Double Tank - Revision history

ClemensZeile: /* Variants */

2020-03-14T07:34:17Z

Variants

← Older revision		Revision as of 07:34, 14 March 2020
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	* different control functions for the two tanks, see [[Double Tank multimode problem]]		* different control functions for the two tanks, see [[Double Tank multimode problem]]
			* a three tank problem with 3 modes, see [[Three Tank multimode problem]]

	== Source Code ==		== Source Code ==

JohnHedengren: /* Reference Solution */

2019-03-14T17:07:54Z

Reference Solution

← Older revision		Revision as of 17:07, 14 March 2019
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	== Reference Solution ==		== Reference Solution ==
	The problem ~~was~~ solved in Matlab (version 2014b) with [[:Category: switch \| Switch]]. The switch codes rely on [http://homepages.laas.fr/henrion/software/gloptipoly3/ GloptiPoly 3] and the solver [http://sedumi.ie.lehigh.edu/ SeDuMi 1.3] (optimization over symmetric cones).		The problem is solved in Python with [[:Category: Gekko \| GEKKO]] or Matlab (version 2014b) with [[:Category: switch \| Switch]]. The switch codes rely on [http://homepages.laas.fr/henrion/software/gloptipoly3/ GloptiPoly 3] and the solver [http://sedumi.ie.lehigh.edu/ SeDuMi 1.3] (optimization over symmetric cones).
	By introducing the lifts <math>l_i=\sqrt{x_i}</math>, algebraically constrained as <math>l_i^2=x_i, \; l_i\geq 0,</math> the problem is recast with polynomial data. In this way way switch in connection with GloptiPoly3 can be applied. GloptiPoly 3 is a Matlab package developed by Didier Henrion for generalized problems of moments and needs polynomial input data. The exact code used to solve the problem can be found under [[Double Tank (switch)]]. The calculated objective is 4.7296 with the following trajectories of states and controls:		By introducing the lifts <math>l_i=\sqrt{x_i}</math>, algebraically constrained as <math>l_i^2=x_i, \; l_i\geq 0,</math> the problem is recast with polynomial data. In this way way switch in connection with GloptiPoly3 can be applied. GloptiPoly 3 is a Matlab package developed by Didier Henrion for generalized problems of moments and needs polynomial input data. The exact code used to solve the problem can be found under [[Double Tank (switch)]]. The calculated objective is 4.7296 with the following trajectories of states and controls:
	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">

JohnHedengren at 17:04, 14 March 2019

2019-03-14T17:04:26Z

← Older revision		Revision as of 17:04, 14 March 2019
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	Model descriptions are available in		Model descriptions are available in

			* [[:Category:Gekko \| GEKKO Python code]] at [[Double Tank (GEKKO)]]
	* [[:Category: switch \| switch code]] at [[Double Tank (switch)]]		* [[:Category: switch \| switch code]] at [[Double Tank (switch)]]

ClemensZeile at 15:03, 10 January 2018

2018-01-10T15:03:37Z

← Older revision		Revision as of 15:03, 10 January 2018
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	Image:Control_sequence_Tank.jpg\| Control sequence over time.		Image:Control_sequence_Tank.jpg\| Control sequence over time.
	</gallery>		</gallery>

			== Variants ==

			There are several alternative formulations and variants of the above problem, in particular

			* different control functions for the two tanks, see [[Double Tank multimode problem]]


	== Source Code ==		== Source Code ==

FelixMueller at 22:10, 27 June 2016

2016-06-27T22:10:00Z

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	[[Category: Tracking objective]]		[[Category: Tracking objective]]
	[[Category: State dependent switches]]		[[Category: State dependent switches]]
			[[Category:Bang bang]]
			[[Category:Path-constrained arcs]]

FelixMueller at 17:37, 3 March 2016

2016-03-03T17:37:23Z

← Older revision		Revision as of 17:37, 3 March 2016
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	}}		}}

	The double tank problem is a basic example for a switching system. It contains the dynamics of an upper and a lower tank, connected to each other with a pipe. The goal is to minimize the deviation of a certain fluid level <math>k_2</math> in the lower tank. The problem was introduced and discussed in a variety of publications for the optimal control of constrained switched systems, e.g. [http://homepages.laas.fr/henrion/papers/switch.pdf Henrion et al.] and [http://epubs.siam.org/doi/pdf/10.1137/120901507 Vasudevan et al.]		The double tank problem is a basic example for a switching system. It contains the dynamics of an upper and a lower tank, connected to each other with a pipe. The goal is to minimize the deviation of a certain fluid level <math>k_2</math> in the lower tank. The problem was introduced and discussed in a variety of publications for the optimal control of constrained switched systems, e.g. <bib id="Henrion2014" /> (Link: [http://homepages.laas.fr/henrion/papers/switch.pdf Henrion et al.]) and <bib id="Vasudevan2013" /> (Link: [http://epubs.siam.org/doi/pdf/10.1137/120901507 Vasudevan et al.]).

