# Algorithms for Fractional Dynamical Behaviors Modelling Using Non-Singular Rational Kernels
## 4 Conclusions  
In this paper, several algorithms are proposed to fit a fractional dynamical behavior using a convolution integral involving a rational kernel. In the case of a pure power law function, a first algorithm based on an interlacing distribution of poles and zeros in the rational kernel is given. Using a logarithmic scale on the abscissa and ordinate axes, a power law function of order \(\
u\) (\(Kt^{-\
u}\)) appears as a line of slope \(-\
u\). It is thus possible to approximate it by interlacing poles and zeros which, respectively, induce in this same system of coordinates, an asymptotic behavior of slope \(-1\) and \(1\). It is then demonstrated that the values of the poles and zeros are linked by geometric ratios that are computed by the algorithm. This fitting solution appears to be similar to the one commonly used in the frequency domain to approximate the behavior of a fractional integrator or differentiator.  
Figure 8: Comparison of \(\eta(t)\) and \(\eta_{a}(t)\) for the impulse response of model (27) with \(\delta=0.2\). **left column:** asymptotic diagram built by the algorithm, **right column:** obtained function \(\eta_{a}(t)\).  
In fact, it is observed that the solution proposed here in the time domain leads to fewer oscillations for the same number of poles and zeros.  
Inspired by the idea of interlacing of poles and zeros, a second algorithm is proposed to fit the impulse response of a system with any fractional dynamical behavior. First, we propose a solution based on a least squares method to derive the sampled-data impulse response of a dynamical system from input-output data. Then, the solution to obtain an approximant of this impulse response in the form of a rational function consists of adding poles or zeros in the approximant so that the error remains bounded in absolute value. This algorithm adds a pole if the approximant asymptotic diagram induced by the pole or the zeros previously added intersects the selected upper bound, and vice versa, a zero is added if the asymptotic diagram of the approximant induced by the pole or the zero previously added intersects the lower bound. In the applications proposed in the paper that illustrate the algorithm's efficiency, a constant error bound is imposed over the whole approximation time interval. However, this algorithm also works by taking different upper and lower bounds and functions of time. This algorithm is able to place the poles and zeros of the approximation and limit the approximation error between an upper and lower bound, but this placement is sub-optimal with regard to the quadratic error between the impulse response and its approximation. Moreover, this sequence of two algorithms (kernel samples computation using the algorithm described in Section 3.1 and then fitting the kernel samples with a rational kernel using Algorithm 2) can lead to the accumulation of errors. Furthermore, although Algorithm 2, which is recursive, has a low chance of leading to numerical instabilities, the algorithm used in Section 3.1 may have some. On the other hand, the solution we propose is very simple to implement and can constitute a solution for initializing future algorithms which will be developed by the authors. The authors have also now to consider additional real-life fractional behavior in order to clearly identify the limits of the use of non-singular rational kernels in modeling situations and to consider possible significant changes over time in the modelled system behavior.  
This work is an additional illustration that fractional behaviors can be modelled using other tools than the ones based on fractional calculus.  
Conceptualization, J.S. and C.F.; methodology, J.S. and C.F.; validation, J.S. and C.F.; writing--original draft preparation, writing--review and editing, J.S. and C.F. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Data available on request as no repository were created. The authors declare no conflict of interest.