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

root exclusion from complex polydomains and some of its applications abstract a solution to an open research problem related to testing root exclusion of a real n dimensional polynomial n 1 from a subspace of n defined as a cartesian product n times of c c is given in this paper the solution is presented in two theorems which have application to several research problems related to robust relative multidimensional stability among them are the testing of robust relative stability of linear differential systems the testing of robust relative stability of linear delay differential systems and a complementary test for a robust stabilization algorithm several examples are presented to illustrate some of these applications

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

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

characterization of zero locations of polytopes of real polynomials given a stability region of a polytope of polynomials the edge theorem can be used to verify certain root clustering properties of the polytope the authors consider the inverse problem starting with a polytope of real polynomials they present a result that allows the construction of the smallest set of regions in the complex plane which characterizes its zero location

a note on the edge theorem abstract we show that an additional condition is required before we can infer the stability of a polytope of polynomials from its edges

extending representations of banach algebras to their biduals we show that a representation of a banach algebra a on a banach space x can be extended to a canonical representation of a on x if and only if certain orbit maps a rightarrow x are weakly compact when this is the case we show that the essential space of the representation is complemented if a has a bounded left approximate identity this provides a tool to disregard the difference between degenerate and nondegenerate representations our results have interesting consequences both in c algebras and in abstract harmonic analysis for example a c algebra a has an isometric representation on an l p space for p in 1 infty setminus 2 if and only if a is commutative moreover the l p operator algebra of a locally compact group is universal with respect to arbitrary representations on l p spaces

robust stability of an interval plant with respect to a convex region in the complex plane the determination of robust stability of interval polynomials with respect to an arbitrary closed convex contour in the left half complex plane is considered it is shown that the problem can be solved by checking only the root locations of a subset of the exposed edges of the family the number of exposed edges to be checked depends on the damping margin defined by the contour and the degree of interval polynomials but does not depend on the exact shape of the contour a simple easy to apply graphical approach determines the subset of the exposed edges an example illustrates the merit of the approach

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

directional stability radius a stability analysis tool for uncertain polynomial systems coefficients of characteristic polynomials for stable parametrically uncertain systems are allowed to perturb to some extent for stability stability radius is a useful tool to assess the allowance of the stability for the systems to enhance its usefulness we modify stability radius so that it takes into account of given restricted perturbations which we call directional stability radius for an application we show shifted hurwitz stability conditions and a stability analysis method for interval polynomial systems using the directional stability radii

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

new results on the separation of matrix eigenvalues and the clustering of matrix elements the problem of separation of matrix eigenvalues is considered in this paper necessary and sufficient conditions are given for a matrix to have all eigenvalues contained in an open set defined on the complex plane by a separable polynomial spl gamma z sub 1 z sub 2 the problem of clustering of matrix elements is also considered in this case a set of matrices is identified having all eigenvalues in a given subset of the complex plane the presented results can be useful in both robust stability analysis and the design of control systems

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

banach algebras generated by an invertible isometry of an l p space abstract we provide a complete description of those banach algebras that are generated by an invertible isometry of an l p space together with its inverse examples include the algebra p f p z of p pseudofunctions on z the commutative c algebra c s 1 and all of its quotients as well as uncountably many exotic banach algebras we associate to each isometry of an l p space a spectral invariant called spectral configuration which contains considerably more information than its spectrum as an operator it is shown that the spectral configuration describes the isometric isomorphism type of the banach algebra that the isometry generates together with its inverse it follows from our analysis that these algebras are semisimple with the exception of p f p z they are all closed under continuous functional calculus and their gelfand transform is an isomorphism as an application of our results we show that banach algebras that act on l 1 spaces are not closed under quotients this answers the case p 1 of a question asked by le merdy 20 years ago

stability of a family of polynomials with coefficients bounded in a diamond the robust stability property is examined for family of nth order real polynomials where the coefficients are bounded within a diamond in the n 1 dimensional space it is shown that such a family of polynomials is hurwitz if and only if four specially selected edge polynomials are hurwitz

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

a homotopy method for solving bilevel programming problem abstract a homotopy method for solving bilevel programming problem blpp is proposed the algorithm is based on constructing the homotopy equation for blpp one satisfiable sufficient condition is given to ensure that the point of homotopy path globally converges to a k k t point of blpp numerical results are presented to show the effectiveness of this method

a priori estimates for the zeros of interval polynomials the aim of this paper is to study the zeros of interval polynomials the characterization of such zeros is given as a function of the interval coefficients and estimates for such zeros are established interval polynomials of degree two three and four are considered

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

the design of optimal reduced order stochastic observers for discrete time linear systems the minimum variance state estimation of linear discrete time systems with random white noise input and partially noisy measurements is investigated an observer of minimal order is found which attains the minimum variance estimation error the structure of this observer is shown to depend strongly on the geometry of the system this geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate the transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise an explicit expression is found for the transfer function matrix of this observer from which a simple solution to the linear discrete time singular optimal filtering problem is obtained

a simple proof of kharitonov s theorem an alternative proof of kharitonov s theorem on the stability of linear time invariant continuous systems under parameter variations is presented

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

damping margins of interval polynomials the zero locations of interval polynomials are examined in particular it is shown that a family of interval polynomials will have zeros only in the left sector if the real and imaginary parts of four specially constructed complex polynomials have an interlacing real zero property this is significant for the analysis of uncertain systems as the computation cost associated with checking the zero locations of interval polynomials will be greatly reduced the results presented can be readily extended to more general stability regions where the real and imaginary parts of the polar plot are polynomial functions

pairwise compatibility graphs a survey a graph g v e is a pairwise compatibility graph pcg if there exists an edge weighted tree t and two nonnegative real numbers d min and d max such that each leaf u of t is a node of v and there is an edge u v in e if and only if d min leq d t u v leq d max where d t u v is the sum of weights of the edges on the unique path from u to v in t in this article we survey the state of the art concerning this class of graphs and some of its subclasses

on the computation of solutions of systems of interval polynomial equations systems of algebraic equations with interval coefficients are very common in several areas of engineering sciences the computation of the solution of such systems is a central problem when the characterization of the variables related by such systems is desired in this paper we characterize the solution of systems of algebraic equations with real interval coefficients the characterization is obtained considering the approach introduced in j comput appl math 136 2001 271

constructing convex directions for stable polynomials new constructions of convex directions for hurwitz stable polynomials are obtained the technique is based on interpretations of the phase derivative conditions in terms of the sensitivity of the root locus associated with the even and odd parts of a polynomial

linear system stability under parameter uncertainties a sufficient condition for linear system stability under parameter variations is presented it yields parameter bounds which are not conservative and the test is easily implemented on a digital computer

a generalization of the routh hurwitz stability criteria and an application to a problem in robust controller design the classical results of routh and hurwitz on the stability of polynomials are generalized from the open left half plane to certain other subregions g of the complex plane by our method arbitrary conic sections can be handled the results are interpreted as criteria for g stability of linear systems a numerical algorithm to compute the criteria is given an application to a problem in robust controller design is outlined

improved finite word length characteristics in digital control using delta operators this paper examines some of the consequences of finite word lengths in digital control it is shown that in many cases of practical importance the usual shift operator formulation is inferior to an alternative formulation which we designate the delta operator approach this latter approach is shown to give better coefficient representation and less roundoff noise in many cases we thus argue that the shift operator and its associated z transform can be replaced by delta operators and their associated transform which we designate a transform an added advantage of this approach is that discrete designs and transforms converge to their continuous time counterparts as the sampling rate is increased

quotients of banach algebras acting on lp spaces we show that the class of banach algebras that can be isometrically represented on an lp space for p 2 is not closed under quotients this answers a question asked by le merdy 20 years ago our methods are heavily reliant on our earlier study of banach algebras generated by invertible isometries of lp spaces

root clustering of interval polynomials in the left sector abstract we examine the root clustering property of interval polynomials in particular we shall show that a family of interval polynomials will have only zeros within the left sector if and only if a finite number of specially chosen vertex polynomials have only zeros in the left sector the number of required vertex polynomials is a linear function of the orders of the polynomial but is independent of the damping margins of the left sector

