Van der Pol OED: Difference between revisions

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

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Revision as of 08:08, 27 January 2026

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

$\begin{array}{rcl} \dot{x_{1}} (t) & = & (p_{1} - x_{2} (t)^{2}) \cdot x_{1} (t) - x_{2} (t) + u (t), & t \in [0, t_{f}], x_{1} (0) = 0, \\ \dot{x_{2}} (t) & = & p_{2} + x_{1} (t), & t \in [0, t_{f}], x_{2} (0) = 1 . \end{array}$

Additionally, we add the constraint $x_{1} (t) \geq - 0.5, t \in [0, t_{f}] .$

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 $θ : = (p_{1}, p_{2})$ :

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

The evolution of the symmetric matrix $F : [0, t_{f}] \to ℝ^{2 \times 2}$ 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

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
States, control, and sampling functions for a local optimum.

References

There were no citations found in the article.

@@ Line 27: / Line 27: @@
 The initial values and <math>t_f = 10</math> are fixed. We are interested in how to choose the control <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w^1(\cdot)</math> and <math>w^2(\cdot)</math> in the same experimental setting. This can be seen either as a two-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.
-Now we formulate the OED problem:
+Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:
 <p>
 <math>
@@ Line 33: / Line 33: @@
   \displaystyle \min_{y,G,F,z,w} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
   \text{subject to} \\
-\quad \dot{y}(t) & = & f(y(t),\theta) \\
+\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
-\quad \dot{G}(t) & = & f_y(y(t),\theta) G(t) + f_\theta(y(t),\theta) \\
+\quad \dot{G}(t) & = & f_x(x(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
-\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{F}(t) & = & \sum_{i=1}^{n_o} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
 \quad \dot{z}(t) & = & w(t), \\
-\quad y(0) & = & y_0 \\
+\quad x(0) & = & x_0 \\
-\quad G(0) & = & \frac{\partial y(0)}{\partial \theta} \\
+\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
 \quad F(0) & = & 0, \\
 \quad z(0) & = & 0 \\
+\quad u(t) & \in & \mathcal{U} \\
 \quad w(t) & \in & \mathcal{W} \\
 \quad z_i(t_f) & \leq & M_i
@@ Line 48: / Line 49: @@
 The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{2 \times 2}</math> is given by the weighted sum of observability Gramians
-<math>h^i_y (y(t)) G(t), \ i = 1,2</math> for each observed function of states.
+<math>h^i_x (x(t)) G(t), \ i = 1,2</math> for each observed function of states.
 == Parameters ==