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30 Dielectrophoresis Particle Control
Time-Optimal Control of a Particle in a Dielectrophoretic System, Dong Eui Chang, Nicolas Petit, and Pierre Rouchon
30.1 Problem Description
Find u over t in [0; t_F ] to minimize:
subject to:
Reference: [12]
30.2 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;
30.3 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
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
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: 0.281250 sec. Elapsed time: 0.281000 sec.
30.4 Plot result
figure(1);
plot(t,x,'*-',t,y,'*-',t,u,'*-');
legend('x','y','u');
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