on computing puiseux series for multiple imaginary characteristic roots of lti systems with commensurate delays obtaining the puiseux series of multiple imaginary characteristic roots mirs is a fundamental issue in the stability analysis of time delay systems however to the best of the authors knowledge this issue has not been fully investigated up to date this note focuses on the puiseux series expansion of mirs of linear time invariant systems including commensurate delays for an mir of any multiplicity we propose an algorithm for defining the structure of the puiseux series as well as the explicit computation of the corresponding coefficients by using the proposed method we can find all the puiseux series corresponding to all the root loci

nilpotent singularities in generic 4 parameter families of 3 dimensional vector fields abstract this paper deals with singularities of vector fields in r 3having a 1 jet linear conjugate toy x z y they first occur in generic 3 parameter families in codimension 3 all such singularities are mutuallyc0equivalent we give a proof of this provide a good normal form for 3 parameter unfoldings and show that all non wandering behaviour in such an unfolding is of small amplitude we also show that for codimension 4 there are exactly 5 types of singularities forc0equivalence the proof relies on normal form theory blowing up and estimation of abelian integrals

delay independent stability test for systems with multiple time delays delay independent stability dis of a general class of linear time invariant lti multiple time delay system mtds is investigated in the entire delay parameter space stability of such systems may be lost only if their spectrum lies on the imaginary axis for some delays we build an analytical approach that requires the inspection of the roots of finite number of single variable polynomials in order to detect if the spectrum ever lies on the imaginary axis for some delays excluding infinite delays the approach enables to test the necessary and sufficient conditions of dis of lti mtds technically known as weak dis as well as the robust stability of single delay systems against all variations in delay ratios the proposed approach which does not require any parameter sweeping and graphical display becomes possible by establishing a link between the infinite spectrum and algebraic geometry case studies are provided to show the effectiveness of the approach

invariance properties for a class of quasipolynomials when a time delay system involves multiple imaginary roots mirs the stability analysis will become much more complicated than that in the case with only simple imaginary roots sirs an mir may exhibit different splitting behaviors and to the best of the authors knowledge their properties have not been fully investigated in this paper we focus on characterizing the invariance properties for mirs with any multiplicity furthermore the proposed methodology makes it possible to also cover some degenerate cases already encountered and discussed in the literature in addition we propose an easily implemented frequency sweeping method making it possible to derive the asymptotic behavior without invoking the puiseux series

nondispersive solutions to the l 2 critical half wave equation we consider the focusing l2 critical half wave equation in one space dimension i partial t u d u u 2 u where d denotes the first order fractional derivative standard arguments show that there is a critical threshold m 0 such that all h1 2 solutions with u l 2 0 and the linear momentum p 0 in mathbb r in particular our main result and its proof can be seen as a model scenario of minimal mass blowup for l2 critical nonlinear pdes with nonlocal dispersion

delay independent stability of a special sequence of neutral difference differential equations with one delay abstract the equations of the title for the state x t r l x d a d a n x t b d b d b n x t t 0 d d dt 1 are considered in the parameter space a b b n p where p is defined by re a a is a second real point a a for a r a is the complex conjugate to a c the same notation applies for b by a tedious but elementary analysis the set s0 p for which 1 has for all t 0 only asymptotically stable solutions in the sense of liapunov is explicitly determined in the form b n a b 1 2 the stability boundary decays exponentially with respect to the multiplicity n of the zeros of the coefficient polynomials of 1 we generalize a theorem of f brauer j differential equations 69 1987 185 191 dealing with delay independent stability for characteristic equations of the form h z a z b z exp zt t 0 3 where a z b z are holomorphic in re z 0 in such a way that the neutral case can be dealt with

an analytic stability test for a class of time delay systems stability properties of linear dynamic systems with characteristic equations of the form sigma min i 0 max n p i s e shi 0 where all the polynomials p i are of order leq r in s can be determined from a polynomial equation in s of order leq 2n r

a new method for computing delay margins for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems we consider delay systems of retarded type modeled both as a high order scalar differential difference equation and as a set of first order differential difference equations expressed in state space form we provide a computational method that can be used to compute a delay interval such that the system under consideration is stable for all delay values that lie in the computed interval this method requires computing only the eigenvalues and generalized eigenvalues of certain constant matrices and it can be implemented efficiently based on this method we further state a simple necessary and sufficient condition concerning stability independent of delay for each of the two types of the models

wellposedness for semirelativistic schrodinger equation with power type nonlinearity abstract in this paper we employ bourgain type norm to investigate the wellposedness for the cauchy problem of one dimensional semirelativistic schrodinger equation with cubic nonlinearity in space h s we extend the previous existence result to the space regularity s 1 4 moreover we point out the persistence of the solution in different regularity space is uniform by an iterate argument additionally we are also able to say more for semirelativistic schrodinger equation with general power type nonlinearity

