determinative vertices of interval family with ωθ stability abstract in this article we pay attention to the stability of interval polynomials with regard to the region an analysis on how to generate the finite set whose stability implies robust stability of the entire family is given those results are intuitive and convenient to operate

on robust hurwitz polynomials in this note kharitonov s theorem on robust hurwitz polynomials is simplified for low order polynomials specifically for n 3 4 and 5 the number of polynomials required to check robust stability is one two and three respectively instead of four furthermore it is shown that for n geq 6 the number of polynomials for robust stability checking is necessarily four thus further simplification is not possible the same simplifications arise in robust schur polynomials by using the bilinear transformation applications of these simplifications to two dimensional polynomials as well as to robustness for single parameters are indicated

r norm bounds and metric properties for zero loci of real analytic functions abstract we consider the problem of deciding whether or not a zero locus x of multivariate real analytic functions crosses a given r norm ball in the real n dimensional affine space we perform a local study of the problem and we provide both necessary and sufficient conditions to answer the question our conditions derive from the analysis of differential geometric properties of x at the center of the ball an algorithm to evaluate r norms distances is proposed

necessary and sufficient conditions for robust stability of discrete systems with coefficients depending continuously on two interval parameters abstract let a real polynomial in a complex variable whose coefficients are any given continuous functions of two real interval parameters be given necessary and sufficient conditions are derived for the polynomial to have all its zeros outside or inside the unit circle of the complex variable plane in a multi polynomial dependence of the coefficients on parameters the conditions involve in addition to a finite number of algebraic steps only resolving single variable fixed coefficients real polynomials

a necessary and sufficient condition for schur invariance and generalized stability of polytopes of polynomials polytopes of characteristic polynomials arise when systems experience real parameter variations a necessary and sufficient condition requiring only a finite number of computations is presented for determining whether all the roots of a polytope of polynomials are contained in a desired region this can be any region that is the image of the open left half plane under a real linear fractional transformation since the open unit cycle is such a region this result establishes a finite algorithm for strict schur invariance finite algorithms for generalized stability regions such as maximum bandwidth and maximum setting time are also established

pi fuzzy controllers for integral plants to ensure robust stability new pi fuzzy controllers comprising takagi sugeno and mamdani fuzzy systems to control a class of integral plants specific to servo systems are proposed in this paper linear pi controllers are designed in the first phase they are tuned using the extended symmetrical optimum method to ensure the desired overshoot and settling time with respect to step setpoint modifications and three types of load disturbance inputs this type of pi controller design guarantees robust stability of the closed loop system in response to parametric variations in a controlled plant following this based on the results from the linear case and the modal equivalence principle an original development method for pi fuzzy controllers is proposed some experimental results are included to illustrate the effectiveness of the design process

kharitonov s theorem and the second method of lyapunov in this paper kharitonov s theorem for the robust stability of interval polynomials is proved using the second method of lyapunov the hermite matrix is taken as the matrix of the quadratic form which is used as a lyapunov function to prove hurwitz stability it is shown that if the four hermite matrices corresponding to the four kharitonov extreme polynomials are positive definite the hermite matrix of any polynomial of the polynomial family remains positive definite

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

perturbed polynomials with zeros in the left sector abstract we investigate the directional perturbation of a real nominal polynomial we propose an algorithm for computing the largest allowable perturbation bound which will simultaneously maintain the zeros of the perturbed polynomial within the left sector in the complex plane

on the stability of low order perturbed polynomials it is shown that for low order perturbed continuous system polynomials n 6 stability can be guaranteed by checking very simple conditions based on the hermite biehler theorem this result is analogous to a recent result by anderson et al based on kharitonov s conditions

new stability conditions via reflection coefficients of polynomials the geometry of stable discrete polynomials using their coefficients and reflection coefficients is investigated starting from so called barycentric simplex some necessary stability conditions in terms of unions of polytopes are obtained by splitting the unit hypercube of reflection coefficients sufficient stability conditions in terms of linear covers of reflection vectors of a family of stable polynomials improve the cohn stability criterion

