Parallel Control With Adaptive Critic-Actor Learning Implementation for State and Input Time-Delayed Nonlinear Continuous-Time Systems

This study seeks to develop a constructive approach that settles the optimal control issue for nonlinear systems with known time delays. The feedback system, which depends on the state and control input, is built to identify the actual control rule utilizing the backstepping integral technique. Optimal control of the augmented system established on parallel control delivers a solution for nonlinear time-delayed systems. At the cost of the modified $L_{2}$ gain condition, the value function is specified in terms of the state and input delays, transforming the optimal control issue into a minimax task. Then, the critic-actor framework is employed to reconstruct the cost function and control rule while maintaining the persistently exciting (PE) condition so that the online optimal control algorithm is investigated. In addition, the Lyapunov proof discusses the system’s stability. Ultimately, the remarkable properties become visible through experimental findings.