Dielectrophoretic Particle OED

Dielectrophoretic Particle OED
State dimension:	1
Differential states:	13
Discrete control functions:	3

The Dielectrophoretic Particle OED problem is a variation of the Dielectrophoretic Particle problem. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale ODE model. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

Mathematical formulation

We are interested in estimating the parameters $α$ and $c$ of the initial value problem

$\begin{array}{rcl} \dot{x_{1}} (t) & = & x_{2} (t) \cdot u (t) + α \cdot u (t)^{2}, & t \in [0, t_{f}], x_{1} (0) = 1, \\ \dot{x_{2}} (t) & = & - c \cdot x_{2} (t) + u (t), x_{2} (0) = 0 . \end{array}$

The initial values and $t_{f} = 10$ are fixed. We are interested in how to fish and when to measure, with an upper bound $M$ on the measuring time. We can measure the states directly, $h^{1} (x (t)) = x_{1} (t)$ and $h^{2} (x (t)) = x_{2} (t)$ . We use two different sampling functions, $w^{1} (\cdot)$ and $w^{2} (\cdot)$ in the same experimental setting. This can be seen either as a two-dimensional measurement function $h (x (t))$ , or as a special case of a multiple experiment, in which $u (\cdot), x (\cdot)$ , and $G (\cdot)$ are identical.

Now we formulate the OED problem:

$\begin{array}{lll} \min_{y, G, F, z, w} & trace (F^{- 1} (t_{f})) \\ subject to \\ \dot{y} (t) & = & f (y (t), θ) \\ \dot{G} (t) & = & f_{y} (y (t), θ) G (t) + f_{θ} (y (t), θ) \\ \dot{F} (t) & = & \sum_{i = 1}^{n_{o}} w_{i} (t) (h_{y}^{i} (y (t)) G (t))^{T} (h_{y}^{i} (y (t)) G (t)) \\ \dot{z} (t) & = & w (t), \\ y (0) & = & y_{0} \\ G (0) & = & \frac{\partial y (0)}{\partial θ} \\ F (0) & = & 0, \\ z (0) & = & 0 \\ w (t) & \in & 𝒲 \\ z_{i} (t_{f}) & \leq & M_{i} \end{array}$

The evolution of the symmetric matrix $F : [0, t_{f}] \to ℝ^{2 \times 2}$ is given by the weighted sum of observability Gramians $h_{y}^{i} (y (t)) G (t), i = 1, 2$ for each observed function of states.

Parameters

We use $t_{f} = 8$ , $α = - 0.75$ , and $c = 1$ . The upper bound on the measurement time intervals is chosen as $M_{1} = M_{2} = 2$ .

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
States, control, and sampling functions for a local optimum.

References

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