Filtering and Control, 1989, 400 pp., ISBN 0-911575-21-9, $85.00

A comprehensive treatment of Stochastic Filtering and Control problems emphasizing engineering aspects and including algorithms. Discusses application to Tracking, Guidance and Navigation. Treats many nonstandard problems. Presents new results in Decomposition and Adaptive Filtering. Much material not available in English. Edited by A.V. Balakrishnan.
 

Preface ..... xiii

Introduction ..... xvii

1.1. Optimal Feedback Control Problem with Incomplete Information ..... 1

1.2. Models of Control Systems. Perturbing Noise, and Measurement Errors ..... 3

1.3. The Influence of Feedback on Control Performance ..... 8

1.4. Fundamental Lemmas ..... 10

1.5. Optimality Conditions: Complete Information on State Variables ..... 12

1.6. Optimality Conditions: Incomplete Information on State Variables. Separation Principle ..... 14

1.7. Optimal State Estimation and Dual Control ..... 18

1.8. Optimality Conditions with Energy Constraints ..... 21

1.9. Optimality Conditions with Energy Constraints and the Fixed Number of Control Intervals ..... 24

1.10 Optimality Conditions for Random Measurement Process Cut-off ..... 26

1.11. Optimality Conditions with Unconstrained Terminal Control ..... 28

1.12. Optimal Control Problem for Random Terminal Time ..... 29

1.13. Optimization Problem for Random and Controlled Terminal Time ..... 31
 

2.1. General Scheme of Numerical Optimization ..... 35

2.2. Linear Multidimensional Interpolation ..... 36

2.3. Definite Matrices and Convex Functions ..... 38

2.4. Estimation of Error Accumulation in the Interpolation Process ..... 40

2.5. Some Properties of Conditional Risk Functions ..... 44

2.6. Nonlinear and Stochastic Programming Methods ..... 47

2.7. Optimization and Stochastic Quasigradient Computation ..... 56

2.8. Stochastic Gradient Optimization of Terminal (Final Value) Control ..... 60

2.9. The Statement of the Optimization Problem Using Nonlinear Programming Methods ..... 64

2.10. Multidimensional Gaussian Distribution ..... 65

2.11. Recursive Equations of Statistical Characteristics in Gaussian Approximation ..... 66

2.12. A General Numerical Method for Statistical Characteristic Determination ..... 70

2.13. Square Root matrix and Improvement of the Covariance Matrix ..... 77

2.14. Control Optimization Algorithm ..... 80

2.15. Parametric Optimization of Nonlinear Automatic Control Systems ..... 82
 

3.1. The Control System ..... 83

3.2. Predictable State Coordinates ..... 86

3.3. Symmetrization of Admissible Control Regions ..... 88

3.4. Optimization by the Stochastic Programming Method ..... 88

3.5. Optimization by the Nonlinear Programming Method ..... 91

3.6. Conditional Risk Functions for Terminal Control ..... 92

3.7. Optimal Terminal Control ..... 99

3.8. Optimal Scalar Stochastic Control ..... 103

3.9. Terminal Control Numerical Optimization ..... 105

3.10. Reachability Regions of Deterministic Terminal Control ..... 105

3.11. Estimation of Random Displacement Regions ..... 108

3.12. Determination of Optimization Regions  ..... 109

3.13. One-Dimensional Control Optimization ..... 111

3.14. Terminal Control Optimization with the Fixed Number of Control Intervals ..... 113

3.15. Terminal Control Optimization with Random Time of Measurement Cut-off ..... 122

3.16. "Robustness" Regions of Terminal Control with Energy Loss ..... 124

3.17. Unconstrained Control Optimization for Quadratic Loss Functions ..... 129
 

4.1. Fundamental Assumptions ..... 133

4.2. Conditional Gaussian Distribution ..... 134

4.3. Stochastic Observability ..... 140

4.4. Recursive Algorithm ..... 141

4.5. The Algorithm for the Elimination of the Covariance Matrix Singularity ..... 143

4.6. Conditional Distributions of a Markov Sequence and Arbitrary Criterion Estimation ..... 144

4.7. The Markov Sequence of Sufficient Statistics ..... 149

