root locus approach to the stability analysis of interval matrices a new approach called the root locus approach is developed for the stability analysis of perturbed matrices the basic idea of this approach is to ensure that the root loci of a continuously perturbed matrix remain in the open left half complex s plane based on this approach new techniques are presented to analyse the stability of interval matrices examples are given to demonstrate the merit of the proposed theorems

stability robustness of interval matrices via lyapunov quadratic forms a sufficient condition for the robust stability of a class of interval matrices is derived using the lyapunov approach the matrices considered have elements which are nonlinear functions of a vector of independent and bounded parameters the robust stability condition requires that a quadratic form be positive definite in a finite number of conspicuous points of an enlarged parameter space

q deformed harmonic and clifford analysis and the q hermite and laguerre polynomials we define a q deformation of the dirac operator inspired by the one dimensional q derivative this implies a q deformation of the partial derivatives by taking the square of this dirac operator we find a q deformation of the laplace operator this allows us to construct q deformed schrodinger equations in higher dimensions the equivalence of these schrodinger equations with those defined on q euclidean space in quantum variables is shown we also define the m dimensional q clifford hermite polynomials and show their connection with the q laguerre polynomials these polynomials are orthogonal with respect to an m dimensional q integration which is related to integration on q euclidean space the q laguerre polynomials are the eigenvectors of an suq 1 1 representation

stability analysis of gray discrete time systems the stability of discrete time system whose state matrix is an interval matrix is discussed in terms of lyapunov stability theory some sufficient conditions are obtained which guarantee the stability of gray discrete time systems or interval discrete time systems illustrative examples are given

sufficient regularity conditions for complex interval matrices and approximations of eigenvalues sets abstract in this paper two approaches are described to establish verifiable sufficient regularity conditions of complex interval matrices in the first approach a complex interval matrix is mapped to a real block interval matrix and then its sufficient regularity conditions are obtained in the second approach a necessary condition for the singularity of a complex interval matrix is derived and used to get its sufficient regularity conditions as an application the above derived sufficient regularity conditions are used to investigate the location of the outer approximations of individual eigenvalue sets of complex interval matrices two algorithms are proposed and results obtained are compared with those obtained by earlier methods and monte carlo simulations the advantages of these algorithms are that they can detect gaps in between the approximations of the whole eigenvalue sets the second algorithm is very effective compared to the first algorithm from the computational time point of view several numerical examples and statistical experiments are worked out to validate and demonstrate the efficacy of our work

an approach to robust stability of matrix polytopes through copositive homogeneous polynomials an approach to guaranteeing the stability of a polytope of matrices is proposed previous results in the literature have made the obvious connection between various robust stability problems and a test for positivity of a multivariable polynomial such results are extended in order to demonstrate that all matrices within a polytope are stable is and only if an associated homogeneous polynomial is strictly copositive the additional structure obtained by exploiting homogeneity of this multivariable polynomial leads to several computationally tractable sufficiency tests for establishing either robust stability or instability of a polytope of matrices

linear rank statistics finite sampling permutation tests and winsorizing asymptotic normality and a representation of all possible subsequential limiting distributions of a simple linear rank statistic are obtained this is then applied to finite sampling and permutation tests for slope coefficients the effects of winsorizing in these situations are considered carefully of particular interest regarding slope coefficients is that either using normal score regression constants or winsorizing slowly increasing numbers of the population values will guarantee asymptotic normality

stability analysis of interval matrices another sufficient condition a sufficient condition for the stability of interval matrices is presented based on a lyapunov approach the condition while requiring the solution of a lyapunov matrix equation removes the restrictions imposed by heinen 1984 examples given illustrate the improvement of the proposed condition over the ones given by heinen and bialas 1983 and daoyi 1985

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

the quantum group structure of 2d gravity and minimal models i on the unit circle an infinite family of chiral operators is constructed whose exchange algebra is given by the universalr matrix of the quantum groupsl 2 q this establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group the method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofsl 2 q the formulae so derived which generalize an earlier particular case worked out by babelon are remarkably compact and may be entirely written in terms of q deformed factorials and binomial coefficients

