some problems in the stability of networked control systems with periodic scheduling this article addresses three stability problems related to networked control systems ncss with periodic scheduling where control systems may have multiple samplings in a hyperperiod a hyperperiod is a periodically repeated scheduling sequence for all tasks in an ncs as expected the analysis of a system with multiple samplings is much richer than the case with single sampling for example a system with two samplings may be stable unstable even if it is unstable stable when sampled by either sampling in this context it is important to understand how network induced delays and multiple samplings affect the system s stability in this article three particular stability problems involving constant and or time varying parameters are investigated and the corresponding stability regions are derived numerical examples and various discussions complete the presentation

on stability of lms methods and characteristic roots of delay differential equations we investigate the use of linear multistep lms methods for computing characteristic roots of systems of linear delay differential equations ddes with multiple fixed discrete delays these roots are important in the context of stability and bifurcation analysis we prove convergence orders for the characteristic root approximations and analyze under what condition for the steplength the discrete integration scheme retains certain delay independent stability properties of the original equations unlike existing results we concentrate on the recovery of both stability and instability we illustrate our findings with a number of numerical test results

lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

on stabilization of unstable steady states of autonomous ordinary differential equations via delayed feedback controls abstract the paper discusses stabilizing effects of some time delayed feedback controls applied to unstable steady states of an autonomous system of ordinary differential equations first we derive explicit delay dependent stability conditions that are applicable to a family of time delayed systems with simultaneously triangularizable system matrices then using this criterion and other argumentation we employ diagonal delayed feedback controls of conventional and pyragas type to stabilize unstable steady states of the studied autonomous system more precisely we formulate explicit non improvable and immediately applicable conditions on time delay and feedback strength that enable such a stabilization as an illustration we stabilize the unstable steady states of the rossler dynamical system considered under the standard choice of entry parameters when the uncontrolled system displays a chaotic behavior also we consider a non diagonal feedback control whose rotational gain matrix involving a feedback strength and phase commutes with the jacobi matrix of the uncontrolled system and show its larger stabilization potential with respect to the appropriate diagonal control the obtained results are tested by numerical experiments and confronted with the existing results as a supplement we provide matlab codes supporting theoretical conclusions

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

extended kronecker summation for cluster treatment of lti systems with multiple delays a new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant lti dynamics with multiple delays these hypersurfaces as they have very recently been introduced in a concept paper r sipahi and n olgac automatica 41 2005 pp 1413 1422 form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots ctcr in fact these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots to determine the kernel and offspring we use the extraordinary features of the extended kronecker summation operation in this paper the end result is that the infinite dimensional problem reduces to a finite dimensional one and preferably into an eigenvalue problem following the procedure described in this paper we are able to shorten the computational time considerably in determining these hypersurfaces we demonstrate these concepts via some example case studies one of the examples treats a 3 delay system for this case another interesting perspective called the building block is also utilized to display the kernel in three dimensional space in the domain of spectral delays

formality theorem for gerbes abstract the main result of the present paper is an analogue of kontsevich formality theorem in the context of the deformation theory of gerbes we construct an l deformation of the schouten algebra of multi vectors which controls the deformation theory of a gerbe

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

semi discretization method for delayed systems summary the paper presents an ecient numerical method for the stability analysis of linear delayed systems the method is based on a special kind of discretization technique with respect to the past eect only the resulting approximate system is delayed and also time periodic but still it can be transformed analytically into a high dimensional linear discrete system the method is applied to determine the stability charts of the mathieu equation with continuous time delay copyright 2002 john wiley sons ltd

a consensus protocol under directed communications with two time delays and delay scheduling this paper studies a consensus protocol over a group of agents driven by second order dynamics the communication among members of the group is assumed to be directed and affected by two rationally independent time delays one in the position and the other in the velocity information channels these delays are unknown but considered to be constant and uniform throughout the system the stability of the consensus protocol is studied using a simplifying factorisation procedure and deploying the cluster treatment of characteristic roots ctcr paradigm this effort results in a unique depiction of the exact stability boundaries in the domain of the delays the ctcr requires the knowledge of the potential stability switching loci exhaustively within this domain the creation of these loci is an important contribution of this work it is done in a new surrogate coordinate system called the spectral delay space the relative stability of the system that is the speed to reach consensus is also investigated fo

