There has been a great deal of excitement over the last few years concerning the emergence of new mathematical techniques for the analysis and control of nonlinear systems: witness the emergence of a set of simplified tools for the analysis of bifurcations, chaos and other complicated dynamical behaviour and the development of a comprehensive theory of nonlinear control. Coupled with this set of analytic advances has been the vast increase in computational power available both for the simulation of nonlinear systems as well as for the implementation in real time of sophisticated, real-time nonlinear control laws. Thus, technological advances have bolstered the impact of analytic advances and produced a tremendous variety of new problems and applications which are nonlinear in an essential way.
This book lays out in a concise mathematical framework the tools and methods of analysis which underlie this diversity of applications.
The material presented in this book is culled from different 1st year graduate courses that the author has taught at MIT and at Berkeley.
Author(s): Shankar Sastry
Publisher: Springer
Year: 1999
Language: English
Pages: 697
Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 16
Acknowledgments......Page 12
Standard Notation......Page 24
1.1 Nonlinear Models.......Page 28
1.2 Complexity in Nonlinear Dynamics.......Page 31
1.2.1 Subtleties of Nonlinear Systems Analysis......Page 37
1.2.2 Autonomous Systems and Equilibrium Points.......Page 39
1.3.1 The Tunnel Diode Circuit.......Page 41
1.3.2 An Oscillating Circuit: Due to van der Pol......Page 43
1.3.3 The Pendulum: Due to Newton.......Page 44
1.3.4 The Buckling Beam: Due to Euler......Page 46
1.3.5 The Volterra-Lotka Predator-Prey Equations.......Page 49
1.4.2 Bowing of a Violin String: Due to Rayleigh......Page 50
1.5 Summary......Page 52
1.6 Exercises.......Page 53
2.2 Linearization About Equilibria of Second-Order Nonlinear Systems.......Page 58
2.2.1 Linear Systems in the Plane......Page 59
2.2.2 Phase Portraits near Hyperbolic Equilibria......Page 63
2.3 Closed Orbits of Planar Dynamical Systems.......Page 68
2.4 Counting Equilibria: Index Theory......Page 76
2.5 Bifurcations......Page 78
2.6 Bifurcation Study of Josephson Junction Equations......Page 82
2.7 The Degenerate van der Pol Equation.......Page 87
2.8.1 Fixed Points and the Hartman-Grobman Theorem......Page 90
2.8.2 Period N Points of Maps......Page 92
2.8.3 Bifurcations of Maps......Page 93
2.10 Exercises.......Page 96
3.1 Groups and Fields.......Page 103
3.2 Vector Spaces, Algebras, Norms, and Induced Norms......Page 104
3.3 Contraction Mapping Theorems......Page 109
3.3.1 Incremental Small Gain Theorem......Page 111
3.4 Existence and Uniqueness Theorems for Ordinary Differential Equations.......Page 113
3.4.2 Circuit Simulation by Waveform Relaxation.......Page 117
3.5 Differential Equations with Discontinuities.......Page 120
3.6 Carleman Linearization......Page 126
3.7 Degree Theory......Page 128
3.8 Degree Theory and Solutions of Resistive Networks.......Page 132
3.9.1 Smooth Manifolds and Smooth Maps......Page 134
3.9.2 Tangent Spaces and Derivatives.......Page 137
3.9.3 Regular Values......Page 140
3.9.4 Manifolds with Boundary.......Page 142
3.10 Summary......Page 145
3.11 Exercises......Page 146
4 Input-Output Analysis......Page 154
4.1 Optimal Linear Approximants to Nonlinear Systems......Page 155
4.1.1 Optimal Linear Approximations for Memoryless, Time-Invariant Nonlinearities......Page 159
4.1.2 Optimal Linear Approximations for Dynamic Nonlinearities: Oscillations in Feedback Loops......Page 162
4.1.3 Justification of the Describing Function.......Page 165
