\geq \beta$.} \\
    0, & \text{if $x \leq \alpha$.} \\
    y_p, & \text{if $ \alpha < x < \beta $.}
    \end{cases}$$

![Relay hysteron [@RelayHyst1][]{data-label="fig:preisachRelay"}](Preisach_RelaySVG){width="0.82\linewidth"}

The value $y_p$ always corresponds to the prior value of $y(t)$. The complex shape of a magnetic hysteresis curve can be achieved by superposition of many elementary relay hysterons.

Modern approaches of hysteresis modeling are discussed in[@Hassani.2014]. The Krasnosel’skii–Pokrovskii (KP) model[@Krasnoselskii.1989] formalizes Preisach’s model in a mathematical way and allows for different operators to be included. Next to operator-based approaches, hysteresis can also be described by differential equations, e.g. using the Bouc–Wen model[@wen1976method] or the extended Bouc-Wen model, which requires 13 parameters to approximate the complex shape of the hysteresis[@Zhang.2001]. The here presented work utilizes the concept of hysterons which is the basis of Preisach’s model.

System Identification
---------------------

The SINDy framework is based upon the concepts of the Koopman operator and of sparsity. This section will give a brief introduction to these concepts, as well as present an overview of the SINDy methodology and its applications. In the following Section \[sec:Concept\], a concept is shown which implements the hysteresis model into the given framework.

### Koopman operator.

The Koopman operator $\mathcal{K}$[@Koopman.1931] is a linear but 