« Previous « Start » Next »
25 Coloumb Friction 1
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
25.1 Problem Formulation
Find u over t in [0; t_F ] to minimize
subject to:
Reference: [15]
25.2 Problem setup
toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 60);
setPhase(p);
tomStates q qdot
tomControls u
% Initial guess
x0 = {t_f == 1, icollocate(q == -t)};
% Box constraints
cbox = {-2 <= collocate(u) <= 2
0.001 <= t_f};
% Boundary constraints
cbnd = {initial({q == 0; qdot == 1}), final({q == -1, qdot == 0})};
% ODEs and path constraints
ceq = collocate({
dot(q) == qdot
dot(qdot) == u-sign(qdot)});
objective = t_f;
25.3 Solve the problem
options = struct;
options.name = 'Coloumb Friction 1';
constr = {cbox, cbnd, ceq};
solution = ezsolve(objective, constr, x0, options);
t = subs(collocate(p,t),solution);
q = subs(collocate(p,q),solution);
qdot = subs(collocate(p,qdot),solution);
u = subs(collocate(p,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: Coloumb Friction 1 f_k 2.070229757012032500
sum(|constr|) 0.000000001043060926
f(x_k) + sum(|constr|) 2.070229758055093200
f(x_0) 1.000000000000000000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 130 ConJacEv 130 Iter 23 MinorIter 676
CPU time: 0.609375 sec. Elapsed time: 0.641000 sec.
25.4 Plot result
subplot(2,1,1)
plot(t,q,'*-',t,qdot,'*-');
legend('q','qdot');
title('Coloumb Friction 1 state variables');
subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Coloumb Friction 1 control');
« Previous « Start » Next »
TOMLAB
