Two Stage CSTR

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

Section 6.3.1 Nonlinear two-stage CSTR system

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

Contents

Problem Description

The system consists of a series of two CSTRs, where there is a transportation delay tau = 0.1 from the first tank to the second. A truncated Taylor series expansion for the time delay.

Find u over t in [0; 2 ] to minimize

$$ J = x_5(t_F) $$

subject to:

$$ \frac{dx_1}{dt} = f_1 $$

$$ \frac{dx_2}{dt} = f_2 $$

$$ \frac{dx_3}{dt} = x_1 - x_3 - tau*f_1 - R_2 + 0.25 $$

$$ \frac{dx_4}{dt} = x_2 - 2*x_4 - u_2*(x_4 + 0.25) - tau*f_2 + R_2 - 0.25 $$

$$ \frac{dx_5}{dt} = x_1^2+ x_2^2 + x_3^2 + x_4^2 + 0.1*(u_1^2 + u_2^2) $$

$$ f_1 = 0.5 - x_1 - R_1 $$

$$ f_2 = -2*(x_2 + 0.25) - u_1*(x_2 + 0.25) + R_1 $$

$$ R_1 = (x_1 + 0.5)*exp(25*\frac{x_2}{x_2 + 2}) $$

$$ R_2 = (x_3 + 0.25)*exp(25*\frac{x_4}{x_4 + 2}) $$

The state variables x1 and x3 are normalized concentration variables in tanks 1 and 2, respectively, and x2 and x4 are normalized temperature variables in tanks 1 and 2, respectively. The variable x5 is introduced to provide the performance index to be minimized.

The initial condition are:

$$ x(0) = [0.15 \ -0.03 \ 0.10 \ 0 \ 0] $$

$$ -0.5 <= u1 <= 0.5 $$

$$ -0.5 <= u2 <= 0.5 $$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 2, 20);
setPhase(p);

tomStates x1 x2 x3 x4 x5
tomControls u1 u2

xi = [0.15;-0.03;0.10;0;0];

% Initial guess
x0 = {icollocate({x1 == xi(1); x2 == xi(2)
    x3 == xi(3); x4 == xi(4); x5 == xi(5)})
    collocate({u1 == 0; u2 == 0})};

% Box constraints
cbox = collocate({-0.5 <= u1 <= 0.5
    -0.5 <= u2 <= 0.5});

% Boundary constraints
cbnd = initial({x1 == xi(1); x2 == xi(2)
    x3 == xi(3); x4 == xi(4); x5 == xi(5)});

% ODEs and path constraints
R1 = (x1 + 0.5).*exp(25*x2./(x2 + 2));
R2 = (x3 + 0.25).*exp(25*x4./(x4 + 2));
f1 = 0.5 - x1 - R1;
f2 = -2*(x2 + 0.25) - u1.*(x2 + 0.25) + R1;
tau = 0.1;
ceq = collocate({
    dot(x1) == f1; dot(x2) == f2
    dot(x3) == x1-x3-tau*f1-R2+0.25
    dot(x4) == x2-2*x4-u2.*(x4+0.25)-tau*f2+R2-0.25
    dot(x5) == x1.^2+ x2.^2+x3.^2+x4.^2+0.1*(u1.^2+u2.^2)});

% Objective
objective = final(x5);

Solve the problem

options = struct;
options.name = 'Two Stage 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: Two Stage CSTR                 f_k       0.023238023992802687
                                       sum(|constr|)      0.000000172957107737
                              f(x_k) + sum(|constr|)      0.023238196949910424
                                              f(x_0)      0.000000000000000000

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

FuncEv    1 ConstrEv   26 ConJacEv   26 Iter   22 MinorIter  108
CPU time: 0.203125 sec. Elapsed time: 0.203000 sec.

Plot result

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

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