On stability and the Lyapunov equation for n-dimensional digital systems

The discrete-time bounded-real lemma is further developed for nonminimal digital systems. Based on this lemma, rigorous necessary and sufficient conditions for the existence of positive definite solutions to the Lyapunov equation for n-dimensional (n-D) digital systems are proposed. These new conditions are improvements and extensions of earlier conditions and can be applied to n-D digital systems with characteristic polynomials involving 1-D factor polynomials. Further, the results in this paper show that the positive definite solutions to the n-D Lyapunov equation of a 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