A necessary and sufficient `extreme point´ solution for checking robust stability of interval matrices

Addresses the issue of developing a finitely computable necessary and sufficient test for checking the robust stability of an interval matrix and provides a complete solution to the problem in the form of an `extreme point´ result. The result uses 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 concept labeled `real axis nonsingularity´. Another important concept introduced is that of `virtual matrix family´ which indirectly captures the `interior´ of the uncertain matrix family. Using measures labeled `weighted real axis determinant´ and `real axis nonsingularity scalar´ which are positive for an asymptotically stable matrix, the proposed necessary and sufficient condition involves checking if a set of `real axis nonsingularity matrices´ (formed in terms of the `vertex´ matrices in the `tilde´ space) possess any positive real eigenvalues or not. This condition thus involves the eigenvalue information of the higher dimensional matrices in the `tilde´ space. The proposed methodology is illustrated with a variety of examples. The importance of this result and the possible extensions are discussed