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In this paper, convergent property of a saturation-strategy-based iterative learning control (ILC) law is first investigated for a class of two-dimensional linear discrete first Fornasini–Marchesini model (2-D LDFFM) with input saturation. A three-dimensional dynamical process is transformed into a 2-D dynamical process by row scanning approach or column scanning approach. As a result, it is theoretical proved no matter which method is adopted, perfect tracking on the desired reference surface is accomplished by virtue of the 2-D linear inequality theory. Numerical simulation on a practical thermal process is used to illustrate the effectiveness and feasibility of the designed ILC law. In addition, ILC convergence analysis for 2-D LDFFM with input delay and input saturation is discussed. [ABSTRACT FROM AUTHOR]