Exponential OED

Exponential OED
State dimension:	1
Differential states:	2
Discrete control functions:	1

The Exponential OED problem was formulated as minimal design problem that to highlight one important difference between single and multiple shooting. We are interested in finding an optimal experimental design to determine the 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 $\dot{x}(t)=x(t)\cdot (x(t)+p),t\in [0,t_f],x(0)=x_0.$

We furthermore restrict the state to be in the interval $[0,200]$. We assume that we have one measurement at the end time point $t_f$. This allows to eliminate sampling function directly from the control problem and to use the objective function

$\text{trace}C(t_f)=\frac{1}{D(t_f)}=\frac{1}{H(0)+G(t_f{)}^2\cdot w_1}=\frac{1}{G(t_f{)}^2}$

Applying our transformation, we obtain the following experimental design control problem:

$\begin{array}{lll}\underset{q}{\min }& & 1/G(t_f{)}^2\\ \text{subject to}\\ \dot{x}(t)& =& x(t)\cdot (x(t)+p),\\ \dot{G}(t)& =& (p+2\cdot x(t))\cdot G(t)+x(t),\\ \dot{F}(t)& =& w(t)\cdot G(t{)}^2,\\ \dot{z}(t)& =& w(t),\\ x(0)& =& q,\\ G(0)& =& \epsilon ,\\ x(t)& \le & 200,\\ q& \in & {𝒟}\end{array}$

Parameters

These fixed values are used within the model:

$p=1;t_f=0.6;\epsilon =10^{-3}.$

Reference Solutions

As can be seen, the optimal solution will maximize the value of $G(t_f)$ and hence also of $x(t_f)$. For $t_f=0.6$ the optimal initial value is given by $q^{*}=1.203$ leading to state values of $x(t_f)=200$ and $G(t_f)=6088$ and an objective value of ${\phi }^{*}=2.7\cdot 10^{-8}$. The main problem with direct single shooting is that a large part of the feasible domain ${𝒟}$ of $q$ will cause the integrator to run into a singularity before $t_f$. Hence only initial guesses for the optimization variable $q$ that are below a critical value of $\approx 1.23$ will give rise to a successful optimization. For multiple shooting the situation is different, due to the decoupling of the integration

• Reference solution plots
• 
  Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values $x(t_i)=2,G(t_i)=10^{-3}$. Note that the solution to the initial value problem on $[0,0.6]$ with $x(t_0)=q=2$ does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.

Miscellaneous and Further Reading

The Toy OED problem was introduced by Körkel et al. in [1], which contains further details.

References

[1] "A Multiple Shooting Formulation for Optimum Experimental Design" by S. Körkel, A. Potschka, H.G. Bock, and Sebastian Sager