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

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

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

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

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

multiterminal source coding with high resolution we consider separate encoding and joint decoding of correlated continuous information sources subject to a difference distortion measure we first derive a multiterminal extension of the shannon lower bound for the rate region then we show that this shannon outer bound is asymptotically tight for small distortions these results imply that the loss in the sum of the coding rates due to the separation of the encoders vanishes in the limit of high resolution furthermore lattice quantizers followed by slepian wolf lossless encoding are asymptotically optimal we also investigate the high resolution rate region in the remote coding case where the encoders observe only noisy versions of the sources for the quadratic gaussian case we establish a separation result to the effect that multiterminal coding aimed at reconstructing the noisy sources subject to the rate constraints followed by estimation of the remote sources from these reconstructions is optimal under certain regularity conditions on the structure of the coding scheme

constrained optimization framework for interface aware sub scale dynamics discrete closure model for multimaterial cells in lagrangian cell centered hydrodynamics abstract we present the new discrete optimization based interface aware sub scale dynamics ia ssd closure model for multimaterial cells for lagrangian cell centered hydrodynamics for the multimaterial cell the kinematic and thermodynamic properties e g velocity density pressure and internal energy will typically vary between the materials the discrete closure model is responsible for an accurate update of the thermodynamic states of the individual material components in the multimaterial cell and for determining the nodal forces that move the vertices of the cell the ia ssd closure model consists of two stages a bulk stage followed by a sub scale stage during the bulk stage the total change in the volume of the cell total force applied to the cell and total work done on the cell are distributed between the materials to update their volume velocity and total energy this distribution is performed using volume fractions of the materials during the second stage sub scale interactions of the materials inside the multimaterial cell are taken into account at this stage information about the topology of the materials inside the multimaterial cell is used allowing the orientations of internal interfaces to be included in the model each material interacts in a pair wise fashion with the materials with which it has a common boundary the interactions are based on the solution of the acoustic riemann problem between each pair of materials and are limited using physically justified constraints positivity of volume positivity of internal energy and controlled rate of pressure relaxation to determine the values of the limiter coefficients a constrained optimization framework is employed using a quadratic objective function with linear constraints it is a first of its kind application of constrained optimization to develop discrete closure models in a more rigorous fashion the pair wise interaction between materials is essentially one dimensional in the direction that is normal to interface for this reason we demonstrate in this paper the performance of our new model on one dimensional numerical examples

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

stability analysis of a predictor multi corrector method for staggered grid lagrangian shock hydrodynamics this article presents the complete von neumann stability analysis of a predictor multi corrector scheme derived from an implicit mid point time integrator often used in shock hydrodynamics computations in combination with staggered spatial discretizations it is shown that only even iterates of the method yield stable computations while the odd iterates are in the most general case unconditionally unstable these findings are confirmed by and illustrated with a number of numerical computations dispersion error analysis is also presented

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

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

robust schur stability of control systems with interval plants the main contribution of this paper is a generalization of the box theorem which was originally introduced for the hurwitz case for determining the robust schur stability of linear time invariant discrete time control systems containing an interval plant this generalization provides necessary and sufficient conditions for the stability of a family of closed loop characteristic polynomials z q1 z p1 z qm z pm z where the qis are fixed controller and then pis are interval plant polynomials whose coefficients vary within a prescribed interval this method requires checking a set of prescribed segment polynomials whose number is considerably less than that of the edges required by the edge theorem a summary of the robust schur stability of discrete time interval polynomials is presented

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

the construction of compatible hydrodynamics algorithms utilizing conservation of total energy the principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought however in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost or obeyed only to an approximation that often becomes worse with time this is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system although a finite truncation error is inherent to all discretization methods it is possible to satisfy certain global properties such as conservation of mass momentum and total energy to numerical roundoff error the purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties in particular it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in lagrangian form and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system for staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy this allows for the specification of forces that can be of an arbitrary complexity such as those derived from an artificial viscosity or subzonal pressures these forces originate only in discrete form nonetheless the change in internal energy caused by them is still completely determined the procedure given here is compared to the method of support operators to which it is closely related difficulties with conservation of momentum volume and entropy are also discussed the proper treatment of boundary conditions and differencing with respect to time are detailed

