Fault detection for hydraulic pump based on chaotic parallel RBF network
 Chen Lu^{1, 2}Email author,
 Ning Ma^{2, 3} and
 Zhipeng Wang^{1, 2}
https://doi.org/10.1186/16876180201149
© Lu et al; licensee Springer. 2011
Received: 27 December 2010
Accepted: 30 August 2011
Published: 30 August 2011
Abstract
In this article, a parallel radial basis function network in conjunction with chaos theory (CPRBF network) is presented, and applied to practical fault detection for hydraulic pump, which is a critical component in aircraft. The CPRBF network consists of a number of radial basis function (RBF) subnets connected in parallel. The number of input nodes for each RBF subnet is determined by different embedding dimension based on chaotic phasespace reconstruction. The output of CPRBF is a weighted sum of all RBF subnets. It was first trained using the dataset from normal state without fault, and then a residual error generator was designed to detect failures based on the trained CPRBF network. Then, failure detection can be achieved by the analysis of the residual error. Finally, two case studies are introduced to compare the proposed CPRBF network with traditional RBF networks, in terms of prediction and detection accuracy.
Keywords
Introduction
Fault detection is becoming important because of the complexity of modern industrial systems and growing demands on quality, cost efficiency, reliability, and safety. Early fault detection is an essential prerequisite for further development of automatic supervision. The interest on fault detection techniques would be increasing correspondingly.
Hydraulic pump is the power source of a hydraulic system in aircraft. Its performance has a direct impact on the stability of the hydraulic system and even on the entire system. It has been proved based on statistical data that hydraulic pump has a higher fault probability over other mechanical systems, thus, it is specifically necessary to investigate and conduct fault detection techniques for hydraulic pump. In this article, considering the complexity of hydraulic system and its severe working conditions, the datadriven fault detection method is suggested and applies to its online fault detection.
Generally, datadriven based fault detection consists of the following aspects: data measurement, data processing, data comparison, and data assessment [1]. Usually, the vibration signal of hydraulic pump is used for fault detection in practice, and artificial neural network (ANN) models have also been widely applied to intelligent fault diagnosis owing to their intrinsic parallel, adaptability, and robustness [2, 3].
Current datadriven based fault detection methods for hydraulic pump pay more attentions to not only linear characteristics but also nonlinear ones. In addition, owing to the universal presence of chaotic phenomena and the intrinsic characteristics and complex operation conditions of hydraulic system, strong nonlinearity and chaotic features can be clearly found from the vibration signals of hydraulic pump. Therefore, the research works on chaosbased fault detection for hydraulic pump should have a high engineering application value. Currently, chaotic correlation dimension has been applied well for condition monitoring and fault diagnosis of hydraulic pump. In addition, some research works based on Duffing oscillator and Lyapunov exponent have been employed to qualitatively or quantitatively solve the incipient fault recognition for hydraulic pump, with good diagnosis performance. However, the method based on neural network in conjunction with chaos theory has rarely appeared, especially for the fault detection of hydraulic pump [4–7].
Among several types of neural networks, radial basis function (RBF) network has relatively high convergence speed, and can approximate to any nonlinear functions. It has been proved that RBF network has a very high performance, in terms of nonlinear time series prediction, fault diagnosis in industrial systems, sensor and flight control systems, etc. [8–14].
