Delay Compensation for Nonlinear, Adaptive, and Pde Systems

by Krstic, Miroslav

• ISBN13: 9780817648763
• ISBN10: 0817648763
• Format: Hardcover
• Copyright: 2009-10-30
• Publisher: Birkhauser

Summary

Some of the most common dynamic phenomena that arise in engineering practice, actuator and sensor delays, fall outside the scope of the standard finite-dimensional system. The first attempt at infinite-dimensional feedback design in the field of control systems---the Smith predictor---has remained limited to linear finite-dimensional plants over the last five decades. Shedding light on new opportunities in predictor feedback, this book significantly broadens the set of techniques available to a mathematician or engineer working on delay systems. The book is collection of tools and techniques that make predictor feedback ideas applicable to nonlinear systems, systems modeled by PDEs, systems with highly uncertain or completely unknown input/output delays, and systems whose actuator or sensor dynamics are modeled by more general hyperbolic or parabolic PDEs, rather than by pure delay. Specific features include:* A construction of explicit Lyapunov functionals, which can be used in control design or stability analysis, lead to a resolution of several long-standing problems in predictor feedback* A detailed treatment of individual classes of problems (nonlinear ODEs, parabolic PDEs, first-order hyperbolic PDEs, second-order hyperbolic PDEs, known time-varying delays, unknown constant delays) will help the reader master the techniques* Numerous examples ease a student new to delay systems into the topic* The only prerequisites are the basics of function spaces and Lyapunov theory for ODEs* The basics of Poincaré and Agmon inequalities, Lyapunov input-to-state stability, parameter projection for adaptive control, and Bessell functions are summarized in appendices for the reader's convenience. Delay Compensation for Nonlinear, Adaptive, and PDE Systems is an excellent reference guide for graduate students, researchers, and professionals in mathematics, systems control, as well as chemical, mechanical, electrical, computer, aerospace, and civil/structural engineering. Parts of the book may also be used in graduate courses on general distributed parameter systems, linear delay systems, PDEs, nonlinear control, state estimator and observers, adaptive control, robust control, or linear time-varying systems.

