By Xu-Guang Li, Silviu-Iulian Niculescu, Arben Cela
In this short the authors determine a brand new frequency-sweeping framework to unravel the total balance challenge for time-delay platforms with commensurate delays. The textual content describes an analytic curve viewpoint 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 homes invariant with appreciate to the hold up parameters. This asymptotic habit is proven to be similar via one other novel idea, the twin Puiseux sequence which is helping make frequency-sweeping curves precious within the examine of normal time-delay structures. The comparability of Puiseux and twin Puiseux sequence ends up in 3 vital results:
- an particular functionality of the variety of volatile roots simplifying research and layout of time-delay platforms in order that to some extent they're handled as finite-dimensional systems;
- categorization of all time-delay structures into 3 varieties in accordance with their final balance houses; and
- a easy frequency-sweeping criterion permitting asymptotic habit research of serious imaginary roots for all optimistic severe delays via observation.
Academic researchers and graduate scholars drawn to time-delay structures and practitioners operating in quite a few fields – engineering, economics and the lifestyles sciences related to move of fabrics, strength or details that are inherently non-instantaneous, will locate the consequences awarded the following precious in tackling a number of the complex difficulties posed by way of delays.
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Additional resources for Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays
In Chap. 10, we extend the proposed frequency-sweeping framework to the timedelay systems of neutral type by taking care of an additional necessary condition for the stability of such time-delay systems. Finally, some concluding remarks and future perspectives are given in Chap. 11. Chapter 2 Introduction to Analytic Curves The study of analytic curves, which at first sight appears to be unrelated to the stability analysis of time-delay systems, will be extremely helpful for addressing the stability problem.
However, we here point out that some key information may be hidden behind the characteristic function f (λ, τ ) (or, its explicit form), as demonstrated below. 2) π2 3π 5 π 2 4 5π 3 π 2 2 7π λ − λ + λ − λ + λ− + 1. 3), λ = j is a triple critical imaginary root at τ = π . However, the asymptotic behavior of the critical pair ( j, π ) is quite different for the quasipolynomials above, which can be observed from the root loci near ( j, π ) shown in Figs. 3. 3), as shown in Fig. 3, exhibit a nearly symmetric structure with respect to the critical imaginary root.
2 If the power series Φ(y, x) are convergent, all the y-roots for Φ(y, x) = 0 are convergent series. 2, the convergence of all the Puiseux series used in this book associated with the complete stability problem for time-delay systems with commensurate delays (including the Puiseux series for studying the asymptotic behavior of the critical imaginary roots as well as the dual Puiseux series, to be proposed later in this book, for studying the asymptotic behavior of the frequency-sweeping curves) can be guaranteed.