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Zhang, X.M., Ma, Z.M., Fang, M.F., et al. (2025) Fault Diagnosis of Motorized Spindle Based on Lumped Parameter Model and Wasserstein Generative Adversarial Network. Mechanical Systems and Signal Processing, 230, Article 112668.
https://doi.org/10.1016/j.ymssp.2025.112668

has been cited by the following article:

• TITLE: Single Internal Fault Diagnosis for Unstable 1D Wave Equation under Boundary Disturbance

  AUTHORS: Zhenxiang Cao

  KEYWORDS: Wave Equation, Fault Detection, Adaptive Threshold, Lyapounov Function

  JOURNAL NAME: Engineering, Vol.18 No.7, July 7, 2026

  ABSTRACT: This paper proposes a fault detection and estimation method for an unstable one-dimensional wave equation based on an adaptive extended state observer (ESO), which relies solely on boundary displacement and velocity measurements. First, an infinite-dimensional naive observer is designed without using modal approximations, paired with a corresponding fault detection filter (FDF) and a residual evaluation mechanism to achieve reliable fault identification. Subsequently, once a fault occurs, an adaptive update law is embedded into the ESO to enable online estimation of the fault amplitude, requiring prior knowledge of the fault shape function. Finally, numerical simulation results verify that the proposed method can rapidly detect the occurrence of faults and accurately track their dynamic evolution.