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
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
|
SUBROUTINE AB05RD( FBTYPE, JOBD, N, M, P, MV, PZ, ALPHA, BETA, A,
$ LDA, B, LDB, C, LDC, D, LDD, F, LDF, K, LDK,
$ G, LDG, H, LDH, RCOND, BC, LDBC, CC, LDCC,
$ DC, LDDC, IWORK, DWORK, LDWORK, 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 construct for a given state space system (A,B,C,D) the closed-
C loop system (Ac,Bc,Cc,Dc) corresponding to the mixed output and
C state feedback control law
C
C u = alpha*F*y + beta*K*x + G*v
C z = H*y.
C
C ARGUMENTS
C
C Mode Parameters
C
C FBTYPE CHARACTER*1
C Specifies the type of the feedback law as follows:
C = 'I': Unitary output feedback (F = I);
C = 'O': General output feedback.
C
C JOBD CHARACTER*1
C Specifies whether or not a non-zero matrix D appears
C in the given state space model:
C = 'D': D is present;
C = 'Z': D is assumed a zero matrix.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The dimension of state vector x, i.e. the order of the
C matrix A, the number of rows of B and the number of
C columns of C. N >= 0.
C
C M (input) INTEGER
C The dimension of input vector u, i.e. the number of
C columns of matrices B and D, and the number of rows of F.
C M >= 0.
C
C P (input) INTEGER
C The dimension of output vector y, i.e. the number of rows
C of matrices C and D, and the number of columns of F.
C P >= 0 and P = M if FBTYPE = 'I'.
C
C MV (input) INTEGER
C The dimension of the new input vector v, i.e. the number
C of columns of matrix G. MV >= 0.
C
C PZ (input) INTEGER.
C The dimension of the new output vector z, i.e. the number
C of rows of matrix H. PZ >= 0.
C
C ALPHA (input) DOUBLE PRECISION
C The coefficient alpha in the output feedback law.
C
C BETA (input) DOUBLE PRECISION.
C The coefficient beta in the state feedback law.
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 system state transition matrix A.
C On exit, the leading N-by-N part of this array contains
C the state matrix Ac of the closed-loop 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 system input matrix B.
C On exit, the leading N-by-M part of this array contains
C the intermediary input matrix B1 (see METHOD).
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 system output matrix C.
C On exit, the leading P-by-N part of this array contains
C the intermediary output matrix C1+BETA*D1*K (see METHOD).
C
C LDC INTEGER
C The leading dimension of array C.
C LDC >= MAX(1,P) if N > 0.
C LDC >= 1 if N = 0.
C
C D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
C On entry, if JOBD = 'D', the leading P-by-M part of this
C array must contain the system direct input/output
C transmission matrix D.
C On exit, the leading P-by-M part of this array contains
C the intermediary direct input/output transmission matrix
C D1 (see METHOD).
C The array D is not referenced if JOBD = 'Z'.
C
C LDD INTEGER
C The leading dimension of array D.
C LDD >= MAX(1,P) if JOBD = 'D'.
C LDD >= 1 if JOBD = 'Z'.
C
C F (input) DOUBLE PRECISION array, dimension (LDF,P)
C If FBTYPE = 'O', the leading M-by-P part of this array
C must contain the output feedback matrix F.
C If FBTYPE = 'I', then the feedback matrix is assumed to be
C an M x M order identity matrix.
C The array F is not referenced if FBTYPE = 'I' or
C ALPHA = 0.
C
C LDF INTEGER
C The leading dimension of array F.
C LDF >= MAX(1,M) if FBTYPE = 'O' and ALPHA <> 0.
C LDF >= 1 if FBTYPE = 'I' or ALPHA = 0.
C
C K (input) DOUBLE PRECISION array, dimension (LDK,N)
C The leading M-by-N part of this array must contain the
C state feedback matrix K.
C The array K is not referenced if BETA = 0.
C
C LDK INTEGER
C The leading dimension of the array K.
C LDK >= MAX(1,M) if BETA <> 0.
C LDK >= 1 if BETA = 0.
C
C G (input) DOUBLE PRECISION array, dimension (LDG,MV)
C The leading M-by-MV part of this array must contain the
C system input scaling matrix G.
C
C LDG INTEGER
C The leading dimension of the array G. LDG >= MAX(1,M).
