Optimal Control An Introduction to the Theory and Its Applications

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ISBN: 9780486453286 | 0486453286
Cover: Paperback
Copyright: 12/29/2006

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Geared toward advanced undergraduates and graduate engineering students, this text introduces the theory and applications of optimal control. It serves as a bridge to the technical literature, enabling students to evaluate the implications of theoretical control work, and to judge the merits of papers on the subject. Michael Athans is Professor Emeritus of Electrical Engineering-Computer Science at MIT.

Sections indicated with a star (*) were treated in the classroom, and the remaining sections were assigned reading.

Preface to the Dover Edition

vii

Preface

CHAPTER 1 Introduction

1-1 Introduction

1-2 The System Design Problem

1-3 The Control Problem

1-4 Historical Perspective

1-5 Aims of This Book

1-6 General Comments on the Structure of This Book

1-7 Chapter Description

1-8 Prerequisites and Study Suggestions

CHAPTER 2 Mathematical Preliminaries: Algebra

2-1 Introduction

2-2 Sets

2-3 Operations on Sets

2-4 Functions

*2-5 Vector Spaces

*2-6 Linear Combinations and Bases

*2-7 Linear Algebra: Linear Transformations and Matrices

*2-8 Linear Algebra: Operations on Linear Transformations and Matrices

*2-9 Linear Algebra: Linear Transformations of V into V

*2-10 Linear Algebra: Eigenvectors and Eigenvalues

*2-11 Euclidean Spaces: Inner Products

2-12 Euclidean Spaces: The Schwarz Inequality

2-13 Euclidean Spaces: Orthogonality and Norm

2-14 Euclidean Spaces: Some Properties of the Scalar Product on

2-15 Euclidean Spaces: Some Properties of Symmetric Matrices

CHAPTER 3 Mathematical Preliminaries: Analysis

3-1 Introduction

*3-2 Distance and Related Notions: Definition

*3-3 Distance and Related Notions: Spheres and Limits

*3-4 Distance and Related Notions: Open and Closed Sets

3-5 Distance and Related Notions: Completeness and Contractions

3-6 Properties of Sets in R.: Compactness

3-7 Properties of Sets in R.: Hyperplanes and Cones

*3-8 Properties of Sets in R.: Convexity

*3-9 Vector Functions: Introductory Remarks

*3-10 Vector Functions: Continuity

3-11 Vector Functions: Piecewise Continuity

*3-12 Vector Functions: Derivatives

3-13 Vector Functions: Smooth Sets in R.

*3-14 Vector Functions: Integrals

*3-15 Vector Functions: Function Spaces

100

*3-16 Vector Functions: Functionals

108

*3-17 Differential Equations: Introductory Remarks

110

3-18 Differential Equations: The Basic Existence Theorem

115

*3-19 Linear Differential Equations: Basic Concepts

122

*3-20 Linear Differential Equations: The Fundamental Matrix

125

*3-21 Time-invariant Systems: The Exponential of At

132

3-22 Time-invariant Systems: Reduction to Canonical Form

135

*3-23 Time-invariant Systems: Evaluation of the Fundamental Matrix by Means of the Laplace Transform

138

*3-24 Time-invariant Systems: The nth-order System

143

3-25 The Adjoint System

147

3-26 Stability of Linear Time-invariant Systems

149

CHAPTER 4 Basic Concepts

151

*4-1 Introduction

151

*4-2 An RL Network

151

4-3 A Multivariable System

157

*4-4 Dynamical Systems: Introductory Remarks

159

4-5 Dynamical Systems: Formal Definition

163

*4-6 Dynamical Systems: The Systems Considered in This Book

168

*4-7 Linear Dynamical Systems

171

4-8 Input-Output Relationships and the Transfer Function

173

*4-9 Finding the State (or Dynamical-system) Representation of a Plant Whose Transfer Function Contains Only Poles

175

*4-10 Finding the State (or Dynamical-system) Representation of a Plant Whose Transfer Function Contains Poles and Zeros

