故障树分析的若干关键问题研究
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摘要
故障树分析(Fault Tree Analysis, FTA)技术是可靠性分析及故障诊断的有效工具,它在分析系统故障模式、寻找薄弱环节、指导故障维修等工作中具有重要的参考价值。
本文主要研究故障树分析理论中的若干关键问题及其实现。研究关键问题的目的在于:一方面,拓展故障树分析的工具,优化故障树结构,以提高故障树分析的效率;另一方面,拓展底事件的分析途径,寻找新的估计方法和评判策略,以改进底事件的分析手段。论文围绕下列线索展开:故障树分析→故障树优化→获取底事件的失效分布。就研究方式而言,主要涉及理论分析、算法实现和验证。就研究内容而言,主要包括:故障树的模块化分解;分析和拓展二元决策图(Binary Decision Diagram, BDD)的核心技术;获取非单调关联故障树的质蕴含集(Prime Implicant Set, PIS)及最小形“最小割集(Minimal Cut Set, MCS)”;改进底事件的失效分布判断及参数估计;基于模糊理论的底事件失效分析。
针对直接分析大型故障树存在效率低的弊端,研究了故障树的模块化分解技术。模块化的关键技术在于:用深度优先搜索(Depth-First Searching, DFS)技术获取所有节点的搜索步数,重点区分中间节点(逻辑门)和底事件的搜索方式。获取节点的搜索步数是模块形成的重要基础,因为只需判断任意节点(公共祖先)的时间区间是否包含其子节点中的最大时间区间,则可得到判断结果。
针对传统故障树分析方法存在繁杂、费时的缺点,研究了BDD技术用于故障树分析的原理和方法,其主要内容是:用递归方法实现子BDD间的连接规则;提出用“继承”技术实现不交化割集(关键技术在于:所有节点都直接“继承”其父节点所继承的信息,而以不同的方式“继承”父节点的信息);在分析最优指标顺序原理的基础上,实现了理论最优BDD结构的核心技术(基于指标代价恒定原理的最小节点代价的指标排序),阐明了存储目标节点所对应的真值表存在浪费时间和空间的缺点,分析了原方法只能获得唯一最优解的弊端,进而对最优指标顺序的实现加以改进(即:删除存储目标节点的真值表,改变获取最优解的设置,使算法能够获得所有最优解);作为最优BDD技术的一个不可分割的部分,研究了形式最优BDD结构的实现方法,主要任务是删除两个子节点等价情况下的父节点,其关键技术是:用深度优先的递归方法搜索BDD结构,通过判断BDD结构中的三个元素的状态(主要获取第二个元素和第三个元素的长度,判断这两个元素的值是否相等)来确定是否分支搜索、删除和重构BDD;为了提高故障树分析的效率,在分析最简割集BDD原理的基础上,研究了最优路径BDD结构的实现方法(实现冗余路径的“去除”操作),阐明了最优(节点数最少)BDD技术和最简割集(路径数最少)BDD技术的本质区别。
获取最小形“MCS”是获取非单调关联故障树的PIS的前提。在分析降阶递归结构函数方法的基础上,阐明了“非正规”BDD结构可能存在掩盖“非单调”底事件作用的弊端(在路径搜索中,不能区分某底事件到底是“单调”正常而不必考虑为故障模式的成员,还是“非单调”而必须考虑为故障模式的成员)。提出统一编码所有底事件的策略(即:将对立事件也考虑成单调状态),进而用“正规”BDD技术来获取非单调关联故障树的最小形“MCS”。
在底事件失效分布的判断上,直觉判断树叶图方法存在粗略的弊端。提出用相关系数方法来改进树叶图的直觉判断,使底事件失效分布的判断有了量的依据。其核心技术是:对原始实验数据进行小波变换,以提取低频轮廓,从而强化样本的真实分布和其它分布之间的差异(使真实分布对应的相关系数明显大于其余分布所对应的相关系数),有利于提高判断的准确性。单纯图估计或线性回归估计方法都存在着参数估计粗略的弊端(前者主观随意性强,后者具有“平均”成分),提出用智能优化算法来实现Weibull分布参数的线性回归——极大似然估计(Maximum Likelihood Estimation, MLE)。参数估计的关键技术在于:用粒子群优化(Particle Swarm Optimization, PSO)算法作为工具,联合估计分布参数(线性回归估计为PSO算法提供初始解,PSO算法则获取MLE的数字解)。
基于专家判断的模糊数能够在一定程度上克服底事件无失效数据的困难。为了量化专家判断的可信度,采用层次分析(Analytical Hierarchy Process, AHP)法确定专家的权重,从而代替专家判断的等权策略。在模糊判断的合成中,采用的关键技术是:以多元扩展原理为基础,用α-截集区间合成模糊数。
Fault Tree Analysis (FTA) has become an effective tool in field of reliability analysis and fault diagnosis, and it has important reference value in course of analyzing failure modes, looking for weaknesses and guiding failure repairing.