C
C H (input) DOUBLE PRECISION array, dimension (LDH,P)
C The leading PZ-by-P part of this array must contain the
C system output scaling matrix H.
C
C LDH INTEGER
C The leading dimension of the array H. LDH >= MAX(1,PZ).
C
C RCOND (output) DOUBLE PRECISION
C The reciprocal condition number of the matrix
C I - alpha*D*F.
C
C BC (output) DOUBLE PRECISION array, dimension (LDBC,MV)
C The leading N-by-MV part of this array contains the input
C matrix Bc of the closed-loop system.
C
C LDBC INTEGER
C The leading dimension of array BC. LDBC >= MAX(1,N).
C
C CC (output) DOUBLE PRECISION array, dimension (LDCC,N)
C The leading PZ-by-N part of this array contains the
C system output matrix Cc of the closed-loop system.
C
C LDCC INTEGER
C The leading dimension of array CC.
C LDCC >= MAX(1,PZ) if N > 0.
C LDCC >= 1 if N = 0.
C
C DC (output) DOUBLE PRECISION array, dimension (LDDC,MV)
C If JOBD = 'D', the leading PZ-by-MV part of this array
C contains the direct input/output transmission matrix Dc
C of the closed-loop system.
C The array DC is not referenced if JOBD = 'Z'.
C
C LDDC INTEGER
C The leading dimension of array DC.
C LDDC >= MAX(1,PZ) if JOBD = 'D'.
C LDDC >= 1 if JOBD = 'Z'.
C
C Workspace
C
C IWORK INTEGER array, dimension (LIWORK)
C LIWORK >= MAX(1,2*P) if JOBD = 'D'.
C LIWORK >= 1 if JOBD = 'Z'.
C IWORK is not referenced if JOBD = 'Z'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= wspace, where
C wspace = MAX( 1, M, P*MV, P*P + 4*P ) if JOBD = 'D',
C wspace = MAX( 1, M ) if JOBD = 'Z'.
C For best performance, LDWORK >= MAX( wspace, N*M, N*P ).
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: if the matrix I - alpha*D*F is numerically singular.
C
C METHOD
C
C The matrices of the closed-loop system have the expressions:
C
C Ac = A1 + beta*B1*K, Bc = B1*G,
C Cc = H*(C1 + beta*D1*K), Dc = H*D1*G,
C
C where
C
C A1 = A + alpha*B*F*E*C, B1 = B + alpha*B*F*E*D,
C C1 = E*C, D1 = E*D,
C
C with E = (I - alpha*D*F)**-1.
C
C NUMERICAL ASPECTS
C
C The accuracy of computations basically depends on the conditioning
C of the matrix I - alpha*D*F. If RCOND is very small, it is likely
C that the computed results are inaccurate.
C
C CONTRIBUTORS
C
C A. Varga, German Aerospace Research Establishment,
C Oberpfaffenhofen, Germany, and V. Sima, Katholieke Univ. Leuven,
C Belgium, Nov. 1996.
C
C REVISIONS
C
C January 14, 1997, February 18, 1998.
C V. Sima, Research Institute for Informatics, Bucharest, July 2003,
C Jan. 2005.
C
C KEYWORDS
C
C Multivariable system, state-space model, state-space
C representation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER FBTYPE, JOBD
INTEGER INFO, LDA, LDB, LDBC, LDC, LDCC, LDD, LDDC,
$ LDF, LDG, LDH, LDK, LDWORK, M, MV, N, P, PZ
DOUBLE PRECISION ALPHA, BETA, RCOND
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), BC(LDBC,*), C(LDC,*),
$ CC(LDCC,*), D(LDD,*), DC(LDDC,*), DWORK(*),
$ F(LDF,*), G(LDG,*), H(LDH,*), K(LDK,*)
C .. Local Scalars ..
LOGICAL LJOBD, OUTPF, UNITF
INTEGER LDWP
C .. External functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External subroutines ..
EXTERNAL AB05SD, DGEMM, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN
C
C .. Executable Statements ..
C
C Check the input scalar arguments.