182

*4-11 The Control Problem: Introductory Remarks

190

*4-12 The Control Problem: Formal Definition

191

4-13 The Control Problem: Important Special Cases

195

*4-14 The Set of Reachable States

197

*4-15 Controllability and Observability: Definition

200

*4-16 Controllability for Linear Time-invariant Systems

202

*4-17 Observability for Linear Time-invariant Systems

207

4-18 Practical Implications: Regulating the Output

211

4-19 Practical Implications: The Effect of Canceling a Pole with a Zero

215

4-20 Practical Implications: An Example

215

*4-21 Normal Linear Time-invariant Systems

217

CHAPTER 5 Conditions for Optimality: The Minimum Principle and the Hamilton-Jacobi Equation

221

5-1 Introduction

221

5-2 Ordinary Minima

222

5-3 Ordinary Minima with Constraints: A Simple Problem

228

5-4 Ordinary Minima with Constraints: Necessary Conditions and the Lagrange Multipliers

233

5-5 Some Comments

236

*5-6 An Example

241

*5-7 The Variational Approach to Control Problem 1: Necessary Conditions for a Free-end-point Problem

254

5-8 The Variational Approach to Control Problem 2: Sufficiency Conditions for the Free-end-point Problem

267

5-9 The Variational Approach to Control Problem 3: A Fixed-end-point Problem

272

*5-10 Comments on the Variational Approach

275

*5-11 The Minimum Principle of Pontryagin: Introduction

284

*5-12 The Minimum Principle of Pontryagin: Restatement of the Control Problem

284

*5-13 The Minimum Principle of Pontryagin

288

*5-14 The Minimum Principle of Pontryagin: Changes of Variable

291

*5-15 Proof of the Minimum Principle: Preliminary Remarks

304

*5-16 A Heuristic Proof of the Minimum Principle

308

*5-17 Some Comments on the Minimum Principle

347

*5-18 Sufficient Conditions for Optimality: Introductory Remarks

351

*5-19 Sufficient Conditions for Optimality: An Equation for the Cost

353

*5-20 A Sufficient Condition for Optimality

355

5-21 Some Comments on the Sufficiency Conditions

361

CHAPTER 6 Structure and Properties of Optimal Systems

364

6-1 Introduction

364

*6-2 Minimum-time Problems 1: Formulation and Geometric Interpretation

365

*6-3 Minimum-time Problems 2: Application of the Minimum Principle

374

*6-4 Minimum-time Problems 3: Comments

393

*6-5 Minimum-time Problems 4: Linear Time-invariant Systems

395

6-6 Minimum-time Problems 5: Structure of the Time-optimal Regulator and the Feedback Problem

407

6-7 Minimum-time Problems 6: Geometric Properties of the Time-optimal Control

411

6-8 Minimum-time Problems 7: The Existence of the Time-optimal Control

419

6-9 Minimum-time Problems 8: The Hamilton-Jacobi Equation

421

6-10 Minimum-time Problems 9: Comments and Remarks

426

*6-11 Minimum-fuel Problems 1: Introduction

428

*6-12 Minimum-fuel Problems 2: Discussion of the Problem and of the Constraints

428

*6-13 Minimum-fuel Problems 3: Formulation of a Problem and Derivation of the Necessary Conditions

430

*6-14 Minimum-fuel Problems 4: Linear Time-invariant Systems

440

6-15 Minimum-fuel Problems 5: Additional Formulations and Cost Functionals

451

6-16 Minimum-fuel Problems 6: Comments

457

*6-17 Minimum-energy Problems 1: Introduction

458

*6-18 Minimum-energy Problems 2: The Linear Fixed-end-point, Fixed-time Problem

460

*6-19 Minimum-energy Problems 3: An Example

466

6-20 Minimum-energy Problems 4: Magnitude Constraints on the Control Variables

475

*6-21 Singular Problems 1: The Hamiltonian a Linear Function of the Control

481

6-22 Singular Problems 2: The Hamiltonian a Linear Function of the Control and of Its Absolute Value