double hopf bifurcation for van der pol duffing oscillator with parametric delay feedback control the stability and bifurcation of a van der pol duffing oscillator with the delay feedback are investigated in which the strength of feedback control is a nonlinear function of delay a geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and hopf bifurcations the hopf bifurcation curves and multi stable regions are obtained as two parameters vary some weak resonant and non resonant double hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the death island regions simultaneously by applying the center manifold theory the normal forms near the double hopf bifurcation points as well as classifications of local dynamics are analyzed furthermore some quasi periodic and chaotic motions are verified in both theoretical and numerical ways

reversals in stability of linear time delay systems a finer characterization abstract in most of the numerical examples of time delay systems proposed in the literature the number of unstable characteristic roots remains positive before and after a multiple critical imaginary root cir appears as the delay seen as a parameter increases this fact may lead to some misunderstandings i a multiple cir may at most affect the instability degree ii it cannot cause any stability reversals stability transitions from instability to stability as far as we know whether the appearance of a multiple cir can induce stability is still unclear in fact when a cir generates a stability reversal has not been specifically investigated in this paper we provide a finer analysis of stability reversals and some new insights into the classification the link between the multiplicity of a cir and the asymptotic behavior with the stabilizing effect based on these results we present an example illustrating that a multiple cir s asymptotic behavior is able to cause a stability reversal to the best of the authors knowledge such an example is a novelty in the literature on time delay systems

a conformal boundary for space times based on light like geodesics the 3 dimensional case a new causal boundary which we will term the l boundary inspired by the geometry of the space of light rays and invariant by conformal diffeomorphisms for space times of any dimension m 3 proposed by one of the authors r j low the space of null geodesics and a new causal boundary lecture notes in physics 692 springer 2006 pp 35 50 is analyzed in detail for space times of dimension 3 under some natural assumptions it is shown that the completed space time becomes a smooth manifold with boundary and its relation with geroch kronheimer penrose causal boundary is discussed a number of examples illustrating the properties of this new causal boundary as well as a discussion on the obtained results will be provided

controller design for delay independent stability of linear time invariant vibration systems with multiple delays abstract one of the critical parameters that can deteriorate the effectiveness of active vibration control avc is the delay in sensors especially in remote sensing where delays are large and in high speed applications with even small delays instability can be inevitable this paper presents algebraic approaches to design controllers in order to achieve stability regardless of the amount of delays for avc applications modeled by linear time invariant systems with multiple constant delays the approaches are based on a nonconservative framework and can identify the regions in the controller gain space where delay independent stability dis is achievable with these controllers we demonstrate via simulations that vibration suppression within certain excitation frequency bands can be improved or be as effective as those in avc applications without delays

an exact method for the stability analysis of time delayed linear time invariant lti systems a general class of linear time invariant systems with time delay is studied recently they attracted considerable interest in the systems and control community the complexity arises due to the exponential type transcendental terms in their characteristic equation the transcendentality brings infinitely many characteristic roots which are cumbersome to elaborate as evident from the literature a number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay this study offers an exact structured and robust methodology to bring a closure to the question at hand ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions pockets in the domain of time delay but it also declares the number of unstable characteristic roots at any given pocket the method starts with the determination of all possible purely imaginary resonant characteristic roots for any positive time delay to achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation it is proven that the number of such resonant roots for a given dynamics is finite each one of these roots is created by infinitely many time delays which are periodically distributed an interesting property is also claimed next that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency these two unique findings facilitate a simple and practical stability method which is the highlight of the work

the routh hurwitz method for stability determination of linear differential difference systems abstract the characteristic quasipolynomial for a linear differential difference system is replaced by a regular polynomial to which standard stability tests for finite dimensional systems can be applied the routh hurwitz stability test is then used to determine necessary and sufficient conditions for asymptotic stability of time delay systems independent of the length of the delay

technical communique on stability of second order quasi polynomials with a single delay in this note the stability of a second order quasi polynomial with a single delay is studied although there is a vast literature on this problem most available solutions are limited to some particular cases moreover some published results on this subject appear to contain imprecise or even wrong conditions the purpose of this note is to show that by accurate application of known theories a complete explicit characterization of stability regions can be derived in a most general case as a byproduct of the proposed analysis we show that in the high order case the quasi polynomial is delay independent unstable whenever its delay free version has an odd number of unstable roots or equivalently a negative static gain

passivity based pi control of first order systems with i o communication delays a frequency domain analysis abstractin this paper the proportional integral pi control of first order linear passive systems through a delayed communication channel is revisited in light of the relative stability concept called stability treating the delayed communication channel as a transport partial differential equation pde the passivity of the overall control loop is guaranteed resulting in a closed loop system of neutral nature spectral methods are then applied to the system to obtain a complete stability map in particular we perform the d subdivision method to declare the exact stability regions in the space of pi parameters this framework is then utilised to analytically determine the maximum achievable exponential decay rate d of the system while achieving the pi tuning as explicit function of d and system parameters

