%URTS_SMOOTH1 Additive form Unscented Rauch-Tung-Striebel smoother
%
% Syntax:
% [M,P,D] = URTS_SMOOTH1(M,P,f,Q,[f_param,alpha,beta,kappa,mat,same_p])
%
% In:
% M - NxK matrix of K mean estimates from Unscented Kalman filter
% P - NxNxK matrix of K state covariances from Unscented Kalman Filter
% f - Dynamic model function as a matrix A defining
% linear function f(x) = A*x, inline function,
% function handle or name of function in
% form a(x,param) (optional, default eye())
% Q - NxN process noise covariance matrix or NxNxK matrix
% of K state process noise covariance matrices for each step.
% f_param - Parameters of f. Parameters should be a single cell array,
% vector or a matrix containing the same parameters for each
% step, or if different parameters are used on each step they
% must be a cell array of the format { param_1, param_2, ...},
% where param_x contains the parameters for step x as a cell array,
% a vector or a matrix. (optional, default empty)
% alpha - Transformation parameter (optional)
% beta - Transformation parameter (optional)
% kappa - Transformation parameter (optional)
% mat - If 1 uses matrix form (optional, default 0)
% same_p - If 1 uses the same parameters
% on every time step (optional, default 1)
%
% Out:
% M - Smoothed state mean sequence
% P - Smoothed state covariance sequence
% D - Smoother gain sequence
%
% Description:
% Unscented Rauch-Tung-Striebel smoother algorithm. Calculate
% "smoothed" sequence from given Kalman filter output sequence by
% conditioning all steps to all measurements.
%
% Example:
% m = m0;
% P = P0;
% MM = zeros(size(m,1),size(Y,2));
% PP = zeros(size(m,1),size(m,1),size(Y,2));
% for k=1:size(Y,2)
% [m,P] = ukf_predict1(m,P,a,Q);
% [m,P] = ukf_update1(m,P,Y(:,k),h,R);
% MM(:,k) = m;
% PP(:,:,k) = P;
% end
% [SM,SP] = urts_smooth(MM,PP,a,Q);
%
% See also:
% URTS_SMOOTH2, UKF_PREDICT1, UKF_UPDATE1, UKF_PREDICT2, UKF_UPDATE2,
% UKF_PREDICT3, UKF_UPDATE3, UT_TRANSFORM, UT_WEIGHTS, UT_MWEIGHTS,
% UT_SIGMAS
% Copyright (C) 2006 Simo S�rkk�
%
% $Id: urts_smooth1.m 764 2011-08-12 10:47:38Z asolin $
%
% This software is distributed under the GNU General Public
% Licence (version 2 or later); please refer to the file
% Licence.txt, included with the software, for details.
function [M,P,D] = urts_smooth1(M,P,f,Q,f_param,alpha,beta,kappa,mat,same_p)
%
% Check which arguments are there
%
if nargin < 4
error('Too few arguments');
end
if nargin < 5
f_param = [];
end
if nargin < 6
alpha = [];
end
if nargin < 7
beta = [];
end
if nargin < 8
kappa = [];
end
if nargin < 9
mat = [];
end
if nargin < 10
same_p = 1;
end
%
% Apply defaults
%
if isempty(f)
f = eye(size(M,1));
end
if isempty(Q)
Q = zeros(size(M,1));
end
if isempty(mat)
mat = 0;
end
%
% Extend Q if NxN matrix
%
if size(Q,3)==1
Q = repmat(Q,[1 1 size(M,2)]);
end
%
% Run the smoother
%
D = zeros(size(M,1),size(M,1),size(M,2));
for k=(size(M,2)-1):-1:1
if isempty(f_param)
params = [];
elseif same_p
params = f_param;
else
params = f_param{k};
end
tr_param = {alpha beta kappa mat};
[m_pred,P_pred,C] = ...
ut_transform(M(:,k),P(:,:,k),f,params,tr_param);
P_pred = P_pred + Q(:,:,k);
D(:,:,k) = C / P_pred;
M(:,k) = M(:,k) + D(:,:,k) * (M(:,k+1) - m_pred);
P(:,:,k) = P(:,:,k) + D(:,:,k) * (P(:,:,k+1) - P_pred) * D(:,:,k)';
end