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

comment on the q analogues of the harmonic oscillator the heisenberg relation for the q analogues of the quantum harmonic oscillator introduced independently by macfarlane and biedenharn 1989 for the quantum group su 2 q is derived by a method analogous to that used by schwinger 1968 for the su 2 case the author also speculates on the possible use of the quantum group as a generalisation of quantum mechanics

characterization of robust controllers abstract this paper presents a procedure for designing a robust controller which is able to stabilize a multivariable system with interval uncertainties in the system parameters the uncertain parameters are assumed to lie within known intervals the fundamental idea behind the procedure is to characterize the set of robust controllers as a convex set by the new matrix inequalities result a suitable controller can then be chosen by means of an optimization algorithm the controller thus chosen will be able to ensure that the closed loop matrix lies within desired intervals for all the prescribed interval uncertainties in the system parameters

another sufficient condition for the stability of interval matrices a sufficient condition for the stability of interval matrices is presented which is based on the condition given by yedavalli 1985 b examples are given which illustrate the improvement of the proposed condition over that given by yedavalli 1986

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

sufficient conditions on stability of interval matrices connections and new results the stability properties of interval matrices are discussed sufficient conditions are given for these properties based upon gershgorin s theorem and its extension and they are shown to improve previous results the derivation helps clarify relations between certain previously available results and shows they can be derived using gershgorin s theorem furthermore the problem of determining the stability of interval matrices is related to that of characterizing certain nonsingular m matrices

bootstrapping empirical functions in our previous sections we have established weak convergence of mean residual life total time of test lorenz and goldie processes to appropriate gaussian processes apart from a few special cases cf section 8 these limiting processes are functions of the underlying distributions consequently when testing for statistical hypotheses for example one would have to compute the resulting limiting distribution for each f of interest the same is true when trying to use our results for constructing confidence bands for the theoretical functionals of the said empirical processes this type of problems can be solved by adapting the bootstrap method proposed by efron 1979 1982 to the present situation as we will see the bootstrap method is simply a monte carlo simulation determined by the given observations

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

counter example to a recent result on the stability of interval matrices by s bialas abstract in bialas 1983 a condition is given which is supposedly necessary and sufficient for the stability of interval matrices we demonstrate via a counter example that the condition given is in fact not sufficient

stabilization of perturbed systems via linear optimal regulator linear quadratic state feedback regulators make the resulting closed loop systems stable enough i e they realize robust stabilization many attempts at robust stabilization using linear quadratic regulators have been reported one of the major trends of formulating uncertainty in systems is to express perturbed parameters as the sum of two terms i e nominal values and the deviation from them in this paper it is assumed that the upper and lower bounds for each uncertain parameter can somehow be estimated this enables us to dispense with nominal values the main aim is to contrive a robust feedback stabilization law for systems with parameters falling into certain ranges via a linear quadratic regulator based only upon information on their bounds the systems under consideration are therefore those having interval system matrices in which each element has the above mentioned two sided bounds a certain feedback law is a stabilizing law for a system with an interval system matrix if and only if the

identification and application of bounded parameter models abstract the place of parameter bounding algorithms in identification methodology is discussed such algorithms provide a radical alternative to the computation of parameter point estimates and covariances instead of p d f or mean and covariance for the noise and prior parameter estimates they require bounds in an essentially deterministic model formulation the paper is a preliminary review of some standard tasks namely experiment design testing for outliers toleranced prediction and worst case control design in the context of parameter bounding identification computational requirements of these tasks are examined and the limitations of existing parameter bounding algorithms are noted

convergence condition for convex combinations of two polynomials a necessary and sufficient condition is derived for a convex combination of two given polynomials to be a convergent polynomial i e a polynomial having roots all within the unit circle in the complex plane this can be regarded as a discrete counterpart to the existing result for the continuous case a simple numerical example is provided to demonstrate the theorem obtained

criteria for robust absolute stability of time varying nonlinear continuous time systems the present paper establishes results for the robust absolute stability of a class of nonlinear continuous time systems with time varying matrix uncertainties of polyhedral type and multiple time varying sector nonlinearities by using the variational method and the lyapunov second method criteria for robust absolute stability are obtained in different forms for the given class of systems specifically the parametric classes of lyapunov functions are determined which define the necessary and sufficient conditions of robust absolute stability the piecewise linear lyapunov functions of the infinity vector norm type are applied to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations several simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for special cases two examples are presented as applications of the present results to a particular second order system and to a specific class of systems with time varying interval matrices in the linear part

stability analysis of interval matrices improved bounds new sufficient conditions for the stability of interval matrices are presented based on the lyapunov stability approach as well as on the frequency domain approach the conditions help to overcome the conservatism of the existing criteria for the stability of interval matrices based on a matrix approach several numerical examples are given to demonstrate the new conditions and to compare them with results previously reported

