By Xu-Guang Li, Silviu-Iulian Niculescu, Arben Cela
In this short the authors determine a brand new frequency-sweeping framework to unravel the entire balance challenge for time-delay structures with commensurate delays. The textual content describes an analytic curve standpoint which permits a deeper figuring out of spectral homes concentrating on the asymptotic habit of the attribute roots situated at the imaginary axis in addition to on houses invariant with appreciate to the hold up parameters. This asymptotic habit is proven to be comparable via one other novel suggestion, the twin Puiseux sequence which is helping make frequency-sweeping curves valuable within the learn of normal time-delay structures. The comparability of Puiseux and twin Puiseux sequence ends up in 3 very important results:
- an specific functionality of the variety of volatile roots simplifying research and layout of time-delay structures in order that to some extent they are handled as finite-dimensional systems;
- categorization of all time-delay structures into 3 kinds in accordance with their final balance homes; and
- a uncomplicated frequency-sweeping criterion permitting asymptotic habit research of severe imaginary roots for all optimistic severe delays by means of observation.
Academic researchers and graduate scholars attracted to time-delay platforms and practitioners operating in numerous fields – engineering, economics and the existence sciences concerning move of fabrics, strength or details that are inherently non-instantaneous, will locate the implications offered right here valuable in tackling a few of the complex difficulties posed through delays.
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Additional info for Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays
In this case, the stability analysis requires to know the variation directions of the critical roots with respect to the unit circle ∂D, based on the Puiseux series. For instance, if for a critical root its variation direction points to the outside (inside) of the unit circle ∂D, it implies that the critical root becomes an unstable (stable) root. It was already pointed out that we only adopt some preliminary results on the singularities of analytic curves and one will find that they are not hard to follow.
The point (x = 0, y = 0)) in the C2 plane. In other words, we study how y varies near “0” with respect to an infinitesimal variation of x near “0”. Such a local study will be extremely useful in the subsequent study of the asymptotic behavior of time-delay systems. Throughout this book, we define the notation ord( · ) as follows. 1 For a function ϕ(x), ord(ϕ(x)) = κ for x = x ∗ denotes that 0 (i = 0, . . , κ − 1) and that d κ ϕ(x) dxκ = 0 when x = x ∗. d i ϕ(x) dxi = Furthermore, for simplicity, we denote by ord y and ordx , respectively, the values of ord( (y, 0)) when y = 0 and ord( (0, x)) when x = 0.
First, a new frequency-sweeping mathematical framework is established based on the embryonic form given in Chap. 7. Next, a series of new mathematical properties regarding the asymptotic behavior of the critical imaginary roots and the frequency-sweeping curves are found. Finally, the general invariance property is confirmed in virtue of these new properties. In Chap. 9, we give a systematic approach, the frequency-sweeping approach, to study the complete stability of time-delay systems with commensurate delays.
Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays by Xu-Guang Li, Silviu-Iulian Niculescu, Arben Cela