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1 ## Copyright (C) 1996, 1997 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 |
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7 ## the Free Software Foundation; either version 2, or (at your option) |
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8 ## any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License 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, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {@var{x} =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{opt}) |
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22 ## |
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23 ## Return the solution, @var{x} of the discrete-time algebraic Riccati |
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24 ## equation |
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25 ## @iftex |
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26 ## @tex |
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27 ## $$ |
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28 ## A^TXA - X + A^TXB (R + B^TXB)^{-1} B^TXA + Q = 0 |
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29 ## $$ |
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30 ## @end tex |
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31 ## @end iftex |
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32 ## @ifinfo |
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33 ## @example |
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34 ## a' x a - x + a' x b (r + b' x b)^(-1) b' x a + q = 0 |
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35 ## @end example |
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36 ## @end ifinfo |
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37 ## @noindent |
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38 ## |
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39 ## @strong{Inputs} |
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40 ## @table @var |
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41 ## @item a |
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42 ## @var{n} by @var{n} matrix; |
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43 ## |
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44 ## @item b |
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45 ## @var{n} by @var{m} matrix; |
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46 ## |
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47 ## @item q |
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48 ## @var{n} by @var{n} matrix, symmetric positive semidefinite, or a @var{p} by @var{n} matrix, |
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49 ## In the latter case @math{q:=q'*q} is used; |
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50 ## |
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51 ## @item r |
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52 ## @var{m} by @var{m}, symmetric positive definite (invertible); |
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53 ## |
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54 ## @item opt |
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55 ## (optional argument; default = @code{"B"}): |
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56 ## String option passed to @code{balance} prior to ordered @var{QZ} decomposition. |
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57 ## @end table |
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58 ## |
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59 ## @strong{Output} |
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60 ## @table @var |
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61 ## @item x |
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62 ## solution of @acronym{DARE}. |
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63 ## @end table |
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64 ## |
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65 ## @strong{Method} |
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66 ## Generalized eigenvalue approach (Van Dooren; @acronym{SIAM} J. |
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67 ## Sci. Stat. Comput., Vol 2) applied to the appropriate symplectic pencil. |
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68 ## |
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69 ## See also: Ran and Rodman, @cite{Stable Hermitian Solutions of Discrete |
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70 ## Algebraic Riccati Equations}, Mathematics of Control, Signals and |
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71 ## Systems, Vol 5, no 2 (1992), pp 165--194. |
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72 ## @seealso{balance, are} |
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73 ## @end deftypefn |
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74 |
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75 ## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu> |
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76 ## Created: August 1993 |
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77 ## Adapted-By: jwe |
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78 |
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79 function x = dare (a, b, q, r, opt) |
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80 |
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81 if (nargin == 4 | nargin == 5) |
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82 if (nargin == 5) |
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83 if (opt != "N" || opt != "P" || opt != "S" || opt != "B") |
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84 warning ("dare: opt has an invalid value -- setting to B"); |
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85 opt = "B"; |
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86 endif |
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87 else |
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88 opt = "B"; |
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89 endif |
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90 |
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91 |
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92 if ((p = issquare (q)) == 0) |
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93 q = q'*q; |
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94 endif |
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95 |
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96 ##Checking positive definiteness |
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97 if (isdefinite(r)<=0) |
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98 error("dare: r not positive definite"); |
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99 end |
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100 if (isdefinite(q)<0) |
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101 error("dare: q not positive semidefinite"); |
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102 end |
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103 |
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104 |
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105 ## Check r dimensions. |
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106 [n,m] = size(b); |
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107 if ((m1 = issquare (r)) == 0) |
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108 error ("dare: r is not square"); |
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109 elseif (m1 != m) |
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110 error ("b,r are not conformable"); |
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111 endif |
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112 |
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113 s1 = [a, zeros(n) ; -q, eye(n)]; |
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114 s2 = [eye(n), (b/r)*b' ; zeros(n), a']; |
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115 [c,d,s1,s2] = balance(s1,s2,opt); |
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116 [aa,bb,u,lam] = qz(s1,s2,"S"); |
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117 u = d*u; |
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118 n1 = n+1; |
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119 n2 = 2*n; |
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120 x = u (n1:n2, 1:n)/u(1:n, 1:n); |
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121 else |
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122 usage ("x = dare (a, b, q, r {,opt})"); |
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123 endif |
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124 |
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125 endfunction |
