Lyapunov Stability Analysis of Higher Order 2D Systems

Description/Abstract

We prove a necessary and sufficient condition for the asymptotic stability of a 2D system described by a system of higher-order linear partial difference equations. We use the definition of asymptotic stability given by Valcher in “Characteristic Cones and Stability Properties of Two-Dimensional Autonomous Behaviors”, IEEE Trans. Circ. Syst. — Part I: Fundamental Theory and Applications, vol. 47, no. 3, pp. 290–302, 2000. This property is shown to be equivalent to the existence of a vector Lyapunov functional satisfying certain positivity conditions together with its divergence along the system trajectories. We use the behavioral framework and the calculus of quadratic difference forms based on four variable polynomial algebra.