Ducted Fan - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T10:43:01Z

Reference Solutions

← Older revision		Revision as of 10:43, 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:Ducted_Fan.png\| States and discretized control for a local optimum.		Image:Ducted_Fan.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Mathematical formulation */

2025-11-24T12:09:50Z

Mathematical formulation

← Older revision		Revision as of 12:09, 24 November 2025
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	\quad x(0) &=& (0, 0, 0, 0, 0, 0)^T, \\		\quad x(0) &=& (0, 0, 0, 0, 0, 0)^T, \\
	\quad x(t_\mathrm{f}) &=& (1, 0, 0, 0, 0, 0)^T, \\		\quad x(t_\mathrm{f}) &=& (1, 0, 0, 0, 0, 0)^T, \\
	\quad u_1(t) & \in & [-5, 5] \ ~~\quad~~ \forall t \in [0,t_\mathrm{f}], \\		\quad u_1(t) & \in & [-5, 5] \ & \forall t \in [0,t_\mathrm{f}], \\
	\quad u_2(t) & \in & [0, 17] \ ~~\quad~~ \forall t \in [0,t_\mathrm{f}], \\		\quad u_2(t) & \in & [0, 17] \ & \forall t \in [0,t_\mathrm{f}], \\
	\quad \alpha(t) & \in & [-30, 30] \ ~~\quad~~ \forall t \in [0,t_\mathrm{f}]		\quad \alpha(t) & \in & [-30, 30] \ & \forall t \in [0,t_\mathrm{f}]
	\end{array}		\end{array}
	</math>		</math>

RobertLampel at 10:35, 24 November 2025

2025-11-24T10:35:40Z

← Older revision		Revision as of 10:35, 24 November 2025
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	{{Dimensions		{{Dimensions
	\|nd = 1		\|nd = 1
	\|nx = 4		\|nx = 6
	\|nw = 1		\|nw = 3
	}}		}}

RobertLampel: /* References */

2025-11-24T10:34:53Z

References

← Older revision		Revision as of 10:34, 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/~~double_oscillator~~.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/ducted_fan.md<br>


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

RobertLampel: /* Reference Solutions */

2025-11-24T10:34:35Z

Reference Solutions

← Older revision		Revision as of 10:34, 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:~~Double_Oscillator~~.png\| States and discretized control for a local optimum.		Image:Ducted_Fan.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Parameters */

2025-11-24T10:17:02Z

Parameters

← Older revision		Revision as of 10:17, 24 November 2025
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	\| align=center \| <math>r</math> \|\| align=right \| 0.2 <math>\mathrm{m}</math>		\| align=center \| <math>r</math> \|\| align=right \| 0.2 <math>\mathrm{m}</math>
	\|-		\|-
	\| align=center \| <math>\mathrm{mg}</math> \|\| align=right \| 0.4 <math>\mathrm{N}</math>		\| align=center \| <math>\mathrm{mg}</math> \|\| align=right \| 4 <math>\mathrm{N}</math>
	\|-		\|-
	\| align=center \| <math>\mu</math> \|\| align=right \| 1		\| align=center \| <math>\mu</math> \|\| align=right \| 1

RobertLampel: /* Parameters */

2025-11-24T10:16:29Z

Parameters

← Older revision		Revision as of 10:16, 24 November 2025
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	\| align=center \| <math>\mu</math> \|\| align=right \| 1		\| align=center \| <math>\mu</math> \|\| align=right \| 1
	\|}		\|}

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

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

RobertLampel: /* Parameters */

2025-11-24T10:15:48Z

Parameters

← Older revision		Revision as of 10:15, 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 ~~!! Description~~		! Symbol !! Value
	\|-		\|-
	\| align=center \| <math>~~m_1~~</math> \|\| align=right \| ~~100~~ <math>\mathrm{kg}~~</math> \|\| First mass directly affected by <math>F( t )~~</math>		\| align=center \| <math>m</math> \|\| align=right \| 2.2 <math>\mathrm{kg}</math>
	\|-		\|-
	\| align=center \| <math>~~m_2~~</math> \|\| align=right \| 2 <math>\mathrm{kg}</math>~~\|\| Second mass influenced by damping control~~		\| align=center \| <math>J</math> \|\| align=right \| 0.05 <math>\mathrm{kg \cdot m^2}</math>
	\|-		\|-
	\| align=center \| <math>~~k_1~~</math> \|\| align=right \| ~~100~~ <math>\mathrm{N/m}</math>~~\|\| Spring connecting first mass to reference~~		\| align=center \| <math>r</math> \|\| align=right \| 0.2 <math>\mathrm{m}</math>
	\|-		\|-
	\| align=center \| <math>~~k_2~~</math> \|\| align=right \| 3 <math>\mathrm{N/m}</math>~~\|\| Coupling spring between the two masses~~		\| align=center \| <math>\mathrm{mg}</math> \|\| align=right \| 0.4 <math>\mathrm{N}</math>
	\|-		\|-
	\| align=center \| <math>~~c</math> \|\| align=right \| 0.5 <math>\mathrm{Ns/m}</math>\|\| Damping affecting second mass~~		\| align=center \| <math>\mu</math> \|\| align=right \| 1
	\|-
	~~\| align=center \| <math>T</math> \|\| align=right \| <math>2~~ \~~pi</math> \|\| Duration of the motion~~
	\|-
	~~\| align=center \| <math>u~~</math> \|\| align=right \| ~~- \|\| Modulates the damping of the second mass~~
	\|}		\|}

