On Critical Delays for Linear Neutral Delay Systems
In this work we address the problem of finding the critical delays of a
linear neutral delay system, i.e. the delays such that the system has a purely
imaginary eigenvalue. Even though neutral delay systems exhibit some
discontinuity properties with respect to changes in the delays, an
essential part in a non-conservative stability analysis with respect to
changes in the delays, is the computation of the critical delays.
We generalize, under some minor assumptions on the delay system,
previous results on critical delays and stability switches for retarded
time-delay systems.
The work starts with stating a general equivalence theorem with the
spectrum and the matrix function condition. We show how this
theorem can be applied to the commensurate time-delay system to compute
the critical delays. It turns out that the resulting method is a
closely related to parts
of the results of Fu, Niculescu and Chen\cite{Fu:2006:NEUTRAL}. For the
incommensurate case we present a scheme which allows the computation of
the critical curves, i.e. the points in delay-space for which the
system has a purely imaginary.
We apply the method to previously investigated examples, and hence
provide a verification of the results, as well as some
examples for which the stability picture is not believed to be known.