Nonlinear and Optimal Control Systems

, by Vincent, Thomas L.; Grantham, Walter J.

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ISBN: 9780471042358 | 0471042358
Cover: Hardcover
Copyright: 6/23/1997

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Designed for one-semester introductory senior-or graduate-level course, the authors provide the student with an introduction of analysis techniques used in the design of nonlinear and optimal feedback control systems. There is special emphasis on the fundamental topics of stability, controllability, and optimality, and on the corresponding geometry associated with these topics. Each chapter contains several examples and a variety of exercises.

Preface

xiii

1 Nonlinear Dynamical Systems

(83)

1.1 Fundamentals

(4)

System Models

(3)

Control System Concepts

(1)

1.2 Properties of Solutions

(14)

Admissible Controls

(1)

Solutions for Nonautonomous Systems

(1)

Autonomous Systems

(1)

Continuity with Respect to Initial State

(2)

Retro-Time Systems

(1)

Discontinuous Ordinary Differential Equations

(6)

Filippov Velocity

(3)

1.3 Numerical Solution Algorithms

(10)

Runge-Kutta Integration Algorithms

(3)

Gear's Integration Algorithm

(3)

Livermore Differential Equation Solver (LSODAR)

(1)

Root-Finding Algorithms

(2)

Plotting Function Contours

(1)

Plotting Three-Dimensional Trajectories

(2)

1.4 Nonlinear Systems Phenomena

(18)

No Superposition Principle

(2)

Multiple Separated Equilibriums

(1)

Finite Escape Time

(1)

Limit Cycles

(4)

Stability and Controllability Boundaries

(2)

Chaos and Strange Attractors

(9)

1.5 Classical Analysis Techniques

(31)

State Perturbation Equations

(2)

Eigenvalues

(1)

Classification of Two-Dimensional Equilibrium Points

(3)

Separatrix and the Brouwer Fixed-Point Theorem

(4)

Limit Cycles in Two Dimensions

(5)

Control System Limit Cycles

(2)

Lindstedt-Poincare Perturbation Method

(7)

Harmonic Balance and Describing Functions

(8)

Exercises

(6)

2 Nonlinear Control Systems

(36)

2.1 Admissible Controls

(6)

Constrained Nonlinear Control Systems

(1)

u-Vectogram

(1)

Corners and "Bang-Bang" Controllers

(1)

Piecewise Differentiable Feedback Controllers

(4)

2.2 Controllability Boundaries

(14)

Controllable Set

(1)

Linear Systems with Unbounded Control

(7)

Linear Systems Under Bounded Control

(4)

Nonlinear Systems

102

(2)

2.3 Domain of Attraction Boundaries

104

(5)

Domain of Attraction

104

(1)

Linear Systems with Unbounded Control

105

(1)

Linear Systems Under Bounded Control

106

(1)

Nonlinear Systems

107

(2)

2.4 Control of Nonlinear Systems

109

(8)

Control Based on Linearization

109

(5)

Control of Chaotic Systems

114

(3)

Exercises

117

(3)

3 Nonlinear Optimization

120

(49)

3.1 Constrained Function Minimization

120

(2)

Definition of a Minimum

120

(1)

Constraints

121

(1)

Parameter Optimization Problem

122

(1)

3.2 Tangent Vectors to a Set

122

(3)

3.3 Necessary Conditions for a Local Minimum

125

(9)

Basic Necessary Condition for Tangent Vectors

125

(1)

Farkas' Lemma

126

(1)

General Necessary Conditions

127

(5)

Linear-Quadratic Parameter Optimization

132

(2)

3.4 Min-Max Parametric Games

134

(5)

Two-Player Zero-Sum Games

134

(2)

Parametric Min-Max Necessary Conditions

136

(3)

3.5 Numerical Optimization Methods

139

(17)

Sequential Quadratic Programming

140

(3)

Trajectory Following

143

(13)

3.6 Calculus of Variations

156

(7)

Simplest Problem

156

(2)

Euler's Equations

158

(5)

Exercises

163

(6)

4 Lyapunov Stability

169

(97)

4.1 Stability Concepts

169

(14)

Gradient Systems

169

(3)

Lyapunov Stability

172

(7)

Lagrange Stability

179

(1)

Orbital Stability

180

(2)

Structural Stability

182

(1)

4.2 Local Stability: Lyapunov's First Method

183

(26)

Linearization and the State Transition Matrix

183

(9)

Stability of an Equilibrium Point

192

(3)

Center Manifold Theory

195

(5)

Orbital Stability and Floquet Theory

200

(5)

Lyapunov Exponents and Chaotic Systems

205

(4)

4.3 Lyapunov's Second Method

209

(12)

Lyapunov Functions

210

(7)

Lyapunov Sufficiency Conditions

217

(4)

4.4 Constructing Lyapunov Functions

221

(18)

Matrix Lyapunov Equation for Linear Systems

222

(5)

Numerical Solution of the Matrix Lyapunov Equation

227

(4)

Lur'e's Integral Lyapunov Function

231

(2)

Krasovskii's Method

233

(2)

Variable Gradient Method

235

(4)

4.5 Estimating Asymptotic Stability Regions

239

(21)

Lyapunov Estimates

239

(5)

An Optimization Procedure for Lyapunov Estimates

244

(3)

