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
|
SUBROUTINE TF01MY( N, M, P, NY, A, LDA, B, LDB, C, LDC, D, LDD,
$ U, LDU, X, Y, LDY, 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 compute the output sequence of a linear time-invariant
C open-loop system given by its discrete-time state-space model
C (A,B,C,D), where A is an N-by-N general matrix.
C
C The initial state vector x(1) must be supplied by the user.
C
C This routine differs from SLICOT Library routine TF01MD in the
C way the input and output trajectories are stored.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
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 NY (input) INTEGER
C The number of output vectors y(k) to be computed.
C NY >= 0.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,N)
C The leading N-by-N part of this array must contain the
C state matrix A of the system.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,M)
C The leading N-by-M part of this array must contain the
C input matrix B of the system.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (input) DOUBLE PRECISION array, dimension (LDC,N)
C The leading P-by-N part of this array must contain the
C output matrix C of the system.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C D (input) DOUBLE PRECISION array, dimension (LDD,M)
C The leading P-by-M part of this array must contain the
C direct link matrix D of the system.
C
C LDD INTEGER
C The leading dimension of array D. LDD >= MAX(1,P).
C
C U (input) DOUBLE PRECISION array, dimension (LDU,M)
C The leading NY-by-M part of this array must contain the
C input vector sequence u(k), for k = 1,2,...,NY.
C Specifically, the k-th row of U must contain u(k)'.
C
C LDU INTEGER
C The leading dimension of array U. LDU >= MAX(1,NY).
C
C X (input/output) DOUBLE PRECISION array, dimension (N)
C On entry, this array must contain the initial state vector
C x(1) which consists of the N initial states of the system.
C On exit, this array contains the final state vector
C x(NY+1) of the N states of the system at instant NY+1.
C
C Y (output) DOUBLE PRECISION array, dimension (LDY,P)
C The leading NY-by-P part of this array contains the output
C vector sequence y(1),y(2),...,y(NY) such that the k-th
C row of Y contains y(k)' (the outputs at instant k),
C for k = 1,2,...,NY.
C
C LDY INTEGER
C The leading dimension of array Y. LDY >= MAX(1,NY).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C
C LDWORK INTEGER
C The length of the array DWORK. LDWORK >= N.
C For better performance, LDWORK should be larger.
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 Given an initial state vector x(1), the output vector sequence
C y(1), y(2),..., y(NY) is obtained via the formulae
C
C x(k+1) = A x(k) + B u(k)
C y(k) = C x(k) + D u(k),
C
C where each element y(k) is a vector of length P containing the
C outputs at instant k and k = 1,2,...,NY.
C
C REFERENCES
C
C [1] Luenberger, D.G.
C Introduction to Dynamic Systems: Theory, Models and
C Applications.
C John Wiley & Sons, New York, 1979.
C
C NUMERICAL ASPECTS
C
C The algorithm requires approximately (N + M) x (N + P) x NY
C multiplications and additions.
C
C FURTHER COMMENTS
C
C The implementation exploits data locality and uses BLAS 3
C operations as much as possible, given the workspace length.
C
C CONTRIBUTOR
C
C V. Sima, Research Institute for Informatics, Bucharest, Mar. 2001.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Discrete-time system, multivariable system, state-space model,
C state-space representation, time response.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDU, LDWORK, LDY, M,
$ N, NY, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), U(LDU,*), X(*), Y(LDY,*)
C .. Local Scalars ..
INTEGER IK, IREM, IS, IYL, MAXN, NB, NS
DOUBLE PRECISION UPD
C .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
C .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DGEMV, DLASET, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN
C .. Executable Statements ..
C
INFO = 0
C
C Test the input scalar arguments.
C
MAXN = MAX( 1, N )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 ) THEN
INFO = -3
ELSE IF( NY.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAXN ) THEN
INFO = -6
ELSE IF( LDB.LT.MAXN ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -10
ELSE IF( LDD.LT.MAX( 1, P ) ) THEN
INFO = -12
ELSE IF( LDU.LT.MAX( 1, NY ) ) THEN
INFO = -14
ELSE IF( LDY.LT.MAX( 1, NY ) ) THEN
INFO = -17
ELSE IF( LDWORK.LT.N ) THEN
INFO = -19
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'TF01MY', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( MIN( NY, P ).EQ.0 ) THEN
RETURN
ELSE IF ( N.EQ.0 ) THEN
C
C Non-dynamic system: compute the output vectors.
C
IF ( M.EQ.0 ) THEN
CALL DLASET( 'Full', NY, P, ZERO, ZERO, Y, LDY )
ELSE
CALL DGEMM( 'No transpose', 'Transpose', NY, P, M, ONE,
$ U, LDU, D, LDD, ZERO, Y, LDY )
END IF
RETURN
END IF
C
C Determine the block size (taken as for LAPACK routine DGETRF).
