Stability independent of delay using rational functions

This paper is concerned with the problem of assessing the stability of linear systems with a single time-delay. Stability analysis of linear systems with time-delays is complicated by the need to locate the roots of a transcendental characteristic equation. In this paper we show that a linear system with a single time-delay is stable independent of delay if and only if a certain rational function parameterized by an integer kk and a positive real number TT has only stable roots for any finite T≥0T≥0 and any k≥2k≥2. We then show how this stability result can be further simplified by analyzing the roots of an associated polynomial parameterized by a real number δδ in the open interval (0,1)(0,1). The paper is closed by showing counterexamples where stability of the roots of the rational function when k=1k=1 is not sufficient for stability of the associated linear system with time-delay. We also introduce a variation of an existing frequency-sweeping necessary and sufficient condition for stability independent of delay which resembles the form of a generalized Nyquist criterion. The results are illustrated by numerical examples.