$title Hang Glider COPS 2.0 #11 (GLIDER,SEQ=239) $onText Maximize the final horizontal position 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. Keywords: nonlinear programming, engineering, hang glider, trajectory optimization optimal control $offText $if not set nh $set nh 50 Set c 'coordinates' / x 'distance', y 'altitude' / h 'intervals' / h0*h%nh% /; Scalar 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 /; Parameter 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 /; Variable 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 Variable step; Equation 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); Alias (h,i); 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]); 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; * Boundary Conditions 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 /; glider.workSpace = 5; solve glider maximizing final_x using nlp;