1. Introduction
Mechatronic systems, which involve the synergistic integration of mechanical and electrical structures, are essential parts of modern industrial systems [
1,
2,
3,
4,
5]. Recently, with the increasing requirements for the reliability of mechatronic systems in industrial applications, fault diagnosis and prognosis, as an important technique to ensure the operational safety and stability of systems, has been a popular topic for researchers and practitioners [
6,
7].
In recent decades, research into fault diagnosis for mechatronic systems has had many achievements [
6,
7,
8,
9,
10,
11,
12,
13,
14]. Fault diagnosis can generally be classified into data-driven methods and model-based methods. The data-driven diagnosis methods do not need to construct accurate physical models of systems; feature data under normal and faulty conditions, extracted from sensor measurements, are needed to implement the fault diagnosis procedure. However, in some cases, the application of data-driven methods may be limited, as it is difficult to obtain feature data under faulty conditions. Compared with data-driven methods, model-based methods can achieve better diagnostic accuracy due to the employment of physical models. However, system modeling is usually not a trivial task. Fortunately, among the various system modeling methods, the bond graph (BG) is an efficient and graphical modeling tool, which can model complex systems with multiple energy domains. The BG technique has been widely applied for fault diagnosis in mechatronic systems, and many significant results have been obtained [
13,
14]. In [
13], the BG and analytical redundancy relations (ARRs)-based fault diagnosis method for continuous systems is extended to hybrid systems (including continuous dynamics and discrete modes) by introducing the concepts of hybrid BG and global ARRs, where both discrete faults and continuous faults can be detected and isolated. It is noteworthy that the aforementioned model-based fault diagnosis methods are developed based on the centralized architecture or global system model, which may lead a heavy computational burden in the centralized fault estimation with increases in system scale. To address this problem, the structural-model-decomposition-based distributed fault estimation method is developed in [
14,
15], where a set of local submodels are decomposed from the global model that is suitable for estimation. The computationally independent local estimation is formed based on these local submodels, resulting in a scalable distributed estimation approach that allows for the local sub-problems to be solved in parallel, thus decreasing the computational burden.
Differing from the relatively mature fault diagnosis technology for mechatronic systems, the research on prognosis is still in the development stage. Some relevant works can be found in [
15,
16,
17,
18,
19,
20,
21,
22]. The prognosis methods can be divided into two strategies, i.e., model-based methods and data-driven methods. The model-based methods typically attempt to construct mathematical models to describe the degradation process of faulty components. In [
16], the improved Wiener degradation process is proposed for the prognosis of incipient faults in the hybrid mechatronic system. In [
18], an adaptive hybrid differential evolution algorithm is used to identify the degradation behavior of incipient faults, by which the remaining useful life (RUL) of faulty components can be predicted. However, in real systems, it is difficult or even impossible to accurately establish physical degradation models for faulty components, which limits the applications of the model-based prognostic method. Unlike the model-based methods, data-driven methods do not need to establish an accurate mathematical model of the monitoring object. Based on the collected system historical data, mining the hidden information in the data for prognosis is a more practical method. At present, neural networks, which can predict the future evolutionary trend according to historical degradation data when the physical degradation model of the faulty component is unavailable, have gradually become popular methods in the data-driven prognosis field [
21,
22]. Among the various neural networks, the extreme learning machine (ELM) possesses the merits of good generalization and a fast learning ability [
23,
24]. Therefore, ELM has been used to solve many prognosis problems [
25,
26,
27]. For example, in [
26], an enhanced multi-sensor prognostic model based on Kalman filter-online sequential ELM and logistic regression model is designed for the RUL prediction of an aircraft engine. It is notable that the aforementioned works mainly focus on the prognosis for permanent faults, while intermittent faults, which are also common in mechatronic systems, are not discussed. Unlike permanent faults, intermittent faults possess discontinuity and stochasticity. If the effective prognosis approach cannot quickly be implemented for the predictive maintenance purpose in the early stage of intermittent faults, intermittent faults may evolve into permanent faults. Recently, a method to address the prognosis of the electric scooter system with intermittent faults was introduced in [
28]. However, this work only solves the problem of RUL prediction under the assumption of the monotonic degradation of intermittent fault magnitude, and does not concern the stochasticity of intermittent faults.. Moreover, the RUL prediction research for intermittently faulty components in [
28] does not consider the fact that the degradation model of the faulty component is usually unknown in real applications.
