allowing a representation resembling an FIR-filter. The state at time $k$ can be expressed by

$$\begin{aligned}
\begin{split}
    X(k) =& [x_1(k), ... , x_n(k), H_1(k-1), \bar{H}_1(k-1), ...
    \\& \bar{H}_m(k-1), x_1(k-1), ... , \bar{H}_m(k-q)]
\end{split}\end{aligned}$$

![The SINDy algorithm with tailored basis functions for identification of hysteretic systems[]{data-label="fig:sindyFlow"}](sindyFlowSVG){width="0.6\linewidth"}

As for the design of the library $\Theta$, polynomial basis functions are useful and easy to include. The function $\varphi_{poly}(x_1, x_2, ..., x_n)$ evaluates all polynomials between $x_1$ to $x_n$ up to a predefined polynomial degree. Hysterons are evolved by $\varphi_{relay}(x_1, x_2, ..., x_n, H_i)$. After evaluating all basis functions $\phi$, all basis functions that are unaffected by hysterons are multiplied with the updated hysterons. The finished library $\Theta$ consists of three parts: One without hysterons, the second with cross terms of hysterons and non hysteretic terms and the third of only the updated hysterons. Once the library is evaluated, a sequential least squares regression with a tuning parameter $\lambda$ is used to solve $ X(k+1) = \Theta(X(k)) \Xi$ for the sparse coefficients $\Xi$, as proposed in[@BruntonSINDy.2016].

For a library consisting of only polynomial basis functions and tailored basis functions for the propagation of hysterons, the evaluation and calculation of the next step is shown 