PROPT Linear Gas Absorber - Revision history

Per: /* Plot result */

2012-02-14T09:22:04Z

Plot result

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	[[File:linearGasAbsorber_01.png]]		[[File:linearGasAbsorber_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:09, 9 November 2011

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	<math> J = \int_{0}^{10} x'x + u'u \mathrm{d}t </math>		<math> J = \int_{0}^{10} x'x + u'u \mathrm{d}t </math>


	subject to:		subject to:

	<math> \frac{dx}{dt} = Ax + Bu </math>		<math> \frac{dx}{dt} = Ax + Bu </math>


	A = tridiag(0.538998 -1.173113 0.634115)		A = tridiag(0.538998 -1.173113 0.634115)
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	<math> 0 <= u1 <= inf </math>		<math> 0 <= u1 <= inf </math>

	<math> 0 <= u2 <= inf </math>		<math> 0 <= u2 <= inf </math>


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

Mbot at 13:06, 8 November 2011

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	<pre>		<pre>
	Problem type appears to be: qp		Problem type appears to be: qp
	Time for symbolic processing: 0.~~059353~~ seconds		Time for symbolic processing: 0.058971 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 15 GradEv 15 ConstrEv 15 Iter 15		FuncEv 15 GradEv 15 ConstrEv 15 Iter 15
	CPU time: 0.~~031200~~ sec. Elapsed time: 0.~~019000~~ sec.		CPU time: 0.015600 sec. Elapsed time: 0.011000 sec.

	</pre>		</pre>

Mbot at 18:24, 2 November 2011

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

	ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS		ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
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	title('Linear Gas Absorber control');		title('Linear Gas Absorber control');
	</source>		</source>

			[[File:linearGasAbsorber_01.png]]

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS Section 7.4.4 Gas absorber with a large number of plates CHAP..."

2011-11-02T15:29:47Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS Section 7.4.4 Gas absorber with a large number of plates CHAP..."

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

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

Section 7.4.4 Gas absorber with a large number of plates

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

==Problem description==

A general case of an n-plate gas absorber controlled by inlet feed stream concentrations

Find u over t in [0; 10 ] to minimize

<math> J = \int_{0}^{10} x'*x + u'*u \mathrm{d}t </math>

subject to:

<math> \frac{dx}{dt} = A*x + B*u </math>

A = tridiag(0.538998 -1.173113 0.634115)

B = [0.538998 0 ... 0 0; 0 0 ... 0 0.634115]

The initial condition is chosen as:

xi(0) = -0.0307-(i-1)/(n-1)*(0.1273-0.0307), i= 1, 2, ... , n. where n is the number of stages

<math> 0 <= u1 <= inf </math>
<math> 0 <= u2 <= inf </math>

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

==Problem setup==

<source lang="matlab">
toms t
n = 10;
t_f = 10;

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

x = tomState(p, 'x', n, 1);
u = tomControl(p, 'u', 2, 1);

i = (1:n)';
x0i = -0.0307-(0.1273-0.0307)/(n-1)*(i-1);

% Initial guess
% Note: The guess for t_f must appear in the list before expression involving t.
guess = icollocate(x == x0i);

% Box constraints
cbox = {0 <= collocate(u)};

% Initial conditions
cinit = initial(x == x0i);

% Various constants and expressions
A = spdiags([0.538998*ones(n,1) ...
-1.173113*ones(n,1) 0.634115*ones(n,1)],-1:1,n,n);
B = sparse(n,2);
B(1,1) = 0.538998;
B(end,2) = 0.634115;

% ODEs and path constraints
ceq = collocate(dot(x) == A*x+B*u);

% Objective
objective = integrate(x'*x+u'*u);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Linear Gas Absorber';
solution = ezsolve(objective, {cbox, ceq, cinit}, guess, options);

% Extract optimal states and controls from solution
</source>

<pre>
Problem type appears to be: qp
Time for symbolic processing: 0.059353 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: 1: Linear Gas Absorber f_k 0.328245958214707810
sum(|constr|) 0.000000000000012839
f(x_k) + sum(|constr|) 0.328245958214720630
f(x_0) 0.000000000000000000

Solver: CPLEX. EXIT=0. INFORM=1.
CPLEX Barrier QP solver
Optimal solution found

FuncEv 15 GradEv 15 ConstrEv 15 Iter 15
CPU time: 0.031200 sec. Elapsed time: 0.019000 sec.

</pre>

==Plot result==

<source lang="matlab">
subplot(2,1,1)
ezplot(x);
legend('x1','x2','x3','x4','x5','x6','x7','x8','x9','x10')
title('Linear Gas Absorber state variables');

subplot(2,1,2)
ezplot(u);
legend('u1', 'u2');
title('Linear Gas Absorber control');
</source>