**1. Introduction**

Fluid plays an important role in the power transmission of the hydraulic system or component [1,2]. Because of hydraulic impact and mechanical faults, fluid flow pressure is very complex and nonstationary in hydraulic systems or components. Meanwhile, there are many background noises due to high and low pressure conversion, fluid pressure impact, cavitation phenomenon, fluid pulsation, and so on. The above aforementioned noises can cause problems in the fault feature extraction of the hydraulic systems or components based on the fluid pressure signal. Much running condition feature information is contained in the fluid pressure signal. Therefore, pressure fluctuation can reflect the running condition of the hydraulic system or component [3–8]. Fault diagnosis for the hydraulic systems or components based on the fluid pressure signal has been studied by many scholars at home and abroad [9–16]. Zhang put forward a method of flow measurement based on the new sensor, where the flow rate could be measured in hydraulic system by applying the mathematical model, and then flow detection of a 7 piston-pump was realized based on the sensor signal [9]. Goharrizi utilized the Hilbert- Huang transform to decompose the pressure signal of hydraulic actuators, and the first intrinsic mode function was used as a data source, and then the root mean square extracted from it could be adopted to detect internal leakage and its severity effectively. This was done without requiring prior knowledge about the model of the actuator or leakage [10]. Vásquez proposed an active model-based algorithm of fault detection and isolation. With the help of frequency-domain estimators, continuous-time models in a user-defined frequency band were identified. Then, a method for fault detection and isolation was adopted to diagnose early faults in hydraulic actuators based

on the fluid pressure signal [11]. Aiming to resolve the serious influences of pressure fluctuation and other noises in the pressure signal, Tang applied a method of wavelet theory to decompose the signal, obtaining the wavelet energy of fault feature information can be got. Then, the inner leakage fault of the hydraulic cylinder could be diagnosed [12]. In order to diagnose the faults in reciprocating pumps, a fluid pressure signal in pump cylinder was analyzed. Frequency energy was extracted for the feature vectors, and then the improved neural network was used to diagnose the pump fault successfully [13]. Guo proposed a pre-filter combined with threshold self-learning wavelet algorithm. The denoising threshold could be self-learnt in the steady flow state, and its noise suppression effect was better than that of the traditional wavelet algorithms based on fluid pressure signals of hydraulic pipeline [14]. You proposed a fusion method using the hybrid particle swarm optimization algorithm and wavelet packet energy entropy. Neural network weights and threshold were optimized by the algorithm, and wavelet packet energy entropy was used for the eigenvector, and then the fault of hydraulic system could be diagnosed effectively [15]. In contrast to the traditional way of detecting the ship fault in the time domain, Li presented a novel method in the frequency domain. The method decomposed the hydraulic pressure signal using the wavelet-transform technique, and reconstructed it at the low-frequency region; thus, the ship fault could be diagnosed effectively [16].

The hydraulic pump is an important power component, and it has been applied in the fields of robotics [17], engineering machinery [18], underwater machinery [19,20], and wind power machinery [21]. These fields often involve working conditions with high temperature, high pressure, high speed, high humidity, and heavy load, leading to a high failure rate and significant casualties and economic losses [22–33].

In 2018, the Italian scholars Ali Moshrefzadeh and Alessandro Fasana proposed a new method named Autogram based on unbiased autocorrelation (AC) [34]. It was applied on rolling element bearings. Fault feature information of the inner race, outer race and rolling element could be thus extracted effectively. The Autogram method possesses some advantages. Firstly, heavy Gaussian and non-Gaussian background noise have a very bad influence on fault feature extraction, and the procedure of unbiased AC has overcome this disadvantage. Secondly, because of down-sampling operation, the length of the time history halves at each level of decomposition, which can limit the ability to investigate the traditional wavelet transform coefficients. Furthermore, the transform may be interfered with selection of a signal starting point. To resolve these problems, maximal overlap (undecimated) discrete wavelet packet transform (MODWPT) can be used to remove the down-sampling step in discrete traditional wavelet packet transform (DWPT) [35]. Thirdly, there is no need to obtain the prior morphological feature knowledge of a signal. Because Autogram was proposed in 2018, few scholars have studied the method at a global scale.

In this paper, we firstly introduce Autogram into the fault feature extraction for the fluid pressure signal of a hydraulic pump successfully. Based on three kinds of kurtosis and threshold values, we find that only standard Autogram can select the optimal frequency band, and rich feature information of center spring wear fault can be extracted effectively without processing of the threshold value. Application of Autogram is extended to hydraulic pump from rolling element bearings, and the acquired results can provide a theoretical basis for the fault feature extraction of the hydraulic pump. It can also provide an important basis for the further study of multiple and single faults diagnosis of the hydraulic pump and other rotating machinery.

The organization of this paper is as follows: In Section 2, the algorithm of Autogram is introduced; In Section 3, the flowchart of Autogram is described. In Section 4, some examples of simulation experiment validation are presented. In Section 5, the experimental results are demonstrated by applying the Autogram to the fault signal of center spring wear of hydraulic pump. In Section 6, the conclusions of this investigation are summarized.

#### **2. Algorithm of Autogram**

In order to be readable, the nomenclature is given as follows:


The fast kurtogram (FK) is adopted to select the signal with the most impulsive frequency band; and it has been a significant method for fault diagnosis of rotating machinery for many years [36]. However, in some harsh backgrounds with low signal-to-noise ratio (SNR), strong non-Gaussian noise, or randomly distributed impulses, its extraction ability is much reduced.

