Investigations on generalized Hjorth's parameters for machine performance degradation assessment
Introduction
The main tasks of machine performance degradation assessment (MPDA) are to detect the time of incipient fault initiation as early as possible and then continuously track machine degradation evolution by using health indicators [1]. The time of incipient fault initiation is also termed as incipient fault time or first prediction time (FPT), which provides prior knowledge for machine fault diagnostics and prognostics. It means that once the FPT is confirmed, timely fault diagnosis methods can be adopted to establish specific fault types. Meanwhile, degradation assessment and remaining useful life (RUL) prediction can be triggered to further evaluate machine severity states [2]. Thus, MPDA is vital to plan and schedule maintenance events at a right time and it plays an essential role in machine diagnostics and prognostics.
Health indicator construction based on statistical features is one of the commonly used methodologies for MPDA [3], [4]. For example, root-mean-square (RMS), kurtosis, and their variants are widely applied as health indicators for MPDA due to their simplicity and interpretability [5], [6], [7]. However, they have inherent limitations. RMS that characterizes energy changes of a signal can only be used to represent a monotonic degradation trajectory whilst it is insensitive to early faults. Kurtosis that reflects the impulsiveness of a signal can detect the time of FPT whilst it cannot describe a monotonic machine degradation trend. Li et al. [8] utilized the advantages of kurtosis and RMS to respectively complete the tasks of early fault monitoring and degradation assessment. Recently, some famous sparsity measures have been extracted as health indicators for MPDA due to their ability to characterize the impulsiveness and cyclo-stationarity of fault signals [9]. For example, Antoni et al. [10], [11] investigated the mathematical properties of spectral kurtosis and used spectral kurtosis for machine health monitoring. Subsequently, more sparsity measures such as spectral Gini index [12], spectral negative entropy [13], and spectral smoothness index [14] were extracted from time-domain or frequency-domain-based envelope signals as health indicators for MPDA. Numerous applications of these sparsity measures revealed that they are sensitive to the time of FPT due to their inherent ability to characterize sparse signals. Nevertheless, they depict strong fluctuations in machine degradation stages, which is inappropriate to degradation assessment and RUL prediction [15].
Health indicator construction based on the fusion of multiple sources and intelligent algorithms is an enabling strategy to integrate various degradation information [16]. Generally, feature extraction, feature selection, and feature fusion are standard procedures to design data-fusion-based health indicators [17]. Time-based, frequency-based, and time–frequency-based features are commonly extracted from multiple sensors [18], [19], [20]. Afterward, extracted features can be selected based on the criteria of monotonicity [21], [22], correlation [23], robustness [24], trendability [25], Fisher’s ratio measure [26] etc. Finally, selected features that are related to a degradation process are fused into health indicators based on various algorithms and techniques, such as self-organizing map (SOM) [27], Gaussian mixture models (GMM) [28], Long-Short Term Memory (LSTM) [29], [30], fuzzy inference system (ANFIS) model [31], DNN [32], and support vector data description (SVDD) [33]. However, these methodologies have weak interpretability in construction sources and the construction process of health indicators. Aiming at these issues, genetic programming (GP) has been introduced to construct health indicators due to its explicit expressions and flexible tree structures [34]. Liao et al. [35] utilized GP to evolve a monotonic health indicator based on the fusion of various fault features. Wang et al. [36] proposed an improved version of GP for health indicator construction. Besides health indicator construction [37], GP has been widely applied to machinery fault diagnosis [38], [39], RUL prediction [40], and feature design [41], [42]. However, most GP-based health indicators mainly focus on enhancing the monotonicity of health indicators while its ability to distinguish between normal and abnormal states is seldom studied.
