A Generalized Lyapunov Stability Theorem for Discrete-time Systems based on Quadratic Difference Forms

In this paper, we consider the generalized Lyapunov stability analysis for a discrete-time system described by a high order difference-algebraic equation. In the behavioral approach, a Lyapunov function is characterized in terms of a quadratic difference form. As a main result, we derive a generalized Lyapunov stability theorem that the asymptotic stability of a behavior is equivalent to the solvability of the two-variable polynomial Lyapunov equation (TVPLE) whose solution induces the Lyapunov function. Moreover, we derive another asymptotic stability condition by using a polynomial matrix solution of the one-variable dipolynomial Lyapunov equation which is reduced from the TVPLE.