the internal consistency stability and accuracy of the discrete compatible formulation of lagrangian hydrodynamics this work explores the somewhat subtle meaning and consequences of the salient properties of the discrete compatible formulation of lagrangian hydrodynamics in particular since this formulation preserves total energy to roundoff error the amount of error in the conservation of total energy cannot be used to gauge the internal consistency of calculations as is often done with the older forms of this algorithm however the compatible formulation utilizes two definitions of zone volume the first is the usual definition whereby the volume of a zone is defined as some prescribed function of the coordinates of the points that define it the second is given as the integration in time of the continuity equation for zone volume as expressed in lagrangian form it is the use of this latter volume in the specific internal energy equation that enables total energy to be exactly conserved these two volume definitions are generally not precisely equal it is the analysis of this difference that forms the first part of this study it is shown that this difference in zone volumes can be used to construct a practical internal consistency measure that not only takes the place of the lack of total energy conservation of the older forms of lagrangian hydrodynamics but is more general in that it can be defined on a single zone basis it can also be used to ascertain the underlying spatial and temporal order of accuracy of any given set of calculations the difference in these two definitions of zone volume may be interpreted as a type of entropy error however this entropy error is found to be significant only when a given calculation becomes numerically unstable otherwise it remains at or far beneath truncation error levels in fact it can be utilized to provide an upper bound on the size of the spatial truncation error for a stable computation it is also shown how this volume difference can be used as an indicator of numerical difficulties since exact local conservation of total energy does not guarantee numerical stability or the quality of any numerical calculation the discrete compatible formulation of lagrangian hydrodynamics utilizes a two level predictor corrector type of time integration scheme a stability analysis both analytical and numerical is given this analysis reveals a novel stability diagram that has not been heretofore published and gives definitive information as to how the stabilizing corrector step should be centered in time

short note volume consistency in a staggered grid lagrangian hydrodynamics scheme staggered grid lagrangian schemes for compressible hydrodynamics involve a choice of how internal energy is advanced in time the options depend on two ways of defining cell volumes an indirect one that guarantees total energy conservation and a direct one that computes the volume from its definition as a function of the cell vertices it is shown that the motion of the vertices can be defined so that the two volume definitions are identical a so modified total energy conserving staggered scheme is applied to the coggeshall adiabatic compression problem and now also entropy is basically exactly conserved for each lagrangian cell and there is increased accuracy for internal energy the overall improvement as the grid is refined is less than what might be expected

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

energy preservation and entropy in lagrangian space and time staggered hydrodynamic schemes usual space and time staggered sts leap frog lagrangian hydrodynamic schemes such as von neumann richtmyer s 1950 wilkins 1964 and their variants are widely used for their simplicity and robustness despite their known lack of exact energy conservation since the seminal work of trulio and trigger 1950 and despite the later corrections of burton 1991 it is generally accepted that these schemes cannot be modified to exactly conserve energy while retaining all of the following properties sts stencil with velocities half time centered with respect to positions explicit second order algorithm locally implicit for internal energy and definite positive kinetic energy it is shown here that it is actually possible to modify the usual sts hydrodynamic schemes in order to be exactly energy preserving regardless of the evenness of their time centering assumptions and retaining their simple algorithmic structure burton s conservative scheme 1991 is found as a special case of time centering which cancels the term here designated as incompatible displacements residue in contrast von neumann richtmyer s original centering can be preserved provided this residue is properly corrected these two schemes are the only special cases able to capture isentropic flow with a third order entropy error instead of second order in general the momentum equation is presently obtained by application of a variational principle to an action integral discretized in both space and time the internal energy equation follows from the discrete conservation of total energy entropy production by artificial dissipation is obtained to second order by a prediction correction step on the momentum equation the overall structure of the equations explicit for momentum locally implicit for internal energy remains identical to that of usual sts leap frog schemes though complementary terms are required to correct the effects of time step changes and artificial viscosity updates in deriving these schemes an apparently novel approach of flux in time was introduced to correct numerical residues and ensure energy conservation this method can be applied to essentially any numerical scheme whenever required or desired provided space and time numerical consistency is preserved numerical test cases are presented confirming the conservative character of the new csts schemes down to computer round off errors and showing various improvements compared to the standard von neumann richtmyer and wilkins sts schemes mostly on shock levels shock velocities singularity induced distortions and cfl stability limits

