文章快速检索 高级检索

Reliability evaluation of radar power amplification system considering epistemic uncertainty
PAN Gang , SHANG Chaoxuan , LIANG Yuying , CAI Jinyan , MENG Yafeng
Department of Electronic and Optic Engineering, Ordnance Engineering College, Shijiazhuang 050003, China
Received: 2015-06-15; Accepted: 2015-08-17; Published online: 2016-06-20 12: 00
Foundation item: National Natural Science Foundation of China (61372039,61271153)
Abstract: There is epistemic uncertainty in degradation law of a high-reliability component because it is hard to obtain its adequate performance data within a short time,and system reliability cannot be accurately estimated. For the purpose of accurate estimation of system reliability,assuming that the component performance distribution parameter was the interval variable,a component' performance parameter distribution model was built based on interval parameter variable and a computational method of the interval-valued state probability was provided. The boundary of the state performance interval was compensated,the interval-valued universal generating function and its algorithm were defined,an assessment method of interval-valued reliability of the multi-state system in consideration of epistemic uncertainty was proposed,and the reliability of a power amplification system was analyzed. This method not only avoids the lack of distribution information of the performance parameters,but also accurately builds a distribution model of the performance parameters. Therefore,it is widely used in engineering.
Key words: epistemic uncertainty     universal generating function     multi-state system     power amplification system     reliability evaluation

20世纪70年代多态系统的概念被提出后[3-4],关于其可靠性的研究就得到了学者的广泛关注。在理论方法方面,文献[1-2, 5-6]对基本概念、评估方法和优化设计等进行了详细阐述。文献[7]对基于不完全维修的多态系统更换维修决策问题进行了深入的研究。在工程应用方面,相关理论已经应用到电力[8-9]、机械[10-11]等领域。

1 多态系统状态分析 1.1 多态部件状态分析

1.1.1 部件性能分布模型

1) 部件只有一个性能参数x,对应一个性能退化过程,且退化过程不可逆。

2) 在任意时刻t,假设部件的性能参数为x(t),服从均值为μx(t)、方差为σ2x(t)的正态分布,μx(t)σ2x(t)分别为在[μ—(t),μ(t)]和[σ2(t),σ2(t)]上服从均匀分布的区间变量,x(t)为独立同分布。

3) 部件在任意给定时刻t,性能参数x(t)的分布参数为服从均匀分布的区间变量,则此时部件的性能分布函数为

 (1)

1.1.2 部件性能区间划分准则

1) 将部件i的性能参数区间划分为Mi=3个连续的子区间,以此为例进行说明。

(1) 在给定t时刻,部件i的性能参数服从均值为μi(t)、方差为σ2i(t)的正态分布,μi(t)=μci(t),σ2i(t)=σ2,ci(t)分别为区间[μi(t)]和[σ2i(t)]的中值。以标准差σi(t)为间隔将部件的整个性能参数分布区间划分为8部分,具体如图 1所示。

 图 1 部件性能参数分布 Fig. 1 Performance parameter distribution of components

(2) 根据系统的最小任务性能区间需求和系统结构函数确定分配到部件的近似最小任务性能参数区间为[wx],且其中值为wcx,并在该区域附近尽量将其细化,增大系统状态性能与系统的最小任务性能需求区分度,提高系统可靠度的计算精度。

(3) 假定部件i的最大性能参数为xi,max,为便于表示,假定xi,ε1=μi－3σi,xi,ε2=μi－2σi,xi,ε3=μiσi,xi,ε4=μi,xi,ε5=μi+σi,xi,ε6=μi+2σi,xi,ε7=μi+3σi。假定部件的性能参数x的区间族由3个连续的性能区间构成,对部件的性能参数区间进行划分时,结合(1)和(2),根据wxc与{xi,ε1,xi,ε2，…,xi,ε7}中各元素的大小关系进行划分。

