Convergence analysis of a state feedback-based iterative learning control scheme for a class of nonlinear systems without the global Lipschitz condition

The global Lipschitz condition, commonly adopted in the theoretical study of iterative learning control (ILC) for nonlinear systems, provides analytical convenience but limits applicability to practical systems. To address this limitation, this paper investigates the convergence of a nonlinear state feedback-based ILC scheme for a class of discrete-time nonlinear multi-input–multi-output (MIMO) systems without relying on the global Lipschitz condition. Firstly, under the assumption that the system nonlinearities are continuous, the convergence of the state, input, and output sequences is rigorously proved by combining mathematical induction with input space transformation and matrix blocking techniques, showing that convergence occurs if and only if the spectral radius condition is satisfied. Secondly, when the system nonlinearities are locally Lipschitz, convergence is analysed using the contraction mapping method together with matrix transformation and the λ-norm techniques, demonstrating that the state, input, and output sequences converge exponentially if and only if the spectral radius condition is satisfied. In addition, a sufficient condition is provided to guarantee monotonic convergence of the output tracking error sequence for locally Lipschitz nonlinearities. Finally, numerical examples are provided to validate the theoretical analysis.