-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathORBIT2.SIF
901 lines (772 loc) · 25.9 KB
/
ORBIT2.SIF
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
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
NAME ORBIT2
*
* A reformulation of the discretized optimal control problem
* ORBIT
*
* Consider minimising the time needed for a spacecraft to
* move from one circular orbit around the earth to another.
* This problem can be put in the following form:
*
* let (x1,x2,x3) be the position and (x4,x5,x6) the velocity
* let (u1,u2,u3) be the driving force vector
* let q be the time required, then our problem becomes:
*
* MINIMISE q
*
* with the equations of motion:
*
* dx1/dt = hv*q*x4
* dx2/dt = hv*q*x5
* dx3/dt = hv*q*x6
* dx4/dt = hv*q*(hg*x1/r^3-hf*u(1)/m) (1)
* dx5/dt = hv*q*(hg*x1/r^3-hf*u(2)/m)
* dx6/dt = hv*q*(hg*x1/r^3-hf*u(3)/m)
*
* with m = m0-hm*q*t ( - the mass variation)
*
* r = sqrt( x1^2 + x2^2 + x3^2 ) - dist. from the center
* of the earth
*
* 't' is a rescaled time varying between 0 and 1, and
*
* 'hv,hf,hg,hm' are scaling constants.
*
* the driving force is bounded by
*
* u1^2 + u2^2 + u3^2 <= 1 (2)
*
* (the rather arbitrary no. '1' representing the max. power
* of the spacecraft).
* We choose the initial conditions:
*
* x1 = x2 = 0 , x3 = 1 - initial position
*
* x5 = x6 = 0 , x4 = Vorb - corresponding orbital
* speed
*
* and the final conditions are:
*
* x1^2 + x2^2 + x3^2 = Rf^2 - final orbit radius
*
* x4^2 + x5^2 + x6^2 = Vforb^2 - corresponding orbital
* speed
*
* x1*x4 + x2*x5 + x3*x6 = 0 - direction must be parallel
* to the velocity
*
*
* we have chosen the constants hg,hf,hv,hm so that the x,u vectors
* are of order one. These correspond to an initial orbit at 150 km
* above the earth's surface, and a final orbit at 250 km above the
* earth's surface.
* The reduction to an NLP is done in that same way as for CAR.SIF
*
* The time taken should be:
*
* q = 315 secs
*
* SIF input: Thomas Felici, Universite de Nancy, France,
* October 1993, with modifications to compute derivatives
* by Nick Gould
* classification LOR1-RN-V-V
********************************************
* M = Number of time nodes.
* Change this for different resolution
*IE M 3 $-PARAMETER n= 25, m= 18
*IE M 10 $-PARAMETER n= 88, m= 67
*IE M 30 $-PARAMETER n=268, m=207 original value
IE M 300 $-PARAMETER n=2698, m=2097
********************************************
IE NEQ 6
IE NCON 1
IE NLIM 3
IE NLINEQ 0
IE NLINCON 0
IE NLINLIM 0
I+ NTEQ NEQ NLINEQ
I+ NTCON NCON NLINCON
I+ NTLIM NLIM NLINLIM
IE NX 6