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

strict aperiodic property of polynomials with perturbed coefficients let a family of polynomials be p s t sub 0 s sup n t sub 1 s sup n or sup 1 t sub n where 0

on damping ratio of polynomials with perturbed coefficients c b soh and c s berger 1988 derived a sufficient condition for a family of interval polynomials to have a damping ratio of phi using kharitonov s theorem for complex polynomials this paper points out that the transformations used by soh and berger to obtain the sufficient conditions also guarantee a simplification of kharitonov s theorem for complex polynomials that is the number of required polynomials to be hurwitz is half the number specified by soh and berger

nonconvex kharitonov regions an interesting nonconvex kharitonov region considered by m fu 1991 is reviewed it is shown that by means of a counterexample that the region identified is not a kharitonov region a different nonconvex kharitonov region is proposed

on extended p 4 reducible and extended p 4 sparse graphs abstract a graph g was defined in 16 as p 4 reducible if no vertex in g belongs to more than one chordless path on four vertices or p 4 a graph g is defined in 15 as p 4 sparse if no set of five vertices induces more than one p 4 in g p 4 sparse graphs generalize both p 4 reducible and the well known class of p 4 free graphs or cographs in an extended abstract in 11 the first author introduced a method using the modular decomposition tree of a graph as the framework for the resolution of algorithmic problems this method was applied to the study of p 4 sparse and extended p 4 sparse graphs in this paper we begin by presenting the complete information about the method used in 11 we propose a unique tree representation of p 4 sparse and a unique tree representation of p 4 reducible graphs leading to a simple linear recognition algorithm for both classes of graphs in this way we simplify and unify the solutions for these problems presented in 16 19 the tree representation of an n vertex p 4 sparse or a p 4 reducible graph is the key for obtaining o n time algorithms for the weighted version of classical optimization problems solved in 20 these problems are np complete on general graphs finally by relaxing the restriction concerning the exclusion of the c 5 cycles from p 4 sparse and p 4 reducible graphs we introduce the class of the extended p 4 sparse and the class of the extended p 4 reducible graphs we then show that a minimal amount of additional work suffices for extending most of our algorithms to these new classes of graphs

closed loop hyperstability of interval plants this note examines the conditions for robust bounded input bounded output closed loop stability of interval plants when there is a nonlinear element in the feedback path specifically the authors present the robust generalized circle criterion and popov s criterion

comments with reply on stability of a polytope of matrices counterexamples by b r barmish et al the commenters show that the counterexample given in the above mentioned paper ibid vol 33 no 6 p 569 72 june 1988 does not invalidate the conjecture about the necessary and sufficient conditions for the stability of interval matrices comments on the relationship between interval matrices and polytopes of polynomials are also given the authors reply that their counterexample is valid and attribute the misunderstanding to insufficiently precise wording in their paper

comments on a note on two methods related to stability robustness of polynomials in a sector relative stability in this paper it is shown that the results of the above mentioned paper katbab and jury 1993 are incorrect for the case of complex coefficients a counterexample is provided an alternative approach is presented for the calculation of sector stability hypersolids in the coefficient space

recognizing p 4 sparse graphs in linear time a graph g is p 4 sparse if no set of five vertices in g induces more than one chordless path of length three p 4 sparse graphs generalize both the class of cographs and the class of p 4 reducible graphs one remarkable feature of p 4 sparse graphs is that they admit a tree representation unique up to isomorphism it has been shown that this tree representation can be obtained in polynomial time this paper gives a linear time algorithm to recognize p 4 sparse graphs and shows how the data structures returned by the recognition algorithm can be used to construct the corresponding tree representation in linear time

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

generalization of strong kharitonov theorems to the left sector the zero locations of interval polynomials are examined in particular it is shown that a family of interval polynomials will have only zeros in a certain class of left sector if and only if a finite number of specially chosen vertex polynomials have only zeros in the left sector this finite number of vertex polynomials is dependent on the damping margins of the left sector but is independent of the orders of the polynomials the exact number of vertex polynomials to be checked will depend on the damping margins but it will be independent of the orders of the polynomials these specially chosen vertex polynomials are independent of the order of the polynomials the importance of the result lies with the great reduction in computation cost associated with checking the zero locations of interval polynomials

column and row operator spaces over qslp spaces and their use in abstract harmonic analysis the notions of column and row operator space were extended by a lambert from hilbert spaces to general banach spaces in this paper we use column and row spaces over quotients of subspaces of general lp spaces to equip several banach algebras occurring naturally in abstract harmonic analysis with canonical yet not obvious operator space structures that turn them into completely bounded banach algebras we use these operator space structures to gain new insights on those algebras

algebraic and invariance properties of the group of isometries abstract in this paper we investigate the algebraic structure of the isometry group of several classical banach spaces namely symmetric sequence spaces c p h 1 p p 2 l l p l r 1 p r and bounded operators l h endowed with chan li tui unitarily invariant c p norms we also identify the isometrically invariant subspaces for each of theses settings

