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This brief proposes novel linear matrix inequalities‐based criteria to investigate the asymptotic convergence of states to an overflow oscillations‐free ellipsoidal region for a two‐dimensional Roesser digital model subjected to quantization and overflow effects of a digital hardware. Existence of a novel region for two‐dimensional Rosser systems is shown in the present study, in which overflow oscillations can be completely removed for realization of a system. New realization conditions for Roesser systems in the absence and presence of external interference along with stability and overflow‐free regional behavior are investigated by application of regional analysis, convex routines, and Lyapunov redesign. In contrast to existing literature that primarily focus on a specific type of quantitation and global asymptotic stability, the conditions in the presented work are derived by considering the generalized quantization arithmetic and for guaranteeing convergence of states in a convex oscillations‐free region (in the steady‐state) in both the absence and presence of bounded interference. The localized stable regions can be directly associated with the word‐length employed to realize the two‐dimensional systems on a digital hardware. Simulation results are also furnished to validate the efficacy of the proposed approach. [ABSTRACT FROM AUTHOR]