PROPT Orbit Raising Maximum Radius - Revision history

Per: /* Plot result */

2012-02-14T09:33:15Z

Plot result

← Older revision		Revision as of 04:33, 14 February 2012
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	[[File:orbitRaisingMaxRadius_01.png]]		[[File:orbitRaisingMaxRadius_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:10, 9 November 2011

2011-11-09T12:10:40Z

← Older revision		Revision as of 07:10, 9 November 2011
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	<math> J = r(t_f) </math>		<math> J = r(t_f) </math>


	subject to the dynamic constraints		subject to the dynamic constraints

	<math> \frac{d_r}{dt} = u </math>		<math> \frac{d_r}{dt} = u </math>

	<math> \frac{du}{dt} = \frac{v^2}{r}-\frac{mmu}{r^2}+T*\frac{w1}{m} </math>		<math> \frac{du}{dt} = \frac{v^2}{r}-\frac{mmu}{r^2}+T*\frac{w1}{m} </math>

	<math> \frac{dv}{dt} = -u\frac{v}{r}+T\frac{w2}{m} </math>		<math> \frac{dv}{dt} = -u\frac{v}{r}+T\frac{w2}{m} </math>

	<math> \frac{dm}{dt} = -\frac{T}{g0*ISP} </math>		<math> \frac{dm}{dt} = -\frac{T}{g0*ISP} </math>


	the boundary conditions		the boundary conditions

	<math> r(t_0) = 1 </math>		<math> r(t_0) = 1 </math>

	<math> u(t_0) = 1 </math>		<math> u(t_0) = 1 </math>

	<math> u(t_f) = 0 </math>		<math> u(t_f) = 0 </math>

	<math> v(t_0) = (\frac{mmu}{r(t_0)})^{0.5} </math>		<math> v(t_0) = (\frac{mmu}{r(t_0)})^{0.5} </math>

	<math> v(t_f) = (\frac{mmu}{r(t_f)})^{0.5} </math>		<math> v(t_f) = (\frac{mmu}{r(t_f)})^{0.5} </math>

	<math> m(t_0) = 1 </math>		<math> m(t_0) = 1 </math>


	and the path constraint		and the path constraint

	<math> w_1^2+w_2^2 = 1 </math>		<math> w_1^2+w_2^2 = 1 </math>


	The control pitch angel is not being used in this formulation. Instead two control variables (w1,w2) are used to for the thrust direction. A path constraint ensures that (w1,w2) is a unit vector.		The control pitch angel is not being used in this formulation. Instead two control variables (w1,w2) are used to for the thrust direction. A path constraint ensures that (w1,w2) is a unit vector.

Mbot at 13:07, 8 November 2011

2011-11-08T13:07:15Z

← Older revision		Revision as of 08:07, 8 November 2011
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	<pre>		<pre>
	Problem type appears to be: lpcon		Problem type appears to be: lpcon
	Time for symbolic processing: 0.~~24619~~ seconds		Time for symbolic processing: 0.22212 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 1 ConstrEv 59 ConJacEv 59 Iter 36 MinorIter 346		FuncEv 1 ConstrEv 59 ConJacEv 59 Iter 36 MinorIter 346
	CPU time: 0.~~249602~~ sec. Elapsed time: 0.~~241000~~ sec.		CPU time: 0.202801 sec. Elapsed time: 0.226000 sec.

	</pre>		</pre>

Mbot at 18:25, 2 November 2011

2011-11-02T18:25:20Z

← Older revision		Revision as of 13:25, 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]]}}

	==Problem description==		==Problem description==
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	title('Orbit Raising Problem Max Radius control');		title('Orbit Raising Problem Max Radius control');
	</source>		</source>

			[[File:orbitRaisingMaxRadius_01.png]]

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} ==Problem description== Maximize: $J = r(t_f)$ subject to the dynamic constraints
2011-11-02T15:30:17Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} ==Problem description== Maximize: <math> J = r(t_f) </math> subject to the dynamic constraints <math..."

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

==Problem description==

Maximize:

<math> J = r(t_f) </math>

subject to the dynamic constraints

<math> \frac{d_r}{dt} = u </math>
<math> \frac{du}{dt} = \frac{v^2}{r}-\frac{mmu}{r^2}+T\frac{w1}{m} </math>
<math> \frac{dv}{dt} = -u\frac{v}{r}+T\frac{w2}{m} </math>
<math> \frac{dm}{dt} = -\frac{T}{g0ISP} </math>

the boundary conditions

<math> r(t_0) = 1 </math>
<math> u(t_0) = 1 </math>
<math> u(t_f) = 0 </math>
<math> v(t_0) = (\frac{mmu}{r(t_0)})^{0.5} </math>
<math> v(t_f) = (\frac{mmu}{r(t_f)})^{0.5} </math>
<math> m(t_0) = 1 </math>

and the path constraint

<math> w_1^2+w_2^2 = 1 </math>

The control pitch angel is not being used in this formulation. Instead two control variables (w1,w2) are used to for the thrust direction. A path constraint ensures that (w1,w2) is a unit vector.

<source lang="matlab">
% Copyright (c) 2007-2009 by Tomlab Optimization Inc.
</source>

==Problem setup==

<source lang="matlab">
t_F = 3.32;
mmu = 1;
m_0 = 1; r_0 = 1; u_0 = 0;
u_f = 0; v_0 = sqrt(mmu/r_0); rmin = 0.9;
rmax = 5; umin = -5; umax = 5;
vmin = -5; vmax = 5; mmax = m_0;
mmin = 0.1;

T = 0.1405;
Ve = 1.8758;

toms t
p1 = tomPhase('p1', t, 0, t_F, 40);
setPhase(p1);

tomStates r u v m
tomControls w1 w2

% Initial guess
x0 = {icollocate({
r == r_0
u == u_0 + (u_f-u_0)t/t_F
v == v_0
m == m_0})
collocate({w1 == 0; w2 == 1})};

% Boundary constraints
cbnd = {initial({
r == r_0
u == u_0
v == v_0
m == m_0
})
final({u == u_f
v - sqrt(mmu/r) == 0})};

% Box constraints
cbox = {
rmin <= icollocate(r) <= rmax
umin <= icollocate(u) <= umax
vmin <= icollocate(v) <= vmax
mmin <= icollocate(m) <= mmax
-1 <= collocate(w1) <= 1
-1 <= collocate(w2) <= 1};

% ODEs and path constraints
ceq = collocate({
dot(r) == u
dot(u) == v^2/r-mmu/r^2+Tw1/m
dot(v) == -uv/r+Tw2/m
dot(m) == -T/Ve
w1^2+w2^2 == 1});

% Objective
objective = -final(r);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Orbit Raising Problem Max Radius';
options.solver = 'snopt';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
</source>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.24619 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Orbit Raising Problem Max Radius f_k -1.518744202740535800
sum(|constr|) 0.000017265830060524
f(x_k) + sum(|constr|) -1.518726936910475200
f(x_0) -0.999999999999991120

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

FuncEv 1 ConstrEv 59 ConJacEv 59 Iter 36 MinorIter 346
CPU time: 0.249602 sec. Elapsed time: 0.241000 sec.

</pre>

==Plot result==

<source lang="matlab">
subplot(2,1,1)
ezplot([r u v m]);
legend('r','u','v','m');
title('Orbit Raising Problem Max Radius state variables');

subplot(2,1,2)
ezplot([w1 w2 atan2(w1,w2)])
legend('w_1', 'w_2', '\phi');
title('Orbit Raising Problem Max Radius control');
</source>