zbMATH — the first resource for mathematics

On convex directions for stable polynomials. (English. Russian original) Zbl 0928.93044
Autom. Remote Control 58, No. 3, Pt. 1, 394-402 (1997); translation from Avtom. Telemekh. 1997, No. 3, 81-92 (1997).
Given two extreme polynomials $$p_0(s)$$ and $$p_1(s)$$ and the “direction” $$q(s)= p_1(s)- p_0(s)$$, the authors introduce a less conservative notion of local convex direction, namely: each polynomial which is a convex direction exhibits the following property: for any stable $$p_0(s)$$ $$(\deg(p_0)> \deg(q))$$ the set $$\{\mu\in \mathbb{R}$$; $$p_0+\mu q$$ stable polynomials} is an interval. The convexity of the intersection of $$p_0+ \mu q$$, $$\mu\in\mathbb{R}$$, with a family of stable polynomials is exploited. The notion of convex direction of a fixed polynomial is introduced and an appropriate graphical test is presented. Finally, Schur stable polynomials are considered.
Reviewer: M.Voicu (Iaşi)

MSC:
 93D09 Robust stability 93C80 Frequency-response methods in control theory 26C10 Real polynomials: location of zeros