**1. Introduction**

The flywheel [1] system is a key actuator for spacecraft attitude control, which is widely used in the aerospace field. The normal operation of a flywheel system is very important for spacecraft. However, the spacecraft environment where the flywheel system is located has a harsh operating environment and complex structure. Once a failure occurs, it will pose a grea<sup>t</sup> threat to space safety. Therefore, to ensure the reliability and orderly operation of the flywheel system, it is of grea<sup>t</sup> significance to diagnose the faults of the flywheel system quickly and accurately.

Many scholars have carried out a lot of research on the fault diagnosis of flywheel systems. Changrui Chen et al. [2] proposed a 3D associated dimension diagnosis method, it is improved by K-Medoids clustering technology for different typical states of satellite flywheel bearings and verified the feasibility of the method through experiments. Xinchang Zhang et al. [3] developed a set of methods for inputting correct premises, and based on consistency test results, presented a fault diagnosis model based on finite state machines, which could locate and diagnose some faults. Junweir Lin et al. [4] proposed a new fault diagnosis scheme for linear analog circuits. The author constructs a diagnostic evaluator, which can diagnose faults through digital signals and diagnose media after analyzing and

**Citation:** Cheng, X.; Liu, S.; He, W.; Zhang, P.; Xu, B.; Xie, Y.; Song, J. A Model for Flywheel Fault Diagnosis Based on Fuzzy Fault Tree Analysis and Belief Rule Base. *Machines* **2022**, *10*, 73. https://doi.org/10.3390/ machines10020073

Academic Editors: Hongtian Chen, Kai Zhong, Guangtao Ran, ChaoChengand DavideAstolfi

Received: 1 December 2021 Accepted: 11 January 2022 Published: 20 January 2022

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modeling the components. Bo Chen et al. [5] studied the distributed fault diagnosis technology and combined it with software technology, computer network, artificial intelligence and fault diagnosis to improve the self-fault diagnosis function of an expert system. Zijian Qiao et al. [6] proposed a second-order stochastic resonance method based on fractional derivative enhancement, which uses strong background noise to enhance the weak fault characteristics. It is used for mechanical fault diagnosis. Wenjun Sun et al. [7] studied a deep neural network based on a sparse self-code device for induction motor fault diagnosis. This method is used in the sparse automatic process to add noise encoding using the sparse automatic learning feature, which is the unsupervised feature learning that is required to measure the data without marking. Yao Cheng et al. [8] studied a set of combined fault diagnoses based on observer redundancy in the background of a satellite attitude control system. The modified scheme can solve actuator and sensor faults that are difficult to solve by traditional methods.

It can be seen from the above, most of the existing flywheel fault diagnosis schemes are designed on the basis of the data-driven method [9]. However, the current flywheel fault diagnosis still lacks an effective diagnosis scheme for the following two problems: First, the model accuracy cannot be guaranteed under small sample data. It is difficult to obtain accurate diagnosis results by using small sample data in actual fault diagnosis. This is because in the system life cycle, it is difficult to obtain a large number of flywheel fault samples, and more difficult to obtain fault samples under different fault modes; second, the black box model has the disadvantage of unexplainable diagnostic processes.

BRB (belief rule base) is a general rule-based reasoning method proposed by Yang Jianbo et al. [10] on the basis of evidentiary reasoning, which has important applications in mechanism analysis [11], health status assessment [12,13] and fault diagnosis [14]. BRB is suitable for flywheel systems, mainly reflected in three aspects: First, BRB can effectively describe the uncertainty of flywheel systems; second, the BRB modeling method is suitable for flywheel systems. It uses expert knowledge for modeling and data for model training; third, BRB has shown to be a good treatment effect for small sample problems. However, applying BRB to the actual fault diagnosis of the flywheel system cannot solve problems such as the difficulty in constructing an expert knowledge base, the unclear logical relationship between the flywheel fault events and the unclear fault index. FFTA (fuzzy fault tree analysis) [15,16] enables the logical relationship between different events to be clearly expressed. This is because FFTA can present the cause of failure and events caused by this cause in the form of a fault tree from the perspective of the fault mechanism. At the same time, FFTA makes the occurrence probability of each event in the fault tree better describe the uncertainty, because it introduces the theory of fuzzy mathematics. The combination of FFTA and BRB not only enables the fault index to be clearly established and the event fuzziness to be better described, but also enables the advantages of BRB to be applied in the fault diagnosis of the flywheel system, which makes comprehensive use of the advantages of the two. Therefore, this paper establishes the FFBRB (fuzzy fault tree analysis and belief rule base) model, which makes full use of the FFTA and BRB's advantages.

The main contributions of the FFBRB model proposed in this paper are as follows: (1) The way FFTA is used to build the initial BRB model. In this paper, the FFTA mechanism is used to expand the BRB knowledge base and solve the problem of constructing an expert knowledge base of complex flywheel system; (2) A new flywheel fault diagnosis model based on BRB is proposed. This model can obtain relatively accurate data even with a small number of samples and has higher applicability. It uses expert knowledge to construct the initial parameters of the model and uses training samples to optimize the model parameters.

The main structure of this paper is as follows: In the first part, the fault diagnosis model of the original flywheel system is analyzed and discussed. On the basis of revealing the shortcomings of the original model, the fault diagnosis model of the FFBRB flywheel system is proposed; In the second part, it describes the problems that need to be solved in the process of flywheel system modeling and gives the general solution diagram; In the third part, it defines and describes the fault diagnosis model of FFBRB flywheel system, and describes its transformation mechanism and inference optimization process in detail; In the fourth part, this paper uses a concrete example to verify the method in this paper and gives the experimental conclusion; In the fifth part, it gives the summary of this thesis.
