Modeling and Analog Realization of the Fundamental Linear Fractional Order Differential Equation

Abstract

This paper provides a rational function approximation of the irrational transfer function of the fundamental linear fra- ctional order differential equation, namely, $(\tau _0)^m \frac{{d^m x(t)}}{{dt^m }} + x(t) = e(t)$ whose transfer function is given by $G(s) = \frac{{X(s)}}{{E(s)}} = \frac{1}{{[1 {+} (\tau _0 s)^m ]}}$ for $0 < m < 2$ for 0<m<2. Simple methods, useful in system and control theory, which consists of approximating, for a given frequency band, the transfer function of this fractional order system by a rational function are presented. The impulse and step responses of this system are derived and simple analog circuit which can serve as fundamental analog fractional order system is also obtained. Illustrative examples are presented to show the exactitude and the usefulness of the approximation methods.