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
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
|
SUBROUTINE AB05MD( UPLO, OVER, N1, M1, P1, N2, P2, A1, LDA1, B1,
$ LDB1, C1, LDC1, D1, LDD1, A2, LDA2, B2, LDB2,
$ C2, LDC2, D2, LDD2, N, A, LDA, B, LDB, C, LDC,
$ D, LDD, 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 obtain the state-space model (A,B,C,D) for the cascaded
C inter-connection of two systems, each given in state-space form.
C
C ARGUMENTS
C
C Mode Parameters
C
C UPLO CHARACTER*1
C Indicates whether the user wishes to obtain the matrix A
C in the upper or lower block diagonal form, as follows:
C = 'U': Obtain A in the upper block diagonal form;
C = 'L': Obtain A in the lower block diagonal form.
C
C OVER CHARACTER*1
C Indicates whether the user wishes to overlap pairs of
C arrays, as follows:
C = 'N': Do not overlap;
C = 'O': Overlap pairs of arrays: A1 and A, B1 and B,
C C1 and C, and D1 and D (for UPLO = 'L'), or A2
C and A, B2 and B, C2 and C, and D2 and D (for
C UPLO = 'U'), i.e. the same name is effectively
C used for each pair (for all pairs) in the routine
C call. In this case, setting LDA1 = LDA,
C LDB1 = LDB, LDC1 = LDC, and LDD1 = LDD, or
C LDA2 = LDA, LDB2 = LDB, LDC2 = LDC, and LDD2 = LDD
C will give maximum efficiency.
C
C Input/Output Parameters
C
C N1 (input) INTEGER
C The number of state variables in the first system, i.e.
C the order of the matrix A1. N1 >= 0.
C
C M1 (input) INTEGER
C The number of input variables for the first system.
C M1 >= 0.
C
C P1 (input) INTEGER
C The number of output variables from the first system and
C the number of input variables for the second system.
C P1 >= 0.
C
C N2 (input) INTEGER
C The number of state variables in the second system, i.e.
C the order of the matrix A2. N2 >= 0.
C
C P2 (input) INTEGER
C The number of output variables from the second system.
C P2 >= 0.
C
C A1 (input) DOUBLE PRECISION array, dimension (LDA1,N1)
C The leading N1-by-N1 part of this array must contain the
C state transition matrix A1 for the first system.
C
C LDA1 INTEGER
C The leading dimension of array A1. LDA1 >= MAX(1,N1).
C
C B1 (input) DOUBLE PRECISION array, dimension (LDB1,M1)
C The leading N1-by-M1 part of this array must contain the
C input/state matrix B1 for the first system.
C
C LDB1 INTEGER
C The leading dimension of array B1. LDB1 >= MAX(1,N1).
C
C C1 (input) DOUBLE PRECISION array, dimension (LDC1,N1)
C The leading P1-by-N1 part of this array must contain the
C state/output matrix C1 for the first system.
C
C LDC1 INTEGER
C The leading dimension of array C1.
C LDC1 >= MAX(1,P1) if N1 > 0.
C LDC1 >= 1 if N1 = 0.
C
C D1 (input) DOUBLE PRECISION array, dimension (LDD1,M1)
C The leading P1-by-M1 part of this array must contain the
C input/output matrix D1 for the first system.
C
C LDD1 INTEGER
C The leading dimension of array D1. LDD1 >= MAX(1,P1).
C
C A2 (input) DOUBLE PRECISION array, dimension (LDA2,N2)
C The leading N2-by-N2 part of this array must contain the
C state transition matrix A2 for the second system.
C
C LDA2 INTEGER
C The leading dimension of array A2. LDA2 >= MAX(1,N2).
C
C B2 (input) DOUBLE PRECISION array, dimension (LDB2,P1)
C The leading N2-by-P1 part of this array must contain the
C input/state matrix B2 for the second system.
C
C LDB2 INTEGER
C The leading dimension of array B2. LDB2 >= MAX(1,N2).
C
C C2 (input) DOUBLE PRECISION array, dimension (LDC2,N2)
C The leading P2-by-N2 part of this array must contain the
C state/output matrix C2 for the second system.
C
C LDC2 INTEGER
C The leading dimension of array C2.
C LDC2 >= MAX(1,P2) if N2 > 0.
C LDC2 >= 1 if N2 = 0.
C
C D2 (input) DOUBLE PRECISION array, dimension (LDD2,P1)
C The leading P2-by-P1 part of this array must contain the
C input/output matrix D2 for the second system.
C
C LDD2 INTEGER
C The leading dimension of array D2. LDD2 >= MAX(1,P2).
