[7] | 1 | /*! |
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| 2 | \file |
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| 3 | \brief Bayesian Filtering for linear Gaussian models (Kalman Filter) and extensions |
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| 4 | \author Vaclav Smidl. |
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| 5 | |
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| 6 | ----------------------------------- |
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| 7 | BDM++ - C++ library for Bayesian Decision Making under Uncertainty |
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| 8 | |
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| 9 | Using IT++ for numerical operations |
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| 10 | ----------------------------------- |
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| 11 | */ |
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| 12 | |
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| 13 | #ifndef KF_H |
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| 14 | #define KF_H |
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| 15 | |
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[262] | 16 | |
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[22] | 17 | #include "../stat/libFN.h" |
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[28] | 18 | #include "../stat/libEF.h" |
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[37] | 19 | #include "../math/chmat.h" |
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[7] | 20 | |
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[283] | 21 | namespace bdm { |
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[7] | 22 | |
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| 23 | /*! |
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[8] | 24 | * \brief Basic Kalman filter with full matrices (education purpose only)! Will be deleted soon! |
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[7] | 25 | */ |
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[32] | 26 | |
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| 27 | class KalmanFull { |
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[62] | 28 | protected: |
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[7] | 29 | int dimx, dimy, dimu; |
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| 30 | mat A, B, C, D, R, Q; |
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[32] | 31 | |
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| 32 | //cache |
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[7] | 33 | mat _Pp, _Ry, _iRy, _K; |
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| 34 | public: |
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[32] | 35 | //posterior |
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[7] | 36 | //! Mean value of the posterior density |
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| 37 | vec mu; |
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| 38 | //! Variance of the posterior density |
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| 39 | mat P; |
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| 40 | |
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[37] | 41 | bool evalll; |
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| 42 | double ll; |
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[7] | 43 | public: |
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[32] | 44 | //! Full constructor |
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| 45 | KalmanFull ( mat A, mat B, mat C, mat D, mat R, mat Q, mat P0, vec mu0 ); |
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[7] | 46 | //! Here dt = [yt;ut] of appropriate dimensions |
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[32] | 47 | void bayes ( const vec &dt ); |
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[33] | 48 | //! print elements of KF |
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[7] | 49 | friend std::ostream &operator<< ( std::ostream &os, const KalmanFull &kf ); |
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[62] | 50 | //! For EKFfull; |
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[283] | 51 | KalmanFull() {}; |
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[7] | 52 | }; |
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| 53 | |
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| 54 | |
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| 55 | /*! |
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[22] | 56 | * \brief Kalman filter with covariance matrices in square root form. |
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[33] | 57 | |
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[283] | 58 | Parameter evolution model:\f[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \f] |
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[33] | 59 | Observation model: \f[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \f] |
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[283] | 60 | Where $e_t$ and $w_t$ are independent vectors Normal(0,1)-distributed disturbances. |
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[7] | 61 | */ |
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| 62 | template<class sq_T> |
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[32] | 63 | |
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| 64 | class Kalman : public BM { |
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[22] | 65 | protected: |
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[33] | 66 | //! Indetifier of output rv |
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[32] | 67 | RV rvy; |
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[33] | 68 | //! Indetifier of exogeneous rv |
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[32] | 69 | RV rvu; |
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[33] | 70 | //! cache of rv.count() |
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| 71 | int dimx; |
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| 72 | //! cache of rvy.count() |
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| 73 | int dimy; |
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| 74 | //! cache of rvu.count() |
