证据理论及其复杂系统可靠性分析方法与应用研究
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摘要
随着现代工业系统和制造装备逐渐向着大型化、复杂化和精密化的方向发展,其可靠性问题日益凸显。科学的分析和提高系统的可靠性已成为人们在现代工业系统和制造装备研制和使用过程中亟待解决的重要问题。然而,由于系统结构和性能机理的复杂性以及客观条件(人力、物力和时间等)的局限,在开展复杂系统的可靠性分析过程中往往难以得到充足的数据和信息,并且可得到的有限的信息中常常又包含着大量的主客观不确定性,传统的基于概率论或统计理论的可靠性分析方法在表征和量化这类不确定性时存在诸多局限,使可靠性分析结果缺乏可信性。这就迫使人们对复杂系统可靠性工程中主客观不确定性分析方法展开系统地研究,以满足现代工程中大型复杂系统和装备在可靠性分析方面的迫切需求。
本论文在国家自然科学基金项目“基于可能性和证据理论的机械系统可靠性分析和设计优化(50775026)”与国家863计划项目“数据不足时的重大装备可靠性分析与设计技术(2007AA04Z403)”的资助下,针对概率可靠性方法在理论和工程应用中存在的局限性,在研究证据理论本身存在问题的基础上,深入研究证据理论在复杂系统可靠性工程中主客观不确定性的量化与传递问题,建立相应的不确定性的量化框架和传递模型。
基于基础理论研究和模型方法的建立,结合航空发动机可靠性工程中不确定性的定量和定性分析,本论文的主要工作如下：
(1)提出了多源证据的量化分类算法。针对合成悖论处理方法的共同前提“在证据合成之前,冲突证据是明确知道的”这一不足,提出了多源证据的量化分类算法,以避免合成悖论的出现。量化分类算法以Jousselme证据距离为标准,描述证据差异性,并以核心向量为基准对多源证据进行量化分类。分析了基本信任分配函数的随机属性,建立了识别框架的可测空间,证明了识别框架的可测空间对可列交运算封闭,得到了识别框架的可测空间对有限并运算、有限交运算和差运算封闭的性质,定义了基本信任函数向量和基本信任函数矩阵。研究结果表明,本文所提出的多源证据的量化分类算法能有效地对多源证据进行分类,避免了直接应用Dempster合成规则出现悖论的情况。
(2)提出了新的证据冲突度量因子和证据相容合成方法。本论文针对证据理论中冲突因子不能完全描述证据冲突程度的情况,分析了目前几种合成悖论处理方法的不足,提出新的证据冲突度量因子,并证明其满足非负性、对称性和有界性。建立了证据间的相容度因子,根据多源证据间的相容程度分别确定证据的相容度向量和证据的修正因子,并对多源证据分别进行了修正。在充分应用冲突信息的基础上,提出了解决合成悖论的新证据合成方法。研究结果表明,本文所提出的新证据冲突度量因子能有效地度量证据冲突程度,并合理地处理合成悖论问题,在合成冲突证据与非冲突证据的信任“聚焦”能力方面具有明显优势。
(3)提出了基于证据理论的多源不确定性风险评估与排序方法。针对故障模式、影响和致命度分析中多个故障模式包含不确定性时的风险分析,提出了应用修正的证据理论模型来融合多个专家对同一故障模式的不同风险因子的不同评估的不确定性信息,建立了基于证据理论的风险优先数模型,给出了在不同情况下故障模式的每一个风险因子风险等级简化后的识别框架及相应的幂集。基本信任分配函数考虑了专家不同权重的修正,建立了相应的专家权重矩阵,构建了多专家对故障模式三个风险因子的不同评价信息的合成公式,并把合成后的每一故障模式的不同风险因子当作离散的随机变量,应用风险优先数的数学期望对多个故障模式进行风险评估排序。研究结果表明,本文所提出的基于证据理论的多源不确定性的风险评估与排序方法能有效地合成故障模式风险评估中的多专家不确定性评估信息。
(4)分析了基于证据理论和证据网络的复杂系统可靠性中不确定性的问题。针对故障树分析,特别是在设计阶段的故障树分析中,由于产品知识、实验数据和失效数据的缺少,很难准确地得到基本事件发生率的精确值的问题,考虑基本事件发生率的不确定性,应用证据理论来融合多源的不确定性评估信息。采用证据网络模型来量化和传递故障树中的不精确性和不确定性,发展了故障树转化成证据网络的模型,分别建立了故障树中一些逻辑门转化成证据网络的详细转化过程。给出了一些逻辑门在证据网络中的真值表和条件基本信任分配函数表。建立了故障树转化成证据网络的方法与步骤。给出了证据网络模型中,故障树概率重要度、结构重要度的计算方法,并提出了一种新的不精确重要度。针对某型航空发动机减速器传动系统可靠性预计中的不确定性,应用证据网络模型来量化和传递在数据信息缺少情况下行星齿轮系的失效率的不确定性,建立了该系统相应的证据网络模型。研究结果表明,证据网络能很好地量化和传递子部件的主客观不确定性,其中并联系统可以降低并联系统子部件所引起的系统的不确定性,而串联系统却放大了串联系统子部件引起的系统的不确定性。同时,当串联系统中只一个部件含有不确定性时,系统总的不确定性与其它部件无关。该结论为包含不确定性时的工程可靠性分析提供了定性的分析依据。
With the tendencies of great size and large tonnage, complexity and preciseness of the industrial systems and manufacturing equipments, their reliability performances are gaining an increasing concern. In order to ensure the general operations of the system, it reliability has become one of the key issues. During reliability analysis of complex systems, it is very difficult to attain enough statistical data because of its complexity of system structure and mechanism with the limits of the daily conditions (human sources, economic resources and time, etc.). Moreover, the initial data contain a great deal of epistemics and uncertainties. The traditional analysis method based on the probability theory has the obviously restrictions. Consequently, there is an urgent need for investigating the uncertainty in reliability engineering of complex system to facilitate the reliability assessment and enhancement for the complex systems.
