On the links between limit characteristic zeros and stability properties of linear time-invariant systems with point delays and their delay-free counterparts

Abstract

We investigate the relationships between the infinitely many characteristic zeros (or modes) of linear systems subject to point delays and their delay-free counterparts based on algebraic results and theory of analytic functions. The cases when the delay tends to zero or to infinity are emphasized in the study. It is found that when the delay is arbitrarily small, infinitely many of those zeros are located in the stable region with arbitrarily large modulus, while their contribution to the system dynamics becomes irrelevant. The remaining finite characteristic zeros converge to those of the delay-free nominal system. When the delay tends to infinity, infinitely many zeros are close to the origin. Furthermore, there exist two auxiliary delay-free systems which describe the relevant dynamics in both cases for zero and infinite delays. The maintenance of the delay-free system stability in the presence of sufficiently small delayed dynamics is also discussed in light of $H_{\infty }$-theory. The main mathematical arguments used to derive the results are based on the theory of analytic functions.