; K(t)), D(x; K(t)). Through the transformation (13.38), we convert (13.33)–(13.37) into the following target system: ˜ ˙ + B ηˆ(0, t), X(t) = Am X(t) − B K(t)X(t) ˆ + p1 ηˆ(0, t), α(0, ˆ t) = (C + p1 D(0; K(t)))X(t) α ˆ t (x, t) = − q1 (x)ˆ αx (x, t) + c1 (x)ˆ α(x, t) − J(x, 0)q1 (0)CX(t), ˆ˙ ηx (x, t) + c4 (x)ˆ η (x, t) − K(t)R(x, t) ηˆt (x, t) = q2 (x)ˆ 

ˆ ˆ ˜ − K(t)D ˆ˙ (x; K(t)) X(t), + D(x; K(t))B K(t) ˆ K(t) ηˆ(l(t), t) = 0,

(13.40) (13.41) (13.42)

(13.43) (13.44)

where ˜ = K − K(t), ˆ K(t) and



x

(13.45)

ˆ t)dy ˆ ˆ (x, y; K(t)) ˆ N β(y, K(t) x y ˆ ˆI (y, σ; K(t))ˆ ˆ ˆ NK(t) N = (x, y; K(t)) ηˆ(y, t) − η (σ, t)dσ ˆ 0 0

ˆ dy. (13.46) − DI (y; K(t))X(t)

R(x, t) =

0

ADAPTIVE EVENT-TRIGGERED CONTROL

383

The partial derivatives appearing in (13.43)–(13.46), respectively, are ˆ ∂D(x; K(t)) ˆ DK(t) (x; K(t)) = ˆ ˆ ∂ K(t) and

ˆ (x, y; K(t)) ˆ ∂N ˆ ˆ (x, y; K(t)) ˆ N = . K(t) ˆ ∂ K(t)

By matching (13.33)–(13.37) and (13.40)–(13.44) via (13.38) (see part C in appendix ˆ ˆ 5.7 for the details), the conditions of the kernels N (x, y; K(t)), D(x; K(t)) in (13.38) are determined as ˆ ˆ D(0; K(t)) = K(t), ˆ ˆ ˆ + D(x; K(t))(A −q2 (x)D (x; K(t)) m − c4 (x)In − B K(t)) x ˆ (x, y; K(t))F ˆ N (y, 0)q1 (0)Cdy = 0, + F (x, 0)q1 (0)C −

(13.47)

(13.48)

0

ˆy (x, y; K(t)) ˆ q2 (y)N ˆx (x, y; K(t)) ˆ ˆ (x, y; K(t)) ˆ + q2  (y)N = 0, + q2 (x)N

(13.49) ˆ ˆ ˆ q2 (0)N (x, 0; K(t)) = D(x; K(t))B, (13.50)

where In is an identity matrix with dimension n. The equation set (13.47)–(13.50) is a transport PDE-ODE coupled system consisting of the transport PDE (13.49) with the boundary condition (13.50) on {(x, y)|0 ≤ y ≤ x ≤ l(t)} and the ODE (13.48) ˆ with the initial value (13.47) on {0 ≤ x ≤ l(t)}. It should be noted that K(t) is a parameter rather than a variable in the transport PDE (13.49), (13.50) with respect to the independent variables x, y and in the ODE (13.47), (13.48) with respect to the independent variable x. In the study of the well-posedness of (13.47)–(13.50), ˆ (x, y; K(t)) ˆ the transport PDE state N can be represented by its boundary value ˆ ˆ (x, y; K(t)), ˆ D(x; K(t))B. Substituting the result into ODE (13.48) to replace N ˆ ˆ the unique and continuous solution of the first-order ODE D(x; K(t)) (13.48) can be obtained. Then the unique and continuous solution of the transport PDE ˆ (x, y; K(t)) ˆ N in (13.49), (13.50) is obtained because of the well-defined and continuous input signal in (13.50). Following section 9.2 in chapter 9, the kernels ˆI (x, y; K(t)), ˆ ˆ N DI (x; K(t)) in the inverse transformation can then be determined. Adaptive Update Laws The objective in this section is to build adaptive update laws to obtain self-tuning of ˆ = [kˆ1 (t), . . . , kˆn (t)], where normalization and projection operthe control gains K(t) ators are used to guarantee boundedness, as is typical in adaptive control designs. ˆ = [kˆ1 , . . . , kˆn ] is of the form The adaptive update law K(t)

 ˙ kˆi (t) = Proj[ki ,k¯i ] τi (t), kˆi (t) . (13.51) While projection is applicable for arbitrary convex sets, the set within which the control gain vector K should reside in a hyperrectangle or, as is colloquially said, ˆ the estimate K(t) should be maintained within box constraints. Given the hyperrectangular set for the feedback gains, for any m ≤ M and any r, p, Proj[m,M ] is

CHAPTER THIRTEEN

384 defined as the operator given by ⎧ ⎨0, Proj[m,M ] (r, p) = 0, ⎩ r,

if p = m and r < 0, if p = M and r > 0, else.

So the projection operator is to keep the scalar components of parameter estimate ˆ = [kˆ1 , . . . , kˆn ] bounded within the interval [k , k¯i ]. The bounds k and k¯i vector K(t) i i are determined from the bounds on the unknown parameters in A using assumption 13.1, as well as (13.7) and (13.10). We choose the parameter update rate functions τi in (13.51) as Γc [τ1 (t), . . . , τn (t)]T = − 2X(t)B T P X(t) 1 + Ω(t) − μm md (t)

l(t) T ˆ eδx ηˆ(x, t)X(t)B T D(x; K(t)) dx , (13.52) + ra 0

where md (t) < 0, a dynamic variable in the event-triggering mechanism, will be defined in the next section, and the adaptation gain matrix is Γc = diag{γc1 , . . . , γcn },

(13.53)

and where Ω(t) is defined as l(t) 1 Ω(t) = X(t) P X(t) + ra eδx ηˆ(x, t)2 dx 2 0 l(t) 1 + rb e−δx α ˆ (x, t)2 dx. 2 0 T

(13.54)

The determination of the positive constants δ, ra , and rb will be shown in the next section. The matrix P = P T > 0 is the unique solution to the Lyapunov equation P Am + ATm P = −Q

(13.55)

for some Q = QT > 0. It should be noted that P is known since Am is known (chosen by the user). We introduce the normalization Ω(t) + 1 in the denominator in (13.52) ˆ˙ in order to keep the rate of change of the parameter estimate K(t) bounded, which will be used in the following ETM design and stability analysis. The functions ηˆ(x, t) and α ˆ (x, t) in (13.52)–(13.54) can be represented by the observer states through (13.31), (13.32), (13.38). The complete set of positive design parameters in the parameter update law is defined as ζa = {Γc , δ, ra , rb , μm }.

(13.56)

The update law designs in this section will be chosen with the help of a Lyapunov analysis in section 13.5. Continuous-in-Time Control Law The continuous-in-time adaptive backstepping control law is derived in this section. For (13.44) to hold, using (13.37), recalling (13.38) and (13.32), we get

ADAPTIVE EVENT-TRIGGERED CONTROL

U (t) =

l(t) 0



¯ (l(t), x; K(t))ˆ ˆ ˆ M z (x, t)dx + D(l(t); K(t))X(t)

l(t)

+ 0

385

¯ (l(t), x; K(t)) ˆ N w(x, ˆ t)dx,

(13.57)

¯ (l(t), x; K(t)), ˆ ¯ (l(t), x; K(t)) ˆ where M N are ¯ (l(t), x; K(t)) ˆ M = F (l(t), x) −

l(t)

ˆ (l(t), y; K(t))F ˆ N (y, x)dy,

(13.58)

x

¯ (l(t), x; K(t)) ˆ ˆ (l(t), x; K(t)) ˆ N = N (l(t), x) + N l(t) ˆ (l(t), y; K(t))N ˆ − N (y, x)dy.

(13.59)

x

In the output-feedback adaptive backstepping control law (13.57), the states w(x, ˆ t), ˆ , D are derived zˆ(x, t) are from the observer (13.14)–(13.17). The kernels J, F, G, N, N from the backstepping process in this section. The state X(t) is the measurement. ˆ The row vector K(t) is the adaptive estimate defined in (13.51), (13.52).

13.4

EVENT-TRIGGERING MECHANISM

In this section, we introduce an observer-based event-triggered control scheme for the stabilization of plant (13.1)–(13.5). It relies on both the continuous-in-time adaptive control signal U (t) (13.57) and a dynamic ETM that determines the triggering times tk (integer k ≥ 0 and t0 = 0) when the control signal is updated. In other words, the event-triggered control signal Ud (t) is the frozen value of the continuousin-time U (t) at the time instants tk , that is, Ud (t) = U (tk ),

t ∈ [tk , tk+1 ).

(13.60)

Inserting Ud (t) into (13.17), we obtain w(l(t), ˆ t) = Ud (t).

(13.61)

A deviation d(t) between the continuous-in-time adaptive control signal and the event-based control signal is given as d(t) =U (t) − Ud (t).

(13.62)

w(l(t), ˆ t) = U (t) − d(t).

(13.63)

Then (13.61) can be written as

Recalling the backstepping transformations and designs of U (t) in section 13.3, the target system becomes (13.40)–(13.43) with the right boundary condition ηˆ(l(t), t) = −d(t).

(13.64)

The ETM to determine the triggering times of Ud is designed, as in [57], using the dynamic triggering condition

CHAPTER THIRTEEN

386

tk+1 = inf{t ∈ R+ |t > tk |d(t)2 ≥ θΦ(t) − md (t)},

(13.65)

where the internal dynamic variable md (t) satisfies the ODE m ˙ d (t) = − ηmd (t) + λd d(t)2 − σΦ(t) − κ1 α ˆ (l(t), t)2 − κ2 ηˆ(0, t)2 − κ3 α ˆ (0, t)2 ,

(13.66)

whose initial condition md (0) should be chosen to be negative, and which is driven by the norm Φ(t) = |X(t)|2 + ˆ η (·, t)2 + ˆ α(·, t)2 .

(13.67)

The signals in (13.67) can be replaced by the observer states via (13.31), (13.32), (13.38). The complete set of ETM parameters is ζe = {θ, η, λd , σ, κ1 , κ2 , κ3 }.

(13.68)

These positive parameters are to be determined later. The reason for introducing an internal dynamic variable md (t) into the event˙ of the deviation between triggering condition (13.65) is that the changing rate d(t) U (t) and Ud (t), upon which the dwell time relies, includes as the last three terms in (13.66), the boundary states α(l(t), ˆ t) ηˆ(0, t), α ˆ (0, t), whose integration should be incorporated into the event-triggering condition (13.65) to avoid the Zeno phenomenon. The internal dynamic variable md (t) is kept negative by the choice of θ. The explanation in this paragraph is formalized through the following three lemmas. Lemma 13.2. For d(t) defined in (13.62), there exists a positive constant λa dependent only on the plant parameters ζp and the design parameters g¯i in Am in (13.7), such that  ˙ 2 ≤ λa (ζp , g¯i ) d(t)2 + α(l(t), d(t) ˆ t)2 + ηˆ(0, t)2 + α(0, ˆ t)2 + m3 (ζp , ζa , g¯i )ˆ α(·, t)2 + m3 (ζp , ζa , g¯i )ˆ η (·, t)2  + m3 (ζp , ζa , g¯i )|X(t)|2

(13.69)

for t ∈ (tk , tk+1 ), where m3 is a positive constant dependent only on the plant parameters ζp , the adaptive law parameters ζa , and the design parameters g¯i ’s in Am . Proof. The proof is shown in appendix 13.7A. In the proof of lemma 13.2 and the remaining text in this chapter, a constant followed by (·) denotes a constant that depends only on the parameters in the parentheses. For conciseness, after the first appearance of the constant, (·) will be omitted when unnecessary. Lemma 13.3. Choosing θ≤

σ λd

(13.70)

for the internal dynamic variable md (t) defined in (13.66), it holds that md (t) < 0.

ADAPTIVE EVENT-TRIGGERED CONTROL

387

Proof. The proof is shown in appendix 13.7B. Lemma 13.4. For some κ1 , κ2 , κ3 , there is a minimal dwell time between two triggering times, which is equal to or greater than a positive constant Tmin , which depends only on the parameters of the plant and the choices of the design parameters. Proof. Introduce a function ψ(t), 2

ψ(t) =

d(t) + 12 md (t) , θΦ(t) − 12 md (t)

(13.71)

which is proposed in [57]. We have ψ(tk+1 ) = 1 because the event condition in (13.65) is triggered, and ψ(tk ) < 0 because md (t) < 0 (lemma 13.3) and d(tk ) = 0. The function ψ(t) is continuous on [tk , tk+1 ] due to the continuity and well-posedness of this class of 2 × 2 hyperbolic PDE-ODE system according to [48]. By the intermediate value theorem, there exists t∗ > tk such that ψ(t) ∈ [0, 1] when t ∈ [t∗ , tk+1 ]. The minimal Tmin can be found as the minimal time it takes for ψ(t) to go from 0 to 1—that is, the reciprocal of the absolute value of the maximum changing rate of ψ(t). Taking the derivative of (13.71); recalling lemma 13.2 and (13.66), (13.67); and choosing κ1 ≥ max{2λa (ζp , g¯i ), 2θλp (ζp )},

(13.72)

κ2 ≥ max{2λa (ζp , g¯i ), 2θλp (ζp )},

(13.73)

κ3 ≥ max{2λa (ζp , g¯i ), 2θλp (ζp )},

(13.74)

for some positive λp shown in appendix 13.7C; through a calculation process in appendix 13.7C, we get ˙ ≤ n1 ψ(t)2 + n2 ψ(t) + n3 ψ(t)

(13.75)

with 1 n1 = λd + θλp (ζp ) > 0, 2

(13.76)

n2 = 1 + λa (ζp , g¯i ) + λd + θλp (ζp ) + η + f1 (σ, μ0 (ζp , ζa , g¯i ), θ) > 0,

(13.77)

1 λa (ζp , g¯i )m3 (ζp , ζa , g¯i ) + η > 0, n3 = 1 + λd + λa (ζp , g¯i ) + 2 θ

(13.78)

which are positive constants where  1 μ0 (ζp , ζa , g¯i ) − 2θ σ, f1 = 0,

if σ < 2θμ0 (ζp , ζa , g¯i ), if σ ≥ 2θμ0 (ζp , ζa , g¯i ).

CHAPTER THIRTEEN

388

Then it follows that the least time needed by ψ(t) to go from 0 to 1 is Tmin =

1 >0 n1 + n2 + n3

˙ because the maximum changing rate ψ(t) is n1 + n2 + n3 for ψ(t) ∈ [0, 1] according to (13.75). The proof of this lemma is complete.

13.5

STABILITY ANALYSIS OF THE CLOSED-LOOP SYSTEM

The expression of the final adaptive event-triggered control law Ud is Ud (t) =

l(tk ) 0



¯ (l(tk ), x; K(t ˆ k ))ˆ M z (x, tk )dx

l(tk )

+ 0

¯ (l(tk ), x; K(t ˆ k ))w(x, ˆ k ))X(tk ) N ˆ tk )dx + D(l(tk ); K(t

(13.79)

for t ∈ [tk , tk+1 ), recalling (13.57) and (13.60). The triggering times tk (for integer k ≥ 0) are determined by the ETM in (13.65), (13.66). In (13.79), zˆ, w ˆ are states ˆ is the adaptive update law (13.51), (13.52), from the observer (13.14)–(13.17), K and X is the ODE measurement. The scalars l(tk ) are the values of the time-varying ¯,N ¯ function l(t), which is known ahead of time, at the times tk , the functions M are given in (13.58), (13.59), and D is defined in (13.47)–(13.50). Lemma 13.5. For all initial values (ˆ α(·, 0), ηˆ(·, 0)) ∈ L2 (0, L), X(0) ∈ Rn , md (0) < 0, the event-based target system (13.40)–(13.43), (13.64) is asymptotically stable in the sense of 2

lim (ˆ α(·, t)2 + ˆ η (·, t)2 + |X(t)| + |md (t)|) = 0.

t→∞

Proof. Define q1 = min {q1 (x)}, 0≤x≤L

q1 = max {|q1 (x)|}, 0≤x≤L

q2 = min {q2 (x)}, 0≤x≤L

q2 = max {|q2 (x)|}, 0≤x≤L

q1 = max {q1 (x)}, 0≤x≤L

q2 = max {q2 (x)}, 0≤x≤L

c1 = max {|c1 (x)|}, 0≤x≤L

c4 = max {|c4 (x)|}. 0≤x≤L

ADAPTIVE EVENT-TRIGGERED CONTROL

389

Step 1. Choose the Lyapunov function as 1˜ −1 ˜ V (t) = ln (1 + Ω(t) − μm md (t)) + K(t)Γ K(t)T , c 2

(13.80)

˜ where the terms K(t), md (t) are related to the adaptive law and ETM. Because of md (t) < 0, we have 1 + Ω(t) − μm md (t) > 0. According to (13.54) and (13.67), we obtain μ1 Φ(t) ≤ Ω(t) ≤ μ2 Φ(t),

(13.81)

with positive μ1 , μ2 as 1 min{ra , rb e−δL , λmin (P )}, 2 1 μ2 = max{ra eδL , rb , λmax (P )}, 2

μ1 =

(13.82) (13.83)

where λmin and λmax denote the minimum and maximum eigenvalues of the corresponding matrix. Taking the derivative of (13.80) along (13.40)–(13.43) with (13.64) and (13.66) and applying the Young and Cauchy-Schwarz inequalities, through a process in appendix 13.7D, we arrive at  7 1 ¯2 ˙ λmin (Q) − q1 (0)rb D V (t) ≤ − 1 + Ω(t) − μm md (t) 8  1 2 2 2 2 2 ¯ ¯ − q1 (0) Lrb J |C| − μm σ − μm κ3 D |X(t)| 2  1 q2 (0)ra − (q1 (0)rb + μm κ3 )p21 − 2  8 2 |P B| − μm κ2 ηˆ(0, t)2 − λmin (Q) 

l(t)  1 1 δq2 − c4 − q2 eδx − μm σ − ra ηˆ(x, t)2 dx 2 2 0

1 ˙ b e−δl(t) − μm κ1 α ˆ (l(t), t)2 − (q1 (l(t)) − l)r 2  

l(t) 1 q1 1 −δx q1 δ − c 1 − − − rb − μm σ α ˆ (x, t)2 dx e 2 2 2 0   l(t) ˆ˙ ˆ ˆ˙ − ra eδx ηˆ(x, t) K(t)D (x; K(t))X(t) + K(t)R(x, t) dx ˆ K(t) 0

  1 (13.84) − μm λd − (q2 + vm )ra eδL d(t)2 + μm ηmd (t) , 2 where

J¯ = max {|J(x, 0)|}, 0≤x≤L

¯= D

max

ˆi (t)≤k ¯i k i ≤k

ˆ {|C + p1 D(0; K(t))|},

and (13.41), (13.51), (13.52), (13.64) have been used.

CHAPTER THIRTEEN

390 Inserting (13.70) into (13.72)–(13.74) to replace θ by condition λd ≥ 1,

σ λd

and adding additional

we summarize the conditions of the design parameters in adaptive event-triggered backstepping control systems as min{κ1 , κ2 , κ3 } ≥ max{2λa , 2σλp },   2c4 + q2 2c1 + 1 + q1 , , δ > max q2 q1 rb


where σ, η are free parameters. Applying the Young and Cauchy-Schwarz inequalities, recalling (13.46), we get

− ra ≤

 ˆ˙ ˆ X(t) + KR(x, ˆ˙ eδx ηˆ(x, t) KD t) dx K(t) 0 

√ 2 max {γci } nλb (ζp , δ, ra , rb , μm , g¯i ) |X(t)| + ˆ η 2 , l(t)

i∈{1,...,n}

(13.92)

where the positive constant λb (ζp , δ, ra , rb , μm , g¯i ) > 0 depends only on the plant parameters ζp and the choices of ra , rb , δ, μm and the g¯i ’s. By choosing parameters as (13.85)–(13.91) and inserting (13.92), the inequality (13.84) becomes  1 2 V˙ (t) ≤ − λc (ζp , ζe , δ, ra , rb , μm , g¯i ) |X(t)| 1 + Ω(t) − μm md (t)  + ηˆ(0, t)2 + ˆ η (·, t)2 + α ˆ (l(t), t)2 + ˆ α(·, t)2 + d(t)2 + μm ηmd (t)

√ 2 2 η (·, t) ) , + max {γci } nλb (|X(t)| + ˆ i∈{1,...,n}

ADAPTIVE EVENT-TRIGGERED CONTROL

391

where the constant λc (ζp , ζe , δ, ra , rb , μm , g¯i ) > 0 depends only on the plant parameters ζp , the design parameters g¯i ’s in Am , the event-triggering mechanism parameters ζe , and δ, ra , rb , μm in the adaptive law parameters. The coefficients γci are independent of λb and λc . Choose maxi∈{1,...,n} {γci } to satisfy λc (ζp , ζe , δ, ra , rb , μm , g¯i ) . max {γci } < √ nλb (ζp , δ, ra , rb , μm , g¯i )

(13.93)

i∈{1,...,n}

Finally, we arrive at ¯ c , η}  − min{λ 2 V˙ (t) ≤ η (·, t)2 |X(t)| + ηˆ(0, t)2 + ˆ 1 + Ω(t)

 + d(t)2 + α(l(t), ˆ t)2 + ˆ α(·, t)2 + μm |md (t)| ≤ 0,

where

¯ c = λc − λ

(13.94)

√ max {γci } nλb > 0

i∈{1,...,n}

and ¯ c , η} λall := min{λ

(13.95)

is related to the convergence rate of the closed-loop system. d d Step 2. Boundedness analysis of dt |X(t)|2 , dt ˆ η (·, t)2 , According to (13.94) obtained in step 1, we thus have

d dt |md (t)|:

d α(·, t)2 , dt ˆ

and

V (t) ≤ V (0), ∀t ≥ 0. ˜ One easily gets that |K(t)|, ˆ η (·, t), ˆ α(·, t), |X(t)|, |md (t)| are uniformly bounded and also that Φ(t) is bounded according to (13.67). Recalling the invertibility of the backstepping transformations (13.31), (13.32), (13.38), the boundedness of the signals ˆ z (·, t), w(·, ˆ t), |X(t)| is obtained. Therefore, U (t) is bounded according to (13.57). It follows that d(t) is bounded via (13.62). Taking the time derivaα(·, t)2 , ˆ η (·, t)2 , and |md (t)| along (13.40)–(13.44), (13.66), we tive of |X(t)|2 , ˆ obtain d ˆ ˜ |X(t)|2 = 2X T (t)(Am X(t) + B ηˆ(0, t) − B KX(t)), (13.96) dt d 2 ˙ ˆ η (·, t)2 = (q2 (l(t)) + l(t))d(t) − q2 (0)ˆ η (0, t)2 dt l(t) − (q2 (x) − 2c4 (x))ˆ η (x, t)2 dx 0



l(t)

+2 0

ˆ ˜ ηˆ(x, t) D(x; K(t))B K(t)

˙ ˙ ˆ ˆ ˆ (x; K(t))X(t) − K(t)R(x, t) dx, − K(t)DK(t) ˆ d ˙ α(l(t), t)2 + q1 (0)ˆ ˆ α(·, t)2 = − (q1 (l(t)) − l(t))ˆ α(0, t)2 dt l(t) + (q1 (x) + 2c1 (x))ˆ α(x, t)2 dx 0

(13.97)

CHAPTER THIRTEEN

392 −2

l(t) 0

α ˆ (x, t)J(x, 0)q1 (0)CX(t)dx,

d |md (t)| = − m ˙ d (t) = ηmd (t) − λd d(t)2 + σΦ(t) dt ˆ (l(t), t)2 + κ2 ηˆ(0, t)2 + κ3 α ˆ (0, t)2 . + κ1 α

(13.98)

(13.99)

Recalling the boundedness results proved above and (13.64), (13.62), (13.60), (13.57), we obtain the boundedness of ηˆ(l(t), t). We then find that ηˆ(0, t) is bounded as a ˆ˙ in (13.115) result of the transport PDE (13.43), with recalling the boundedness of K in appendix 13.7A. The signal α ˆ (0, t) is bounded as well due to (13.41), and then α ˆ (l(t), t) is bounded as a result of the transport PDE (13.42). Therefore, by applying the Young and Cauchy-Schwarz inequalities to (13.96)–(13.99), with the bounded2 d ˙ in assumption 13.3, we get that d |X(t)|2 , d ˆ ness of l(t) α(·, t)2 , and dt dt η (·, t) , dt ˆ d dt |md (t)| are uniformly bounded. Finally, integrating (13.94) obtained in step 1 from 0 to ∞, it follows that α(·, t)2 , ˆ η (·, t)2 , and |md (t)| are integrable. Then using the results |X(t)|2 , ˆ obtained in steps 1 and 2, according to Barbalat’s lemma, we have that |X(t)|2 , η (·, t)2 , and |md (t)| tend to zero as t → ∞. ˆ α(·, t)2 , ˆ The following theorem establishes that, in the closed-loop system, no Zeno phenomenon takes place, namely that lim tk = +∞,

k→∞

(13.100)

and the states and the control signal are convergent to zero. The well-posedness of the event-based closed-loop system can be studied in a similar manner as in the proof of property (1) of theorem 11.1. Theorem 13.1. For all initial values (z(·, 0), w(·, 0)) ∈ L2 (0, L), X(0) ∈ Rn , (ˆ z (·, 0), w(·, ˆ 0)) ∈ L2 (0, L), and md (0) < 0, with the design parameters satisfying (13.85)– (13.91), (13.93), the closed-loop system—that is, the plant (13.1)–(13.5) with the proposed observer-based adaptive event-triggered controller (13.79), which consists of the observer (13.14)–(13.17), the adaptive update law (13.51), (13.52), and the ETM (13.65), (13.66)—has the following properties: 1) There exists a positive constant Tmin that depends only on the parameters of the plant and the choices of the control parameters such that min{tk+1 − tk } ≥ Tmin . k≥0

(13.101)

2) In the closed-loop system, the states are asymptotically convergent to zero in the sense of lim (|X(t)|2 + z(·, t)2 + w(·, t)2

t→∞

+ ˆ z (·, t)2 + w(·, ˆ t)2 + |md (t)|) = 0.

(13.102)

3) The adaptive event-triggered control signal is convergent to zero: lim Ud (t) = 0.

t→∞

(13.103)

ADAPTIVE EVENT-TRIGGERED CONTROL

393

Proof. 1) Recalling lemma 13.4, property (1) is obtained. 2) Recalling the asymptotic stability result proved in lemma 13.5 and considering the invertibility and continuity of the backstepping transformations (13.31), (13.32), (13.38), we obtain the asymptotic convergence to zero of ˆ z (·, t)2 + 2 2 w(·, ˆ t) + |X(t)| + |md (t)|. Recalling lemma 13.1 and (13.18) and applying the separation principle, we obtain property (2). 3) Recalling (13.57) and property (2), we know that the continuous-in-time control signal U (t) is asymptotically convergent to zero. According to the definition (13.60) and property (1), property (3) is then obtained.  Conditions on all the control parameters (13.85)–(13.91), (13.93) are cascaded rather than mutually dependent. The optimal choices of these parameters are not studied in this book.

13.6

APPLICATION IN THE FLEXIBLE-GUIDE MINING CABLE ELEVATOR

Simulation Model According to section 5.1, the wave PDE modeled lateral vibration dynamics of the mining cable elevator are ρutt = T (x)uxx (x, t) + T  (x)ux (x, t) − c¯ut (x, t), Mc utt (0, t) = −kc u(0, t) − cd ut (0, t) + T (0)ux (0, t), −T (l(t))ux (l(t), t) = U (t),

(13.104) (13.105) (13.106)

where u(x, t) denotes the lateral vibration displacements along the cable shown in figure 13.1, and x ∈ [0, l(t)] are the positions along the cable in a moving coordinate system associated with the motion l(t), whose origin is located at the cage. The function T (x) = Mc g + xρg is the static tension along the cable, and ρ is the linear density of the cable. The constant c¯ is the material damping coefficient of the cable. The values of the physical parameters of the mining cable elevator tested in the simulation are shown in table 13.1. The constants kc , cd are the unknown equivalent stiffness and damping coefficients of the viscoelastic guide. Through applying the Riemann transformations  T (x) ux (x, t), (13.107) z(x, t) = ut (x, t) − ρ  w(x, t) = ut (x, t) +

T (x) ux (x, t), ρ

and defining X(t) = [x1 (t), x2 (t)]T = [u(0, t), ut (0, t)]T ,

(13.108)

CHAPTER THIRTEEN

394 Control force

x = l(t)

Hydraulic actuator

X(t) Vibration displacement x1(t) = u(0, t) u(x, t)

Integration

Vibration velocity x2(t) = ut (0, t)

Approximated flexible guideway

Integration

Accelerometer

kc x=0 cd

Cage

Figure 13.1. Mining cable elevator with viscoelastic guideways.

Table 13.1. Physical parameters of the descending mining cable elevator Parameters (units)

Values

Initial length L0 (m) Final length (m) Cable linear density ρ (kg/m) Total hoisted mass Mc (kg) Gravitational acceleration g (m/s2 ) Maximum hoisting velocities v¯max (m/s) Total hoisting time tf (s) Cable material damping coefficient c¯ (N·s/m)

300 2460 8.1 15000 9.8 18 150 0.4

which physically means the lateral displacement and velocity of the cage, we convert (13.104)–(13.106) into a 2 × 2 coupled transport PDE-ODE model in the form of (13.1)–(13.5) with the following coefficients:  T (x) T  (x) −¯ c , c1 (x) = c3 (x) = −  , (13.109) q1 (x) = q2 (x) = ρ 2ρ 4 ρT (x) c2 (x) = c4 (x) =

T  (x) −¯ c +  , p1 = −1, 2ρ 4 ρT (x)

(13.110)

ADAPTIVE EVENT-TRIGGERED CONTROL

395 20 15

1750

. l(t) (m/s)

l(t) (m)

2500

10 1000 250

5 0

40

80 Time (s)

0

120

˙ Figure 13.2. Time-varying domain l(t) and the according velocity l(t).

1 A= Mc



0 −kc

M √c −cd − Mc ρg



,B =

0

ρg Mc

 , C = [0, 2].

(13.111)

For the lateral vibration model of the mining cable elevator (13.104)–(13.106) with the Riemann transformations (13.107), (13.108), the condition of the controlled boundary in (13.1)–(13.5) should have a simple augmentation, as follows: w(l(t), t) = − 

2 ρT (l(t))

U (t) + z(l(t), t).

