Abstract
Fault diagnosis of rotating machineries is becoming important because of the complexity of modern industrial systems and the increasing demands for quality, cost efficiency, reliability, and safety. In this study, an informationgeometric support vector machine used in conjunction with chaos theory (chaotic IGSVM) is presented and applied to practical fault diagnosis of hydraulic pumps, which are critical components of aircraft. First, the phasespace reconstruction of chaos theory is used to determine the dimensions of input vectors for IGSVM, which uses information geometry to modify SVM and improves performance in a datadependent manner without prior knowledge or manual intervention. Chaotic IGSVM is trained by using the dataset from the normal state without fault, and a residual error generator is then designed to detect failures based on the trained chaotic IGSVM. Failures can be diagnosed by analyzing residual error. Chaotic IGSVM can then be used for fault clustering by analyzing residual error. Finally, two case studies are presented, and the performance and effectiveness of the proposed method are validated.
1. Introduction
To reduce costs and shorten repair time, technologies for machine maintenance, diagnostics, and prognostics have received significant attention. Fault diagnosis is an essential prerequisite for the further development of automatic supervision. Realtime condition monitoring that can detect, classify, and predict impending faults is critical to reduce operating and maintenance costs [1]. Moreover, condition monitoring is important to increase machinery availability and improve manufacturing process productivity and reliability [2].
Hydraulic pumps are the power sources of hydraulic systems in aircraft. The performance of these pumps directly affects the stability of the hydraulic system and even that of the entire system. Statistical data show that hydraulic pumps have higher fault probability than other mechanical systems. Therefore, diagnosing pump health in real time is an important factor to increase the reliability and performance of hydraulic systems. If a faultdetection scheme that provides early warning of component failures can be developed, then repairs or replacements can be carried out at the earliest or most convenient time with minimum productivity loss [3]. However, hydraulic pumps are complex and have a high degree of coupling [4]. Considering complexity and severe working conditions, a datadriven faultdetection method is typically applied to online fault diagnosis. Many datadriven methods have been proposed, such as wavelet decomposition [5], artificial neural networks (ANNs) [6, 7], fuzzy logic, kernel principal component analysis [8], and DS evidence theory [9].
Given the universal presence of chaotic phenomena and the intrinsic characteristics and complex operating conditions of hydraulic systems, strong nonlinearity and chaotic features can clearly be observed in the vibration signals of hydraulic pumps [7]. Therefore, chaos theory is valuable for the fault diagnosis of hydraulic pumps [10].
A support vector machine (SVM), as a datadriven method, has been widely applied. Compared with ANNs, SVM overcomes numerous defects such as overfitting and local convergence. In addition, SVM has advantages over ANNs in terms of robustness and in preventing the curse of dimensionality. SVM has been applied to many fields, such as in pattern recognition and fault diagnosis [1].
Despite its excellent applicability, the performance of SVM largely depends on the kernel [11, 12]. Kernel functions are mostly chosen based on experience. However, unsuitably chosen kernel functions may significantly impair performance [13]. No systematic approach for choosing appropriate kernel functions has yet been introduced [14]. Choosing a kernel corresponds to a smoothness assumption of the discriminant function of the classifier. When we have prior knowledge, we can use it to choose a kernel [15, 16]. In practice, however, prior knowledge is typically unavailable. Therefore, the kernel should be optimized in a datadependent manner. An informationgeometric method is employed in the present study. Based on the structure of the Riemannian geometry induced in the input space by the kernel, SVM can be modified in a datadependent manner, and informationgeometric SVM (IGSVM) can be obtained.
This paper is divided into three sections. Section 2 describes chaos theory on phasespace reconstruction, proposes a new IGSVM for chaotic timeseries prediction, and describes the designed residual error generator. Sections 3 and 4 present several case studies, including the simulation results of a onestep iterative prediction and the experimental results of fault detection for a hydraulic pump. The feasibility and efficiency of the method are validated via a plunger pump test bed.