493

6-23 Some Remarks Concerning the Existence and Uniqueness of the Optimal and Extremal Controls

496

*6-24 Fixed-boundary-condition vs. Free-boundary-condition Problems

498

6-25 Concluding Remarks

503

CHAPTER 7 The Design of Time-optimal Systems

504

*7-1 Introduction

504

*7-2 Time-optimal Control of a Double-integral Plant

507

7-3 Time-optimal Control of Plants with Two Time Constants

526

*7-4 Time-optimal Control of a Third-order Plant with Two Integrators and a Single Time Constant

536

7-5 Time-optimal Control of Plants with N Real Poles

551

7-6 Some Comments

565

*7-7 Time-optimal Control of the Harmonic Oscillator

568

7-8 Time-optimal Control of a Stable Damped Harmonic Oscillator

590

7-9 Time-optimal Control of a Harmonic Oscillator with Two Control Variables

595

7-10 Time-optimal Control of First-order Nonlinear Systems

610

*7-11 Time-optimal Control of a Class of Second-order Nonlinear Systems

614

*7-12 Time-optimal Control of a Double-integral Plant with a Single Zero

622

7-13 Time-optimal Control of a Double-integral Plant with Two Zeros

638

7-14 General Results on the Time-optimal Control of Plants with Numerator Dynamics

647

7-15 Concluding Remarks

660

CHAPTER 8 The Design of Fuel-optimal Systems

662

8-1 Introduction

662

8-2 First-order Linear Systems: The Integrator

664

8-3 First-order Linear Systems: Single-time-constant Case

671

*8-4 Fuel-optimal Control of the Double-integral Plant: Formulation

675

*8-5 Fuel-optimal Control of the Double-integral Plant: Free Response Time

676

*8-6 Fuel-optimal Control of the Double-integral Plant: Fixed or Bounded Response Time

684

8-7 Fuel-optimal Control of the Double-integral Plant: Response Time Bounded by a Multiple of the Minimum Time

693

*8-8 Minimization of a Linear Combination of Time and Fuel for the Double-integral Plant

703

8-9 Minimization of a Linear Combination of Time and Fuel for the Integral-plus-time-constant Plant

710

*8-10 Minimization of a Linear Combination of Time and Fuel for a Nonlinear Second-order System

726

*8-11 Comments and Generalizations

745

CHAPTER 9 The Design of Optimal Linear Systems with Quadratic Criteria

750

*9-1 Introduction

750

*9-2 Formulation of the Problem

752

*9-3 The State-regulator Problem

756

9-4 Discussion of the Results and Examples

766

*9-5 The State-regulator Problem: Time-invariant Systems; T = infinity

771

9-6 Analysis of a First-order System

776

9-7 The Output-regulator Problem

782

*9-8 The Output-regulator Problem for Single-input-Single-output Systems

788

*9-9 The Tracking Problem

793

9-10 Approximate Relations for Time-invariant Systems

801

*9-11 Tracking Problems Reducible to Output-regulator Problems

804

9-12 Analysis of a First-order Tracking System

806

*9-13 Some Comments

813

CHAPTER 10 Optimal-control Problems when the Control Is Constrained to a Hypersphere

815

10-1 Introduction

815

10-2 Discussion of the Constraint ||u(t)|| less than or equal to m

816

10-3 Formulation of a Time-optimal Problem

817

10-4 Analytical Determination of the Time-optimal Control for a Class of Nonlinear Systems

821

10-5 Discussion of the Results

824

10-6 The Optimal Control of Norm-invariant Systems

829

10-7 Time-optimal Velocity Control of a Rotating Body with a Single Axis of Symmetry

841

10-8 Suggestions for Further Reading

855

References

857

Index

867

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Optimal Control An Introduction to the Theory and Its Applications

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