① 当wxcxi,ε1时,

② 当xijwxcxi,εj+1时,

③ 当xi,ε7wxcxi,max时,

④ 当wxcxi,max时,表明部件i的性能参数大小已不满足系统的需求,对其进行性能参数区间的划分已经没有实际的工程意义,故此时可认为部件i失效。

2) 部件i的性能参数区间个数为Mi=2时,

3) 部件i的性能区间个数为Mi时,Mi＞3,

1.2 部件状态概率分析

1.2.1 部件状态性能区间定义

1) 当部件i单独构成分系统l,且分系统l可以与其他分系统近似地认为是串联结构,可得

2) 当ni个部件i构成分系统l时,且分系统l的性能输出为ni个部件i输出和时,分系统l可以与其他分系统近似地认为是串联结构,可得

1.2.2 部件状态概率求解

 (2)

2 运算法则

2.1 状态性能区间边界补偿

1) 状态性能水平区间上边界补偿。假定t时刻部件i的第ki个状态性能水平区间为[gi,ki]=[gi,ki,gi,ki],则性能区间上边界为gi,ki,根据概率论中参数区间估计方法,在方差σ2已知的情况下,将补偿半径定义为

2) 状态性能水平区间下边界补偿。同理,根据状态性能水平区间上边界补偿分析,可得状态性能水平区间下边界gi,ki的最大补偿半径为

3) 状态性能水平区间中值补偿。同理,根据状态性能水平区间上边界补偿分析,可得状态性能水平区间中值gi,kic的最大补偿半径为

2.2 区间通用生成函数定义

 (3)
2.3 区间通用生成函数运算法则

 (4)

1) 当[gks]为[gi,ki]与[gj,kj]的和时,定义δ1运算符如下：

2) 当[gks]为[gi,ki]与[gj,kj]的乘积时,定义δ2运算符如下：

3) 当[gks]为[gi,ki]与[gj,kj]的最小值时,定义δ3运算符如下：

3 多态系统可靠性分析

 (5)

1) 若，则

2) 若,则

3) 若,可定义[gks]－[w]的可能度为

 (6)
4 多态系统平均瞬态区间性能分析

 (7)

5 算例分析

 图 2 某型雷达功率放大分系统 Fig. 2 Radar power amplification subsystem

 部件 分布类型 性能参数分布/d 部件状态水平/% 分系统结构函数 1∶4功率分配器 指数分布 λ=[6.25,8.25]× 10-5 0,100 分系统2 G2(t)=X2(t) 4∶1功率合成器 指数分布 λ=[6.25,8.25]× 10-5 0,100 分系统4 G4(t)=X4(t)

 部件 部件退化性能参数分布 分系统结构函数 系统结构函数(状态水平) 预放大器 组件 x1服从[0,6]范围内的正态分布,其中, μ1(t)=([5.445,5.565]-[1.05,1.25])10-4t, σ1(t)=([0.105,0.135]-[1.65,1.85])10-5t 分系统1 G1(t)=X1(t) 分系统3 G3(t)=X3(t)+X4(t)+…+ X6(t) Gsys(t)=min{G2(t), Gsub13(t),G4(t)} 功率放大 器组件 x3服从[0,75]范围内的正态分布,其中, μ3(t)=([68.55,69.55]-[0.85,1.05])10-3t, σ3(t)=([1.25,1.45]-[1.85,2.05])10-4t 分系统13 Gsub13(t)=G1(t)·G3(t)

5.1 基于状态性能区间边界补偿的系统可靠性评估

5.1.1 部件区间通用生成函数

t=1000d时,根据第2.2节性能水平划分规则,假定预放大器组件的性能参数区间族为{x1}={[x1,1],[x1,2],[x1,3],[x1,4],[x1,5]},其中,性能参数区间分别为[x1,1]=[0,4.9775],[x1,2]=[4.9775,5.1150],[x1,3]=[5.1150,5.2525],[x1,4]=[5.2525,5.3900],[x1,5]=[5.3900,6.0000]。

