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

Received May 10, 2018, accepted June 7, 2018, date of publication July 2, 2018, date of current version July 19, 2018. Digital Object Identifier 10.1109/ACCESS.2018.2851453 Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays: A Literature Overview of Some Recent Results LIBOR PEKAŘ 1AND QINGBIN GAO 2 1Faculty of Applied Informatics, Tomas Bata University in Zlín, 76005 Zlín, Czech Republic 2Department of Mechanical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA Corresponding author: Libor Pekař ([email protected]) This work was supported in part by the European Regional Development Fund through the Project CEBIA-Tech Instrumentation under Grant CZ.1.05/2.1.00/19.0376 and in part by the National Sustainability Program Project under Grant LO1303 (MSMT-7778/2014). ABSTRACT 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, sampleddata, 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. INDEX TERMS Delay systems, eigenvalues and eigenfunctions, literature review, stability. I. INTRODUCTION Systems evincing delays or after-effect phenomenon emerge in a wide range of human activities and can be observed in many applications including but not limited to engineering, economy and biology. For instance, the cognitive delays represented by past states can be observed in predator-pray models [1] which are very early real-life-inspired time-delay systems (TDSs). The human nervous system represents another common biological example. The associated cognition-toreaction delay may cause balancing problems [2], [3] or traffic jams [4]. The effects of delays on the physiological systems were analyzed in [5]. Engineers from various industrial branches must deal with the after-effect phenomena in daily practice, such as delays in metallurgic processes [6], delays in the distribution of heat or power [7], [8], delays incurred by the mass flow in the production of sugar [9], etc. In machining processes, the milling dynamics models are usually represented by delay differential equations [10], and the machine tool chatters can be modeled by the so-called regenerative delays [11]. All in all, the modern world is full of various networks, with their different components communicating with one another, the processes of which are affected by the so-called communication latencies inevitably [12], [13]. Because of a very high interdependence with the everyday routine and practice, TDSs have attracted researchers and engineers since the nascence of modern systems and control theory. This is supported by the fact that the existence of delays may cause much worse dynamical properties of the system (especially, their stability and periodicity), and the attractiveness also increases because of that the inclusion of delays often leads to more realistic (infinite-dimensional) models but it significantly complicates the analysis and synthesis [14]. In the recent decades, an increasing number of international meetings and research articles have dealt with VOLUME 6, 2018 2169-3536 2018 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 35457 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays FIGURE 1. The total number of recently published research articles dealing with delay(s) listed in Scopus and WoS indexing databases. TDSs; it is worth noting that some important recent books summarize the up-to-date knowledge [15]–[22]. In addition, the top world’s leading scientific, research and development publication indexing databases, Elsevier’s Scopus and Web of Science (WoS) by Clarivate Analytics, would yield over 200 thousand results if delay is included in the searching title or keyword. Fig. 1 shows that the total number of such publications has been increasing over the past ten years, with an approximate annual increment of 14,000 publications. Therefore, it is almost impossible nowadays to address all the most recent advances in the field by only one person. The knowledge of the loci of the characteristic roots, i.e. system poles, constitutes crucial information about the systems’ stability and dynamics. It is well-known that a TDS has an infinite number of characteristic roots, i.e. its spectrum is infinite, and therefore the complete image of their loci is very hard to analyze, if possible at all. The methods and techniques of the spectral analysis of the TDSs deal with the aforementioned negative effects of the delays on the overall performance including stability and dynamics, and they have yielded many practical impacts and conclusions. Although researchers have found many new results on a variety of complex systems with different kinds of delays, including but not limited to time-invariant, time-variant, nonlinear, chaotic or discrete-time ones, even more studies have been published in the field of continuous linear time-invariant (LTI) TDSs which represents the most common yet still a very attractive class of TDSs. In spite of the relative simplicity of LTI-TDS models, an enormous number of books, conferences and journal papers indicates that there are many unsolved problems related to these systems. For example, the very famous work of Richard [23] provided an overview of various modeling, stability analysis and control approaches for LTI-TDSs and discussed some open problems as well, e.g. the control using delayed information. In addition, Gu and Niculescu [24] presented a broad overview of the stability and control of TDSs concerning practical problems and engineering applications. Various research related to literature overview in the field of TDSs has been done under different names with different aims. For instance, a delay jitter problem in the packet network based telephony was discussed in details in [25] where an overview of various attempting methods was also provided. Wang et al. [26] provided a survey on recent progress in the measurement of two types of the end-to-end latencies (round trip delay and one-way delay) over the Internet. Chen et al. [27] presented a systematic overview of the timedelay-estimation algorithms ranging from the simple crosscorrelation method to the advanced blind channel identification based techniques. Fruchard and Schäfke [28] provided an overview of the problem of bifurcation delay from its appearance in France at the end of the eighties to the most recent contributions. A research focused on the performance of domestic and international transport and logistics systems as perceived by Chinese importers and exporters was provided by Zhang and Figliozzi [29] along with a broad literature review of the meteoric logistics industrial development in China. A survey on the virtual-environment-based and bilateral teleoperations of space robots with time delay effects was addressed in [30]. Wang [31] presented a review of the recent advances in delay-time-based maintenance modeling, which is one of the mathematical techniques for optimizing inspection planning and related problems. A step-bystep introduction to the notion of time-delay in classical and quantum mechanics, aiming at clarifying its foundation at a conceptual level, was made by Sassoli de Bianchi [32]. Cui et al. [33] gave a survey on several major systematic approaches in dealing with delay-aware control problems, namely the equivalent rate constraint approach, the Lyapunov stability drift approach, and the approximate Markov decision process approach using stochastic learning. The research of Wang et al. [34] focused on a review of some design and tuning methods of active disturbance rejection control methodology for TDS with its applications as well. Flunkert et al. [35] provided an overview of the effect of delayed coupling and feedback on dynamical systems. A comprehensive and excellent survey on stochastic hybrid systems by Teel et al. [36] addressed results on the stability of delayed ones as well. Similarly, Tao [37] included TDSs into his overview of some fundamental theoretical aspects and technical issues of the multivariable adaptive control. Doudou et al. [38] provided a comprehensive review and taxonomy of the state-of-the-art synchronous medium access control protocols with respect to sender-receiver latencies. Lu and Shen [39] provided an overview of the development in the area of scaling laws for the throughput capacity and the delay in wireless networks. Liu and Yang [40] summarized the progress of grey system research from 2004 to 2014 including the problems of robust stability for grey stochastic time-delay systems of neutral type, distributed-delay type and neutral distributeddelay type. An overview of the basic results and methods for the stability investigations of higher-order autonomous linear difference equations, with a special emphasis on delay differenceequationswaspublishedbyČermák [41].Zhu et al. [42] presented a variety of stability conditions for TDSs with 35458 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays varying delays in the form of scaled small-gain conditions. In the broad introduction in [43], Liao et al. summarizes the recent advances in optimal, robust and preview control for systems with input delays. Mahmoud [44] provided an overview of the research investigations in the field of networked control systems (NCSs) that include discussions on addressing communication delays. A similar topic was reviewed in [45], where the authors focused on time-delay fuzzy-model-based nonlinear NCSs as well. To name just one publication example from medicine, Wechkunanukul et al. [46] provided a summarized review on a range of countries describing the differences in time taken to seek medical care for chest pain and the factors which contribute to delay times. Besides journal review articles, several books summarizing the up-to-date knowledge about TDSs [15], [16], [19], [47], and those representing a compiled set of selected papers from the most important control conferences and specialized workshops within the research area [17], [18], [20]–[22] have been published as well. Currently, a family of LTI-TDSs is lacking a holistic literature overview that covers and summarizes most of the spectral analysis techniques for systems with constant (known or unknown) delays. This