PROPT Bryson-Denham Problem (Detailed) - Revision history

Per at 08:55, 14 February 2012

2012-02-14T08:55:03Z

← Older revision		Revision as of 03:55, 14 February 2012
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			[[Category:PROPT Examples]]

Mbot at 13:03, 8 November 2011

2011-11-08T13:03:56Z

← Older revision		Revision as of 08:03, 8 November 2011
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	FuncEv 35 GradEv 33 ConstrEv 33 ConJacEv 33 Iter 32 MinorIter 170		FuncEv 35 GradEv 33 ConstrEv 33 ConJacEv 33 Iter 32 MinorIter 170
	CPU time: 0.062400 sec. Elapsed time: 0.~~068000~~ sec.		CPU time: 0.062400 sec. Elapsed time: 0.064000 sec.

	</pre>		</pre>

Mbot at 18:22, 2 November 2011

2011-11-02T18:22:22Z

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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==

Mbot at 14:54, 2 November 2011

2011-11-02T14:54:26Z

← Older revision		Revision as of 09:54, 2 November 2011
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			{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT \|PROPT Manual]]}}

	==Problem description==		==Problem description==
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	A detailed version of the Bryson-Denham problem		A detailed version of the Bryson-Denham problem

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

	==Define the independent variable, and phase:==		==Define the independent variable, and phase:==
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	It is common for the independent variable to be named "t", but any legal tomSym name is possible. The length of the time-interval is variable, so another tomSym symbol is created for that.		It is common for the independent variable to be named "t", but any legal tomSym name is possible. The length of the time-interval is variable, so another tomSym symbol is created for that.

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	toms t t_f		toms t t_f
	p = tomPhase('p', t, 0, t_f, 30);		p = tomPhase('p', t, 0, t_f, 30);
	setPhase(p);		setPhase(p);
	</~~syntaxhighlight~~>		</source>

	==A few constants.==		==A few constants.==

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	% The name on the mathematics:		% The name on the mathematics:
	options = struct;		options = struct;
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	x1max = 1/9;		x1max = 1/9;
	tfmax = 50;		tfmax = 50;
	</~~syntaxhighlight~~>		</source>

	==Define a list of states==		==Define a list of states==
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	After a phase has been defined, states can be created Note that the states can be given meaningful names, even though they are simply named x1...x3 in this particular problem.		After a phase has been defined, states can be created Note that the states can be given meaningful names, even though they are simply named x1...x3 in this particular problem.

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	tomStates x1 x2 x3		tomStates x1 x2 x3

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	-10 <= icollocate(x2) <= 10		-10 <= icollocate(x2) <= 10
	-10 <= icollocate(x3) <= 10};		-10 <= icollocate(x3) <= 10};
	</~~syntaxhighlight~~>		</source>

	==Adding the control variable and equations==		==Adding the control variable and equations==

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	tomControls u		tomControls u

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	cbox = {cbox		cbox = {cbox
	-5000 <= collocate(u) <= 5000};		-5000 <= collocate(u) <= 5000};
	</~~syntaxhighlight~~>		</source>

	==Equations==		==Equations==
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	The equations are on a nonlinear DAE form, i.e. the states and their derivatives can be combined into arbitrary equations. Equations can also be specified point-wise, or using integrals. Integrals can be achieved by using the function integral(). Each equation must contain one or more equals (==) signs, or one or more greater than (>=) signs, or one or more less than (<=) signs.		The equations are on a nonlinear DAE form, i.e. the states and their derivatives can be combined into arbitrary equations. Equations can also be specified point-wise, or using integrals. Integrals can be achieved by using the function integral(). Each equation must contain one or more equals (==) signs, or one or more greater than (>=) signs, or one or more less than (<=) signs.

