Exact Upper and Lower Bounds of Crossing Frequency Set and Delay Independent Stability Test for Multiple Time Delayed Systems

A general class of linear time invariant (LTI) multiple time-delay system is investigated in order to (i) obtain exact lower and upper bounds of crossing frequency set (CFS), and (ii) test the necessary and sufficient conditions of delay independent stability (DIS). The method commences by deploying Rekasius substitution for the transcendental terms in the characteristic function, reducing it into a finite dimensional one. After substitution, utilization of elimination theory allows one to achieve the two nontrivial objectives (i)-(ii). The approach is new and novel as it does not require any parameter sweeping and graphical display; it is exact and can test necessary and sufficient conditions of DIS over only a single variable polynomial. A case study is provided to show the effectiveness of the proposed method.