C
C N (output) INTEGER
C The number of state variables (N1 + N2) in the resulting
C system, i.e. the order of the matrix A, the number of rows
C of B and the number of columns of C.
C
C A (output) DOUBLE PRECISION array, dimension (LDA,N1+N2)
C The leading N-by-N part of this array contains the state
C transition matrix A for the cascaded system.
C If OVER = 'O', the array A can overlap A1, if UPLO = 'L',
C or A2, if UPLO = 'U'.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N1+N2).
C
C B (output) DOUBLE PRECISION array, dimension (LDB,M1)
C The leading N-by-M1 part of this array contains the
C input/state matrix B for the cascaded system.
C If OVER = 'O', the array B can overlap B1, if UPLO = 'L',
C or B2, if UPLO = 'U'.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N1+N2).
C
C C (output) DOUBLE PRECISION array, dimension (LDC,N1+N2)
C The leading P2-by-N part of this array contains the
C state/output matrix C for the cascaded system.
C If OVER = 'O', the array C can overlap C1, if UPLO = 'L',
C or C2, if UPLO = 'U'.
C
C LDC INTEGER
C The leading dimension of array C.
C LDC >= MAX(1,P2) if N1+N2 > 0.
C LDC >= 1 if N1+N2 = 0.
C
C D (output) DOUBLE PRECISION array, dimension (LDD,M1)
C The leading P2-by-M1 part of this array contains the
C input/output matrix D for the cascaded system.
C If OVER = 'O', the array D can overlap D1, if UPLO = 'L',
C or D2, if UPLO = 'U'.
C
C LDD INTEGER
C The leading dimension of array D. LDD >= MAX(1,P2).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C The array DWORK is not referenced if OVER = 'N'.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX( 1, P1*MAX(N1, M1, N2, P2) ) if OVER = 'O'.
C LDWORK >= 1 if OVER = 'N'.
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
C METHOD
C
C After cascaded inter-connection of the two systems
C
C X1' = A1*X1 + B1*U
C V = C1*X1 + D1*U
C
C X2' = A2*X2 + B2*V
C Y = C2*X2 + D2*V
C
C (where ' denotes differentiation with respect to time)
C
C the following state-space model will be obtained:
C
C X' = A*X + B*U
C Y = C*X + D*U
C
C where matrix A has the form ( A1 0 ),
C ( B2*C1 A2)
C
C matrix B has the form ( B1 ),
C ( B2*D1 )
C
C matrix C has the form ( D2*C1 C2 ) and
C
C matrix D has the form ( D2*D1 ).
C
C This form is returned by the routine when UPLO = 'L'. Note that
C when A1 and A2 are block lower triangular, the resulting state
C matrix is also block lower triangular.
C
C By applying a similarity transformation to the system above,
C using the matrix ( 0 I ), where I is the identity matrix of
C ( J 0 )
C order N2, and J is the identity matrix of order N1, the
C system matrices become
C
C A = ( A2 B2*C1 ),
C ( 0 A1 )
C
C B = ( B2*D1 ),
C ( B1 )
C
C C = ( C2 D2*C1 ) and
C
C D = ( D2*D1 ).
C
C This form is returned by the routine when UPLO = 'U'. Note that
C when A1 and A2 are block upper triangular (for instance, in the
C real Schur form), the resulting state matrix is also block upper
C triangular.
C
C REFERENCES
C
C None
C
C NUMERICAL ASPECTS
C
C The algorithm requires P1*(N1+M1)*(N2+P2) operations.
C
C CONTRIBUTORS
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, and
C A. Varga, German Aerospace Research Establishment,
C Oberpfaffenhofen, Germany, Nov. 1996.
C Supersedes Release 2.0 routine AB05AD by C.J.Benson, Kingston
C Polytechnic, United Kingdom, January 1982.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, July 2003,
C Feb. 2004.
C
C KEYWORDS
C
C Cascade control, continuous-time system, multivariable
C system, state-space model, state-space representation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER OVER, UPLO
INTEGER INFO, LDA, LDA1, LDA2, LDB, LDB1, LDB2, LDC,
$ LDC1, LDC2, LDD, LDD1, LDD2, LDWORK, M1, N, N1,
$ N2, P1, P2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), A1(LDA1,*), A2(LDA2,*), B(LDB,*),
$ B1(LDB1,*), B2(LDB2,*), C(LDC,*), C1(LDC1,*),
$ C2(LDC2,*), D(LDD,*), D1(LDD1,*), D2(LDD2,*),
$ DWORK(*)
C .. Local Scalars ..
