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dlqe.m

## Copyright (C) 1993, 1994, 1995, 2000, 2005, 2007 Auburn University
##
##
## This program 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.
##
## This program 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 this program; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {[@var{l}, @var{m}, @var{p}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{sigw}, @var{sigv}, @var{z})
## Construct the linear quadratic estimator (Kalman filter) for the
## discrete time system
## @iftex
## @tex
## $$
##  x_{k+1} = A x_k + B u_k + G w_k
## $$
## $$
##  y_k = C x_k + D u_k + v_k
## $$
## @end tex
## @end iftex
## @ifinfo
##
## @example
## x[k+1] = A x[k] + B u[k] + G w[k]
##   y[k] = C x[k] + D u[k] + v[k]
## @end example
##
## @end ifinfo
## where @var{w}, @var{v} are zero-mean gaussian noise processes with
## respective intensities @code{@var{sigw} = cov (@var{w}, @var{w})} and
## @code{@var{sigv} = cov (@var{v}, @var{v})}.
##
## If specified, @var{z} is @code{cov (@var{w}, @var{v})}.  Otherwise
## @code{cov (@var{w}, @var{v}) = 0}.
##
## The observer structure is
## @iftex
## @tex
## $$
##  z_{k|k} = z_{k|k-1} + l (y_k - C z_{k|k-1} - D u_k)
## $$
## $$
##  z_{k+1|k} = A z_{k|k} + B u_k
## $$
## @end tex
## @end iftex
## @ifinfo
##
## @example
## z[k|k] = z[k|k-1] + L (y[k] - C z[k|k-1] - D u[k])
## z[k+1|k] = A z[k|k] + B u[k]
## @end example
## @end ifinfo
##
## @noindent
## The following values are returned:
##
## @table @var
## @item l
## The observer gain,
## @iftex
## @tex
## $(A - ALC)$.
## @end tex
## @end iftex
## @ifinfo
## (@var{a} - @var{a}@var{l}@var{c}).
## @end ifinfo
## is stable.
##
## @item m
## The Riccati equation solution.
##
## @item p
## The estimate error covariance after the measurement update.
##
## @item e
## The closed loop poles of
## @iftex
## @tex
## $(A - ALC)$.
## @end tex
## @end iftex
## @ifinfo
## (@var{a} - @var{a}@var{l}@var{c}).
## @end ifinfo
## @end table
## @end deftypefn

## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
## Created: August 1993
## Modified for discrete time by R. Bruce Tenison (btenison@eng.auburn.edu)
## October, 1993
## Modified by Gabriele Pannocchia <pannocchia@ing.unipi.it>
## July 2000

function [l, m, p, e] = dlqe (a, g, c, sigw, sigv, s)

  if (nargin != 5 && nargin != 6)
    error ("dlqe: invalid number of arguments");
  endif

  ## The problem is dual to the regulator design, so transform to dlqr call.

  if (nargin == 5)
    [k, m, e] = dlqr (a', c', g*sigw*g', sigv);
  else
    [k, m, e] = dlqr (a', c', g*sigw*g', sigv, g*s);
    warning ("dlqe: use dkalman when there is a cross-covariance term");
  endif

  l = m*c'/(c*m*c'+sigv);
  p = m - m*c'/(c*m*c'+sigv)*c*m;

endfunction


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