survey paper aims to provide the reader with a literature overview of such recently published results. The exigency of such an up-to-date survey can be considered as the greatest contribution of this work. However, because of a huge number of published results in this area it does not allow to cover all the ideas, therefore this study does not claim to be exhaustive. It should be noted that this study is primarily not interested in controloriented studies, and that the reader are referred to the cited literature for more details – especially, regarding complete mathematical issues. Throughout the paper, C,R,Zand Ndenote the set of, respectively, complex numbers, real numbers, integers and non-negative integers. The closed unit disk, the unit circle and the imaginary axis are denoted as D,T, and C0, respectively. Rndenotes the n-dimensional Euclidean space (a column vector), Rn×mis the set of all real-valued matrices of the dimension n×m. For s∈C, Re(s)and Im(s)denote, respectively, the real and imaginary parts of s;C− 0:= {s∈C|Re(s)<0}, C+=C\C− 0, the set of real polynomials is denoted as R[s]. The superscript T stands for the matrix transpose. For F(s):s∈C7→ F∈C, define sets H2:= F(s):Z∞ −∞ F(jω)F(jω)dω < ∞, H∞:= (F(s):kF(s)k∞:= sup s∈C+|F(s)|<∞), and let Lp:= (f(t):Zt 0|f(t)|pdt1/p <∞,p∈N) for f(t):t∈R7→ f∈R. We use L(·)for the Laplace transform of (·). The unit matrix of the dimension nis denoted as In. The rest of the paper is organized as follows: In the preliminary Section II, the research field of LTI-TDSs is specified and their basic spectral and stability properties are introduced. Section III concisely explains the methodology of the literature review used herein the study. Section IV provides a brief overview of some earlier important and famous methods in the field of the spectrum analysis of LTI-TDSs and related stability issues. The main contribution of the paper is given in Sections V and VI. In Section V, a detailed review of recently published papers with a theoretical contribution to pole loci is presented. Section VI includes selected recent results on the stability analysis based on the knowledge of the spectrum. In Section VII recent papers on related practical problems and engineering applications are outlined. Section VIII lists our research questions and points out unexplored areas inthe field of spectrum analysis of LTI-TDSs aiming at providing directions for the future research. Finally, Section IX concludes the paper. II. PRELIMINARIES In this section, LTI-TDSs are defined to determine the field of study; then their essential spectral properties and stability issues related to the eigenvalue spectrum are introduced. A. LTI-TDS MODEL A general continuous-time LTI-TDS with constant delays can be formulated by state and output functional differential equations (FDEs) as ˙ x(t)+XnH i=1Hi˙ xt−τH,i+ZL 0 Hd(τ)˙ x(t−τ)dτ =A0x(t)+XnA i=1Aixt−τA,i+B0u(t) +XnB i=1Biut−τB,i+ZL 0[Ad(τ)x(t−τ) +Bd(τ)u(t−τ)]dτ y(t)=Cx (t)+XnC i=1Cixt−τC,i +ZL 0 Cd(τ)x(t−τ)dτ(1) where x(t)∈Rn,u(t)∈Rm,y(t)∈Rl, stand for the vector of state variables, inputs and outputs, respectively, ˙ x(t)= dx(t)/dt,Ai,Ad(τ),Bi,Bd(τ),C,Cd(τ),H,Hd(τ)are real-valued matrices of compatible dimensions, 0 < τ·,1< τ·,2< . . . ≤Lexpress lumped (point-wise) delays and convolution integrals characterize distributed delays. If τ·,i= n·,ihfor all iwhere n·,i∈N, with the base delay, h∈R, delays are called commensurate. The initial condition is a function segment x(θ)=ϕ(θ), θ ∈[−L,0],ϕ∈`([−L,0],Rn), where `means the Banach space of continuous-time functions mapping θ∈[−L,0]into Rnequipped with the supreme norm k·ks. For t≥0, denote xt=xt(θ, ϕ)= x(t+θ), θ ∈[−L,0]for the initial data ϕ. If ∃iso that VOLUME 6, 2018 35459 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays Hi6= 0or Hd(τ)6= 0, the system is of neutral type; otherwise, it is of retarded type. For the sake of this review, retarded and neutral systems without distributed delays are abbreviated as RTDSs and NTDSs, respectively, and those with (non-lumped) delay distribution let be dRTDSs and dNTDSs, respectively. Note that an LTI-TDS can also be represented in the context of the Hilbert space as follows. The homogeneous state equation has the form ˙ x(t)=_ `(t)xt,t≥0 where _ `(t): R+×X→Rn,X=([−L,0],Rn)with Xbeing the Hilbert space of continuous functions on the interval [−L,0]. For further details, the reader is referred e.g. to [48]. By using the Laplace transform, one can construct the characteristic function 1(s)=det      sI+PnH i=1Hiexp−sτH,i +RL 0Hd(τ)exp(−sτ)dτ −A0−PnA i=1Aiexp−sτA,i −RL 0Ad(τ)exp(−sτ)dτ       (2) For RTDSs and NTDSs, (2) is a quasipolynomial; however, due to distributed delays it takes a form of a quasipolynomial fraction where some numerator/denominator roots can be mutually algebraically canceled. Systempoles (eigenvalues or characteristic values) constituting the spectrum 6of (1) satisfy 6:= {s:1(s)=0}.(3) Let the numerator (if applicable) of 1(s)be denoted as ¯ 1(s):= Pn1 i=0di(s)siwhere diare exponential polynomials and n1≥n. Then dn1(s)=dn1,0+P˜n i=1dn1,iexp(−s˜τi) with dn1,·∈R,˜n≥nH,˜τi≤L+PnH i=1τH,iconstitutes the associated characteristic exponential polynomial. The essential spectrum of a (d)NTDS reads 6ess := s:dn1(s)=0.(4) B. BASIC SPECTRAL PROPERTIES Let us introduce some very basic properties of 6,6ess as follows, leaving proofs to references: Proposition 1 [14]–[16]: For a (d)RTDS it can be deduced that: (i) If the numerator of ¯ 1(s)is a quasipolynomial not a polynomial, then |6|= ∞; (ii) All sk∈6are isolated; (iii) There are only finitely many characteristic roots sk∈ 6in the strip {β1<Re(sk)< β2}⊂C; (iv) For any β∈Rwith β < 0, only finitely many poles are located in the half-plane Res > β, while infinitely many ones are located to the left-hand side of Res =β; (v) Isolated poles behave continuously and smoothly with respect to τand parameters on C. Proposition 2 [14], [24], [48], [51]: For a (d)NTDS the following statements are true: (i) Let sk∈˘ 6⊆6,se,l∈˘ 6ess ⊆6ess with |sk−1|<|sk|, se,l−1<se,l. Then for every ε > 0, there exist K,L, such that sk−se,l< ε for k>K,l>L. It means that both system and essential spectra constitute vertical strips at high frequencies which converge to each other; (ii) Define γ:= sup{Re(6ess)}, then limk→∞ Im(sk)= ∞for |sk−1|<|sk|,sk∈˘ 6as limk→∞ Re(sk)=γ; (iii) In the half-plane with Re(s)> γ , there may lie infinitely many system poles; (iv) The value of γis not continuous with respect to τ (where τ∈Rnτ>0 represents the vector of all system delays; (v) Define ˜γ:= sup{γ(τ+δτ): ∀kδτk< ε, ε > 0}, the safe upper bound estimation ¯c(that is continuous with respect to delays) on ˜γcan be calculated from ¯c=c∈R:X˜n i=1 dn1,i dn1,0 exp(c˜τi)=1,(5) see e.g. [52]. Only a finite number of poles is located in the half-plane with Re(s)>¯c≥ ˜γand they are isolated. The spectral abscissa is the function α(p):= p7→ sup {Re6}(6) where pis a system parameter (including delay). Proposition 3 [53]: For α(τ)of a (d)NTDS (1) the following two claims are valid: (i) The function may be non-smooth and hence not differentiable; e.g. in points with more than one real pole or conjugate pairs with the same maximum real part; (ii) It is non-Lipschitz; for instance, at points where the maximum real part has multiplicity greater than one. C. LTI-TDS STABILITY In this subsection, stability issues of delayed systems, which are closely related to their eigenvalue spectrum, are concisely addressed. Tasks of stability analysis as well as those attempting to guarantee the stable control system constitute the primary problems solved when dealing with the system spectrum. Definition 1 [14]–[16], [48], [54]: The system (1) (or more precisely, its null solution) is said to be stable, if for any ε > 0, there exists δ(ε)>0 such that kϕks:= maxθ∈[−L,0]kϕ(θ)k< δ implies that kx(t)k< ε for any t≥0 where k·kdenotes the Euclidean norm. If, moreover, for any ϕ∈`([−L,0],Rn)holds that limt→∞ x(t)=0, the system is asymptotically stable. System (1) is said to be exponentially stable if there exist a>0, µ > 0 such that kxt(θ, ϕ)ks≤aexp(−µt)kϕks,∀t≥0 for all ϕ∈ `([−L,0],Rn). Proposition 4 [14], [15], [24], [48]: A (d)RTDS (1) is (i) asymptotically and exponentially stable if and only if α(·)<0, (ii) stable if and only if α(·)≤0 and for any pole sk∈C0 it holds that rankskI−A0−PnA i=1Aiexp−skτA,i −RL 0Ad(τ)exp(−skτ)dτ=n−qk(7) where qkis the algebraic multiplicity of sk. 35460 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays Whereas asymptotic and exponential stability coincide for a RTDS, it is not the same case for a NTDS. The most delicate is the case of poles in one chain of neutral-type given by (ii) in Proposition 2 shaped asymptotically to the imaginary axis yet located in C− 0(with no counterpart in C+) Proposition 5 [14], [48], [51], [55]: For a (d)NTDS given by (1), the following statements hold: (i) It is exponentially stable if and only if there exists ε > 0 such that α(·)<−ε; (ii) If α(·)<0 and there is a critical chain of poles close to C0, asymptotic stability may or may not occur. For instance, if H16= 0,Hi=Hd=0,i>1, it depends on geometric and algebraic multiplicities of eigenvalues of H1. The family of (d)NTDSs is equipped with the following two specific stability issues. Definition 2 [16], [56]: A (d)NTDS is said to be formally stable if rank"I+XnH i=1Hiexp−sτH,i +RL 0Hd(τ)exp(−sτ)dτ#=n,∀s∈C+.