Nonlinear Control Systems Analysis and Design

by Má; rquez, Horacio

• ISBN13: 9780471427995
• ISBN10: 0471427993
• Edition: 1st
• Format: Hardcover
• Copyright: 2003-04-25
• Publisher: Wiley-Interscience

Summary

Provides complete coverage of both the Lyapunov and Input-Output stability theories, ina readable, concise manner. * Supplies an introduction to the popular backstepping approach to nonlinear control design * Gives a thorough discussion of the concept of input-to-state stability * Includes a discussion of the fundamentals of feedback linearization and related results. * Details complete coverage of the fundamentals of dissipative system's theory and its application in the so-called L2gain control prooblem, for the first time in an introductory level textbook. * Contains a thorough discussion of nonlinear observers, a very important problem, not commonly encountered in textbooksat this level.*An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.

Author Biography

HORACIO J. MARQUEZ is currently Associate Professor and Associate Chair of Graduate Studies in the Department of Electrical and Computer Engineering at the University of Alberta, Canada. Dr. Marquez has published numerous articles in journals and conference proceedings on subjects including control theory, nonlinear systems, robust control, and gain scheduling. He received the University of Alberta McCalla Research Professorship for 2003—04.

Table of Contents

Introduction	1	(31)
Linear Time-Invariant Systems	1	(2)
Nonlinear Systems	3	(2)
Equilibrium Points	5	(1)
First-Order Autonomous Nonlinear Systems	5	(3)
Second-Order Systems: Phase-Plane Analysis	8	(2)
Phase-Plane Analysis of Linear Time-Invariant Systems	10	(8)
Phase-Plane Analysis of Nonlinear Systems	18	(2)
Limit Cycles	18	(2)
Higher-Order Systems	20	(2)
Chaos	21	(1)
Examples of Nonlinear Systems	22	(5)
Magnetic Suspension System	23	(2)
Inverted Pendulum on a Cart	25	(1)
The Ball-and-Beam System	26	(1)
Exercises	27	(4)
Mathematical Preliminaries	31	(34)
Sets	31	(1)
Metric Spaces	32	(1)
Vector Spaces	32	(7)
Linear Independence and Basis	34	(2)
Subspaces	36	(1)
Normed Vector Spaces	37	(2)
Matrices	39	(5)
Eigenvalues, Eigenvectors, and Diagonal Forms	40	(1)
Quadratic Forms	41	(3)
Basic Topology	44	(1)
Basic Topology in Rn	44	(1)
Sequences	45	(1)
Functions	46	(3)
Bounded Linear Operators and Matrix Norms	48	(1)
Differentiability	49	(3)
Some Useful Theorems	51	(1)
Lipschitz Continuity	52	(2)
Contraction Mapping	54	(2)
Solution of Differential Equations	56	(3)
Exercises	59	(6)
Lyapunov Stability I: Autonomous Systems	65	(42)
Definitions	65	(4)
Positive Definite Functions	69	(2)
Stability Theorems	71	(3)
Examples	74	(3)
Asymptotic Stability in the Large	77	(3)
Positive Definite Functions Revisited	80	(2)
Exponential Stability	82	(1)
Construction of Lyapunov Functions	82	(3)
The Invariance Principle	85	(8)
Region of Attraction	93	(3)
Analysis of Linear Time-Invariant Systems	96	(4)
Linearization of Nonlinear Systems	99	(1)
Instability	100	(2)
Exercises	102	(5)
Lyapunov Stability II: Nonautonomous Systems	107	(30)
Definitions	107	(3)
Positive Definite Functions	110	(3)
Examples	111	(2)
Stability Theorems	113	(2)
Proof of the Stability Theorems	115	(4)
