﻿ 基于聚类状态主控边界点的单调多态关联系统可靠性分析
 文章快速检索 高级检索

1. 安徽工业大学 数理科学与工程学院, 马鞍山 243002;
2. 南京航空航天大学 民航学院, 南京 211106;
3. 扬州大学 机械工程学院, 扬州 225127

Reliability analysis for multi-state coherent system with monotonic components based on pivotal boundary points of clustering states
ZHANG Yongjin1, SUN Youchao2, ZHANG Yanjun3
1. School of Mathematics and Physics, Anhui University of Technology, Maanshan 243002, China;
2. College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China;
3. College of Mechanical Engineering, Yangzhou University, Yangzhou 225127, China
Received: 2016-10-19; Revised: 2017-02-26; Accepted: 2017-04-05; Published online: 2017-04-19 13:31
Foundation item: National Natural Science Foundation of China (U1333119, 60979019, 60572171, 51605424, 71601002); CAAC Science and Technology Project (MHRD201123, MHRD200908, MHRD0722); Natural Science Foundation of Jiangsu Province (BK20150455)
Corresponding author. SUN Youchao, E-mail: sunyc@nuaa.edu.cn
Abstract: Considering the monotone and coherence of the multi-state system, the multiple discrete function theory is introduced to describe the structure function of system state. The logic approaches for the equivalence class of the component state which control the state vector of system are proposed, and the expressions for the state structure function, reliability and expected states are derived for the multi-state coherent system. To avoid the complexity of computation caused by the number of the state, the Demogen law and the new block diagram algorithm are developed to simplify the expression for the system reliability. An illustrative example of a certain type of aero engine verifies the effectiveness of the logic vector measure controlling the state equivalence class and the block diagram algorithm. It provides theoretical basis for reliability design and reliability management of system engineering.
Key words: pivotal boundary point     discrete function     multi-state coherent system     monotonicity     reliability

1 单调多态关联系统

φ为系统状态的结构函数。

x为系统各元件所处状态构成的系统状态向量，ΩCi为第i个部件状态空间，ΩS为系统状态空间，ΩCn为系统元件的状态空间，φ为构成系统的元件状态组合到系统状态的映射，则有相关标记

① 规范性(边界条件)。若系统所有部件失效，则系统处于完全失效状态；若系统所有部件处于完美状态，则系统处于完美状态，即有

② 可达性。对于系统的每一个状态k，至少有一个部件状态向量x=[x1, x2, …, xn]满足

③ 关联性。系统中每个元件均具有其特定作用，不存在无关元件，即系统中不存在不关联元件。若标记元件状态向量

④ 单调性。部件性能改进不会导致系统性能恶化，即系统元件的状态函数φ(x)是单调增的。若2个部件状态向量满足序xy，则系统状态向量满足φ(x) > φ(y)；若向量序xy，则系统状态向量满足φ(x)≤φ(y)。

2 可靠状态类与边界

3 单调MSCS可靠性

3.1 离散函数析取运算式

 （1）

 （2）

 （3）

3.2 边界点结构函数

 （4）

 （5）

 （6）

 （7）

3.3 MSCS可靠度计算

 （8）

 （9）

 （10）

 （11）

 （12）

 （13）

 （14）

3.4 系统性能效用

 （15）

 （16）

 （17）

4 算例分析

 图 1 双转子航空发动机系统 Figure 1 Twin-rotor aero engine system

 State description Boost ratio Contributions State x1 Probability Total failure 0~7 0 0 0.02 Partial failure 8~9 100 1 0.04 Perfect function 10~12 200 2 0.94

 State description Failure mechanism Contri-butions State x2 Proba-bility Total failure Failed ignition 0 0 0.01 Partial function Successful ignition/Unreliable combustion 60 1 0.11 Perfect function Successful ignition success/Reliable combustion 120 2 0.88

 State description Failure mechanism Contri-butions State x3 Proba-bility Total failure Failed ignition 0 0 0.01 Partial function Successful ignition/Unreliable combustion 80 1 0.08 Perfect function Successful ignition/Reliable combustion 160 2 0.91

 State description Failure mode Contri-butions State x4 Proba-bility Total failure Creep/High and low cycle fatigue fracture 0 0 0.01 High failure Blade tips drop/Abrasion/Corrosion/Oxidation 70 1 0.06 Low failure Low abrasion/Corrosion/Oxidation 150 2 0.13 Perfect function No surface and global failure 200 3 0.80