In this dissertation, several key problems and their implementations of FTA theory have been researched. The purposes of researching key details include two sides: One is expanding FTA tool and optimizing fault tree structure to improve analysis efficiency, and the other is expanding ways of analyzing bottom event and looking for new estimation methods and evaluation strategies to improve means of analyzing bottom event. Dissertation expands around the following clues: to make FTA→to research fault tree optimization→to research how to get failure distribution of bottom event. For research methodology, it mainly involves theoretical analysis, algorithm implementation and verification. For research content, it mainly relates to making modular decomposition to fault tree, analyzing and expanding key technologies of Binary Decision Diagram (BDD), making Prime Implicant Set (PIS) and“Minimal Cut Set”(“MCS”) from Non-Coherent Fault Tree (NCFT), improving determining method about failure distribution of bottom event and making parameter estimation, making failure analysis of bottom event based on fuzzy theory.
Aiming at the disadvantages of low efficiency to directly analyze large-scale fault tree, the modular decomposition technology applied to fault tree has been researched. Taken as a key technology, the method of Depth-First Searching (DFS) has been used to search steps of all nodes, and the key point is distinguishing searching mode to intermediate nodes (logic gates) and bottom events. Getting node steps is an important foundation for forming modules, for, any node whether a module or not can be determined only by means of judging inclusion relations between time interval of common ancestor and maximum time interval of its child nodes.
Due to inherent drawbacks of complexity and time-consuming character in method of traditional FTA, the principle and method of BDD technology used for FTA have been researched. The main contents are shown as follow: Connection rules and their recursive implementation between two sub-BDDs have been researched; A sort of technique called“inheritance”to make non-intersection cut set has been put forward (The key technology is that all nodes directly“inherit”node’s information his father-node has inherited, but“inherit”its father-node’s information with different ways); On the basis of analyzing principle of the best index order, the core technology to form theory optimal BDD structure has been achieved (to get index order of nodes with minimum cost based on principle of constant index cost), and at the same time, the disadvantage that storing truth table of target node will waste time and space has been stated, and the drawback that original method can only get one optimal solution has been analyzed too. The implementation of optimal index order has been improved (i.e. the truth table used for storing target node’s Boolean value has been deleted, and the setting of getting optimal solution of original method has been changed to obtain all the optimal solutions); As an integral part of optimal BDD, the implementation method of formal optimal BDD has been researched. Its main task is to remove father-node whose two child-nodes are equivalent, and the recursive method with DFS technology has been used for searching BDD structure. In this key technology, the states of three elements in BDD structure have been judged (mainly to get the length of the second and third element respectively, and to judge whether the values of the two elements are equal or not) to determine whether making branch searching, deleting and restructuring BDD or not; In order to improve the efficiency of FTA, the implementation of optimal path BDD has been researched, which is based on analyzing the principle of the simplest cut set BDD, and the essential differences between the optimal BDD (the least number of nodes) and the simplest cut set BDD (the least number of paths) have been illustrated.
Getting“MCS”is the premise to obtain PIS of NCFT. On the basis of analyzing the method of recursive structure function, the drawback that“irregular”BDD structure may cover up the function of“non-monotonic”bottom event has been analyzed (Searching can not distinguish whether one bottom event is“monotonous”and need not consider it a member of failure mode or is“non-monotonous”and must be considered as a member of failure mode). A strategy by means of coding all bottom events with unified method has been proposed, and then the“regular”BDD technology has been used to obtain“MCS”of NCFT.
Because intuitively judging leaf-figure has drawback with roughness in determining failure distribution of bottom event, a method of correlation coefficient has been put forward to improve the intuitive judgment, which make determining failure distribution has basis of amount. Taken as a core technology, wavelet transform is applied to original experimental data to extract low frequency contour, so that the difference between true distribution and other distributions is strengthened (i.e. to make correlation coefficient correspond to true distribution be significantly larger than others), which can increase the correctness of judgment. Because the figure estimation or linear regression estimation has drawback with roughness (The former has subjective and arbitrary factor, and the latter has“average”component), the Weibull distribution parameters have been estimated with the method of linear regression-Maximum Likelihood Estimation (MLE) by means of intelligent optimization algorithm. In this method, the Particle Swarm Optimization (PSO) algorithm is taken as a key tool to jointly estimate parameters (The linear regression estimation offers initial solution for PSO algorithm, and the PSO algorithm is used for getting digital solutions of MLE).
The Fuzzy numbers based on expert judgments may overcome the difficulties that the bottom events have no failure data. To quantify the experts’credibility, the method of Analytical Hierarchy Process (AHP) has been used to determine experts’weights to replace strategy of average weight. In course of pooling fuzzy judgments, theα-cut set has been used to form Fuzzy numbers, which is based on multiple extension principle.
