Robust stability and eigenvectors of uncertain matrices

Abstract

This paper reveals the relationship between eigenvectors and robust Schur stability of uncertain matrices, including the relationship between eigenvectors and Schur stability of matrices. The results are derived and valid for robust pole clustering in a general circle region for uncertain system matrices of discrete-time systems or continuous-time systems. Robust Schur stability is dealt as a special case of robust pole clustering in a general circle region. The conditions on eigenvectors for robust pole clustering (or pole clustering) within a general circle are necessary and sufficient conditions. These conditions of eigenvector directions are viewed in some fixed basis which is constituted by the orthonormal eigenvectors of a symmetric matrix, called a criterion matrix. Three types of criterion matrices are adopted: the direct symmetric criterion matrix, the similarity transformed criterion matrix and the Lyapunov-type criterion matrix. The concerned uncertainties include both structured/unstructured uncertainties.