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

limitations of robust stability of a linear delayed feedback control in this paper we consider the issue of robust stability of a linear delayed feedback control dfc mechanism in particular we consider a dfc for stabilizing fixed points of a smooth function f mathbb r m rightarrow mathbb r m of the form x k 1 f x k u k where u k is given by the formula u k sum j 1 n 1 epsilon j x k j x k j 1 we associate with each fixed point of f an explicit polynomial whose schur stability corresponds to the local asymptotic stability of the dfc at that fixed point this polynomial is the characteristic polynomial of the jacobian matrix of an auxiliary map from mathbb r mn to mathbb r mn and may be given in terms of the eigenvalues of the jacobian of f at the fixed point this enables us to evaluate the robustness of the control by considering over what possible sets of eigenvalues of the jacobian of f the associated characteristic polynomials are schur stable we will show that for a given control of the above form sta

perturbation of polynomials and applications to the hough transform abstract let f and g be complex polynomials of the same degree we provide a new lower bound on the euclidean distance of points belonging to their zero loci in terms of bombieri s norm we also present a minimization of the bombieri s norm of the difference g f for c in the real case we apply the results above in the setting of the hough transform a standard technique to detect curves in images suggesting a bombieri s norm based recognition algorithm

estimating the matrixp norm the holderp norm of anm n matrix has no explicit representation unlessp 1 2 or it is shown here that thep norm can be estimated reliably ino mn operations a generalization of the power method is used with a starting vector determined by a technique with a condition estimation flavour the algorithm nearly always computes ap norm estimate correct to the specified accuracy and the estimate is always within a factorn 1 1 p of a p as a by product a new way is obtained to estimate the 2 norm of a rectangular matrix this method is more general and produces better estimates in practice than a similar technique of cline conn and van loan

stability of perturbed polynomials based on the argument principle and nyquist criterion the stability robustness of the characteristic polynomial with perturbed coefficients for linear time invariant systems is studied the schur strictly hurwitz and g stability properties of perturbed polynomials are all considered with a unified approach new upper bounds on the allowable coefficient perturbation of a polynomial for keeping one of the stability properties are obtained the proposed upper bounds are directly formulated in terms of the polynomial coefficients and can be computed easily we also provide a sufficient condition for the discrete stability of interval polynomials and an algorithm for testing the g stability of polynomials with constant coefficients illustrative examples are given to show the applicability of our results especially in determining measures of stability robustness for any schur polynomial subject to coefficient perturbation

robust stability of control systems extreme point results for the stability of edges for the investigation of the robust stability of control systems with structured uncertainties many results have been presented recently that lead to stability tests of the edges of a polytope in this paper some results are discussed where the stability of an edge is guaranteed by the stability of the vertices of the edge given the characteristic polynomial of a closed loop system with structured uncertainties in the parameters for special types of uncertainties the family of polynomials is defined by a polytope in the space of parameters in the frequency domain the corresponding value set of the family for s jw fixed is a polygon whose edges are formed by the so called exposed edges of the polytope for the investigation of the stability of the polynomial family these exposed edges must be tested kraus and truol 1991 a this test can be simplified if an edge has the vertex property i e if the stability of the edge is guaranteed by the stability of the two vertices

on kharitonov type results for complex coefficient interval schur polynomials a kharitonov type result for the stability analysis of real schur polynomials that have been transformed by a new transformation technique has been proposed and the necessary and sufficient conditions for the stability of the transformed polynomials were developed by p p vaidyanathan see ieee trans acoust speech signal process vol 38 no 2 p 277 85 1990 these results are generalized to the case of the complex coefficient and the stability of the whole transformed family of interval polynomials is proved the sufficiency conditions of this test for the stability of the original interval polynomial family is commented on and checking the stability of the required polynomials for low order cases is addressed some illustrative examples are given the results may be found useful to testing the interval stability of two dimensional digital filters

a necessary and sufficient condition for the stability of nonnegative interval discrete systems the stability of nonnegative discrete systems with interval uncertainty is discussed for this class of system a necessary and sufficient condition for the stability is provided when all elements of the system matrix are assumed to be specified by intervals an illustrative example is used as a guideline to demonstrate the effectiveness of the result simple computational tests are suggested and their usefulness in this framework is discussed the possible extension of this result to more general classes of systems is addressed

robust controller synthesis using kharitonov s theorem the authors present a method based upon kharitonov s theorem for synthesizing robustly stabilizing feedback controllers the approach is based upon a multiple plant model formulation with a quadratic cost functional sufficient conditions are obtained for characterizing robustly stabilizing static output feedback proportional controllers for mimo multiple input multiple output plants with denominator polynomial uncertainty

