PROPT Missile Intercept

This page is part of the PROPT Manual. See PROPT Manual.

Egwald Mathematics: Optimal Control, Intercept Missile, Elmer G. Wiens

Problem Description

Find scalar w over t in [0; t_F ] to minimize:

$J=0$

subject to:

${\frac {dx_{1}}{dt}}=x_{3}$

${\frac {dx_{2}}{dt}}=x_{4}$

${\frac {dx_{3}}{dt}}=cos(w)$

${\frac {dx_{4}}{dt}}=sin(w)$

$x(t_{0})=[0\ 0\ 0\ 0]$

$x(t_{f})=[4+1.5*t_{f}\ 1]$

$0<=w<={\frac {pi}{4}}$

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

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;

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
Time for symbolic processing: 0.1299 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Missile Intercept              f_k       0.000000000000000000
                                       sum(|constr|)      0.000000010884798781
                              f(x_k) + sum(|constr|)      0.000000010884798781
                                              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: 0.015600 sec. Elapsed time: 0.014000 sec. 

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]);