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| 75 | int dimu; |
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| 76 | //! Matrix A |
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| 77 | mat A; |
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| 78 | //! Matrix B |
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[283] | 79 | mat B; |
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[33] | 80 | //! Matrix C |
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| 81 | mat C; |
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| 82 | //! Matrix D |
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| 83 | mat D; |
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| 84 | //! Matrix Q in square-root form |
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| 85 | sq_T Q; |
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| 86 | //! Matrix R in square-root form |
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| 87 | sq_T R; |
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[32] | 88 | |
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| 89 | //!posterior density on $x_t$ |
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[28] | 90 | enorm<sq_T> est; |
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| 91 | //!preditive density on $y_t$ |
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| 92 | enorm<sq_T> fy; |
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[32] | 93 | |
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[33] | 94 | //! placeholder for Kalman gain |
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[8] | 95 | mat _K; |
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[33] | 96 | //! cache of fy.mu |
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[83] | 97 | vec& _yp; |
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[33] | 98 | //! cache of fy.R |
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[83] | 99 | sq_T& _Ry; |
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[33] | 100 | //!cache of est.mu |
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[83] | 101 | vec& _mu; |
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[33] | 102 | //!cache of est.R |
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[83] | 103 | sq_T& _P; |
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[32] | 104 | |
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[7] | 105 | public: |
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[22] | 106 | //! Default constructor |
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[270] | 107 | Kalman ( ); |
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[32] | 108 | //! Copy constructor |
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| 109 | Kalman ( const Kalman<sq_T> &K0 ); |
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| 110 | //! Set parameters with check of relevance |
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[271] | 111 | void set_parameters ( const mat &A0,const mat &B0,const mat &C0,const mat &D0,const sq_T &Q0,const sq_T &R0 ); |
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[28] | 112 | //! Set estimate values, used e.g. in initialization. |
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[32] | 113 | void set_est ( const vec &mu0, const sq_T &P0 ) { |
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[283] | 114 | sq_T pom ( dimy ); |
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[32] | 115 | est.set_parameters ( mu0,P0 ); |
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[283] | 116 | P0.mult_sym ( C,pom ); |
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[32] | 117 | fy.set_parameters ( C*mu0, pom ); |
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| 118 | }; |
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| 119 | |
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[7] | 120 | //! Here dt = [yt;ut] of appropriate dimensions |
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[32] | 121 | void bayes ( const vec &dt ); |
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[33] | 122 | //!access function |
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[271] | 123 | const epdf& posterior() const {return est;} |
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[200] | 124 | const enorm<sq_T>* _e() const {return &est;} |
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[51] | 125 | //!access function |
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| 126 | mat& __K() {return _K;} |
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| 127 | //!access function |
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| 128 | vec _dP() {return _P->getD();} |
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[7] | 129 | }; |
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| 130 | |
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[37] | 131 | /*! \brief Kalman filter in square root form |
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[271] | 132 | |
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| 133 | Trivial example: |
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| 134 | \include kalman_simple.cpp |
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[283] | 135 | |
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[37] | 136 | */ |
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[283] | 137 | class KalmanCh : public Kalman<chmat> { |
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[37] | 138 | protected: |
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| 139 | //! pre array (triangular matrix) |
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[283] | 140 | mat preA; |
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[37] | 141 | //! post array (triangular matrix) |
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[283] | 142 | mat postA; |
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[37] | 143 | |
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| 144 | public: |
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[283] | 145 | //! copy constructor |
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| 146 | BM* _copy_() const { |
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| 147 | KalmanCh* K=new KalmanCh; |
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| 148 | K->set_parameters ( A,B,C,D,Q,R ); |
