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

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

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

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

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

blowup and self similar solutions for two component drift diffusion systems abstract we discuss asymptotic properties of solutions of two component parabolic drift diffusion systems coupled through an elliptic equation in two space dimensions in particular conditions for finite time blowup versus the existence of forward self similar solutions are studied

an iterative multi level algorithm for solving nonlinear ill posed problems the convergence analysis of landweber s iteration for the solution of nonlinear ill posed problem has been developed recently by hanke neubauer and scherzer in concrete applications sufficient conditions for convergence of the landweber iterates developed there although quite natural turned out to be complicated to verify analytically however in numerical realizations when discretizations are considered sufficient conditions for local convergence can usually easily be stated this paper is motivated by these observations initially a discretization is fixed and a discrete landweber iteration is implemented until an appropriate stopping criterion becomes active the output is used as an initial guess for a finer discretization an advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces the numerical performance of this multi level algorithm is compared with landweber s iteration

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

extremal robustness properties of multilinear interval systems this paper deals with robustness properties of a control system subject to parameter uncertainty when the parameters appear affine multilinearly in the coefficients of the characteristic polynomial the robust hurwitz stability of such a multilinear interval polynomial family reduces to that of the stability of a set of manifolds derived from the kharitonov polynomials several extremal properties of these stability manifolds are derived these properties are useful in determining the worst case parametric h and nonlinear sector bounded stability margins for control systems containing interval parametric uncertainty they generalize previous results obtained by the authors for the linear case

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

robust output tracking control of siso plants with multiple operating points and with parametric and unstructured uncertainties the robust output tracking control problem is considered for single input single output plants with multiple operating points and with both parametric and unstructured uncertainties i e interval parameter uncertainties and norm bounded unmodelled dynamics the parametric uncertainties can involve parameter perturbations as well as degree variations the plant with uncertainties is assumed to be of minimum phase but is not required to have a fixed number of unstable poles a new method is presented by which a linear time invariant robust controller is designed in two steps first a nominal controller is designed to get exact output tracking for the nominal plant then a robust compensator is added to achieve robust properties it is shown that robust stability robust static state property robust transient property and robust tracking property can be achieved simultaneously a computationally tractable design procedure is presented a salient feature of our results shown in the present paper is th

the method of conjugate residuals for solving the galerkin equations associated with symmetric positive semidefinite ill posed problems for the numerical solution of the galerkin equations associated with linear ill posed problems that are symmetric and positive semidefinite the method of conjugate residuals is considered an a posteriori stopping rule is introduced and associated estimates for the approximations are provided which are order optimal with respect to noise in the right hand side and with respect to the discretization error

generalized cross validation applied to conjugate gradient for discrete ill posed problems in this paper we propose a new method to apply the generalized cross validation gcv as a stopping rule for the conjugate gradient cg in general to apply gcv to an iterative method one must estimate the trace of the so called influence matrix which appears in the denominator of the gcv function in the case of cg unlike what happens with stationary iterative methods the regularized solution has a nonlinear dependence on the noise which affects the data of the problem this fact is often pointed out as a cause of poor performance of gcv to overcome this drawback our proposal linearizes the dependence by computing the derivatives through iterative formulas we compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation on both 1d and 2d test problems

accelerated landweber iterations for the solution of ill posed equations in this paper the potentials of so calledlinear semiiterative methods are considered for the approximate solution of linear ill posed problems and ill conditioned matrix equations several efficient two step methods are presented most of which have been introduced earlier in the literature stipulating certain conditions concerning the smoothness of the solution a notion of optimal speed of convergence may be formulated various direct and converse results are derived to illustrate the properties of this concept if the problem s right hand side data are contaminated by noise semiiterative methods may be used asregularization methods assuming optimal rate of convergence of the iteration for the unperturbed problem the regularized approximations will be of order optimal accuracy to derive these results specific properties of polynomials are used in connection with the basic theory of solving ill posed problems rather recent results onfast decreasing polynomials are applied to answer an open question of brakhage numerical examples are given including a comparison to the method of conjugate gradients