4.8. A priori and a posteriori Accuracy of the ORF Algorithm ..... 150

4.9. Stochastic Observability for Successive Measurements ..... 154

4.10 The ORF Algorithm for Model 1 Measurements (Kalman Algorithm) ..... 156

4.11. Sufficient Statistics for Model 1 Measurements ..... 163

4.12. Nonsingularity Conditions of the Conditional Covariance Matrix ..... 164

4.13. Stochastic Observability for Model 1 Measurements ..... 166

4.14. The Model 1 Measurement Rate ..... 168

4.15. Limiting Conditional Distribution ..... 169

4.16. The Convergence of the ORF "Estimator" Algorithm for Model 1 Measurements ..... 174

4.17. The ORF Algorithm for Model 2 Measurements ..... 180

4.18. The Convergence of the ORF Algorithm for Model 2 Measurements ..... 184

4.19. The Generating Filter ..... 186

4.20. The ORF Algorithm for Dependent Measurement Errors ..... 189

4.21. Sufficient Statistics for Model 2 Measurements ..... 196

4.22. Recursive Filtering for Mixed Model Measurements ..... 198

4.23. The Verification of the ORF Algorithm ..... 199
 

5.1. Vector Components of Measurements ..... 201

5.2. The Problem of Mathematical Consistency of the Coordinates (Mathematical Formulation) ..... 202

5.3. The Problem of Mathematical Consistency for the Mixed Measurement Model ..... 210

5.4. Alighment of Geographic Coordinates with Gyroplatform Coordinates ..... 211
 

6.1. A priori and a posteriori Estimation Accuracy with Statistical Characteristics Errors ..... 215

6.2. NORF Algorithm Convergence for Model 1 Measurements ..... 221

6.3. The Effect of the "Zero-Drift" Vector: the Generating Filter of Random Perturbations ..... 224

6.4. The Effect of the "Zero-Drift" Vector: the Modeling Method ..... 227

6.5. The Effect of Random Measurement Errors: the Generating Filter Model ..... 228

6.6. The Effect of Dynamic System Model Errors ..... 230

6.7. The Effect of Computational Errors of the Fundamental Matrix of the Model Equations ..... 233

6.8. Estimation of the Effect of Computational Errors ..... 234

6.9. The "Outlier" Problem ..... 238
 

7.1. Fundamentals ..... 241

7.2. Order Reduction by Going from Model 1 to Model 2 Measurements ..... 241

7.3. Order Reduction by Measurement Vector Transformation ..... 247

7.4. The QORF Algorithm Robust with Respect to "Zero-Drift" of the Data Sensors ..... 250

7.5. Double-Frequency Recursive Filtering ..... 253

7.6. Summation: Primary Data Processing ..... 263

7.7. Analog-Discrete Recursive Filtering ..... 264

7.8. Two Structures of the QORF Algorithms ..... 268

7.9. The Modeling Problem for the Inertial-Doppler Navigation ..... 271
 

8.1. Equations and Methods of Optimization ..... 277

8.2. Terminal Control Optimization ..... 281

8.3. Analytic Solution of the Stochastic Control Synthesis Problem ..... 284

8.4. The Problem of Actual Consistency of the Dependent Coordinate System ..... 286

8.5. Numerical Synthesis of the Optimal Control for = 2 (The Soft Landing Model) ..... 287

8.7. Stochastic Control: Effect of Erros in a priori Statistics ..... 299

8.8. Stochastic Dual Control Synthesis ..... 303
 

9.1. The Nonlinear Filtering Problem ..... 307

9.2. The NLRF Algorithm in Gaussian Approximation ..... 308

9.3. The Adaptive Estimation Algorithm: Gaussian Approximation ..... 313

9.4. Moments and Semi-Invariants ..... 318

9.5. Conditional Distribution Parameters in Nonlinear Approximation ..... 321

9.6. Recursive Equations for Statistical Characteristics in Non-Gaussian Approximation ..... 326

9.7. The Approximation of Probability Density of a State Variable ..... 328

9.8. The NLRF Algorithm: Non-Gaussian Approximation ..... 337

9.9. The Adaptive Algorithm: Non-Gaussian Approximation ..... 338

9.10. The Algorithm of Finite-Valued Adaptation and Quasi-Optimal Control for Multiple Hypotheses ..... 341

9.11. The Minimax Recursive Filtering Algorithm ..... 350
 

Appendix: Filtering, Control and Detection for Sudden Disruption ..... 365

References ..... 375

Index ..... 379