By Xu-Guang Li, Silviu-Iulian Niculescu, Arben Cela

In this short the authors determine a brand new frequency-sweeping framework to unravel the full 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 homes targeting the asymptotic habit of the attribute roots positioned at the imaginary axis in addition to on homes invariant with recognize to the hold up parameters. This asymptotic habit is proven to be comparable through one other novel thought, the twin Puiseux sequence which is helping make frequency-sweeping curves worthy within the examine of common time-delay platforms. The comparability of Puiseux and twin Puiseux sequence results in 3 vital results:

- an specific functionality of the variety of volatile roots simplifying research and layout of time-delay structures in order that to a point they're handled as finite-dimensional systems;
- categorization of all time-delay structures into 3 forms in response to their final balance houses; and
- a uncomplicated frequency-sweeping criterion permitting asymptotic habit research of severe imaginary roots for all optimistic severe delays via observation.

Academic researchers and graduate scholars drawn to time-delay platforms and practitioners operating in various fields – engineering, economics and the lifestyles sciences related to move of fabrics, power or details that are inherently non-instantaneous, will locate the implications offered right here invaluable in tackling a few of the advanced difficulties posed by means of delays.

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**Additional info for Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays**

**Example text**

1002. 7178). 9913); . .. As the invariance property has been proved for the simple critical imaginary root case, we have stronger results: Each time τ increases near some τ0,k (τ1,k ), two roots cross C0 from right to left (from left to right). Moreover, such results can be directly known from the frequency-sweeping curve. More precisely, for this timedelay system we have two fascinating properties [70]: (1) The frequency-sweeping curve crosses ℑ1 from below to above (from above to below) if and only if the corresponding critical imaginary root crosses C0 from left to right (from right to left).

13 The notation ΔNU (τ ) has been largely used in the literature. However, at a critical delay, the system may happen to have more than one critical imaginary root. 8). 1 will be studied in detail in Chap. 4. 1) in the presence of any finitely large τ (see Sect. 3). It is worth mentioning that such a stability result is accurate without any conservatism. For general linear time-delay systems with commensurate delays, this result represents a new contribution. However, it is still not sufficient for solving the complete stability problem.

4, we will provide some details on how to acquire the above Puiseux series solutions. For simplicity, we here give two examples where Φ(y, x) are polynomials, which represent a specific form of power series. The approach applies to the general power series equations. Historically, the study of the singularities of analytic curves stemmed from solving the polynomial equations. 3 Convergence of Puiseux Series Before discussing deeper the way to derive the Puiseux series, it is necessary to pay attention to the corresponding convergence property.