Dielectrophoretic Particle - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T10:44:01Z

Reference Solutions

← Older revision		Revision as of 10:44, 28 November 2025
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	Here is one local solution to the above control problem.		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Dielectrophoretic_Particle.png\| States and discretized control for a local optimum. The control <math>t_f</math> represents the scaling of the time interval, where the base time interval is [0,5].		Image:Dielectrophoretic_Particle.png\| States and discretized control for a local optimum. The control <math>t_f</math> represents the scaling of the time interval, where the base time interval is [0,5].
	</gallery>		</gallery>

2025-11-24T15:04:39Z

Mathematical formulation

← Older revision		Revision as of 15:04, 24 November 2025
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	\quad \dot{x_0}(t) & = & x_1(t) \cdot u(t) + \alpha \cdot u(t)^2,\\		\quad \dot{x_0}(t) & = & x_1(t) \cdot u(t) + \alpha \cdot u(t)^2,\\
	\quad \dot{x_1}(t) & = & -c \cdot x_1(t) + u(t), \\		\quad \dot{x_1}(t) & = & -c \cdot x_1(t) + u(t), \\
	\quad x(0) &=& x_0, \\		\quad x_0(0) &=& x_0, \\
	\quad y(0) &=& 0, \\		\quad x_1(0) &=& 0, \\
	\quad x(t_f) &=& x_f, \\		\quad x_0(t_f) &=& x_f, \\
	\quad t_f & \geq & 0, \\		\quad t_f & \geq & 0, \\
	\quad u(t) & \in & [-1, 1] \ \quad \forall t \in [0,t_f]		\quad u(t) & \in & [-1, 1] \ \quad \forall t \in [0,t_f]

2025-11-24T15:04:14Z

← Older revision		Revision as of 15:04, 24 November 2025
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	}}		}}

	The '''Dielectrophoretic Particle problem''' is a classical time-optimal control benchmark for microfluidic particle manipulation. This description is taken from [[#OCPjl \| [1]]]		The '''Dielectrophoretic Particle problem''' is a classical time-optimal control benchmark for microfluidic particle manipulation. This description is taken from [[#OCPjl \| [1]]].

	It models the motion of a particle under a dielectrophoretic force, where the control voltage applied to electrodes directly influences the particle trajectory.		It models the motion of a particle under a dielectrophoretic force, where the control voltage applied to electrodes directly influences the particle trajectory.

2025-11-24T12:12:35Z

Mathematical formulation

← Older revision		Revision as of 12:12, 24 November 2025
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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{u} && t_f \\		\displaystyle \min_{t_f, u} && t_f \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x_0}(t) & = & x_1(t) \cdot u(t) + \alpha \cdot u(t)^2,\\		\quad \dot{x_0}(t) & = & x_1(t) \cdot u(t) + \alpha \cdot u(t)^2,\\

2025-11-24T09:23:02Z

References

2025-11-24T09:22:23Z

2025-11-24T09:21:43Z

Miscellaneous and Further Reading

← Older revision		Revision as of 09:21, 24 November 2025
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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	This formulation and a detailed description can be found in [[#~~openmdao~~\|[1]]].		This formulation and a detailed description can be found in [[#OCPjl\|[1]]].

	== References ==		== References ==

2025-11-24T09:21:22Z

Reference Solutions

← Older revision		Revision as of 09:21, 24 November 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:Dielectrophoretic_Particle.png\| States and discretized control for a local optimum.		Image:Dielectrophoretic_Particle.png\| States and discretized control for a local optimum. The control <math>t_f</math> represents the scaling of the time interval, where the base time interval is [0,5].
	</gallery>		</gallery>

2025-11-24T09:19:17Z

Reference Solutions

← Older revision		Revision as of 09:19, 24 November 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:~~Mountain_Car~~.png\| States and discretized control for a local optimum.		Image:Dielectrophoretic_Particle.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

2025-11-24T09:11:00Z

Parameters

← Older revision		Revision as of 09:11, 24 November 2025
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	== Parameters ==		== Parameters ==
	These fixed values are used within the model:		These fixed values are used within the model:
	~~<p>~~
	~~<math>~~
	~~y_0 = 0; \quad t_f = 40; \quad \mathcal{W} = [0,1]; \quad M = 5; \quad p = (0.05884,4.298,21.80)~~
	~~</math>~~
	~~</p>~~


	{\| border="1" align="center" cellpadding="5" cellspacing="0"		{\| border="1" align="center" cellpadding="5" cellspacing="0"
	\|- bgcolor=#c7c7c7		\|- bgcolor=#c7c7c7
	! Symbol !! Value !! Description		! Symbol !! Value !! Description
			\|-
			\| align=center \| <math>x_0</math> \|\| align=right \| 1 \|\| Initial particle position
	\|-		\|-
	~~\| align=center \| <math>x_0</math> \|\| align=right \| 1 \|\| Initial particle position~~
	\|-
	\| align=center \| <math>x_f</math> \|\| align=right \| 2 \|\| Final particle position		\| align=center \| <math>x_f</math> \|\| align=right \| 2 \|\| Final particle position
	\|-		\|-
	\| align=center \| <math>\alpha</math> \|\| align=right \| -0.75 \|\| Nonlinear coefficient		\| align=center \| <math>\alpha</math> \|\| align=right \| -0.75 \|\| Nonlinear coefficient
	\|-		\|-
	\| align=center \| <math>c</math> \|\| align=right \| 1 \|\| Damping coefficient \|\|		\| align=center \| <math>c</math> \|\| align=right \| 1 \|\| Damping coefficient
	\|}		\|}