Van der Pol OED

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

The Van der Pol 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:

$\begin{array}{lll}\underset{y,G,F,z,w}{\min }& & \text{trace}(F^{-1}(t_f))\\ \text{subject to}\\ \dot{y}(t)& =& f(y(t),\theta )\\ \dot{G}(t)& =& f_y(y(t),\theta )G(t)+f_{\theta }(y(t),\theta )\\ \dot{F}(t)& =& \sum _{i=1}^{n_o}{w}_i(t)(h_y^i(y(t))G(t){)}^T(h_y^i(y(t))G(t))\\ \dot{z}(t)& =& w(t),\\ y(0)& =& y_0\\ G(0)& =& \frac{\partial y(0)}{\partial \theta }\\ F(0)& =& 0,\\ z(0)& =& 0\\ w(t)& \in & {𝒲}\\ z_i(t_f)& \le & M_i\end{array}$

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 $h_y^i(y(t))G(t),i=1,2$ for each observed function of states.

Parameters

We use $t_f=10$ and $p_1=p_2=1$. The upper bound on the measurement time intervals is chosen as $M_1=M_2=2$.

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
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  States, control, and sampling functions for a local optimum.

References

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