(13.112)

The additional term z(l(t), t) can be canceled at the drum (see figure 2.1), so in the simulation, we consider the controlled boundary as (13.5), that is, w(l(t), t) = U (t), where the designed control input, based on (13.1)–(13.5) with the above coefficients √ ρT (l(t))

to convert the input signal (13.109)–(13.111), should be multiplied by − 2 computed based on (13.5) into the control force at the head sheave in the mining cable elevator—that is, into the control signal U (t) in (13.112). In the practical mining cable elevator, l(t) is obtained by the product of the radius and the angular displacement of the rotating drum driving the cable, where the angular displacement is measured by the angular displacement sensor at the drum. In the simulation, the unknown damping and stiffness coefficients of the flexible guide are set, respectively, as cd = 0.4 and kc = 1000. The target system matrix of the ODE is set as   0 1 Am = . (13.113) −2.2 −5.8 The unknown target control parameters k1 , k2 are sought online by the adaptive mechanism to achieve the target system matrix Am . The bounds of the unknown control parameters k1 , k2 in the adaptive estimates are defined as [−50, 0], [−100, 0]. ˙ are shown in figThe time-varying cable length l(t) and its changing rate l(t) ure 13.2. The maximum velocity of the moving boundary—that is, the maximum hoisting velocity v¯max = 18 m/s—satisfies the limit of the changing rate of the timevarying domain proposed in assumption 13.3. The initial conditions of the plant (13.1)–(13.5) are defined as w(x, 0) = 0.2 sin(2πx/L0 ),

π z(x, 0) = 0.4 sin 3πx/L0 + , 6

CHAPTER THIRTEEN

396

Table 13.2. Parameters of the proposed adaptive event-based control system Parameters

Values

In adaptive law In the ETM

γc1 = 0.95, γc2 = 0.46, δ = 3, rb = 0.06, ra = 4, μm = 0.00002 θ = 0.118, η = 41, λd = 1.3, σ = 0.5, κ1 = κ2 = κ3 = 4

Self-tuned gains

0 kˆ1

–20 –31.3 –40

kˆ2

–60 –78.2 –80 0

50

100

150

Time (s)

Figure 13.3. Self-tuned control gains kˆ1 , kˆ2 , whose target values are k1 = −31.3, k2 = −78.2.

x2 (0) = 0.5w(0, 0) + 0.5z(0, 0), x1 (0) = 0.1. The initial value md (0) in the ETM is chosen as −0.03. Simulation Results The design parameters in the proposed adaptive event-based control system, including the design parameters in the adaptive law and ETM, are shown in table 13.2. According to the system matrix, the input matrix in (13.111), and the target system matrix Am (13.113), we know that the ideal control parameters k1 , k2 are −31.3, −78.2, respectively. Figure 13.3 shows that our adaptive design can adjust the control parameters kˆ1 (t), kˆ2 (t) in an online fashion to approach the ideal values. It often happens in adaptive control that, even though the estimates do not exactly arrive at their actual values, the state convergence is achieved in the closed-loop system, which will be seen shortly. The proposed adaptive event-based control input and the continuous-in-time adaptive control input are shown in figure 13.4. The internal dynamic variable md (t) in the ETM is shown in figure 13.5. The PDE states used in the control law are from the observer (13.14)–(13.17), and we can see from the observer error shown in figure 13.6 that observer errors at the midpoint of the time-varying spatial domain are convergent to zero after t = 6.4 s. The responses of the PDE states and the ODE state are shown in figures 13.7– 13.10, where the proposed controller can quickly suppress to zero the oscillations appearing in the open-loop system. The fact that oscillation amplitude decreases in the open-loop system is due to the fact that material damping of the cable is considered in the simulation. Figures 13.7 and 13.8 show that the PDE states at

Control input signal

ADAPTIVE EVENT-TRIGGERED CONTROL

397

0.5 0 –0.5 Event-based Continuous-in-time

–1 0

50

100

150

Time (s)

Figure 13.4. Adaptive event-based control input and the continuous-in-time adaptive control input. 0

md (t)

–0.01

–0.02

–0.03

0

50

100

150

Time (s)

Figure 13.5. Dynamic internal variable md (t) in the ETM.

Observer error

0.1 0 –0.1 l(t) w˜ (—, t) 2

–0.2

l(t) ˜z (—, t) 2

–0.3 0 6.4 s

50

150

100 Time (s)

Figure 13.6. Observer errors at the midpoint of the time-varying spatial domain.

l(t) w(—, t) 2

0.5

0 With control Without control

–0.5 0

50

100 Time (s)

Figure 13.7. Responses of w



l(t) 2 ,t

150

 .

CHAPTER THIRTEEN

398

0.4

With control Without control

l(t) z(—, t) 2

0.2 0 –0.2 –0.4

0

50

100 Time (s)



Figure 13.8. Responses of z

l(t) 2 ,t

150

 .

x1(t) (m)

0.1 0 –0.1

With control Without control

–0.2 0

50

100

150

Time (s)

Figure 13.9. Responses of x1 (t).

x2(t) (m/s)

0.03 0.015 0 –0.015 –0.03

With control Without control 0

50

100

150

Time (s)

Figure 13.10. Responses of x2 (t).

the midpoint of the time-varying spatial domain are reduced to zero. Figures 13.9 and 13.10 show that the responses of the ODE state X(t) = [x1 (t), x2 (t)]T , which physically represents the displacement and velocity of the lateral vibrations of the cage moving along flexible guideways, are suppressed to zero under the proposed controller. Represent the responses of z(x, t), w(x, t) in the original cable model (13.104)– 1 (13.106) by using (13.107), (13.108), obtaining the norm (ut (·, t)2 + ux (·, t)2 ) 2 , which physically represents the vibration energy of the cable. A comparison of the performance of the proposed controller with that of a traditional proportionalderivative (PD) controller Upd (t) = 2x1 (t) + 1.2x2 (t),

399

1

– (|| ut (., t) ||2 + || ux (., t) ||2) 2

ADAPTIVE EVENT-TRIGGERED CONTROL

Without control PD control Proposed control

0.3 0.2 0.1 0 0

50

100

150

Time (s) 1

Figure 13.11. Time evolution of the norm (ut (·, t)2 + ux (·, t)2 ) 2 , which physically reflects the vibration energy of the cable modeled by (13.104)–(13.106). where the PD parameters are chosen by trial and error over many tests, is shown in figure 13.11. From the comparison we observe that both controllers reduce the vibrations compared to the result without control. Even though the vibration energy under the proposed controller is larger at the beginning, which is due to the fact that the self-tuned control gains kˆ1 , kˆ2 start to search for the target values from bad initial values of zero (see figure 13.3), the proposed controller reduces the vibration energy to a much smaller range around zero as time goes on.

13.7

APPENDIX

A. Proof of lemma 13.2 The following notation is used: ¯ (l(t), x; K(t)) ˆ ∂M ¯ l(t) (l(t), x; K(t)) ˆ M . = ∂l(t) Because U˙ d = 0 for t ∈ (tk , tk+1 ), recalling (13.62) and taking the time derivative along (13.13)–(13.17), we obtain ˙ 2 = U˙ (t)2 d(t)

 ˙ − q1 (l(t)) M ¯ (l(t), l(t), K(t))ˆ ˆ = l(t) z (l(t), t) ¯ (l(t), 0; K(t))q ˆ +M z (0, t) 1 (0)ˆ 

˙ ¯ (l(t), l(t), K(t)) ˆ N w(l(t), ˆ t) + q2 (l(t)) + l(t)   ˆ ¯ (l(t), 0; K(t))q ˆ ˆ t) + D(l(t); K(t))B −N 2 (0) w(0,    ˙ ˆ˙ ˆ ˆ ˆ + K(t)D (l(t); K(t)) + l(t)D (l(t); K(t)) + D(l(t); K(t))A X(t) ˆ K(t)

l(t)

+ 0



 ¯ ˆ ¯ (l(t), x; K(t))q ˆ (M 1 (x)) + M (l(t), x; K(t))c1 (x)

˙ ¯ ¯ (l(t), x; K(t))c ˆ ˆ +N 3 (x) + l(t)Ml(t) (l(t), x; K(t))

CHAPTER THIRTEEN

400  ˙ ˆ ¯ ˆ + K(t)MK(t) (l(t), x; K(t)) zˆ(x, t)dx ˆ

l(t)

+ 0



 ¯ (l(t), x; K(t))c ˆ ¯ ˆ N 4 (x) − (N (l(t), x; K(t))q2 (x))

ˆ˙ N ¯ ˆ (l(t), x; K(t)) ˆ ¯ (l(t), x; K(t))c ˆ + K(t) +M 2 (x) K(t) 

2 ˙ N ˆ ¯l(t) (x, l(t), K(t)) w(x, ˆ t)dx + l(t)  ≤ λ0 (ζp , g¯i ) w(l(t), ˆ t)2 + zˆ(l(t), t)2 + w(0, ˆ t)2 + zˆ(0, t)2 + m3 (ζp , ζa )ˆ z (·, t)2 + m3 (ζp , ζa )w(·, ˆ t)2 + m3 (ζp , ζa )|X(t)|2 ,

(13.114)

t ∈ (tk , tk+1 ) for some positive λ0 , which depends only on the plant parameters ζp and the design parameters g¯i ’s (the bounds of all kernels depend on the plant ˆ depend on the bounds of the unknown parameters ζp , and the bounds of K(t) parameters gi ’s in A and the design parameters g¯i ’s in Am , as mentioned in section 13.1). According to (13.51), (13.52), we know that    ˆ˙ 2 K(t) ≤ m3 (ζp , ζa , g¯i ),

(13.115)

where m3 is a positive constant dependent only on the plant parameters ζp , the adaptive law parameters ζa , and the design parameters g¯i ’s in Am . (Once again, in this chapter a constant followed by (·) denotes a constant that depends only on the parameters in the parentheses, as in (13.115).) Recalling the invertibility of the backstepping transformations (ˆ z , w, ˆ X(t)) ↔ (ˆ α, ηˆ, X(t)) and inserting (13.64), we get (13.69). The proof is complete. B. Proof of lemma 13.3 According to (13.65), events are triggered to guarantee d(t)2 ≤ θΦ(t) − md (t).

(13.116)

Inserting (13.116) into (13.66), one obtains m ˙ d (t) ≤ −(η + λd )md (t) + (λd θ − σ)Φ(t) − κ1 α ˆ (l(t), t)2 − κ2 ηˆ(0, t)2 − κ3 α ˆ (0, t)2 ≤ −(η + λd )md (t) − κ1 α ˆ (l(t), t)2 − κ2 ηˆ(0, t)2 − κ3 α ˆ (0, t)2 ,

t≥0

(13.117)

by using (13.70). Hence, by the comparison principle and md (0) < 0, we conclude that md (t) < 0. The proof is complete.

ADAPTIVE EVENT-TRIGGERED CONTROL

401

C. Calculation of (13.75) Taking the derivative of (13.71), using lemma 13.2, and inserting the inequality ˙ −Φ(t) ≤ μ0 (ζp , ζa , g¯i )Φ(t) + λp (ζp )[ˆ α(l(t), t)2 + ηˆ(0, t)2 + d(t)2 + α(0, ˆ t)2 ], (13.118) which is obtained by taking the derivative of Φ(t) along (13.40)–(13.44) for all √ t∈ ˜ ˜ [t∗ , tk+1 ] and recalling the boundedness of K(t) (specifically, |K(t)| ≤ n× ˆ˙ maxi∈{1,...,n} (|k i − k i |)) and the bound on K(t) in (13.115), where the constant μ0 (ζp , ζa , g¯i ) > 0 in (13.118) only depends on the plant parameters ζp , the adaptive law parameters ζa , and the design parameters g¯i ’s, and the constant λp (ζp ) > 0 in the same inequality depends only on the plant parameters ζp , we get 1 ˙ ˙ ˙ d (t) (θΦ(t) ˙ d (t)) − 12 m ˙ = 2d(t)d(t) + 2 m ψ(t) − ψ(t) 1 1 θΦ(t) − 2 md (t) θΦ(t) − 2 md (t)  1 ≤ ˆ t)2 + ηˆ(0, t)2 λa d(t)2 + α(l(t), θΦ(t) − 12 md (t)

+ α(0, ˆ t)2 + m3 (ζp , ζa , g¯i )ˆ α(·, t)2 + m3 (ζp , ζa , g¯i )ˆ η (·, t)2

 1 2 ˙ d (t) + m3 (ζp , ζa , g¯i )|X(t)|2 + d(t) + m 2  1 − ˆ t)2 θ − μ0 (ζp , ζa , g¯i )Φ(t) − λp α(l(t), θΦ(t) − 12 md (t)

 1 ˙ d (t) ψ(t). − λp ηˆ(0, t)2 − λp d(t)2 − λp α(0, ˆ t)2 − m 2

Inserting (13.66) to rewrite m ˙ d (t) and recalling (13.67), we obtain     1 1 1 2 ˙ ≤ λ κ + 1 + + λ − d(t) α(l(t), t)2 ψ(t) λ a d a 1 2 2 θΦ(t) − 12 md (t)



κ3  κ2  α(0, t)2 + λa − ηˆ(0, t)2 + λa m3 (ζp , ζa , g¯i )Φ(t) + λa − 2 2 1 1 − ηmd (t) − − θμ0 (ζp , ζa , g¯i )Φ(t) 2 θΦ(t) − 12 md (t)     1 1 − θλp − κ1 α(l(t), t)2 − θλp − κ2 ηˆ(0, t)2 2 2     1 1 − θλp + λd d(t)2 − θλp − κ3 α ˆ (0, t)2 2 2

1 1 (13.119) + ηmd (t) + σΦ(t) ψ(t). 2 2 Inserting (13.72)–(13.74), (13.119) becomes   1 1 2 ˙ ≤ λ + 1 + ¯i )m3 (ζp , ζa , g¯i )Φ(t) ψ(t) λ a d d(t) + λa (ζp , g 2 θΦ(t) − 12 md (t) 

 1 1 1 − ηmd (t) − σ Φ(t) (ζ , ζ , g ¯ ) − − θμ 0 p a i 2 2 θΦ(t) − 12 md (t) 

 1 1 2 − θλp + λd d(t) + ηmd (t) ψ(t). (13.120) 2 2

CHAPTER THIRTEEN

402 Applying, in (13.120), the inequalities −

1 2 ηmd (t) θΦ(t) − 12 md (t)

≤−

1 2 ηmd (t) − 12 md (t)

= η,

1 Φ(t) Φ(t) = , ≤ θΦ(t) − 12 md (t) θΦ(t) θ d(t)2 + 12 md (t) − 12 md (t) d(t)2 = ≤ ψ(t) + 1, 1 θΦ(t) − 2 md (t) θΦ(t) − 12 md (t) which hold because md (t) < 0, we obtain (13.75). D. Calculation of (13.84) Taking the derivative of (13.80) along (13.40)–(13.43), employing (13.54), (13.55), ˜˙ ˆ˙ (13.64), (13.66), and applying K(t) = −K(t), we obtain V˙ (t) =

1 − X T (t)QX(t) 1 + Ω(t) − μm md (t) + 2X T P B ηˆ(0, t) +

˙ l(t) rb e−δl(t) α ˆ (l(t), t)2 2

1 δl(t) ˙ + (q2 (l(t)) + l(t))r ηˆ(l(t), t)2 ae 2 l(t) 1 1 2 − q2 (0)ra ηˆ(0, t) − δra eδx q2 (x)ˆ η (x, t)2 dx 2 2 0 l(t) l(t) 1 δx  2 − ra e q2 (x)ˆ η (x, t) dx + ra c4 (x)eδx ηˆ(x, t)2 dx 2 0 0 1 1 ˆ t)2 + q1 (0)rb α ˆ (0, t)2 − q1 (l(t))rb e−δl(t) α(l(t), 2 2 l(t) 1 − δrb e−δx q1 (x)ˆ α(x, t)2 dx 2 0 l(t) l(t) 1 −δx  2 + rb e q1 (x)ˆ α(x, t) dx + rb c1 (x)e−δx α ˆ (x, t)2 dx 2 0 0   l(t) ˙ ˙ δx ˆ ˆ − ra e ηˆ(x, t) K(t)DK(t) X(t) + KR(x, t) dx ˆ − rb

0

l(t) 0

e−δx α ˆ (x, t)J(x, 0)q1 (0)CX(t)dx

+ μm ηmd (t) − μm λd d(t)2 + μm σΦ(t) ˆ (l(t), t)2 + μm κ2 ηˆ(0, t)2 + μm κ3 α ˆ (0, t)2 + μ m κ1 α

ADAPTIVE EVENT-TRIGGERED CONTROL

403

ˆ˙ T + ˜ − K(t) Γc −1 K(t)

1 1 + Ω(t) − μm md (t)   l(t) T δx T T ˆ × 2X(t)B P X(t) − ra e ηˆ(x, t)X(t)B D(x; K(t)) dx . 0

(13.121) Applying (13.64), inserting the adaptive laws (13.51), (13.52) into (13.121), and using Young’s inequality, we get 1 7 2 ˙ V (t) ≤ − λmin (Q)|X(t)| 1 + Ω(t) − μm md (t) 8 8 1 2 |P B| ηˆ(0, t)2 − q2 (0)ra ηˆ(0, t)2 λmin (Q) 2 l(t) l(t) 1 1 − δra eδx q2 (x)ˆ η (x, t)2 dx − ra eδx q2  (x)ˆ η (x, t)2 dx 2 2 0 0 l(t) + ra c4 (x)eδx ηˆ(x, t)2 dx +

0

1 −δl(t) ˙ α ˆ (l(t), t)2 − (q1 (l(t)) − l(t))r be 2 l(t) 1 1 + q1 (0)rb α ˆ (0, t)2 − δrb e−δx q1 (x)ˆ α(x, t)2 dx 2 2 0 l(t) l(t) rb + e−δx q1 (x)ˆ α(x, t)2 dx + rb c1 (x)e−δx α(x, ˆ t)2 dx 2 0 0 l(t)

 ˆ˙ ˆ ˆ˙ − ra eδx ηˆ(x, t) K(t)D (x; K(t))X(t) + K(t)R(x, t) dx ˆ K(t) − rb

0

l(t) 0

e−δx α ˆ (x, t)J(x, 0)q1 (0)CX(t)dx + μm ηmd (t)

  1 − μm λd − (q2 + vm )ra eδL d(t)2 2 + μm σΦ(t) + μm κ1 α ˆ (l(t), t)2 2

2

+ μm κ2 ηˆ(0, t) + μm κ3 α ˆ (0, t) .

(13.122)

Recalling (13.41) and (13.67) applying the Young and Cauchy-Schwarz inequalities, we get (13.84).

13.8

NOTES

The adaptive control of hyperbolic PDEs is a popular research topic, on which recent results are presented in [9], which is limited to continuous inputs and identification. In this chapter, we presented the event-triggered adaptive output-feedback

404

CHAPTER THIRTEEN

boundary control design of a coupled hyperbolic PDE-ODE system where the control input is piecewise-constant. Another triggered-type adaptive control where the identification employs piecewise-constancy instead of the control input is presented in chapter 14. Developed by nontrivially integrating the adaptive control designs in this chapter (piecewise-constant control inputs) and chapter 14 (piecewise-constant identification), chapter 15 presents the adaptive control of coupled hyperbolic PDEs with piecewise-constancy in both inputs and identification. Recent results on the triggered-type adaptive control of PDEs are summarized in the notes section of chapter 15.

Chapter Fourteen Adaptive Control with Regulation-Triggered Parameter Estimation of Hyperbolic PDEs

The event-triggered adaptive control design in chapter 13 employed a continuousin-time parameter estimator, feeding an event-triggered control law, to produce a piecewise-constant input signal. In this chapter, following a recent approach by Karafyllis and coworkers [104], we pursue an event-triggered adaptive design that is a complement of the one in chapter 13. Here, we use a continuous-in-time control law that is fed piecewise-constant parameter estimates from an event-triggered parameter update law that applies a least-squares estimator to data “batches” collected over time intervals between the triggers. A parameter update is triggered by an observed growth in the norm of the partial differential equation (PDE) state. Since the PDE state is being adaptively regulated, this adaptive control approach is called regulation-triggered. Since the parameter updates are done on batches of data using least-squares identification, this identifier is referred to as a BaLSI. As in all conventional adaptive control, in chapters 5, 8, and 13 the adaptive control designs achieve only asymptotic convergence of the plant states and are not guaranteed to identify the true parameters exactly. The regulation-triggered BaLSI adaptive control scheme in this chapter guarantees the exponential regulation of plant states to zero and the finite-time exact identification of the unknown parameters from all but a negligible set of initial conditions, for hyperbolic PDE-ODE (ordinary differential equation) systems with unknown transport speeds. We start this chapter by formulating the problem in section 14.1 and presenting the nominal control design based on the basic backstepping design in section 14.2. In section 14.3, we propose the regulation-triggered adaptive control scheme, including a certainty-equivalence controller and a least-squares identifier updated in a sequence of times, which are determined by an event trigger designed based on the progress of the regulation of the states. In section 14.4, we prove that the proposed triggering-based adaptive control guarantees: 1) no Zeno phenomenon occurs; 2) parameter estimates are convergent to the true values in finite time (from most initial conditions); 3) the plant states are exponentially regulated to zero. The effectiveness of the proposed design is illustrated with a simulation in section 14.5.

14.1

PROBLEM FORMULATION

In this chapter, we consider the class of plants ˙ = (a − q1 bc)ζ(t) + b(q2 + q1 p)w(0, t), ζ(t) zt (x, t) = −q1 zx (x, t),

x ∈ [0, 1], t ≥ 0,

t ≥ 0,

(14.1) (14.2)

CHAPTER FOURTEEN

406 wt (x, t) = q2 wx (x, t),

x ∈ [0, 1], t ≥ 0,

z(0, t) = cζ(t) − pw(0, t), w(1, t) =

t ≥ 0,

q1 c¯ U (t) + z(1, t), q2 q2

t≥0

(14.3) (14.4) (14.5)

with initial conditions w(x, 0) = w0 (x) for x ∈ [0, 1), z(x, 0) = z0 (x) for x ∈ (0, 1], ζ(0) = ζ0 , where ζ(t) is a scalar ODE state, and scalar z(x, t), w(x, t) are PDE states. The boundary condition (14.5) contains the control input U (t). The class of (14.1)– (14.4) is motivated by a wave PDE converted to Riemann variables. It is through such a transformation process that possibly unmotivated-looking coefficients a − q1 bc and b(q2 + q1 p) in (14.1) arise. It is the parameters q1 and q2 , which appear both as transport speeds and in the ODE (14.1) and the boundary condition (14.5), that we consider unknown. The speed q2 is arbitrary and, of course, positive. The constants a, b, c, p are arbitrary and positive as well. The constant c¯ is arbitrary and nonzero. To make the problem as nontrivial as we can within this class, we only consider the case where the ODE (14.1) is unstable, with a − q1 bc > 0—that is, the case where the unknown propagation speed q1 satisfies 0 < q1
0, q¯2 > 0 and lower bounds q 1 > 0, q 2 > 0, respectively. The bounds q 2 , q¯2 are arbitrary, in addition to satisfying the obvious relation q 2 < q¯2 . For the bounds q 1 , q¯1 , the following two assumptions are made. Assumption 14.1. The upper bound q¯1 satisfies q¯1
q1 rb p + , q2 2m rb
0,

(14.11)

and the constant κ appearing in (14.11), and to be used later in control design, is chosen to satisfy 

 ⎧ ⎫ cb(¯ q2 +¯ q1 p) ⎨ (a − q bc) q¯2 + (¯ ⎬ q − q ) + p 1 a−¯ q bc a 1 1 1 κ < min , q1 c − . (14.12) ⎩ −q 2 b b⎭ The purpose of assumption 14.2—that is, (14.8)—will become evident in section 14.2, with the purpose of (14.8) becoming evident specifically in inequality (14.28), whose role is in estimating the exponential decay rate under nominal feedback. This assumption is not required in the BaLSI design and the exact parameter estimation. It is used in the stability analysis by the Lyapunov method in section 14.4. If there is no unknown parameter staying with the proximal reflection term z(1, t) in (14.5), assumption 14.2—(14.8)—is not required. Assumption 14.2 is difficult to verify a priori for two reasons: First, because the unknown q1 and q2 appear in (14.8), (14.9), (14.10), (14.11). Second, because q¯1 − q 1 appears both on the left of (14.8) and on the right of (14.12). But assumption 14.2 can unquestionably be satisfied for sufficiently small q¯1 − q 1 . Unfortunately, very small q¯1 − q 1 essentially means that the transport speed q1 is known. Let us now recap that q2 , which appears in the actuated w-PDE in (14.3) and is the transport speed in the direction downstream from the input, is arbitrary (positive), whereas the unactuated z-PDE in (14.2) may have to be nearly perfectly known. The parameters q1 , q2 appear in both the PDE as well as the ODE. The structure of the plant and the conditions of the plant parameters come, at least in principle, and as we already indicated above, from writing a wave PDE-ODE coupled model in Riemann coordinates. If the wave PDE’s Young modulus were unknown, the transformation into the Riemann variables would contain such an unknown quantity, which would render z(x, t) and w(x, t) unmeasurable. We proceed with an adaptive design for the class of systems (14.2), (14.3) with the expectation that applications do exist in which the transformation step into (14.2), (14.3) is not needed and (z, w) are measurable. If the original plant were a wave PDE, the ODE (14.1) would be driven by the wave PDE’s boundary state of the Neumann type, multiplied by a coefficient associated with the wave PDE’s propagation velocity, while the opposite boundary of the wave PDE would be actuated using Neumann actuation with a coefficient associated with the wave PDE’s propagation velocity as well. One physical model of this type of system is an oil well-drilling model [155], where  GJ Id 2 ca , b= , q1 = q 2 = , c¯ = , p = 1, c = 2, a= IB 2IB Id Id with Id the moment of inertia per unit of length, G the shear modulus, J the geometric moment of inertia of the drill pipe, ca the anti-damping coefficient at the bit due to the stick-slip instability, and IB the moment of inertia of the bottom-hole assembly.

CHAPTER FOURTEEN

408

As in the previous chapters, we adopt the following notation: • The symbol Z+ denotes the set of all nonnegative integers and R+ := [0, +∞). • Let U ⊆ Rn be a set with a nonempty interior, and let Ω ⊆ R be a set. By C 0 (U ; Ω), we denote the class of continuous mappings on U , which take values in Ω. • We use the notation N for the set {1, 2, · · · }—that is, the natural numbers without 0. • For an I ⊆ R+ , the space C 0 (I; L2 (0, 1)) is the space of continuous mappings I  t → u[t] ∈ L2 (0, 1). • Let u : R+ × [0, 1] → R be given. We use the notation u[t] to denote the profile of u at certain t ≥ 0—that is, (u[t])(x) = u(x, t), for all x ∈ [0, 1]. 14.2

NOMINAL CONTROL DESIGN

We introduce the following backstepping transformation: α(x, t) = z(x, t), β(x, t) = w(x, t) −

(14.13)



x 0

φ(x, y)w(y, t)dy − λ(x)ζ(t),

(14.14)

where 1 κ e q2 (a−q1 bc)x , q1 p + q2 κ 1 φ(x, y; q1 , q2 ) = e q2 (a−q1 bc)(x−y) b, q2

λ(x; q1 , q2 ) =

(14.15) (14.16)

and κ is a design parameter, first mentioned in assumption 14.2, and to be chosen according to (14.12). Writing q1 , q2 after “; ” in λ(x; q1 , q2 ) and φ(x, y; q1 , q2 ) emphasizes the fact that φ(x, y), λ(x) depend on the unknown parameters q1 , q2 . By applying the backstepping transformations (14.13), (14.14), we convert the plant (14.1)–(14.5) to the target system ˙ = −mζ(t) + b(q1 p + q2 )β(0, t), ζ(t)

(14.17)

α(0, t) = c0 ζ(t) − pβ(0, t), αt (x, t) = −q1 αx (x, t),

(14.18) (14.19)

βt (x, t) = q2 βx (x, t), β(1, t) = 0,

(14.20) (14.21)

where c0 = c − pλ(0; q1 , q2 ). The control input U (t) is chosen as    1 1 U (t) = − q1 z(1, t) − q2 φ(1, y; q1 , q2 )w(y, t)dy − q2 λ(1; q1 , q2 )ζ(t) c¯ 0 to ensure (14.21).

(14.22)

(14.23)

REGULATION-TRIGGERED ESTIMATION

409

Define Ω(t) = z[t] 2 + w[t] 2 + ζ(t)2

(14.24)

and a vector θ containing the two unknown parameters as θ = [q1 , q2 ]T .

(14.25)

Through Lyapunov analysis for the target system (14.17)–(14.21) and applying the invertibility of the backstepping transformation, the estimate Ω(t) ≤ Υθ Ω(0)e−λ1 t ,

t ≥ 0,

is obtained, where the decay rate λ1 is   1 (q1 bc − a − bκ), δq2 , δq1 λ1 = min 2 with the analysis parameter δ > 0 selected as    q2 q1 rb 1 δ ≤ ln q¯1 − q 1 2ra

(14.26)

(14.27)

(14.28)

in order to meet the needs of the Lyapunov analysis, which will become evident in section 14.4. For the right-hand side of (14.28) to be positive, we need  q2 q1 rb 1 > 1. (14.29) (¯ q1 − q 1 ) 2ra This is ensured by assumption 14.2. To recap, δ in (14.28) is only an analysis parameter, which influences the decay rate in (14.26). The overshoot coefficient Υθ obtained in (14.26), through the straightforward and omitted Lyapunov analysis, is Υθ =

ξ2 ξ4 , ξ1 ξ3

(14.30)

where the positive constants ξ1 , ξ2 , ξ3 , ξ4 are   1 1 −δ 1 ra , rb e , ξ1 = min , 2 2 2   1 1 1 ra eδ , rb , , ξ2 = max 2 2 2 1  , ξ3 = 3κ2 b2 3κ2 2 2 +1 max 3 + q2 m

¯ , (q2 +q

m

¯ 2 1 p) 2   2 2 3κ b 3κ2 2 2

¯ n

,

¯ n

+ 1 , ξ4 = max 3 + q22 (q2 + q1 p)2

(14.31) (14.32) (14.33)

(14.34)

with m(x) ¯ =e

a−q1 bc+bκ x q2

,

1

n ¯ (x) = e q2 (a−q1 bc)x

(14.35)

CHAPTER FOURTEEN

410

and with the positive constants ra , rb required in assumption 14.2 to satisfy (14.10), (14.9). The relations (14.30)–(14.35), (14.9), (14.10), (14.28) will be used in the proofs of the main results in section 14.4. We refer to the controller U (t) in (14.23) as the nominal feedback, which requires the knowledge of the values of the parameters q1 , q2 . The adaptive scheme working with the nominal feedback (14.23) and guaranteeing exponential regulation is presented in the next section.

14.3

REGULATION-TRIGGERED ADAPTIVE CONTROL

The regulation-triggered adaptive control includes a certainty-equivalence controller and a least-squares identifier that is updated in a sequence of time instants. The Certainty-Equivalence Controller The control action in the interval between two consecutive events is the result of replacing the unknown parameters q1 , q2 in the nominal control law (14.23) by their estimates qˆ1 , qˆ2 at the beginning of the interval, with the estimates qˆ1 , qˆ2 kept constant during the interval. In other words, the adaptive version of (14.23) is given by   1 1 ˆ i ))w(y, t)dy φ(1, y; θ(τ U (t) = − qˆ1 (τi )z(1, t) − qˆ2 (τi ) c¯ 0  ˆ (14.36) − qˆ2 (τi )λ(1; θ(τi ))ζ(t) , t ∈ [τi , τi+1 ), i ∈ Z+ , ˆ = (ˆ ˆ i ), θ(t) q1 (t), qˆ2 (t))T = (ˆ q (τi ), qˆ2 (τi ))T = θ(τ

t ∈ [τi , τi+1 ),

i ∈ Z+ ,

(14.37)

where {τi ≥ 0}∞ i=0 is the sequence of time instants, which, along with the estimates ˆ i ), is defined next. θ(τ The Event Trigger The sequence of time instants {τi ≥ 0}∞ i=0 is chosen to satisfy τi+1 = min{τi + T, ri }, i ∈ Z+

(14.38)

with τ0 = 0. The constant T > 0 is a design parameter with the purpose of avoiding a low update frequency and, more importantly, ri > τi is a time instant determined by an event trigger which is designed next. The trigger was introduced in [106] and is based on the progress of the regulation of the states. The event trigger sets ri > τi to be the smallest value of time t > τi for which Ω(t) = Υθ(τ ¯)Ω(τi ) ˆ i ) (1 + a

(14.39)

for Ω(τi ) = 0, where Υθ(τ ˆ i ) ≥ 1 is the coefficient defined by (14.30) with q1 , q2 replaced by qˆ1 , qˆ2 , the design parameter a ¯ is positive, and Ω is defined by (14.24) with the solutions of (14.1)–(14.5) under (14.36). In simple terms, the parameter estimate update is triggered if the plant norm has grown by a certain factor, specifically, by

REGULATION-TRIGGERED ESTIMATION

411

Υθ(τ ¯). Since Υθ(τ ˆ i ) (1 + a ˆ i ) is the overshoot coefficient already associated with the system transient in accordance with the estimate (14.26), the real net growth factor that triggers the update is 1 + a ¯ for any a ¯ > 0 chosen by the user. If a time t > τi satisfying (14.39) does not exist, we set ri = +∞. For the case that Ω(τi ) = 0, we set ri := τi + T . Therefore, the event trigger ri is built as ri : = inf{t > τi : Ω(t) = Υθ(τ ¯)Ω(τi )}, Ω(τi ) = 0, ˆ i ) (1 + a

(14.40)

ri : = τi + T, Ω(τi ) = 0.

(14.41)

The following lemma shows that the event trigger is well-defined and produces an increasing sequence of events. Lemma 14.1. The event trigger (14.38), (14.40), (14.41) is well-defined—that is, τi+1 > τi , for all i ∈ Z+ . Proof. If Ω(τi ) = 0, it follows from (14.38), (14.41) that τi+1 = τi + T . If Ω(τi ) = 0 and ri defined in (14.40) is less than τi + T , the dwell time τi+1 − τi is greater than ¯)Ω(τi ) > Ω(τi ) and Ω(t) defined in (14.24) is a zero because Ω(τi+1 ) = Υθ(τ ˆ i ) (1 + a continuous function on t ∈ [τi , τi+1 ]. If ri ≥ τi + T or ri is infinite, it follows from (14.38) that τi+1 = τi + T . The above lemma allows us to define the solution on the interval [0, limi→∞ (τi )). Least-Squares Identifier The least-squares identifier activated by the trigger defined by (14.38)–(14.41) is designed in this subsection. The design idea of the identifier follows from [106]. According to the considered dynamic model, by applying integration, formulating a cost function, and using Fermat’s theorem, we construct a matrix equation, with an unknown vector of plant parameters and with the equation’s coefficients being the plant states over a time interval. The parameter estimation is then treated as a convex optimization problem with linear equality constraints. By virtue of (14.1)–(14.5), we get for τ > 0 and n = 1, 2, . . . that d dτ

 1 cos(xπn)w(x, τ )dx + ζ(τ ) b 0 0  1 = −q1 (−1)n z(1, τ ) + q1 z(0, τ ) − q1 πn sin(xπn)z(x, τ )dx 



1

1

cos(xπn)z(x, τ )dx +

0



+ q2 (−1)n w(1, τ ) − q2 w(0, τ ) + q2 πn a + ζ(τ ) + q2 w(0, τ ) − q1 z(0, τ ) b  1  = −q1 πn sin(xπn)z(x, τ )dx + q2 πn 0

a + (−1)n c¯U (τ ) + ζ(τ ), b

1 0

sin(xπn)w(x, τ )dx

1 0

sin(xπn)w(x, τ )dx

where (14.4) was inserted into (14.1) to replace q1 bcζ(t) and to yield

(14.42)

CHAPTER FOURTEEN

412 d ζ(τ ) = aζ(τ ) + b(q2 w(0, τ ) − q1 z(0, τ )). dτ

(14.43)

Integrating (14.42) from μi+1 to t yields fn (t, μi+1 ) = q1 gn,1 (t, μi+1 ) + q2 gn,2 (t, μi+1 ),

(14.44)

where  1 cos(xπn)w(x, t)dx + ζ(t) b 0 0  1   1 1 − cos(xπn)z(x, μi+1 )dx + cos(xπn)w(x, μi+1 )dx + ζ(μi+1 ) b 0 0  t

a − (−1)n c¯U (τ ) + ζ(τ ) dτ, b μi+1  t  1 gn,1 (t, μi+1 ) = − πn sin(xπn)z(x, τ )dxdτ, (14.45) 



1

fn (t, μi+1 ) =

1

cos(xπn)z(x, t)dx +



μi+1 t

gn,2 (t, μi+1 ) =

πn μi+1

0



1 0

sin(xπn)w(x, τ )dxdτ

(14.46)

for n = 1, 2, . . . . The time μi+1 introduced in [104] is ˜ T }, μi+1 := min{τf : f ∈ {0, . . . , i}, τf ≥ τi+1 − N

(14.47)

˜ ≥ 1 is a design parameter. In practice, a larger N ˜ can where the positive integer N reduce the effect of measurement noise on the precision of estimation, with a cost of larger computation [104]. Equation (14.44) is written as fn (t, μi+1 ) = ηn (t, μi+1 )θ,

(14.48)

  ηn (t, μi+1 ) = gn,1 (t, μi+1 ), gn,2 (t, μi+1 ) ,

(14.49)

where

and θ is defined in (14.25). Define the function hi,n : R2 → R+ by the formula 

τi+1

hi,n () =

(fn (t, μi+1 ) − ηn (t, μi+1 ))2 dt

(14.50)

μi+1

for n = 1, 2, . . .,  = [1 , 2 ]T , i ∈ Z+ . According to (14.48), the function hi,n () (14.50) has a global minimum hi,n (θ) = 0. We get from Fermat’s theorem (vanishing gradient at extrema) that the following equations hold for every i ∈ Z+ and n = 1, 2, . . . : Hn,1 (μi+1 , τi+1 ) = q1 Qn,1 (μi+1 , τi+1 ) + q2 Qn,2 (μi+1 , τi+1 ),

(14.51)

Hn,2 (μi+1 , τi+1 ) = q1 Qn,2 (μi+1 , τi+1 ) + q2 Qn,3 (μi+1 , τi+1 ),

(14.52)

REGULATION-TRIGGERED ESTIMATION

where



413

τi+1

Hn,1 (μi+1 , τi+1 ) =

gn,1 (t, μi+1 )fn (t, μi+1 )dt,

(14.53)

gn,2 (t, μi+1 )fn (t, μi+1 )dt,

(14.54)

gn,1 (t, μi+1 )2 dt,

(14.55)

gn,1 (t, μi+1 )gn,2 (t, μi+1 )dt,

(14.56)

gn,2 (t, μi+1 )2 dt.

(14.57)

μi+1  τi+1

Hn,2 (μi+1 , τi+1 ) = μi+1  τi+1

Qn,1 (μi+1 , τi+1 ) = μi+1  τi+1

Qn,2 (μi+1 , τi+1 ) = μi+1  τi+1

Qn,3 (μi+1 , τi+1 ) = μi+1

Indeed, (14.51), (14.52) are obtained by differentiating the functions hi,n () defined by (14.50) with respect to 1 , 2 , respectively, and evaluating the derivatives at the position of the global minimum (1 , 2 ) = (q1 , q2 ). The equations (14.51), (14.52) are organized as Zn (μi+1 , τi+1 ) = Gn (μi+1 , τi+1 )θ,

(14.58)

where Zn (μi+1 , τi+1 ) = [Hn,1 (μi+1 , τi+1 ), Hn,2 (μi+1 , τi+1 )]T ,   Qn,1 (μi+1 , τi+1 ) Qn,2 (μi+1 , τi+1 ) . Gn (μi+1 , τi+1 ) = Qn,2 (μi+1 , τi+1 ) Qn,3 (μi+1 , τi+1 )

(14.59) (14.60)

The parameter update law is defined as   2 ˆ ˆ θ(τi+1 ) = argmin | − θ(τi )| :  ∈ Θ, Zn (μi+1 , τi+1 ) = Gn (μi+1 , τi+1 ), n = 1, 2, . . . , (14.61) where Θ = { ∈ R2 : q 1 ≤ 1 ≤ q¯1 , q 2 ≤ 2 ≤ q¯2 }. The estimates are updated at τi+1 — ˆ i+1 ) = [ˆ that is, θ(τ q1 (τi+1 ), qˆ2 (τi+1 )]T —using the plant states over the time interval ˜ in (14.47). [μi+1 , τi+1 ], where the length of the data acquisition can be adjusted by N The initial values of the estimates qˆ1 (0), qˆ2 (0) are chosen as qˆ1 (0) = q 1 , qˆ2 (0) = q 2 , making qˆ1 (τi ) ≤ q1 and qˆ2 (τi ) ≤ q2 , which will be seen more clearly later. If a more robust identifier with respect to random measurement noise is required, the identifier can be designed in a double integral form as in [104]. For the solution notion, according to definition A.5 in [16], we give the following weak solution definition. Definition 14.1. Consider the system (14.62) Rt + Λ(x)Rx + M (x)R = 0, t ∈ [0, ∞), x ∈ [0, L],  +   +   +    L + R (t, 0) R (t, L) N F (x) =K + Rdx, (14.63) X+ − N− (x) R− (t, L) R− (t, 0) F 0 dX = E + R+ (t, L) + E − R+ (t, 0) + E0 X, X ∈ Rp , (14.64) dt

CHAPTER FOURTEEN

414 R(0, x) = R0 (x), X(0) = X0 ,

(14.65)

where R : [0, +∞) × [0, L] → Rn , M : [0, L] → Mn,n (R), and the symbol Mn,n (R), as usual, denotes the set of n × n real matrices, F + : [0, L] → Mm,n (R), F − : [0, L] → Mn−m,n (R), and Λ(x)  diag{Λ+ (x), Λ− (x)} such that Λ+ (x)  diag{λ1 (x), . . . , λm (x)},

(14.66)

−

Λ (x)  −diag{λm+1 (x), . . . , λn (x)},

(14.67)

with λi (x) > 0, ∀x ∈ [0, L], and where   K00 K01 , K00 ∈ Mm,m (R), K01 ∈ Mm,n−m (R), K K10 K11

(14.68)

K10 ∈ Mn−m,m (R), K11 ∈ Mn−m,n−m (R), +

m×p

+

p×m

N ∈R E ∈R

−

(n−m)×p

, N ∈R −

, E ∈R

p×(n−m)

(14.69)

,

(14.70)

, E0 ∈ R

p×p

.

(14.71)

A solution R: (0, +∞) × (0, L) → Rn , X : (0, ∞) → Rp of the system (14.62)–(14.65) is a map R ∈ C 0 ([0, +∞); L2 (0, 1); Rn ), X ∈ C 0 ([0, +∞); Rp ) satisfying (14.65) such that for every T > 0, every ψ ∈ C 1 ([0, T ] × [0, L]; Rn ), and every η ∈ C 1 ([0, T ]; Rp ) satisfying    +  + T + T − Λ (0) Λ+ (L)−1 K10 Λ (L) Λ (L)−1 K00 ψ (t, L) = T + T − ψ − (t, 0) Λ (0) Λ− (0)−1 K11 Λ (L) Λ− (0)−1 K01  +   +  ψ (t, 0) Λ (L)E +T × + η (14.72) − ψ (t, L) Λ+ (0)E −T we have  L  L ψ(T, x)T R(T, x)dx − ψ(0, x)T R0 (x) + η T (T )X(T ) − η T (0)X(0) 0 0  T  L ψtT + ψxT Λ + ψ T (Λx − M ) + ψ −T (t, L)Λ− (L)F − = 0 0  +T + + + ψ (t, 0)Λ (0)F Rdxdt 

T

+ 0

 ηtT

T

+ η E0 + ψ

−T

−

−

(t, L)Λ (L)N + ψ

+T

+

(t, 0)Λ (0)N

+

 Xdt. (14.73)

Proposition 14.1. For every (z0 , w0 )T ∈ L2 ((0, 1); R2 ), ζ0 ∈ R and θˆ0 ∈ Θ, the initial boundary value problem (14.1)–(14.5) with (14.36), (14.37), (14.38), (14.40), ˆ = (14.41), (14.47), (14.61) and initial conditions w[0] = w0 , z[0] = z0 , ζ(0) = ζ0 , θ(0) T 0 2 ˆ θ0 , has a unique (weak) solution ((z, w) , ζ) ∈ C ([0, limk→∞ (τk )); L (0, 1); R2 ) × C 0 ([0, limk→∞ (τk )); R). Proof. The proof is shown in appendix 14.6A. The flowchart of the mechanism of the regulation-triggered adaptive controller is shown in figure 14.1, and some system properties are given in the following

REGULATION-TRIGGERED ESTIMATION

415

Plant states

Ω(t) t ≥ τi

Ω(τi) ≠ 0

N

Y

Ω(t) = ϒθˆ(τ ) (1+ā)Ω(τi)

N

i

Y

t = τi + T

N

Y τi+1 = τi + T

Update θˆ(τi+1) : activating the least-squares identifier using the measurements of the plant states on the interval [μi+1, τi+1]

The adaptive certaintyequivalence controller

Figure 14.1. The adaptive certainty-equivalence control scheme with regulationtriggered batch least-squares identification. lemmas. In the rest of this chapter, when we say that z[t], w[t] are equal to zero for x ∈ [0, 1], t ∈ [μi+1 , τi+1 ], or not identically zero on the same domain, we mean “except possibly for finitely many discontinuities of the functions w[t], z[t].” These discontinuities are isolated curves in the rectangle [0, 1] × [μi+1 , τi+1 ]. Lemma 14.2. The sufficient and necessary condition of Qn,1 (μi+1 , τi+1 ) = 0 (or Qn,3 (μi+1 , τi+1 ) = 0) for n = 1, 2, . . . is z[t] = 0 (or w[t] = 0) on t ∈ [μi+1 , τi+1 ]. Proof. Necessity: if Qn,1 (μi+1 , τi+1 ) = 0 for n = 1, 2, . . . , then the definition (14.55) in conjunction with the continuity of gn,1 (t, μi+1 ) for t ∈ [μi+1 , τi+1 ] (a consequence of definition (14.45) and the fact that z ∈ C 0 ([μi+1 , τi+1 ]; L2 (0, 1))) implies gn,1 (t, μi+1 ) = 0, t ∈ [μi+1 , τi+1 ].

(14.74)

According to the definition (14.45) and the continuity of the mapping τ → 1 sin(xπn)z[τ ]dx (a consequence of the fact that z ∈ C 0 ([μi+1 , τi+1 ]; L2 (0, 1)), 0 (14.74) implies  1 sin(xπn)z(x, τ )dx = 0, τ ∈ [μi+1 , τi+1 ] (14.75) 0

√ for n = 1, 2, . . . . Since the set { 2 sin(nπx) : n = 1, 2, . . .} is an orthonormal basis of L2 (0, 1), we have z[t] = 0 for t ∈ [μi+1 , τi+1 ].

416

CHAPTER FOURTEEN

Similarly, if Qn,3 (μi+1 , τi+1 ) = 0 for n = 1, 2, . . . , then w[t] = 0 on t ∈ [μi+1 , τi+1 ], recalling the definitions √(14.57), (14.46) and the fact that w ∈ C 0 ([μi+1 , τi+1 ]; L2 (0, 1)), and the set { 2 sin(nπx) : n = 1, 2, . . .} being an orthonormal basis of L2 (0, 1). Sufficiency: if z[t] = 0 on t ∈ [μi+1 , τi+1 ] (or w[t] = 0 on t ∈ [μi+1 , τi+1 ]), then Qn,1 (μi+1 , τi+1 ) = 0 (or Qn,3 (μi+1 , τi+1 ) = 0) for n = 1, 2, . . . is obtained directly, according to (14.45), (14.55) and (14.46), (14.57). The proof of lemma 14.2 is complete. Lemma 14.3. For the adaptive estimates defined by (14.61) based on the data in the interval t ∈ [μi+1 , τi+1 ], the following statements hold: 1) If z[t] is not identically zero and w[t] is identically zero on t ∈ [μi+1 , τi+1 ], then qˆ1 (τi+1 ) = q1 , qˆ2 (τi+1 ) = qˆ2 (τi ). 2) If w[t] is not identically zero and z[t] is identically zero on t ∈ [μi+1 , τi+1 ], then qˆ1 (τi+1 ) = qˆ1 (τi ), qˆ2 (τi+1 ) = q2 . 3) If w[t], z[t] are identically zero on t ∈ [μi+1 , τi+1 ], then qˆ1 (τi+1 ) = qˆ1 (τi ), qˆ2 (τi+1 ) = qˆ2 (τi ). 4) If both w[t] and z[t] are not identically zero on t ∈ [μi+1 , τi+1 ], then qˆ1 (τi+1 ) = q1 , qˆ2 (τi+1 ) = q2 . Moreover, if qˆ1 (τi ) = q1 (or qˆ2 (τi ) = q2 ) for certain i ∈ Z+ , then qˆ1 (t) = q1 (or qˆ2 (t) = q2 ) for all t ∈ [τi , limk→∞ (τk )). Proof. Define the following set   Si :=  ∈ Θ : Zn (μi+1 , τi+1 ) = Gn (μi+1 , τi+1 ), n = 1, 2, . . . .

(14.76)

If Si is a singleton, then it is nothing else but the least-squares estimate of the unknown vector of parameters q1 , q2 on the interval [μi+1 , τi+1 ]. 1) Because z[t] is not identically zero and w[t] is identically zero on t ∈ [μi+1 , τi+1 ], there exists n ∈ N such that Qn,1 (μi+1 , τi+1 ) = 0 recalling lemma 14.2. Define the index set I to be the set of all n ∈ N with Qn,1 (μi+1 , τi+1 ) = 0. According to (14.46) and w[t] being identically zero on t ∈ [μi+1 , τi+1 ], we know that gn,2 (t, μi+1 ) = 0 on t ∈ [μi+1 , τi+1 ] for n = 1, 2, . . . . It follows that Qn,2 (μi+1 , τi+1 ) = 0, Qn,3 (μi+1 , τi+1 ) = 0, Hn,2 (μi+1 , τi+1 ) = 0 for n = 1, 2, . . . , recalling (14.56), (14.57) and (14.54). Then Hn,1 (μi+1 ,τi+1 ) (14.76) implies Si = {(1 , 2 ) ∈ Θ : 1 = Qn,1 (μi+1 ,τi+1 ) , n ∈ I}, recalling (14.59), (14.60). Because (q1 , q2 ) ∈ Si according to (14.58), it follows that Si = {(q1 , 2 ) ∈ Θ : q 2 ≤ 2 ≤ q¯2 }. Therefore, (14.61) shows that qˆ1 (τi+1 ) = q1 and qˆ2 (τi+1 ) = qˆ2 (τi ). 2) The proof of (2) is very similar to the proof of (1), and thus it is omitted. 3) Because w[t], z[t] are identically zero on t ∈ [μi+1 , τi+1 ], then Qn,1 (μi+1 , τi+1 ) = 0, Qn,2 (μi+1 , τi+1 ) = 0, Qn,3 (μi+1 , τi+1 ) = 0, Hn,1 (μi+1 , τi+1 ) = 0, Hn,2 (μi+1 , τi+1 ) = 0 for n = 1, 2, . . . according to (14.45), (14.46), (14.53)–(14.57). It follows that Si = Θ, and then (14.61) shows that qˆ1 (τi+1 ) = qˆ1 (τi ), qˆ2 (τi+1 ) = qˆ2 (τi ). 4) Because w[t] (or z[t]) is not identically zero on t ∈ [μi+1 , τi+1 ], there exists n ∈ N such that Qn,3 (μi+1 , τi+1 ) = 0 (or Qn,1 (μi+1 , τi+1 ) = 0) recalling lemma 14.2. Define the index set I1 to be the set of all n ∈ N with Qn,1 (μi+1 , τi+1 ) = 0, and define the index set I2 to be the set of all n ∈ N with Qn,3 (μi+1 , τi+1 ) = 0. Denote the elements in I1 as n1 ∈ N and those in I2 as n2 ∈ N—that is, Qn1 ,1 (μi+1 , τi+1 ) = 0, Qn2 ,3 (μi+1 , τi+1 ) = 0.

REGULATION-TRIGGERED ESTIMATION

417

From (14.76), recalling (14.59)–(14.60), we obtain   Qn ,2 (μi+1 , τi+1 ) Hn1 ,1 (μi+1 , τi+1 ) Si ⊆ S¯ai := (1 , 2 ) ∈ Θ : 1 = − 2 1 , n1 ∈ I1 , Qn1 ,1 (μi+1 , τi+1 ) Qn1 ,1 (μi+1 , τi+1 ) (14.77)   Qn2 ,2 (μi+1 , τi+1 ) Hn2 ,2 (μi+1 , τi+1 ) ¯ − 1 , n2 ∈ I2 . Si ⊆ Sbi := (1 , 2 ) ∈ Θ : 2 = Qn2 ,3 (μi+1 , τi+1 ) Qn2 ,3 (μi+1 , τi+1 ) (14.78) We next prove by contradiction that Si = {(q1 , q2 )}. Suppose that on the contrary Si = {(q1 , q2 )}; that is, Si defined by (14.76) is not a singleton, which implies the sets S¯ai , S¯bi defined by (14.77), (14.78) are not singletons (because either of S¯ai , S¯bi being a singleton implies that Si is a singleton). It follows that there exist constants ¯ ∈ R, λ ¯ 1 ∈ R such that λ Qn1 ,2 (μi+1 , τi+1 ) ¯ = λ1 , n1 ∈ I1 , (14.79) Qn1 ,1 (μi+1 , τi+1 ) Qn2 ,2 (μi+1 , τi+1 ) ¯ = λ, n2 ∈ I2 Qn2 ,3 (μi+1 , τi+1 ) because if there were two different indices k1 , k2 ∈ I2 with Qk2 ,2 (μi+1 ,τi+1 ) Qk2 ,3 (μi+1 ,τi+1 ) ,

(14.80) Qk1 ,2 (μi+1 ,τi+1 ) Qk1 ,3 (μi+1 ,τi+1 )

=

then the set S¯bi defined by (14.78) would be a singleton, and the same would be the case with S¯ai defined by (14.77) if there were two different Q ¯ (μi+1 ,τi+1 ) Q ¯ (μi+1 ,τi+1 ) = k2 ,2 . indices k¯1 , k¯2 ∈ I1 with k1 ,2 Qk ¯ ,1 (μi+1 ,τi+1 ) 1

Qk ¯ ,1 (μi+1 ,τi+1 ) 2

Moreover, since Si is not a singleton, definition (14.76) implies Qn,2 (μi+1 , τi+1 )2 = Qn,1 (μi+1 , τi+1 )Qn,3 (μi+1 , τi+1 )

(14.81)

for all n ∈ I1 ∪I2 ((14.81) naturally holds for n ∈ / I1 ∪ I2 if N {I1 ∪ I2 } = ∅, because both sides of (14.81) are zero) by recalling (14.60) (because if (14.81) does not hold, it follows from (14.60) that there exists n ∈ I1 ∪ I2 such that det(Gn (μi+1 , τi+1 )) = 0, which implies Si defined by (14.76) is a singleton: a contradiction). According to (14.81), (14.55)–(14.57) and the fact that the Cauchy-Schwarz inequality holds as equality only when two functions are linearly dependent, we obtain the existence ˇ n ∈ R such that ˆ n ∈ R, λ of constants λ 1 2 ˆ n gn ,1 (t, μi+1 ), n1 ∈ I1 , gn1 ,2 (t, μi+1 ) = λ 1 1 ˇ gn2 ,1 (t, μi+1 ) = λn2 gn2 ,2 (t, μi+1 ), n2 ∈ I2

(14.82) (14.83)

for t ∈ [μi+1 , τi+1 ] (notice that gn1 ,1 (t, μi+1 ) and gn2 ,2 (t, μi+1 ) are not identically zero on t ∈ [μi+1 , τi+1 ] because Qn1 ,1 (μi+1 , τi+1 ) = 0 and Qn2 ,3 (μi+1 , τi+1 ) = 0). Recalling (14.79), (14.80), we obtain from (14.55)–(14.57) and (14.82), (14.83) that ¯ 1 gn ,1 (t, μi+1 ), n1 ∈ I1 , gn1 ,2 (t, μi+1 ) = λ 1 ¯ n ,2 (t, μi+1 ), n2 ∈ I2 gn2 ,1 (t, μi+1 ) = λg 2

(14.84) (14.85)

for t ∈ [μi+1 , τi+1 ]. Equations (14.84), (14.85) holding is a necessary condition of the hypothesis that Si is not a singleton. The remaining proof of case 4 is divided into the following three claims.

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418

¯ = 0, λ ¯ 1 = 0 and λ ¯ = ¯1 in (14.84), Claim 14.1. If Si is not a singleton, then λ λ1 (14.85). Proof. The proof is shown in appendix 14.6B. ¯ = 0, λ ¯ 1 = 0 and λ ¯ = ¯1 ) hold if and only Claim 14.2. Equations (14.84), (14.85) (λ λ1 ¯ ¯ = 0) for t ∈ [μi+1 , τi+1 ]. if z[t] + λw[t] = 0 (λ Proof. The proof is shown in appendix 14.6C. ¯ ¯ = 0) is not identically zero for Claim 14.3. The function z[t] + λw[t] (λ t ∈ [μi+1 , τi+1 ]. Proof. The proof is shown in appendix 14.6D. Recalling claims 14.1–14.3, we know that (14.84), (14.85), which is a necessary condition of the hypothesis that Si not be a singleton, does not hold. Consequently, Si is a singleton—that is, Si = {(q1 , q2 )}. Therefore, (14.61) shows that qˆ1 (τi+1 ) = q1 , qˆ2 (τi+1 ) = q2 . The proof of case 4 is complete. If qˆ1 (τi ) = q1 (or qˆ2 (τi ) = q2 ) for certain i ∈ Z+ , recalling (14.61) and the analysis in the above four cases, we have qˆ1 (ti+1 ) = q1 (or qˆ2 (ti+1 ) = q2 ). Repeating the above process, we then have qˆ1 (t) = q1 (or qˆ2 (t) = q2 ) for all t ∈ [τi , limk→∞ (τk )). The proof of lemma 14.3 is complete.

14.4

MAIN RESULT

Theorem 14.1. With arbitrary initial data (z0 , w0 )T ∈ L2 ((0, 1); R2 ), ζ0 ∈ R, and θˆ0 = (q 1 , q 2 )T , for the plant (14.1)–(14.5) under the adaptive certainty-equivalence boundary controller (14.36) where the regulation-triggered BaLSI is defined by (14.37), (14.61) with (14.38), (14.40), (14.41), (14.47), the closed-loop system satisfies the following properties: 1) The Zeno phenomenon does not occur—that is, lim τi = +∞,

i→∞

(14.86)

and the closed-loop system is well-posed. 2) If the finite-time convergence of parameter estimates to the true values does not occur, Ω(t) reaches zero in finite time q11 —that is, Ω(t) ≡ 0 on t ∈ [ q11 , ∞). 3) If the parameter estimates converge to the true values in finite time, there exist positive constants Mθ,θ(0) ˆ , λ1 such that −λ1 t Ω(t) ≤ Mθ,θ(0) , t ≥ 0, ˆ Ω(0)e

(14.87)

is a family of constants parameterized where Ω(t) is given in (14.24), and Mθ,θ(0) ˆ by the positive constants q1 , q2 , qˆ1 (0), qˆ2 (0). The decay rate λ1 is the same as the nominal control result in (14.26). Proof. First, we propose the following claim about the sufficient and necessary condition of the finite-time convergence of parameter estimates to the true values.

REGULATION-TRIGGERED ESTIMATION

419

Claim 14.4. When qˆ1 (0) = q1 (or qˆ2 (0) = q2 ), the estimate qˆ1 (t) (or qˆ2 (t)) reaches the actual value q1 (or q2 ) in finite time if and only if z[t] (or w[t]) is not identically zero on t = [0, limi→∞ (τi )). Proof. The proof is shown in appendix 14.6E. 1) Now we prove the first of the three portions of the theorem. First, if the estimates qˆ1 (t), qˆ2 (t) reach the true values in finite time τε , we have τj = τε + (j − ε)T , j ∈ Z+ , j > ε. The proof of this is shown next. We prove by induction that τi+1 = τi + T for i ≥ ε. Let i ≥ ε be an integer. Notice that (14.23) holds for all t ∈ [τi , τi+1 ). Assume that Ω(τi ) = 0. By virtue of (14.26) and since (14.23) holds, we have Ω(t) ≤ Υθ(τ ˆ i ) Ω(τi )

(14.88)

for all t ∈ [τi , τi+1 ). It follows that Ω(t) ≤ Υθ(τ ¯)Ω(τi ) ˆ i ) Ω(τi ) < Υθ(τ ˆ i ) (1 + a

(14.89)

¯ is positive. Therefore, we get from (14.38), (14.40) that for t ∈ [τi , τi+1 ), where a τi+1 = τi + T for i ≥ ε. The same conclusion follows from (14.38), (14.41) if Ω(τi ) = 0. Therefore, limi→∞ (τi ) = +∞. If the finite-time convergence of the parameter estimates to the true values is not achieved, the proof is divided into the three cases. Case 1: We suppose that the estimate qˆ2 (t) does not reach q2 in finite time, but qˆ1 (t) does reach q1 in finite time. The fact that qˆ2 (t) does not reach q2 in finite time implies w[t] ≡ 0 on t ∈ [0, limi→∞ (τi )) according to claim 14.4, and qˆ2 (t) = qˆ2 (0) = q2 on t ∈ [0, limi→∞ (τi )) according to lemma 14.3. The fact that qˆ1 (t) reaches q1 in finite time implies q˜1 (t) ≡ 0 after a certain τf . Inserting (14.36) into (14.5), we obtain  q2 w(1, t) = qˆ2 (0)

1 0

φ(1, y; qˆ1 (t), qˆ2 (0))w(y, t)dy

+ qˆ2 (0)λ(1; qˆ1 (t), qˆ2 (0))ζ(t) + q˜1 (t)z(1, t).

