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