Dominant pole analysis of stable time‐delay positive systems

This study is concerned with the dominant pole analysis of asymptotically stable time‐delay positive systems (TDPSs). It is known that a TDPS is asymptotically stable if and only if its corresponding delay‐free system is asymptotically stable, and this property holds irrespective of the length of delays. However, convergence performance (decay rate) should degrade according to the increase of delays and this intuition motivates us to analyse the dominant pole of TDPSs. As a preliminary result, in this study, the authors show that the dominant pole of a TDPS is always real. They also construct a bisection search algorithm for the dominant pole computation, which readily follows from recent results on α ‐exponential stability of asymptotically stable TDPSs. Then, they next characterise a lower bound of the dominant pole as an explicit function of delays. On the basis of the lower bound characterisation, they finally show that the dominant pole of an asymptotically stable TDPS is affected by delays if and only if associated coefficient matrices satisfy eigenvalue‐sensitivity condition to be defined in this study. Moreover, they clarify that the dominant pole goes to zero (from negative side) as time‐delay goes to infinity if and only if the coefficient matrices are eigenvalue‐sensitive.