PROPT Singular Arc Problem

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

Problem 3: Miser3 manual

Problem Formulation

Find u(t) over t in [0; t_f ] to minimize

$J=t_{f}$

subject to:

${\frac {dx_{1}}{dt}}=u$

${\frac {dx_{2}}{dt}}=cos(x_{1})$

${\frac {dx_{3}}{dt}}=sin(x_{1})$

$x_{2}(t_{f})=x_{3}(t_{f})=0$

$|u|<=2$

$x(0)=[{\frac {pi}{2}}\ 4\ 0]$

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 60);
setPhase(p);

tomStates x1 x2 x3
tomControls u

% Initial guess
x0 = {t_f == 20
    icollocate({
    x1 == pi/2+pi/2*t/t_f
    x2 == 4-4*t/t_f; x3 == 0})
    collocate(u == 0)};

% Box constraints
cbox = {2 <= t_f <= 1000
    -2 <= collocate(u) <= 2};

% Boundary constraints
cbnd = {initial({x1 == pi/2; x2 == 4; x3 == 0})
    final({x2 == 0; x3 == 0})};

% ODEs and path constraints
ceq = collocate({dot(x1) == u
    dot(x2) == cos(x1); dot(x3) == sin(x1)});

% Objective
objective = t_f;

Solve the problem

options = struct;
options.name = 'Singular Arc';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
u  = subs(collocate(u),solution);

Problem type appears to be: lpcon
Time for symbolic processing: 0.1042 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Singular Arc                   f_k       4.321198387073171600
                                       sum(|constr|)      0.000000179336713690
                              f(x_k) + sum(|constr|)      4.321198566409885100
                                              f(x_0)     20.000000000000000000

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

FuncEv    1 ConstrEv   77 ConJacEv   77 Iter   70 MinorIter  377
CPU time: 0.483603 sec. Elapsed time: 0.473000 sec. 

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-');
legend('x1','x2','x3');
title('Singular Arc state variables');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Singular Arc control');