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ON OPTIMAL CONTROL OF STOCHASTIC DYNAMICAL SYSTEMS

by

BAN-BARU VENKATA RAJARAO B.E.(Hons), M.Tech.

Thesis submitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy

to the

Indian Institute of Technology, Delhi

Department of Electrical Engineering Indian Institute of Technology

New Delhi-29 INDIA

December 1970

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PREFACE

The thesis is based on results of the

investigations carried out by tl-e author at the Indian Institute of Technology, Delhi, over the period

December 1967 to November 1970. All of these studies have been concerned with problems of optimal control

of stochastic dynamical systems. Except the investigations reported in chapter 4, all the other studies are concerned with nonlinear systems. Attempts have been made to

develop suboptimal solutions for the nonlinear control problem.In chapter 4, however, an optimal solution is presented for linear stochastic systems involving process time delay.

I would like to express my gratitude to my thesis supervisor Professor A. K. Mahalanabis, whose work. and encouragement provided the motivation and support for the thesis. I would also like to thank my colleagues Mr. S. Purkayastha and Mr. H. S. Kang for their suggestions and useful discussions. Thanks are also due to the Head of Electrical Engineering Department for

providing the facilities in the Department.

Finally I wish to thank my wife Lakshmi for her patient understanding, help and encouragement throughout the course of this endeavour.

AX4-4P

New Delhi-29 B.V. RAJA RAO

December 12, 1970.

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ONTENT . 3

CHAPTER I INTRODUCTION 1 1.1 Some results in Deterministic Optimal

Control Theory. 2

1.2 Optimal Stochastic Control 7 1.3 Statement of the Problems 19 1.4 Organisation of the Thesis 24 CHAPTER II MINIMUM VARIANCE CONTROL OF DISCRETE

TIME NONLINEAR STOCHASTIC SYSTEMS 29 2.1 General Discussion 29 2.2 Problem Formulation 30 2.3 Method of Solution 32 2.4 Suboptimal Solutions 36

2.5 Examples 44

2.6 Conclusions 47

CHAPTER III MINIMUM VARIANCE CONTROL OF NONLINEAR CONTINUOUS TIME STOCHASTIC SYSTEMS 54 3.1 General Discussion 54 3.2 Problem Formulation 56 3.3 Method of Solution 58 3.4 Results based on Taylor's Series 62 3.5 An Alternative Approach 67

3.6 Examples 70

3.7 Conclusions 73

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CHAPTER IV MINIMUM VARIANCE CONTROL OF LINEAR DISCRETE TIME STOCHASTIC SYSTEMS WITH TIME DELAY 77 4.1 General Discussion 77 4.2 Problem Formulation 78 4.3 Method of solution El 4.4 An Alternative Approach 84

4.5 An Example 88

4.6 Conclusions 89

Appendix 90

CHAPTER V ON OPTIMIZATION OF NONLIYEAR STOCHASTIC

SYSTEMS SUBJECT TO QUADRATIC CRITERION 97 5.1 General Discussion 97 5.2 Problem Formulation 98 5.3 Linear systems 99 5.4 Statistical Linearization 104 5.5 Stochastic Quadratic Approximation 107

5.6 Examples 110

5.7 Conclusions 111

CHAPTER VI ON OPTIMIZATION OF LINEAR STOCHASTIC

SYSTEMS WITH NONLINEAR NOISY MEASUREMENTS

SUBJECT TO QUADRATIC CRITERION 117 6.1 General Discussion 11?

6.2 Problem Formulation 118 6.3 Method of Solution 119 6.4 Solution of the Filtering Problem 121 6.5 Solution of the suboptimal Control Problem 123

6.6 An Example 125

6.7 Conclusions 127

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CHAPTER VII CONCLUSIONS AND SUGGESTIONS 130

7.1

Conclusions 130 7.2 Suggestions for further work 132

REFERENCES 124

References

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