Van der Pol OED: Difference between revisions

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

Latest revision as of 09:25, 26 March 2026

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

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

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 $θ : = (p_{1}, p_{2})$ :

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

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

These fixed values are used within the model:

Symbol	Value	Description
$p_{1}$	1	Unknown parameter
$p_{2}$	1	Unknown parameter
$t_{f}$	10	Horizon of the control problem
$ε_{r e g}$	0.1	Regularization of Fisher matrix
$𝒳$	[-0.5, $\infty$ ]	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

There were no citations found in the article.

@@ Line 2: / Line 2: @@
 |nd        = 1
 |nx        = 11
-|nw        = 2
+|nw        = 3
-|nc        = 4
-|nre       = 11
 }}
-The '''Lotka Experimental Design problem''' looks for an optimal fishing strategy to be performed on a fixed time horizon to control the biomasses of both predator as prey fish as described in [[Lotka Volterra fishing problem]]. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. In addition to the fishing decision the question when measurements of the two state variables are performed becomes a degree of freedom.
+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 [[:Category:ODE model|ODE model]]. The ODE from [[Lotka Volterra fishing problem]] is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.
+The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.
-The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.
 == Mathematical formulation ==
-We are interested in estimating the parameters <math>p_2</math> and <math>p_4</math> of the Lotka-Volterra type predator-prey fish initial value problem
+We are interested in estimating the parameters <math>p_1</math> and <math>p_2</math> of the initial value problem
 <p>
 <math>
 \begin{array}{rcl}
-\dot{x_1}(t) &=& p_1 \; x_1(t) - p_2 x_1(t) x_2(t) - p_5 u(t) x_1(t), \; t \in [0,t_f], \quad x_1(0) = 0.5, \\
+\dot{x_1}(t) &=& (p_1 - x_2(t)^2) \cdot x_1(t) - x_2(t) + u(t), && t \in [0,t_f], \quad x_1(0) = 0, \\
-\dot{x_2}(t) &=& - p_3 \; x_2(t) + p_4 x_1(t) x_2(t) - p_6 u(t) x_2(t), \; t \in [0,t_f], \quad x_2(0) = 0.7,
+\dot{x_2}(t) &=& p_2 + x_1(t), && t \in [0,t_f], \quad x_2(0) = 1.
 \end{array}
 </math>
 </p>
-where <math>u(\cdot)</math> is a fishing control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 12</math> are fixed. We are interested in how to fish 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. The experimental design problem then reads
+Additionally, we add the constraint
+<math>
+ x_1(t) \geq -0.5, \; t\in [0,t_f].
+</math>
+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 with <math>\theta := (p_1, p_2)</math>:
 <p>
 <math>
   \begin{array}{lll}
-  \displaystyle \min_{x,G,F,z^1,z^2,u,w^1,w^2} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
+  \displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
   \text{subject to} \\
-\quad \dot{x_1}(t) & = &  p_1 \; x_1(t) - p_2 x_1(t) x_2(t) - p_5 u(t) x_1(t),\\
+\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
-\quad \dot{x_2}(t) & = &  - p_3 \; x_2(t) + p_4 x_1(t) x_2(t) - p_6 u(t) x_2(t),\\
+\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
-\quad \dot{G_{11}}(t) & = & f_{x11}(\cdot) \; G_{11}(t) + f_{x12}(\cdot) \; G_{21}(t) + f_{p12}(\cdot), \\
+\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{G_{12}}(t) & = & f_{x11}(\cdot) \; G_{12}(t) + f_{x12}(\cdot) \; G_{22}(t), \\
+\quad \dot{z}(t) & = & w(t), \\
-\quad \dot{G_{21}}(t) & = & f_{x21}(\cdot) \; G_{11}(t) + f_{x22}(\cdot) \; G_{21}(t), \\
+\quad x(0) & = & x_0 \\
-\quad \dot{G_{22}}(t) & = & f_{x21}(\cdot) \; G_{12}(t) + f_{x22}(\cdot) \; G_{22}(t) + f_{p24}(\cdot), \\
+\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