LOGICAL LOVER, LUPLO
INTEGER I, I1, I2, J, LDWN2, LDWP1, LDWP2
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN
C .. Executable Statements ..
C
LOVER = LSAME( OVER, 'O' )
LUPLO = LSAME( UPLO, 'L' )
N = N1 + N2
INFO = 0
C
C Test the input scalar arguments.
C
IF( .NOT.LUPLO .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOVER .AND. .NOT.LSAME( OVER, 'N' ) ) THEN
INFO = -2
ELSE IF( N1.LT.0 ) THEN
INFO = -3
ELSE IF( M1.LT.0 ) THEN
INFO = -4
ELSE IF( P1.LT.0 ) THEN
INFO = -5
ELSE IF( N2.LT.0 ) THEN
INFO = -6
ELSE IF( P2.LT.0 ) THEN
INFO = -7
ELSE IF( LDA1.LT.MAX( 1, N1 ) ) THEN
INFO = -9
ELSE IF( LDB1.LT.MAX( 1, N1 ) ) THEN
INFO = -11
ELSE IF( ( N1.GT.0 .AND. LDC1.LT.MAX( 1, P1 ) ) .OR.
$ ( N1.EQ.0 .AND. LDC1.LT.1 ) ) THEN
INFO = -13
ELSE IF( LDD1.LT.MAX( 1, P1 ) ) THEN
INFO = -15
ELSE IF( LDA2.LT.MAX( 1, N2 ) ) THEN
INFO = -17
ELSE IF( LDB2.LT.MAX( 1, N2 ) ) THEN
INFO = -19
ELSE IF( ( N2.GT.0 .AND. LDC2.LT.MAX( 1, P2 ) ) .OR.
$ ( N2.EQ.0 .AND. LDC2.LT.1 ) ) THEN
INFO = -21
ELSE IF( LDD2.LT.MAX( 1, P2 ) ) THEN
INFO = -23
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -26
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -28
ELSE IF( ( N.GT.0 .AND. LDC.LT.MAX( 1, P2 ) ) .OR.
$ ( N.EQ.0 .AND. LDC.LT.1 ) ) THEN
INFO = -30
ELSE IF( LDD.LT.MAX( 1, P2 ) ) THEN
INFO = -32
ELSE IF( ( LOVER.AND.LDWORK.LT.MAX( 1, P1*MAX( N1, M1, N2, P2 )) )
$.OR.( .NOT.LOVER.AND.LDWORK.LT.1 ) ) THEN
INFO = -34
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB05MD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( MAX( N, MIN( M1, P2 ) ).EQ.0 )
$ RETURN
C
C Set row/column indices for storing the results.
C
IF ( LUPLO ) THEN
I1 = 1
I2 = MIN( N1 + 1, N )
ELSE
I1 = MIN( N2 + 1, N )
I2 = 1
END IF
C
LDWN2 = MAX( 1, N2 )
LDWP1 = MAX( 1, P1 )
LDWP2 = MAX( 1, P2 )
C
C Construct the cascaded system matrices, taking the desired block
C structure and possible overwriting into account.
C
C Form the diagonal blocks of matrix A.
C
IF ( LUPLO ) THEN
C
C Lower block diagonal structure.
C
IF ( LOVER .AND. LDA1.LE.LDA ) THEN
IF ( LDA1.LT.LDA ) THEN
C
DO 20 J = N1, 1, -1
DO 10 I = N1, 1, -1
A(I,J) = A1(I,J)
10 CONTINUE
20 CONTINUE
C
END IF
ELSE
CALL DLACPY( 'F', N1, N1, A1, LDA1, A, LDA )
END IF
IF ( N2.GT.0 )
$ CALL DLACPY( 'F', N2, N2, A2, LDA2, A(I2,I2), LDA )
ELSE
C
C Upper block diagonal structure.
C
IF ( LOVER .AND. LDA2.LE.LDA ) THEN
IF ( LDA2.LT.LDA ) THEN
C
DO 40 J = N2, 1, -1
DO 30 I = N2, 1, -1
A(I,J) = A2(I,J)
30 CONTINUE
40 CONTINUE
C
END IF
ELSE
CALL DLACPY( 'F', N2, N2, A2, LDA2, A, LDA )
END IF
IF ( N1.GT.0 )
$ CALL DLACPY( 'F', N1, N1, A1, LDA1, A(I1,I1), LDA )
END IF
C
C Form the off-diagonal blocks of matrix A.