本文主要研究故障树分析理论中的若干关键问题及其实现。研究关键问题的目的在于:一方面,拓展故障树分析的工具,优化故障树结构,以提高故障树分析的效率;另一方面,拓展底事件的分析途径,寻找新的估计方法和评判策略,以改进底事件的分析手段。论文围绕下列线索展开:故障树分析→故障树优化→获取底事件的失效分布。就研究方式而言,主要涉及理论分析、算法实现和验证。就研究内容而言,主要包括:故障树的模块化分解;分析和拓展二元决策图(Binary Decision Diagram, BDD)的核心技术;获取非单调关联故障树的质蕴含集(Prime Implicant Set, PIS)及最小形“最小割集(Minimal Cut Set, MCS)”;改进底事件的失效分布判断及参数估计;基于模糊理论的底事件失效分析。
针对直接分析大型故障树存在效率低的弊端,研究了故障树的模块化分解技术。模块化的关键技术在于:用深度优先搜索(Depth-First Searching, DFS)技术获取所有节点的搜索步数,重点区分中间节点(逻辑门)和底事件的搜索方式。获取节点的搜索步数是模块形成的重要基础,因为只需判断任意节点(公共祖先)的时间区间是否包含其子节点中的最大时间区间,则可得到判断结果。
针对传统故障树分析方法存在繁杂、费时的缺点,研究了BDD技术用于故障树分析的原理和方法,其主要内容是:用递归方法实现子BDD间的连接规则;提出用“继承”技术实现不交化割集(关键技术在于:所有节点都直接“继承”其父节点所继承的信息,而以不同的方式“继承”父节点的信息);在分析最优指标顺序原理的基础上,实现了理论最优BDD结构的核心技术(基于指标代价恒定原理的最小节点代价的指标排序),阐明了存储目标节点所对应的真值表存在浪费时间和空间的缺点,分析了原方法只能获得唯一最优解的弊端,进而对最优指标顺序的实现加以改进(即:删除存储目标节点的真值表,改变获取最优解的设置,使算法能够获得所有最优解);作为最优BDD技术的一个不可分割的部分,研究了形式最优BDD结构的实现方法,主要任务是删除两个子节点等价情况下的父节点,其关键技术是:用深度优先的递归方法搜索BDD结构,通过判断BDD结构中的三个元素的状态(主要获取第二个元素和第三个元素的长度,判断这两个元素的值是否相等)来确定是否分支搜索、删除和重构BDD;为了提高故障树分析的效率,在分析最简割集BDD原理的基础上,研究了最优路径BDD结构的实现方法(实现冗余路径的“去除”操作),阐明了最优(节点数最少)BDD技术和最简割集(路径数最少)BDD技术的本质区别。
获取最小形“MCS”是获取非单调关联故障树的PIS的前提。在分析降阶递归结构函数方法的基础上,阐明了“非正规”BDD结构可能存在掩盖“非单调”底事件作用的弊端(在路径搜索中,不能区分某底事件到底是“单调”正常而不必考虑为故障模式的成员,还是“非单调”而必须考虑为故障模式的成员)。提出统一编码所有底事件的策略(即:将对立事件也考虑成单调状态),进而用“正规”BDD技术来获取非单调关联故障树的最小形“MCS”。
在底事件失效分布的判断上,直觉判断树叶图方法存在粗略的弊端。提出用相关系数方法来改进树叶图的直觉判断,使底事件失效分布的判断有了量的依据。其核心技术是:对原始实验数据进行小波变换,以提取低频轮廓,从而强化样本的真实分布和其它分布之间的差异(使真实分布对应的相关系数明显大于其余分布所对应的相关系数),有利于提高判断的准确性。单纯图估计或线性回归估计方法都存在着参数估计粗略的弊端(前者主观随意性强,后者具有“平均”成分),提出用智能优化算法来实现Weibull分布参数的线性回归——极大似然估计(Maximum Likelihood Estimation, MLE)。参数估计的关键技术在于:用粒子群优化(Particle Swarm Optimization, PSO)算法作为工具,联合估计分布参数(线性回归估计为PSO算法提供初始解,PSO算法则获取MLE的数字解)。
基于专家判断的模糊数能够在一定程度上克服底事件无失效数据的困难。为了量化专家判断的可信度,采用层次分析(Analytical Hierarchy Process, AHP)法确定专家的权重,从而代替专家判断的等权策略。在模糊判断的合成中,采用的关键技术是:以多元扩展原理为基础,用α-截集区间合成模糊数。
Fault Tree Analysis (FTA) has become an effective tool in field of reliability analysis and fault diagnosis, and it has important reference value in course of analyzing failure modes, looking for weaknesses and guiding failure repairing.