2. Methodology
2.1. Phasespace reconstruction of a chaotic time series
Phasespace reconstruction theory regards a 1D chaotic time series as the compressed information of highdimensional space. The Takens embedding theorem [17] suggests that a dependable phasespace reconstruction of a dynamic system can be obtained if:
where $m$ is the system embedding dimension, and $D$ is the dimension of the system attractor. To obtain a correct system embedding dimension, $D$ should be estimated starting from the time series.
The correlation dimension, as defined by Grassberger and Procaccia, is a popular definition because of its calculation simplicity [18]. The correlation integral ${C}_{m}\left(r\right)$ is defined as:
where $H$ is the Heaviside function, $m$ is the embedding dimension, and $N$ is the number of vectors in the reconstructed phase space. If $r$ is sufficiently small and $N$ is sufficiently large, then the correlation dimension $D$ is equal to:
The algorithm plots a cluster of $\text{ln}{(C}_{m}\left(r\right))\text{ln}(r)$ curves by increasing $m$ until the slope of the linear part of the curve is nearly constant. The correlation dimension D can then be estimated.
2.2. Modified SVM that uses information geometry
The SVM proposed by Vapnik [16] aims to minimize an upper bound of the generalization error by maximizing the margin between the separating hyperplane and the data. Consider a pattern classifier that uses a hyperplane to separate two classes of patterns based on given examples $=\left\{\left({x}_{1},{y}_{1}\right),\dots ,\left({x}_{i},{y}_{i}\right)\right\}$, where $x$ is a vector in the input space $S={R}^{d}$, $y\in \{\mathrm{1,1}\}$ is a class label, and $i=$1,…, $l$. A nonlinear SVM maps the input data $x$ onto a highdimensional feature space $F={R}^{n}$ ($N$ may be infinite) via nonlinear mapping $\varphi \left(x\right)$. Then, SVM searches for a linear discriminant function, that is:
The basic concepts of SVM theory are comprehensively explained in [13]. Once the correlation dimension $D$ is obtained by using the GrassbergerProcaccia (GP) algorithm, the number of input nodes in SVM can be determined as:
To modify the SVM kernel by using information geometry, the geometrical structure induced in the input space by a kernel should be analyzed as follows [19].
Mapping $\varphi \left(x\right)$ defines an embedding of $S$ into $F$ as a curved submanifold. When $F$ is a Euclidean or Hilbert space, a Riemannian metric is induced in space $S$, wherein the length of a small line element $dx$ in $S$ is defined by the length in larger space $F$.
$z$ denotes the mapped pattern of $x$ in the feature space, i.e., $z=\varphi \left(x\right)$. A small vector $dx$ is mapped onto:
where:
The squared length of $dz=d{z}_{\alpha}$ is written in quadratic form as:
where:
The dot denotes the summation over index $\alpha $ of $\varphi \text{.}$ The $n\times n$ positivedefinite matrix $G\left(x\right)={g}_{ij}\left(x\right)$ is the Riemannian metric tensor induced in $S$. This matrix shows that the metric is directly derived from the kernel.
The following is a theorem presented in [19]:
the proof of which is:
This proof verifies Eq. (10) [11].
Based on the preceding analysis, forecasting precision in regression problems can be improved if a special nonlinear map $\varphi $ is constructed, such that ${g}_{ij}\left(x\right)$ is reduced around the neighboring areas of the hyperplane $\leftyf\left(x\right)b\right=\xi $, which is contrary to the method used by Amari [11] in classification problems. This concept can be implemented by a conformal transformation of the kernel, that is:
with a properly positive scalar function $c\left(x\right)$. $\stackrel{~}{K}\left(x,{x}^{\text{'}}\right)$ is called the conformal transformation of a kernel by factor $c\left(x\right)$. The nonlinear mapping $\varphi \left(x\right)$ can be regarded as being modified to $\stackrel{~}{\varphi}\left(x\right)=c\left(x\right)\varphi \left(x\right)$, thus satisfying the Mercer positivity condition.
