# Copyright (C) 1993 John W. Eaton
# 
# This file is part of Octave.
# 
# Octave 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 2, or (at your option) any
# later version.
# 
# Octave is distributed in the hope that it will be useful, but WITHOUT
# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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# for more details.
# 
# You should have received a copy of the GNU General Public License
# along with Octave; see the file COPYING.  If not, write to the Free
# Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.

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

# Usage: [l, m, p, e] = dlqe (A, G, C, SigW, SigV {,Z})
#
# Linear quadratic estimator (Kalman filter) design for the 
# discrete time system
#
#  x[k+1] = A x[k] + B u[k] + G w[k]
#    y[k] = C x[k] + D u[k] + w[k]
#
# where w, v are zero-mean gaussian noise processes with respective
# intensities SigW = cov (w, w) and SigV = cov (v, v).
#
# Z (if specified) is cov(w,v); otherwise cov(w,v) = 0.
#
# Observer structure is 
#     z[k+1] = A z[k] + B u[k] + k(y[k] - C z[k] - D u[k]).
#
# Returns:
#
#   l = observer gain, (A - A L C) is stable
#   m = Ricatti equation solution
#   p = the estimate error covariance after the measurement update
#   e = closed loop poles of (A - A L C)

# Written by A. S. Hodel (scotte@eng.auburn.edu) August, 1993.
# Modified for discrete time by R. Bruce Tenison (btenison@eng.auburn.edu)
# October, 1993

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

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

  if (nargin == 5)
    [k, p, e] = dlqr (a', c', g*sigw*g', sigv);
    m = p';
    l = (m*c')/(c*m*c'+sigv);
  else
    [k, p, e] = dlqr (a', c', g*sigw*g', sigv, g*zz);
    m = p';
    l = (m*c'+a\g*t)/(c*m*c'+sigv);
    a = a-g*t/sigv*c;
    sigw = sigw-t/sigv*t';
  endif

  p = a\(m-g*sigw*g')/a';

endfunction