Glider : Maximize the Final Horizontal Position of a Hang Glider While in the Presence of a Thermal Updraft

Reference

  • Neculai Andrei, Nonlinear Optimization Applications Using the GAMS Technology,Springer Optimization and Its Applications, Model Glider (5.40) in chapter Applications of Mechanical Engineering , 2013

Category : GAMS NOA library


Mainfile : glider.gms

$ontext
Maximize the final horizontal position of a hang glider while in the
presence of a thermal updraft.

This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.

The number of discretization points can be specified using the GAMS user1
parameter.
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.

Bulirsch, R, Nerz, E, Pesch, H J, and von Stryk, O, Combining Direct and
Indirect Methods in Nonlinear Optimal Control: Range Maximization of a
Hang Glider. In Bulirsch, R, Miele, A, Stoer, J, and Well, K H, Eds,
Optimal Control. Birkhauser Verlag, 1993, pp. 273-288.
$offtext


$if set n $set nh %n%
$if not set nh $set nh 800
sets c coordinates / x distance
                     y altitude /
h intervals        / h0 * h%nh%/ ;

alias(h,i);

scalars nh   Number of intervals in mesh  / %nh% /
cL_min       bound on control variable    / 0.0 /
cL_max       bound on control variable    / 1.4 /
u_c   / 2.5 /
r_0   / 100 /
m     / 100 /
g     / 9.81 /
c0    / 0.034 /
c1    / 0.069662 /
S     / 14 /
rho   / 1.13 / ;

parameters c_0(c) initial position  / x 0, y 1000/
           v_0(c) initial velocity  / x 13.23, y -1.288/
           c_f(c) final position    / y 900/
           v_f(c) final velocity    / x 13.23, y -1.288/ ;

variables t_f
          pos(c,h)   position x distance y altitude
          vel(c,h)   velocity x distance y altitude
          cl(h)      control variables
          r(h)       the r function
          u(h)       the u function
          w(h)       the w function
          v(h)       the v function
          D(h)       the D function
          L(h)       the L function
          v_dot(c,h)
          final_x
          step       step size

positive variables step;

equations tf_eqn
          rdef(h)
          udef(h)
          wdef(h)
          vdef(h)
          Ddef(h)
          Ldef(h)
          vx_dot_def(h)
          vy_dot_def(h)
          obj
          pos_eqn(c,h)
          vel_eqn(c,h);

tf_eqn.. t_f =e= step*nh;

rdef(i).. r[i] =e= sqr(pos['x',i]/r_0 - 2.5);

udef(i).. u[i] =e= u_c*(1-r[i])*exp(-r[i]);

wdef(i).. w[i] =e= vel['y',i] - u[i];

vdef(i).. v[i] =e= sqrt(sqr(vel['x',i]) + sqr(w[i]));

Ddef(i).. D[i] =e= .5*(c0+c1*sqr(cL[i]))*rho*S*sqr(v[i]);

Ldef(i).. L[i] =e= .5* cL[i] *rho*S*sqr(v[i]);

vx_dot_def(i).. v_dot['x',i] =e= (-L[i]*w[i]/v[i] - D[i]*vel['x',i]/v[i])/m;

vy_dot_def(i).. v_dot['y',i] =e= ( L[i]*vel['x',i]/v[i] - D[i]*w[i]/v[i])/m - g;

obj.. final_x =e= pos('x','h%nh%');

pos_eqn(c,i-1).. pos[c,i] =e= pos[c,i-1] + .5*step*(vel[c,i] + vel[c,i-1]);

vel_eqn(c,i-1).. vel[c,i] =e= vel[c,i-1] + .5*step*(v_dot[c,i] + v_dot[c,i-1]);

* Boundary Conditions
cl.lo(h) = cL_min;
cl.up(h) = cL_max;
pos.lo('x',h) = 0;
vel.lo('x',h) = 0;
v.lo(h) = 0.01;

* Fixed values
pos.fx(c,'h0') = c_0(c);
pos.fx('y','h%nh%') = c_f('y');
vel.fx(c,'h0') = v_0(c);
vel.fx(c,'h%nh%') = v_f(c);

* Initial point
pos.l('x',h) = c_0('x') + v_0('x')*((ord(h)-1)/nh);
pos.l('y',h) = c_0('y') + ((ord(h)-1)/nh)*(c_f('y') - c_0('y'));
vel.l(c,h) = v_0(c);
cL.l(h) = cL_max/2;
step.l = 1.0/nh;

* Initial values for intermediate variables
t_f.l = step.l*nh;
r.l[i] = sqr(pos.l['x',i]/r_0 - 2.5);
u.l[i] = u_c*(1-r.l[i])*exp(-r.l[i]);
w.l[i] = vel.l['y',i] - u.l[i];
v.l[i] = sqrt(sqr(vel.l['x',i]) + sqr(w.l[i]));
D.l[i] = .5*(c0+c1*sqr(cL.l[i]))*rho*S*sqr(v.l[i]);
L.l[i] = .5* cL.l[i] *rho*S*sqr(v.l[i]);
v_dot.l['x',i] = (-L.l[i]*w.l[i]/v.l[i] - D.l[i]*vel.l['x',i]/v.l[i])/m;
v_dot.l['y',i] = ( L.l[i]*vel.l['x',i]/v.l[i] - D.l[i]*w.l[i]/v.l[i])/m - g;

model glider /all/;

$iftheni x%mode%==xbook
$onecho >bench.opt
  solvers conopt knitro
$offecho
glider.optfile=1;

option nlp=conopt;

glider.workspace = 40;
$endif

solve glider maximizing final_x using nlp;

$iftheni x%mode%==xbook
file res /g8.dat/;
put res
loop(h, put vel.l('y',h):10:7, put/)
$endif


*     --------------------------- Numerical Experiments -------------------
*                                   January 25, 2011
*
* For nh=1200 I got:
* CONOPT: 1547 iterations, 217.242 seconds, vfo=1247.9859370
* KNITRO: 604  iterations, 631.468 seconds, vfo=1247.9858984
*------
* End Glider