Complexity

Volume 2019, Article ID 1576817, 18 pages

https://doi.org/10.1155/2019/1576817

## Multistep Degradation Tendency Prediction for Aircraft Engines Based on CEEMDAN Permutation Entropy and Improved Grey–Markov Model

^{1}Faculty of Mechanical and Material Engineering, Huaiyin Institute of Technology, Huai’an 223003, China^{2}School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, China^{3}Nondestructive Detection and Monitoring Technology for High Speed Transportation Facilities, Key Laboratory of Ministry of Industry and Information Technology, Nanjing 210016, China

Correspondence should be addressed to Wei Jiang; moc.361@tsuh_wj

Received 16 May 2019; Revised 25 September 2019; Accepted 11 October 2019; Published 31 October 2019

Academic Editor: Yong Xu

Copyright © 2019 Wei Jiang 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

As an essential component and core power source of aircraft, the operational stability of aeroengine has important impact on system safety and reliability. Accurate degradation tendency prediction on an engine can not only improve its operational stability but also significantly reduce the maintenance costs. In this paper, a novel forecasting method that combines CEEMDAN permutation entropy and improved Grey–Markov model is proposed to perform multistep degradation tendency prediction of aircraft engines. In order to accurately quantify the degradation level of engines, a new integrated degradation index (IDI) is innovatively designed by multidimensional sensory data. And then, because of high speed and excellent performance, CEEMDAN algorithm is specifically employed to decompose the generated IDI series to eliminate the potential influence of stochastic fluctuations. Aiming at the complexity of intrinsic mode functions (IMFs) generated by CEEMDAN, an IMFs reconstruction strategy based on permutation entropy is developed to better characterize the degradation states. Finally, on the basis of above achievements and for higher forecasting efficiency and accuracy, an improved Grey–Markov model combined with the moving window algorithm, which is unique, is constructed to realize multistep degradation trend prediction of engines. The proposed method is applied to the degradation tendency prediction of aircraft engines. The experimental results validate the effectiveness and superiority of the proposed method, and it is more suitable for engineering applications in comparison with other methods.

#### 1. Introduction

With the gradual improvement of mechanical system’s integration and complexity, reasonable and complete health state monitoring is of great significance for ensuring stable and reliable operation of equipment [1–3]. As an essential component and core power source of aircraft, the operational stability of aeroengine has important impact on system safety and maintenance costs [4–6]. In recent years, with the proposal of condition-based maintenance (CBM), the fault response mode has converted from passive treatment to active prevention [7, 8]. CBM contributes to identify the operational status of equipment and avoid unnecessary downtime maintenance. Because of these advantages, it gradually becomes one of the most commonly used maintenance pattern and attracts more and more focuses of researchers [9, 10]. More specifically, degradation tendency prediction plays an important role in the implementation of CBM, which is helpful to discover abnormal operation states before fault occurs and effectively decrease the failure rate and maintenance costs [11, 12].

Generally, the implementation of degradation tendency prediction can be mainly divided into two stages, i.e., degradation indicator construction and development trend forecasting. Throughout the entire process of prediction, an appropriate indicator needs to be constructed to quantify the degradation levels of an aircraft engine, and it can be regarded as the foundation of the subsequent degradation trend forecasting. With the accumulation of running time, large amounts of sensory data from different positions are collected for analysis [13, 14]. On this basis, how to build a degradation indicator by adequately utilizing these data is the primary problem to be solved in the task of tendency prediction. For this reason, some researches focusing on constructing a suitable index that can effectively indicate the extent of deterioration have been carried out. For instance, Kral first pointed out that the status of used oil can be used to analyze the operation conditions of the vehicle cooling system [15]. Volponi utilized the fuel flow rate to evaluate the health state of gas turbine [16]. Gebraeel analyzed the states of rolling element bearing by the collected vibration signals [17]. However, these indicators are mainly based on single sensory signal, which means that some important signals containing abundant degradation information would be ignored. Besides, because of the growing diversity of collected signals, it is difficult to select a representative signal from large amounts of sensor signals that can accurately reflect the degradation state of equipment. Thus, the construction of an excellent degradation index fully utilizing different sensory data is still a difficult point in the research of degradation tendency prediction.

