This study is the first time devoted to seek an online optimal tracking solution for unknown nonlinear singularly perturbed systems based on single network adaptive critic (SNAC) design. Firstly, a novel identifier with more efficient parametric multi-time scales differential neural network (PMTSDNN) is developed to obtain the unknown system dynamics. Then, based on the identification results, the online optimal tracking controller consists of an adaptive steady control term and an optimal feedback control term is developed by using SNAC to solve the Hamilton–Jacobi–Bellman (HJB) equation online. New learning law considering filtered parameter identification error is developed for the PMTSDNN identifier and the SNAC, which can realize online synchronous learning and fast convergence. The Lyapunov approach is synthesized to ensure the convergence characteristics of the overall closed loop system consisting of the PMTSDNN identifier, the SNAC and the optimal tracking control policy. Three examples are provided to illustrate the effectiveness of the investigated method.

本研究首次致力于基于单网络自适应批评（SNAC）设计寻求未知非线性奇异摄动系统的在线最优跟踪解。首先，开发了一种具有更有效的参数多时间尺度差分神经网络（PMTSDNN）的新型识别器来获取未知系统动力学。然后，基于辨识结果，通过使用SNAC在线求解Hamilton-Jacobi-Bellman（HJB）方程，开发了由自适应稳定控制项和最优反馈控制项组成的在线最优跟踪控制器。针对PMTSDNN识别器和SNAC，提出了考虑过滤参数识别误差的新学习律，可以实现在线同步学习和快速收敛。综合Lyapunov方法来保证由PMTSDNN标识符、SNAC和最优跟踪控制策略组成的整个闭环系统的收敛特性。提供了三个例子来说明所研究方法的有效性。