4.2 Input Output Stability.......Page 170
4.3 Applications of the Small Gain Theorems......Page 175
4.3.1 Robustness of Feedback Stability.......Page 177
4.3.2 Loop Transformation Theorem.......Page 179
4.4 Passive Nonlinear Systems......Page 180
4.5 Input-Output Stability of Linear Systems.......Page 183
4.6 Input-Output Stability Analysis of Feedback Systems......Page 187
4.6.1 The Lur'e Problem......Page 189
4.7 Volterra Input-Output Representations.......Page 194
4.7.1 Homogeneous, Polynomial and Volterra Systems in the Time Domain.......Page 195
4.7.2 Volterra Representations from Differential Equations......Page 197
4.7.3 Frequency Domain Representation of Volterra Input Output Expansions.......Page 200
4.8 Summary......Page 201
4.9 Exercises.......Page 202
5.1 Introduction......Page 209
5.2.1 The Lipschitz Condition and Consequences......Page 210
5.3.1 Energy-Like Functions......Page 215
5.3.2 Basic Theorems......Page 216
5.3.3 Examples of the Application of Lyapunov's Theorem......Page 219
5.3.4 Exponential Stability Theorems.......Page 222
5.4 LaSalle's Invariance Principle......Page 225
5.5 Generalizations of LaSalle's Principle.......Page 231
5.6 Instability Theorems......Page 233
5.7 Stability of Linear Time-Varying Systems......Page 234
5.7.1 Autonomous Linear Systems......Page 236
5.7.2 Quadratic Lyapunov Functions for Linear Time Varying Systems......Page 239
5.8 The Indirect Method of Lyapunov......Page 241
5.9 Domains of Attraction.......Page 244
5.11 Exercises......Page 250
6.1 Feedback Stabilization.......Page 262
6.2 The Lur'e Problem, Circle and Popov Criteria......Page 264
6.2.1 The Circle Criterion......Page 267
6.2.2 The Popov Criterion......Page 272
6.3 Singular Perturbation.......Page 274
6.3.1 Nonsingular Points, Solution Concepts, and Jump Behavior......Page 277
6.4.1 Dynamics of Nonlinear Circuits.......Page 279
6.4.2 Dynamics of Power Systems.......Page 282
6.5 Adaptive Identification of Single-Input Single-Output Linear Time-Invariant Systems.......Page 283
6.5.1 Linear Identifier Stability......Page 289
6.5.2 Parameter Error Convergence.......Page 292
6.6 Averaging.......Page 293
6.7 Adaptive Control......Page 300
6.8 Back-stepping Approach to Stabilization.......Page 302
6.9 Summary......Page 304
6.10 Exercises......Page 305
7.1 Qualitative Theory......Page 314
7.2 Nonlinear Maps......Page 318
7.3.1 The Poincar6 Map and Closed Orbits.......Page 321
7.3.2 The Poincare Map and Forced Oscillations......Page 324
7.4 Structural Stability......Page 329
7.5 Structurally Stable Two Dimensional Flows.......Page 332
7.6.1 Center Manifolds for Flows......Page 336
7.6.2 Center Manifolds for Flows Depending on Parameters......Page 340
7.6.3 Center Manifolds for Maps......Page 341
7.7 Bifurcation of Vector Fields: An Introduction.......Page 342
7.8.1 Single, Simple Zero Eigenvalue......Page 344
7.8.2 Pure Imaginary Pair of Eigenvalues: Poincar6-Andronov-Hopf Bifurcation......Page 349
7.9 Bifurcations of Maps.......Page 351
7.9.1 Single Eigenvalue 1: Saddle Node, Transcritical and Pitchfork......Page 352
7.9.2 Single Eigenvalue -1: Period Doubling.......Page 354
7.9.3 Pair of Complex Eigenvalues of Modulus 1: Naimark-Sacker Bifurcation......Page 355
7.10.1 Bifurcations of Equilibria and Fixed Points: Catastrophe Theory......Page 356
7.10.2 Singular Perturbations and Jump Behavior of Systems......Page 362
7.10.3 Dynamic Bifurcations: A Zoo.......Page 364
7.11 Routes to Chaos and Complex Dynamics.......Page 367
7.12 Exercises.......Page 368
8.1 Tangent Spaces......Page 376
8.1.1 Vector Fields, Lie Brackets, and Lie Algebras......Page 379
