Van der Pol Oscillator (binary variant) - Revision history

RobertLampel at 10:36, 2 December 2024

2024-12-02T10:36:26Z

← Older revision		Revision as of 10:36, 2 December 2024
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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\min\limits_{x,y,w} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2 dt\\		\min\limits_{x,y,w} & \int\limits_{t_0}^{t_f} & x(t)^2+y(t)^2 dt\\
	s.t. & \dot x & = y,\\		s.t. & \dot x & = y,\\
	& \dot y & = \sum\limits_{i=1}^{3} c_{i}\; w_i \;(1-x^2) y-x,\\		& \dot y & = \sum\limits_{i=1}^{3} c_{i}\; w_i \;(1-x^2) y-x,\\

ClemensZeile at 17:05, 10 January 2018

2018-01-10T17:05:20Z

← Older revision		Revision as of 17:05, 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:~~VanderpolCIA_6000_100_1~~.pdf\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=60</math>.		Image:VanderpolRelaxed 6000 100 1.pdf\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=60</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=6000, \, n_u=60</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.		Image:VanderpolCIA_6000_100_1.pdf\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=60</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	</gallery>		</gallery>



			== Source Code ==

			Model description is available in
			* [[:Category:AMPL \| AMPL code]] at [[Van der Pol Oscillator binary variant(AMPL)]]

ClemensZeile at 16:58, 10 January 2018

2018-01-10T16:58:11Z

← Older revision		Revision as of 16:58, 10 January 2018
Line 14:		Line 14:
	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\min\limits_{x,y,w} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2~~+u(t)^2)~~ dt\\		\min\limits_{x,y,w} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2 dt\\
	s.t. & \dot x & = y,\\		s.t. & \dot x & = y,\\
	& \dot y & = \sum\limits_{i=1}^{3} c_{i}\; w_i \;(1-x^2) y-x,\\		& \dot y & = \sum\limits_{i=1}^{3} c_{i}\; w_i \;(1-x^2) y-x,\\
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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=6000, \, n_u=60 </math> is <math>1.30167235</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>1.30273681</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:VanderpolCIA_6000_100_1.pdf\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=60</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:MmlotkaCIA 12000 30 1.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=60</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	</gallery>		</gallery>

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

2018-01-10T16:42:16Z

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

New page

{{Dimensions
|nd = 1
|nx = 2
|nw = 3
|nre = 2
}}This site describes a Van der Pol Oscillator variant with three binary controls instead of only one continuous control.

== Mathematical formulation ==

The mixed-integer optimal control problem is given by

<math>
\begin{array}{lll}
\min\limits_{x,y,w} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2+u(t)^2) dt\\
s.t. & \dot x & = y,\\
& \dot y & = \sum\limits_{i=1}^{3} c_{i}\; w_i \;(1-x^2) y-x,\\
& x(0) & =1,\\
& y(0) & =0,\\
& 1 &= \sum\limits_{i=1}^{3}w_i(t), \\
& w_i(t) &\in \{0, 1\}, \quad i=1\ldots 3.
\end{array}</math>

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

<math>[t_0,t_f]=[0,20], c_1=-1, c_2=0.75, c_3=-2.</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:Chattering]]
[[Category:Sensitivity-seeking arcs]]