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

multiterminal source coding achievable rates and reliability some open problems concerning classical multiterminal configuration in which two correlated sources x and y are encoded separately and decoded by common decoder with respect to a fidelity criterion for messages of both sources are resolved the results of berger and yeung 1989 are expanded in two directions first we determine the rate distortion region r spl delta of achievable rates r sub x r sub y for a given spl delta spl delta sub x spl delta sub y with spl delta sub x the permissible distortion level for reproduction of x and spl delta sub y for y thus we give a full solution of the problem noted by kaspi and berger 1982 they constructed the inner bound for r spl delta and solved by berger and yeung for the case of one distortion second we introduce the notion of rate reliability distortion region for the multiterminal source encoding problem more precisely the region of admissible rates of codes ensuring exponential decrease with a given exponent e 0 of the probability of exceeding the distortion levels spl delta sub x or spl delta sub y the inner and outer bounds for this region r e spl delta are constructed the analytical representation obtained for r spl delta proved to be somewhat simpler than that considered by kaspi and berger and by berger and yeung ours uses only variables pertaining to the problem statement with no auxiliary ones

elimination of artificial grid distortion and hourglass type motions by means of lagrangian subzonal masses and pressures the bane of lagrangian hydrodynamics calculations is the premature breakdown of grid topology that results in severe degradation of accuracy and run termination often long before the assumption of a lagrangian zonal mass has ceased to be valid at short spatial grid scales this is usually referred to by the terms hourglass mode or keystone motion associated in particular with underconstrained grids such as quadrilaterals and hexahedrons in two and three dimensions respectively at longer spatial lengths relative to the grid spacing there is what is referred to ubiquitously as spurious vorticity or the long thin zone problem in both cases the result is anomalous grid distortion and tangling that has nothing to do with the actual solution as would be the case for turbulent flow in this work we show how such motions can be eliminated by the proper use of subzonal lagrangian masses and associated densities and pressures these subzonal pressures give rise to forces that resist these spurious motions the pressure is no longer a constant in a zone it now accurately reflects the density gradients that can occur within a zone due to its differential distortion subzonal lagrangian masses can be choosen in more than one manner to obtain subzonal density and pressure variation however these masses arise in a natural way from the intersection of the lagrangian contours through which no mass flows that are associated with both the lagrangian zonal and nodal masses in a staggered spatial grid hydrodynamics formulation this is an extension of the usual lagrangian assumption that is often applied to only the zonal mass we show that with a proper discretization of the subzonal forces resulting from subzonal pressures hourglass motion and spurious vorticity can be eliminated for a very large range of problems

universal coding for the slepian wolf data compression system and the strong converse theorem universal coding for the slepian wolf 1973 data compression system is considered we shall demonstrate based on a simple observation that the error exponent given by csiszar and korner 1980 for the universal coding system can strictly be sharpened in general for a region of relatively higher rates this kind of observation can be carried over also to the case of lower rates outside the slepian wolf region which establishes the strong converse along with the optimal exponent

aperiodic property of interval polynomials an examination is made of the aperiodic property of interval polynomials the authors also characterize the zero locations of aperiodic interval polynomials examples are given that clarify the points made

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

the force work differencing of exceptional points in the discrete compatible formulation of lagrangian hydrodynamics this study presents the force and mass discretization of exceptional points in the compatible formulation of lagrangian hydrodynamics it concludes a series of papers that develop various aspects of the theoretical exposition and the operational implementation of this numerical algorithm exceptional points are grid points at the termination of lines internal to the computational domain and where boundary conditions are therefore not applied these points occur naturally in most applications in order to ameliorate spatial grid anisotropy and the consequent timestep reduction that will otherwise arise for grids with highly tapered regions or a center of convergence they have their velocity enslaved to that of neighboring points in order to prevent large excursions of the numerical solution about them how this problem is treated is given herein for the aforementioned numerical algorithm such that its salient conservation properties are retained in doing so the subtle aspects of this algorithm that are due to the interleaving of spatial contours that occur with the use of a spatially staggered grid mesh are illuminated these contours are utilized to define both forces and the work done by them and are the central construct of this type of finite volume differencing additionally difficulties that occur due to uncertainties in the specification of the artificial viscosity are explored and point to the need for further research in this area

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

multiterminal source encoding with one distortion criterion the authors unify earlier investigations concerning the encoding of two correlated sources x sub k y sub k by means of separate encoders decoding is done by a single decoder which receives the outputs from both encoders the reconstruction of x sub k is required to be perfect in the usual shannon sense the authors determine the admissible rate region r d where d is the distortion of the reconstruction of y sub k the binary hamming case is investigated explicitly