moore penrose approach in the hough transform framework abstract let f x a be a real polynomial in two sets of variables x and a that is linear with respect to one of the variable sets say a in this paper we deal with two of the main steps of the hough transform framework for the pattern recognition technique to detect loci in images more precisely we present an algorithmic process based on the moore penrose pseudo inverse to provide a region of analysis in the parameter space in addition we state an upper bound for the sampling distance of the discretization of the parameter space region

kharitonov regions it suffices to check a subset of vertex polynomials the authors deal with the d stability property of interval polynomials in particular they show that certain d domains are kharitonov regions that is the d stability of interval polynomials is implied by the d stability of all its vertex polynomials they then proceed to show that it suffices to check the d stability of a subset of the vertex polynomials

on the strict and wide sense stability robustness of uncertain systems application of a new frequency criterion abstract in this paper we discuss a new frequency domain graphical approach for robust stability of uncertain systems this method is based on mikhailov s criterion and it requires the plotting of only one hodograph and evaluation of two boundary conditions moreover this approach is extended to a more general robust stability sense i e extended and wide sense hurwitz stability which may be applied to a number of robustness problems such as positivity nonnegativity spectral analysis and array processing

comments on sufficient and necessary condition for the asymptotic stability of discrete linear interval systems abstract in a recent paper by jiang 1988 a necessary and sufficient condition for asymptotic stability of the interval matrix of discrete time systems is presented in this contribution it is shown that this condition is not true in general a counter example is given and additional remarks are made

a class of stability regions for which a kharitonov like theorem holds families of complex polynomials whose coefficients lie within given intervals are discussed in particular the problem of determining if all polynomials in a family have the property that all of their roots lie within a given region is discussed towards this end a notion of a kharitonov region is defined roughly speaking a kharitonov region is a region in the complex plane with the following property given any suitable family of polynomials in order to determine if all polynomials in the family have all of their roots in the region it suffices to check only the vertex polynomials of the family the main result is a sufficient condition for a given region to be a kharitonov region

robust strict positive realness new results for interval plant controller families a frequency domain approach is used to derive several robust strict positive realness results for interval plants and interval plant plus controller families of transfer functions based on simple frequency domain properties of transfer functions the approach provides a framework for obtaining new results and constructing easy proofs of several important existing results on robust strict positive realness the main new result states that the minimum of the real part of the transfer functions belonging to an interval plant controller family is achieved on one of the 32 kharitonov segments of the interval plant the argument used in the proof is of wider interest and suggests easy ways of proving that robustness of other different frequency properties of interval plant plus controller families of transfer functions such as robust stability of h sub infinity norm computation can be deduced from a fixed number of segments of transfer functions of the family

counter example to sufficient and necessary condition for the asymptotic stability of discrete linear interval systems abstract in this note the interval matrix result for discrete linear interval system by jiang 1988 is shown to be incorrect i e all eigenvalues of each matrix a e a 1 an interval matrix have magnitudes less than 1 if and only if all the eigenvalues of every extreme matrix of the interval matrix a 1 have magnitudes less than 1

on the robust stability of polynomials with no cross coupling between the perturbations in the coefficients of even odd powers abstract some criteria are presented for the stability in hurwitz sense of a family of polynomials with even and odd coefficients subject to uncoupled perturbations it is shown that the stability of an appropriate subset of extreme polynomials guarantees the stability of the entire family in particular a polytope of polynomials with the even odd uncoupling property is stable provided a certain subset of its vertices is for the case of an arbitrary set of polynomials this result often gives a less conservative sufficient condition than that provided by kharitonov s theorem

application of kharitonov s theorem to mechanical systems kharitonov s theorem is used to derive a robust stability condition for proportional integral derivative pid controlled multi degree of freedom mechanical systems the characteristic equation of such a system is given as a determinant of a third order polynomial with matrix coefficients from which a scalar interval polynomial is obtained a simple procedure for designing pid controllers for these mechanical systems is described and a new kharitonov like result which states roughly that a controller designed for an upper bounding inertia matrix results in stable set point regulation for all other inertias is proved

