|
3431
|
1 ## Copyright (C) 1993, 1994, 1995 Auburn University. All rights reserved. |
|
|
2 ## |
|
|
3 ## This file is part of Octave. |
|
|
4 ## |
|
|
5 ## Octave is free software; you can redistribute it and/or modify it |
|
|
6 ## under the terms of the GNU General Public License as published by the |
|
|
7 ## Free Software Foundation; either version 2, or (at your option) any |
|
|
8 ## later version. |
|
|
9 ## |
|
|
10 ## Octave is distributed in the hope that it will be useful, but WITHOUT |
|
|
11 ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
|
|
12 ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
|
|
13 ## for more details. |
|
|
14 ## |
|
|
15 ## You should have received a copy of the GNU General Public License |
|
|
16 ## along with Octave; see the file COPYING. If not, write to the Free |
|
|
17 ## Software Foundation, 59 Temple Place, Suite 330, Boston, MA 02111 USA. |
|
|
18 |
|
|
19 ## -*- texinfo -*- |
|
3500
|
20 ## @deftypefn {Function File} {} dlyap (@var{a}, @var{b}) |
|
3431
|
21 ## Solve the discrete-time Lyapunov equation |
|
|
22 ## |
|
|
23 ## @strong{Inputs} |
|
|
24 ## @table @var |
|
|
25 ## @item a |
|
|
26 ## @var{n} by @var{n} matrix |
|
|
27 ## @item b |
|
|
28 ## Matrix: @var{n} by @var{n}, @var{n} by @var{m}, or @var{p} by @var{n}. |
|
|
29 ## @end table |
|
|
30 ## |
|
|
31 ## @strong{Outputs} |
|
|
32 ## @var{x}: matrix satisfying appropriate discrete time Lyapunov equation. |
|
|
33 ## Options: |
|
|
34 ## @itemize @bullet |
|
|
35 ## @item @var{b} is square: solve @code{a x a' - x + b = 0} |
|
|
36 ## @item @var{b} is not square: @var{x} satisfies either |
|
|
37 ## @example |
|
|
38 ## a x a' - x + b b' = 0 |
|
|
39 ## @end example |
|
|
40 ## @noindent |
|
|
41 ## or |
|
|
42 ## @example |
|
|
43 ## a' x a - x + b' b = 0, |
|
|
44 ## @end example |
|
|
45 ## @noindent |
|
|
46 ## whichever is appropriate. |
|
|
47 ## @end itemize |
|
|
48 ## |
|
|
49 ## @strong{Method} |
|
|
50 ## Uses Schur decomposition method as in Kitagawa, |
|
3502
|
51 ## @cite{An Algorithm for Solving the Matrix Equation @math{X = F X F' + S}}, |
|
3431
|
52 ## International Journal of Control, Volume 25, Number 5, pages 745--753 |
|
|
53 ## (1977). |
|
|
54 ## |
|
|
55 ## Column-by-column solution method as suggested in |
|
|
56 ## Hammarling, @cite{Numerical Solution of the Stable, Non-Negative |
|
|
57 ## Definite Lyapunov Equation}, IMA Journal of Numerical Analysis, Volume |
|
|
58 ## 2, pages 303--323 (1982). |
|
|
59 ## @end deftypefn |
|
|
60 |
|
|
61 ## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu> |
|
|
62 ## Created: August 1993 |
|
|
63 |
|
|
64 function x = dlyap (a, b) |
|
|
65 |
|
4030
|
66 if ((n = issquare (a)) == 0) |
|
3431
|
67 warning ("dlyap: a must be square"); |
|
|
68 endif |
|
|
69 |
|
4030
|
70 if ((m = issquare (b)) == 0) |
|
3431
|
71 [n1, m] = size (b); |
|
|
72 if (n1 == n) |
|
|
73 b = b*b'; |
|
|
74 m = n1; |
|
|
75 else |
|
|
76 b = b'*b; |
|
|
77 a = a'; |
|
|
78 endif |
|
|
79 endif |
|
|
80 |
|
|
81 if (n != m) |
|
|
82 warning ("dlyap: a,b not conformably dimensioned"); |
|
|
83 endif |
|
|
84 |
|
|
85 ## Solve the equation column by column. |
|
|
86 |
|
|
87 [u, s] = schur (a); |
|
|
88 b = u'*b*u; |
|
|
89 |
|
|
90 j = n; |
|
|
91 while (j > 0) |
|
|
92 j1 = j; |
|
|
93 |
|
|
94 ## Check for Schur block. |
|
|
95 |
|
|
96 if (j == 1) |
|
|
97 blksiz = 1; |
|
|
98 elseif (s (j, j-1) != 0) |
|
|
99 blksiz = 2; |
|
|
100 j = j - 1; |
|
|
101 else |
|
|
102 blksiz = 1; |
|
|
103 endif |
|
|
104 |
|
|
105 Ajj = kron (s (j:j1, j:j1), s) - eye (blksiz*n); |
|
|
106 |
|
|
107 rhs = reshape (b (:, j:j1), blksiz*n, 1); |
|
|
108 |
|
|
109 if (j1 < n) |
|
|
110 rhs2 = s*(x (:, (j1+1):n) * s (j:j1, (j1+1):n)'); |
|
|
111 rhs = rhs + reshape (rhs2, blksiz*n, 1); |
|
|
112 endif |
|
|
113 |
|
|
114 v = - Ajj\rhs; |
|
|
115 x (:, j) = v (1:n); |
|
|
116 |
|
|
117 if(blksiz == 2) |
|
|
118 x (:, j1) = v ((n+1):blksiz*n); |
|
|
119 endif |
|
|
120 |
|
|
121 j = j - 1; |
|
|
122 |
|
|
123 endwhile |
|
|
124 |
|
|
125 ## Back-transform to original coordinates. |
|
|
126 |
|
|
127 x = u*x*u'; |
|
|
128 |
|
|
129 endfunction |
