Nonlinear Control

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.

Table of Contents:
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