Oscillating OED: Difference between revisions

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

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Revision as of 14:25, 25 August 2025

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 $\dot{x} (t) = : f (t) = 0.2 + 0.8 \cdot t + 0.3 \cdot (\sin (p \cdot t) + \cos (p \cdot t) \cdot p \cdot t) - 2.5 \cdot \sin (50 \cdot t), x (0) = x_{0} .$

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 as described in [2].

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

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 $p = 15$ .

Miscellaneous and Further Reading

The Toy OED problem was introduced by Sebastian Sager in [Sager2013]Author: Sager, S.
Journal: SIAM Journal on Control and Optimization
Number: 4
Pages: 3181--3207
Title: Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin's Maximum Principle
Url: http://mathopt.de/PUBLICATIONS/Sager2013.pdf
Volume: 51
Year: 2013
, which contains further details.

References

There were no citations found in the article.

@@ Line 12: / Line 12: @@
 For a single parameter <math>p</math> the original initial value problem is given by
 <math>
-   \dot{x}(t) = 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), \quad x(0) = x_0.
+   \dot{x}(t) =: f(t) = 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), \quad x(0) = x_0.
 </math>
-We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>. Thus, the experimental design problem simplifies to:
+We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>.
+Now we formulate the OED problem as described in [[#OEDUDE | [2]]].
 <p>
 <math>
   \begin{array}{lll}
-  \displaystyle \min_{x,G,F,z,w} && 1 / F(t_f) \\
+  \displaystyle \min_{y,G,F,z,w} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
   \text{subject to} \\
-\quad \dot{x}(t) & = &  p \cdot x(t),\\
+\quad \dot{y}(t) & = & f(y(t),\theta) \\
-\quad \dot{G}(t) & = & p \cdot G(t) + x(t), \\
+\quad \dot{G}(t) & = & f_y(y(t),\theta) G(t) + f_\theta(y(t),\theta) \\
-\quad \dot{F}(t) & = & w(t) \cdot G(t)^2, \\
+\quad \dot{F}(t) & = & \sum_{i=1}^{n_o} w_i(t)(h^i_y(y(t))G(t))^T(h^i_y(y(t))G(t)) \\
 \quad \dot{z}(t) & = & w(t), \\
-\quad x(0) &=& x_0, \\
+\quad y(0) & = & y_0 \\
-\quad G(0) &=& F(0) = z(0) = 0, \\
+\quad G(0) & = & \frac{\partial y(0)}{\partial \theta} \\
-\quad w(t) &\in& \mathcal{W}, \\
+\quad F(0) & = & 0, \\
-\quad 0    & \le & M - z(t_f)
+\quad z(0) & = & 0 \\
+\quad w(t) & \in & \mathcal{W} \\
+\quad z_i(t_f) & \leq & M_i
    \end{array}
 </math>