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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. [ABSTRACT FROM AUTHOR]

This technical note considers the problem of reducing the computational complexity associated with the Sum-of-Squares approach to stability analysis of time-delay systems. Specifically, this technical note considers systems with a large state-space but where delays affect only certain parts of the system. This yields a coefficient matrix of the delayed state with low rank—a common scenario in practice. The technical note uses the general framework of coupled differential-difference equations with delays in feedback channels. This framework includes systems of both the neutral and retarded-type. The approach is based on recent results which introduced a new Lyapunov–Krasovskii structure which was shown to be necessary and sufficient for stability of this class of systems. This technical note shows how exploiting the structure of the new functional can yield dramatic improvements in computational complexity. Numerical examples are given to illustrate this improvement. [ABSTRACT FROM AUTHOR]