Nonlinear Systems

by Khalil, Hassan K.

• ISBN13: 9780130673893
• ISBN10: 0130673897
• Edition: 3rd
• Format: Hardcover
• Copyright: 2001-12-18
• Publisher: Pearson

Summary

This book is written is such a way that the level of mathematical sophistication builds up from chapter to chapter. It has been reorganized into four parts: basic analysis, analysis of feedback systems, advanced analysis, and nonlinear feedback control.Updated content includes subjects which have proven useful in nonlinear control design in recent yearsnew in the 3rd edition are: expanded treatment of passivity and passivity-based control; integral control, high-gain feedback, recursive methods, optimal stabilizing control, control Lyapunov functions, and observers.For use as a self-study or reference guide by engineers and applied mathematicians.

Table of Contents

Preface	xiii
Introduction	1	(34)
Nonlinear Models and Nonlinear Phenomena	1	(4)
Examples	5	(19)
Pendulum Equation	5	(1)
Tunnel-Diode Circuit	6	(2)
Mass-Spring System	8	(3)
Negative-Resistance Oscillator	11	(3)
Artificial Neural Network	14	(2)
Adaptive Control	16	(2)
Common Nonlinearities	18	(6)
Exercises	24	(11)
Second-Order Systems	35	(52)
Qualitative Behavior of Linear Systems	37	(9)
Multiple Equilibria	46	(5)
Qualitative Behavior Near Equilibrium Points	51	(3)
Limit Cycles	54	(5)
Numerical Construction of Phase Portraits	59	(2)
Existence of Periodic Orbits	61	(8)
Bifurcation	69	(7)
Exercises	76	(11)
Fundamental Properties	87	(24)
Existence and Uniqueness	88	(7)
Continuous Dependence on Initial Conditions and Parameters	95	(4)
Differentiability of Solutions and Sensitivity Equations	99	(3)
Comparison Principle	102	(3)
Exercises	105	(6)
Lyapunov Stability	111	(84)
Autonomous Systems	112	(14)
The Invariance Principle	126	(7)
Linear Systems and Linearization	133	(11)
Comparison Functions	144	(3)
Nonautonomous Systems	147	(9)
Linear Time-Varying Systems and Linearization	156	(6)
Converse Theorems	162	(6)
Boundedness and Ultimate Boundedness	168	(6)
Input-to-State Stability	174	(7)
Exercises	181	(14)
Input-Output Stability	195	(32)
L Stability	195	(6)
L Stability of State Models	201	(8)
L2 Gain	209	(8)
Feedback Systems: The Small-Gain Theorem	217	(5)
Exercises	222	(5)
Passivity	227	(36)
Memoryless Functions	228	(5)
State Models	233	(4)
Positive Real Transfer Functions	237	(4)
L2 and Lyapunov Stability	241	(4)
Feedback Systems: Passivity Theorems	245	(14)
Exercises	259	(4)
Frequency Domain Analysis of Feedback Systems	263	(40)
Absolute Stability	264	(16)
Circle Criterion	265	(10)
Popov Criterion	275	(5)
The Describing Function Method	280	(16)
Exercises	296	(7)
Advanced Stability Analysis	303	(36)
The Center Manifold Theorem	303	(9)
Region of Attraction	312	(10)
Invariance-like Theorems	322	(7)
Stability of Periodic Solutions	329	(5)
Exercises	334	(5)
Stability of Perturbed Systems	339	(42)
Vanishing Perturbation	340	(6)
Nonvanishing Perturbation	346	(4)
Comparison Method	350	(5)
Continuity of Solutions on the Infinite Interval	355	(3)
Interconnected Systems	358	(7)
Slowly Varying Systems	365	(7)
Exercises	372	(9)
Perturbation Theory and Averaging	381	(42)
The Perturbation Method	382	(11)
Perturbation on the Infinite Interval	393	(4)
Periodic Perturbation of Autonomous Systems	397	(5)
Averaging	402	(9)
Weakly Nonlinear Second-Order Oscillators	411	(2)
General Averaging	413	(6)
Exercises	419	(4)
Singular Perturbations	423	(46)
The Standard Singular Perturbation Model	424	(6)
Time-Scale Properties of the Standard Model	430	(9)
Singular Perturbation on the Infinite Interval	439	(4)
Slow and Fast Manifolds	443	(6)
Stability Analysis	449	(11)
Exercises	460	(9)
Feedback Control	469	(36)
Control Problems	469	(6)
Stabilization via Linearization	475	(3)
Integral Control	478	(3)
Integral Control via Linearization	481	(4)
Gain Scheduling	485	(14)
Exercises	499	(6)
Feedback Linearization	505	(46)
Motivation	505	(4)
Input-Output Linearization	509	(12)
Full-State Linearization	521	(9)
State Feedback Control	530	(14)
Stabilization	530	(10)
Tracking	540	(4)
Exercises	544	(7)
Nonlinear Design Tools	551	(96)
Sliding Mode Control	552	(27)
Motivating Example	552	(11)
Stabilization	563	(9)
Tracking	572	(3)
Regulation via Integral Control	575	(4)
Lyapunov Redesign	579	(10)
Stabilization	579	(9)
Nonlinear Damping	588	(1)
Backstepping	589	(15)
Passivity-Based Control	604	(6)
High-Gain Observers	610	(15)
Motivating Example	612	(7)
Stabilization	619	(4)
Regulation via Integral Control	623	(2)
Exercises	625	(22)
A Mathematical Review	647	(6)
B Contraction Mapping	653	(4)
C Proofs	657	(62)
Proof of Theorems 3.1 and 3.2	657	(2)
Proof of Lemma 3.4	659	(2)
Proof of Lemma 4.1	661	(1)
Proof of Lemma 4.3	662	(1)
Proof of Lemma 4.4	662	(1)
Proof of Lemma 4.5	663	(2)
Proof of Theorem 4.16	665	(4)
Proof of Theorem 4.17	669	(6)
Proof of Theorem 4.18	675	(1)
Proof of Theorem 5.4	676	(1)
Proof of Lemma 6.1	677	(3)
Proof of Lemma 6.2	680	(4)
Proof of Lemma 7.1	684	(4)
Proof of Theorem 7.4	688	(2)
Proof of Theorems 8.1 and 8.3	690	(9)
Proof of Lemma 8.1	699	(1)
Proof of Theorem 11.1	700	(6)
Proof of Theorem 11.2	706	(2)
Proof of Theorem 12.1	708	(1)
Proof of Theorem 12.2	709	(1)
Proof of Theorem 13.1	710	(2)
Proof of Theorem 13.2	712	(1)
Proof of Theorem 14.6	713	(6)
Note and References	719	(5)
Bibliography	724	(16)
Symbols	740	(2)
Index	742

