Reliability analysis of complex systems based on fault tree decomposition and Copula modeling
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摘要:
针对复杂控制系统在多个子系统之间的强耦合特性展开研究，发现传统基于独立性假设的可靠性建模方法易导致系统可靠性高估问题。针对该问题，提出一种基于故障树解构与Copula建模的可靠性分析方法：通过故障树对复杂控制系统进行结构解构，识别关键失效模式及最小割集；利用Copula函数刻画子系统间的非独立相关关系，推导系统整体可靠度表达式；以某型复杂控制系统为案例，对比独立假设与Copula建模下的可靠性结果。研究表明：考虑耦合效应后，该型复杂控制系统在任务执行期间，可靠度下降幅度较传统计算方法扩大得更大，证实传统基于独立性假设的建模方法存在显著高估风险。所提方法为复杂控制系统的可靠性评估与优化设计提供有效的理论支撑。
Abstract:Traditional reliability modeling techniques based on the independence assumption tend to overstate system reliability, according to this study, which examines the strong coupling characteristics among several subsystems in complex control systems. To address this issue, a reliability analysis method combining fault tree decomposition and Copula modeling is proposed. Firstly, the complex control system is structurally decomposed using a fault tree to identify key failure modes and minimal cut sets. Secondly, the Copula function is employed to characterize the non-independent correlation among subsystems, and the expression for the overall system reliability is derived. Finally, a specific type of complex control system is taken as a case study, and the reliability results under the independence assumption and Copula modeling are compared. The study confirms that the traditional modeling approach based on the independence assumption has a significant overestimation risk by demonstrating that the reliability decline of this kind of complex control system during task execution is more significant after taking the coupling effect into account than that calculated by traditional methods. This method provides effective theoretical support for the reliability assessment and optimal design of complex control systems.
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图 5 5种典型二维Copula函数概率密度特征
Figure 5. Five typical two-dimensional Copula function probability density characteristics
图 6 典型情况下基于5种不同Copula函数下通信组合子系统可靠度曲线
Figure 6. Reliability curves of communication integrated subsystem under five typical Copula functions in typical cases
图 7 典型情况下基于5种不同Copula函数下各子系统可靠度曲线
Figure 7. Reliability curves of each subsystem based on five different Copula functions in typical cases
表 1 系统解构
Table 1. System Deconstruction
子系统 编号 功能 通信组合子系统 A 负责数据传输与外部信息交互等 综合控制子系统 B 实现指令生成、分配与反馈调控等 集控装置子系统1 C 完成任务操作与物理动作输出等 集控装置子系统2 D 包括传感、散热、冗余模块等，提升系统
整体适应性与容错性等配电子系统 E 为各子系统提供持续稳定的能源支持 
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表 2 通信组合子系统组成结构
Table 2. The structure of the communication combinatorial subsystem
部件 编号 电源模块 AA $\vdots $ $\vdots $ 网络安全接入模块 AM
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表 3 典型二维Copula函数
Table 3. Typical two-dimensional Copula function
Copula函数类型 表达式 Kendall秩相关系数 Clayton $ {C}_{\theta }(u,v)={\left({u}^{-\theta }+{v}^{-\theta }-1\right)}^{-1/\theta } $ τ=θ/(2+θ) Gumbel ${C_\theta }(u,v) = \exp \left\{ { - {{\left[ {{{( - \ln u)}^\theta } + {{( - \ln v)}^\theta }} \right]}^{1/\theta }}} \right\},\;\;\;{\mkern 1mu} {\mkern 1mu} \theta \geqslant 1$ τ=1−1/θ Frank $ {C}_{\theta }(u,v)=-\dfrac{1}{\theta }\ln \left(1+\dfrac{({\mathrm{e}}^{-\theta u}-1)({\mathrm{e}}^{-\theta v}-1)}{{\mathrm{e}}^{-\theta }-1}\right), \theta \neq 0 $ $ \tau =1-\dfrac{4}{\theta }\left(1-\dfrac{{D}_{1}(\theta )}{\theta }\right) $ Gaussian $ {C}_{R}({u}_{1},{u}_{2})={\mathit{\Phi }}_{R}\left({\mathit{\Phi }}^{-1}({u}_{1}),{\mathit{\Phi }}^{-1}({u}_{2})\right) $ $ \tau =\dfrac{2}{{\text{π}} }\arcsin R $ t $ {C}_{R,\nu }({u}_{1},{u}_{2})={t}_{R,\nu }\left(t_{\nu }^{-1}({u}_{1}),t_{\nu }^{-1}({u}_{2})\right) $ $ \tau =\dfrac{2}{{\text{π}} }\arcsin R $
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表 4 第1次嵌套子系统间的Kendall秩相关系数
Table 4. The Kendall rank correlation coefficient between the first nested subsystems
τ A B C D E A 1.000 0 0.750 0 0.600 0 0.800 0 0.550 0 B 0.750 0 1.000 0 0.650 0 0.700 0 0.500 0 C 0.600 0 0.650 0 1.000 0 0.620 0 0.450 0 D 0.800 0 0.700 0 0.620 0 1.000 0 0.580 0 E 0.550 0 0.500 0 0.450 0 0.580 0 1.000 0
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表 5 第2次嵌套子系统间的Kendall秩相关系数
Table 5. The Kendall rank correlation coefficient between the second nested subsystems
τ A B C DE A 1.000 0 0.720 0 0.580 0 0.650 0 B 0.720 0 1.000 0 0.630 0 0.520 0 C 0.580 0 0.630 0 1.000 0 0.450 0 DE 0.650 0 0.520 0 0.450 0 1.000 0
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表 6 第3次嵌套子系统间的Kendall秩相关系数
Table 6. Kendall rank correlation coefficient between the third nested subsystems
τ A B CDE A 1.000 0 0.580 0 0.532 0 B 0.580 0 1.000 0 0.763 0 CDE 0.532 0 0.763 0 1.000 0
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表 7 第4次嵌套子系统间的Kendall秩相关系数
Table 7. Kendall rank correlation coefficient between the forth nested subsystems
τ B ACDE B 1.000 0 0.363 0 ACDE 0.363 0 1.000 0
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表 8 每次嵌套子系统间的Kendall秩相关系数
Table 8. The Kendall rank correlation coefficient between nested subsystems each time
AIC(102) Clayton Gumbel Frank Gaussian t D-E −2.84 −3.31 −3.30 −3.18 −2.85 C-DE −3.54 −4.10 −4.25 −3.52 −3.75 A-CDE −4.15 −5.25 −5.55 −4.01 −5.53 B-ACDE −6.36 −6.96 −5.86 −5.26 −6.16
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表 9 五维相关系统可靠度
Table 9. Reliability of 5-dimensional related systems
分析方法 可靠度 理论值 0.982 1 嵌套Copula 0.9806 Clayton Copula 0.973 8 Frank Copula 0.979 1 Gaussian Copula 0.977 5 t Copula 0.971 8 Gumbel Copula 0.979 5
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