On the Characterization and Formation of Local Convex Directions for Hurwitz Stability
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.
KeywordsConvex Combination Positive Realness Real Polynomial Stable Polynomial Single Input Single Output
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