By Xu-Guang Li, Silviu-Iulian Niculescu, Arben Cela
In this short the authors identify a brand new frequency-sweeping framework to resolve the total balance challenge for time-delay structures with commensurate delays. The textual content describes an analytic curve viewpoint which permits a deeper figuring out of spectral houses 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 similar through one other novel notion, the twin Puiseux sequence which is helping make frequency-sweeping curves beneficial within the learn of common time-delay platforms. The comparability of Puiseux and twin Puiseux sequence ends up in 3 vital results:
- an particular functionality of the variety of risky roots simplifying research and layout of time-delay platforms in order that to a point they're handled as finite-dimensional systems;
- categorization of all time-delay structures into 3 varieties in keeping with their final balance houses; and
- a uncomplicated frequency-sweeping criterion permitting asymptotic habit research of serious imaginary roots for all optimistic serious delays through observation.
Academic researchers and graduate scholars attracted to time-delay structures and practitioners operating in various fields – engineering, economics and the lifestyles sciences concerning move of fabrics, strength or info that are inherently non-instantaneous, will locate the consequences awarded right here helpful in tackling many of the complex difficulties posed via delays.
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Extra info for Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays
1 If there exists a point (y ∗ , x ∗ ) other than (0, 0) such that Φ(y ∗ , x ∗ ) = 0, we may obtain a new power series with a zero constant term. More precisely, we may define two new variables x = x − x ∗ and y = y − y ∗ . As a result, we obtain a new power series Φ(y, x) satisfying that Φ(0, 0) = 0 from the original power series equation Φ(y ∗ , x ∗ ) = 0 and the local behavior of the original equation Φ(y, x) = 0 as y → y ∗ and x → x ∗ is reflected by that of the new one Φ(y, x) = 0 as y → 0 and x → 0.
3) f (λ, τ ) = a0 (λ) + a1 (λ)e−τ λ + · · · + aq (λ)e−qτ λ . -G. 1007/978-3-319-15717-7_3 27 28 3 Analytic Curve Perspective for Time-Delay Systems As discussed in Chap. 1), for a critical imaginary root we need to know its asymptotic behavior at a critical delay. 1. 1. 1 A Motivating Example It is true that the asymptotic behavior of a critical imaginary root with respect to a critical delay is fully determined by the characteristic function f (λ, τ ). In fact, most of the existing results are based on a direct study of f (λ, τ ).
However, they exhibit different types of Puiseux series. Next, we present an example with g > n. To the best of the authors’ knowledge, such cases have not been sufficiently discussed in the literature. Finally, we consider an example with a critical imaginary root at the origin. 1)). For τ = π , λ = j is a triple critical imaginary root ( f λ ( j, π ) = f λλ ( j, π ) = 0 and f λ3 ( j, π ) = 0). As 40 4 Computing Puiseux Series for a Critical Pair f τ ( j, π ) = f τ τ ( j, π ) = 0 and f τ 3 ( j, π ) = 0, g = 3.
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