on robust stability of linear systems abstract in this paper the robust stability of linear continuous and discrete systems in the frequency domain is considered giving necessary and sufficient conditions based on that the method of tsypkin and polyak 1991 is proved in a simple way extensions to discrete systems in the z domain and sampled data systems in the delta domain are done using two ways of parametrization

derivation of new and existing discrete time kharitonov theorems based on discrete time reactances digital signal processing the author first uses a discrete time reactance approach to give a second proof of existing discrete time kharitonov type results 1979 he then uses the same reactance language to derive a new discrete time kharitonov type theorem which in some sense is a very close analog to the continuous time case he also points out the relation between discrete time reactances and the technique of line spectral pairs lsp used in speech compression

robust stability of polynomials with multilinear parameter dependence abstract the problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way we show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials obtained by setting parameters at their extreme values while checking for unstable complex roots normally involves examining the real solutions of up to m 1 simultaneous polynomial equations where m is the number of parameters when m 2 this is an especially simple task

strictly hurwitz property invariance of quartics under coefficient perturbation in this note it is shown how the maximum interval centered around each of the coefficients of a prescribed fourth degree strictly hurwitz polynomial may be determined so that the strictly hurwitz property remains invariant when each coefficient might be perturbed to any arbitrary value within its permissible interval

a dual result to kharitonov s theorem a dual problem to kharitonov s problem involving a diamond instead of a rectangle is considered the results show that a family of polynomials with coefficients varying in the diamond is strictly hurwitz if and only if eight one dimensional exposed edges of the diamond are strictly hurwitz

vector quantization of image subbands a survey subband and wavelet decompositions are powerful tools in image coding because of their decorrelating effects on image pixels the concentration of energy in a few coefficients their multirate multiresolution framework and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system vector quantization vq provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces since 1988 a growing body of research has examined the use of vq for subband wavelet transform coefficients we present a survey of these methods

strict hurwitz property of polynomials under coefficient perturbation a simple procedure for computing the allowable interval coefficient perturbations of a strictly stable polynomial is presented the amount of computations involved is very little while the procedure is able to handle the case where the relative uncertainty of the polynomial coefficients are not known it can also handle the case of linearly dependent parameter uncertainties the procedure can also be modified to treat the case of simple nonlinear dependent parameter uncertainties the details of the modified procedure will depend on the nature of the nonlinearity

strong kharitonov theorems for low order polynomials the root clustering property of low order interval polynomials is examined and it is shown that the number of polynomials required to check root clustering within the left sector may be reduced if the order of the interval polynomial is low specifically the reduction is possible if n phi

fast computation of the multivariable stability margin for real interrelated uncertain parameters an alternative implementation of an algorithm for the computation of the multivariable stability margin to check the robust stability of feedback systems with real parametric uncertainty is proposed this method eliminates the need for the frequency search by reducing it to the testing of a finite number of conditions these conditions have a special structure that allows a significant improvement in the speed of computation

robustness measures for linear systems with application to stability radii of hurwitz and schur polynomials in this paper we consider robustness measures stability radii for system matrices which are subjected to structured real and complex perturbations of the form a a bdc where b c are given matrices our object is twofold a to present a number of new results mainly concerning the real stability radius and its differences from the complex one b to give an overview of our approach to the robustness analysis of linear state space systems including basic properties and characterizations of the complex stability radius applying the results to the special case where a is in companion form and b 0 0 0 1 t we are able to determine stability radii for hurwitz and schur polynomials under arbitrary complex and real affine perturbations of the coefficient vector computable formulae are obtained and illustrated by several examples

stability issues on takagi sugeno fuzzy model parametric approach the main objective of this research is to provide a systematic stability analysis technique for fuzzy systems using well established results from robust stability of a family of polynomials the model based takagi sugeno 1985 fuzzy control system is considered and shown to have a dependent coefficient structure involving firing strengths two kharitonov regions via overbounding are established to facilitate the extreme results for stability test in the space of polynomials as well as the complex plane the criterion obtained is a sufficient condition for a fuzzy logic controller three nonlinear examples are demonstrated to show that the systematic approach proposed here can be used to solve fuzzy stability problems fuzzy users concerned over stability issues may find these analytic tools useful