hochschild cohomology of abelian categories and ringed spaces this paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors we show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of hochschild cohomology for abelian categories we then show that this hochschild cohomology coincides with the one defined by gerstenhaber schack and swan in the case of module categories over diagrams and schemes and also with the hochschild cohomology for exact categories introduced recently by keller in addition we show in complete generality that hochschild cohomology satisfies a mayer vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf

stability criteria for certain third order delay differential equations in this paper we study the asymptotic stability of the zero solution of third order linear delay differential equations of the form y t p1y t p2y t q1y t q2y t v1y t v2y t where p1 p1 q1 q2 v1 and v2 are certain constants here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

critical delays and polynomial eigenvalue problems in this work we present a new method to compute the delays of delay differential equations ddes such that the dde has a purely imaginary eigenvalue for delay differential equations with multiple delays the critical curves or critical surfaces in delay space that is the set of delays where the dde has a purely imaginary eigenvalue are parameterized we show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value i e commensurate delays the parameterization is done by solving a quadratic eigenvalue problem which is constructed from the vectorization of a matrix equation and hence typically of large size for commensurate delay differential equations the corresponding equation is a polynomial eigenvalue problem as a special case of the proposed method we find a closed form for a parameterization of the critical surface for the scalar case we provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems

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

analysis and optimization of feedback control algorithms for data transfers in high speed networks two linear feedback control algorithms for handling and preventing congestion in high speed networks are proposed and analyzed the fluid approximation model is described with a continuous time system of delay differential equations the algorithms are asymptotically stable and the transient processes are nonoscillatory the control parameters are locally optimal optimality is based on the asymptotic rate of convergence the results of numerical experiments suggest that these parameters are globally optimal as well the dependence of the quality of service on the duration of the control intervals is analyzed and the performance of algorithms in a nonstationary environment is addressed

on the comparison of semi analytical methods for the stability analysis of delay differential equations abstract this paper provides the first comparison of the semi discretization spectral element and legendre collocation methods each method is a technique for solving delay differential equations ddes as well as determining regions of stability in the dde parameter space we present the necessary concepts assumptions and equations required to implement each method to compare the relative performance between the methods the convergence rate and computational time for each method is compared in three numerical studies consisting of a ship stability example the delayed damped mathieu equation and a helicopter rotor control problem for each study we present one or more stability diagrams in the parameter space and one or more convergence plots the spectral element method is demonstrated to have the quickest convergence rate while the legendre collocation method requires the least computational time the semi discretization method on the other hand has both the slowest convergence rate and requires the most computational time

algebraic k theory of stable c algebras abstract let a be a unital c algebra and let l denote the calkin algebra the bounded operators on a separable hilbert space modulo the compact operators k we prove the following conjecture of m karoubi the algebraic and topological k theory groups of the tensor product c algebra a l are equal the algebra a l may be regarded as a suspension of the more elementary c algebra a k thus karoubi s conjecture asserts roughly speaking that the algebraic and topological k theories of stable c algebras agree

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

critical frequencies and parameters for linear delay systems a lyapunov matrix approach abstract a new method based on recent results concerning the construction of the lyapunov matrix of delay systems reveals conditions under which the characteristic function has a root s 0 such that s 0 is also a root as the pure imaginary roots which are crucial in the stability analysis of time delay systems are of that type this method gives rise to a novel approach for the delay independent and delay dependent stability analyses of systems with delay that are multiple of a basic delay distributed delays and of a class of neutral type time delay systems a number of examples are given to illustrate the approach and to show its strength

an improved procedure in detecting the stability robustness of systems with uncertain delay a very efficient and reliable shortcut is explained here on the procedure which resolves the stability robustness of lti dynamics against an uncertain delay this procedure reduces the numerical complexity of the solution by an order of magnitude this note is in response to a number of user requests on the practicality of the mentioned methodology

deformations of sheaves of algebras abstract a construction of the tangent dg lie algebra of a sheaf of operad algebras on a site is presented the requirements on the site are very mild the requirements on the algebra are more substantial a few applications including the description of deformations of a scheme and equivariant deformations are considered the construction is based upon a model structure on the category of presheaves which should be of an independent interest

