Oscillating OED - Revision history

RobertLampel: /* Mathematical formulation */

2026-01-27T08:24:20Z

Mathematical formulation

← Older revision		Revision as of 08:24, 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} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{y}(t) & = & f(t, p) \\		\quad \dot{x}(t) & = & f(t, p) \\
	\quad \dot{G}(t) & = & f_p(y(t),p) \\		\quad \dot{G}(t) & = & f_p(x(t),p) \\
	\quad \dot{F}(t) & = & w(t)(~~h_y~~(y(t))G(t))^T(~~h_y~~(y(t))G(t)) \\		\quad \dot{F}(t) & = & w(t)(h_x(x(t))G(t))^T(h_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) & = & 0 \\		\quad G(0) & = & 0 \\
	\quad F(0) & = & 0, \\		\quad F(0) & = & 0, \\

RobertLampel: /* Parameters */

2026-01-27T08:23:18Z

Parameters

← Older revision		Revision as of 08:23, 27 January 2026
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	== Parameters ==		== Parameters ==
	These fixed values are used within the model:		These fixed values are used within the model:
	<p>
	<math>		{\| border="1" align="center" cellpadding="5" cellspacing="0"
	~~x_0~~ = 0.1; \~~quad t_f~~ = 2~~; \quad~~ \mathcal{W} = [0,1]~~; \quad M~~ = ~~0.2; \quad p = 15~~		\|- bgcolor=#c7c7c7
	</math>		! Symbol !! Value !! Description
	</p>		\|-
			\| align=center \| <math>x_0</math> \|\| align=right \| 0.1 \|\| Initial value for <math>x</math>
			\|-
			\| align=center \| <math>p</math> \|\| align=right \| 15 \|\| Unknown parameter
			\|-
			\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 2 \|\| Horizon of the control problem
			\|-
			\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| [0,1] \|\| Bounds of measurement function
			\|-
			\| align=center \| <math>M</math> \|\| align=right \| 0.2 \|\| Maximum measurement time
			\|}

	== Reference Solutions ==		== Reference Solutions ==

RobertLampel: /* Mathematical formulation */

2026-01-27T08:19:13Z

Mathematical formulation

← Older revision		Revision as of 08:19, 27 January 2026
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	For a single parameter <math>p</math> the original initial value problem is given by		For a single parameter <math>p</math> the original initial value problem is given by
	<math>		<math>
	\dot{x}(t) =: f(t, p) = 0.2 + 0.8 \cdot t + 0.3 \cdot (\sin(p \cdot t) + \cos(p \cdot t) \cdot p \cdot t) - 2.5 \cdot \sin(50 \cdot t), \quad x(0) = x_0.		\dot{x}(t) = f(t, p) = 0.2 + 0.8 \cdot t + 0.3 \cdot (\sin(p \cdot t) + \cos(p \cdot t) \cdot p \cdot t) - 2.5 \cdot \sin(50 \cdot t), \quad x(0) = x_0.
	</math>		</math>

RobertLampel: /* Reference Solutions */

2025-11-28T10:44:49Z

Reference Solutions

← Older revision		Revision as of 10:44, 28 November 2025
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	Here is one local solution to the above control problem.		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Oscillating OED.png\| States and measurement control for <math>p=15</math>. The time <math>t</math> was added as an additional state.		Image:Oscillating OED.png\| States and measurement control for <math>p=15</math>. The time <math>t</math> was added as an additional state.
	</gallery>		</gallery>

RobertLampel at 11:48, 4 September 2025

2025-09-04T11:48:53Z

← Older revision		Revision as of 11:48, 4 September 2025
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	}}		}}

	The '''Oscillating OED problem''' looks for an optimal measurement strategy to determine a single parameter in a one-dimensional [[:Category:ODE model\|ODE model]], where can directly measure the single state.		The '''Oscillating OED problem''' looks for an optimal measurement strategy to determine a single parameter in a one-dimensional [[:Category:ODE model\|ODE model]], where we can directly measure the single state.

