Online learning and control of nonlinear delayed systems with efficient signal history encoding

Modeling and control of nonlinear systems with time delays presents two fundamental challenges: representing the extended history of past signals needed for delay compensation and capturing complex nonlinear dynamics that can vary over time. This paper presents a novel method that addresses both challenges by combining efficient signal history compression using Legendre Delay Networks (LDN) with adaptive nonlinear learning through Kernel Recursive Least Squares Tracker (KRLST). By compressing temporal information prior to nonlinear modeling, the approach achieves accurate predictions while maintaining computational feasibility for online adaptation. The method supports multiple configurations, including state-dependent and state-independent variants, and integrates naturally with Smith predictor control architectures. Comprehensive simulation experiments evaluate both open-loop modeling accuracy and closed-loop control performance under various operating conditions. The results demonstrate that state-dependent variants achieve superior accuracy under nominal conditions, while state-independent variants provide more consistent performance under uncertainties and disturbances. Several tests with increasing delay times, parameter variations, and disturbance measurement show robust performance of the method even with significant uncertainties in both delay estimates and system parameters. This combination of efficient temporal compression, online nonlinear learning, and flexible configuration options provides a practical approach to controlling nonlinear delayed systems across diverse operating scenarios.