# Reliability Analysis of Vehicle Braking System Based on Hyperellipsoidal Dynamic Bayesian Network
## 2 Basic Theory of Reliability Analysis
### Hyperellipsoid Model  
As a significant subcategory of convex set models within the domain of nonprobabilistic reliability, hyperellipsoid models possess the distinctive advantage of being constructable in a time-efficient manner, contingent upon the availability of a clearly delineated set of uncertain event boundaries [10]. In comparison to other complex models, the hyperellipsoid model is distinguished by its simplicity of structure and moderate demand for uncertain information. Of particular significance is the model's capacity to capture and reflect the correlation characteristics among variables, which is a crucial aspect in mitigating the potential for over-conservative results that fuzzy Bayesian networks may produce when performing system analysis or evaluation [3]. The fundamental premise of the hyperellipsoid model is the utilization of distance as a metric for deviation. In particular, it quantifies uncertainty by mapping random vectors to a normalized space and calculating the distance from the origin of these vectors to the surface of the failed structure in unit sphere coordinates. The specific formulation of the hyperellipsoid model is as follows:  
\[\left(\frac{P_{h}\left(x_{1}^{a_{1}}\right)-P_{m}\left(x_{1}^{a_{1}}\right)}{ P_{r}\left(x_{1}^{a_{1}}\right)},\ldots,\frac{P_{h}\left(x_{2}^{a_{n}}\right)-P_{m} \left(x_{2}^{a_{n}}\right)}{P_{r}\left(x_{2}^{a_{n}}\right)}\right)\cdot\left( \frac{P_{h}\left(x_{1}^{a_{1}}\right)-P_{m}\left(x_{1}^{a_{1}}\right)}{P_{r} \left(x_{1}^{a_{1}}\right)},\ldots,\frac{P_{h}\left(x_{2}^{a_{n}}\right)-P_{m }\left(x_{2}^{a_{n}}\right)}{P_{r}\left(x_{2}^{a_{n}}\right)}\right)^{T} \tag{3}\]  
where \(P_{h}\left(x_{i}^{a_{i}}\right)\) is the failure interval, \(P_{m}\left(x_{i}^{a_{i}}\right)=\frac{P_{h}\left(x_{i}^{a_{i}}\right)+P_{ll} \left(x_{i}^{a_{i}}\right)}{2}\) is the nominal value of \(P_{h}\left(x_{i}^{a_{i}}\right)\), and \(P_{r}\left(x_{i}^{a_{i}}\right)=\frac{P_{h}\left(x_{i}^{a_{i}}\right)-P_{ll} \left(x_{i}^{a_{i}}\right)}{2}\) is the deviation of \(P_{h}\left(x_{i}^{a_{i}}\right)\). In comparison to the fuzzy Bayesian network model, the hyper-ellipsoid model lacks certain extreme regions that are unlikely to occur in reality. The probability of occurrence is inversely proportional to the number of root nodes. The two-dimensional hyper-ellipsoid model is contrasted with the interval model in Figure 1[11]. The vector is introduced with the objective of ensuring that the failure probability of the root node falls within the hyperellipsoid domain of Equation (3), which is expressed as follows:  
\[z=\mathbf{D}^{-1}\mathbf{P} \tag{4}\]  
Figure 1: Comparison of the 2D superellipsoid model with the interval model. \[z=(Z_{1},Z_{2},\ldots,Z_{n})^{\top} \tag{5}\]  
\[D=diag(P_{r}(x_{1}^{a_{1}}),P_{r}(x_{2}^{a_{2}}),\ldots,P_{r}(x_{n}^{a_{n}})) \tag{6}\]  
\[P=(P_{h}(x_{1}^{a_{1}}),P_{h}(x_{2}^{a_{2}}),\ldots,P_{h}(x_{n}^{a_{n}})) \tag{7}\]  
\[(z-z_{0})^{T}(z-z_{0})\leq 1 \tag{8}\]  
\[z_{0}=\left(\frac{P_{m}\big{(}x_{1}^{a_{1}}\big{)}}{P_{r}\big{(}x_{1}^{a_{1}} \big{)}},\frac{P_{m}\big{(}x_{2}^{a_{2}}\big{)}}{P_{r}\big{(}x_{2}^{a_{2}} \big{)}},\ldots,\frac{P_{m}\big{(}x_{n}^{a_{n}}\big{)}}{P_{r}\big{(}x_{n}^{a_{n }}\big{)}}\right) \tag{9}\]  
From Equation (8), the root node failure interval is randomly valued inside the hyperellipsoid space of \(\Delta z=z-z_{0}\). If \((r,\theta_{1},\theta_{2},\ldots,\theta_{n-1}),r\in[0,1],\ \theta_{i}\in[0,2\pi]\) are the coordinates of the unit hyperellipsoid. The Equation is formulated as a new hyperellipsoid model:  
\[z-z_{0}= \begin{bmatrix}r\cos\theta_{1}\\ r\sin\theta_{1}\cos\theta_{2}\\ \cdots\\ r\sin\theta_{1}\sin\theta_{2}\ldots\sin\theta_{n-2}\cos\theta_{n-1}\\ r\sin\theta_{1}\sin\theta_{2}\ldots\sin\theta_{n-2}\sin\theta_{n-1}\end{bmatrix}  \tag{10}\]  
From Equations (3)-(10), the failure probability of \(x_{i}\) is expressed as the following:  
\[