A sufficiently large time delay in feedback loop must destroy exponential stability of any decay rate

Abstract

A linear time-invariant system with time delays is considered. It is assumed that the delays are nontrivial in the sense that they affect the characteristic equation (i.e., the characteristic quasi-polynomial is not a polynomial of one complex variable). It is proved that if any of such delays grows to infinity then either the system becomes unstable or, at least, some of its eigenvalues approach the imaginary axis. In the latter case, of course, the system may become extremely sensitive to changes of other parameters, which means practical instability. The theoretical results of this paper support well-known real-world observations concerning the stability of systems with large delays in the feedback loop.