Three Tank OED - Revision history

RobertLampel: /* Mathematical formulation */

2026-03-26T10:15:58Z

Mathematical formulation

← Older revision		Revision as of 10:15, 26 March 2026
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	</math>		</math>

	The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>~~w^i~~(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-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 = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w_i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-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>:		Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

RobertLampel: /* Reference Solutions */

2026-03-26T10:12:54Z

Reference Solutions

← Older revision		Revision as of 10:12, 26 March 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:~~Van_der_Pol_OED~~.png\| States, control, and sampling functions for a local optimum.		Image:Three_Tank_OED.png\| States, control, and sampling functions for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Mathematical formulation */

2026-03-26T10:03:49Z

Mathematical formulation

← Older revision		Revision as of 10:03, 26 March 2026
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	\quad z(0) & = & 0 \\		\quad z(0) & = & 0 \\
	\quad \sum_{i=1}^3 u_i(t) & = & 1 \\		\quad \sum_{i=1}^3 u_i(t) & = & 1 \\
	~~\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: /* Mathematical formulation */

2026-03-26T10:02:39Z

Mathematical formulation

← Older revision		Revision as of 10:02, 26 March 2026
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	\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\		\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
	\quad z(0) & = & 0 \\		\quad z(0) & = & 0 \\
	\quad \sum_{i=1}^3 u_i(t) & = & 1		\quad \sum_{i=1}^3 u_i(t) & = & 1 \\
	\quad x_1(t) & \in & \mathcal{X} \\		\quad x_1(t) & \in & \mathcal{X} \\
	\quad u(t) & \in & \mathcal{U} \\		\quad u(t) & \in & \mathcal{U} \\

RobertLampel: /* Mathematical formulation */

2026-03-26T10:02:18Z

Mathematical formulation

← Older revision		Revision as of 10:02, 26 March 2026
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	\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\		\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
	\quad z(0) & = & 0 \\		\quad z(0) & = & 0 \\
			\quad \sum_{i=1}^3 u_i(t) & = & 1
	\quad x_1(t) & \in & \mathcal{X} \\		\quad x_1(t) & \in & \mathcal{X} \\
	\quad u(t) & \in & \mathcal{U} \\		\quad u(t) & \in & \mathcal{U} \\

RobertLampel: /* Mathematical formulation */

2026-03-26T09:58:29Z

Mathematical formulation

← Older revision		Revision as of 09:58, 26 March 2026
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	\quad \dot{F}(t) & = & \sum_{i=1}^{3} 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}^{3} 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 x(0) & = & (2,2,2) \\		\quad x(0) & = & (2,2,2)^T \\
	\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\		\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
	\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\		\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\

RobertLampel: /* Mathematical formulation */

2026-03-26T09:58:00Z

Mathematical formulation

← Older revision		Revision as of 09:58, 26 March 2026
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	\quad \dot{G}(t) & = & f_x(x(t),u(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}^{3} 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}^{3} 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 x(0) & = & (2,2,2) \\		\quad x(0) & = & (2,2,2) \\
	\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\		\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\

RobertLampel: /* Parameters */

2026-03-26T09:52:20Z

Parameters

← Older revision		Revision as of 09:52, 26 March 2026
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	\| align=center \| <math>\mathcal{U}</math> \|\| align=right \| <math>[0,1]^3</math> \|\| Bounds of control function		\| align=center \| <math>\mathcal{U}</math> \|\| align=right \| <math>[0,1]^3</math> \|\| Bounds of control function
	\|-		\|-
	\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| [0,1] \|\| Bounds of measurement function		\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| <math>[0,1]^3</math> \|\| Bounds of measurement function
	\|-		\|-
	\| align=center \| <math>M_1, M_2, M_3</math> \|\| align=right \| 2 \|\| Maximum measurement time		\| align=center \| <math>M_1, M_2, M_3</math> \|\| align=right \| 2 \|\| Maximum measurement time

RobertLampel: /* Parameters */

2026-03-26T09:51:59Z

Parameters

← Older revision		Revision as of 09:51, 26 March 2026
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	\| align=center \| <math>\varepsilon_{\mathrm{reg}}</math> \|\| align=right \| 0.1 \|\| Regularization of Fisher matrix		\| align=center \| <math>\varepsilon_{\mathrm{reg}}</math> \|\| align=right \| 0.1 \|\| 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 \| <math>[0,1]^3</math> \|\| Bounds of control function
	\|-		\|-
	\| 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

RobertLampel: /* Parameters */

2026-03-26T09:49:25Z

Parameters

← Older revision		Revision as of 09:49, 26 March 2026
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	! Symbol !! Value !! Description		! Symbol !! Value !! Description
	\|-		\|-
	\| align=center \| <math>~~p_1~~</math> \|\| align=right \| 1 \|\| Unknown parameter		\| align=center \| <math>c_1</math> \|\| align=right \| 1 \|\| Unknown parameter
	\|-		\|-
	\| align=center \| <math>~~p_2~~</math> \|\| align=right \| 1 \|\| Unknown parameter		\| align=center \| <math>c_2</math> \|\| align=right \| 2 \|\| Unknown parameter
			\|-
			\| align=center \| <math>c_3</math> \|\| align=right \| 0.8 \|\| Unknown parameter
	\|-		\|-
	\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 10 \|\| Horizon of the control problem		\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 12 \|\| Horizon of the control problem
	\|-		\|-
	\| align=center \| <math>\varepsilon_{\mathrm{reg}}</math> \|\| align=right \| 0.1 \|\| 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{U}</math> \|\| align=right \| [0,1] \|\| Bounds of control function
	\|-
	~~\| 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>\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		\| align=center \| <math>M_1, M_2, M_3</math> \|\| align=right \| 2 \|\| Maximum measurement time
	\|}		\|}