Complete Stability of Linear Fractional Order Time Delay Systems: A Unified Frequency-Sweeping Approach

This paper is dedicated to the complete stability analysis of linear fractional order time delay systems (FOTDSs) with commensurate delays of retarded type, inspired by that the complete stability of integer order time delay systems was recently solved within a new frequency-sweeping framework [14]. We are hence motivated to extend the methodology to FOTDSs. The complete stability problem has been solved for some specific types of FOTDSs. However, for the general FOTDSs, it still remains open. In this paper, two technical aspects (such as the asymptotic behavior of critical imaginary roots (CIRs) and the invariance property for CIRs) of the complete stability problem for FOTDSs will be studied. Furthermore, an explicit expression of the number of the unstable roots at any given finitely large time delay will be obtained. As a consequence, the frequency-sweeping framework is proved to be a unified approach for the complete stability of FOTDSs. Finally, illustrative examples are given.