(8) It is said to be strongly stable if ˜γ < 0. To rephrase the definition, formal stability means that system (1) has only a finite number of poles in C+, whereas the strong one expresses that it remains formally stable under small delay deviations (see items (iv) and (v) of Proposition 1 and Proposition 2). Other commonly concerned stability issues, such as BIBO (Bounded-Input Bounded-Output) and H∞stabilities cannot be investigated solely from pole loci. Note that, in the singleinput single-output (SISO) case – for the simplicity – BIBO means that |u(t)|<M1,M1>0 implies |y(t)|<M2, M2>0 for t≥0; or equivalently, the system impulse response g(t)∈L1. The system is H∞stable if its transfer function satisfies G(s)∈H∞(i.e. it is a bounded analytic function in C+); or equivalently, u(t)∈L2implies y(t)∈L2. For instance, let G1(s)=1/(s+sexp(−s)+1),G2(s)= G1(s)/(s+1),G3(s)=G1(s)/(s+1)4. All the three transfer functions have poles asymptotically approaching C0 from the left at infinity. It can be proved that G1/∈H∞, yet G2,G3∈H∞, and G1,G2are not BIBO stable, unlike G3[57]. Definition 3: System (1) is said to be (weakly) delayindependent stable (DIS) if it remains stable for any τ∈ Rnτ>0. It is said to bedelay-dependent stable (DDS) if it is stable for a disjoint set Iτ,iof regions in the delay space. Note that DIS and DDS according to Definition 3 are usually considered in terms of exponential or asymptotic stability. III. LITERATURE REVIEW METHODOLOGY The purpose of a systematic review is to summarize the best available research on a particular topic. It should transparently collect, analyze and synthesize the results of relevant research. As stated above, the authors do not dare to call this survey as systematic due to the enormous number of results on the addressed topic. However, they attempted to follow the principles of the systematic review. Four steps of the literature review are implemented in this research: planning, searching, screening, and extraction. In the planning phase, the problem to be addressed is specified in the form of clear research questions. Based on the preliminary section the following research questions are considered here: What is the current status of research on the spectrum analysis for LTI-TDSs with constant delays? What are the open problems in this field? In early 2018, research papers related to the spectrum analysis for LTI-TDSs with constant delays were searched and collected by the authors. Note that nonlinear, time-varying, stochastic, non-integer-order systems, etc., are not within the scope of this study. Inclusion and exclusion criteria are defined and applied to the found results in the screening phase. To present upto-date results, the publication period was determined to be within the last five years (i.e., from 2013 to 2017). It is noteworthy that the authors of this review are by no means intended to claim that earlier results are less important – yet, this paper is primarily aimed at giving an overview of some most recent results. However, some others (out of this period) of critical importance (i.e. those not within the period) are also included when appropriate, and a brief overview of the famous earlier methods is presented as well. A total amount of 137 results covered by the aforementioned period have been finally selected to provide some significant insights into the considered research questions, based on the screening of abstracts and the linkage of particular cited papers and citing sources; the relevance to the topic of this review has been the most important selection criterion. For instance, methods based on Lyapunov stability theory and linear matrix inequalities (LMIs) are mostly out of this scope of this study since they usually provide only the estimate of the exponential decay, i.e. the spectral abscissa, not a deeper insight into the spectrum. Finally, in the extraction phase, open research questions arising from the analysis of selected papers are concisely introduced and discussed (see Section VIII). IV. SOME WELL-ESTABLISHED METHODS OF LTI-TDS SPECTRUM ANALYSIS In this section, we briefly summarize selected significant and famous well-established direct methods on LTI-TDS spectrum analysis and related stability issues. A. SPECTRUM APPROXIMATION BY SPECTRAL AND PSEUDOSPECTRAL METHODS This methodology attempts to estimate the infinitedimensional spectrum by a sufficiently accurate finitedimensional one by means of a discretized state-space formulation (i.e., the discretization of the so-called solution operator or the infinitesimal generator). VOLUME 6, 2018 35461 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays The solution operator (t)is defined by the relation (t)ϕ=xt(ϕ),t≥0.(9) The infinitesimal generator :D( ) ⊆`→`of (t) has domain D( )=ϕ∈`, ϕ0:=dϕ(θ)/dθ∈`:Fl(ϕ, 0)=Fr(ϕ, 0) (10) where Fl(ϕ, 0),Fr(ϕ, 0)express left-hand and right-hand sides, respectively, of the first equality in (1) where xis substituted by ϕ,t=0 and u(·)≡0. The generator acts as ϕ=ϕ, ϕ ∈D( ). Then (1) can be treated as an abstract Cauchy problem given by the following operator (ordinary) differential equation ˙ xt=xt,t≥0 x0=ϕ. (11) It holds that sk=1 tlnµ, µ ∈σ(t)/{0} sk∈σ( ).(12) where sk∈6, and σ(·)denotes the matrix spectrum. Hence, the problem can be transformed into a suitable matrix discretization of the solution operator or its infinitesimal generator. Many of these methods discretize xtby a finitedimensional approximation in the form of a block vector Xt∈`Nwith components ˜ xtθN,iwhere `Nis the space of discrete functions defined on the grid N= θN,i,i=0,1,...,N,θN,0=0, θN,i> θN,i+1,θN,N= −L, and ˜ xtis the approximation of xt. Then (11) can be approximated as ˙ Xt=ANXt AN∈Rn(N+1)×n(N+1).(13) and the solution operator (9) is usually computed by the following one-step approximation Xt+1t=TN(1t)Xt,TN∈Rn(N+1)×n(N+1) 1t=θN,i+1−θN,i.(14) where different techniques are used to determine ANor TN(1t), see e.g. [14], [58]–[60]. B. FREQUENCY-DOMAIN APPROACHES TO GET FINITE-DIMENSIONAL MODEL REDUCTION Whenever an LTI model (1) is represented by the transfer function, one may apply a rational approximation of exponential terms to obtain a finite-dimensional model, the spectrum of which can be easily computed [61], [62]. However, some artificial roots appear after the approximation. For instance, the nth order Padé, diagonal Padé, Laguerre and Kautz shift approximations can be expressed as exp (−τs)≈ p(−s)/p(s)where, respectively, p(s)=Xn k=0n k(2n−k)! (2n)!(τs)k, p(s)=Xn k=0 (2n−k)! k!(n−k)!(−sτ)k, p(s)=lim n→∞1+τs 2nn . p(s)=1+τs 2n+1 2τs 2n2n . Theses approximations play an essential role in some other delay approximation concepts. For instance, a famous result based on the Trotter-Kato approximation theorem [63] for strongly continuous semigroups was developed in [64]. Using this general framework, two families of particular approximation schemes were constructed. Approximation of the state is done by functions which are piecewise polynomials on a mesh (m-th order splines of deficiency m). C. LAMBERT W FUNCTION The use of the Lambert W function yields the analytical solution of the pole loci. However, it is only applicable to systems with commensurate delays. The Lambert function W(z)is a multivalued complex function defined as z=W(z)exp(W(z)) (15) where its solution constitutes an infinite set of branches and zcan be a scalar or a matrix [67]. We refer to Wk(z)as the k-th branch of the Lambert W function of z. The main idea is to express the solution of (3) by means of the Lambert W function. For instance, let the scalar system be ˙x(t)= a0x(t)+a1x(t−L), then the characteristic equation can be written as (s−a0)exp(sL)=a1which is equivalent to the identity L(s−a0)=W(a1Lexp(−a0L)), i.e., z= a1Lexp(−a0L), and hence the solution of the characteristic equation reads s=W(a1Lexp(−a0L)) /L+a0. D. ARGUMENT PRINCIPLE TECHNIQUE If the task is to decide on the number NDof poles located inside a region D⊂Cgiven by the closed positive Jordan curve ϕ+, it holds that ND=1 2πjZ φ+ 10(s) 1(s)ds=1 2π1φ+arg1(s),(16) where 1(s)is a retarded characteristic quasipolynomial, and 10(s)=d1(s)/ds[65]. From (16) it follows that, if 1(0)> 0, 1(s)6= 0 for any s=jω, ω ≥0, then =n 2−1 π1arg1(s) s=jω,ω∈[0,∞) where means the number of poles in C+. In [66], the result was extended to NTDSs as follows: Consider a strongly stable 35462 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays system with 1(0)>0, 1(s)6= 0 for any s=jω, ω ≥0, then (1) is asymptotically stable if and only if nπ 2−8≤1arg1(s) s=jω,ω∈[0,∞)≤nπ 2+8, 8=arcsinX˜n i=1dn1,i. As clear from Proposition 4 and Proposition 5, exponential stability can effectively be studied by the determination of system (or even delay) parameters such that its spectrum crosses imaginary axis. A bunch of well-established techniques follows. The techniques rely on two important fundamental theoretical results; namely, the associated D-decomposition/τ-decomposition theorem [68], [69], and the continuity property of system eigenvalues with respect to system parameters [70]. By D-decomposition/ τ-decomposition theorem, one can find countably many regions in the parameter space, where in each region the system can possess only a finite number of . The system is stable for all the points in that region whenever =0, and the boundaries that separate these regions are formed by some critical parameter values that impart imaginary-axis poles. E. REKASIUS SUBSTITUTION Rekasius [71] proposed the following exact (unlike the Padé one) transformation implemented to calculate the stableunstable regions of an LTI-TDS exp(−τijω)→1−jTiω 1+jTiω,i=1,2, . . . nτ(17) where Ti∈Rare so-called pseudo-delays. As a consequence, a new characteristic polynomial in ωparameterized by Tiis obtained.Thesubstitution(17)isvalid and exactifτcomplies with τi,k=2ω−1tan−1(ωTi)+kπ,k∈Z.(18) That is, every pair satisfying (17) with s=jωis mapped to delays according to (18). These points bisect the delay parameter space into intervals, where in each interval the system is either stable or unstable. F. ELIMINATION OF TRANSCENDENTAL TERMS IN THE CHARACTERISTIC EQUATION (DIRECT METHOD) The core idea of the direct method [72] lies in the iterative elimination of exponential terms in 1(s)with commensurate delays based on the fact that whenever 1(jω)=0, it also holds that 1(−jω)=0 (the complex conjugate symmetry of complex roots). Let 1(s)=P2 i=0di(s)siexp(−ihs),h>0, then construct 1(1)(s):= d0(−s)1(s)−d2(s)1(−s)exp(−2hs) =d0(−s)d0(s)−d2(−s)d2(s) +(d0(−s)d1(s)−d1(−s)d2(s)) exp(−hs) =: d(1) 0(s)+d(1) 1(s)exp(−hs).(19) The second iteration reads 1(2)(s):= d(1) 0(s)d(1) 0(−s)−d(1) 1(s)d(1) 1(−s) =: d(2) 0(s).(20) Since (20) has no exponential terms and it holds that the elimination procedure preserves the imaginary solutions sc= ±jωcof the original characteristic equation, one can write the following polynomial equation to get the imaginary poles Wω2 c=1(2)(jωc)=0.(21) The set of potential base delays is then computed by hk=ω−1 c tan−1  Red(1) 0(jωc)/d(1) 1(jωc) Im−d(1) 0(jωc)/d(1) 1(jωc) +2kπ , k∈N0.