Robust Hurwitz stability via sign-definite decomposition

In this paper, we first consider the problem of determining the robust positivity of a real function f(x) as the real vector x varies over a box X∈R^l. We show that, it is sufficient to check a finite number of specially constructed points. This is accomplished by using some results on sign-definite decomposition. We then apply this result to determine the robust Hurwitz stability of a family of complex polynomials whose coefficients are polynomial functions of the parameters of interest. We develop an eight polynomial vertex stability test that is a sufficient condition of Hurwitz stability of the family. This test reduces to Kharitonov’s well known result for the special case where the parameters are just the polynomial coefficients. In this case, the result is tight. This test can be recursively and modularly used to construct an approximation of arbitrary accuracy to the actual stabilizing set. The result is illustrated by examples.