Double Oscillator - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T10:43:26Z

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:Double_Oscillator.png\| States and discretized control for a local optimum.		Image:Double_Oscillator.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

2025-11-26T15:16:13Z

Mathematical formulation

← Older revision		Revision as of 15:16, 26 November 2025
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	<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} && \frac{1}{2}\int_0^T \left( x_0(t)^2 + x_1(t)^2 + u(t)^2 \right) \mathrm{d}t \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x_0}(t) & = & x_2(t),\\		\quad \dot{x_0}(t) & = & x_2(t),\\

2025-11-26T15:15:39Z

Mathematical formulation

← Older revision		Revision as of 15:15, 26 November 2025
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	</math>		</math>
	</p>		</p>

			The control <math>u</math> modulates the damping of the second mass.

	== Parameters ==		== Parameters ==

2025-11-26T15:15:05Z

Parameters

← Older revision		Revision as of 15:15, 26 November 2025
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	\|-		\|-
	\| align=center \| <math>T</math> \|\| align=right \| <math>2 \pi</math> \|\| Duration of the motion		\| 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~~
	\|}		\|}

2025-11-24T15:09:35Z

Mathematical formulation

← Older revision		Revision as of 15:09, 24 November 2025
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	\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) & = & - \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_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{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 x_0(0) &=& 0, \\
	\quad x_1(0) &=& 0, \\		\quad x_1(0) &=& 0, \\
	~~\quad x_2(0) &=& 0, \\~~
	\quad u(t) & \in & [-1, 1] \ \quad \forall t \in [0,T]		\quad u(t) & \in & [-1, 1] \ \quad \forall t \in [0,T]
	\end{array}		\end{array}

2025-11-24T09:58:49Z

← Older revision		Revision as of 09:58, 24 November 2025
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	}}		}}

	The '''Double Oscillator problem''' is a benchmark in constrained optimal control illustrating the control of coupled mechanical systems with damping and stiffness effects. This description is taken from [[#OCPjl \| [1]]]		The '''Double Oscillator problem''' is a benchmark in constrained optimal control illustrating the control of coupled mechanical systems with damping and stiffness effects. This description is taken from [[#OCPjl \| [1]]].

	It consists of two masses connected by springs and a damper, with one mass directly influenced by an external periodic force and the other influenced indirectly through the coupling and a controlled damping term.		It consists of two masses connected by springs and a damper, with one mass directly influenced by an external periodic force and the other influenced indirectly through the coupling and a controlled damping term.

2025-11-24T09:58:28Z

References

← Older revision		Revision as of 09:58, 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/~~dielectrophoretic_particle~~.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/double_oscillator.md<br>


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

2025-11-24T09:58:09Z

Reference Solutions

← Older revision		Revision as of 09:58, 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:~~Dielectrophoretic_Particle~~.png\| States and discretized control for a local optimum~~. The control <math>t_f</math> represents the scaling of the time interval, where the base time interval is [0,5]~~.		Image:Double_Oscillator.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

2025-11-24T09:37:39Z

Parameters

← Older revision		Revision as of 09:37, 24 November 2025
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	! Symbol !! Value !! Description		! Symbol !! Value !! Description
	\|-		\|-
	\| align=center \| <math>~~x_0~~</math> \|\| align=right \| 1 \|\| ~~Initial particle position~~		\| 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>~~x_f~~</math> \|\| align=right \| 2 \|\| ~~Final particle position~~		\| align=center \| <math>m_2</math> \|\| align=right \| 2 <math>\mathrm{kg}</math>\|\| Second mass influenced by damping control
	\|-		\|-
	\| align=center \| <math>~~\alpha~~</math> \|\| align=right \| ~~-0.75~~ \|\| ~~Nonlinear coefficient~~		\| align=center \| <math>k_1</math> \|\| align=right \| 100 <math>\mathrm{N/m}</math>\|\| Spring connecting first mass to reference
	\|-		\|-
	\| align=center \| <math>c</math> \|\| align=right \| 1 \|\| Damping ~~coefficient~~		\| align=center \| <math>k_2</math> \|\| align=right \| 3 <math>\mathrm{N/m}</math>\|\| Coupling spring between the two masses
			\|-
			\| align=center \| <math>c</math> \|\| align=right \| 0.5 <math>\mathrm{Ns/m}</math>\|\| Damping affecting second mass
			\|-
			\| 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
	\|}		\|}

2025-11-24T09:30:53Z

Mathematical formulation