	The optimal integer control functions shows [[:Category:Chattering\|bang bang]] behavior.		The optimal integer control functions shows [[:Category:Chattering\|bang bang]] behavior.

RobertLampel at 13:42, 26 August 2025

2025-08-26T13:42:31Z

← Older revision		Revision as of 13:42, 26 August 2025
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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	~~The Toy OED~~ problem was introduced by Sebastian Sager ~~in <bib id="Sager2013" />, which contains further details~~.		This problem was introduced by [[User:SebastianSager \| Sebastian Sager]].

	~~== References ==~~
	~~<biblist />~~

	<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->		<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->

RobertLampel: /* Mathematical formulation */

2025-08-26T06:15:26Z

Mathematical formulation

← Older revision		Revision as of 06:15, 26 August 2025
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	For a single parameter <math>p</math> the original initial value problem is given by		For a single parameter <math>p</math> the original initial value problem is given by
	<math>		<math>
	\dot{x}(t) =: f(t) = 0.2 + 0.8 \cdot t + 0.3 \cdot (\sin(p \cdot t) + \cos(p \cdot t) \cdot p \cdot t) - 2.5 \cdot \sin(50 \cdot t), \quad x(0) = x_0.		\dot{x}(t) =: f(t, p) = 0.2 + 0.8 \cdot t + 0.3 \cdot (\sin(p \cdot t) + \cos(p \cdot t) \cdot p \cdot t) - 2.5 \cdot \sin(50 \cdot t), \quad x(0) = x_0.
	</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),p) \\		\quad \dot{y}(t) & = & f(t, p) \\
	\quad \dot{G}(t) & = & ~~f_y(y(t),p) G(t) +~~ f_p(y(t),p) \\		\quad \dot{G}(t) & = & f_p(y(t),p) \\
	\quad \dot{F}(t) & = & w(t)(h_y(y(t))G(t))^T(h_y(y(t))G(t)) \\		\quad \dot{F}(t) & = & w(t)(h_y(y(t))G(t))^T(h_y(y(t))G(t)) \\
	\quad \dot{z}(t) & = & w(t), \\		\quad \dot{z}(t) & = & w(t), \\

RobertLampel: /* Reference Solutions */

2025-08-25T14:30:36Z

Reference Solutions

← Older revision		Revision as of 14:30, 25 August 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:Oscillating OED.png\| States and measurement control for <math>p=15</math>.		Image:Oscillating OED.png\| States and measurement control for <math>p=15</math>. The time <math>t</math> was added as an additional state.
	</gallery>		</gallery>

RobertLampel: /* Mathematical formulation */

2025-08-25T14:28:57Z

Mathematical formulation

← Older revision		Revision as of 14:28, 25 August 2025
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	We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>.		We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>.

	Now we formulate the OED problem ~~as described in [[#OEDUDE \| [2]]].~~		Now we formulate the OED problem:
	<p>		<p>
	<math>		<math>

RobertLampel: /* Mathematical formulation */

2025-08-25T14:28:13Z

Mathematical formulation

← Older revision		Revision as of 14:28, 25 August 2025
Line 23:		Line 23:
	\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{y}(t) & = & f(y(t),p) \\
	\quad \dot{G}(t) & = & f_y(y(t),~~\theta~~) G(t) + ~~f_\theta~~(y(t),~~\theta~~) \\		\quad \dot{G}(t) & = & f_y(y(t),p) G(t) + f_p(y(t),p) \\
	\quad \dot{F}(t) & = & w(t)(h_y(y(t))G(t))^T(h_y(y(t))G(t)) \\		\quad \dot{F}(t) & = & w(t)(h_y(y(t))G(t))^T(h_y(y(t))G(t)) \\
	\quad \dot{z}(t) & = & w(t), \\		\quad \dot{z}(t) & = & w(t), \\