Double Tank multimode problem - Revision history

ClemensZeile at 07:37, 14 March 2020

2020-03-14T07:37:53Z

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	{{Dimensions		{{Dimensions
	\|nd = 1		\|nd = 1
	\|nx = 3		\|nx = 2
	\|nw = 3		\|nw = 3
	\|nre = 2		\|nre = 2

ClemensZeile at 15:32, 10 January 2018

2018-01-10T15:32:22Z

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			== Source Code ==

			Model description is available in
			* [[:Category:AMPL \| AMPL code]] at [[Double Tank multimode problem (AMPL)]]


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	[[Category:Chattering]]		[[Category:Chattering]]
	[[Category:Sensitivity-seeking arcs]]		[[Category:Sensitivity-seeking arcs]]
	~~[[Category:Population dynamics]]~~

ClemensZeile: /* Reference Solutions */

2018-01-10T15:29:49Z

Reference Solutions

← Older revision		Revision as of 15:29, 10 January 2018
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:~~MmlotkaRelaxed_12000_120_1~~.png\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=100</math>.		Image:MmdoubletankRelaxed 12000 120 1.png\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=100</math>.
	Image:MmdoubletankCIA_12000_120_1.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=100</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.		Image:MmdoubletankCIA_12000_120_1.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=100</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	</gallery>		</gallery>

ClemensZeile at 15:25, 10 January 2018

2018-01-10T15:25:31Z

← Older revision		Revision as of 15:25, 10 January 2018
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	</math>		</math>


	Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between three different fishing options.

	== Parameters ==		== Parameters ==
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	These fixed values are used within the model.		These fixed values are used within the model.

	<math>		<math> T=10, c_1=1, c_2=0.5, c_3=2, k_1=2, k_2=3. </math>
	~~\begin{array}{rcl}~~
	~~[t_0~~, ~~t_f] &~~=~~& [0, 12],\\~~
	~~(c_{0,~~1}, ~~c_{1,1}) &~~=~~& (0.2,~~ 0.~~1),\\~~
	~~(c_{0~~,2}, ~~c_{1,2}) &~~=~~& (0.4, 0.~~2~~),\\~~
	~~(c_{0~~,3~~}, c_{1,3}) &=& (0.01, 0.1)~~.
	~~\end{array}~~
	</math>

	== Reference Solutions ==		== Reference Solutions ==
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	If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.		If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.

	The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=~~400~~ </math> is <math>~~x_2(t_f) =1~~.~~82875272~~</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>~~x_2(t_f) =1~~.~~82878681~~</math>.		The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=100 </math> is <math>2.59106823</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>2.59121008</math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:~~MmlotkaRelaxed_12000_30_1~~.png\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=~~400~~</math>.		Image:MmlotkaRelaxed_12000_120_1.png\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=100</math>.
	Image:~~MmlotkaCIA 12000 30 1~~.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=~~400~~</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.		Image:MmdoubletankCIA_12000_120_1.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=100</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	</gallery>		</gallery>

ClemensZeile: Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 3 |nre = 2 }}This site describ..."

2018-01-10T15:14:43Z

Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 3 |nre = 2 }}This site describ..."

New page

{{Dimensions
|nd = 1
|nx = 3
|nw = 3
|nre = 2
}}This site describes a Double tank problem variant with three binary controls instead of only one control.

== Mathematical formulation ==

The mixed-integer optimal control problem is given by

<math>
\begin{array}{llll}
\displaystyle \min_{x,w} & \displaystyle \int_{0}^{T} & k_1(x_2-k_2)^2 \; \text{d}t\\[1.5ex]
\mbox{s.t.} & \dot{x}_1 & = \sum\limits_{i=1}^{3} c_{i}\; w_i,-\sqrt{x_1}, \\[1.5ex]
& \dot{x}_2 & = \sqrt{x_1}-\sqrt{x_2}, \\[1.5ex]
& x(0) & = (2,2)^T, \\[1.5ex]
& 1 & = \sum\limits_{i=1}^{3}w_i(t), \\
& w_i(t) &\in \{0, 1\}, \quad i=1\ldots 3.
\end{array}
</math>

Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between three different fishing options.

== Parameters ==

These fixed values are used within the model.

<math>
\begin{array}{rcl}
[t_0, t_f] &=& [0, 12],\\
(c_{0,1}, c_{1,1}) &=& (0.2, 0.1),\\
(c_{0,2}, c_{1,2}) &=& (0.4, 0.2),\\
(c_{0,3}, c_{1,3}) &=& (0.01, 0.1).
\end{array}
</math>

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=400 </math> is <math>x_2(t_f) =1.82875272</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_2(t_f) =1.82878681</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:MmlotkaRelaxed_12000_30_1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=400</math>.
Image:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]
[[Category:Population dynamics]]