On τ-decomposition frequency-sweeping strategy to rational-order fractional systems with commensurate delays

This study presents an effective analysis procedure to give the bounded input bounded output stability intervals of time delay for a class of commensurate delayed fractional-order systems with rational order. First, it is proposed that the analysis on multi-valued characteristic function of delayed fractional-order systems can be conducted equivalent on the consideration of the principal branch. The statement makes the generalisation of the τ-decomposition method in integer order case possible. With the convenience in studying delayed systems, the frequency-sweeping method is applied logically, and the PIRs and the cross-direction around them can be determined through the frequency-sweeping curves simultaneously. Moreover, the verification of the neutral stability condition of fractional-order delayed systems is considered in a systematic manner. At last, some examples show that the proposed strategy is practically useful in the analysis and design of feedback control for both integer and fractional-order systems with time delays.