Table of Contents

Preface	p. v
Introduction	p. 1
Delay Systems	p. 1
How Does the Difficulty of Delay Systems Compare with PDEs?	p. 2
A Short History of Backstepping	p. 3
From Predictor Feedbacks for LTI-ODE Systems to the Results in This Book	p. 4
Organization of the Book	p. 4
Use of Examples	p. 5
Krasovskii Theorem or Direct Stability Estimates?	p. 7
DDE or Transport PDE Representation of the Actuator/Sensor State?	p. 9
Notation, Spaces, Norms, and Solutions	p. 9
Beyond This Book	p. 11
Linear Delay-ODE Cascades
Basic Predictor Feedback	p. 17
Basic Idea of Predictor Feedback Design for ODE Systems with Actuator Delay	p. 18
Backstepping Design Via the Transport PDE	p. 19
On the Relation Among the Backstepping Design, the FSA/Reduction Design, and the Original Smith Controller	p. 22
Stability of Predictor Feedback	p. 23
Examples of Predictor Feedback Design	p. 27
Stability Proof Without a Lyapunov function	p. 30
Backstepping Transformation in the Standard Delay Notation	p. 36
Notes and References	p. 39
Predictor Observers	p. 41
Observers for ODE Systems with Sensor Delay	p. 41
Example: Predictor Observer	p. 44
On Observers That Do Not Estimate the Sensor State	p. 46
Observer-Based Predictor Feedback for Systems with Input Delay	p. 48
The Relation with the Original Smith Controller	p. 48
Separation Principle: Stability Under Observer-Based Predictor Feedback	p. 49
Notes and References	p. 52
Inverse Optimal Redesign	p. 53
Inverse Optimal Redesign	p. 54
Is Direct Optimally Possible Without Solving Operator Riccati Equations?	p. 59
Disturbance Attenuation	p. 60
Notes and References	p. 63
Robustness to Delay Mismatch	p. 65
Robustness in the L2 Norm	p. 65
Aside: Robustness to Predictor for Systems That Do Not Need It	p. 72
Robustness in the H1 Norm	p. 73
Notes and References	p. 83
Time-Varying Delay	p. 85
Predictor Feedback Design with Time-Varying Actuator Delay	p. 85
Stability Analysis	p. 88
Observer Design with Time-Varying Sensor Delay	p. 96
Examples	p. 97
Notes and References	p. 101
Adaptive Control
Delay-Adaptive Full-State Predictor Feedback	p. 107
Categorization of Adaptive Control Problems with Actuator Delay	p. 109
Delay-Adaptive Predictor Feedback with Full-State Measurement	p. 110
Proof of Stability for Full-State Feedback	p. 112
Simulations	p. 117
Notes and References	p. 119
Delay-Adaptive Predictor with Estimation of Actuator State	p. 121
Adaptive Control with Estimation of the Transport PDE State	p. 121
Local Stability and Regulation	p. 123
Simulations	p. 131
Notes and References	p. 131
Trajectory Tracking Under Unknown Delay and ODE Parameters	p. 135
Problem Formulation	p. 135
Control Design	p. 137
Simulations	p. 140
Proof of Global Stability and Tracking	p. 140
Notes and References	p. 149
Nonlinear Systems
Nonlinear Predictor Feedback	p. 153
Predictor Feedback Design for a Scalar Plant with a Quadratic Nonlinearity	p. 155
Nonlinear Infinite-Dimensional "Backstepping Transformation" and Its Inverse	p. 157
Stability	p. 159
Failure of the Uncompensated Controller	p. 165
What Would the Nonlinear Version of the Standard "Smith Predictor" Be?	p. 168
Notes and References	p. 169
Forward-Complete Systems	p. 171
Predictor Feedback for General Nonlinear Systems	p. 171
A Categorization of Systems That Are Globally Stabilizable Under Predictor Feedback	p. 173
The Nonlinear Backstepping Transformation of the Actuator State	p. 176
Lyapunov Functions for the Autonomous Transport PDE	p. 178
Lyapunov-Based Stability Analysis for Forward-Complete Nonlinear Systems	p. 181
Stability Proof Without a Lyapunov function	p. 187
Notes and References	p. 190
Strict-Feedforward Systems	p. 191
Example: A Second-Order Strict-Feedforward Nonlinear System	p. 192
General Strict-Feedforward Nonlinear Systems: Integrator Forwarding	p. 197
Predictor for Strict-Feedforward Systems	p. 199
General Strict-Feedforward Nonlinear Systems: Stability Analysis	p. 201
Example of Predictor Design for a Third-Order System That Is Not Linearizable	p. 207
An Alternative: A Design with Nested Saturations	p. 211
Extension to Nonlinear Systems with Time-Varying Input Delay	p. 212
Notes and References	p. 214
Linearizable Strict-Feedforward Systems	p. 217
Linearizable Strict-Feedforward Systems	p. 218
Integrator Forwarding (SJK) Algorithm Applied to Linearizable Strict-Feedforward Systems	p. 218
Two Sets of Linearizing Coordinates	p. 219
Predictor Feedback for Linearizable Strict-Feedforward Systems	p. 220
Explicit Closed-Loop Solutions for Linearizable Strict-Feedforward Systems	p. 223
Examples with Linearizable Strict-Feedforward Systems	p. 227
Notes and References	p. 230
PDE-ODE Cascades
ODEs with General Transport-Like Actuator Dynamics	p. 235
First-Order Hyperbolic Partial Integra-Differential Equations	p. 235
Examples of Explicit Design	p. 242
Korteweg-de Vries-like Equation	p. 243
Simulation Example	p. 246
ODE with Actuator Dynamics Given by a General First-Order Hyperbolic PIDE	p. 246
An ODE with Pure Advection-Reaction Actuator Dynamics	p. 250
Notes and References	p. 251
ODEs with Heat PDE Actuator Dynamics	p. 253
Stabilization with Full-State Feedback	p. 254
Example: Heat PDE Actuator Dynamics	p. 261
Robustness to Diffusion Coefficient Uncertainty	p. 262
Expressing the Compensator in Terms of Input Signal Rather Than Heat Equation State	p. 264
On Differences Between Compensation of Delay Dynamics and Diffusion Dynamics	p. 264
Notes and References	p. 266
ODEs with Wave PDE Actuator Dynamics	p. 269
Control Design for Wave PDE Compensation with Neumann Actuation	p. 270
Stability of the Closed-Loop System	p. 277
Robustness to Uncertainty in the Wave Propagation Speed	p. 283
An Alternative Design with Dirichlet Actuation	p. 290
Expressing the Compensator in Terms of Input Signal Rather Than Wave Equation State	p. 294
Examples: Wave PDE Actuator Dynamics	p. 297
On the Stabilization of the Wave PDE Alone by Neumann and Dirichlet Actuation	p. 302
Notes and References	p. 304
Observers for ODEs Involving PDE Sensor and Actuator Dynamics	p. 305
Observer for ODE with Heat PDE Sensor Dynamics	p. 306
Example: Heat PDE Sensor Dynamics	p. 309
Observer-Based Controller for ODEs with Heat PDE Actuator Dynamics	p. 310
Observer for ODE with Wave PDE Sensor Dynamics	p. 316
Example: Wave PDE Sensor Dynamics	p. 320
Observer-Based Controller for ODEs with Wave PDE Actuator Dynamics	p. 322
Notes and References	p. 327
Delay-PDE and PDE-PDE Cascades
Unstable Reaction-Diffusion PDE with Input Delay	p. 331
Control Design for the Unstable Reaction-Diffusion PDE with Input Delay	p. 331
The Baseline Design (D = 0) for the Unstable Reaction-Diffusion PDE	p. 334
Inverse Backstepping Transformations	p. 335
Stability of the Target System (w, z)	p. 336
Stability of the System in the Original Variables (u, v)	p. 339
Estimates for the Transformation Kernels	p. 341
Explicit Solutions for the Control Gains	p. 349
Explicit Solutions of the Closed-Loop System	p. 350
Notes and References	p. 354
Antistable Wave PDE with Input Delay	p. 357
Control Design for Antistable Wave PDE with Input Delay	p. 357
The Baseline Design (D = 0) for the Antistable Wave PDE	p. 363
Explicit Gain Functions	p. 365
Stability of the Target System (w, z)	p. 370
Stability in the Original Plant Variables (u, v)	p. 377
Notes and References	p. 383
Other PDE-PDE Cascades	p. 385
Antistable Wave Equation with Heat Equation at Its Input	p. 385
Unstable Reaction-Diffusion Equation with a Wave Equation at Its Input	p. 388
Notes and References	p. 391
Poincaré, Agmon, and Other Basic Inequalities	p. 393
Input-Output Lemmas for LTI and LTV Systems	p. 397
Lyapunov Stability and ISS for Nonlinear ODEs	p. 403
Lyapunov Stability and Class-K Functions	p. 403
Input-to-State Stability	p. 406
Bessel Functions	p. 413
Bessel Function Jn	p. 413
Modified Bessel Function In	p. 414
Parameter Projection	p. 417
Strict-Feedforward Systems: A General Design	p. 421
The Class of Systems	p. 421
The Sepulchre-Jankovic-Kokotovic Algorithm	p. 422
Strict-Feedforward Systems: A Linearizable Class	p. 425
Linearizabiiity of Feedforward Systems	p. 425
Algorithms for Linearizable Feedforward Systems	p. 428
Strict-Feedforward Systems: Not Linearizable	p. 441
Algorithms for Nonlinearizable Feedforward Systems	p. 441
Block-Forwarding	p. 444
Interlaced Feedforward-Feedback Systems	p. 448
References	p. 453
Index	p. 465
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