Inverse Laplace Transforms of the Fractional Order Transfer Functions

The history of the fractional calculus goes back to approximately 300 years. In the recent years, it is common to come across fractional calculus in many publications of control systems. The systems with non-integer order of derivative in their differential equations are called fractional order systems. The Laplace transformation of such systems results fractional order transfer functions. However, the inverse Laplace transformation of a fractional order transfer function and its time responses can present a challenge to express analytically. In this work, the first order transfer functions and their fractional cases are considered. Furthermore, approximate inverse Laplace transformation, i.e., time response of the system, is obtained analytically by using MATLAB curve fitting method. These analytical equations are presented in a table for the interval of the fractional orders 0.1 <; α<; 0.9. Then the calculations of approximate inverse Laplace transform of a particular transfer function are presented numerically.