approximation accuracy of the krylov subspaces for linear discrete ill posed problems abstract for the large scale linear discrete ill posed problem min a x b or a x b with b contaminated by gaussian white noise the lanczos bidiagonalization based krylov solver lsqr and its mathematically equivalent cgls the conjugate gradient cg method implicitly applied to a t a x a t b are most commonly used and cgme the cg method applied to min a a t y b or a a t y b with x a t y and lsmr which is equivalent to the minimal residual minres method applied to a t a x a t b have also been choices these methods exhibit typical semi convergence feature and the iteration number k plays the role of the regularization parameter however there has been no definitive answer to the long standing fundamental question can lsqr and cgls find 2 norm filtering best possible regularized solutions the same question is for cgme and lsmr too at iteration k lsqr cgme and lsmr compute different iterates from the same k dimensional krylov subspace a first and fundamental step towards answering the above question is to accurately estimate the accuracy of the underlying k dimensional krylov subspace approximating the k dimensional dominant right singular subspace of a assuming that the singular values of a are simple we present a general sin theorem for the 2 norm distances between these two subspaces and derive accurate estimates on them for severely moderately and mildly ill posed problems we also establish some relationships between the smallest ritz values and these distances numerical experiments justify the sharpness of our results

robust stability and performance of interval plants abstract in this paper we consider the problem of robust performance for systems with unstructured and parametric perturbations the parametric uncertainty is modeled through a single input single output interval system description and the unstructured uncertainty through a norm bounded perturbation for this type of parametric robust performance we show that in order to synthesize a controller that achieves robust performance for a family of interval systems it is sufficient to synthesize that controller for a much smaller subset of the interval family regardless of the number of uncertain parameters in the original problem this reduced set is composed of thirty two one parameter families of systems a special class of controllers with stronger results is also presented

a generalization of strong kharitonov theorems to polytopes of polynomials the root clustering property of a certain class of polytopes of polynomials is examined it is shown that the polytopes of polynomials have only zeros within the left sector if and only if a subset of their edge polynomials has only zeros within the left sector certain conditions are derived under which the root clustering property of the polytopes of polynomials can be inferred from a subset of their vertices

brief paper simultaneous schur stability of interval matrices interval matrix structures are ubiquitous in nature and engineering ordinarily in an uncertain system there is associated with a set of coupled interval matrices a basic issue of exploring its asymptotic stability here we introduce the notion of simultaneous schur stability by linking the concepts of the majorant and the joint spectral radius and prove the asymptotic stability of a set of interval matrices governed by simultaneous schur stability the present result may lead to the stability analysis of discrete dynamical interval systems

representations of robustly stabilizing controllers for systems with parameter uncertainty in this paper we consider the problem of robust stabilization for systems with parameter uncertainty we first formulate a design methodology and then demonstrate how it leads to analytical and graphical solutions we show that in the case of siso plant families with linear uncertainty and two parameter compensators one can obtain a representation of all robustly stabilizing controllers of a fixed order by introducing a rectangular overbound of value sets in the complex plane we also show how one can obtain representations of m parameter robustly stabilizing compensators of a fixed order

comments on extreme point results for robust stabilization of interval plants with first order compensators the authors point out a simple alternative proof of the main result theorem 1 of the paper by barmish hollot kraus and tempo 1992 this proof follows immediately on applying the vertex result of hollot and yang 1990 to the generalization of kharitonov s theorem given by chapellat and bhattacharyya 1988 1989 the authors also point out that more general vertex results are known

an overview of extremal properties for robust control of interval plants abstract this paper provides an overview of recent results on the problem of robust stability and performance of feedback control systems in the presence of plant perturbations the widely studied class of interval plants is considered and an effort is made to cover different types of plant perturbations by incorporating in addition to the interval plant description unstructured norm bounded perturbations extremality results for these classes of uncertain systems are provided in a unifying framework robust performance of control systems is addressed in the paper and a number of extremal results are given for the computation of the structured singular value function for feedback systems with interval plants also classical frequency response analysis for interval plant controller families of open loop transfer functions is surveyed with the aim of providing extremal results for the computation of specification parameters such as phase or gain margins and sensitivity and complementary sensitivity peaks

real and complex stability radii of polynomial matrices analytic expressions are derived for the complex and real stability radii of non monic polynomial matrices with respect to an arbitrary stability region of the complex plane numerical issues for computing these radii for different perturbation structures are also considered with application to robust stability of hurwitz and schur polynomial matrices c 2002 elsevier science inc all rights reserved

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

frequency response of uncertain systems with interval plants frequency domain properties of uncertain rational functions depending on two independent interval polynomials are investigated several simplifications for efficient computation of the envelope of the nyquist plots of the uncertain family are reported stronger computational reduction results are given for the bode magnitude and phase plots of an interval plant controller family of transfer functions these results allow for a clear understanding of the extremality properties of several frequency domain performance specifications commonly used in control system design

