A Finite Zero Exclusion Principle
Anders Rantzer · Birkhäuser Boston eBooks · 1990
The paper shows that the “frequency sweep” in the well known “zero exclusion principle” for checking robust stability of linear systems, can be avoided. In fact, we derive a simple sufficient condition for zero exclusion in an entire frequency interval. The main idea is that a polynomial p of degree n is Hurwitz if and only if $$\sum olimits_{k = 2}^N {\arg (p(i{\omega _k})/p(i{\omega _{k - 1}}))} > (n - 1)\pi$$ for some ω l < . . . < ω N . As an application, we consider polynomials of degree n with coefficients depending linearly on m parameters in the interval [0,1]. The number of calculations for checking Hurwitz stability of the complete family then grows only as n 2 m log m, but depends also on the “stability margin” of the family. The test is also applied to problems with multilinear parameter dependence, in particular checking positive realness of a rational function, whose numerator and denominator depend linearly on interval bounded parameters.