With the aim of resolving the above problems, Italian scholars proposed a method named Autogram to enhance the feature extraction ability in heavy Gaussian and non-Gaussian background noise in 2018. It is an effective tool for processing the impulsive fault signal, and no prior knowledge of the signal is needed [34].

The algorithm is described as follows:

(1) Decomposition of maximal overlap (undecimated) discrete wavelet packet transform (MODWPT)

According to a dyadic tree structure, a fault signal is divided in frequency bands by means of the wavelet transform. The MODWPT removes the down-sampling step of the discrete wavelet packet transform (DWPT), and then it is used as a filter; the signals are consequently produced at each level of decomposition. The signals in each decomposition level correspond to a frequency band and central frequency, known as the node.

(2) Calculating unbiased AC of the squared envelope for each node

Unbiased AC analysis of the (periodic) instantaneous autocovariance of the signal *Rxx*(*ti*, 0) is calculated and shown in Equation (1).

$$\mathcal{R}\_{\text{xx}}(\tau) = \frac{1}{n-q} \sum\_{i=1}^{n-q} \mathbf{x}(t\_i)\mathbf{x}(t\_i + \tau) \tag{1}$$

where *x* is squared envelope of the signal filtered by MODWPT at Step 1, τ = *q*/*fs* is the delay factor, *fs* is sampling frequency, and *q* = 0, ... ,*n*−1.

The advantage of AC is that it filters out the uncorrelated components within the fault feature information, i.e. noise and random impulsive contents. Furthermore, fault feature information can be made more obvious. It is even more effective, because it is done for each node separately rather than on the complete original signal, so that SNR for each demodulated band signal is increased.

It can be seen form Equation (1) that the point number of node will decrease with an increment of τ, and therefore AC will have an estimation error, thus the first half of the AC is used in the paper. With the help of the MODWPT, AC can make the diagnostic process more accurate.

(3) Kurtosis of the AC

This step is to select the node, and let the node be data source for further fault feature extraction. The proposed method is different from FK, because the kurtosis of Autogram is computed based on the AC of the node of each level. Subsequently, the kurtosis values of all nodes, similar to FK, are presented in a colormap for which the color scale is proportional to the kurtosis value, and the vertical and horizontal axis represent the level of the MODWPT decomposition and frequency, respectively.

Kurtosis aims to quantify the impulsivity of the AC of each node. Three kinds of equations are illustrated in Equations (2)–(4):

$$\text{Kurtosis}(\mathbf{x}) = \frac{\sum\_{i=1}^{n/2} \left[ \mathcal{R}\_{\text{xx}}(i) - \min(\mathcal{R}\_{\text{xx}}(\tau)) \right]^4}{\left[ \sum\_{i=1}^{n/2} \left[ \mathcal{R}\_{\text{xx}}(i) - \min(\mathcal{R}\_{\text{xx}}(\tau)) \right]^2 \right]} \tag{2}$$

$$\text{Kurtosis}\_{\boldsymbol{u}}(\mathbf{x}) = \frac{\sum\_{i=1}^{n/2} \left| \mathcal{R}\_{\text{xx}}(i) - \overline{\mathbf{x}}\_{T}(i) \right|\_{+}^{4}}{\left[ \sum\_{i=1}^{n/2} \left| \mathcal{R}\_{\text{xx}}(i) - \overline{\mathbf{x}}\_{T}(i) \right|\_{+}^{2} \right]} \tag{3}$$

$$\text{Kurtosis}(\mathbf{x}) = \frac{\sum\_{i=1}^{n/2} \left| \hat{\mathcal{R}}\_{\mathbf{x}\mathbf{x}}(i) - \overline{\mathbf{x}}\_{T}(i) \right|\_{-}^{4}}{\left[ \sum\_{i=1}^{n/2} \left| \hat{\mathcal{R}}\_{\mathbf{x}\mathbf{x}}(i) - \overline{\mathbf{x}}\_{T}(i) \right|\_{-}^{2} \right]} \tag{4}$$

where *n* is node length, operator of |-| <sup>+</sup> or |-| <sup>−</sup> illustrate that only positive or negative value is adopted respectively, and the other values are set to 0. *xT*(*i*) is the threshold value, and it can be obtained based on the moving mean value of AC.

$$
\overline{\mathfrak{X}}\_T(i) = \frac{1}{k} \sum\_{j=i}^{i+k-1} \hat{\mathcal{R}}\_{\text{xx}}(i) \tag{5}
$$

where *k* is length of the windowed signal to be averaged.

Colormap presentations, based on the Equations (2)–(4), are denominated standard Autogram, upper Autogram, and lower Autogram, respectively.

Ultimately, the node associated with the largest kurtosis value is considered for further investigation.

(4) Spectrum analysis based on the threshold value

Based on the data source obtained in Step (3), the Fourier transform of the squared envelope based on the no threshold value (original node), the smaller than threshold value, and the larger than threshold value can be obtained. Thus, their spectrums are known as the no threshold spectrum, the lower threshold spectrum, and upper threshold spectrum, respectively.

#### **3. Flowchart of the proposed method**

Fluid pressure signal of center spring wear fault is sampled. The MODWPT is adopted to decompose the signal, and some nodes at each level of decomposition can be obtained. The AC of each node is computed, and the node that corresponding to the biggest AC value is selected as the data source for further investigation. Then, the spectrum of data source can be acquired, and the fault feature information can be extracted. The flowchart of the Autogram is shown in Figure 1.

**Figure 1.** The flowchart of the proposed method.
