## 1 Introduction  
According to Boeing, Airbus, EASA (European Union Aviation Safety Agency), and ICAO (International Civil Aviation Organization) [1; 2; 3; 4], more than 50% of fatal accidents occur during final approach and landing. Due to its irreplaceable nature, landing gear performance directly affects aircraft takeoffs and landings. A significant part of the landing gear retraction/extension process is the retraction/extension (R/E) hydraulic system. Hydraulic transmissions, in comparison with gear mechanical transmissions, are able to achieve a wide range of stepless speed regulation, which is essential to the smooth retraction/extension of landing gears within the specified timeframe [5; 6]. Fault detection, on the other hand, is the first step in implementing fault diagnosis technology. Unless the detection is accurate and effective, it will be difficult to isolate faults in the future, and it will also affect subsequent health assessment and maintenance. It is therefore essential to detect faults accurately. Research methods for fault diagnosis of complex nonlinear systems such as hydraulic systems fall into three categories: knowledge-based, data-driven, and model-based [7]. Inknowledge-based methods, knowledge reasoning and long-term accumulated experience knowledge are mainly used [8; 9; 10]. In data-driven analysis, a large amount of online and offline data from the system is analyzed [11; 12; 13; 14]. [PERSON] and others [15] have made a detailed review of fault failure analysis in the traditional mechanical field from the perspective of artificial intelligence, involving artificial neural networks, Bayesian networks, genetic algorithms, fuzzy logic, and case-based reasoning. Based on a model-based approach, this work examines whether there is a consistency criterion between the system's actual output and its expected output. State estimation methods, parameter estimation methods, and equivalent space methods are kinds of model-based methods. Generally, state estimation methods compare the state output of the system model with the output of the actual system to generate system residuals that are used as consistency criteria. Observer and filter methods are used, with Kalman filter being the most commonly used. Most of the studied systems are nonlinear systems, which are inevitably affected by various factors such as the uncertainty of model parameters, the interaction between components and systems, the interaction between environment and systems, and noise. Therefore, most experts and scholars mainly focus on the robust state estimation of nonlinear systems. In the research of observer methods, [PERSON] [16] proposed nonlinear adaptive unknown input observer, [PERSON] [17] studied the sliding mode control of rolling mills using a disturbance observer, and [PERSON] [18] studied the sliding mode control of excavator based on a high-gain observer. In the research of the Kalman filter, various improved Kalman filters or their variants have been produced, such as the extended Kalman filter [19], multiple outlier robust Kalman filter [20], federated Kalman filter [21], unscented Kalman filter [22], distributed Kalman filter [23], etc. In the parameter estimation method, the estimated physical parameters are compared with the nominal values in order to generate the system residual. This serves as a consistency criterion for the core idea. There are two common estimation methods: joint estimation and minimum error estimation. In the joint estimation method [24; 25], the estimated parameters are used as the extension of the state, and the state estimation method is used to estimate the state and parameter. Despite its simplicity and speed, this method may result in dimension disaster if the space dimension is too high after expansion. Compared to the extended state space method, the method based on multi structure observer and filter [26] has stronger robustness and lower computational difficulty.