Van der Pol Oscillator
|Van der Pol Oscillator|
|Continuous control functions:||1|
|Interior point equalities:||2|
The Van der Pol Oscillator is an oscillating system with non-linear damping and self regulation. The System was first introduced by the Dutch physician Balthasar Van der Pol in 1927. The aim is to control the oscillation such that the system stays in a mean position.
The Van der Pol Oscillator evolves over time according to the second order differential equation:
where is the position coordinate, which is a function of the time , and is a scalar parameter indicating the non-linearity and the strength of the damping.
For the oscillator is being damped, whereas for energy is added to the system.
Based on the transformation the problem can be reformulated:
The Optimal Control Problem arises by adding the objective function:
Optimal Control Problem
The Optimal Control Problem with the aim to minimize the deflection can be formulated as follows:
These fixed values are used within the model:
The following reference solution was generated using JModelica with the automatic differentiation tool CasADI and the solver IPOPT. The Optimica code used to solve the problem can be found under Van der Pol Oscillator (JModelica) The optimal value of the objective function is 3.1762.
Model descriptions are available in:
- JModelica code at Van der Pol Oscillator (JModelica)
- JuMP code (of a slightly modified Van der Pol oscillator problem) at Van der Pol Oscillator (Jump)
There are several alternative formulations and variants of the above problem, in particular
- a variant where partial outer convexification is applied on the control and the continous control is replaces by binary controls, see also Van der Pol Oscillator (binary variant),