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Acknowledgments.
Acronyms.
List of algorithms.
Introduction. PART I INTRODUCTORY MATERIAL.
1 Linear systems theory.
1.1 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.
1.2 Linear systems.
1.3 Nonlinear systems.
1.4 Discretization.
1.5 Simulation.
1.5.1 Rectangular integration.
1.5.2 Trapezoidal integration.
1.5.3 RungeKutta integration.
1.6 Stability.
1.6.1 Continuous-time systems.
1.6.2 Discretetime systems.
1.7 Controllability and observability.
1.7.1 Controllability.
1.7.2 Observability.
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.
2.5 Stochastic Processes.
2.6 White noise and colored noise.
2.7 Simulating correlated noise.
2.8 Summary. Problems.

3 Least squares estimation.
3.1 Estimation of a constant.
3.2 Weighted least squares estimation.
3.3 Recursive least squares estimation.
3.3.1 Alternate estimator forms.
3.3.2 Curve fitting. 3.4 Wiener filtering.
3.4.1 Parametric filter optimization.
3.4.2 General filter optimization.
3.4.3 Noncausal filter optimization.
3.4.4 Causal filter optimization.
3.4.5 Comparison.
3.5 Summary. Problems.

4 Propagation of states and covariances.
4.1 Discretetime systems.
4.2 Sampled-data systems.
4.3 Continuous-time systems.
4.4 Summary. Problems.

PART II THE KALMAN FILTER.
5 The discrete-time Kalman filter.
5.1 Derivation of the discrete-time Kalman filter.
5.2 Kalman filter properties.
5.3 One-step Kalman filter equations.
5.4 Alternate propagation of covariance.
5.4.1 Multiple state systems.
5.4.2 Scalar systems.
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 The square root time-update equation.
6.3.3 Potter's square root measurement-update equation.
6.3.4 Square root measurement update via triangularization.
6.3.5 Algorithms for orthogonal transformations.
6.4 U-D filtering.
6.4.1 U-D filtering: The measurement-update equation.
6.4.2 U-D filtering: The time-update equation.
6.5 Summary. Problems.

7 Kalman filter generalizations.
7.1 Correlated process and measurement noise.
7.2 Colored process and measurement noise.
7.2.1 Colored process noise.
7.2.2 Colored measurement noise: State augmentation.
7.2.3 Colored measurement noise: Measurement differencing.
7.3 Steady-state filtering.
7.3.1 a-P filtering.
7.3.2 a-P-y filtering.
7.3.3 A Hamiltonian approach to steady-state filtering.
7.4 Kalman filtering with fading memory.
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 Kalman filter.
8.1 Discrete-time and continuous-time white noise.
8.1.1 Process noise.
8.1.2 Measurement noise.
8.1.3 Discretized simulation of noisy continuous-time systems.
8.2 Derivation of the continuous-time Kalman filter.
8.3 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.
8.4 Generalizations of the continuous-time filter.
8.4.1 Correlated process and measurement noise.
8.4.2 Colored measurement noise
8.5 The steady-state continuous-time Kalman filter
8.5.1 The algebraic Riccati equation.
8.5.2 The Wiener filter is a Kalman filter.
8.5.3 Duality.
8.6 Summary. Problems.

9 Optimal smoothing.
9.1 An alternate form for the Kalman filter.
9.2 Fixed-point smoothing.
9.2.1 Estimation improvement due to smoothing.
9.2.2 Smoothing constant states.
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.

10 Additional topics in Kalman filtering.
10.1 Verifying Kalman filter performance.
10.2 Multiple-model estimation.
10.3 Reduced-order Kalman filtering.
10.3.1 Anderson's approach to reduced-order filtering.
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.

