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

Contents

Problem Formulation

Find u over t in [0; tF ] to minimize

$$ J = t_f $$

subject to:

$$ m_1*\frac{d^{2}q_1}{dt^{2}} = (-k_1-k_2)*q_1 + k_2*q_2 - mu*sign(\frac{dq_1}{dt}) +u_1 $$

$$ m_2*\frac{d^{2}q_2}{dt^{2}} = k_2*q_1 - k_2*q_2 - mu*sign(\frac{dq_2}{dt}) +u_2 $$

$$ q_{1:2}(0) = [0 \ 0] $$

$$ \frac{dq_{1:2}}{dt}_0 = [-1 \ -2] $$

$$ q_{1:2}(t_f) = [1 \ 2] $$

$$ \frac{dq_{1:2}}{dt}_{t_f} = [0 \ 0] $$

$$ -4 <= u_{1:2} <= 4 $$

$$ k_{1:2} = [0.95 \ 0.85] $$

$$ m_{1:2} = [1.1 \ 1.2] $$

$$ mu = 1.0 $$

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t
toms tf

p = tomPhase('p', t, 0, tf, 40, [], 'gauss');
setPhase(p);

tomStates q1 q1dot q2 q2dot
tomControls u1 u2

% Initial guess
x0 = {tf == 1};

% Box constraints
cbox = {1.8 <= tf <= 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;
mu = 1;

% ODEs and path constraints
ceq = collocate({dot(q1) == q1dot
    m1*dot(q1dot) == (-k1-k2)*q1+k2*q2-mu*sign(q1dot)+u1
    dot(q2)       == q2dot
    m2*dot(q2dot) == k2*q1-k2*q2-mu*sign(q2dot)+u2});

% Objective
objective = tf;

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
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Coloumb Friction 2             f_k       2.125397797774969600
                                       sum(|constr|)      0.000000000012176456
                              f(x_k) + sum(|constr|)      2.125397797787146100
                                              f(x_0)      1.800000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   31 ConJacEv   31 Iter   27 MinorIter  389
CPU time: 0.484375 sec. Elapsed time: 0.485000 sec.

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');