Abstract
One of cogent ways to express uncertain systems is to describe them by polytopes of polynomials. Uncertainities are represented in this case as a convex hull of the upper or lower bound for the range of coefficients of characteristic polynomials of the systems. It is known that stability of polytopes of polynomials can be determined by that of edge polynomials, i.e., polynomials that span the convex hull. This gives rise to the stability test of convex combinations of two polynomials. Several methods are currently available for that purpose and they have their own advantages and disadvantages.
In this paper, new alternative stability criteria are provided for convex combinations of two polynomials, covering some disadvantages of the existing methods. The criteria are algebraic and required computation terminates with finite steps resorting neither to matrix inversions nor to solving polynomial equations as demanded in the existing methods. The second feature is that the case where degree difference exists between the two vertex polynomials can be naturally incorporated without demanding specific considerations. The criteria can also yield analytic conditions for low-degree polynomials. It is furthermore pointed out that allowable relative degree for stability is only two.
