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comparison scripts/control/base/dlqr.m @ 3431:99ab64f4a09d

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[project @ 2000-01-14 03:53:03 by jwe]
author jwe
date Fri, 14 Jan 2000 04:12:41 +0000
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3430:65b3519ac3a1 3431:99ab64f4a09d
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 -*-
20 ## @deftypefn {Function File} {[@var{k}, @var{p}, @var{e}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{z})
21 ## Construct the linear quadratic regulator for the discrete time system
22 ## @iftex
23 ## @tex
24 ## $$
25 ## x_{k+1} = A x_k + B u_k
26 ## $$
27 ## @end tex
28 ## @end iftex
29 ## @ifinfo
30 ##
31 ## @example
32 ## x[k+1] = A x[k] + B u[k]
33 ## @end example
34 ##
35 ## @end ifinfo
36 ## to minimize the cost functional
37 ## @iftex
38 ## @tex
39 ## $$
40 ## J = \sum x^T Q x + u^T R u
41 ## $$
42 ## @end tex
43 ## @end iftex
44 ## @ifinfo
45 ##
46 ## @example
47 ## J = Sum (x' Q x + u' R u)
48 ## @end example
49 ## @end ifinfo
50 ##
51 ## @noindent
52 ## @var{z} omitted or
53 ## @iftex
54 ## @tex
55 ## $$
56 ## J = \sum x^T Q x + u^T R u + 2 x^T Z u
57 ## $$
58 ## @end tex
59 ## @end iftex
60 ## @ifinfo
61 ##
62 ## @example
63 ## J = Sum (x' Q x + u' R u + 2 x' Z u)
64 ## @end example
65 ##
66 ## @end ifinfo
67 ## @var{z} included.
68 ##
69 ## The following values are returned:
70 ##
71 ## @table @var
72 ## @item k
73 ## The state feedback gain,
74 ## @iftex
75 ## @tex
76 ## $(A - B K)$
77 ## @end tex
78 ## @end iftex
79 ## @ifinfo
80 ## (@var{a} - @var{b}@var{k})
81 ## @end ifinfo
82 ## is stable.
83 ##
84 ## @item p
85 ## The solution of algebraic Riccati equation.
86 ##
87 ## @item e
88 ## The closed loop poles of
89 ## @iftex
90 ## @tex
91 ## $(A - B K)$.
92 ## @end tex
93 ## @end iftex
94 ## @ifinfo
95 ## (@var{a} - @var{b}@var{k}).
96 ## @end ifinfo
97 ## @end table
98 ## @strong{References}
99 ## @enumerate
100 ## @item Anderson and Moore, Optimal Control: Linear Quadratic Methods,
101 ## Prentice-Hall, 1990, pp. 56-58
102 ## @item Kuo, Digital Control Systems, Harcourt Brace Jovanovich, 1992,
103 ## section 11-5-2.
104 ## @end enumerate
105 ## @end deftypefn
106
107 ## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
108 ## Created: August 1993
109 ## Converted to discrete time by R. B. Tenison
110 ## (btenison@eng.auburn.edu) October 1993
111
112 function [k, p, e] = dlqr (a, b, q, r, s)
113
114 if (nargin != 4 && nargin != 5)
115 error ("dlqr: invalid number of arguments");
116 endif
117
118 ## Check a.
119 if ((n = is_square (a)) == 0)
120 error ("dlqr: requires 1st parameter(a) to be square");
121 endif
122
123 ## Check b.
124 [n1, m] = size (b);
125 if (n1 != n)
126 error ("dlqr: a,b not conformal");
127 endif
128
129 ## Check q.
130 if ((n1 = is_square (q)) == 0 || n1 != n)
131 error ("dlqr: q must be square and conformal with a");
132 endif
133
134 ## Check r.
135 if((m1 = is_square(r)) == 0 || m1 != m)
136 error ("dlqr: r must be square and conformal with column dimension of b");
137 endif
138
139 ## Check if n is there.
140 if (nargin == 5)
141 [n1, m1] = size (s);
142 if (n1 != n || m1 != m)
143 error ("dlqr: z must be identically dimensioned with b");
144 endif
145
146 ## Incorporate cross term into a and q.
147
148 ao = a - (b/r)*s';
149 qo = q - (s/r)*s';
150 else
151 s = zeros (n, m);
152 ao = a;
153 qo = q;
154 endif
155
156 ## Check that q, (r) are symmetric, positive (semi)definite
157 if (is_symmetric (q) && is_symmetric (r) ...
158 && all (eig (q) >= 0) && all (eig (r) > 0))
159 p = dare (ao, b, qo, r);
160 k = (r+b'*p*b)\b'*p*a + r\s';
161 e = eig (a - b*k);
162 else
163 error ("dlqr: q (r) must be symmetric positive (semi) definite");
164 endif
165
166 endfunction