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

frequency sweeping tests for stability independent of delay in this paper we study the stability properties of linear time invariant delay systems given in a state space form we consider specifically the notion of asymptotic stability independent of delay systems with both commensurate and noncommensurate delays are investigated we present for each class of systems a necessary and sufficient condition in terms of structured singular values and further we demonstrate how these conditions may be extended to study stability independent of delay for uncertain systems our results consist of several frequency sweeping tests that can be systematically implemented and that should complement the previous work

robust fuzzy controller design for nonlinear multiple time delay systems by dithers this study presents a robustness design of fuzzy control for nonlinear systems with multiple time delays first the neural network nn model is employed to approximate the nonlinear multiple time delay nmtd plant then a linear differential inclusion ldi state space representation is established for the dynamics of the nn model according to the ldi state space representation a robustness design of fuzzy control is proposed to overcome the effect of modeling errors between the nmtd plant and the nn model if the designed fuzzy controller cannot stabilize the nmtd system a dither as an auxiliary of the fuzzy controller is simultaneously introduced to stabilize the nmtd system based on the relaxed method the nmtd system can be stabilized by regulating appropriately the parameters of dither if the frequency of dither is high enough the trajectory of the dithered system and that of its corresponding mathematical model the relaxed system can be made as close as desired this fact enables us to get a rigorous prediction of stability of the dithered system by establishing the stability of the relaxed system

complete stability robustness of third order lti multiple time delay systems a unique procedure is presented in this paper for a complete stability robustness of the third order lti multiple time delay systems lti mtds the uniqueness of the treatment is simply due to the fact that there is no comparable methodology presently in the literature the end result of this procedure is an exhaustive and precise determination of the stable regions in the domain of time delays the backbone of the method is a novel framework called the cluster treatment of characteristic roots ctcr ctcr is constructed over two fundamental propositions the first proposition claims the existence of a bounded number of so called kernel curves where the only imaginary characteristic roots occur the second proposition is on an interesting directional invariance property of the crossing tendencies of these imaginary roots for simplicity of conveyance and without loss of generality the number of time delays is taken as two in this document the new methodology is expandable to higher order dynamics with more time delays than two as the authors intend to demonstrate in future publications publications

a frequency sweeping framework for stability analysis of time delay systems for time delay systems the asymptotic behavior analysis of the critical imaginary roots w r t the infinitely many critical delays is an open problem in order to find a general solution we will exploit the link between the asymptotic behavior of critical imaginary roots and the asymptotic behavior of frequency sweeping curves from a new analytic curve perspective as a consequence we will establish a frequency sweeping framework with three main results 1 a finer regularity singularity classification for time delay systems will be obtained 2 the general invariance property will be proved and hence the asymptotic behavior of the critical imaginary roots w r t the infinitely many critical delays can be adequately studied 3 the complete stability problem can be fully solved moreover the frequency sweeping framework is extended to cover a broader class of time delay systems finally the geometric insights of frequency sweeping curves are investigated consequently some deeper properties on the asymptotic behavior of time delay systems and the link to frequency sweeping curves are found

brief pi stabilization of first order systems with time delay this paper considers the problem of stabilizing a first order plant with dead time using a pi controller using a version of the hermite biehler theorem applicable to quasipolynomials a complete and constructive characterization of all stabilizing pi gain values is obtained

an eigenvalue perturbation approach to stability analysis part ii when will zeros of time delay systems cross imaginary axis this paper presents an application of the eigenvalue series developed in part i j chen et al siam j control optim 48 2010 pp 5564 5582 to the study of linear time invariant delay systems focusing on the asymptotic behavior of critical characteristic zeros on the imaginary axis we consider systems given in state space form and as quasi polynomials and we develop an eigenvalue perturbation analysis approach which appears to be both conceptually appealing and computationally efficient our results reveal that the zero asymptotic behavior of time delay systems can in general be characterized by solving a simple eigenvalue problem and additionally when described by a quasi polynomial by computing the derivatives of the quasipolynomial

output feedback stabilisation of single input single output linear systems with i o network induced delays an eigenvalue based approach this work addresses the output feedback stabilisation problem for a class of linear single input single output systems subject to i o network delays more precisely we are interested in the characterisation of the set of delay and gain parameters guaranteeing the stability of the closed loop system to perform such an analysis we adopt an eigenvalue perturbation based approach various illustrative numerical examples complete the presentation

tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials a vandermonde based approach a standard approach in analyzing dynamical systems consists in identifying and understanding the eigenvalues bifurcations when crossing the imaginary axis efficient methods for crossing imaginary roots identification exist however to the best of the author s knowledge the multiplicity of such roots was not deeply investigated in recent papers by the authors 1 2 it is emphasized that the multiplicity of the zero spectral value can exceed the number of the coupled scalar delay differential equations and a constructive approach vandermonde based allowing to an adaptive bound for such a multiplicity is provided namely it is shown that the zero spectral value multiplicity depends on the system structure number of delays and number of non zero coefficients of the associated quasipolynomial rather than the degree of the associated quasipolynomial 3 this technical note extends the constructive approach in investigating the multiplicity of crossing imaginary roots j omega where omega neq 0 and establishes a link with a new class of functional confluent vandermonde matrices a symbolic algorithm for computing the lu factorization for such matrices is provided as a byproduct of the proposed approach a bound sharper than the polya szego generic bound arising from the principle argument is established

comments on stability of linear time invariant fractional delay systems of retarded type in the space of delay parameters automatica 49 2013 1287 1294 this correspondence points out that some results reported in a recent paper mesbahi and haeri 2013 are not correct mainly due to an insufficient consideration of the degenerate case three counterexamples to these results are presented

colored stability crossing sets for siso delay systems this paper considers the stability of a single input single output siso system with an input output i o delay called siso delay system hereafter we first derive an algebraic criterion to solve the complete stability w r t the delay tau next the stability analysis is extended to the case with additional free plant and or controller parameters denoted by a vector x by adopting a mathematical tool for polynomial algebra the discrimination system the complete classification of the critical imaginary roots cirs in the whole x tau parameter space may be obtained furthermore with the above results we present the stability crossing sets scss in a new form called the colored scss for each point on the scs the asymptotic behavior information of the corresponding cir can be coded by a certain color red green or yellow hence the stability in the x tau space can be analyzed easily and intuitively moreover the geometry of the scss can be investigated from a new perspective which will help not only to better understand the stability of the siso delay system but also to considerably simplify the procedure for generating the colored scss

inverted pendulum stabilization characterization of codimension three triple zero bifurcation via multiple delayed proportional gains the paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin the controller that we propose uses multiple delayed measurements instead of derivative terms doing so we increase the performances of the closed loop in presence of system uncertainties and or noisy measurements the problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally on one hand we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension three triple zero bifurcation on the other hand we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium the aim of this analysis is to characterize the possible local bifurcations finally the proposed control scheme is numerically illustrated and discussed

stability analysis of polynomially dependent systems by eigenvalue perturbation in this technical note we present a stability analysis approach for polynomially dependent one parameter systems the approach which appears to be conceptually appealing and computationally efficient and is referred to as an eigenvalue perturbation approach seeks to characterize the analytical and asymptotic properties of eigenvalues of matrix valued functions or operators the essential problem dwells on the asymptotic behavior of the critical eigenvalues on the imaginary axis that is on how the imaginary eigenvalues may vary with respect to the varying parameter this behavior determines whether the imaginary eigenvalues cross from one half plane into another and hence plays a critical role in determining the stability of such systems our results reveal that the eigenvalue asymptotic behavior can be characterized by solving a simple generalized eigenvalue problem leading to numerically efficient stability conditions

stability of linear time invariant fractional delay systems of retarded type in the space of delay parameters this paper presents a new method for assessing the bounded input bounded output stability of a class of fractional delay systems with commensurate orders and multiple commensurate delays of retarded type in the proposed method first by mapping the principal sheet of the riemann surface and a pseudo delay transformation an auxiliary polynomial is generated then this auxiliary polynomial is employed to find roots of the characteristic equation on the imaginary axis the properties of the root path close to these roots are used to identify intervals of delay values in which the system is stable the obtained results are illustrated via some numerical examples

brief paper extraction of 3d stability switching hypersurfaces of a time delay system with multiple fixed delays stability switching hypersurfaces ssh of a general class of linear time invariant lti multiple time delay systems mtds are extracted the number of delays and system order n 0 in this problem are given but they are arbitrarily large hence the extraction of ssh becomes non trivial a novel and efficient procedure reveals the 3d projections of dimensional ssh without the need for finding the ssh in achieving this computation times required remain in the same order of magnitude independently of and n 0 case studies demonstrate the effectiveness of the approach

the geometry of the space of null geodesics the topology and geometry of the space of null geodesics n of a space time m are used to study the causal structure of the space time itself in particular the question of whether the topology of n is hausdorff or admits a compatible manifold structure carries information on the global structure of m and the transversality properties of the intersections of skies of points tell whether the points are conjugate points on a null geodesic

normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation abstract the paper addresses for retarded functional differential equations fdes with parameters the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity the analysis is based on the theory previously developed for fdes without parameters by considering an extension of a phase space appropriate to the computation of these normal forms as an application the generic hopf bifurcation for scalar retarded fdes is considered

