Optimal State Estimation Kalman H Infinity and Nonlinear Approaches
Optimal state estimation Kalman, Hoo, and nonlinear approaches Dan simon Cleveland State University TERSCIENCE A JOHN WILEY sons INc publication Copyright 2006 by John Wiley sons, Inc. 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ISBN-13978-0471-708582 ISBN-100471-70858-5 Printed in the United States of America 10987654321 CONTENTS Acknowledgments Acronyms xV List of algorithms Introduction PART INTRODUCTORY MATERIAL 1 Linear systems theory 1.1 Matrix algebra and matrix calculus 4 1. 1. 1 Matrix algebra 1.1.2 The matrix inversion lemma 1.1.3 Matrix calculus 14 1.1.4 The history of matrices 7 1.2 Linear systems 8 1.3 Nonlinear systems 2 1.4 Discretization 26 1.5 Simulation 27 1.5. 1 Rectangular integration 29 1.5.2 Trapezoidal integration 1.5.3 Runge-Kutta integration 31 1.6 Stabilit 33 CONTENTS 1.6.1 Continuous-time systems 33 1.6.2 Discrete-time systems 37 1.7 Controllability and observability 1.7.1 Controllability 1.7.2 Observability 1.7.3 Stabilizability and detectability 43 8 Summary 45 Problems 2 Probability theory 49 2.1 Probability 50 2.2 Random variables 53 2.3 Transformations of random variables 59 2.4 Multiple random variables 61 2.4.1 Statistical independence 62 2.4.2 Multivariate statistics 65 2.5 Stochastic Processes 2.6 White noise and colored noise 2.7 Simulating correlated noise 2. 8 Summary Problems 75 3 Least squares estimation 79 3.1 Estimation of a constant 80 3.2 Weighted least squares estimation 82 3. 3 Recursive least squares estimation 84 .3.1 Alternate estimator forms 86 3.3.2 Curve fitting 92 3.4 Wiener filtering 94 3.4.1 Parametric filter optimization 3.4.2 General filter optimizatio 97 3.4.3 Noncausal filter optimization 98 3. 4.4 Causal filter optimization 100 3.4.5 Comparison 101 3.5 Summary 102 Proble 102 4 Propagation of states and covariances 107 4.1 Discrete-time system 07 4.2 Sampled-data systems 111 4.3 Continuous-time systems 114 CONTENTS 4.4 Summary 117 Problems 117 PART‖ THE KALMAN FILTER 5 The discrete-time Kalman filter 1?2 123 5.1 Derivation of the discrete-time Kalman filter 124 5.2 Kalman filter properties 5.3 One-step Kalman filter equations 131 5.4 Alternate propagation of covariance 135 5.4. 1 Multiple state systems 135 5.4.2 Scalar systems 37 5.5 Divergence issues 139 5.6 Summary 144 Problems 145 6 Alternate Kalman filter formulations 6. 1 Sequential Kalman filtering 150 6.2 Information filtering 56 6. 3 Square root filtering 158 6.3. 1 Condition number 159 6.3.2 The square root time-update equation 162 6. 3. 3 Potter's square root measurement-update equation 165 6.3.4 Square root measurement update via triangularization 169 6.3.5 Algorithms for orthogonal transformations 171 6.4 U-D fltering 174 6.4.1 U-D Altering: The measurement-update equation 174 6.4.2 U-D filtering: The time-update equation 176 6.5 Summary Problems 179 7 Kalman filter generalizations 183 7.1 Correlated process and measurement noise 184 7.2 Colored process and measurement noise 88 7.2.1 Colored process noise 188 7.2.2 Colored measurement noise State augmentation 189 7.2.3 Colored measurement noise: Measurement differencing 190 7. 3 Steady-state filtering 193 7.3.1 a-B filtering 199 7.3.2 a-B-y filtering 202 7. 3. 3 A Hamiltonian approach to steady-state filtering 203 7.4 Kalman filtering with fading memory 208 CONTENTS 7.5 Constrained Kalman filtering 12 21 7.5.1 Model reduction 212 7.5.2 Perfect measurements 213 7.5. 3 Projection approaches 214 7.5.4 A pdf truncation approach 218 7.6 Summary 223 Problems 225 8 The continuous-time Kalman filter 229 8.1 Discrete-time and continuous-time white noise 230 8. 1. 