Add a useful package (from Source forge) for octave

[CreaPhase.git] / octave_packages / control-2.3.52 / dlqe.m

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+## Copyright (C) 2012   Lukas F. Reichlin
+## Copyright (C) 2012   Megan Zagrobelny
+##
+## 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{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r})
+## @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}, @var{s})
+## @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r})
+## @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}, @var{s})
+## Kalman filter for discrete-time systems.
+##
+## @example
+## @group
+## x[k] = Ax[k] + Bu[k] + Gw[k]   (State equation)
+## y[k] = Cx[k] + Du[k] + v[k]    (Measurement Equation)
+## E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S
+## @end group
+## @end example
+##
+## @strong{Inputs}
+## @table @var
+## @item a
+## State transition matrix of discrete-time system (n-by-n).
+## @item g
+## Process noise matrix of discrete-time system (n-by-g).
+## If @var{g} is empty @code{[]}, an identity matrix is assumed.
+## @item c
+## Measurement matrix of discrete-time system (p-by-n).
+## @item q
+## Process noise covariance matrix (g-by-g).
+## @item r
+## Measurement noise covariance matrix (p-by-p).
+## @item s
+## Optional cross term covariance matrix (g-by-p), s = cov(w,v).
+## If @var{s} is empty @code{[]} or not specified, a zero matrix is assumed.
+## @end table
+##
+## @strong{Outputs}
+## @table @var
+## @item m
+## Kalman filter gain matrix (n-by-p).
+## @item p
+## Unique stabilizing solution of the discrete-time Riccati equation (n-by-n).
+## Symmetric matrix.
+## @item z
+## Error covariance (n-by-n), cov(x(k|k)-x)
+## @item e
+## Closed-loop poles (n-by-1).
+## @end table
+##
+## @strong{Equations}
+## @example
+## @group
+## x[k|k] = x[k|k-1] + M(y[k] - Cx[k|k-1] - Du[k])
+##
+## x[k+1|k] = Ax[k|k] + Bu[k] for S=0
+##
+## x[k+1|k] = Ax[k|k] + Bu[k] + G*S*(C*P*C' + R)^-1*(y[k] - C*x[k|k-1]) for non-zero S
+##          
+##
+## E = eig(A - A*M*C) for S=0
+## 
+## E = eig(A - A*M*C - G*S*(C*P*C' + Rv)^-1*C) for non-zero S
+##  
+## @end group
+## @end example
+## @seealso{dare, care, dlqr, lqr, lqe}
+## @end deftypefn
+
+## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
+## Created: April 2012
+## Version: 0.1
+
+function [m, p, z, e] = dlqe (a, g, c, q, r, s = [])
+
+  if (nargin < 5 || nargin > 6)
+    print_usage ();
+  endif
+
+  if (isempty (g))
+    [p, e] = dare (a.', c.', q, r, s);   # dlqe (a, [], c, q, r, s), g=I
+  elseif (columns (g) != rows (q) || ! issquare (q))
+    error ("dlqe: matrices g(%dx%d) and q(%dx%d) have incompatible dimensions", \
+            rows (g), columns (g), rows (q), columns (q));
+  elseif (isempty (s))
+    [p, e] = dare (a.', c.', g*q*g.', r);
+  elseif (columns (g) != rows (s))
+    error ("dlqe: matrices g(%dx%d) and s(%dx%d) have incompatible dimensions", \
+            rows (g), columns (g), rows (s), columns (s));
+  else
+    [p, e] = dare (a.', c.', g*q*g.', r, g*s);
+  endif
+    
+  m = p*c.' / (c*p*c.' + r);
+
+  z = p - m*c*p;
+  z = (z + z.') / 2;
+
+endfunction