PROPT Flight Path Tracking - Revision history

Per: /* Plot result */

2012-02-14T09:02:22Z

Plot result

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	[[File:flightPathTracking_01.png]]		[[File:flightPathTracking_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:07, 9 November 2011

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	<math> J = \int_0^{100} (x_3^2) \mathrm{d}t </math>		<math> J = \int_0^{100} (x_3^2) \mathrm{d}t </math>


	subject to:		subject to:

Mbot at 13:05, 8 November 2011

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	<pre>		<pre>
	Problem type appears to be: qpcon		Problem type appears to be: qpcon
	Time for symbolic processing: 0.~~22655~~ seconds		Time for symbolic processing: 0.20327 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 1 ConstrEv 453 ConJacEv 453 Iter 415 MinorIter 709		FuncEv 1 ConstrEv 453 ConJacEv 453 Iter 415 MinorIter 709
	CPU time: 5.~~304034~~ sec. Elapsed time: 5.~~287000~~ sec.		CPU time: 4.945232 sec. Elapsed time: 4.963000 sec.

	</pre>		</pre>

Mbot at 18:23, 2 November 2011

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	{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT \|PROPT Manual]]}}		{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT\|PROPT Manual]]}}

	Minimum Cost Optimal Control: An Application to Flight Level Tracking, John Lygeros, Department of Engineering, University of Cambridge, Cambridge, UK.		Minimum Cost Optimal Control: An Application to Flight Level Tracking, John Lygeros, Department of Engineering, University of Cambridge, Cambridge, UK.
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	title('Thrust')		title('Thrust')
	</source>		</source>

			[[File:flightPathTracking_01.png]]

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2011-11-02T15:29:15Z

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{{Part Of Manual|title=the PROPT Manual|link=[[PROPT |PROPT Manual]]}}

Minimum Cost Optimal Control: An Application to Flight Level Tracking, John Lygeros, Department of Engineering, University of Cambridge, Cambridge, UK.

==Problem Description==

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

<math> J = \int_0^{100} (x_3^2) \mathrm{d}t </math>

subject to:

Equations in the code.

<source lang="matlab">
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
</source>

==Problem setup==

<source lang="matlab">
toms t
p = tomPhase('p', t, 0, 100, 60);
setPhase(p);

tomStates x1s x2 x3s
tomControls u1s u2

x1 = x1s*100;
x3 = x3s*10;
u1 = u1s*10e3;
cr2d = pi/180;

% Box constraints
cbox = {
92 <= icollocate(x1) <= 170
-20*cr2d <= icollocate(x2) <= 25*cr2d
-150 <= icollocate(x3) <= 150
60e3 <= collocate(u1) <= 120e3
-150 <= collocate(u2) <= 150};

% Boundary constraints
cbnd = initial({x1 == 153.73
x2 == 0; x3 == 0});

L = 65.3;
D = 3.18;
m = 160e3;
g = 9.81;
c = 6;

% ODEs and path constraints
ceq = collocate({
dot(x1) == (-D/m*x1.^2-g*sin(x2)+u1/m)
dot(x2) == L/m*x1.*(1-c*x2)-g*cos(x2)./x1+L*c/m*u2
dot(x3) == (x1.*sin(x2))});

% Objective
objective = integrate(x3.^2);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Flight Path Tracking';
solution = ezsolve(objective, {cbox, cbnd, ceq}, [], options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);
</source>

<pre>
Problem type appears to be: qpcon
Time for symbolic processing: 0.22655 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Flight Path Tracking f_k 0.000000009806762749
sum(|constr|) 0.000000000018231012
f(x_k) + sum(|constr|) 0.000000009824993762
f(x_0) 0.000000000000000000

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

FuncEv 1 ConstrEv 453 ConJacEv 453 Iter 415 MinorIter 709
CPU time: 5.304034 sec. Elapsed time: 5.287000 sec.

</pre>

==Plot result==

<source lang="matlab">
figure(1);
subplot(2,3,1);
plot(t,x3,'-');
title('Alt')
subplot(2,3,2);
plot(t,x1,'-');
title('Vel')
subplot(2,3,3);
plot(t,x2,'-');
title('Gamma')
subplot(2,3,4);
plot(t,u2,'-');
title('Angle')
subplot(2,3,5);
plot(t,u1,'-');
title('Thrust')
</source>