static output feedback stabilization necessary conditions for multiple delay controllers this note focuses on the static output feedback stabilization problem for a class of single input single output systems when the control law includes multiple distinct delays we are interested in giving necessary conditions for the existence of such stabilizing controllers illustrative examples second order system chain of integrators or chain of oscillators are presented and discussed

delay dependent energy to peak stability of 2 d time delay roesser systems with multiplicative stochastic noises this paper is concerned with the problem of energy to peak stochastic stability epss of two dimensional 2 d roesser systems in the presence of state time varying delays and multiplicative noises first a scheme that ensures a 2 d stochastic time delay system is stochastically stable with an attenuation performance is proposed the scheme presented in this paper can be regarded as an extension of the lyapunov krasovskii functional method for 2 d stochastic time delay systems focusing on the epss problem the proposed scheme is then utilized to derive delay dependent epss conditions in terms of tractable linear matrix inequalities a numerical example is given to illustrate the effectiveness of the derived stability conditions

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

on null and causal pseudoconvex space times the condition of pseudoconvexity has important implications for geodesic structures of space time low showed that this condition guarantees the existence of a smooth structure of the space of geodesics recently the place of this property within the causality ladder is in the attention there is a conjecture which states that every strongly causal space time is causally simple if and only if it is null pseudoconvex in this paper we study the difference of causal and null pseudoconvexity in space times and introduce the limit geodesic segment as an equivalence condition to the pseudoconvexity then we show that in the case of open subspaces of n minkowski space time this is causal geodesic connectedness finally we prove a refined form of the conjecture that says every strongly causal space time is causally simple if and only if it is maximal null pseudoconvex

stability with respect to the delay on a paper of k l cooke and p van den driessche a proposal for the correction of the main result of the title s paper is made some accompanying comments on other points are given so a simpler central result is derived an explicit representation of the set of stable delays is achieved

bounds of imaginary spectra of lti systems in the domain of two of the multiple time delays the stability of linear time invariant lti systems with independent multiple time delays and the cluster treatment of characteristic roots ctcr paradigm are investigated from a new perspective it is known that for such systems all the imaginary characteristic roots can be detected completely on a small set of hypersurfaces in the domain of the delays sipahi and olgac 2006a they are called kernel hypersurfaces kh the complete description of kh is the only prerequisite for the ctcr stability assessment procedure as the number of delays increases however their evaluation becomes infeasible instead we present a procedure to extract the 2 d cross sections of these hypersurfaces in the domain of any two of the delays by fixing the remaining delays in the 2 delay domain of interest the exact upper and lower bounds of the imaginary spectra are determined for this a combination of half angle tangent representation of the characteristic equation and the dixon resultant theory is used as the main contributions of this paper the complete kh are obtained by sweeping the root crossing frequency in this interval using this knowledge ctcr creates the cross section of the stability map in the domain of the two arbitrarily selected delays we demonstrate the effectiveness of this methodology over an example case study with three independent delays and two commensurate ones

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

input delay compensation of linear systems with both state and input delays by nested prediction abstract in this paper we consider the input delay compensation of linear systems with both state and input delays under the condition that the open loop time delay systems in the absence of input delay can be stabilized by delayed state feedback a nested predictor based approach is established to predict the future states so that the delays in the input are compensated it is shown that the compensated closed loop system in the presence of input delay possesses the same characteristic equation as the closed loop system in the absence of input delay the proposed approach can be generalized to general linear systems having multiple state delays and or multiple input delays to implement the nested predictor based controllers by applying numerical integration approaches an augmented nested predictor based approach by adding input filters is proposed two numerical examples are worked out to show the effectiveness of the proposed approach

technical communique invariance properties in the root sensitivity of time delay systems with double imaginary roots if i w ir is an eigenvalue of a time delay system for the delay t 0 then i w is also an eigenvalue for the delays t k t 0 k2 p w for any k z we investigate the sensitivity periodicity and invariance properties of the root i w for the case that i w is a double eigenvalue for some t k it turns out that under natural conditions the condition that the root exhibits the completely regular splitting property if the delay is perturbed the presence of a double imaginary root i w for some delay t 0 implies that i w is a simple root for the other delays t k k 0 moreover we show how to characterize the root locus around i w the entire local root locus picture can be completely determined from the square root splitting of the double root we separate the general picture into two cases depending on the sign of a single scalar constant the imaginary part of the first coefficient in the square root expansion of the double eigenvalue

stability analysis of time delay systems with incommensurate delays using positive polynomials we present a new stability condition for linear time delay systems tds with multiple incommensurate delays based on the rekasius substitution and positive polynomials the condition is checked by testing the positivity of multivariate polynomials in certain domains for this purpose we propose two alternative algorithms based on linear programming and sum of squares methods respectively the efficiency and accuracy of the algorithms is compared in an example to alternative stability conditions taken from the literature

