Complexity

Volume 2018, Article ID 9154682, 11 pages

https://doi.org/10.1155/2018/9154682

## Fault Diagnosis of Electromechanical Actuator Based on VMD Multifractal Detrended Fluctuation Analysis and PNN

^{1}School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China^{2}Science & Technology on Reliability & Environmental Engineering Laboratory, Beijing 100191, China

Correspondence should be addressed to Jian Ma; nc.ude.aaub@77990

Received 10 February 2018; Revised 10 June 2018; Accepted 13 June 2018; Published 1 August 2018

Academic Editor: Minvydas Ragulskis

Copyright © 2018 Hongmei Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Electromechanical actuators (EMAs) are more and more widely used as actuation devices in flight control system of aircrafts and helicopters. The reliability of EMAs is vital because it will cause serious accidents if the malfunction of EMAs occurs, so it is significant to detect and diagnose the fault of EMAs timely. However, EMAs often run under variable conditions in realistic environment, and the vibration signals of EMAs are nonlinear and nonstationary, which make it difficult to effectively achieve fault diagnosis. This paper proposed a fault diagnosis method of electromechanical actuators based on variational mode decomposition (VMD) multifractal detrended fluctuation analysis (MFDFA) and probabilistic neural network (PNN). First, the vibration signals were decomposed by VMD into a number of intrinsic mode functions (IMFs). Second, the multifractal features hidden in IMFs were extracted by using MFDFA, and the generalized Hurst exponents were selected as the feature vectors. Then, the principal component analysis (PCA) was introduced to realize dimension reduction of the extracted feature vectors. Finally, the probabilistic neural network (PNN) was utilized to classify the fault modes. The experimental results show that this method can effectively achieve the fault diagnosis of EMAs even under diffident working conditions. Simultaneously, the diagnosis performance of the proposed method in this paper has an advantage over that of EMD-MFDFA method for feature extraction.

#### 1. Introduction

Although most aircrafts and helicopters still adopt hydraulic actuation systems, electromechanical actuators have increasingly been applied as the key actuators for flight control systems of advanced aircrafts and helicopters in recent years. The main reason is that electromechanical actuator (EMA) has more superiorities in terms of reliability, economy, and other aspects than traditional hydraulic actuator. However, aircrafts and helicopters often perform mission under variable complex environments, and it will cause serious consequences when the faults of EMAs appear. Therefore, fault detection and diagnosis of EMAs in various working conditions play a vital role in the normal operation of aircrafts and helicopters. More and more researches have been done about the function of EMAs, but few are about fault. Consequently, it is very meaningful to carry out the research on fault diagnosis algorithms of EMAs under variable working conditions.

NASA Ames Research Center’s researchers conducted failure mode and effect analysis of EMAs through extensive literature investigation, and the main fault modes of EMAs were obtained [1]. The researchers built the flyable electromechanical actuator (FLEA) test-bed, so that the normal data and fault data of EMA can be obtained through a large number of experiments [2]. A method based on neural network was proposed to realize the diagnosis for critical failure modes of EMAs [3]. Narasimhan et al. implemented the degeneration trend prognostics of EMAs by using the Gaussian process regression algorithm [4]. A method based on WPD-STFT time-frequency entropy and PNN was presented by Jing et al., which achieved the accurate diagnosis of EMAs [5]. At present, there are relatively few researches on fault diagnosis methods of EMAs under variable working conditions at home and abroad.

The vibration signal of rotating machinery contains abundant information about the running state of the equipment. And extracting the fault feature which represents the fault information of the equipment is the most important step in fault diagnosis. However, vibration signal generally has the characteristics of nonlinear and nonstationary, and there are external disturbances such as noise, so that extracting features from vibration signal is the key problem for researchers. The commonly used methods for processing vibration signal to extract fault features include short-time Fourier transform (STFT), wavelet transform (WT), empirical mode decomposition (EMD), and local mean decomposition (LMD). STFT can depict signal in both time domain and frequency domain at the same time and can reflect the time-varying characteristics of the signal frequency spectrum. But the window function of STFT is fixed, so it is not suitable to analyze strong time-varying and nonstationary signal [6]. WT can realize the multiresolution analysis of signal, but its resolution in the frequency domain is not adjustable at the same scale, and it needs to preselect the basis function according to the characteristic of the signal [7]. EMD decomposes the signal into a finite number of single-component signals which are called intrinsic mode functions (IMFs). It has great potential for analyzing the nonlinear and nonstationary signal. However, EMD has a series of problems such as end effects, modal confusion, over-envelope and under-envelope, negative frequency, and lacking theoretical basis [8]. LMD is an adaptive time-frequency analysis method which is proposed on the basis of EMD. It can decompose the complex signal into several product functions (PFs). However, LMD also has the problem of end effects, modal confusion, and large amounts of calculation [9]. In addition, fault diagnosis methods based on various multidisciplinary algorithms have been studied in recent years. A rotating machinery fault diagnosis method combining bispectrum and image processing algorithm was proposed, and its validity was proved by experiments of hydraulic pump and centrifugal pump [10]. A method based on narrowband demodulation with frequency shift and spectrum edit was used to achieve the fault diagnosis of gears [11]. Variational mode decomposition (VMD) is a new signal processing method which has a different theoretical framework with EMD [12]. VMD transforms signal decomposition into nonrecursive and variational mode decomposition problem which has theoretical foundation. It shows better noise robustness and can reduce the sampling effect and modal confusion.

