本书较为全面的介绍了最优状态估计的方法，适合作为最优状态估计相关领域工程技术人员参考著作，也可作为相关课程的高年级本科生或研究生教材
Optimal State Estimation This page Intentionally left blank Optimal State Estimation Kalman, Hoo, and Nonlinear approaches Dan simon Cleveland State University 少 WILEY INTERSCIENCE A JoHN WiLEY sons INc. publication Copyright o 2006 by John Wiley sons, Inc. 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ISBN-139780471-708582 ISBN-100-471-70858-5 Printed in the United States of america 10987654321 CONTENTS Acknowledgments XIIl Acronyms XV List of algorithms XVIl Introduction PART INTRODUCTORY MATERIAL 1 Linear systems theory 1.1 Matrix algebra and matrix calculus 4 1. 1. 1 Matrix algebra 6 1.1.2 The matrix inversion lemma 1.1.3 Matrix calculus 14 1.1.4 The history of matrices 17 1.2 Linear systems 18 1.3 Nonlinear systems 22 1.4 Discretization 26 1.5 Simulation 27 1.5. 1 Rectangular integration 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 38 1.7, 1 Controllability 1.7.2 Observabili 40 1.7. 3 Stabilizability and detectability 43 1.8 Summary 45 Proble 2 Probability theory 49 2.1 Probabilit 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 2.5 Stochastic Processes 68 2.6 White noise and colored noise 71 2.7 Simulating correlated noise 73 2.8 Summary 74 Problems 75 3 Least squares estimation 3.1 Estimation of a constant 80 3.2 Weighted least squares estimation 82 3.3 Recursive least squares estimation 84 3.3.1 Alternate estimator forms 3.3.2 Curve fitt 92 3.4 Wiener fltering 94 3.4. 1 Parametric filter optimization 3.4.2 General filter optimization 97 3.4. 3 Noncausal filter optimization 98 3.4.4 Causal filter optimization 3.4.5 Comparison 101 3.5 Summary 102 Problems 102 4 Propagation of states and covariances 107 4.1 Discrete-time systems 107 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 123 5.1 Derivation of the discrete-time Kalman filter 124 5.2 Kalman filter properties 129 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 137 5.5 Divergence issues 139 5.6 Summary 144 Problems 145 6 Alternate Kalman filter formulations 149 6. 1 Sequential Kalman filtering 150 6.2 Information filtering 156 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 Potters square root measurement-update equation 165 6.3.4 Square root measurement update via triangularization 169 6.3.5 Algorithms for orthogonal transformations 6.4 U-D filtering 174 6.4.1 U-D iltering: The measurement-update equation 174 6.4.2 U-D filtering: The time-update equation 176 6.5 Summary 178 Problems 179 7 Kalman filter generalizations 183 7. 1 Correlated process and measurement noise 184 7. 2 Colored process and measurement noise 188 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-By 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 212 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 Summar 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 flter 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 flter 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 flter 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 9.5 Summary 294 Problems 294