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

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

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

proof of a conjecture about atomic and molecular cores related to scott s correction a great deal is known about the ground states of large atoms in the framework of the non relativistic schrodinger equation with fixed i e infinitely massive nuclei the leading term in powers of the nuclear charge z is given exactly by thomas fermi theory as was proved by lieb and simon 12 see 11 for a review this leading term in the energy is proportional to z 7 3 with the proportionality constant depending on the ratio of n z which is assumed to be held fixed as z here n is the electron number neutrality i e n z is not required even though it is the case of primary physical interest the characteristic length scale for the electron density in the sense that all the electrons can be foun on this scale in the limit z is z 1 3 the fact that the true quantum mechanical electron density d converges after suitable scaling to the thomas fermi density tf as z with n z fixed was proved in 12 the chemical radius which is another length altogether is believed but not proved to be order z 0 as z

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

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

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

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

lp parameter perturbation and design of robust controllers for linear systems in this paper linear systems with characteristic equations that are linearly dependent on some uncertain parameters are considered a new procedure is proposed for the calculation of d stability margins in the parameter space in general lp norms the procedure is simple and computationally efficient for several important regions of interest in the complex plane e g left hand plane sectors and circles centred at the origin it requires the solution of only one equation based on this result a new method is presented for the design of robust controllers robust placement of the closed loop poles inside the regions of interest is addressed the method provides a simple and computationally tractable solution to the robust controller design problem it can be used to synthesize robust controllers with general rational transfer functions for plants modelled as the ratio of two uncertain polynomials subjected to lp parameter perturbation

asymptotic behavior of the one particle density matrix of atoms at distances z 1 from the nucleus we prove that the suitably rescaled density matrix of ground states of atomic schrodinger operators with nuclear chargez converges on the scale 1 z to the projection of the negative spectral subspace of the schrodinger operator of the hydrogen atom z 1

an atomic energy lower bound that agrees with scott s correction abstract in this paper we prove that the energy of an atom with nuclear charge z is greater than c tf z 7 3 q 8 z 2 o z 17 9 log z atomic units as z q is the number of spin states q 2 for electrons and c tf is the thomas fermi constant for such systems

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

a note on kharitonov type results in the space of markov parameters c v hollot ibid vol 34 no 5 p 536 may 1989 used markov s theorem of determinants to obtain the largest stability box around a nominal hurtwitz polynomial under coefficient perturbation in the space of markov s parameters in the present work the authors use some previously established related research results to considerably simplify the proposed technique and make it computationally more efficient these results are also extended to the discrete time case and some illustrative examples are given

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

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

doubly cuspidal cohomology for principal congruence subgroups of gl 3 z the cohomology of arithmetic groups is made up of two pieces the cuspidal and noncuspidal parts within the cuspidal cohomology is a subspacethe f cuspidal cohomology spanned by the classes that generate representations of the associated finite lie group which are cuspidal in the sense of finite lie group theory few concrete examples of f cuspidal cohomology have been computed geometrically outside the cases of rational rank 1 or where the symmetric space has a hermitian structure this paper presents new computations of the f cuspidal cohomology of principal congruence subgroups f p of gl 3 2 of prime level p we show that the f cuspidal cohomology of f p vanishes for all p 19 with p 11 but that it is nonzero for p 11 we give a precise description of the f cuspidal cohomology for jt 11 in terms of the f cuspidal representations of the finite lie group gl 3 e 11 we obtained the result ultimately by proving that a certain large complex matrix m is rank deficient computation with the svd algorithm gave strong evidence that m was rank deficient but to prove it we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields

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