By Xu-Guang Li, Silviu-Iulian Niculescu, Arben Cela
In this short the authors determine a brand new frequency-sweeping framework to resolve the whole balance challenge for time-delay structures with commensurate delays. The textual content describes an analytic curve point of view which permits a deeper knowing of spectral houses targeting the asymptotic habit of the attribute roots situated at the imaginary axis in addition to on houses invariant with recognize to the hold up parameters. This asymptotic habit is proven to be comparable via one other novel notion, the twin Puiseux sequence which is helping make frequency-sweeping curves helpful within the research of basic time-delay platforms. The comparability of Puiseux and twin Puiseux sequence ends up in 3 very important results:
- an particular functionality of the variety of risky roots simplifying research and layout of time-delay structures in order that to some extent they're handled as finite-dimensional systems;
- categorization of all time-delay structures into 3 kinds in response to their final balance houses; and
- a easy frequency-sweeping criterion permitting asymptotic habit research of severe imaginary roots for all confident serious delays through observation.
Academic researchers and graduate scholars drawn to time-delay structures and practitioners operating in a number of fields – engineering, economics and the existence sciences related to move of fabrics, power or info that are inherently non-instantaneous, will locate the implications provided right here necessary in tackling a number of the complex difficulties posed by means of delays.
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Extra info for Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays
The concept of Puiseux series is not new in mathematics. It was first introduced by Issac Newton in his correspondence with Leibniz and Oldenburg in 1676  and further developed by Victor Puiseux in 1850 . The naming of the series after Puiseux rather than Newton is based upon the fact that Puiseux investigated this series expansion more thoroughly. The above information can be found in . 3 Unlike the well-known Taylor series, the exponents of a Puiseux series are allowed to be positive fractional numbers.
In our opinion, the analytic curve idea in fact may be used for a broader range of stability and stabilization problems in the area of control, as it is applicable to both continuous-time and discrete-time systems. For continuous-time systems (including the time-delay systems considered in the forthcoming chapters), we are concerned with the variation of the critical roots with respect to the imaginary axis C0 as some system parameters vary. Recall that for a continuous-time system a critical root refers to a characteristic root located on the imaginary axis C0 .
In this chapter, we start by presenting some fundamentals concerning analytic curves. Especially, as an important tool for studying analytic curves, the Puiseux series will be introduced and discussed in detail. In Sect. 1, we will first present the related concepts on analytic curves and show that an analytic curve can be understood in an intuitive manner. In Sect. 2, the Puiseux series will be introduced for describing and analyzing an analytic curve. The convergence of the Puiseux series will be discussed in Sect.