# 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

## Problem Formulation

Find u over t in [0; tF ] to minimize: (the state variables are moved to bounds)

subject to:        The initial condition are:  % 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'); 