Xia Yanjun, Ding Linfei, Liu Pan, Tang Zhangchun
School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China.
Nanjing Research Institute of Simulation Technology, Nanjing 210016, China.
Materials (Basel). 2023 Apr 25;16(9):3367. doi: 10.3390/ma16093367.
Various uncertain factors exist in the practical systems. Random variables, uncertain-but-bounded variables and fuzzy variables are commonly employed to measure these uncertain factors. Random variables are usually employed to define uncertain factors with sufficient samples to accurately estimate probability density functions (PDFs). Uncertain-but-bounded variables are usually employed to define uncertain factors with limited samples that cannot accurately estimate PDFs but can precisely decide variation ranges of uncertain factors. Fuzzy variables can commonly be employed to define uncertain factors with epistemic uncertainty relevant to human knowledge and expert experience. This paper focuses on the practical systems subjected to epistemic uncertainty measured by fuzzy variables and uncertainty with limited samples measured by uncertain-but-bounded variables. The uncertainty propagation of the systems with fuzzy variables described by a membership function and uncertain-but-bounded variables defined by a multi-ellipsoid convex set is investigated. The combination of the membership levels method for fuzzy variables and the non-probabilistic reliability index for uncertain-but-bounded variables is employed to solve the uncertainty propagation. Uncertainty propagation is sued to calculate the membership function of the non-probabilistic reliability index, which is defined by a nested optimization problem at each membership level when all fuzzy variables degenerate into intervals. Finally, three methods are employed to seek the membership function of the non-probabilistic reliability index. Various examples are utilized to demonstrate the applicability of the model and the efficiency of the proposed method.
实际系统中存在各种不确定因素。随机变量、有界不确定变量和模糊变量通常用于度量这些不确定因素。随机变量通常用于定义具有足够样本的不确定因素，以便准确估计概率密度函数（PDF）。有界不确定变量通常用于定义样本有限的不确定因素，这些样本无法准确估计PDF，但可以精确确定不确定因素的变化范围。模糊变量通常可用于定义与人类知识和专家经验相关的认知不确定性的不确定因素。本文关注由模糊变量度量认知不确定性以及由有界不确定变量度量有限样本不确定性的实际系统。研究了由隶属函数描述的模糊变量和由多椭球凸集定义的有界不确定变量的系统的不确定性传播。采用模糊变量的隶属度水平方法和有界不确定变量的非概率可靠性指标相结合来解决不确定性传播问题。利用不确定性传播来计算非概率可靠性指标的隶属函数，当所有模糊变量退化为区间时，该隶属函数在每个隶属度水平上由一个嵌套优化问题定义。最后，采用三种方法来求解非概率可靠性指标的隶属函数。通过各种示例来证明模型的适用性和所提方法的有效性。
