Stability Boundary Mapping for Discrete Time Linear Systems

In this study, a new method is proposed in order to analytically determine the stability boundaries of discrete time linear systems. The stability conditions of discrete time systems are reformulated in this method to derive an equation that includes the products of all eigenvalues, that can be checked more easily than conditions on every single eigenvalue. Thus, the time for calculating the space of all stable parameters of a given system can be greatly reduced. By further analysis, redundant products in the proposed method are eliminated in order to reduce the computational complexity more. The new procedure avoids bilinear transformations, decoupling at singular frequencies and discretization of the parameter space and doesn't include any conservatism. Further, interpretation of the proposed method in context of the Lyapunov stability is given. Different case studies are included in the paper to prove correctness of the results and to illustrate the advantages of the proposed approach.