Dielectrophoretic Particle: Difference between revisions

Dielectrophoretic Particle
State dimension:	1
Differential states:	2
Discrete control functions:	2

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Revision as of 09:01, 24 November 2025

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

It models the motion of a particle under a dielectrophoretic force, where the control voltage applied to electrodes directly influences the particle trajectory. Both the particle position and an auxiliary state related to its dipole moment, as well as the control voltage, are decision variables. The objective is to transfer the particle from an initial position to a target position in minimal time, while satisfying bounds on the control input and maintaining the auxiliary state dynamics.

Mathematical formulation

$\begin{array}{lll} \min_{u} & t_{f} \\ subject to \\ \dot{x_{0}} (t) & = & x_{1} (t) \cdot u (t) + α \cdot u (t)^{2}, \\ \dot{x_{1}} (t) & = & - c \cdot x_{1} (t) + u (t), \\ x (0) & = & x_{0}, \\ y (0) & = & 0, \\ x (t_{f}) & = & x_{f}, \\ t_{f} & \geq & 0, \\ u (t) & \in & [- 1, 1] \forall t \in [0, t_{f}] \end{array}$

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
States and discretized control for a local optimum.

Miscellaneous and Further Reading

This formulation and a detailed description can be found in [1].

References

[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/dielectrophoretic_particle.md

@@ Line 17: / Line 17: @@
   \displaystyle \min_{u} && t_f \\
   \text{subject to} \\
-\quad \dot{x}(t) & = & v(t),\\
+\quad \dot{x_0}(t) & = & x_1(t) \cdot u(t) + \alpha \cdot u(t)^2,\\
-\quad \dot{v}(t) & = & 0.001 \cdot u(t) - 0.0025 \cdot \cos(3 \cdot x(t)), \\
+\quad \dot{x_1}(t) & = & -c \cdot x_1(t) + u(t), \\
-\quad x(0) &=& -0.5, \\
+\quad x(0) &=& x_0, \\
-\quad v(0) &=& 0, \\
+\quad y(0) &=& 0, \\
-\quad x(t_f) &=& 0.5, \\
+\quad x(t_f) &=& x_f, \\
-\quad v(t_f) & \geq & 0, \\
+\quad t_f & \geq & 0, \\
 \quad u(t) & \in & [-1, 1] \ \quad \forall t \in [0,t_f]
    \end{array}