(22) The crossing property is then checked by a nonzero RT value calculated via RT =sgn nd(1) 0(jωc)dWω2 c/dω2 co. G. FREQUENCY SWEEPING This is very simple yet effective technique to determine stability border in the parameter (or delay) space see e.g. [73]. However, it can be used only if 1(s|p)is linear with respect to the unknown parameters p. The leading idea was based on the partition of 1(s,p)|s=jωinto real and imaginary parts, i.e., Re(1(jω, p)) and Im(1(jω, p)), respectively. Then, the common solution of Re (1(jω, p)) =Im(1(jω, p)) =0 can be plotted in the parameter space for ω∈[0, ωmax], where ωmax means a particularly selected maximum frequency. The generated plot in the parameter space represents a potential stability boundaries and they determine regions that can be further tested in order to identify the stable ones (for instance, via the D-decomposition procedure). H. SCHUR-COHN PROCEDURE This procedure can be used to compute the stability margin for LTI-TDSs with commensurate delays [15], [74]. The original Schur-Cohn procedure computes the determinant of a partitioned matrix S=31(s)32(s) 3∗ 2(s)3∗ 1(s) where 31, 32∈Cn×n[s]are appropriate matrices over the ring of polynomial in s, and the asterisk denotes the particular Hermitian. In order to determine all the imaginary roots of the characteristic quasipolynomial 1(s)with the base delay τ0and the commensuracy degree nC, the variable q=exp(−τ0s) is introduced and 1(s)is rewritten as a polynomial p(s,q) in two unknowns over R. Then, the Schur-Cohn criterion solves the problem of computing the values of ωsuch that p(jω, q)=0 by multiplying p(jω, q)by q−i,i= 0,1, . . . nC−1, and the complex conjugate ¯p(jω, q)by qi,i= 1, . . . nC. This yields a system of 2nChomogenous equations with the unknowns qi. VOLUME 6, 2018 35463 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays I. KRONECKER SUM AND MATRIX PENCIL METHOD To introduce these techniques concisely, let A,B∈Rn×n, then the spectrum of the Kronecker sum A⊕B=A⊗I+I⊗B consists of all sums in the form of λ+µ, where λ, µ belong to the spectrum of A,B, respectively, and Iis the identity matrix [15], [74]–[76]. For instance, consider the following simple system ˙ x(t)=A0x(t)+A1x(t−L).(23) If some sk,−skare zeros of 1(s)for (23), then skbelongs to the spectrum of A0+A1z, while −sklies in the spectrum of A0+A1z−1, where z=exp(−sL); hence 1ass,1(z):= det(A0+A1z)⊕A0+A1z−1=0.(24) If skis a purely imaginary system pole, (24) must hold. The transcription of the Kronecker sum into the matrix pencil form reduces the problem to the existence of the unit circle of generalized eigenvalues of the corresponding matrix pencil. Crossing delays are then given by τk= ω−1(arg(z)+2kπ),k∈Z. J. KRONECKER MULTIPLICATION METHOD The main result that characterizes this framework can be expressed by the following theorem. Theorem 1 [77]: Let 5be the set of all eigenvalues of the matrix 5=000 −32−31∈Rn2×n2(25) where 00=I⊗I,01=I⊗A0−A0⊗I, 02=A1⊗A1−A0⊗A0, 31=0−1 001,32=0−1 002.(26) Then ⊆5where := Im6∩C0. Last but not least, let us introduce two well-established frameworks from the field that enables to decide about the exponential stability, i.e., whether all characteristic roots are located in C− 0. K. EXTENDED HERMITE-BIEHLER THEOREM The famous work of Bellman and Cooke [78] applied the extension of the Hermite-Biehler theorem to RTDSs based on the earlier work of Pontryagin [79]. This theorem yields the necessary and sufficient conditions for the stability of 1(s). Let 1∗(s)=1(s)exp(sL)(having the same roots as 1(s)), 1∗ R(s)=Re(1∗(s)),1∗ I(s)=Im(1∗(s)), then the system is stable if and only if 1∗ R(s)and 1∗ I(s)have only simple real roots, these roots interlace, and 1∗ I(ω0)01∗ R(ω0)− 1∗ I(ω0)1∗ R(ω0)0>0 for some ω0∈(−∞,∞). The crucial problem is to make sure that 1∗ R(s),1∗ I(s)have only real roots, see e.g. [80] for details. L. CLUSTER TREATMENT OF CHARACTERISTIC ROOTS The famous Cluster Treatment of Characteristic Roots (CTCR) paradigm was developed more than a decade ago, see e.g. [81], and it originally consists of the following basic steps: First, the characteristic quasipolynomial is transformed to the corresponding polynomial p(ω, T) including pseudo-delays Ti(via the Rekasius substitution). Second, the eventual polynomial is subjected to the Routh array scheme to get the so-called potential stability switching hypersurfaces (or, the kernel curves in the 2D delay space) defined as ℘0(τ):= ττ:1(s, τ)=0,s=jω, ω ∈c(27) where τrepresents all kernel points in the delay space and cmeans the set of all possible corresponding crossing frequencies. The necessary crossing condition is that the root tendency RT := sgnReds/dτi|s=jω,i=1,2, .., nτ, is nonzero, where nτdenotes the number of independent delays in 1(s). As the third step, the kernel curves (or, potential stability switching hypersurfaces)together with the crossing frequencies give rise to the so-called offspring hypersurfaces ℘off (τ)which are generated based on the knowledge that whenever there exists an imaginary pole sc= ±jωc for some τ0, the same pole exists for τi=τ0,i+2πki/ωc, τ0,i−2π/ωc≤0, i=1,2, .., nτ;ki∈N, as well. Finally, the D-subdivision method [68] is deployed to determine the number of unstable poles on the right half-plane, starting from the delay-free case. V. OVERVIEW OF RECENT THEORETICAL RESEARCH ON POLE LOCI CALCULATION, COMPUTATION AND APPROXIMATION This section is dedicated to the review of recent theoretical research on the analysis of the eigenvalue (characteristic roots) spectrum 6of LTI-TDSs with constant parameters and delays. Methods for the root loci computation or approximation and those determining a part of 6within the specified region in Care presented. Methods for the computation of the characteristic roots covered herein can be divided into the following categories: (i) Numerical methods based on the approximation of the solution operator associated with the system (1) or its infinitesimal generator via spectral (or pseudospectral) method; (ii) Semidiscretization and full-discretization methods; (iii) Numerical integration and differential quadrature methods; (iv) Contour integral method; (v) Lambert W function; (vi) Special numerical, semi-analytic and analytic methods. A. SPECTRAL AND PSEUDOSPECTRAL METHODS In the literature, Wu and Michiels [60] summarized the methodology introduced in (9)-(14) to get ANconcerning the 35464 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays computation of all characteristic roots in a given right halfplane and they also provided a procedure for the automatic selection of N; Chebyshev polynomials of the first kind were utilized to approximate xtby ˜ xt– the so-called pseudospectral collocation (PsC) method is then obtained (the term pseudospectral means that the solution is approximated in a finite dimensional subspace). Another point of view was presented by Lehotzky and Insperger [48] where a detailed comparison of a family of weighted residual-type methods to approximate the operator differential equation (11) in the Hilbert space (see after (1)) was provided. In more detail, let the solution of (11) be approximated by ˜ xt(θ)=PNa j=1φj(θ)aj(t)where θ∈ [−L,0], and aj(t)are unknown variables spanned by the basis φjNa j=1, then the residual function reads rt(θ)=˙ ˜ xt− ˜ x0t. The objective is to determine aj(t)to get ˜ xt→xtas close as possible. Methods of weighted residuals yield rt, ψj:= R0 −Lrt(θ)ψj(θ)dθ=0, j=1,2,...,Na[48], [82], where ψj(θ)are test functions, which can also be represented in a matrix form as N˙ a(t)=Ma (t).(28) where a(t)=aj(t)Na j=1,N,M∈RnNa×nNa, and entries of these matrices include inner products φj, ψj. In the so-called pseudospectral tau approximation (PsT), the rough solution segment is given by ˜ xt(θ)=XNa j=1φj(θ)xt(θ).(29) where φj(θ)are Lagrange base polynomials, i.e. φjθN,i= 1 for i=j, else φjθN,i=1. The barycentric formula of Lagrange polynomials used in [48] is more numerically stable and derivatives of φjare then less complicated. The PsC method introduced above (also called the Chebyshev spectral continuous-time approximation) reduces the error ˜ xt(θ)−xt(θ)by means of the selection of Chebyshev nodes for the nodes of interpolation in (29). The spectral Legendre tau (SLT) method employs Legendre polynomials as base functions in (29). The tau approximation (TA) uses (29) to get (28) in the form of (13) where a time-dependent matrix G(t) appears instead of AN. This matrix contains φj, ψjwhich can be approximated by using quadrature methods. Unlike the aforementioned methods (PsC, PsT, SLT, TA), the spectral element (SE) method (called the time finite elements method as well), which was also included in the comparative study [48], approximates (t)to obtain the discrete mapping. The idea lies in the splitting of the history segment [−L,0]into Enumber of temporary elements with the length 1t. Each element contains Nsinner points. An approximate solution is then sought in each element according to ˜ xk(t)=XNs j=1φj(t)xktk,j,t∈[−k1t,−(k−1)1t], k=0,1,...,E(30) where tk,j∈[−k1t,−(k−1)1t]are inner time instants and φjrepresent trial test functions. Then (12) is used to compute the system spectrum. Note that in [48], φjare Lagrange base polynomials, and the abovementioned principles were compared inter alia by using the loci of the rightmost poles here. Vyasarayani et al. [83] compared the spectral tau (ST) and the spectral least-square (SLS) methods. The former one computes N,Min (28) simply using N= R0 −Lϕ(θ)ϕT(θ)dθ, M=R0 −Lϕ(θ)ϕ0T(θ)dθwhere ϕ= φ1, φ2,...φNaT; whereas, the latter one attempts to solve the following constrained optimization problem min 0.5˙ a(t)Z0 −L r2 t(θ)dθ s.t. FlϕT,0˙ a(t)=FrϕT,0a(t).