(14.90)

Considering w[t] ≡ 0 and q˜1 (t) ≡ 0 on t ∈ [τf , limi→∞ (τi )), we obtain from (14.90) that ζ(t) ≡ 0 on t ∈ [τf , limi→∞ (τi )) because qˆ2 (0) = 0 and λ(1; qˆ1 (t), qˆ2 (0)) = 0. Recalling (14.1), considering w[t] ≡ 0 on t ∈ [0, limi→∞ (τi )) and ζ(t) ≡ 0 on t ∈ [τf , limi→∞ (τi )), it further follows that ζ(0) = 0, that is, ζ(t) ≡ 0 on t ∈ [0, limi→∞ (τi )), which means that z(0, t) ≡ 0 on t ∈ [0, limi→∞ (τi )). It follows that Ω(t) defined in (14.24) is nonincreasing on t ∈ [0, q11 ]. Therefore, we have limi→∞ (τi ) > q11 according to the definition of triggering times (14.38), (14.40), (14.41). By virtue of (14.2), (14.4) and z(0, t) ≡ 0 on t ∈ [0, limi→∞ (τi )), we have z[t] ≡ 0 for t ∈ [ q11 , limi→∞ (τi )). If z[t] is identically zero on t ∈ [0, q11 ], then qˆ1 (0) = q1 (because z[t] ≡ 0 on t ∈ [0, limi→∞ (τi )), and qˆ1 (t) reaches q1 in finite time only when the initial estimate is the true value according to claim 14.4). If z[t] is not identically zero in t ∈ [0, q11 ), it implies that the initial condition z(x, 0) is not identically zero for x ∈ (0, 1], moreover, that τf must be less than q11 and the function z(x, 0) is not identically zero on the interval (0, 1 − q1 τf ] for x (otherwise w[t] is not identically zero according to (14.90): a contradiction). The state z(x, t) propagates from its initial condition

CHAPTER FOURTEEN

420

z(x, 0), which is possibly not identically zero only on (0, 1 − q1 τf ], toward the boundary x = 1 and finally vanishes not later than t = q11 (z(1, t) = 0 for t ∈ [0, τf ) and the nonzero values of z(1, t) on t ∈ [τf , q11 ] are eliminated by q˜1 (t) = 0 in (14.90)). Together with w[t] ≡ 0, ζ(t) ≡ 0 on t ∈ [0, limi→∞ (τi )), we conclude that Ω(t) is nonincreasing in t ∈ [0, q11 ], and Ω(t) ≡ 0 for t ∈ [ q11 , limi→∞ (τi )). Therefore, τj = jT , j ∈ Z+ , according to the definition of triggering times (14.38), (14.40), (14.41). Case 2: We suppose that the estimate qˆ1 (t) does not reach q1 in finite time, but qˆ2 (t) does reach q2 in finite time. The fact that qˆ1 (t) does not reach q1 in finite time implies that z(x, t) ≡ 0 on t ∈ [0, limi→∞ (τi )) according to claim 14.4, and qˆ1 (t) = qˆ1 (0) = q1 on t ∈ [0, limi→∞ (τi )) according to lemma 14.3. Recalling (14.36), then (14.1)–(14.5) become c

ζ(t) = e[(a−q1 bc)+b(q2 +q1 p) p ]t ζ(0),

(14.91)

wt (x, t) = q2 wx (x, t), (14.92) c (14.93) w(0, t) = ζ(t), p   1 qˆ2 (t) w(1, t) = φ(1, y; qˆ1 (0), qˆ2 (t))w(y, t)dy + λ(1; qˆ1 (0), qˆ2 (t))ζ(t) , (14.94) q2 0 t ∈ [0, limi→∞ (τi )). If ζ(0) = 0, then z[t], w[t], ζ(t) are identically zero on t ∈ [0, limi→∞ (τi )) according to (14.91)–(14.93). Next, we discuss the case of ζ(0) = 0. Considering z[t] ≡ 0 on t ∈ [0, limi→∞ (τi )), the dynamics for w[t], ζ(t) given as (14.91)–(14.94), and the definition of triggering times (14.38), (14.40), (14.41), we have that limi→∞ (τi ) > q12 . The equation w(0, t) = pc ζ(t) (14.93) holding for t ∈ [0, limi→∞ (τi )) requires the initial condition of w to be c 1 c w(x, 0) = e[(a−q1 bc)+b(q2 +q1 p) p ] q2 x ζ(0) p

for ensuring that (14.93) holds on t ∈ [0, q12 ] and w(1, t) to be c 1 c w(1, t) = e[(a−q1 bc)+b(q2 +q1 p) p ] q2 ζ(t), t ∈ [0, lim (τi )) i→∞ p

(14.95)

for ensuring that (14.93) holds on t ∈ [ q12 , limi→∞ (τi )). Comparing (14.95), where w(1, t) is a continuous function by virtue of (14.91), with (14.94) which includes possible discontinuities in qˆ2 , the necessary condition for the equation (14.93) to hold on t ∈ [0, limi→∞ (τi )) is that w(1, t) is a continuous function. In other words, there is no discontinuity in case 2. Considering that the state of the w-PDE in (14.92) propagates from x = 1 to x = 0 with the propagation speed q2 , by representing the function w(y, t) as the future value of ζ(t) and using the expression for ζ(t) given by (14.91), we write the relation (14.94) as w(1, t) =

qˆ2 (t) q2



1

c 1 c φ(1, y; qˆ1 (0), qˆ2 (t)) e[(a−q1 bc)+b(q2 +q1 p) p ] q2 y dy p 0  (14.96) + λ(1; qˆ1 (0), qˆ2 (t)) ζ(t), t ≥ 0.

Comparing (14.95) and (14.96), applying (14.15), (14.16), the necessary condition of w(0, t) = pc ζ(t) (14.93) always holds on t ∈ [ q12 , limi→∞ (τi )) (when ζ(0) = 0), is

REGULATION-TRIGGERED ESTIMATION

1 q2



421

1

cκb qˆ 1(t) (a−ˆq1 (0)bc)(1−y) [(a−q1 bc)+b(q2 +q1 p) pc ] q1 y 2 dy e 2 e p 0  1 c 1 c qˆ2 (t)κ (a−ˆ q1 (0)bc) e qˆ2 (t) ≡ e[(a−q1 bc)+b(q2 +q1 p) p ] q2 + qˆ1 (0)p + qˆ2 (t) p

(14.97)

on t ∈ [0, limi→∞ (τi )). The right-hand side of (14.97) is constant, while the left-hand side of (14.97) includes qˆ2 (t), whose potential values are q2 , qˆ2 (0) because of lemma 14.3. If the left-hand side of (14.97) is varying with qˆ2 (t), then (14.97) does not hold. If the left-hand side of (14.97) is kept constant with qˆ2 (t) = q2 and qˆ2 (t) = qˆ2 (0) (such as qˆ2 (0) = q2 ), since, as we mentioned above, there is no discontinuity in case 2, (14.97) holds only when the design parameter κ is equal to κ∗ , where κ∗ =

cq2 [(a−q1 bc)+b(q2 +q1 p) pc ] q1 2 e p 1  1  bc q1 (a−ˆq1 (0)bc)(1−y) [(a−q1 bc)+b(q2 +q1 p) pc ] q1 y q2 e q2 (a−ˆq1 (0)bc) 2 dy + e 2 ÷ e , qˆ1 (0)p + q2 0 p (14.98)

where the symbol ÷ denotes division. The constant κ∗ is positive because b > 0, c > 0, p > 0, q1 > 0, q2 > 0, qˆ1 (0) > 0. The positivity of κ = κ∗ contradicts κ < 0. Therefore, case 2 would happen only when ζ(0) = 0, where Ω(t) ≡ 0 on t ∈ [0, limi→∞ (τi )). Therefore, τj = jT , j ∈ Z+ . Case 3. If neither qˆ1 (t) nor qˆ2 (t) reaches q1 , q2 , it follows that z[t], w[t], ζ(t) are identically zero on t ∈ [0, limi→∞ (τi ))—that is, Ω(t) ≡ 0 on t ∈ [0, limi→∞ (τi )), according to claim 14.4 and (14.4). Therefore, τj = jT , j ∈ Z+ according to (14.38), (14.41). By virtue of the results in the above discussions, we have limi→∞ (τi ) = +∞. The well-posedness of the closed-loop system is then obtained by recalling proposition 14.1 and limi→∞ (τi ) = +∞. This completes the proof of portion (1) of the theorem. The fact that limi→∞ (τi ) = +∞ allows the solution to be defined on R+ . 2) Now we prove the second of the three portions of the theorem. Recalling the results in the discussions in cases 1–3 in the proof of portion (1) and limi→∞ (τi ) = +∞, we conclude that Ω(t) reaches zero not later than q11 —that is, Ω(t) ≡ 0 on t ∈ [ q11 , ∞)—when the finite-time convergence of the parameter estimates to the true values is not achieved. Thus, portion (2) of the theorem is obtained. 3) Finally, we prove the last of the three portions of the theorem—that is, establish the exponential regulation result when estimates (ˆ q1 (t), qˆ2 (t)) reach the true values (q1 , q2 ) in finite time τε , that is, when qˆ1 (t) = q1 , qˆ2 (t) = q2 , t ≥ τε . Define a Lyapunov function  1  1 1 1 1 δx 2 V (t) = ra e β(x, t) dx + rb e−δx α(x, t)2 dx + ζ(t)2 , t ≥ 0, 2 2 2 0 0

(14.99)

(14.100)

where the positive constants ra , rb , δ are constrained through the inequalities (14.10), (14.9), (14.28). Denoting Ω1 (t) = α[t] 2 + β[t] 2 + ζ(t)2 ,

CHAPTER FOURTEEN

422 we obtain ξ1 Ω1 (t) ≤ V (t) ≤ ξ2 Ω1 (t), t ≥ 0,

(14.101)

where the positive constants ξ1 , ξ2 are shown in (14.31), (14.32). Define the errors between the gains in the nominal control law (14.23) and those in the certainty-equivalence controller (14.36), caused by the parameter estimate errors, as q˜1 (t) = q1 − qˆ1 (t), ˜ R1 (y, t) = q2 φ(1, y; q1 , q2 ) − qˆ2 (t)φ(1, y; qˆ1 (t), qˆ2 (t)), ˜ 2 (t) = q2 λ(1; q1 , q2 ) − qˆ2 (t)λ(1; qˆ1 (t), qˆ2 (t)), R

(14.102) (14.103) (14.104)

where φ(1, y; qˆ1 (t), qˆ2 (t)), λ(1; qˆ1 (t), qˆ2 (t)) are the results of replacing q1 , q2 with ˜ 1 (·, t), R ˜ 2 (t) qˆ1 (t), qˆ2 (t) in φ(1, y; q1 , q2 ) and λ(1; q1 , q2 ). Because of (14.99), q˜1 (t), R are zero for t ≥ τε . Applying the adaptive control law (14.36), recalling (14.23), the boundary condition (14.21) in the target system (14.17)–(14.21) becomes    1 1 ˜ ˜ R1 (y, t)w(y, t)dy − R2 (t)ζ(t) . β(1, t) = q˜1 (t)z(1, t) − q2 0

(14.105)

Applying the Cauchy-Schwarz inequality into the backstepping transformation (14.13), (14.14) and its inverse z(x, t) = α(x, t), w(x, t) = β(x, t) −

(14.106)



x 0

m κb qm (x−y) κ e q2 x ζ(t), e 2 β(y, t)dy − q2 (q2 + q1 p)

(14.107)

we have that Ω1 (t) is bounded by ξ3 Ω(t) ≤ Ω1 (t) ≤ ξ4 Ω(t), t ≥ 0,

(14.108)

where the positive constants ξ3 , ξ4 are defined by (14.33)–(14.35). Taking the derivative of (14.100) along (14.17)–(14.21), (14.105) and applying Young’s inequality and the Cauchy-Schwarz inequality, we get     q2 ra 1 (q1 p + q2 )2 b2 2 m − q1 rb c20 ζ(t) − − q1 rb p2 − V˙ (t) ≤ − β(0, t)2 2 2 2m  1  1 1 1 δx 2 − ra δq2 e β(x, t) dx − rb δq1 e−δx α(x, t)2 dx 2 2 0 0 1 1 − q1 rb e−δ α(1, t)2 + q2 ra eδ β(1, t)2 (14.109) 2 2 for t ≥ 0. Recalling (14.9) and (14.10), then (14.109) becomes 1 V˙ (t) ≤ −λ1 V (t) + q2 ra eδ β(1, t)2 2

(14.110)

for t ≥ 0, where the positive constant λ1 is shown in (14.27). Multiplying both sides of (14.110) by eλ1 t yields

REGULATION-TRIGGERED ESTIMATION

423

d(V (t)eλ1 t ) 1 λ1 t ≤ e q2 ra eδ β(1, t)2 , t ≥ 0, dt 2 and then, integrating from τε to t, we obtain  t 1 λ1 ς e q2 ra eδ β(1, ς)2 dς, t ≥ τε . V (t)eλ1 t − V (τε )eλ1 τε ≤ 2 τε

(14.111)

(14.112)

˜ 1 (t), R ˜ 2 (t) are identically zero Recalling (14.105) and the fact that q˜1 (t), R for t ≥ τ , we get β(1, t) ≡ 0 for t ≥ τε according to (14.105). Therefore, the term  t 1 λ ςε e 1 q2 ra eδ β(1, ς)2 dς in (14.112) is zero. Multiplying both sides of (14.112) by τε 2 e−λ1 t yields V (t) ≤V (τε )e−λ1 (t−τε ) ,

t ≥ τε .

(14.113)

Recalling (14.101), we get Ω1 (t) ≤

ξ2 Ω1 (τε )e−λ1 (t−τε ) , t ≥ τε . ξ1

Recalling (14.108), we further have that Ω(t) ≤ Υθ Ω(τε )e−λ1 (t−τε ) , t ≥ τε ,

(14.114)

where the overshoot coefficient Υθ is shown in (14.30). If τε = 0, we obtain directly from (14.114) that Ω(t) ≤ Υθ Ω(0)e−λ1 t , t ≥ 0. Next, we conduct an analysis for t ∈ [0, τε ] when τε = 0. Recalling (14.13), (14.105), (14.109), (14.110), we obtain   1 1 −δ δ 2 ˙ V (t) ≤ − λ1 V (t) − q1 rb e − ra e (¯ q1 − q 1 ) α(1, t)2 2 q2  1  9ra eδ 2 2 2 2 ˜ ˜ R1 (y, t) w(y, t) dy + R2 (t) ζ(t) + (14.115) 2q2 0 for t ∈ [0, τε ]. Recalling (14.28), which makes the coefficient in the parentheses in ˜ 1 (y, t), R ˜ 2 (t), defined by (14.103), (14.104), front of α(1, t)2 positive, and recalling R where qˆ1 (t) is equal to either qˆ1 (0) or q1 and qˆ2 (t) is equal to either qˆ2 (0) or q2 in t ∈ [0, τε ] according to lemma 14.3, as well as applying (14.107) and the CauchySchwarz inequality, we obtain from (14.115) that V˙ (t) ≤ − λ1 V (t) + Q(ˆ q1 (0), qˆ2 (0), q1 , q2 )V (t),

t ∈ [0, τε ],

(14.116)

where the positive constant Q(ˆ q1 (0), qˆ2 (0), q1 , q2 ), obtained by bounding the last line of (14.115), is a family of constants parameterized by the positive constants q1 , q2 , qˆ1 (0), qˆ2 (0). q1 (0), qˆ2 (0), q1 , q2 ), then by defining a positive constant If λ1 < Q(ˆ λ2 (ˆ q1 (0), qˆ2 (0), q1 , q2 ) = Q(ˆ q1 (0), qˆ2 (0), q1 , q2 ) − λ1 and multiplying both sides of (14.116) by e−λ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 )t we obtain

(14.117)

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424

q1 (0), qˆ2 (0), q1 , q2 )V (t)e−λ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 )t ≤ 0 V˙ (t)e−λ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 )t − λ2 (ˆ (14.118) for t ∈ [0, τε ], that is, d(V (t)e−λ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 )t ) ≤0 dt

(14.119)

for t ∈ [0, τε ]. Then integrating from 0 to t yields V (t) ≤ V (0)eλ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 )t , t ∈ [0, τε ].

(14.120)

Recalling (14.101), (14.108), we get Ω(t) ≤ Υθ Ω(0)eλ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 )t , t ∈ [0, τε ].

(14.121)

If λ1 ≥ Q(q1 (0), q2 (0), q1 , q2 ), then by defining a positive constant λ3 (ˆ q1 (0), qˆ2 (0), q1 , q2 ) = λ1 − Q(ˆ q1 (0), qˆ2 (0), q1 , q2 )

(14.122)

we obtain from (14.116) that Ω(t) ≤ Υθ Ω(0)e−λ3 (ˆq1 (0),ˆq2 (0),q1 ,q2 )t , t ∈ [0, τε ].

(14.123)

Comparing (14.121) and (14.123), we obtain Ω(τε ) ≤ Υθ eλ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 )τε Ω(0), t ∈ [0, τε ].

(14.124)

Let us now recap that assumption 14.2—that is, (14.8)—is only used to ensure the existence of a positive δ satisfying (14.28), which enables going from (14.115) to (14.116), with the purpose of arriving at (14.124). In plain words, assumption 14.2 is only used in the Lyapunov analysis when the estimate qˆ1 (t) has not reached the true value q1 , in order to ensure (14.124). Recalling (14.114), we conclude that Ω(t) ≤ Υ2θ eλ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 )τε eλ1 τε Ω(0)e−λ1 t ,

t ≥ 0.

(14.125)

ˆ reaches θ at τε , then τε ≤ max{ 1 + T, 2T }. Claim 14.5. If θ(t) q2 Proof. The proof is shown in appendix 14.6F. Applying claim 14.5, (14.125) is written as 1

1

Ω(t) ≤ Υ2θ eλ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 ) max{ q2 +T,2T } eλ1 max{ q2 +T,2T } Ω(0)e−λ1 t

(14.126)

for t ≥ 0. Denoting 1

1

Mθ,θ(0) = Υ2θ eλ2 (ˆq1 (0),ˆq2 (0),q1 ,q2 ) max{ q2 +T,2T } eλ1 max{ q2 +T,2T } , ˆ we obtain (14.87). This completes the proof of portion (3) of the theorem. With the proof of theorem 14.1 completed, we thank the reader for sticking with us for the nearly six-page ride and commend the reader’s stamina.

REGULATION-TRIGGERED ESTIMATION

14.5

425

SIMULATION

The simulation is conducted for the plant (14.1)–(14.5) with the model parameters taken as a = 13, b = 1, c = 2, p = 0.5,

(14.127)

c¯ = 1, q1 = 4, q2 = 6,

(14.128)

where q1 , q2 are treated as unknown, with the known bounds q¯1 = 6, q 1 = 2, q¯2 = 7, q 2 = 3. The initial values for the test are chosen as π

z(x, 0) = cos πx + + x3 , 4 π

+ x2 , w(x, 0) = sin 1.5πx + 3 1 p ζ(0) = z(0, 0) + w(0, 0). c c

(14.129)

(14.130) (14.131) (14.132)

The finite-difference method is adopted to conduct the simulation with the time and space steps of 0.0001 and 0.01, respectively. For the regulation-triggered BaLSI defined by (14.37), (14.38), (14.40), (14.41), (14.47), (14.61), we choose n = 1, 2, . . . , 7 and ˜ = 1, T = 8 a ¯ = 0.8, N

(14.133)

and take the initial values of the estimates as qˆ1 (0) = q 1 qˆ2 (0) = q 2 —namely, we start the parameter estimates from their lower bounds. Using the given bounds in (14.129) to determine the gain κ in the controller (14.36) by (14.12), a control gain satisfying κ < −267 is needed. For κ = −320, recalling the model parameters in (14.127), (14.128), we obtain rb < 0.079, ra > 0.06 according to (14.9), (14.10), which indicates that q¯1 − q 1 needs to be smaller than 4.01 according to assumption 14.2 (satisfied by the known bounds of q1 given in (14.129)). The source of the high gain κ is the first term of (14.12) which is used in claim 14.3 to exclude some rare ¯ for x ∈ [0, 1], t ∈ [μi+1 , τi+1 ] and extreme situations (w(x, t) = M and z(x, t) = −λM ¯ where M , λ are nonzero constants) affecting the exact parameter estimation. In the simulation, we find that the high gain is actually not needed (the aforementioned extreme situations do not happen), and κ = −6 derived from the second term in (14.12) is perfectly sufficient to achieve a satisfactory result. According to the initial by (14.30)– values of the estimates qˆ1 (0) qˆ2 (0), we get a large initial value Υθ(0) ˆ (14.35), (14.9), (14.10), which is a conservative value obtained by the stability analysis in section 14.4. In this simulation, guided by reason rather than by a highly conservative estimate, we adopt a smaller initial value as Υθ(0) = 5.5, which prevents ˆ the activation of the identifier from being extremely late (particularly relative to T ). From figure 14.2, we observe that the estimates qˆ1 , qˆ2 reach the exact values of the unknown parameters q1 = 4, q2 = 6 at t = 0.13 s, in just one trigger. In figure 14.3, the nominal control input applied at the boundary x = 1 goes through the PDE domain and reaches the boundary x = 0, starting to regulate the ODE state ζ(t) at t = q12 ≈ 0.17 s. As shown in figure 14.2, the estimates reach the true values

CHAPTER FOURTEEN

426 7 6 5 4 3

1

qˆ1 qˆ2

t = 0.13

2 0

1

2

3

4

5

Time (s)

Figure 14.2. Parameter estimates qˆ1 (t), qˆ2 (t).

3

t = 0.3

Nominal control Adaptive regulation-triggered control

|ζ(t)|

2

1

0

0 t = 0.17 1

2

3

4

5

Time (s)

Figure 14.3. The evolution of |ζ(t)| under the nominal control (14.23) and the proposed adaptive regulation-triggered control (14.36).

7 Nominal control Adaptive regulation-triggered control

6 1–

Ω(t) 2

5 4 3 2 1 0

0

1

2

3

4

5

Time (s) 1

Figure 14.4. The evolution of Ω(t) 2 under the nominal control (14.23) and the proposed adaptive regulation-triggered control (14.36). and update the certainty-equivalence controller at t = 0.13 s. Then it takes 1/q2 ≈ 0.17 s for the updated control signal to travel to the ODE; that is, the updated control signal starts to properly regulate the ODE state ζ(t), as intended by the nominal controller, at t = 0.3 s. For the remaining time, as shown in figure 14.3, the performance of the proposed adaptive controller coincides with the nominal feedback, and |ζ(t)| converges to zero. Similar results are observed in figure 14.4, 1 which shows the evolution of Ω(t) 2 defined by (14.24) under the nominal control and the proposed adaptive regulation-triggered control. Figures 14.5 and 14.6 show that

w(x, t)

REGULATION-TRIGGERED ESTIMATION

427

4 2 0 –2 –4 –6 –8 1 0.5 x

0 0

2

1

3 t

5

4

z(x, t)

Figure 14.5. The evolution of w(x, t) under the proposed adaptive regulationtriggered control (14.36).

4 2 0 –2 –4 –6 –8 1 0.5 x

0 0

2

1

3 t

5

4

Figure 14.6. The evolution of z(x, t) under the proposed adaptive regulationtriggered control (14.36).

40

U(t)

20 0 –20 Nominal control Adaptive regulation-triggered control

–40 0

1

2

3

4

5

Time (s)

Figure 14.7. The control signals of the nominal control (14.23) and the proposed adaptive regulation-triggered control (14.36).

the PDE states z(x, t), w(x, t) are regulated to zero under the proposed adaptive regulation-triggered controller. The adaptive regulation-triggered control law and the nominal control law are shown in figure 14.7. At the end of this section, and this chapter, let us reiterate that, as announced at the beginning of this chapter, the BaLSI identifier has ensured the perfect identification of the unknown parameters in finite time and has enabled the regulation-triggered adaptive backstepping controller to achieve exponential regulation, with a decay rate matching the rate corresponding to the case of known parameters.

CHAPTER FOURTEEN

428 14.6

APPENDIX

A. Proof of proposition 14.1 Inserting (14.36) into (14.5) yields the closed-loop system ˙ = (a − q1 bc)ζ(t) + b(q2 + q1 p)w(0, t), ζ(t) zt (x, t) = − q1 zx (x, t), wt (x, t) = q2 wx (x, t),

(14.134) (14.135) (14.136)

z(0, t) = cζ(t) − pw(0, t),  1 1 ˆ i ))w(y, t)dy w(1, t) = qˆ2 (τi ) φ(1, y; θ(τ q2 0 1 ˆ i ))ζ(t) + q1 − qˆ1 (τi ) z(1, t) + qˆ2 (τi )λ(1; θ(τ q2 q2

(14.137)

(14.138)

ˆ i ) = (ˆ q1 (τi ), qˆ2 (τi ))T is constant and for t ∈ [τi , τi+1 ), x ∈ [0, 1], i ∈ Z+ , where θ(τ defined by (14.37), (14.38), (14.40), (14.41), (14.47), (14.61). With the purpose of decoupling the ODE and the PDEs, we introduce two transformations. The first is the following Volterra transformation:  x ¯ − y)z(y, t)dy − ϕ(x)ζ(t), z¯(x, t) = z(x, t) − φ(x ¯ (14.139) 0

where the functions ϕ¯ and φ¯ satisfy   b ¯ = 0, q1 ϕ¯ (x) + (a − q1 bc) + (q2 + q1 p)c ϕ(x) p ϕ(0) ¯ = c, ¯ = 1 ϕ(x)b(q ¯ φ(x) 2 + q1 p). q1 p

(14.140) (14.141) (14.142)

Through the transformation (14.139), (14.149), the system (14.134)–(14.138) is converted to ˙ = (a − q1 bc)ζ(t) + b(q2 + q1 p)w(0, t), ζ(t) z¯t (x, t) = − q1 z¯x (x, t), wt (x, t) = q2 wx (x, t), z¯(0, t) = − pw(0, t), w(1, t) =

(14.143) (14.144) (14.145)



(14.146) 1

q1 − qˆ1 (τi ) 1 ˆ i ))w(y, t)dy z¯(1, t) + qˆ2 (τi ) φ(1, y; θ(τ q2 q2 0  q1 − qˆ1 (τi ) 1 ¯ − ψ(1 − y)¯ z (y, t)dy q2 0   1 ˆ i )) − q1 − qˆ1 (τi ) γ¯ (1) ζ(t) + qˆ2 (τi )λ(1; θ(τ (14.147) q2 q2

for t ∈ [τi , τi+1 ), x ∈ [0, 1]. The conditions (14.140)–(14.142) of the functions ϕ¯ and φ¯ in the transformation (14.139), (14.149) are obtained through matching (14.134)– (14.138) and (14.143)–(14.147), as follows. Inserting (14.139) into (14.144) and using

REGULATION-TRIGGERED ESTIMATION

429

(14.134), (14.135), (14.137), we obtain z¯t (x, t) + q1 z¯x (x, t)  x ¯ − y)zx (y, t)dy − ϕ(x)(a = zt (x, t) + q1 ¯ − q1 bc)ζ(t) φ(x 0

¯ − ϕ(x)b(q ¯ 2 + q1 p)w(0, t) + q1 zx (x, t) − q1 φ(0)z(x, t)  x − q1 φ¯ (x − y)z(y, t)dy − q1 ϕ¯ (x)ζ(t) 0

¯ ¯ ¯ = q1 φ(0)z(x, t) − q1 φ(x)z(0, t) − q1 φ(0)z(x, t) − q1 ϕ¯ (x)ζ(t) 1 1 ¯ ¯ − ϕ(x)(a ¯ − q1 bc)ζ(t) − ϕ(x)b(q 2 + q1 p)cζ(t) + ϕ(x)b(q 2 + q1 p)z(0, t) p p   1 ¯ ϕ(x)b(q ¯ = 2 + q1 p) − q1 φ(x) z(0, t) p   1  ¯ + q p)c + q ϕ ¯ (x) ζ(t) = 0. (14.148) − ϕ(x)(a ¯ − q1 bc) + ϕ(x)b(q 2 1 1 p For (14.148) to hold, we obtain the conditions (14.140), (14.142). Matching (14.137) and (14.146), we get the condition (14.141). Because φ¯ is a continuous function, we find that the inverse transformation  x ¯ − y)¯ ψ(x z (y, t)dy − γ¯ (x)ζ(t) (14.149) z(x, t) = z¯(x, t) − 0

¯ γ¯ is ensured exists (see, e.g., chapter 9.9 in [175]), where the well-posedness of ψ, by the well-posedness of (14.140)–(14.142). Applying the second transformation  1  1 K1i (x)¯ z (x, t)dx − K2i (x)w(x, t)dx (14.150) χ(t) = ζ(t) − 0

0

for t ∈ [τi , τi+1 ), i ∈ Z+ , where the functions K1i , K2i satisfy the well-posed firstorder ODEs 

 ˆ i )) K1i (x) − q1 K1i  (x) (a − q1 bc) + K2i (1) (q1 − qˆ1 (τi ))¯ γ (1) − qˆ2 (τi )λ(1; θ(τ ¯ − x), (14.151) = −K2i (1)(q1 − qˆ1 (τi ))ψ(1 

 ˆ i )) K2i (x) + q2 K2i  (x) (a − q1 bc) + K2i (1) (q1 − qˆ1 (τi ))¯ γ (1) − qˆ2 (τi )λ(1; θ(τ ˆ i )), = K2i (1)ˆ q2 (τi )φ(1, x; θ(τ

(14.152)

K1i (1)q1 = K2i (1)(q1 − qˆ1 (τi )), K2i (0)q2 = −b(q2 + q1 p) − pK1i (0)q1 ,

(14.153) (14.154)

transforms the system (14.143)–(14.147) to χ(t) ˙ = Ai χ(t), z¯t (x, t) = − q1 z¯x (x, t),

(14.155) (14.156)

wt (x, t) = q2 wx (x, t), z¯(0, t) = − pw(0, t),

(14.157) (14.158)

CHAPTER FOURTEEN

430 q1 − qˆ1 (τi ) w(1, t) = z¯(1, t) + q2





1 0

D1i (x)w(x, t)dx +

1 0

D2i (x)¯ z (x, t)dx + D3i χ(t), (14.159)

for t ∈ [τi , τi+1 ), x ∈ [0, 1], where ˆ i )), Ai = a − q1 bc + K2i (1)(q1 − qˆ1 (τi ))¯ γ (1) − K2i (1)ˆ q2 (τi )λ(1; θ(τ   1 1 q1 − qˆ1 (τi ) ˆ ˆ qˆ2 (τi )λ(1; θ(τi )) − γ¯ (1) K1i (x), D1i (x) = qˆ2 (τi )φ(1, x; θ(τi )) + q2 q2 q2   1 q1 − qˆ1 (τi ) ¯ q1 − qˆ1 (τi ) ˆ qˆ2 (τi )λ(1; θ(τi )) − γ¯ (1) K2i (x), D2i (x) = − ψ(1 − x) + q2 q2 q2 D3i =

