Convex combinations of stable polynomials

Convex combinations of G-stable polynomials

“…Under some mild additional assumptions concerning f(s), namely [Sondergeld 1983, (2.7)-(2.9)] we have Fu and Barmish 1987;Bialas and Garloff 1985;Bialas 1985].…”

“…The corresponding Hurwitz matrices can be directly determined. To apply the Bialas criterion [Fu and Barmish 1987;Bialas and Garloff 1985;Bialas 1985] we have to calculate the eigenvalues of the matrix ttf~I-If2. Some basic calculations show a(H~IHr2) : {1, 0.8, 0.5} (41) and the convex combination of the two polynomials under consideration is G-stable.…”

“…However, effective criteria exist to test G-stability of the convex combinations of polynomials pi(s), pj (s) in (1) only when G is either the left halfplane Fu and Barmish 1987;Bialas and Garloff 1985; or the interior of the unit circle centered at zero Ackerman and Barmish 1988]. The existing criteria can be extended to the case of complex polynomials [Gantmacher 1977, Chapter XV, §18] and also are available in ].…”

Hadamard Factorization of Stable Polynomials

“…Other questions about interval polynomials can be consulted in [9] and [10]. Since the set of Hurwitz polynomials is not convex, the stability of segments of polynomials has also been investigated (see for instance [2], [3], [8] [18] and [24]). Excellent references about Hurwitz polynomials reported during 1987-1991 can be seen in [11].…”

To stabilize an interval plant family it suffices to simultaneously stabilize 64 polynomials

“…Note that if n(s) is other than a constant the family in (1) is not interval. C l e a r l y s t a b i l i t y of f a m i l y ( 2 ) implies s t a b i l i t y of family ( 1 ) . W e can show t h e following r e s u l t :…”