.. _harmonic_oscillator_fit:

Optimization of resonant system based on Gaussian fit
==========================================================================================


:Driver: :ref:`ActiveLearning`


**Download script**: :download:`harmonic_oscillator_fit.m`


The target of the study is to tune a resonant system. For simplicity, the resonant system is a harmonic oscillator with eigenfrequency :math:`\omega_0` and damping :math:`\gamma` driven with force :math:`F` at frequency :math:`\omega`. The amplitude of the oscillator is

.. math::

   a_{\rm harm}(\omega) = \frac{F}{\sqrt{(2\omega\omega_0\gamma)^2 + (\omega_0^2 - \omega)^2}} .

We assume the system as a black box with unknown resonant behavior. To capture the resonant feature, it is fitted against a Gaussian plus a linear function

.. math::

   a_{\rm fit}(\omega) = A \exp\left(-\frac{1}{2}\frac{(\omega-\tau)^2}{\sigma^2}\right) + B\omega + C .
   
The target is to tune :math:`F, \omega_0, \gamma` such that the fitted amplitude is :math:`A = 10`, the resonance frequency is :math:`\tau = 2` and the resonance width :math:`\sigma` is as small as possible.



.. literalinclude:: ./harmonic_oscillator_fit.m
   :language: matlab
   :linenos: