Discrete-time noise-tolerant Z-type model for online solving nonlinear time-varying equations in the presence of noises

Abstract

Nonlinear time-varying equation problems (NTVEPs), a core mathematical problem in engineering applications and scientific computing fields, have been widely researched in recent years. In this paper, the zeroing-dynamic design formula and continuous-time Z-type model are revisited for solving NTVEPs. Then, a modified Z-type design formula is developed to address NTVEPs in the presence of noises. Specifically, a novel class of discrete-time noise-tolerant Z-type model with ψτ(χ(τ),τ) known (DTNTZTM-K) and discrete-time noise-tolerant Z-type model with ψτ(χ(τ),τ) unknown (DTNTZTM-U) models are first proposed and investigated for online solving NTVEPs with different measurement noises. Furthermore, general-type DTNTZTM-K and DTNTZTM-U models (termed as GDTNTZTM-K and GDTNTZTM-U models) with different activation function are proposed to verify the robustness and superiority. In addition, theoretical analyses demonstrate that the presented DTNTZTM-K and DTNTZTM-U models are 0-stable, consistent and convergent. Besides, it further indicates that different activation functions can be utilized to accelerate the convergent speed of a class of general discrete-time noise-tolerant Z-type models, which demonstrates their high efficiency and robustness. Ultimately, numerical results show the efficacy and superiority of the proposed DTNTZTM-K, DTNTZTM-U, GDTNTZTM-K and GDTNTZTM-U models for noise-polluted NTVEPs compared with classical methods.