frequency domain criterion for robust stability of interval time delay systems abstract in this paper we characterize the boundary f b of the image f b of a box b in r m under a nonlinear mapping f r m c we generalize results recently reported by polyak and kogan 1993 necessary and sufficient conditions for robust stability of multiaffine systems mathematics research report 93 06 university of maryland baltimore county for multiaffine mappings and provide computationally tractable necessary and sufficient robust stability conditions for quasipolynomials with interval coefficients and interval delays a numerical stability verification for a quasipolynomial family with two interval delays is presented

robust stability of neutral delay differential systems abstract in this paper we consider the robust stability problem for a neutral delay differential system where the characteristic equations involve a polytope of quasi polynomials first the edge theorem is developed for robust stability analysis of a polytope of neutral quasi polynomials based on the edge theorem the robust stability of neutral systems with commensurate delays will be reduced to the checking of the h hurwitz stability of all the edges of a polytope of quasi polynomials and the schur stability of all the edges of the subpolytope of neutral terms then an effective graphical test is made for checking the robust stability of a polytope of neutral quasi polynomials finally we show that checking vertic quasi polynomials is sufficient for the asymptotic stability of the entire family of first order quasi polynomials of the neutral type

some existence results for a periodic problem with non smooth potential abstract in this paper two existence results for a class of second order periodic boundary value problems with non smooth potential are obtained we extend the castro lazer thews reduction method to non smooth functionals the obtained result is then exploited to prove the existence of a nontrivial solution furthermore we prove the existence of multiple solutions by using a multiplicity result based on local linking

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

brief paper preservation of exponential stability for linear non autonomous functional differential systems we consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation as particular cases the system includes models with concentrated delays and systems of integrodifferential equations our method is based on bohl perron type theorems

robust stability of quasi polynomials and the finite inclusions theorem a semianalytic tool for the study of the robust stability of quasi polynomials of retarded type is presented the approach is based on graphical methods the finite nyquist theorem for polynomials is extended to the case of quasi polynomials with the help of a novel approach the finite inclusions theorem for quasi polynomials is derived it allows to conclude on the stability of polytopic families of quasi polynomials from a finite number of testing values

on nontrivial solutions of a dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue let be a bounded domain in rn n 2 with smooth boundary let p r r be a continuous function such that p 0 0 and limt p t t p exist we discuss the existence of nontrivial solutions of the dirichlet problem u p u in u 0 on our hypotheses relate the position of the point p p in the plane with the extended spectrum of they seem to allow noticeably greater freedom for the numbers p than is usually found in the literature in the sense that the interval with end points p is here permitted to contain in its interior an eigenvalue of arbitrarily multiplicity of it also appears that we significantly weaken some technical conditions on p of another paper which treats the crossing of a simple eigenvalue

robust stability analysis of linear time delay systems by lambert w function some extreme point results robust stability conditions are derived for linear time delay systems using lambert w function the characteristic quasi polynomials of the systems are assumed to be factorized it is proven that if uncertainties in the coefficients of the quasi polynomial are set in appropriate regions in the complex plane we can enjoy extreme point results finite number of stability checks at some points of the boarder of the regions suffice the strength of lambert w function approach lies in the fact that the function is implemented on some standard software packages such as mathematica maple or matlab which afford to compute the function value very easily the above two points make the stability test for the class of uncertain time delay systems quite practical

basis of the identities of the matrix algebra of order two over a field of characteristic p 2 abstract in this paper we prove that the polynomial identities of the matrix algebra of order 2 over an infinite field of characteristic p 2 admit a finite basis we exhibit a finite basis consisting of four identities and in almost all cases for p we describe a minimal basis consisting of two identities the only possibilities for p where we do not exhibit minimal bases of these identities are p 3 and p 5 we show that when p 3 one needs at least three identities and we conjecture a minimal basis in this case in the course of the proof we construct an explicit basis of the vector space of the central commutator polynomials modulo the ideal of the identities of the matrix algebra of order two

