Steering : Minimize the Time Taken for a Particle, Acted upon by a Thrust of Constant Magnitude, to Achieve a Given Altitude and Terminal Velocity


  • Neculai Andrei, Nonlinear Optimization Applications Using the GAMS Technology,Springer Optimization and Its Applications, Model Steering (5.39) in chapter Applications of Mechanical Engineering , 2013

Category : GAMS NOA library

Mainfile : steering.gms

Minimize the time take for a particle, acted upon by a thrust of constant
magnitude, to achieve a given altitude and terminal velocity.

This model is from the COPS benchmarking suite.

The number of discretization points can be specified using the command
line parameter --nh.
COPS performance tests have been reported for nh = 50, 100, 200, 400

Dolan, E D, and More, J J, Benchmarking Optimization Software with COPS.
Tech. rep., Mathematics and Computer Science Division, 2000.

Betts, J, Eldersveld, S, and Huffman, W, Sparse Nonlinear Programming Test
Problems. Tech. rep., Boeing Computer Services, 1993.

Bryson, A, and Ho, Y, Applied Optimal Control: Optimization, Estimation,
and Control. John Wiley and Sons, 1975.

$if set n $set nh %n%
$if not set nh $set nh 800

sets h intervals / h0 * h%nh% /
c coordinates /
      y1 first position coordinate
      y2 second position coordinate
      y3 first velocity coordinate
      y4 second velocity coordinate /

scalars pi

nh number of intervals / %nh% /

a magnitude of force / 100.0 / ;

variables u(h)            control
          y(c,h)          coordinates
          tf              final time ;

positive variables step   step size ;

y.l('y2',h) = 5*(ord(h)-1)/nh;
y.l('y3',h) = 45*(ord(h)-1)/nh;
step.l = 1.0/nh;

equations tf_eqn, pos_eqn(c,h), velo1_eqn(h), velo2_eqn(h);

tf_eqn.. tf =e= step*nh;

pos_eqn(c+2,h+1).. y(c,h+1) =e= y(c,h) + 0.5*step*(y(c+2,h) + y(c+2,h+1));

velo1_eqn(h+1).. y('y3',h+1) =e= y('y3',h) + 0.5*step*(a*cos(u(h)) +

velo2_eqn(h+1).. y('y4',h+1) =e= y('y4',h) + 0.5*step*(a*sin(u(h)) +

pi = 2*arctan(inf);

u.lo(h) = -pi/2;
u.up(h) = pi/2;

y.fx(c,'h0') = 0;
y.fx('y2','h%nh%') = 5;
y.fx('y3','h%nh%') = 45;
y.fx('y4','h%nh%') = 0;

model steering /all/;

$iftheni x%mode%==xbook
$onecho >bench.opt
  solvers knitro

option nlp=bench;

solve steering using nlp minimizing tf;

$iftheni x%mode%==xbook
*file res /g7.dat/;
*put res
*loop(h, put u.l(h):10:7, put/)

*End steering