C
UNITF = LSAME( FBTYPE, 'I' )
OUTPF = LSAME( FBTYPE, 'O' )
LJOBD = LSAME( JOBD, 'D' )
C
INFO = 0
C
IF( .NOT.UNITF .AND. .NOT.OUTPF ) THEN
INFO = -1
ELSE IF( .NOT.LJOBD .AND. .NOT.LSAME( JOBD, 'Z' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( P.LT.0 .OR. UNITF.AND.P.NE.M ) THEN
INFO = -5
ELSE IF( MV.LT.0 ) THEN
INFO = -6
ELSE IF( PZ.LT.0 ) THEN
INFO = -7
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -11
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( ( N.GT.0 .AND. LDC.LT.MAX( 1, P ) ) .OR.
$ ( N.EQ.0 .AND. LDC.LT.1 ) ) THEN
INFO = -15
ELSE IF( ( LJOBD .AND. LDD.LT.MAX( 1, P ) ) .OR.
$ ( .NOT.LJOBD .AND. LDD.LT.1 ) ) THEN
INFO = -17
ELSE IF( ( OUTPF .AND. ALPHA.NE.ZERO .AND. LDF.LT.MAX( 1, M ) )
$ .OR. ( ( UNITF .OR. ALPHA.EQ.ZERO ) .AND. LDF.LT.1 ) ) THEN
INFO = -19
ELSE IF( ( BETA.NE.ZERO .AND. LDK.LT.MAX( 1, M ) ) .OR.
$ ( BETA.EQ.ZERO .AND. LDK.LT.1 ) ) THEN
INFO = -21
ELSE IF( LDG.LT.MAX( 1, M ) ) THEN
INFO = -23
ELSE IF( LDH.LT.MAX( 1, PZ ) ) THEN
INFO = -25
ELSE IF( LDBC.LT.MAX( 1, N ) ) THEN
INFO = -28
ELSE IF( ( N.GT.0 .AND. LDCC.LT.MAX( 1, PZ ) ) .OR.
$ ( N.EQ.0 .AND. LDCC.LT.1 ) ) THEN
INFO = -30
ELSE IF( ( ( LJOBD .AND. LDDC.LT.MAX( 1, PZ ) ) .OR.
$ ( .NOT.LJOBD .AND. LDDC.LT.1 ) ) ) THEN
INFO = -32
ELSE IF( ( LJOBD .AND. LDWORK.LT.MAX( 1, M, P*MV, P*P + 4*P ) )
$ .OR. ( .NOT.LJOBD .AND. LDWORK.LT.MAX( 1, M ) ) ) THEN
INFO = -35
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB05RD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( MAX( N, MIN( M, P ), MIN( MV, PZ ) ).EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
C
C Apply the partial output feedback u = alpha*F*y + v1
C
CALL AB05SD( FBTYPE, JOBD, N, M, P, ALPHA, A, LDA, B, LDB, C,
$ LDC, D, LDD, F, LDF, RCOND, IWORK, DWORK, LDWORK,
$ INFO )
IF ( INFO.NE.0 ) RETURN
C
C Apply the partial state feedback v1 = beta*K*x + v2.
C
C Compute Ac = A1 + beta*B1*K and C1 <- C1 + beta*D1*K.
C
IF( BETA.NE.ZERO .AND. N.GT.0 ) THEN
CALL DGEMM( 'N', 'N', N, N, M, BETA, B, LDB, K, LDK, ONE, A,
$ LDA )
IF( LJOBD )
$ CALL DGEMM( 'N', 'N', P, N, M, BETA, D, LDD, K, LDK, ONE,
$ C, LDC )
END IF
C
C Apply the input and output conversions v2 = G*v, z = H*y.
C
C Compute Bc = B1*G.
C
CALL DGEMM( 'N', 'N', N, MV, M, ONE, B, LDB, G, LDG, ZERO, BC,
$ LDBC )
C
C Compute Cc = H*C1.
C
IF( N.GT.0 )
$ CALL DGEMM( 'N', 'N', PZ, N, P, ONE, H, LDH, C, LDC, ZERO, CC,
$ LDCC )
C
C Compute Dc = H*D1*G.
C
IF( LJOBD ) THEN
LDWP = MAX( 1, P )
CALL DGEMM( 'N', 'N', P, MV, M, ONE, D, LDD, G, LDG, ZERO,
$ DWORK, LDWP )
CALL DGEMM( 'N', 'N', PZ, MV, P, ONE, H, LDH, DWORK, LDWP,
$ ZERO, DC, LDDC )
END IF
C
RETURN
C *** Last line of AB05RD ***
END
|
