# Fault Diagnosis of a Solenoid Valve Based on Multi-Feature Fusion
## 1 Introduction
Since time-domain analysis directly analyzes the system in the time-domain, it has the advantages of a large amount of information, intuitiveness and accuracy. Despite that, it is not trivial to identify the relationship between the extracted information and potential faults. Principal component analysis (PCA) is a kind of multivariate statistical analysis technique, and it can obtain a smaller number of principal components on the basis of the limited matrix generated on the basis of a time series. PCA can be used for data dimensionality reduction and feature recognition of data [9; 10; 11]. The core idea of PCA is to find a set of new variables that are a linear combination of the original ones. The number of new variables is less than the original variables, and the valuable information content of the original variables is preserved to the maximum extent. The new variables are unrelated, achieving dimensionality reduction [12]. Frequency domain analysis is a classic method for studying control systems and is an engineering method that uses graphical analysis to evaluate system performance in the frequency domain [13]. This analysis aims to approach the analysis problem from the frequency perspective and to complement time-domain analysis. The significant advantage of this method is guiding the analysis from the signal's surface to its essence and revealing the signal's components. This is important, as understanding the structure of the signal affords its optimum exploitation. The frequency-domain analysis method also has shortcomings, as it is not intuitive, and it is challenging to comprehend. It requires a calculation to obtain the frequency spectrum or to restore the frequency spectrum to a time-domain signal, and its sine wave component cannot reflect the moment of their occurrence [14]. Spurred by the shortcomings of time-domain and frequency-domain analyses, researchers have proposed the time-frequency analysis methods. This method shows how the energy of a signal is distributed on a two-dimensional time-frequency plane, and it is intended for non-stationary signals [15; 16]. Standard time-frequency analysis methods include the short-time Fourier transform, continuous wavelet transform, Wigner-Ville distribution, Hilbert-Huang Transform (HHT) and Laplace Transform (S transform), which present their limitations. For example, the short-time Fourier transform does not consider the requirements of frequency and time resolution. The continuous wavelet transform needs to ensure that it has an inverse transform, and the Wigner-Ville distribution interferes with the cross term when analyzing multi-component signals. Accordingly, the cubic spline interpolation method in HHT poses overshooting and undershooting problems when the envelope or mean line is fitted, and the S transform imposes a significant computational burden. Thus, on the basis of wavelet transform and multi-resolution analysis, researchers proposed wavelet packet decomposition. Wavelet packet decomposition is a development of wavelet decomposition that offers a richer range of possibilities for signal analysis. This analysis is obtained as a result of successive time localizations of frequency sub-bands generated by a tree of low-pass and high-pass filtering operations [17]. Alternatively, the energy-fault characteristic value extraction method can be used to establish the mapping relationship between the energy value of each frequency band on the second-order rate of the current and each fault state of the solenoid valve [18]. Wavelet packet decomposition plays a vital role in signal denoising, filtering, compression and fault diagnosis [19]. Traditional classification algorithms, on the basis of classical statistics, assume that sufficient training samples exist. The advantage of such approaches requires a large number of training samples [20]. Nevertheless, given that the fault diagnosis of hydraulic systems affords a small sample dataset, these methods are not suitable for this type of problem [21].