IE NU 3
IE NQ 1
* DEFINE REQUIRED INTEGERS
IE 1 1
IE 2 2
IE 3 3
IE 4 4
IE 5 5
IE 6 6
I- M-1 M 1
* DEFINE TIME NODES
RI RM M-1
DO I 1 M
I- I-1 I 1
RI RI1 I-1
A/ T(I) RI1 RM
ND
* Orbit related parameters
* Earth radius in KM
RE RT 6371.0D0
R* RT*RT RT RT
* Gravitation constant
RM MU RT*RT 9.81D-3
* pi
RE ONE 1.0D+0
R( PI/4 ARCTAN ONE
RM PI PI/4 4.0D+0
* Height of initial orbit (KM)
RA R0 RT 1.5D+2
R* R0*R0 R0 R0
* Speed on initial circular orbit = VITB
R/ MU/R0 MU R0
R( VORB SQRT MU/R0
* Height of final orbit
RA RF RT 2.5D+2
* Mass of spacecraft (in KG)
RE M0 3.0D+3
* Debit total QT (in KG/S)
RE QT 3.333D+0
* Speed of gas (in KM/S)
RE VG 3.0D+0
* Typical time scale
RE TS 1.0D+0
* Scaling the constants used in the equations of motion
* (so that x,v are of order 1)
R* HG VORB R0*R0
R/ HG MU HG
R* HG TS HG
R/ HV VORB R0
R* HV TS HV
R* HF VORB M0
R/ HF VG HF
R* HF QT HF
R* HF TS HF
R/ HM QT M0
R* HM TS HM
R/ VF MU RF
R( VF SQRT VF
R/ VF VF VORB
R* VFVF VF VF
R/ RF RF R0
R* RFRF RF RF
VARIABLES
DO I 1 M
DO J 1 NX
X X(I,J)
ND
DO I 1 M-1
DO J 1 NU
X U(I,J)
ND
DO J 1 NQ
X Q(J) 'SCALE' 1.0D+2
OD
GROUPS
XN OBJ Q(1) 1.0
* The differential equation constraints
DO I 1 M-1
DO J 1 NTEQ
XE K(I,J)
ND
* The final condition constraints
DO J 1 NTLIM
XE L(J)
ND
* The driving force constraints
DO I 1 M
DO J 1 NTCON
*XL G(I,J) 'SCALE' 1.0D+3
XL G(I,J) 'SCALE' 1.0D+2
ND
CONSTANTS
Z ORBIT2 L(1) RFRF
Z ORBIT2 L(3) VFVF
DO I 1 M
X ORBIT2 G(I,1) 1.0D+0
ND
BOUNDS
XL BNDS Q(1) .10000E+03
XU BNDS Q(1) .40000E+26
* AT T=0, X IS GIVEN
XX BNDS X(1,1) .00000E+00
XX BNDS X(1,2) .00000E+00
XX BNDS X(1,3) .10000E+01
XX BNDS X(1,4) .10000E+01
XX BNDS X(1,5) .00000E+00
XX BNDS X(1,6) .00000E+00
* X FOR T>0 IS FREE
DO I 2 M
XL BNDS X(I,1) -.10000E+26
XU BNDS X(I,1) .10000E+26
XL BNDS X(I,2) -.10000E+26
XU BNDS X(I,2) .10000E+26
XL BNDS X(I,3) -.10000E+26
XU BNDS X(I,3) .10000E+26
XL BNDS X(I,4) -.10000E+26
XU BNDS X(I,4) .10000E+26
XL BNDS X(I,5) -.10000E+26
XU BNDS X(I,5) .10000E+26
XL BNDS X(I,6) -.10000E+26
XU BNDS X(I,6) .10000E+26
ND
* THE CONTROLS ARE BOUNDED : -1 < U1,U2,U3 < 1
* these bounde are already taken into account by the
* nonlinear constraint:
*
* g = u1^2 + u2^2 + u3^2 <= 1
*
* But they restrain the feasible space for SBMIN
DO I 1 M-1
XL BNDS U(I,1) -.10000E+01
XU BNDS U(I,1) .10000E+01
XL BNDS U(I,2) -.10000E+01
XU BNDS U(I,2) .10000E+01
XL BNDS U(I,3) -.10000E+01
XU BNDS U(I,3) .10000E+01
ND
START POINT
* Initially, we set X(T) = X(0) (given already in BOUNDS)
* and U(T) = (1,1,1)
* these are the values I used for NPSOL ....