5.1.2 基于状态性能区间中值补偿的多态系统可靠性评估

5.2 对比分析

5.2.1 基于二态模型的系统可靠性评估

t=1000d时,系统的可靠度为 Rtrda(t)=0.5221

5.2.2 基于性能区间方法的系统可靠性评估

5.2.3 基于Monte Carlo仿真的系统可靠性评估

t=1000d时,采用Monte Carlo仿真方法进行分析。在分析过程中,因分系统3所含部件较多,部件的状态数Ncomponet3≥20,分系统的状态数Nsubsys3≥204,仿真规模巨大,使得后续对系统可靠性的分析面临严重的挑战,为了降低计算的复杂度和仿真规模,假设Ncomponet3=20,为了充分表述Monte Carlo仿真方法在每次仿真计算过程中所得计算结果的可能性,取L=500次仿真结果的平均值作为Monte Carlo仿真方法的结果,可得可靠度为0.8070,其中,实现L=500次仿真时间大概为7980s,而本文方法耗时大约为40s。

 图 3 不同方法所得可靠度结果对比 Fig. 3 Comparison of reliability results respectively obtained by different methods

1) 采用本文所提状态性能区间边界补偿方法,分别对状态性能区间下边界、中值和上边界

2) 将传统可靠性方法、状态性能区间中值补偿方法和Monte Carlo仿真方法三者所得结果进行对比,传统可靠性方法与后两者相比误差较大。在传统可靠性方法中,部件失效阈值的选取对系统可靠性的影响较为敏感,因此其成为影响系统可靠度评估精度的薄弱环节,进一步体现了多态系统理论在描述系统可靠性时的准确性优势。

3) 状态性能区间中值补偿方法、性能区间方法和Monte Carlo仿真方法三者相比,状态性能区间中值补偿方法的结果与Monte Carlo仿真方法的结果更加接近,虽然随着时间的推移,误差具有增大的趋势,但实际工程中,雷达装备每隔2～3a(700～1200d)进行一次大修,状态性能区间中值补偿方法的分析精度已经满足预防性维修的需求,因此其克服了Monte Carlo仿真方法仿真规模大、耗时长的不足。

5.3 雷达功率放大分系统平均瞬态性能分析

 方法 t/d 400 600 800 1000 1200 上边界补偿方法 [81.66%,84.77%] [78.43%,81.51%] [75.26%,78.83%] [68.53%,71.44%] [65.20%,68.05%] 中值补偿方法 [72.68%,79.42%] [73.66%,76.61%] [69.89%,72.77%] [66.01%,68.80%] [62.02%,64.72%] 下边界补偿方法 [71.91%,74.75%] [67.23%,67.97%] [62.13%,64.74%] [56.78%,59.24%] [51.20%,53.48%] 性能区间方法 [73.25%,83.14%] [69.06%,78.57%] [65.00%,74.62%] [60.50%,69.64%] [56.54%,65.70%]

 图 4 不同方法所得平均瞬态性能结果对比 Fig. 4 Comparison of mean instantaneous performance respectively obtained by different methods

1) 采用本文所提状态性能区间边界补偿方法,分别对状态性能区间下边界、中值和上边界进行补偿,所得平均瞬态性能结果整体上呈增大的趋势。

2) 状态性能区间方法所得平均瞬态性能结果介于状态性能区间下边界补偿方法和状态性能区间上边界补偿方法之间,与状态性能区间中值相近。

3) 采用上述方法所得的平均瞬态性能结果与可靠性评估结果是相对应的,即其所求平均瞬态性能的值较大时,其对应的可靠度的值较大,因此平均瞬态性能可作为可靠度分析结果的有效补充,也可作为评价系统可靠性的指标。

