Double Oscillator

Double Oscillator
State dimension:	1
Differential states:	4
Discrete control functions:	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 [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. Both the state trajectory $x (\cdot)$ and the control $u (\cdot)$ are decision variables. The aim is to minimise a quadratic cost that balances state deviations and control effort, subject to input constraints and the system dynamics.

Mathematical formulation

$\begin{array}{lll} \min_{u} & \frac{1}{2} \int_{0}^{T} (x_{0} (t)^{2} + x_{1} (t)^{2} + u (t)^{2}) \\ subject to \\ \dot{x_{0}} (t) & = & x_{2} (t), \\ \dot{x_{1}} (t) & = & x_{3} (t), \\ \dot{x_{2}} (t) & = & - \frac{k_{1} + k_{2}}{m_{1}} \cdot x_{0} + \frac{k_{2}}{m_{1}} \cdot x_{1} + \frac{1}{m_{1}} \sin (\frac{2 π}{T} \cdot t), \\ \dot{x_{3}} (t) & = & \frac{k_{2}}{m_{2}} x_{0} (t) - \frac{k_{2}}{m_{2}} x_{1} (t) - \frac{c (1 - u)}{m_{2}} x_{3} (t), \\ x_{1} (0) & = & 0, \\ x_{2} (0) & = & 0, \\ u (t) & \in & [- 1, 1] \forall t \in [0, T] \end{array}$

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$m_{1}$	100 $k g$	First mass directly affected by $F (t)$
$m_{2}$	2 $k g$	Second mass influenced by damping control
$k_{1}$	100 $N / m$	Spring connecting first mass to reference
$k_{2}$	3 $N / m$	Coupling spring between the two masses
$c$	0.5 $N s / m$	Damping affecting second mass
$T$	$2 π$	Duration of the motion
$u$	-	Modulates the damping of the second mass

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
States and discretized control for a local optimum. The control $t_{f}$ represents the scaling of the time interval, where the base time interval is [0,5].

Miscellaneous and Further Reading

This formulation and a detailed description can be found in [1].

References

[1] 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