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

polytopes of polynomials with zeros in a prescribed set in the publication by a c bartlett c v holot and h lin proc amer contr conf minneapolis mn 1987 a fundamental result is established on the zero locations of a family of polynomials it is shown that the zeros of a polytope p of nth order real polynomials are contained in a simply connected set d if and only if the zeros of all polynomials along the edges of p are contained in d the present authors are motivated by the fact that the requirement of simple connectedness of d may be too restrictive and applications such as dominant pole assignment and filter design where the separation of zeros is required they extend the edge criterion of bartlett et al to handle any set d whose complement d sup c has the following property every point d in d sup c lies on some continuous path which remains within d sup c and is unbounded this requirement is typically verified by inspection and allows for a large class of disconnected sets

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

strict positive realness of a family of polytopic plants some strict positive realness results for a family of polytopic plants are presented more specifically it is shown that the strict positive real property of such a family of plants can be ascertained from its vertex plants

a note on two methods related to stability robustness of polynomials in a sector relative stability two recent results on the stability robustness of uncertain systems polynomials are addressed one gives a vertex test for left sector relative stability where the number of the required vertices is four for a real case and eight for a complex case the other gives an elegant frequency domain graphical hodograph approach for left plane stability where the plotting of only one two for a complex case hodograph plus simple boundary conditions is both necessary and sufficient to obtain the maximum coefficient perturbation bounds it is shown that a similar graphical test exists for the left sector stability case

a fully dynamic algorithm for the recognition of p4 sparse graphs in this paper we solve the dynamic recognition problem for the class of p 4 sparse graphs the objective is to handle edge vertex additions and deletions to recognize if each such modification yields a p 4 sparse graph and if yes to update a representation of the graph our approach relies on maintaining the modular decomposition tree of the graph which we use for solving the recognition problem we establish properties for each modification to yield a p 4 sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in o 1 time and vertex modifications in o d time for a vertex of degree d thus our algorithm implies an optimal edges only dynamic algorithm and a new optimal incremental algorithm for p 4 sparse graphs

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

state space measures for stability robustness stability robustness measures for a perturbed linear feedback system are derived based on state space models of the system the system may be a continuous time or discrete time system the perturbations are modeled as additive perturbation matrices necessary and sufficient conditions for the stability of the perturbed closed loop system for all perturbations of norm bounded by some positive number are obtained the destabilizing perturbations of minimal norm are characterized it is shown by an example that there are cases when the destabilizing perturbations of minimal norm are all complex the results are expressed in terms of induced operator norms these are later specialized to the euclidean norm and expressed in terms of singular values a simple example is also included to illustrate an application of the results of this note

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

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

a new extension to kharitonov s theorem an extension to a well known theorem due to kharitonov is presented kharitonov s theorem gives a necessary and sufficient condition for all polynomials in a given family to be hurwitz stable in kharitonov s theorem the family of polynomials considered is obtained by allowing each of the polynomial coefficients to vary independently within an interval kharitonov s theorem shows that stability of this family of polynomials can be determined by looking at the stability of four specially constructed vertex polynomials kharitonov s theorem is extended to allow for more general families of polynomials and to allow a given margin of stability to be guaranteed for the family of polynomials

vertices and segments of interval plants are not sufficient for step response analyses abstract interval plants are of interest in control theory as models of uncertain systems they are useful because many worst case analyses of these models are simple to carry out for example robust stability of an interval plant can be determined by investigating only the four kharitonov vertices of the denominator polynomial also the maximum peak of the bode magnitude plot can be found using just 16 special plant vertices these 16 vertices are connected by 32 special line segments most stability and frequency domain analyses that cannot be done using only the special vertices can be carried out using just the segments from these results it is tempting to conjecture that the 16 vertices or at least the 32 segments are adequate for step response analyses this paper presents examples showing that these conjectures are not true

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