An `extreme point´ sufficient condition for checking interior instability of interval matrices in the march towards a necessary and sufficient result

This paper addresses a current research topic of high interest, namely the issue of developing a finitely computable necessary and sufficient test for checking the robust stability of an interval matrix. In this paper, errors in two previous papers by the author on this difficult problem (Proceedings of IEEE CDC, p.3284, 1997; Proceedings of IEEE ACC, p.2309, 1998) are identified and those errors are corrected leading to interesting new paths to possible solution to the problem. The logic revolves around the fact that the robust stability problem can be converted to a robust nonsingularity problem involving the original matrix and the associated bialternate sum matrix (which we label as the `tilde´ matrix). The special nature of the `tilde´ matrix is exploited with the introduction of concepts labeled `real axis nonsingularity´, `virtual matrix family´ and `weighted real axis determinant´ and `real axis nonsingularity scalar´. As a prelude to a necessary and sufficient condition, in this paper an `extreme point´ sufficient condition for checking instability in the interior of interval matrices is provided. The proposed methodology is illustrated with a variety of examples. The possibility of obtaining a necessary and sufficient extreme point result using the proposed concepts is explored