AN ENGINEERING
APPROACH TO OPTIMAL
CONTROL AND
ESTIMATION THEORY
GEORGE M. SIOURIS
Air Force Institute of Technology
Wright-Patterson AFB, Ohio

CONTENTS
PREFACE
CHAPTER 1 INTRODUCTION AND HISTORICAL PERSPECTIVE 1
CHAPTER 2 MATHEMATICAL PRELIMINARIES 6
2.1 Random Variables 6
2.2 Expectations and Moments 10
2.2.1 Statistical Averages (Means) 11
2.2.2 Moments 12
2.2.3 Conditional Mean 17
2.3 The Chebychev and Schwarz Inequalities 22
2.4 Covariance 23
2.5 The Autocorrelation Function and Power Spectral Density 25
2.6 Linear Systems 31
2.7 The Classical Wiener Filter 38
2.8 White Noise 42
2.9 System Input Error Models 47
Problems 53
CHAPTER 3 LINEAR REGRESSION; LEAST-SQUARES
AND MAXIMUM-LIKELIHOOD ESTIMATION 63
3.1 Introduction 63
3.2 Simple Linear Regression 64
3.3 Least-Squares Estimation 64
3.3.1 Recursive Least-Squares Estimator 71
3.4 The Maximum-Likelihood Estimator (MLE) 73
3.4.1 Recursive Maximum-Likelihood Estimator 80
3.5 Bayes Estimation 82
3.6 Concluding Remarks 83
Problems 85
HAPTER 4 THE KALMAN FILTER 92
4.1 The Continuous-Time Kalman Filter 93
4.2 Interpretation of the Kalman Filter 109
4.3 The Discrete-Time Kalman Filter 111
4.3.1 Real-WorId Model Errors 136
4.4 The State Transition Matrix 141
4.5 Controllability and Observability 159
4.5.1 Observers 163
4.6 Divergence 169
4.7 The U-D Covariance Algorithm in Kalman Filters 171
4.8 The Extended Kalman Filter 190
4.9 Shaping Filters and Colored Noise 201
4.10 Concluding Remarks 204
Problems 205
CHAPTER 5 LINEAR REGULATORS 219
5.1 Introduction 219
5.2 The Role of the Calculus of Variations in Optimal
Control 221
5.3 The Continuous-Time Linear-Quadratic Regulator 232
5.4 The Discrete-Time Linear-Quadratic Regulator 253
5.5 Optimal Linear-Quadratic-Gaussian Regulators 256
5.5.1 Introduction 256
5.5.2 Properties of the LQG Regulators 257
5.6 Pontryagin's Minimum Principle 265
5.7 Dynamic Programming and the Hamilton-Jacobi Equation 274
5.8 Concluding Remarks 291
Problems 294
HAPTER 6 COVARIANCE ANALYSIS AND SUBOPTIMAL
FILTERING 302
6.1 Covariance Analysis 302
6.1.1 Concluding Remarks 319
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