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57 Linear Tangent Steering Problem
Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABORATORY
57.1 Problem Formulation
Find u(t) over t in [0; t_F ] to minimize
subject to:
The following transformation gives a new formulation:
Reference: [14]
57.2 Problem setup
toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 30);
setPhase(p);
tomStates x1 x2 x3 x4
tomControls u
% Initial guess
x0 = {t_f == 1
icollocate({
x1 == 12*t/t_f
x2 == 45*t/t_f
x3 == 5*t/t_f
x4 == 0})};
% Box constraints
cbox = {sqrt(eps) <= t_f
-pi/2 <= collocate(u) <= pi/2};
% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0; x3 == 0; x4 == 0})
final({x2 == 45; x3 == 5; x4 == 0})};
% ODEs and path constraints
a = 100;
ceq = collocate({dot(x1) == x2
dot(x2) == a*cos(u)
dot(x3) == x4
dot(x4) == a*sin(u)});
% Objective
objective = t_f;
57.3 Solve the problem
options = struct;
options.name = 'Linear Tangent Steering';
options.solver = 'knitro';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
x4 = subs(collocate(x4),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: Linear Tangent Steering f_k 0.554570876848855420
sum(|constr|) 0.000053410901111122
f(x_k) + sum(|constr|) 0.554624287749966530
f(x_0) 1.000000000000000000
Solver: KNITRO. EXIT=0. INFORM=0.
Default NLP KNITRO
Locally optimal solution found
FuncEv 14 GradEv 152 ConstrEv 13 ConJacEv 152 Iter 11 MinorIter 12
CPU time: 0.312500 sec. Elapsed time: 0.312000 sec.
57.4 Plot result
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Linear Tangent Steering state variables');
subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Linear Tangent Steering control');
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