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