Stability and the lyapunov equation for n-dimensional digital systems

The discrete-time bounded-real lemma for nonminimal discrete systems is presented. Based on this lemma, rigorous necessary and sufficient conditions for the existence of positive definite solutions to the Lyapunov equation for ra-dimensional (n-D) digital systems are proposed. These new conditions can be applied to n-D digital systems with n-D characteristic polynomials involving factor polynomials of any dimension, 1-D to n-D. Further, the results in this paper show that the positive definite solutions to the n-D Lyapunov equation of an n-D system with characteristic polynomial involving 1-D factors can be obtained from the solutions of a k-D (0 < k < n) subsystem and m (1 < m < n) 1-D subsystems. This could significantly simplify the complexity of solving the n-D Lyapunov equation for such cases.