Performance assessment and fault classification for hydraulic pump based on LMD and LR

2.1. Procedures of the methodology

There are two major steps of the methodology. Step 1: feature extraction and selection that extracts and determines appropriate features for health assessment and root cause classification as well as reducing the search space for fast computation. Step 2: health assessment and root cause classification by means of the logistic regression method. The process of the methodology can be summarized as Fig. 1.

Fig. 1Process of the methodology

2.2. Feature extraction using local mean decomposition

The LMD scheme involves separating a frequency modulated signal from an amplitude modulated envelope signal. The process of the separation includes smoothing the original signal, subtracting the smoothed signal from the original signal and using an envelope estimate to amplitude demodulate the result. Product function is obtained through multiplying the frequency modulated signal by the final envelope estimate. The whole process is repeated on the resulting signal, to acquire a second product function with an associated envelope and instantaneous frequency. By the time the remaining signal contains no more oscillations, the decomposition is finally completed [4].

The feature vector of the hydraulic pump vibration signals contains twelve components. The feature vector $T$ is:

where the $g_{11}\text{,}{g}_{21}\text{,}\dots \text{,}{g}_{m1}$ and $g_{21}\text{,}{g}_{22}\text{,}\dots \text{,}{g}_{m2}$ are the skewness coefficients and kurtosis coefficients of the product functions, given by:

The energy value of each product function is given by:

In consideration of the energy values are always large, for the sake of processing and analysis, the energy vector is normalized as [5]:

2.3. Feature reduction based on PCA

Assuming that an operation stage $S$ (in this paper, only the normal machine behavior and machine operation with a worn piston shoe and swashplate/valve plate rotor) of the hydraulic pump is given and its features $X$ are characterized by the multivariate Gaussian distribution with mean $\stackrel{-}{{\mu }_s}$ and covariance matrix$K_s$. The symmetric matrix $K_s$ is given as:

where $r$ is the rank of $K_s$, ${\lambda }_i$ ($i=$ 1, 2,…,$r$) are the non-zero eigenvalues of $K_s$ and $\stackrel{-}{v_i}$ are the corresponding unit norm eigen-vectors. The definition of $V$ and $\mathrm{\Lambda }$ are given by:

Each eigenvalue ${\lambda }_i$ ($i=$ 1, 2,…, $r$) depicts the amount of the covariance matrix energy projected in the direction of the corresponding eigenvector. When there exists a high degree of correlation among the components of $X$, only a few of the eigenvalues in $\mathrm{\Lambda }$ account for most of the energy in the covariance matrix $K_s\text{.}$ Thus, assuming that eigenvalues ${\lambda }_i$ ($i=$ 1, 2,…, $r$) are arranged in descending order, Eq. (6) can be represented as:

where:

where $p$ is the number of the principal components of $K_s$, ${\lambda }_i$ ($i=$1, 2,…,$p$) are the largest $p$ eigenvalues of $K_s$, and $\stackrel{-}{v_i}$ are the corresponding unit norm eigenvectors.

A query item $\tilde{X}$ can now be transformed into a $p$ component random variable $\tilde{Y}$ given as:

If $\tilde{X}$ belongs to the class of signals from centrifugal pump state $S$, then $\tilde{Y}$ should be normally distributed with zero mean and variance $I_p$, where $I_p$ is the unity matrix of order $p$. Thus, for each query item $\tilde{X}$, its adherence to the class $S$ can be assessed through the Euclidean norm of the vector $\tilde{X}$, which in turn corresponds to assessment and classification based on the logistic regression of the query item from the training classes [6].

2.4. Logistic regression method

The machine condition description from daily maintenance records/logs is a dichotomous problem (either normal or failed) which can be represented using a logistic regression function [1]. The goal of logistic regression is to find the best fitting model to describe the relationship between the categorical characteristic of dependent variable (the probability of an event, constrained between 0 and 1) and a set of independent variables. The logistic function is:

The logistic model is:

where $g\left(x\right)$ is a linear combination of the independent variables $x_1\text{,}{x}_2\text{,…,}{x}_k$.

The pre-condition for figuring out $P\left(x\right)$ is determining parameters $\alpha$ and ${\beta }_1\text{,…,}{\beta }_k$ in advance. Using the maximum likelihood method, estimation in logistic regression parameters is chosen [7]. Then, the probability of failure for each input vector $x$ can be calculated according to Eq. (11).

3.1. Data acquisition system description

In the experiment, two commonly occurring faults in the hydraulic pump were set, namely, slipper loosing and valve plate wear. The vibration signal of the hydraulic pump was acquired from the end face with a stabilized 528 rpm motor speed and 1000 Hz sampling rate.

3.2. Feature extraction and reduction

First, each vibration signal was processed by local mean decomposition to acquire the product functions. Fig. 2(a) shows the result of the local mean decomposition of the normal condition signal while Fig. 2(b) shows the result of the local mean decomposition of the signal under slipper loosing condition. Second, the skewness coefficients, the kurtosis coefficients and the normalized energy values of the product functions was calculated to form the feature vector. Then PCA was used to determine a subset of feature components. In this case, a 3-dimension feature vector was finally used for the health assessment and fault mode classification.

Fig. 2Local mean decomposition results of normal and slipper loosing condition

a)

b)

3.3. Logistic regression models training

3.4. Validation

For each fault mode, one set data of the hydraulic pump were used for validation.

Set 1 (fault mode 1): 40 sets of data under normal condition versus 80 sets of valve plate wearing data;

Set 2 (fault mode 2): 40 sets of data under normal condition versus 80 sets of slipper loosing data.

Fig. 3Health assessment result of two fault modes

a)

b)

Fig. 4Probability of fault modes 1 and 2

a)

b)

The confidence value (CV) is defined as CV = 1 – $P\left(x\right)$ where $P\left(x\right)$ is the probability of failure. When the pump is running normally, CV is very close to 1. However when the pump is going to fail, the CV will approach 0 correspondingly. In the course of the CV falling, as long as the value is less than 0.7, a prefixed threshold, the fault mode classification model will be triggered to input the features into fault classification modes to get the probability of each fault. Fig. 3(a) and (b) shows the result of overall health assessment of the two sets vibration data of hydraulic pump using model LR1. Fig. 4(a) and (b) shows the different fault modes’ probability got from LR2 and LR3.

In Fig. 3, both faults can be detected from the CV drops, however, clarifying the difference and the reasons of the drops is still very hard. To solve this problem, a root cause classification module is used. As long as the confidence value is below 0.7, a prefixed threshold, the probability of fault modes will be calculated by inputting the corresponding features into the trained models (LR2 and LR3). In Fig. 4(a) and (b), the probability of fault 1 (valve plate fault) and fault 2 (slipper loosing) is very high from sample 40. Consequently the minor probability points of corresponding fault are showed in Fig. 4.