g(x(t), \mu(t)). \label{eq:MuDerivativeSimple}$$

Therefore it is sufficient to model the change of $\mu$ by current values of $x$ and $\mu$, even for hysteretic systems. The question of how to choose a proper function $g$ will be addressed next.

Modeling discrete states {#modelingDiscreteStates}
------------------------

In order to find a suited function $g$ from Eq. to model changes of discrete systems, one has to consider the way the change of states takes place. The transition can for example be either abrupt or smooth, following a certain shape. In the case of two-point controlled systems the state transition is abrupt: As soon as a certain threshold gets passed, the system behavior changes more or less instantaneously. This assumption holds for many hybrid systems.

In \[hysteresisModeling\] we discussed the Preisach model for identification of hysteresis, which uses the relay operator $R_{\alpha, \beta}$. Based upon the idea that state transitions in many hybrid systems happen abruptly, the relay hysteron seems to be a suited candidate for the function $g$.

Preparing the data
------------------

The first step of data processing is affine scaling of all data to a sensible and comparable range, e.g. from -1 to 1. This promotes an equal treatment of all time series, 