1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
|
## Copyright (C) 1996, 2000, 2004, 2005, 2006, 2007
## Auburn University. All rights reserved.
##
##
## This program is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
## your option) any later version.
##
## This program is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{retsys}, @var{nc}, @var{no}] =} sysmin (@var{sys}, @var{flg})
## Returns a minimal (or reduced order) system
##
## @strong{Inputs}
## @table @var
## @item sys
## System data structure
## @item flg
## When equal to 0 (default value), returns minimal system,
## in which state names are lost; when equal to 1, returns system
## with physical states removed that are either uncontrollable or
## unobservable (cannot reduce further without discarding physical
## meaning of states).
## @end table
## @strong{Outputs}
## @table @var
## @item retsys
## Returned system.
## @item nc
## Number of controllable states in the returned system.
## @item no
## Number of observable states in the returned system.
## @item cflg
## @code{is_controllable(retsys)}.
## @item oflg
## @code{is_observable(retsys)}.
## @end table
## @end deftypefn
## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
function [retsys, nc, no, cflg, oflg] = sysmin (sys, flg)
switch (nargin)
case 1
flg = 0;
case 2
jnk = flg; # dummy operation
otherwise,
print_usage ();
endswitch
dflg = is_digital (sys, 2);
[n, nz, m, p] = sysdimensions (sys);
if (n*nz > 0)
# both continuous and discrete states
[aa, bb, cc, dd, tsam, n, nz, stnam, innam, outnam, yd] = sys2ss (sys);
crng = 1:n;
drng = n+(1:nz);
# get minimal realization of continuous part
Ac = aa(crng,crng);
Acd = aa(crng,drng);
Adc = aa(drng,crng);
Ad = aa(drng,drng);
Bc = bb(crng,:);
Bd = bb(drng,:);
Cc = cc(:,crng);
Cd = cc(:,drng);
cstnam = stnam(crng);
dstnam = stnam(drng);
cinnam = __sysconcat__ (innam, stnam(drng));
coutnam = __sysconcat__ (outnam, stnam(drng));
csys = ss (Ac, [Bc, Acd], [Cc; Adc]);
csys = syssetsignals (csys, "st", cstnam);
csys = syssetsignals (csys, "in", cinnam);
csys = syssetsignals (csys, "out", coutnam);
# reduce continuous system, recombine with discrete part
csys = sysmin (csys, flg);
cn = sysdimensions (csys);
if (cn == 0)
# continuous states are removed; just reduce the discrete part
sys = sysprune (sys, 1:p, 1:m, drng);
retsys = sysmin (sys, flg);
else
# extract updated parameters from reduced continuous system
[caa, cbb, ccc, cdd, ctsam, cn, cnz, cstnam, cinnam, coutnam] ...
= sys2ss (csys);
crng = 1:cn;
Ac = caa;
Bc = cbb(:,1:m);
Acd = cbb(:,m+(1:nz));
Cc = ccc(1:p,:);
Adc = ccc(p + (1:nz),:);
# recombine to reduce discrete part of the system
dinnam = __sysconcat__ (innam, cstnam);
doutnam = __sysconcat__ (outnam, cstnam);
dsys = ss (Ad, [Bd, Adc], [Cd; Acd], [], tsam);
dsys = syssetsignals (dsys, "st", dstnam);
dsys = syssetsignals (dsys, "in", dinnam);
dsys = syssetsignals (dsys, "out", doutnam);
# reduce discrete subsystem
dsys = sysmin (dsys);
[n1, nz] = sysdimensions (dsys);
if (nz == 0)
# discrete subsystem is not needed
retsys = sysprune (csys, 1:p, 1:m);
else
# combine discrete, continuous subsystems
[Ad, dbb, dcc] = sys2ss (dsys);
dstnam = sysgetsignals (dsys, "st");
Bd = dbb(:,1:m);
Adc = dbb(:,m+(1:cn));
Cd = dcc(1:p,:);
Acd = dcc(p+(1:cn),:);
stnam = __sysconcat__ (cstnam, dstnam);
aa = [Ac, Acd; Adc, Ad];
bb = [Bc; Bd];
cc = [Cc, Cd];
retsys = ss ([Ac, Acd; Adc, Ad], [Bc ; Bd], [Cc, Cd], dd, tsam,
cn, nz, stnam, innam, outnam, find(yd == 1));
endif
endif
else
Ts = sysgettsam (sys);
switch (flg)
case 0
## reduce to a minimal system
[aa, bb, cc, dd] = sys2ss (sys);
[cflg, Uc] = is_controllable (aa, bb);
if (! cflg)
## reduce to controllable states
if (! isempty (Uc))
aa = Uc'*aa*Uc;
bb = Uc'*bb;
cc = cc*Uc;
else
aa = bb = cc = [];
endif
endif
if (! isempty (aa))
[oflg, Uo] = is_observable (aa, cc);
if (! oflg)
if (! isempty (Uo))
aa = Uo'*aa*Uo;
bb = Uo'*bb;
cc = cc*Uo;
else
aa = bb = cc = [];
endif
endif
endif
switch (dflg)
case 0
nc = no = nn = columns (aa);
nz = 0;
case 1
nc = no = nz = columns (aa);
nn = 0;
endswitch
innam = sysgetsignals (sys, "in");
outnam= sysgetsignals (sys, "out");
retsys = ss (aa, bb, cc, dd, Ts, nn, nz, [], innam, outnam);
case 1
## reduced model with physical states
[cflg, Uc] = is_controllable (sys);
xc = find (max (abs (Uc')) != 0);
[oflg, Uo] = is_observable (sys);
xo = find (max (abs (Uo')) != 0);
xx = intersection (xc, xo);
## signal no states in reduced model
if (isempty (xx))
xx = 0;
endif
retsys = sysprune (sys, [], [], xx);
otherwise
error ("invalid value of flg = %d", flg);
endswitch
if (sysdimensions (retsys, "st") > 0)
[cflg, Uc] = is_controllable (retsys);
nc = columns (Uc);
[oflg, Uo] = is_observable (retsys);
no = columns (Uo);
else
nc = no = 0;
endif
endif
endfunction
|
