## 2 Related Works  
### Mixed Uncertainties Quantification Based on Probability Theory and Convex Set Theory  
In order to make the best use of uncertainties, both the probability and convex set theories are employed to quantify the aleatory and epistemic uncertainties respectively. The multi-ellipsoid convex model [16] is adopted to quantify the epistemic uncertainty. The epistemic uncertainty involved in this paper refers to the interval uncertainty. When there are enough data or information for the design variables, they can be modeled using the probability theory. The normal random variable \(\mathbf{x}\) can be transformed into a standard normal random variables \(\mathbf{u}\) by:  
\[\mathbf{u}=\frac{(\mathbf{x}-\overline{\mathbf{x}})}{\sigma} \tag{1}\]  
where \(\overline{\mathbf{x}}\) and \(\sigma\) are the mean value and the standard deviation of \(\mathbf{x}\), respectively. Generally, the aleatory variables \(\mathbf{x}\) can be transformed to a set of uncorrelated normal variables via the Rosenblatt transformation [17]. This transformation from \(x\) to \(u\) is based on the condition that the cumulative distribution functions (CDF) of the random variables remain the same before and after the transformation. The transformation can be expressed by  
\[F_{X_{i}}(x_{i})=\Phi(u_{i}),i=1,2,\cdots,n \tag{2}\]  
where \(\mathbf{\Phi(\bullet)}\) is the (cumulative distribution functions) CDF of the standard normal distribution. Then, the transformed standard normal variable can be denoted as  
\[u_{i}=\Phi^{-1}\big{[}F_{X_{i}}(\mathbf{x}_{i})\big{]} \tag{3}\]  
For example, the normally distributed random variable \(\mathbf{x}_{i}\)-\(N(\mu_{i},\sigma_{i})\) and the transformed \(u_{i}\) can be obtained with Equation (4). \[u_{i}=\Phi^{-1}\big{[}F_{X_{i}}(\mathbf{x}_{i})\big{]}=\Phi^{-1}\bigg{[}\Phi \bigg{(}\frac{\mathbf{x}_{i}-\mu_{i}}{\sigma_{i}}\bigg{)}\bigg{]}=\frac{ \mathbf{x}_{i}-\mu_{i}}{\sigma_{i}} \tag{4}\]  
where \(\Phi^{-1}(\bullet)\) is the inverse standard normal cumulative distribution function. As a result, all the random variables can be transformed into independent standard normal ones in \(\mathbf{u}\) space. For the quantification of epistemic uncertainties, scholars have also made a lot of attempts and innovations. In this paper, a convex model, which has a clear concept, simple model and straightforward multiple variables, is used to quantify epistemic uncertainties. When dealing with variables, convex model requires less uncertain information and only needs to know its disturbance range. Ellipsoidal convex model, referred to as ELP model, can be expressed as follows:  
\[\Omega_{SELP}=\Big{\{}\boldsymbol{\alpha}(t)\in R^{r}:[\boldsymbol{\alpha}- \overline{\boldsymbol{\alpha}}]^{T}\mathbf{W}[\boldsymbol{\alpha}-\overline {\boldsymbol{\alpha}}]\leq\theta^{2}\Big{\}} \tag{5}\]  
where \(\boldsymbol{\alpha}\) is the uncertainty parameter to be described, \(\mathbf{W}\) is a weighted matrix, which reflects the correlation between different variables; \(\theta\) is the radius of the ellipsoid, which reflects the disturbance range and degree of uncertainties; \(\overline{\boldsymbol{\alpha}}\) express the nonzero principal value (or mean value) of \(\boldsymbol{\alpha}\). For the convenience of processing, the epistemic uncertainty variables quantized by convex model can be standardized as follows. Firstly, the positive definite matrix \(\mathbf{W}\) is Eigen decomposed:  
\[\mathbf{Q}^{T}\mathbf{W}\mathbf{Q}=\Lambda \tag{6}\]where \(\mathbf{Q}^{\mathrm{T}}\mathbf{Q}=\mathbf{I}\); \(\Lambda\) is the characteristic matrix of \(\mathbf{W}\). \[\mathbf{v}=\frac{1}{\varepsilon}\sqrt{\Lambda}\mathbf{Q}^{\mathrm{T}}(\mathbf{ \alpha}-\mathbf{\overline{\alpha}}) \tag{7}\]  
The original convex model can be transformed into:  
\[\mathbf{E}_{c}=\left\{\mathbf{v}\Big{|}\mathbf{v}^{\mathrm{T}}\mathbf{v}\leq 1\right\} \tag{8}\]  
\(\mathbf{E}_{c}\) is a set of unit multiellipsoids in standard space.