Jackson - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T10:42:40Z

Reference Solutions

← Older revision		Revision as of 10:42, 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:Jackson.png\| States and discretized control for a local optimum.		Image:Jackson.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

2025-11-25T07:12:57Z

Mathematical formulation

← Older revision		Revision as of 07:12, 25 November 2025
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	\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\		\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\
	\quad x(0) &=& (1, 0, 0)^T, \\		\quad x(0) &=& (1, 0, 0)^T, \\
	\quad x(t) & \in & [0, 1.1] \ & \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\}, \\		\quad x_i(t) & \in & [0, 1.1] \ & \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\}, \\
	\quad u (t) & \in & [0, 1] \ & \forall t \in [0,t_\mathrm{f}]		\quad u (t) & \in & [0, 1] \ & \forall t \in [0,t_\mathrm{f}]
	\end{array}		\end{array}

2025-11-24T12:09:10Z

Mathematical formulation

← Older revision		Revision as of 12:09, 24 November 2025
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	\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\		\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\
	\quad x(0) &=& (1, 0, 0)^T, \\		\quad x(0) &=& (1, 0, 0)^T, \\
	\quad x(t) & \in & [0, 1.1] \ & \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\} \\		\quad x(t) & \in & [0, 1.1] \ & \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\}, \\
	\quad u (t) & \in & [0, 1] \ & \forall t \in [0,t_\mathrm{f}]		\quad u (t) & \in & [0, 1] \ & \forall t \in [0,t_\mathrm{f}]
	\end{array}		\end{array}

2025-11-24T12:08:53Z

Mathematical formulation

← Older revision		Revision as of 12:08, 24 November 2025
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	\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\		\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\
	\quad x(0) &=& (1, 0, 0)^T, \\		\quad x(0) &=& (1, 0, 0)^T, \\
	\quad x(t) & \in & [0, 1.1] \ ~~\quad~~ \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\} \\		\quad x(t) & \in & [0, 1.1] \ & \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\} \\
	\quad u (t) & \in & [0, 1] \ ~~\quad~~ \forall t \in [0,t_\mathrm{f}]		\quad u (t) & \in & [0, 1] \ & \forall t \in [0,t_\mathrm{f}]
	\end{array}		\end{array}
	</math>		</math>

2025-11-24T10:50:52Z

2025-11-24T10:50:08Z

References

← Older revision		Revision as of 10:50, 24 November 2025
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	== References ==		== References ==
	<span id="OCPjl">[1]</span> Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/~~ducted_fan~~.md<br>		<span id="OCPjl">[1]</span> Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/jackson.md<br>


	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]

2025-11-24T10:49:51Z

Reference Solutions

← Older revision		Revision as of 10:49, 24 November 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:~~Ducted_Fan~~.png\| States and discretized control for a local optimum.		Image:Jackson.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

2025-11-24T10:44:15Z

Parameters

← Older revision		Revision as of 10:44, 24 November 2025
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	{\| border="1" align="center" cellpadding="5" cellspacing="0"		{\| border="1" align="center" cellpadding="5" cellspacing="0"
	\|- bgcolor=#c7c7c7		\|- bgcolor=#c7c7c7
	! Symbol !! Value		! Symbol !! Value !! Description
	\|-		\|-
	\| align=center \| <math>m</math> \|\| align=right \| ~~2.2~~ <math>~~\mathrm{kg}~~</math>		\| align=center \| <math>k_1</math> \|\| align=right \| 1 \|\| Interaction between <math>x_1</math> and <math>x_2</math>
	\|-		\|-
	\| align=center \| <math>J</math> \|\| align=right \| ~~0.05~~ <math>~~\mathrm{kg \cdot m^2}~~</math>		\| align=center \| <math>k_2</math> \|\| align=right \| 10 \|\| Interaction between <math>x_1</math> and <math>x_2</math>
	\|-		\|-
	\| align=center \| <math>r</math> \|\| align=right \| ~~0.2 <math>\mathrm{m}</math>~~		\| align=center \| <math>k_3</math> \|\| align=right \| 1 \|\| Growth of <math>x_3</math> under complementary control
	\|-
	~~\| align=center~~ \| ~~<math>\mathrm{mg}</math> \|\| align=right~~ \| 4 <math>~~\mathrm{N}~~</math>
	\|-		\|-
	\| align=center \| <math>\mu</math> \|\| align=right \| 1		\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 1 \|\| Horizon of the control problem
	\|}		\|}

	~~The weight <math>\mu</math> balances control effort and transition time.~~

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

2025-11-24T10:40:16Z

Mathematical formulation

← Older revision		Revision as of 10:40, 24 November 2025
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	\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\		\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\
	\quad x(0) &=& (1, 0, 0)^T, \\		\quad x(0) &=& (1, 0, 0)^T, \\
	\quad x_(t) & \in & [0, 1.1] \ \quad \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\} \\		\quad x(t) & \in & [0, 1.1] \ \quad \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\} \\
	\quad u (t) & \in & [0, 1] \ \quad \forall t \in [0,t_\mathrm{f}]		\quad u (t) & \in & [0, 1] \ \quad \forall t \in [0,t_\mathrm{f}]
	\end{array}		\end{array}

2025-11-24T10:40:03Z

Mathematical formulation

← Older revision		Revision as of 10:40, 24 November 2025
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	\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\		\quad \dot{x_3}(t) & = & (1-u(t)) k_3 x_2(t), \\
	\quad x(0) &=& (1, 0, 0)^T, \\		\quad x(0) &=& (1, 0, 0)^T, \\
	~~\quad x(t_\mathrm{f}) &=& (1, 0, 0, 0, 0, 0)^T, \\~~
	\quad x_(t) & \in & [0, 1.1] \ \quad \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\} \\		\quad x_(t) & \in & [0, 1.1] \ \quad \forall t \in [0,t_\mathrm{f}], \ i \in \{1,2,3\} \\
	\quad u (t) & \in & [0, 1] \ \quad \forall t \in [0,t_\mathrm{f}]		\quad u (t) & \in & [0, 1] \ \quad \forall t \in [0,t_\mathrm{f}]