6 结论

1) 利用本文定义的连续状态部件的状态划分规则和给出的部件状态及其对应概率随时间变化关系的求解方法,解决了给定状态和对应状态概率对可靠性分析的影响,得到了多态系统可靠性随时间的动态变化关系。

2) 通过解析法获得部件状态及其对应概率随时间变化关系,采用区间通用生成函数对系统可靠性进行分析,克服了贝叶斯等方法对专家知识的依赖性。

3) 本文方法适用于连续状态系统的可靠性分析,是对多态系统可靠性分析方法的有效补充。

4) 通过算例分析验证了本文方法的正确性和有效性,且结果表明所提方法相比于传统可靠性方法、性能区间方法和Monte Carlo仿真方法具有一定优势。

 [1] LISNIANSKI A, LEVITIN G. Multi-state system reliability:Assessment,optimization and applications[M]. Singapore: World Scientific, 2003 : 1 -2. Click to display the text [2] 李春洋.基于多态系统理论的可靠性分析与优化设计方法研究[D].长沙:国防科学技术大学,2010: 1-5.LI C Y. Research on reliability analysis and optimization based on the multi-state system theory[D].Changsha:National University of Defense Technology,2010:1-5(in Chinese). (in Chinese). Cited By in Cnki (0) | Click to display the text [3] 李春洋, 陈循, 易晓山, 等. 基于向量通用生成函数的多性能参数多态系统可靠性分析[J]. 兵工学报,2010, 31 (12) : 1604 –1610. C Y, CHEN X, YI X S, et al. Reliability analysis of multi-state system with multiple performance parameters based on vector universal generating function[J]. Acta Armamentarii,2010, 31 (12) : 1604 –1610. (in Chinese). Cited By in Cnki (0) | Click to display the text [4] BARLOW R E, WU A S. Coherent systems with multi-state components[J]. Mathematics of Operations Research,1978, 3 (4) : 275 –281. Click to display the text [5] NATVIG B. Multi-state systems reliability theory with applications[M]. New York: Wiley, 2011 : 11 -30. Click to display the text [6] LEVITIN G. The universal generating function in reliability analysis and optimization[M]. London: Springer, 2005 : 99 -262. Click to display the text [7] 刘宇.多状态复杂系统可靠性建模及维修决策[D].成都:电子科技大学,2011:48-100.LIU Y.Multi-state complex system reliability modeling and maintenance decision[D].Chengdu:University of Electronic Science and Technology of China,2011:48-100(in Chinese). (in Chinese). Cited By in Cnki (0) | Click to display the text [8] LI Y, ZIO E. A multi-state model for the reliability assessment of a distributed generation system via universal generating function[J]. Reliability Engineering and System Safety,2012, 106 : 28 –36. Click to display the text [9] TABOADA H A, ESPIRITU J F, COIT D W. Design allocation of multi-state series-parallel systems for power systems planning:A multiple objective evolutionary approach[J]. Journal of Reliability and Safety,2008, 222 (3) : 381 –391. Click to display the text [10] 史新红, 齐先军, 王治国. 基于UGF的发电系统区间可靠性评估及其仿射算法改进[J]. 合肥工业大学学报(自然科学版),2014, 37 (3) : 286 –291. X H, QI X J, WANG Z G. Interval reliability estimation of power generating system based on UGF method and its modification by using affine arithmetic[J]. Journal of Hefei University of Technology(Natural Science),2014, 37 (3) : 286 –291. (in Chinese). Cited By in Cnki (0) | Click to display the text [11] 尚彦龙, 蔡琦, 赵新文, 等. 基于UGF和Semi-Markov方法的反应堆泵机组多状态可靠性分析[J]. 