Subway ride - Revision history

FelixMueller: /* Mathematical formulation */

2016-02-22T16:21:14Z

Mathematical formulation

← Older revision		Revision as of 16:21, 22 February 2016
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	<center>		<center>
	<math>		<math>
	\begin{array}{~~llcl~~}		\begin{array}{llll}
	\displaystyle \min_{x, w} & \int_{0}^{t_f} & L(x, w) \; \mathrm{d} t \\[1.5ex]		\displaystyle \min_{x, w} & \int_{0}^{t_f} & L(x, w) \; \mathrm{d} t \\[1.5ex]
	\mbox{s.t.} & \dot{x}_0 & = & x_1, \\		\mbox{s.t.} & \dot{x}_0 & = & x_1, \\

FelixMueller: /* Mathematical formulation */

2016-02-22T16:20:32Z

Mathematical formulation

← Older revision		Revision as of 16:20, 22 February 2016
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	<math>		<math>
	\begin{array}{llcl}		\begin{array}{llcl}
	\displaystyle \min_{x, w} & \int_{0}^{t_f} L(x, w) \; \mathrm{d} t \\[1.5ex]		\displaystyle \min_{x, w} & \int_{0}^{t_f} & L(x, w) \; \mathrm{d} t \\[1.5ex]
	\mbox{s.t.} & \dot{x}_0 & = & x_1, \\		\mbox{s.t.} & \dot{x}_0 & = & x_1, \\
	& \dot{x}_1 & = & f_1(x,w), \\		& \dot{x}_1 & = & f_1(x,w), \\

FelixMueller: /* Mathematical formulation */

2016-02-22T16:19:42Z

Mathematical formulation

← Older revision		Revision as of 16:19, 22 February 2016
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	<math>		<math>
	\begin{array}{llcl}		\begin{array}{llcl}
	\displaystyle \min_{x, w} & & \int_{0}^{t_f} L(x, w) \; \mathrm{d} t \\[1.5ex]		\displaystyle \min_{x, w} & \int_{0}^{t_f} L(x, w) \; \mathrm{d} t \\[1.5ex]
	\mbox{s.t.} & \dot{x}_0 & = & x_1, \\		\mbox{s.t.} & \dot{x}_0 & = & x_1, \\
	& \dot{x}_1 & = & f_1(x,w), \\		& \dot{x}_1 & = & f_1(x,w), \\

FelixMueller: /* Mathematical formulation */

2016-02-22T16:19:32Z

Mathematical formulation

← Older revision		Revision as of 16:19, 22 February 2016
Line 16:		Line 16:
	<math>		<math>
	\begin{array}{llcl}		\begin{array}{llcl}
	\displaystyle \min_{x, w} & & & \int_{0}^{t_f} L(x, w) \; \mathrm{d} t \\[1.5ex]		\displaystyle \min_{x, w} & & \int_{0}^{t_f} L(x, w) \; \mathrm{d} t \\[1.5ex]
	\mbox{s.t.} & \dot{x}_0 & = & x_1, \\		\mbox{s.t.} & \dot{x}_0 & = & x_1, \\
	& \dot{x}_1 & = & f_1(x,w), \\		& \dot{x}_1 & = & f_1(x,w), \\

FelixMueller: /* Mathematical formulation */

2016-02-22T16:19:12Z

Mathematical formulation

← Older revision		Revision as of 16:19, 22 February 2016
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	\begin{array}{llcl}		\begin{array}{llcl}
	\displaystyle \min_{x, w} & & & \int_{0}^{t_f} L(x, w) \; \mathrm{d} t \\[1.5ex]		\displaystyle \min_{x, w} & & & \int_{0}^{t_f} L(x, w) \; \mathrm{d} t \\[1.5ex]
	\mbox{s.t.} & \dot{x}_0~~(t)~~ & = & x_1~~(t)~~, \\		\mbox{s.t.} & \dot{x}_0 & = & x_1, \\
	& \dot{x}_1~~(t)~~ & = & f_1(x,w), \\		& \dot{x}_1 & = & f_1(x,w), \\
	& x(0) &=& (0, 0)^T, \\		& x(0) &=& (0, 0)^T, \\
	& x(t_f) &=& (2112, 0)^T, \\		& x(t_f) &=& (2112, 0)^T, \\
	& w(t) &\in& \{1, 2, 3, 4\}~~, \quad t \in [0, t_f]~~.		& w(t) &\in& \{1, 2, 3, 4\}.
	\end{array}		\end{array}
	</math>		</math>