1 Process noise 230 8.1.2 Measurement noise 232 8. 1.3 Discretized simulation of noisy continuous-time systems 232 8.2 Derivation of the continuous-time Kalman filter 233 8. 3 Alternate solutions to the Riccati equation 238 8.3. 1 The transition matrix approach 238 8.3.2 The Chandrasekhar algorithm 242 8.3.3 The square root filter 246 8.4 Generalizations of the continuous-time filter 247 8.4.1 Correlated process and measurement noise 248 8. 4.2 Colored measurement noise 249 8.5 The steady-state continuous-time Kalman filter 252 8.5. 1 The algebraic Riccati equation 253 8.5.2 The Wiener filter is a Kalman filter 257 8.5.3 Duality 258 8.6 Summary 259 Problems 260 9 Optimal smoothing 263 9.1 An alternate form for the Kalman filter 265 9.2 Fixed-point smoothing 267 9.2.1 Estimation improvement due to smoothing 270 9.2.2 Smoothing constant states 274 9.3 Fixed-lag smoothing 274 9.4 Fixed-interval smoothing 279 9.4.1 Forward-backward smoothing 280 9.4.2 RTS smoothing 286 9.5 Summary 294 Problems 294 CONTENTS X 10 Additional topics in Kalman filtering 297 10.1 Verifying Kalman filter performance 298 10.2 Multiple-model estimation 301 10.3 Reduced-order Kalman filtering 305 10.3.1 Andersons approach to reduced-order fltering 306 10.3.2 The reduced-order Schmidt-Kalman filter 309 10.4 Robust Kalman filtering 312 10.5 Delayed measurements and synchronization errors 317 10.5.1 A statistical derivation of the Kalman filter 318 10.5.2 Kalman filtering with delayed measurements 320 10.6S 325 Problems 326 PART川THEH。 FILTER 11 The Ho filter 333 11.1 Introduction 334 11.1.1 An alternate form for the kalman flter 334 11.1.2 Kalman flter limitations 336 11.2 Constrained optimization 337 11.2.1 Static constrained optimization 337 11.2.2 Inequality constraints 39 11. 2. 3 Dynamic constrained optimization 341 11. 3 A game theory approach to Hoo filtering 343 11. 3.1 Stationarity with respect to o and wk 345 11. 3.2 Stationarity with respect to i and y 347 11.3.3 A comparison of the Kalman and hoo filters 354 11.3.4 Steady-state Hoo filtering 354 11.3.5 The transfer function bound of the hoo filter 357 11.4 The continuous-time hm flter 61 11.5 Transfer function approaches 11.6 Summary 367 Problems 369 12 Additional topics in Hoo filtering 373 12.1 Mixed Kalman Hoo filtering 374 12.2 Robust Kalman/ Hoo filtering 377 12.3 Constrained Hoo filtering 381 12.4 Summary 388 Problems 389 CONTENTS PART V NONLINEAR FILTERS 13 Nonlinear Kalman filtering 395 13. 1 The linearized Kalman filter 397 13.2 The extended Kalman filter 400 13.2.1 The continuous-time extended Kalman filter 13. 2.2 The hybrid extended Kalman filter 403 13.2.3 The discrete-time extended Kalman filter 407 13.3 Higher-order approaches 410 13.3.1 The iterated extended Kalman filter 410 13.3.2 The second-order extended Kalman fter 413 3.3.3 Other approaches 420 13.4 Parameter estimation 22 13.5 Summary 425 Problems 426 14 The unscented Kalman filter 433 14.1 Means and covariances of nonlinear transformations 434 14.1.1 The mean of a nonlinear transformation 14.1.2 The covariance of a nonlinear transformation 437 14.2 Unscented transformations 441 14.2.1 Mean approximation 441 14.2.2 Covariance approximation 444 14.3 Unscented Kalman filtering 447 14.4 Other unscented transformations 452 14.4.1 General unscented transformations 452 14.4.2 The simplex unscented transformation 454 14.4.3 The spherical unscented transformation 455 4.5 Summary 457 Problems 458 15 The particle filter 461 15. 1 Bayesian state estimation 462 15.2 Particle filtering 466 15.3 Implementation issues 469 15.3. 1 Sample impoverishment 469 15.3.2 Particle filtering combined with other filters 477 15.4 Summary 480 Problems 481