1 ˆ i )) − q1 − qˆ1 (τi ) γ¯ (1). qˆ2 (τi )λ(1; θ(τ q2 q2

The conditions (14.151)–(14.154) of K1i (x), K2i (x) are defined by matching (14.143)– (14.147) and (14.155)–(14.159), as follows. Inserting (14.150) into (14.155) and using (14.143)–(14.147), we obtain χ(t) ˙ − Ai χ(t)



ˆ i )) χ(t) = χ(t) ˙ − (a − q1 bc)χ(t) − K2i (1)(q1 − qˆ1 (τi ))¯ γ (1) − K2i (1)ˆ q2 (τi )λ(1; θ(τ  1  1 ˙ K1i (x)¯ zt (x, t)dx − K2i (x)wt (x, t)dx = ζ(t) − 0



− (a − q1 bc)ζ(t) + (a − q1 bc)

0 1

0

K1i (x)¯ z (x, t)dx

 1 + (a − q1 bc) K2i (x)w(x, t)dx 0

ˆ i )) χ(t) − K2i (1)(q1 − qˆ1 (τi ))¯ γ (1) − K2i (1)ˆ q2 (τi )λ(1; θ(τ  1 K1i (x)¯ zx (x, t)dx = b(q2 + q1 p)w(0, t) + q1  − q2

1 0

0



K2i (x)wx (x, t)dx + (a − q1 bc)

1 0

K1i (x)¯ z (x, t)dx

 1 + (a − q1 bc) K2i (x)w(x, t)dx 0

ˆ i )) χ(t) γ (1) − K2i (1)ˆ q2 (τi )λ(1; θ(τ − K2i (1)(q1 − qˆ1 (τi ))¯  1 q1 K1i  (x)¯ z (x, t)dx = b(q2 + q1 p)w(0, t) + K1i (1)q1 z¯(1, t) − K1i (0)q1 z¯(0, t) −  − K2i (1)q2 w(1, t) + K2i (0)q2 w(0, t) +

1 0

0

q2 K2i  (x)w(x, t)dx

 1  1 K1i (x)¯ z (x, t)dx + (a − q1 bc) K2i (x)w(x, t)dx + (a − q1 bc) 0 0

ˆ i )) χ(t) γ (1) − K2i (1)ˆ q2 (τi )λ(1; θ(τ − K2i (1)(q1 − qˆ1 (τi ))¯

REGULATION-TRIGGERED ESTIMATION



1 qˆ2 (τi ) q2



431

1

ˆ i ))w(y, t)dy = − K2i (1)q2 φ(1, y; θ(τ 0   1 q1 − qˆ1 (τi ) ˆ qˆ2 (τi )λ(1; θ(τi )) − γ¯ (1) ζ(t) + q2 q2   q1 − qˆ1 (τi ) q1 − qˆ1 (τi ) 1 ¯ + z¯(1, t) − ψ(1 − y)¯ z (y, t)dy + K1i (1)q1 z¯(1, t) q2 q2 0 + (K2i (0)q2 + b(q2 + q1 p) + pK1i (0)q1 ) w(0, t)  1   q2 K2i  (x) + (a − q1 bc)K2i (x) w(x, t)dx + 

0

1

+



 (a − q1 bc)K1i (x) − q1 K1i  (x) z¯(x, t)dx

0

ˆ i )) χ(t) − K2i (1)(q1 − qˆ1 (τi ))¯ γ (1) − K2i (1)ˆ q2 (τi )λ(1; θ(τ = (K1i (1)q1 − K2i (1)(q1 − qˆ1 (τi ))) z¯(1, t) + (K2i (0)q2 + b(q2 +q1 p) + pK1i (0)q1 ) w(0, t)  1 ˆ i )) q2 (τi )φ(1, x; θ(τ + q2 K2i  (x) + (a − q1 bc)K2i (x) − K2i (1)ˆ 0    ˆ γ (1) − K2i (1)ˆ q2 (τi )λ(1; θ(τi )) K2i (x) w(x, t)dx + K2i (1)(q1 − qˆ1 (τi ))¯ 

1



(a − q1 bc)K1i (x) − q1 K1i  (x)  ¯ − x) + K2i (1)(q1 − qˆ1 (τi ))¯ + K2i (1)(q1 − qˆ1 (τi ))ψ(1 γ (1)   ˆ i )) K1i (x) z¯(x, t)dx = 0. q2 (τi )λ(1; θ(τ − K2i (1)ˆ +

0

(14.160)

For (14.160) to hold, the conditions (14.151)–(14.154) are obtained. The equation set (14.156)–(14.159), where χ(t) is a well-defined external signal generated by (14.155), has an analogous structure with (5), (6) in [35]. According to the result in part 1 of the appendix in [35], the system (14.155)–(14.159) has a unique solution on t ∈ [τi , τi+1 ) for all (w[τi ], z¯[τi ])T ∈ L2 ((0, 1); R2 ), χ(τi ) ∈ R. By virtue of the transformations (14.150), (14.149), we see that, for given (w[τi ], z[τi ])T ∈ L2 ((0, 1); R2 ), ζ(τi ) ∈ R, the system (14.134)–(14.138) has a unique solution for t ∈ [τi , τi+1 ). Recalling the definition of the weak solution in definition ˆ i) ∈ 14.1, we find that for every (z[τi ], w[τi ])T ∈ L2 ((0, 1); R2 ), ζ(τi ) ∈ R, and θ(τ Θ, there exists a unique (weak) solution ((z, w)T , ζ) ∈ C 0 ([τi , τi+1 ]; L2 (0, 1); R2 ) × C 0 ([τi , τi+1 ]; R) to the system (14.1)–(14.5) with (14.36), (14.37), (14.38), (14.40), (14.41), (14.47), (14.61). For every (z0 , w0 )T ∈ L2 ((0, 1); R2 ), ζ0 ∈ R, and θˆ0 ∈ Θ, through iterative constructions between successive triggering times, the proposition is thus obtained. B. Proof of claim 14.1 ¯ = 0, it means that Qn ,2 (μi+1 , τi+1 ) = 0 by recalling (14.85), (14.56), and then If λ 2 H ,2 (μi+1 ,τi+1 ) in (14.78). Together with (14.77), we get 2 = Qnn2 ,3 (μi+1 ,τi+1 ) 2

CHAPTER FOURTEEN

432

  Hn1 ,1 (μi+1 , τi+1 ) Hn2 ,2 (μi+1 , τi+1 ) Qn1 ,2 (μi+1 , τi+1 ) Hn2 ,2 (μi+1 , τi+1 ) − , Si = , Qn1 ,1 (μi+1 , τi+1 ) Qn2 ,3 (μi+1 , τi+1 ) Qn1 ,1 (μi+1 , τi+1 ) Qn2 ,3 (μi+1 , τi+1 ) (14.161) ¯ 1 = 0, it means that Qn ,2 which is a singleton: a contradiction. Similarly, if λ 1 H ,1 (μi+1 ,τi+1 ) (μi+1 , τi+1 ) = 0 by recalling (14.84), (14.56), and then 1 = Qnn1 ,1 in (14.77). (μi+1 ,τi+1 ) 1 Together with (14.78), we get   Hn1 ,1 (μi+1 , τi+1 ) Hn2 ,2 (μi+1 , τi+1 ) Hn1 ,1 (μi+1 , τi+1 ) Qn2 ,2 (μi+1 , τi+1 ) , − Si = , Qn1 ,1 (μi+1 , τi+1 ) Qn2 ,3 (μi+1 , τi+1 ) Qn1 ,1 (μi+1 , τi+1 ) Qn2 ,3 (μi+1 , τi+1 ) (14.162) ¯ = 0, λ ¯ 1 = 0. According to (14.45), which is a singleton: a contradiction. Therefore, λ ¯ ¯ (14.46) and λ = 0, λ1 = 0, we find from (14.84), (14.85) that 



t

πn1 μi+1

1 =−¯ λ1  t





πn1 

1 0

μi+1

μi+1



sin(xπn1 )z(x, τ )dxdτ

0

t

πn2 ¯ = −λ

1

sin(xπn1 )w(x, τ )dxdτ, n1 ∈ I1 ,

(14.163)

1 0

t

πn2 μi+1

sin(xπn2 )z(x, τ )dxdτ 

1 0

sin(xπn2 )w(x, τ )dxdτ, n2 ∈ I2

(14.164)

1 for t ∈ [μi+1 , τi+1 ]. According to the continuity of the mappings τ → 0 sin(xπn) 1 z[τ ]dx and τ → 0 sin(xπn)w[τ ]dx, n ∈ N (a consequence of the fact that z ∈ C 0 ([μi+1 , τi+1 ]; L2 (0, 1)) and w ∈ C 0 ([μi+1 , τi+1 ]; L2 (0, 1)), (14.163), (14.164) imply 

1 0



 1 sin(xπn1 ) z(x, τ ) + ¯ w(x, τ ) dx = 0, n1 ∈ I1 , λ1  1 ¯ sin(xπn2 )(z(x, τ ) + λw(x, τ ))dx = 0, n2 ∈ I2

(14.165) (14.166)

0

for τ ∈ [μi+1 , τi+1 ]. We then prove I1 = I2 in (14.165), (14.166). If I2 includes ele1 / I1 such that 0 sin(xπn2 ) ments not belonging to I1 , there exists n2 ∈ I2 with n2 ∈ / I1 z(x, τ )dx = 0 on τ ∈ [μi+1 , τi+1 ] due to the fact that Qn,1 (μi+1 , τi+1 ) = 0 for n ∈ by recalling (14.55), (14.45), and then 

1 0

 ¯ sin(xπn2 )(z(x, τ ) + λw(x, τ ))dx =

1 0

¯ sin(xπn2 )λw(x, τ )dx,

(14.167)

which is not identically zero on τ ∈ [μi+1 , τi+1 ] because of Qn2 ,3 (μi+1 , τi+1 ) = 0 ¯ = 0. This contradicts (14.166). Similarly, if together with (14.57), (14.46) and λ / I2 such that I1 includes elements not belonging to I2 , there exists n1 ∈ I1 with n1 ∈    1  1 1 sin(xπn1 ) z(x, τ ) + ¯ w(x, τ ) dx = sin(xπn1 )z(x, τ )dx, (14.168) λ1 0 0

REGULATION-TRIGGERED ESTIMATION

433

which is not identically zero on τ ∈ [μi+1 , τi+1 ] because of Qn1 ,1 (μi+1 , τi+1 ) = 0 1 together with (14.55), (14.45), where 0 sin(xπn1 ) λ¯11 w(x, τ )dx = 0 on τ ∈ [μi+1 , τi+1 ] is due to the fact that Qn,3 (μi+1 , τi+1 ) = 0 for n ∈ / I2 by recalling (14.57), (14.46). This contradicts (14.165). Therefore, we conclude that I1 = I2 in (14.165), (14.166). ¯ − ¯1 = 0, recalling I1 = I2 and ¯ = ¯1 by contradiction. If λ We then prove λ λ1 λ1 (14.166), then we obtain  1 sin(xπn1 ) z(x, τ ) + ¯ w(x, τ ) dx λ1 0      1 1 ¯+λ ¯ w(x, τ ) dx sin(xπn1 ) z(x, τ ) + ¯ − λ = λ1 0    1  1 1 ¯ ¯ w(x, τ )dx = sin(xπn2 )(z(x, τ ) + λw(x, τ ))dx + sin(xπn2 ) ¯ − λ λ1 0 0   1 1 ¯ = ¯ −λ sin(xπn2 )w(x, τ )dx, (14.169) λ1 0 



1

which is not identically zero for τ ∈ [μi+1 , τi+1 ] because of Qn2 ,3 (μi+1 , τi+1 ) = 0 ¯ − ¯1 = 0. Claim 14.1 with (14.57), (14.46), which contradicts (14.165). Therefore, λ λ1 is proven. C. Proof of claim 14.2 ¯= ¯ 1 = 0 and λ ¯= According to (14.45), (14.46), the equations (14.84), (14.85) (λ  0, λ 1 ¯ 1 ) are equivalent to λ 

1 0

¯ sin(xπn)(z(x, τ ) + λw(x, τ ))dx = 0, n ∈ I2 ∪ I1 .

(14.170)

If N = I2 ∪ I1 , it means that (14.170) holds for all n ∈ N. If I2 ∪ I1 ⊂ N, recalling the 1 1 definitions of I1 , I2 , we know that 0 sin(xπn)z(x, τ )dx = 0 sin(xπn)w(x, τ )dx = 0 for n ∈ N {I1 ∪ I2 } on τ ∈ [μi+1 , τi+1 ], and thus (14.170) is equivalent to 

1 0

¯ sin(xπn)(z(x, τ ) + λw(x, τ ))dx = 0, n = 1, 2, . . .

(14.171)

√ on τ ∈ [μi+1 , τi+1 ]. Since the set { 2 sin(nπx) : n = 1, 2, . . .} is an orthonormal basis ¯ t) = 0 for t ∈ [μi+1 , τi+1 ]. of L2 (0, 1), if (14.171) holds, it follows that z(x, t) + λw(x, ¯ If z(x, t) + λw(x, t) = 0 for t ∈ [μi+1 , τi+1 ], then (14.171) and (14.84), (14.85) ¯ = ¯1 ), naturally hold. Claim 14.2 is proven. ¯ = 0, λ ¯ 1 = 0 and λ (λ λ1 D. Proof of claim 14.3 ¯ ¯ = 0) to hold on The necessary condition for the equation z(x, t) + λw(x, t) = 0 (λ x ∈ [0, 1], t ∈ [μi+1 , τi+1 ] is that z(x, t), w(x, t) are kept constant on x ∈ [0, 1], t ∈ [μi+1 , τi+1 ] excluding finitely many possible points of discontinuity; that is, w(x, t) = ¯ on x ∈ [0, 1], t ∈ [μi+1 , τi+1 ] excluding finitely many possible M and z(x, t) = −λM points of discontinuity, where M is a nonzero constant (because z[t], w[t] are not identically zero on t ∈ [μi+1 , τi+1 ]). We prove this by contradiction next.

CHAPTER FOURTEEN

434

Taking a spatial interval [x1 , x ¯2 ] ∈ [0, 1] with x ¯2 − x1 ≤ (q1 + q2 )(τi+1 − μi+1 ) (the ¯2 ] is arbitrary on [0, 1], and (x1 , μi+1 ), (¯ x2 , μi+1 ) are position of the interval [x1 , x not points of discontinuity of the functions w(x, t), z(x, t)), suppose that there exist ¯2 ] with xa , xb (without loss of generality we assume xa < xb ) in the interval [x1 , x w(xa , μi+1 ) = w(xb , μi+1 ), where (xa , μi+1 ), (xb , μi+1 ) are not points of discontinu¯ ity of the functions w(x, t), z(x, t). Also, we know that z(xa , μi+1 ) = −λw(x a , μi+1 ) ¯ according to z(x, t) + λw(x, t) = 0 always holding on x ∈ [0, 1], t ∈ [μi+1 , τi+1 ]. Because the state of the w-PDE propagates from x = 1 to x = 0 and the state of the z-PDE propagates from x = 0 to x = 1, with the respective propagation speeds q1 , q2 , according to the statement in page 60 in [16], which indicates that the system (14.2), (14.3) is equivalent to a pair of scalar delay equations even if the solutions are not differentiable and even not continuous with respect to t and x, we get the following relationships: w(xb − s1 q2 , μi+1 + s1 ) = w(xb , μi+1 ),

(14.172)

z(xa + s1 q1 , μi+1 + s1 ) = z(xa , μi+1 )

(14.173)

a for s1 ∈ [0, min{ xq2b , 1−x q1 }], where (xb − s1 q2 , μi+1 + s1 ), (xa + s1 q1 , μi+1 + s1 ) are not the points of discontinuity because (xb , μi+1 ), (xa , μi+1 ) are not the points of a discontinuity. There exists a s1 = xqb1−x +q2 such that xb − s1 q2 = xa + s1 q1 = xc , and then we obtain

w(xc , tc ) = w(xb , μi+1 ), z(xc , tc ) = z(xa , μi+1 ),

(14.174)

2 xa a where xc = q1 xqb1+q ∈ (xa , xb ), tc = μi+1 + xqb1−x +q2 +q2 ∈ (μi+1 , τi+1 ], recalling xb − xa ≤ x ¯2 − x1 ≤ (q1 + q2 )(τi+1 − μi+1 ). Because

¯ ¯ z(xa , μi+1 ) = −λw(x a , μi+1 ) = −λw(xb , μi+1 ), ¯ = 0 and the hypothesis that w(xa , μi+1 ) = w(xb , μi+1 ) and using (14.174), recalling λ we have ¯ z(xc , tc ) = −λw(x c , tc ) with xc ∈ [0, 1], tc ∈ (μi+1 , τi+1 ]: a contradiction. Therefore, the hypothesis that there ¯2 ] such that w(xa , μi+1 ) = w(xb , μi+1 ) ((xa , μi+1 ), exist xa , xb in the interval [x1 , x (xb , μi+1 ) are not points of discontinuity) does not hold, and we conclude that ¯2 ] excluding finitely many posw(x, μi+1 ), z(x, μi+1 ) are kept constant on x ∈ [x1 , x ¯2 ] is arbitrary sible points of discontinuity. Because the position of the interval [x1 , x x2 , μi+1 ) not points of discontinuity of the functions on [0, 1] (with (x1 , μi+1 ), (¯ w(x, t), z(x, t)), we find that w(x, μi+1 ), z(x, μi+1 ) are kept constant for x ∈ [0, 1] excluding finitely many possible points of discontinuity. Taking a time increment s 1 , we have with 0 < s ≤ 2 max{q 1 ,q2 } w(x, μi+1 + s) = w(x + q2 s, μi+1 ) = w(x, μi+1 ) for x ∈ [0, 12 ], excluding the points of discontinuity of the functions w(x, t), z(x, t) 1 ] ensures x + q2 s ∈ (0, 1]. We also along x ∈ [0, 1], t = μi+1 , where s ∈ (0, 2 max{q 1 ,q2 } get z(x, μi+1 + s) = z(x − q1 s, μi+1 ) = z(x, μi+1 )

REGULATION-TRIGGERED ESTIMATION

435

for x ∈ [ 12 , 1], excluding the points of discontinuity of the functions w(x, t), z(x, t) 1 ] ensures x − q1 s ∈ [0, 1). Recalling along x ∈ [0, 1], t = μi+1 , where s ∈ (0, 2 max{q 1 ,q2 } ¯ that z(x, t) + λw(x, t) = 0 always holds on x ∈ [0, 1], t ∈ [μi+1 , τi+1 ], we know that 1 ] excluding finitely z, w are kept constant in x ∈ [0, 1], t ∈ [μi+1 , μi+1 + 2 max{q 1 ,q2 } 1 many possible points of discontinuity. If μi+1 + 2 max{q1 ,q2 } ≥ τi+1 , we directly obtain ¯ the necessary condition for z(x, t) + λw(x, t) = 0 to hold on x ∈ [0, 1], t ∈ [μi+1 , τi+1 ] 1 < τi+1 , mentioned at the beginning of the proof of claim 14.3. If μi+1 + 2 max{q 1 ,q2 } then by repeatedly taking the time increments s and conducting the above process k times, based on the fact that w, z are kept constant for x ∈ [0, 1] at the beginning of each time increment, with excluding finitely many possible points of disconk ≥ τi+1 , we also obtain the necessary condition for tinuity until μi+1 + 2 max{q 1 ,q2 } ¯ z(x, t) + λw(x, t) = 0 to hold on x ∈ [0, 1], t ∈ [μi+1 , τi+1 ] mentioned at the beginning of the proof—namely, that z(x, t), w(x, t) are kept constant on x ∈ [0, 1], t ∈ [μi+1 , τi+1 ], excluding finitely many possible points of discontinuity: ¯ w(x, t) = M, z(x, t) = −λM, (x, t) ∈ ([0, 1] × [μi+1 , τi+1 ])\Id ,

(14.175)

where Id denotes a set of finitely many possible points of discontinuity of the functions w(x, t), z(x, t) in x ∈ [0, 1], t ∈ [μi+1 , τi+1 ], and where M is a nonzero constant (because z[t], w[t] are not identically zero on t ∈ [μi+1 , τi+1 ]). ¯ The situation in which w(x, t) = M and z(x, t) = −λM for (x, t) ∈ ([0, 1] × ¯ (−λ+p) [μi+1 , τi+1 ])\Id means ζ(t) = c M on t ∈ [μi+1 , τi+1 ] according to (14.4), excluding finitely many possible points of discontinuity on t ∈ [μi+1 , τi+1 ]. Recalling (14.1), ¯ it then must be that (a − q1 bc) (−λ+p) + b(q2 + q1 p) = 0. It follows that c ¯ = cb(q2 + q1 p) + p > 0 λ a − q1 bc

(14.176)

because the constants c, b, q1 , q2 , p, and a − q1 bc are positive. Inserting the control input (14.36) into the right boundary condition (14.5), recalling (14.15), (14.16) and ¯ ζ(t) = (−λ+p) M , we know that a necessary condition of w(x, t) = M and z(x, t) = c ¯ for (x, t) ∈ ([0, 1] × [μi+1 , τi+1 ])\Id is for the following equation to hold −λM   ¯ [q2 + (q1 − qˆ1 (τi ))λ]M = κ qˆ2 (τi )

1

1 b (a−ˆ q1 (τi )bc)(1−y) e qˆ2 (τi ) dy q ˆ (τ ) 2 i 0  ¯ + p) 1 qˆ2 (τi )(−λ (a−ˆ q1 (τi )bc) e qˆ2 (τi ) + M, c(ˆ q1 (τi )p + qˆ2 (τi ))

(14.177)

that is,  q2 + (q1 − qˆ1 (τi ))

cb(q2 + q1 p) +p a − q1 bc



1 1    (a−ˆ q1 (τi )bc) (a−ˆ q1 (τi )bc) 1 − e qˆ2 (τi ) e qˆ2 (τi ) (q2 + q1 p) = −κ qˆ2 (τi )b + (14.178) a − qˆ1 (τi )bc qˆ1 (τi )p + qˆ2 (τi ) (a − q1 bc)

because M = 0. Recalling qˆ1 (0) = q 1 and qˆ2 (0) = q 2 , we have 0 < q 1 ≤ qˆ1 (τi ) ≤ q1 , 0 < q 2 ≤ qˆ2 (τi ) ≤ q2 (the consequence of (14.61) and θ ∈ Si defined by (14.76)), which

CHAPTER FOURTEEN

436 implies

q2 +q1 p a−q1 bc

(τi )+ˆ q1 (τi )p ≥ qˆ2a−ˆ q1 (τi )bc . We thus have 1

(a−ˆ q (τ )bc)

1 i 1 − e qˆ2 (τi ) a − qˆ1 (τi )bc 1

1

(a−ˆ q (τ )bc)

1 i e qˆ2 (τi ) (q2 + q1 p) + qˆ1 (τi )p + qˆ2 (τi ) (a − q1 bc)

(a−ˆ q (τ )bc)

1 i 1 − e qˆ2 (τi ) ≥ a − qˆ1 (τi )bc 1 > 0. ≥ a − qˆ1 (τi )bc

1

(a−ˆ q (τ )bc)

1 i e qˆ2 (τi ) (ˆ q2 (τi ) + qˆ1 (τi )p) + qˆ1 (τi )p + qˆ2 (τi ) (a − qˆ1 (τi )bc)

(14.179)

Therefore, recalling that the constants c, b, q1 , q2 , p, a − q1 bc and qˆ1 (τi ), qˆ2 (τi ) are positive, the right-hand side of (14.178) is greater than zero because of κ < 0 and (14.179), and the left-hand side of (14.178) is also greater than zero because of ¯ for (x, t) ∈ qˆ1 (τi ) ≤ q1 . The necessary condition of w(x, t) = M and z(x, t) = −λM ([0, 1] × [μi+1 , τi+1 ])\Id with M = 0 becomes that the design parameter κ is equal to κ ¯ defined as    cb(q2 + q1 p) +p κ ¯ = − q2 + (q1 − qˆ1 (τi )) a − q1 bc 1 1    (a−ˆ q1 (τi )bc) (a−ˆ q1 (τi )bc) 1 − e qˆ2 (τi ) e qˆ2 (τi ) (q2 + q1 p) ÷ qˆ2 (τi )b + < 0. (14.180) a − qˆ1 (τi )bc qˆ1 (τi )p + qˆ2 (τi ) (a − q1 bc) ¯ defined by (14.180) According to (14.179) and qˆ1 (τi ) ≥ q 1 , qˆ2 (τi ) ≥ q 2 , we know that κ is in the following range: 2 +q1 p) (a − q 1 bc)[q2 + (q1 − q 1 )( cb(q a−q1 bc + p)]

−q 2 b

≤κ ¯ ≤ 0.

(14.181)

Recalling the first term in (14.12), we know that κ = κ ¯ . We thus conclude that ¯ with M = 0 on (x, t) ∈ ([0, 1] × [μi+1 , τi+1 ])\Id does w(x, t) = M and z(x, t) = −λM not hold. Claim 14.3 is proven. E. Proof of claim 14.4 We first prove sufficiency. If z[t] (or w[t]) are not identically zero for t = [0, limi→∞ (τi )), there exists an interval [μi+1 , τi+1 ] on which z[t] (or w[t]) are not identically zero. It follows that qˆ1 (τi+1 ) = q1 (or qˆ2 (τi+1 ) = q2 ), recalling lemma 14.3. Next, we prove necessity. When qˆ1 (0) = q1 (or qˆ2 (0) = q2 ), if the estimate reaches the true value at an instant τi+1 —that is, qˆ1 (τi+1 ) = q1 (or qˆ2 (τi+1 ) = q2 )—it follows there exists n ∈ N such that Qn,1 (μi+1 , τi+1 ) = 0 (or Qn,3 (μi+1 , τi+1 ) = 0). (This is true because if Qn,1 (μi+1 , τi+1 ) = 0 (or Qn,3 (μi+1 , τi+1 ) = 0) for all n ∈ N, it would also be true that gn,1 (t, μi+1 ) = 0 (or gn,2 (t, μi+1 ) = 0) for all n ∈ N on t ∈ [μi+1 , τi+1 ], according to (14.55), (14.57). It follows that Qn,2 (μi+1 , τi+1 ) = 0, Hn,1 (μi+1 , τi+1 ) = 0 (or Qn,2 (μi+1 , τi+1 ) = 0, Hn,2 (μi+1 , τi+1 ) = 0) for all n ∈ N according to (14.53)–(14.57). Consequently, we have from (14.61) that qˆ1 (τi+1 ) = qˆ1 (τi ) = q1 (or qˆ2 (τi+1 ) = qˆ2 (τi ) = q2 ) by recalling (14.59)–(14.60)). We then conclude that z[t] (or w[t]) are not identically zero on t ∈ [μi+1 , τi+1 ] according to lemma 14.2. That is, z[t] (or w[t]) are not identically zero on t = [0, limi→∞ (τi )). The proof of claim 14.4 is complete.

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437

F. Proof of claim 14.5 We prove this claim by estimating the largest convergence time of parameter estimates τε in various situations of the initial conditions z[0], w[0], ζ(0). Inserting (14.36) into (14.5), we get  q2 w(1, t) = qˆ2 (t)

1 0

ˆ ˆ φ(1, y; θ(t))w(y, t)dy + qˆ2 (t)λ(1; θ(t))ζ(t) + q˜1 (t)z(1, t). (14.182)

Case 1: z[0] = 0, w[0] = 0, ζ(0) = 0. According to lemma 14.3, we have qˆ1 (τ1 ) = q1 and q˜1 (t) ≡ 0 for t ≥ τ1 . If w[t] ≡ 0 on t ∈ [0, τ1 ], then w[t] and ζ(t) are identically zero on t ≥ 0 according to (14.1), (14.3), and (14.182), with q˜1 (t) ≡ 0 for t ≥ τ1 . If qˆ2 (0) = q2 , it follows that qˆ2 (t) cannot reach the true value q2 according to claim 14.4 with property (1): a contradiction with the fact that qˆ2 (t) would reach q2 in finite time. Thus, w[t] is not identically zero on t ∈ [0, τ1 ] if qˆ2 (0) = q2 . It follows that qˆ2 (t) can reach q2 not later than τ1 according to lemma 14.3. We know from (14.38) that the dwell time is less than or equal to T . Therefore, τε ≤ T . Case 2: w[0] = 0, z[0] = 0, ζ(0) = 0. The maximum time taken by the nonzero values of w[0] to propagate to x = 0 and enter z(0, t) is q12 . Therefore, the estimate qˆ1 (t) would reach the true value q1 not later than τf = min{τf : f ∈ Z+ , τf > q12 } according to lemma 14.3. Because of w[0] = 0, we have qˆ2 (τ1 ) = q2 . It follows that τε ≤ q12 + T because the dwell time is less than or equal to T . Case 3: ζ(0) = 0, z[0] = 0, w[0] = 0. According to (14.182) and (14.4), we know that w[t], z[t] are not identically zero on t ∈ [0, τ1 ], which implies that the estimates ˆ reach the true values θ not later than τ1 according to lemma 14.3. Therefore, θ(t) τε ≤ τ 1 ≤ T . Case 4: ζ(0) = 0, w[0] = 0, z[0] = 0. The necessary condition of the fact that z[t] is identically zero (i.e., w(0, t) = pc ζ(t) always holds) for t ∈ [0, τf ] where τf = min{τf : f ∈ Z+ , τf > q12 } is κ > 0, according to the analysis in case 2 in the proof of portion (1) of the theorem. Recalling κ < 0 in (14.12), we know that z[t] is not identically zero on t ∈ [0, τf ], which implies that the estimate qˆ1 (t) reaches the true value q1 not later than τf according to lemma 14.3. Because w[0] = 0, we have qˆ2 (τ1 ) = q2 . Therefore, τε ≤ q12 + T . Case 5: ζ(0) = 0, z[0] = 0, w[0] = 0. According to lemma 14.3, we have qˆ1 (τ1 ) = q1 and q˜1 (t) ≡ 0 for t ≥ τ1 . If w[t] ≡ 0 on t ∈ [0, τ2 ], it follows from (14.1) that ζ(t) = e(a−q1 bc)t ζ(0) is not identically zero on t ∈ [τ1 , τ2 ]. From (14.182) we know that w(1, t) is not identically zero on t ∈ [τ1 , τ2 ]: a contradiction. Therefore, w[t] are not identically zero on t ∈ [0, τ2 ], which implies that the estimate qˆ2 (t) reaches the true value q2 not later than τ2 according to lemma 14.3. Therefore, we find that τε ≤ τ2 ≤ 2T . Case 6: ζ(0) = 0, z[0] = 0, w[0] = 0 and case 7: ζ(0) = 0, z[0] = 0, w[0] = 0. According to lemma 14.3, we see that τε ≤ τ1 ≤ T . Case 8: z[0] = 0, w[0] = 0, ζ(0) = 0. According to the plant (14.1)–(14.5) with the control input (14.36), we know that z[t], w[t], ζ(t) are identically zero for t ≥ 0. The estimates reach the true values in finite time only when qˆ1 (0) = q1 , qˆ2 (0) = q2 —that is, τε = 0. In summary, we have proved for all eight cases that τε ≤ max{ q12 + T, 2T }. This completes the proof of claim 14.5.

