This paper considers the robust stability verification of linear time-
invariant systems admitting a class of nonlinear parametric perturbati
ons, The general setting is one of determining the closed-loop stabili
ty of systems whose open-loop transfer functions consist of powers, pr
oducts, and ratios of polytopes of polynomials, Apart from this genera
l setting, two special cases of independent interest are also consider
ed. The first special case concerns uncertainties in the open-loop gai
n and real poles and zeros, while the second special case treats uncer
tainties in the open-loop gain and complex poles and zeros, In light o
f the zero exclusion principle, robust stability is equivalent to zero
exclusion of the value sets of the system characteristic function (a
value set consists of the values of the characteristic functions at a
fixed frequency), The main results of this paper are as follows, 1) Th
e value set of the characteristic function at each fixed frequency is
determined by the edges and some frequency-dependent internal line seg
ments, 2) Consequently, Hurwitz invariance verification simplifies to
that of checking certain continuous scalar functions for avoidance of
the negative real axis, 3) For the case of real zero-pole-gain variati
ons, the critical lines are all frequency independent, and therefore,
the determination of the robust stability is even simpler, 4) For the
case of complex zero-pole gain variations, the critical internal lines
are shown to be either frequency independent or to be confined in cer
tain (two-dimensional) planes or (three-dimensional) boxes,