PROPT Dielectrophoresis Particle Control: Difference between revisions
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{{Part Of Manual|title=the PROPT | {{Part Of Manual|title=the PROPT Manual|link=[[PROPT|PROPT Manual]]}} | ||
Time-Optimal Control of a Particle in a Dielectrophoretic System, Dong Eui Chang, Nicolas Petit, and Pierre Rouchon | Time-Optimal Control of a Particle in a Dielectrophoretic System, Dong Eui Chang, Nicolas Petit, and Pierre Rouchon | ||
Line 8: | Line 8: | ||
<math> J = t_f </math> | <math> J = t_f </math> | ||
subject to: | subject to: | ||
<math> \frac{dx}{dt} = y*u+alpha*u^2 </math> | <math> \frac{dx}{dt} = y*u+alpha*u^2 </math> | ||
<math> \frac{dy}{dt} = -c*y+u </math> | <math> \frac{dy}{dt} = -c*y+u </math> | ||
<math> |u| <= 1 </math> | <math> |u| <= 1 </math> | ||
<math> alpha = -\frac{3}{4} </math> | <math> alpha = -\frac{3}{4} </math> | ||
<math> c = 1 </math> | <math> c = 1 </math> | ||
<math> [x_0 \ y_0] = [1 \ 0]</math> | <math> [x_0 \ y_0] = [1 \ 0]</math> | ||
<math> x_{t_f} = 2 </math> | <math> x_{t_f} = 2 </math> | ||
<source lang="matlab"> | <source lang="matlab"> | ||
Line 76: | Line 84: | ||
<pre> | <pre> | ||
Problem type appears to be: lpcon | Problem type appears to be: lpcon | ||
Time for symbolic processing: 0. | Time for symbolic processing: 0.10788 seconds | ||
Starting numeric solver | Starting numeric solver | ||
===== * * * =================================================================== * * * | ===== * * * =================================================================== * * * | ||
Line 91: | Line 99: | ||
FuncEv 1 ConstrEv 26 ConJacEv 26 Iter 25 MinorIter 218 | FuncEv 1 ConstrEv 26 ConJacEv 26 Iter 25 MinorIter 218 | ||
CPU time: 0.093601 sec. Elapsed time: 0. | CPU time: 0.093601 sec. Elapsed time: 0.096000 sec. | ||
</pre> | </pre> | ||
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legend('x','y','u'); | legend('x','y','u'); | ||
</source> | </source> | ||
[[File:dielectrophoresisProblem_01.png]] | |||
[[Category:PROPT Examples]] |
Latest revision as of 04:57, 14 February 2012
This page is part of the PROPT Manual. See PROPT Manual. |
Time-Optimal Control of a Particle in a Dielectrophoretic System, Dong Eui Chang, Nicolas Petit, and Pierre Rouchon
Problem Description
Find u over t in [0; t_F ] to minimize:
subject to:
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
Problem setup
toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 60);
setPhase(p);
tomStates x y
tomControls u
% Initial guess
x0 = {t_f == 10
icollocate({
x == 1+1*t/t_f
y == t/t_f
})
collocate(u == 1)};
% Box constraints
cbox = {
sqrt(eps) <= icollocate(x)
sqrt(eps) <= collocate(y)
1 <= t_f <= 100
-1 <= collocate(u) <= 1};
% Boundary constraints
cbnd = {initial({x == 1; y == 0})
final({x == 2})};
% ODEs and path constraints
ceq = collocate({
dot(x) == y.*u-3/4*u.^2
dot(y) == -y+u});
% Objective
objective = t_f;
Solve the problem
options = struct;
options.name = 'Dielectrophoresis Control';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x = subs(collocate(x),solution);
y = subs(collocate(y),solution);
u = subs(collocate(u),solution);
Problem type appears to be: lpcon Time for symbolic processing: 0.10788 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Dielectrophoresis Control f_k 7.811292811901784800 sum(|constr|) 0.000001365448751008 f(x_k) + sum(|constr|) 7.811294177350536200 f(x_0) 10.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 26 ConJacEv 26 Iter 25 MinorIter 218 CPU time: 0.093601 sec. Elapsed time: 0.096000 sec.
Plot result
figure(1);
plot(t,x,'*-',t,y,'*-',t,u,'*-');
legend('x','y','u');