%0 Journal Article
%T On Stability of Parametrized Families of Polynomials and Matrices
%A Handan Akyar
%A Taner B¨¹y¨¹kk£¿ro£¿lu
%A Vak£¿f Dzhafarov
%J Abstract and Applied Analysis
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/687951
%X The Schur and Hurwitz stability problems for a parametric polynomial family as well as the Schur stability problem for a compact set of real matrix family are considered. It is established that the Schur stability of a family of real matrices is equivalent to the nonsingularity of the family if has at least one stable member. Based on the Bernstein expansion of a multivariable polynomial and extremal properties of a multilinear function, fast algorithms are suggested. 1. Introduction Let ( ) be the set of real vectors (numbers), the set of complex numbers. Let a polynomial family be defined by where the uncertainty vector belongs to a box Denote the set of all polynomials by , that is, The family is said to be Schur (Hurwitz) stable if every polynomial in this family is Schur (Hurwitz) stable, that is, all roots lie in the open unit disc (open left half plane). A similar definition is valid for a matrix family where the word ¡°roots¡± is replaced by ¡°eigenvalues¡±. If then the family (1.3) is called an interval polynomial family. The Hurwitz stability problem of interval polynomials is solved by Kharitonov Theorem [1]. The Schur stability problem of interval polynomials has been studied in many works (see [2¨C5] and references therein). In [3, 5] using techniques from complex analysis, necessary conditions for the Schur stability of interval polynomials are obtained. A function is said to be multilinear if it is affine-linear with respect to each component of . The polynomial (1.1) is called multilinear if all coefficient functions are multilinear. The family (1.1) is called polynomially parameter dependent if all coefficient functions are depending polynomially on parameters , . In [6] an algorithm for the robust Schur stability verification of polynomially parameter dependent families is given. This algorithm relies on the Bernstein expansion of a multivariable polynomial and is based on the decomposition of a polynomial into its symmetric and antisymmetric parts, and on the Chebyshev polynomials of the first and second kinds. In this paper we investigate the robust Schur stability of polynomially dependent families without employing Chebyshev polynomials (cf. [6]) (see Sections 2 and 4). The following theorems express the well-known properties of a multilinear function defined on a box. Theorem 1.1 (see [2]). Suppose that is a box with extreme points , and is multilinear. Then both the maximum and the minimum of are attained at extreme points of . That is, Theorem 1.1 leads to the following sufficient condition for stability of the multilinear family
%U http://www.hindawi.com/journals/aaa/2010/687951/