A CPRBF network for fault detection of hydraulic pump is presented in this article. This CPRBF network was first trained using the dataset from the normal state without fault of hydraulic pump, and then a residual error generator was designed to detect several types of failures of hydraulic pump based on the trained CPRBF network with onestep prediction of chaotic time series. The proposed model, based on Camastra and Colla's approach [15] and Yang et al.' method [16], is able to reduce the effect of cumulative error and improve the prediction accuracy of RBF.
This article is divided into three sections as follows: Section "Phasespace reconstruction of chaotic time series" describes the chaotic theory on phase space reconstruction employed to obtain the estimation of correlation dimension. Section "Model of chaotic time series prediction and fault detection" proposes a new CPRBF network for chaotic time series prediction, and a residual error generator based on CPRBF network was also designed to detect fault. Then, Section "Case studies" gives several case studies, including simulation results of onestep iterative prediction and experimental results of fault detection for hydraulic pump.
Phasespace reconstruction of chaotic time series
where D is the dimension of system attractor. In order to obtain a correct system embedding dimension, starting from the time series, it is necessary to estimate the attractor dimension D.
The algorithm plots a cluster of lnC_{ m }(r)ln(r) curves through increasing m until the slope of the curve's linear part is almost constant. Then, the correlation dimension estimation D can be attained using least square regression.
Model of chaotic time series prediction and fault detection
In practice, it is difficult to get the exact estimation value of the minimum embedding dimension through GP algorithm. Furthermore, a single RBF network uses the estimation value of minimum embedding dimension as the number of its input, usually resulting in an inaccurate output due to the inaccurate estimation of embedding dimension from human factor. Therefore, a PRBF network consisting of multiple RBF subnets is proposed to increase the system performance with decreased error.
Structure of CPRBF
The CPRBF consists of n RBF subnets, which are denoted as subRBF_{ i }(i = 1,2,...,n), respectively. Each subRBF subnet realizes onestep prediction independently at t + 1. After the training of subRBF by historical dataset, onestep predicted value ${\widehat{x}}_{i}\left(t+1\right)$ can be obtained. The final predicted value$\widehat{x}\left(t+1\right)$ of PRBF can be achieved through proper weighted combination of ${\widehat{x}}_{i}\left(t+1\right)$.
Input nodes of subnet
Estimation value of the minimum embedding dimension is regarded as the number of input nodes in the central subnet, and each of other subnets uses different numbers (calculated based on m) as its input size.
In this article, each subnet RBF_{ i }uses the default parameters: the number of hidden layer is one, and the number of hidden nodes is equal to the number of input vectors.
Calculation of weighted factors
where N is the number of samples.
Residual error generator
Evaluation of residual error
Residual error evaluation is an important step of fault detection. In this article, threshold selector is adopted to evaluate the residual error. The concept of threshold selector is firstly introduced systematically in [20] to solve the residual error evaluation problem of LTI systems with model uncertainty. The diagnostic decision is obtained based on the following rule:
r_{eval} > J_{ th } → fault state detected
r_{eval} ≤ J_{ th } → normal state
where r_{eval} is a function related to residual error signal and employed to measure its deviation value, J_{th} is the threshold.
The corresponding standard of threshold value can also be determined based on diagnostic experiences in conjunction with different working conditions.
Process of fault detection
The detailed process is described as below:

Step 1. Normalize the original time series from diagnosed system

Step 2. Determine the number of input nodes of each subnet in CPRBF according to GP algorithm and Takens' theory

Step 3. Determine weighted factor ω based on the onestep prediction result of each subnet

Step 4. Calculate the final onestep prediction output of CPRBF

Step 5. Construct a residual error generator, and calculate the residual error according to the predicted output and the corresponding system output

Step 6. Choose a residual error evaluation function with a threshold standard

Step 7. Fault can be detected based on the evaluation function, with a fault alarm, once the residual error exceeds the threshold value.
Case studies
Verification results of onestep iterative prediction
Considering the lack of practicability from a common onestep prediction method, onestep iterative prediction should be adopted to verify the prediction performance instead. In general, each predicted result at Step 4 is consecutively used as the next input data to achieve onestep iterative prediction. The future trend of actual case (Lorenz's attractor, hydraulic pump) can be obtained gradually with the repetition of Steps 3 and 4, and the loop times depends on the length of actual expected data.
Simulation of Lorenz's attractor
where σ = 16, r = 45.95, b = 4. 1,000 points of Xcomponent Lorenz time series data were first normalized and used for the following prediction.

Onestep iterative prediction based on CPRBF network has good prediction performance

Comparing with RBF network, CPRBF network has better performance on iterative prediction, in terms of convergence and stability.
Experimental result using real data of hydraulic pump
Lyapunovs of hydraulic pump's sample data
Data  λ_{max} 

Data1  0.0508 
Data2  0.0744 
Data3  0.0435 
Comparing with RBF network, PRBF model has higher prediction accuracy, without the effect of error accumulation.
Experimental results of fault detection for hydraulic pump
Construction of detection model using CPRBF
Residual error signals of hydraulic pump based on CPRBF network
Wear fault of valve plate
Dry friction is probably caused by fatigue crack, surface wear, or cavitation erosion, etc. In case of this failure, with the increasing of moment coefficient between rotor and valve plate, contact stress grows and oil film becomes thinner. Further, as a repetitive impact of the contact stress, the surface of valve plate is fatigued and spalls. As a result, dry friction appears, with an increment of motion gap of hydraulic pump and a decrement of volumetric efficiency. Meanwhile, the dry friction inevitably generates additional vibration signals in the valve plate's shell near the high pressure chamber.
Wear fault between swash plate and slipper
Dry friction, caused by oil impurities or small holes on plunger ball, etc., usually results in wear or burnout of the faying surface between swash plate and slipper, which probably causes the falling of slipper, and affects the performance of hydraulic pump.
Fault detection
Threshold value is a key point in fault decisionmaking, due to uncertainties in practical and external disturbances. The rate of failtoreport increases if the threshold is too large, vice versa, the rate of false alarm would increase. Appropriate threshold should be selected according to the analysis, with the support of residual error evaluation function proposed, on hydraulic pump's normal and faulty data.
Variance values of residual error series
State  Residual error signal  

Normal (e006)  Fault (e004)  
Data  Normal 1  Normal 2  Fault 1  Fault 2  Fault 3  Fault 4  Fault 5  Fault 6 
Variance  3.0529  4.7796  1.3400  1.2244  1.0634  1.0831  2.6596  2.5383 
Mean of variance  3.91625  1.651467 
It can be seen obviously from Table 2 that, two magnitude levels of residual error's variance values between normal and fault states are clearly distinct. According to experience, the threshold can be determined with a standard of 10 times higher than the mean of variances under normal states. Here, J_{th} = 3.196e005. It should be also noticed that, the threshold standard must be readjusted according to different working conditions.
The variance values of the above two cases are 3.7781e004 and 1.7305e004, respectively. These values are greater than J_{th}, thus, the fault can be detected based on the variance of residual error signal.
Conclusions
It is shown from the simulation results that, CPRBF network model, in conjunction with phase space reconstruction, show better capabilities and reliability in predicting chaotic time series, as well as a high performance of convergence ability and prediction precision on shortterm prediction of chaotic time series.
The experimental results show that, CPRBF model has high ability in approximation to the output and state of a normal system, which is useful for fault detection. The CPRBF network can memorize various nonlinear states or interferences of a system with normal states, therefore, the actual system output will be different with the predicted output of CPRBF network once any anomaly occurs, and the system can be regarded as faulty state if the residual error exceeds the threshold. Thus, CPRBF network based method is effective to realtime fault detection. However, it is also shown from the experiments that different types of faults might represent the same fault form, accordingly, the proposed method is not suitable for performing fault location but for conducting condition monitoring. Further work will focus on how to isolate any type of fault and identify its fault classification.
This article mainly aims to discuss the feasibility and possibility of practical fault detection for hydraulic pump using neural network in conjunction with chaos theory. A commonly used neural network in the past and now, namely, RBF network was employed for fault detection for hydraulic pump in conjunction with chaos theory. Certainly, methods using chaos theory combined with other popular ANNs should be also our emphasis in the following works. As known, support vector machine (SVM) has been widely applied in many fields. Compared with other ANNs, SVM overcomes many defects, such as overfitting, local convergence. In addition, SVM has advantages over other ANNs, in terms of robustness and prevention of curse of dimensionality, etc. Thus, our further work will focus on SVM in conjunction with chaos theory, especially for those modified SVM.
Declarations
Acknowledgements
The research is supported by the National Natural Science Foundation of China (Grant Nos. 61074083, 50705005), as well as the Technology Foundation Program of National Defense (Grant No. Z132010B004). The authors are also very grateful to the reviewers and the editor for their valuable suggestions.
Authors’ Affiliations
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