invariance of a strict hurwitz property for uncertain polynomials with dependent coefficients this note is concerned with the invariance of strict hurwitz property for uncertain polynomials having dependent coefficients a feasible test criterion is provided which reduces the conservatism of the kharitonov theorem

stability analysis of dynamic interval systems abstract the stability analysis of interval matrices with certain stability margins is considered by means of similarity transformation and gershgorin s theorem new criteria are developed and illustrated by examples we also demonstrate a counter example to a recent result by argoun 1986

perturbation bounds for robust stability in linear state space models abstract this paper addresses the aspect of stability robustness of linear systems in the time domain upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability both for structured as well as unstructured perturbations using the lyapunov approach for structured perturbation the resulting bound is such that it garners the structural information about the nominal as well as the perturbation matrix into a single unified expression in the case of unstructured perturbations special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound without the need to solve the lyapunov equation improvement of the proposed measures is illustrated with the help of examples

sufficient conditions for stability of interval matrices extremely simple sufficient conditions are presented for the stability of interval matrices these complement the necessary and sufficient conditions developed by bialas 1983 which become unwieldy for large matrices 4 4 and above

robustness analysis of interval matrices based on kharitonov s theorem this paper deals with the robustness analysis problem of interval matrices expansion of det a b is used to get characteristic polynomials of two corresponding systems kharitonov s theorem is applied to test hurwitz properties of the two polynomials a new sufficient condition for all eigenvalues of the interval matrix to lie in a damping cone on complex plan is derived illustrative examples are given

simple criteria for stability of interval matrices in this paper the simple sufficient conditions are presented for the stability of a constant real matrix whose elements are not known precisely but we know the upper and lower bounds of intervals in which the elements of matrix a are confined these conditions are also applied in determining the stability and the degree of stability of a matrix

stability robustness analysis of digital control systems in state space models the stability robustness of discrete linear time invariant systems in state space models is analysed based on a root locus approach and from the stable eigenvalue viewpoint fundamental criteria for testing the stability robustness of autonomous systems are derived and applied to the robustness analysis of multivariable feedback systems both the norm bound and the element bounds for the allowed perturbations are obtained a stability robustness index is denned which is useful both for the analysis and synthesis of control systems

comments on sufficient condition for the asymptotic stability of interval matrices abstract recently jiang 1987 has shown that a set of interval matrices is stable if the symmetric part of the vertex matrices are stable this result of jiang 1987 is shown to be only applicable to matrices with all the diagonal elements negative

sufficient conditions for the stability of interval matrices the stability of interval dynamical systems is studied sufficient conditions for the polytope of interval matrices are examined and some of the proofs are greatly simplified more generally sufficient conditions are obtained and the approach taken in the new proofs has the potential for further generalizations of the result obtained in this paper

a necessary and sufficient condition for the stability of interval matrices consider the linear dynamical system described by the equation xdot t ax t where a is a constant real matrix with dimensions n n and the elements of matrix a are not known precisely we know only the upper and lower bounds of the intervals in which the elements of matrix a are confined in this paper there is proved a necessary and sufficient condition for the stability of solutions of eqn 1 with incomplete information about the elements of matrix a

embedding the finite sampling process at a rate a huge body of if and only if theorems can be obtained based on certain strong embedding theorems for the partial sum process mathbb s n and the uniform empirical and quantile processes mathbb u n and mathbb v n this embedding was accomplished in 1986 by m csorgo s csorgo l horvath and d mason their embedding is beautifully formulated so that many necessary and sufficient type results can be established using it it is worthwhile to have an accessible proof indeed these authors have since produced two papers that obtain the essential form of their result without appealing to earlier work of j komlos p major and g tusnady indeed this present paper does this for the finite sampling process mathbb r n and the weighted empirical process mathbb w n these latter results are entirely new and are the main objectives of the present paper the applications of these latter results will appear elsewhere

how do bootstrap and permutation tests work resampling methods are frequently used in practice to adjust critical values of nonparametric tests in the present paper a comprehensive and unified approach for the conditional and unconditional analysis of linear resampling statistics is presented under fairly mild assumptions we prove tightness and an asymptotic series representation for their weak accumulation points from this series it becomes clear which part of the resampling statistic is responsible for asymptotic normality the results leads to a discussion of the asymptotic correctness of resampling methods as well as their applications in testing hypotheses they are conditionally correct iff a central limit theorem holds for the original test statistic we prove unconditional correctness iff the central limit theorem holds or when symmetric random variables are resampled by a scheme of asymptotically random signs special cases are the m n out of k n bootstrap the weighted bootstrap the wild bootstrap and all kinds of permutation statistics the program is carried out for convergent partial sums of rowwise independent infinitesimal triangular arrays in detail these results are used to compare power functions of conditional resampling tests and their unconditional counterparts the proof uses the method of random scores for permutation type statistics

