# Nonlinear Control

Published Date: Feb 6, 2014

## Description

For a first course on nonlinear control that can be taught in one semester

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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.

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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.

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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