(31) Base functions φjwere considered as mixed Fourier basis, shifted Legendre polynomials and shifted Chebyshev polynomials, in [83]. The authors compared the methods via pole loci and stated that the ST method is easy to code and understand, and performs better than the SLS method. To overcome computational burdens of the eigenvalues associated with the sparse AN, an iterative PsC method for RTDSs was presented by Ye et al. [84]. The sparsity of ANis explored by reformulating its blocks into Kronecker products as follows AN=RN MN⊗In(32) where RN=PnA i=0Li⊗Aiwith LT i∈RN+1being constant Lagrange vectors. Then, the shift operation ˜s= s−ssh,ssh >0, is utilized to get eigenvalues of matrix ˜ ANwith the largest modulus, followed by the computation of ˜ AN−1. This shift-invert preconditioning transformation technique is used for sparse eigenvalue computations with a reduced dimension by means of the implicitly restarted Arnoldi algorithm (IRA) [85] to get Krylov sequences {qk} via qk+1=(AN)−1qk. A different technique was presented by Ye et al. [86] where ˜ AN−1=(0N)−15N, in which 5Nis a highly sparse companion-type constant matrix and ˜ 0N=eT 1⊗In−PnA i=0Li⊗Ai. In a similar manner, a pseudospectral discretization of T(t)was published in [87]. The authors utilize a technique introduced in [88]; however, matrices forming T·(1t)are reformulated by using Kronecker products to reduce their dimensions. Then, the rotation-andamplification operation ˜s=αexp (−θj)s, α > 1, θ > 0, is made prior to the generation of Krylov vectors via the IRA, to accelerate its convergence rate. This operation implies that system poles are first rotated by θand then amplified by α. Fabiano [89] suggested the approximation of A(therein called a semidiscrete approximation) for a NTDS via linear spline functions where he proved Trotter-Kato type semigroup convergence [63] for this scheme as well. The same author used this scheme to investigate the DIS problem in [90]. In both papers, the exact eigenvalues of ANwere used to measure the accuracy of the approximation. VOLUME 6, 2018 35465 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays FIGURE 2. G-sector and S-sector [159]–[161]. done via advanced computer algebra technique, the so-called rational univariate representation [158], which is a one-to-one mapping between the solutions of the polynomial system and the roots of a univariate polynomial. It is worth noting that the following form of the Möbius transformation (mapping ω∈R∪{∞}to the unit circle T:= {x∈C:|x|=1}) was used exp(−τjω)→x−j x+j.(48) The critical pairs can be computed by solving the following identities Re{p(ω, x)}=Im {p(ω, x)}=0 where p(ω, x)is the bivariate polynomial corresponding to 1(s). The critical delays are then obtained as follows τk=ω−1tan−12x x2−1+kπ,k∈Z.(49) Note also that the well-known Rekasius substitution (29) is obtained by the setting x= −Tω, T∈R. An alternative Möbius mapping reads exp(−τjω)→z=u+jv∈T. The latter sub-result of [157] included an efficient algorithm to compute the different terms in the Puiseux series. The movement of double, triple and quadruple imaginary poles when two delays are subjected to small deviations was analytically studied without using the Puiseux series in [159]–[161], respectively. Two sectors, great (G) and small (S), were introduced in the neighborhood of the point in the delay parameter space that causes the multiple imaginary root, see Fig. 2. This critical point constitutes a cusp in the stability crossing curve. When the delay parameters move into the G-sector, one root (two roots) move(s) to C+, and the other one (two others) move(s) to C−for the double (quadruple) imaginary pole. If the parameters move into the S-sector, then one (three) of the roots move(s) to one halfplane, and the remaining root moves to the other half-plane. For the triple pole, it was proved in the cited paper that the stability crossing curves are smooth - two roots move to one half-plane and one root goes to the other half-plane. B. H∞AND BIBO STABILITY As mentioned above, Bonnet et al. [51] provided a thorough H∞analysis by means of estimating the pole loci asymptotic to C0which was then implemented by using YALTA software package by Avanessoff et al. [120]. H∞stability of some classes of TDSs with multiple chains of poles asymptotic to the same set of points on C0was studied in [117] and [118]. Similar approximation tools were also utilized to analyze poles of a ‘‘small’’ modulus and the corresponding BIBO and H∞stability for a NTDS with a single delay in [161]. C. STRONG AND ROBUST STABILITY Strong stability has been introduced in Definition 2. By the notion of robust stability we mean the ability of a system to remain stable (in some sense) with respect to parameters fluctuations, or under model or parameter uncertainties. A strong stability criterion for NTDSs was presented in [103], the derivation of which was made by means of the DQ method. Rabah et al. [119], inter alia studied conditions under which a mixed RTDS/NTDS is strongly stable. Du et al. [163] presented necessary and sufficient conditions for exponential stability of TDS governed by differential-algebraic equations. In particular, the robustness of this type of stability was studied when the equation is subject to structured perturbations. A computable formula for the structured stability radius was also derived. Otten and Mönnigmann [164] proposed an optimization method for parametrically uncertain delay differential equations with state-dependent delays. The central idea of the optimization is to stay off the stability boundaries in the parameter space. As a result, dynamical properties such as stability can be guaranteed in spite of parametric uncertainties in the model under the optimization. The so-called fold bifurcation expressing the situation when a real pole crosses the imaginary axis played a crucial role in this research. D. DDS Stability issues depended on the value of τcan be investigated using several methods and techniques. Two basic families of DDS methods for computing the delay stability margins prevail in the literature; namely, time-domain indirect and frequency-domain direct methods. In this survey (dealing inherently with the latter group), research results utilize the following methods for the delay-margin computation: (i) CTCR; (ii) Direct method; (iii) Argument principle (Cauchy theorem) method; (iv) Schur-Cohn criterion; (v) Kronecker sum and matrix pencil methods; (vi) Lyapunov matrix (Kronecker multiplication) approaches; (vii) Other numerical, semi-analytic and analytic methods. The key idea lies in the determination of all stability switching system poles (i.e. the characteristic quasipolynomial zeros) located exactly on C0, which can be used to determine the stability margin. In fact, only the rightmost subset of the spectrum makes the system switch from stability to instability or vice versa. Techniques included in all the above items (except for (iii)) are based on the elimination of 35472 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays exponential terms from 1(s); however, only bivariate polynomials include the information of the critical delay values explicitly. This can be done by the direct replacement of the exponential terms by using, e.g., the Möbius or Rekasius transformation (substitution) according to (48) and (49). A common alternative way is to apply the half-angle tangent substitution as follows: exp(−τjω)→cos(υ)−jsin(υ), υ =jω, cos(υ)=1−z2 1+z2,sin(υ)=2z 1+z2,z=tanυ 2, τk=ω−12tan−1(z)+kπ, k∈Z,tan1(·)∈[0, π).(50) 1) CTCR Some research results have extended or improved the original CTCR concept. Sipahi and Delice [165] focused on the socalled core hypersurfaces and showed some their features for the case when nτ>0. The core hypersurfaces mean the image of ℘0(τ)computed from the corresponding multivariate algebraic polynomial p(ω, T)in the parameter space of pseudo-delays. The authors were concerned with the identification of the asymptotic directions of the delays on the potential stability switching hypersurfaces approaching infinity. These results can also be used in connection with [166] to study strong DIS by covering both finite and infinite delays. Jesintha Mary and Rangarajan [167] applied the resultant theory introduced in [165] in order to investigate a new flexible methodology for stability analysis of in load frequency control scheme with delays in the transmission of control signals from the control center to generating unit. The proposed method offered larger delay margin and takes less computation time compared to some existing methods. A recent work of Sipahi’s [168] utilized the knowledge (based on the above-introduced research) that it is possible to compute the exact range of the imaginary spectrum of such systems to design imaginary poles with the objective to manipulate stability regions in the delay space. In addition, Kammer and Olgac [169] studied stability of dRTDSs via the CTCR paradigm by means of the equivalence of a general class of distributed delay system to a discrete-delay system with multiple independent delays. A comparison between delay space (represented by ℘0(τ), ℘off (τ)) and the spectral delay space was presented in [170]. The latter domain contains pointwise frequency information as well as the delay and it was preferred here for its advantageous boundedness properties and the simple construction of stability transition boundaries. Gao and Olgac [171]–[173] investigated the bounds of the imaginary spectra via the substitution (49) and by deploying the Dixon resultant theory [174] for a RTDS with an arbitrary number of delays. Consequently, the proof of the differentiability of the crossing-frequency variations dω/dzi was provided to investigate the bounds. As a result, 2D crosssections of the hypersurfaces were extracted. The concept of the so-called 3D building blocks in the spectral delay space [175] was utilized to meet this objective. Theexactdelayboundforaconsensusoflinearmulti-agent systems with a fixed and uniform communication time delay was determined by Cepeda-Gomez in [176] in an efficient manner by using the CTCR methodology. A state transformation was performed to decouple the system and simplify the problem prior to the stability analysis. 2) DIRECT METHOD The ‘‘direct’’ refers to the method introduced by (19)-(22). Sönmez et al. [177] studied the DDS problem for load frequency control systems with constant communication delays of the commensurate degree of one and two. However, the complete stability windows were not considered because only the minimum positive value of (22) with RT = +1 was taken as the unique delay margin. 