1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159
|
## Copyright (C) 1993, 1994, 1995, 2000, 2002, 2004, 2005, 2007
## Auburn University. All rights reserved.
##
##
## 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} {} dlyap (@var{a}, @var{b})
## Solve the discrete-time Lyapunov equation
##
## @strong{Inputs}
## @table @var
## @item a
## @var{n} by @var{n} matrix;
## @item b
## Matrix: @var{n} by @var{n}, @var{n} by @var{m}, or @var{p} by @var{n}.
## @end table
##
## @strong{Output}
## @table @var
## @item x
## matrix satisfying appropriate discrete time Lyapunov equation.
## @end table
##
## Options:
## @itemize @bullet
## @item @var{b} is square: solve
## @iftex
## @tex
## $$ axa^T - x + b = 0 $$
## @end tex
## @end iftex
## @ifinfo
## @code{a x a' - x + b = 0}
## @end ifinfo
## @item @var{b} is not square: @var{x} satisfies either
## @iftex
## @tex
## $$ axa^T - x + bb^T = 0 $$
## @end tex
## @end iftex
## @ifinfo
## @example
## a x a' - x + b b' = 0
## @end example
## @end ifinfo
## @noindent
## or
## @iftex
## @tex
## $$ a^Txa - x + b^Tb = 0, $$
## @end tex
## @end iftex
## @ifinfo
## @example
## a' x a - x + b' b = 0,
## @end example
## @end ifinfo
## @noindent
## whichever is appropriate.
## @end itemize
##
## @strong{Method}
## Uses Schur decomposition method as in Kitagawa,
## @cite{An Algorithm for Solving the Matrix Equation @math{X = F X F' + S}},
## International Journal of Control, Volume 25, Number 5, pages 745--753
## (1977).
##
## Column-by-column solution method as suggested in
## Hammarling, @cite{Numerical Solution of the Stable, Non-Negative
## Definite Lyapunov Equation}, @acronym{IMA} Journal of Numerical Analysis, Volume
## 2, pages 303--323 (1982).
## @end deftypefn
## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
## Created: August 1993
function x = dlyap (a, b)
if (nargin != 2)
print_usage ();
endif
if ((n = issquare (a)) == 0)
warning ("dlyap: a must be square");
endif
if ((m = issquare (b)) == 0)
[n1, m] = size (b);
if (n1 == n)
b = b*b';
m = n1;
else
b = b'*b;
a = a';
endif
endif
if (n != m)
warning ("dlyap: a,b not conformably dimensioned");
endif
## Solve the equation column by column.
[u, s] = schur (a);
b = u'*b*u;
j = n;
while (j > 0)
j1 = j;
## Check for Schur block.
if (j == 1)
blksiz = 1;
elseif (s (j, j-1) != 0)
blksiz = 2;
j = j - 1;
else
blksiz = 1;
endif
Ajj = kron (s(j:j1,j:j1), s) - eye (blksiz*n);
rhs = reshape (b (:,j:j1), blksiz*n, 1);
if (j1 < n)
rhs2 = s*(x(:,(j1+1):n) * s(j:j1,(j1+1):n)');
rhs = rhs + reshape (rhs2, blksiz*n, 1);
endif
v = - Ajj\rhs;
x(:,j) = v (1:n);
if (blksiz == 2)
x (:, j1) = v ((n+1):blksiz*n);
endif
j = j - 1;
endwhile
## Back-transform to original coordinates.
x = u*x*u';
endfunction
|