Hanging chain problem (TACO) - Revision history

Ckirches: /* AMPL */

2011-09-29T18:25:53Z

AMPL

← Older revision		Revision as of 18:25, 29 September 2011
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	var L >= 0, <= 10;		var L >= 0, <= 10;
	var E >= 0, <= 10;		var E >= 0, <= 10;
	var u >= -10, <= 20;		var u >= -10, <= 20 suffix type "u1";
	~~let u.~~type := "u1";

	param a := 1;		param a := 1;
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	dx: diff(x,t) = u;		dx: diff(x,t) = u;
	dE: diff(E,t) = x*sqrt(1+u^2);		dE: diff(E,t) = x*sqrt(1+u^2);
	dL: diff(L,t) = sqrt(1+u^2);		dL: diff(L,t) = sqrt(1+u^2);

	x0: eval(x,0) = a;		x0: eval(x,0) = a;

Ckirches: /* AMPL */

2011-09-29T18:24:52Z

AMPL

← Older revision		Revision as of 18:24, 29 September 2011
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	include OptimalControl.mod;		include OptimalControl.mod;

	~~param nh > 0, integer; # number of subintervals~~		var t;
	~~param L > 0; # length of the suspended chain~~		var tf := 1;
	~~param a~~; ~~# height of the chain at t=0 (left)~~
	~~param b; # height of the chain at t~~=1 ~~(right)~~

	~~param tf~~; ~~# ODEs defined in [~~0,~~tf]~~		var x >= 0, <= 10;
	~~param h~~ := ~~tf/nh~~; ~~# uniform interval length~~		var L >= 0, <= 10;
			var E >= 0, <= 10;
			var u >= -10, <= 20;
			let u.type := "u1";

	~~# x[*,~~1~~] - height of the chain~~		param a := 1;
	~~# x[*,2] - potential energy of the chain~~		param b := 3;
	~~# x[*,3] - lenght of the chain~~		param Lp := 4;

	~~param tmin~~ :~~= if b > a then 0.25 else 0.75;~~;		minimize energy: eval(E,tf);

			subject to

	~~var~~ x~~{k in 0..nh~~,~~j in~~ 1~~..3}~~; ~~# state variables~~		dx: diff(x,t) = u;
	~~var~~ u~~{k in 0..nh}~~; ~~# derivative of x~~		dE: diff(E,t) = x*sqrt(1+u^2);
			dL: diff(L,t) = sqrt(1+u^2);

	~~minimize potential_energy~~: ~~x[nh,2];~~		x0: eval(x,0) = a;
			x1: eval(x,1) = b;
	~~subject to de1 {j in 0..nh-1}:~~		E0: eval(E,0) = 0;
	~~x[j+1,1] = x[j,1] + 0.5h(u[j] + u[j+1]);~~		L0: eval(L,0) = 0;
			L1: eval(L,1) = Lp;
	~~subject to de2 {j in 0..nh-1}:~~
	x[j+1,2] = x[j,2] + 0.5h(x~~[j,1]sqrt(1+u[j]^2) + x[j+1,1]sqrt(1+u[j+1]^2));~~

	~~subject to de3 {j in 0..nh-1}:~~
	~~x[j+1,3] = x[j~~,~~3] +~~ 0~~.5h(sqrt(1+u[j]^2) + sqrt(1+u[j+1]^2)~~);

	~~# Boundary conditions~~

	~~subject to bc1: x[0,1]~~ = a;
	~~subject to bc2~~: x~~[nh~~,1] = b;
	~~subject to bc3~~: x[0~~,2]~~ = 0.0;
	~~subject to bc4~~: x[0~~,3]~~ = 0.0;
	~~subject to bc5~~: ~~x[nh~~,3] = L;

	option solver ...;		option solver ...;

Ckirches: Hanging chain problem (TACO)

2011-09-29T18:23:46Z

Hanging chain problem (TACO)

New page

This page contains a model of the [[Hanging chain problem]] in [http://www.ampl.org AMPL] format, making use of the TACO toolkit for AMPL control optimization extensions. The original model using a collocation formulation can be found in the [http://www.mcs.anl.gov/~more/cops/ COPS library].
Note that you will need to include a generic [[support AMPL files|AMPL/TACO support file]], OptimalControl.mod.
To solve this model, you require an optimal control or NLP code that uses the TACO toolkit to support the AMPL optimal control extensions.

=== AMPL ===

This is the source file hangchain_taco.mod
<source lang="AMPL">
# ----------------------------------------------------------------
# Hanging chain problem using AMPL and TACO
# (c) Christian Kirches, Sven Leyffer
#
# Source: COPS 3.1 collocation formulation - March 2004
# Alexander S. Bondarenko - Summer 1998
# ----------------------------------------------------------------
include OptimalControl.mod;

param nh > 0, integer; # number of subintervals
param L > 0; # length of the suspended chain
param a; # height of the chain at t=0 (left)
param b; # height of the chain at t=1 (right)

param tf; # ODEs defined in [0,tf]
param h := tf/nh; # uniform interval length

# x[*,1] - height of the chain
# x[*,2] - potential energy of the chain
# x[*,3] - lenght of the chain

param tmin := if b > a then 0.25 else 0.75;;

var x{k in 0..nh,j in 1..3}; # state variables
var u{k in 0..nh}; # derivative of x

minimize potential_energy: x[nh,2];

subject to de1 {j in 0..nh-1}:
x[j+1,1] = x[j,1] + 0.5*h*(u[j] + u[j+1]);

subject to de2 {j in 0..nh-1}:
x[j+1,2] = x[j,2] + 0.5*h*(x[j,1]*sqrt(1+u[j]^2) + x[j+1,1]*sqrt(1+u[j+1]^2));

subject to de3 {j in 0..nh-1}:
x[j+1,3] = x[j,3] + 0.5*h*(sqrt(1+u[j]^2) + sqrt(1+u[j+1]^2));

# Boundary conditions

subject to bc1: x[0,1] = a;
subject to bc2: x[nh,1] = b;
subject to bc3: x[0,2] = 0.0;
subject to bc4: x[0,3] = 0.0;
subject to bc5: x[nh,3] = L;

option solver ...;

solve;

</source>

== Other Descriptions ==

Other descriptions of this problem are available in

* Mathematical notation at [[Hanging chain problem]]
* [[:Category:AMPL | AMPL]] (using a fixed discretization) at the [http://www.mcs.anl.gov/~more/cops/ COPS library]

[[Category:AMPL/TACO]]