Stability Crossing Set for Systems With Three Delays

This article describes the stability crossing set for linear time-delay systems of arbitrary order with three delays. The crossing frequency set, which consists of all frequencies where a pair of zeros of the characteristic quasipolynomial may cross the imaginary axis, is partitioned to Grashof sets and Non-Grashof sets of various types. It was found that the general characteristics of the stability crossing set is completely determined by the partition structure of the crossing frequency set. With the exception of degenerate cases, this article provides a method of explicit and complete parameterization and geometric characterization of the stability crossing set of linear systems with three delays. With the well-known method of finding the crossing directions, this provides a powerful method of finding parameter regions of stable systems. Extension to systems with additional fixed delays is also discussed.