Van der Pol Oscillator - Revision history

RobertLampel: /* Parameters */

2026-03-26T09:22:31Z

Parameters

← Older revision		Revision as of 09:22, 26 March 2026
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	<math>[t_0,t_f]=[0,20]</math>		<math>[t_0,t_f]=[0,20]</math>

			== Variants ==
			The following modified version is used in [https://github.com/rlampel/pysherloc PySHeRLOC]:

			<math>
			\begin{array}{lll}
			\min\limits_{u} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2+u(t)^2) dt\\
			s.t. & \dot x(t) & = (1 - y(t)^2) \cdot x(t) - y(t) + u(t),\\
			& \dot y(t) & = x(t),\\
			& x(0) & = 0,\\
			& y(0) & = 1,\\
			& x(t) & \geq -0.25,\\
			& u(t) & \in [-1,1].\\
			\end{array}</math>

	== Reference Solution ==		== Reference Solution ==

2018-01-10T16:27:47Z

← Older revision		Revision as of 16:27, 10 January 2018
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	* [[:Category: JModelica \| JModelica code]] at [[Van der Pol Oscillator (JModelica)]]		* [[:Category: JModelica \| JModelica code]] at [[Van der Pol Oscillator (JModelica)]]
	* [[:Category: Julia/JuMP \| JuMP code]] (of a slightly modified Van der Pol oscillator problem) at [[Van der Pol Oscillator (Jump)]]		* [[:Category: Julia/JuMP \| JuMP code]] (of a slightly modified Van der Pol oscillator problem) at [[Van der Pol Oscillator (Jump)]]

			== Variants ==

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

			* a variant where partial outer convexification is applied on the control and the continous control is replaces by binary controls, see also [[Van der Pol Oscillator (binary variant)]],


	== References ==		== References ==

2016-07-27T08:34:16Z

2016-06-27T22:13:13Z

2016-06-27T22:12:41Z

← Older revision		Revision as of 22:12, 27 June 2016
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	The Problem can be found under the following [https://en.wikipedia.org/wiki/Van_der_Pol_oscillator link] or in the [http://www.jmodelica.org/api-docs/usersguide/1.4.0/ch08s02.html JModelica Users Guide].		The Problem can be found under the following [https://en.wikipedia.org/wiki/Van_der_Pol_oscillator link] or in the [http://www.jmodelica.org/api-docs/usersguide/1.4.0/ch08s02.html JModelica Users Guide].

	[[Category:MIOCP]] [[Category: ~~Tracking objective~~]]		[[Category:MIOCP]]
			[[Category:Bang bang]]
			[[Category:Path-constrained arcs]]
			[[Category:ODE model]]

2016-02-22T15:45:46Z

Optimal Control Problem

2016-02-22T15:45:32Z

Optimal Control Problem

← Older revision		Revision as of 15:45, 22 February 2016
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	The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:		The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

	<math> \begin{array}{lll}		<math>
	\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\		\begin{array}{lll}
	s.t. & \dot x &= y\\		\min\limits_{u} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2+u(t)^2) dt\\
	& \dot y &= u(1-x^2) y-x\\		s.t. & \dot x & = y,\\
	& x(0)&=1\\		& \dot y & = u(1-x^2) y-x,\\
	& y(0)&=0\\		& x(0) & =1,\\
	& u &\le 0.75\\		& y(0) & =0,\\
			& u & \le 0.75.\\
			\end{array}</math>

	\end{array}</math>

	== Parameters ==		== Parameters ==

2016-02-22T15:44:44Z

Optimal Control Problem

← Older revision		Revision as of 15:44, 22 February 2016
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	<math> \begin{array}{lll}		<math> \begin{array}{lll}
	\min\limits_{u} & \int\limits_{t_0}^{t_f}&(x(t)^2+y(t)^2+u(t)^2) dt\\		\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\
	s.t. & \dot x &= y\\		s.t. & \dot x &= y\\
	& \dot y &= u(1-x^2) y-x\\		& \dot y &= u(1-x^2) y-x\\

2016-02-22T15:44:26Z

Optimal Control Problem

← Older revision		Revision as of 15:44, 22 February 2016
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	The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:		The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:

	<math> \begin{array}{ll}		<math> \begin{array}{lll}
	\min\limits_{u} & \int\limits_{t_0}^{t_f}(x(t)^2+y(t)^2+u(t)^2) dt\\		\min\limits_{u} & \int\limits_{t_0}^{t_f}&(x(t)^2+y(t)^2+u(t)^2) dt\\
	s.t. & \dot x = y\\		s.t. & \dot x &= y\\
	& \dot y = u(1-x^2) y-x\\		& \dot y &= u(1-x^2) y-x\\
	& x(0)=1\\		& x(0)&=1\\
	& y(0)=0\\		& y(0)&=0\\
	& u\le 0.75\\		& u &\le 0.75\\



	\end{array}</math>		\end{array}</math>


	== Parameters ==		== Parameters ==

2016-01-27T17:21:17Z

← Older revision		Revision as of 17:21, 27 January 2016
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	The Problem can be found under the following [https://en.wikipedia.org/wiki/Van_der_Pol_oscillator link] or in the [http://www.jmodelica.org/api-docs/usersguide/1.4.0/ch08s02.html JModelica Users Guide].		The Problem can be found under the following [https://en.wikipedia.org/wiki/Van_der_Pol_oscillator link] or in the [http://www.jmodelica.org/api-docs/usersguide/1.4.0/ch08s02.html JModelica Users Guide].

	[[Category:MIOCP]]		[[Category:MIOCP]] [[Category: Tracking objective]]