combination of sign inverting and delay scheduling control concepts for multiple delay dynamics abstract in this paper we treat a novel combination of two controversial control concepts sign inverting control sic and delay scheduling ds for systems with multiple independent and large delays sic suggests the inversion of the control polarity and ds prolongs the existing delays the combined scheme functions with a single requirement that the union of the control schemes provides a larger stable operating region than each of its components does in the domain of the delays the critical knowledge that is needed to execute such a unified control strategy is the crisp description of these stable regions for each time delayed control scheme this need can be fulfilled using the recent cluster treatment of characteristic roots ctcr paradigm which establishes the stable regions exhaustively and non conservatively the resulting options in selecting operating modes render more robust control performance against much larger delay variations than each of the schemes we also investigate the disturbance rejection speeds within these enlarged stable regions in order to improve the control performance even further such multi faceted paradoxical combinations provide previously unexplored tools to control designers experimental validations of these novel concepts are presented on a simple setup with a single axis manipulator

absolute stability in delay equations abstract an equilibrium of a delay equation is said to be absolutely stable if it is asymptotically stable for all delays this is equivalent to a certain characteristic equation s having all its roots in the left half plane for all delays we obtain a simple necessary and sufficient condition for absolute stability of a class of delay equations

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

new results on eigenvalue distribution and controller design for time delay systems this paper considers the eigenvalue distribution of a linear time invariant lti system with commensurate time delays and its application to proportional integral derivative pid controller design for a delay plant via dominant eigenvalue assignment a new result on the root distribution of a quasi polynomial is first produced by applying part of pontryagin s conclusions this result which gives a necessary and sufficient condition can be directly used to judge the number of the right half plane eigenvalues of the characteristic equation of a time delay system based on the proposed result necessary and sufficient conditions on dominant eigenvalue assignment for pid control of time delay systems are presented and an algorithm is then provided to determine the pid controller gains the proposed approaches can assign some one or two eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones which enables us to design the controller according to the desired performance indexes for a standard first order system or a standard second order system in addition to stability the method is effective for the closed loop characteristic equation being retarded type or neutral type the controller gains to achieve the control objective can be characterized by a straightforward computation further a result on degradation to proportional integral pi control of time delay systems is given

computational scheme of a center manifold for neutral functional differential equations abstract this work addresses a computational algorithm of terms of a center manifold for neutral functional differential equations the bogdanov takens and the hopf singularities are considered finally as an illustration of our scheme we give an example where the second term of a center manifold is explicitly determined

stability of linear neutral time delay systems exact conditions via matrix pencil solutions in this note we study the stability properties of linear neutral delay systems we consider systems described by both neutral differential difference and state space equations and we seek to determine the delay margin of such systems that is the largest range of delay values for which a neutral delay system may preserve its stability in both cases we show that the delay margin can be found by computing the eigenvalues and generalized eigenvalues of certain constant matrices which can be executed efficiently and with high precision the results extend previously known work on retarded systems and demonstrate that similar stability tests exist for neutral systems in particular the tests require essentially the same amount of computation required for retarded systems

stabilizing a chain of integrators using multiple delays this note addresses the output feedback stabilization problem of a chain of integrators using multiple delays we shall prove that either n distinct delays or a proportional delay compensator with n 1 distinct delays are sufficient to stabilize a chain including n integrators we present two different approaches both are constructive and rely on frequency domain techniques on a derivative feedback approximation idea and a pole placement idea respectively an illustrative example triple integrator is presented

convergence factors of newton methods for nonlinear eigenvalue problems abstract consider a complex sequence k k 0 convergent to c with order p n the convergence factor is typically defined as the fraction c k k 1 k p in the limit k in this paper we prove formulas characterizing c k in the limit k for two different newton type methods for nonlinear eigenvalue problems the formulas are expressed in terms of the left and right eigenvectors the two treated methods are called the method of successive linear problems mslp and augmented newton and are widely used in the literature we prove several explicit formulas for c k for both methods formulas for both methods are found for simple as well as double eigenvalues in some cases we observe in examples that the limit c k as k does not exist for cases where this limit does not appear to exist we prove other limiting expressions such that a characterization of c k in the limit is still possible

advanced clustering with frequency sweeping methodology for the stability analysis of multiple time delay systems a new methodology advanced clustering with frequency sweeping acfs is introduced for the stability analysis of the most general class of linear time invariant time delay systems with multiple delays acfs is a cross fertilization of algebraic geometry frequency sweeping technique and the root clustering paradigm and it can directly extract the 2 d cross sections of the stability views in any two delay domain in this domain acfs can also test the delay independent stability based on necessary and sufficient conditions and can compute for the delay dependent case the precise lower and upper bounds of the only parameter the frequency that acfs sweeps case studies that are prohibitive to analyze with the existing analytical methods are presented to demonstrate the strengths of the method