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	usquared = u^2;		usquared = u^2;

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	% If costs are defined as many parts they should be added together.		% If costs are defined as many parts they should be added together.
	objective = final(x3);		objective = final(x3);
	</~~syntaxhighlight~~>		</source>

	==Build the .m files and general TOMLAB problem==		==Build the .m files and general TOMLAB problem==

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	Prob = sym2prob('con',objective,{cbox, ceq, cbnd},x0,options);		Prob = sym2prob('con',objective,{cbox, ceq, cbnd},x0,options);

	% Solve the problem using any TOMLAB solver		% Solve the problem using any TOMLAB solver
	Result = tomRun('snopt', Prob, 1);		Result = tomRun('snopt', Prob, 1);
	</~~syntaxhighlight~~>		</source>

	<pre>		<pre>

Mbot: Created page with " ==Problem description== A detailed version of the Bryson-Denham problem % Copyright (c) 2007-2008 by Tomlab Optimization Inc.
2011-11-02T11:42:10Z

Created page with " ==Problem description== A detailed version of the Bryson-Denham problem <syntaxhighlight lang="matlab"> % Copyright (c) 2007-2008 by Tomlab Optimization Inc. </syntaxhighlight..."

New page

==Problem description==

A detailed version of the Bryson-Denham problem

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

==Define the independent variable, and phase:==

It is common for the independent variable to be named "t", but any legal tomSym name is possible. The length of the time-interval is variable, so another tomSym symbol is created for that.

<syntaxhighlight lang="matlab">
toms t t_f
p = tomPhase('p', t, 0, t_f, 30);
setPhase(p);
</syntaxhighlight>

==A few constants.==

<syntaxhighlight lang="matlab">
% The name on the mathematics:
options = struct;
options.name = 'Bryson Denham Detailed';

x1max = 1/9;
tfmax = 50;
</syntaxhighlight>

==Define a list of states==

After a phase has been defined, states can be created Note that the states can be given meaningful names, even though they are simply named x1...x3 in this particular problem.

<syntaxhighlight lang="matlab">
tomStates x1 x2 x3

% Initial guess
% The guess for t_f must appear first in the list
x0 = {t_f == 0.5
icollocate({
x1 == 0
x2 == 1-2t/t_f
x3 == 0
})};

cbox = {0 <= t_f <= 50
-10 <= icollocate(x1) <= 10
-10 <= icollocate(x2) <= 10
-10 <= icollocate(x3) <= 10};
</syntaxhighlight>

==Adding the control variable and equations==

<syntaxhighlight lang="matlab">
tomControls u

x0 = {x0
collocate(u == 0)};

cbox = {cbox
-5000 <= collocate(u) <= 5000};
</syntaxhighlight>

==Equations==

The equations are on a nonlinear DAE form, i.e. the states and their derivatives can be combined into arbitrary equations. Equations can also be specified point-wise, or using integrals. Integrals can be achieved by using the function integral(). Each equation must contain one or more equals (==) signs, or one or more greater than (>=) signs, or one or more less than (<=) signs.

<syntaxhighlight lang="matlab">
usquared = u^2;

ceq = collocate({
dot(x1) == x2
dot(x2) == u
dot(x3) == 0.5usquared
0 <= x1 <= x1max % Path constraint
});

cbnd = {initial({
x1 == 0; x2 == 1; x3 == 0})
final({x1 == 0; x2 == -1})};

% The "objective" function to minimize. Cost can be specified at a point,
% or integrated over time by using the function integral.
% If costs are defined as many parts they should be added together.
objective = final(x3);
</syntaxhighlight>

==Build the .m files and general TOMLAB problem==

<syntaxhighlight lang="matlab">
Prob = sym2prob('con',objective,{cbox, ceq, cbnd},x0,options);

% Solve the problem using any TOMLAB solver
Result = tomRun('snopt', Prob, 1);
</syntaxhighlight>

<pre>
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Bryson Denham Detailed f_k 3.998132387937467900
sum(|constr|) 0.000000810449850953
f(x_k) + sum(|constr|) 3.998133198387318700
f(x_0) 0.000000000000000000

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

FuncEv 35 GradEv 33 ConstrEv 33 ConJacEv 33 Iter 32 MinorIter 170
CPU time: 0.062400 sec. Elapsed time: 0.068000 sec.

</pre>