computing the frequency response of linear systems with parametric perturbation abstract the problem of computing the frequency response of linear systems with parametric perturbation is addressed under the assumption that the coefficients of the system transfer function depend on the perturbation parameters in a linear manner we provide results for plotting the frequency response of the perturbed transfer function these results are useful in determining the h norm gain margin and phase margin and in improving the diagonal dominance of multi input multi out put uncertain systems an illustrative example is provided to demonstrate the results

qft template generation for time delay plants based on zero inclusion test abstract plant template generation is the key step in applying quantitative feedback theory qft to design robust control for uncertain systems in this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients the main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f t q f q t q q k 0 m 1 q k q k where f q g q h q e j g q and h q are both complex valued affine functions of the m dimensional real vector q and is a fixed frequency based on this condition an efficient algorithm which involves in the worst case evaluation of m algebraic inequalities and solution of m2m 1 one variable quadratic equations is developed for testing the zero inclusion of the value set f t q this zero inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution the proposed template generation technique has a linear computational complexity in resolution and is therefore more efficient than the parameter gridding and interval methods a numerical example illustrating the proposed technique and its computational superiority over the interval method is included

preservation of stability in delay equations under delay perturbations we consider a class of linear delay equations with perturbed time lags and present conditions which guarantee that the asymptotic stability of the trivial solution of the equation at hand is preserved under these perturbations as an example we show how our results can be used to obtain an estimate on the maximum allowable sampling interval in the stabilization of a hybrid system with feedback delays we also present applications of our perturbation theorem to obtain stability conditions for delay equations with multiple delays

feedback controllers for a wind tunnel model involving a delay analytical design and numerical simulation an application of a finite spectrum assignment method for time delay systems to a feedback control of mach number in a wind tunnel is presented the linearized model of mach number control is a system of three state equations with a delay in one of the state variables the proposed feedback is a linear combination of state variables and weighted integrals of some of the state variables over a period equal to time delay the spectrum of the closed loop system is finite and consists of three eigenvalues that can be placed arbitrarily four possible variants of the feedback control law are presented the calculation of feedback coefficients is very simple which makes it possible to update the controller s parameters on line with the changes of the operating point an extensive numerical simulation of the system dynamics and the feedback controllers was done and led to several conclusions in particular a very good settling time under problem constraints has been obtained and several nice features of the proposed feedback controller have been demonstrated

multiplicity of nontrivial solutions for quasilinear elliptic equation two nontrivial solutions are obtained by the reduction method for the nonautonomous semilinear elliptic dirichlet boundary value problem some well known multiplicity results are generalized

delay dependent stability analysis by using delay independent integral evaluation for the stability analysis of time delay systems the available methods usually require the exact evaluation of some quantities the definite integral stability method originated from the argument principle or the cauchy theorem is effective because it only requires a rough estimation of the testing integral over a finite interval to judge stability however no general rule is given in the literature for properly choosing the upper limit of the testing integral in this paper two simple algorithms are presented for finding the parameter dependent critical upper limit and a parameter independent upper limit without any restriction on the number of time delays these results improve and complete the definite integral stability method as illustrated by the numerical examples the proposed algorithms work effectively and accurately

robust stability of quasi polynomials frequency sweeping conditions and vertex tests in this paper we study the robust stability of uncertain time delay systems we consider uncertain quasi polynomials whose coefficients may vary in a certain prescribed range our goal is to derive necessary and sufficient conditions for such uncertain quasi polynomials to maintain stability independent of delay parameters our primary contributions are frequency sweeping conditions for interval diamond and spherical quasi polynomial families which can be readily checked requiring only the computation of two simple frequency dependent functions additionally we also obtain vertex and edge type results in the spirit of the kharitonov approach known in robust stability analysis showing that the stability of interval and diamond quasi polynomials can be ascertained by checking the stability of certain special vertex and or edge members in those families both type of results provide necessary and sufficient conditions for the quasi polynomial families to be robustly stable independent of delay

robust hurwitz stability test for linear systems with uncertain commensurate time delays the note deals with the robust hurwitz stability of a class of linear systems with uncertain commensurate time delays which is formulated as the robust hurwitz stability of a nonpolytopic family of quasi polynomials a np hard problem from the viewpoint of computational complexity on the basis of the edge theorem and sylvester resultant elimination a necessary and sufficient condition is developed for the robust stability of the whole family and it yields an effective testing procedure an illustrative example is given to show the effectiveness of the method

