PROPT Nondifferentiable system

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

Global Optimization of Chemical Processes using Stochastic Algorithms 1996, Julio R. Banga, Warren D. Seider

Case Study IV: Optimal control of a nondifferentiable system

Problem description

This problem has been studied by Thomopoulos and Papadakis who report convergence difficulties using several optimization algorithms and by Luus using Iterative Dynamic Programming. The optimal control problem is:

Find u(t) to minimize:

$J=x_{3}(t_{f})$

Subject to:

${\frac {dx_{1}}{dt}}=x_{2}$ ${\frac {dx_{2}}{dt}}=-x_{1}-x_{2}+u+d$ ${\frac {dx_{3}}{dt}}=5*x_{1}^{2}+2.5*x_{2}^{2}+0.5*u^{2}$

with

$d=100*[U(t-0.5)-U(t-0.6)]$

where U(t-alpha) is the unit function such that U = 0 for t - alpha < 0 and U = 1 for t - alpha > 0. Hence d is a rectangular pulse of magnitude 100 from t=0.5 until t=0.6. These authors consider t_f = 2.0s and the initial conditions:

$x(t_{0})=[0\ 0\ 0]'$

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

Problem setup

toms t1
p1 = tomPhase('p1', t1, 0, 0.5, 30);
toms t2
p2 = tomPhase('p2', t2, 0.5, 0.1, 20);
toms t3
p3 = tomPhase('p3', t3, 0.6, 2-0.6, 50);

x1p1 = tomState(p1,'x1p1');
x2p1 = tomState(p1,'x2p1');
x3p1 = tomState(p1,'x3p1');
up1 = tomControl(p1,'up1');

x1p2 = tomState(p2,'x1p2');
x2p2 = tomState(p2,'x2p2');
x3p2 = tomState(p2,'x3p2');
up2 = tomControl(p2,'up2');

x1p3 = tomState(p3,'x1p3');
x2p3 = tomState(p3,'x2p3');
x3p3 = tomState(p3,'x3p3');
up3 = tomControl(p3,'up3');

% Initial guess
x0 = {icollocate(p1,{x1p1 == 0;x2p1 == 0;x3p1 == 0})
    icollocate(p2,{x1p2 == 0;x2p2 == 0;x3p2 == 0})
    icollocate(p3,{x1p3 == 0;x2p3 == 0;x3p3 == 0})
    collocate(p1,up1==-4-8*t1/0.5)
    collocate(p2,up2==-12)
    collocate(p3,up3==-12+14*t3/2)};

% Box constraints
cbox = {
    icollocate(p1,{-100 <= x1p1 <= 100
    -100 <= x2p1 <= 100
    -100 <= x3p1 <= 100})
    icollocate(p2,{-100 <= x1p2 <= 100
    -100 <= x2p2 <= 100
    -100 <= x3p2 <= 100})
    icollocate(p3,{-100 <= x1p3 <= 100
    -100 <= x2p3 <= 100
    -100 <= x3p3 <= 100})
    collocate(p1,-15 <= up1 <= 2)
    collocate(p2,-15 <= up2 <= 2)
    collocate(p3,-15 <= up3 <= 2)};

% Boundary constraints
cbnd = {initial(p1,{x1p1 == 0;x2p1 == 0;x3p1 == 0})
    final(p3,x3p3) <= 60};

% ODEs and path constraints
ceq = {collocate(p1,{
    dot(p1,x1p1) == x2p1
    dot(p1,x2p1) == -x1p1-x2p1+up1
    dot(p1,x3p1) == 5*x1p1.^2+2.5*x2p1.^2+0.5*up1.^2})
    collocate(p2,{
    dot(p2,x1p2) == x2p2
    dot(p2,x2p2) == -x1p2-x2p2+up2+100
    dot(p2,x3p2) == 5*x1p2.^2+2.5*x2p2.^2+0.5*up2.^2})
    collocate(p3,{
    dot(p3,x1p3) == x2p3
    dot(p3,x2p3) == -x1p3-x2p3+up3
    dot(p3,x3p3) == 5*x1p3.^2+2.5*x2p3.^2+0.5*up3.^2})};

% Objective
objective = final(p3,x3p3);

% Link phase
link = {final(p1,x1p1) == initial(p2,x1p2)
    final(p1,x2p1) == initial(p2,x2p2)
    final(p1,x3p1) == initial(p2,x3p2)
    final(p2,x1p2) == initial(p3,x1p3)
    final(p2,x2p2) == initial(p3,x2p3)
    final(p2,x3p2) == initial(p3,x3p3)};

Solve the problem

options = struct;
options.name = 'Nondiff System';
constr = {cbox, cbnd, ceq, link};
solution = ezsolve(objective, constr, x0, options);

t = subs(collocate(p1,t1),solution);
t = [t;subs(collocate(p2,t2),solution)];
t = [t;subs(collocate(p3,t3),solution)];
x1 = subs(collocate(p1,x1p1),solution);
x1 = [x1;subs(collocate(p2,x1p2),solution)];
x1 = [x1;subs(collocate(p3,x1p3),solution)];
x2 = subs(collocate(p1,x2p1),solution);
x2 = [x2;subs(collocate(p2,x2p2),solution)];
x2 = [x2;subs(collocate(p3,x2p3),solution)];
x3 = subs(collocate(p1,x3p1),solution);
x3 = [x3;subs(collocate(p2,x3p2),solution)];
x3 = [x3;subs(collocate(p3,x3p3),solution)];
u = subs(collocate(p1,up1),solution);
u = [u;subs(collocate(p2,up2),solution)];
u = [u;subs(collocate(p3,up3),solution)];

Problem type appears to be: lpcon
Time for symbolic processing: 0.40681 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Nondiff System                 f_k      58.065028469582899000
                                       sum(|constr|)      0.000030760924864613
                              f(x_k) + sum(|constr|)     58.065059230507764000
                                              f(x_0)      0.000000000000000000

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

FuncEv    1 ConstrEv   45 ConJacEv   45 Iter   38 MinorIter 1056
CPU time: 0.561604 sec. Elapsed time: 0.576000 sec. 

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-');
legend('x1','x2','x3');
title('Nondiff System state variable');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Nondiff System control');