Van der Pol OED - Revision history

RobertLampel: /* Parameters */

2026-03-26T09:25:34Z

Parameters

← Older revision		Revision as of 09:25, 26 March 2026
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	\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 10 \|\| Horizon of the control problem		\| 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.~~001~~ \|\| Regularization of Fisher matrix		\| 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{X}</math> \|\| align=right \| [-0.5,<math>\infty</math>] \|\| Bounds of <math>x_1</math>

RobertLampel: /* Parameters */

2026-02-05T14:10:50Z

Parameters

← Older revision		Revision as of 14:10, 5 February 2026
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	\|-		\|-
	\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 10 \|\| Horizon of the control problem		\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 10 \|\| Horizon of the control problem
			\|-
			\| align=center \| <math>\varepsilon_{\mathrm{reg}}</math> \|\| align=right \| 0.001 \|\| 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{X}</math> \|\| align=right \| [-0.5,<math>\infty</math>] \|\| Bounds of <math>x_1</math>

RobertLampel: /* Mathematical formulation */

2026-02-05T14:09:56Z

Mathematical formulation

← Older revision		Revision as of 14:09, 5 February 2026
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	\quad x(0) & = & x_0 \\		\quad x(0) & = & x_0 \\
	\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\		\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
	\quad F(0) & = & 0, \\		\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
	\quad z(0) & = & 0 \\		\quad z(0) & = & 0 \\
	\quad x_1(t) & \in & \mathcal{X} \\		\quad x_1(t) & \in & \mathcal{X} \\

RobertLampel at 08:15, 27 January 2026

2026-01-27T08:15:00Z

← Older revision		Revision as of 08:15, 27 January 2026
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	}}		}}

	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 '''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]]. 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.

RobertLampel: /* Mathematical formulation */

2026-01-27T08:14:20Z

Mathematical formulation

← Older revision		Revision as of 08:14, 27 January 2026
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	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\		\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
	\quad \dot{G}(t) & = & f_x(x(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\		\quad \dot{G}(t) & = & f_x(x(t),u(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_x(x(t))G(t))^T(h^i_x(x(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 \dot{z}(t) & = & w(t), \\

RobertLampel: /* Parameters */

2026-01-27T08:13:01Z

Parameters

← Older revision		Revision as of 08:13, 27 January 2026
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	! Symbol !! Value !! Description		! Symbol !! Value !! Description
	\|-		\|-
	\| align=center \| <math>p_1</math> \|\| align=right \| ~~-0.75~~ \|\| Unknown parameter		\| 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>p_2</math> \|\| align=right \| 1 \|\| Unknown parameter

RobertLampel: /* Parameters */

2026-01-27T08:12:45Z

Parameters

← Older revision		Revision as of 08:12, 27 January 2026
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	== Parameters ==		== Parameters ==

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

	These fixed values are used within the model:		These fixed values are used within the model:

RobertLampel: /* Mathematical formulation */

2026-01-27T08:12:11Z

Mathematical formulation

← Older revision		Revision as of 08:12, 27 January 2026
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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{y,G,F,z,w} && \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} \\		\text{subject to} \\
	\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\		\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
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	\quad F(0) & = & 0, \\		\quad F(0) & = & 0, \\
	\quad z(0) & = & 0 \\		\quad z(0) & = & 0 \\
			\quad x_1(t) & \in & \mathcal{X} \\
	\quad u(t) & \in & \mathcal{U} \\		\quad u(t) & \in & \mathcal{U} \\
	\quad w(t) & \in & \mathcal{W} \\		\quad w(t) & \in & \mathcal{W} \\

RobertLampel: /* Parameters */

2026-01-27T08:11:13Z

Parameters

← Older revision		Revision as of 08:11, 27 January 2026
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	We use <math>t_f=10</math> and <math>p_1 = p_2 = 1</math>. The upper bound on the measurement time intervals is chosen as <math>M_1 = M_2 = 2</math>.		We use <math>t_f=10</math> and <math>p_1 = p_2 = 1</math>. The upper bound on the measurement time intervals is chosen as <math>M_1 = M_2 = 2</math>.

			These fixed values are used within the model:

			{\| border="1" align="center" cellpadding="5" cellspacing="0"
			\|- bgcolor=#c7c7c7
			! Symbol !! Value !! Description
			\|-
			\| align=center \| <math>p_1</math> \|\| align=right \| -0.75 \|\| 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>\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 ==		== Reference Solutions ==

RobertLampel: /* Mathematical formulation */

2026-01-27T08:08:01Z

Mathematical formulation

← Older revision		Revision as of 08:08, 27 January 2026
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	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.		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>		<p>
	<math>		<math>
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	\displaystyle \min_{y,G,F,z,w} && \text{trace} \; \left( F^{-1}(t_f) \right) \\		\displaystyle \min_{y,G,F,z,w} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
	\text{subject to} \\		\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 \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 F(0) & = & 0, \\
	\quad z(0) & = & 0 \\		\quad z(0) & = & 0 \\
			\quad u(t) & \in & \mathcal{U} \\
	\quad w(t) & \in & \mathcal{W} \\		\quad w(t) & \in & \mathcal{W} \\
	\quad z_i(t_f) & \leq & M_i		\quad z_i(t_f) & \leq & M_i
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	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		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 ==		== Parameters ==