On Stability of Parametrized Families of Polynomials and Matrices

and Applied Analysis 3 a splitting procedure of the box of reflection coefficients, new conditions for the Schur stability are given. The application of Theorem 1.2 for determining the stability of polynomials with multilinear coefficients yield conservative results. In 8, 9 sufficient conditions are given for ensuring that the image of a multilinear function over the box Q is a convex polygon whose edges are images of the edges of the box Q. In this case the stability can be tested by the Edge Theorem 2 . In 10 , the notion of the principal point of Q is introduced, and it is shown that for a multilinear mapping f : Q → C, the boundary of f Q is a subset of f P , where P is the set of all principal points. Application of the results obtained to stability of one important subclass of multilinear systems is given in 10 . In 11 , using the notion of generalized principal point the characterization of the smallest set of regions in the complex plane within which the roots of 1.1 lie, is given. As far as the recent works on the stability of multilinear families are concerned, we can refer the reader to 12, 13 . In 13 , a multilinear family which can be expressed as the product of independent linear structures is considered. The paper suggests an elimination approach which eliminates the vertices q ∈ Q that are not useful for the construction of the boundary of the value set f Q . In 12 a sufficient condition for the zero inclusion of the value set f Q is given, where f : Q → C is multilinear. On the basis of this condition a numerical procedure for testing the whether or not f Q includes the origin is given. The procedure uses the iterative subdivision of the box Q. In this paper we suggest a new simple algorithm for testing Schur stability of a multilinear family. This algorithm is based on Theorem 1.1 and is sufficiently fast. The Schur stability rather than Hurwitz stability has the following advantage. In the obtained results, the segment −2, 2 arises naturally see Theorem 1.1, Algorithm 3.1, Theorem 5.1 whereas the cutoff frequencies should be calculated in the Hurwitz stability problems. On the other hand the Hurwitz stability can also be tested by this algorithm, since by using the well-known transformation s z 1 / z − 1 , the Hurwitz stability problem can be transformed into Schur stability problem see Example 3.3 taken from 14 . In the second part of the paper, we consider the application of our approach to matrix Schur stability problem. Stability problem of matrix families has been studied in many works see 2, 4, 15–18 and references therein . Naturally, a great deal of research has been devoted to interval matrices. Interval matrix structures are ubiquitous in nature and engineering. In 15, 18 extreme point results for Hurwitz stability are obtained which expresses the stability conditions in terms of extreme matrices. In 17 , using the notion of a block P-matrix a characterization of the Schur stability of all convex combinations of Schur stable matrices is derived. We consider the Schur stability problem for a familyA, which is a pathwise connected real matrix family. We show that Schur stability ofA is equivalent to the nonsingularity of an extended family. A similar problem for the Hurwitz stability is considered in 16 whereA is a polytope. As pointed out above this paper addresses the following points: 1 Robust Schur stability of polynomially dependent polynomials without involving Chebyshev polynomials see 2.3 . 2 A new algorithm multilinearization for a multilinear family Algorithm 3.1 . 3 Robust Schur stability criteria for a real matrix family via nonsingularity of an extended family. 4 Abstract and Applied Analysis In the computational procedures, we use Theorem 1.1 and the Bernstein expansion of a multivariable polynomial developed in 19–21 . The examples were run on a 2.53 GHz Intel Core2 Duo with 4GB of RAM. 2. Stability of a Polynomially Dependent Family Consider the family 1.1 , where ai q are polynomials. Here we give two polynomial equations defined on a box and show that the Schur stability is equivalent to the nonexistence of common solutions to these equations Theorem 2.1 . Suppose that the points z ±1 are not the roots of P and P has at least one stable member. Suppose that the family 1.3 is not Schur stable. Then, by continuity of roots cf. page 52 in 2 , there exists θ ∈ 0, π such that z e is a root of P, where j2 −1. Then z e−jθ is also a root, and there exist b0, b1, . . . , bn−2 such that a0 q a1 q z · · · an q z ( z − e )( z − e−jθ ) · ( b0 b1z · · · bn−2zn−2 ) ( z2 − 2 cos θz 1 ) · ( b0 b1z · · · bn−2zn−2 ) 2.1 is valid. Taking t 2 cos θ in 2.1 , it follows that the equalities b0 a0 q b1 − tb0 a1 q b2 − tb1 b0 a2 q b3 − tb2 b1 a3 q .. .. .. bk − tbk−1 bk−2 ak q .. .. .. bn−2 − tbn−3 bn−4 an−2 q bn−3 − tbn−2 an−1 q bn−2 an q 2.2