Degradation tendency prediction aims at obtaining the evolution of degradation in the future and supplying adequate data basis for decision-making. In general, the current models for trend prediction can be divided into three types, including knowledge-based models, physics-based models, and data-driven models. In particular, the practical application of former two models would face more restrictions due to the difficulties in obtaining relevant knowledge and establishing a suitable physical model. Data-driven models, which fully utilize the acquired monitoring data, can effectively achieve the purpose of degradation trend prediction without the help of domain knowledge and physical rules [18, 19]. Ma used a stacked sparse autoencoder with multilayer self-learning to forecast the remaining useful life of engine unit [4]. Fu adopted an improved least squares support vector machine model to predict the state development trend of hydroelectric generating unit [20]. Grey–Markov model, one of data-driven models, is widely applied in the prediction problem of systems with uncertain structure or characteristics due to its simple principle and excellent performance. Zhou presented the application of a Grey–Markov model with incidence analysis in the degradation trend forecasting for energy conversion equipment [21]. The results confirmed that the proposed method realized the satisfied performance. In Reference [22], an improved Grey–Markov model based on wavelet transform was developed to achieve accurate prediction of China’s energy supply and demand. Until now, the advantages of Grey–Markov over other prediction models, such as convenient parameters training, low computing time, and high forecasting accuracy, have been validated by massive experimental researches [23, 24]. Based on this, the Grey–Markov model is conducive to obtain optimal solutions. However, there are still two inherent drawbacks in this prediction model. Firstly, the Grey–Markov model is established based on the complete training samples, which would lead to lower accuracy with the increasing of prediction time and is unreasonable when there are exponential and chaotic data in training sets [25]. In addition, the single-step prediction pattern of Grey–Markov may cause the decrease of computation efficiency. Secondly, due to stochastic volatility and inherent complexity of original sensory data, it is difficult that relying on just the single Grey–Markov model to accurately forecast the degradation trend. In order to obtain better predicted results, it is necessary to first analyze the characteristics of raw data. Thus, the multiscale decomposition algorithm is introduced into Grey–Markov to improve the prediction performance of the single model. Various decomposition methods, such as wavelet transform (WT) [26], empirical mode decomposition (EMD) [27], and ensemble empirical mode decomposition (EEMD) [28], are adopted to decompose the original data series to reduce the influence of irregular volatility on forecasting results. Compared with the approaches above, a new type of decomposition method, named “complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN),” has attracted a huge amount of attention due to its excellent performance and high efficiency [29–31]. Qu developed a wind speed forecasting method based on CEEMDAN and an improved backpropagation neural network (BPNN), and the experiment results indicated that CEEMDAN could efficiently solve the problem of data fluctuations [30]. Therefore, it is a valuable subject that a novel prediction method combining the merits of CEEMDAN and the Grey–Markov model should be constructed to further enhance the forecasting performance.

The main contributions of this work is the development of a multistep degradation tendency prediction method for aircraft engines based on CEEMDAN permutation entropy (PE) and the improved Grey–Markov model with moving window (IGMMW). And on the subject of detail, a new integrated degradation index (IDI) is constructed by multidimensional sensory data for the accurate quantification of engine degradation levels. Meanwhile, CEEMDAN algorithm is first used to decompose the IDI series to eliminate the effect of data fluctuations. Then, an intrinsic mode functions (IMFs) reconstruction strategy based on PE theory is innovatively designed to reduce the complexity of decomposed components. Finally, for the sake of higher forecasting efficiency and accuracy, a novel prediction model, namely, IGMMW, is developed to forecast the degradation trend of engine units. The general implementation of the proposed method can be divided into four steps, i.e., IDI series construction, series decomposition, IMFs reconstruction, and tendency prediction. Firstly, an IDI for the measurement of engine deterioration levels is built by using different sensor data and an appropriate data fusion method. Compared with the single signal adopted in [15–17], the generated IDI fully retains valuable degradation information contained in various signals and achieves the mapping from high-dimensional signal space to one-dimensional index space. Secondly, the CEEMDAN is utilized to adaptively decompose the generated IDI series to further eliminate the potential influence of stochastic fluctuations, which is a remarkable improvement in comparison with other algorithms adopted in [26–28] due to its excellent decomposition performance. Subsequently, a PE-based reconstruction strategy is designed to achieve the reduction of IMFs’ complexity, i.e., several IMFs to fewer reconstructed IMFs (RIMFs). With the idea of aggregation based on PE values, there will be fewer decomposition components and the forecasting accuracy and efficiency will also be improved. Finally, based on the obtained RIMFs, an IGMMW prediction model is developed to efficiently forecast the future degradation trend of engines. Because of the combination of moving window method, the problem of circular update for sequences being modeled, which occurs in [21–24], can be solved well. Besides, the adaptive parameter in moving window, i.e., the step size, is helpful to the implementation of multistep prediction to further improve the computational efficiency. The proposed method is used for the degradation trend prediction of aircraft engines, in which the sensory signals are measured from different parts of engine units. The experimental results confirm the effectiveness and superiority of the proposed method, and it is more suitable for engineering applications in comparison with other methods.

