ABSTRACT
Stability analysis of 2-D systems and filters is a technically challenging problem as the zeros of the denominator polynomial of a general 2-D transfer function define algebraic curves as opposed to isolated points in the 1-D case. In the first part of the paper, Gutman's déterminent condition and a technique of linearizing matrix polynomials are used to deduce a new stability criterion that reduces the 2-D stability verification problem to a generalized eigenvalue problem. The matrices involved in the generalized eigenvalue problem are of large size even for systems of moderate order, but they are very sparse. In the second part of the paper, two algorithms for testing 2-D stability are presented. The first algorithm is based on a singular-value perturbation analysis of a parameterized matrix, and the second is based on a 1-D robust stability analysis through a Lyapunov approach. It is shown that both algorithms can be considerably enhanced by a pre-processing procedure that reduces the norm of the system matrix. A direct method based on unconstrained optimization and an indirect method based on state-variable transformation are proposed for the norm reduction. A case study is given to illustrate the stability anslysis algorithms proposed and the critical role that the pre-processing for norm-reduction may play in the algorithms' implementation.