Supported by the NSFC project:'Reliability analysis and design optimization of mechanical systems based on the possibility and evidence theory (50775026)'and the NHTRDP (863program):'Reliability analysis and design techniques of major equipment under the lack of data (2007AA04Z403)', in this dissertation, some drawbacks of probability theory and epistemics and random uncertainties quantification and propagation in reliability analysis has been considered based on the investigating the question of evidence theory. The questions of using evidence theory to quantify and propagate the uncertainties of complex system are studied. The corresponding quantitative framework and propagating model are constructed respectively.
In combination with theoretical researches and methodological method, integrating qualitative and quantitative uncertainty analysis of the aeroengine system, the dissertation proposes the following contributions:
(1) The algorithm of quantification classification of multiple sources of evidence is proposed. The stochastic interpretation of basic probability assignment function is analyzed. The measurable space of frame of discernment is constructed. It is proved that countable intersection of the measurable space of the frame of discernment is close. This space is satisfied with the property of closed under finite countable intersection, finite countable union and difference operation. The vector and matrix of basic probability assignment function are defined. A common precondition underlying methods of the combination paradox is that conflict evidence has been known and existed, which, however, is not always true. Moreover, it has been verified that the conflict factor cannot accurately characterize the degree of conflict. In order to avoid the counter-intuitive results, multiple sources of evidence should be classified firstly. This paper proposes a novel algorithm for quantification classification of multiple sources of evidence based on a core vector method and the Jousselme distance has been regarded as quantification criterion for the degree of conflict because of its promising properties. Demonstrated by numerical studies and examples, the proposed methodology can classify the multi-sources evidence effectively and avoid the paradox of combination using the Dempster combination rule.
(2) A combination rule of evidence compatibility is proposed based on the novel conflict degree of evidence. For the conflict factor of evidence can not fully describe the level of conflict between two pieces of evidence in Dempster-Shafer (Dempster) evidence theory, the deficiencies of present several methods to describe the conflict degree of evidence are analyzed and a novel conflict factor is proposed in this dissertation. The non-negativity, symmetry and boundedness of novel conflict factor are confirmed respectively. Based on the researches above, compatibility degree between two pieces of evidence is constructed according to the mutual compatibility degree among all pieces of evidence, the vector of compatibility degree is constructed. Then a correction factor of evidence is determined. The correction factor is used to modify the evidence respectively. A novel combination rule to deal with the conflict evidence is proposed. Based upon the proposed method, the novel combination rule can deal with the combination paradox effectively. When the novel combination rule is used to combine the evidence, the efficiency of combination to multi-sources evidence is better with comparison to other rules.