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

the instability of some gradient methods for ill posed problems for the solution of linear ill posed problems some gradient methods like conjugate gradients and steepest descent have been examined previously in the literature it is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a priori parameter choice regularization methods upon them

absolute stability of systems with parametric uncertainty and nonlinear feedback this paper gives an extension of the absolute stability analysis of systems whose forward path contains a linear time invariant subsystem with uncertain coefficients and whose feedback path contains a conic nonlinearity the results provide a simple frequency domain interpretation of stability that can be of importance in compensator design and robust stability analysis in nonlinear systems

frequency response of interval plant controller families abstract in this paper a general frequency domain result for families of interval plant with a fixed linear controller is given it is shown that the locus of the polar diagrams of frequency responses of the transfer functions of an interval plant controller family is bounded by the polar plots relative to the 32 segments of transfer functions of the interval plant family easy proofs of several important results such as the generalization of the theorem of kharitonov for feedback systems with interval plants or the robust version of the small gain theorem for the same class of systems are constructed by using the general result more importantly an immediate consequence of the main theorem is that extremal phase gain margins or sensitivity complementary sensitivity peaks for systems of the family can be deduced from those of the 32 segments of the interval plant family

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

robust stability of a family of disc polynomials in his well known theorem v l kharitonov established that hurwitz stability of a set f1 of interval polynomials with complex coefficients polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane is equivalent to the hurwitz stability of only eight polynomials in this set in this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fd of disc polynomials characterized by the fact that each coefficient of a typical element p s in fd can be any complex number in an arbitrary but fixed disc of the complex plane our result shows that the entire set is hurwitz stable if and only if the center polynomial is stable and the h norms of two specific stable rational functions are less than one our result can be readily extended to deal with the schur stability problem and the resulting condition is equally simple

robust strict positive real stabilization and asymptotic hyperstability robustness abstract this paper is devoted to the study of robust strict positive real stabilization of siso uncertain systems i e finding a single controller that not only robustly stabilizes a plant with uncertainties under unity feedback but also makes the closed loop system robustly strictly positive real it is proved that there must exist a stable controller achieving robust strict positive real stabilization of uncertain systems with both parameter variations and unstructured uncertainties under some reasonable assumptions the propemess of the controller is guaranteed when the largest relative degree of an uncertain system is retricted by applying these results to nonlinear systems some significant conclusions on asymptotic hyperstability robustness are given

brief condition and algorithm for simultaneous stabilization of linear plants this paper is devoted to the problem of simultaneous stabilization of linear plants it is shown that four given plants p i s i 1 2 3 4 are simultaneously stabilizable if and only if there exist four stable polynomials h i s i 1 2 3 4 which simultaneously satisfy two polynomial equations if we can only find four stable h i s i 1 2 3 4 to satisfy one of the two polynomial equations then two controllers c 1 s and c 2 s having a common denominator or numerator can be obtained such that c 1 s simultaneously stabilizes p 1 s and p 3 s and c 2 s simultaneously stabilizes p 2 s and p 4 s which is called the simultaneous stabilization in groups based on such two controllers a design algorithm for the simultaneous stabilization of three plants is induced

stability bounds of singularity perturbed systems a necessary and sufficient condition is developed for computing the upper bound in sub 0 of the parasitic parameter in in a singularly perturbed system which ensures stability of such a system if 0

robust parametric classical control design this paper extends well known graphical design tools such as the nyquist plot bode plot and nichols charts to the domain of control systems containing parameter uncertainty we have specifically considered linear interval control systems namely those where the uncertain parameters appear linearly in the characteristic polynomial and vary in intervals because in these cases exact construction of frequency envelopes is possible using these developments we calculate various extremal stability margins for such systems and these are used with standard classical control design techniques to deal with robustness with respect to parametric uncertainty in addition we give an effective technique to iteratively select values of the design parameters within given ranges so that the closed loop system produces improved gain or phase margin

robust stability of multivariable interval control systems in this paper we will give necessary and sufficient conditions for the hurwitz stability of mimo interval systems as a consequence of this result we will give a precise bound on the h norm of unstructured perturbations that can be tolerated by mimo interval systems connected in feedback with these perturbations we will also consider the robust version of the popov problem that considers interval mimo systems subject to nonlinear time invariant sector bounded perturbations and in turn obtain a robust stability condition of the popov problem with mixed perturbations

extreme point results for robust stability of interval plants beyond first order compensators motivated by robust stabilization of interval plants this paper concentrates on a fundamental problem involving a family of polynomials obtained via convex combination of extreme members the objective is to provide conditions under which stability of the extremes implies stability of the entire family although the existing literature contains rather strong extreme point results involving frequency domain analysis there is a paucity of more direct coefficient space criteria this paper deals exclusively with the coefficient space problem to this end we introduce the so called alternating hurwitz minor condition ahmc and show how it can be used to enlarge the class of polynomial families for which extreme point results can be obtained subsequently the ramifications on robust stability of interval plants are fully explained in contrast to existing literature for first order compensators the ahmc makes it possible to deal with classes of higher order compensators in an extreme point context

