Singular CSTR

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

10.4 Nonlinear two-stage CSTR problem

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

Contents

Problem Formulation

Find u over t in [0; tF ] to minimize:

$$ J = x(t_F)'*x(t_F) + t_F $$

(the state variables are moved to bounds)

subject to:

$$ \frac{dx_1}{dt} = -3*x_1+g_1 $$

$$ \frac{dx_2}{dt} = -11.1558*x_2+g_1-8.1558*(x_2+0.1592)*u_1 $$

$$ \frac{dx_3}{dt} = 1.5*(0.5*x_1-x_3)+g_2 $$

$$ \frac{dx_4}{dt} = 0.75*x_2-4.9385*x_4+g_2-3.4385*(x_4+0.122)*u_2 $$

$$ g_1 = 1.5e7*(0.5251-x_1)*exp(-\frac{10}{x_2+0.6932})- $$

$$ 1.5e10*(0.4748+x_1)*exp(-\frac{15}{x_2+0.6932}) - 1.4280 $$

$$ g_2 = 1.5e7*(0.4236-x_2)*exp(-\frac{10}{x_4+0.6560})- $$

$$ 1.5e10*(0.5764+x_3)*exp(-\frac{15}{x4+0.6560}) - 0.5086 $$

The initial condition are:

$$ x(0) = [0.1962 \ -0.0372 \ 0.0946 \ 0] $$

$$ -1 <= u(1:2) <= 1 $$

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

Problem setup

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

tomStates x1 x2 x3 x4
tomControls u1 u2

% Initial guess
x0 = {tf == 0.3
    icollocate({x1 == 0.1962; x2 == -0.0372
    x3 == 0.0946; x4 == 0})
    collocate({u1 == 0; u2 == 0})};

% Box constraints
cbox = {0.1 <= tf <= 100
    -1 <= collocate(u1) <= 1
    -1 <= collocate(u2) <= 1};

% Boundary constraints
cbnd = {initial({x1 == 0.1962; x2 == -0.0372
    x3 == 0.0946; x4 == 0})
    final({x1 == 0; x2 == 0
    x3 == 0; x4 == 0})};

% ODEs and path constraints
g1 = 1.5e7*(0.5251-x1).*exp(-10./(x2+0.6932)) ...
    - 1.5e10*(0.4748+x1).*exp(-15./(x2+0.6932)) - 1.4280;
g2 = 1.5e7*(0.4236-x2).*exp(-10./(x4+0.6560)) ...
    - 1.5e10*(0.5764+x3).*exp(-15./(x4+0.6560)) - 0.5086;

ceq = collocate({
    dot(x1) == -3*x1+g1
    dot(x2) == -11.1558*x2+g1-8.1558*(x2+0.1592).*u1
    dot(x3) == 1.5*(0.5*x1-x3)+g2
    dot(x4) == 0.75*x2-4.9385*x4+g2-3.4385*(x4+0.122).*u2});

% Objective
objective = tf;

Solve the problem

options = struct;
options.name = 'Singular CSTR';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
x4 = subs(collocate(x4),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Singular CSTR                  f_k       0.324402684069356020
                                       sum(|constr|)      0.000000010809182049
                              f(x_k) + sum(|constr|)      0.324402694878538070
                                              f(x_0)      0.299999999999999990

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

FuncEv    1 ConstrEv   75 ConJacEv   75 Iter   42 MinorIter  478
CPU time: 0.546875 sec. Elapsed time: 0.562000 sec.

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Singular CSTR state variables');

subplot(2,1,2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Singular CSTR control');