Per: /* Plot result */

2012-02-14T09:29:32Z

Plot result

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	[[File:singularCSTR_01.png]]		[[File:singularCSTR_01.png]]

			[[Category:PROPT Examples]]

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{{Part Of Manual|title=the PROPT Manual|link=[[PROPT|PROPT Manual]]}}

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; t_F ] to minimize:

<math> J = x(t_F)'*x(t_F) + t_F </math>

(the state variables are moved to bounds)

subject to:

<math> \frac{dx_1}{dt} = -3*x_1+g_1 </math>
<math> \frac{dx_2}{dt} = -11.1558*x_2+g_1-8.1558*(x_2+0.1592)*u_1 </math>
<math> \frac{dx_3}{dt} = 1.5*(0.5*x_1-x_3)+g_2 </math>
<math> \frac{dx_4}{dt} = 0.75*x_2-4.9385*x_4+g_2-3.4385*(x_4+0.122)*u_2 </math>

<math> g_1 = 1.5e7*(0.5251-x_1)*exp(-\frac{10}{x_2+0.6932})- </math>
<math> 1.5e10*(0.4748+x_1)*exp(-\frac{15}{x_2+0.6932}) - 1.4280 </math>
<math> g_2 = 1.5e7*(0.4236-x_2)*exp(-\frac{10}{x_4+0.6560})- </math>
<math> 1.5e10*(0.5764+x_3)*exp(-\frac{15}{x4+0.6560}) - 0.5086 </math>

The initial condition are:

<math> x(0) = [0.1962 \ -0.0372 \ 0.0946 \ 0] </math>
<math> -1 <= u(1:2) <= 1 </math>

<source lang="matlab">
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
</source>

==Problem setup==

<source lang="matlab">
toms t t_f
p = tomPhase('p', t, 0, t_f, 30);
setPhase(p)

tomStates x1 x2 x3 x4
tomControls u1 u2

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

% Box constraints
cbox = {0.1 <= t_f <= 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 = t_f;
</source>

==Solve the problem==

<source lang="matlab">
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);
</source>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.51719 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Singular CSTR f_k 0.324402684069356740
sum(|constr|) 0.000000010809237995
f(x_k) + sum(|constr|) 0.324402694878594740
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 427
CPU time: 0.187201 sec. Elapsed time: 0.192000 sec.

</pre>

==Plot result==

<source lang="matlab">
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');
</source>

[[File:singularCSTR_01.png]]

← Older revision		Revision as of 07:12, 9 November 2011
Line 12:		Line 12:

	<math> J = x(t_F)'*x(t_F) + t_F </math>		<math> J = x(t_F)'*x(t_F) + t_F </math>


	(the state variables are moved to bounds)		(the state variables are moved to bounds)
Line 18:		Line 19:

	<math> \frac{dx_1}{dt} = -3*x_1+g_1 </math>		<math> \frac{dx_1}{dt} = -3*x_1+g_1 </math>

	<math> \frac{dx_2}{dt} = -11.1558x_2+g_1-8.1558(x_2+0.1592)*u_1 </math>		<math> \frac{dx_2}{dt} = -11.1558x_2+g_1-8.1558(x_2+0.1592)*u_1 </math>

	<math> \frac{dx_3}{dt} = 1.5(0.5x_1-x_3)+g_2 </math>		<math> \frac{dx_3}{dt} = 1.5(0.5x_1-x_3)+g_2 </math>

	<math> \frac{dx_4}{dt} = 0.75x_2-4.9385x_4+g_2-3.4385(x_4+0.122)u_2 </math>		<math> \frac{dx_4}{dt} = 0.75x_2-4.9385x_4+g_2-3.4385(x_4+0.122)u_2 </math>


	<math> g_1 = 1.5e7(0.5251-x_1)exp(-\frac{10}{x_2+0.6932})- </math>		<math> g_1 = 1.5e7(0.5251-x_1)exp(-\frac{10}{x_2+0.6932})- </math>

	<math> 1.5e10(0.4748+x_1)exp(-\frac{15}{x_2+0.6932}) - 1.4280 </math>		<math> 1.5e10(0.4748+x_1)exp(-\frac{15}{x_2+0.6932}) - 1.4280 </math>

	<math> g_2 = 1.5e7(0.4236-x_2)exp(-\frac{10}{x_4+0.6560})- </math>		<math> g_2 = 1.5e7(0.4236-x_2)exp(-\frac{10}{x_4+0.6560})- </math>

	<math> 1.5e10(0.5764+x_3)exp(-\frac{15}{x4+0.6560}) - 0.5086 </math>		<math> 1.5e10(0.5764+x_3)exp(-\frac{15}{x4+0.6560}) - 0.5086 </math>


	The initial condition are:		The initial condition are:

	<math> x(0) = [0.1962 \ -0.0372 \ 0.0946 \ 0] </math>		<math> x(0) = [0.1962 \ -0.0372 \ 0.0946 \ 0] </math>

	<math> -1 <= u(1:2) <= 1 </math>		<math> -1 <= u(1:2) <= 1 </math>


	<source lang="matlab">		<source lang="matlab">

PROPT Singular CSTR - Revision history

Per: /* Plot result */

Mbot at 12:12, 9 November 2011

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS 10.4 Nonlinear two-stage CSTR problem CHAPMAN & HALL/CRC Monogra..."