3) ARGUMENT PRINCIPLE METHOD This definite integral stability method, originated from the argument principle (or the Cauchy theorem), is effective because it only requires a rough estimation of the testing integral over a finite interval to judge DDS. Consider the socalled testing integral F(, a)defined in Theorem 2. Xu and Wang [106] proved that if 1(s)of a NTDS has no imaginary roots and the condition (36) is satisfied, then there exists a sufficiently large 0>0 so that for all  > 0it holds that ∈−F(, 0) π+n−1 2,−F(, 0) π+n+1 2(51) where is the number of poles in C+. Two DDS algorithms, for finding the parameter (delay)-dependent critical upper limit and a parameter (delay)-independent upper limit without any restriction on the number of time delays, were presented by Xu et al. [178] who proved the following theorem. Theorem 4: Assume that 1(s)has no roots on C0and (36) holds. Let 0(τ)=max(ωR,0)where ωRstands for the maximal positive root of R(ω):= Rej−n11(jω). Then (37) and (51) are true for all >0. 4) SCHUR-COHN METHOD Mulero-Martínez [179] presented a modified Schur-Cohn criterion for RTDSs with commensurate delays that requires seeking real roots only, which is comparable to the Rekasius substitution criterion. In contrast to the classical Schur-Cohn criterion, the approach is based on the application of triangular matrices over a polynomial ring in a similar way as in the Jury test of stability for discrete systems, and it halves the dimension of the subjected polynomial. It starts with the construction of a bivariate polynomial r∈C[ω, z],r(ω, z)= PnC i=1b(ω)zifrom 1(s)=PnC i=0d(s)exp(−shi)where b(ω)=d(jω),z=exp(−sh),nCis the commensuracy degree, and hstands for the base delay. Then, two associated triangular matrices are assembled, from which the VOLUME 6, 2018 35473 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays determinant polynomial ζ(ω)is calculated. As a core of the approach, the following theorem holds Theorem 5: Let ωc>0 be a real root of ζ√s, then ±j√ωcis a pair of poles of the RTDS. 5) KRONECKER SUM AND MATRIX PENCIL METHODS Ma et al. [180] studied DDS of a NTDS with a single delay H0˙ x(t)+H1˙ x(t−τ)=A0x(t)+A1x(t−τ)(52) by applying the matrix pencil and the linear operator methods. The main result of the method regarding the DDS problem was enshrined in the following theorem. Theorem 6: If skis a purely complex root of 1(s), it is also a zero of 1ass,2(s):= det(sH0−A0)⊗(sH0+A0) −(sH1−A1)⊗(sH1+A1),(53) see also Louisell [77]. 6) LYAPUNOV MATRIX APPROACHES Consider the system (23) again, one approach is based on the fact that any purely imaginary root of 1(s)is also a root of the polynomial 1ass,3(s):= detsI+AT 0⊗(sI−A0)−AT 1⊗A1 (54) that is also the characteristic polynomial of the system X00(θ)=X0(θ)A0+X−1(θ)A1 X01(θ)= −AT 1X0(θ)−AT 0X−1(θ)(55) see (53) for the comparison. System (55) can be then subjected to the computation of Lyapunov matrices. Once the spectrum of (55) is computed, critical values of the delay can be obtained by substituting these roots into 1(s). Ochoa et al. [181] adopted the above idea to derive explicit relations between the spectrum of an original dRTDS and NTDS, and that the delay-free system (55), which constituted a bridge between time-domain and spectral approaches. Delay-dependent stability regions were determined as well. Since 1ass,3(s)has only even powers of s, the searching of imaginary poles was reduced to the computation of real roots of 1ass,3(λ)λ=s2. To solve this task, the authors utilized Sturm‘s theorem that is based on the computation of sign changes of the Sturm sequence. Another technique based on the Lyapunov–Krasovskii methodology to investigate delay-dependent (robust) exponential stability of a RTDS was derived by Cao [182]. The author used LMIs and slack matrices to get the upper bound of the exponential decay rate. The given criterion provides the computation method of the value of Lmax, so that the system is (robustly) exponential stable for L∈(0,Lmax], i.e. no other stability windows were considered. A comparison with some other methods was also given to the reader. Sun et al. [183] derived the sufficient condition for the delaydependent asymptotic stability of the closed-loop power system with prescribed degree of stability α(i.e., the decay rate or a spectral abscissa) based on the Lyapunov stability theory and transformation operation in complex plane, and presented a method based on LMIs to calculate the delay margin of the closed-loop system considering the prescribed value of α. 7) OTHER METHODS Regarding the research results already introduced in this survey, Wang et al. [146] calculated mutual delay values satisfying exponential stability of a RTDS with example case studies supporting their method. Complete stability intervals for the base delay of a system with commensurate delays were determined by means of the frequency sweeping method and the Puiseux series in the works of Li et al. [151]–[154]. The singular value decomposition technique was used by Ramachandran and Ram [184] to determine critical delays of a single-input multi-output (SIMO) system. The leading idea was as follows: Let 1(s)=det (A−sB+exp(−sτ)H) where A,B∈R2n×2n, and His a rank-one matrix subject to the singular value decomposition H=U6V,6= diag σ0. . . 0. After some algebraic operations, the condition 1(s)|s=jω=0 can be expressed as N(sk)¯ N(sk)−D(sk)¯ D(sk)=0 (56) where N(sk)=detQ,D(sk)=σdetQ1,Q= UT(A−skB)V,Q1stands for the (2n−1)×(2n−1)trailing submatrix of Q1,skis a purely imaginary root, and the bar expresses the complex conjugate, i.e. the problem is reduced to the task of finding the roots of a polynomial. Nevertheless, the technique cannot be used for a multi-input multi-output (MIMO) system. In this case, the authors separated the matrix eigenvalue problem into its real and imaginary components, so that the problem of determining the critical delay was transformed to P(τ, sk)z=0 (57) where skis a purely imaginary repeated eigenvalue with a multiplicity larger than one. Since the Jacobian matrix associated with (57) is singular in the neighborhood of the solution, the convergence of Newton’s method is linear; hence, a bisection algorithm for solving the problem was developed. Pontes Duff et al. [185] solved the model reduction problem (45) for RTDSs with multiple delays via the so-called TFIRKA algorithm [186] giving rise to the finite-dimensional model E˙ x(t)=Ax (t)+Bu (t),y(t)=Cx (t).(58) The obtained model was then used to estimate stability regions. An interesting comparative study on the stability analysis of DCPPS was presented by Gao et al. [187] where three methods were adopted; namely, a Padé approximation based method [123], the explicit infinitesimal generator discretization-based method [84] and a DDS technique [188] that allows for the determination of the maximal delay such 35474 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays that a DCPPS remain stable by using an LMI technique (see [182] for the comparison). The conservativeness of the DDS method was analyzed via an example and it was inter alia observed that the Padé approximation based method is sufficiently accurate even for a low approximation order. A simply implementable gridding delay-discretization DDS technique was published in [189] and [190]. This numerical concept is based on the recursive approximation of 1(s)by the associate polynomial in the vicinity of the current estimation of the rightmost pole s0in every node of the grid in the delay space. While in [189], the associated polynomial 1ass,4(s|s0)was obtained from the Taylor series expansion, the bilinear transformation together with the pre-warping technique were utilized in [190] to get 1ass,5(z|s0,Ts)where Tsexpresses the sampling time. The leading zero of 1ass,4(s|s0)or 1ass,5(z|s0,Ts)then yielded the eventual rightmost pole estimation for the next grid node. The estimation of the switching poles was further enhanced by using the average of RT values (a combined Newton’s technique) and by the linear interpolation, respectively. The concept clearly utilizes the root continuity property (see item (v) of Proposition 1); however, one has to be careful in the neutral delay case (Proposition 2 (iv), Proposition 3). A combination of three techniques to determine the delay stability margin for wide-area measurement systems (WAMS) – modeled by RTDSs with multiple delays – was proposed in [191]. Namely, matrices Aiin (1) are standardized into the Jordan form first, yielding a new state vector z(t). Second, the Taylor expansion is applied to separate the connection between z(t)and z(t−τi). Finally, the Schur simplification [192] is implemented to reduce the number of state variables. Roales and Ródriguez [193] studied the existence of stability switches and Hopf bifurcations (i.e. the periodic stability boundary) for the second-order scalar delay differential equation ¨x(t)+a˙x(t−τ)+bx (t)=0, t, τ > 0, in which a,b∈C. The presented analytic derivations were based on the theorem established in [194] that characterizes, for the critical values τisuch that 1(jω, τi)=0, the variation of the number of zeros with nonnegative real parts of 1(s, τ) in terms of the order and sign of the first nonzero derivate of F(ω):= |Re(1(jω))|2−|Im(1(jω))|2. E. DIS Frequency-based DIS methods are generally built on the verification of the non-existence of purely imaginary system poles for arbitrary delay values. This task is usually achieved by transforming 1(s)into associated (auxiliary) polynomial 1ass (s)or 1ass (z), which is completely free of delays and can be uni-, bior even multivariate, and then by proving that there is no zero of 1ass (s)lying exactly on the imaginary axis, or no zero zkof 1ass (z)such that zk∈T. Delice and Sipahi [166] used the technique of computing the resultant and consequently that of the iterated discriminant [68] to eliminate pseudo-delays from p(ω, T)(see the description of the CTCR concept above), which allowed one to construct a single-variable function D(ω)to be equal to zero. Then, the non-existence of any positive real root of D(ω)- which is a sufficient DIS condition - was proved by the Déscartes