Stability Theory pp 173-180 | Cite as

# On the Characterization and Formation of Local Convex Directions for Hurwitz Stability

## Abstract

Let *P* _{ o }(
*s*) be a real given Hurwitz polynomial. A local convex direction with respect to *P* _{ o }(
*s*) is a polynomial *P* _{
1}(
*s*), such that all polynomials which belong to the convex combination of *P* _{ o }(
*s*) and *P* _{1}(
*s*) are Hurwitz. The main result of this paper is the characterization of local convex directions, which enables the derivation of a continuum of pertinent real polynomials *P* _{1}(
*s*), by a very simple algorithm. In arriving at this result, first a sufficient condition and then a necessary and sufficient condition are derived for a polynomial *P* _{1}(
*s*) to be a local convex direction of a given Hurwitz polynomial *P* _{ o }(
*s*). These conditions are related to the property of strict positive realness of a rational function, and can be tested by tractable methods.

## Keywords

Convex Combination Positive Realness Real Polynomial Stable Polynomial Single Input Single Output## Preview

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