Mathematical Problems in Engineering

Volume 2018, Article ID 2126049, 5 pages

https://doi.org/10.1155/2018/2126049

## Complexity-Entropy Causality Plane Based on Return Intervals: A Useful Approach to Quantify the Aeroengine Gas Path Parameters

College of Science, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to Keqiang Dong; nc.ude.utjb@09912160

Received 2 January 2018; Accepted 20 March 2018; Published 24 April 2018

Academic Editor: Oluwole D. Makinde

Copyright © 2018 Keqiang Dong and Linan Long. 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

The complexity-entropy causality plane, as a powerful tool for discriminating Gaussian from non-Gaussian process, has been recently introduced to describe the complexity among time series. We propose to use this method to distinguish the stage of climb-cruise-decline of aeroengine. Our empirical results demonstrate that this statistical physics approach is useful. Further, the return intervals based complexity-entropy causality plane is introduced to describe the complexity of aeroengine fuel flow time series. The results can infer that the cruise process has lowest complexity and the decline process has highest complexity.

#### 1. Introduction

The understanding and analysis of aeroengine time series, especially the evolution of gas path system sequences, has been attracting the attention of mathematicians and physicists for many years. Doel and Urban modeled the gas path system by linear approaches [1, 2], such as weighted-least-squares [3], filtering approaches [4, 5], where the accuracy and reliability were limited. For improving calculation accuracy and reliability, nonlinear techniques, such as adaptive modeling [6, 7], neural networks [8–11], and genetic algorithms [12–15], were introduced to investigate the gas path system. The existence of autocorrelation between gas path system variables reveals the aeroengine efficiency because past observations can help to predict future variables. This question motivates the research on the subject, especially by aeroengine managers and analysts, trying to save aeroengine maintenance costs. For bearing fault detection, Liu has applied the detrended fluctuation analysis to analyze gas path system correlation [16]. Dong et al. have found that the exhaust gas temperature (EGT), the low-spool rotor speed (N1), the high-spool rotor speed (N2), and the fuel flow (FF) have greater correlation and cross-correlation than other observations, suggesting more predictability [17]. It was also shown that the EGT, N1, N2, and FF play an important role in understanding gas path system. Thus, the analysis of EGT, N1, N2, and FF seems to be more efficient than other observations.

Nowadays, it is clear that the design of aeroengine gas path system is a typical complex system, which involves many subcomponents. The computing ranges of aeroengine were chosen from the process of take-off-climb-cruise-decline-approach-land of aeroengine [18, 19]. To distinguish the process of take-off-climb-cruise-decline-approach-land of aeroengine depends on the airplane’s altitude in previous analysis. In this paper, we propose to test the gas path system of aeroengine time series by employing a recently introduced statistical tool: the complexity-entropy causality plane. It is shown that this plane allows distinguishing cruise and decline process.

To make a distinction between climb, cruise, and decline process, the return intervals of aeroengine time series are proposed. The central quantities here are the time interval between successive events above (or below) some threshold . By studying the statistics of the return intervals for gas path system time series, one aims to find out the laws distinguishing the climb, cruise, and decline process throughout the flight.

The organization of this paper is as follows. In the next section, we simply present the aeroengine gas path performance parameters employed in this paper. We show the complexity-entropy causality plane and empirical results in Section. In Section, we introduce a technique named return intervals based complexity-entropy causality plane enabling us to estimate the complexity characteristic for aeroengine gas path system. In particular, the ability to identify the complexity in gas path system is demonstrated. Finally, we draw some conclusions in Section.

#### 2. The Dataset

In this paper, the complexity-entropy causality plane will be used to examine flight course of aeroengine. Here we make a brief description for gas path performance parameters, offered by Aircraft Maintenance and Engineering Corporation. In civil aviation flight management system, flight data are acquired from on-board flight data recorders, which are part of the Aircraft Condition Monitoring System (ACMS) such as Smart ACMS Recorder (SAR) and Quick Access Recorder (QAR) [20, 21]. The QAR data is more comprehensive where data includes an extensive list of flight parameters recorded at specific sampling intervals which are set by the manufacturer. Therefore, the QAR data is applied in this paper.

Previously, researches demonstrate that the parameters including EGT, N1, N2, and FF play an important role in understanding aeroengine system [17, 22]. For this reason, the parameters EGT, N1, N2, and FF are usually selected to amplify the study of aeroengine gas path system. Here, the parameter FF is selected as an example in this paper.

#### 3. Complexity-Entropy Causality Plane and Empirical Results

##### 3.1. Complexity-Entropy Causality Plane Method

For measuring the information content of aeroengine system, a typical method is to evaluate probability distribution function, describing the distribution of some measurable or observable property. Therefore, the Shannon entropy, which can be of great help when analyzing aeroengine system data since it captures the uncertainty and disorder of the time series without imposing any limitations on the theoretical probability distribution [23–25], is used as a first natural approach. For a time series with probability distribution , the Shannon entropy is given by [26–28]

The Shannon entropy , if the time series is certain which of the possible outcomes whose probabilities are given by will actually take place. The Shannon entropy should be maximal, if all the outcomes are equally likely (uncertainty is highest when all possible events are equiprobable).

For differentiating different degrees of periodicity and chaos, nevertheless, Lamberti et al. [29] proposed a statistical complexity measure (SCM) method. This method offers significant additional information regarding the peculiarities of the underlying probability distribution, not detected by the entropy. It is defined as follows: where is the normalized Shannon entropy, with , , and .

with and is defined in terms of the extensive Jensen-Shannon divergence, and is a normalization constant, equal to the inverse of maximum possible value of .

The diagram of versus is the complexity-entropy causality plane defined in [23, 30]. Statistical complexity measure was recently shown to be necessary because it captures the property of organization [31]. SCM has been successfully used to research changes in system dynamics originated by modifications of some characteristic parameters [32–34].

##### 3.2. Complexity-Entropy Causality Plane Results

In this section, we analyze the gas path system fuel flow (FF) parameters for 10 aeroengines. The data of the FF variables for climb-cruise-decline process are presented in Figure 1. We employ gas path parameters over climb-cruise-decline process, where the cruise process has lowest complexity (see Figure 1). In order to make comparisons, all aeroengines gas path parameters are studied for the same time length.