-\quad \dot{F_{11}}(t) & = & w^1(t) G_{11}(t)^2 + w^2(t) G_{21}(t)^2, \\
+\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
-\quad \dot{F_{12}}(t) & = & w^1(t) G_{11}(t) G_{12}(t) + w^2(t) G_{21}(t) G_{22}(t), \\
+\quad z(0) & = & 0 \\
-\quad \dot{F_{22}}(t) & = & w^1(t) G_{12}(t)^2 + w^2(t) G_{22}(t)^2, \\
+\quad x_1(t) & \in & \mathcal{X} \\
-\quad \dot{z^1}(t) & = & w^1(t), \\
+\quad u(t) & \in & \mathcal{U} \\
-\quad \dot{z^2}(t) & = & w^2(t), \\[1.5ex]
+\quad w(t) & \in & \mathcal{W} \\
-\quad x(0) &=& (0.5, 0.7), \\
+\quad z_i(t_f) & \leq & M_i
-\quad G(0) &=& F(0) = 0, \\
-\quad z^1(0) &=& z^2(0) = 0, \\ [1.5ex]
-\quad u(t) & \in & \mathcal{U}, \; w^1(t) \in \mathcal{W}, \; w^2(t) \in \mathcal{W}, \\
-\quad 0    & \le & M - z(t_f)
    \end{array}
 </math>
 </p>
-with
+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>f_{x11}(\cdot) = \partial f_1(\cdot) / \partial x_1 = p_1 - p_2 x_2(t) - p_5 u(t)</math>,
+<math>h^i_x (x(t)) G(t), \ i = 1,2</math> for each observed function of states.
-<math>f_{x12}(\cdot) = - p_2 x_1(t)</math>,
-<math>f_{x21}(\cdot) = p_4 x_2(t)</math>,
-<math>f_{x22}(\cdot) = -p_3 + p_4 x_1(t) - p_6 u(t)</math>, and
-<math>f_{p12}(\cdot) = \partial f_1(\cdot) / \partial p_2 = -x_1(t) x_2(t)</math>,
-<math>f_{p24}(\cdot) = \partial f_2(\cdot) / \partial p_4 = x_1(t) x_2(t)</math>.
-Note that the state <math>F_{21}(\cdot) = F_{12}(\cdot)</math> has been left out for reasons of symmetry.
 == Parameters ==
+These fixed values are used within the model:
-We use <math>t_f=12</math>, <math>p_1 = p_2 = p_3 = p_4 = 1</math>, and <math>p_5 = 0.4</math>, <math>p_6 = 0.2</math>. The upper bound on the measurement time intervals is chosen as <math>M=4</math>.
+{| border="1" align="center" cellpadding="5" cellspacing="0"
+|- bgcolor=#c7c7c7
+! Symbol !! Value !! Description
+|-
+| align=center | <math>p_1</math> || align=right | 1  || Unknown parameter
+|-
+| align=center | <math>p_2</math> || align=right | 1 || Unknown parameter
+|-
+| align=center | <math>t_\mathrm{f}</math> || align=right | 10 || Horizon of the control problem
+|-
+| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
+|-
+| align=center | <math>\mathcal{X}</math> || align=right | [-0.5,<math>\infty</math>] || Bounds of <math>x_1</math>
+|-
+| align=center | <math>\mathcal{U}</math> || align=right | [-1,1] || Bounds of control function
+|-
+| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
+|-
+| align=center | <math>M_1, M_2</math> || align=right | 2 || Maximum measurement time
+|}
 == Reference Solutions ==
@@ Line 69: / Line 80: @@
 Here is one local solution to the above control problem.
-<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
+<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
-  Image:vplotkaUStates.png| States and fishing control.
+  Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
- Image:vplotkaUSensitivities.png| Sensitivities G().
- Image:vplotkaUMeas1.png| Sampling function for first state.
- Image:vplotkaUMeas2.png| Sampling function for second state.
 </gallery>
-== Source Code ==
-Model descriptions are available in
-* [[:Category:AMPL | AMPL]] at [[Lotka Experimental Design (AMPL)]]
-* [[:Category: VPLAN | VPLAN code]] at [[Lotka Experimental Design (VPLAN)]]
-== Variants ==
-There are several alternative formulations and variants of the above problem, in particular
-* a prescribed time grid for the control function <bib id="Sager2006" />, see also [[Lotka Experimental Design (AMPL)]],
-* no fishing, i.e., <math>u \equiv 0</math>,
-* different fishing control functions for the two species,
-* different parameters and start values.
-== Miscellaneous and Further Reading ==
-The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.
 == References ==
@@ Line 103: / Line 92: @@
 [[Category:ODE model]]
 [[Category:Bang bang]]
-[[Category:Population dynamics]]