# Reliability Analysis of Vehicle Braking System Based on Hyperellipsoidal Dynamic Bayesian Network
## 1 Introduction  
The braking system directly affected the safety and efficiency of the vehicle operational. It is imperative not to underestimate the potential risks associated with brake system failure, as it may result in emergency braking failure, operational delays, and even safety accidents. These incidents could have grave consequences for the lives and properties of passengers and the general public. Therefore, it is of paramount importance to conduct a comprehensive and in-depth reliability analysis of the braking system of rail vehicles. Traditional reliability analysis methods, such as fault tree analysis (FTA), have demonstrated certain advantages in evaluating the reliability of general complex systems. However, the limitations of FTA are increasingly evident when applied to highly complex systems with diverse failure modes, such as vehicle braking systems. In particular, based on the two-state assumption, it is challenging to comprehensively capture the intricate and variable fault states in the braking system and their dynamic effects on one another. Furthermore, the limitations of FTA in addressing dynamic systems have limited its extensive application in braking system reliability assessment. To address these challenges, researchers have actively sought to develop enhanced methodologies, such as the T-S fuzzy fault tree analysis approach. This method employs fuzzy theory to provide a fuzzy description of the failure rate and probability of failure and introduces a polymorphic representation through the use of T-S gates, thereby offering a novel perspective for the reliability analysis of complex systems [1]. However, the T-S fuzzy fault tree still encounters significant challenges in terms of computational complexity and the ability to conduct dynamic analysis when dealing with complex systems [2]. In order to significantly improve the precision and processing efficiency of reliability analysis, particularly when confronted with high-dimensional and nonlinear data, this paper proposes the hyperellipsoid model. As an efficient data fusion and dimensionality reduction technique, the hyperellipsoid model can accurately capture the intrinsic structure of the data, construct a multi-dimensional hyperellipsoid space, and map the original complex high-dimensional data into this space. The shape, size, direction, and other geometric characteristics of the superellipsoid effectively characterize the essential features of the data. In the reliability analysis of the braking system of rail vehicles, the multidimensional fault data of the braking system is processed in depth to accurately extract the key fault characteristics and significantly reduce the data dimensionality, thereby enhancing the efficiency of the analysis while improving the accuracy of the data analysis [3]. At present, the T-S fault tree model is the prevailing approach in the field of reliability analysis. Although the T-S fault tree model has been demonstrated to be effective in a multitude of application scenarios, there are still some limitations, such as insufficient ability to capture the dynamic characteristics of complex systems, limitations in uncertainty and ambiguity handling, and the potential for computational complexity and loss of accuracy when dealing with high-dimensional data. To address these issues, this paper proposes a hyperellipsoidal dynamic Bayesian modeling approach to assess system reliability accurately. The vehicle braking system is used as an illustrative example to demonstrate how the limitations of the traditional FTA method, specifically the three-state assumption and multistate representation, can be overcome. Additionally, the proposed approach addresses the assessment challenges associated with the high complexity and dynamic characteristics of the braking system.