A centrifugal pump fault diagnosis approach based on LCD-ApEn and PNN

2.1. Local characteristics-scale decomposition method

The LCD method can self-adaptively decomposed a complex signal into a sum of ISCs and a residue with the assumption that any two ISCs are independent of each other. The ISC component must meet the following two conditions [1]:

1) in the entire data set, the signal is monotonic between any two adjacent extreme points;

2) among the whole data set, suppose the number of extrema of the signal $x\left(t\right)$ is $M$, and determine all the extrema $(t_k,x_k)$, any two adjacent maxima (minima), $(t_k,x_k)$ and $(t_{k+1},x_{k+1})$, are connect by a straight line, as follows:

Fig. 1The process of the methodology

In order to assure the smoothness and symmetry of the ISC, the proportions of and remain constant, as follows:

Generally, $a$ is set as 0.5, thus, $A_k/x_k=$ –1. At the time, $x_k$ and $A_k$ are symmetrical for the $x$-axis, which ensure the symmetry and smoothness of the ISC component waveform [4].

Based on the definition of the ISC component, a complex signal $x\left(t\right)$ ($t>$ 0), can be decomposed into a sum of ISCs using the LCD method as follows:

1) Suppose the number of extrema of the signal $x\left(t\right)$ is $M$, and the corresponding time of any extrema is $t_k$, which determine all the extrema $(t_k,x_k)$ ($k=\text{1,…,}M\text{)}$ of the signal $x\left(t\right)$.

2) Compute $A_k$ ($k=\text{2,…}\text{,}M-1$) according to Eq. (1). Based on the following equation, the corresponding $L_k$ ($k=\text{2,…}\text{,}M-1\text{)}$ can be calculate:

Since $k$ is from 2 to $M-1$, the values of $L_1$ and $L_M$ should be calculate by extent by extending the boundary of the data, which can be fulfilled by different methods [5-7].

$A_1(t_0,x_0)$ and $A_M(t_{M+1},x_{M+1})$ can be obtained by the extension two end extrema, thus, the values of $L_1$ and $L_M$ can be get [8].

3) All the $L_k$$\text{(}k=\text{1,…}\text{,}M$) are connected by the cubic as spline line $S{L}_1\left(t\right)$, which is defined as the base line of the LCD.

4) The difference between the data and the base line $S{L}_1\left(t\right)$ is the first component $h_1\left(t\right)$:

5) Separate the first $IS{C}_1$ from the initial data $x\left(t\right)$:

6) Take the residue $r_1$ as the original signal to be processed. Repeat the above process until the residue are constant or monotonic function or a function with no more than twenty extrema. $IS{C}_1$,…, $IS{C}_n$ and the residue are obtained, as follows:

where $c_i\left(t\right)$ is the $i$th ISC and $r_n\left(t\right)$ is the residue.

The LCD method is applied to decompose the original signal into several ISCs with different time scale and the characteristics. Then, the approximate entropy (ApEn) is used to characterize the probability of the new model of the signal. That is, the values can be change when the centrifugal pump working under different conditions, which can reflect the different fault mode of centrifugal pump. Thus, the LCD-ApEn can be applied to obtain the feature vector.

2.2. A description probabilistic neural network

The Probabilistic Neural Network (PNN) is a supervised neural network with the advantage of fast training, high stability, strong supplement capability, good diagnosis robustness, which was widely used in fault diagnosis [9].

PNN is a feedforward neural network with a parallel 4-year structure: input layer, pattern layer, summation layer and output layer, as showed in Fig. 2. The first layer is the input layer, the number of whose neurons is determined by feature vector, and with the function to transferring samples to each node of model layer [10]. The pattern layer contains nodes with exponential function, whose nodes are as many as training samples. The ability of summation layer is to sum up the input from the pattern layer and obtains the estimated probability densities, the output layer calculate the probability of each mode, the one with the maximum probability density correspond to the kind of recognition model of sample category is 1, all others neurons output is 0 [11].

Fig. 2Probabilistic neural network architecture

3.1. Feature extraction based on LCD-ApEn

To acquire the fault feature vectors, each vibration signal was processed by local characteristic-scale decomposition to acquire the intrinsic scale components. Fig. 4(a) shows the result of the local characteristic-scale decomposition of the inner race wearing while Fig. 4(b) shows the result of normal. Each signal was decomposed into five ISCs.

Fig. 4Time domain waveform of ISCs of vibration signal of inner race wearing a) and normal b)

a)

b)

Then as shown in Tables 1-5, the ApEn of each ICS for each fualt mode is respectively computed to further extract the fault feature with the established parameter values of $m=\text{2}$ and $r=\text{0.25}$. For convenience, only the first five groups of values of Apens was listed in the table under different fault conditions. For each fault mode, the ApEn of each ICS component decreases as the number of decomposition layers increase. Compared with each condition, we can see that the ApEns are different, which can reflect their different nonlinear generation mechanism However, to recognize the fault mode, the feature vector has to be effectively integrated with classifiers.

Fig. 5a), b) The result of network training and c) fault diagnosis

a) Training result of PNN

b) Training error of PNN

c) Diagnosis result of PNN

3.2. Fault diagnosis based on PNN

In this section, the Probabilistic Neural Network (PNN) was applied to recognize the states of the centrifugal pump based on the feature vectors obtained by LCD-ApEn.

For each fault mode, 40 samples were acquired and 20 sets of data were used for training network. The total number of training samples is 100. Each group of data was decomposed by LCD and the approximate entropy of the five ISCs was calculated for training as a network input. Then the remaining samples were selected for testing as PNN input. The output results of testing samples are shown as Fig. 5. The classification result No. 1-5 respectively represent the bearing roller wearing, the bearing inner race wearing, the normal condition, the centrifugal pump impeller wearing and the bearing outer race wearing. As showed in Fig. 5, the accuracy rate of fault diagnostic can reach 100 %, which verifies the effectiveness of the proposed feature extraction method.