Analysis of Linear Time-Varying Systems	119	(3)
The Linearization Principle	120	(2)
Perturbation Analysis	122	(3)
Converse Theorems	125	(1)
Discrete-Time Systems	126	(1)
Discretization	127	(3)
Stability of Discrete-Time Systems	130	(3)
Definitions	130	(1)
Discrete-Time Positive Definite Functions	131	(1)
Stability Theorems	132	(1)
Exercises	133	(4)
Feedback Systems	137	(18)
Basic Feedback Stabilization	138	(3)
Integrator Backstepping	141	(4)
Backstepping: More General Cases	145	(6)
Chain of Integrators	145	(3)
Strict Feedback Systems	148	(3)
Example	151	(2)
Exercises	153	(2)
Input--Output Stability	155	(28)
Function Spaces	156	(3)
Extended Spaces	157	(2)
Input--Output Stability	159	(5)
Linear Time-Invariant Systems	164	(2)
Lp Gains for LTI Systems	166	(2)
L∞ Gain	166	(1)
L2 Gain	167	(1)
Closed-Loop Input--Output Stability	168	(3)
The Small Gain Theorem	171	(3)
Loop Transformations	174	(4)
The Circle Criterion	178	(2)
Exercises	180	(3)
Input-to-State Stability	183	(18)
Motivation	183	(2)
Definitions	185	(1)
Input-to-State Stability (ISS) Theorems	186	(5)
Examples	189	(2)
Input-to-State Stability Revisited	191	(4)
Cascade-Connected Systems	195	(3)
Exercises	198	(3)
Passivity	201	(22)
Power and Energy: Passive Systems	201	(3)
Definitions	204	(4)
Interconnections of Passivity Systems	208	(3)
Passivity and Small Gain	210	(1)
Stability of Feedback Interconnections	211	(3)
Passivity of Linear Time-Invariant Systems	214	(3)
Strictly Positive Real Rational Functions	217	(3)
Exercises	220	(3)
Dissipativity	223	(32)
Dissipative Systems	224	(1)
Differentiable Storage Functions	225	(1)
Back to Input-to-State Stability	226	(1)
QSR Dissipativity	226	(3)
Examples	229	(2)
Mass-Spring System with Friction	229	(2)
Mass--Spring System without Friction	231	(1)
Available Storage	231	(2)
Algebraic Condition for Dissipativity	233	(4)
Special Cases	235	(2)
Stability of Dissipative Systems	237	(2)
Feedback Interconnections	239	(4)
Nonlinear L2 Gain	243	(4)
Linear Time-Invariant Systems	245	(1)
Strictly Output Passive Systems	246	(1)
Some Remarks about Control Design	247	(4)
Nonlinear L2-Gain Control	251	(2)
Exercises	253	(2)
Feedback Linearization	255	(36)
Mathematical Tools	255	(10)
Lie Derivative	256	(1)
Lie Bracket	257	(2)
Diffeomorphism	259	(1)
Coordinate Transformations	259	(2)
Distributions	261	(4)
Input--State Linearization	265	(5)
Systems of the Form x = Ax + Bω(x)[u -- φ(x)]	265	(2)
Systems of the Form x = f(x) + g(x)u	267	(3)
Examples	270	(3)
Conditions for Input--State Linearization	273	(2)
Input--Output Linearization	275	(5)
The Zero Dynamics	280	(7)
Conditions for Input--Output Linearization	287	(1)
Exercises	287	(4)
Nonlinear Observers	291	(16)
Observers for Linear Time-Invariant Systems	291	(5)
Observability	292	(2)
Observer Form	294	(1)
Observers for Linear Time-Invariant Systems	294	(1)
Separation Principle	295	(1)
Nonlinear Observability	296	(2)
Nonlinear Observers	298	(1)
Observers with Linear Error Dynamics	298	(3)
Lipschitz Systems	301	(2)
Nonlinear Separation Principle	303	(4)
A Proofs	307	(30)
Chapter 3	307	(6)
Chapter 4	313	(2)
Chapter 6	315	(5)
Chapter 7	320	(4)
Chapter 8	324	(4)
Chapter 9	328	(2)
Chapter 10	330	(7)
Bibliography	337	(8)
List of Figures	345	(4)
Index	349