[272] | 1 | /*! |
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| 2 | \page kalman Examples of (extended) Kalman filtering |
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| 3 | |
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| 4 | Kalman filtering and Extended Kalman filtering are special cases of Bayesian filtering. |
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| 5 | The Kalman filter is optimal for linear state space model with Gaussian disturbances, the extended Kalman filter |
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| 6 | is derived as linearization of non-linear state space models with Gaussian noises. Hence it is only sub-optimal filter. |
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| 7 | |
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| 8 | More advanced filtering algorithms for non-linear non-Gaussian models can be derived, see ... |
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| 9 | |
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| 10 | \section klm Kalman Filtering |
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| 11 | Kalman filtering is optimal estimation procedure for linear state space model: |
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| 12 | \f{eqnarray} |
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| 13 | x_t &= &A x_{t-1} + B u_{t} + v_t,\\ |
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| 14 | y_t &= &C x_{t} + D u_{t} + w_t, |
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| 15 | \f} |
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| 16 | where \f$ x_t \f$ is the state, \f$ y_t \f$ is the system output, \f$ A, B, C, D\f$ are state matrices of appropriate dimensions, \f$v_t, w_t\f$ are zero mean Gaussian noises with covariance matrices \f$Q, R\f$, respectively. |
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| 17 | |
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| 18 | Both prior and posterior densities on the state are Gaussian, i.e. of the class enorm. |
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| 19 | |
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| 20 | There is a range of classes that implements this functionality, namely: |
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| 21 | - bdm::KalmanFull which implements the estimation algorithm on full matrices, |
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| 22 | - bdm::KalmanCh which implements the estimation algorithm using choleski decompositions and QR algorithm. |
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| 23 | |
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| 24 | \section ekf Extended Kalman Filtering |
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| 25 | Extended Kalman filtering arise by linearization of non-linear state space model: |
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| 26 | \f{eqnarray} |
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| 27 | x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ |
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| 28 | y_t &= &h( x_{t} , u_{t}) + w_t, |
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| 29 | \f} |
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| 30 | where \f$ g(), h() \f$ are general non-linear functions which have finite derivatives. Remaining variables have the same meaning as in the Kalman Filter. |
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| 31 | |
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| 32 | In order to use this class, the non-linear functions and their derivatives must be defined as an instance of class \c diffbifn. |
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| 33 | |
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| 34 | Two classes are defined: |
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| 35 | - bdm::EKFfull on full size matrices, |
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| 36 | - bdm::EKFCh on Choleski decompositions and using QR algorithm. |
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| 37 | |
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| 38 | \section exa Examples of Use |
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| 39 | |
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[661] | 40 | The classes can be used directly in C++ or via User Info. The latter example is illustrated in file \subpage user_guide2. A very short example of the former follows: |
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[272] | 41 | |
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| 42 | \include kalman_simple.cpp |
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| 43 | |
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| 44 | */ |
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