Optimal Control with a Worst-Case Performance Criterion and Applications

144 Pages·1990·1.809 MB·English

by  M. Bala Subrahmanyam (eds.)

Lecture Notes ni
Control and
noitamrofnI Sciences
detidE yb amohT.M dna renyW.A
541
.M .B Subrahmanyam
Optimal Control with a
Worst-Case Performance
Criterion and Applications
galreV-regnirpS
Berlin Heidelberg New York London
Series Editors
M. Thoma • A. Wyner
Advisory Board
D. L. Davisson • G.A . .J MacFarlane • H. Kwakernaak
.J L. Massey Ya • Z. Tsypkin • A. .J Viterbi
Author
M. Bala Subrahmanyam
Flight Control, Code 6012
Air Vehicle & Crew Systems Technology Dept.
Naval Air Development Center
Warminster, AP 18974-5000
USA
ISBN 3-540-52822-9 galreV-regndpS Heidelberg NewYork Berlin
ISBN 0-387-52822-9 gatreV-regndpS NewYork Heidelberg Berlin
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© Spdnger-Vedag Bedin, Heidelberg 1990
Printed in Germany
The use of registered names, trademarks, etc. in this publication does not imply, even in the
absence of a specific statement, that such names are exempt from the relevant protective laws
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Printing: Meroedes-Druck, Berlin
Binding: B. Helm, Berlin
This monograph is lovingly dedicated to
SUSEJ ~TSIRHC
God's gift of love.
[1 JOHN 4:10]
PREFACE
This monograph deals with optimal control problems in which the cost functional is
a product of powers of definite integrals. In particular, we consider cost functionals of
the form of a quotient of definite integrals and their relation to finite-interval Hoo control,
performance robustness, and model reduction.
The material in the book is taken from a collection of research papers written by the
author. These research papers are reproduced here without much alteration. Thus there
is some duplication of material in the various chapters and the chapters are reasonably
independent of one another. As far as further research is concerned, I feel that there is a
need in the areas of performance robustness and computational algorithms.
In our compuations, we found the function which represents the worst-case perfor-
mance measure (denoted by ~) in the various chapters) to be a difficult function to evaluate.
A vast amount of computer time was spent in evaluating this function for given control
parameters. Further research needs to be done to find an alternate method to evaluate
more accurately and efficiently. We found it especially difficult to evaluate the global
maximum of this function because of the existence of a large number of local maxima.
However, in most cases it is adequate to obtain a good local maximum. The area of model
reduction is a notable exception. In this case one needs to look for the global maximum.
Also, inaccurate evaluation of k, may prematurely terminate the optimization procedure
which seeks to maximize .k, In any case, it is hoped that this monograph will stimulate
additional work in the area of evaluation of k, and its optimization. When progress is made
in that area, the methodology can be applied to time-varying examples.
The book is organized as follows. Chapter 1 treats nonlinear control problems in
which the cost functional is of the form of a quotient or a product of powers of definite
iv PREFACE
integrals. In Chapter 2, necessary conditions for optimality are developed in the case of
linear control problems, with the cost functional being a quotient of definite integrals.
Also, an existence theorem is given for the attainment of the optimal cost in a specialized
case. Chapter 3 shows the relationship of such cost functionals to the finite-interval ooH
problem and the application otfh e results to optimal disturbance rejection. An expression
for the variation tohfe worst-case performance with system parameter variations is derived.
The results are developed in the case of time-varying systems. Chapter 4 derives certain
necessary conditions that need to be satisfied by the controller which yields maximum
disturbance rejection. In Chapter ,5 the tools developed in the previous chapters are
applied to the finite-interval ooH problem, making use of observer-based parametrization
of all stabilizing controllers. In Chapter 6, a generalized finite-interval ooH problem is
treated, and necessary conditions and existence results are given. Also, an expression for
the worst-case performance variation is derived in terms of system parameter variations.
Finally, Chapter 7 treats the problem of optimal reduction of a high order system to a low
order one. The cost functional involved in such reduction is of the form of a quotient of
definite integrals. Only Chapters 5 and 7 make use of figures and these are given at the
end of the respective chapters.
I taket his opportunity to express my gratitude to Mr. F~ed Kuster, Head of the Flight
Control Branch at the Naval Air Development Center, for his encouragement and support
of this research. Thanks are also duc to my colleague Marc Steinberg for providing the
necessary computational assistance in connection with Chapter .7 Finally, I thank my wife
Carol for her love and support.
Warminster, Pennsylvania