RobertLampel: /* Mathematical formulation */

2025-11-24T10:12:35Z

Mathematical formulation

← Older revision		Revision as of 10:12, 24 November 2025
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	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x_1}(t) & = & v_1(t),\\		\quad \dot{x_1}(t) & = & v_1(t),\\
	\quad \dot{v_1}(t) & = & \frac{1}{m} \left( u_1 \cos \alpha - u_2 \sin \alpha \right),		\quad \dot{v_1}(t) & = & \frac{1}{m} \left( u_1 \cos \alpha - u_2 \sin \alpha \right), \\
	\quad \dot{x_2}(t) & = & v_2(t), \\		\quad \dot{x_2}(t) & = & v_2(t), \\
	\quad \dot{v_2}(t) & = & \frac{1}{m} \left( -\mathrm{mg} + u_1 \sin \alpha + u_2 \cos \alpha \right), \\		\quad \dot{v_2}(t) & = & \frac{1}{m} \left( -\mathrm{mg} + u_1 \sin \alpha + u_2 \cos \alpha \right), \\

RobertLampel: /* Mathematical formulation */

2025-11-24T10:12:19Z

Mathematical formulation

← Older revision		Revision as of 10:12, 24 November 2025
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	== Mathematical formulation ==		== Mathematical formulation ==
			We summarize the states as <math>x := (x_1, v_1, x_2, v_2, \alpha, v_\alpha)</math>.

	<p>		<p>
	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{u} && \frac{1}{2}\int_0^T \left( ~~x_0(t)~~^2 ~~+ x_1~~(t)^2 ~~+ u~~(t)^2 \right) \\		\displaystyle \min_{u, t_\mathrm{f}} && \frac{1}{t_\mathrm{f}} \int_0^{t_\mathrm{f}} \left( 2 u_1^2(t) + u_2^2(t) \right) \mathrm{d}t + \mu \, t_\mathrm{f} \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{~~x_0~~}(t) & = & ~~x_2~~(t),\\		\quad \dot{x_1}(t) & = & v_1(t),\\
	\quad \dot{~~x_1~~}(t) & = & ~~x_3~~(t), \\		\quad \dot{v_1}(t) & = & \frac{1}{m} \left( u_1 \cos \alpha - u_2 \sin \alpha \right),
	\quad \dot{x_2}(t) & = & - \~~frac{k_1 + k_2}{m_1}~~ \~~cdot x_0 +~~ \~~frac~~{~~k_2~~}~~{m_1} \cdot x_1 +~~ \frac{1}{~~m_1~~} ~~\sin~~\left(\~~frac~~{2\~~pi}{T}~~ \~~cdot t~~\right), \\		\quad \dot{x_2}(t) & = & v_2(t), \\
	\quad \dot{~~x_3~~}~~(t)~~ & = & \~~frac~~{~~k_2~~}~~{m_2} x_0(t) -~~ \frac{~~k_2~~}{~~m_2~~} ~~x_1~~(t) - \~~frac{c~~(~~1-u)}~~{~~m_2~~} ~~x_3~~(t), \\		\quad \dot{v_2}(t) & = & \frac{1}{m} \left( -\mathrm{mg} + u_1 \sin \alpha + u_2 \cos \alpha \right), \\
	\quad ~~x_1~~(0) &=& 0, \\		\quad \dot{\alpha} & = & v_\alpha, \\
	\quad ~~x_2~~(0) &=& 0, \\		\quad \dot{v}_\alpha & = & \frac{r}{J} u_1, \\
	\quad u(t) & \in & [-1, 1] \ \quad \forall t \in [0,T]		\quad x(0) &=& (0, 0, 0, 0, 0, 0)^T, \\
			\quad x(t_\mathrm{f}) &=& (1, 0, 0, 0, 0, 0)^T, \\
			\quad u_1(t) & \in & [-5, 5] \ \quad \forall t \in [0,t_\mathrm{f}], \\
			\quad u_2(t) & \in & [0, 17] \ \quad \forall t \in [0,t_\mathrm{f}], \\
			\quad \alpha(t) & \in & [-30, 30] \ \quad \forall t \in [0,t_\mathrm{f}]
	\end{array}		\end{array}
	</math>		</math>