**1. Introduction**

Rotating machines are critical elements of almost all forms of mechanical assemblies, which play an important role in today's industrial applications. Unexpected failures and unscheduled system interruptions caused by harsh working environments may result in costly lapse in production or even catastrophic incidents. Accurate/timely prediction and health assessment of rotating machines are of grea<sup>t</sup> significance in the functionality and performance of these equipment, especially in the early stages of failure [1–3].

Accordingly, in recent years, a plethora of research works have proposed tracking the degradation trend of rotating machines and predicting their remaining useful life (RUL), wherein condition-based prognostics maintenance (CBPM) has become an efficient strategy for the PHM of rotating machines. The CBPM is based on the data information collected via embedded sensors to assess and judge the

real-time state of the mechanical assemblies, and then predict their degradation trend to develop appropriate decisions on maintenance activities before failure occurs. More advanced prognostics techniques are focused on degradation assessment of the systems, but it is also a relatively complex problem due to the non-stationary and nonlinear characteristic of data information [4–6]. Generally, to fulfill the goal of prognostics, three crucial steps are needed:


Among these three steps, model establishment and trajectory estimation are the most important and challenging steps because of the stochastic nature and nonlinear characteristics of the health indicators. Therefore, many researchers have laid their emphasis on model foundation in recent years. From the view of application, roughly, the existing prediction approaches can be divided into two categories: (a) parametric-based methods and (b) nonparametric-based methods. In the literature, parametric-based methods mainly include time-series methods such as autoregressive moving average (ARMA) [7,8], fractional autoregressive integrated moving average (FARIMA) [9,10], fractional Brownian motion (FBM) [11,12], hidden Markov model (HMM) [13,14] and grey theoretical model (GTM) [15,16], etc. Generally, the parametric-based methods overcome the hurdle of predictive availability during long-term prediction (according to needs) which assumes the model's parameters to be constants in the predicted region. However, due to the stochastic, complex and nonlinear properties of health indicators time series (HITS), the trade-off of parametric-based method is that only accurate prediction with real-time can help adjust the system model. Otherwise, the prediction performance may not be ensured. In addition, the parametric-based methods are often developed case by case, thus model parameters identification also requires extensive experiments, and therefore, it is usually not desirable in practice.

To address this issue, much attention has recently been focused on the nonparametric-based methods in the degradation prognostic of rotating machinery. For instance, artificial neural network (ANN) [17–20], fuzzy logic [21], and deep learning network (DPN) [22,23], etc., could be considered as successful nonparametric-based approaches in the PHM field. The benefit of using nonparametric-based methods lies in their ability to model the evolution of complex multi-dimensional degradation data, which can effectively extract latent features such as the spatio-temporal correlations (STC) among historical data. In addition, for the nonparametric-based methods, it is not necessary to establish an accurate linkage between the reliability health indicators and physical degradation such as crack propagation in an individual element. However, the common drawbacks associated with these nonparametric-based techniques are time-consuming and a large number of training samples are required in advance.

Due to time-variant and non-stationary characteristics of the health indicators, and the disadvantages of the model, which is sometimes case-specific, it is difficult to determine the best prognostic model. Therefore, it is intuitive to develop a fusion idea via combining parametric-based methods and nonparametric-based methods to blend their merits and enhance the prediction performance of degradation trajectories. However, it is not clear how to fuse parametric-based and

nonparametric-based models to improve the prediction accuracy, especially when health indicators become abundant and redundant.

It is generally thought that the health indicators time series (HITS) is commonly composed of different sub-components which correspond to different time-variant characteristics of the systems. In this work, we assume that the HITS is mainly composed of high frequency component (HFC) and low frequency component (LFC). The HFC part corresponds to the stable change around the zero line of the HITS which frequently occurs; in contrast, the LFC part is essentially related to the evolutionary trend of the HITS which rarely occurs in practice, because the stable running trend occupies most of the life time, only the drastic trend can be clearly emerged when the failures are serious. This can be explained using degradation trajectories of several rotating machines (e.g., rolling bearings) as shown in Figure 1, where the degradation process of the bearings generally consists of two phases, i.e., phase I: the normal operation phase and phase II: the failure phase. The degradation health indicator in phase I is stable, which means that the bearing is normal. The degradation process changes from phase I to phase II once a fault occurs, where the HITS generally increases as the fault gets worse, the curve that rises steadily could be assumed as the LFC; in contrast, the volatile components with large and small amplitude could be treated as the HFC. To ensure better prediction accuracy, it is necessary to isolate the LFC associated with bearing defects and HRC associated with structural vibration from the degradation trajectories, and then develop a degradation algorithm based on both components.

**Figure 1.** Typical bearing degradation trajectories described by the bearing health indicators.

To overcome this problem, a sparse low-rank matrix decomposition (SLMD) method is introduced in this research. In recent years, many researchers have laid their emphasis on fault diagnosis and sparse filtering using SLMD algorithm. For instance, in ref. [24], Selesnick et al. developed a convex low-rank matrix decomposition approach based on non-convex bivariate penalty function for sparse one-dimensional deconvolution. In ref. [25], He et al. proposed convex sparsity-based regularization scheme to extract multiple faults of motor bearing and locomotive bearing. In refs. [26,27], Li et al. proposed a bicomponent sparse low-rank matrix separation and group sparsity total variation de-noising approach to extract transient fault impulses from the noisy vibration signals, using the rolling bearing and gearbox as example. In ref. [28], Wang et al. utilized a sparse low-rank matrix decomposition method based on generalized minimax-concave (GMC) penalty to explore and diagnosis the localized faults in rolling bearings. In refs. [29,30], Parekh et al. proposed an enhanced low-rank matrix approximation and its effectiveness was verified by the synthetic data and nonlocal self-similarity based image. In ref. [31], Ding et al. proposed the sparse low-rank matrix decomposition (SLMD) algorithm based on split augmented Lagrangian shrinkage (SALS) algorithm

with majorization-minimization (MM) to detect and extract the periodical oscillatory features of gearbox of an oil refinery.

Unfortunately, despite the effectiveness of the above approaches, the SLMD approaches still suffer from the following several challenges and their application development has been restricted:


To overcome these limitations and to robustly separate the LFC and HFC from degradation trajectories of health indicators, a novel asymmetric penalty sparse decomposition (APSD) algorithm with non-convex sparsity constraint is proposed in this work. The resulting non-convex regularization problem will be efficiently handled using the majorization-minimization (MM) method. Having the LFC and HFC separated, the LFC and HFC components (e.g., last 100 points) will be predicted using the wavelet neural network (WNN) and ARMA combined with recursive least squares algorithm (ARMA-RLS) methods, respectively. In an effort to establish a comprehensive assessment of degradation processes, the proposed fusion prognostics framework combines the WNN (a nonparametric-based approach) and the ARMA-RLS (a parametric-based approach) in an attempt to improve the prediction performance, in which the physical information of the degenerative process and massive sample data are not considered and accommodated. The final predicted data could be generated by combining the predicted LFC and HFC components. Finally, we demonstrate the proposed approach by four case studies of accelerated aging tests of rolling bearing.

The main contributions of this paper are summarized as follows:


The frame of this paper is structured as follows. Section 2 contains the description of the asymmetric nonconvex sparse decomposition (APSD) algorithm. Section 3 provides the algorithms of degradation modeling, i.e., wavelet neural network and ARMA combined with recursive least squares algorithm (ARMA-RLS) methods. Experimental results and discussion are given in Section 4. Finally, Section 5 concludes with a discussion of future research.