X INITS Q(1) .10000E+03
DO I 1 M
X INITS X(I,1) .00000E+00
X INITS X(I,2) .00000E+00
X INITS X(I,3) .10000E+01
X INITS X(I,4) .10000E+01
X INITS X(I,5) .00000E+00
X INITS X(I,6) .00000E+00
ND
DO I 1 M-1
X INITS U(I,1) .10000E+01
X INITS U(I,2) .10000E+01
X INITS U(I,3) .10000E+01
ND
ELEMENT TYPE
* Square
EV SQR X
* Product
EV PROD X Y
* ODE`S
EV Ktyp XI1 XF1
EV Ktyp XI2 XF2
EV Ktyp XI3 XF3
EV Ktyp XI4 XF4
EV Ktyp XI5 XF5
EV Ktyp XI6 XF6
EV Ktyp UV1
EV Ktyp UV2
EV Ktyp UV3
EV Ktyp QV1
EP Ktyp TN
EP Ktyp TN1
EP Ktyp ID
ELEMENT USES
* ODEs
DO I 1 M-1
IA S I 1
DO J 1 NTEQ
RI ReJ J
XT Ke(I,J) Ktyp
ZP Ke(I,J) TN T(I)
ZP Ke(I,J) TN1 T(S)
ZP Ke(I,J) ID ReJ
ZV Ke(I,J) XI1 X(I,1)
ZV Ke(I,J) XF1 X(S,1)
ZV Ke(I,J) XI2 X(I,2)
ZV Ke(I,J) XF2 X(S,2)
ZV Ke(I,J) XI3 X(I,3)
ZV Ke(I,J) XF3 X(S,3)
ZV Ke(I,J) XI4 X(I,4)
ZV Ke(I,J) XF4 X(S,4)
ZV Ke(I,J) XI5 X(I,5)
ZV Ke(I,J) XF5 X(S,5)
ZV Ke(I,J) XI6 X(I,6)
ZV Ke(I,J) XF6 X(S,6)
ZV Ke(I,J) UV1 U(I,1)
ZV Ke(I,J) UV2 U(I,2)
ZV Ke(I,J) UV3 U(I,3)
ZV Ke(I,J) QV1 Q(1)
ND
* constraints
DO J 1 NTCON
DO I 1 M-1
XT Ge(I,J,1) SQR
ZV Ge(I,J,1) X U(I,1)
XT Ge(I,J,2) SQR
ZV Ge(I,J,2) X U(I,2)
XT Ge(I,J,3) SQR
ZV Ge(I,J,3) X U(I,3)
ND
DO J 1 NTCON
XT Ge(M,J,1) SQR
ZV Ge(M,J,1) X U(M-1,1)
XT Ge(M,J,2) SQR
ZV Ge(M,J,2) X U(M-1,2)
XT Ge(M,J,3) SQR
ZV Ge(M,J,3) X U(M-1,3)
ND
* limit conditions
XT Le(1,1) SQR
ZV Le(1,1) X X(M,1)
XT Le(1,2) SQR
ZV Le(1,2) X X(M,2)
XT Le(1,3) SQR
ZV Le(1,3) X X(M,3)
XT Le(2,1) PROD
ZV Le(2,1) X X(M,1)
ZV Le(2,1) Y X(M,4)
XT Le(2,2) PROD
ZV Le(2,2) X X(M,2)
ZV Le(2,2) Y X(M,5)
XT Le(2,3) PROD
ZV Le(2,3) X X(M,3)
ZV Le(2,3) Y X(M,6)
XT Le(3,1) SQR
ZV Le(3,1) X X(M,4)
XT Le(3,2) SQR
ZV Le(3,2) X X(M,5)
XT Le(3,3) SQR
ZV Le(3,3) X X(M,6)
GROUP USES
* ODE`S
DO I 1 M-1
DO J 1 NTEQ
XE K(I,J) Ke(I,J)
ND
* constraints
DO I 1 M
DO J 1 NTCON
XE G(I,J) Ge(I,J,1) Ge(I,J,2)
XE G(I,J) Ge(I,J,3)
ND
* limit conditions
DO I 1 NTLIM
XE L(I) Le(I,1) Le(I,2)
XE L(I) Le(I,3)
ND
ENDATA
********************************************************
* DEFINITION OF THE VARIOUS NONLINEAR FUNCTIONS *
********************************************************
ELEMENTS ORBIT2
TEMPORARIES
R F
R GXI1
R GXI2
R GXI3
R GXI4
R GXI5
R GXI6
R GXF1
R GXF2
R GXF3
R GXF4
R GXF5
R GXF6
R GUV1
R GUV2
R GUV3
R GQV1
R COLLOC
F COLLOC
INDIVIDUALS
* Square group type
T SQR
F X * X
G X X + X
H X X 2.0
* Product group type
T PROD
F X * Y
G X Y
G Y X
H X Y 1.0
* ODE`S
T Ktyp
F COLLOC(
F+ XI1, XI2, XI3, XI4, XI5, XI6,
F+ XF1, XF2, XF3, XF4, XF5, XF6,
F+ UV1, UV2, UV3, QV1,
F+ GXI1, GXI2, GXI3, GXI4, GXI5, GXI6,
