A Comprehensive Framework for Generating Adaptive Arbitrarily Predefined-Time Convergent RNNs for Dynamic Zero-Finding Problem Applied to Circuits and Robotics

Recurrent neural networks (RNNs) with predefined-time convergence capabilities are among the most powerful solvers for time-varying zero-finding problems (TVZFPs). However, a comprehensive design framework for such neural networks has not yet been well established. To address this gap, this article presents a comprehensive framework for generating a series of adaptive arbitrarily predefined-time convergent RNNs (A-APTC-RNNs). Compared with most existing RNNs, the A-APTC-RNNs generated using the proposed comprehensive framework exhibit the following distinctive features: 1) owing to a novel piecewise evolution formula, their convergence time can be arbitrarily predefined; 2) owing to a proportional-integral-derivative regulatory mechanism, they achieve lower steady-state residual errors after convergence; and 3) owing to a novel adaptive parameter initialization scheme, they are able to automatically determine their own model parameters. Theoretical analysis rigorously demonstrates the stability and arbitrarily predefined-time convergence (APTC) capability of the A-APTC-RNNs. Various experiments (i.e., numerical simulations, alternating-current estimation, chaotic synchronization of Chua's circuit, and motion generation for dual-arm robots) demonstrate the state-of-the-art convergence performance of the A-APTC-RNNs generated by the proposed comprehensive framework.