mesoscopic simulations of systolic flow in the human abdominal aorta the complex nature of blood flow in the human arterial system is still gaining more attention as it has become clear that cardiovascular diseases localize in regions of complex geometry and complex flow fields in this article we demonstrate that the lattice boltzmann method can serve as a mesoscopic computational hemodynamic solver we argue that it may have benefits over the traditional navier stokes techniques the accuracy of the method is tested by studying time dependent systolic flow in a 3d straight rigid tube at typical hemodynamic reynolds and womersley numbers as an unsteady flow benchmark simulation results of steady and unsteady flow in a model of the human aortic bifurcation reconstructed from magnetic resonance angiography are presented as a typical hemodynamic application

stability and bifurcations of equilibria in a multiple delayed differential equation the influence of multiple negative delayed feedback loops on the stability of a single action mechanism are considered a characteristic equation for the linearized stability of the equilibrium is completely analyzed as a function of two parameters describing a delay in one loop and a ratio of the gains in the two feedback loops the bifurcations occurring as the linear stability is lost are analyzed by the construction of a centre manifold in particular the nature of hopf and more degenerate higher codimension bifurcations are explicitly determined

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

critical effects of the polarity change in delayed states within an lti dynamics with multiple delays this note concerns the polarity inversion of the delayed states in linear time invariant multiple time delay systems lti mtds to start with for such systems the assessment of asymptotic stability is complicated due to the infinite dimensionality introduced by the delays it is shown that the mentioned polarity inversion influences the respective delay dependent stability maps in interesting ways there are some intriguing invariant characteristics and correspondences between the original and inverted systems which form the primary contributions of this note these findings are shown to be the enabling features to relate the stability maps of the two systems the cluster treatment of characteristic roots ctcr paradigm is utilized to identify these features an example case study is provided to illustrate the claims

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

lattice boltzmann method for parallel simulations of cardiac electrophysiology using gpus this work presents the lattice boltzmann method lbm for computational simulations of the cardiac electrical activity using monodomain model an optimized implementation of the lattice boltzmann method is presented which uses a collision model with multiple relaxation parameters in order to consider the anisotropy of the cardiac tissue with focus on fast simulations of cardiac dynamics due to the high level of parallelism present in the lbm a gpu parallelization was performed and its performance was studied under regular and irregular three dimensional domains the results of our optimized lattice boltzmann parallel implementation for cardiac simulations have shown acceleration factors as high as 500 for the overall simulation and for the lbm a performance of 419 mega lattice updates per second was achieved with near real time simulations in a single computer equipped with a modern gpu these results show that the proposed framework is a promising approach for application in a clinical workflow

efficient computation of characteristic roots of delay differential equations using lms methods we aim at the efficient computation of the rightmost stability determining characteristic roots of a system of delay differential equations the approach we use is based on the discretization of the time integration operator by a linear multistep lms method the size of the resulting algebraic eigenvalue problem is inversely proportional to the steplength we summarize theoretical results on the location and numerical preservation of roots furthermore we select nonstandard lms methods which are better suited for our purpose we present a new procedure that aims at computing efficiently and accurately all roots in any right half plane the performance of the new procedure is demonstrated for small and large scale systems of delay differential equations

the asymptotic stability of linear autonomous systems with commensurate time delays the asymptotic stability of linear systems with commensurate time delays is studied algorithms are given to check if a given system is asymptotically stable independent of delay or to determine the largest delay time the system can tolerate to preserve its asymptotic stability these algorithms are simple to apply since they require only to find the eigenvalues of two specially constructed matrices and some simple mathematical calculations

pseudospectral differencing methods for characteristic roots of delay differential equations in d breda s maset and r vermiglio ima j numer anal 24 2004 pp 1 19 and d breda the infinitesimal generator approach for the computation of characteristic roots for delay differential equations using bdf methods research report udmi rr17 2002 dipartimento di matematica e informatica universita degli studi di udine udine italy 2002 the authors proposed to compute the characteristic roots of delay differential equations ddes with multiple discrete and distributed delays by approximating the derivative in the infinitesimal generator of the solution operator semigroup by runge kutta rk and linear multistep lms methods respectively in this work the same approach is proposed in a new version based on pseudospectral differencing techniques we prove the spectral accuracy convergence behavior typical of pseudospectral schemes as also illustrated by some numerical experiments

