PROPT Goddard Rocket, Maximum Ascent, Final time fixed, Singular solution

This page is part of the PROPT Manual. See PROPT Manual.

Example 7.2 (ii) from the paper: H. Maurer, "Numerical solution of singular control problems using multiple shooting techniques", Journal of Optimization Theory and Applications, Vol. 18, No. 2, 1976, pp. 235-257

Remark: You can vary the fixed final time t_f to obtain Fig. 8 in the paper

L.G. van Willigenburg, W.L. de Koning

Copyright (c) 2007-2009 by Tomlab Optimization Inc.

Problem setup

toms t

% Parameters
aalpha = 0.01227; bbeta = 0.145e-3; c = 2060; g0 = 9.81;
r0 = 6.371e6; r02=r0*r0; m0 = 214.839; mf = 67.9833; Fm = 9.525515;
t_f = 100; %Paper value 206.661;

Solve the problem, using a successively larger number of collocation points

nvec = [20 40 60];
for i=1:length(nvec);

    n = nvec(i);

    p = tomPhase('p', t, 0, t_f, n);
    setPhase(p);
    tomStates h v m
    tomControls F

    % Initial guess
    if i==1
        x0 = {icollocate({v == 10*t; h == 10*t^2
            m == m0+(t/t_f)*(mf-m0)})
            collocate(F == Fm)};
    else
        x0 = {icollocate({v == vopt; h == hopt
            m == mopt})
            collocate(F == Fopt)};
    end

    % Box constraints
    cbox = {icollocate({0 <= v; 0 <= h
        mf <= m <= m0
        0 <= F <= Fm})};

    % Boundary constraints
    cbnd = {initial({v == 0; h == 0; m == m0})
        final({m == mf})};

    D = aalpha*v.^2.*exp(-bbeta*h);
    g = g0; % or g0*r02./(r0+h).^2;

    % ODEs and path constraints
    ceq = collocate({dot(h) == v
        m*dot(v) == F*c-D-g*m
        dot(m) == -F});

    % Objective
    objective = -final(h);

Solve the problem

    options = struct;
    options.name = 'Goddard Rocket';
    if i==1
        options.solver = 'multiMin';
        options.xInit = 20;
    end
    %options.scale = 'auto'
    solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

    % Optimal v and more to use as starting guess
    vopt = subs(v, solution);
    hopt = subs(h, solution);
    mopt = subs(m, solution);
    Fopt = subs(F, solution);

Problem type appears to be: lpcon
Time for symbolic processing: 0.12794 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Goddard Rocket - Trial 11      f_k -108076.040001956850000000
                                       sum(|constr|)      0.000047359539114820
                              f(x_k) + sum(|constr|)-108076.039954597310000000

Solver: multiMin with local solver snopt.  EXIT=0.  INFORM=0.
Find local optima using multistart local search
Did 20 local tries. Found 1 global, 1 minima. TotFuncEv 20. TotConstrEv 3120

FuncEv   20 ConstrEv 3120 ConJacEv   35 Iter 1429 
CPU time: 2.199614 sec. Elapsed time: 2.181000 sec. 

Problem type appears to be: lpcon
Time for symbolic processing: 0.12647 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Goddard Rocket                 f_k -108220.931727365110000000
                                       sum(|constr|)      0.000379507699263397
                              f(x_k) + sum(|constr|)-108220.931347857400000000
                                              f(x_0)-108076.040001956280000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   20 ConJacEv   20 Iter   19 MinorIter  536
CPU time: 0.078001 sec. Elapsed time: 0.072000 sec. 

Problem type appears to be: lpcon
Time for symbolic processing: 0.12744 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Goddard Rocket                 f_k -108245.171657268930000000
                                       sum(|constr|)      0.000109411007603940
                              f(x_k) + sum(|constr|)-108245.171547857930000000
                                              f(x_0)-108220.931727364890000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   31 ConJacEv   31 Iter   25 MinorIter 1128
CPU time: 0.280802 sec. Elapsed time: 0.272000 sec. 

end

t = subs(collocate(t),solution);
v = subs(collocate(vopt),solution);
h = subs(collocate(hopt),solution);
m = subs(collocate(mopt),solution);
F = subs(collocate(Fopt),solution);

Plot result

subplot(2,1,1)
plot(t,v/1e3,'*-',t,h/1e5,'*-',t,m/1e2,'*-');
legend('v','h','m');
title('Goddard Rocket state variables');

subplot(2,1,2)
plot(t,F,'+-');
legend('F');
title('Goddard Rocket control');