Stability and resonance conditions of the non-commensurate elementary fractional transfer functions of the second kind

This paper deals with stability and resonance conditions of the non-commensurate elementary fractional transfer function of the second kind. This transfer function is a generalization of the elementary fractional transfer function of the second kind to an arbitrary order. It is written in the canonical form and characterized by a non-commensurate order, a pseudo-damping factor and a natural frequency. Stability and resonance analysis is done in terms of the pseudo-damping factor and the non-commensurate order. Also, an overall study of frequency-domain and time-domain performances of the considered system is done. Therefore many time-domain and frequency-domain curves are presented to help obtaining system parameters for a specified fractional order. Many illustrative examples show the efficiency of this study. Also, an application to the control of a spherical tank is also presented to show the usefulness of this study.