(3) The method of risk evaluation and ranking for failure modes and effects analysis is proposed using Dempster-Shafer evidence theory under uncertainty. Dempster evidence theory is adopted to aggregate the risk evaluation information of multiple experts, which may be inconsistent, imprecise and uncertain. The modified evidence theory is proposed for dealing with different opinions of multiple experts, multiple failure modes and three risk factors in risk priority number analysis of failure mode and effects analysis. In this method, the simplified frames of discernment are provided according to our practical engineering application. The different frame of discernment and the corresponding power set is constructed respectively. The basic probability assignment function of risk rank of different risk factor is modified by the expert weights weight matrix. The combination rule of different evaluation information of multiple experts on the three risk factors of failure modes is proposed. Meanwhile, the fused three risk factors are regarded as the discrete random variables. Consequently, the risk priority number is a function of the discrete random variable. The mean value of risk priority number is utilised to the risk priority ranking of failure modes. The proposed method is demonstrated by an application of risk priority ranking of failure modes in failure mode and effects analysis of compressor blades of an aeroengine. The consequence is demonstrated that the novel method can manage the uncertainty risk evaluation and ranking to failure modes in failure mode and effects analysis.
(4) Uncertainties of reliability analysis of complexity system based on the evidence network has been analyzed. Fault tree analysis, as one of the powerful tools in reliability engineering, has been widely used to enhance system quality attributes. In most cases of fault tree analyses, precise values are adopted to represent the probabilities of occurrence of those events. Due to the lack of sufficient data or imprecision of existing data at the early stage of product design, it is usually difficult to estimate the failure rates of individual events or the probabilities of occurrence of the events accurately. Therefore, such imprecision and uncertainty need to be taken into account in reliability analysis. The evidential networks is employed to quantify and propagate the aforementioned uncertainty and imprecision in fault tree analysis. The detailed conversion processing of some logiec gates to evidential networks is developed, respectively. The figures of the logic gates and the converted equivalent evidential networks, together with the associated truth tables and the conditional belief mass tables, are also presented in this work. The step of fault tree mapping into evidential networks is constructed. The computing of probability importance, structure importance is illustrated by evidential networks respectively. The novel uncertainty importance is proposed. For the design of the planetary gear is improve in the reducer, the information about experimental observations or tests is limited. Evidence network is adopted to quantify and propagate the uncertainty and imprecise probability in reliability prediction of improved aero-engine reducer. The evidence network of the system is constructed. The range of system reliability is attained by evidence theory. From the consequence of the computing, it is can be attained that the parallel system can decrease the uncertainty of the system because the sub-components includes the uncertainty in the parallel system. The series system can increase the uncertainty of the system because the sub-components involve the uncertainty in the series system. When one component has uncertainty in the series system, the overall uncertainty of the component is unrelated to the other components in system. This conclusion provides the qualitative criterion for uncertainty analysis in reliability engineering.
