PROPT Linear Problem with Bang Bang Control

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

Problem 5a: Miser3 manual

Problem description

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

$J=\int _{0}^{1}-6*x_{1}-12*x_{2}+3*u_{1}+u_{2}\mathrm {d} t$

subject to:

${\frac {dx_{1}}{dt}}=u_{2}$

${\frac {dx_{2}}{dt}}=-x_{1}+u_{1}$

$x_{1}(0)=1$

$x_{2}(0)=0$

$|u|<=10$

% Copyright (c) 2007-2010 by Tomlab Optimization Inc.

Problem setup

toms t
p = tomPhase('p', t, 0, 1, 30);
setPhase(p);

tomStates x1 x2
tomControls -integer u1 u2

% Initial guess
x0 = {icollocate({x1 == 1; x2 == 0})
    collocate({u1 == 0; u2 == 0})};

% Box constraints
cbox = {-10 <= icollocate(x1) <= 10
    -10 <= icollocate(x2) <= 10
    -10 <= collocate(u1)  <= 10
    -10 <= collocate(u2)  <= 10};

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

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

% Objective
objective = integrate(-6*x1-12*x2+3*u1+u2);

Solve the problem

options = struct;
options.name = 'Linear Problem Bang';
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);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);

Problem type appears to be: mip
Time for symbolic processing: 0.04476 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Linear Problem Bang            f_k     -41.377652164326122000
                                       sum(|constr|)      0.000000200077248423
                              f(x_k) + sum(|constr|)    -41.377651964248876000
                                              f(x_0)     -5.999999999999976900

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

FuncEv   30 GradEv   27 HessEv   19 
CPU time: 0.046800 sec. Elapsed time: 0.168000 sec. 

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-');
legend('x1','x2');
title('Linear Problem Bang state variables');

subplot(2,1,2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Linear Problem Bang control');