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

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

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

stability crossing set for systems with two scalar delay channels the characteristic function of a system with three scalar delay channels contains cross terms of different delays this article studies the parameterization and geometric structure of the stability crossing set the set of delay combinations with at least one characteristic root on the imaginary axis for such systems understanding the structure of this set is crucial to the identification of stable regions in the delay parameter space using the d subdivision method the presence of the cross terms significantly complicates the analysis and requires a quite different method than the case without these cross terms and it involves more numerical computation

deformation quantization of gerbes this is the first in a series of articles devoted to deformation quantization of gerbes we introduce basic definitions interpret deformations of a given stack as maurer cartan elements of a differential graded lie algebra dgla and classify deformations of a given gerbe in terms of maurer cartan elements of the dgla of hochschild cochains twisted by the cohomology class of the gerbe we also classify all deformations of a given gerbe on a symplectic manifold as well as provide a deformation theoretic interpretation of the first rozansky witten class

deformation theory of abelian categories in this paper we develop the basic infinitesimal deformation theory of abelian categories this theory yields a natural generalization of the well known deformation theory of algebras developed by gerstenhaber as part of our deformation theory we define a notion of flatness for abelian categories we show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems

multiple relaxation time lattice boltzmann model for the convection and anisotropic diffusion equation a lattice boltzmann model with a multiple relaxation time mrt collision operator for the convection diffusion equation is presented the model uses seven discrete velocities in three dimensions d3q7 model the off diagonal components of the relaxation time matrix which originate from the rotation of the principal axes enable us to take into account full anisotropy of diffusion an asymptotic analysis of the model equation with boundary rules for the dirichlet and neumann type specified flux conditions is carried out to show that the model is first and second order accurate in time and space respectively the results of the analysis are verified by several numerical examples it is also shown numerically that the error of the mrt model is less sensitive to the variation of the relaxation time coefficients than that of the classical bgk model in addition an alternative treatment for the neumann type boundary condition that improves the accuracy on a curved boundary is presented along with a numerical example of a spherical boundary

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

a fundamental control performance limit for a class of positive nonlinear systems abstract a fundamental performance limit is derived for a class of positive nonlinear systems the performance limit describes the achievable output response in the presence of a positive disturbance and subject to a sign constraint on the allowable input an explicit optimal input is derived which minimises the maximum output response whilst ensuring that the minimum output response does not fall below a pre specified lower bound the result provides a fundamental performance standard against which all control policies including closed loop schemes can be compared implications of the result are examined in the context of blood glucose regulation for type 1 diabetes

absolute stability in linear delay differential systems ill posedness and robustness the author considers matrix delay differential systems which are polynomial in several delay operators using a necessary and sufficient criterion for stability independent of delay or absolute stability the author shows that system stability for all values of the delay vector lying in a sector will imply absolute stability we then show that absolute stability is ill posed with respect to arbitrarily small perturbations of the delay ratios if a certain extended delay differential system which is formed from the original is not also absolutely stable

consensus analysis with large and multiple communication delays using spectral delay space concept in this study we consider the consensus problem for a group of second order agents interacting under a fixed undirected communication topology communication lines are affected by two rationally independent delays the first delay is assumed to be in the position information channels whereas the second delay is in the velocity information exchange the delays are assumed to be large and uniform throughout the entire network the stability analysis of such systems becomes quickly intractable as the number of agents increases and the delays enlarge to resolve this dilemma we first reduce the complexity of the problem dramatically by decomposing the characteristic equation of the system into a set of second order factors then we assess the stability of the resulting subsystems exactly and exhaustively in the domain of the time delays using the cluster treatment of characteristic roots ctcr paradigm ctcr requires the determination of all the potential stability switching loci in the domain of the d