on the graded identities for elementary gradings in matrix algebras over infinite fields we find a basis for the g graded identities of the n n matrix algebra mn k over an infinite field k of characteristic p 0 with an elementary grading such that the neutral component corresponds to the diagonal of mn k

technical communique stabilizability considerations and design of rational controllers for a class of time delay systems the problem of stabilization of a linear system characterized by its rational transfer function with an input time delay by a rational controller is considered if the delay is not known at all its interval is infinite the necessary and sufficient condition for the existence of a rational controller is stability of the open loop rational plant moreover the existence of a rational stabilizing controller implies the existence of a constant gain stabilizing controller if the delay is known to lie in a given finite interval two non existence theorems are derived one for the constant gain stabilizing controller and one for any rational stabilizing controller a design method for stabilizing rational controllers based on first order all pass filters cascaded by a constant gain and on violating the conditions for the non existence theorem of the constant gain stabilizer is presented

on stability of a weighted diamond of real quasi polynomials this paper deals with the stability problem for a weighted diamond of real quasi polynomials we show that under certain conditions on the weights and coefficients in the exponents the stability of the weighted diamond follows from the stability of eight one parameter families edges of quasi polynomials in order to check the diamond for stability it is sufficient to examine only eight one parameter families of quasi polynomials in contrast to the case of a rectangle of quasi polynomials which requires checking the stability of an exponential number of one parameter families of quasi polynomials

stabilization of vibration systems via delayed state difference feedback this paper presents a study of the stabilization problem via delayed state feedback of pyragas type for unstable vibration systems that have even number of characteristic roots with positive real parts on the basis of stability switches with respect to the time delay a new stabilization criterion is established and an effective procedure for determining the admissible values of the feedback gains and the delay is given two examples are given in detail to demonstrate the efficiency of the theory

multiple solutions for resonant elliptic equations via local linking theory and morse theory abstract we consider two classes of elliptic resonant problems first by local linking theory we study the double double resonant case and obtain three solutions second we introduce some new conditions and compute the critical groups both at zero and at infinity precisely combining morse theory we get three solutions for the completely resonant case

an effective analytical criterion for stability testing of fractional delay systems this paper investigates the bibo stability of a class of fractional delay systems with rational orders a stability that holds true if all the characteristic roots have negative real parts only based on the argument principle for complex functions as well as hassard s technique for ordinary time delay systems an explicit formula is established for calculating the number of characteristic roots lying in the closed right half complex plane of the first sheet of the riemann surface and in turn a sufficient and necessary condition is obtained for testing the bibo stability of fractional delay systems as shown in the illustrative examples this stability criterion involves easy computation and works effectively

the ring of generic matrices in chapter 3 we defined a single n x n generic matrix we now introduce the ring generated by a set of n x n generic matrices

on stability of quasipolynomials with annular uncertainties and interval commensurate delays the paper concerns the hurwitz stability of a family of quasipolynomials with commensurate delays each coefficient of the quasipolynomials belongs to a prescribed annulus in the complex plane and the delay belongs to a prescribed real interval a computationally tractable robust stability criterion is the main result of the paper

nonlinear elliptic boundary value problems at resonance nor do we assume more smoothness than mere continuity of g x t in t in particular no assumption concerning the modulus of continuity of g x t in i is made castro and lazer l studied 0 1 under the assumptions that a a 5 g x tyt i a a and ag x t is increasing in it i ambrosetti and mancini 2 consider the situation when g x t r j x r with f x f uniformly bounded thews 3 assumed 1j h x f f 5 g x t t 5 it hz x t t where the hi are bounded and satisfy

on nontrivial solutions of an asymptotically linear dirichlet problem let be a bounded domain in rn n 2 with smooth boundary let r r be a continuous function such that 0 0 and lim exist we discuss the existence of nontrivial solutions of the dirichlet problem u u in u 0 on our hypotheses seem to allow significantly greater freedom for the numbers lim than is usually found in the literature in that the interval with end points is here permitted to contain an eigenvalue of subject to zero dirichlet data furthermore we do not require any differentiability of and we weaken other technical conditions on that are generally assumed in other papers dealing with this problem