Based on the above discussions, the prognosis of intermittent faults is still a challenging issue. Specifically, there are two major problems to be solved. Firstly, considering the discontinuity of intermittent faults and the randomness of fault appearance and disappearance, the construction of intermittent fault features based on the fault estimation results is an essential problem. Secondly, the degradation models of intermittently faulty components are usually unknown in practical applications. Thus, without the exact degradation model, predicting the RUL of the intermittently faulty component based on established intermittent fault degradation features is challenging.
An electric scooter is an essential vehicular transportation mode for people with different mobility difficulties when travelling. Note that various electrical and mechanical components in the electric scooter may suffer from intermittent faults due to aging and frequent usage. It is easy to neglect the influence of intermittent faults on the system’s normal operation at the early stage. If an effective diagnosis and prognosis scheme is not predesigned for intermittent faults in the electric scooter, and with the continuous degradation of intermittently faulty components due to frequent usage, intermittent faults may eventually evolve into permanent faults, which will lead to system failure, and disastrous consequences. Therefore, it is necessary to develop a prognosis method for an electric scooter with intermittent faults. Therefore, an adaptive Cuckoo search extreme learning machine (ACS-ELM)-based prognosis method for an electric scooter with intermittent faults is proposed in this paper. The main contributions of this work are twofold:
(1) An integrated condition-monitoring framework combining distributed model-based diagnosis and data-driven prognosis (which contains the merits of both methods) is developed. On the one hand, the BG-based structural model decomposition is used to build submodels from the global model, based on which the distributed intermittent fault estimation can be implemented with less computational burden. On the other hand, considering the fact that the physical degradation models are usually unknown in practice, the data-driven prognosis method is developed to predict the RULs of intermittently faulty components.
(2) As intermittent faults gradually deteriorate, and possesses discontinuity and stochasticity, the intermittent fault features are captured with the aid of tumbling window (TW). Then, the ACS-ELM is proposed to model the intermittent fault feature evolutionary trend, as well as the RUL prediction of the intermittently faulty component, where ACS-ELM is developed by introducing adaptive Cuckoo search (ACS) into the ELM to optimize input weights and hidden layer biases.
This paper is organized as follows.
Section 2 presents the FDI framework under intermittent fault for an electric scooter based on a diagnostic bond graph (DBG) model.
Section 3 discusses the distributed intermittent fault estimation based on structural model decomposition.
Section 4 proposes the prognosis method for intermittently faulty components using ACS-ELM.
Section 5 analyzes the simulation and presents the experimental results. Finally,
Section 6 concludes this paper.
3. Distributed Intermittent Fault Estimation
3.1. Parameterization of Intermittently Faulty Component
The intermittent fault estimation aims to identify the intermittent fault magnitude, with appearing and disappearing instants for possible faulty components in SPF. Thus, the value change in
(
represents the parameter or efficiency factor in
Table 2) under intermittent faults in the time interval
can be described by the following function:
where
is the unit step function,
is the nominal value of
,
is the fault magnitude vector,
is the fault appearing instant vector,
is the fault disappearing instant vector,
. Thus,
is the parameterized function of the faulty component with three sets of variables (i.e.,
, and
) to be identified. Based on (
4), the value changes in all possible faulty components in SPF can be described by parameterization functions.
3.2. Construction of Submodels by Structural Model Decomposition
Since a large number of unknown variables need to be identified under the multiple intermittent faults condition (i.e., one has to identify the fault magnitude vector, fault-appearing instant vector and fault-disappearing instant vector for each fault candidate in SPF), there is a heavy computational burden if the centralized fault estimation method, based on the global model, is used. Therefore, the use of a distributed fault estimation technique is recommended to achieve better computational efficiency. The distributed fault estimation is accomplished based on the submodels that were decomposed from the global model using structural model decomposition [
14,
15]. For illustration, the global model and the submodel can be defined as follows.
Definition I: (Global model) A global model is represented by , where U and Y are the sets of inputs and outputs of the global model, respectively, is the set of parameters and efficiency factors.
Definition II: (Submodel) The ith submodel is represented by , where and are the sets of local inputs and local output of the ith submodel, respectively, is the set of parameters and efficiency factors in the ith submodel.
Theoretically, the number of submodels is determined by the number of sensors in the global model. Therefore, three submodels, as shown in
Figure 3, are built from the global DBG model of the electric scooter system. Based on
Figure 3, the submodel MARRs can be derived as follows.
