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## Copyright (C) 2009-2016 Lukas F. Reichlin
##
## This file is part of LTI Syncope.
##
## LTI Syncope is free software: you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## LTI Syncope is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with LTI Syncope. If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{Q}, @var{R})
## @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{Q}, @var{R}, @var{S})
## @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{Q}, @var{R}, @var{[]}, @var{sensors}, @var{known})
## @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{Q}, @var{R}, @var{S}, @var{sensors}, @var{known})
## @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{Q}, @var{R}, @var{[]}, @var{sensors}, @var{known}, @var{type})
## @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{Q}, @var{R}, @var{S}, @var{sensors}, @var{known}, @var{type})
## Design Kalman estimator for @acronym{LTI} systems.
##
## @strong{Inputs}
## @table @var
## @item sys
## Nominal plant model.
## @item q
## Covariance of white process noise.
## @item r
## Covariance of white measurement noise.
## @item s
## Optional cross term covariance. Default value is 0.
## @item sensors
## Indices of measured output signals y from @var{sys}. If omitted or empty, all outputs are measured.
## @item known
## Indices of known input signals u (deterministic) to @var{sys}. All other inputs to @var{sys}
## are assumed stochastic. If argument @var{known} is omitted or empty, the first m-l inputs to @var{sys}
## are known, where m is the total number of inputs to @var{sys} and l is the size of the quadratic
## matrix @var{Q}.
## @item type
## Type of the estimator for discrete-time systems. If set to 'delayed' the current
## estimation is based on y(k-1), if set to 'current' the current estimation is
## based on the lates mesaruement y(k). If omitted, the 'delayed' version is created.
## @end table
##
## @strong{Outputs}
## @table @var
## @item est
## State-space model of the Kalman estimator.
## @item g
## Estimator gain.
## @item x
## Solution of the Riccati equation.
## @end table
##
## @strong{Block Diagram}
## @example
## @group
## u +-------+ ^
## +---------------------------->| |-------> y
## | +-------+ + y | est | ^
## u ----+--->| |----->(+)------>| |-------> x
## | sys | ^ + +-------+
## w -------->| | |
## +-------+ | v
##
## Q = cov (w, w') R = cov (v, v') S = cov (w, v')
## @end group
## @end example
##
## @seealso{care, dare, estim, lqr}
## @end deftypefn
## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
## Created: November 2009
## Version: 0.3
function [est, K, X] = kalman (sys, Q, R, varargin)
if (nargin < 3 || nargin > 7 || ! isa (sys, "lti"))
print_usage ();
endif
## System in state space
[A, B, C, D, E] = dssdata (sys, []);
## Optional parameters:
## The new default for known inputs (deterministic) are not
## backward compatible because previously, all inputs are
## assumed to be stochastic if is empty. However, this would
## result in an error if the number if the number of stochastic
## inputs would not match the dimension of Q. For all valid cases
## the new version is compatibel to the old one.
S = [];
sensors = 1 : rows (C);
deterministic = 1 : columns (B) - size (Q,1); # first inputs deterministic
type = 'delayed';
varidxoff = 3; # offset no. of variable and fixed input arguments
argidx = varidxoff; # index of last input argument
if (nargin > argidx++)
S = varargin{argidx-varidxoff};
if (nargin > argidx++)
if (! isempty (varargin{argidx-varidxoff}))
sensors = varargin{argidx-varidxoff};
endif
if (nargin > argidx++)
if (! isempty (varargin{argidx-varidxoff}))
deterministic = varargin{argidx-varidxoff};
endif
if (nargin > argidx++)
if (isct (sys))
warning ("kalman: ignoring 'type' parameter for continuous-time estimator\n");
else
type = varargin{argidx-varidxoff};
endif
endif
endif
endif
endif
m = columns (B);
m_d = length (deterministic);
m_s = size (Q,1);
p = length (sensors);
p_s = size (R,1);
## plausibility check for parameters
if (! issquare (Q))
error ("kalman: second argument Q must be square\n");
endif
if (! issquare (R))
error ("kalman: third argument R must be square\n");
endif
if (m_s != m - m_d)
error ("kalman: number of stochastic inputs (%d) does not match size %d of Q\n",...
m - m_d, m_s);
endif
if (p_s != p)
error ("kalman: size %d of measurment noise does not match size %d of R\n",...
p, p_s);
endif
if ((! isempty (S)) && (size (S) != [m_s,p_s]))
error ("kalman: size [%d,%d] of S does not match size %d of Q and %d of R\n",...
size(S,1), size(S,2), m_s, p_s);
endif
## matrices for Kalman filter design
stochastic = setdiff (1 : columns (B), deterministic);
C = C(sensors, :);
G = B(:, stochastic);
H = D(sensors, stochastic);
if (isempty (S))
Rbar = R + H*Q*H.';
Sbar = G * Q*H.';
else
Rbar = R + H*Q*H.'+ H*S + S.'*H.';
Sbar = G * (Q*H.' + S);
endif
if (isct (sys))
[X, L, K] = care (A.', C.', G*Q*G.', Rbar, Sbar, E.');
else
[X, L, K] = dare (A.', C.', G*Q*G.', Rbar, Sbar, E.');
endif
K = K.';
est = estim (sys, K, sensors, deterministic, type);
endfunction
%!test
%! sys = ss (-2, 1, 1, 3);
%! [est, g, x] = kalman (sys, 1, 1, 1);
%! [a, b, c, d] = ssdata (est);
%! m = [a, b; c, d];
%! m_exp = [-2.25, 0.25; 1, 0; 1, 0];
%! g_exp = 0.25;
%! x_exp = 0;
%! assert (m, m_exp, 1e-2);
%! assert (g, g_exp, 1e-2);
%! assert (x, x_exp, 1e-2);
%!shared n, nw, A, B, C, D, Bw, Dw, sys, Q, R, N, Pinf, K
%! n = 3;
%! nw = 2;
%!
%! A = [1.1269 -0.4940 0.1129
%! 1.0000 0 0
%! 0 1.0000 0];
%! B = [-0.3832
%! 0.5919
%! 0.5191];
%! Bw = rand(n,nw);
%! B = [B Bw];
%! C = [1 0 0];
%! D = [1];
%! Dw = zeros(1,nw);
%! D = [D Dw];
%! sys = ss(A,B,C,D,1);
%! Q = eye(nw,nw);
%! R = 1;
%! N = [];
%! PP = eye(3,3);
%! # asymptotic filter equations
%! for i = 1:10
%! Pinf = A*PP*A' + Bw*Q*Bw';
%! K = Pinf*C'*inv(C*Pinf*C'+R);
%! PP = (eye(3,3)-K*C)*Pinf;
%! endfor;
%!test
%! [kalmf,L,P] = kalman(sys,Q,R,N,1,1);
%! assert (Pinf, P, 4e-4);
%! assert (inv(A)*L, K, 2e-4);
%! assert (kalmf.a, A-L*C, 1e-4);
%!test
%! [kalmfc,Lc,Pc] = kalman(sys,Q,R,N,1,1,'current');
%! assert (Pinf, Pc, 4e-4);
%! assert (inv(A)*Lc, K, 2e-4);
%! assert (kalmfc.a, A-A*K*C, 1e-4);
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