## 3 Flight Control System Health Assessment under Uncertainty
### Uncertainty Quantification Based on \(\lambda-PDF\)  
The propagation problem based on probability theory needs to first know the probability distribution of model inputs. However, in actuality, the model input parameters are not continuous, or their distribution is not a common probability distribution, such as uniform distribution, normal distribution, etc. In view of this, an uncertainty measurement method based on \(\lambda-PDF\)[28] is proposed to describe the unknown distribution type. \(\lambda-PDF\) is a class of bounded, unimodal and symmetrical probability density functions (PDF), which can be used to approximate a variety of common distributions in a specified interval. Its specific form is:  
\[f_{\xi}^{\lambda}(\xi)=\left\{ \begin{array}{ll}\kappa_{\lambda}\left(1-\xi^ {2}\right)^{\lambda-0.5}&,|\xi|\leq 1\\ 0&,|\xi|\geq 1\end{array} \right., \tag{11}\]  
where \(\lambda\geq 0\).The value of the constant,\(\kappa_{\lambda}\), is:  
\[\kappa_{\lambda}=\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+1/2)\Gamma(1/2)}, \tag{12}\]  
where \(\Gamma()\) is the gamma function. Figure 2: Flight control system health assessment process under certain conditions. The continuous unimodal symmetrical distribution with a mean value of 0 in the interval (-1,1) can be approximated by Equation (11) with different value of \(\lambda\). However, the \(\lambda-PDF\) method is only suitable for parameter with a symmetrical distribution in (-1,1), which is different winfactual flight control system parameters. Therefore, in order to quantifythese parameters' distribution, the \(\lambda-PDF\) method needs to be extended. First, the following linear expansion is used to enlarge the random variable's interval:  
\[x=b_{0}+b_{1}\xi, \tag{13}\]  
where \(x\) is a new constructed random variable and \(b_{0}\) and \(b_{1}\) are related parameters for the first expansion. It can be rewritten as:  
\[\xi=\frac{x-b_{0}}{b_{1}}, \tag{14}\]  
Substitute Equation (14) into Equation (11), and the expanded probability density function is:  
\[f_{x}^{\lambda}(x)=\frac{\kappa_{\lambda}}{b_{1}}\Bigg{(}1-\bigg{(}\frac{x-b_{ 0}}{b_{1}}\bigg{)}^{2}\Bigg{)}^{\lambda-1/2}x\in[b_{0}-b_{1},b_{0}+b_{1}], \tag{15}\]  
That is, with the values of \(b_{0}\) and \(b_{1}\) variationin the extended model, the range in the initial \(\lambda-PDF\) method can be extended from (-1,1) to any finite interval. Further, for the quantification of unimodal, asymmetric and continuously distributed random variables, a quadratic extended \(\lambda-PDF\) method is proposed to fit it. Consider the following second expansion:  
\[x=b_{0}+b_{1}\xi+b_{2}\xi^{2}, \tag{16}\]  
It can be solved by Equation (16):  
\[\xi=\frac{-b_{1}\mp\sqrt{b_{1}^{2}-4b_{2}(b_{0}-x)}}{2b_{2}}, \tag{17}\]  
Substitute Equation (17) into Equation (11), when \(b_{2}\geq 0\) and \(b_{1}\geq 2b_{2}\), and the PDF is:  
\[f_{x}^{\lambda}(x)=\frac{\kappa_{\lambda}}{\sqrt{b_{1}{}^{2}-4b_{2}(b_{0}-x)} }\cdot\Bigg{(}1-\bigg{(}\frac{-b_{1}+\sqrt{b_{1}{}^{2}-4b_{2}(b_{0}-x)}}{2b_{ 2}}\bigg{)}^{2}\Bigg{)}^{\lambda-0.5}, \tag{18}\]  
where the value range of \(x\) is \([b_{0}+b_{2}-b_{1},b_{0}+b_{2}+b_{1}]\). When \(b_{2}<0\) and \(b_{1}\leq 2b_{2}\), the PDF is:  
\[f_{x}^{\lambda}(x)=\frac{\kappa_{\lambda}}{\sqrt{b_{1}{}^{2}-4b_{2}(b_{0}-x)} }\cdot\Bigg{(}1-\bigg{(}\frac{-b_{1}-\sqrt{b_{1}{}^{2}-4b_{2}(b_{0}-x)}}{2b_{ 2}}\bigg{)}^{2}\Bigg{)}^{\lambda-0.5}, \tag{19}\]  
where the value range of \(x\) is \([b_{0}+b_{2}+b_{1},b_{0}+b_{2}-b_{1}]\). The \(\lambda-PDF\) method after the first and second expansion can be used to describe the distribution of continuous, unimodal, normal or skewed parameters on any finite interval. Forthe first and second expansion \(\lambda-PDF\) methods, the unknown parameters are \(b_{i}\) and \(\lambda\). Theoretically, the common methods for parameter estimation include moment estimation, least square estimation and likelihood estimation.