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438 14.7

NOTES

Adaptive control with a regulation-triggered BaLSI was originally introduced in [102, 104]. In this chapter, we designed an adaptive certainty-equivalence controller with regulation-triggered batch least-squares identification for a hyperbolic PDEODE system where the unknown parameters are transport speeds. It would be appropriate to regard this chapter as the hyperbolic equivalent of the paper [106] for a parabolic PDE, where the unknown parameters are the reaction coefficient and the high-frequency gain. In chapter 13, triggering was employed for the control law instead of the parameter estimator and only asymptotic convergence to zero of the plant states was achieved. However, this chapter’s adaptive control design employed triggering for the parameter update law rather than the control input and achieved finite-time exact identification of the unknown parameters from most initial conditions and, as a result of the finite-time settling of the parameter estimates, exponential regulation of the plant states to zero.

Chapter Fifteen Adaptive Control of Hyperbolic PDEs with Piecewise-Constant Inputs and Identification

The event-triggered adaptive control design in chapter 13 employs triggering for the control law instead of the parameter estimator, whereas the design in chapter 14 employs triggering for the parameter estimator and not the control law. In this chapter, we pursue the design of an event-triggered adaptive controller where triggering is employed for updating both the parameter estimator and the plant states in the control law simultaneously. As a result, both the parameter estimates and the control input employ piecewise-constant values. The controller consists of a nominal continuous-in-time backstepping feedback law and a triggered batch leastsquares identifier (BaLSI). The triggering mechanism is designed based on both evaluating the actuation deviation caused by the difference between the plant states and their sampled values and on evaluating the growth of the plant norm. When either condition is met, recomputing of the parameter estimator and resampling of the plant states in the feedback are done simultaneously. The problem formulation is presented in section 15.1. The nominal continuousin-time control design is presented in section 15.2. The design of event-triggered control with piecewise-constant parameter identification is proposed in section 15.3. The results, including the absence of a Zeno phenomenon, parameter convergence, and exponential regulation of the states, are proved in section 15.4. The effectiveness of the proposed design is illustrated with an application in the axial vibration control of a mining cable elevator in section 15.5.

15.1

PROBLEM FORMULATION

We conduct the control design based on the following 2 × 2 hyperbolic partial differential equation-ordinary differential equation (PDE-ODE) system: ˙ = aζ(t) + bw(0, t), ζ(t) zt (x, t) = −q1 zx (x, t) + d1 z(x, t) + d2 w(x, t), wt (x, t) = q2 wx (x, t) + d3 z(x, t) + d4 w(x, t), z(0, t) = cζ(t) − pw(0, t), w(1, t) = c0 U (t)

(15.1) (15.2) (15.3) (15.4) (15.5)

for x ∈ [0, 1], t ∈ [0, ∞), where ζ(t) is a scalar ODE state, and scalar functions z(x, t) and w(x, t) are PDE states. The function U (t) is the control input to be designed. The positive constants q1 , q2 are transport speeds, and p = 0, c0 = 0 are arbitrary

CHAPTER FIFTEEN

440

constants. The ODE system parameter a and the coefficients d2 , d3 of the PDE in-domain couplings are unknown. To make the problem as nontrivial as we can within this class, we focus on the case where in-domain instability exists—that is, d3 = 0. Other parameters are arbitrary and satisfy assumptions 15.1, 15.2. The connection between the general model (15.1)–(15.5) and the mining cable elevator is illustrated in section 15.5. Additionally, the plant (15.1)–(15.5) is inspired by the suppression of vibrations of airplane wings with aeroelastic instability at high Mach numbers [161, 196], which are described by a wave PDE with the in-domain fluid pressure represented by positive stiffness and anti-damping; that is, utt (x, t) = c2w uxx (x, t) + μw ut (x, t) + vw ux (x, t), where u(x, t) denotes the membrane displacements, and the physical meanings of the positive coefficients cw , μw , vw are given in [161]. Applying the Riemann transformations z(x, t) = ut (x, t) − cw ux (x, t), w(x, t) = ut (x, t) + cw ux (x, t), the airplane wing vibration dynamics are transformed into a 2 × 2 hyperbolic PDE system, covered by the considered plant (15.1)–(15.5), where the ODE (15.5) can describe a mass at the wing tip. Assumption 15.1. The parameter b is nonzero. For c = 0, b is not equal to − pa c . Assumption 15.2. For a = 0, the parameters d1 , d2 , d3 , d4 satisfy   bc 2 d3 ( bc bc a + p) − (d4 − d1 )( a + p) − d2 + p + d4 = 0. − d3 q2 a ( bc a + p)(q1 + q2 ) For a = 0 and c = 0, the parameters d1 , d2 , d3 , d4 satisfy q2

d3 p2 − (d4 − d1 )p − d2 − d3 p + d4 = 0. p(q1 + q2 )

For the cases not mentioned in assumptions 15.1 and 15.2, no restrictions are imposed for the corresponding parameters. The generically satisfied assumptions 15.1 and 15.2 act as sufficient (but not necessary) conditions for the parameter convergence of the estimates. Assumption 15.3. Bounds on the unknown parameters d3 , d2 , a are known though ¯, where d¯3 , d¯2 , a ¯ are arbitrary positive arbitrary; that is, |d3 | ≤ d¯3 , |d2 | ≤ d¯2 , |a| ≤ a constants whose values are known. We adopt the following notation. • The symbol Z+ denotes the set of all nonnegative integers, and R+ := [0, +∞) and R− := (−∞, 0]. • Let U ⊆ Rn be a set with a nonempty interior, and let Ω ⊆ R be a set. By C 0 (U ; Ω), we denote the class of continuous mappings on U , which take values in Ω. By C k (U ; Ω), where k ≥ 1, we denote the class of continuous functions on U , which have continuous derivatives of order k on U and take values in Ω. • We use the notation L2 (0, 1) for the standard space of the equivalence class of square-integrable, measurable functions defined on (0, 1), and f  =  12  1 2 f (x) dx < +∞ for f ∈ L2 (0, 1). 0 • We use the notation N for the set {1, 2, · · · }—that is, the natural numbers without 0. • For an I ⊆ R+ , the space C 0 (I; L2 (0, 1)) is the space of continuous mappings I t → u[t] ∈ L2 (0, 1).

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441

• Let u : R+ × [0, 1] → R be given. We use the notation u[t] to denote the profile of u at certain t ≥ 0—that is, (u[t])(x) = u(x, t), for all x ∈ [0, 1].

15.2

NOMINAL CONTROL DESIGN

We introduce the following backstepping transformation [48]  x α(x, t) = z(x, t) − φ(x, y)z(y, t)dy 0  x ϕ(x, y)w(y, t)dy − γ(x)ζ(t), − 0  x Ψ(x, y)z(y, t)dy β(x, t) = w(x, t) − 0  x − Φ(x, y)w(y, t)dy − λ(x)ζ(t),

(15.6)

(15.7)

0

and its inverse



x

¯ y)α(y, t)dy z(x, t) = α(x, t) − φ(x, 0  x ϕ(x, ¯ y)β(y, t)dy − γ¯ (x)ζ(t), − 0  x ¯ Ψ(x, y)α(y, t)dy w(x, t) = β(x, t) − 0  x ¯ ¯ Φ(x, y)β(y, t)dy − λ(x)ζ(t) −

(15.8)

(15.9)

0

to convert the plant (15.1)–(15.5) to the target system ˙ = −am ζ(t) + bβ(0, t), ζ(t) α(0, t) = −pβ(0, t),

(15.10) (15.11)

αt (x, t) = −q1 αx (x, t) + d1 α(x, t),

(15.12)

βt (x, t) = q2 βx (x, t) + d4 β(x, t),

(15.13)

β(1, t) = 0,

(15.14)

where am = bκ − a > 0 is ensured by a design parameter κ satisfying a ¯ κ > , ∀b > 0, b

a ¯ κ < , ∀b < 0. b

(15.15)

The conditions on the kernels φ(x, y), ϕ(x, y), γ(x), Ψ(x, y), Φ(x, y), λ(x) and ¯ y), ϕ(x, ¯ ¯ ¯ φ(x, ¯ y), γ¯ (x), Ψ(x, y), Φ(x, y), λ(x) in the backstepping transformations (15.6)–(15.9), which are obtained by matching the original system (15.1)–(15.5) and the target system (15.10)–(15.14), are shown in appendix 15.6A, and the wellposedness has been proved in theorem 4.1 of [48]. The nominal continuous-in-time control is obtained from the right boundary condition (15.14) in the target system as follows:

CHAPTER FIFTEEN

442 1 U (t) = c0 +



1 0

1 c0

Ψ(1, y; θ)z(y, t)dy 

1 0

Φ(1, y; θ)w(y, t)dy +

1 λ(1; θ)ζ(t). c0

(15.16)

Writing θ = [d3 , d2 , a]T after “; ” in Ψ(x, y; θ), Φ(x, y; θ), and λ(x; θ) emphasizes the fact that Ψ(x, y), Φ(x, y), λ(x) depend on the unknown parameters d3 , d2 , a, due to the fact that Ψ, Φ, λ is the solution of the kernel PDE set, shown in appendix 15.6A, with the unknown coefficients d3 , d2 , a. Under suitable conditions on (a, b, c), the explicit design from [176] using the generalized Marcum Q-functions of the first order could be employed in the controller (15.16).

15.3

EVENT-TRIGGERED CONTROL DESIGN WITH PIECEWISE-CONSTANT PARAMETER IDENTIFICATION

Based on the nominal continuous-in-time feedback (15.16), we give the form of an adaptive event-triggered control law Ud as follows:  1 1 ˆ i ))z(y, ti )dy Ψ(1, y; θ(t Ud (t) = c0 0  1 1 ˆ i ))w(y, ti )dy + 1 λ(1; θ(t ˆ i ))ζ(ti ) + Φ(1, y; θ(t (15.17) c0 0 c0 for t ∈ [ti , ti+1 ), where θˆ = [dˆ3 , dˆ2 , a ˆ]T is an estimate, generated with a triggered BaLSI, of the three unknown parameters d3 , d2 , a. The identifier is presented shortly, + is the sequence of time instants, which, along with the and {ti ≥ 0}∞ i=0 , i ∈ Z parameter estimates and sampled states in (15.17), is defined in the next subsection. When we mention the continuous-in-state control input Uc , we refer to the control input consisting of triggered parameter estimates and continuous states— that is,  1 1 ˆ i ))z(y, t)dy Ψ(1, y; θ(t Uc (t) = c0 0  1 1 ˆ i ))w(y, t)dy + 1 λ(1; θ(t ˆ i ))ζ(t) + Φ(1, y; θ(t (15.18) c0 0 c0 for t ∈ [ti , ti+1 ). Define the difference between the continuous-in-state control input Uc in (15.18) and the event-triggered control input Ud in (15.17) as d(t), given by d(t) = Uc (t) − Ud (t)  1 1 ˆ i ))(z(y, t) − z(y, ti ))dy = Ψ(1, y; θ(t c0 0  1 1 ˆ i ))(w(y, t) − w(y, ti ))dy + Φ(1, y; θ(t c0 0 1 ˆ i ))(ζ(t) − ζ(ti )), t ∈ [ti , ti+1 ), + λ(1; θ(t c0 which reflects the deviation of the plant states from their sampled values.

(15.19)

PIECEWISE-CONSTANT INPUT/ESTIMATE

443

Define the difference between the continuous-in-state control input Uc (t) in (15.18) and the nominal continuous-in-time control input U (t) in (15.16) as ξ(t), given by ξ(t) = U (t) − Uc (t)  1 1 ˆ i )))z(y, t)dy (Ψ(1, y; θ) − Ψ(1, y; θ(t = c0 0  1 1 ˆ i )))w(y, t)dy + (Φ(1, y; θ) − Φ(1, y; θ(t c0 0 1 ˆ i )))ζ(t), t ∈ [ti , ti+1 ), + (λ(1; θ) − λ(1; θ(t c0

(15.20)

which reflects the deviation of the estimates from the actual unknown parameters. The deviations d(t) and ξ(t) will be used in the following design and analysis. Triggering Mechanism Before showing the triggering mechanism that determines the sequence of time + ¯ instants {ti ≥ 0}∞ i=0 , i ∈ Z , we introduce the two sets Si and Si , which will be used in the triggering mechanism to judge whether the exact identification of the unknown parameters has been achieved. The set Si is defined as  ¯ ¯ n (μi+1 , ti+1 ) , Si := ¯= ( 1 , 2 )T ∈ Θ1 : Z¯n (μi+1 , ti+1 ) = G n = 1, 2, . . . , , i ∈ Z+ , where

(15.21)

Θ1 = { ¯∈ R2 : | 1 | ≤ d¯3 , | 2 | ≤ d¯2 },

¯ n associated with the plant states over a time interval [μi+1 , ti+1 ] (the and Z¯n , G time instant μi+1 is defined shortly) are Z¯n (μi+1 , ti+1 ) = [Hn,1 (μi+1 , ti+1 ), Hn,2 (μi+1 , ti+1 )]T ,

 Qn,1 (μi+1 , ti+1 ) Qn,2 (μi+1 , ti+1 ) ¯ Gn (μi+1 , ti+1 ) = Qn,2 (μi+1 , ti+1 ) Qn,3 (μi+1 , ti+1 ) with



(15.22) (15.23)

ti+1

Hn,1 (μi+1 , ti+1 ) =

gn,1 (t, μi+1 )fn (t, μi+1 )dt,

(15.24)

gn,2 (t, μi+1 )fn (t, μi+1 )dt,

(15.25)

gn,1 (t, μi+1 )2 dt,

(15.26)

gn,1 (t, μi+1 )gn,2 (t, μi+1 )dt,

(15.27)

gn,2 (t, μi+1 )2 dt,

(15.28)

μi+1  ti+1

Hn,2 (μi+1 , ti+1 ) = μi+1  ti+1

Qn,1 (μi+1 , ti+1 ) = μi+1  ti+1

Qn,2 (μi+1 , ti+1 ) = μi+1  ti+1

Qn,3 (μi+1 , ti+1 ) = μi+1

CHAPTER FIFTEEN

444 where  fn (t, μi+1 ) =



1

sin(xπn)z(x, t)dx +

0



−

1 0

 





t

1

μi+1 0  1 t 0

μi+1 t



gn,1 (t, μi+1 ) = μi+1  t

gn,2 (t, μi+1 ) = μi+1

0

sin(xπn)w(x, t)dx 

sin(xπn)z(x, μi+1 )dx −

− πn −

1



1 0

sin(xπn)w(x, μi+1 )dx

cos(xπn)(q1 z(x, τ ) − q2 w(x, τ ))dxdτ

sin(xπn)(d1 z(x, τ ) + d4 w(x, τ ))dxdτ,

(15.29)

1 0

sin(xπn)z(x, τ )dxdτ,

(15.30)

sin(xπn)w(x, τ )dxdτ

(15.31)

1 0

for n = 1, 2, · · · . The set S¯i is defined as  ¯ Si := 3 ∈ [−¯ a, a ¯] : H3 (μi+1 , ti+1 ) = Q4 (μi+1 , ti+1 ) 3 , , i ∈ Z+ ,

(15.32)

where the function H3 is defined as  H3 (μi+1 , ti+1 ) =

(15.33)

ti+1

ga (t, μi+1 )fa (t, μi+1 )dt, μi+1

and the function Q4 is defined as 

ti+1

Q4 (μi+1 , ti+1 ) =

ga (t, μi+1 )2 dt

(15.34)

μi+1

for i ∈ Z+ , with 

t

fa (t, μi+1 ) =ζ(t) − ζ(μi+1 ) − b 

w(0, τ )dτ,

(15.35)

μi+1 t

ga (t, μi+1 ) =

ζ(τ )dτ.

(15.36)

μi+1

Based on the above definitions, we introduce the triggering mechanism next. The sequence of time instants {ti ≥ 0}∞ i=0 (t0 = 0) is defined as ti+1 = min{τi , ri },

(15.37)

τi = inf{t > ti : d(t)2 ≥ ϑV (t) − m(t)}

(15.38)

where τi is given by

and where ri is defined next. If i ≥ 1 and there exists a singleton Sj for a certain j (j ∈ Z+ , j ≤ i − 1) and a singleton S¯k for a certain k (k ∈ Z+ , k ≤ i − 1), then ri is set as

PIECEWISE-CONSTANT INPUT/ESTIMATE

445

ri := +∞,

(15.39)

ri = max ti + r, min inf{t > ti : Ω(t) = (1 + δ)Ω(ti )},

ti + T , Ω(ti ) = 0, ri = ti + T, Ω(ti ) = 0,

(15.40) (15.41)

or as

where the design parameters δ, T are positive and free, and r is a positive constant satisfying 0 < r < T,

(15.42)

and where the function Ω(t) is defined as Ω(t) = z[t]2 + w[t]2 + ζ(t)2 .

(15.43)

The Lyapunov function V (t) in (15.38) is given as  1 1 1 2 V (t) = ζ(t) + ra eδ1 x β(x, t)2 dx 2 2 0  1 1 −δ2 x + e α(x, t)2 dx, 2 0

(15.44)

where the positive constants ra , δ1 , δ2 , which are design parameters, are chosen to satisfy δ1 >

2|d4 | , q2

(15.45)

δ2 >

2|d1 | , q1

(15.46)

ra >

p2 q 1 b2 + , q2 (bκ − a ¯) q2

(15.47)

which depend only on the design parameter κ in (15.15), the known plant parameters, and the known bounds of the unknown parameters in assumption 15.3. The internal dynamic variable m(t) in (15.38) satisfies the ordinary differential equation m(t) ˙ = − ηm(t) + λd d(t)2 − σV (t) − κ1 α(1, t)2 − κ2 β(0, t)2

(15.48)

with the initial condition m(0) < 0. Choose the positive design parameters κ1 , κ2 in (15.48) to satisfy κ1 ≥ 2 max{λa , λα },

(15.49)

κ2 ≥ 2 max{λa , λβ },

(15.50)

where the positive constants λα , λβ , λa are

CHAPTER FIFTEEN

446 1 1 1 λα = q1 e−δ2 , λβ = q2 ra + |b|, 2 2 2  6 λa = 2 max |M1 (y, θ1 , θ2 )|2 , |M2 (y, θ1 , θ2 )|2 , c0 y∈[0,1],θ1 ∈Θ,θ2 ∈Θ

 |M3 (θ1 , θ2 )|2 , |M4 (θ1 , θ2 )|2 , |M5 (θ1 , θ2 )|2 , |M6 (θ1 , θ2 )|2 .

(15.51)

(15.52)

The functions M1 , . . . , M6 are given in appendix 15.6B, and λa depends only on the design parameter κ, the known plant parameters, and the known bounds of the unknown parameters in assumption 15.3. Choose the design parameters λd , σ, η in (15.48) such that they satisfy 1 q2 ra eδ1 c20 , μ νa 0 νa − μσ,

λd ≥

where the positive constants μ, νa are  q2 ra p2 q 1 1 b2 − μ ≤ min − , q1 e−δ2 , 2κ2 2κ2 (bκ − a ¯) 2κ2 2κ1    1 1 1 1 (bκ − a ¯), δ1 q2 ra − ra |d4 |, δ2 q1 − |d1 | e−δ2 νa = min ξ2 2 2 2

(15.53) (15.54) (15.55)

(15.56) (15.57)

with ξ2 =

1 min{1, ra eδ1 } > 0. 2

(15.58)

The final design parameter ϑ in (15.38) is chosen in accordance with the following condition:  σ . (15.59) 0 < ϑ ≤ max 1, λd The time instant μi+1 used in (15.21), (15.32) is defined as ˜ T }, μi+1 := min{tf : f ∈ {0, . . . , i}, tf ≥ ti+1 − N

(15.60)

˜ ≥ 1 is a design parameter, and the according to [104], where the positive integer N positive constant T is the maximum dwell time according to (15.37)–(15.41). The flowchart of implementing the designed triggering mechanism (15.37)–(15.41) is shown in figure 15.1, and the proof that the triggering mechanism (15.37)–(15.41) is well defined and produces an increasing sequence of events will be shown in lemma 15.3. Design rationale of the triggering mechanism (15.37)–(15.41): According to (15.19), (15.20)—that is, two deviation signals affecting the stability obtained under the nominal continuous-in-time control—we know the size of d(t) is associated with the deviation of the plant states from their sampled values, and the size of ξ(t) is associated with the deviation of the parameter estimates from the actual unknown parameters. According to the form of the event trigger presented in [57], the triggering condition (15.38) is designed, based on the evolution of the square of

N

Uc(θˆ(ti+1), z[t], w[t], ς(t))

Y

d(t)2 ≥ ϑV(t) – m(t)

Y

i ≥ 1, and there exist singletons – Sj, Sk, j, k ϵ Z+, j, k ≤ i – 1

Measurements at the current time t > ti and data in [0,ti]

N Y

t ≥ ri N Y

d(t)2 ≥ ϑV(t) – m(t)

N

ri = ti + ṟ or ri = ti + T N Y

t ≥ ti + T

N

Y

Y

t ≥ ti + ṟ

Decision

N

Initial ri := +∞

Data

Ω(t) < (1 + δ)Ω(ti)

Figure 15.1. Implementing the triggering mechanism (15.37)–(15.41).

Ud (θˆ(ti+1), z[ti+1], w[ti+1], ς(ti+1))

Update estimates θˆ(ti+1) and states z[ti+1], w[ti+1], ς(ti+1) in the control signal

ri = ti + T

Y

Ω(ti) = 0 N

Y

ri = ti + ṟ

Process

N

CHAPTER FIFTEEN

448

d(t), to guarantee that d(t)2 is bounded by the plant norm, and an internal dynamic variable m(t) defined in (15.48) (the introduction of m(t) is to ensure that no Zeno phenomenon occurs, as will be seen in claim 15.2 in lemma 15.3). Even if d(t) is ensured to be small enough in the closed-loop system, a large growth of the plant norm still may appear under a large ξ(t) (as in the analysis of property (3) in theorem 15.1). Therefore, in addition to the triggering condition (15.38) designed to bound d(t), another triggering condition in (15.40) is designed based on evaluating the growth of the plant norm [104], where the updates are triggered if the plant norm has grown by a certain factor, to avoid a large overshoot. Introducing the design parameters r, T in (15.40), which set the lower and upper bounds of the interval between ri , enables the user to adjust the number of ri . As will be seen later, the condition below (15.38), that there exist the singleton sets Sj , S¯k , implies that the exact identification of θ has been achieved at ti for i ≥ 1. Therefore, in this case we set ri := +∞ in (15.39) in order to avoid unnecessary updates of the control signal after the exact parameter identification has been achieved (when initial estimates are not the true values). Introducing a sufficiently small positive constant r in (15.40) is to ensure that no Zeno phenomenon occurs (as will be seen in claim 15.1 in lemma 15.3). The first triggering condition (15.38) is for resampling the states z[t], w[t], ζ(t), and the second triggering condition (15.40), (15.41) is for recomputing the estimate ˆ θ(t). The synchronous triggering is ensured by the fact that if either condition ˆ and the resampling of z[t], w[t], ζ(t) are done is met, both the recomputing of θ(t) simultaneously, due to (15.37). All design parameters are κ, r, δ1 , δ2 , ra , κ1 , κ2 , λd , σ, η, θ, whose conditions are cascaded rather than mutually dependent, and only defined by known parameters. They can be solved in the sequence (15.15), (15.42), (15.45), (15.46), (15.47), (15.49), (15.50), (15.53), (15.54), (15.55), (15.59). The motivation for defining these conditions for the design parameters will become clear in the rest of this chapter. Least-Squares Identifier According to (15.2), (15.3), we get the following for τ > 0 and n = 1, 2, · · · : d dτ





1 0

sin(xπn)z(x, τ )dx + 

= −q2 πn  + d1

1 0

+ d3

 1

0

0

0

sin(xπn)w(x, τ )dx

cos(xπn)w(x, τ )dx 

sin(xπn)z(x, τ )dx + d2

+ q1 πn 

1



1

sin(xπn)w(x, τ )dx

0

1 0

1

cos(xπn)z(x, τ )dx

sin(xπn)z(x, τ )dx + d4



1 0

sin(xπn)w(x, τ )dx.

(15.61)

Integrating (15.61), (15.1) from μi+1 to t yields fn (t, μi+1 ) = d3 gn,1 (t, μi+1 ) + d2 gn,2 (t, μi+1 ),

(15.62)

fa (t, μi+1 ) = aga (t, μi+1 ),

(15.63)

PIECEWISE-CONSTANT INPUT/ESTIMATE

449

where fn , gn,1 , gn,2 are given in (15.29)–(15.31), and fa , ga are defined in (15.35), (15.36). Define the function hi,n : R3 → R+ by the formula 

ti+1

hi,n ( ) =

[(fn (t, μi+1 ) − 1 gn,1 (t, μi+1 ) − 2 gn,2 (t, μi+1 ))2

μi+1

+ (fa (t, μi+1 ) − 3 ga (t, μi+1 ))2 ]dt, i ∈ Z+

(15.64)

for n = 1, 2, · · · , = [ 1 , 2 , 3 ]T . According to (15.62), (15.63), the function hi,n ( ) in (15.64) has a global minimum hi,n (θ) = 0. We get from Fermat’s theorem (vanishing gradient at extrema) that the following matrix equation holds for every i ∈ Z+ and n = 1, 2, · · · : Zn (μi+1 , ti+1 ) = Gn (μi+1 , ti+1 )θ, where

⎡

Qn,1 ⎢ T Zn = [Hn,1 , Hn,2 , H3 ] , Gn = ⎣ Qn,2 0

Qn,2 Qn,3 0

(15.65) ⎤ 0 ⎥ 0 ⎦, Q4

(15.66)

and Hn,1 (μi+1 , ti+1 ), Hn,2 (μi+1 , ti+1 ), H3 (μi+1 , ti+1 ), Qn,1 (μi+1 , ti+1 ), Qn,2 (μi+1 , ti+1 ), Qn,3 (μi+1 , ti+1 ), Q4 (μi+1 , ti+1 ) are given in (15.24)–(15.28), (15.33), (15.34). Indeed, (15.65), (15.66) are obtained by differentiating the functions hi,n ( ) defined by (15.64) with respect to 1 , 2 , 3 , respectively, and evaluating the derivatives at the position of the global minimum ( 1 , 2 , 3 ) = (d3 , d2 , a). The parameter estimator (update law) is defined as  2  ˆ i ) : ∈ Θ, ˆ θ(ti+1 ) = argmin  − θ(t (15.67) Zn (μi+1 , ti+1 ) = Gn (μi+1 , ti+1 ) , n = 1, 2, · · · , ¯}. where Θ = { ∈ R3 : | 1 | ≤ d¯3 , | 2 | ≤ d¯2 , | 3 | ≤ a Proposition 15.1. For every (z[ti ], w[ti ])T ∈ L2 ((0, 1); R2 ), ζ(ti ) ∈ R, m(ti ) ∈ R− , ˆ i ) ∈ Θ, there exists a unique (weak) solution ((z, w)T , ζ) ∈ C 0 ([ti , ti+1 ]; and θ(t L2 (0, 1); R2 ) × C 0 ([ti , ti+1 ]; R), m ∈ C 0 ([ti , ti+1 ]; R− ) to the system (15.1)–(15.5), (15.17), (15.48), and θˆ ∈ Θ in (15.67), between two time instants ti , ti+1 . Proof. The proof is similar to that of proposition 14.1 in chapter 14, and thus it is omitted.

15.4

MAIN RESULT

ˆ ∈ Θ, Theorem 15.1. For all initial data (z[0], w[0])T ∈ C 1 ([0, 1]), ζ(0) ∈ R, θ(0) m(0) < 0, the closed-loop system (15.1)–(15.5) under the controller (15.17) with the triggering mechanism (15.37)–(15.41) and the least-squares identifier defined by (15.67) has the following properties:

CHAPTER FIFTEEN

450

1) No Zeno phenomenon occurs—that is, limi→∞ ti = +∞, and the closed-loop system is well-posed. 2) If the finite-time convergence of parameter estimates to the true values is not achieved, it implies Ω(t) ≡ 0 on t ∈ [ q11 , ∞), and m(t) is exponentially convergent ˆ reaches the true value θ in finite time— to zero. If the parameter estimate θ(t) that is, ˆ = θ, θ(t)

∀t ≥ tε

(15.68)

for certain ε ∈ Z+ , then tε ≤

1 1 + + T. q1 q2

(15.69)

3) If the finite-time convergence of parameter estimates to the true values is achieved, the exponential regulation of the closed-loop system is obtained in the sense that there exist the positive constants M, λ1 such that −λ1 t ¯ ≤ M Ω(0)e ¯ Ω(t) , t ≥ 0,

(15.70)

¯ = z[t]2 + w[t]2 + ζ(t)2 + |m(t)| Ω(t)

(15.71)

where

ˆ and where M is related to the initial estimate θ(0). The proof of theorem 15.1 is based on the following technical lemmas. First, we present the two lemmas that will be used in the analysis of the lower bound of the minimal dwell time. Lemma 15.1. For d(t) defined in (15.19), there exists a positive constant λa such that  ˙ 2 ≤ λa d(t)2 + α(1, t)2 + β(0, t)2 d(t)  + α(·, t)2 + β(·, t)2 + ζ(t)2 (15.72) for t ∈ (ti , ti+1 ), where the positive constant λa is given in (15.52), which depends only on the design parameter κ in (15.15), the known plant parameters, and the known bounds a ¯, d¯3 , d¯2 of the unknown parameters. Proof. Inserting Ud (t) into (15.5), we have w(1, t) = c0 Ud (t).

(15.73)

Applying (15.19), (15.20) allows us to rewrite (15.73) as w(1, t) = c0 (Uc (t) − d(t)) = c0 U (t) − c0 ξ(t) − c0 d(t). Inserting (15.8), (15.9) into (15.20), we obtain

(15.74)

PIECEWISE-CONSTANT INPUT/ESTIMATE

1 ξ(t) = c0



1 0



1

+ 0

451

ˆ i ), θ)α(y, t)dy ˜ 1 (y; θ(t K 

ˆ i ), θ)β(y, t)dy + K ˆ i ), θ)ζ(t) ˜ 2 (y; θ(t ˜ 3 (θ(t K

(15.75)

˜ 1, K ˜ 2, K ˜ 3 are shown in appendix 15.6B. for t ∈ [ti , ti+1 ), where the functions K Applying the backstepping control design in section 15.2 into (15.1)–(15.4), (15.74), the right boundary condition of the target system (15.10)–(15.14) becomes β(1, t) = −c0 ξ(t) − c0 d(t).

