Two-Phase Schwartz

Users Guide for dyn.Opt, Example 4

Schwartz, A. L., Theory and Implementation of Numerical Methods based on Runge-Kutta Integration for Solving Optimal Control Problems. Ph.D. Dissertation, University of California, Berkeley, 1989

Contents

Problem Formulation

Find u over t in [0; 2.9 ] to minimize

$$ J = 5*(x_1(t_F)^2 + x_2(t_F)^2) $$

subject to:

$$ \frac{dx_1}{dt} = x_2 $$

$$ \frac{dx_2}{dt} = u - 0.1*(1+2*x_1^2)*x_2 $$

$$ x(0) = [1 \ 1] $$

and path constraints for t<1:

$$ 1-9*(x_1-1)^2-(\frac{x_2-0.4}{0.3})^2 <=0 $$

$$ -0.8-x_2 <= 0 \ --> \ -0.8 <= x_2$$

$$ -1 <= u <= 1, (t<1) $$

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

Problem setup

toms t1
p1 = tomPhase('p1', t1, 0, 1, 25);
toms t2
p2 = tomPhase('p2', t2, 1, 1.9, 25);

setPhase(p1);
tomStates x1p1 x2p1
tomControls up1

setPhase(p2);
tomStates x1p2 x2p2
tomControls up2

setPhase(p1);
% Initial guess
x01 = {icollocate({x1p1 == 1; x2p1 == 1})
    collocate(up1==0)};

% Box constraints
cbox1 = {-0.8 <= icollocate(x2p1)
    -1 <= collocate(up1) <= 1};

% Boundary constraints
cbnd1 = initial({x1p1 == 1; x2p1 == 1});

% ODEs and path constraints
ceq1 = collocate({
    dot(x1p1) == x2p1
    dot(x2p1) == up1 - 0.1*(1+2*x1p1.^2).*x2p1
    1-9*(x1p1-1).^2-((x2p1-0.4)/0.3).^2 <= 0});

setPhase(p2);
% Initial guess
x02 = {icollocate({x1p2 == 1; x2p2 == 1})
    collocate(up2==0)};

% Box constraints
cbox2 = {-50 <= collocate(up2) <= 50};

% ODEs and path constraints
ceq2 = collocate({
    dot(x1p2) == x2p2
    dot(x2p2) == up2-0.1*(1+2*x1p2.^2).*x2p2});

% Link phase
link = {final(p1,x1p1) == initial(p2,x1p2)
    final(p1,x2p1) == initial(p2,x2p2)};

% Objective
objective = 5*(final(p2,x1p2)^2+final(p2,x2p2)^2);

Solve the problem

options = struct;
options.name = 'Two Phase Schwartz';
constr = {cbox1, cbnd1, ceq1, cbox2, ceq2, link};
solution = ezsolve(objective, constr, {x01, x02}, options);

t = subs(collocate(p1,t1),solution);
t = [t;subs(collocate(p2,t2),solution)];
x1 = subs(collocate(p1,x1p1),solution);
x1 = [x1;subs(collocate(p2,x1p2),solution)];
x2 = subs(collocate(p1,x2p1),solution);
x2 = [x2;subs(collocate(p2,x2p2),solution)];
u = subs(collocate(p1,up1),solution);
u = [u;subs(collocate(p2,up2),solution)];

Problem type appears to be: qpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Two Phase Schwartz             f_k       0.000000000000034700
                                       sum(|constr|)      0.000000000045020400
                              f(x_k) + sum(|constr|)      0.000000000045055100
                                              f(x_0)     10.000000000000014000

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

FuncEv    1 ConstrEv   19 ConJacEv   19 Iter   18 MinorIter  260
CPU time: 0.125000 sec. Elapsed time: 0.125000 sec.

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-');
legend('x1','x2');
title('Two Phase Schwartz state variables');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Two Phase Schwartz control');