Mathematical Problems in Engineering

Volume 2015, Article ID 532684, 11 pages

http://dx.doi.org/10.1155/2015/532684

## Spatial Information Entropy and Its Application in the Degradation State Identification of Hydraulic Pump

Shijiazhuang Mechanical Engineering College, No. 97, Heping West Road, Shijiazhuang 050003, China

Received 18 April 2015; Accepted 25 May 2015

Academic Editor: Michael Small

Copyright © 2015 Yu-kui Wang 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

The degradation state identification is a key step of the condition based maintenance of hydraulic pump. In this paper, spatial information entropy (SIE) as a novel degradation feature of pump is proposed based on the study of permutation entropy (PE) algorithm. The fundamental principle of SIE is introduced and contrasted with PE. Different parameters used in the calculation of SIE are discussed and meaningful conclusion is gained. The results of simulation analysis not only checked the rationality of SIE but also demonstrated the availability and superiority of adopting SIE as the degradation feature. Based on simulation analysis, SIE and PE are united and used as degradation feature vector of pump. FCM algorithm is employed to diagnose the degradation state of pump. The analysis results of practical signal testified the rationality and availability of the proposed method.

#### 1. Introduction

As the heart of hydraulic system, the condition of hydraulic pump is important to the whole system. For the heavy load and complex working condition, pump is also the unit that usually fails in function [1]. As the failure will result in high industrial cost or even disaster, considerable attention has been paid to the condition monitoring and fault diagnosis of pump for a long time [2]. But most of the researches concentrate on the fault recognition and fault location [1, 3]. Researches on the degradation state identification of pump are rarely reported [4]. The method based on vibration signal analysis is extensively used for pump because of the advantages of fitness and validity [3].

With the development of maintenance theory and relative technologies, condition based maintenance is getting more and more attention [5–7]. A lot of work has been done about degradation state identification (DSI) as it is the fundament of condition based maintenance [8–11]. DSI has two important steps: one is extracting appropriate features which can reflect comprehensive degradation degree from the raw vibration signal; the other is building an effective intelligent model which is used to assess the state of equipment [12]. Proper feature extraction is the key step of DSI as it affects the precision of final state identification. Feature extraction is the process of converting original data into relevant information of the equipment. Traditional features contain three categories: time domain, frequency domain, and time-frequency domain. They are sensitive for fault identification and are widely used in the fault diagnosis of mechanical equipment [13]. However, they also have defects; for example, their stability is not good enough; as a result, they cannot indicate the fault degree of equipment [14]. As the vibration signal displays strong nonlinear feature when the machinery is broken-down [15], a lot of nonlinear methods are applied to the vibration signal processing with the development of nonlinear theory, such as fractal dimension, approximate entropy, and sample entropy [16, 17]. Permutation entropy (PE) was proposed by Bandt et al. [18, 19] as a complexity indicator of time series. For its advantages of clear concept and simple calculation, it is widely used in the mutation detection of electroencephalogram, heart interbeat signal, and geomagnetic storms as well as mechanical signal [20–23]. From the researches mentioned above, it can be concluded that the research of degradation feature extraction acquired great progress in recent years.

In this paper, the degradation state identification of pump which reflects the relationship between its fault degree and the degradation features is researched. And PE is used as the degradation feature of it. Based on the study of PE algorithm, we found that it only explored the sorting relationship of the elements in certain refactoring component on account of their value [18]. The spatial location information of each element in the original time series is not considered. In order to reveal the distribution information of refactoring component’s elements in the original time series, the spatial information entropy (SIE) is put forward. And it is used as another degradation feature of pump. The relationship between degradation feature and fault degrees is nonlinear; therefore, the nonlinear model should be founded in order to obtain better result [4]. Furthermore, the fuzzy recognition method should be selected because the degradation process of pump is gradual and the description of its fault degree by degradation features is fuzzy and uncertain. The methods such as rough set theory, D-S evidence theory, and fuzzy set theory are widely used in the fuzzy system analysis and fuzzy information processing; their validity and superiority are demonstrated. However, they also have the shortage of requiring large amount prior knowledge and experiences [24]. Fuzzy C-means (FCM) clustering is a traditional but excellent algorithm for pattern recognition, with some advantages such as being simple, intuitive, and explicable [8]. In addition, FCM clustering transforms clustering into the nonlinear programming with constraints; therefore, just small amount training sample is needed in the clustering process and high precision ratio also can be obtained. It is widely used in fault diagnosis, image processing, and also other fields. In this paper, FCM is used to identify the degradation states of pump.

The remainder of this paper is organized as follows. In Section 2, the basic theory of PE is simply introduced. The proposed SIE scheme is detailed at first in Section 3, and the selection of its parameters is analyzed, based on which both PE and SIE are used to analyze the degradation simulation signal of pump, and the excellent performance of SIE is checked. In Section 4, the FCM clustering algorithm and the degradation state identification strategy are introduced. Section 5 introduces the pump degradation state experiment at first, and then the collected vibration signals are employed to evaluate the proposed method. Our conclusions are drawn in Section 6.

#### 2. Permutation Entropy

Permutation entropy (PE) represents a new way to assess the complexity of nonlinear time series. PE has some advantages as compared to other entropy measures, since it is an ordinal measure. Indeed, PE decomposes the time series into a series of ordinal patterns describing the order relations between the present and a fixed number of equidistant past values at a given time. The mathematical theorem of permutation entropy was described in detail in [18, 19]. The PE of a time series can be calculated as follows [18].

(1) Reconstruct the time series and its phase space can be obtained as follows:where is embedded dimension and is delay time, , . Each row of the reconstructed matrix is a refactoring component.

(2) Extract the rank numbers of elements and name them as their labels.

(3) Arrange real values contained in refactoring component in increasing order.

(4) A symbol series can be obtained by using the labels instead of their real values.

(5) Extract the symbol series of all refactoring components by steps (1–4). Then, count the number of each existing symbol series and calculate their probability.

(6) If the probability are denoted by , , the PE of this time series can be defined as

As each refactoring component contains elements, the largest number of symbol series is . The maximum value of can be obtained as when all the symbol series have the same probability distribution . Therefore, the PE can be normalized with

The value of represents the randomness and complexity of the time series , and it also describes local order structure of the time series. The smallest possible value of is zero, which means that the time series is very regular [18]. The largest possible value of is 1, which can be realized when all symbol series have equal probability. When the pump has certain fault, the more serious the fault is, the more regular the components will exist in its vibration signal, which results in smaller randomness and lower complexity of the signal and also smaller PE of it. On the contrary, when the pump is normal, the randomness, complexity, and also PE of its signal reach their maximum. Therefore, PE can be used as degradation feature of pump to indicate its fault degree.

#### 3. Spatial Information Entropy

PE can sensitively reflect the variation of the randomness and complexity of time series, so it is usually used to indicate the system dynamics mutations [18]. As it adopts the sorting variation of elements in certain refactoring component to reflect the dynamics variation of the time series [19], the spatial location information of each element in the original time series is not considered. When the time series varies, the spatial location information of each element is sure to change; in order to reflect this change, the spatial information entropy is proposed; the feasibility and availability of using it to indicate the dynamics variation of time series are also analyzed in this section.

##### 3.1. Principle

SIE of time series can be calculated with the following steps.

(1) Find the maximum value and the minimum value of the time series, and divide the time series into regions, which are denoted by . Figure 1 shows the partition state of dividing a signal into 7 regions.