Discretization of Fractional Order Operator in Delta Domain

Abstract

The fractional order operator is the backbone of the fractional order system (FOS). The fractional order operator (FOO) is generally represented as s^(±μ) (0<μ<1). Discrete time FOS can be obtained through the discretization of the fractional order operator. The FOO is the general form of either fractional order differentiator (FOD) or integrator (FOI) depending upon the values of μ. Out of the two discretization methods, direct discretization outperforms the method of indirect discretization. The mapping between the continuous time and discrete time domain is done with the development of generating function. Continuous fraction expansion (CFE) is used expand the generating function for the rational approximation of the FOO. There is an inherent problem associated with the discretization of FOO in discrete z-domain particularly at very fast sampling rate. In the other hand, discretization using delta operator parameterization provides the continuous time and discrete time results in hand to hand, when the continuous time systems are sampled at very fast sampling rate and circumventing the problem with shift operator parameterization at fast sampling rate. In this work, a new generating function is proposed to discretize the FOO using the Gauss-Legendre 3-point quadrature rule and generating function is expanded using the CFE to form rational approximation of the FOO in delta domain. The benchmark fractional order systems are considered in this work for the simulation purpose and comparison of results are made to prove the efficacy of the proposed method using MATLAB.