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| 149 | K->set_statistics ( _mu,_P ); |
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| 150 | return K; |
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| 151 | } |
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[37] | 152 | //! Set parameters with check of relevance |
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[271] | 153 | void set_parameters ( const mat &A0,const mat &B0,const mat &C0,const mat &D0,const chmat &Q0,const chmat &R0 ); |
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| 154 | void set_statistics ( const vec &mu0, const chmat &P0 ) { |
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[37] | 155 | est.set_parameters ( mu0,P0 ); |
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| 156 | }; |
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[283] | 157 | |
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| 158 | |
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[37] | 159 | /*!\brief Here dt = [yt;ut] of appropriate dimensions |
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[283] | 160 | |
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[37] | 161 | The following equality hold::\f[ |
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[283] | 162 | \left[\begin{array}{cc} |
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| 163 | R^{0.5}\\ |
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| 164 | P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ |
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| 165 | & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} |
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| 166 | R_{y}^{0.5} & KA'\\ |
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| 167 | & P_{t+1|t}^{0.5}\\ |
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| 168 | \\\end{array}\right]\f] |
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[37] | 169 | |
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[283] | 170 | Thus this object evaluates only predictors! Not filtering densities. |
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[37] | 171 | */ |
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| 172 | void bayes ( const vec &dt ); |
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| 173 | }; |
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| 174 | |
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[22] | 175 | /*! |
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[62] | 176 | \brief Extended Kalman Filter in full matrices |
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| 177 | |
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| 178 | An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean. |
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| 179 | */ |
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| 180 | class EKFfull : public KalmanFull, public BM { |
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[317] | 181 | protected: |
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[62] | 182 | //! Internal Model f(x,u) |
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| 183 | diffbifn* pfxu; |
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| 184 | //! Observation Model h(x,u) |
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| 185 | diffbifn* phxu; |
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[283] | 186 | |
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| 187 | enorm<fsqmat> E; |
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[62] | 188 | public: |
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| 189 | //! Default constructor |
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[270] | 190 | EKFfull ( ); |
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[62] | 191 | //! Set nonlinear functions for mean values and covariance matrices. |
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| 192 | void set_parameters ( diffbifn* pfxu, diffbifn* phxu, const mat Q0, const mat R0 ); |
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| 193 | //! Here dt = [yt;ut] of appropriate dimensions |
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| 194 | void bayes ( const vec &dt ); |
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| 195 | //! set estimates |
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[283] | 196 | void set_est ( vec mu0, mat P0 ) {mu=mu0;P=P0;}; |
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[62] | 197 | //!dummy! |
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[283] | 198 | const epdf& posterior() const{return E;}; |
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| 199 | const enorm<fsqmat>* _e() const{return &E;}; |
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| 200 | const mat _R() {return P;} |
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[62] | 201 | }; |
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| 202 | |
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| 203 | /*! |
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[32] | 204 | \brief Extended Kalman Filter |
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[22] | 205 | |
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| 206 | An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean. |
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| 207 | */ |
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| 208 | template<class sq_T> |
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[51] | 209 | class EKF : public Kalman<fsqmat> { |
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[22] | 210 | //! Internal Model f(x,u) |
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| 211 | diffbifn* pfxu; |
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| 212 | //! Observation Model h(x,u) |
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| 213 | diffbifn* phxu; |
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[32] | 214 | public: |
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[22] | 215 | //! Default constructor |
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[32] | 216 | EKF ( RV rvx, RV rvy, RV rvu ); |
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[283] | 217 | //! copy constructor |
