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.
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