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
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
|
<HTML>
<HEAD><TITLE>AB09AX - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="AB09AX">AB09AX</A></H2>
<H3>
Balance & Truncate model reduction for stable systems with state matrix in real Schur canonical form
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute a reduced order model (Ar,Br,Cr) for a stable original
state-space representation (A,B,C) by using either the square-root
or the balancing-free square-root Balance & Truncate model
reduction method. The state dynamics matrix A of the original
system is an upper quasi-triangular matrix in real Schur canonical
form. The matrices of the reduced order system are computed using
the truncation formulas:
Ar = TI * A * T , Br = TI * B , Cr = C * T .
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE AB09AX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
$ C, LDC, HSV, T, LDT, TI, LDTI, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDT, LDTI, LDWORK,
$ M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*),
$ T(LDT,*), TI(LDTI,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'N': use the balancing-free square-root
Balance & Truncate method.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
is the desired order on entry and NMIN is the order of a
minimal realization of the given system; NMIN is
determined as the number of Hankel singular values greater
than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL,N*EPS*HNORM(A,B,C)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A in a real Schur
canonical form.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.
T (output) DOUBLE PRECISION array, dimension (LDT,N)
If INFO = 0 and NR > 0, the leading N-by-NR part of this
array contains the right truncation matrix T.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
TI (output) DOUBLE PRECISION array, dimension (LDTI,N)
If INFO = 0 and NR > 0, the leading NR-by-N part of this
array contains the left truncation matrix TI.
LDTI INTEGER
The leading dimension of array TI. LDTI >= MAX(1,N).
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
If ORDSEL = 'A', TOL contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL <= 0 on entry.
If ORDSEL = 'F', the value of TOL is ignored.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOB = 'B', or
LIWORK = N, if JOB = 'N'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(MAX(N,M,P)+5) + N*(N+1)/2).
For optimum performance LDWORK should be larger.
</PRE>
<B>Warning Indicator</B>
<PRE>
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is
set automatically to a value corresponding to the
order of a minimal realization of the system.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2: the computation of Hankel singular values failed.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09AX determines for
the given system (1), the matrices of a reduced NR order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) (2)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C) and (Ar,Br,Cr), respectively, and INFNORM(G) is the
infinity-norm of G.
If JOB = 'B', the square-root Balance & Truncate method of [1]
is used and, for DICO = 'C', the resulting model is balanced.
By setting TOL <= 0, the routine can be used to compute balanced
minimal state-space realizations of stable systems.
If JOB = 'N', the balancing-free square-root version of the
Balance & Truncate method [2] is used.
By setting TOL <= 0, the routine can be used to compute minimal
state-space realizations of stable systems.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[2] Varga A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.),
Vol. 2, pp. 42-46.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
<HR>
<A HREF=support.html><B>Return to Supporting Routines index</B></A></BODY>
</HTML>
|