	== Mathematical formulation ==		== Mathematical formulation ==

FelixMueller at 13:55, 23 February 2016

2016-02-23T13:55:29Z

← Older revision		Revision as of 13:55, 23 February 2016
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	[[Category:ODE model]]		[[Category:ODE model]]
	[[Category: Tracking objective]]		[[Category: Tracking objective]]
			[[Category: State dependent switches]]

FelixMueller: /* Reference Solution */

2016-02-17T13:12:07Z

Reference Solution

← Older revision		Revision as of 13:12, 17 February 2016
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	== Reference Solution ==		== Reference Solution ==
	The problem was solved in Matlab (version 2014b) with [[:Category: switch \| Switch]]. The switch codes rely on [http://homepages.laas.fr/henrion/software/gloptipoly3/ GloptiPoly 3] and the solver [http://sedumi.ie.lehigh.edu/ SeDuMi 1.3] (optimization over symmetric cones).		The problem was solved in Matlab (version 2014b) with [[:Category: switch \| Switch]]. The switch codes rely on [http://homepages.laas.fr/henrion/software/gloptipoly3/ GloptiPoly 3] and the solver [http://sedumi.ie.lehigh.edu/ SeDuMi 1.3] (optimization over symmetric cones).
	By introducing the lifts <math>l_i=\sqrt{x_i}</math>, algebraically constrained as <math>l_i^2=x_i, \; l_i\geq 0,</math> the problem is recast with polynomial data. In this way way switch in connection with GloptiPoly3 can be applied. GloptiPoly 3 is a Matlab package developed by Didier Henrion for generalized problems of moments and needs polynomial input data. The exact code used to solve the problem can be found under [[Double Tank ~~Problem~~ (switch)]]. The calculated objective is 4.7296 with the following trajectories of states and controls:		By introducing the lifts <math>l_i=\sqrt{x_i}</math>, algebraically constrained as <math>l_i^2=x_i, \; l_i\geq 0,</math> the problem is recast with polynomial data. In this way way switch in connection with GloptiPoly3 can be applied. GloptiPoly 3 is a Matlab package developed by Didier Henrion for generalized problems of moments and needs polynomial input data. The exact code used to solve the problem can be found under [[Double Tank (switch)]]. The calculated objective is 4.7296 with the following trajectories of states and controls:
	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:State_sequence_Tank.jpg\| State trajectory over time.		Image:State_sequence_Tank.jpg\| State trajectory over time.

FelixMueller: /* Parameters */

2016-02-17T13:11:39Z

Parameters

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 == Parameters ==
 In an exemplary test, the parameters were chosen to be:
 {| class="wikitable"
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 |<math>10</math>
 |}
 == Reference Solution ==

FelixMueller: /* Mathematical formulation */

2016-02-17T13:11:02Z

Mathematical formulation

← Older revision		Revision as of 13:11, 17 February 2016
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	<math>		<math>
	\begin{array}{llll}		\begin{array}{llll}
	\displaystyle \min_{\sigma} & \int_{0}^{T} & k_1(x_2-k_2)^2 \; \text{d}t\\[1.5ex]		\displaystyle \min_{\sigma} & \displaystyle \int_{0}^{T} & k_1(x_2-k_2)^2 \; \text{d}t\\[1.5ex]
	\mbox{s.t.} & \dot{x}_1 & = c_{\sigma(t)}-\sqrt{x_1}, \\[1.5ex]		\mbox{s.t.} & \dot{x}_1 & = c_{\sigma(t)}-\sqrt{x_1}, \\[1.5ex]
	& \dot{x}_2 & = \sqrt{x_1}-\sqrt{x_2}, \\[1.5ex]		& \dot{x}_2 & = \sqrt{x_1}-\sqrt{x_2}, \\[1.5ex]