Table of Contents

Preface ix Acknowledgments xiii List of Abbreviations xv 1 General Information 1 1.1 On Kalman Filtering, 1 1.2 On Optimal Estimation Methods, 5 1.3 On the Notation Used In This Book, 23 1.4 Summary, 25 Problems, 26 2 Linear Dynamic Systems 31 2.1 Chapter Focus, 31 2.2 Dynamic System Models, 36 2.3 Continuous Linear Systems and Their Solutions, 40 2.4 Discrete Linear Systems and Their Solutions, 53 2.5 Observability of Linear Dynamic System Models, 55 2.6 Summary, 61 Problems, 64 3 Random Processes and Stochastic Systems 67 3.1 Chapter Focus, 67 3.2 Probability and Random Variables (RVs), 70 3.3 Statistical Properties of RVs, 78 3.4 Statistical Properties of Random Processes (RPs), 80 3.5 Linear RP Models, 88 3.6 Shaping Filters and State Augmentation, 95 3.7 Mean and Covariance Propagation, 99 3.8 Relationships Between Model Parameters, 105 3.9 Orthogonality Principle, 114 3.10 Summary, 118 Problems, 121 4 Linear Optimal Filters and Predictors 131 4.1 Chapter Focus, 131 4.2 Kalman Filter, 133 4.3 Kalman-Bucy Filter, 144 4.4 Optimal Linear Predictors, 146 4.5 Correlated Noise Sources, 147 4.6 Relationships Between Kalman-Bucy and Wiener Filters, 148 4.7 Quadratic Loss Functions, 149 4.8 Matrix Riccati Differential Equation, 151 4.9 Matrix Riccati Equation In Discrete Time, 165 4.10 Model Equations for Transformed State Variables, 170 4.11 Application of Kalman Filters, 172 4.12 Summary, 177 Problems, 179 5 Optimal Smoothers 183 5.1 Chapter Focus, 183 5.2 Fixed-Interval Smoothing, 189 5.3 Fixed-Lag Smoothing, 200 5.4 Fixed-Point Smoothing, 213 5.5 Summary, 220 Problems, 221 6 Implementation Methods 225 6.1 Chapter Focus, 225 6.2 Computer Roundoff, 227 6.3 Effects of Roundoff Errors on Kalman Filters, 232 6.4 Factorization Methods for Square-Root Filtering, 238 6.5 Square-Root and UD Filters, 261 6.6 Other Implementation Methods, 275 6.7 Summary, 288 Problems, 289 7 Nonlinear Filtering 293 7.1 Chapter Focus, 293 7.2 Quasilinear Filtering, 296 7.3 Sampling Methods for Nonlinear Filtering, 330 7.4 Summary, 345 Problems, 350 8 Practical Considerations 355 8.1 Chapter Focus, 355 8.2 Detecting and Correcting Anomalous Behavior, 356 8.3 Prefiltering and Data Rejection Methods, 379 8.4 Stability of Kalman Filters, 382 8.5 Suboptimal and Reduced-Order Filters, 383 8.6 Schmidt-Kalman Filtering, 393 8.7 Memory, Throughput, and Wordlength Requirements, 403 8.8 Ways to Reduce Computational Requirements, 409 8.9 Error Budgets and Sensitivity Analysis, 414 8.10 Optimizing Measurement Selection Policies, 419 8.11 Innovations Analysis, 424 8.12 Summary, 425 Problems, 426 9 Applications to Navigation 427 9.1 Chapter Focus, 427 9.2 Host Vehicle Dynamics, 431 9.3 Inertial Navigation Systems (INS), 435 9.4 Global Navigation Satellite Systems (GNSS), 465 9.5 Kalman Filters for GNSS, 470 9.6 Loosely Coupled GNSS/INS Integration, 488 9.7 Tightly Coupled GNSS/INS Integration, 491 9.8 Summary, 507 Problems, 508 Appendix A MATLAB Software 511 A.1 Notice, 511 A.2 General System Requirements, 511 A.3 CD Directory Structure, 512 A.4 MATLAB Software for Chapter 2, 512 A.5 MATLAB Software for Chapter 3, 512 A.6 MATLAB Software for Chapter 4, 512 A.7 MATLAB Software for Chapter 5, 513 A.8 MATLAB Software for Chapter 6, 513 A.9 MATLAB Software for Chapter 7, 514 A.10 MATLAB Software for Chapter 8, 515 A.11 MATLAB Software for Chapter 9, 515 A.12 Other Sources of Software, 516 Appendix B A Matrix Refresher 519 B.1 Matrix Forms, 519 B.2 Matrix Operations, 523 B.3 Block Matrix Formulas, 527 B.4 Functions of Square Matrices, 531 B.5 Norms, 538 B.6 Cholesky Decomposition, 541 B.7 Orthogonal Decompositions of Matrices, 543 B.8 Quadratic Forms, 545 B.9 Derivatives of Matrices, 546 Bibliography 549 Index 565

About the Author

Mohinder S. Grewal, PhD, PE, is Professor of Electrical Engineering in the College of Engineering and Computer Science at California State University, Fullerton. He has more than thirty-five years of experience in inertial navigation and control, and his mechanizations are currently used in commercial and military aircraft, surveillance satellites, missile and radar systems, freeway traffic control, and the Global Navigation Satellite System. Angus P. Andrews, PhD, is a retired senior scientist from the Rockwell Science Center. His experience with aerospace systems analysis and design using Kalman filters began with his involvement in the Apollo moon project, and he is credited with the discovery of unknown landmark tracking as an orbital navigation method.