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1、Optimal State Estimation Optimal State Estimation Kalman,H,and Nonlinear Approaches Dan Simon Cleveland State University A JOHN WILEY&SONS,INC.,PUBLICATION Copyright 6 2006 by John Wiley&Sons,Inc.All rights reserved.Published by John Wiley&Sons,Inc.,Hoboken,New Jersey.Published simultaneously in Can
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6、 not be suitable for your situation.You should consult with a professional where appropriate.Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages,including but not limited to special,incidental,consequential,or other damages.For general information
7、on our other products and services or for technical support,please contact our Customer Care Department within the US.at(800)762-2974,outside the US.at(317)572-3993 or fax(317)572-4002.Wiley also publishes its books in a variety of electronic formats.Some content that appears in print may not be ava
8、ilable in electronic format.For information about Wiley products,visit our web site at .Library of Congress Cataloging-in-Publication is available.ISBN-13 978-0-471-70858-2 ISBN-10 0-47 1-7085 8-5 Printed in the United States of America.10 9 8 7 6 5 4 3 2 1 CONTENTS Acknowledgments Acronyms List of
9、algorithms Introduction PART I INTRODUCTORY MATERIAL 1 Linear systems theory 1.1 1.2 1.3 1.4 1.5 1.6 Matrix algebra and matrix calculus 1.1.1 Matrix algebra 1.1.2 The matrix inversion lemma 1.1.3 Matrix calculus 1.1.4 The history of matrices Linear systems Nonlinear systems Discretization Simulation
10、 1.5.1 Rectangular integration 1.5.2 Trapezoidal integration 1.5.3 Rung-Kutta integration Stability xiii xv xvii xxi 3 4 6 11 14 17 18 22 26 27 29 29 31 33 V vi CONTENTS 2 1.6.1 Continuous-time systems 1.6.2 Discret6time systems 1.7 Controllability and observability 1.7.1 Controllability 1.7.2 Obser
11、vability 1.7.3 Stabilizability and detectability 1.8 Summary Problems Probability theory 2.1 Probability 2.2 Random variables 2.3 Transformations of random variables 2.4 Multiple random variables 2.4.1 Statistical independence 2.4.2 Multivariate statistics White noise and colored noise 2.5 Stochasti
12、c Processes 2.6 2.7 Simulating correlated noise 2.8 Summary Problems 3 Least squares estimation 3.1 3.2 3.3 3.4 3.5 Estimation of a constant Weighted least squares estimation Recursive least squares estimation 3.3.1 Alternate estimator forms 3.3.2 Curve fitting Wiener filtering 3.4.1 Parametric filt
13、er optimization 3.4.2 General filter optimization 3.4.3 Noncausal filter optimization 3.4.4 Causal filter optimization 3.4.5 Comparison Summary Problems 4 Propagation of states and covariances 4.1 Discretetime systems 4.2 Sampled-data systems 4.3 Continuous-time systems 33 37 38 38 40 43 45 45 49 50
14、 53 59 61 62 65 68 71 73 74 75 79 80 82 84 86 92 94 96 97 98 100 101 102 102 107 107 111 114 CONTENTS vii 4.4 Summary Problems 117 117 PART II THE KALMAN FILTER 5 The discretetime Kalman filter 5.1 5.2 Kalman filter properties 5.3 One-step Kalman filter equations 5.4 Alternate propagation of covaria
15、nce 5.4.1 Multiple state systems 5.4.2 Scalar systems Derivation of the discretetime Kalman filter 5.5 Divergence issues 5.6 Summary Problems 6 Alternate Kalman filter formulations 6.1 Sequential Kalman filtering 6.2 Information filtering 6.3 Square root filtering 6.3.1 Condition number 6.3.2 6.3.3
16、6.3.4 6.3.5 Algorithms for orthogonal transformations 6.4.1 6.4.2 6.5 Summary Problems The square root time-update equation Potters square root measurement-update equation Square root measurement update via triangularization 6.4 U-D filtering U-D filtering:The measurement-update equation U-D filteri
17、ng:The timeupdate equation 7 Kalman filter generalizations 7.1 7.2 Correlated process and measurement noise Colored process and measurement noise 7.2.1 Colored process noise 7.2.2 7.2.3 7.3 Steady-state filtering 7.3.1 a-/I filtering 7.3.2 a-p-y filtering 7.3.3 Kalman filtering with fading memory Co
18、lored measurement noise:State augmentation Colored measurement noise:Measurement differencing A Hamiltonian approach to steady-state filtering 7.4 123 124 129 131 135 135 137 139 144 145 149 150 156 158 159 162 165 169 171 174 174 176 178 179 183 184 188 188 189 190 193 199 202 203 208 viii CONTENTS
19、 7.5 Constrained Kalman filtering 7.5.1 Model reduction 7.5.2 Perfect measurements 7.5.3 Projection approaches 7.5.4 A pdf truncation approach 7.6 Summary Problems 8 The continuous-time Kalrnan filter 8.1 Discretetime and continuous-time white noise 8.1.1 Process noise 8.1.2 Measurement noise 8.1.3
