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26 Coloumb Friction 2
Minimum-Time Control of Systems With Coloumb Friction: Near Global Optima Via Mixed Integer Linear Programming, Brian J. Driessen, Structural Dynamics Department, Sandia National Labs.
4. Numerical Examples
26.1 Problem Formulation
Find u over t in [0; t_F ] to minimize
subject to:
m1* | | = (−k1−k2)*q1 + k2*q2 − mmu*sign( | | ) +u1 |
m2* | | = k2*q1 − k2*q2 − mmu*sign( | | ) +u2 |
Reference: [15]
26.2 Problem setup
toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 40, [], 'gauss');
setPhase(p);
tomStates q1 q1dot q2 q2dot
tomControls u1 u2
% Initial guess
x0 = {t_f == 1};
% Box constraints
cbox = {1.8 <= t_f <= 4
-4 <= collocate(u1) <= 4
-4 <= collocate(u2) <= 4};
% Boundary constraints
cbnd = {initial({q1 == 0; q1dot == -1
q2 == 0; q2dot == -2})
final({q1 == 1; q1dot == 0
q2 == 2; q2dot == 0})};
k1 = 0.95; k2 = 0.85;
m1 = 1.1; m2 = 1.2;
mmu = 1;
% ODEs and path constraints
ceq = collocate({dot(q1) == q1dot
m1*dot(q1dot) == (-k1-k2)*q1+k2*q2-mmu*sign(q1dot)+u1
dot(q2) == q2dot
m2*dot(q2dot) == k2*q1-k2*q2-mmu*sign(q2dot)+u2});
% Objective
objective = t_f;
26.3 Solve the problem
options = struct;
options.name = 'Coloumb Friction 2';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
q1 = subs(collocate(q1),solution);
q2 = subs(collocate(q2),solution);
q1dot = subs(collocate(q1dot),solution);
q2dot = subs(collocate(q2dot),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);
q1dot_f = q1dot(end);
q2dot_f = q2dot(end);
Problem type appears to be: lpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
Problem: --- 1: Coloumb Friction 2 f_k 2.125397251986161700
sum(|constr|) 0.000006640472891742
f(x_k) + sum(|constr|) 2.125403892459053300
f(x_0) 1.800000000000000000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 33 ConJacEv 32 Iter 27 MinorIter 388
CPU time: 0.531250 sec. Elapsed time: 0.531000 sec.
26.4 Plot result
subplot(2,1,1)
plot(t,q1,'*-',t,q2,'*-');
legend('q1','q2');
title(sprintf('Coloumb Friction 2, q1dot_f = %g, q2dot_f = %g',q1dot_f,q2dot_f));
subplot(2,1,2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Coloumb Friction 2 controls');
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