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68 Missile Intercept
Egwald Mathematics: Optimal Control, Intercept Missile, Elmer G. Wiens
68.1 Problem Description
Find scalar w over t in [0; t_F ] to minimize:
subject to:
Reference: [35]
68.2 Problem setup
toms t t_f w
p = tomPhase('p', t, 0, t_f, 10);
setPhase(p);
tomStates x1 x2 x3 x4
% Initial guess
x0 = {t_f == 10; w == 0.2};
% Box constraints
cbox = {1 <= t_f <= 1e4
0 <= w <= pi/4};
% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0
x3 == 0; x4 == 0})
final({x1 == 4+1.5*t_f; x2 == 1})};
% ODEs and path constraints
ceq = collocate({dot(x1) == x3; dot(x2) == x4
dot(x3) == cos(w); dot(x4) == sin(w)});
% Objective
objective = 0;
68.3 Solve the problem
options = struct;
options.name = 'Missile Intercept';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
w = subs(w,solution);
t_f = subs(t_f,solution);
Problem type appears to be: lpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
Problem: --- 1: Missile Intercept f_k 0.000000000000000000
sum(|constr|) 0.000000010885019417
f(x_k) + sum(|constr|) 0.000000010885019417
f(x_0) 0.000000000000000000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 13 ConJacEv 13 Iter 10 MinorIter 43
CPU time: 0.031250 sec. Elapsed time: 0.031000 sec.
68.4 Plot result
figure(1);
plot(x1,x2,'*-');
hold on
plot(4+1.5*t,ones(length(t)),'-*r');
legend('x1 vs x2','Missile path');
title(sprintf('Missile Intercept, t_F=%g, w=%g',t_f,w));
ylim([0 1.1]);
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