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| 218 | EKF<sq_T>* _copy_() const { return new EKF<sq_T>(this); } |
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[33] | 219 | //! Set nonlinear functions for mean values and covariance matrices. |
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[32] | 220 | void set_parameters ( diffbifn* pfxu, diffbifn* phxu, const sq_T Q0, const sq_T R0 ); |
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[22] | 221 | //! Here dt = [yt;ut] of appropriate dimensions |
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[32] | 222 | void bayes ( const vec &dt ); |
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[22] | 223 | }; |
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| 224 | |
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[32] | 225 | /*! |
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[62] | 226 | \brief Extended Kalman Filter in Square root |
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[37] | 227 | |
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| 228 | An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean. |
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| 229 | */ |
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| 230 | |
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| 231 | class EKFCh : public KalmanCh { |
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[283] | 232 | protected: |
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[37] | 233 | //! Internal Model f(x,u) |
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| 234 | diffbifn* pfxu; |
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| 235 | //! Observation Model h(x,u) |
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| 236 | diffbifn* phxu; |
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| 237 | public: |
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[283] | 238 | //! copy constructor duplicated - calls different set_parameters |
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| 239 | BM* _copy_() const { |
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| 240 | EKFCh* E=new EKFCh; |
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| 241 | E->set_parameters ( pfxu,phxu,Q,R ); |
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| 242 | E->set_statistics ( _mu,_P ); |
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| 243 | return E; |
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| 244 | } |
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[37] | 245 | //! Set nonlinear functions for mean values and covariance matrices. |
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| 246 | void set_parameters ( diffbifn* pfxu, diffbifn* phxu, const chmat Q0, const chmat R0 ); |
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| 247 | //! Here dt = [yt;ut] of appropriate dimensions |
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| 248 | void bayes ( const vec &dt ); |
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| 249 | }; |
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| 250 | |
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| 251 | /*! |
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[32] | 252 | \brief Kalman Filter with conditional diagonal matrices R and Q. |
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| 253 | */ |
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| 254 | |
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[283] | 255 | class KFcondQR : public Kalman<ldmat> { |
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[32] | 256 | //protected: |
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| 257 | public: |
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[283] | 258 | void condition ( const vec &QR ) { |
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| 259 | it_assert_debug ( QR.length() == ( dimx+dimy ),"KFcondRQ: conditioning by incompatible vector" ); |
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[32] | 260 | |
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[283] | 261 | Q.setD ( QR ( 0, dimx-1 ) ); |
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| 262 | R.setD ( QR ( dimx, -1 ) ); |
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| 263 | }; |
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[32] | 264 | }; |
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| 265 | |
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| 266 | /*! |
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| 267 | \brief Kalman Filter with conditional diagonal matrices R and Q. |
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| 268 | */ |
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| 269 | |
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[283] | 270 | class KFcondR : public Kalman<ldmat> { |
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[32] | 271 | //protected: |
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| 272 | public: |
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[33] | 273 | //!Default constructor |
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[283] | 274 | KFcondR ( ) : Kalman<ldmat> ( ) {}; |
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[32] | 275 | |
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[283] | 276 | void condition ( const vec &R0 ) { |
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| 277 | it_assert_debug ( R0.length() == ( dimy ),"KFcondR: conditioning by incompatible vector" ); |
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| 278 | |
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| 279 | R.setD ( R0 ); |
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| 280 | }; |
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| 281 | |
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[32] | 282 | }; |
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| 283 | |
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[7] | 284 | //////// INstance |
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| 285 | |
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| 286 | template<class sq_T> |
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[270] | 287 | Kalman<sq_T>::Kalman ( const Kalman<sq_T> &K0 ) : BM ( ),rvy ( K0.rvy ),rvu ( K0.rvu ), |
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| 288 | dimx ( K0.dimx ), dimy ( K0.dimy ),dimu ( K0.dimu ), |
