Keywords

70.1 Introduction

As the rapid development of mechanical technology, the equipments become more sophisticated. The real-time status of equipment is important problem in condition monitoring. Spectrometric analysis of oil liquid model is a valid method in condition monitoring (Zheng et al. 2009; Li et al. 2009; Hongbo et al. 2006; Bing et al. 2006).

PSST is sophisticated equipment which is often used in hostile environment. The configuration of PSST is shown in Fig. 70.1. The failure of any part may leads to very serious consequences. Therefore, people want timely to know the status of PSST. From the lubrication system of the equipment regularly to obtain oil samples and spectral analysis, we can get the quantitative information of component wear without dismantling and identify the potential failures and timely maintenance. Therefore, we want to build a fault classification model using the samples that have been obtained during the running of PSST.

Fig. 70.1
figure 1

The configuration of PSST

Full size image

In this paper, a fault classification model of power-shift steering transmission based on v-Support Vector Machine is proposed. The correct rate of the model has been proved to be very high with limited fault samples.

70.2 v-Support Vector Machine

Basic classification approach of SVM is an issue of dichotomies (Boser et al. 1992; Vapnik 1998). For nonlinear problems, C, a penalty factor is introduced in order to obtain relatively good classification performance. In SVM training process, C, the penalty factor plays two roles: maximize classification interval and minimize training errors (Li et al. 2003; Zhai et al. 2003). The aforementioned roles are competing in the way that maximizing classification interval will inevitably lead to the increased training errors while minimizing training mistakes will result in the declined classification interval. In the training process, C mainly plays a compromised function (Hu et al. 2007). As a result, difficulty is in existence for selection of C. Hence, Schölkoph et.al. (2000) propose an improved approach, namely, substituting parameter v for C.

Assume that sample set X = {xi} ∈ Rd is composed of two types of points. In case of xi belongs to the type 1, it can be concluded that yi = 1. In case of xi belongs to the type 2, it can be concluded that yi = −1. Therefore, the training sample set {xi, yi} can be obtained, among which i = 1,2,3…n. Map x from original space to F, a high-dimensional feature space through applying a nonlinear mapping function:

$$ \phi :x \to \phi (x),\,R^{d} \to F $$
(70.1)

Thus, corresponded optimization problem of dichotomies issue is:

$$ \hbox{min} \;J\left( {\omega ,\xi ,\rho } \right) = \frac{1}{2}\omega^{T} \cdot \omega - \nu \rho + \frac{1}{n}\sum\limits_{i = 1}^{n} {\xi_{i} } $$
(70.2)
$$ {\text{s}} . {\text{t}} .\;y_{i} \left[ {\left( {\omega^{T} \cdot \phi \left( {x_{i} } \right)} \right) + b} \right] \ge \rho - \xi_{i} $$
(70.3)
$$ \xi_{i} \ge 0,\,i = 1,\;2,\; \ldots ,\;n,\;\rho \ge 0 $$
(70.4)

where ξ i reflects distance between y i , actually indicates value and SVM output. Compared with C-SVM, there contains no parameter C, which is replaced by parameter v. In addition, the parameter of ρ is also added in. In case that ξ i  = 0, condition (70.3) means that the two types of points are separated by the interval of \( 2\rho /\left\| \omega \right\| \) (Boser et al. 1992; Naiyang and Yingjie 2004).

The following function of formula (70.2) is obtained through introducing Lagrange function:

$$ L(\omega ,\xi ,b ,\rho ,\alpha ,\beta ,\delta ) = \frac{1}{2}\left\| {\left. \omega \right\|} \right.^{2} - \nu \rho + \frac{1}{n}\sum\limits_{i = 1}^{n} {\xi_{i} } - \sum\limits_{i = 1}^{n} {\left\{ {\alpha_{i} \left[ {y_{i} \left[ {\left( {\omega^{T} \cdot \phi (x_{i} )} \right) + b} \right] - \rho + \xi_{i} } \right] + \beta_{i} \xi_{i} } \right\}} - \delta \rho $$
(70.5)

Partial derivative of formula (70.5) is solved and the result is introduced into the function. Maximum value of α is worked out, and then dual problem of optimization issue (70.2) is obtained as follows:

$$ \hbox{max} \; - \frac{1}{2}\sum\limits_{i}^{n} {\sum\limits_{j}^{n} {\alpha_{i} } } \alpha_{j} y_{i} y_{j} \left( { \phi \left( {x_{i} } \right)^{T} \cdot \phi \left( {x_{j} } \right)} \right) $$
(70.6)

In other words, the minimum value of the following dual problem needs to be solved:

$$ \hbox{min} \;\frac{1}{2}\sum\limits_{i}^{n} {\sum\limits_{j}^{n} {\alpha_{i} } } \alpha_{j} y_{i} y_{j} \left( { \phi \left( {x_{i} } \right)^{T} \cdot \phi \left( {x_{j} } \right)} \right) $$
(70.7)