C
NB = ILAENV( 1, 'DGETRF', ' ', NY, MAX( M, P ), -1, -1 )
C
C Find the number of state vectors that can be accommodated in
C the provided workspace and initialize.
C
NS = MIN( LDWORK/N, NB*NB/N, NY )
C
IF ( NS.LE.1 .OR. NY*MAX( M, P ).LE.NB*NB ) THEN
C
C LDWORK < 2*N or small problem:
C only BLAS 2 calculations are used in the loop
C for computing the output corresponding to D = 0.
C One row of the array Y is computed for each loop index value.
C
DO 10 IK = 1, NY
CALL DGEMV( 'No transpose', P, N, ONE, C, LDC, X, 1, ZERO,
$ Y(IK,1), LDY )
C
CALL DGEMV( 'No transpose', N, N, ONE, A, LDA, X, 1, ZERO,
$ DWORK, 1 )
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB, U(IK,1), LDU,
$ ONE, DWORK, 1 )
C
CALL DCOPY( N, DWORK, 1, X, 1 )
10 CONTINUE
C
ELSE
C
C LDWORK >= 2*N and large problem:
C some BLAS 3 calculations can also be used.
C
IYL = ( NY/NS )*NS
IF ( M.EQ.0 ) THEN
UPD = ZERO
ELSE
UPD = ONE
END IF
C
CALL DCOPY( N, X, 1, DWORK, 1 )
C
DO 30 IK = 1, IYL, NS
C
C Compute the current NS-1 state vectors in the workspace.
C
CALL DGEMM( 'No transpose', 'Transpose', N, NS-1, M, ONE,
$ B, LDB, U(IK,1), LDU, ZERO, DWORK(N+1), MAXN )
C
DO 20 IS = 1, NS - 1
CALL DGEMV( 'No transpose', N, N, ONE, A, LDA,
$ DWORK((IS-1)*N+1), 1, UPD, DWORK(IS*N+1), 1 )
20 CONTINUE
C
C Initialize the current NS output vectors.
C
CALL DGEMM( 'Transpose', 'Transpose', NS, P, N, ONE, DWORK,
$ MAXN, C, LDC, ZERO, Y(IK,1), LDY )
C
C Prepare the next iteration.
C
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB,
$ U(IK+NS-1,1), LDU, ZERO, DWORK, 1 )
CALL DGEMV( 'No transpose', N, N, ONE, A, LDA,
$ DWORK((NS-1)*N+1), 1, UPD, DWORK, 1 )
30 CONTINUE
C
IREM = NY - IYL
C
IF ( IREM.GT.1 ) THEN
C
C Compute the last IREM output vectors.
C First, compute the current IREM-1 state vectors.
C
IK = IYL + 1
CALL DGEMM( 'No transpose', 'Transpose', N, IREM-1, M, ONE,
$ B, LDB, U(IK,1), LDU, ZERO, DWORK(N+1), MAXN )
C
DO 40 IS = 1, IREM - 1
CALL DGEMV( 'No transpose', N, N, ONE, A, LDA,
$ DWORK((IS-1)*N+1), 1, UPD, DWORK(IS*N+1), 1 )
40 CONTINUE
C
C Initialize the last IREM output vectors.
C
CALL DGEMM( 'Transpose', 'Transpose', IREM, P, N, ONE,
$ DWORK, MAXN, C, LDC, ZERO, Y(IK,1), LDY )
C
C Prepare the final state vector.
C
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB,
$ U(IK+IREM-1,1), LDU, ZERO, DWORK, 1 )
CALL DGEMV( 'No transpose', N, N, ONE, A, LDA,
$ DWORK((IREM-1)*N+1), 1, UPD, DWORK, 1 )
C
ELSE IF ( IREM.EQ.1 ) THEN
C
C Compute the last 1 output vectors.
C
CALL DGEMV( 'No transpose', P, N, ONE, C, LDC, DWORK, 1,
$ ZERO, Y(IK,1), LDY )
C
C Prepare the final state vector.
C
CALL DCOPY( N, DWORK, 1, DWORK(N+1), 1 )
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB,
$ U(IK,1), LDU, ZERO, DWORK, 1 )
CALL DGEMV( 'No transpose', N, N, ONE, A, LDA,
$ DWORK(N+1), 1, UPD, DWORK, 1 )
END IF
C
C Set the final state vector.
C
CALL DCOPY( N, DWORK, 1, X, 1 )
C
END IF
C
C Add the direct contribution of the input to the output vectors.
C
CALL DGEMM( 'No transpose', 'Transpose', NY, P, M, ONE, U, LDU,
$ D, LDD, ONE, Y, LDY )
C
RETURN
C *** Last line of TF01MY ***
END
|
