Spectral properties of uncertain matrices

In this thesis, asymptotic stability of continuous and discrete uncertain linear systems are discussed. Basic definitions and theorems for both systems are summarized. The location of eigenvalues of a matrix in the complex plane characterizes the asymptotic stability of the system given by that matrix. The existence of positive definite solution of Lyapunov (Stein) matrix inequality corresponding to the system is equivalent to the asymptotic stability of that system. The concept of common solution of Lyapunov (Stein) matrix inequality and related results for a finite number of matrices are discussed. A condition is given the common solution of the Stein inequality for two Schur stable matrices whose difference is rank one. One way to determine the common solution of Lyapunov's inequality for a two-dimensional finite number of square matrices is to work with an algorithm based on the centroid on a two-dimensional unit disk. In this thesis, it has been shown that it is sufficient to work on a square containing this disk instead of the unit disk used in the current algorithm. In addition, an algorithm has been developed for the common solution of a finite number of two -dimensional square complex matrices. Examples of the results obtained are given.