Calculating problem with similar shape of \( \frac{1}{2}\sum\nolimits_{i,j = 1}^{n} {\alpha_{i} \alpha_{j} y_{i} y_{j} \varphi \left( {x_{i} } \right)^{T} \varphi \left( {x_{j} } \right)} \) will be encountered when solving optimization issue (70.7) with SVM. Direct calculation will be quite complicated due to the fact that φ function is nonlinear. In order to avoid the problem, Vapnik (Boser et al. 1992) puts forward kernel function method, namely, replace \( \varphi \left( {x_{i} } \right)^{T} \varphi \left( {x_{j} } \right) \) calculation with \( K\left( {x_{i} ,x_{j} } \right) = \varphi \left( {x_{i} } \right)^{T} \varphi \left( {x_{j} } \right) \). RBF kernel function, which currently enjoys the most extensive application, is employed in the research with the following form:

$$ K\left( {x_{i} ,x_{j} } \right) = \exp \left( { - \frac{{\left| {x_{i} - x_{j} } \right|^{2} }}{{\sigma^{2} }}} \right) $$
(70.8)

After introducing kernel function, dual problem of dichotomies issue can be demonstrated as follows:

$$ \hbox{min} \;\frac{1}{2}\sum\limits_{i}^{n} {\sum\limits_{j}^{n} {\alpha_{i} } } \alpha_{j} y_{i} y_{j} K\left( {x_{i} ,x_{j} } \right) $$
(70.9)
$$ {\text{s}} . {\text{t}} .\;\sum\limits_{i = 1}^{n} {\alpha_{i} y_{i} } = 0 $$
(70.10)
$$ 0 \le \alpha_{i} \le \frac{1}{n},\;i = 1,2, \ldots ,n $$
(70.11)
$$ \sum\limits_{i = 1}^{n} {\alpha_{i} } \ge \nu $$
(70.12)

70.3 Experimentation Research

Spectrometric oil analysis technology mainly detects the type and content of elements of oil. Different components may contain the same elements. According to Spectrometric analysis method, the same component elements are classified as the same type. The fault classification model of PSST is shown in Table 70.1.

Table 70.1 The fault classification model of PSST
Full size table

The spectrometric analysis of oil liquid data shows in Table 70.2. We obtain these data in the vehicle road test and optimize these samples. Each group contains 8 elements such as Fe, Cr, etc.

Table 70.2 Spectrometric analysis of oil liquid data (unit: μg/mL)
Full size table

70.3.1 Selection of Parameters

Parameter selection occupies a significant status in SVM model training. RBF kernel function is put into application in the process of model training. Therefore, two parameters: σ and v need to be confirmed in the process. Related researches indicate that numeric area of ν is supposed to be (0, 1) (Li and Xu 2005; Pawlak 1997; Swiniarski and Skowron 2003).

Figure 70.2 indicates variation trend of classification accuracy rate with parameters of σ and ν. According to the figure, when σ enjoys certain numeric area, no significant variation of classification accuracy rate is witnessed with the increasing of ν; when ν enjoys certain numeric area, classification accuracy rate decreases in a gradual manner with the increasing of σ. It can be seen from calculation that when σ = 0.003 and numeric area of ν is 0.36–0.5, the accuracy rate is 0.9, which also applies in case of σ = 0.045 and numeric area of ν is 0.42–0.55. However, another problem needs to be noticed, namely, number of support vector, for numeric area of ν is influenced by number of support vector.

Fig. 70.2
figure 2

Relationship of accuracy rate and parameters

Full size image

Figure 70.3 indicates variation trend of support vector number with parameters of σ and ν. #SV represents number of support vector. The figure shows that parameter σ can pose significant influence on the number of support vector. No matter what the numeric area of ν is, number of support vector always increases sharply with the increasing of σ; no significant variation of support vector number is witnessed when σ enjoys certain values, no matter how ν changes. With the increasing of σ, number of support vector also increases, while accuracy rate of the model declines, thus making popularization capacity of the model decrease. Therefore, selection of the parameter of σ is of great significance. Selection of the parameter of ν is supposed to be affected by the number of support vector; otherwise, it will pose influence on popularization capacity of the model.

Fig. 70.3
figure 3

Relationship of support vector number and parameters

Full size image

Taking into comprehensive consideration influence of model parameters on accuracy rate and number of support vectors, the paper ultimately selects (σ, ν) = (0.045, 0.42) as the parameter for SVM model.

70.3.2 Experimentation

Through selecting, we pick out 33 group data for training and 10 group data as testing samples. During the training, the model makes only one error of judgment in test samples. The correct rate of judgment for the test group data is 90 %. Table 70.3 show that, comparing with C-support vector machine and BP neural network, the v-support vector machine has good properties in research of fault classification of PSST.

Table 70.3 Correct rate of different classifiers
Full size table

In most cases, the oil samples of the lubrication system of the equipment are normal. But, the components of the PSST may have a potential fault that could not be detected. The model of PSST’ fault classification based on v-SVM can judge the status of PSST. The correct rate of the model has been proved to be very high with small samples.

70.4 Conclusion

In this paper, a model of PSST’ fault classification based on v-SVM is proposed. The fundamental of v-SVM is researched. The influence of model parameters for performance of v-SVM is analyzed. The correct rate of the model has been proved to be very high with limited fault samples.