Grusins Metric

A Short Introduction to Optimal Control, Ugo Boscain, SISSA, Italy

3.4 A Detailed Application: the Grusin's Metric

Contents

Problem Description

Find u over t in [0; 1 ] to minimize:

$$ J = \int_0^{1} u_1^2+u_2^2 \mathrm{d}t $$

subject to:

$$ \frac{dx_1}{dt} = u_1 $$

$$ \frac{dx_2}{dt} = u_2*x_1 $$

$$ x(t_0) = [0 \ 0 ] $$

$$ x(t_f) = [-0.001 \ -1 ] $$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 1, 60);
setPhase(p);

tomStates x1 x2
tomControls u1 u2

% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0})
    final({x1 == -0.01; x2 == -1})};

% ODEs and path constraints
ceq = collocate({dot(x1) == u1
    dot(x2) == u2.*x1});

% Objective
objective = integrate(u1.^2+u2.^2);

Solve the problem

options = struct;
options.name = 'Grusins Metric';
solution = ezsolve(objective, {cbnd, ceq}, [], options);
t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);

Problem type appears to be: qpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Grusins Metric                 f_k       6.333416558770673900
                                       sum(|constr|)      0.000000007863215256
                              f(x_k) + sum(|constr|)      6.333416566633888900
                                              f(x_0)      0.000000000000000000

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

FuncEv    1 ConstrEv   81 ConJacEv   81 Iter   74 MinorIter  326
CPU time: 1.062500 sec. Elapsed time: 1.062000 sec.

Plot result

subplot(2,1,1)
plot(x1,x2,'*-');
legend('x1 vs x2');
title('Grusins Metric state variables');

subplot(2,1,2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Grusins Metric control');