Stability analysis of multiple time-delayed system

A general class of linear time-invariant systems with time delays is studied. A number of methodologies have been suggested to assess the stability in the parametric domain of time delay or coefficient. This study offers an exact, structured and robust methodology to determine the stability regions of uncertain parameters in both time-delay space and coefficient space. The Rekasius transformation is used as a connection between time-delay space and coefficient space. An explicit analytical expression in terms of the system parameters which reveals the stability regions(pockets) in the domain of time delay and coefficient is presented. The method starts with the determination of all possible values of uncertain parameters which result in purely imaginary characteristic roots. In addition, some special stability boundaries are also discussed. After generating stability boundaries in parametric space, the two-step determination procedure is proposed to determine the actual stability regions. Such an approach can be used to determine the stability regions of any uncertain parameters of any retarded time-delay system. A complete example case study is also provided.