Fault diagnosis for hydraulic pump based on intrinsic time-scale decomposition and softmax regression

2.1. Procedures of the method

The methodology has two main steps as below:

The first step involves the time-frequency-energy analysis of the vibration signals based on ITD. Proper rotation components are decomposed from original signals and the form of feature vector is based on energy extraction;

In the second step, softmax regression model is employed to realize the fault mode classification. The training dataset of former feature vector is used to train the fault classification model, which is tested by the testing dataset then. The procedure of the methodology can be summarized as Fig. 1.

Fig. 1The procedure of the proposed methodology

2.2. Feature extraction by using intrinsic time-scale decomposition

The ITD method decomposes a vibration signal into several proper rotation components and a trend component. For the signal $X_t$, $\xi$ is defined as the baseline extraction operator. After extracting a baseline from the signal $X_t$, the rest of the signal becomes an inherent rotation. The first decomposition of $X_t$ is as follows:

where $L_t=\xi X_t$ is the baseline signal and $H_t=\left(1-\xi \right)X_t$ is the proper rotation component.

Decomposition can extract a baseline signal and a proper rotation component, the latter denoting the local relative high frequency of the decomposed signal. Taking the baseline signal as the input signal, the decomposition is continued until a monotone signal is obtained. Using this method, the original signal is decomposed into several proper rotation components of frequency ranges from high to low and a monotone trend component. The whole process can be expressed as:

where $\psi {\xi }^k{X}_t$ is the $\left(k+1\right)$th layer of the proper rotation and ${\xi }^p{X}_t$ is a monotone trend or an extracted lowest frequency baseline when the decomposition is stopped before the monotone trend is obtained.

The decomposed components from the ITD method have certain physical meanings. They can effectively reflect the characteristics of the original signal and are suitable for analyzing the amplitude and frequency modulation of the signal [5].

At last, proper rotation components $IT{D}_1$,…, $IT{D}_n$ and the trend component are obtained, as follows:

where $c_i\left(t\right)$ is the $i$th ITD component and $r_n\left(t\right)$ is the trend component.

The ITD method is applied to decompose the original signal into several proper rotation components with intrinsic time scales. Then, the energy values are calculated as features of the components. Each proper rotation component $c_i\left(t\right)$ is processed as following:

where $E_i$ is the energy of Each proper rotation component, and then normalized as below:

where $T_i$ is the normalized energy feature vector which contains key information of original signal. The feature value can change when the hydraulic pump works under different conditions, which can reflect the different fault mode of the pump. Thus, the ITD method can be applied to obtain the feature vector effectively.

2.3. Softmax regression method

Softmax regression is an expansion of logistic regression. Different from traditional logistic regression, softmax regression focuses on multi-class classification. It generalizes logistic regression to the multi-class classification setting, where the classes are mutually exclusive. Given an input $x$, softmax regression estimates the probability of the class label y taking on each of the k different possible values with a $k$-dimensional vector generated from the hypothesis function.

For a test input $x$, the probability that $P(y=j|x)$ for each value of$\mathrm{}j=1,\dots ,k$ is estimated by softmax regression. Assume the training set $\left\{\left(x^{\left(1\right)},y^{\left(1\right)}\right),\cdots ,\left(x^{\left(k\right)},y^{\left(k\right)}\right)\right\}$ and labels $y^{\left(i\right)}\in \left\{\mathrm{1,2},\dots ,k\right\}$, the hypothesis outputs a $k$-dimensional vector (whose elements sum to 1) and also provides k estimated probabilities. The hypothesis $h_{\theta }\left(x\right)$ takes the form:

Here ${\theta }_1,{\theta }_2,\dots ,{\theta }_k\in R^{n+1}$ are the parameters of the model. Normalized by $1/{\sum }_{j=1}^k{e}^{{\theta }_j^{T}x\left(i\right)}$, the hypothesis for$\mathrm{}{x}^{\left(i\right)}$ are then equaled to one.

The cost function will be as follows:

where $1\left\{·\right\}$is the indicator function so that 1{a true statement} = 1, and 1{a false statement} = 0. Then it is plugged into a gradient descent algorithm and get a minimum $J\left(\theta \right)$ which helps to get the final classification results:

Besides, the cost function is modified by adding a weight decay term $\mathrm{}\frac{\lambda }{2}\sum _{i=1}^k\sum _{j=0}^n{\theta }_{ij}^2$, which penalizes large values of the parameters since softmax regression has superfluous parameters and is easy to be over parameterized [7-8].

3.1. Feature extraction of the vibration signals

In the process of feature extraction, each raw data was decomposed to acquire the proper rotation components by ITD. Figs. 2-4 shows the results of the intrinsic time-scale decomposition of the normal condition signal and two fault condition signals. Each signal was decomposed into four proper rotation components. Then the normalized energy values of the proper rotation components were extracted to form the feature vector with the assumption that energy of the trend component can be ignored.

Fig. 4ITD result of a valve plate wear signal

3.2. Softmax regression model training for fault classification

In this section, the softmax regression model was applied to recognize the states of the hydraulic pump based on the feature vectors obtained by ITD. For each fault mode, the fault classification model was trained by 250 sets of fault state data$\mathrm{}\left(P\left(y=1|x\right)=1\right)$ versus 250 sets of normal data$\left(P\left(y=1|x\right)=0\right)$. And the gradient descent method was employed to determine the parameters of the softmax regression model, which was then used to classify the faults of the hydraulic pump.

The training datasets which contained 40 sets of data under normal condition, 40 sets of data under fault mode 1(slipper loosing), and 40 sets of data under fault mode 2(valve plate wear), were used for validating the trained classification model. The classifying results expressed by the different fault mode probabilities were shown in Fig. 5.

Fig. 5Probabilities of fault modes 1 and 2

As is shown in Fig. 5, the probabilities of each corresponding fault was much higher than those under other states. For sample (a) in Fig. 5, the probabilities of samples from 41 to 80 was much higher than the samples between 1 and 40 (normal state) together with samples 81 and 120 (fault mode 2), thus drawing the conclusion that the pump was diagnosed with fault mode 1. The result can demonstrate the effectiveness and practicality of fault diagnosis using softmax regression.