A computing method on stability intervals of time-delay for fractional-order retarded systems with commensurate time-delays

This paper investigates the stability intervals of time-delays for fractional-order retarded time-delay systems. By the Orlando formula, the existence of the crossing frequencies is brought to verify the stability related to the commensurate time-delay. For each crossing frequency, the corresponding critical time-delays are determined by the generalized eigenvalues of two matrices constructed by the crossing frequency, the commensurate fractional-order and the coefficients of the characteristic function. The root tendency (RT) is defined to provide a method to analyze the number of the unstable roots for a given crossing frequency and critical time-delay. Based on the RT values and the number of the unstable roots for fractional-order systems with no time-delay, a computing method on the stability intervals of time-delay is proposed in this paper. Finally, a numerical example is offered to validate the effectiveness of this method.