On the approximate inverse Laplace transform of the transfer function with a single fractional order

Abstract

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of $\alpha \in \{0.1,0.2,0.3,\dots ,0.9\}$ . Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for $1/(s^{\alpha }+1)$ . By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of $\alpha$ for $0<\alpha <1$ . Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.