The rest of this paper is organized as follows. In Section 2, the essential background knowledge about CEEMDAN, Grey theory, and Markov chain modeling mechanism is reviewed. The proposed method is introduced in Section 3. In Section 4, the proposed method is used to predict the degradation trend of aircraft engines and the experimental results are analyzed and discussed in detail. Finally, general conclusions are given in Section 5.

#### 2. Preliminaries

##### 2.1. Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

In order to deal with the analysis of nonstationary signal, an empirical mode decomposition (EMD) algorithm was proposed by Huang in 1998 [32]. The method is an adaptive signal time-frequency-domain analysis technique and decomposes the raw signals into a series of IMFs. Specifically, each IMF component reflects the different characteristics of raw signal at different time scales, which usually satisfies two conditions below: (a) in the entire time series, the number of extreme points is always equal to that of zero-crossings or the difference of number being not more than one; (b) the average value of envelope at any time point, defined by the local maxima and the local minimum, should be zero [30]. With the definition of IMFs mentioned above and relevant hypothesis, the original signal can be decomposed into several IMFs and one residue component by EMD:where denotes the raw signal, is the *i*-th IMF component obtained by EMD method, is the number of IMFs, and is the final residue function, which represents the mean tendency of data sequence.

Although EMD method has significant advantages in analyzing the nonstationary signal, there are some inherent limitations that make great influence on the performance of EMD, such as the mode mixing problem and the end-point effect [31]. In order to eliminate these problems in EMD, a noise-assisted signal analysis approach, named “ensemble empirical mode decomposition (EEMD),” was developed by Wu and Huang in 2009 [33]. However, the EEMD algorithm cannot eliminate the effect of Gaussian white noise on the reconstructed signal, and the high computational costs due to the added noise make a great restriction to the application of the decomposition method. Because of the above defects in EEMD, the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) algorithm was designed to improve the performance of EEMD [34]. The method effectively eliminates the mode mixing phenomenon in the IMFs, and the reconstruction error is always zero. Meanwhile, compared with EEMD, the computation time of CEEMDAN can be decreased remarkably. Suppose is the operator of *j*-th IMF component that is decomposed by EMD, and let be Gaussian white noise with zero mean and homogeneity of variance. The general procedures of CEEMDAN are given as follows:(1)Decompose each using EMD to obtain the first IMF, in which is a noise coefficient. Meanwhile, define the first mode component decomposed by CEEMDAN as(2)Calculate the first residue :(3)Decompose residue to calculate the second mode denoted by :(4)Repeat steps (2) and (3) until all of the IMFs are analyzed. The final residue can be defined as:where *m* is the total number of IMFs decomposed by CEEMDAN. Therefore, based on the steps mentioned above, the original signal can be expressed as

Based on equation (6), the signal can be decomposed into *m* IMFs and a residue, which provides an exact way of reconstruction for raw data. It is worth noting that the obtained IMFs reflect the characteristics of original signal at different timescales, and the residue is smoother and contributes to the reduction of prediction error.

##### 2.2. Grey System Modeling and Prediction

Grey theory is usually used to deal with the prediction problems which have insufficient and uncertain information [35]. On the basis of Grey theory, the Grey prediction model has been developed and applied in different fields successfully [36, 37]. In general, the Grey model can be presented as , where is the order of the differential equation and is the number of variables. Because of superior computation efficiency, is the most widely used in practical applications [37]. The general steps of are illustrated as follows:(1)Suppose the raw series can be written aswhere is the system output at time .(2)Based on the raw series and the accumulated generating operator (AGO), a new series can be generated to show the tendency of exponential growth [35]:where(3)The whitening differential equation of can be built as follows [35]:where is the so-called development coefficient and is known as the endogenous control coefficient, and they are two constants determined by the original series.

With the generated series and the least squares method, the coefficients of equation (10) can be estimated aswhere(4)Based on equation (10), the solution of can be obtained as(5)Finally, with the subtraction operation, the predicted value of raw data point can be acquired as follows:

##### 2.3. Markov Chain Modeling Mechanism

Without the consideration of random fluctuations in the original series, the forecasting accuracy of is low when the sequence being modeled fluctuates sharply [25]. In order to reveal the inherent laws of data fluctuation and improve the prediction precision, Markov chain, a particular stochastic process theory, is used to modify the prognostic residue series of and effectively improve the prediction accuracy [38, 39]. The detailed modeling mechanism of Markov chain is described below.

