Van der Pol OED

Van der Pol OED
State dimension:	1
Differential states:	11
Discrete control functions:	3

The Van der Pol OED problem is a variation of the Van der Pol Oscillator 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 $p_1$ and $p_2$ of the initial value problem

$\begin{array}{rclll}\dot{x_1}(t)& =& (p_1-x_2(t{)}^2)\cdot x_1(t)-x_2(t)+u(t),& & t\in [0,t_f],x_1(0)=0,\\ \dot{x_2}(t)& =& p_2+x_1(t),& & t\in [0,t_f],x_2(0)=1.\end{array}$

Additionally, we add the constraint $x_1(t)\ge -0.5,t\in [0,t_f].$

The initial values and $t_f=10$ are fixed. We are interested in how to choose the control $u$ 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 with $\theta :=(p_1,p_2)$:

$\begin{array}{lll}\underset{x,G,F,z,w,u}{\min }& & \text{trace}(F^{-1}(t_f))\\ \text{subject to}\\ \dot{x}(t)& =& f(x(t),u(t),\theta )\\ \dot{G}(t)& =& f_x(x(t),u(t),\theta )G(t)+f_{\theta }(x(t),u(t),\theta )\\ \dot{F}(t)& =& \sum _{i=1}^{n_o}{w}_i(t)(h_x^i(x(t))G(t){)}^T(h_x^i(x(t))G(t))\\ \dot{z}(t)& =& w(t),\\ x(0)& =& x_0\\ G(0)& =& \frac{\partial x(0)}{\partial \theta }\\ F(0)& =& I\cdot {\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{g}},\\ z(0)& =& 0\\ x_1(t)& \in & {𝒳}\\ u(t)& \in & {𝒰}\\ w(t)& \in & {𝒲}\\ z_i(t_f)& \le & M_i\end{array}$

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

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$p_1$	1	Unknown parameter
$p_2$	1	Unknown parameter
$t_{\mathrm{f}}$	10	Horizon of the control problem
${\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{g}}$	0.1	Regularization of Fisher matrix
${𝒳}$	[-0.5,$\mathrm{\infty }$]	Bounds of $x_1$
${𝒰}$	[-1,1]	Bounds of control function
${𝒲}$	[0,1]	Bounds of measurement function
$M_1,M_2$	2	Maximum measurement time

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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