is optimal in $\text{L}_1$-sense and sparse within the space spanned by the columns of $\theta$, i.e. the basis functions. $$\label{eq:l1regression}
s = \operatorname*{arg\,min}_{s^{\prime}} \| \theta s^{\prime} - y \|_2 + \lambda \| s^{\prime} \|_1.$$

### Sparse Identification of Nonlinear Dynamics (SINDy).

Since 2015, Nathan Kutz, Steven Brunton and Joshua Proctor have developed the Sparse Identification of Nonlinear Dynamics method[@BruntonSINDy.2016], which is the basis for our data-driven approach to identify hysteretic behavior. First, a library $\Theta(X)$ with possible basis functions is built. SINDy aims to find the minimal amount of those functions, with which a given signal can be approximated. In contrast to DMD, which results in a linear model, SINDy can include arbitrary nonlinearities. As a further development, SINDYc can include feedback and control[@BRUNTON_SINDYc]. A successful application of SINDy in model predictive control is reported in [@Kaiser.].

The main principle of SINDy is the approximation of signals using only few basis functions, i.e. *sparse* solutions. The vector $x(t) = \begin{bmatrix}
x_1(t) &    x_2(t)  & \dots & x_n(t)
\end{bmatrix}^T \in \mathbb{R}^n $ represents all states of the system at time $t$. Measurement data of these states $x(t)$ and their derivatives $\dot{x}(t)$ at the time-steps $t_1, t_2, \dots , t_m$ form two 