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

on the largest polytope of polynomials with a precise number of distinct real zeros abstract this paper presents a procedure to construct the largest polytope of polynomials with every polynomial having a precise number of distinct real zeros in a specific real line segment this polytope can be specified from the zeros of a finite number of 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

a compatible energy and symmetry preserving lagrangian hydrodynamics algorithm in three dimensional cartesian geometry this work presents a numerical algorithm for the solution of fluid dynamics problems with moderate to high speed flow in three dimensions cartesian geometry is chosen owing to the fact that in this coordinate system no curvature terms are present that break the conservation law structure of the fluid equations written in lagrangian form these equations are discretized utilizing compatible control volume differencing with a staggered grid placement of the spatial variables the concept of compatibility means that the forces used in the momentum equation to advance velocity are also incorporated into the internal energy equation so that these equations together define the total energy as a quantity that is exactly conserved in time in discrete form multiple pressures are utilized in each zone they produce forces that resist spurious vorticity generation this difficulty can severely limit the utility of the lagrangian formulation in two dimensions and make this representation otherwise virtually useless in three dimensions an edge centered artificial viscosity whose magnitude is regulated by local velocity gradients is used to capture shocks the particular difficulty of exactly preserving one dimensional spherical symmetry in three dimensional geometry is solved this problem has both practical and pedagogical significance the algorithm is suitable for both structured and unstructured grids limitations that symmetry preservation imposes on the latter type of grids are delineated

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

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

graph decomposition a new key to coding theorems a new and simple method is proposed for finding good encoders both for channels and for sources with side information this method relies on the continuous version of a graph decomposition result of lovasz the presently known best exponential error bounds for both problems follow in a unified manner with an improvement on the source coding bound the previous bounds for universal codes of the authors and marton are also improved

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

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

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

invariance of the aperiodic property for polynomials with perturbed coefficients a procedure for obtaining the maximal intervals of the coefficients for a perturbed polynomial to be still strictly aperiodic is presented lemmas theorems and proofs are given an illustrative example is presented

some results on robust stability of discrete time systems the theorem of kharitonov on the hurwitz property of interval families of polynomials cannot be extended in genera to obtain sufficient conditions for the stability of families of characteristic polynomials of discrete time systems necessary and sufficient conditions for this stability problem are given such conditions naturally give rise to a computationally efficient stability test which requires the solution of a one parameter optimization problem and which can be considered as a counterpart to the kharitonov test for continuous time systems at the same time the method used to derive the stability conditions provides a procedure for solving another stability robustness problem i e the estimation of the largest domain of stability with a rectangular box shape around given nominal values of the polynomial coefficients

high order staggered schemes for compressible hydrodynamics weak consistency and numerical validation abstract staggered grids schemes formulated in internal energy are commonly used for cfd applications in industrial context here we prove the consistency of a class of high order lagrange remap staggered schemes for solving the euler equations in 1d and 2d on cartesian grids the main result of the paper is that using an a posteriori internal energy corrector the lagrangian schemes are proved to be conservative in mass momentum and total energy and to be weakly consistent with the 1d lagrangian formulation of the euler equations extension in 2d is done using directional splitting methods and face staggering numerical examples in both 1d and 2d illustrate the accuracy the convergence and the robustness of the schemes

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

stability conditions for polytopes of polynomials in considering the stability or performance of linear systems with uncertain parameters one is led to consider the simultaneous stability of families of polynomials for a general stability region in the complex plane the author gives a stability criterion for polytopes of polynomials in terms of the stability of a minimal number of corners and edges of the polytope the testing set of edges and corners depends entirely on the edge directions of the polytope hence the results are particularly applicable to problems with fixed directions but varying lengths and positions of the edges applications to robust pole placement of uncertain systems and computation of worst case h sub infinity norms are demonstrated

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

robust pole assignment abstract this paper presents new theorems on the theory of interval matrix inequalities and the theory of polynomials with interval roots and applies them to the problem of robust pole placement we formulate optimization problems and derive convergent iterative algorithms which allow the designer to find controllers that place closed loop poles within desired intervals for plants with unknown but bounded parameter uncertainties the algorithms are computationally reasonable and provide a useful addition to currently existing control cad tools

rate distortion for correlated sources with partially separated encoders we resolve some open problems concerning the encoding of a pair of correlated sources with respect to a fidelity criterion two encoders are used one encoder observes only the output of the first source while the other is supplied both with the second source output and with partial information about the first source at some predetermined rate a general coding theorem is proved which established that cal s ast a certain region defined in terms of single letter information theoretic quantities is an inner bound to the region of all attainable vectors of rates and distortions for certain cases of interest the converse is proved too thereby establishing the rate distortion region for these cases

