Output optimization of scalar and 2‐dimension time‐varying nonlinear systems using zeroing dynamics

Abstract

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.

From Penrose Equations to Zhang Neural Network, Getz–Marsden Dynamic System, and DDD (Direct Derivative Dynamics) Using Substitution Technique

“…ird, the higher-order discrete algorithm (i.e., the ten-instant discrete algorithm) is more easily affected by the rounding error disturbance [44]. us, the experimental results of the DDD model in this study showing the divergence are actually complementing the previous research studies, in addition to the confirmation of [47] about divergence. erefore, we summarize that the DDD model is generally less effective, with further in-depth investigation being also a future research direction.…”

“…According to the assumption of the matrix inverse problem, we can substitute A − 1 (t) with X(t) in (47) and thus obtain…”

“…Note that the convergence of DDD models is shown in [12,47], where the DDD models are utilized to solve the time-varying nonlinear optimization problems.…”