From Finite Through Fixed to Arbitrary-Time Convergent Zeroing Neurodynamics for Time-Varying Optimization With Nonlinear Equation Constraint

Solving the time-varying optimization with nonlinear equation constraint (TVONEC) remains a significant challenge due to the complexities introduced by temporal variability and nonlinearity, which is less explored in existing research. To address this challenge, a novel nonlinear activation function (AF), the sign-exponential AF (SEAF), is first designed in this article. Building on the SEAF, we propose two advanced zeroing neurodynamics (ZN) models: the finite and fixed-time convergent ZN (FFTCZN) and the arbitrary-time convergent ZN (ATCZN). Unlike traditional models, the FFTCZN model reaches a convergence state within a fixed and finite time frame, while the ATCZN model achieves arbitrary-time convergence, enabling it to reach the convergence state within an arbitrary time frame. Their respective convergence properties, including superior convergence speed and robustness, are verified through rigorous mathematical analyses and experimental validations. Furthermore, two novel controllers are developed to achieve the drift-free path tracking for the manipulator, exhibiting excellent tracking precision and strong resilience to persistent noise.