Neural network-based adaptive reinforcement learning for optimized backstepping tracking control of nonlinear systems with input delay - Zhu, Boyan; Karimi, Hamid Reza; Zhang, Liang; Zhao, Xudong (Applied intelligence (Dordrecht, Netherlands), 2025)
In this paper, the problem of adaptive optimized tracking control design is addressed for a class of nonlinear systems in strict-feedback form. The system under consideration contains input delay and has unmeasurable and restricted states within predefined compact sets. First, neural networks (NNs) are employed to approximate the unknown nonlinear dynamics, and an adaptive neural network (NN) state observer is constructed to compensate for the absence of state information. Additionally, by utilizing an auxiliary system compensation method alongside the backstepping technique, the impact of input delay is eliminated, and the generation of intermediate variables is prevented. Second, tan-type barrier optimal cost functions are established for each subsystem within the backstepping method to prevent the state variables from exceeding preselected sets. Moreover, by establishing both actor and critic NNs to execute a reinforcement learning algorithm, the optimal controller and optimal performance index function are evaluated, while relaxing the persistence of excitation condition. According to the Lyapunov stability theorem, it is demonstrated that all signals in the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB), and the output signal accurately tracks a reference trajectory with the desired precision. Finally, a practical simulation example is provided to verify the effectiveness of the proposed control strategy, demonstrating its potential for real-world implementation.