technical communique on minimal order stabilization of minimum phase plants in this note the problem of minimal order stabilization in the case where the plant is minimum phase is studied a low bound on the order of stabilizers is derived and a set of minimal order stabilizers are characterized the low bound is related to the number and location of the plant s unstable and lightly damped poles and the number of zeros how to construct a minimal order or low order stabilizer for a general case is also discussed and the algorithm is provided numerical examples are given to illustrate the proposed method

robustness of sampled data control systems having parametric uncertainty a conic sector approach the robustness of sampled data control systems when the open loop system is modeled by a family of plants having affine or interval uncertainty structure is considered using the conic sector approach as adapted to sampled data systems the authors give a sufficient condition for robust stability that depends only on edge parameters in the case of affine uncertainty and on vertex parameters when interval uncertainty is assumed

on possibilities of the extension of kharitonov s stability test for interval polynomials to the discrete time case it is shown that the kharitonov test for hurwitz stability of an interval polynomial does not extend in general to the discrete time case unless the degree n of the polynomial is not greater than two for n leq 3 a given monic interval polynomial has all roots inside the unit disk if all 2 3 8 extreme polynomials have that property instead of only four polynomials in kharitonov s test for n 4 it is shown by a counterexample that discrete time stability of all extreme polynomials does not guarantee the stability of the interval polynomial

robust stability of continuous time difference systems in this paper some special results about the robust stability of a linear continuoustime difference system are given these results may be treated as a natural generalization of the well known results on the schur stability of polynomial families defined by convex polytopes in the coefficient space

stability tests of interval time delay systems abstract this paper presents some sufficient conditions for robust stability independent of delay and decaying rate for uncertain time delay systems the obtained robustness results are extended to cover the class of interval time delay systems sufficient conditions are also derived for checking the instability of this class of systems the results are given in terms of norm and matrix measure easy to check a comparison between the derived conditions is given numerical computations are performed to illustrate the results

some results for semilinear differential equations at resonance where the linear differential operator a is self adjoint with a nontrivial kernel on l o and the nonlinear map f x 0 satisfies some growth conditions for large values of r the starting of the problem is the wellknown paper of landesman and lazer 11 they assume that f x as t a and x co and are able to prove necessary and sufficient conditions for 1 1 to be solvable since then many works have been done on the programs we refer to the extensive bibliographies of the paper by brezis and nirenberg 2 and the survey paper by fucik 3 this paper is stimulated by the work of amann and mancini 14 in 4 they present a very general existence theorem for the case where the nonlinearity does not cross an eigenvalue that is f satisfies

on an elliptic boundary value problem at double resonance abstract let be a bounded domain in r n n 2 with smooth boundary we discuss the existence and multiplicity of solutions of the boundary value problem u p x u in u 0 on roughly speaking under the assumption that l lim t p x t t l 1 for two distinct consecutive eigenvalues l l 1 of the first boundary value problem of

robust stability of time delay systems with an uncertain time delay constant frequency domain methods are described for testing the robust stability of a class of time delay systems with an uncertain time constant both parametric and non parametric uncertainties are considered our technique determines without a sweep of time delay the maximum time delay constant which preserves the robust stability

a graphical test for the interval stability of fractional delay systems this paper investigates the bibo bounded input and bounded output interval stability testing of fractional delay systems a problem of justifying the bibo stability of a polytopic family of functions involving fractional order powers as well as exponential powers it proves that the bibo stability of the polytope is governed by the bibo stability of the edges of the polytope and the latter can be tested graphically via frequency response plots the main results generalize some results in the literature

