9780130673893

Nonlinear Systems

by
  • ISBN13:

    9780130673893

  • ISBN10:

    0130673897

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 12/18/2001
  • Publisher: Pearson

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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

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