Nonlinear Control, 1st edition
Published by Pearson (February 6, 2014) © 2015
- Hassan K. Khalil Michigan State University, East Lansing
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For a first course on nonlinear control that can be taught in one semester
This book emerges from the award-winning book, Nonlinear Systems, but has a distinctly different mission and organization. While Nonlinear Systems was intended as a reference and a text on nonlinear system analysis and its application to control, this streamlined book is intended as a text for a first course on nonlinear control. In Nonlinear Control, author Hassan K. Khalil employs a writing style that is intended to make the book accessible to a wider audience without compromising the rigor of the presentation.
Teaching and Learning Experience
This program will provide a better teaching and learning experience–for you and your students. It will help:
- Provide an Accessible Approach to Nonlinear Control: This streamlined book is intended as a text for a first course on nonlinear control that can be taught in one semester.
- Support Learning: Over 250 end-of-chapter exercises give students plenty of opportunities to put theory into action.
Provide an Accessible Approach to Nonlinear Control
- A New Approach from an Award-winning Author: This book emerges from the award-winning book, Nonlinear Systems, but has a distinctly different mission and organization. While Nonlinear Systems was intended as a reference and a text on nonlinear system analysis and its application to control, this streamlined book is intended as a text for a first course on nonlinear control.
- Designed for the First Course on Nonlinear Control:Â Nonlinear Control is intended for use in a first course on nonlinear control that can be taught in one semester (forty lectures).
- Accessible Writing that Resonates with Students: The writing style is accessible to a wider audience without compromising the rigor of the presentation.
- Streamlined Material for a One-semester Course: Proofs are included only when they are needed to understand the material; otherwise references are given. In a few cases when it is not convenient to find the proofs in the literature, they are included in the appendix.
Support Learning
- End-of-chapter Exercises: Over 250 end-of-chapter exercises give students plenty of opportunities to put theory into action; almost all of the exercises are different from Nonlinear Systems. Many exercises require computer simulation using MATLAB or SIMULINK.
- Instructor Solution Manual: An electronic solution manual is available to instructors from the publisher.
- Companion Website: The homepage of the book includes, among other things, an errata sheet, a link to report errors and typos, PDF slides of the course, and Simulink models of selected examples and exercises.
1 Introduction 1
1.1 Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Nonlinear Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Two-Dimensional Systems 15
2.1 Qualitative Behavior of Linear Systems . . . . . . . . . . . . . . . . . . 17
2.2 Qualitative Behavior Near Equilibrium Points . . . . . . . . . . . . . . 21
2.3 Multiple Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Numerical Construction of Phase Portraits . . . . . . . . . . . . . . . . 31
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Stability of Equilibrium Points 37
3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Lyapunov’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 The Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Region of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.7 Converse Lyapunov Theorems . . . . . . . . . . . . . . . . . . . . . . . 68
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Time-Varying and Perturbed Systems 75
4.1 Time-varying Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Perturbed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Boundedness and Ultimate Boundedness . . . . . . . . . . . . . . . . . 85
4.4 Input-to-State Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 Passivity 103
5.1 Memoryless Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 State Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3 Positive Real Transfer Functions . . . . . . . . . . . . . . . . . . . . . 112
5.4 Connection with Lyapunov Stability . . . . . . . . . . . . . . . . . . . 115
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Input-Output Stability 121
6.1 L Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 L Stability of State Models . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 L2 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Stability of Feedback Systems 141
7.1 Passivity Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2 The Small-Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Absolute Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3.1 Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.3.2 Popov Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8 Special Nonlinear Forms 171
8.1 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.2 Controller Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.3 Observer Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
9 State Feedback Stabilization 197
9.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.3 Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.4 Partial Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . 207
9.5 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
9.6 Passivity-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . 217
9.7 Control Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . 222
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10 Robust State Feedback Stabilization 231
10.1 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10.2 Lyapunov Redesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
10.3 High-Gain Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
11 Nonlinear Observers 263
11.1 Local Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
11.2 The Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . 266
11.3 Global Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
11.4 High-Gain Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
12 Output Feedback Stabilization 281
12.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
12.2 Passivity-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . 283
12.3 Observer-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . 286
12.4 High-Gain Observers and the Separation Principle . . . . . . . . . . . . 288
12.5 Robust Stabilization of Minimum Phase Systems . . . . . . . . . . . . 296
12.5.1 Relative Degree One . . . . . . . . . . . . . . . . . . . . . . . 296
12.5.2 Relative Degree Higher Than One . . . . . . . . . . . . . . . 298
12.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
13 Tracking and Regulation 307
13.1 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
13.2 Robust Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
13.3 Transition Between Set Points . . . . . . . . . . . . . . . . . . . . . . 314
13.4 Robust Regulation via Integral Action . . . . . . . . . . . . . . . . . . 318
13.5 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
A Examples 329
A.1 Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
A.2 Mass—Spring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
A.3 Tunnel-Diode Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
A.4 Negative-Resistance Oscillator . . . . . . . . . . . . . . . . . . . . . . 335
A.5 DC-to-DC Power Converter . . . . . . . . . . . . . . . . . . . . . . . . 337
A.6 Biochemical Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
A.7 DC Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
A.8 Magnetic Levitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
A.9 Electrostatic Microactuator . . . . . . . . . . . . . . . . . . . . . . . . 342
A.10 Robot Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
A.11 Inverted Pendulum on a Cart . . . . . . . . . . . . . . . . . . . . . . . 344
A.12 Translational Oscillator with Rotating Actuator . . . . . . . . . . . . . 347
B Mathematical Review 349
C Composite Lyapunov Functions 355
C.1 Cascade Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
C.2 Interconnected systems . . . . . . . . . . . . . . . . . . . . . . . . . . 357
C.3 Singularly Perturbed Systems . . . . . . . . . . . . . . . . . . . . . . . 359
D Proofs 363
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