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

Abstract

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.