(15.76)

Inserting the inverse backstepping transformations into Uc (t) in (15.18) to replace the original states by the states in the target system, we obtain  1 1 ˆ i ), θ)α(y, t)dy K1 (y, θ(t Uc (t) = c0 0  1 1 ˆ i ), θ)β(y, t)dy + 1 K3 (θ(t ˆ i ), θ)ζ(t) K2 (y, θ(t (15.77) + c0 0 c0 for t ∈ [ti , ti+1 ), where the functions K1 , K2 , K3 are shown in appendix 15.6B. The event-triggered control input Ud is constant on t ∈ (ti , ti+1 )—that is, U˙ d (t) = 0. Taking the time derivative of (15.19) and recalling (15.75), (15.76), (15.77), we obtain ˙ = U˙ c (t) d(t)

 1 1 ˆ i ), θ)α(y, t)dy = M1 (y, θ(t c0 0  1 ˆ i ), θ)β(y, t)dy − M3 (θ(t ˆ i ), θ)ζ(t) − M2 (y, θ(t 0

 ˆ ˆ ˆ + M4 (θ(ti ), θ)β(0, t) − M5 (θ(ti ), θ)α(1, t) − M6 (θ(ti ), θ)d(t) ,

(15.78)

where the functions M1 , . . . , M6 are shown in appendix 15.6B. Applying the CauchySchwarz inequality into (15.78), we obtain (15.72). The proof is complete. Lemma 15.2. For the internal dynamic variable m(t) defined in (15.48), it holds that m(t) < 0. Proof. According to (15.38), events are triggered to guarantee that d(t)2 ≤ ϑV (t) − m(t). Recalling (15.48), we then have m(t) ˙ ≤ −(η + λd )m(t) + (λd ϑ − σ)V (t) − κ1 α(1, t)2 − κ2 β(0, t)2 ≤ −(η + λd )m(t),

t ≥ 0,

(15.79)

where ϑ ≤ λσd obtained from (15.59) has been used. Together with m(0) < 0, we conclude that m(t) < 0. Relying on lemmas 15.1 and 15.2, we present a lemma that shows that the event trigger is well defined and produces an increasing sequence of events.

CHAPTER FIFTEEN

452

Lemma 15.3. Under the triggering mechanism defined by (15.37)–(15.41), the minimal dwell time exists, in the sense of ti+1 − ti ≥ min{τ , r} > 0, ∀i ∈ Z+

(15.80)

for some positive τ . Proof. At a time instant ti , the next time instant ti+1 is triggered by either ri in (15.40), (15.41) or τi in (15.38). Next we present two claims regarding the lower bound of the minimal dwell time in the two situations in which ti+1 is triggered by either ri or τi , respectively. Claim 15.1. For an arbitrary time instant ti , if the next time instant ti+1 is triggered by ri in (15.40), (15.41), then ti+1 − ti ≥ r > 0. Proof. If Ω(ti ) = 0, the interval ti+1 − ti is greater than r > 0 by virtue of (15.40). If Ω(ti ) = 0, according to (15.41), the interval ti+1 − ti is equal to the positive constant T , which is larger than r according to (15.42). Claim 15.2. For an arbitrary time instant ti , if the next time instant ti+1 is triggered by τi in (15.38), there exists a minimal dwell time τ > 0 such that ti+1 − ti ≥ τ . Proof. We know from (15.38) that the events are triggered to guarantee d(t)2 ≤ ϑV (t) − m(t) for all t ≥ 0. We introduce the following function ψ(t), which is similar to (13.71) in chapter 13: 2

ψ(t) =

d(t) + 12 m(t) , ϑV (t) − 12 m(t)

(15.81)

where ψ(ti+1 ) = 1 because the event is triggered, and where ψ(ti ) < 0 because of m(t) < 0 proved in lemma 15.2 and d(ti ) = 0 according to (15.19). The function ψ(t) is a continuous function on [ti , ti+1 ] recalling proposition 15.1. By the intermediate value theorem, there exists t∗ > ti such that ψ(t) ∈ [0, 1] when t ∈ [t∗ , ti+1 ]. The minimal dwell time can be found as the minimal time it takes for ψ(t) from 0 to 1. Defining Ω0 (t) = α(·, t)2 + β(·, t)2 + ζ(t)2

(15.82)

and recalling (15.44), we see that the following inequality holds ξ1 Ω0 (t) ≤ V (t) ≤ ξ2 Ω0 (t), where the positive constant ξ1 is ξ1 = min



1 1 1 , ra , e−δ2 2 2 2

(15.83)



and where the positive constant ξ2 is defined in (15.58). Taking the time derivative of V (t) in (15.44) and applying (15.10)–(15.13), using integration by parts, we

PIECEWISE-CONSTANT INPUT/ESTIMATE

453

obtain 1 V˙ (t) = − am ζ(t)2 + ζ(t)bβ(0, t) + q2 ra eδ1 β(1, t)2 2  1  1 1 − δ1 q2 ra eδ1 x β(x, t)2 dx + ra d4 eδ1 x β(x, t)2 dx 2 0 0 1 1 1 − q1 e−δ2 α(1, t)2 + q1 α(0, t)2 − q2 ra β(0, t)2 2 2 2  1  1 1 − δ 2 q1 e−δ2 x α(x, t)2 dx + d1 e−δ2 x α(x, t)2 dx. 2 0 0

(15.84)

We then have V˙ (t) ≥ − μ0 V − λα α(1, t)2 − λβ β(0, t)2 , where the positive constant μ0 is  1 1 1 δ1 δ1 1 δ1 q2 ra e + ra |d4 |e , δ2 q1 + |d1 |, am + |b| μ0 = max ξ1 2 2 2

(15.85)

(15.86)

and where λα , λβ are given in (15.51). Taking the derivative of (15.81) for all t ∈ (t∗ , ti+1 ), applying Young’s inequality, using (15.72) in lemma 15.1, inserting (15.48), (15.85), and applying (15.82), (15.83) to rewrite α(·, t)2 + β(·, t)2 + ζ(t)2 , we have ˙ + 1 m(t) 2d(t)d(t) (ϑV˙ (t) − 12 m(t)) ˙ 2 ˙ ψ˙ = − ψ 1 1 ϑV (t) − 2 m(t) ϑV (t) − 2 m(t)   

 1 1 1 2 ≤ λa + 1 + λd d(t) + λa − κ1 α(1, t)2 2 2 ϑV (t) − 12 m(t)    1 λa 1 + λa − κ2 β(0, t)2 + V (t) − ηm(t) 2 ξ1 2  

1 1 2 − κ V (t) − ϑλ − − ϑμ 0 α 1 α(1, t) 2 ϑV (t) − 12 m(t)    1 1 1 1 − ϑλβ − κ2 β(0, t)2 − λd d(t)2 + ηm(t) + σV (t) ψ. 2 2 2 2

(15.87)

Applying κ1 ≥ max{2λa , 2ϑλα }, κ2 ≥ max{2λa , 2ϑλβ }, which are ensured by (15.49), (15.50), (15.59), and applying the following inequalities −

ϑV

1 2 ηm(t) (t) − 12 m(t)

≤−

1 2 ηm(t) − 12 m(t)

= η,

1 V (t) V (t) = , ≤ ϑV (t) − 12 m(t) ϑV (t) ϑ

d(t)2 + 12 m(t) − 12 m(t) d(t)2 = ≤ ψ(t) + 1, ϑV (t) − 12 m(t) ϑV (t) − 12 m(t)

(15.88) (15.89) (15.90)

CHAPTER FIFTEEN

454 which hold because m(t) < 0, then (15.87) becomes ˙ ≤ 1 λd ψ(t)2 + (λa + 1 + λd + η + μ0 )ψ(t) ψ(t) 2 λa 1 + η. + 1 + λd + λa + 2 ξ1 ϑ

(15.91)

The differential inequality (15.91) has the form ψ˙ ≤ n1 ψ 2 + n2 ψ + n3 , where 1 n1 = λd , 2 n2 = λa + 1 + λd + η + μ0 , 1 λa +η n3 = 1 + λd + λa + 2 ξ1 ϑ are positive constants. It follows that the time needed by ψ to go from 0 to 1 is at least  1 1 τ= ds > 0. (15.92) n + n s + n3 s2 1 2 0 The proof of this claim is complete. According to claims 15.1 and 15.2, the proof of lemma 15.3 is complete. Next, we present the following lemmas regarding parameter convergence. In the rest of this chapter, when we say that z[t], w[t] are equal to zero for x ∈ [0, 1], t ∈ [μi+1 , ti+1 ], or not identically zero on the same domain, we mean “except possibly for finitely many discontinuities of the functions w[t], z[t].” These discontinuities are isolated curves in the rectangle [0, 1] × [μi+1 , ti+1 ]. Lemma 15.4. The sufficient and necessary conditions of Qn,1 (μi+1 , ti+1 ) = 0, Qn,3 (μi+1 , ti+1 ) = 0 for n = 1, 2, . . . are z[t] = 0, w[t] = 0 on t ∈ [μi+1 , ti+1 ], respectively. The sufficient and necessary condition of Q4 (μi+1 , ti+1 ) = 0 is ζ(t) = 0 on t ∈ [μi+1 , ti+1 ]. Proof. The proof that the sufficient and necessary conditions of Qn,1 (μi+1 , ti+1 ) = 0, Qn,3 (μi+1 , ti+1 ) = 0 for n = 1, 2, . . . are z[t] = 0, w[t] = 0 on t ∈ [μi+1 , ti+1 ], respectively, is the same as the proof of lemma 14.2 in chapter 14, where √ the fact that z ∈ C 0 ([ti , ti+1 ]; L2 (0, 1)), w ∈ C 0 ([ti , ti+1 ]; L2 (0, 1)), and the set { 2 sin(nπx) : n = 1, 2, . . .} is an orthonormal basis of L2 (0, 1) has been used. By recalling (15.34), (15.36), it is straightforward to see that the sufficient and necessary condition of Q4 (μi+1 , ti+1 ) = 0 is ζ(t) = 0 on t ∈ [μi+1 , ti+1 ]. The proof of lemma 15.4 is complete. Lemma 15.5. For the adaptive estimates defined by (15.67) based on the data in the interval t ∈ [μi+1 , ti+1 ], the following statements hold: If z[t] (or w[t] or ζ(t)) is not identically zero for t ∈ [μi+1 , ti+1 ], then dˆ3 (ti+1 ) = ˆ(ti+1 ) = a, respectively). d3 (or dˆ2 (ti+1 ) = d2 or a

PIECEWISE-CONSTANT INPUT/ESTIMATE

455

If z[t] (or w[t] or ζ(t)) is identically zero for t ∈ [μi+1 , ti+1 ], then dˆ3 (ti+1 ) = ˆ ˆ(ti+1 ) = a ˆ(ti ), respectively). d3 (ti ) (or dˆ2 (ti+1 ) = dˆ2 (ti ) or a Proof. We prove the following five results, from which the statements in this lemma are obtained. 1) Result 1: If z[t] is not identically zero and w[t] is identically zero on t ∈ [μi+1 , ti+1 ], then dˆ3 (ti+1 ) = d3 , dˆ2 (ti+1 ) = dˆ2 (ti ); if w[t] is not identically zero and z[t] is identically zero on t ∈ [μi+1 , ti+1 ], then dˆ3 (ti+1 ) = dˆ3 (ti ), dˆ2 (ti+1 ) = d2 . The proof of result 1 is very similar to the proof of cases 1 and 2 in lemma 14.3 in chapter 14, and thus they are omitted. 2) Result 2: If w[t], z[t] are identically zero on t ∈ [μi+1 , ti+1 ], then dˆ3 (ti+1 ) = dˆ3 (ti ), dˆ2 (ti+1 ) = dˆ2 (ti ). The proof of result 2 is shown as follows. In this case, Qn,1 (μi+1 , ti+1 ) = 0, Qn,2 (μi+1 , ti+1 ) = 0, Qn,3 (μi+1 , ti+1 ) = 0, Hn,1 (μi+1 , ti+1 ) = 0, Hn,2 (μi+1 , ti+1 ) = 0 for n = 1, 2, . . . according to (15.30), (15.31), (15.24)–(15.28). It follows that Si = Θ, and then (15.67) shows that dˆ3 (ti+1 ) = dˆ3 (ti ), dˆ2 (ti+1 ) = dˆ2 (ti ). 3) Result 3: If both w[t] and z[t] are not identically zero on t ∈ [μi+1 , ti+1 ], then dˆ3 (ti+1 ) = d3 , dˆ2 (ti+1 ) = d2 . The proof of result 3 is shown as follows. According to lemma 15.4, there exists n ∈ N such that Qn,3 (μi+1 , ti+1 ) = 0 (or Qn,1 (μi+1 , ti+1 ) = 0). Define the index set I1 to be the set of all n ∈ N with Qn,1 (μi+1 , ti+1 ) = 0, and define the index set I2 to be the set of all n ∈ N with Qn,3 (μi+1 , ti+1 ) = 0. Denote the elements in I1 as n1 ∈ N and those in I2 as n2 ∈ N—that is, Qn1 ,1 (μi+1 , ti+1 ) = 0, Qn2 ,3 (μi+1 , ti+1 ) = 0. For the set Si defined in (15.21), by virtue of (15.65)–(15.67), if Si is a singleton then it is nothing but the least-squares estimate of the unknown vector of parameters (d3 , d2 ) on the interval [μi+1 , ti+1 ], and Si = {(d3 , d2 )} according to (15.65), (15.66). From (15.21)–(15.23), we have  Hn1 ,1 (μi+1 , ti+1 ) Si ⊆ Sai := ( 1 , 2 ) ∈ Θ1 : 1 = Qn1 ,1 (μi+1 , ti+1 ) Qn ,2 (μi+1 , ti+1 ) − 2 1 , n1 ∈ I1 , (15.93) Qn1 ,1 (μi+1 , ti+1 )  Hn2 ,2 (μi+1 , ti+1 ) Si ⊆ Sbi := ( 1 , 2 ) ∈ Θ1 : 2 = Qn2 ,3 (μi+1 , ti+1 ) Qn ,2 (μi+1 , ti+1 ) − 1 2 , n2 ∈ I2 . (15.94) Qn2 ,3 (μi+1 , ti+1 ) We next prove by contradiction that Si = {(d3 , d2 )}. Suppose that on the contrary Si = {(d3 , d2 )}—that is, Si defined by (15.21) is not a singleton, which implies that the sets Sai , Sbi defined by (15.93), (15.94) are not singletons (because either Sai or Sbi being a singleton implies that Si is a singleton). It follows that there exist ¯ ∈ R, λ ¯ 1 ∈ R such that the constants λ Qn1 ,2 (μi+1 , ti+1 ) ¯ Qn2 ,2 (μi+1 , ti+1 ) ¯ = λ1 , n1 ∈ I1 , = λ, n2 ∈ I2 Qn1 ,1 (μi+1 , ti+1 ) Qn2 ,3 (μi+1 , ti+1 ) because if there were two different indices k1 , k2 ∈ I2 with Qk1 ,2 (μi+1 , ti+1 ) Qk2 ,2 (μi+1 , ti+1 ) = , Qk1 ,3 (μi+1 , ti+1 ) Qk2 ,3 (μi+1 , ti+1 )

(15.95)

CHAPTER FIFTEEN

456

then the set Sbi defined by (15.94) would be a singleton, and the same would be the case with Sai defined by (15.93) if there were two different indices k¯1 , k¯2 ∈ I1 with Qk¯1 ,2 (μi+1 , ti+1 ) Qk¯2 ,2 (μi+1 , ti+1 ) = . Qk¯1 ,1 (μi+1 , ti+1 ) Qk¯2 ,1 (μi+1 , ti+1 ) Moreover, since Si is not a singleton, the definition (15.21) implies Qn,2 (μi+1 , ti+1 )2 = Qn,1 (μi+1 , ti+1 )Qn,3 (μi+1 , ti+1 )

(15.96)

for all n ∈ I1 ∪I2 by recalling (15.23) (because if (15.96) does not hold, it follows from ¯ n (μi+1 , ti+1 )) = 0, which implies (15.23) that there exists n ∈ I1 ∪ I2 such that det(G Si defined by (15.21) is a singleton: a contradiction). According to (15.96), (15.26)– (15.28) and the fact that the Cauchy-Schwarz inequality holds as an equality only when two functions are linearly dependent, we obtain the existence of the constants ˇ n ∈ R such that ˆ n ∈ R, λ λ 1 2 ˆ n gn ,1 (t, μi+1 ), n1 ∈ I1 , gn1 ,2 (t, μi+1 ) = λ 1 1 ˇ n gn ,2 (t, μi+1 ), n2 ∈ I2 gn2 ,1 (t, μi+1 ) = λ 2 2

(15.97) (15.98)

for t ∈ [μi+1 , ti+1 ] (notice that gn1 ,1 (t, μi+1 ) and gn2 ,2 (t, μi+1 ) are not identically zero on t ∈ [μi+1 , ti+1 ] because of Qn1 ,1 (μi+1 , ti+1 ) = 0 and Qn2 ,3 (μi+1 , ti+1 ) = 0). Recalling (15.95), we obtain from (15.26)–(15.28) and (15.97), (15.98) that ¯ 1 gn ,1 (t, μi+1 ), n1 ∈ I1 , gn1 ,2 (t, μi+1 ) = λ 1 ¯ n ,2 (t, μi+1 ), n2 ∈ I2 gn2 ,1 (t, μi+1 ) = λg 2

(15.99) (15.100)

for t ∈ [μi+1 , ti+1 ]. ¯= ¯ 1 = 0 and λ ¯ = ¯1 in (15.99),  0, λ Claim 15.3. If Si is not a singleton, then λ λ1 (15.100). Proof. The proof is very similar to that of claim 14.1 in chapter 14, and thus it is omitted. ¯ = 0, λ ¯ 1 = 0 and λ ¯ = ¯1 ) hold if and Claim 15.4. Equations (15.99), (15.100) (λ λ1 ¯ ¯ = 0) for t ∈ [μi+1 , ti+1 ], x ∈ [0, 1]. only if z(x, t) − λw(x, t) = 0 (λ Proof. The proof is very similar to the proof of claim 14.2 in chapter 14, and thus it is omitted. Claim 15.5. If w(x, t) is not identically zero on x ∈ [0, 1], t ∈ [μi+1 , ti+1 ], the func¯ ¯ = 0) is not identically zero on x ∈ [0, 1], t ∈ [μi+1 , ti+1 ]. tion z(x, t) − λw(x, t) (λ Proof. The proof is shown in appendix 15.6C. Recalling claims 15.3, 15.4, and 15.5, we know that the equation set (15.99), ¯ = 0, λ ¯ 1 = 0 and λ ¯ = ¯1 ), which is a necessary condition of the hypothesis (15.100) (λ λ1 that Si is not a singleton, does not hold. Consequently, Si is a singleton—that is, Si = {(d3 , d2 )}. Therefore, dˆ3 (ti+1 ) = d3 , dˆ2 (ti+1 ) = d2 . 4) Result 4: If ζ(t) is not identically zero for t ∈ [μi+1 , ti+1 ], then a ˆ(ti+1 ) = a. The proof of result 4 is shown as follows. By virtue of (15.65)–(15.67), if S¯i defined in

PIECEWISE-CONSTANT INPUT/ESTIMATE

457

(15.32) is a singleton, then a ˆ(ti+1 ) = a. According to lemma 15.4, Q4 (μi+1 , ti+1 ) = 0. ˆ(ti+1 ) = a. It follows that S¯i is a singleton. Therefore, we obtain a ˆ(ti+1 ) = a ˆ(ti ). 5) Result 5: If ζ(t) is identically zero for t ∈ [μi+1 , ti+1 ], then a The proof of result 5 is shown as follows. According to (15.33), (15.34), (15.36), ¯}. Q4 (μi+1 , ti+1 ) = 0, H3 (μi+1 , ti+1 ) = 0. We then find that the set S¯i = {| 3 | ≤ a ˆ(ti ). Recalling (15.67), we have a ˆ(ti+1 ) = a From the above five results, we obtain lemma 15.5. Lemma 15.6. If dˆ3 (ti ) = d3 (or dˆ2 (ti ) = d2 or a ˆ(ti ) = a) for certain i ∈ Z+ , then ˆ(t) = a, respectively) for all t ∈ [ti , limk→∞ (tk )). dˆ3 (t) = d3 (or dˆ2 (t) = d2 or a Proof. According to lemma 15.5, dˆ3 (ti+1 ) is equal to either d3 or dˆ3 (ti ). Therefore, if dˆ3 (ti ) = d3 , then dˆ3 (t) = d3 for all t ∈ [ti , limk→∞ (tk )). The same is true of dˆ2 and a ˆ. The proof is complete. We are now ready to provide the proof of theorem 15.1. Proof. 1) We now prove the first of the three portions of the theorem. Recalling lemma 15.3, we know that ti ≥ min{r, τ }(i − 1),

i ≥ 1,

(15.101)

which yields limi→∞ (ti ) = +∞. Then the well-posedness of the closed-loop system is obtained by recalling proposition 15.1. The property (1) is thus obtained. The fact that limi→∞ (ti ) = +∞ allows the solution to be defined on R+ in the following analysis. 2) We now prove the second of the three portions of the theorem. First, we propose the following claim about the sufficient and necessary condition of the finite-time convergence of parameter estimates to the true values. ˆ(0) = a), the estimate dˆ3 (t) (or Claim 15.6. When dˆ3 (0) = d3 (or dˆ2 (0) = d2 or a ˆ d2 (t) or a ˆ(t)) reaches the actual value d3 (or d2 or a) in finite time if and only if z[t] (or w[t] or ζ(t), respectively) is not identically zero on t ∈ [0, ∞). Proof. The proof is shown in appendix 15.6D. Claim 15.7. If any one of the three estimates dˆ3 (t), dˆ2 (t), a ˆ(t) does not reach the true value in finite time, it implies that Ω(t) ≡ 0 on t ∈ [ q11 , ∞) and that m(t) is exponentially convergent to zero. Proof. The proof is shown in appendix 15.6E. Next, we estimate the maximum convergence time of the parameter estimates when they reach the true values. ˆ reaches the true value θ in finite time— Claim 15.8. If the parameter estimate θ(t) that is, tε —then tε ≤ q11 + q12 + T . Proof. The proof is shown in appendix 15.6F. According to claims 15.7 and 15.8, the proof of property (2) is complete.

CHAPTER FIFTEEN

458

3) We now prove the last of the three portions of the theorem. Define a Lyapunov function as Va (t) = V (t) − μm(t),

(15.102)

where m(t) is defined in (15.48) (m(t) < 0 is shown in lemma 15.2), the positive constant μ is defined in (15.56), and V (t) is given in (15.44). Recalling (15.82), (15.83) and applying the Cauchy-Schwarz inequality into the backstepping transformation (15.6), (15.7) and its inverse (15.8), (15.9), we have ¯ ≤ Va (t) ≤ ξ4 Ω(t) ¯ ξ3 Ω(t)

(15.103)

¯ for some positive ξ3 , ξ4 , where the definition of Ω(t) is given in (15.71). Taking the derivative of (15.102) along (15.10)–(15.13), recalling the equalities (15.48), (15.84), inserting (15.76), and applying Young’s inequality and the CauchySchwarz inequality, we have   1 −δ2 1 q1 e V˙ a (t) ≤ − am ζ(t)2 − − μκ1 α(1, t)2 2 2   1 b2 p2 q2 ra − − − q1 − μκ2 β(0, t)2 2 2am 2   δ1 2 + q2 ra e c0 − μλd d(t)2 + q2 ra eδ1 c20 ξ(t)2   1 1 δ1 q2 ra − ra |d4 | − eδ1 x β(x, t)2 dx + μηm(t) 2 0   1 1 δ2 q1 − |d1 | − e−δ2 x α(x, t)2 dx + μσV (t) (15.104) 2 0 for t ≥ 0. Because of (15.68), according to (15.20), we have ξ(t) ≡ 0, t ∈ [tε , ∞).

(15.105)

Recalling the conditions of δ1 , δ2 , ra , λd , μ in (15.45)–(15.47), (15.53), (15.56), we arrive at V˙ a (t) ≤ − (νa − μσ)V + μηm(t) = − (νa − μσ)Va + [−μ(νa − μσ) + μη]m(t)

(15.106)

for t ≥ tε , where (15.82), (15.83), (15.102), (15.105) and the fact that am ≥ bκ − a ¯>0 have been used and where the positive constant νa is given in (15.57). Recalling σ, η in (15.54), (15.55), we arrive at V˙ a (t) ≤ −λ1 Va (t), t ≥ tε ,

(15.107)

where λ1 = νa − μσ > 0. Multiplying both sides of (15.107) by eλ1 t and integrating both sides of (15.107) from tε to t, we obtain Va (t) ≤ Va (tε )e−λ1 (t−tε ) , t ≥ tε . Recalling (15.103), we have ¯ ≤ Υθ Ω(t ¯ ε )e−λ1 (t−tε ) , t ≥ tε , Ω(t)

(15.108)

PIECEWISE-CONSTANT INPUT/ESTIMATE

459

where the positive constant Υθ is Υθ =

ξ4 . ξ3

(15.109)

If tε = 0, property (3)—that is, (15.70)—is obtained directly. Next we conduct an analysis for t ∈ [0, tε ] with tε = 0. Recalling (15.45)–(15.47), (15.53), (15.54), (15.55), (15.56), (15.82), (15.83), (15.102), (15.104), we obtain ˆ V˙ a (t) ≤ − λ1 Va (t) + Q(θ(0))V a (t),

t ∈ [0, tε ],

(15.110)

ˆ where the positive constant Q(θ(0)) is obtained by bounding ξ(t)2 given by (15.75), ˆ ˆ and the value of Q(θ(0)) depends on the initial estimate θ(0). We thus obtain ˆ λ2 (θ(0))t ¯ ≤ Υθ Ω(0)e ¯ Ω(t) , t ∈ [0, tε ],

(15.111)

ˆ ˆ where λ2 (θ(0)) = |Q(θ(0)) − λ1 | > 0, and the positive constant Υθ is given in (15.109). Therefore, we have ˆ ε ¯ ¯ ε ) ≤ Υθ eλ2 (θ(0))t Ω(t Ω(0).

(15.112)

Inserting (15.112) into (15.108) and applying claim 15.8 yields ˆ 1 )( q + q +T ) ¯ ¯ ≤ Υ2 e(λ2 (θ(0))+λ 1 2 Ω(t) Ω(0)e−λ1 t , θ 1

Denoting

1

ˆ

1

t ≥ 0.

(15.113)

1

M = Υ2θ e(λ2 (θ(0))+λ1 )( q1 + q2 +T ) , we obtain (15.70). This completes the proof of property (3) of the theorem.

15.5

APPLICATION IN THE MINING CABLE ELEVATOR

In this section, the proposed boundary controller, where both the parameter estimates and the control input employ piecewise-constant values, is applied to the axial vibration control of a mining cable elevator that is 2000 m deep, where the damping coefficients of the cable and the cage are unknown. Model The axial vibration model of the mining cable elevator is described by the following wave PDE-ODE model with in-domain damping [185]: πRd2 Euxx (x, t) − dc ut (x, t), 4 πRd2 Mc utt (0, t) = −cL ut (0, t) − Eux (0, t), 4 ρutt (x, t) =

πRd2 Eux (L, t) = Ud (t), 4

(15.114) (15.115) (15.116)

where the cable length is considered constant. The PDE state u(x, t) denotes the distributed axial vibration displacements along the cable, and the boundary state

CHAPTER FIFTEEN

460

Table 15.1. Physical parameters of the mining cable elevator Parameters (units)

Values

Cable length L (m) Cable diameter Rd (m) Cable effective Young’s modulus E (N/m2 ) Cable linear density ρ (kg/m) Mass of cage Mc (kg) Damping coefficient of cage cL Cable material damping coefficient dc Gravitational acceleration g (m/s2 )

2000 0.2 1.02×109 8.1 15000 0.4 0.5 9.8

u(0, t) represents the axial vibration displacement of the cage. The physical parameters in (15.114)–(15.116) are shown in table 15.1. According to [149], we apply the Riemann transformations  z(x, t) = ut (x, t) −  w(x, t) = ut (x, t) +

Eπ Rd ux (x, t), ρ 2

(15.117)

Eπ Rd ux (x, t) ρ 2

(15.118)

and define the new variable ζ(t) = ut (0, t), which allows us to rewrite (15.114)– (15.116) as (15.1)–(15.5) with the coefficients  q1 = q2 =

Eπ Rd −dc , d1 = d 2 = d 3 = d 4 = , ρ 2 2ρ

p = 1, c = 2, c0 =

4 √ , Rd Eπρ

√ √ −cL Rd Eπρ Rd Eπρ a= + ,b=− , Mc 2Mc 2Mc

(15.119) (15.120)

(15.121)

where a reflection term z(L, t) appearing at the controlled boundary (15.5) is considered to be compensated by the control input at the drum [178] independent of the head sheave control input designed in this chapter. The unknown physical parameters are damping coefficients of the cable and the cage, dc , cL , which leads to the fact that a is unknown, and d1 = d2 = d3 = d4 are unknown in the 2 × 2 hyperbolic PDE-ODE system. The designs in this chapter are directly applicable to this problem (with only one slight modification: removing the last term in (15.29) and multiplying (15.30), (15.31) by 2), and the outputs of the identifier are the estimates of d2 , d3 . According to the values in table 15.1, together with (15.119)–(15.121), we know that assumptions 15.1, 15.2 are satisfied. The bounds a ¯, d¯3 , d¯2 of the unknown parameters a = 1.07, d3 = d2 = −0.025 are set as, 3, 0.4, 0.5, respectively. The initial conditions of z(x, t) and w(x, t) are defined as z(x, 0) = 0.5 sin(2πx/L + π/6), w(x, 0) = 0.5 sin(3πx/L)

Control forces (N)

PIECEWISE-CONSTANT INPUT/ESTIMATE

461

2000

Uc Ud

0

2000

–4000

t = 0.51 s

0

–2000

–2000 0.4

0

2

4

6

0.6

0.8

8

10

Time (s)

Figure 15.2. The continuous-in-state control signal Uc (t) in (15.18) and the piecewise-constant control signal Ud (t) in (15.17).

and

1 ζ(0) = (pw(0, 0) + z(0, 0)) 2 according to (15.4). We pick the initial value of m(t) as m(0) = −10.