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

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

a delay dependent stability criterion for systems with uncertain time invariant delays this paper provides a new stability criterion for systems with time invariant uncertain delays based on an improved upper bound for the inner product of two vectors a new delay dependent robust stability criterion is derived which is shown by an example less conservative than existing stability criteria

stability of linear time periodic delay differential equations via chebyshev polynomials this paper presents a new technique for studying the stability properties of dynamic systems modeled by delay differential equations ddes with time periodic parameters by employing a shifted chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period the dynamic system can be reduced to a set of linear difference equations for the chebyshev expansion coefficients of the state vector in the previous and current intervals this defines a linear map which is the infinite dimensional floquet transition matrix u two different formulas for the computation of the approximate u whose size is determined by the number of polynomials employed are given the first one uses the direct integral form of the original system in state space form while the second uses a convolution integral variation of parameters formulation additionally a variation on the former method for direct application to second order systems is also shown an error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance an extension of the method to the case where the delay and parametric periods are commensurate is also shown stability charts are produced for several examples of time periodic ddes including the delayed mathieu equation and a model for regenerative chatter in impedance modulated turning the results indicate that this method is an effective way to study the stability of time periodic ddes

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

exact stability analysis of second order leaderless and leader follower consensus protocols with rationally independent multiple time delays abstract an investigation of double integrator agents with directed asymmetric consensus protocols and multiple rationally independent time delays is presented in this paper from two novel perspectives first we complement the group consensus literature on crucial stability analysis using a recent technique called the cluster treatment of characteristic roots ctcr for the first time on this class of time delayed systems the ctcr paradigm is pursued after a block diagonalization mode decoupling transformation on the system this treatment produces some unique stability tables for the dynamics in the space of the delays which are non conservative and exhaustive second a novel concept of spectral delay space is presented as an overture to the ctcr for the determination of the complete set of stability crossing switching hypersurfaces in the delay space examples are provided to display the strengths and efficiency of this new stability analysis mechanism

mapping based algorithm for large scale computation of quasi polynomial zeros a method for computing all zeros of a retarded quasi polynomial that are located in a large region of the complex plane is presented the method is based on mapping the quasi polynomial and on utilizing asymptotic properties of the chains of zeros first the asymptotic exponentials of the chains are determined based on the distribution diagram of the quasi polynomial secondly large regions free of zeros are defined finally the zeros are located as the intersection points of the zero level curves of the real and imaginary parts of the quasi polynomial which are evaluated over the areas of the region outside those free of zeros

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

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

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties

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 in linear delay equations abstract for linear autonomous differential difference equations of retarded or neutral type necessary and sufficient conditions are given for the zero solution to be stable hyperbolic for all values of the delays

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

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

technical communique on delay robustness analysis of a simple control algorithm in high speed networks this paper focuses on the robustness analysis of a simple second order control algorithm for data transfer in high speed networks the model under consideration derived using a fluid approximation technique can be found in izmailov siam j contr optimiz 34 1996 1767 the novelty of the approach lies in characterizing the stability of the scheme in the delay parameter space where the delays correspond to the control time interval and to the round trip time respectively thus we shall compute some delay insensitive measures for the algorithm which give upper and lower bounds for the uncertainty in the knowledge of the round trip times

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

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

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

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

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

algebroid prestacks and deformations of ringed spaces for a ringed space x o we show that the deformations of the abelian category mod o of sheaves of o modules lowen and van den bergh 2006 are obtained from algebroid prestacks as introduced by kontsevich in case x is a quasi compact separated scheme the same is true for qch o the category of quasi coherent sheaves on x it follows in particular that there is a deformation equivalence between mod o and qch o

delayed controller design in communication networks via positive output feedback this paper addresses delayed controller design in communication networks cns with unstable open loop system employing the nyquist curve approach the sufficient and necessary conditions of the gains and the delays respectively are obtained the controllers are designed by using delayed positive output feedback some examples are presented to illustrate the effectiveness and simplicity of the designed controllers in both the time and frequency domains

deformations of algebroid stacks abstract in this paper we consider deformations of an algebroid stack on an etale groupoid we construct a differential graded lie algebra dgla which controls this deformation theory in the case when the algebroid is a twisted form of functions we show that this dgla is quasiisomorphic to the twist of the dgla of hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe