1, & \text{if } h > h_{max}. \\
        0, & \text{if } h < h_{min}. \\
        H_{1}(k-1), & \text{else}.
        \end{cases} \end{aligned}$$

![image](samplingrateProxHystSVG){width="\linewidth"} \[fig:exp3\_proxi\_sim\_h\]

Effect of noisy data
--------------------

Beside an insufficient sampling rate, noise is a typical problem for data-driven approaches. In this experiment we investigate its influence with the SNR values 1000, 100, 50 and 10. The results are depicted in Fig.\[fig:exp4\_sim\_h\].

![image](snrSimHSVG){width="\linewidth"} \[fig:exp4\_sim\_h\]

It can be seen that noise influences the identifcation quality very harshly. The approach works properly for an SNR of 1000. For an SNR of 100, the slope of the dynamics is not modeled accurately, resulting in a phase shift. The model for SNR 50 apparently identified two discrete states, but the state transitions as well as the dynamics within the states are faulty. For SNR 10, the measurements are too noisy to identify discrete states. As such, the model tries to satisfy all measurements by converging to the average.

Discussion {#sec:discussion}
==========

The results show, that SINDy with taylored basis functions in form of hysterons (SINDyHybrid) is a suitable method for the identification of hysteresis-controlled systems. The continuous system dynamics and the discrete states do not have do be modeled separately. With given signals 