A new approach for direct discretization of fractional order operator in delta domain

The fractional order system (FOS) comprises fractional order operator. In
 order to obtain the discretized version of the fractional order system, the
 first step is to discretize the fractional order operator, commonly
 expressed as s?m, 0 < m < 1. The fractional order operator can be used as
 fractional order differentiator or integrator, depending upon the values of
 . In general, there are two approaches for discretization of fractional
 order operator, one is indirect method of discretization and another is
 direct method of discretization. The direct discretization method
 capitalizes the method of formation of generating function where fractional
 order operator s?m is expressed as a function of Z in the shift operator
 parameterization and continued fraction expansion (CFE) method is then
 utilized to get the corresponding discrete domain rational transfer
 function. There is an inherent problem with this discretization method using
 shift operator parameterization (discrete Z-domain). At fast sampling time,
 the discretized version of the continuous time operator or system should
 resemble that of the continuous time counterpart if the sampling theorem is
 satisfied. At very high sampling rate, the shift operator parameterized
 system fails to provide meaningful information due to its numerical ill
 conditioning. To overcome this problem, Delta operator parameterization for
 discretization is considered in this paper, where at fast sampling rate, the
 continuous time results can be obtained from the discrete time experiments
 and therefore a unified framework can be developed to get the discrete time
 results and continuous time results hand to hand. In this paper a new
 generating function is proposed to discretize the fractional order operator
 using the Gauss-Legendre 2-point quadrature rule. Additionally, the function
 has been expanded using the CFE in order to obtain rational approximation of
 the fractional order operator. The detailed mathematical formulations along
 with the simulation results in MATLAB, with different fractional order
 systems are considered, in order to prove the newness of this formulation
 for discretization of the FOS in complex Delta domain.