On linearized stability of differential repetitive processes and iterative learning control

Repetitive processes are two dimensional (2D) systems that arise in the modeling of engineering applications such as additive manufacturing, in which information propagation occurs along two axes of independent variables. While the existing literature on repetitive processes is predominantly on linear systems, recent work highlights the need to develop rigorous tests for stability of nonlinear processes. Using existing results from linear repetitive process theory, we establish a differential repetitive process analogue of the well known result that the stability of a nonlinear feedback system can be verified by the stability of the linearized dynamics. In particular, we employ a 2D Lyapunov equation to show that the feasibility of a linear matrix inequality, combined with 2 small gain conditions, can guarantee stability locally around an equilibrium. Finally, we apply this result to the design and stability analysis of iterative learning control (ILC) systems, and discuss implications in the context of nonlinear ILC.