M. BALA SUBRAHMANYAM
January, 1990
TABLE OF CONTENTS
CHAPTER 1
Necessary Conditions rof Optimality ni Problems with
Nonstandard Cost Functionals
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
.3 NECESSARY CONDITIONS FOR OPTIMALITY ............... 5
.4 COST FUNCTIONAL OF THE FORM OF A PRODUCT ............ 10
5. CERTAIN GENERALIZATIONS . . . . . . . . . . . . . . . . . . . . . 10
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
CHAPTER 2
Linear Control Problems and an Existence Theorem
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2. AN EXISTENCE THEOREM . . . . . . . . . . . . . . . . . . . . . . 14
3. NECESSARY CONDITIONS FOR OPTIMALITY . . . . . . . . . . . . . . . 17
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
CHAPTER 3
Optimal Disturbance Rejection and Performance Robustness
in Linear Systems
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2. PROBLEM FORMULATION . . . . . . . . . . . . . . . . . . . . . . . 28
3. EXISTENCE OF THE WORST EXOGENOUS INPUT . . . . . . . . . . . . . 29
viii TABLE OF CONTENTS
4. EVALUATION OF ~) . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5. SOME APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 37
6. PERFORMANCE ROBUSTNESS . . . . . . . . . . . . . . . . . . . . . 40
7'. NAVION WING LEVELER SYSTEM . . . . . . . . . . . . . . . . . . . 43
8. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
CHAPTER 4
Necessary Conditions for Optimal Disturbance Rejection
in Linear Systems
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2. NECESSARY CONDITIONS FOR h FIXED C(t) . . . . . . . . . . . . . . 52
3. A NECESSARY CONDITION FOR THE MAXIMIZATION OF ~) . . . . . . . . 54
4. PROBLEM FORMULATION FOR AN OBSERVER-BASED CONTROLLER .... 57
5. NECESSARY CONDITIONS FOR A FIXED C($) IN THE
CASE OF AN OBSERVER-BASED CONTROLLER . . . . . . . . . . . . . . 58
6. MAXIMIZATION OF A IN THE CASE OF AN
OBSERVER-BASED CONTROLLER . . . . . . . . . . . . . . . . . . . . 59
7. REDUCED ORDER OBSERVER CASE . . . . . . . . . . . . . . . . . . 62
8. SUFFICIENCY THEORY . . . . . . . . . . . . . . . . . . . . . . . . 65
9. AN EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
10. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
TABLE OF CONTENTS ix
CHAPTER 5
Synthesis of Finite-Interval H¢o Controllers by
State Space Methods
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2. STATE SPACE FORMULATION OF THE Hoo PROBLEM . . . . . . . . . . . 72
3. OPTIMALITY CONDITIONS . . . . . . . . . . . . . . . . . . . . . . 76
4. OPTIMALITY CONDITIONS FOR THE MAXIMIZATION OF ~ . . . . . . . . . 79
5. A NUMERICAL EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . 81
6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
CHAPTER 6
Worst-Case Performance Measures for
Linear Control Problems
TCARTSBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2. EXISTENCE OF THE WORST EXOGENOUS INPUT . . . . . . . . . . . . . 91
3. CHARACTERIZATION OF THE OPTIMAL VALUE . . . . . . . . . . . . . 94
4, ROBUSTNESS CONSIDERATIONS . . . . . . . . . . . . . . . . . . . . 98
5. VARIATION OF PERFORMANCE WITH CONTROL PARAMETERS . . . . . 102
6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
x TABLE OF CONTENTS
CHAPTER 7
Model Reduction with a Finite-Interval
H~ Criterion
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2. COMPUTATION OF ~) FOR A GIVEN REDUCED ORDER MODEL . . . . . . 109
3. ROBUST MODEL REDUCTION . . . . . . . . . . . . . . . . . . . . 112
4. NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . . . . . . . 116
5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
CHAPTER. 1
Necessary Conditions for Optimality in
Problems with Nonstandard Cost Funetionals
1. INTRODUCTION
Usual formulation of optimal control problems involves minimization of a cost func-
tional which is of the form of a definite integral. In this chapter, we develop necessary
conditions for an optimal control in the case of problems in which the cost functional is
either a quotient or a product of definite integrals. We call such funetionals nonstandard.
Preliminary results for problems having a fixed final time and free terminal state are in
[1]. Related results can also be found in [2,3]. In this monograph, we consider only fixed
final time problems. Problems in which the final time is frce are treated in [4]. In Section
5, we discuss the relation of our results to those in [2,3].
In this chapter, results will be derived for nonlinear systems, although the emphasis
in susequent chapters is on linear systems. We make use of the Dubovitskii-Milyutin
formalism [5,6] to derive the necessary conditions. This formalism is narrated in Section
2. We will not give detailed proofs of the results in Section 2 since a very lucid treatment
of the theory is given in [6].
We consider problems where the cost functional is of the form of a quotient in Section
3, and of the form of a product in Section 4. Finally, certain generalizations are considered
in Section 5.
2. PRELIMINARIES
Throughout this section, unless otherwise stated, E denotes a linear topological space
[7]. Let F(x) be a real-~ued function defined on E.