SebastianSager: Moved C to Muscod

2016-01-28T09:23:18Z

Moved C to Muscod

← Older revision		Revision as of 09:23, 28 January 2016
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	Model descriptions are available in		Model descriptions are available in

	* [[:Category:C \| C code]] at [[Subway ride (C)]]		* [[:Category:Muscod \| Muscod code]] at [[Subway ride (Muscod)]]

	== References ==		== References ==

JonasSchulze: Text replacement - "\(.*)\<\/bibref\>" to ""

2015-12-30T20:32:10Z

Text replacement - "\<bibref\>(.*)\<\/bibref\>" to "<bib id="$1" />"

← Older revision		Revision as of 20:32, 30 December 2015
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	}}		}}

	The ''subway ride'' optimal control problem goes back to work of <~~bibref>~~Bock1982</~~bibref~~> for the city of New York. In an extension, also velocity limits that lead to path-constrained arcs appear.		The ''subway ride'' optimal control problem goes back to work of <bib id="Bock1982" /> for the city of New York. In an extension, also velocity limits that lead to path-constrained arcs appear.
	The aim is to minimize the energy used for a subway ride from one station to another, taking into account boundary conditions and a restriction on the time.		The aim is to minimize the energy used for a subway ride from one station to another, taking into account boundary conditions and a restriction on the time.

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	</math></center>		</math></center>

	Details about the derivation of this model and the assumptions made can be found in <~~bibref>~~Bock1982</~~bibref~~> or in <~~bibref>~~Kraemer-Eis1985</~~bibref~~>.		Details about the derivation of this model and the assumptions made can be found in <bib id="Bock1982" /> or in <bib id="Kraemer-Eis1985" />.

	== Parameters ==		== Parameters ==
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	== Reference Solutions ==		== Reference Solutions ==

	The optimal trajectory for this problem has been calculated by means of an indirect approach in <~~bibref>~~Bock1982</~~bibref~~><~~bibref>~~Kraemer-Eis1985</~~bibref~~>, and based on the direct multiple shooting method in <~~bibref>~~Sager2009</~~bibref~~>.		The optimal trajectory for this problem has been calculated by means of an indirect approach in <bib id="Bock1982" /><bib id="Kraemer-Eis1985" />, and based on the direct multiple shooting method in <bib id="Sager2009" />.

JonasSchulze: Text replacement - "" to ""

2015-12-30T20:28:26Z

Text replacement - "<bibreferences/>" to "<biblist />"

← Older revision		Revision as of 20:28, 30 December 2015
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	== References ==		== References ==
	<~~bibreferences~~/>		<biblist />

	<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->		<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->

SebastianSager: Added solutions

2010-11-21T17:52:39Z

Added solutions

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 The optimal trajectory for this problem has been calculated by means of an indirect approach in <bibref>Bock1982</bibref><bibref>Kraemer-Eis1985</bibref>, and based on the direct multiple shooting method in <bibref>Sager2009</bibref>.
 == Variants ==
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 The additional path constraint changes the qualitative behavior of the relaxed solution. While all solutions considered this far were bang-bang and the main work consisted in finding the switching points, we now have a path-constrained arc. The optimal solutions for refined grids yield a series of monotonically decreasing objective function values, where the limit is the best value that can be approximated by an integer feasible solution. In our case we obtain 1.33108, 1.31070, 1.31058, 1.31058, ...
 == Source Code ==

SebastianSager: Next table

2010-11-21T17:38:32Z

Next table

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 </math></center>
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 |}
+{| border="1" align="center" cellpadding="5" cellspacing="0"
-Details about the derivation of this model and the assumptions made can be found in <bibref>Bock1982</bibref> or in <bibref>Kraemer-Eis1985</bibref>.
+|- bgcolor=#c7c7c7
 == Reference Solutions ==