A Self-Learning Noise-Resistant Zeroing Neural Network for Dynamic Equations and Its Applications

Abstract

Dynamic equations provide mathematical frameworks to capture the evolving behavior of systems, which is essential in various fields. While the zeroing neural network (ZNN) is one of the most effective real-time solvers for dynamic equations, it is highly susceptible to noise interference, which reduces the precision and reliability of solutions. Current research struggles to address more complex noise disturbances, particularly complex-valued and random noise. To overcome this limitation, this article introduces a set of self-learning operators with real-time correction ability to counteract noise interference and obtain a new self-learning noise-resistant ZNN (SLNR-ZNN). The operators within the SLNR-ZNN model adaptively learn the physical forms of noise, utilizing the noise’s derivative properties, through continuous system oscillations to enhance noise tolerance and improve the accuracy of dynamic equation resolution. Theoretical analysis and experimental validation show that SLNR-ZNN effectively resolves linear and nonlinear dynamic equations under various types of noise, including constant, harmonic, complex spectral, and Gaussian white noise. Compared to existing ZNN models, SLNR-ZNN achieves comparable convergence rates and simultaneously maintains significantly lower steady-state errors, which are often reduced by nearly an order of magnitude under noise. Furthermore, simulation experiments demonstrate that the SLNR-ZNN-based control protocol achieves state consensus in leader-following multiagent systems and enables trajectory tracking in the UR5 robotic arm with millimeter-level accuracy, even under composite disturbances. These results highlight its practical value and robustness in robotic control applications.