F+ GXF1, GXF2, GXF3, GXF4, GXF5, GXF6,
F+ GUV1, GUV2, GUV3, GQV1,
F+ TN, TN1, ID, .FALSE. )
A F COLLOC(
A+ XI1, XI2, XI3, XI4, XI5, XI6,
A+ XF1, XF2, XF3, XF4, XF5, XF6,
A+ UV1, UV2, UV3, QV1,
A+ GXI1, GXI2, GXI3, GXI4, GXI5, GXI6,
A+ GXF1, GXF2, GXF3, GXF4, GXF5, GXF6,
A+ GUV1, GUV2, GUV3, GQV1,
A+ TN, TN1, ID, .TRUE. )
G XI1 GXI1
G XI2 GXI2
G XI3 GXI3
G XI4 GXI4
G XI5 GXI5
G XI6 GXI6
G XF1 GXF1
G XF2 GXF2
G XF3 GXF3
G XF4 GXF4
G XF5 GXF5
G XF6 GXF6
G UV1 GUV1
G UV2 GUV2
G UV3 GUV3
G QV1 GQV1
ENDATA
*****************************
* EXTERNAL FUNCTIONS *
*****************************
DOUBLE PRECISION FUNCTION COLLOC(
* X1, X2, X3, X4, X5, X6, XP1, XP2, XP3, XP4, XP5, XP6,
* U1, U2, U3, Q1, GX1, GX2, GX3, GX4, GX5, GX6,
* GXP1, GXP2, GXP3, GXP4, GXP5, GXP6, GU1, GU2, GU3, GQ1,
* T, T1, DIM, DERIV )
C Compute a finite-difference discretization to the system of
C ordinary differential equations
C
C dX/dT - F( X, U, Q, T ) = 0
C
C involving the values X, U, Q and XP = X + (T1 - T)
C The returned value will be the approximation function
C COLLOC( X, XP, U, Q ) and, if DERIV is true, its derivative
C with respect to each parameter.
C
INTEGER ISTART, ID, I, J , NE, NX, NU, NQ, NP
DOUBLE PRECISION X1, X2, X3, X4, X5, X6,
* XP1, XP2, XP3, XP4, XP5, XP6, U1, U2, U3, Q1
DOUBLE PRECISION GX1, GX2, GX3, GX4, GX5, GX6,
* GXP1, GXP2, GXP3, GXP4, GXP5, GXP6, GU1, GU2, GU3, GQ1
DOUBLE PRECISION T, T1, DIM, TOR, TC, C1, C2
C
C Parameters:
C NE = Number of differential equations
C NX = Number of variables X in the equations
C NU = Number of variables U in the equations
C NQ = Number of variables Q in the equations
C NP = Number of fixed parameters in the equations
C
PARAMETER ( NE = 6, NX = 6, NU = 3, NQ = 1, NP = 4 )
DOUBLE PRECISION X( NX ), XP( NX ), XC( NX ), U( NU ), Q( NQ )
DOUBLE PRECISION F( NX ), FP( NX ), FC( NX ), FCPRED( NX )
DOUBLE PRECISION DX( NX ), DXP( NX ), DU( NU ), DQ( NQ )
DOUBLE PRECISION DFDX ( NX, NE ), DFDU ( NU, NE )
DOUBLE PRECISION DFDQ ( NQ, NE ), DFDQP( NQ, NE )
DOUBLE PRECISION DFDXP( NX, NE ), DFDUP( NU, NE )
DOUBLE PRECISION DFDXC( NX, NE ), DFDUC( NU, NE )
DOUBLE PRECISION DFDQC( NQ, NE ), PARAM( NP )
DOUBLE PRECISION HALF, FOURTH, EIGHTH, ONEPT5
DOUBLE PRECISION HM, HV, HG, HF
PARAMETER ( HALF = 5.0D-1 , FOURTH = 2.5D-1 )
PARAMETER ( EIGHTH = 1.25D-1, ONEPT5 = 1.5D+0 )
LOGICAL DERIV
COMMON / STRTUP / ISTART
DOUBLE PRECISION MU, M0, RT, R0, VORB, QT, VG, TS
INTRINSIC SQRT
SAVE RT, MU, R0, VORB, M0, QT, VG, TS, HM, HV, HG, HF
C
C Initiialize constants.
