Cushioned Oscillation - Revision history

RobertLampel: /* Model formulation */

2026-02-19T13:26:49Z

Model formulation

← Older revision		Revision as of 13:26, 19 February 2026
Line 25:		Line 25:
	The goal is to reset position and velocity of the object as fast as possible, meaning:		The goal is to reset position and velocity of the object as fast as possible, meaning:

	<math>(x(~~t_f~~),v(~~t_f~~)) = (0,0)</math>,		<math>(x(t\mathrm{f}),v(t_\mathrm{f})) = (0,0)</math>,

	with the objective function:		with the objective function:

	<math>\min\limits_{~~t_f~~} ~~t_f~~</math>		<math>\min\limits_{t_\mathrm{f}} t_\mathrm{f}</math>

	== Optimal Control Problem Formulation ==		== Optimal Control Problem Formulation ==

2026-02-19T13:26:03Z

Parameters and Reference Solution

← Older revision		Revision as of 13:26, 19 February 2026
Line 53:		Line 53:
	== Parameters and Reference Solution ==		== Parameters and Reference Solution ==

	The following parameters were used, to create the reference solution below, with an almost optimal final time <math> ~~t_f~~ = 8.98 s</math>:		The following parameters were used, to create the reference solution below, with an almost optimal final time <math> t_\mathrm{f} = 8.98 s</math>:

	<math> m=5, </math>		<math> m=5, </math>
Line 59:		Line 59:
	<math> x_0=2, </math>		<math> x_0=2, </math>
	<math> v_0=5, </math>		<math> v_0=5, </math>
	<math> u_{mm}=5.</math>		<math> u_\mathrm{mm}=5.</math>

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

2026-02-19T13:25:43Z

Optimal Control Problem Formulation

← Older revision		Revision as of 13:25, 19 February 2026
Line 37:		Line 37:
	<math> \begin{array}{llll}		<math> \begin{array}{llll}

	\min\limits_{x,v,u,~~t_f~~} & ~~t_f~~ & & \\		\min\limits_{x,v,u,t_\mathrm{f}} & t_\mathrm{f} & & \\

	s.t. & \dot x & = v,\\		s.t. & \dot x & = v,\\
Line 45:		Line 45:
	& x(0) & = x_0,\\		& x(0) & = x_0,\\
	& v(0) & = v_0,\\		& v(0) & = v_0,\\
	& x(~~t_f~~) & = 0,\\		& x(t_\mathrm{f}) & = 0,\\
	& v(~~t_f~~) & = 0,\\		& v(t_\mathrm{f}) & = 0,\\
	& \|u\| & \le u_{mm}.\\		& \|u\| & \le u_\mathrm{mm}.\\

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

2016-06-30T09:06:27Z

← Older revision		Revision as of 09:06, 30 June 2016
Line 3:		Line 3:
	\|nx = 2		\|nx = 2
	\|nu = 1		\|nu = 1
			\|nc = 2
	\|nre = 4		\|nre = 4
	}}The Cushioned Oscillation is a simplified model of time optimal "stopping" of an oscillating object attached to a spring by applying a control and moving it back into the relaxed position and zero velocity.		}}The Cushioned Oscillation is a simplified model of time optimal "stopping" of an oscillating object attached to a spring by applying a control and moving it back into the relaxed position and zero velocity.

2016-06-27T22:02:44Z

2016-02-14T17:57:04Z

Optimal Control Problem Formulation

2016-02-14T17:56:25Z

Optimal Control Problem Formulation

← Older revision		Revision as of 17:56, 14 February 2016
Line 34:		Line 34:
	The above results in the following OCP		The above results in the following OCP

	<math> \begin{array}{llll}		<math> \begin{array}{llll}
	\min\limits_{x,v,u,t_f} & t_f & &\\
			\min\limits_{x,v,u,t_f} & t_f & & \\

	s.t. & \dot x & = & v,\\		s.t. & \dot x & = & v,\\

	& \dot v & = &\frac{1}{m}(u - c \cdot x),\\		& \dot v & = & \frac{1}{m}(u - c \cdot x),\\
	\\		\\
	& x(0) & = & x_0,\\		& x(0) & = & x_0,\\
	& v(0) & = & v_0,\\		& v(0) & = & v_0,\\

2016-02-14T17:55:18Z

Optimal Control Problem Formulation

2016-02-14T17:53:39Z

Optimal Control Problem Formulation

2016-02-14T17:53:19Z

Optimal Control Problem Formulation

← Older revision		Revision as of 17:53, 14 February 2016
Line 34:		Line 34:
	The above results in the following OCP		The above results in the following OCP

	<math> \begin{array}{~~llr~~}		<math> \begin{array}{llll}
	\min\limits_{x,v,u,t_f} & t_f\\		\min\limits_{x,v,u,t_f} & t_f\\

	s.t. & \dot x = v\\		s.t. & \dot x & = & v,\\

	& \dot v= \frac{1}{m}(u - c \cdot x)\\		& \dot v & = &\frac{1}{m}(u - c \cdot x),\\
	\\		\\
	& x(0)=x_0,\\		& x(0) & = & x_0,\\
	& v(0)=v_0,\\		& v(0) & = & v_0,\\
	& x(t_f)=0,\\		& x(t_f) & = & 0,\\
	& v(t_f)=0,\\		& v(t_f) & = & 0,\\
	\\		& \|u\| & \le % u_{mm}.\\
	& \|u\| \le u_{mm}\\
	~~& -u_{mm} \le u \le u_{mm}\\~~
	\end{array}</math>		\end{array}</math>