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This paper is concerned with iterative learning control (ILC) algorithms for two-dimensional (2-D) linear discrete systems described by the first Fornasini–Marchesini model (FMMI) with iteration-varying reference trajectories/profiles. The variation of reference trajectories in iteration domain is represented by a high-order internal model (HOIM) formula. Robustness and convergence of two types of HOIM-based ILC laws with different boundary conditions are investigated, respectively. A strategy employed in this paper is to reconstruct the HOIM-based ILC process of the 2-D linear FMMI system into a set of 2-D linear inequalities or a 2-D linear Roesser model such that sufficient robustness/convergence conditions of the HOIM-based ILC laws are obtained. Under random boundary conditions, the designed ILC law (9) is capable to drive the ILC tracking error into a bounded range. Moreover, under the HOIM-based boundary conditions, a perfect tracking to the iteration-varying reference trajectories can be achieved by utilizing the proposed ILC law (32). Two simulation examples are given to validate the effectiveness of the two proposed ILC algorithms. [ABSTRACT FROM AUTHOR]