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 we 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:

$\begin{array}{lll}\underset{x,G,F,z,w}{\min }& & \text{trace}(F^{-1}(t_f))\\ \text{subject to}\\ \dot{x}(t)& =& f(t,p)\\ \dot{G}(t)& =& f_p(x(t),p)\\ \dot{F}(t)& =& w(t)(h_x(x(t))G(t){)}^T(h_x(x(t))G(t))\\ \dot{z}(t)& =& w(t),\\ x(0)& =& x_0\\ G(0)& =& 0\\ F(0)& =& 0,\\ z(0)& =& 0\\ w(t)& \in & {𝒲}\\ z(t_f)& \le & M\end{array}$

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$x_0$	0.1	Initial value for $x$
$p$	15	Unknown parameter
$t_{\mathrm{f}}$	2	Horizon of the control problem
${𝒲}$	[0,1]	Bounds of measurement function
$M$	0.2	Maximum measurement time

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.