Oscillating OED

Oscillating OED
State dimension:	1
Differential states:	4
Discrete control functions:	1

The Oscillating OED problem looks for an optimal measurement strategy to determine a single parameter in a one-dimensional ODE model, where can directly measure the single state.

The optimal integer control functions shows bang bang behavior.

Mathematical formulation

For a single parameter $p$ the original initial value problem is given by $\dot{x} (t) = : f (t, p) = 0.2 + 0.8 \cdot t + 0.3 \cdot (\sin (p \cdot t) + \cos (p \cdot t) \cdot p \cdot t) - 2.5 \cdot \sin (50 \cdot t), x (0) = x_{0} .$

We assume both $x_{0}$ and $t_{f}$ to be fixed and are only interested in when to measure, with an upper bound $M$ on the measuring time. We can measure the state directly, i.e. $h (x (t)) = x (t)$ .

Now we formulate the OED problem:

$\begin{array}{lll} \min_{y, G, F, z, w} & trace (F^{- 1} (t_{f})) \\ subject to \\ \dot{y} (t) & = & f (t, p) \\ \dot{G} (t) & = & f_{p} (y (t), p) \\ \dot{F} (t) & = & w (t) (h_{y} (y (t)) G (t))^{T} (h_{y} (y (t)) G (t)) \\ \dot{z} (t) & = & w (t), \\ y (0) & = & y_{0} \\ G (0) & = & 0 \\ F (0) & = & 0, \\ z (0) & = & 0 \\ w (t) & \in & 𝒲 \\ z (t_{f}) & \leq & M \end{array}$

Parameters

These fixed values are used within the model:

$x_{0} = 0.1; t_{f} = 2; 𝒲 = [0, 1]; M = 0.2; p = 15$

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
States and measurement control for $p = 15$ . The time $t$ was added as an additional state.

Miscellaneous and Further Reading

This problem was introduced by Sebastian Sager.