thermodynamics of boson systems related to dunkl differential difference operators abstract we study the thermodynamics of systems based on a fock space representation inspired by the differential difference operators proposed in dunkl 1989 we calculate thermodynamic functions as the entropy and heat capacity and compare them with the standard boson case a calculation of the second virial coefficient and the scalar curvature in two and three dimensions show that these systems become repulsive within an interval of negative values of the reflection operator parameter in addition the stability of this system is examined as a function of

non commutative differential calculus and q analysis starting from the formulation of covariant noncommutative differential calculus recently given by wess and zumino 1990 the author constructs a deformation of the virasoro algebra which allows him to identify the variables and differential operators on the quantum plane r2q to those on the classical plane r2 this correspondence indicates how noncommutative geometry can be understood in terms of q analysis on the commutative plane he generalizes this result to the general n dimensional case and discuss some of its consequences

q deformation of radial problems the simple harmonic oscillator in two dimensions the possibility of q deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem the algebraic structure of the d dimensional simple harmonic oscillator is analysed from this point of view in the two dimensional case q deformed bosons are introduced for the radial degrees of freedom which together with the standard angular operators yield the so 2 soq 2 1 dynamical algebra the corresponding hilbert space is identified and possible q deformed hamiltonians described

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

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

necessary and sufficient conditions for stability of a class of interval matrices in this paper some necessary and sufficient conditions for stability complete instability and composite stability type of a class of interval matrices are given

comments on a necessary and sufficient condition for the stability of interval matrices in a recent paper by bialas a new proof of a known result concerning interval polynomials was given this result is summarized here and a much stronger result is mentioned bialas main theorem on the stability of interval matrices is shown to be false

an algorithm for the computation of the structured complex stability radius abstract perturbed state space models of the form x a bdc x are considered where a b c are given matrices a is stable and d is an unknown complex perturbation matrix a reliable and efficient algorithm for the computation of the norm of the minimal norm destabilizing matrix d is presented the performance of the algorithm and possible applications are illustrated

necessary and sufficient conditions for stability of time varying discrete interval matrices sufficient conditions for the stability of time varying discrete interval systems are presented the spectral radius of the maximum absolute matrix of all interval matrices is employed as opposed to utilizing the matrix norm or lyapunov theorem for special cases our result becomes a necessary and sufficient condition for stability furthermore the proposed conditions are inclusive of the previous results of stability of time invariant interval systems

stability analysis of a family of matrices the authors extend the use of the lyapunov equation method to check the stability of a family of matrices the family of interval matrices is defined as the convex hull of a finite number of matrices and is a special case where each element has independent perturbation while the stability of interval matrices cannot be inferred from the stability of its vertex matrices the authors show that the family of matrices will be stable if the symmetric matrices associated with the vertex matrices have matrix measures less than two

the q deformed poisson bracket levi civita symbol and poincare algebra in this paper we use a special choice of the glx qij 2n quantum plane and its differential calculus for a q deformed phase space to define a modified poisson bracket and construct the contraction rule for the q deformed levi civita symbol we find the q deformed phase space variable realization of the soq 3 and q deformed poincare algebras

a method for estimating upper and lower bounds of eigenvalues of closed loop systems with uncertain parameters using the convex model theory the vibration control problem of structures with uncertain parameters is discussed which is approximated by a deterministic one a method for estimating the upper and lower bounds of eigenvalues of the closed loop systems is presented by combining the matrix perturbation and optimization the present method is applied to a vibration system to illustrate the application the numerical results show that the present method is effective

interval finite element method for dynamic response of closed loop system with uncertain parameters in practical engineering it is difficult to obtain all possible solutions of dynamic responses with sharp bounds even if an optimum scheme is adopted where there are many uncertain parameters in this paper using the interval finite element ife method and precise time integration pti method we discuss the dynamic response of vibration control problem of structures with interval parameters with matrix perturbation theory and interval arithmetic the algorithm for estimating upper and lower bounds of dynamic response of the closed loop system is developed directly from the interval parameters two numerical examples are given to illustrate the application of the present method the example 1 is used to show the applicability of the present method the example 2 is used to show the validity of the present method by comparing the results with those obtained by the classical random perturbation method copyright 2006 john wiley sons ltd