Limited frequency band diffusive representation for nabla fractional order transfer functions

Abstract

Though infinite-dimensional characteristic is the natural property of nabla fractional order systems and it is the foundation of stability analysis, controller synthesis and numerical realization, there are few research focusing on this topic. Under this background, this paper concerns the diffusive representation of nabla fractional order systems. Firstly, several variants are developed for the elementary equality in frequency domain, i.e. $\frac{1}{s^\alpha} = \int_0^{ + \infty } {\frac{{{\mu _\alpha }( \omega )}}{{s + \omega }}{\rm{d}}\omega }$. Afterwards, the limited frequency band diffusive representation and the unit impulse response are derived for a series of nabla fractional order transfer functions. Finally, an attempt to find the diffusive representation for general nabla fractional order transfer functions is made. Some conclusions are presented.