20、Derivation of the continuous-time Kalman filter Alternate solutions to the Riccati equation 8.3.1 The transition matrix approach 8.3.2 The Chandrasekhar algorithm 8.3.3 The square root filter Generalizations of the continuous-time filter 8.4.1 8.4.2 Colored measurement noise The steady-state continu
21、ous-time Kalman filter 8.5.1 The algebraic Riccati equation 8.5.2 8.5.3 Duality 8.6 Summary Problems Discretized simulation of noisy continuous-time systems 8.2 8.3 8.4 Correlated process and measurement noise 8.5 The Wiener filter is a Kalman filter 9 Optimal smoothing 9.1 9.2 Fixed-point smoothing
22、 An alternate form for the Kalman filter 9.2.1 9.2.2 Smoothing constant states Estimation improvement due to smoothing 9.3 Fixed-lag smoothing 9.4 Fixed-interval smoothing 9.4.1 Forward-backward smoothing 9.4.2 RTS smoothing 9.5 Summary Problems 212 212 213 214 218 223 225 229 230 230 232 232 233 23
23、8 238 242 246 247 248 249 252 253 257 258 259 260 263 265 267 270 274 274 279 280 286 294 294 CONTENTS iX 10 Additional topics in Kalman filtering 10.1 Verifying Kalman filter performance 10.2 Multiplemodel estimation 10.3 Reduced-order Kalman filtering 10.3.1 Andersons approach to reduced-order fil
24、tering 10.3.2 The reduced-order Schmidt-Kalman filter 10.4 Robust Kalman filtering 10.5 Delayed measurements and synchronization errors 10.5.1 A statistical derivation of the Kalman filter 10.5.2 Kalman filtering with delayed measurements 10.6 Summary Problems PART 111 THE H,FILTER 11 The H,filter 1
25、1.1 Introduction 11.1.1 An alternate form for the Kalman filter 11.1.2 Kalman filter limitations 11.2.1 Static constrained optimization 11.2.2 Inequality constraints 11.2.3 Dynamic constrained optimization 11.3 A game theory approach to H,filtering 11.3.1 Stationarity with respect to 20 and Wk 11.3.
26、2 Stationarity with respect to 2 and y 11.3.3 A comparison of the Kalman and H,filters 11.3.4 Steady-state H,filtering 11.3.5 The transfer function bound of the H,filter 11.2 Constrained optimization 11.4 The continuous-time H,filter 11.5 Transfer function approaches 11.6 Summary Problems 12 Additio
27、nal topics in H,filtering 12.1 Mixed Kalman/H,filtering 12.2 Robust Kalman/H,filtering 12.3 Constrained H,filtering 12.4 Summary Problems 297 298 301 305 306 309 312 317 318 320 325 326 333 334 334 336 337 337 339 341 343 345 347 354 354 357 361 365 367 369 373 374 377 381 388 389 X CONTENTS PART IV
28、 NONLINEAR FILTERS 13 Nonlinear Kalman filtering 13.1 The linearized Kalman filter 13.2 The extended Kalman filter 13.2.1 The continuous-time extended Kalman filter 13.2.2 The hybrid extended Kalman filter 13.2.3 The discretetime extended Kalman filter 13.3 Higher-order approaches 13.3.1 The iterate
29、d extended Kalman filter 13.3.2 The second-order extended Kalman filter 13.3.3 Other approaches 13.4 Parameter estimation 13.5 Summary Problems 14 The unscented Kalman filter 14.1 Means and covariances of nonlinear transformations 14.1.1 The mean of a nonlinear transformation 14.1.2 The covariance o
30、f a nonlinear transformation 14.2 Unscented transformations 14.2.1 Mean approximation 14.2.2 Covariance approximation 14.3 Unscented Kalman filtering 14.4 Other unscented transformations 14.4.1 General unscented transformations 14.4.2 The simplex unscented transformation 14.4.3 The spherical unscent
31、ed transformation 14.5 Summary Problems 15 The particle filter 15.1 Bayesian state estimation 15.2 Particle filtering 15.3 Implementation issues 15.3.1 Sample impoverishment 15.3.2 Particle filtering combined with other filters 15.4 Summary Problems 395 397 400 400 403 407 410 410 413 420 422 425 42
32、6 433 434 434 437 441 44 1 444 447 452 452 454 455 457 458 461 462 466 469 469 477 480 481 Appendix A:Historical perspectives Appendix B:Other books on Kalman filtering Appendix C:State estimation and the meaning of life References Index CONTENTS xi 485 489 493 501 521 ACKNOWLEDGMENTS The financial
33、support of Sanjay Garg and Donald Simon(no relation to the au-thor)at the NASA Glenn Research Center was instrumental in allowing me to pursue research in the area of optimal state estimation,and indirectly led to the idea for this book.I am thankful to Eugenio Villaseca,the Chair of the Depart-ment
34、 of Electrical and Computer Engineering at Cleveland State University,for his encouragement and support of my research and writing efforts.Dennis Feucht and Jonathan Litt reviewed the first draft of the book and offered constructive criticism that made the book better than it otherwise would have be
35、en.I am also indebted to the two anonymous reviewers of the proposal for this book,who made suggestions that strengthened the material presented herein.I acknowledge the work of Sandy Buettner,Joe Connolly,Classica Jain,Aaron Radke,Bryan Welch,and Qing Zheng,who were students in my Optimal State Est
