Eigenvectors and robust pole clustering in general subregions of the complex plane for uncertain matrices

This paper reveals the relationship between eigen, vector and robust pole clustering in general subregions of the complex plane for uncertain matrices. It presents a set of new necessary and sufficient conditions for robust pole clustering in general regions for uncertain matrices via their eigenvector directions. The eigenvector directions are viewed and measured in a fixed basis in the vector space. The basis is constituted by the orthonormal eigenvectors of a Symmetric matrix, called a criterion matrix f(A,A*) = Sigma(p,q)c(p,q)A*(q)A(P), related to the nominal It will not be changed when uncertain matrix parameters change because it is related only to the nominal matrix. The considered robust pole clustering regions include Omega-transformable regions and non-R-transformable regions. The concerned uncertainties include both structured and unstructured uncertainties. The results may be applied to control systems for robust pole clustering analysis and design. The interesting thing is that the robust stability and robust performance, i.e., robust pole clustering, are represented and achieved via their eigenvector directions.