In
Figure 3 and (5)–(7),
and
denote that ∗ is treated as the local input and output of the submodel
,
, respectively. Note that the output (i.e., sensor measurement) in the global model may be treated as a local input or local output in different submodels. However, for the faulty sensor, regardless of the function it plays in the submodel (i.e., local input or local output), the efficiency factor should remain the same to ensure consistent detection results from different submodels.
In the electric scooter system, multiple intermittent faults are considered. Two typical cases are discussed in detail as follows.
Case I: Intermittent faults occur in and , CV = [1 0 1] and SPF = can be obtained by implementing the FDI procedure. Based on the SPF, and are located at submodel , while and are located at submodel . Therefore, the -based local estimator and the -based local estimator can be implemented in parallel to identify and , respectively.
Case II: The intermittent fault occurs in , CV = [1 1 0] and SPF = are obtained. and are located at submodel , while exists in both submodels, and . Since the submodel contains all possible faulty components in SPF, the -based local estimator can be used to identify , and .
3.3. Distributed Fault Estimation via ACS Algorithm
Since the
of the submodel
is the function of
, and
can be represented as the function of the unknown variables to be identified based on (
4). The distributed fault estimation problem for submodel
can be considered as the optimization problem using the following fitness function:
where
R is the number of samples. The possible faults in an
need to be represented by (
4), such that the corresponding unknown variables can be obtained by the optimization algorithm, and
is a small constant to avoid zero division.
After the fitness function of each submodel is obtained, the submodel-based local estimators that are affected by faults can be activated. ACS is utilized for fault parameter identification in the local estimators, while ACS is developed by introducing the adaptive step-size scaling factor into the standard Cuckoo search (CS). The CS, as a natural heuristic algorithm, is proposed to be inspired by brood parasitism and Levy flight (LF) foraging behaviors of cuckoos [
29]. Suppose that
and
denote the positions of the
lth and the
th generations of the cuckoo
d, respectively. Then, the LF-based position updating formulation is expressed as
where
is the step-size scaling factor, ⊗ denotes the entry-wise multiplication,
represents the LF random search path, and the random step-size obeys the Levy distribution as follows:
The Mantegna algorithm, which can achieve a symmetric Levy stable distribution, is usually an effective means of generating a random step-size that obeys the Levy distribution. Specifically, the step-size
s is calculated via two variables with Gaussian distribution, as follows:
where
The Levy index
and the step-size scaling factor
are default setups for the standard CS. However, the standard CS lacks the dynamical adaptability of search step-size, which may cause difficulties in algorithm convergence and lower estimation accuracy [
30]. Therefore, the ACS is proposed to alleviate this problem, where the dynamic adaptive strategy of the step-size scaling factor based on (
13) is introduced to the original CS. Using (
13), the dynamic adaptive strategy is expressed by a nonlinear piece-wise function, where the larger
at the early searching stage helps the algorithm to converge to near the optimal solution quickly, while the smaller
at a later stage can achieve fine-tuning near the optimal solution.
where
,
, and
represent the maximum, the minimum and the
lth generation step-size scaling factor of the algorithm, respectively.
l is the current iteration number, and
is the maximum number of iterations.
6. Conclusions
In this paper, an ACS-ELM-based prognosis method is developed for an electric scooter system with intermittent faults. The FDI framework helps to find possible faulty components. Based on the model’s structural decomposition and ACS algorithm, distributed fault estimation was implemented to identify the magnitude and the appearing and disappearing instants of each intermittent fault for faulty components. For the prognosis of intermittently faulty components, ACS-ELM was proposed to model the degradation process of intermittent fault features and predict the RUL of intermittently faulty components. A series of simulation and experiment results verified the effectiveness of the proposed methods. Through experiment and comparison studies, it is concluded that the ACS-ELM-based RUL prediction results of intermittently faulty components are accurate, and the ACS-ELM performs better than traditional ELM and PSO-ELM for prognosis under intermittent faults.
This work provides an effective method for the RUL prediction of intermittently faulty components under the condition of unavailable degradation models. Several challenging issues still need to be addressed. Future research directions will focus on the following two aspects. First, this work only considers the RUL prediction method of the intermittent fault magnitude degradation process based on ACS-ELM, while intermittent fault degradation can also be reflected in terms of duration. It is necessary to apply the proposed method to RUL predictions of the intermittent fault duration degradation process. Secondly, as the system working conditions (road conditions, system input, system mode, etc.) often change in practice, RUL predictions of intermittent faults under variations in system working conditions should be considered in future work.