Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results

In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity, and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampled-data, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles-which are most closely related to the research area-are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic and Second, to suggest possible future research directions to be tackled by scientists and engineers in the field. © 2013 IEEE.MSMT-7778/2014, FEDER, European Regional Development Fund; LO1303, FEDER, European Regional Development Fund; CZ.1.05/2.1.00/19.0376, FEDER, European Regional Development FundEuropean Regional Development Fund through the Project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; National Sustainability Program Project [LO1303 (MSMT-7778/2014)

Migration of double imaginary characteristic roots under small deviation of two delay parameters

International audience— This paper studies the migration of double imaginary roots of the characteristic equation for systems with two delays when the delay parameters are subjected to small deviations. As the double roots are not differentiable with respect to the delay parameters, Puiseux series is often used in such a situation in the literature. In this article, we study the " least degenerate " case, and a more traditional analysis was used without Puiseux series. It was found that the local stability crossing curve has a cusp at the point in the parameter space that causes the double root, and it divides the neighborhood of this point into a G-sector and an S-sector. When the parameters move into the G-sector, one of the roots moves to the right half plane, and the other moves to the left half plane. When the parameters move into the S-sector, both roots move either to the left half plane or the right half plane depending on the sign of some value explicitly expressed in terms of derivatives of the characteristic function up to the third order

On computing Puiseux series for multiple imaginary characteristic roots of LTI delay systems with commensurate delays

International audienc