In this dissertation, several key problems and their implementations of FTA theory have been researched. The purposes of researching key details include two sides: One is expanding FTA tool and optimizing fault tree structure to improve analysis efficiency, and the other is expanding ways of analyzing bottom event and looking for new estimation methods and evaluation strategies to improve means of analyzing bottom event. Dissertation expands around the following clues: to make FTA→to research fault tree optimization→to research how to get failure distribution of bottom event. For research methodology, it mainly involves theoretical analysis, algorithm implementation and verification. For research content, it mainly relates to making modular decomposition to fault tree, analyzing and expanding key technologies of Binary Decision Diagram (BDD), making Prime Implicant Set (PIS) and“Minimal Cut Set”(“MCS”) from Non-Coherent Fault Tree (NCFT), improving determining method about failure distribution of bottom event and making parameter estimation, making failure analysis of bottom event based on fuzzy theory.
Aiming at the disadvantages of low efficiency to directly analyze large-scale fault tree, the modular decomposition technology applied to fault tree has been researched. Taken as a key technology, the method of Depth-First Searching (DFS) has been used to search steps of all nodes, and the key point is distinguishing searching mode to intermediate nodes (logic gates) and bottom events. Getting node steps is an important foundation for forming modules, for, any node whether a module or not can be determined only by means of judging inclusion relations between time interval of common ancestor and maximum time interval of its child nodes.
Due to inherent drawbacks of complexity and time-consuming character in method of traditional FTA, the principle and method of BDD technology used for FTA have been researched. The main contents are shown as follow: Connection rules and their recursive implementation between two sub-BDDs have been researched; A sort of technique called“inheritance”to make non-intersection cut set has been put forward (The key technology is that all nodes directly“inherit”node’s information his father-node has inherited, but“inherit”its father-node’s information with different ways); On the basis of analyzing principle of the best index order, the core technology to form theory optimal BDD structure has been achieved (to get index order of nodes with minimum cost based on principle of constant index cost), and at the same time, the disadvantage that storing truth table of target node will waste time and space has been stated, and the drawback that original method can only get one optimal solution has been analyzed too. The implementation of optimal index order has been improved (i.e. the truth table used for storing target node’s Boolean value has been deleted, and the setting of getting optimal solution of original method has been changed to obtain all the optimal solutions); As an integral part of optimal BDD, the implementation method of formal optimal BDD has been researched. Its main task is to remove father-node whose two child-nodes are equivalent, and the recursive method with DFS technology has been used for searching BDD structure. In this key technology, the states of three elements in BDD structure have been judged (mainly to get the length of the second and third element respectively, and to judge whether the values of the two elements are equal or not) to determine whether making branch searching, deleting and restructuring BDD or not; In order to improve the efficiency of FTA, the implementation of optimal path BDD has been researched, which is based on analyzing the principle of the simplest cut set BDD, and the essential differences between the optimal BDD (the least number of nodes) and the simplest cut set BDD (the least number of paths) have been illustrated.
Getting“MCS”is the premise to obtain PIS of NCFT. On the basis of analyzing the method of recursive structure function, the drawback that“irregular”BDD structure may cover up the function of“non-monotonic”bottom event has been analyzed (Searching can not distinguish whether one bottom event is“monotonous”and need not consider it a member of failure mode or is“non-monotonous”and must be considered as a member of failure mode). A strategy by means of coding all bottom events with unified method has been proposed, and then the“regular”BDD technology has been used to obtain“MCS”of NCFT.
Because intuitively judging leaf-figure has drawback with roughness in determining failure distribution of bottom event, a method of correlation coefficient has been put forward to improve the intuitive judgment, which make determining failure distribution has basis of amount. Taken as a core technology, wavelet transform is applied to original experimental data to extract low frequency contour, so that the difference between true distribution and other distributions is strengthened (i.e. to make correlation coefficient correspond to true distribution be significantly larger than others), which can increase the correctness of judgment. Because the figure estimation or linear regression estimation has drawback with roughness (The former has subjective and arbitrary factor, and the latter has“average”component), the Weibull distribution parameters have been estimated with the method of linear regression-Maximum Likelihood Estimation (MLE) by means of intelligent optimization algorithm. In this method, the Particle Swarm Optimization (PSO) algorithm is taken as a key tool to jointly estimate parameters (The linear regression estimation offers initial solution for PSO algorithm, and the PSO algorithm is used for getting digital solutions of MLE).
The Fuzzy numbers based on expert judgments may overcome the difficulties that the bottom events have no failure data. To quantify the experts’credibility, the method of Analytical Hierarchy Process (AHP) has been used to determine experts’weights to replace strategy of average weight. In course of pooling fuzzy judgments, theα-cut set has been used to form Fuzzy numbers, which is based on multiple extension principle.
引文
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