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| 289 | A ( K0.A ), B ( K0.B ), C ( K0.C ), D ( K0.D ), |
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[283] | 290 | Q ( K0.Q ), R ( K0.R ), |
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| 291 | est ( K0.est ), fy ( K0.fy ), _yp ( fy._mu() ),_Ry ( fy._R() ), _mu ( est._mu() ), _P ( est._R() ) { |
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[32] | 292 | |
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[37] | 293 | // copy values in pointers |
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[270] | 294 | // _mu = K0._mu; |
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| 295 | // _P = K0._P; |
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| 296 | // _yp = K0._yp; |
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| 297 | // _Ry = K0._Ry; |
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[32] | 298 | |
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| 299 | } |
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| 300 | |
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| 301 | template<class sq_T> |
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[283] | 302 | Kalman<sq_T>::Kalman ( ) : BM (), est ( ), fy (), _yp ( fy._mu() ), _Ry ( fy._R() ), _mu ( est._mu() ), _P ( est._R() ) { |
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[22] | 303 | }; |
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| 304 | |
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| 305 | template<class sq_T> |
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[271] | 306 | void Kalman<sq_T>::set_parameters ( const mat &A0,const mat &B0, const mat &C0, const mat &D0, const sq_T &Q0, const sq_T &R0 ) { |
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| 307 | dimx = A0.rows(); |
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| 308 | dimy = C0.rows(); |
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| 309 | dimu = B0.cols(); |
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[283] | 310 | |
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[32] | 311 | it_assert_debug ( A0.cols() ==dimx, "Kalman: A is not square" ); |
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| 312 | it_assert_debug ( B0.rows() ==dimx, "Kalman: B is not compatible" ); |
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| 313 | it_assert_debug ( C0.cols() ==dimx, "Kalman: C is not square" ); |
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| 314 | it_assert_debug ( ( D0.rows() ==dimy ) || ( D0.cols() ==dimu ), "Kalman: D is not compatible" ); |
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| 315 | it_assert_debug ( ( R0.cols() ==dimy ) || ( R0.rows() ==dimy ), "Kalman: R is not compatible" ); |
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| 316 | it_assert_debug ( ( Q0.cols() ==dimx ) || ( Q0.rows() ==dimx ), "Kalman: Q is not compatible" ); |
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[7] | 317 | |
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| 318 | A = A0; |
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| 319 | B = B0; |
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| 320 | C = C0; |
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| 321 | D = D0; |
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| 322 | R = R0; |
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[32] | 323 | Q = Q0; |
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[7] | 324 | } |
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| 325 | |
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| 326 | template<class sq_T> |
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[32] | 327 | void Kalman<sq_T>::bayes ( const vec &dt ) { |
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| 328 | it_assert_debug ( dt.length() == ( dimy+dimu ),"KalmanFull::bayes wrong size of dt" ); |
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[7] | 329 | |
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[283] | 330 | sq_T iRy ( dimy ); |
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[32] | 331 | vec u = dt.get ( dimy,dimy+dimu-1 ); |
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| 332 | vec y = dt.get ( 0,dimy-1 ); |
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[7] | 333 | //Time update |
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[83] | 334 | _mu = A* _mu + B*u; |
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[7] | 335 | //P = A*P*A.transpose() + Q; in sq_T |
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[83] | 336 | _P.mult_sym ( A ); |
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| 337 | _P +=Q; |
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[7] | 338 | |
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| 339 | //Data update |
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| 340 | //_Ry = C*P*C.transpose() + R; in sq_T |
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[83] | 341 | _P.mult_sym ( C, _Ry ); |
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| 342 | _Ry +=R; |
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[7] | 343 | |
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[83] | 344 | mat Pfull = _P.to_mat(); |
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[32] | 345 | |
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[83] | 346 | _Ry.inv ( iRy ); // result is in _iRy; |
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| 347 | _K = Pfull*C.transpose() * ( iRy.to_mat() ); |
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[32] | 348 | |
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| 349 | sq_T pom ( ( int ) Pfull.rows() ); |
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[83] | 350 | iRy.mult_sym_t ( C*Pfull,pom ); |
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[283] | 351 | ( _P ) -= pom; // P = P -PC'iRy*CP; |
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| 352 | ( _yp ) = C* _mu +D*u; //y prediction |
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| 353 | ( _mu ) += _K* ( y- _yp ); |
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[32] | 354 | |
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| 355 | |
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| 356 | if ( evalll==true ) { //likelihood of observation y |
