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This paper investigates the asymptotic stability of discrete dynamical systems in a class of two-dimensional (2-D) systems whose dynamical parts are described in the Fornasini-Marchesini model along with a standard saturation operator on the state space. Under the assumption that the stability of the nominal system is ensured by the solvability of a linear matrix inequality, two techniques are introduced for checking the tolerance of the system against saturation effects. One is a quadratic method that accounts for the stability margin of a system. Another is a non-quadratic method that uses the asymptotic property of a majorant nonnegative system. These techniques are useful to improve upon previously known results. Two theorems are introduced in this paper in different manners. The first theorem provides a plain interpretation on the stability condition; however, it requires a two-step process to search for a solution. The second theorem is expressed as a linear matrix inequality, which is the dual statement of the first theorem. These results can be naturally modified for 1-D systems, 2-D systems in the Roesser's model, and multidimensional systems. Illustrative examples show that the two techniques adopted in this paper have different effectiveness in enlarging the scope of application.