Traditionally, Hjorth's parameters including Activity, Mobility, and Complexity have been commonly used for the analysis of electroencephalography (EEG) signals and electro-oculogram (EOG) signals for pattern recognition in the field of medical science and kinematics [43], [44]. To authors’ best knowledge, applications of Hjorth's parameters to MPDA are still limited and not fully explored. Caesarendra and Tjahjowidodo [45] considered using Hjorth's parameters to assess low-speed slew bearing degradation performance. Based on experimental studies, they concluded that Activity is the most suitable to track bearing degradation. Nevertheless, their study lacked theoretical investigations on Hjorth's parameters for MPDA. Grover et al. [46] combined Hjorth’s parameters with empirical mode decomposition for bearing fault diagnosis. In the newest work, Cocconcelli et al. [47] studied the characteristics of Hjorth's parameters for MPDA based on simulation studies and a new health indicator, namely detectivity was developed based on a linear combination of Activity, Mobility, and Complexity. However, the theoretical study of Hjorth’s parameters for MPDA still lacks and only a linear combination of Hjorth’s parameters was considered. In a nutshell, theoretical and experimental investigations on Hjorth's parameters for MPDA are still ongoing research.
In this study, the characteristics of Hjorth's parameters for MPDA are theoretically studied and generalized Hjorth's parameters are accordingly constructed based on Genetic programming (GP). The main contributions of this paper are summarized as follows. Firstly, the characteristics of Hjorth's parameters including Activity, Mobility, and Complexity for MPDA are theoretically and experimentally studied. A new theorem of Hjorth's parameters for MPDA is proposed. Based on the results of theoretical studies, it is verified that Activity is more suitable to MPDA than Mobility and Complexity. Secondly, Hjorth's parameters including Activity, Mobility, and Complexity are reformulated as a unified mathematical framework. A new composite fitness function for GP is designed to tailor for the aforementioned two successive tasks of MPDA. Afterward, generalized Hjorth's parameters are constructed by integrating the unified mathematical framework with the newly designed fitness function of GP.
The structure of this paper is organized as follows. Section 2 reports the unified mathematical framework of Hjorth's parameters and their theoretical investigations for MPDA. A new composite fitness function of GP and the proposed generalized Hjorth's parameters for online MPDA are proposed in Section 3. Section 4 studies three datasets including two bearing run-to-failure datasets and a gear run-to-failure dataset to verify the effectiveness of the proposed framework. Conclusions are drawn in the final section.
Section snippets
Unified mathematical framework and theoretical investigation on Hjorth's parameters for MPDA
This section begins with the definition of Hjorth's parameters including Activity, Mobility, Complexity, and their mathematical formulations. Then, three basic elements, namely Hjorths_1 to 3 are introduced and their mathematical expressions are given. Afterward, the unified mathematical framework of Hjorth's parameters is established based on these three basic elements. Finally, theoretical investigations on Hjorth’s parameters for MPDA are thoroughly investigated, which is further studied and
Generalized Hjorth’s parameters for MPDA
Inspired by the unified mathematical framework of Hjorth’s Parameters in Fig. 1, generalized Hjorth’s parameters for MPDA are constructed based on GP. Herein, a composite fitness function of GP is designed, which simultaneously integrates two essential properties of health indicators. Then Hjorth_1, Hjorth_2, and Hjorth_3 are used as inputs to GP to obtain a tree structure of generalized Hjorth’s parameters for MPDA. Herein, the tree structure can be understood as a fusion rule. Once the tree
Case studies
In this section, three run-to-failure datasets including two bearings and one gear are used to validate the feasibility of the proposed methodology for MPDA.
Conclusions
In this study, a data-driven methodology was proposed to construct generalized Hjorth's parameters for MPDA. Firstly, Hjorth's Parameters were reformulated into a unified mathematical framework and their performances for MPDA were theoretically and experimentally investigated. Besides, a new theorem of Hjorth's parameters for MPDA was proposed. Aiming at the limitations of Hjorth's Parameters, a newly composite fitness function of GP was designed to develop generalized Hjorth's parameters for
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The research work was fully supported by the National Natural Science Foundation of China under Grant No. 51975355 and Grant No. 11632011, Grant No: 12121002, Ministry of Education-China Mobile Research Foundation (CMHQ-JS-201900003), and Natural Science Foundation of Shanghai (20ZR1428600). The authors would like to thank three reviewers for their constructive and valuable comments on our manuscript.
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