Excerpts

This text is intended for a first-year graduate-level course on nonlinear systems or control. It may also be used for self study or reference by engineers and applied mathematicians. It is an outgrowth of my experience teaching the nonlinear systems course at Michigan State University, East Lansing. Students taking this course have had background in electrical engineering, mechanical engineering, or applied mathematics. The prerequisite for the course is a graduate-level course in linear systems, taught at the level of the texts by Antsaklis and Michel 9, Chen 35, Kailath 94), or Rugh 158. The linear systems prerequisite allowed me not to worry about introducing the concept of "state" and to refer freely to "transfer functions," "state transition matrices," and other linear system concepts. The mathematical background is the usual level of calculus, differential equations, and matrix theory that any graduate student in engineering or mathematics would have. In the Appendix, I have collected a few mathematical facts that are used throughout the book. I have written the text in such a way that the level of mathematical sophistication increases as we advance from chapter to chapter. This is why the second chapter is written in an elementary context. Actually, this chapter could be taught at senior, or even junior, level courses without difficulty. This is also the reason I have split the treatment of Lyapunov stability into two parts. In Sections 4.1 through 4.3, I introduce the essence of Lyapunov stability for autonomous systems where I do not have to worry about technicalities such as uniformity, classkfunctions, etc. In Sections 4.4 through 4.6, I present Lyapunov stability in a more general setup that accommodates nonautonomous systems and allows for a deeper look into advanced aspects of the stability theory. The level of mathematical sophistication at the end of Chapter 4 is the level to which I like to bring the students, so that they can comfortably read the rest of the text. There is yet a higher level of mathematical sophistication that is assumed in writing the proofs in the Appendix. These proofs are not intended for classroom use. They are included to make the text on one hand, self contained, and, on the other, to respond to the need or desire of some students to read such proofs, such as students continuing on to conduct Ph.D. research into nonlinear systems or control theory. Those students can continue to read the Appendix in a self-study manner. This third edition has been written with the following goals in mind: To make the book (especially the early chapters) more accessible to first-year graduate students. As an example of the changes made toward that end, note the change in Chapter 3: All the material on mathematical background, the contraction mapping theorem, and the proof of the existence and uniqueness theorem have been moved to the Appendix. Several parts of the books have been rewritten to improve readability. To reorganize the book in such a way that makes it easier to structure nonlinear systems or control courses around it. In the new organization, the book has four parts, as shown in the flow chart. A course on nonlinear systems analysis will cover material from Parts 1, 2, and 3, while a course on nonlinear control will cover material from Parts 1, 2, and 4. To update the material of the book to include topics or results that have proven to be useful in nonlinear control design in recent years. New to the third addition are the: expanded treatment of passivity and passivity-based control, integral control, sliding mode control, and high-gain observers. Moreover, bifurcation is introduced in the context of second-order systems. On the technical side, the reader will find Kurzweil's converse Lyapunov theorem, nonlocal results in Chapters 10 and 11, and new results on integral; control and gain scheduling. To update the exercises. More