C
IF ( MIN( N1, N2 ).GT.0 ) THEN
CALL DLASET( 'F', N1, N2, ZERO, ZERO, A(I1,I2), LDA )
CALL DGEMM ( 'No transpose', 'No transpose', N2, N1, P1, ONE,
$ B2, LDB2, C1, LDC1, ZERO, A(I2,I1), LDA )
END IF
C
IF ( LUPLO ) THEN
C
C Form the matrix B.
C
IF ( LOVER .AND. LDB1.LE.LDB ) THEN
IF ( LDB1.LT.LDB ) THEN
C
DO 60 J = M1, 1, -1
DO 50 I = N1, 1, -1
B(I,J) = B1(I,J)
50 CONTINUE
60 CONTINUE
C
END IF
ELSE
CALL DLACPY( 'F', N1, M1, B1, LDB1, B, LDB )
END IF
C
IF ( MIN( N2, M1 ).GT.0 )
$ CALL DGEMM ( 'No transpose', 'No transpose', N2, M1, P1,
$ ONE, B2, LDB2, D1, LDD1, ZERO, B(I2,1), LDB )
C
C Form the matrix C.
C
IF ( N1.GT.0 ) THEN
IF ( LOVER ) THEN
C
C Workspace: P1*N1.
C
CALL DLACPY( 'F', P1, N1, C1, LDC1, DWORK, LDWP1 )
CALL DGEMM ( 'No transpose', 'No transpose', P2, N1, P1,
$ ONE, D2, LDD2, DWORK, LDWP1, ZERO, C, LDC )
ELSE
CALL DGEMM ( 'No transpose', 'No transpose', P2, N1, P1,
$ ONE, D2, LDD2, C1, LDC1, ZERO, C, LDC )
END IF
END IF
C
IF ( MIN( P2, N2 ).GT.0 )
$ CALL DLACPY( 'F', P2, N2, C2, LDC2, C(1,I2), LDC )
C
C Now form the matrix D.
C
IF ( LOVER ) THEN
C
C Workspace: P1*M1.
C
CALL DLACPY( 'F', P1, M1, D1, LDD1, DWORK, LDWP1 )
CALL DGEMM ( 'No transpose', 'No transpose', P2, M1, P1,
$ ONE, D2, LDD2, DWORK, LDWP1, ZERO, D, LDD )
ELSE
CALL DGEMM ( 'No transpose', 'No transpose', P2, M1, P1,
$ ONE, D2, LDD2, D1, LDD1, ZERO, D, LDD )
END IF
C
ELSE
C
C Form the matrix B.
C
IF ( LOVER ) THEN
C
C Workspace: N2*P1.
C
CALL DLACPY( 'F', N2, P1, B2, LDB2, DWORK, LDWN2 )
IF ( MIN( N2, M1 ).GT.0 )
$ CALL DGEMM ( 'No transpose', 'No transpose', N2, M1, P1,
$ ONE, DWORK, LDWN2, D1, LDD1, ZERO, B(I2,1),
$ LDB )
ELSE
CALL DGEMM ( 'No transpose', 'No transpose', N2, M1, P1,
$ ONE, B2, LDB2, D1, LDD1, ZERO, B, LDB )
END IF
C
IF ( MIN( N1, M1 ).GT.0 )
$ CALL DLACPY( 'F', N1, M1, B1, LDB1, B(I1,1), LDB )
C
C Form the matrix C.
C
IF ( LOVER .AND. LDC2.LE.LDC ) THEN
IF ( LDC2.LT.LDC ) THEN
C
DO 80 J = N2, 1, -1
DO 70 I = P2, 1, -1
C(I,J) = C2(I,J)
70 CONTINUE
80 CONTINUE
C
END IF
ELSE
CALL DLACPY( 'F', P2, N2, C2, LDC2, C, LDC )
END IF
C
IF ( MIN( P2, N1 ).GT.0 )
$ CALL DGEMM ( 'No transpose', 'No transpose', P2, N1, P1,
$ ONE, D2, LDD2, C1, LDC1, ZERO, C(1,I1), LDC )
C
C Now form the matrix D.
C
IF ( LOVER ) THEN
C
C Workspace: P2*P1.
C
CALL DLACPY( 'F', P2, P1, D2, LDD2, DWORK, LDWP2 )
CALL DGEMM ( 'No transpose', 'No transpose', P2, M1, P1,
$ ONE, DWORK, LDWP2, D1, LDD1, ZERO, D, LDD )
ELSE
CALL DGEMM ( 'No transpose', 'No transpose', P2, M1, P1,
$ ONE, D2, LDD2, D1, LDD1, ZERO, D, LDD )
END IF
END IF
C
RETURN
C *** Last line of AB05MD ***
END
|