C
IF ( ISTART .NE. 123 ) THEN
ISTART = 123
C
C....................Earth radius in KM
C
RT = 6371.0D0
C
C....................Gravitation constant
C
MU = 9.81D-3 * RT * RT
C
C...... Height of initial orbit (KM)
C
R0 = RT + 150.0D0
C
C---- Speed on initial circular orbit = VITB
C
VORB = SQRT( MU / R0 )
C
C------ Mass of spacecraft (in KG) [3000.]
C
M0 = 3000.0D0
C
C------ Debit total QT (in KG/S) [3.]
C
QT = 3.333D0
C
C------ Speed of gas (in KM/S) [3.333]
C
VG = 3.0D0
C
C ----- Typical time scale
C
TS = 1.0D0
C
C ----- Scaling constants used in the equations of motion
C ------ (so that x,v are of order 1)
C
HM = TS * QT / M0
HV = TS * VORB / R0
HG = TS * MU / ( VORB * R0 ** 2 )
HF = TS * QT * VG / ( VORB * M0 )
END IF
PARAM( 1 ) = HM
PARAM( 2 ) = HV
PARAM( 3 ) = HG
PARAM( 4 ) = HF
C
C Place scalar input in vectors, for convenience.
C
ID = DIM
X ( 1 ) = X1
X ( 2 ) = X2
X ( 3 ) = X3
X ( 4 ) = X4
X ( 5 ) = X5
X ( 6 ) = X6
XP( 1 ) = XP1
XP( 2 ) = XP2
XP( 3 ) = XP3
XP( 4 ) = XP4
XP( 5 ) = XP5
XP( 6 ) = XP6
U ( 1 ) = U1
U ( 2 ) = U2
U ( 3 ) = U3
Q ( 1 ) = Q1
C
C Calculate the values of F, F and FP, at X and XP, respectively.
C
TOR = T1 - T
CALL FSYS( DERIV, F , X , U, T , Q, PARAM,
* DFDX , DFDU , DFDQ )
CALL FSYS( DERIV, FP, XP, U, T1, Q, PARAM,
* DFDXP, DFDUP, DFDQP )
C
C Calculate the corrector value XC and the predicted value of F,
C FCPRED, at XC.
C
C1 = EIGHTH * TOR
C2 = ONEPT5 / TOR
DO 10 J = 1, NX
XC ( J ) = HALF * ( X( J ) + XP( J ) ) +
* C1 * ( F( J ) - FP( J ) )
FCPRED( J ) = C2 * ( XP( J ) - X( J ) ) -
* FOURTH * ( F( J ) + FP( J ) )
10 CONTINUE
C Calculate the actual value of F, FC, at XC.
TC = HALF * ( T1 + T )
CALL FSYS( DERIV, FC, XC, U, TC, Q, PARAM,
* DFDXC, DFDUC, DFDQC )
C
C The objective function value is the difference between the predicted
C and actual values of F at XC.
C
COLLOC = FCPRED( ID ) - FC( ID )
IF ( .NOT. DERIV ) RETURN
C
C Obtain the gradient values of the IDth term. First, include the
C contributions from the FC term.
C
DO 20 I = 1, NX
DX ( I ) = - FOURTH * DFDX ( I, ID )
DXP( I ) = - FOURTH * DFDXP( I, ID )
20 CONTINUE
DO 30 I = 1, NU
DU( I ) = - FOURTH * ( DFDU( I, ID ) + DFDUP( I, ID ) )
30 CONTINUE
DO 40 I = 1, NQ
DQ( I ) = - FOURTH * ( DFDQ( I, ID ) + DFDQP( I, ID ) )
40 CONTINUE
DX ( ID ) = DX ( ID ) - C2
DXP( ID ) = DXP( ID ) + C2
C
C Now include the contributions from the FCF term, using the
C chain rule for partial differentiation.
C
C Derivatives with respect to X and XP.
C
DO 80 I = 1, NX
DO 50 J = 1, NX
DX ( I ) = DX ( I ) - DFDXC( J, ID ) * C1 * DFDX ( I, J )
DXP( I ) = DXP( I ) + DFDXC( J, ID ) * C1 * DFDXP( I, J )
50 CONTINUE
DX ( I ) = DX ( I ) - DFDXC( I, ID ) * HALF
DXP( I ) = DXP( I ) - DFDXC( I, ID ) * HALF
80 CONTINUE
C
C Derivatives with respect to U.