rules of signs. However, infinite delays were omitted in this technique. Asymptotic directions of the delays on the potential stability switching hypersurfaces approaching infinity derived by Sipahi and Delice [165] linked DDS and the strong DIS problem (here, the authors used the term ‘‘strong’’ for DIS including infinity delays). Concerning multiple-delay cases, the comprehensive study by Nia and Sipahi [195] also utilized the Rekasius transformation (29) and the resultant theory to investigate DIS in the delay space and the controller parameters space for active vibration control systems. In addition, Sturm sequences were applied to establish the necessary and sufficient conditions in identifying the number of distinct positive real roots of D(ω). A matrix pencil methodology along with an algebraic method were utilized by Ma et al. [180] to investigate the DIS problem via 1ass,2(s)as in (53). Ergenc [196] presented a method for determining the DIS zones of a general RTDS with multiple delays against parametric uncertainties. This method adopted the Kronecker summation scheme 1ass,1(z)as in (24) expressed by means of the Kronecker multiplication operators. The system is DIS if Re(s:1(s,p)=0)<0 (59) and all zeros zkof 1ass,1(z,p)satisfy zk/∈Twhere p represents a vector of unknown parameters. In fact, 1ass,1z1,z2,...,znτ,p =Xm j=1bjz1,z2,...,zi−1,zi+1,...,znτ,pzj i is a self-inversive (symmetric) multivariable polynomial satisfying 1ass,1(zi)=znτ i1ass,1(1/zi)for i=1,2,...,nτ. The following unique property of self-inversive polynomials was utilized: β=(2µ+1)−mwhere βis the number of its zeros lying on T, and µis the number of zeros inside D(including multiplicity). With the combination of this property and another general polynomial property (Pellet’s theorem), the following sufficient condition for DIS was presented: Theorem 7: The system is DIS if (59) holds and bµz1,z2,...,zi−1,zi+1,...,znτ,p >Xm j=1,j6=µbjz1,z2,...,zi−1,zi+1,...,znτ,p(60) for µ≤m/2−1, i=1,2,...,nτ. An experimental study verifying this result was presented in [197]. The methodology was further improved by Alikoç and Ergenc [198] where the Bistritz tabulation method [199] was used to determine the location of zeros with respect to the unit circle for a single delay RTDS. The method is based on a three-term recursion of symmetric polynomials and the number of sign variations of these polynomials at z=1; namely, the sign variation in a sequence of numbers VOLUME 6, 2018 35475 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays obtained by the solution of recursive equations calculated from the polynomial D(z)=d1ass,1(z)/dzis evaluated instead of the condition (60), which enables the use of real arithmetic operations. Alikoç and Ergenc [200] extended this technique to multiple incommensurate delays. These results can be utilized, for instance, when determining the controller parameters’ set robustness with respect to delay values. Recall that the DIS problem was investigated also by the semi-discrete approximation of by means of linear spline functions presented by Fabiano [89]. The argument principle (or, contour integral) method [107], [178], was used to deal with the DIS as well. Consider Theorem 4, in which ωRis the maximal positive root of RL(ω)instead of R(ω). Polynomial RL(ω)is constructed from R(ω), the coefficients of which are substituted by their infima that are independent of delay values. Assuming the reign of spectral methods, marginal yet interesting results were presented by Li et al. [201] where the strong DIS condition via LMIs was analyzed using frequency domain discretization into several sub-intervals and the piecewise constant Lyapunov matrices. A series of proposed stability criteria yield necessary and sufficient strong DIS conditions for RTDSs with a single delay which is less conservative than some typical sufficient LMI conditions. It is worth noting that the notion of strong DIS introduced there is rather different than that in [165]. Namely, consider a RTDS with commensurate delays and the base delay h, the system is strongly DIS if 1(s,z)6= 0 for all s∈C+and z∈Dwhere z=exp(−sh). This property is robust against perturbations of parameters in the state matrices in (1), see [202] for details. F. PARAMETER-DEPENDENT STABILITY By parameter-dependent stability we mean the stability investigation with respect to system parameters except for delays, i.e., in the non-delay parameter space. Recalling research results already introduced above again, Dong et al. [103] evaluated the optimal parameters for the controller design by searching the global minimum of the spectral radius of the transition matrix that was obtained by means of the DQ method. In order to solve such optimization problems using gradient descent algorithms, the gradient of the spectral radius of transition matrix with respect to the concerned parameters was analytically formulated. The Vandermonde/Birkhoff matrix DDS approach for multiple purely imaginary poles by Boussaada and Niculescu has also included non-delay parameters while studying parameter-dependent exponential stability [148], [149]. Otten and Mönnigmann [164] proposed an optimization scheme that was based on the solution of the H2minimization problem in the parameter space subject to the manifold of the critical parameter values. In addition, the normal vector has to be solved to enforce a robust distance between any candidate optimal steady state and the critical manifold. Argument principle based DDS methodology by Xu et al. [178] can be applied to stability analysis with respect to non-delay parameter values as well. FIGURE 3. Function φ7→ LmaxωLmax [204]. A simple systematic frequency sweeping procedure for solving the exact stability boundaries in the parameter plane p=(p1,p2)for RTDSs was proposed by Perng [203]. Note that the methodology has also been used to solve the DDS problem for systems with commensurate delays as follows: Let exp(−sh)=exp (−jωh)=cos (ωh)−j sin (ωh)= p1−jp2, then after some algebraic operations on goniometric functions, the potential stability boundary plots in p1−p2space can be obtained again. Since it must hold that |exp(−sh)|=1, the boundary must intersect the unit circle in the parameter plane for admissible solutions. The exact maximum delay value for asymptotic stability then reads h=ω−1cos−1(p1)=ω−1sin−1(p2). If no intersection is found, the system is DIS (or unstable). The design of parameters p=(p1,p2)such that the system represented by (23) is asymptotically stable for L∈(0,Lmax] with a predetermined (known and fixed) value Lmax was presented by Sipahi [204]. The author used the Rekasius substitution and introduced the sweeping parameter φ=ωTin an interval φ∈[φmin, φmax]. Then, it can be computed from (48) that exp−jωLmaxLmax=(1−jφ)/(1+jφ)to get a polynomial pωLmax, φ, pinstead of the quasipolynomial 1jωLmax,pwhere the corresponding frequency reads ωLmax =2 Lmax tan−1(φ)−(sgn(φ)−1)π 2, φ6= 0,0< ωLmax ≤2π/Lmax,(61) see Fig. 3. Then, for these fixed values, one should simultaneously solve the set of equations Re(p(p)) =Im(p(p)) =0. However, the feasibility of this solution must be verified by the computation of the number of imaginary crossing, M, by means of Theorem 1. Hence, it is necessary to compute the eigenvalues of 5as in Theorem 1, followed by the verification where these values are included in the system spectrum . Note that M≤n2(for  > 0) and in the referred research study, the author enforced M=1 initially. Schrödel et al. [205] presented a comparative overview of four existing frequency-based methods for the stability boundary calculation problem in the parameter space, namely, the Rekasius substitution method, the direct 35476 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays method [72], the Kronecker multiplication method [77] (see also Ma et al. [180]) and the so-called matrix sum method [74]. The last one of the methods is based on the elimination of ωfrom the characteristic equation and the solution of the associated equation for z∈T. The characteristic equation can be rewritten as 1ass,4(s,z):= det((sI−M(z))) =0 (62) where z=exp(−sh)and M(z)is apparent from (2) for commensurate delays. Crossing poles satisfy zc∈Tand sc= ±jωc. Equation(62)can also be reformulated as det ((zU−V)) = 0 where U,Vare matrices that include Kronecker sum and multiplication operations (see [205] for more detail). By solving this equation for z∈T, the crossing frequencies can be obtained from 1ass,4(ω)=1ass,4(jω, zc)=0 and the corresponding delays via τk=ω−1(arg(z)+2kπ),k∈Z. In [205], a generalization of the problem of calculating the stability region for TDSs in the delay and non-delay parameter space (which is very close to the CTCR paradigm) was also given to the reader. In addition, three types of stability boundaries were introduced. It is worth noting that most of the results on parameterdependent stability were obtained for control systems and related tasks of controller parameters tuning, see e.g. [119], [124], [125], [146], [165], [195], which goes, however, beyond the objective of this survey that is aimed at system analysis. Selected results from the general part of this section (and also from the previous one in some cases) are summarized in Table 2 to provide the reader with an overview of the theoretical stability studies. VII. ENGINEERING APPLICATIONS AND CASE STUDIES This section is focused on the commented list of academic or practical applications of the methods for LTITDSs spectrum analysis. Note that this section is summarized in Table 3. Method approximating or (t), introduced herein in sections from IV to VI, can be found in the literature as favorite tools for the analysis of milling processes – unfortunately, these models of type (23) usually include a time-dependent state matrix A1. Recall that Tang et al. [98] predicted milling stability via an improved FD method with Lagrange polynomial interpolation, and the authors presented a comparison with SD and NI techniques as well. The same problem was solved using FD and NI methods in [95] and [99], respectively. The DQ method utilized for the stability analysis of milling processes was presented by Ding et al. [102]. Quo