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[211] | 357 | ll=fy.evallog ( y ); |
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[7] | 358 | } |
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[32] | 359 | |
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| 360 | //cout << "y: " << y-(*_yp) <<" R: " << _Ry->to_mat() << " iR: " << _iRy->to_mat() << " ll: " << ll <<endl; |
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| 361 | |
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[7] | 362 | }; |
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| 363 | |
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[62] | 364 | |
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[283] | 365 | |
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[22] | 366 | //TODO why not const pointer?? |
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[7] | 367 | |
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[22] | 368 | template<class sq_T> |
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[37] | 369 | EKF<sq_T>::EKF ( RV rvx0, RV rvy0, RV rvu0 ) : Kalman<sq_T> ( rvx0,rvy0,rvu0 ) {} |
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[28] | 370 | |
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| 371 | template<class sq_T> |
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[32] | 372 | void EKF<sq_T>::set_parameters ( diffbifn* pfxu0, diffbifn* phxu0,const sq_T Q0,const sq_T R0 ) { |
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| 373 | pfxu = pfxu0; |
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| 374 | phxu = phxu0; |
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[7] | 375 | |
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[32] | 376 | //initialize matrices A C, later, these will be only updated! |
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[83] | 377 | pfxu->dfdx_cond ( _mu,zeros ( dimu ),A,true ); |
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[33] | 378 | // pfxu->dfdu_cond ( *_mu,zeros ( dimu ),B,true ); |
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| 379 | B.clear(); |
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[83] | 380 | phxu->dfdx_cond ( _mu,zeros ( dimu ),C,true ); |
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[33] | 381 | // phxu->dfdu_cond ( *_mu,zeros ( dimu ),D,true ); |
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| 382 | D.clear(); |
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[22] | 383 | |
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[32] | 384 | R = R0; |
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| 385 | Q = Q0; |
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[22] | 386 | } |
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| 387 | |
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| 388 | template<class sq_T> |
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[32] | 389 | void EKF<sq_T>::bayes ( const vec &dt ) { |
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| 390 | it_assert_debug ( dt.length() == ( dimy+dimu ),"KalmanFull::bayes wrong size of dt" ); |
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[22] | 391 | |
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[283] | 392 | sq_T iRy ( dimy,dimy ); |
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[32] | 393 | vec u = dt.get ( dimy,dimy+dimu-1 ); |
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| 394 | vec y = dt.get ( 0,dimy-1 ); |
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[22] | 395 | //Time update |
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[83] | 396 | _mu = pfxu->eval ( _mu, u ); |
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| 397 | pfxu->dfdx_cond ( _mu,u,A,false ); //update A by a derivative of fx |
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[32] | 398 | |
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[22] | 399 | //P = A*P*A.transpose() + Q; in sq_T |
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[83] | 400 | _P.mult_sym ( A ); |
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| 401 | _P +=Q; |
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[22] | 402 | |
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| 403 | //Data update |
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[83] | 404 | phxu->dfdx_cond ( _mu,u,C,false ); //update C by a derivative hx |
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[22] | 405 | //_Ry = C*P*C.transpose() + R; in sq_T |
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[83] | 406 | _P.mult_sym ( C, _Ry ); |
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| 407 | ( _Ry ) +=R; |
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[22] | 408 | |
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[83] | 409 | mat Pfull = _P.to_mat(); |
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[32] | 410 | |
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[83] | 411 | _Ry.inv ( iRy ); // result is in _iRy; |
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| 412 | _K = Pfull*C.transpose() * ( iRy.to_mat() ); |
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[32] | 413 | |
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| 414 | sq_T pom ( ( int ) Pfull.rows() ); |
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[83] | 415 | iRy.mult_sym_t ( C*Pfull,pom ); |
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[283] | 416 | ( _P ) -= pom; // P = P -PC'iRy*CP; |
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[83] | 417 | _yp = phxu->eval ( _mu,u ); //y prediction |
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| 418 | ( _mu ) += _K* ( y-_yp ); |
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[32] | 419 | |
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[211] | 420 | if ( evalll==true ) {ll+=fy.evallog ( y );} |
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[22] | 421 | }; |
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| 422 | |
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| 423 | |
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[254] | 424 | } |
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[7] | 425 | #endif // KF_H |
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| 426 | |
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