核动力工程,2012, 33 (1) : 117 –123. Y L, CAI Q, ZHAO X W, et al. Multi-state reliability for pump group in nuclear power system based on UGF and Semi-Markov process[J]. Nuclear Power Engineering,2012, 33 (1) : 117 –123. (in Chinese). Cited By in Cnki (0) | Click to display the text [12] DING Y, ZUO M J, LISNIANSKI A, et al. Fuzzy multi-state systems:General definitions,and performance assessment[J]. IEEE Transactions on Reliability,2008, 57 (4) : 589 –594. Click to display the text [13] DING Y, LISNIANSKI A. Fuzzy universal generating functions for multi-state system reliability assessment[J]. Fuzzy Sets and Systems,2008, 159 (3) : 307 –324. Click to display the text [14] LI C Y, CHEN X, YI X S. Interval-valued reliability analysis of multi-state systems[J]. IEEE Transactions on Reliability,2011, 60 (1) : 323 –330. Click to display the text [15] DESTERCKE S, SALLAK M. An extension of universal generating function in multi-state systems considering epistemic uncertainties[J]. IEEE Transactions on Reliability,2013, 62 (2) : 504 –514. Click to display the text [16] 鄢民强, 杨波, 王展. 不完全覆盖的模糊多状态系统可靠性计算方法[J]. 西安交通大学学报,2011, 45 (10) : 109 –114. M Q, YANG B, WANG Z. Reliability assessment for multi-state system subject to imperfect fault coverage[J]. Journal of Xi'an Jiaotong University,2011, 45 (10) : 109 –114. (in Chinese). Cited By in Cnki (0) | Click to display the text [17] LIU Y, HUANG H. Reliability assessment for fuzzy multi-state system[J]. International Journal of Systems Science,2010, 41 (4) : 365 –379. Click to display the text [18] 陈东宁, 姚成玉. 基于模糊贝叶斯网络的多态系统可靠性分析及在液压系统中的应用[J]. 机械工程学报,2012, 48 (16) : 175 –183. D N, YAO C Y. Reliability analysis of multi-state system based on fuzzy Bayesian networks and application in hydraulic system[J]. Chinese Journal of Mechanical Engineering,2012, 48 (16) : 175 –183. (in Chinese). Cited By in Cnki (0) | Click to display the text [19] 马德仲, 周真, 于晓洋, 等. 基于模糊概率的多状态贝叶斯网络可靠性分析[J]. 系统工程与电子技术,2012, 34 (12) : 2607 –2611. D Z, ZHOU Z, YU X Y, et al. Reliability analysis of multi-state Bayesian networks based on fuzzy probability[J]. Systems Engineering and Electronics,2012, 34 (12) : 2607 –2611. (in Chinese). Cited By in Cnki (0) | Click to display the text [20] 陈东宁, 姚成玉, 党振. 基于T-S模糊故障树和贝叶斯网络的多态液压系统可靠性分析[J]. 中国机械工程,2013, 24 (7) : 899 –905. D N, YAO C Y, DANG Z. Reliability analysis of multi-state hydraulic system based on T-S fuzzy fault tree and Bayesian network[J]. China Mechanical Engineering,2013, 24 (7) : 899 –905. (in Chinese). Cited By in Cnki (0) | Click to display the text [21] 邱志平. 非概率集合理论凸方法及其应用[M]. 北京: 国防工业出版社, 2005 : 30 -43. Z P. Convex method of non-probabilistic set-theory and its application[M]. Beijing: National Defense Industry Press, 2005 : 30 -43. (in Chinese). Cited By in Cnki (0) | Click to display the text [22] 高峰记. 可能度及区间数综合排序[J]. 系统工程理论与实践,2013, 33 (8) : 2033 –2040. F J. Possibility degree and comprehensive priority of interval numbers[J]. Systems Engineering-Theory and Practice,2013, 33 (8) : 2033 –2040. (in Chinese). Cited By in Cnki (0) | Click to display the text

#### 文章信息

PAN Gang, SHANG Chaoxuan, LIANG Yuying, CAI Jinyan, MENG Yafeng

Reliability evaluation of radar power amplification system considering epistemic uncertainty

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(6): 1185-1194
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0390