本论文在国家自然科学基金项目“基于可能性和证据理论的机械系统可靠性分析和设计优化(50775026)”与国家863计划项目“数据不足时的重大装备可靠性分析与设计技术(2007AA04Z403)”的资助下,针对概率可靠性方法在理论和工程应用中存在的局限性,在研究证据理论本身存在问题的基础上,深入研究证据理论在复杂系统可靠性工程中主客观不确定性的量化与传递问题,建立相应的不确定性的量化框架和传递模型。
基于基础理论研究和模型方法的建立,结合航空发动机可靠性工程中不确定性的定量和定性分析,本论文的主要工作如下：
(1)提出了多源证据的量化分类算法。针对合成悖论处理方法的共同前提“在证据合成之前,冲突证据是明确知道的”这一不足,提出了多源证据的量化分类算法,以避免合成悖论的出现。量化分类算法以Jousselme证据距离为标准,描述证据差异性,并以核心向量为基准对多源证据进行量化分类。分析了基本信任分配函数的随机属性,建立了识别框架的可测空间,证明了识别框架的可测空间对可列交运算封闭,得到了识别框架的可测空间对有限并运算、有限交运算和差运算封闭的性质,定义了基本信任函数向量和基本信任函数矩阵。研究结果表明,本文所提出的多源证据的量化分类算法能有效地对多源证据进行分类,避免了直接应用Dempster合成规则出现悖论的情况。
(2)提出了新的证据冲突度量因子和证据相容合成方法。本论文针对证据理论中冲突因子不能完全描述证据冲突程度的情况,分析了目前几种合成悖论处理方法的不足,提出新的证据冲突度量因子,并证明其满足非负性、对称性和有界性。建立了证据间的相容度因子,根据多源证据间的相容程度分别确定证据的相容度向量和证据的修正因子,并对多源证据分别进行了修正。在充分应用冲突信息的基础上,提出了解决合成悖论的新证据合成方法。研究结果表明,本文所提出的新证据冲突度量因子能有效地度量证据冲突程度,并合理地处理合成悖论问题,在合成冲突证据与非冲突证据的信任“聚焦”能力方面具有明显优势。
(3)提出了基于证据理论的多源不确定性风险评估与排序方法。针对故障模式、影响和致命度分析中多个故障模式包含不确定性时的风险分析,提出了应用修正的证据理论模型来融合多个专家对同一故障模式的不同风险因子的不同评估的不确定性信息,建立了基于证据理论的风险优先数模型,给出了在不同情况下故障模式的每一个风险因子风险等级简化后的识别框架及相应的幂集。基本信任分配函数考虑了专家不同权重的修正,建立了相应的专家权重矩阵,构建了多专家对故障模式三个风险因子的不同评价信息的合成公式,并把合成后的每一故障模式的不同风险因子当作离散的随机变量,应用风险优先数的数学期望对多个故障模式进行风险评估排序。研究结果表明,本文所提出的基于证据理论的多源不确定性的风险评估与排序方法能有效地合成故障模式风险评估中的多专家不确定性评估信息。
(4)分析了基于证据理论和证据网络的复杂系统可靠性中不确定性的问题。针对故障树分析,特别是在设计阶段的故障树分析中,由于产品知识、实验数据和失效数据的缺少,很难准确地得到基本事件发生率的精确值的问题,考虑基本事件发生率的不确定性,应用证据理论来融合多源的不确定性评估信息。采用证据网络模型来量化和传递故障树中的不精确性和不确定性,发展了故障树转化成证据网络的模型,分别建立了故障树中一些逻辑门转化成证据网络的详细转化过程。给出了一些逻辑门在证据网络中的真值表和条件基本信任分配函数表。建立了故障树转化成证据网络的方法与步骤。给出了证据网络模型中,故障树概率重要度、结构重要度的计算方法,并提出了一种新的不精确重要度。针对某型航空发动机减速器传动系统可靠性预计中的不确定性,应用证据网络模型来量化和传递在数据信息缺少情况下行星齿轮系的失效率的不确定性,建立了该系统相应的证据网络模型。研究结果表明,证据网络能很好地量化和传递子部件的主客观不确定性,其中并联系统可以降低并联系统子部件所引起的系统的不确定性,而串联系统却放大了串联系统子部件引起的系统的不确定性。同时,当串联系统中只一个部件含有不确定性时,系统总的不确定性与其它部件无关。该结论为包含不确定性时的工程可靠性分析提供了定性的分析依据。
With the tendencies of great size and large tonnage, complexity and preciseness of the industrial systems and manufacturing equipments, their reliability performances are gaining an increasing concern. In order to ensure the general operations of the system, it reliability has become one of the key issues. During reliability analysis of complex systems, it is very difficult to attain enough statistical data because of its complexity of system structure and mechanism with the limits of the daily conditions (human sources, economic resources and time, etc.). Moreover, the initial data contain a great deal of epistemics and uncertainties. The traditional analysis method based on the probability theory has the obviously restrictions. Consequently, there is an urgent need for investigating the uncertainty in reliability engineering of complex system to facilitate the reliability assessment and enhancement for the complex systems.
Supported by the NSFC project:'Reliability analysis and design optimization of mechanical systems based on the possibility and evidence theory (50775026)'and the NHTRDP (863program):'Reliability analysis and design techniques of major equipment under the lack of data (2007AA04Z403)', in this dissertation, some drawbacks of probability theory and epistemics and random uncertainties quantification and propagation in reliability analysis has been considered based on the investigating the question of evidence theory. The questions of using evidence theory to quantify and propagate the uncertainties of complex system are studied. The corresponding quantitative framework and propagating model are constructed respectively.
