## 4 Analysis of Systems Reliability  
Reliability is defined by AFNOR (French Association of Normalisation) as the ability of systems to assure a required function in given conditions within a specified period [27]. In the case of complex systems and mechatronic systems, reliability is a challenge for developers and industrialists today, because reliability analysis represents an important step in the development process of many systems and different structures. Evaluating the reliability \(R\) is related to the evaluation of the failure probability \(P_{f}\) using the following expression:  
\[R=1-P_{f}. \tag{3}\]  
The precedent expression serves to evaluate reliability from the value of the probability of failure. In addition, the probability of failure is related to several parameters:  
\[P_{f}=P(G(X)\leq 0)=\int\limits_{G(X)\leq 0}f_{X}(x)dx, \tag{4}\]  
where \(G(X)\) is the limit state function (performance function), it is a function of random variable \(X\) such that:  
* \(G(X)>0\) : the security domain of the system. * \(G(X)=0\) : the boundary of state surface. * \(G(X)<0\) : the failure domain of the system. To evaluate the failure probability \(P_{f}\) or the reliability \(R\), it is necessary to calculate the integral in the Expression (4). However, the calculation is analytically difficult and mostly impossible, especially if the random variable \(X\) has a large dimension. For this reason, we should use approximation methods [21; 27]. There are two methods that are very efficient for evaluating the reliability of systems: the First Order Reliability Method (FORM) and the Second Order Reliability Method (SORM) are based principally on the search for the Most Probable Failure Point (MPFP). With FORM and SORM, we can calculate the failure probability and reliability from the reliability index. The FORM method allows us to calculate \(P_{f}\) using this expression:  
\[P_{f}=\phi(-\beta), \tag{5}\]  
where \(\phi\) represents the reduced centered normal distribution function. The FORM method consists of replacing the boundary state surface by the tangent hyperplane at the design point (Figure 5). However, when the the performance function is strongly nonlinear, this approximation can result in an incorrect probability of failure. The SORM method makes it possible to calculate \(P_{f}\), but differently from the FORM method, using the following formula:  
\[P_{f}=\Phi(-\beta)\left(\prod_{i=1}^{n-1}\frac{1}{\sqrt{1+\beta k_{i}}}\right), \tag{6}\]  
where \(k_{i}\) are the main curves of the performance function \(G\) at the MPFP (Most Probable Failure Point). The SORM method consists of replacing the limit state surface at the most probable point of failure by the quadratic surface (Figure 6). The reliability index can be evaluated with different methods. [PERSON] and [PERSON] have proposed to estimate \(\beta\) in the space of the reduced centered normal and statistically independent random variables [27]. For this reason, the random vector \(X\) will be transformed into the random vector \(U\), such as:  
\[U_{i}=T(X_{i}).\]The random variables follow the reduced centered normal distribution and, regardless of \(i=j\), \(U_{i}\) and \(U_{j}\), are independent of each other. The probabilistic transformation T necessitates knowing the statistical distributions of every one of the random variables. The performance function after transformation becomes:  
\[H(U)=G[X(U)]. \tag{8}\]  
The reliability index of [PERSON] and [PERSON] \(\beta_{HL}\) can be defined as the Euclidean distance between the origin of the standard normal space and the boundary state surface \(H(u)=0\). We consider \(u\) a realization of the random vector \(U\), with \(u=(u_{1},u_{2},\ldots,u_{m})^{T}\). Therefore, \(H(u)\) will be a realization of the random variable \(H(U)\). To evaluate the reliability index, it is necessary to solve this minimization problem with constraint:  
\[\left\{ \begin{array}{c}\beta_{HL}=\min\sqrt{u^{T}\cdot u}\\ u\in R^{m}:\\ H(u)=0.\end{array} \right. \tag{9}\]  
Figure 5: Principle of the FORM method.