8.2 Distributions and Codistributions.......Page 383
8.3 Frobenius Theorem......Page 386
8.4 Matrix Groups......Page 389
8.4.1 Matrix Lie Groups and Their Lie Algebras.......Page 394
8.4.2 The Exponential Map.......Page 396
8.4.3 Canonical Coordinates on Matrix Lie Groups......Page 397
8.4.4 The Campbell-Baker-Hausdorff Formula......Page 398
8.5.1 Frenet-Serret Equations: A Control System on SE (3)......Page 402
8.5.2 The Wei-Norman Formula......Page 405
8.7 Exercises.......Page 406
9.1 Introduction.......Page 411
9.2.1 Input-Output Linearization.......Page 412
9.2.2 Zero Dynamics for SISO Systems.......Page 425
9.2.3 Inversion and Exact Tracking.......Page 429
9.2.4 Asymptotic Stabilization and Tracking for SISO Systems.......Page 431
9.3.1 MIMO Systems Linearizable by Static State Feedback.......Page 434
9.3.2 Full State Linearization of MIMO Systems......Page 438
9.3.3 Dynamic Extension for MIMO Systems......Page 441
9.4 Robust Linearization......Page 444
9.5.1 SISO Sliding Mode Control......Page 450
9.6 Tracking for Nonminimum Phase Systems.......Page 452
9.6.1 The Method of Devasia, Chen, and Paden......Page 453
9.6.2 The Bymes-Isidori Regulator......Page 456
9.7 Observers with Linear Error Dynamics......Page 460
9.8 Summary......Page 466
9.9 Exercises.......Page 467
10.1 Introduction.......Page 476
10.2 The Ball and Beam Example :......Page 477
10.2.2 Exact Input-Output Linearization......Page 478
10.2.3 Full State Linearization.......Page 480
10.2.4 Approximate Input Output Linearization......Page 481
10.2.5 Switching Control of the Ball and Beam System......Page 485
10.3 Approximate Linearization for Nonregular SISO Systems......Page 487
10.4 Nonlinear Flight Control.......Page 495
10.4.1 Force and Moment Generation.......Page 496
10.4.2 Simplification to a Planar Aircraft.......Page 499
10.4.3 Exact Input-Output Linearization of the PVTOL Aircraft System.......Page 500
10.4.4 Approximate Linearization of the PVTOL Aircraft......Page 503
10.5.1 Single Input Single Output (SISO) Case.......Page 508
10.5.2 Generalization to MIMO Systems.......Page 511
10.6.1 SISO Singularly Perturbed Zero and Driven Dynamics......Page 517
10.6.2 MIMO Singularly Perturbed Zero and Driven Dynamics.......Page 520
10.7 Summary......Page 525
10.8 Exercises.......Page 526
11.1 Controllability Concepts.......Page 537
11.2 Drift-Free Control Systems......Page 540
11.3 Steering of Drift-Free Nonholonomic Systems......Page 545
11.4 Steering Model Control Systems Using Sinusoids......Page 547
11.5.1 Fourier Techniques......Page 556
11.5.2 Optimal Steering of Nonholonomic Systems.......Page 562
11.5.3 Steering with Piecewise Constant Inputs......Page 566
11.5.4 Control Systems with Drift.......Page 572
11.6 Observability Concepts......Page 576
11.7 Zero Dynamics Algorithm and Generalized Normal Forms......Page 578
11.8 Input-Output Expansions for Nonlinear Systems......Page 587
11.9 Controlled Invariant Distributions and Disturbance Decoupling......Page 591
11.10 Summary......Page 593
11.11 Exercises......Page 594
12.1 Introduction.......Page 601
12.2 Introduction to Exterior Differential Systems.......Page 602
12.2.1 Multilinear Algebra......Page 603
12.2.2 Forms......Page 619
12.2.3 Exterior Differential Systems.......Page 627
12.3.1 The Goursat Normal Form......Page 633
12.3.2 The n-trailer Pfaffian System.......Page 641
12.3.3 The Extended Goursat Normal Form.......Page 650
12.4 Control Systems.......Page 656
12.5 Summary.......Page 662
12.6 Exercises......Page 663
13.1 Embedded Control and Hybrid Systems......Page 668
13.2 Multi-Agent Systems and Hybrid Systems......Page 669
References......Page 672
Index......Page 688
