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
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
|
SUBROUTINE AB09AD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, A, LDA,
$ B, LDB, C, LDC, HSV, TOL, IWORK, DWORK, LDWORK,
$ IWARN, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To compute a reduced order model (Ar,Br,Cr) for a stable original
C state-space representation (A,B,C) by using either the square-root
C or the balancing-free square-root Balance & Truncate (B & T)
C model reduction method.
C
C ARGUMENTS
C
C Mode Parameters
C
C DICO CHARACTER*1
C Specifies the type of the original system as follows:
C = 'C': continuous-time system;
C = 'D': discrete-time system.
C
C JOB CHARACTER*1
C Specifies the model reduction approach to be used
C as follows:
C = 'B': use the square-root Balance & Truncate method;
C = 'N': use the balancing-free square-root
C Balance & Truncate method.
C
C EQUIL CHARACTER*1
C Specifies whether the user wishes to preliminarily
C equilibrate the triplet (A,B,C) as follows:
C = 'S': perform equilibration (scaling);
C = 'N': do not perform equilibration.
C
C ORDSEL CHARACTER*1
C Specifies the order selection method as follows:
C = 'F': the resulting order NR is fixed;
C = 'A': the resulting order NR is automatically determined
C on basis of the given tolerance TOL.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the original state-space representation, i.e.
C the order of the matrix A. N >= 0.
C
C M (input) INTEGER
C The number of system inputs. M >= 0.
C
C P (input) INTEGER
C The number of system outputs. P >= 0.
C
C NR (input/output) INTEGER
C On entry with ORDSEL = 'F', NR is the desired order of the
C resulting reduced order system. 0 <= NR <= N.
C On exit, if INFO = 0, NR is the order of the resulting
C reduced order model. NR is set as follows:
C if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
C is the desired order on entry and NMIN is the order of a
C minimal realization of the given system; NMIN is
C determined as the number of Hankel singular values greater
C than N*EPS*HNORM(A,B,C), where EPS is the machine
C precision (see LAPACK Library Routine DLAMCH) and
C HNORM(A,B,C) is the Hankel norm of the system (computed
C in HSV(1));
C if ORDSEL = 'A', NR is equal to the number of Hankel
C singular values greater than MAX(TOL,N*EPS*HNORM(A,B,C)).
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading N-by-N part of this array must
C contain the state dynamics matrix A.
C On exit, if INFO = 0, the leading NR-by-NR part of this
C array contains the state dynamics matrix Ar of the reduced
C order system.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
C On entry, the leading N-by-M part of this array must
C contain the original input/state matrix B.
C On exit, if INFO = 0, the leading NR-by-M part of this
C array contains the input/state matrix Br of the reduced
C order system.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
C On entry, the leading P-by-N part of this array must
C contain the original state/output matrix C.
C On exit, if INFO = 0, the leading P-by-NR part of this
C array contains the state/output matrix Cr of the reduced
C order system.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C HSV (output) DOUBLE PRECISION array, dimension (N)
C If INFO = 0, it contains the Hankel singular values of
C the original system ordered decreasingly. HSV(1) is the
C Hankel norm of the system.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C If ORDSEL = 'A', TOL contains the tolerance for
C determining the order of reduced system.
C For model reduction, the recommended value is
C TOL = c*HNORM(A,B,C), where c is a constant in the
C interval [0.00001,0.001], and HNORM(A,B,C) is the
C Hankel-norm of the given system (computed in HSV(1)).
C For computing a minimal realization, the recommended
C value is TOL = N*EPS*HNORM(A,B,C), where EPS is the
C machine precision (see LAPACK Library Routine DLAMCH).
C This value is used by default if TOL <= 0 on entry.
C If ORDSEL = 'F', the value of TOL is ignored.
C
C Workspace
C
C IWORK INTEGER array, dimension (LIWORK)
C LIWORK = 0, if JOB = 'B';
C LIWORK = N, if JOB = 'N'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX(1,N*(2*N+MAX(N,M,P)+5)+N*(N+1)/2).
C For optimum performance LDWORK should be larger.
C
C Warning Indicator
C
C IWARN INTEGER
C = 0: no warning;
C = 1: with ORDSEL = 'F', the selected order NR is greater
C than the order of a minimal realization of the
C given system. In this case, the resulting NR is
C set automatically to a value corresponding to the
C order of a minimal realization of the system.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value;
C = 1: the reduction of A to the real Schur form failed;
C = 2: the state matrix A is not stable (if DICO = 'C')
C or not convergent (if DICO = 'D');
C = 3: the computation of Hankel singular values failed.
C
C METHOD
C
C Let be the stable linear system
C
C d[x(t)] = Ax(t) + Bu(t)
C y(t) = Cx(t) (1)
C
C where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
C for a discrete-time system. The subroutine AB09AD determines for
C the given system (1), the matrices of a reduced order system
C
C d[z(t)] = Ar*z(t) + Br*u(t)
C yr(t) = Cr*z(t) (2)
C
C such that
C
C HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
C
C where G and Gr are transfer-function matrices of the systems
C (A,B,C) and (Ar,Br,Cr), respectively, and INFNORM(G) is the
C infinity-norm of G.
