Derivation of Analytical Inverse Laplace Transform for Fractional Order Integrator

Abstract

There is considerable interest in the study of fractional order derivative integrator but obtaining analytical impulse and step responses is a difficult problem. Therefore all methods reported on to date use approximations for the fractional derivative/integrator both for analytical based computations and more relevantly in simulation studies. In this paper, an analytical formula is first derived for the inverse Laplace transform of fractional order integrator, 1/sα where α∈R and 0<α<1 using Stirling’s formula and Gamma function. Then, the analytical step response of fractional integrator has been computed from the derived impulse response of 1/sα. The obtained analytical formulas for impulse and step responses of fractional order integrator are exact results except the very small error due to the neglected terms of Stirling’s series. The results are compared with some well known integer order approximation methods and Grunwald-Letnikov (GL) approximation technique. It has been shown via numerical examples that the presented method is very successful according to other methods.