Robust stability and eigenvectors of uncertain matrices

This paper presents the relationship between eigenvectors and Hurwitz stability of uncertain matrices. First, it reveals new necessary and sufficient conditions for stability of a nominal matrix A through the relationship of its eigenvectors and its symmetric criterion matrices. Three types of criterion matrices are adopted, i.e., the direct symmetric matrix A_s=(A+A*)/2, the similarity transformed symmetric matrix and the Lyapunov-type symmetric matrix. Then, the necessary and sufficient conditions for robust stability of uncertain matrices are provided by using their eigenvector directions with respect to a basis constituted by their symmetric criterion matrix eigenvectors. The concerned uncertainties include both structured and unstructured uncertainties. The results may be applied to control systems for robust stability analysis and design