Applications of robust stability results to the analysis of multidimensional discrete-time systems

Abstract

Investigates the problem of asymptotic stability for a class of linear time-invariant multidimensional discrete systems. We establish the equivalence of the present problem and the problem of robust stability for a class of linear time-varying one-dimensional discrete systems with the matrix uncertainty set defined by the coefficient matrices of the original multidimensional system. On the basis of this equivalence, by using the variational method and the Lyapunov second method, necessary and sufficient conditions for asymptotic stability are obtained in different forms for the class or multidimensional systems considered in the present paper. The parametric classes of Lyapunov functions which define the necessary and sufficient conditions of asymptotic stability are determined. We use the piecewise linear polyhedral Lyapunov functions of the infinity vector norm type to derive an algebraic criterion for asymptotic stability of the given class of multidimensional systems in the form of solvability conditions of a set of matrix equations.