On Stability of Parametrized Families of Polynomials and Matrices

Abstract

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.

Poli̇nom Ai̇leleri̇ İçi̇n Kesme Frekansi Hesabi

“…We show that the nonexistence of such a root is equivalent to the nonexistence of a common solution of two polynomial equations defined on a box. Similar system of two equations for discrete polynomial families has been obtained in [5].…”

“…This polynomial family is multilinear. Calculation by (5) gives = 269.1. Now apply Algortihm 1 to reduce .…”

“…Calculation of the cutoff frequency by (5) gives large value as usually. Therefore, the minimization of becomes an important problem the consideration of which is the subject of the next section.…”

“…The system (8) is almost multilinear and the variables ( , ) vary on the box [ 1 2 , 2 2 ] × . This system can be multilinearized by introducing new variables (see [5]). Indeed, if system (8) contains as the greatest power of , we can replace by the product 1 2 … where 1 , 2 , …, , are new variables and add new equations 2 − 1 = 0, 3 − 1 = 0, …, − 1 = 0 to the system (8) (we set 1 = ).…”

Bernstein Polynomials

“…, q m ) in the tensor-product Bernstein basis over the m-dimensional volume [ q 1 , q 1 ] × • • • × [ q m , q m ] and exploiting the convex hull, variation-diminishing, and subdivision properties. For more comprehensive details on this approach, the reader may consult the papers [6,63,92,93,137,184,223] and references therein.…”