PROPT Linear Gas Absorber
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This page is part of the PROPT Manual. See 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
subject to:
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
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
Problem setup
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);
Solve the problem
options = struct;
options.name = 'Linear Gas Absorber';
solution = ezsolve(objective, {cbox, ceq, cinit}, guess, options);
% Extract optimal states and controls from solution
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.
Plot result
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');