## 1 Introduction  
Multi-cylinder parallel balance oil circuit, as a heavy-duty support mechanism, has been widely used in heavy equipment in fields such as construction, port, and marine engineering [1, 2, 3, 4]. Its notable feature is to use multi-hydraulic cylinders in parallel to realize the synchronous support between multiple points, and thus significantly improve the stability and reliability of the equipment. Without loss of generality, an illustration of the dual-cylinder parallel balanced oil circuit is shown in Figure 1, where this mechanism consists of power supply unit, control valve group (Relief valve, Directional valve and Balance valves) and two parallel hydraulic cylinders. However, the severe working environment (water, particle and air contamination, etc.) and long-term continuous load-bearing work cause the failure of this mechanism [5, 6, 7], and the hydraulic cylinder leakage accounts for the main part of the failure [8, 9]. Therefore, driven by the trend of automation and unmanned development of equipment in various industries, the leakage mechanism of the hydraulic cylinder and corresponding fault diagnosis are increasingly being valued [10, 11]. According to the leakage source, the hydraulic cylinder leakage can be generally divided into external leakage and internal leakage [12]. The external leakage is the leakage from the high-pressure chamber to the outside environment or components, such as the leakage at the inlet pipeline and the leakage when the piston rod extending. There are currently some methods that rely on machine vision to monitor this problem [13]. Correspondingly, the internal leakage is the leakage from the high-pressure side to the low-pressure side at the seal of the piston rod, which is inside the hydraulic cylinder and difficult to be detect. The concealment of internal leakage poses a challenge for effective detection in engineering [14]. Especially, there are multiple sources of internal leakage faults in the multi-cylinder parallel balancing oil circuit. Basically, the efforts for the diagnosis of hydraulic cylinder leakage can be divided into two categories: mathematical-model-based methods [15; 16; 17; 18; 19] and data-driven based methods [20; 21; 22]. The mathematical-model-based methods tend to build accurate fault models of internal leakage. Specifically, the extended Kalman filter and the spatial recognition theories have been verified [23; 24; 25], and also some adaptive modification methods for model error are adopted [26]. However, considering that the occurrence and evolution of the internal leakage mechanism are nonlinear, the model-based fault diagnosis method suffers degradation in the robustness and generalizability. The data-driven based methods diagnose the internal leakage by extracting the fault features of historical data, and they have better detection accuracy and generalization performance. Specifically, the data analysis and feature extraction methods of fast fourier transform (FFT), wavelet transform, wavelet packet decomposition and Hilbert Huang transform have been involved [8; 27]. Firstly, the pressure signal features related to the internal leakage were revealed by using the discrete wavelet transform, FFT and Hilbert Huang transform in [28]. Then, this work has been extended to a diagnostic model of internal and external leakage [12; 29]. Moreover, a time-frequency image method has been proposed based on the continuous wavelet transform of the pressure signal [30]. In [14], the pressure signal and displacement signal of the hydraulic cylinder were collected, and five fault features were extracted using wavelet packet analysis. Furthermore, the wavelet packet decomposition was involved to extract the energy features of the pressure, and then the principal component analysis (PCA) was used for dimensionality reduction and extraction of the internal leakage fault features [31]. Figure 1: An illustration of the dual-cylinder parallel balanced oil circuit.