In combination with theoretical researches and methodological method, integrating qualitative and quantitative uncertainty analysis of the aeroengine system, the dissertation proposes the following contributions:
(1) The algorithm of quantification classification of multiple sources of evidence is proposed. The stochastic interpretation of basic probability assignment function is analyzed. The measurable space of frame of discernment is constructed. It is proved that countable intersection of the measurable space of the frame of discernment is close. This space is satisfied with the property of closed under finite countable intersection, finite countable union and difference operation. The vector and matrix of basic probability assignment function are defined. A common precondition underlying methods of the combination paradox is that conflict evidence has been known and existed, which, however, is not always true. Moreover, it has been verified that the conflict factor cannot accurately characterize the degree of conflict. In order to avoid the counter-intuitive results, multiple sources of evidence should be classified firstly. This paper proposes a novel algorithm for quantification classification of multiple sources of evidence based on a core vector method and the Jousselme distance has been regarded as quantification criterion for the degree of conflict because of its promising properties. Demonstrated by numerical studies and examples, the proposed methodology can classify the multi-sources evidence effectively and avoid the paradox of combination using the Dempster combination rule.
(2) A combination rule of evidence compatibility is proposed based on the novel conflict degree of evidence. For the conflict factor of evidence can not fully describe the level of conflict between two pieces of evidence in Dempster-Shafer (Dempster) evidence theory, the deficiencies of present several methods to describe the conflict degree of evidence are analyzed and a novel conflict factor is proposed in this dissertation. The non-negativity, symmetry and boundedness of novel conflict factor are confirmed respectively. Based on the researches above, compatibility degree between two pieces of evidence is constructed according to the mutual compatibility degree among all pieces of evidence, the vector of compatibility degree is constructed. Then a correction factor of evidence is determined. The correction factor is used to modify the evidence respectively. A novel combination rule to deal with the conflict evidence is proposed. Based upon the proposed method, the novel combination rule can deal with the combination paradox effectively. When the novel combination rule is used to combine the evidence, the efficiency of combination to multi-sources evidence is better with comparison to other rules.
(3) The method of risk evaluation and ranking for failure modes and effects analysis is proposed using Dempster-Shafer evidence theory under uncertainty. Dempster evidence theory is adopted to aggregate the risk evaluation information of multiple experts, which may be inconsistent, imprecise and uncertain. The modified evidence theory is proposed for dealing with different opinions of multiple experts, multiple failure modes and three risk factors in risk priority number analysis of failure mode and effects analysis. In this method, the simplified frames of discernment are provided according to our practical engineering application. The different frame of discernment and the corresponding power set is constructed respectively. The basic probability assignment function of risk rank of different risk factor is modified by the expert weights weight matrix. The combination rule of different evaluation information of multiple experts on the three risk factors of failure modes is proposed. Meanwhile, the fused three risk factors are regarded as the discrete random variables. Consequently, the risk priority number is a function of the discrete random variable. The mean value of risk priority number is utilised to the risk priority ranking of failure modes. The proposed method is demonstrated by an application of risk priority ranking of failure modes in failure mode and effects analysis of compressor blades of an aeroengine. The consequence is demonstrated that the novel method can manage the uncertainty risk evaluation and ranking to failure modes in failure mode and effects analysis.
(4) Uncertainties of reliability analysis of complexity system based on the evidence network has been analyzed. Fault tree analysis, as one of the powerful tools in reliability engineering, has been widely used to enhance system quality attributes. In most cases of fault tree analyses, precise values are adopted to represent the probabilities of occurrence of those events. Due to the lack of sufficient data or imprecision of existing data at the early stage of product design, it is usually difficult to estimate the failure rates of individual events or the probabilities of occurrence of the events accurately. Therefore, such imprecision and uncertainty need to be taken into account in reliability analysis. The evidential networks is employed to quantify and propagate the aforementioned uncertainty and imprecision in fault tree analysis. The detailed conversion processing of some logiec gates to evidential networks is developed, respectively. The figures of the logic gates and the converted equivalent evidential networks, together with the associated truth tables and the conditional belief mass tables, are also presented in this work. The step of fault tree mapping into evidential networks is constructed. The computing of probability importance, structure importance is illustrated by evidential networks respectively. The novel uncertainty importance is proposed. For the design of the planetary gear is improve in the reducer, the information about experimental observations or tests is limited. Evidence network is adopted to quantify and propagate the uncertainty and imprecise probability in reliability prediction of improved aero-engine reducer. The evidence network of the system is constructed. The range of system reliability is attained by evidence theory. From the consequence of the computing, it is can be attained that the parallel system can decrease the uncertainty of the system because the sub-components includes the uncertainty in the parallel system. The series system can increase the uncertainty of the system because the sub-components involve the uncertainty in the series system. When one component has uncertainty in the series system, the overall uncertainty of the component is unrelated to the other components in system. This conclusion provides the qualitative criterion for uncertainty analysis in reliability engineering.
引文
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