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

$\begin{array}{lll}\underset{u}{\min }& & \frac{1}{2}\underset{0}{\overset{T}{\int }}(x_0(t{)}^2+x_1(t{)}^2+u(t{)}^2)\\ \text{subject to}\\ \dot{x_0}(t)& =& x_2(t),\\ \dot{x_1}(t)& =& x_3(t),\\ \dot{x_2}(t)& =& -\frac{k_1+k_2}{m_1}\cdot x_0+\frac{k_2}{m_1}\cdot x_1+\frac{1}{m_1}\sin (\frac{2\pi }{T}\cdot t),\\ \dot{x_3}(t)& =& \frac{k_2}{m_2}{x}_0(t)-\frac{k_2}{m_2}{x}_1(t)-\frac{c(1-u)}{m_2}{x}_3(t),\\ x_0(0)& =& 0,\\ x_1(0)& =& 0,\\ u(t)& \in & [-1,1]\mathrm{\forall }t\in [0,T]\end{array}$

The control $u$ modulates the damping of the second mass.

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$m_1$	100 $\mathrm{k}\mathrm{g}$	First mass directly affected by $F(t)$
$m_2$	2 $\mathrm{k}\mathrm{g}$	Second mass influenced by damping control
$k_1$	100 $\mathrm{N}/\mathrm{m}$	Spring connecting first mass to reference
$k_2$	3 $\mathrm{N}/\mathrm{m}$	Coupling spring between the two masses
$c$	0.5 $\mathrm{N}\mathrm{s}/\mathrm{m}$	Damping affecting second mass
$T$	$2\pi$	Duration of the motion

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States and discretized control for a local optimum.

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/double_oscillator.md