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50  High Dimensional Control

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.

50.1  Problem description

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

∫

0.2 0

5.8*(q*x1−u4)−3.7*u1−4.1*u2+

q*(23*x4+11*x5+28*x6+35*x7)−5.0*u32−0.099 dt

subject to:

dx1 dt

= u4−q*x1−17.6*x1*x2−23*x1*x6*u3

dx2 dt

= u1−q*x2−17.6*x1*x2−146*x2*x3

dx3 dt

= u2−q*x3−73*x2*x3

dx4 dt

= −q*x4+35.2*x1*x2−51.3*x4*x5

dx5 dt

= −q*x5+219*x2*x3−51.3*x4*x5

dx6 dt

= −q*x6+102.6*x4*x5−23*x1*x6*u3

dx7 dt

= −q*x7+46*x1*x6*u3

where

q = u1+u2+u4

x(0) = [0.1883  0.2507  0.0467  0.0899  0.1804  0.1394  0.1046]′

0 <= u1 <= 20

0 <= u2 <= 6

0 <= u3 <= 4

0 <= u4 <= 20

Reference: [13]

50.2  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));

50.3  Solve the problem

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

Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: High Dim Control               f_k     -21.834326989498084000
                                       sum(|constr|)      0.000000000215730497
                              f(x_k) + sum(|constr|)    -21.834326989282353000
                                              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  488
CPU time: 1.234375 sec. Elapsed time: 1.250000 sec.

50.4  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');

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