How BIBO stability of LTI fractional-order time delayed systems relates to their approximated integer-order counterparts

Time delay generates exponential transcendental terms in characteristic equations. Subsequently, the applied methods are narrow to assess the stability map of delayed fractional-order systems with vanishing fractions. In this case, it is convenient to approximate these equations with its integer-order counterparts to design controllers or investigate features of the systems. However, delay can cause enormous differences between characteristic of these two equations. This study offers a method to survey analytically the stability of fractional-order time delayed linear time invariant systems with infinitesimal fractions. In this method, equations are transferred to an explicit expression which enables one to analyse stability of retarded, neutral and even integer-order systems simultaneously.