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

robust stability under structured and unstructured perturbations the problem of robust stability for linear time invariant single output control systems subject to both structured parametric and unstructured h sub infinity perturbations is studied a generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time invariant dynamic system under perturbations of mixed type is presented the solution involves calculating the h sub infinity norm of a finite number of extremal plants the problem of calculating the exact structured and unstructured stability margins is then constructively solved a feedback control system containing a linear time invariant plant which is subject to both structured and unstructured perturbations is considered the case where the system to be controlled is interval is treated and a nonconservative easily verifiable necessary and sufficient condition for robust stability is given the solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered

convergence rate results for steepest descent type method for nonlinear ill posed equations convergence rate result for a modified steepest descent method and a modified minimal error method for the solution of nonlinear ill posed operator equation have been proved with noisy data to our knowledge convergence rate result for the steepest descent method and minimal error method with noisy data are not known we provide a convergence rate results for these methods with noisy data the result in this paper are obtained under less computational cost when compared to the steepest descent method and minimal error method we present an academic example which satisfies the assumptions of this paper

control of discrete time hybrid stochastic systems a realistic stochastic control problem for hybrid systems with markovian jump parameters can have switching parameters in both the state and the measurement equations furthermore both the base and jump states in general are not perfectly observed there are only two existing controllers for this problem both with complexity exponentially increasing with time the authors present another control algorithm for stochastic systems with markovian jump parameters this algorithm is derived through the use of stochastic dynamic programming and is designed to be used for realistic stochastic control problems i e with noisy state observations this scheme has fixed computational requirements at each stage and a natural parallel implementation simulation results are used to compare the algorithm with previous schemes

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

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

structured singular values and stability analysis of uncertain polynomials part 2 a missing link abstract in part 1 of this paper a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one in part 2 of this series we demonstrate that the framework developed in part 1 is particularly suitable for a class of stability problems and its unifies several frequency based stability conditions obtained via alternative approaches generally these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion several stability conditions are derived readily from solution of the structured singular value presented in part 1 which either extend or coincide with previously known results in particular for family of interval and diamond polynomials we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of kharitonov theorem lead to one another thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series a related robust stabilization problem is also investigated in this framework

mixed deterministic randomized methods for fixed order controller design in this paper we propose a general methodology for designing fixed order controllers for single input single output plants the controller parameters are classified into two classes randomized and deterministically designed for the first class we study randomized algorithms in particular we present two low complexity algorithms based on the chernoff bound and on a related bound often called ldquolog over logrdquo bound which is generally used for optimization problems secondly for the deterministically designed parameters we reformulate the original problem as a set of linear equations then we develop a technique which efficiently solves it using a combination of matrix inversions and sensitivity methods a detailed complexity analysis of this technique is carried on showing its superiority from the computational point of view to existing algorithms based on linear programming in the second part of the paper these results are extended to h infin performance one of the contributions is to prove that the deterministically designed parameters enjoy a special convex characterization this characterization is then exploited in order to design fixed order controllers efficiently we then show further extensions of these methods for stabilization of interval plants in particular we derive a simple one parameter formula for computing the so called critical frequencies which are required by the algorithms

a unified approach for the stability robustness of polynomials in a convex set abstract consider a polynomial p s k which is affine in the parameter k assume that the vector k is uncertain but belongs to a bounded closed convex set which contains the origin as an interior point and call a polynomial stable if all of its roots are contained in a pre specified stability region in the complex plane then the stability robustness of p s k can be measured by the maximal nonnegative number with the property that if the gauge or the minkowski functional of k with respect to the convex set is less than the polynomial p s k is always stable this paper develops a unified approach to compute the robustness measure using the framework of convex analysis the approach is very general because of two reasons i the stability region in the complex plane can be an arbitrary open set which includes the open left half plane and the open unit disc as special cases ii the convex set in which k is contained can be assumed to have an arbitrary shape which includes polytopes and ellipsoids as special cases the computational procedure to compute which results from this approach is easy to implement an example is included to illustrate the type of results which may be obtained the approach developed in this paper when specialized leads to some of the available results in the literature moreover it generates many new and interesting results another important feature of this approach is that it provides a rich mathematical insight to the stability robustness problem of polynomials

