Path: Common/Estimation

% Implement an Unscented Kalman Filter.
 The filter uses numerical integration to propagate the state.
 The filter propagates sigma points, points computed from the
 state plus a function of the covariance matrix. For each state
 there are two sigma states.

 The default integrator is RK4, or you cab specify an integrator
 with the field

 d.integrator = @myIntegratorName

 To initialize UKF call

 d = UKF( 'initialize', d );

 To update
 d = UKF( 'update', d, y );

 The filter appends internal data to the data structure.
 d.x gives the current estimated state and d.pXX the state covariance.

 The measurement function is of the form 

 meas = MeasFun( x, dMeasFun );

 The state function is of the form

 xDot = StateFun( x, t, dStateFun );

 You can use either the Cholesky transform or matrix square root to
 form the sigma points.

   d = UKF( action, d, y )
   d = UKF( 'datastructure', nStates, nMeas )

   action    (1,:)  'initialize' or 'update' or 'datastructure'
   d          (.)  UKF data structure
                    .rHSFun        (1,:) Name or handle of RHS function
                    .integrator    (1,:) Handle of integrator function
                    .rHSFunData	  (.)   RHS function data structure
                    .measFun       (1,:) Handle of measurement function
                    .measFunData	  (.)   Measurement function data 
                    .x             (n,1) Initial state vector
                    .p             (n,n) Covariance matrix for x   
                    .pXX           (n,n) Covariance after state update
                    .dY            (1,1) Number of measurements        
                    .dX            (1,1) Number of states        
                    .rP            (n,n) Process noise covariance        
                    .rM            (n,n) Measurement noise covariance
                    .alpha         (1,1) Scaling 1e-4 <= alpha <= 1
                    .kappa         (1,1) Secondary scaling usually 0
                    .beta          (1,1) Prior knowledge of distribution = 2
                                         for Gaussian distributed noise
                    .dT            (1,1) Time step
                    .t             (1,1) Time
                    .sigmaPtAlg    (1,1) 1 = sqrtm, 2 = chol
   y         (m,1)  Measurement vector
   verbose   (1,1)  Flag for verbose output. 1 = warn, 2 = error.

   d          (.)   UKF data structure with the following appended
                    .dX          (1,1) Number of states
                    .n           (1,1) Number of sigma points
                    .y           (m,1) Measurements based on the states
                    .xA          {n}   Sigma points
                    .pP          (n,n) Covariance matrix prior to
                    .xP          (n,1) State vector prior to measurement
                    .res         (:,1) Residuals
   References: Voss, H. U., Timmer, J., Kurths, J., "Nonlinear Dynamical 
               System Identification from Uncertain and Indirect 
               Measurements,"  International Journal of Bifurcation 
               and Chaos, Vol. 14, No. 6, 2005, pp. 1905-1933.
               Van der Merwe, R. and Wan, E., "Sigma-Point Kalman Filters 
               for Probabilistic Inference in Dynamic State-Space Models",


Common: DemoFuns/GXUKF
Common: DemoFuns/RHSUKF
Common: Estimation/SigmaPointsChol
Math: Integration/RK4

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