robust stability of time delay systems there are two fundamental results available when we study stability of a polynomial family that is described by convex polytope in the coefficient space the edge theorem and the theory based on the concept of convex directions many known results can be explained with these two results this paper deals with a generalization of this line of research to the case of quasipolynomials that are entire functions which include both degree of the independent variable and exponential functions the main objects of the paper are the developing of the concept of convex directions for quasipolynomials and exploiting this concept for construction of testing sets for quasipolynomial families one of the primary sources of motivation for the class of problems considered in this paper is derived from process control a typical problem formulation almost always includes a delay element in each subsystem process block when we interconnect a number of such blocks in a feedback system the study of robust stability involves quasipolynomials of the sort considered in this paper

stability crossing set for systems with three delays this article describes the stability crossing set for linear time delay systems of arbitrary order with three delays the crossing frequency set which consists of all frequencies where a pair of zeros of the characteristic quasipolynomial may cross the imaginary axis is partitioned to grashof sets and non grashof sets of various types it was found that the general characteristics of the stability crossing set is completely determined by the partition structure of the crossing frequency set with the exception of degenerate cases this article provides a method of explicit and complete parameterization and geometric characterization of the stability crossing set of linear systems with three delays with the well known method of finding the crossing directions this provides a powerful method of finding parameter regions of stable systems extension to systems with additional fixed delays is also discussed

imaginary axis eigenvalues of matrix delay equations with a certain alternating coefficient structure abstract in this paper we examine the imaginary axis eigenvalues of matrix delay differential equations with coefficients alternating between hermitian and skew hermitian such as occurring in many mechanical systems we show that given this special structure and a certain sign condition the dimension of the eigenvalue problem is greatly reduced we also give some theorems on eigenvalue crossing directions in this context and show how reduction of dimension and crossing theorems work to provide insight on mechanical systems with time delay

well posedness for the cauchy problem for a system of semirelativistic equations the local well posedness for the cauchy problem of a system of semirelativistic equations in one space dimension is shown in the sobolev space hs of order s 0 we apply the standard contraction mapping theorem by using bourgain type spaces xs b we also use an auxiliary space for the solution in l2 h0 we give the global well posedness by this conservation law and the argument of the persistence of regularity

optimal sign inverting control for time delayed systems a concept study with experiments an intriguing control logic sign inverting control sic is considered for control systems with delayed feedback it starts with a nominal control law formulated for non delayed case and simply inverts the sign of the control gains for some surprising benefits when used with the delays this operation sounds paradoxical as the sign inversion potentially harms the stability of the non delayed dynamics however sic with large delays may yield some complementary benefits to the nominal control logic from delay robustness perspective the main question we address in this paper is how to select the nominal control law so that such a contribution can be a feasible b optimal in some sense a structured methodology is proposed to achieve this starting with a linear quadratic regulator based controller a single scaling factor on the corresponding control gains is used for one dimensional optimisation experimental validation of the concept of this optimal sic procedure is also reported on a single axis

stability analysis for lti systems with multiple time delays using the bounds of its imaginary spectra abstract the stability of linear time invariant lti systems with multiple independent time delays and the cluster treatment of characteristic roots ctcr paradigm are investigated from a new perspective any delay composition that results in an imaginary characteristic root lies either on a small number of kernel hypersurfaces kh or their infinitely many offspring hypersurfaces oh the complete description of kh is the only prerequisite for the ctcr based stability assessment procedure as the number of delays increases however the determination of these kh becomes computationally costly instead we present a practical procedure to extract the 2 d cross sections of the kh set in the domain of the two arbitrarily selected delays first we determine the exact upper and lower bounds of the imaginary spectra in this 2 d cross section of interest this process starts with a half angle tangent representation of the characteristic equation followed by the dixon resultant operation based on the full knowledge of kh the ctcr paradigm creates the complete stability map in the domain of these two arbitrarily selected delays we demonstrate the effectiveness of this methodology over an example case study

stability robustness analysis of multiple time delayed systems using building block concept an intriguing perspective is presented in studying the stability robustness of systems with multiple independent and uncertain delays it is based on a holographic mapping which is implemented over the domain of the delays this mapping considerably alleviates the problem which is otherwise known to be notoriously complex it creates a dramatic reduction in the dimension of the problem from infinity to manageably small number ultimately the process is reduced to studying the problem within a finite dimensional cube with edges of length 2pi in the new domain what we call the building block in essence the mapping collapses the entire set of potential stability switching points onto a small upperbounded number of building hypersurfaces we further demonstrate that these building hypersurfaces can be implicitly defined and they are completely isolated within the above mentioned cube it is also shown that the exhaustive detection of these building hypersurfaces is necessary and sufficient in order to arrive at the complete stability robustness picture we seek as a consequence this concept yields a very practical and efficient procedure for the stability assessment of such systems this novel perspective serves very well for the preparatory steps of the authors earlier contribution in the area cluster treatment of characteristic roots ctcr we elaborate on this combination which forms the main contribution of the paper several example case studies are also provided

