Dissipativity-Based Model Reduction for Two-Dimensional Periodic Systems

Abstract

This paper presents a novel and efficient analysis based on linear matrix inequalities (LMIs) to derive optimized reduced models that preserve dissipativity for discrete-time periodic systems described by the two-dimensional (2D) Roesser model. To simplify stability analysis, we assume that the horizontal and vertical directions of the augmented system share the same period. By leveraging periodic Lyapunov functionals, we establish less conservative conditions that guarantee the existence of a 2D periodic reduced model that maintains the fundamental properties of the full-order system, ensuring asymptotic stability and $(O_1,O_2,O_3)$ dissipativity. Furthermore, we examine a specific case of dissipativity related to the $H_{\infty }$ norm, addressing a crucial aspect of system performance. The parameters of the reduced model are determined through convex optimization techniques, and numerical simulations validate the theoretical results, demonstrating the effectiveness of the proposed approach and highlighting the correlation between optimal dissipative performance indices and different Lyapunov functionals.