On the various forms and methods of solution of the Lyapunov equation for 2-D discrete systems

The various formulations of the Lyapunov equation for 2-D systems are reviewed. Recent results [6] on the necessary and sufficient condition for the existence of positive definite solutions to the 2-D Lyapunov equation with constant coefficients are considered and the fact that the existence of positive definite solutions to the 2-D Lyapunov equation is only sufficient for stability is discussed. The extension of these results to the n-D case, as well as algorithms for obtaining positive definite solutions to the 2-D Lyapunov equation, are presented. Furthermore, the formulation of the Lyapunov equation for 2-D systems with frequency as a parameter [3,4] is reviewed and briefly compared to the 2-D Lyapunov equation with constant coefficients.