Some Studies On Spectral Characteristics Of Time-Delay Systems Based On Analytic Curves Framework

Although the stability of linear time-invariant time-delay systems (TDSs) is an old topic and has received a lot of attention, its research is full of complexity and some main problems have not been fully investigated. Unlike a finite-dimensional linear system, a time-delay system is an infinite-dimensional system involving infinitely many characteristic roots, the distribution of which cannot be fully detected by existing mathematical tools. More differently, the critical imaginary roots (CIRs) of a time-delay system correspond to infinitely many critical delays (CDs). With these intricacies the complete stability of linear time-delay systems has remained unsolved so far. The key property for solving complete stability is invariance property, which is proved only for special cases without general confirmation. Including the proof of invariance property, the main research work and conclusions in this thesis is as follows:Based on analytic curves framework, the full proof of invariance property for time-delay systems of Retarded type is finished. By connecting the asymptotic behavior of the critical roots with Puiseux series and connecting the asymptotic behavior of the frequency-sweeping curves with dual Puiseux series similarly, some internal relation between Puiseux series pairs is presented to confirm invariance property. Due to the simplicity of the proof for non-degenerate case, this thesis mainly focuses on the proof for degenerate case via dividing it into three sub-cases, which gives rise to great complexity and tremendous work. It is worth mentioning that the proof in this thesis holds for general case regardless of the multiplicity of the critical roots.Inspired by some homologous results for Retarded time-delay systems, the stability condition of the Neutral operator(as a necessary stability condition additionally required by Neutral systems) is embedded into frequency-sweeping approach and invariance property is generalized to time-delay systems of Neutral type. By observing the frequency-sweeping curves with sufficiently large frequency, the stability of the Neutral operator can be directly examined without transforming the systems in scalar form back to the matrix form. As for the confirmation of invariance property, the same methods used for Retarded type is employed.Using the numerical analysis software Matlab, the frequency-sweeping approach is implemented and the simulation for the asymptotic behavior of critical imaginary roots is finished.