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CORKSCRW.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME CORKSCRW
* Problem :
* *********
* A nonlinear optimal control problem with both state- and
* control constraints.
* The problem is to control (using an applied force of limited
* magnitude) a mass moving in the 3D space, such that its
* trajectory lies within a prescribed distance TOL of the
* corkscreww-like curve defined by
* y = sin(x), z = cos(x),
* and such that it stops at a given abscissa XT in minimum time.
* The mass is initially stationary at (0,0,1).
* Source:
* Ph. Toint, private communication.
* SIF input: Ph. Toint, April 1991.
* classification SOR2-AN-V-V
* Number of time intervals
* The number of variables is 9T+6, of which 9 are fixed.
*IE T 10 $-PARAMETER n = 96 original value
*IE T 50 $-PARAMETER n = 456
*IE T 100 $-PARAMETER n = 906
IE T 500 $-PARAMETER n = 4506
*IE T 1000 $-PARAMETER n = 9006
* Target abscissa
RE XT 10.0 $-PARAMETER target abscissa
* Mass
RE MASS 0.37 $-PARAMETER mass
* Tolerance along the sine trajectory
RE TOL 0.1 $-PARAMETER tolerance
* Constants
IE 0 0
IE 1 1
* Useful parameters
RI RT T
RA T+1 RT 1.0
R/ H XT RT
RD 1/H H 1.0
RM -1/H 1/H -1.0
R/ M/H MASS H
RM -M/H M/H -1.0
R* TOLSQ TOL TOL
R* XTT+1 XT T+1
RM W XTT+1 0.5
DO I 1 T
RI RI I
R* TI RI H
A/ W/T(I) W TI
ND
* Maximal force at any time
R/ FMAX XT RT
RM -FMAX FMAX -1.0
VARIABLES
DO I 0 T
X X(I)
X Y(I)
X Z(I)
X VX(I)
X VY(I)
X VZ(I)
ND
DO I 1 T
X UX(I)
X UY(I)
X UZ(I)
ND
GROUPS
DO I 1 T
ZN OX(I) 'SCALE' W/T(I)
XN OX(I) X(I) 1.0
ND
DO I 1 T
IA I-1 I -1
ZE ACX(I) VX(I) M/H
ZE ACX(I) VX(I-1) -M/H
XE ACX(I) UX(I) -1.0
ZE ACY(I) VY(I) M/H
ZE ACY(I) VY(I-1) -M/H
XE ACY(I) UY(I) -1.0
ZE ACZ(I) VZ(I) M/H
ZE ACZ(I) VZ(I-1) -M/H
XE ACZ(I) UZ(I) -1.0
ZE PSX(I) X(I) 1/H
ZE PSX(I) X(I-1) -1/H
XE PSX(I) VX(I) -1.0
ZE PSY(I) Y(I) 1/H
ZE PSY(I) Y(I-1) -1/H
XE PSY(I) VY(I) -1.0
ZE PSZ(I) Z(I) 1/H
ZE PSZ(I) Z(I-1) -1/H
XE PSZ(I) VZ(I) -1.0
XL SC(I)
ND
CONSTANTS
DO I 1 T
Z CORKSCRW OX(I) XT
Z CORKSCRW SC(I) TOLSQ
ND
BOUNDS
FR CORKSCRW 'DEFAULT'
XX CORKSCRW X(0) 0.0
XX CORKSCRW Y(0) 0.0
XX CORKSCRW Z(0) 1.0
XX CORKSCRW VX(0) 0.0
XX CORKSCRW VY(0) 0.0
XX CORKSCRW VZ(0) 0.0
XX CORKSCRW VX(T) 0.0
XX CORKSCRW VY(T) 0.0
XX CORKSCRW VZ(T) 0.0
DO I 1 T
ZL CORKSCRW UX(I) -FMAX
ZU CORKSCRW UX(I) FMAX
ZL CORKSCRW UY(I) -FMAX
ZU CORKSCRW UY(I) FMAX
ZL CORKSCRW UZ(I) -FMAX
ZU CORKSCRW UZ(I) FMAX
XL CORKSCRW X(I) 0.0
ZU CORKSCRW X(I) XT
ND
START POINT
XV CORKSCRW X(0) 0.0
XV CORKSCRW Y(0) 0.0
XV CORKSCRW Z(0) 1.0
XV CORKSCRW VX(0) 0.0
XV CORKSCRW VY(0) 0.0
XV CORKSCRW VZ(0) 0.0
XV CORKSCRW VX(T) 0.0
XV CORKSCRW VY(T) 0.0
XV CORKSCRW VZ(T) 0.0
DO I 1 T
RI RI I
R* TI RI H
Z CORKSCRW X(I) TI
X CORKSCRW VX(I) 1.0
ND
ELEMENT TYPE
EV ERRSIN X Y
EV ERRCOS X Z
ELEMENT USES
DO I 1 T
XT ES(I) ERRSIN
ZV ES(I) X X(I)
ZV ES(I) Y Y(I)
XT EC(I) ERRCOS
ZV EC(I) X X(I)
ZV EC(I) Z Z(I)
ND
GROUP TYPE
GV L2 GVAR
GROUP USES
DO I 1 T
XT OX(I) L2
XE SC(I) ES(I) EC(I)
ND
OBJECT BOUND
* Solution
*LO SOLTN(10) 1.1601050195
*LO SOLTN(50) 26.484181830
*LO SOLTN(100) 44.368110588
*LO SOLTN(500)
*LO SOLTN(1000)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS CORKSCRW
TEMPORARIES
R SINX
R COSX
R ERR
M SIN
M COS
INDIVIDUALS
T ERRSIN
A SINX SIN(X)
A COSX COS(X)
A ERR Y - SINX
F ERR * ERR
G X - 2.0 * ERR * COSX
G Y 2.0 * ERR
H X X 2.0 * ( COSX**2 + ERR * SINX )
H X Y -2.0 * COSX
H Y Y 2.0
T ERRCOS
A SINX SIN(X)
A COSX COS(X)
A ERR Z - COSX
F ERR * ERR
G X 2.0 * ERR * SINX
G Z 2.0 * ERR
H X X 2.0 * ( SINX**2 + ERR * COSX )
H X Z 2.0 * SINX
H Z Z 2.0
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
GROUPS CORKSCRW
* Least-square groups
INDIVIDUALS
T L2
F GVAR * GVAR
G GVAR + GVAR
H 2.0
ENDATA