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
|
<HTML>
<HEAD><TITLE>SB10HD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="SB10HD">SB10HD</A></H2>
<H3>
H2 optimal state controller for a continuous-time system
</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 compute the matrices of the H2 optimal n-state controller
| AK | BK |
K = |----|----|
| CK | DK |
for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---| ,
| C1 | 0 D12 | | C | D |
| C2 | D21 D22 |
where B2 has as column size the number of control inputs (NCON)
and C2 has as row size the number of measurements (NMEAS) being
provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) The block D11 of D is zero,
(A3) D12 is full column rank and D21 is full row rank.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE SB10HD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ RCOND, TOL, IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 4 )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
RCOND (output) DOUBLE PRECISION array, dimension (4)
RCOND(1) contains the reciprocal condition number of the
control transformation matrix;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix;
RCOND(3) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
RCOND(4) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of the applied
transformations for computing the normalized form in
SLICOT Library routine SB10UD. Transformation matrices
whose reciprocal condition numbers are less than TOL are
not allowed. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (max(2*N,N*N))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
max(max(M2 + NP1*NP1 +
max(NP1*N,3*M2+NP1,5*M2),
NP2 + M1*M1 +
max(M1*N,3*NP2+M1,5*NP2),
N*M2,NP2*N,NP2*M2,1),
N*(14*N+12+M2+NP2)+5),
where M1 = M - M2 and NP1 = NP - NP2.
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
2*Q*(3*Q+2*N)+max(1,Q*(Q+max(N,5)+1),N*(14*N+12+2*Q)+5).
BWORK LOGICAL array, dimension (2*N)
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix D12 had not full column rank in
respect to the tolerance TOL;
= 2: if the matrix D21 had not full row rank in respect
to the tolerance TOL;
= 3: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices D12 or D21).
= 4: if the X-Riccati equation was not solved
successfully;
= 5: if the Y-Riccati equation was not solved
successfully.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine implements the formulas given in [1], [2].
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The accuracy of the result depends on the condition numbers of the
input and output transformations and on the condition numbers of
the two Riccati equations, as given by the values of RCOND(1),
RCOND(2), RCOND(3) and RCOND(4), respectively.
</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>
* SB10HD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
$ LDAK = NMAX, LDBK = NMAX, LDCK = PMAX,
$ LDDK = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*NMAX, NMAX*NMAX ) )
INTEGER MPMX
PARAMETER ( MPMX = MAX( MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*MPMX*( 2*NMAX + 3*MPMX ) +
$ MAX( MPMX*( MPMX + MAX( NMAX, 5 ) + 1 ),
$ NMAX*( 14*NMAX + 12 + 2*MPMX ) + 5 ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, M, N, NCON, NMEAS, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
$ BK(LDBK,MMAX), C(LDC,NMAX), CK(LDCK,NMAX),
$ D(LDD,MMAX), DK(LDDK,MMAX), DWORK(LDWORK),
$ RCOND( 4 )
* .. External Subroutines ..
EXTERNAL SB10HD
* .. 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, NP, NCON, NMEAS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) NCON
ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NMEAS
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) TOL
* Compute the optimal H2 controller
CALL SB10HD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
*
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NMEAS )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NMEAS )
40 CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 4 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10HD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10HD =',I2)
99997 FORMAT (' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (6(1X,F10.4))
99991 FORMAT (5(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
99986 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
SB10HD EXAMPLE PROGRAM DATA
6 5 5 2 2
-1.0 0.0 4.0 5.0 -3.0 -2.0
-2.0 4.0 -7.0 -2.0 0.0 3.0
-6.0 9.0 -5.0 0.0 2.0 -1.0
-8.0 4.0 7.0 -1.0 -3.0 0.0
2.0 5.0 8.0 -9.0 1.0 -4.0
3.0 -5.0 8.0 0.0 2.0 -6.0
-3.0 -4.0 -2.0 1.0 0.0
2.0 0.0 1.0 -5.0 2.0
-5.0 -7.0 0.0 7.0 -2.0
4.0 -6.0 1.0 1.0 -2.0
-3.0 9.0 -8.0 0.0 5.0
1.0 -2.0 3.0 -6.0 -2.0
1.0 -1.0 2.0 -4.0 0.0 -3.0
-3.0 0.0 5.0 -1.0 1.0 1.0
-7.0 5.0 0.0 -8.0 2.0 -2.0
9.0 -3.0 4.0 0.0 3.0 7.0
0.0 1.0 -2.0 1.0 -6.0 -2.0
0.0 0.0 0.0 -4.0 -1.0
0.0 0.0 0.0 1.0 0.0
0.0 0.0 0.0 0.0 1.0
3.0 1.0 0.0 1.0 -3.0
-2.0 0.0 1.0 7.0 1.0
0.00000001
</PRE>
<B>Program Results</B>
<PRE>
SB10HD EXAMPLE PROGRAM RESULTS
The controller state matrix AK is
88.0015 -145.7298 -46.2424 82.2168 -45.2996 -31.1407
25.7489 -31.4642 -12.4198 9.4625 -3.5182 2.7056
54.3008 -102.4013 -41.4968 50.8412 -20.1286 -26.7191
108.1006 -198.0785 -45.4333 70.3962 -25.8591 -37.2741
-115.8900 226.1843 47.2549 -47.8435 -12.5004 34.7474
59.0362 -101.8471 -20.1052 36.7834 -16.1063 -26.4309
The controller input matrix BK is
3.7345 3.4758
-0.3020 0.6530
3.4735 4.0499
4.3198 7.2755
-3.9424 -10.5942
2.1784 2.5048
The controller output matrix CK is
-2.3346 3.2556 0.7150 -0.9724 0.6962 0.4074
7.6899 -8.4558 -2.9642 7.0365 -4.2844 0.1390
The controller matrix DK is
0.0000 0.0000
0.0000 0.0000
The estimated condition numbers are
0.23570D+00 0.26726D+00 0.22747D-01 0.21130D-02
</PRE>
<HR>
<p>
<A HREF=..\libindex.html><B>Return to index</B></A></BODY>
</HTML>
|