Different from time-frequency analysis, fractal analysis can be used to reveal the fractal features of the signal, while fractal features can characterize the different operating states of a complex system. Therefore, fractal features can be utilized as fault features for fault diagnosis. Multifractal analysis can extract fractal features of different local scales, and researchers have applied classical multifractal theory to feature extraction of fault diagnosis in recent years. A method based on wavelet analysis and multifractal spectrum was applied to extract the fault features of hydropower unit [13]. And the multifractal spectrum was combined with PSO-SVM to achieve the fault diagnosis of gearbox [14]. However, the traditional multifractal theory can be easily disturbed by the trend of signal fluctuation and cannot reveal the multifractal characteristics hidden in nonstationary signal accurately. Thus, Liu et al. proposed a method called multifractal detrended fluctuation analysis (MFDFA) combining multifractal (MF) with detrended fluctuation analysis (DFA), which can eliminate the influence of signal fluctuation and can further effectively extract the multifractal characteristics of nonstationary signal. MFDFA has been applied to the field of fault diagnosis for complex system. A method based on MFDFA and local characteristic-scale decomposition-Teager energy operator was proposed to realize the fault diagnosis of rolling bearing [15]. Tang et al. applied MFDFA into the fault diagnosis of nonlinear analog circuit [16].

A fault diagnosis method for EMA based on VMD-MFDFA and PNN is proposed in this paper. Firstly, the vibration signal of the accelerometer is collected. After preprocessing the vibration signal, a series of IMFs are obtained by using the VMD. Then, the multifractal features of IMFs are calculated by MFDFA, and the fault feature vectors are acquired by reducing the dimension with PCA. Finally, PNN model is trained to classify the fault modes.

#### 2. Feature Extraction Method Based on VMD and MFDFA

The vibration signal of the EMA has the characteristics of nonlinear, nonstationary, and strong time-varying. In this paper, the vibration signal is decomposed by VMD, and the feature vectors are extracted by MFDFA to characterize the operating state of the EMA.

##### 2.1. A Description of Variational Mode Decomposition (VMD)

The VMD algorithm can obtain the optimal solution of the constrained variational problem and determine different central frequencies and bandwidths through iteration. The intrinsic mode functions (IMFs) of different frequencies are obtained by nonrecursive decomposition [17]. The implementation of VMD is divided into two parts: the construction of variational problem and the solution of variational problem [18].

The first part is the construction of variational problem. This time-frequency analysis method assumes that the multicomponent signal consisted of intrinsic mode functions with limited bandwidth, and the central frequency of each intrinsic mode function corresponds to .

The analytic signal of each intrinsic mode function is obtained by Hilbert demodulation as the following formula:

A central frequency is estimated as for each analytic signal, and the frequency spectrum of each IMF is modulated to the fundamental frequency band:

The square norm of the above analytic signal gradient is calculated, and the bandwidth of each IMF is estimated. Then, the constrained variational problem is obtained as the following formula: where represents one of the intrinsic mode functions obtained by decomposition and represents the central frequency of each intrinsic mode function.

The second part is the solution of variational problem. In order to obtain the optimal solution of the variational model, Lagrange multiplication operator and quadratic penalty factor need to be introduced to change the constrained variational problem into nonconstrained variational problem. The transformed Lagrange expression is

The saddle point of formula (4) is obtained through iteratively updating , , and by using the alternate direction method of multipliers (ADMM).

The update method of is

The update method of is

The update method of is

The real part after the Fourier transform of combining formula (5) and formula (6) is the intrinsic mode functions .

The specific steps of VMD can be described as follows [19]: (1)Initialize , , , and .(2)Set and begin the circulation.(3)Update and according to formula (5) and formula (6).(4)Set , and repeat step (3) until .(5)Update according to formula (7).(6)Repeat step (3) to step (5), until iteration stop condition is reached.

In the process of decomposition by VMD, the central frequency and bandwidth of each IMF are constantly updated to realize the adaptive decomposition of signal.

##### 2.2. A Description of Multifractal Detrended Fluctuation Analysis (MFDFA)

Multifractal detrended fluctuation analysis can effectively eliminate the effect of signal fluctuation trend and can accurately extract the implied multifractal features of nonlinear signal [20].

The steps of MFDFA can be described as follows [21]: (1)For time series , construct cumulative deviation of the sequence to the mean:(2)The new sequence is divided into nonoverlapping subsequences with a fixed scale :

Then, the sequence is divided into segments by the same scale from the reverse direction of the sequence, and subsequences can be obtained. (3)Fit the polynomial trend of each subsequence by using the least square method, and calculate the variance as follows:where is the fitting polynomial of the subsequence. (4)Calculate the mean value of the -order fluctuation function:where different values of represent different degrees of fluctuation. And when , MFDFA degenerates to DFA. (5)Change the length of the subsequence and repeat steps (2) to (4).(6) is the function of the length of the subsequence and the fractal order and has the following power-law relation with the scale :where is the mean value of -order fluctuation function and is the generalized Hurst exponent.

If is a monofractal time series, is a constant, and if is a multifractal time series, is the function of order . The different order corresponds to the different generalized Hurst exponent.

The generalized Hurst exponent can describe the influence of the past time series on the present and the later time series, and the influences are different under different states of system.

Therefore, the generalized Hurst exponent can be used as the feature vector to describe the multifractal characteristics of the system and can characterize the different states of the system.

#### 3. Fault Classification Based on PNN

The theoretical basis of probabilistic neural network (PNN) is Bayesian minimum risk criterion. PNN directly considers the probability characteristics of the sample space and takes the typical samples of the sample space as the nodes of the hidden layer. There is no need for training anymore once PNN is determined, and it is only necessary to append samples according to actual problems [22]. PNN has the advantages of short training time and global optimization and has great performance for classification.

The network structure of PNN is shown in Figure 1, which consists of the input layer, the pattern layer, the summation layer, and the output layer [23].