Determination of Design Parameters The free design parameters δ, T in (15.40) are selected as δ = 1.8, T = 3, and the positive integer n in the parameter estimator (15.67) is defined as 7. According to (15.119)–(15.121), the parameter values in table 15.1, and the known bounds a ¯, d¯3 , d¯2 defined above, recalling (15.15), (15.45)–(15.47), the design parameters κ, δ1 , δ2 , ra are determined to be κ = −10, δ1 = 0.1, δ2 = 0.1, ra = 1.5. Choose the design parameter r as r = 0.2 via (15.42). Next, choose κ1 = 2100, κ2 = 3000 according to (15.49), (15.50), where λα = 899.6, λβ = 1490.8 are obtained from (15.51), and a conservative estimate λa = 1000 comes from (15.52). Then the design parameters λd , σ, η are selected as λd = 0.005, σ = 60, η = 120 to satisfy (15.53)–(15.55), where μ = 0.15, νa = 9.66 are obtained from (15.56), (15.57). Finally, pick the design parameter ϑ = 12000 via (15.59). Simulation Results We simulate a mining cable elevator running for a short time period of 10 s where the cable length is regarded as constant. The simulation is conducted based on the finite-difference method with a time step of 0.0015 s and a space step of 0.5 m. By employing the finite-difference method as well with a step length of 0.5 m for ˆ i )), ˆ i )), Φ(x, y; θ(t y running from 0 to x, the approximate solutions of Ψ(x, y; θ(t ˆ λ(x; θ(ti )) in the control law (15.17) are obtained from the conditions (15.128)– (15.133), whose unknown coefficients are replaced by the piecewise-constant estimates. The triggering mechanism (15.37)–(15.41) is implemented as the flowchart in figure 15.1, with the selected design parameters. The piecewise-constant control input Ud (t) defined in (15.17) is shown in figure 15.2, where the estimate θˆ is recomputed, and the states z, w, ζ are resampled simultaneously. The first update is triggered at t = 0.51 s, the total number of triggering times is 33, the maximum dwell time is 0.595 s, and the minimal dwell time is 0.105 s, which is much larger than the highly conservative minimal dwell time estimate of 5.75 × 10−4 s obtained from (15.80), (15.92) in lemma 15.3, where the analysis parameter μ0 defined in (15.86) is μ0 = 331. The continuous-in-state control signal

CHAPTER FIFTEEN

462 –0.02 dˆ3, dˆ2

–0.04 t = 0.51 s

–0.06

dˆ3 dˆ2

–0.08 –0.1 0

2

4

6

8

10

Time (s)

Figure 15.3. Parameter estimates dˆ3 and dˆ2 .

â

1.2

t = 0.51 s

0.8

0.4

0

2

4

8

6

10

Time (s)

Figure 15.4. Parameter estimate a ˆ. 6

× 106 d(t)2 ϑV(t) – m(t) Execution times

4 2 0

0

2

4

6

8

10

Time (s)

Figure 15.5. The evolution of d(t)2 and ϑV (t) − m(t) in (15.38).

Uc (t) (15.18) used in the ETM is also shown in figure 15.2. There is a jump in the continuous-in-state signal Uc (t) at t = 0.51 s because Uc (t) defined in (15.18) includes the piecewise-constant parameter estimate θˆ whose evolution is shown in figures 15.3 and 15.4, where, under the nonzero initial conditions defined at the ˆ are updated and reach the true beginning of this section, the estimates dˆ3 , dˆ2 , a values of the unknown parameters d3 = −0.025, d2 = −0.025, a = 1.06 (figures 15.3 and 15.4, dashed line) at the first triggering time, t = 0.51 s, and are kept constant in the subsequent recomputations. Because the estimates have reached the true values at the first triggering time, according to the designed triggering mechanism (15.37)–(15.41), the following execution times are determined by the triggering condition (15.38), which can be seen in figure 15.5 showing the time evolution of the functions in the triggering condition (15.38) and the execution times. Figure 15.5 shows that an event is generated, the control value is updated, and d(t) is reset to zero when the trajectory d(t)2 reaches the trajectory ϑV (t) − m(t). Figure 15.6 demonstrates that the ODE state

0.25 0.2 0.15 0.1 0.05 0 –0.05 –0.1

463 × 106 ζ(t) m(t)

0 –0.5 –1 –1.5

m(t)

ζ(t) (m/s)

PIECEWISE-CONSTANT INPUT/ESTIMATE

–2 0

2

4

6

–2.5 10

8

Time (s)

Figure 15.6. The evolution of ζ(t), m(t) under the control input Ud (t) in (15.17).

w(x, t)

0.6 0.3 0 –0.3 –0.6 2000 1000 x(m)

0

0

8

6

4

2

10

t(s)

Figure 15.7. The evolution of w(x, t) under the control input Ud (t) in (15.17).

z(x, t)

0.6 0.3 0 –0.3 –0.6 2000 1000 x(m)

0 0

2

6

4

8

10

t(s)

Figure 15.8. The evolution of z(x, t) under the control input Ud (t) in (15.17).

ζ(t), whose physical meaning is the axial vibration velocity of the cage, is convergent to zero, and the internal variable m(t) in (15.38) is less than zero all the time and convergent to zero as well. Figures 15.7 and 15.8 show that the PDE states w(x, t), z(x, t) are regulated to zero under the control input Ud (t). Therefore, according to the Riemann transformation (15.117), (15.118), we find that R2 π the vibration energy of the cable 12 ρut (·, t)2 + 8d Eux (·, t)2 also decreases to zero. Finally, we run simulations for eight different initial conditions and compute the inter-execution times between two triggering times. The density of the interexecution times is shown in figure 15.9, from which we know that the prominent inter-execution times are in the range from 0.1 s to 0.2 s.

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464

Density

0.1 0.08 0.06 0.04 0.02 0

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Inter-execution times (s)

0.8

0.9

Figure 15.9. Density of the inter-execution times computed for eight different 2 ¯ , w(x, 0) = initial conditions given by z(x, 0) = 0.5 sin(2¯ nπx/L + π/6) + k(x/L) 2 ¯ ¯ ¯ = 1, 2, m ¯ = 1, 2, k = 0, 1. 0.5 sin(3mπx/L) ¯ + k(x/L) for n 15.6

APPENDIX

A. Conditions of kernels in the backstepping transformation (15.6), (15.7) and its inverse (15.8), (15.9) The conditions of kernels ϕ, φ, γ, Ψ, Φ, λ in the backstepping transformation (15.6), (15.7) are d2 − (q1 + q2 )ϕ(x, x) = 0, −γ(x)b + q2 ϕ(x, 0) + q1 pφ(x, 0) = 0,

(15.122) (15.123)

−q1 φx (x, y) − q1 φy (x, y) − d3 ϕ(x, y) = 0, q2 ϕy (x, y) − q1 ϕx (x, y) − (d4 − d1 )ϕ(x, y) − d2 φ(x, y) = 0,

(15.124) (15.125)

q1 γ  (x) + (a − d1 )γ(x) + q1 cφ(x, 0) = 0, γ(0) = −pκ + c, d3 + (q1 + q2 )Ψ(x, x) = 0,

(15.126) (15.127) (15.128)

−λ(x)b + q2 Φ(x, 0) + q1 pΨ(x, 0) = 0, −q1 Ψy (x, y) + q2 Ψx (x, y) + (d4 − d1 )Ψ(x, y) − d3 Φ(x, y) = 0,

(15.129) (15.130)

q2 Φy (x, y) + q2 Φx (x, y) − d2 Ψ(x, y) = 0, −q2 λ (x) + (a − d4 )λ(x) + q1 cΨ(x, 0) = 0,

(15.131) (15.132)

λ(0) = −κ.

(15.133)

¯ γ¯ , Ψ, ¯ in the inverse backstepping transfor¯ Φ, ¯ λ The conditions of kernels ϕ, ¯ φ, mation (15.8), (15.9) are ¯ −d3 + (q1 + q2 )Ψ(x, x) = 0, ¯ ¯ ¯ λ(x)b − q2 Φ(x, 0) − q1 pΨ(x, 0) = 0, ¯ y) = 0, ¯ y (x, y) + q2 Ψ ¯ x (x, y) + (d4 − d1 )Ψ(x, ¯ −q1 Ψ y) + d3 φ(x, ¯ y (x, y) + q2 Φ ¯ x (x, y) + d3 ϕ(x, ¯ y) = 0, q2 Φ  ¯ ¯ −q2 λ (x) − (bκ − a + d4 )λ(x) − d3 γ¯ (x) = 0, ¯ = κ, λ(0) −d2 − (q1 + q2 )ϕ(x, ¯ x) = 0, ¯ 0) = 0, γ¯ (x)b − q2 ϕ(x, ¯ 0) − q1 pφ(x, ¯ y) = 0, q2 ϕ¯y (x, y) − q1 ϕ¯x (x, y) − (d4 − d1 )ϕ(x, ¯ y) + d2 Φ(x,

(15.134) (15.135) (15.136) (15.137) (15.138) (15.139) (15.140) (15.141) (15.142)

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465

¯ −q1 φ¯x (x, y) − q1 φ¯y (x, y) + d2 Ψ(x, y) = 0,  ¯ γ (x) − d2 λ(x) = 0, q1 γ¯ (x) − (bκ − a + d1 )¯ γ¯ (0) = −pκ − c.

(15.143) (15.144) (15.145)

The equation sets (15.122)–(15.133) and (15.134)–(15.145) are well-known coupled linear heterodirectional hyperbolic PDE-ODE systems whose well-posedness has been proved in theorem 4.1 of [48]. ˜ 1, K ˜ 2, K ˜3 B. Expressions of the functions M1 , . . . , M6 , K1 , K2 , K3 , and K The expressions of the functions M1 , . . . , M6 are ˆ i ), θ) = q1 K1y (y, θ(t ˆ i ), θ) + K1 (y, θ(t ˆ i ), θ)d1 M1 (y, θ(t ˆ i ), θ)K ˆ i ), θ), ˜ 1 (y, θ(t − q2 K2 (1, θ(t

(15.146)

ˆ i ), θ) = q2 K2y (y, θ(t ˆ i ), θ) − K2 (y, θ(t ˆ i ), θ)d4 M2 (y, θ(t ˆ i ), θ)K ˆ i ), θ), ˜ 2 (y, θ(t + q2 K2 (1, θ(t

(15.147)

ˆ i ), θ) = K3 (θ(t ˆ i ), θ)am M3 (θ(t ˆ i ), θ)K ˆ i ), θ), ˜ 3 (θ(t + q2 K2 (1, θ(t

(15.148)

ˆ i ), θ) = K3 (θ(t ˆ i ), θ)b − q2 K2 (0, θ(t ˆ i ), θ) M4 (θ(t ˆ i ), θ), − pq1 K1 (0, θ(t

(15.149)

ˆ i ), θ) = q1 K1 (1, θ(t ˆ i ), θ), M5 (θ(t ˆ i ), θ) = q2 c0 K2 (1, θ(t ˆ i ), θ), M6 (θ(t

(15.150) (15.151)

where the expressions of the functions K1 , K2 , K3 are  1 ˆ i ), θ) = ψ(1, y; θ(t ˆ i )) − ˆ i ))φ(ε, ¯ y; θ)dε ψ(1, ε; θ(t K1 (y, θ(t y



1

−

ˆ i ))ψ(ε, ¯ y; θ)dε, Φ(1, ε; θ(t

y



1

ˆ i ), θ) = Φ(1, y; θ(t ˆ i )) − K2 (y, θ(t

ˆ i ))ϕ(ε, ψ(1, ε; θ(t ¯ y; θ)dε

y



1

−

ˆ i ))Φ(ε, ¯ y; θ)dε, Φ(1, ε; θ(t

y



ˆ i ), θ) = λ(1; θ(t ˆ i )) − K3 (θ(t  −

1 0

1 0

(15.153)

ˆ i ))¯ ψ(1, y; θ(t γ (y; θ)dy

ˆ i ))λ(y; ¯ θ)dy Φ(1, y; θ(t

˜ 2, K ˜ 3 are ˜ 1, K and where the expressions of the functions K ˆ i ), θ) = ψ(1, y; θ) − ψ(1, y; θ(t ˆ i )) ˜ 1 (y, θ(t K  1 ˆ i ))]φ(ε, ¯ y; θ)dε [ψ(1, ε; θ) − ψ(1, ε; θ(t − y

(15.152)

(15.154)

CHAPTER FIFTEEN

466 

1

−

ˆ i ))]ψ(ε, ¯ y; θ)dε, [Φ(1, ε; θ) − Φ(1, ε; θ(t

(15.155)

y

ˆ i ), θ) = Φ(1, y; θ) − Φ(1, y; θ(t ˆ i )) ˜ 2 (y, θ(t K  1 ˆ i ))]ϕ(ε, [ψ(1, ε; θ) − ψ(1, ε; θ(t ¯ y; θ)dε − 

y 1

−

ˆ i ))]Φ(ε, ¯ y; θ)dε, [Φ(1, ε; θ) − Φ(1, ε; θ(t

(15.156)

y

ˆ i ), θ) = λ(1; θ) − λ(1; θ(t ˆ i )) ˜ 3 (θ(t K  1 ˆ i ))]¯ [ψ(1, y; θ)−ψ(1, y; θ(t γ (y; θ)dy −  −

0

0

1

ˆ i ))]λ(y; ¯ θ)dy. [Φ(1, y; θ) − Φ(1, y; θ(t

(15.157)

C. Proof of claim 15.5 ¯ = 0 such that We prove this by contradiction. Suppose there exists a λ ¯ z(x, t) − λw(x, t) ≡ 0, x ∈ [0, 1], t ∈ [μi+1 , ti+1 ).

(15.158)

Taking the time and spatial derivatives of (15.158) yields ¯ t (x, t) = 0, zt (x, t) − λw ¯ x (x, t) = 0 zx (x, t) − λw

(15.159) (15.160)

for x ∈ [0, 1], t ∈ [μi+1 , ti+1 ) (except possibly for finitely many discontinuities of the functions w[t], z[t]). Recalling (15.2), (15.3), we have ¯ − d2 ¯ 2 + (d4 − d1 )λ d3 λ w(x, t), ¯ −λ(q1 + q2 ) ¯ − d2 ¯ 2 + (d4 − d1 )λ d3 λ z(x, t) zx (x, t) = ¯ −λ(q1 + q2 )

wx (x, t) =

(15.161) (15.162)

for x ∈ [0, 1], t ∈ [μi+1 , ti+1 ). The solutions of (15.161), (15.162) are obtained as w(x, t) = w(0, t)e

¯ 2 +(d −d )λ−d ¯ d3 λ 4 1 2 ¯ −λ(q 1 +q2 )

¯ z(x, t) = λw(0, t)e

x

¯ 2 +(d −d )λ−d ¯ d3 λ 4 1 2 ¯ −λ(q 1 +q2 )

,

(15.163)

x

(15.164)

¯ for x ∈ [0, 1], t ∈ [μi+1 , ti+1 ), where z(0, t) = λw(0, t), obtained from the hypothesis (15.158), has been used. By virtue of (15.5), (15.17), we know that w(1, t) is constant on t ∈ [μi+1 , ti+1 ). It implies that w(0, t) is constant on t ∈ [μi+1 , ti+1 ) recalling (15.163)—that is, w(0, t) = w(0, μi+1 ), t ∈ [μi+1 , ti+1 ).

(15.165)

According to the hypothesis (15.158), we also have ¯ z(0, t) = λw(0, μi+1 ), t ∈ [μi+1 , ti+1 ).

(15.166)

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467

By virtue of (15.3), (15.158), (15.161)–(15.166), we obtain q2

¯ − d2 ¯ 2 + (d4 − d1 )λ d3 λ ¯ + d4 = 0, + d3 λ ¯ 1 + q2 ) −λ(q

(15.167)

which is a necessary condition for the hypothesis (15.158) to hold. For the case that c = 0, a = 0, inserting (15.165), (15.166) into (15.4) yields ¯ that ζ(t) = (λ+p) c w(0, μi+1 ) is also constant on t ∈ [μi+1 , ti+1 ). Recalling (15.1), and w(0, μi+1 ) = 0 which is obtained from the fact that w(x, t) is not identically zero in the interval and (15.163), (15.165), it follows that ¯ = − bc − p. λ a

(15.168)

For the cases that c = 0, we have ¯ = −p λ

(15.169)

according to (15.4), (15.165), (15.166), and w(0, μi+1 ) = 0. By virtue of assumption 15.2, (15.168), and (15.169), we know that the necessary condition for the hypothesis (15.158) to hold—that is, (15.167)—does not hold for the cases that a = 0 and the case that a = 0, c = 0. For the case that a = 0, c = 0, inserting (15.165), (15.166) into (15.4), recalling (15.1) and w(0, μi+1 ) = 0, it follows ¯ = 0: contradiction. Therefore, the hypothesis (15.158) does not hold. The that λ proof of this claim is complete. D. Proof of claim 15.6 We first prove sufficiency. If z[t] (or w[t] or ζ(t)) is not identically zero for t ∈ [0, ∞), there exists an interval [μi+1 , ti+1 ] on which z[t] (or w[t] or ζ(t), respectively) is ˆ(τi+1 ) = a, not identically zero. It follows that dˆ3 (ti+1 ) = d3 (or dˆ2 (ti+1 ) = d2 or a respectively), recalling lemma 15.5. ˆ(0) = a), if z[t] Next, we prove necessity. When dˆ3 (0) = d3 (or dˆ2 (0) = d2 or a (or w[t] or ζ(t)) is identically zero for t ∈ [0, ∞), applying lemma 15.5, we have ˆ(t) = a ˆ(0) = a, respectively). Therefore, dˆ3 (t) = dˆ3 (0) = d3 (or dˆ2 (t) = dˆ2 (0) = d2 or a ˆ) reaches the true value, it follows that z[t] (or w[t] or if the estimate dˆ3 (or dˆ2 or a ζ(t), respectively) is not identically zero on t ∈ [0, ∞). The proof of claim 15.6 is complete. E. Proof of claim 15.7 The proof is divided into three cases. Case 1: We suppose that the estimate dˆ2 (t) does not reach d2 in finite time. This implies that w[t] ≡ 0 on t ∈ [0, ∞) according to claim 15.6. By virtue of (15.3) and d3 = 0, we have z[t] ≡ 0 on t ∈ [0, ∞). We then obtain from (15.4) that ζ(t) ≡ 0 on t ∈ [0, ∞). Therefore, Ω(t) ≡ 0. According to w[t] ≡ 0, z[t] ≡ 0, and ζ(t) ≡ 0 on t ∈ [0, ∞), we have that d(t) ≡ 0 according to (15.19), and m(t) is exponentially convergent to zero recalling (15.48) and (15.6), (15.7), (15.44). Case 2: We suppose that the estimate dˆ3 (t) does not reach d3 in finite time. This implies that z[t] ≡ 0 on t ∈ [0, ∞) according to claim 15.6. Then (15.1)–(15.5) become

CHAPTER FIFTEEN

468 c

ζ(t) =e(a+b p )t ζ(0), wt (x, t) =q2 wx (x, t) + d4 w(x, t), c w(0, t) = ζ(t), p w(1, t) =c0 Ud (t),

(15.170) (15.171) (15.172) (15.173)

t ∈ [0, ∞), in the closed-loop system. If ζ(0) = 0, then z[t], w[t], ζ(t) are identically zero on t ∈ [0, ∞) according to (15.170)–(15.172)—that is, Ω(t) ≡ 0. Next we discuss the case when ζ(0) = 0. The equation (15.172) holding for t ∈ [0, ∞) requires the initial condition of w to be d4 c 1 c w(x, 0) = e− q2 x e(a+b p ) q2 x ζ(0) p

(15.174)

to ensure that (15.172) holds on t ∈ [0, q12 ] and w(1, t) to be d4 c c 1 c w(1, t) = e− q2 e(a+b p ) q2 e(a+b p )t ζ(0), t ∈ [0, ∞) p

(15.175)

to ensure that (15.172) holds on t ∈ [ q12 , ∞). If c = 0, we obtain from (15.171), (15.174), (15.175) that w ≡ 0 for t ∈ [0, ∞). Together with z ≡ 0 for t ∈ [0, ∞), recalling (15.17), (15.173), ζ(0) = 0 with (15.170), and λ(1) = 0 (according to (15.15), (15.132) with c = 0, and (15.133)), it follows that w(1, t) is not identically zero on t ∈ [0, ∞): this contradicts (15.175) under c = 0. Next, we discuss the cases with c = 0. According to (15.17) and (15.173), w(1, t) is piecewise-constant, which contradicts (15.175) that is an exponential function (a + b pc = 0 ensured by assumption 15.1). Therefore, case 2 only happens when ζ(0) = 0, that is, z[t], w[t], ζ(t) are identically zero on t ∈ [0, ∞). This implies that Ω(t) is identically zero on t ∈ [0, ∞), and m(t) is exponentially convergent to zero, recalling (15.6), (15.7), (15.19), (15.44), (15.48). Case 3. We suppose that the estimate a ˆ(t) does not reach a in finite time. This implies that ζ[t] ≡ 0 on t ∈ [0, ∞) according to claim 15.6. It means that β(0, t) is identically zero on t ∈ [0, ∞) according to (15.10), and α(0, t) is identically zero on t ∈ [0, ∞) according to (15.11). It follows that β[t] is identically zero on t ∈ [0, ∞), and α[t] is identically zero on t ∈ [ q11 , ∞), recalling (15.12), (15.13). Therefore, for t ∈ [ q11 , ∞), β[t], α[t], ζ(t) are identically zero; that is, w[t], z[t], ζ(t) are identically zero on t ∈ [ q11 , ∞), recalling the inverse transformation (15.8), (15.9). Therefore, Ω(t) is identically zero on t ∈ [ q11 , ∞), and m(t) is exponentially convergent to zero, recalling (15.6), (15.7), (15.19), (15.44), (15.48). The proof of claim 15.7 is complete. F. Proof of claim 15.8 We estimate the maximum value of tε in various situations of initial conditions z[0], w[0], ζ(0). Case 1: z[0] = 0, w[0] = 0, ζ(0) = 0. According to (15.3), z[t], w[t] are not identically zero on t ∈ [0, t1 ] (if w[t] is identically zero on t ∈ [0, t1 ], by virtue of (15.3) and d3 = 0, it implies that z[t] is identically zero on t ∈ [0, t1 ]: contradiction). Suppose

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that ζ(t) is identically zero on t ∈ [0, tf ], where tf = min{tf : f ∈ Z+ , tf > q11 + q12 }. This means that β(0, t) is identically zero on t ∈ [0, tf ] according to (15.10), and α(0, t) is identically zero on t ∈ [0, tf ] according to (15.11). It follows that β[t] is identically zero on t ∈ [0, tf − q12 ], and α[t] is identically zero on t ∈ [ q11 , tf ], recalling (15.12), (15.13). Therefore, for t ∈ [ q11 , tf − q12 ], β[t], α[t], ζ(t) are identically zero, that is, w[t], z[t], ζ(t) are identically zero, recalling the inverse transformation (15.8), (15.9). According to (15.5), Ud in (15.17) is kept constant in a time interval. Therefore, Ud is identically zero on t ∈ [ q11 , tf − q12 ], which implies that Ud is identically zero on t ∈ [ q11 , tf ]. It follows that z(x, t), w(x, t), ζ(t) are identically zero for t ∈ [ q11 , tf ]. By virtue of (15.1)–(15.5), (15.17), through iterative constructions between successive triggering times, we have that Ud ≡ 0 and z(x, t), w(x, t), ζ(t) are identically zero for t ∈ [ q11 , ∞). This implies that ζ(t) ≡ 0 on t ∈ [0, ∞) and that a ˆ does not reach the true value in finite time: contradiction. Therefore, the nonzero values of ζ(t) appear not later than tf . Recalling lemma 15.5, we have dˆ3 (t1 ) = d3 , dˆ2 (t1 ) = d2 , a ˆ(tf ) = a. Therefore, tε ≤ T + q11 + q12 , where T is the maximum dwell time between two triggering times. Case 2: w[0] = 0, z[0] = 0, ζ(0) = 0. The maximum time taken by the nonzero ˆ(t) would values of w[0] propagate to z and ζ(t) is q12 . Therefore, the estimate dˆ3 (t), a reach the true value d3 , a not later than tf = min{tf : f ∈ Z+ , tf > q12 } according to lemmas 15.5 and 15.6. Because of w[0] = 0, we have dˆ2 (t1 ) = d2 . It follows that tε ≤ q12 + T . Case 3: ζ(0) = 0, z[0] = 0, w[0] = 0. If w[t] is identically zero on t ∈ [0, t1 ], this implies that z[t] is identically zero on t ∈ [0, t1 ] according to (15.3) with d3 = 0, which means ζ(t) is identically zero on t ∈ [0, t1 ] by (15.4): contradiction. Therefore, w[t] is not identically zero on t ∈ [0, t1 ]. According to the proof in case 2 of claim 15.7, we know that the necessary condition for z[t] to be identically zero on t ∈ [0, tf ], where tf = min{tf : f ∈ Z+ , tf > q12 }, is that w satisfies (15.174), (15.175), which implies w[0] = 0 (c = 0) or w[t] = 0 for t ∈ [0, t1 ] (c = 0): contradiction. Therefore, z[t] is not ˆ(t1 ) = a, dˆ3 (tf ) = d3 . It follows identically zero on t ∈ [0, tf ]. Therefore, dˆ2 (t1 ) = d2 , a 1 that τε ≤ q2 + T . Case 4: ζ(0) = 0, w[0] = 0, z[0] = 0. Suppose that z[t] is identically zero on t ∈ [0, tf ], where tf = min{tf : f ∈ Z+ , tf > q12 }. According to the proof in case 2 of claim 15.7, if c = 0, a necessary condition of the above hypothesis is (15.174), which means w[0] = 0: contradiction. If c = 0, a necessary condition of the above hypothesis is (15.175), which does not hold, because the control input Ud applied at (15.173) is piecewise-constant while w(1, t) in (15.175) is an exponential function (a + b pc = 0 ensured by assumption 15.1). Therefore, z[t] is not identically zero on t ∈ [0, tf ]. This implies that dˆ3 (τf ) = d3 according to lemmas 15.5 and 15.6. Because of w[0] = 0 and ζ(0) = 0, we have dˆ2 (t1 ) = d2 , a ˆ(t1 ) = a. Therefore, tε ≤ q12 + T . Case 5: ζ(0) = 0, z[0] = 0, w[0] = 0. According to (15.3) with d3 = 0 and the fact that z[t] is not identically zero on t ∈ [0, t1 ], we know that w[t] is not identically ˆ 1 ) = θ. Therefore, tε ≤ T . zero on t ∈ [0, t1 ]. Recalling lemma 15.5, we have θ(t Case 6: ζ(0) = 0, z[0] = 0, w[0] = 0. Following the analysis in case 1, we show that the nonzero values of ζ(t) appear not later than tf , where tf = min{tf : f ∈ Z+ , tf > 1 1 ˆ ˆ ˆ(tf ) = a. q1 + q2 }. Recalling lemmas 15.5 and 15.6, we have d3 (t1 ) = d3 , d2 (t1 ) = d2 , a 1 1 Therefore, tε ≤ T + q1 + q2 .

Hyperbolic PDEs

Hyperbolic PDEs

Parabolic PDEs

Coupled hyperbolic PDEs Coupled hyperbolic PDEs

Coupled hyperbolic PDEs

[11]

[10]

[106]

Chapter 13 Chapter 14

Chapter 15

Types of PDEs Continuous except for finite-time instants Continuous except for finite-time instants Continuous except for finite-time instants Piecewise-constant Continuous except for finite-time instants Piecewise-constant

Types of control inputs

Piecewise-constant

Continuous Piecewise-constant

Piecewise-constant

Piecewise-constant

Piecewise-constant

Types of identification

Table 15.2. Recent results of triggered-type adaptive boundary control of PDEs

Nonequidistant

Nonequidistant Nonequidistant

Nonequidistant

Equidistant

Equidistant

Types of triggering times

PIECEWISE-CONSTANT INPUT/ESTIMATE

471

Case 7: ζ(0) = 0, z[0] = 0, w[0] = 0. According to lemma 15.5, tε ≤ t1 ≤ T . Case 8: z[0] = 0, w[0] = 0, ζ(0) = 0. According to the plant (15.1)–(15.5) with the control input (15.17), we know that z[t], w[t], ζ(t) are identically zero for t ∈ [0, ∞). The estimates reach the true values in finite time only when qˆ1 (0) = q1 , qˆ2 (0) = q2 — that is, τε = 0. In summary, we have proved for all eight cases that tε ≤ q11 + q12 + T . This completes the proof of claim 15.8.

15.7

NOTES

Recent results of triggered-type adaptive boundary control of PDEs are summarized in table 15.2.

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Index

2 × 2 hyperbolic system, 38, 97 4 × 4 hyperbolic system, 139, 169

floating head sheave, 11 frequency domain, 179

accelerometer, 66, 171 active disturbance rejection control, 67 adaptive cancellation, 5, 230

gain kernels, 17 Gaussian white noise, 194 globally Lipschitz, 342

backstepping transformation, 315, 408 bottom-hole assembly, 203 bounded-input bounded-output, 321

Hamilton’s principle, 12 harmonic disturbance, 69 heterodirectional coupled first-order hyperbolic PDE, 4

cage, 11 catenary cable, 11 Cauchy-Schwarz inequality, 75, 88, 417 certainty-equivalence controller, 410 chain of integrators, 275 continuous-in-time control, 183, 320 continuous mappings, 174 continuously differentiable, 342 converging series, 270 convex optimization, 411 cost function, 411 deep-sea construction vessel, 2 delay compensation, 170 disturbance estimator, 67 disturbance rejection, 5 drill bit, 203 drill collar, 203 drilling rig, 203 drill pipe, 203 drum, 11 event-triggering mechanism, 7, 183, 310, 320, 385 exponential regulation, 421, 450 Fermat’s theorem, 449 finite-difference method, 81, 118, 334 finite-time convergence, 418, 457

in-domain couplings, 139, 315, 440 initial boundary value problem, 15 integral equation, 270 intermediate system, 73, 178 intermediate value theorem, 324 internal material damping, 34 inverse transformation, 17, 31 kernel functions, 15 Laplace transform, 177 least-squares identifier, 411, 448 left inversion, 275 linear density, 12 linear equality constraints, 411 linearization, 135 low-pass filter, 319 Lyapunov analysis, 21 Marcum Q-function, 442 method of characteristics, 269, 286 minimum dwell time, 452 mining cable elevator, 1, 11 moving boundary, 15, 342 moving coordinate system, 12 multi-dimensional coupled vibrations, 168

INDEX

488 Neumann interconnection, 14 observer error system, 19 ocean current disturbances, 194 oil drilling, 202 orthonormal basis, 415 output-feedback controller, 20, 21 parameter estimator, 449 regulation-triggered adaptive controller, 414 relative degree, 256, 311, 315 Riemann transformations, 173 sandwich hyperbolic PDE, 233, 276, 341 sensor delay, 171 separation principle, 102, 393 ship-mounted crane, 171 single-cable mining elevator, 11 singleton, 416, 444 spatially varying coefficients, 136 state-feedback controller, 15, 17 state observer, 71 steady state, 135

stick-slip instability, 202 strictly proper transfer function, 177 successive approximations, 270 target system, 15 time-varying domain, 136 transport PDE, 174, 276 transport speeds, 173 triggering mechanism, 443 uniform ultimate boundedness, 222 unmanned aerial vehicles, 2 unmatched disturbances, 6 viscoelastic guide, 95 wave PDE-ODE, 14 wave speed, 15 well-posedness, 15, 237 Young’s inequality, 24 Young’s modulus, 12 Zeno phenomenon, 193, 322, 329, 331, 392