An Inversion-Free Fuzzy Zeroing Neural Network Under Adaptive Input Range Fuzzy Scheme: Design, Analysis, and Application

Abstract

This article proposes a predefined-time robust inversion-free fuzzy zeroing neural network (PRIFZNN) model, which comprehensively considers computational complexity, convergence, and robustness. The PRIFZNN model achieves the lowest computational complexity among zeroing neural network (ZNN) models for solving similar problems. By avoiding the matrix inversion operation, it reduces the computational complexity to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn</i>), whereas other ZNN models typically exhibit a complexity level of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i>(min(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>,<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>)). To enhance the convergence and robustness of the PRIFZNN model, a piecewise predefined-time robust activation function and a novel adaptive input range fuzzy control scheme are proposed. The former provides predefined-time convergence and the ability to resist bounded noise, while the latter is employed to further improve the convergence rate. Moreover, the theoretical analysis section offers a detailed discussion of the convergence properties of the PRIFZNN model. It also derives the upper bound of the convergence time under bounded noise interference, thereby theoretically ensuring the robustness of the PRIFZNN model. Finally, numerical simulation studies on solving the time-varying Sylvester matrix equation show that the PRIFZNN model surpasses the comparative ZNN models in both convergence and robustness, while also validating the correctness of the theoretical analysis. Meanwhile, the PRIFZNN model is successfully applied to the path-tracking tasks of robotic manipulators, demonstrating its excellent practicality.