## 1 Introduction
Nevertheless, the diagnostic method based on current signals requires longer detection time, whereas detecting voltage signals involves adding extra sensors, introducing additional potential fault sources and increasing system costs. The analytical model-based methods construct observers from the system's mathematical model, generating residual signals for diagnosis by comparing observed values with actual outputs. The observers presently used for diagnosis include the Kalman filter [24], Luenberger observer [25], disturbance observer [26], and sliding mode observer (SMO) [27]. Reference [20] proposed a unified diagnosis method for both current sensor faults and switch OC faults based on a Kalman filter. However, it relies on a fixed threshold to determine the faults, which reduces the robustness under load fluctuation or complex working conditions. Reference [21] used the Luenberger observer in conjunction with adaptive thresholding to diagnose OC faults in back-to-back converters. However, the designed observer ignores the effect of actual unknown perturbations, limiting applicability to ideal operating scenarios. In [22], a novel differential current observer was proposed for the variable-speed permanent magnet synchronous motor (PMSM) system. The method can obtain diagnostic results within 9.7% of electrical cycles, but it can only diagnose single-switch faults in three-phase voltage source inverters. SMO has the advantage of being highly adaptable to both nonlinear and incomplete measurement systems. This capability ensures stable and accurate state estimation even in the presence of measurement noise and uncertainties [28]. Moreover, the SMO is extensively utilized in diagnosing OC faults due to its rapid response, straightforward structure, and robustness. In [29], an SMO-based diagnosis method for multi-phase floating interleavedboost converters was proposed. The observer's switching function uses a sign function, but the intrinsic chattering phenomena affect the accuracy of the diagnostic results. To enhance the observation accuracy of SMO, an interval sliding mode observer of three-phase voltage source inverters was constructed in [30]. Although this method demonstrates a rapid fault detection response, the fixed threshold strategy inherently limits robustness under parameter-dependent operating conditions. For multilevel converter, Reference [31] proposed a composite fault diagnosis method based on an interval SMO. This approach targets simultaneous switch and current sensor faults in grid-connected NPC inverters. Reference [32] detects faults via a reduced-order SMO, generating residuals by comparing observed and measured single-phase three-level rectifier currents. Fault location is performed by analyzing the system voltage residuals. Although the aforementioned methods retain efficacy under intricate operational environments, prior studies have primarily concentrated on single-switch fault diagnosis. For the more complex scenario of simultaneous double-switch faults, a comprehensive diagnostic framework has yet to be developed, with deficiencies particularly evident in fault location. Due to the scarcity of research on double-switch faults in multilevel inverters, this paper proposes a multiple OC fault diagnosis method based on an improved adaptive sliding mode observer. The main contributions can be summarized as follows:  
(1) A novel diagnosis method is proposed for single-switch and double-switch faults of T\({}^{2}\)3L inverters. This method accurately localizes same-phase double-switch faults without requiring additional diagnostic variables. Furthermore, the adaptive-threshold-based diagnostic strategy dynamically adjusts thresholds according to operating conditions, enhancing the robustness of fault detection. (2) An improved adaptive sliding mode observer (IASMO) is designed using a hyperbolic tangent function to replace the traditional discontinuous switching function, effectively mitigating the chattering phenomenon.