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
|
<HTML>
<HEAD><TITLE>SB08ED - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="SB08ED">SB08ED</A></H2>
<H3>
Left coprime factorization with prescribed stability degree
</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 construct, for a given system G = (A,B,C,D), an output
injection matrix H and an orthogonal transformation matrix Z, such
that the systems
Q = (Z'*(A+H*C)*Z, Z'*(B+H*D), C*Z, D)
and
R = (Z'*(A+H*C)*Z, Z'*H, C*Z, I)
provide a stable left coprime factorization of G in the form
-1
G = R * Q,
where G, Q and R are the corresponding transfer-function matrices.
The resulting state dynamics matrix of the systems Q and R has
eigenvalues lying inside a given stability domain.
The Z matrix is not explicitly computed.
Note: If the given state-space representation is not detectable,
the undetectable part of the original system is automatically
deflated and the order of the systems Q and R is accordingly
reduced.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE SB08ED( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C, LDC,
$ D, LDD, NQ, NR, BR, LDBR, DR, LDDR, TOL, DWORK,
$ LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, IWARN, LDA, LDB, LDBR, LDC, LDD, LDDR,
$ LDWORK, M, N, NQ, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHA(*), B(LDB,*), BR(LDBR,*),
$ C(LDC,*), D(LDD,*), DR(LDDR,*), DWORK(*)
</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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The dimension of the state vector, i.e. the order of the
matrix A, and also the number of rows of the matrices B
and BR, and the number of columns of the matrix C.
N >= 0.
M (input) INTEGER
The dimension of input vector, i.e. the number of columns
of the matrices B and D. M >= 0.
P (input) INTEGER
The dimension of output vector, i.e. the number of rows
of the matrices C, D and DR, and the number of columns of
the matrices BR and DR. P >= 0.
ALPHA (input) DOUBLE PRECISION array, dimension (2)
ALPHA(1) contains the desired stability degree to be
assigned for the eigenvalues of A+H*C, and ALPHA(2)
the stability margin. The eigenvalues outside the
ALPHA(2)-stability region will be assigned to have the
real parts equal to ALPHA(1) < 0 and unmodified
imaginary parts for a continuous-time system
(DICO = 'C'), or moduli equal to 0 <= ALPHA(2) < 1
for a discrete-time system (DICO = 'D').
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.
On exit, the leading NQ-by-NQ part of this array contains
the leading NQ-by-NQ part of the matrix Z'*(A+H*C)*Z, the
state dynamics matrix of the numerator factor Q, in a
real Schur form. The leading NR-by-NR part of this matrix
represents the state dynamics matrix of a minimal
realization of the denominator factor R.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the input/state matrix of the system.
On exit, the leading NQ-by-M part of this array contains
the leading NQ-by-M part of the matrix Z'*(B+H*D), the
input/state matrix of the numerator factor Q.
The remaining part of this array is needed as workspace.
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 state/output matrix of the system.
On exit, the leading P-by-NQ part of this array contains
the leading P-by-NQ part of the matrix C*Z, the
state/output matrix of the numerator factor Q.
The first NR columns of this array represent the
state/output matrix of a minimal realization of the
denominator factor R.
The remaining part of this array is needed as workspace.
LDC INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P), if N > 0.
LDC >= 1, if N = 0.
D (input) DOUBLE PRECISION array, dimension (LDD,MAX(M,P))
The leading P-by-M part of this array must contain the
input/output matrix. D represents also the input/output
matrix of the numerator factor Q.
This array is modified internally, but restored on exit.
The remaining part of this array is needed as workspace.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,M,P).
NQ (output) INTEGER
The order of the resulting factors Q and R.
Generally, NQ = N - NS, where NS is the number of
unobservable eigenvalues outside the stability region.
NR (output) INTEGER
The order of the minimal realization of the factor R.
Generally, NR is the number of observable eigenvalues
of A outside the stability region (the number of modified
eigenvalues).
BR (output) DOUBLE PRECISION array, dimension (LDBR,P)
The leading NQ-by-P part of this array contains the
leading NQ-by-P part of the output injection matrix
Z'*H, which moves the eigenvalues of A lying outside
the ALPHA-stable region to values on the ALPHA-stability
boundary. The first NR rows of this matrix form the
input/state matrix of a minimal realization of the
denominator factor R.
LDBR INTEGER
The leading dimension of array BR. LDBR >= MAX(1,N).
DR (output) DOUBLE PRECISION array, dimension (LDDR,P)
The leading P-by-P part of this array contains an
identity matrix representing the input/output matrix
of the denominator factor R.
LDDR INTEGER
The leading dimension of array DR. LDDR >= MAX(1,P).
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
The absolute tolerance level below which the elements of
C are considered zero (used for observability tests).
If the user sets TOL <= 0, then an implicitly computed,
default tolerance, defined by TOLDEF = N*EPS*NORM(C),
is used instead, where EPS is the machine precision
(see LAPACK Library routine DLAMCH) and NORM(C) denotes
the infinity-norm of C.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of working array DWORK.
LDWORK >= MAX( 1, N*P + MAX( N*(N+5), 5*P, 4*M ) ).
For optimum performance LDWORK should be larger.
</PRE>
<B>Warning Indicator</B>
<PRE>
IWARN INTEGER
= 0: no warning;
= K: K violations of the numerical stability condition
NORM(H) <= 10*NORM(A)/NORM(C) occured during the
assignment of eigenvalues.