on stability crossing curves for general systems with two delays for the general linear scalar time delay systems of arbitrary order with two delays this article provides a detailed study on the stability crossing curves consisting of all the delays such that the characteristic quasipolynomial has at least one imaginary zero the crossing set consisting of all the frequencies corresponding to all the points in the stability crossing curves are expressed in terms of simple inequality constraints and can be easily identified from the gain response curves of the coefficient transfer functions of the delay terms this crossing set forms a finite number of intervals of finite length the corresponding stability crossing curves form a series of smooth curves except at the points corresponding to multiple zeros and a number of other degenerate cases these curves may be closed curves open ended curves and spiral like curves oriented horizontally vertically or diagonally the category of curves are determined by which constraints are violated at the two ends of the corresponding intervals of the crossing set the directions in which the zeros cross the imaginary axis are explicitly expressed an algorithm may be devised to calculate the maximum delay deviation without changing the number of right half plane zeros of the characteristic quasipolynomial and preservation of stability as a special case

polynomial two parameter eigenvalue problems and matrix pencil methods for stability of delay differential equations several recent methods used to analyze asymptotic stability of delay differential equations ddes involve determining the eigenvalues of a matrix a matrix pencil or a matrix polynomial constructed by kronecker products despite some similarities between the different types of these so called matrix pencil methods the general ideas used as well as the proofs differ considerably moreover the available theory hardly reveals the relations between the different methods in this work a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem the polynomial two parameter eigenvalue problem of which the quadratic two parameter eigenvalue problem is a special case this framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods we also recognize a few new matrix pencil variants to determine dde stability finally the recognition of the new types of eigenvalue problem opens a door to efficient computation of dde stability

robust stability for time delay systems the edge theorem and graphical tests the robust stability problem is discussed for a class of uncertain delay systems where the characteristic equations involve a polytope p of quasi polynomials i e polynomials in one complex variable and exponential powers of the variable given a set d in the complex plane the goal is to find a constructive technique to verify whether all roots of every quasi polynomial in p belong to d that is to verify the d stability of p first it is demonstrated by counterexample that kharitonov s theorem does not hold for general delay systems next it is shown that under a mild assumption on the set d a polytope of quasi polynomials is d stable if and only if the edges of the polytope are d stable this extends the edge theorem for the d stability of a polytope of polynomials the third result gives a constructive graphical test for checking the d stability of a polytope of quasi polynomials which is especially simple when the set d is the open left half plane an application is given to demonstrate the power of the results

brief a practical method for analyzing the stability of neutral type lti time delayed systems a new paradigm is presented for assessing the stability posture of a general class of linear time invariant neutral time delayed systems lti ntds the ensuing method which we name the direct method dm offers several unique features it returns the number of unstable characteristic roots of the system in an explicit and non sequentially evaluated function of time delay t consequently the direct method creates exclusively all possible stability intervals of t furthermore it is shown that this method inherently verifies a widely accepted necessary condition for the t stabilizability of a lti ntds in the core of the dm lie a root clustering paradigm and the strength of rekasius transformation in mapping a transcendental characteristic equation into an equivalent rational polynomial in addition we also demonstrate by an example that dm can tackle systems with unstable starting posture for t 0 only to stabilize for higher values of delay which is rather unique in the literature

a unique methodology for the stability robustness of multiple time delay systems abstract the stability robustness is considered for linear time invariant lti systems with rationally independent multiple time delays against delay uncertainties the problem is known to be notoriously complex primarily because the systems are infinite dimensional due to delays multiplicity of the delays in this study complicates the analysis even further and rationally independent feature of the delays makes the problem prohibitively challenging as opposed to the tds with commensurate time delays where time delays are rationally related a unique framework is described for this broadly studied problem and the enabling propositions are proven we show that this procedure analytically reveals all possible stability regions exclusively in the space of the delays as an added strength it does not require the delay free system under consideration to be stable our methodology offers a resolution to this question which has been studied from variety of directions in the past four decades none of these respectable investigations can however deliver an exact and exhaustive robustness declaration from this stand point the new method has a unique contribution

small gain stability conditions for linear systems with time varying delays abstract in this paper we show that a variety of stability conditions both existing and new can be derived for linear systems subject to time varying delays in a unified manner in the form of scaled small gain conditions from a robust control perspective our development seeks to cast the stability problem as one of robust stability analysis and the resulting stability conditions are also reminiscent of robust stability bounds typically found in robust control theory the development is built on the well known conventional robust stability analysis requiring essentially no more than a straightforward application of the small gain theorem the derived conditions have conceptual appeal and they can be checked using standard robust control toolboxes