On Polynomial Zero Exclusion from an RHP Sector

Simple conditions based on generalisations of the Routh-Hurwitz and Mikhailov criteria that ensure the absence of polynomial roots in an RHP sector straddling the positive real semi-axis ($\mathcal{S}$-stability) are presented. In particular, it is shown that $\mathcal{S}$-stability is ensured if the phase variation of a suitable power of the original $n$ th-degree characteristic polynomial is equal to $n\pi/2$, which implies that the zeros of the real and imaginary parts of this power must satisfy an interlacing property similar to the interlacing property satisfied by Hurwitz polynomials according to the classic Hermite-Biehler theorem. The condition can be checked by means of Sturm sequences. Examples show how the proposed methods operate.