Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems

In this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of s(& alpha;)-domain poles, the corresponding basis functions are found, the kernel of which is the two-parameter Mittag-Leffler function E-& alpha;(,& beta;), & beta; = & alpha;. It is pointed out that there are mutually unambiguous correspondences between the basis functions of FOS and the known basis functions of the integer-order system (IOS) for & alpha; = 1. This correspondence can be used to algorithmically find analytical formulas for the impulse responses of FOS when the formulas for the characteristics of IOS are known. It is shown that all basis functions of FOS can be generated with Podlubny's function of type & epsilon;(k) (t, c; & alpha;, & alpha;), where c and k are the corresponding pole and its multiplicity, respectively.