Abstract
The paper is aimed at a comparative simulation study on three prospective ideas how to approximate a general exponential polynomial by another one having all its exponents in the exp-function as integer multiples of some real number. This work is motivated by spectral properties of neutral time-delay systems (NTDS) and the contemporary state of the knowledge about the spectrum of NTDS with commensurate delays which are characterized by the latter family of exponential polynomials. The three ideas are, namely, those: Taylor series expansion, the interpolation in points given by dominant roots estimates and the special extrapolation technique presented by the authors recently. The goal is to match dominant parts of both the spectra as close as possible. However, some properties from the so called strong stability point of view can not be, in principle, preserved. The presented simulation example demonstrates the accuracy and efficiency of all the methods.
Highlights
The characteristic quasipolynomial of a linear-time invariant time delay system (TDS) gives the fundamental information about the system’s dynamics; for instance, its zeros coincide with system poles [1, 2]
This work is motivated by spectral properties of neutral time-delay systems (NTDS) and the contemporary state of the knowledge about the spectrum of NTDS with commensurate delays which are characterized by the latter family of exponential polynomials
The three ideas are, namely, those: Taylor series expansion, the interpolation in points given by dominant roots estimates and the special extrapolation technique presented by the authors recently
Summary
The characteristic quasipolynomial of a linear-time invariant time delay system (TDS) gives the fundamental information about the system’s dynamics; for instance, its zeros coincide (under some conditions) with system poles (i.e. eigenvalues) [1, 2]. Positions of vertical strips of poles are sensitive to infinitesimal delay changes, which give rise to the notion of strong stability [10, 11] that is affected i.a. by the rational dependence of delays [9]. These infinite vertical strips constitute the so called essential spectrum of a NTDS and they are unambiguously given by roots of the associated exponential polynomial [1]. While there are many analytical results on the (essential and whole) spectrum of systems with commensurate delays in the literature, see e.g. [6, 7], it is extremely difficult to find any exact laws for roots loci in the case of non-commensurate delays
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