The metric ${\stackrel{~}{g}}_{ij}\left(x\right)$ can be obtained as follows:
where ${c}_{i}\left(x\right)=\partial c\left(x\right)/\partial {x}_{i}$ and ${\left.{K}_{i}\left(x,x\right)=\partial K(x,{x}^{\text{'}})/\partial {x}_{i}\right}_{{x}^{\text{'}}=x}$. The last term is zero for the Gaussian radial basis function kernel.
Therefore, if we choose function $c\left(x\right)$, such that its value is large when $x$ is close to the boundary and small otherwise, then we can enlarge the spatial resolution around the boundary [11].
Considering the preceding analysis, $c\left(x\right)$ can be chosen as:
where parameters $m$, ${o}_{i}$, and $\tau $ are the number of partitioning points, the center of the $i$th partition, and the width of the $i$th partition, respectively. Outside the circles, the value of $c\left(x\right)$ and its derivative are extremely small [15]. Therefore, this function satisfies the aforementioned requirement and can be used to modify SVM in a datadependent manner.
2.3. Residual error generator
A residual error generator can be designed for fault diagnosis based on the IGSVM prediction process. The structure is shown in Fig. 1.
In Fig. 1, $x\left(t\right)$ is the time series that can be observed in the actual system, IGSVM is the residual error generator trained by the data from the normal state, $\widehat{x}\left(t\right)$ is the onestep prediction value of IGSVM, and $e\left(t\right)$ is the output of the residual error generator.
The diagnostic decision is obtained based on the following rule:
${r}_{eval}\le {J}_{th}$
where ${r}_{eval}$ is the mean absolute value of the residual error signal, and ${J}_{th}$ is the threshold that can be determined by experience.
Fig. 1Structure of the residual error generator based on IGSVM
3. Simulation results
To verify the proposed method, the simulation result of Lorenz attractor data is provided. Eq. (15) is employed to generate Lorenz timeseries data:
where $\sigma =$10, $r=$28, and $b=$8/3. A total of 1000 points of $X$component Lorenz time series were used for the following prediction. According to the GP algorithm, the embedding dimension can be determined, $m=$6.
Fig. 2Plot of lnCm(r)ln(r) of the Lorenz time series
As a whole, 200 points were divided into two groups (i.e., the training and testing datasets). The first 100 samples were used for SVM training. The next 100 samples were employed to test the prediction accuracy between SVM and IGSVM.
Fig. 3The onestep iterative predicted result of the Lorenz time series
Fig. 4Comparison of APE between SVM and IGSVM
The number of input nodes of SVM was 6, which was obtained by estimating the minimum embedding dimension. Fig. 3 shows the onestep iterative predicted result for the last 100 points of the Lorenz time series by IGSVM. Fig. 4 shows a comparison of the absolute percent error (APE) of the Lorenz time series between SVM and IGSVM. Compared with SVM, IGSVM exhibits better performance in iterative prediction in terms of convergence and stability.
4. Experimental results
In this section, a test rig of an SCY hydraulic plunger pump was evaluated and analyzed to verify the proposed method. In the experiment, two common types of faults in the plunger pump were set: (1) a wear fault between the swash plate and the slipper and (2) a wear fault of the valve plate. Under three different states, including the normal state, a vibration signal was acquired from the end face of the plunger pump at a stabilized motor speed of 528 r/min and a sampling rate of 1000 Hz.
Table 1 shows the corresponding maximum Lyapunov exponents (${\lambda}_{max}$) of the three datasets. Given that all ${\lambda}_{max}$ values are positive, the experimental data can be regarded as a chaotic time series.
Table 1Lyapunov exponents of the datasets of the hydraulic pump
Data  State  ${\lambda}_{max}$ 
Data 1  Normal condition  0.0508 
Data 2  Wear fault between the swash plate and the slipper  0.0744 
Data 3  Wear fault of the valve plate  0.0435 
Following the GP method, a cluster of $\text{ln}{C}_{m}\left(r\right)\text{ln}\left(r\right)$ curves for Data 1 is plotted with the increase in embedding dimension $m$ (Fig. 5). The correlation dimension can be determined correspondingly, $D=$2.2346. According to (1), $m=$6.
