A necessary and sufficient condition for stabilization of decentralized time-delay systems with commensurate delays

This paper investigates the stabilization problem for interconnected linear time-invariant (LTI) time-delay systems by means of linear time-invariant output feedback decentralized controllers. The delays are assumed to be commensurate and can appear in the states, inputs, and outputs of the system. First, the canonical forms for this type of time-delay systems are introduced and centralized fixed modes (CFM) for this type of systems are defined. It is then shown that a time-delay system which is both controllable and observable does not have any CFMs. Furthermore, an efficient technique for characterizing CFMs of any LTI time-delay system with commensurate delays is obtained. Decentralized fixed modes (DFM) are then defined accordingly, and a necessary and sufficient condition for decentralized stabilizability of the interconnected time-delay systems is proposed. Finally, a numerical example is given to illustrate the importance of results.