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

operator splitting and time accuracy in lagrange plus remap solution methods operator splitting and time accuracy in lagrange plus remap solution methods for the hydrodynamics equations are investigated the time accuracy of the common solution approach is shown both analytically and numerically to be limited to first order due to operator splitting errors low order time integration of the remap terms and other postulated first order errors even if the lagrange step is second order accurate in time additional numerical studies are used to demonstrate how these errors can be eliminated with an unsplit treatment that solves the remap terms directly the discontinuous remap method in which a new mesh is generated during the remap step also is shown to be first order accurate in time

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 stability of staggered schemes this paper investigates the theoretical stability bound of a lagrangian staggered scheme used to solve hydrodynamics equations we present the two dimensional 2d wave equation as a possible model for this study and by using the numerical radius of the amplification matrix we prove that the family of schemes defined with two time centering parameters is limited by a nonclassical stability bound limit defined with an analytical curve we further show that 2d numerical experiments agree with this theoretical result

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

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

formulations of artificial viscosity for multi dimensional shock wave computations in this paper we present a new formulation of the artificial viscosity concept physical arguments for the origins of this term are given and a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated the first important property is that by definition a viscosity must always be dissipative transferring kinetic energy into internal energy and must never act as a false pressure the artificial viscous force should be galilean invariant and vary continuously as a function of the criterion used to determine compression and expansion and remain zero for the latter case these requirements significantly constrain the functional form that the artificial viscous force can have in addition an artificial viscosity should be able to distinguish between shock wave and adiabatic compression and not result in spurious entropy production when only the latter is present it must therefore turn off completely for self similiar motion where only a uniform stretching and or a rigid rotation occurs an additional important but more subtle condition where the artificial viscosity should produce no effect is along the direction tangential to a convergent shock front since the velocity is only discontinuous in the normal direction our principal result is the development of a new formulation of an edge centered artificial viscosity that is to be used in conjunction with a staggered spatial placement of variables that meets all of these standards and without the need for problem dependent numerical coefficients that have in the past made the artificial viscosity method appear somewhat arbitrary our formulation and numerical results are given with respect to two spatial dimensions but all of our arguments carry over directly to three dimensions a central feature of our development is the implementation of simple advection limiters in a straightforward manner in more than one dimension to turn off the artificial viscosity for the above mentioned conditions and to substantially reduce its effect when strong velocity gradients are absent

numerical preservation of symmetry properties of continuum problems we investigate the problem of perfectly preserving a symmetry associated naturally with one coordinate system when calculated in a different coordinate system this allows a much wider range of problems that may be viewed as perturbations of the given symmetry to be investigated we study the problem of preserving cylindrical symmetry in two dimensional cartesian geometry and spherical symmetry in two dimensional cylindrical geometry we show that this can be achieved by a simple modification of the gradient operator used to compute the force in a staggered grid lagrangian hydrodynamics algorithm in the absence of the supposed symmetry we show that the new operator produces almost no change in the results because it is always close to the original gradient operator our technique thus results in a subtle manipulation of the spatial truncation error in favor of the assumed symmetry but only to the extent that it is naturally present in the physical situation this not only extends the range of previous algorithms and the use of new ones for these studies but for spherical or cylindrical calculations it reduces the sensitivity of the results to grid setup with equal angular zoning that has heretofore been necessary with these problems

modal control of large flexible space structures using collocated actuators and sensors the authors consider the problem of assigning the eigenvalues associated with critical modes of a large flexible space structure into a specified region in the left half plane via direct velocity feedback control dvfc using collocated actuators and sensors conditions for the existence of a dvfc using collocated actuators and sensors that can achieve the eigenvalue assignment are derived when there exist nonunique feasible dvfcs the one with least frobenius norm feedback gain is determined an experimental four bay truss is used to illustrate 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

stability of an entire polytope of polynomials abstract stability is a central issue for polynomials with variable coefficients the stability of a class of polynomials whose coefficients are characterized by a polytope is studied we show that the stability of this whole family of polynomials is assured if the zeros of the even and odd parts of the vertex polynomials have interlacing properties

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