maximal polytopic stability domains in parameter space for uncertain systems abstract some results on the stability and pole location of families of uncertain systems with characteristic polynomial coefficients affected by structured perturbations are presented the case in which the coefficients are affine in a set of system physical parameters is considered assuming that the parameters are subject to polytopic perturbations of a given structure for these cases which often arise in system stability robustness analysis as for example in the robustification of controllers for multivariable closed loop systems a procedure is given to estimate maximal stability polytopic regions in the space of parameters this result is also used to derive simple necessary and sufficient conditions for the discrete time stability of convex combinations of two polynomials

stability of families of polynomials geometric considerations in coefficient space the stability robustness of both continuous and discrete time dynamical systems is tantamount to the stability of a whole family of characteristic polynomials the authors treat such families as sets in coefficient space and examine some of their geometric properties specifically they examine cases where the stability of the extreme points of a set implies the stability of the complete family they are also concerned with the construction and convexity of the stability region in coefficient space

some discrete time counterparts to extreme point results for robust stability of a diamond of polynomials the main result of the paper is that a family of polynomials with roughly half of all the real coefficients lying in a diamond is robustly schur stable if and only if all the extreme polynomials are schur stable this special polynomial family is called a degenerate diamond of polynomials in addition we obtain the extreme point result for robust schur stability of a so called simplex degenerate diamond polynomial family

robust stability of perturbed systems with time delays abstract motivated by dynamical system considerations a number of new results on robust stability of perturbed polynomials have been recently obtained in the literature under consideration the perturbations are known only within given bounds and enter affine linearly into the polynomial coefficients in this paper this formulation is expanded to include time delays under a properness assumption a necessary and sufficient condition for robust stability is developed an example is given to illustrate application of the main result

schur stability of interval polynomials a method for checking the schur stability of interval polynomials is presented the numbers of critical vertex and edge polynomials that are sufficient for inferring robust schur stability are obtained it is shown that the critical edges can be greatly reduced to the order of 2n sup 2

expansion of det a b and robustness analysis of uncertain state space systems in this note an expansion for the determinant of the sum of two matrices det a b is given a robustness analysis approach for state space systems with uncertain parameters is investigated by using kharitonov s theorem some useful results are obtained several illustrative examples are given

necessary and sufficient conditions for root clustering of a polytope of polynomials in a simply connected domain linear time invariant systems with uncertain parameters are considered a test is given which is both necessary and sufficient for root clustering of a family of polytopic polynomials in a simply connected domain the test does not require checking a continuum of polynomials special cases include stability testing of continuous time systems and of discrete time systems with uncertain parameters as well as relative stability tests of such systems interval polynomials can also be handled as a special case

least change secant update methods for undetermined systems least change secant updates for nonsquare matrices have been addressed recently in 6 here the use of these updates in iterative procedures for the numerical solution of underde retrained systems is considered the model method is the normal flow algorithm used in homotopy or continuation methods for determining points on an implicitly defined curve a kantorovich type local convergence analysis is given which supports the use of least change secant updates in this al gorithm this analysis also provides a kantorovich type local convergence analysis for least change secant update methods in the usual case of an equal number of equations and unknowns this in turn gives a local convergence analysis for augmented jacobian algorithms which use least change secant updates in conclusion the results of some numerical experiments are given

extreme point robust stability results for discrete time polynomials this paper provides some new extreme point robust stability results for discrete time polynomials with special uncertainties in the coefficient space the proofs obtained using the bilinear transformation are simple and the results specialize to existing robust stability results

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

a generalization of kharitonov s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations kharitonov s four polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions to this end a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis the present work is motivated by two fundamental limitations of kharitonov s theorem namely 1 the theorem only applies to polynomials with independent coefficient perturbations and 2 it only applies to zeros in the left hand plane

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

robustness of pole location in perturbed systems abstract in this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients a geometric approach in coefficient space is exploited to derive maximal deviations in a given class of admissible perturbations of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane it is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane this last result can be considered an extension of the result of the well known theorem of kharitonov it is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

a multiresolution approach to computer verification of handwritten signatures we took a multi resolution approach to the signature verification problem the top level representation of signatures was the global geometric features a multi resolution representation of signatures was obtained using the wavelet transformation we built vq and network classifiers to demonstrate the advantages of the multi resolution approach high verification rates were achieved based on a limited database