C
C If JOB = 'B', the square-root Balance & Truncate method of [1]
C is used and, for DICO = 'C', the resulting model is balanced.
C By setting TOL <= 0, the routine can be used to compute balanced
C minimal state-space realizations of stable systems.
C
C If JOB = 'N', the balancing-free square-root version of the
C Balance & Truncate method [2] is used.
C By setting TOL <= 0, the routine can be used to compute minimal
C state-space realizations of stable systems.
C
C REFERENCES
C
C [1] Tombs M.S. and Postlethwaite I.
C Truncated balanced realization of stable, non-minimal
C state-space systems.
C Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
C
C [2] Varga A.
C Efficient minimal realization procedure based on balancing.
C Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
C A. El Moudui, P. Borne, S. G. Tzafestas (Eds.),
C Vol. 2, pp. 42-46.
C
C NUMERICAL ASPECTS
C
C The implemented methods rely on accuracy enhancing square-root or
C balancing-free square-root techniques.
C 3
C The algorithms require less than 30N floating point operations.
C
C CONTRIBUTOR
C
C C. Oara and A. Varga, German Aerospace Center,
C DLR Oberpfaffenhofen, March 1998.
C Based on the RASP routines SRBT and SRBFT.
C
C REVISIONS
C
C May 2, 1998.
C November 11, 1998, V. Sima, Research Institute for Informatics,
C Bucharest.
C
C KEYWORDS
C
C Balancing, minimal state-space representation, model reduction,
C multivariable system, state-space model.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE, C100
PARAMETER ( ONE = 1.0D0, C100 = 100.0D0 )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDWORK, M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
C .. Local Scalars ..
LOGICAL FIXORD
INTEGER IERR, KI, KR, KT, KTI, KW, NN
DOUBLE PRECISION MAXRED, WRKOPT
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL AB09AX, TB01ID, TB01WD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
C .. Executable Statements ..
C
INFO = 0
IWARN = 0
FIXORD = LSAME( ORDSEL, 'F' )
C
C Test the input scalar arguments.
C
IF( .NOT. ( LSAME( DICO, 'C' ) .OR. LSAME( DICO, 'D' ) ) ) THEN
INFO = -1
ELSE IF( .NOT. ( LSAME( JOB, 'B' ) .OR. LSAME( JOB, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT. ( LSAME( EQUIL, 'S' ) .OR.
$ LSAME( EQUIL, 'N' ) ) ) THEN
INFO = -3
ELSE IF( .NOT. ( FIXORD .OR. LSAME( ORDSEL, 'A' ) ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( P.LT.0 ) THEN
INFO = -7
ELSE IF( FIXORD .AND. ( NR.LT.0 .OR. NR.GT.N ) ) THEN
INFO = -8
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -14
ELSE IF( LDWORK.LT.MAX( 1, N*( 2*N + MAX( N, M, P ) + 5 ) +
$ ( N*( N + 1 ) )/2 ) ) THEN
INFO = -19
END IF
C
IF( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB09AD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( MIN( N, M, P ).EQ.0 .OR. ( FIXORD .AND. NR.EQ.0 ) ) THEN
NR = 0
DWORK(1) = ONE
RETURN
END IF
C
C Allocate working storage.
C
NN = N*N
KT = 1
KR = KT + NN
KI = KR + N
KW = KI + N
C
IF( LSAME( EQUIL, 'S' ) ) THEN
C
C Scale simultaneously the matrices A, B and C:
C A <- inv(D)*A*D, B <- inv(D)*B and C <- C*D, where D is a
C diagonal matrix.
C
MAXRED = C100
CALL TB01ID( 'A', N, M, P, MAXRED, A, LDA, B, LDB, C, LDC,
$ DWORK, INFO )
END IF
C
C Reduce A to the real Schur form using an orthogonal similarity
C transformation A <- T'*A*T and apply the transformation to
C B and C: B <- T'*B and C <- C*T.
C
CALL TB01WD( N, M, P, A, LDA, B, LDB, C, LDC, DWORK(KT), N,
$ DWORK(KR), DWORK(KI), DWORK(KW), LDWORK-KW+1, IERR )
IF( IERR.NE.0 ) THEN
INFO = 1
RETURN
END IF
C
WRKOPT = DWORK(KW) + DBLE( KW-1 )
KTI = KT + NN
KW = KTI + NN
C
CALL AB09AX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB, C,
$ LDC, HSV, DWORK(KT), N, DWORK(KTI), N, TOL, IWORK,
$ DWORK(KW), LDWORK-KW+1, IWARN, IERR )
C
IF( IERR.NE.0 ) THEN
INFO = IERR + 1
RETURN
END IF
C
DWORK(1) = MAX( WRKOPT, DWORK(KW) + DBLE( KW-1 ) )
C
RETURN
C *** Last line of AB09AD ***
END
|