C
DO 120 I = 1, NU
DO 90 J = 1, NX
DU ( I ) = DU ( I ) - DFDXC( J, ID ) *
* C1 * ( DFDU ( I, J ) - DFDUP( I, J ) )
90 CONTINUE
DU ( I ) = DU ( I ) - DFDUC( I, ID )
120 CONTINUE
C
C Derivates with respect to Q.
C
DO 160 I = 1, NQ
DO 130 J = 1, NX
DQ ( I ) = DQ ( I ) - DFDXC( J, ID ) *
* C1 * ( DFDQ ( I, J ) - DFDQP( I, J ) )
130 CONTINUE
DQ ( I ) = DQ ( I ) - DFDQC( I, ID )
160 CONTINUE
C
C Assign the array components to their scalar counterparts.
C
GX1 = DX ( 1 )
GX2 = DX ( 2 )
GX3 = DX ( 3 )
GX4 = DX ( 4 )
GX5 = DX ( 5 )
GX6 = DX ( 6 )
GXP1 = DXP( 1 )
GXP2 = DXP( 2 )
GXP3 = DXP( 3 )
GXP4 = DXP( 4 )
GXP5 = DXP( 5 )
GXP6 = DXP( 6 )
GU1 = DU ( 1 )
GU2 = DU ( 2 )
GU3 = DU ( 3 )
GQ1 = DQ ( 1 )
RETURN
END
BLOCK DATA SETDIM
INTEGER ISTART
COMMON / STRTUP / ISTART
DATA ISTART / 0 /
END
C ** Subroutines for user defined optimal control problem
SUBROUTINE FSYS( DERIV, F, X, U, T, Q, PARAM,
* DFDX, DFDU, DFDQ )
C
C Subroutine defining system of ODE's for given X, U, Q and T.
C
C Given a set of differential equations
C
C dX/dT = F( X, U, Q, T, PARAM ),
C
C returns the values of F and, if DERIV is .TRUE., its Jacobians
C dF/dX, dF/dU and dF/dQ for given input X, U, Q, T and PARAM.
C
C DFDX( I, J ) contains the derivative of F(J) w.r.t. X(I)
C DFDU( I, J ) contains the derivative of F(J) w.r.t. U(I)
C DFDQ( I, J ) contains the derivative of F(J) w.r.t. Q(I)
C
INTEGER NE, NX, NU, NQ, NP
PARAMETER ( NE = 6, NX = 6, NU = 3, NQ = 1, NP = 4 )
DOUBLE PRECISION M, R, S, T, D1OVER, ZERO, ONE, TWO
DOUBLE PRECISION HM, HV, HG, HF
DOUBLE PRECISION ONEPT5, TWOPT5
DOUBLE PRECISION F( NE ), X( NX ), U( NU ), Q( NQ ), PARAM( NP )
DOUBLE PRECISION DFDX( NX, NE ), DFDU( NU, NE ), DFDQ( NQ, NE )
LOGICAL DERIV
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
PARAMETER ( ONEPT5 = 1.5D+0, TWOPT5 = 2.5D+0 )
HM = PARAM( 1 )
HV = PARAM( 2 )
HG = PARAM( 3 )
HF = PARAM( 4 )
M = ONE - HM * T * Q( 1 )
S = X( 1 ) ** 2 + X( 2 ) ** 2 + X( 3 ) ** 2
R = S ** ONEPT5
D1OVER = - ONEPT5 / S ** TWOPT5
C equation 1
F( 1 ) = HV * Q( 1 ) * X( 4 )
IF ( DERIV ) THEN
DFDX( 1, 1 ) = ZERO
DFDX( 2, 1 ) = ZERO
DFDX( 3, 1 ) = ZERO
DFDX( 4, 1 ) = HV * Q( 1 )
DFDX( 5, 1 ) = ZERO
DFDX( 6, 1 ) = ZERO
DFDU( 1, 1 ) = ZERO
DFDU( 2, 1 ) = ZERO