For the obtained series by , the relative error between the predicted value and the original value can be calculated:

Then, the range of the relative error can be split into intervals with equal length, which are called as states. Each state is a section of the range , i.e.,where

In the theory of Markov chain modeling, the transition probability from state to by steps can be deduced aswhere is the transition times that occurred from state to state by steps and is the number of data whose relative errors belong to state .

With these -step transition probabilities, the state transition probability matrix can be constructed to effectively restrain the effects of random fluctuations:

Subject to

The probability matrix reveals the transition laws between different states, which is the modeling foundation of Markov chain. Through the more detailed procedures mentioned in [39], the future state transition on the basis of the current state, i.e., state denoted by , can be estimated. The lower boundary and upper boundary of this state are denoted as and , respectively. Finally, the corresponding predicted value by , denoted by , can be modified by Markov chain according to the following formula:

#### 3. The Proposed Degradation Tendency Prediction Method

In this paper, a novel multistep method based on CEEMDAN permutation entropy and IGMMW model, systematically blending the signal decomposition technique and intelligent prediction technology, is proposed for the degradation tendency prediction of aircraft engines. This section includes four parts: the construction of integrated degradation index, the reconstruction of IMFs using PE theory, improved Grey–Markov model with moving window, and the general procedure of the proposed method.

##### 3.1. Construction of Integrated Degradation Index

For the sake of quantifying the degradation degree of engines effectively, a proper degradation index should be constructed. With the help of a large number of available sensory data and the idea of liner transformation, a new integrated degradation index (IDI) is innovatively proposed in this paper, which achieves the mapping from multidimensional data domain to one-dimensional index domain.

Suppose of matrix and of matrix are two groups of multidimensional sensor dataset, which represent the faulty and healthy states of engines, respectively. and are the sizes of datasets under faulty and healthy conditions, and is the dimension of sensory dataset. With and , a matrix can be designed to build the relationship of mapping between multidimensional sensory data and one-dimensional IDI aswhere , , represents a zero vector, and represents a unit vector. With the constructed matrix and the historical dataset collected from the sensor, the IDI denoted by can be obtained:

Note that the value of IDI changes between 0 and 1, and “0” represents healthy state and “1” represents faulty state. The calculation of IDI can be essentially regarded as a process of multidimensional data fusion, and it provides an effective way to accurately describe the degradation levels of aircraft engines.

##### 3.2. Reconstruction of IMFs Using PE Theory

The sensory signals collected from the online monitoring system are susceptible to operation environment and background noise. For this reason, the corresponding decomposition results with CEEMDAN may consist of many IMFs, which will enhance the complexity of model training and reduce the forecasting accuracy. Thus, on the premise of retaining all effective components, an IMFs reconstruction strategy using PE theory is first developed for the decrease of the number of IMFs. The reconstruction process is described in detail below.

For the IMF components after decomposition ( represents the *i*-th IMF component and is the number of IMFs), the corresponding phase space reconstruction vector of can be expressed based on the the Takens–Maine theorem aswhere is the time delay and is the embedded dimension. On this basis, the elements of are rearranged by number of real values in ascending order, which meets

According to the above equation, map into a group of symbols:where and , is one of the arrangements. Calculate the probabilities of these symbols denoted as , and then, the PE value of IMF component can be acquired according to the following formula:where is the regularization coefficient and is bounded in . In essence, PE has significant advantages in measuring the randomness of series. Thus, with the idea of similarity-based combination, the IMFs after reconstruction, expressed as , can be generated according to the designed criterion as follows: With the proposed IMFs reconstruction method, the original IMFs decomposed by CEEMDAN algorithm can be classified into several groups adaptively, and the PE values of IMFs in each group are limited to one specific interval. For this reason, the constructed RIMFs not only preserve all of the components’ information but also obviously reduce the number of IMFs. On this foundation, the RIMFs can be used as the inputs of the prediction model to effectively improve the forecasting efficiency and accuracy.

##### 3.3. Improved Grey–Markov Model with Moving Window

In order to eliminate the influence of exponential and chaotic data on modeling and enhance the prediction performance of Grey–Markov model, an improved Grey–Markov model with moving window (IGMMW) is innovatively proposed to perform the multistep degradation tendency prediction of aircraft engines. With the moving window algorithm, the prediction model can be circularly reconstructed based on adjacent datasets, and thus, the forecasting accuracy can be further improved. The grey modeling mechanism based on moving window is shown in Figure 1, where is the length of original modeling series, is the number of data points to be predicted, is the step size of moving window, and and represent the coefficients depicted in equation (10).