 （18）

4.1 状态类与边界点

 State k Output Equivalence ΩCk 0 0 0 (0000), (0001), (0002), (0003), (0100), (0200), (0010), (0020), (1000), (2000), (0011), (0012), (0013), (0021), (0022), (0023), (0101), (0102), (0103), (0201), (0202), (0203), (0110), (0120), (0210), (0220), (1001), (1002), (1003), (2001), (2002), (2003), (1010), (1020), (2010), (2020), (1100), (1200), (2100), (2200), (0111), (0112), (0113), (0121), (0122), (0123), (0211), (0212), (0213), (0221), (0222), (0223), (1110), (1120), (1210), (1220), (2110), (2120), (2210), (2220) 1 1 60 (1101), (1102), (1103), (2101), (2102), (2103) 2 70 (1011), (1021), (1111), (1121), (1201), (1211), (1221), (2011), (2021), (2111), (2121), (2201), (2211), (2221) 2 3 80 (1012), (1013), (2012), (2013) 4 100 (1022), (1023), (1112), (1113), (1122), (1123), (1202), (1203), (1212), (1213), (1222), (1223) 3 5 120 (2202), (2203) 6 140 (2112), (2113) 4 7 150 (2022), (2122), (2212), (2222) 8 160 (2023) 5 9 200 (2123), (2213), (2223)

 k+ Output Equivalence ΩCk+ 0+ ≥0 ΩC0，ΩC1，ΩC2，ΩC3，ΩC4，ΩC5 1+ ≥70 ΩC1，ΩC2，ΩC3，ΩC4，ΩC5 2+ ≥100 ΩC2，ΩC3，ΩC4，ΩC5 3+ ≥140 ΩC3，ΩC4，ΩC5 4+ ≥160 ΩC4，ΩC5 5+ ≥200 ΩC5

 ΩCk Boundary points ΩCk+ Main boundary ΩC0 (0000) ΩC0+ (0000) ΩC1 (1101), (1011) ΩC1+ (1101), (1011) ΩC2 (1012), (1202) ΩC2+ (1012), (1202) ΩC3 (2202), (2112) ΩC3+ (2202), (2112), (2022) ΩC4 (2022), (2212) ΩC4+ (2022), (2212) ΩC5 (2123), (2213) ΩC5+ (2123), (2213)

4.2 MSCS的结构函数运算

 图 2 新方法框图形式 Figure 2 Diagram form of new approach

 （19）

 （20）

4.3 MSCS的可靠性计算

 Reliability R1 R2 R3 R4 R5 New 0.970 1 0.910 3 0.872 5 0.857 1 0.730 4 L-L 0.970 1 0.910 3 0.872 5 0.857 1 0.730 4

5 结论

1) 鉴于工程复杂系统处于多个状态运行的现实问题，将多元离散函数理论引入结构函数描述系统状态，给出了单调关联系统的数学概念，状态等价类，控制等价类的主导向量等相关定义与逻辑方法。

2) 基于离散函数理论的析取与合取运算的代数规则，将二元逻辑运算推广到多元逻辑运算，给出了多状态系统的析取与合取范式，推导了系统多状态边界点的结构函数。为便于运算，提出了集合运算的德摩根律方法，简化了边界点下系统的结构函数和可靠度表达式。为评估整个系统所处的平均性能，给出了系统的期望状态计算式，考虑系统面向顾客的偏好与需求，建立了基于负效用函数的性能效用模型。

3) 为验证模型的有效性与合理性，将一个某型航空发动机系统进行框图分解，基于各个子单元对系统的贡献度，给出了系统等价类，获取了系统状态相关的上下边界点数据。为便于运算，算例中构建了新型的框图式算法，给出了析取与合取形式下的结构函数表示，大大减少了计算复杂度，增加了问题描述的清晰度。通过新框图算法和德摩根律方法，降低了可靠度的计算复杂程度。

4) 通过算例分析，基于本文提出的状态等价类的主控边界点数据，以L-L方法作为比较对象，提出的方法能够寻找到完全控制等价类中其它向量，而且计算量简洁许多，并不需要计算所有满足≥0状态的所有边界点，只需要计算控制其他边界点的状态等价类中主控边界点即可，大大减少计算复杂度。

5) 本文给出的状态等价类的主控边界点方法，推广了L-L方法，通过实例验证了提出方法的有效性与合理性。然而随着单元数和子单元状态数的增加，复杂度也快速增加，可以进一步考虑使用计算机，给出系统状态和可靠度的计算程序以便于应用。