Acknowledgmentsp. xiii
Acronymsp. xv
List of algorithmsp. xvii
Introductionp. xxi
Introductory Material
Linear systems theoryp. 3
Matrix algebra and matrix calculusp. 4
Matrix algebrap. 6
The matrix inversion lemmap. 11
Matrix calculusp. 14
The history of matricesp. 17
Linear systemsp. 18
Nonlinear systemsp. 22
Discretizationp. 26
Simulationp. 27
Rectangular integrationp. 29
Trapezoidal integrationp. 29
Runge-Kutta integrationp. 31
Stabilityp. 33
Continuous-time systemsp. 33
Discrete-time systemsp. 37
Controllability and observabilityp. 38
Controllabilityp. 38
Observabilityp. 40
Stabilizability and detectabilityp. 43
Summaryp. 45
Problemsp. 45
Probability theoryp. 49
Probabilityp. 50
Random variablesp. 53
Transformations of random variablesp. 59
Multiple random variablesp. 61
Statistical independencep. 62
Multivariate statisticsp. 65
Stochastic Processesp. 68
White noise and colored noisep. 71
Simulating correlated noisep. 73
Summaryp. 74
Problemsp. 75
Least squares estimationp. 79
Estimation of a constantp. 80
Weighted least squares estimationp. 82
Recursive least squares estimationp. 84
Alternate estimator formsp. 86
Curve fittingp. 92
Wiener filteringp. 94
Parametric filter optimizationp. 96
General filter optimizationp. 97
Noncausal filter optimizationp. 98
Causal filter optimizationp. 100
Comparisonp. 101
Summaryp. 102
Problemsp. 102
Propagation of states and covariancesp. 107
Discrete-time systemsp. 107
Sampled-data systemsp. 111
Continuous-time systemsp. 114
Summaryp. 117
Problemsp. 117
The Kalman Filter
The discrete-time Kalman filterp. 123
Derivation of the discrete-time Kalman filterp. 124
Kalman filter propertiesp. 129
One-step Kalman filter equationsp. 131
Alternate propagation of covariancep. 135
Multiple state systemsp. 135
Scalar systemsp. 137
Divergence issuesp. 139
Summaryp. 144
Problemsp. 145
Alternate Kalman filter formulationsp. 149
Sequential Kalman filteringp. 150
Information filteringp. 156
Squ
are root filteringp. 158
Condition numberp. 159
The square root time-update equationp. 162
Potter's square root measurement-update equationp. 165
Square root measurement update via triangularizationp. 169
Algorithms for orthogonal transformationsp. 171
U-D filteringp. 174
U-D filtering: The measurement-update equationp. 174
U-D filtering: The time-update equationp. 176
Summaryp. 178
Problemsp. 179
Kalman filter generalizationsp. 183
Correlated process and measurement noisep. 184
Colored process and measurement noisep. 188
Colored process noisep. 188
Colored measurement noise: State augmentationp. 189
Colored measurement noise: Measurement differencingp. 190
Steady-state filteringp. 193
[alpha]-[beta] filteringp. 199
[alpha]-[beta]-[gamma] filteringp. 202
A Hamiltonian approach to steady-state filteringp. 203
Kalman filtering with fading memoryp. 208
Constrained Kalman filteringp. 212
Model reductionp. 212
Perfect measurementsp. 213
Projection approachesp. 214
A pdf truncation approachp. 218
Summaryp. 223
Problemsp. 225
The continuous-time Kalman filterp. 229
Discrete-time and continuous-time white noisep. 230
Process noisep. 230
Measurement noisep. 232
Discretized simulation of noisy continuous-time systemsp. 232
Derivation of the continuous-time Kalman filterp. 233
Alternate solutions to the Riccati equationp. 238
The transition matrix approachp. 238
The Chandrasekhar algorithmp. 242
The square root filterp. 246
Generalizations of the continuous-time filterp. 247
Correlated process and measurement noisep. 248
Colored measurement noisep. 249
The steady-state continuous-time Kalman filterp. 252
The algebraic Riccati equationp. 253
The Wiener filter is a Kalman filterp. 257
Dualityp. 258
Summaryp. 259
Problemsp. 260
Optimal smoothingp. 263
An alternate form for the Kalman filterp. 265
Fixed-point smoothingp. 267
Estimation improvement due to smoothingp. 270
Smoothing constant statesp. 274
Fixed-lag smoothingp. 274
Fixe
d-interval smoothingp. 279
Forward-backward smoothingp. 280
RTS smoothingp. 286
Summaryp. 294
Problemsp. 294
Additional topics in Kalman filteringp. 297
Verifying Kalman filter performancep. 298
Multiple-model estimationp. 301
Reduced-order Kalman filteringp. 305
Anderson's approach to reduced-order filteringp. 306
The reduced-order Schmidt-Kalman filterp. 309
Robust Kalman filteringp. 312
Delayed measurements and synchronization errorsp. 317
A statistical derivation of the Kalman filterp. 318
Kalman filtering with delayed measurementsp. 320
Summaryp. 325
Problemsp. 326
The H[subscript infinity] Filter
The H[subscript infinity] filterp. 333
Introductionp. 334
An alternate form for the Kalman filterp. 334
Kalman filter limitationsp. 336
Constrained optimizationp. 337
Static constrained optimizationp. 337
Inequality constraintsp. 339
Dynamic constrained optimizationp. 341
A game theory approach to H[subscript infinity] filteringp. 343
Stationarity with respect to x[subscript 0] and w[subscript k]p. 345
Stationarity with respect to x and yp. 347
A comparison of the Kalman and H[subscript infinity] filtersp. 354
Steady-state H[subscript infinity] filteringp. 354
The transfer function bound of the H[subscript infinity] filterp. 357
The continuous-time H[subscript infinity] filterp. 361
Transfer function approachesp. 365
Summaryp. 367
Problemsp. 369
Additional topics in H[subscript infinity] filteringp. 373
Mixed Kalman/H[subscript infinity] filteringp. 374
Robust Kalman/H[subscript infinity] filteringp. 377
Constrained H[subscript infinity] filteringp. 381
Summaryp. 388
Problemsp. 389
Nonlinear Filters
Nonlinear Kalman filteringp. 395
The linearized Kalman filterp. 397
The extended Kalman filterp. 400
The continuous-time extended Kalman filterp. 400
The hybrid extended Kalman filterp. 403
The discrete-time extended Kalman filterp. 407
Higher-order approachesp. 410
The iterated extended Kalman filterp. 410
The second-order
extended Kalman filterp. 413
Other approachesp. 420
Parameter estimationp. 422
Summaryp. 425
Problemsp. 426
The unscented Kalman filterp. 433
Means and covariances of nonlinear transformationsp. 434
The mean of a nonlinear transformationp. 434
The covariance of a nonlinear transformationp. 437
Unscented transformationsp. 441
Mean approximationp. 441
Covariance approximationp. 444
Unscented Kalman filteringp. 447
Other unscented transformationsp. 452
General unscented transformationsp. 452
The simplex unscented transformationp. 454
The spherical unscented transformationp. 455
Summaryp. 457
Problemsp. 458
The particle filterp. 461
Bayesian state estimationp. 462
Particle filteringp. 466
Implementation issuesp. 469
Sample impoverishmentp. 469
Particle filtering combined with other filtersp. 477
Summaryp. 480
Problemsp. 481
Historical perspectivesp. 485
Other books on Kalman filteringp. 489
State estimation and the meaning of lifep. 493
Referencesp. 501
Indexp. 521
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This book offers the best mathematical approaches to estimating the state of a general system. The author presents state estimation theory clearly and rigorously, providing the right amount of advanced material, recent research results, and references to enable the reader to apply state estimation techniques confidently across a variety of fields in science and engineering.