PROPT Linear Tangent Steering Problem

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

Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABORATORY

Problem Formulation

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

$J=t_{f}$

subject to:

${\frac {d^{2}y_{1}}{dt^{2}}}=a*cos(u)$ ${\frac {d^{2}y_{2}}{dt^{2}}}=a*sin(u)$ $|u|<={\frac {pi}{2}}$

$y_{1:2}(0)=0$ ${\frac {dy_{1:2}}{dt}}=0$ $a=1$ $y_{2}(f)=5$ ${\frac {dy_{1:2}}{dt}}(f)=[45\ 0]$

The following transformation gives a new formulation:

$x_{1}=y_{1}$ $x_{2}={\frac {dy_{1}}{dt}}$ $x_{3}=y_{2}$ $x_{4}={\frac {dy_{2}}{dt}}$

${\frac {dx_{1}}{dt}}=x_{2}$ ${\frac {dx_{2}}{dt}}=a*cos(u)$ ${\frac {dx_{3}}{dt}}=x_{4}$ ${\frac {dx_{4}}{dt}}=a*sin(u)$

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

Problem setup

toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 30);
setPhase(p);

tomStates x1 x2 x3 x4
tomControls u

% Initial guess
x0 = {t_f == 1
    icollocate({
    x1 == 12*t/t_f
    x2 == 45*t/t_f
    x3 == 5*t/t_f
    x4 == 0})};

% Box constraints
cbox = {sqrt(eps) <= t_f
    -pi/2 <= collocate(u) <= pi/2};

% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0; x3 == 0; x4 == 0})
    final({x2 == 45; x3 == 5; x4 == 0})};

% ODEs and path constraints
a = 100;
ceq = collocate({dot(x1) == x2
    dot(x2) == a*cos(u)
    dot(x3) == x4
    dot(x4) == a*sin(u)});

% Objective
objective = t_f;

Solve the problem

options = struct;
options.name = 'Linear Tangent Steering';
options.solver = 'knitro';
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);
x4 = subs(collocate(x4),solution);
u  = subs(collocate(u),solution);

Problem type appears to be: lpcon
Time for symbolic processing: 0.12675 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Linear Tangent Steering        f_k       0.554570876848740510
                                       sum(|constr|)      0.000053415505390106
                              f(x_k) + sum(|constr|)      0.554624292354130580
                                              f(x_0)      1.000000000000000000

Solver: KNITRO.  EXIT=0.  INFORM=0.
Default NLP KNITRO
Locally optimal solution found

FuncEv   14 GradEv  152 ConstrEv   13 ConJacEv  152 Iter   11 MinorIter   12
CPU time: 0.093601 sec. Elapsed time: 0.106000 sec. 

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Linear Tangent Steering state variables');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Linear Tangent Steering control');