on the thirty two virtual polynomials to stabilize an interval plant this paper studies the conservatism of the 32 virtual polynomials to stabilize an interval plant it is shown that working with the 32 virtual vertices is generally less conservative than with the kharitonov polynomials of the smallest interval polynomial containing the characteristic polynomial polytope by means of the former it is possible to find all the controllers such that the value set of the polytope of characteristic polynomials is applied in two quadrants as a maximum for each spl omega while using the latter only some of them can be found the cases in which both methods coincide are also analyzed and the conditions on the numerator and denominator of the controller are developed thus this coincidence can be known a priori from the characteristics of the coefficients of the numerator and denominator of the controller it is shown that these conditions are satisfied by the first order controllers

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

calculation of the largest generalized stability hypersphere in the robust stability problem for the maximum setting time and minimum damping ratio cases the results of a procedure which employs a geometric approach to derive the maximal ellipsoidal domains of stability in the physical parameter space for the case in which the characteristic polynomial coefficients depend linearly on the physical parameters of the plant obtained by biernacki et al see ibid vol ac 32 no 6 p 495 505 1987 are extended to a more generalized stability region such as the maximum setting time and the minimum damping ratio cases the method can be used to estimate the largest generalized stable hypersphere in the parameter space for a given controller

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

stability robustness analysis of multiple time delayed systems using building block concept an intriguing perspective is presented in studying the stability robustness of systems with multiple independent and uncertain delays it is based on a holographic mapping which is implemented over the domain of the delays this mapping considerably alleviates the problem which is otherwise known to be notoriously complex it creates a dramatic reduction in the dimension of the problem from infinity to manageably small number ultimately the process is reduced to studying the problem within a finite dimensional cube with edges of length 2pi in the new domain what we call the building block in essence the mapping collapses the entire set of potential stability switching points onto a small upperbounded number of building hypersurfaces we further demonstrate that these building hypersurfaces can be implicitly defined and they are completely isolated within the above mentioned cube it is also shown that the exhaustive detection of these building hypersurfaces is necessary and sufficient in order to arrive at the complete stability robustness picture we seek as a consequence this concept yields a very practical and efficient procedure for the stability assessment of such systems this novel perspective serves very well for the preparatory steps of the authors earlier contribution in the area cluster treatment of characteristic roots ctcr we elaborate on this combination which forms the main contribution of the paper several example case studies are also provided

on the sixty four polynomials of djaferis to stabilize an interval plant this paper shows that 32 of the 64 polynomials developed by djaferis to stabilize an interval plant with a controller are superfluous thus it is sufficient to stabilize 32 polynomials simultaneously moreover the construction developed in this note allows the implementation of these polynomials in a simple way

robust stability manifolds for multilinear interval systems the stability of a class of multilinearly perturbed families of systems is considered it is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved the extremal property of these subsets is established the results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces

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

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

root clustering for convex combination of complex polynomials in some important applications such as the edge theorem it is required that a polynomial is alpha stable along a parameter segment using the critical constraint the necessary and sufficient conditions for root clustering alpha stability of convex combinations of complex polynomials are presented the approach is general and requires that a certain real polynomial has no zeros in the open interval

robust pole assignment via reflection coefficients of polynomials a robust version of the output controller design for discrete time systems is introduced instead of a single stable point a stable polytope or simplex is preselected in the closed loop characteristic polynomial coefficients space a constructive procedure for generating stable polytopes is given starting from the unit hypercube of reflection coefficients of monic polynomials this procedure is quite straightforward because for a special family of polynomials the linear cover of so called reflection vectors is stable the roots placement of reflection vectors is studied if a stable target simplex is preselected then the robust output controller design task is solved by quadratic programming approach

computation of the frequency response of multilinear affine systems deals with the problem of computing the frequency response of an uncertain transfer function whose numerator and denominator polynomials are multiples of independent uncertain polynomials of the form p s q l sub o q l sub 1 q s spl middot spl middot spl middot l sub n q s sup n whose coefficients depend linearly on q q sub 1 q sub 2 q sub q sup t and the uncertainty box is q q q sub i spl epsiv q sub i q sub i i 1 2 q using the geometric structure of the value set of p s q a powerful edge elimination procedure is proposed for computing the bode nyquist and nichols envelopes of these uncertain systems a numerical example is included to illustrate the benefit of the method presented