 [1] CHAHKANDI M, RUGGERI F, SUAREZ L A. A generalized signature of repairable coherent systems[J]. IEEE Transactions on Reliability, 2016, 65(1): 434–445. Click to display the text [2] 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 [3] EI-NEWEIHI E. Multistate coherent systems[J]. Journal of Applied Probability, 1978, 15(4): 675–688. Click to display the text [4] NATRING B. Two suggestions of how to define a multistate coherent system[J]. Advances in Applied Probability, 1982, 14(2): 434–455. Click to display the text [5] HUANG H Z, TONG X, ZUO M J. Posbist fault tree analysis of coherent systems[J]. Reliability Engineering & System Safety, 2004, 84(2): 141–148. Click to display the text [6] AVEN T. On performance measures for multistate monotone systems[J]. Reliability Engineering & System Safety, 1993, 41(3): 259–266. Click to display the text [7] BOEDIGHEIMER R A, KAPUR K C. Customer-driven reliability models for multistate coherent systems[J]. IEEE Transactions on Reliability, 1994, 43(1): 46–50. Click to display the text [8] LUO T, TRIVEDI K S. An improved algorithm for coherent system reliability[J]. IEEE Transactions on Reliability, 1998, 47(1): 73–78. Click to display the text [9] BOUTSIKAS M V, KOUTRAS M V. Generalized reliability bounds for coherent structures[J]. Journal of Applied Probability, 2000, 37(3): 778–794. Click to display the text [10] ESARY J D, PROSCHAN F. A reliability bound for systems of maintained, interdependent components[J]. Journal of the American Statistical Association, 1970, 65(329): 329–338. Click to display the text [11] HSIEH Y C. New reliability bounds for coherent systems[J]. Journal of the Operational Research Society, 2003, 54(9): 995–1001. Click to display the text [12] LI J A, WU Y, LAI K K, et al. Reliability estimation of multi-state components and coherent systems[J]. Reliability Engineering & System Safety, 2005, 88(1): 93–98. Click to display the text [13] XUE J, YANG K. Dynamic reliability analysis of coherent multistate systems[J]. IEEE Transactions on Reliability, 1995, 44(4): 683–688. Click to display the text [14] CUI L, LI H. Analytical method for reliability and MTTF assessment of coherent systems with dependent components[J]. Reliability Engineering & System Safety, 2007, 92(3): 300–307. Click to display the text [15] LIU Y W, KAPUR K C. Reliability measures for dynamic multistate nonrepairable systems and their applications to system performance evaluation[J]. ⅡE Transactions, 2006, 38(6): 511–520. Click to display the text [16] NATVIG B. On the deterioration of non-repairable multistate strongly coherent systems[J]. Journal of Applied Probability, 2014, 51(51): 69–81. Click to display the text [17] KUNDU P, HAZRA N K, NANDA A K. Reliability study of a coherent system with single general standby component[J]. Statistics & Probability Letters, 2016, 110: 25–33. Click to display the text [18] ERYILMAZ S. A new look at dynamic behavior of binary coherent system from a state-level perspective[J]. Annals of Operations Research, 2014, 212(1): 115–125. Click to display the text [19] ZHANG X, WILSON A. System reliability and component importance under dependence:A Copula approach[J]. Technometrics, 2016: 1–28. Click to display the text [20] POLPO A, SINHA D, DE B. Nonparametric Bayesian estimation of reliabilities in a class of coherent systems[J]. IEEE Transactions on Reliability, 2013, 62(2): 455–465. Click to display the text [21] KUNDU P, HAZRA N K, NANDA A K. Reliability study of a coherent system with single general standby component[J]. Statistics & Probability Letters, 2016, 110: 25–33. Click to display the text [22] FRANKO C, OZHKUT M, KAN C. Reliability of coherent systems with a single cold standby component[J]. Journal of Computational & Applied Mathematics, 2015, 281(C): 230–238. Click to display the text [23] GRIFFITH W S. Multistate reliability models[J]. Journal of Applied Probability, 1980, 17(3): 735–744. Click to display the text [24] BUTLER D A. Bounding the reliability of multi-state systems[J]. Operations Research, 1982, 30(3): 530–544. Click to display the text [25] BLOCK H W, SAVITS T H. A decomposition for multistate monotone systems[J]. Journal of Applied Probability, 1982, 19(2): 391–402. Click to display the text [26] HUDSON J C, KAPUR K C. Reliability bounds for multi-state systems with multistate components[J]. Operations Research, 1985, 33(1): 153–160. Click to display the text [27] LISNIANSKI A, LEVITIN G. Multi-state system reliability assessment, optimization, application[M]. Singapore: World Scientific, 2003: 89-153. [28] LIU Y W, KAPUR K C. Customer's cumulative experience measures for reliability of non-repairable aging multistate systems[J]. Quality Technology and Quantitative Management, 2007, 4(2): 225–234. Click to display the text [29] MAGANA S, ALBERTO C. Dynamic reliability based performance measures for multi-state systems[D]. Seattle, WA:University of Washington, 2010:116-130. Click to display the text [30] LIU Y W. Multi-state system reliability:Models, dynamic measures and applications[D]. Seattle, WA:Univesity of Washington, 2006:115-121. Click to display the text [31] XUE J, YANG K. Dynamic reliability analysis of coherent multistate systems[J]. IEEE Transactions on Reliability, 1995, 44(4): 683–688. Click to display the text [32] 刘长富, 邓明. 航空发动机结构分析[M]. 西安: 西北工业大学出版社, 2006: 72-112. LIU C F, DENG M. Structural analysis of aeroengine[M]. Xi'an: Northwestern Polytechnical University Press, 2006: 72-112. (in Chinese)
http://dx.doi.org/10.7527/S1000-6893.2017.220868

0

#### 文章信息

ZHANG Yongjin, SUN Youchao, ZHANG Yanjun

Reliability analysis for multi-state coherent system with monotonic components based on pivotal boundary points of clustering states

Acta Aeronautica et Astronautica Sinica, 2017, 38(8): 220868.
http://dx.doi.org/10.7527/S1000-6893.2017.220868