Zubov's Analytic Continuation Method

247

(13)

Exercises

260

(6)

5 Lyapunov Control System Design

266

(48)

5.1 Unconstrained Linear Systems

266

(4)

5.2 Linear Systems with Bounded Control

270

(3)

5.3 Function Optimizing Feedback Controls

273

(28)

Lyapunov Optimizing Feedback Controls

273

(4)

Steepest Descent Control

277

(6)

Quickest Descent Control

283

(2)

A Comparison of Three Optimizing Feedback Controllers

285

(9)

Lyapunov Optimizing Control Using Trajectory Following

294

(1)

Minimum Cost-Descent Control

295

(6)

5.4 Function Min-Max Feedback Control

301

(7)

A Min-Max Controller for Uncertain Systems

302

(3)

Instantaneous Two-Player Games

305

(3)

Exercises

308

(6)

6 Controllability of Nonlinear Systems

314

(50)

6.1 Controllable Sets and Reachable Sets

314

(9)

Controllable Set

314

(1)

Reachable Set

315

(1)

Forward and Backward Integration

315

(4)

The Intersection of Controllable and Reachable Sets

319

(2)

Controllability Boundaries

321

(2)

6.2 Controllability Minimum Principle

323

(20)

Tangent Planes to a Trajectory Surface

323

(4)

Jump Conditions Crossing a Switching Surface

327

(3)

Controllability Path Necessary Conditions

330

(3)

Control Constraints

333

(2)

Target Conditions

335

(2)

A Controllability Minimum Principle

337

(5)

A Reachability Maximum Principle

342

(1)

6.3 Additional Controllable Set Results

343

(10)

Existence of Boundary Trajectories

343

(2)

Open Controllable Sets

345

(1)

Equilibrium Points in a Controllability Boundary

346

(4)

Boundary Attraction

350

(3)

6.4 Determination of Controllable Sets

353

(7)

Two-Dimensional Systems

353

(5)

Higher-Dimensional Systems

358

(2)

Exercise

360

(4)

7 Optimal Control Systems

364

(86)

7.1 Optimal Control Problems

364

(4)

Performance Measures and Optimal Control Laws

364

(1)

The Integral Cost Problem in Optimal Control

365

(3)

7.2 Pontryagin's Minimum Principle

368

(8)

Transversality Conditions

371

(1)

The Optimal Control H Function

372

(1)

Minimizing the H Function

373

(1)

The Minimum Principle

374

(2)

7.3 Some Applications of the Minimum Principle

376

(28)

7.4 Singular Optimal Controls

404

(6)

7.5 Nonstandard Problems

410

(10)

Terminal Cost Problems

410

(1)

The Problem of Bolza

411

(1)

Problems with an Integral Constraint

412

(1)

Optimal Parametric Controls

413

(7)

7.6 Dynamic Programming

420

(7)

Hamilton-Jacobi-Bellman Equation

420

(2)

Linear-Quadratic Optimal Control

422

(2)

Linear-Quadratic Regulator

424

(3)

7.7 State Constraints

427

(17)

State-Dependent Control Constraints

427

(5)

State Constraints

432

(12)

Exercises

444

(6)

8 Optimal Control Design

450

(41)

8.1 Linear Systems with Linear Cost

450

(16)

Time-Optimal Control

451

(2)

Minimum-Time Control of Second-Order IO Systems

453

(12)

Minimum-Time Control as a Feedback Controller

465

(1)

8.2 Linear Systems with Quadratic Cost

466

(8)

Linear System Quadratic Cost Regulator (LQR)

467

(5)

Two Important Additional Properties of S

472

(1)

Control Saturation

473

(1)

8.3 Abnormal Arcs

474

(5)

Abnormality and Controllability

474

(1)

Linear Systems

475

(4)

8.4 Nonlinear Systems

479

(7)

Goddard's Problem

479

(6)

Optimal Return Function as a Lyapunov Function

485

(1)

Exercises

486

(5)

9 Differebtial Games

491

(52)

9.1 Introduction

491

(3)

The Homicidal Chauffeur Game

492

(2)

9.2 Qualitative Games

494

(9)

The Boundary of the Playable Set

495

(2)

The H Function

497

(1)

Parametric Min-Max Games

498

(1)

A Min-Max Principle for the Playable Set Boundary

499

(4)

9.3 Qualitative Design Concepts

503

(9)

v-Reachable and v-Attractive Sets

504

(1)

Linear Systems with Bounded Control

504

(8)

9.4 Quantitative Games

512

(4)

Zero-Sum Games

513

(1)

Min-Max Cost Surface

514

(2)

9.5 Isaacs' Min-Max Principle

516

(19)

Transversality Conditions

519

(1)

The Differential Game H Function

520

(1)

Min-Max Conditions for the H Function

520

(2)

The Min-Max Principle

522

(4)

9.6 Linear System-Quadratic Cost

526

(8)

Min-Max Control

527

(5)

Connections with HXXX Control

532

(2)

9.7 Missile Guidance

534

(6)

Direct Pursuit

535

(1)

Collision Course

536

(1)

Proportional Navigation

537

(1)

Game-Theoretic Guidance

538

(2)

Exercises

540

(3)

References

543

(8)

Index

551

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Nonlinear and Optimal Control Systems

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