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The goal in this section is to explain the fundamentals of Kalman filter theory by a few illustrative examples. The Kalman filter requires a state space model for describing the signal dynamics. To describe its role, we need a concrete example, so let us return to the target tracking example from Chapter 1. Assume that we want a model with the states z1 = X , x2 = Y, x3 = X och x4 = Y . This is the simplest possible case of state vector used in practice. Before we derive a model in the next section, a few remarks will be given on what role. | Adaptive Filtering and Change Detection Fredrik Gustafsson Copyright 2000 John Wiley Sons Ltd ISBNs 0-471-49287-6 Hardback 0-470-84161-3 Electronic Part IV State estimation Adaptive Filtering and Change Detection Fredrik Gustafsson Copyright 2000 John Wiley Sons Ltd ISBNs 0-471-49287-6 Hardback 0-470-84161-3 Electronic 8 Kalman filtering 8.1. Basics.264 8.2. State space modeling.267 8.2.1. Sampling formula.268 8.2.2. Physical modeling.268 8.2.3. Using known transfer functions.272 8.2.4. Modeling tricks.274 8.3. The Kalman filter.278 8.3.1. Basic formulas.278 8.3.2. Numerical examples.280 8.3.3. Optimality properties.284 8.4. Time-invariant signal model.286 8.4.1. Error sources.287 8.4.2. Observer .288 8.4.3. Frequency response.289 8.4.4. Spectral factorization .289 8.5. Smoothing .290 8.5.1. Fixed-lag smoothing.290 8.5.2. Fixed-interval smoothing .292 8.6. Computational aspects.295 8.6.1. Divergence.296 8.6.2. Cross correlated noise.297 8.6.3. Bias error.298 8.6.4. Sequential processing.299 8.7. Square root implementation.300 8.7.1. Time and measurement updates .301 8.7.2. Kalman predictor.304 8.7.3. Kalman filter.304 8.8. Sensor fusion.306 8.8.1. The information filter.308 8.8.2. Centralized fusion.310 8.8.3. The general fusion formula.310 8.8.4. Decentralized fusion.311 8.9. The extended Kalman filter.313 264 Kalman filtering 8.9.1. Measurement update.313 8.9.2. Time update.314 8.9.3. Linearization error.318 8.9.4. Discretization of state noise.321 8.10. Whiteness based change detection using the Kalman filter.324 8.11. Estimation of covariances in state space models . 326 8.12. Applications .327 8.12.1. DC motor.327 8.12.2. Target tracking.328 8.12.3. GPS.337 8.1. Basics The goal in this section is to explain the fundamentals of Kalman filter theory by a few illustrative examples. The Kalman filter requires a state space model for describing the signal dynamics. To describe its role we need a concrete example so let us return to the target tracking example from
