Optimal Control An Introduction to the Theory and Its Applications

by Athans, Michael; Falb, Peter L.

• ISBN13: 9780486453286
• ISBN10: 0486453286
• Format: Paperback
• Copyright: 2006-12-29
• Publisher: Dover Publications

Summary

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.

Table of Contents

Sections indicated with a star (*) were treated in the classroom, and the remaining sections were assigned reading.
Preface to the Dover Edition	vii
Preface	ix
CHAPTER 1 Introduction	1
1-1 Introduction	1
1-2 The System Design Problem	1
1-3 The Control Problem	3
1-4 Historical Perspective	4
1-5 Aims of This Book	6
1-6 General Comments on the Structure of This Book	7
1-7 Chapter Description	9
1-8 Prerequisites and Study Suggestions	13
CHAPTER 2 Mathematical Preliminaries: Algebra	15
2-1 Introduction	15
2-2 Sets	15
2-3 Operations on Sets	16
2-4 Functions	20
*2-5 Vector Spaces	21
*2-6 Linear Combinations and Bases	24
*2-7 Linear Algebra: Linear Transformations and Matrices	27
*2-8 Linear Algebra: Operations on Linear Transformations and Matrices	28
*2-9 Linear Algebra: Linear Transformations of V into V	32
*2-10 Linear Algebra: Eigenvectors and Eigenvalues	35
*2-11 Euclidean Spaces: Inner Products	40
2-12 Euclidean Spaces: The Schwarz Inequality	42
2-13 Euclidean Spaces: Orthogonality and Norm	43
2-14 Euclidean Spaces: Some Properties of the Scalar Product on	44
2-15 Euclidean Spaces: Some Properties of Symmetric Matrices	48
CHAPTER 3 Mathematical Preliminaries: Analysis	54
3-1 Introduction	54
*3-2 Distance and Related Notions: Definition	54
*3-3 Distance and Related Notions: Spheres and Limits	56
*3-4 Distance and Related Notions: Open and Closed Sets	58
3-5 Distance and Related Notions: Completeness and Contractions	62
3-6 Properties of Sets in R.: Compactness	64
3-7 Properties of Sets in R.: Hyperplanes and Cones	65
*3-8 Properties of Sets in R.: Convexity	69
*3-9 Vector Functions: Introductory Remarks	74
*3-10 Vector Functions: Continuity	75
3-11 Vector Functions: Piecewise Continuity	80
*3-12 Vector Functions: Derivatives	82
3-13 Vector Functions: Smooth Sets in R.	92
*3-14 Vector Functions: Integrals	95
*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