Fig. 5Plot of lnCm(r)ln(r) of data from the hydraulic pump under the normal state
In this case, 200 points of the timeseries data from the normal state were used. The first 100 samples were employed to train SVMs, whereas the last 100 samples were used to test and determine the threshold of fault diagnosis. After training and testing, a prediction model of the normal state is determined. Fig. 6 shows the result of using the onestep, iterative, predictionbased IGSVM.
Fig. 6The onestep iterative predicted result of data from the normal state
4.1. Residual error of the normal state
The residual error of the normal state data can be obtained by using the residual error generator. As mentioned earlier, ${r}_{eval}$, which is the mean absolute value of the residual error, is used for fault clustering. Fig. 7 shows the absolute value of the residual error of the normal state.
Fig. 7Absolute value of the residual error of the normal state
4.2. Residual error of the wear fault between the swash plate and the slipper
In this case, 100 points of the timeseries data from the vibration signal with a wear fault between the swash plate and the slipper were used via the residual error generator. Fig. 8 shows the absolute value of the residual error of the wear fault between the swash plate and the slipper.
Fig. 8Absolute value of the residual error of the wear fault between the swash plate and the slipper
4.3. Residual error of the wear fault of the valve plate
In this case, 100 points of the timeseries data from the vibration signal of the hydraulic pump with a wear fault of the valve plate were used for fault detection. Fig. 9 shows the absolute value of the residual error of the wear fault of the valve plate.
Fig. 9Absolute value of the residual error of the wear fault of the valve plate
4.4. Fault clustering
In this study, two groups of normal data and eight groups of faulty data in different fault types were acquired via the test rig. The residual error series was calculated by using the residual error generator based on IGSVM. The absolute values of the residual error series are shown in Table 2.
Table 2 indicates that different types of data can be distinguished clearly by the absolute value ${r}_{eval}$ of the residual error. The threshold of normal data is typically set to a standard of that is three to five times higher than the means of the absolute values for normal data. In the present case, the threshold of Data1 is ${J}_{th}=$27.23e004. All ${r}_{eval}$ values of faulty data are greater than the ${J}_{th}$ of Data 1, and faults can be detected successfully.
In addition, the ${r}_{eval}$ of Data 2 ranges from 65 to 75, whereas that of Data 3 ranges from 85 to 95. Therefore, the types of faults can also be isolated. The results show a 100 % success rate in correctly detecting and isolating hydraulic pump faults.
Table 2Absolute values of the residual error series
No.  Data 1  Data 2  Data 3  
State  Normal (e004)  Wear fault between the swash plate and the slipper (e004)  Wear fault of the valve plate (e004)  
Data  11  12  21  22  23  24  31  32  33  34 
${r}_{eval}$  5.36  5.53  73.523  72.39  68.11  66.95  92.91  93.13  89.11  87.27 
Mean (${r}_{eval}$)  5.45  70.24  90.60  
Threshold ${J}_{tj}$  27.23  65 to 75  85 to 95 
5. Conclusion
Fault diagnostics is the primary technique used for conditionbased predictive maintenance. This approach can reduce costs and shorten repair time. The strong nonlinearity and chaotic features of the vibration signals of hydraulic pumps result in difficulties in fault diagnosis. This study presents a fault diagnosis method based on chaotic IGSVM, which improves the performance of SVM in a datadependent manner. The simulation results show that IGSVM, when used in conjunction with phasespace reconstruction, exhibits better capability and reliability in predicting a chaotic time series, as well as highperformance convergence ability and prediction precision in the shortterm prediction of a chaotic time series. The experimental results show that the IGSVM model demonstrates a strong ability to approximate the output and state of a normal system, which is useful in fault detection. Additional works are necessary to validate the proposed method further for wider applications. Moreover, determining thresholds automatically is another issue that should be addressed.
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About this article
This research was supported by the National Natural Science Foundation of China (Grant No. 61074083, 50705005 and 51105019), the Technology Foundation Program of National Defense (Grant No. Z132013B002), as well as the Innovation Foundation of BUAA for PhD Graduates.