PROPT High Dimensional Control

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

Problem 7: DYNOPT User's Guide version 4.1.0

M. Cizniar, M. Fikar, M. A. Latifi, MATLAB Dynamic Optimisation Code DYNOPT. User's Guide, Technical Report, KIRP FCHPT STU Bratislava, Slovak Republic, 2006.

Problem description

Find u over t in [0; 0.2 ] to minimize

$\int _{0}^{0.2}5.8*(q*x_{1}-u_{4})-3.7*u_{1}-4.1*u_{2}+$ $q*(23*x_{4}+11*x_{5}+28*x_{6}+35*x_{7})-5.0*u_{3}^{2}-0.099\mathrm {d} t$

subject to:

${\frac {dx_{1}}{dt}}=u_{4}-q*x_{1}-17.6*x_{1}*x_{2}-23*x_{1}*x_{6}*u_{3}$ ${\frac {dx_{2}}{dt}}=u_{1}-q*x_{2}-17.6*x_{1}*x_{2}-146*x_{2}*x_{3}$ ${\frac {dx_{3}}{dt}}=u_{2}-q*x_{3}-73*x_{2}*x_{3}$ ${\frac {dx_{4}}{dt}}=-q*x_{4}+35.2*x_{1}*x_{2}-51.3*x_{4}*x_{5}$ ${\frac {dx_{5}}{dt}}=-q*x_{5}+219*x_{2}*x_{3}-51.3*x_{4}*x_{5}$ ${\frac {dx_{6}}{dt}}=-q*x_{6}+102.6*x_{4}*x_{5}-23*x_{1}*x_{6}*u_{3}$ ${\frac {dx_{7}}{dt}}=-q*x_{7}+46*x_{1}*x_{6}*u_{3}$

where

$q=u_{1}+u_{2}+u_{4}$ $x(0)=[0.1883\ 0.2507\ 0.0467\ 0.0899\ 0.1804\ 0.1394\ 0.1046]'$ $0<=u_{1}<=20$ $0<=u_{2}<=6$ $0<=u_{3}<=4$ $0<=u_{4}<=20$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 0.2, 20);
setPhase(p);

tomStates x1 x2 x3 x4 x5 x6 x7
tomControls u1 u2 u3 u4

x = [x1; x2; x3; x4; x5; x6; x7];
u = [u1; u2; u3; u4];

x0i = [0.1883;0.2507;0.0467;0.0899;0.1804;0.1394;0.1046];
x0 = icollocate({x1==x0i(1),x2==x0i(2),x3==x0i(3),x4==x0i(4),x5==x0i(5),x6==x0i(6),x7==x0i(7)});

% Box constraints and boundary
uL = zeros(4,1); uU = [20;6;4;20];
cbb = {collocate(uL <= u <= uU)
    initial(x == x0i)};

% ODEs and path constraints
q = u(1)+u(2)+u(4);
ceq = collocate({
    dot(x1) == u4-q.*x1-17.6*x1.*x2-23*x1.*x6.*u3;
    dot(x2) == u1-q.*x2-17.6*x1.*x2-146*x2.*x3;
    dot(x3) == u2-q.*x3-73*x2.*x3;
    dot(x4) == -q.*x4+35.2*x1.*x2-51.3*x4.*x5;
    dot(x5) == -q.*x5+219*x2.*x3-51.3*x4.*x5;
    dot(x6) == -q.*x6+102.6*x4.*x5-23*x1.*x6.*u3;
    dot(x7) == -q.*x7+46*x1.*x6.*u3});

% Objective
objective = integrate(-(5.8*(q.*x1-u4)-3.7*u1-4.1*u2+...
    q.*(23*x4+11*x5+28*x6+35*x7)-5.0*u3.^2-0.099));

Solve the problem

options = struct;
options.name = 'High Dim Control';
solution = ezsolve(objective, {cbb, ceq}, x0, options);

Problem type appears to be: qpcon
Time for symbolic processing: 0.57599 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: High Dim Control               f_k     -21.834326989498166000
                                       sum(|constr|)      0.000000000215774218
                              f(x_k) + sum(|constr|)    -21.834326989282392000
                                              f(x_0)      0.000000000000000000

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

FuncEv    1 ConstrEv   95 ConJacEv   95 Iter   88 MinorIter  489
CPU time: 0.358802 sec. Elapsed time: 0.372000 sec. 

Plot result

figure(1)
ezplot(x);
legend('x1','x2','x3','x4','x5','x6','x7');
title('High Dim Control state variables');

figure(2)
ezplot(u);
legend('u1','u2','u3','u4');
title('High Dim Control control');