PROPT Stirred Tank: Difference between revisions
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[[Category:PROPT Examples]] |
Latest revision as of 05:27, 14 February 2012
This page is part of the PROPT Manual. See PROPT Manual. |
Users Guide for dyn.Opt, Example 5a, 5b, 5c
Stirred-Tank Chemical Reactor - Kirk, D. E., Optimal control theory: An introduction, Prentice-Hall, 1970.
5a - unconstrained with terminal penalty 5b - unconstrained 5c - control constraint
Problem Description
Find u over t in [0; 0.78 ] to minimize
Does not say u^2 in text
5a:
5b:
5c:
subject to:
5b, 5c - x(t_F) = [0 0];
5c - <= 1
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
Problem setup
toms t
for i=1:3
p = tomPhase('p', t, 0, 0.78, 40);
setPhase(p);
tomStates x1 x2
tomControls u
% Initial guess
x0 = {icollocate({x1 == 0.05; x2 == 0})
collocate(u == 0)};
% Box constraints
cbox = {-1.99 <= icollocate(x1) <= 100
-100 <= icollocate(x2) <= 100
-1000 <= collocate(u) <= 1000};
% x1 cannot be equal to -2, setting to greater
% to avoid singularity in a2*exp(25.0*x1/a3)
% Boundary constraints
cbnd = initial({x1 == 0.05; x2 == 0});
% ODEs and path constraints
a1 = x1 + 0.25; a2 = x2 + 0.5;
a3 = x1 + 2.0; a4 = a2.*exp(25.0*x1./a3);
ceq = collocate({
dot(x1) == -2.0*a1 + a4 - a1.*u
dot(x2) == 0.5 - x2 - a4});
Solve the problem
options = struct;
if i==1
objective = final(x1)^2+final(x2)^2+...
integrate((x1.^2+x2.^2+0.1*u.^2)/2);
options.name = 'Stirred Tank 5a';
solution1 = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t1 = subs(collocate(t),solution1);
x11 = subs(collocate(x1),solution1);
x21 = subs(collocate(x2),solution1);
u1 = subs(collocate(u),solution1);
elseif i == 2
cbnd = {cbnd; final({x1 == 0; x2 == 0})};
objective = integrate((x1.^2+x2.^2+0.1*u.^2)/2);
options.name = 'Stirred Tank 5b';
solution2 = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
else
cbnd = {cbnd; final({x1 == 0; x2 == 0})};
cbox = {-1.99 <= icollocate(x1) <= 100
-100 <= icollocate(x2) <= 100
-1 <= collocate(u) <= 1};
objective = integrate((x1.^2+x2.^2)/2);
options.name = 'Stirred Tank 5c';
solution3 = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
end
Problem type appears to be: qpcon Time for symbolic processing: 0.20654 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Stirred Tank 5a f_k 0.014213969120012288 sum(|constr|) 0.000000005238862716 f(x_k) + sum(|constr|) 0.014213974358875005 f(x_0) 0.003474999999999964 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 30 ConJacEv 30 Iter 27 MinorIter 113 CPU time: 0.062400 sec. Elapsed time: 0.054000 sec.
Problem type appears to be: qpcon Time for symbolic processing: 0.19774 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Stirred Tank 5b f_k 0.016702811155814214 sum(|constr|) 0.000000899223600415 f(x_k) + sum(|constr|) 0.016703710379414628 f(x_0) 0.000974999999999998 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 18 ConJacEv 18 Iter 16 MinorIter 121 CPU time: 0.031200 sec. Elapsed time: 0.037000 sec.
Problem type appears to be: qpcon Time for symbolic processing: 0.18987 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Stirred Tank 5c f_k 0.000989922252663804 sum(|constr|) 0.000000035597654545 f(x_k) + sum(|constr|) 0.000989957850318349 f(x_0) 0.000974999999999998 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 14 ConJacEv 13 Iter 10 MinorIter 139 CPU time: 0.015600 sec. Elapsed time: 0.028000 sec.
end
Plot result
subplot(2,1,1)
plot(t1,x11,'*-',t1,x21,'*-');
legend('x1','x2');
title('Stirred Tank state variables');
subplot(2,1,2)
plot(t1,u1,'+-');
legend('u');
title('Stirred Tank control');