A numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systems

This paper aims to provide a numerical algorithm able to locate all unstable poles, and therefore the characterization of the stability as a function of the delay, for a class of linear fractional-order neutral systems with multiple commensurate delays. We start by giving the asymptotic position of the chains of poles and the conditions for their stability for a small delay. When these conditions are met, the root continuity argument and some simple substitutions allow us to determine the locations where some roots cross the imaginary axis, providing therefore the complete characterization of the stability windows. The same method can be extended to provide the position of all unstable poles as a function of the delay. © 2012 Elsevier Ltd. All rights reserved

Stability windows and unstable root-loci for linear fractional time-delay systems

The main point of this paper is on the formulation of a numerical algorithm to find the location of all unstable poles, and therefore the characterization of the stability as a function of the delay, for a class of linear fractional-order neutral systems with multiple commensurate delays. We start by the asymptotic position of the chains of poles and conditions for their stability, for a small delay. When these conditions are met, we continue by means of the root continuity argument, and using a simple substitution, we can find all the locations where roots cross the imaginary axis. We can extend the method to provide the location of all unstable poles as a function of the delay. Before concluding, some examples are presented. © 2011 IFAC

Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results

In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity, and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampled-data, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles-which are most closely related to the research area-are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic and Second, to suggest possible future research directions to be tackled by scientists and engineers in the field. © 2013 IEEE.MSMT-7778/2014, FEDER, European Regional Development Fund; LO1303, FEDER, European Regional Development Fund; CZ.1.05/2.1.00/19.0376, FEDER, European Regional Development FundEuropean Regional Development Fund through the Project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; National Sustainability Program Project [LO1303 (MSMT-7778/2014)

A unified approach for the H∞H_\infty-stability analysis of classical and fractional neutral systems with commensurate delays

International audienceWe examine the stability of linear integer-order and fractional-order systems with commensurate delays of neutral type in the sense of $H_\infty$-stability. The systems may have chains of poles approaching the imaginary axis. While several classes of these systems have been previously studied on a case-by-case basis, a unified method is proposed in this paper which allows to deal with all these classes at the same time. Approximation of poles of large modulus is systematically calculated based on a convex hull derived from the coefficients of the system. This convex hull also serves to establish sufficient conditions for instability and necessary and sufficient conditions for stability

The revision and extension of the R-MS ring for time delay systems

This paper is aimed at reviewing the ring of retarded quasipolynomial meromorphic functions (R-MS) that was recently introduced as a convenient control design tool for linear, time-invariant time delay systems (TDS). It has been found by the authors that the original definition does not constitute a ring and has some essential deficiencies, and hence it could not be used for an algebraic control design without a thorough reformulation which i.e. extends the usability to neutral TDS and to those with distributed delays. This contribution summarizes the original definition of RMS, simply highlights its deficiencies via examples, and suggests a possible new extended definition. Hence, the new ring of quasipolynomial meromorphic functions (R-QM) is established to avoid confusion. The paper also investigates and introduces selected algebraic properties supported by some illustrative examples and concisely outlines its use in controller design.European Regional Development Fund under the project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19

A comparison of possible exponential polynomial approximations to get commensurate delays

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.Ministry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme [L01303, MSMT-7778/2014]; European Regional Development Fund [CZ.1.05/2.1.00/03.0089

Gridding discretization-based multiple stability switching delay search algorithm: The movement of a human being on a controlled swaying bow

Delay represents a significant phenomenon in the dynamics of many human-related systems - including biological ones. It has i.a. a decisive impact on system stability, and the study of this influence is often mathematically demanding. This paper presents a computationally simple numerical gridding algorithm for the determination of stability margin delay values in multiple-delay linear systems. The characteristic quasi-polynomial - the roots of which decide about stability - is subjected to iterative discretization by means of pre-warped bilinear transformation. Then, a linear and a quadratic interpolation are applied to obtain the associated characteristic polynomial with integer powers. The roots of the associated characteristic polynomial are closely related to the estimation of roots of the original characteristic quasi-polynomial which agrees with the system's eigenvalues. Since the stability border is crossed by the leading one, the switching root locus is enhanced using the Regula Falsi interpolation method. Our methodology is implemented on - and verified by - a numerical bio-cybernetic example of the stabilization of a human-being's movement on a controlled swaying bow. The advantage of the proposed novel algorithm lies in the possibility of the rapid computation of polynomial zeros by means of standard programs for technical computing; in the low level of mathematical knowledge required; and, in the sufficiently high precision of the roots loci estimation. The relationship to the direct search QuasiPolynomial (mapping) Rootfinder algorithm and computational complexity are discussed as well. This algorithm is also applicable for systems with non-commensurate delays. © 2017 Pekař et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.CZ.1.05/2.1.00/19.0376, ERDF, European Regional Development FundMinistry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme [LO1303 (MSMT-7778/2014)]; European Regional Development Fund under the project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376

Stability of fractional neutral systems with multiple delays and poles asymptotic to the imaginary axis

This paper addresses the H∞-stability of linear fractional systems with multiple commensurate delays, including those with poles asymptotic to the imaginary axis. The asymptotic location of the neutral chains of poles are obtained, followed by the determination of conditions that guarantee a finite H∞ norm for those systems with all poles in the left half-plane of the complex plane. ©2010 IEEE