Jackson OED - Revision history

RobertLampel: /* Mathematical formulation */

2026-03-26T10:17:14Z

Mathematical formulation

← Older revision		Revision as of 10:17, 26 March 2026
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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_x (x(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 ==

RobertLampel: /* Mathematical formulation */

2026-03-26T10:16:59Z

Mathematical formulation

← Older revision		Revision as of 10:16, 26 March 2026
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	</p>		</p>

	The initial values and <math>t_f</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 <math>x_1</math> and <math>x_2</math> 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</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 <math>x_1</math> and <math>x_2</math> 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 := (k_1, k_2)</math>:		Now we formulate the OED problem with <math>\theta := (k_1, k_2)</math>:

RobertLampel: /* Parameters */

2026-01-27T15:47:12Z

Parameters

← Older revision		Revision as of 15:47, 27 January 2026
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	\|-		\|-
	\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 1 \|\| Horizon of the control problem		\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 1 \|\| Horizon of the control problem
			\|-
			\| align=center \| <math>\varepsilon_\mathrm{reg}</math> \|\| align=right \| 0.01 \|\| Regularization of Fisher matrix
	\|-		\|-
	\| align=center \| <math>\mathcal{U}</math> \|\| align=right \| [0,1] \|\| Bounds of control function		\| align=center \| <math>\mathcal{U}</math> \|\| align=right \| [0,1] \|\| Bounds of control function

RobertLampel: /* Mathematical formulation */

2026-01-27T15:46:26Z

Mathematical formulation

← Older revision		Revision as of 15:46, 27 January 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 u(t) & \in & \mathcal{U} \\		\quad u(t) & \in & \mathcal{U} \\

RobertLampel: /* Reference Solutions */

2026-01-27T08:04:41Z

Reference Solutions

← Older revision		Revision as of 08:04, 27 January 2026
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	<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Jackson_OED.png\| States, control, and sampling functions for a local optimum.		Image:Jackson_OED.png\| States, control, and sampling functions for a local optimum. Both measurement functions overlap.
	</gallery>		</gallery>

RobertLampel: /* Reference Solutions */

2026-01-27T08:04:04Z

Reference Solutions

← Older revision		Revision as of 08:04, 27 January 2026
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	<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:~~Dielectrophoretic_Particle_OED~~.png\| States, control, and sampling functions for a local optimum.		Image:Jackson_OED.png\| States, control, and sampling functions for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Mathematical formulation */

2026-01-27T07:58:18Z

Mathematical formulation

← Older revision		Revision as of 07:58, 27 January 2026
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	\displaystyle \min_{x,G,F,z,w,u} && \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),\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),\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 */

2026-01-27T07:55:05Z

Parameters

← Older revision		Revision as of 07:55, 27 January 2026
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	\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| [0,1] \|\| Bounds of measurement function		\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| [0,1] \|\| Bounds of measurement function
	\|-		\|-
	\| align=center \| <math>~~\mathcal{M}~~</math> \|\| align=right \| 0.2 \|\| Maximum measurement time		\| align=center \| <math>M_1, M_2</math> \|\| align=right \| 0.2 \|\| Maximum measurement time
	\|}		\|}

RobertLampel: /* Mathematical formulation */

2026-01-27T07:48:27Z

Mathematical formulation

← Older revision		Revision as of 07:48, 27 January 2026
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	The initial values and <math>t_f</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 <math>x_1</math> and <math>x_2</math> 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</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 <math>x_1</math> and <math>x_2</math> 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 := (k_1, k_2)</math>:
	<p>		<p>
	<math>		<math>

RobertLampel: /* Mathematical formulation */

2026-01-27T07:45:37Z

Mathematical formulation

← Older revision		Revision as of 07:45, 27 January 2026
Line 22:		Line 22:
	</p>		</p>

	The initial values and <math>t_f ~~= 1~~</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 <math>x_1</math> and <math>x_2</math> 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</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 <math>x_1</math> and <math>x_2</math> 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: