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Kalman Filtering Theory and practice Using MatLab® 2nd Edition Mohinder S. Grewal and Angus P. Andrews Navtech Part # 1029 PREFACE ACKNOWLEDGMENTS 1 General Information 1.1 On Kalman Filtering 1.2 On Estimation Methods 1.3 On the Notation Used an flu, Book 1.4 Summary Problems 2 Linear Dynamic Systems 2.1 Chapter Focus 2.2 Dynamic Systems 2.3 Continuous Linear Systems and Their Solutions 2.4 Discrete Linear Systems and Their Solutions 2.5 Observability of Linear Dynamic System Models 2.6 Procedures for Computing Matrix Exponentials 2.7 Summary Problems ix xiii 1 1 5 20 22 23 25 25 26 30 41 42 48 50 53

3 Random Processes and Stochastic Systems 56 3.1 Chapter Focus 56 3.2 Probability and Random Variables 58 3.3 Statistical Properties of Random Variables 66 3.4 Statistical Properties of Random Processes 68 3.5 Linear System Models of Random Processes and Sequences76 3 6 Shaping Filters and State Augmentation 84 3.7 Covariance Propagation Equations 88 3.8 Orthogonality Principle 97 3.9 Summary 102 Problems 104 4 Linear Optimal Filters and Predictors 4.1 Chapter Focus 4.2 Kalman Filter 43 Kalman Bucy Filter 44 Optimal Linear Predictors 4 5 Correlated Noise Sources 4.6 Relationships between Kalman and Wiener Filters 4 7 Quadratic Loss Functions 114 114 116 126 128 129 130 131

4.8 4.9 4.10 4.11 4.12 4.13 4.14

Matrix Riccati Differential Equation 133 Matro Riccati Equation in Discrete Time 148 Relationships between Continuous and Discrete Riccati Equations Model Equations for Transformed State Variables 154 Application of Kalman Filters 155 Smoothers 160 Summary 164 Problems 165 169 169 170 171 171 175 176 178 181 182 184 198 200 202 202 204 209 216 238 252 265 266 270 270 271 294 298 299 309 316 326 332 336 342 346 347

153

5 Nonlinear Applications 5.1 Chapter Focus 5.2 Problem Statement 5.3 Linearization Methods 5 4 Linearization about a Nominal Trajectory 5.5 Linearization about the Estimated Trajectory 5 6 Discrete Linearization and Extended Filtering 5.7 Discrete Extended Kalman Filter 5.8 Continuous Linearized and Extended Filters 5.9 Biased Errors in Quadratic Measurements 5.10 Application of Nonlinear Filters 5.11 Summary Problems 6 Implementation Methods 6.1 Chapter Focus 6.2 Computer Roundoff 6.3 Effect, of Roundoff Errors on Kalman Filters 6 4 Factorization Methods for Kalman Filtering 6.5 Square-Root and UD Filters 6.6 Other Alternative Implementation Methods 6.7 Summary Problems 7 Practical Considerations 7.1 Chapter Focus 7.2 Detecting and Correcting Anomalous Behavior 7.3 Prefiltering and Data Rejection Methods 7.4 Stability of Kalman Filters 7.5 Suboptimal and Reduced-Order Filters 7.6 Schmidt Kalman Filtering 7.7 Memory, Throughput, and Wordlength Requirements 7.8 Ways to Reduce Computational Requirements 7.9 Error Budgets and Sensitivity Analysis 7.10 Optimizing Measurement Selection Policies 7.11 Applicator to Aided Inertial Navigation 7.12 Summary Problems

Appendix A MATLAB Software A.1 Notice A.2 General System Requirements A.3 Diskette Directory Structure A.4 MATLAB Software for Chapter 2 A.5 MATLAB Software for Chapter 4 A.6 MATLAB Software for Chapter 5 A.7 MAT[ AB Software for Chapter 6 A.8 MATLAB Software for Chapter 7 A.9 Other Sources of Software Appendix B A Matrix Refresher B I Matrix Forms B.2 Mann, Operations B.3 Block Matrix Formulas B.4 Functions of Square Matrices B.5 Norms B.6 Cholesky Decomposition B.7 Orthogonal Decompositions of Matrices B 8 Quadratic Forms B 9 Derivatives of Matrices REFERENCES INDEX

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