</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 reduction of A to a real Schur form failed;
= 2: a failure was detected during the ordering of the
real Schur form of A, or in the iterative process
for reordering the eigenvalues of Z'*(A + H*C)*Z
along the diagonal.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The subroutine uses the right coprime factorization algorithm
of [1] applied to G'.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Varga A.
Coprime factors model reduction method based on
square-root balancing-free techniques.
System Analysis, Modelling and Simulation,
vol. 11, pp. 303-311, 1993.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3
The algorithm requires no more than 14N 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>
* SB08ED EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MPMAX
PARAMETER ( MPMAX = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDBR, LDC, LDD, LDDR
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MPMAX,
$ LDD = MPMAX, LDBR = NMAX, LDDR = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX*PMAX + MAX( NMAX*( NMAX + 5 ),
$ 5*PMAX, 4*MMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, IWARN, J, M, N, NQ, NR, P
CHARACTER*1 DICO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALPHA(2), B(LDB,MPMAX),
$ BR(LDBR,PMAX), C(LDC,NMAX), D(LDD,MPMAX),
$ DR(LDDR,PMAX), DWORK(LDWORK)
* .. External Subroutines ..
EXTERNAL SB08ED
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, ALPHA(1), TOL, DICO
ALPHA(2) = ALPHA(1)
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1, N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1, M ), I = 1, N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
* Find a LCF for (A,B,C,D).
CALL SB08ED( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C,
$ LDC, D, LDD, NQ, NR, BR, LDBR, DR, LDDR,
$ TOL, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NQ )
20 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NQ
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
40 CONTINUE
IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NQ )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
70 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99986 )
DO 80 I = 1, NR
WRITE ( NOUT, FMT = 99995 )
$ ( A(I,J), J = 1, NR )
80 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
DO 90 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( BR(I,J), J = 1, P )
90 CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
DO 100 I = 1, P
WRITE ( NOUT, FMT = 99995 )
$ ( C(I,J), J = 1, NR )
100 CONTINUE
WRITE ( NOUT, FMT = 99983 )
DO 110 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, P )
110 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB08ED EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08ED = ',I2)
99996 FORMAT (/' The numerator state dynamics matrix AQ is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The numerator input/state matrix BQ is ')
99992 FORMAT (/' The numerator state/output matrix CQ is ')
99991 FORMAT (/' The numerator input/output matrix DQ is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99986 FORMAT (/' The denominator state dynamics matrix AR is ')
99985 FORMAT (/' The denominator input/state matrix BR is ')
99984 FORMAT (/' The denominator state/output matrix CR is ')
99983 FORMAT (/' The denominator input/output matrix DR is ')
END
</PRE>
<B>Program Data</B>
<PRE>
SB08ED EXAMPLE PROGRAM DATA (Continuous system)
7 2 3 -1.0 1.E-10 C
-0.04165 0.0000 4.9200 0.4920 0.0000 0.0000 0.0000
-5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000
0.5450 0.0000 0.0000 0.0000 0.0545 0.0000 0.0000
0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200
0.0000 0.0000 0.0000 0.0000 0.5210 -12.500 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300
0.0000 0.0000
12.500 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 12.500
0.0000 0.0000
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
</PRE>
<B>Program Results</B>
<PRE>
SB08ED EXAMPLE PROGRAM RESULTS
The numerator state dynamics matrix AQ is
-1.0000 0.0526 -0.1408 -0.3060 0.4199 0.2408 1.7274
-0.4463 -1.0000 2.0067 4.3895 0.0062 0.1813 0.0895
0.0000 0.0000 -12.4245 3.5463 -0.0057 0.0254 -0.0053
0.0000 0.0000 0.0000 -3.5957 -0.0153 -0.0290 -0.0616
0.0000 0.0000 0.0000 0.0000 -13.1627 -1.9835 -3.6182
0.0000 0.0000 0.0000 0.0000 0.0000 -1.4178 5.6218
0.0000 0.0000 0.0000 0.0000 0.0000 -0.8374 -1.4178
The numerator input/state matrix BQ is
-1.1544 -0.0159
-0.0631 0.5122
0.0056 -11.6989
0.0490 4.3728
11.7198 -0.0038
-2.8173 0.0308
3.1018 -0.0009
The numerator state/output matrix CQ is
0.2238 0.0132 -0.0006 -0.0083 0.1279 0.8797 0.3994
0.9639 0.0643 -0.0007 -0.0041 0.0305 -0.2562 0.0122
-0.0660 0.9962 0.0248 -0.0506 0.0000 0.0022 -0.0017
The numerator input/output matrix DQ is
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
The denominator state dynamics matrix AR is
-1.0000 0.0526
-0.4463 -1.0000
The denominator input/state matrix BR is
-0.2623 -1.1297 0.0764
-0.0155 -0.0752 -1.1676
The denominator state/output matrix CR is
0.2238 0.0132
0.9639 0.0643
-0.0660 0.9962
The denominator input/output matrix DR is
1.0000 0.0000 0.0000
0.0000 1.0000 0.0000
0.0000 0.0000 1.0000
</PRE>
<HR>
<p>
<A HREF=..\libindex.html><B>Return to index</B></A></BODY>
</HTML>
|
