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In this brief the authors establish a new frequency-sweeping framework to solve the complete stability problem for time-delay systems with commensurate delays. The text describes an analytic curve perspective which allows a deeper understanding of spectral properties focusing on the asymptotic behavior of the characteristic roots located on the imaginary axis as well as on properties invariant with respect to the delay parameters. This asymptotic behavior is shown to be related by another novel concept, the dual Puiseux series which helps make frequency-sweeping curves useful in the study of general time-delay systems. The comparison of Puiseux and dual Puiseux series leads to three important results:

an explicit function of the number of unstable roots simplifying analysis and design of time-delay systems so that to some degree they may be dealt with as finite-dimensional systems;categorization of all time-delay systems into three types according to their ultimate stability properties; anda simple frequency-sweeping criterion allowing asymptotic behavior analysis of critical imaginary roots for all positive critical delays by observation.

Academic researchers and graduate students interested in time-delay systems and practitioners working in a variety of fields – engineering, economics and the life sciences involving transfer of materials, energy or information which are inherently non-instantaneous, will find the results presented here useful in tackling some of the complicated problems posed by delays.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to Complete Stability of Time-Delay Systems

Abstract
In this introductory chapter of the book, we give the preliminaries and prerequisites regarding the complete stability problem of time-delay systems. First, the basic concepts and definitions of the stability of time-delay systems are recalled. Then we illustrate the intricate spectral characteristics along with some examples and explain the necessity of adopting the so-called \(\tau \)-decomposition idea. Furthermore, the current bottleneck is briefly discussed. Finally, we outline the methodology proposed in this book.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 2. Introduction to Analytic Curves

Abstract
We first recall in this chapter some fundamentals concerning the analytic curves. Then, as an important tool for studying the analytic curves, the Puiseux series will be introduced and discussed in detail (the related definitions, the convergence, and the Newton diagram are included). Finally, we illustrate how to apply the Puiseux series to analyze the local behavior of an analytic curve. It turns out that an analytical curve can be appropriately understood from an intuitive root-locus viewpoint.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 3. Analytic Curve Perspective for Time-Delay Systems

Abstract
In this chapter, we apply the analytic curve perspective to the asymptotic behavior analysis of time-delay systems, using the prerequisites introduced in Chap. 2. Such a new perspective may help us to understand more deeply the spectral characteristics associated with time-delay systems. As illustrated by the motivating examples of this chapter, some key information concerning the spectral characteristics tends to be overlooked by the existing methods in the literature.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 4. Computing Puiseux Series for a Critical Pair

Abstract
In this chapter, we first explain why the existing methods for describing the asymptotic behavior do not work in the general case. Next, we present an algorithm for calculating all the first-order terms of the Puiseux series. Furthermore, we show that the proposed approach can be used to obtain higher order terms of the Puiseux series in order to study appropriately the degenerate case. Finally, some useful properties on the Puiseux series are presented.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 5. Invariance Property: A Unique Idea for Complete Stability Analysis

Abstract
In this chapter, we show that the stability of a system with any finitely large delay can be analyzed by invoking the Puiseux series. However, it is still not sufficient for solving the complete stability problem and there seems to be no routine solution due to the peculiarity that a critical imaginary root has infinitely many critical delays. In order to overcome such a peculiarity, we propose to prove the general invariance property.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 6. Invariance Property for Critical Imaginary Roots with Index $$g=1$$ g = 1

Abstract
In this chapter, we confirm the invariance property for the case where the partial derivative of the characteristic function with respect to delay is nonzero for all the critical pairs. It is proved that the asymptotic behavior of a critical imaginary root with any multiplicity can be determined from the corresponding frequency-sweeping curve. As the frequency-sweeping curves are independent of delay, the invariance property concerning the critical imaginary root’s asymptotic behavior is confirmed. Moreover, from the development in this chapter we come to a useful idea: The asymptotic behavior of the frequency-sweeping curves may be treated as a reference object in studying the invariance property in the forthcoming chapters.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 7. Invariance Property for Critical Imaginary Roots with Index $$n=1$$ n = 1

Abstract
We address in this chapter the invariance property for a specific case where all the critical imaginary roots are simple. Although such an invariance property has been proved in the literature, some new perspectives are obtained. We first study the asymptotic behavior of the frequency-sweeping curves by means of the dual Puiseux series, a new concept proposed in this chapter. Then, a useful equivalence relation between the Puiseux series and the dual Puiseux series is found. Based on it, we give a new proof on the invariance property. The aforementioned new perspectives are crucial to confirm the invariance property in the general case.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 8. A New Frequency-Sweeping Framework and Invariance Property in General Case

Abstract
In this chapter, a new frequency-sweeping framework is established, based on its embryonic form proposed in Chaps. 6 and 7. Using this new frequency-sweeping framework, we confirm the invariance property for the most general case. In other words, for any time-delay system with commensurate delays, the invariance property always holds.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 9. Complete Stability for Time-Delay Systems: A Unified Approach

Abstract
Based on the general invariance property, we systematically solve the complete stability problem for linear time-delay systems with commensurate delays. More precisely, we first study the so-called ultimate stability problem (i.e., the stability property as the delay approaches infinity). Next, we present the analytical expression of the number of unstable roots. As a consequence, the whole stability domain can be easily obtained by using the frequency-sweeping approach proposed in this book. This is an accurate result, i.e., the system is asymptotically stable if and only if the delay lies in the obtained stability domain.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 10. Extension to Neutral Time-Delay Systems

Abstract
In this chapter, we show that the frequency-sweeping framework proposed in this book is applicable to the time-delay systems of neutral type as well. Compared to the time-delay systems of retarded type, the stability of the neutral systems requires an additional necessary condition: the stability of the neutral operator. We see that this necessary condition can be embedded in the frequency-sweeping approach, i.e., we may directly verify this condition from the frequency-sweeping curves. Therefore, the complete stability of neutral time-delay systems can be studied within the frequency-sweeping framework.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Chapter 11. Concluding Remarks and Further Perspectives

Abstract
In this final chapter, some concluding remarks are given. In addition, some potential extensions of the proposed methodology are briefly discussed. For instance, the methodology may be extended for dynamic performance analysis, design problems, other classes of time-delay systems, and other parameter-sweeping approaches, in the future.
Xu-Guang Li, Silviu-Iulian Niculescu, Arben Çela

Backmatter

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