et al. [207] utilized the third order FD method to get the exact stability bounds. Ozoegwu [208] presented a method being very close to the FD one, yet the least squares (also called the hyper third-order) approximation was applied instead of the interpolation procedure. The hyper third-order approximation followed by the NI method was further used by Ozoegwu et. al. [209] and extended third and fourth order vector NI schemes for onedegree-of-freedom and two-degree-of-freedom milling processes in [100]. The same authors also presented the use of the SE method while analyzing the chatter stability of a threetooth plastic end-milling CNC machine [210]. These approximation methods, however, have been applied in other industrial applications as well. Khasawneh [211] utilized the SE method with the barycentric Lagrange formula to analyze the stability of machining processes which may lose stability due to chatter vibrations, i.e., self-excited vibrations due to the surface regeneration effect. A short (in its form) yet comprehensive (in its content) overview of numerical techniques that are based on a finite dimensional approximation of the infinite dimensional system used for the stability prediction of machining processes was presented by Insperger et al. [11]. This type of chatter occurs due to workpiece rotations or dynamic cutting load changings. Kishor et al. [8] discussed stability analysis using spectral discretization of time-delayed electric power systems, namely, the 4-generator and the 14-generator Southeast Australian power systems. The PsC method was used to get the discrete mapping there, and the authors computed the rightmost poles and the spectral abscissa over a wide range of time delays, which characterizes a partial solution of the DDS problem. As introduced above, Ye et al. proposed an iterative PsC method for spectral analysis of large DCPPS to overcome computational problems with sparse matrix approximation of the infinitesimal generator [84], [86], and the solution operator [87]. Milano [138], and Milano and Anghel [144] used the PsC technique to compute poles of a large DCPPS and compared it with some other discretization schemes to get a finitedimensional approximation of the solution operator (t). For these results, see also Table 1. The pseudospectral method by Breda et al. [58] was utilized by Coelho et al. [212] to analyze the spectrum of a single delay RTDS expressing the feedback control system for an islanded microgrid composed of two or more voltage source inverters with communication delays. Sensitivity analysis of the poles was conducted by Zhao [213] in order to reveal the dynamic stability margin and to identify the proper range of the control parameters, for an islanded medium-voltage microgrid placed in the Dongao Island. Unfortunately, the authors did not refer to the used method. Dong et al. [214] proposed a stability analysis method of the hybrid energy storage systems with delays and applied it to a lab-scale DC microgrid. The stability margin (i.e., the maximum stabilizable delay) was computed by the determination of purely imaginary poles. The leading idea of the critical poles computation is based on the assumption that all delays are rational numbers or they can be approximated by the rational numbers. Then, one can rewrite the characteristic equation 1(jωc)=0 so that its solution has a period of 2π. VOLUME 6, 2018 35477 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays TABLE 2. Stability studies related to pole loci – theory (Section vi). 35478 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays TABLE 2. (Continued.) Stability studies related to pole loci – theory (Section vi). VOLUME 6, 2018 35479 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays TABLE 3. Stability studies related to pole loci – applications (Section vii). 35480 VOLUME 6, 2018 L. Pekař, Q. Gao: Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays TABLE 3. (Continued.) Stability studies related to pole loci – applications (Section vii). Hence, the interval [0,2π)is discretized, and system poles are then computed in every single discrete step inside this interval. Although the Lambert W function has a limited utilization due to model restrictions, some engineering applications can be found. For instance, Petit et al. [215] studied reactiondiffusion systems with a time delay considered in the complex networks in the framework of Turing instabilities. Explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time delay, and the network topology were obtained using the scalar Lambert W function. The authors then predicted whether or not the systems would exhibit a wave pattern associated with a Hopf bifurcation, or a stationary Turing pattern. Yi et al. [216] used the function to obtain the rightmost poles of neural networks with time delays and parametric uncertainties modeled by a single delay RTDS. However, note also that more particular applications of the Lambert W function have been made concerning controller design, see e.g. [109], [217], and references therein. Niu et al. [123] used the Padé approximation to estimate the spectrum of a power system with a time delay, see Table 1. Gölgeli and Özbay [218] utilized the YALTA software to investigate the unique local stability by analyzing the impact of the nicotine exposure on the cholesterol biosynthesis. The so-called delay-dependent coupling (DDC) was considered in [219] to prevent instability in a multi-agent system in which agents communicate with each other under homogeneous delays, while attempting to reach consensus. The system model has a simplified form as follows ˙ x(t)=f(L)Ax (t−L)(63) where f(L)represented the DDC as a function of the delay value L. The main idea while designing the stability of (63) was based on the following formula for the computation of the delay margin Lmax. Lmax =1 f(L)min k ηk/2 2αksinηk/2(64) where αk,ηkare related to the particular eigenvalue skof A according to Fig. 4. Trajectories of poles were obtained via TRACE-DDE tool [220]. Note that a multi-agent consensus dynamics under FIGURE 4. Eigenvalues skof A with respect to αk,ηk[219]. a communication delay, the delay margin and the network topologies to reduce the duration to reach consensus were also investigated by means of the rightmost poles e.g. in works [221], [222], which, however, can be considered as control rather than analytic tasks. Sipahi et al. [223] studied how the memory of drivers modeled by distributed delays affects the decision-making process in a car following scenario, in which each driver aims at keeping a fixed time-headway with respect to the preceding vehicle. When analyzing stability, the approximation of delays was done by using the asymptotic (limit) properties of distributed delay terms and the Taylor series expansion. The spectrum was computed by means of the QPmR toolbox. The authors inter alia found that the dynamics can exhibit the spectrum similar to NTDSs for some interconnection schemes, although the model does not fit in the standard NTDSs. 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Zhang, ‘‘The modeling and control of a singular biological economic system with time delay in a polluted environment,’’ Discrete Dyn. Nat. Soc., vol. 2016, Sep. 2016, Art. no. 5036305, doi: 10.1155/2016/5036305. [248] K. Ito and R. Teglas, ‘‘Legendre–Tau approximations for functional differential equations,’’ SIAM J. Control Optim., vol. 24, no. 4, pp. 737–759, 1986, doi: 10.1137/0324046. LIBOR PEKAŘ was born in Zlín, Czech Republic, in 1979. He received the B.S. degree in automation and informatics, the M.S. degree in automation and control engineering in consumption industry, and the Ph.D. degree in technical cybernetics from Tomas Bata University in Zlín, in 2002, 2005, and 2013, respectively. From 2006 to 2013, he was a Junior Lecturer with the Faculty of Applied Informatics, Tomas Bata University in Zlín. From 2013 to 2018, he was a Senior Lecturer with Tomas Bata University in Zlín, where he has been an Associate Professor since 2018. He has authored two book chapters, over 40 journal articles, and over 70 conference papers. His research interests include analysis, modeling, identification, and control of time-delay systems, algebraic control methods, and autotuning and optimization techniques. He has been the Lead Guest Editor of journals Mathematical Problems in Engineering and Advances in Mechanical Engineering, and an Editor of the Mathematical Problems in Engineering since 2018. Dr. Pekař was a recipient of the Rectors’ Award for the best Ph.D. thesis in the Faculty of Applied Informatics, Tomas Bata University in Zlín, in 2013, and the Laureate of the ASR Seminary Instrumentation and Control in 2007 and 2009. QINGBIN GAO received the B.S. degree in mechanical engineering from the Harbin Institute of Technology, China, in 2011, and the Ph.D. degree in mechanical engineering from the University of Connecticut in 2015. He was an Assistant Professor with the Department of Mechanical and Aerospace Engineering, California State University, Long Beach, from 2015 to 2018. Since 2018, he has been an Assistant Professor with the Department of Mechanical Engineering, The University of Alabama. His main research focuses on the stability analysis and control synthesis of time-delay systems with applications to multi-agent systems, manufacturing, connected vehicles, human learning, and power systems and vibrations. He was a recipient of the Best Conference Paper Award of the 19th International Conference on Networking, Sensing, and Control (ICNSC) in 2017 and the 6th American Society of Mechanical Engineers (ASME) Dynamic Systems and Control Conference (DSCC) in 2013. He has served as a Session Chair for 2017 ASME DSCC, 2017 ICNSC, and 2016 IEEE American Control Conference (ACC). He has served as an Associate Editor for 2017 ACC, 2018 ACC, and 2017 ASME DSCC. He has also served as a Reviewer for over 100 papers from over 30 journals, including but not limited to Automatica, the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, and Mechatronics. He is currently a Guest Editor of the IEEE ACCESS and Advances in Mechanical Engineering. VOLUME 6, 2018 35491