Generalized Lyapunov Functions for Discrete-Time Lurie Systems With Slope-Restricted Nonlinearities

A class of Lyapunov functions for discrete-time Lurie systems with monotonic nonlinearities is proposed. The Lyapunov functions are composed of quadratic terms on the states and of the system's nonlinearities as well as Lurie–Postnikov-type integral terms. Crucially, positive definiteness of the matrix in the generalized quadratic form and positivity of the scaling terms of the Lurie–Postnikov integrals are relaxed in the stability conditions. Furthermore, they are used for regional stability analysis and performance assessment. Numerical examples show that the proposed Lyapunov function structure matches or outperforms existing ones for these systems.