The optimal control of time-varying nonlinear systems is an important research topic and faces many difficulties and challenges, but also has substantial benefits. In this paper, the output optimization of scalar time-varying nonlinear (STVN) system and 2-dimension time-varying nonlinear (2DTVN) system is considered. The zeroing dynamics (ZD) method, which can solve the time-varying problems flexibly and effectively, has been extensively used in recent years. By introducing the ZD method, the time-varying optimal system state is attained, and the controller with the optimal output is designed. Meanwhile, the direct dynamics (DD) method is presented to further illustrate the superiority of the ZD method. Theoretical analyses prove that the ZD method has a faster convergence rate (that is, exponential convergence). In addition, the effectiveness and feasibility of the ZD method for solving the output optimization of STVN and 2DTVN systems are verified by numerical experiments, and further substantiated via output optimization of a two-wheeled mobile robot.

中文翻译：

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使用归零动力学对标量和二维时变非线性系统进行输出优化

时变非线性系统的最优控制是一个重要的研究课题，面临着许多困难和挑战，但也有很大的好处。本文考虑了标量时变非线性(STVN)系统和二维时变非线性(2DTVN)系统的输出优化。归零动力学（ZD）方法可以灵活有效地解决时变问题，近年来得到广泛应用。通过引入ZD方法，得到时变最优系统状态，设计出具有最优输出的控制器。同时，提出了直接动力学（DD）方法，以进一步说明ZD方法的优越性。理论分析证明ZD方法具有更快的收敛速度（即指数收敛）。此外，