DFDU( 3, 1 ) = ZERO
DFDQ( 1, 1 ) = HV * X( 4 )
END IF
C equation 2
F( 2 ) = HV * Q( 1 ) * X( 5 )
IF ( DERIV ) THEN
DFDX( 1, 2 ) = ZERO
DFDX( 2, 2 ) = ZERO
DFDX( 3, 2 ) = ZERO
DFDX( 4, 2 ) = ZERO
DFDX( 5, 2 ) = HV * Q( 1 )
DFDX( 6, 2 ) = ZERO
DFDU( 1, 2 ) = ZERO
DFDU( 2, 2 ) = ZERO
DFDU( 3, 2 ) = ZERO
DFDQ( 1, 2 ) = HV * X( 5 )
END IF
C equation 3
F( 3 ) = HV * Q( 1 ) * X( 6 )
IF ( DERIV ) THEN
DFDX( 1, 3 ) = ZERO
DFDX( 2, 3 ) = ZERO
DFDX( 3, 3 ) = ZERO
DFDX( 4, 3 ) = ZERO
DFDX( 5, 3 ) = ZERO
DFDX( 6, 3 ) = HV * Q( 1 )
DFDU( 1, 3 ) = ZERO
DFDU( 2, 3 ) = ZERO
DFDU( 3, 3 ) = ZERO
DFDQ( 1, 3 ) = HV * X( 6 )
END IF
C equation 4
F( 4 ) = Q( 1 ) * ( - HG * X( 1 ) / R - HF * U( 1 ) / M )
IF ( DERIV ) THEN
DFDX( 1, 4 ) = - Q( 1 ) * HG * ( ONE / R +
* X( 1 ) * D1OVER * TWO * X( 1 ) )
DFDX( 2, 4 ) = - Q( 1 ) * HG * X( 1 ) * D1OVER * TWO * X( 2 )
DFDX( 3, 4 ) = - Q( 1 ) * HG * X( 1 ) * D1OVER * TWO * X( 3 )
DFDX( 4, 4 ) = ZERO
DFDX( 5, 4 ) = ZERO
DFDX( 6, 4 ) = ZERO
DFDU( 1, 4 ) = - Q( 1 ) * HF / M
DFDU( 2, 4 ) = ZERO
DFDU( 3, 4 ) = ZERO
DFDQ( 1, 4 ) = - HG * X( 1 ) / R - HF * U( 1 ) / M -
* Q( 1 ) * HF * HM * T * U( 1 ) / M ** 2
END IF
C equation 5
F( 5 ) = Q( 1 ) * ( - HG * X( 2 ) / R - HF * U( 2 ) / M )
IF ( DERIV ) THEN
DFDX( 1, 5 ) = - Q( 1 ) * HG * X( 2 ) * D1OVER * TWO * X( 1 )
DFDX( 2, 5 ) = - Q( 1 ) * HG * ( ONE / R +
* X( 2 ) * D1OVER * TWO * X( 2 ) )
DFDX( 3, 5 ) = - Q( 1 ) * HG * X( 2 ) * D1OVER * TWO * X( 3 )
DFDX( 4, 5 ) = ZERO
DFDX( 5, 5 ) = ZERO
DFDX( 6, 5 ) = ZERO
DFDU( 1, 5 ) = ZERO
DFDU( 2, 5 ) = - Q( 1 ) * HF / M
DFDU( 3, 5 ) = ZERO
DFDQ( 1, 5 ) = - HG * X( 2 ) / R - HF * U( 2 ) / M -
* Q( 1 ) * HF * HM * T * U( 2 ) / M ** 2
END IF
C equation 6
F( 6 ) = Q( 1 ) * ( - HG * X( 3 ) / R - HF * U( 3 ) / M )
IF ( DERIV ) THEN
DFDX( 1, 6 ) = - Q( 1 ) * HG * X( 3 ) * D1OVER * TWO * X( 1 )
DFDX( 2, 6 ) = - Q( 1 ) * HG * X( 3 ) * D1OVER * TWO * X( 2 )
DFDX( 3, 6 ) = - Q( 1 ) * HG * ( ONE / R +
* X( 3 ) * D1OVER * TWO * X( 3 ) )
DFDX( 4, 6 ) = ZERO
DFDX( 5, 6 ) = ZERO
DFDX( 6, 6 ) = ZERO
DFDU( 1, 6 ) = ZERO
DFDU( 2, 6 ) = ZERO
DFDU( 3, 6 ) = - Q( 1 ) * HF / M
DFDQ( 1, 6 ) = - HG * X( 3 ) / R - HF * U( 3 ) / M -
* Q( 1 ) * HF * HM * T * U( 3 ) / M ** 2
END IF
RETURN
END