the edge theorem and graphical tests for robust stability of neutral time delay systems abstract in this paper we consider the robust stability problem for a class of uncertain neutral time delay systems where the characteristic equations involve a poly tope p of quasipoly neutrals of neutral type given a stability region d in the complex plane our goal is to find a constructive technique to verify whether all roots of every quasipolynomial in p belong to d that is to verify the d stability of p our first result is that under a mild assumption on the stability region d p is d stable if and only if the edges of p are d stable hence the d stability problem of a higher dimensional polytope is reduced to the d stability problem of a finite number of pairwise convex combinations of vertices this extends the edge theorem developed by bartlett hollot and lin bhl88 and fu and barmish fb88 for the d stability of a poly tope of polynomials and that of fu olbrot and polis fop88 for a poly tope of quasipolynomials our second result gives effective graphical tests for checking the d stability of a polytope of quasipolynomials of neutral type when the set d is any open left half plane the graphical test is based on the frequency response plots of some transfer functions associated with the vertices

group gradings on full matrix rings we study g gradings of the matrix ring mn k k a field and give a complete description of the gradings where all the elements ei j are homogeneous called good gradings among these we determine the ones that are strong gradings or crossed products if g is a finite cyclic group and k contains a primitive g th root of 1 we show how all g gradings of mn k can be produced in particular we give a precise description of all c2 gradings of m2 k and show that for algebraically closed k any such grading is isomorphic to one of the two good gradings

some discrete time counterparts to kharitonov s stability criterion for uncertain systems in 1 kharitonov gave an elegant and simple stability criterion for continuous time systems this note reports on similar results for discrete time systems

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

existence and multiplicity of solutions of semilinear elliptic equations the existence and multiplicity results of solutions are obtained by the reduction method and the minimax methods for nonautonomous semilinear elliptic dirichlet boundary value problem some well known results are generalized

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

robust stability under a class of nonlinear parametric perturbations this paper considers the robust stability verification of linear time invariant systems admitting a class of nonlinear parametric perturbations the general setting is one of determining the closed loop stability of systems whose open loop transfer functions consist of powers products and ratios of polytopes of polynomials apart from this general setting two special cases of independent interest are also considered the first special case concerns uncertainties in the open loop gain and real poles and zeros while the second special case treats uncertainties in the open loop gain and complex poles and zeros in light of the zero exclusion principle robust stability is equivalent to zero exclusion of the value sets of the system characteristic function a value set consists of the values of the characteristic functions at a fixed frequency the main results of this paper are as follows 1 the value set of the characteristic function at each fixed frequency is determined by the edges and some frequency dependent internal line segments 2 consequently hurwitz invariance verification simplifies to that of checking certain continuous scalar functions for avoidance of the negative real axis 3 for the case of real zero pole gain variations the critical lines are all frequency independent and therefore the determination of the robust stability is even simpler 4 for the case of complex zero pole gain variations the critical internal lines are shown to be either frequency independent or to be confined in certain two dimensional planes or three dimensional boxes

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

graded identities of block triangular matrices abstract let f be an infinite field and u t d 1 d n be the algebra of upper block triangular matrices over f in this paper we describe a basis for the g graded polynomial identities of u t d 1 d n with an elementary grading induced by an n tuple of elements of a group g such that the neutral component corresponds to the diagonal of u t d 1 d n in particular we prove that the monomial identities of such algebra follow from the ones of degree up to 2 n 1 our results generalize for infinite fields of arbitrary characteristic previous results in the literature which were obtained for fields of characteristic zero and for particular g gradings in the characteristic zero case we also generalize results for the algebra u t d 1 d n c with a tensor product grading where c is a color commutative algebra generating the variety of all color commutative algebras

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

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

calculation of the rightmost characteristic root of retarded time delay systems via lambert w function abstract generally it is not easy to analyze the stability of time delay systems especially when the systems are of high order or they have multiple delays for retarded time delay systems the stability can be determined by the rightmost characteristic root this paper presents a case study on the calculation of the rightmost root three practical time delay systems are discussed the first system is an oscillator with delayed state feedback the second one is a delayed neural network based on the fitzhugh nagumo model for neural cells and the third one is a car model of suspension with a delayed sky hook damper by using the lambert w function the rightmost root becomes a root of a function associated with the principal branch of the lambert w function then the rightmost root is located by using newton raphson s scheme or halley s accelerating scheme some suggestions for successful application of the proposed method are given