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1 ## Copyright (C) 1993, 1994, 1995, 2000, 2002, 2003, 2005, 2007 |
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2 ## Auburn University |
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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} {[@var{k}, @var{p}, @var{e}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{z}) |
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22 ## Construct the linear quadratic regulator for the discrete time system |
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23 ## @iftex |
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24 ## @tex |
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25 ## $$ |
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26 ## x_{k+1} = A x_k + B u_k |
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27 ## $$ |
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28 ## @end tex |
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29 ## @end iftex |
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30 ## @ifinfo |
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31 ## |
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32 ## @example |
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33 ## x[k+1] = A x[k] + B u[k] |
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34 ## @end example |
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35 ## |
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36 ## @end ifinfo |
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37 ## to minimize the cost functional |
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38 ## @iftex |
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39 ## @tex |
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40 ## $$ |
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41 ## J = \sum x^T Q x + u^T R u |
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42 ## $$ |
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43 ## @end tex |
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44 ## @end iftex |
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45 ## @ifinfo |
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46 ## |
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47 ## @example |
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48 ## J = Sum (x' Q x + u' R u) |
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49 ## @end example |
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50 ## @end ifinfo |
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51 ## |
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52 ## @noindent |
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53 ## @var{z} omitted or |
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54 ## @iftex |
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55 ## @tex |
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56 ## $$ |
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57 ## J = \sum x^T Q x + u^T R u + 2 x^T Z u |
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58 ## $$ |
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59 ## @end tex |
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60 ## @end iftex |
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61 ## @ifinfo |
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62 ## |
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63 ## @example |
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64 ## J = Sum (x' Q x + u' R u + 2 x' Z u) |
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65 ## @end example |
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66 ## |
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67 ## @end ifinfo |
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68 ## @var{z} included. |
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69 ## |
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70 ## The following values are returned: |
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71 ## |
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72 ## @table @var |
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73 ## @item k |
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74 ## The state feedback gain, |
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75 ## @iftex |
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76 ## @tex |
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77 ## $(A - B K)$ |
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78 ## @end tex |
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79 ## @end iftex |
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80 ## @ifinfo |
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81 ## (@var{a} - @var{b}@var{k}) |
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82 ## @end ifinfo |
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83 ## is stable. |
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84 ## |
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85 ## @item p |
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86 ## The solution of algebraic Riccati equation. |
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87 ## |
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88 ## @item e |
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89 ## The closed loop poles of |
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90 ## @iftex |
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91 ## @tex |
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92 ## $(A - B K)$. |
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93 ## @end tex |
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94 ## @end iftex |
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95 ## @ifinfo |
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96 ## (@var{a} - @var{b}@var{k}). |
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97 ## @end ifinfo |
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98 ## @end table |
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99 ## @end deftypefn |
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100 |
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101 ## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu> |
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102 ## Created: August 1993 |
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103 ## Converted to discrete time by R. B. Tenison |
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104 ## (btenison@eng.auburn.edu) October 1993 |
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105 |
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106 function [k, p, e] = dlqr (a, b, q, r, s) |
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107 |
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108 if (nargin != 4 && nargin != 5) |
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109 error ("dlqr: invalid number of arguments"); |
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110 endif |
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111 |
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112 ## Dimension check is done inside dare.m |
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113 [n,m] = size(b); |
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114 |
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115 ## Check if s is there. |
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116 if (nargin == 5) |
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117 [n1, m1] = size (s); |
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118 if (n1 != n || m1 != m) |
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119 error ("dlqr: z must be identically dimensioned with b"); |
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120 endif |
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121 |
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122 ## Incorporate cross term into a and q. |
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123 ao = a - (b/r)*s'; |
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124 qo = q - (s/r)*s'; |
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125 else |
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126 s = zeros (n, m); |
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127 ao = a; |
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128 qo = q; |
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129 endif |
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130 |
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131 ## Checking stabilizability and detectability (dimensions are checked |
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132 ## inside these calls). |
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133 tol = 200*eps; |
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134 if (is_stabilizable (ao, b, tol, 1) == 0) |
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135 error ("dlqr: (a,b) not stabilizable"); |
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136 endif |
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137 dflag = is_detectable (ao, qo, tol, 1); |
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138 if (dflag == 0) |
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139 warning ("dlqr: (a,q) not detectable"); |
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140 elseif (dflag == -1) |
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141 error ("dlqr: (a,q) has non minimal modes near unit circle"); |
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142 end |
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143 |
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144 ## Compute the Riccati solution |
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145 p = dare (ao, b, qo, r); |
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146 k = (r+b'*p*b)\(b'*p*a + s'); |
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147 e = eig (a - b*k); |
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148 |
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149 endfunction |