36、imation class in Fall 2005.They contributed some of the problems at the end of the chapters and made many suggestions for improvement that helped clarify the subject matter.Finally I acknowledge the love and support of my wife,Annette,whose encouragement of my endeavors has always been above and bey
37、ond the call of duty.D.J.S.xiii ACRONYMS ACR ARE CARE DARE EKF erf FPGA GPS HOT iff INS LHP LTI LTV MCMC MIMO N(a,b)Pdf Acronym Algebraic Riccati equation Continuous ARE Discrete ARE Extended Kalman filter Error function Field programmable gate array Global Positioning System Higher-order terms If a
38、nd only if Inertial navigation system Left half plane Linear t ime-invari ant Linear time-varying Markov chain Monte Carlo Multiple input,multiple output Normal pdf with a mean of a and a variance of b Probability density function xv xvi ha of acronyms PDF QED RHP RMS RPF RTS RV SIR SISO sss SVD TF
39、UKF wss U(a,b)Probability distribution function Quod erat demonstrandum(i.e.,“that which was to be demonstrated”)Right half plane Root mean square Regularized particle filter Rauch-Tung-Striebel Random variable Sampling importance resampling Single input,single output Strict-sense stationary Singula
40、r value decomposition Transfer function Uniform pdf that is nonzero on the domain u,b Unscented Kalman filter Wide-sense stationary LIST OF ALGORITHMS Chapter 1:Linear systems theory Rectangular integration Trapezoidal integration Fourth-order Runge-Kutta integration Chapter 2:Probability theory Cor
41、related noise simulation Chapter 3:Least squares estimation Recursive least squares estimation General recursive least squares estimation Chapter 5:The discrete-time Kalman filter The discrete-time Kalman filter Chapter 6:Alternate Kalman filter formulations The sequential Kalman filter The informat
42、ion filter The Cholesky matrix square root algorithm Potters square root measurement-update algorithm The Householder algorithm The Gram-Schmidt algorithm The U-D measurement update The U-D time update 29 31 32 74 86 88 128 151 156 160 166 171 172 175 177 xvii XViii List of algwithms Chapter 7:Kalma
43、n filter generalizations The general discretetime Kalman filter The discrete-time Kalman filter with colored measurement noise The Hamiltonian approach to steady-state Kalman filtering The fading-inemory filter Chapter 8:The continuous-time Kalman filter The continuous-time Kalman filter The Chandra
44、sekhar algorithm The continuous-time square root Kalman filter The continuous-time Kalman filter with correlated noise The continuous-time Kalman filter with colored measurement noise Chapter 9:Optimal smoothing The fixed-point smoother The fixed-lag smoother The RTS smoother Chapter 10:Additional t
45、opics in Kalman filtering The multiplemodel estimator The reduced-order Schmidt-Kalman filter The delayed-measurement Kalman filter Chapter 11:The H,filter The discretetime H,filter Chapter 12:Additional topics in H,filtering The mixed Kalman/H,filter The robust mixed Kalman/H,filter The constrained
46、 H,filter Chapter 13:Nonlinear Kalman filtering The continuous-time linearized Kalman filter The continuous-time extended Kalman filter The hybrid extended Kalman filter The discretetime extended Kalman filter The iterated extended Kalman filter The second-order hybrid extended Kalman filter The sec
47、ond-order discretetime extended Kalman filter The Gaussian sum filter Chapter 14:The unscented Kalman filter The unscented transformation The unscented Kalman filter The simplex sigma-point algorithm The spherical sigma-point algorithm 186 191 207 210 235 244 247 249 251 269 278 293 302 312 324 353
48、374 378 385 399 401 405 409 41 1 416 419 421 446 448 454 455 Chapter 15:The particle filter The recursive Bayesian state estimator The particle filter Regularized particle filter resampling The extended Kalman particle filter L s t of algorithms XiX 465 468 473 478 INTRODUCTION This book discusses m
49、athematical approaches to the best possible way of estimat-ing the state of a general system.Although the book is firmly grounded in math-ematical theory,it should not be considered a mathematics text.It is more of an engineering text,or perhaps an applied mathematics text.The approaches that we pre
50、sent for state estimation are all given with the goal of eventual implementation in s0ftware.l The goal of this text is to present state estimation theory in the most clear yet rigorous way possible,while providing enough advanced material and ref-erences so that the reader is prepared to contribute