# Algorithms for Fractional Dynamical Behaviors Modelling Using Non-Singular Rational Kernels
## 1 Introduction  
The search for non-singular kernels [1; 2; 3; 4; 5] or alternative solutions to fractional models [6; 7] is gaining momentum in the field of modeling fractional dynamical behaviors. Using such kernels makes it possible to overcome certain limitations of fractional models, such as fractional differential equations or pseudo-state space descriptions, which have recently been highlighted [8; 9; 10; 11; 12; 13; 14]. For instance, it was demonstrated that a fractional integrator operator in the Riemann-Liouville sense [15; 16] involves infinitely large and infinitely small time constants, which makes fractional models doubly infinite models: infinite because distributed, but also infinite as defined on an infinite equivalent spatial domain [17]. In a modelling context, these infinitely large and infinitely small time constants are not required to capture the behavior of a real system that operates on finite spatial and time scales. Moreover, these time constants introduced by fractional models (and that do exist in the modelled system) can greatly complicate the analyses and lead to erroneous conclusions on certain internal properties of the modelled system. This is the case for initialization [18; 19] or observability for example [20]. However, fractional behaviors (produced by physical phenomena) and fractional models should be distinguished. The former designates a property of a physical system while the latter designates a model class, among a set of model classes able to capture fractional dynamical behaviors. Based on this observation, a whole new field opens up, from the search for new models dedicated to capturing fractional behaviors to the study of their properties and the development of identification methods. A few studies undertaken from this perspective are already available in the literature [6; 7], which is also the case in the present article. To overcome the limitations mentioned in the first paragraph, we propose in this paper to use non-singular rational kernels to model fractional behaviors and propose an algorithm to find the parameters in the kernel description. Non-singular rational kernels are considered for the three following reasons:  
* a fractional behavior defined by the power law \(t^{-\
u}\), can be associated with a rational function with an infinite number of interlaced poles and zeros as shown in this paper. the approximation of such a function by rational functions of degree \(n\) leads to a very small approximation error in comparison to polynomials of degree \(n\)[21];rational kernels permit the approximation of fractional behaviors with a reduced number of parameters in comparison to fractional models. For all these reasons, it appeared necessary to develop a simple and efficient algorithm allowing the determination of the coefficients of a rational kernel in a convolution model, which fits a given fractional behavior with a defined given level of accuracy. To the best knowledge of the authors, there is no method that allows a direct estimation of the parameters of a rational kernel in a convolution model. That is why we have defined a two-step method that consists of:  
* first, estimating the time response of the kernel in the convolution model that fits the input-output behavior of the modeled system. * then, estimate the parameters of a rational function that fits the kernel time response. For the second step of this strategy, there are many methods that allow rational interpolation [22; 23; 24; 25; 26]. But these methods are not well adapted to fractional behaviors, that is to say with time behaviors with slow convergence. On the other hand, when working in the logarithmic domain as in the methodology we propose, a time compression occurs which facilitates fitting over large time domains. The paper is thus organized as follows. In Section 2, the approximation of a pure power law function \(Kt^{-\
u}\) by a rational function involving interlaced poles and zeros is considered.