robust schur stability of interval polynomials the investigation of schur stability using a kharitonov parameter box is discussed the discrete counterpart of kharitonov s theorem is obtained the solution is based on the use of the hollot bartlett huang theorem and the hollott bartlett theorem this made it possible to test for schur stability only a subset of the edges the schur testing of the required edges of the cube is performed using three different methods namely the critical edge polynomial edge stability as an eigenvalue problem and edge stability using colinearity conditions comparison of these three methods is presented it is believed that with the schur testing of the minimum number of edges and the use of the critical stability constraints minimum computational effort can be achieved

a root distribution criterion for interval polynomials the problem of finding the conditions under which an interval polynomial has a given number of roots in the open left half plane and the other roots in the open right half plane irrespective of the values of its coefficients is considered a simple criterion is provided to test interval polynomials for the root distribution invariance viewed as an extension of kharitonov s theorem the goal is to provide an alternative theorem and then give an efficient means of checking the root distribution invariance

generalized edge theorem abstract in this paper we shall show that a family of analytic functions constructed from the convex hull of a finite number of vertex functions will have no zeros within a simply connected region in the complex plane if and only if all its edge functions have no zeros within the simply connected region the result is in fact a generalization of the edge theorem which has been derived for polynomial functions 1 2 7

on the stability of a weighted diamond of real polynomials recently a subset of the robust stability literature concentrated on the so called diamond of polynomials in this paper we study the weighted diamond qw defined as qw q rn 1 w0 q0 q0 wn qn qn r where the center q the weights w0 w1 3dot wn 0 and the radius r 0 are given for this family we show that an extreme point result holds if and only if a certain interlacing condition on the weights is satisfied

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

convergence property of interval matrices and interval polynomials in association with robust control system design and analysis the hurwitz property of interval matrices and interval polynomials has recently been actively investigated however its discrete counterpart the convergence property has seemingly not been much discussed in this paper this property is studied in comparison with the hurwitz counterpart some conditions under which interval matrices or interval polynomials are convergent are derived

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

hurwitz testing sets for parallel polytopes of polynomials in considering robustness of linear systems with uncertain paramenters one is lead to consider simultaneous stability of families of polynomials efficient hurwitz stability tests for polytopes of polynomials have earlier been developed using evaluations on the imaginary axis this paper gives a stability criterion for parallel polytopes in terms of hurwitz stability of a number of corners and edges the testing set of edges and corners depends entirely on the edge directions of the polytope hence the results are particularly applicable in simultaneous analysis of several polytopes with equal edge directions it follows as a consequence that kharitonov s four polynomial test for independent coefficient uncertainties is replaced by a test of 2q polynomials when the stability region is a sector eiv 0 r q v and r q is a rational number

adding a single state memory optimally accelerates symmetric linear maps previous papers have proposed to add memory registers to the dynamics of discrete time linear systems in order to accelerate their convergence in particular it has been proved that adding one memory slot per agent allows faster convergence towards average consensus we here prove that this situation cannot be improved by adding more memory slots when the knowledge about the self adjoint linear map to be accelerated reduces to bounds on its extreme eigenvalues

on interval polynomials with no zeros in the unit disc we give a necessary condition for an interval polynomial to have no zeros in the closed unit disc the condition is expressed in terms of the two first intervals

computation of moore penrose generalized inverses of matrices with meromorphic function entries abstract in this paper given a field with an involutory automorphism we introduce the notion of moore penrose field by requiring that all matrices over the field have moore penrose inverse we prove that only characteristic zero fields can be moore penrose and that the field of rational functions over a moore penrose field is also moore penrose in addition for a matrix with rational functions entries with coefficients in a field k we find sufficient conditions for the elements in k to ensure that the specialization of the moore penrose inverse is the moore penrose inverse of the specialization of the matrix as a consequence we provide a symbolic algorithm that given a matrix whose entries are rational expression over c of finitely many meromorphic functions being invariant by the involutory automorphism computes its moore penrose inverve by replacing the functions by new variables and hence reducing the problem to the case of matrices with complex rational function entries

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

a class of weak kharitonov regions for robust stability of linear uncertain systems kharitonov s theorems are generalized to the problem of so called weak kharitonov regions for robust stability of linear uncertain systems given a polytope of characteristic polynomials p and a stability region d in the complex plane p is called d stable if the zeros of every polynomial in p are contained in d it is of interest to know whether the d stability of the vertices of p implies the d stability of p a simple approach is developed which unifies and generalizes many known results on this problem

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed