基于發(fā)生函數(shù)的模糊多狀態(tài)復雜系統(tǒng)可靠性通用評估方法

史躍東, 金家善, 徐一帆, 胡俊波

(1. 海軍工程大學科研部, 湖北 武漢 430033; 2. 海軍工程大學管理工程系, 湖北 武漢 430033)

0 引 言

工程領(lǐng)域的復雜系統(tǒng)多具備組成結(jié)構(gòu)復雜、運行過程復雜、使用環(huán)境復雜等特點,且可靠性、安全性、可用性要求較高,然而鑒于其自身性能潛在的突變性、涌現(xiàn)性以及結(jié)構(gòu)敏感性等原因,前述可靠性、安全性、可用性計算分析工作難度往往較大。為此,探索一類能夠較為準確刻畫復雜系統(tǒng)運行歷程,真實映射其可靠性、安全性和可用性的通用模型和研究方法,一直備受行業(yè)研究人員關(guān)注,特別是在面臨配套保障資源籌劃、維修管理決策等與壽命周期成本密切相關(guān)的現(xiàn)實工程問題時,復雜系統(tǒng)的可靠性預測、評估技術(shù),更是解決相關(guān)問題的關(guān)鍵所在,因此成為研究熱點和難點。

早期受到傳統(tǒng)建模理念和現(xiàn)實解算能力所限,有關(guān)復雜系統(tǒng)的可靠性研究工作,多假定系統(tǒng)性能僅為二元狀態(tài),即“正常工作”狀態(tài)和“完全失效”狀態(tài)。然而,長期的工作實踐表明,僅依賴二元狀態(tài)刻畫評估系統(tǒng)工作性能,獲取的研究結(jié)果多有欠理想,甚至與實際觀測結(jié)果大相徑庭,因此,亟待探索一類能夠較為準確刻畫系統(tǒng)真實運行狀態(tài)的建模方法。自20世紀90年代起,多狀態(tài)理論被引入復雜系統(tǒng)可靠性建模與評估研究領(lǐng)域,并被作為系統(tǒng)成本優(yōu)化、維修決策的重要輔助工具[1-3]。文獻[4-12]利用多狀態(tài)理論開展復雜系統(tǒng)失效機理研究,建立了較為理想的可靠性多狀態(tài)評估模型。文獻[13-14]針對機械多狀態(tài)系統(tǒng),提出了一類離散化的可靠性解析方法。文獻[15-16]基于貝葉斯網(wǎng)絡理論,開展了多狀態(tài)復雜系統(tǒng)的可靠性建模與評估研究。文獻[17-21]針對一類極具代表性的相鄰多狀態(tài)k/n系統(tǒng),給出了通用解算方法。文獻[22-24]則進一步探討了計及部件老化的多狀態(tài)復雜系統(tǒng)的可靠性評估特殊性。

與二元理論相比,多狀態(tài)理論改善了系統(tǒng)建模精度,能夠更為真實地反映復雜系統(tǒng)工作歷程,但同時也在一定程度上加大了模型解算的工作量,對于包括較多構(gòu)件的大型復雜系統(tǒng)來說,解算工作量更是呈幾何級數(shù)增加。為此,一類極具計算優(yōu)勢的通用發(fā)生函數(shù)解算方法,被納入多狀態(tài)復雜系統(tǒng)可靠性評估解算體系。其中,文獻[25-30]在此方面開展了諸多工作;文獻[31-33]則在此基礎上,進一步探討了半馬爾可夫狀態(tài)躍遷假設下的復雜系統(tǒng)多狀態(tài)可靠性評估方法。

近年來,隨著多狀態(tài)復雜系統(tǒng)建模理論的逐步完善,解算方法和解算能力的迅速提升,相關(guān)研究成果已得以成功應用于復雜系統(tǒng)的可靠性優(yōu)化設計、保障資源合理配置、維修活動科學決策等工程領(lǐng)域[34-40]。其中,文獻[41-46]給出了典型多狀態(tài)串并聯(lián)冗余系統(tǒng)的可靠性優(yōu)化分配方法;文獻[47-53]則進一步針對多狀態(tài)復雜系統(tǒng)構(gòu)件失效后的維修策略選擇問題,開展了相關(guān)深入研究。

雖然，以多狀態(tài)理論和發(fā)生函數(shù)算法為核心的復雜系統(tǒng)建模和可靠性分析工作,已有部分研究成果,但隨著現(xiàn)代工業(yè)技術(shù)的飛速發(fā)展,產(chǎn)品研制周期越來越短,壽命周期越來越長,有些復雜系統(tǒng)尚未經(jīng)歷足夠時長的可靠性觀測,就已投入高強度使用,因此,較難獲取足量有效數(shù)據(jù),用以精確評估其不同狀態(tài)性能水平及其狀態(tài)發(fā)生概率。同時,為降低計算載荷,采用的各類復雜系統(tǒng)建模簡化方法、數(shù)值算法,也或多或少地在一定程度上降低了問題解算精度。為此,亟需進一步發(fā)展一類新技術(shù)和新理論,以彌補多狀態(tài)復雜系統(tǒng)建模理論的現(xiàn)存技術(shù)短板。模糊多狀態(tài)建模理論以及數(shù)值處理技術(shù),正是為解決前述相關(guān)問題,被嘗試引入復雜系統(tǒng)可靠性建模、分析與評估領(lǐng)域。目前,此方面的研究工作成果尚不多見[54-55]。

本文以經(jīng)典串并混聯(lián)通用系統(tǒng)模型為研究對象,旨在綜合多態(tài)理論、模糊理論和發(fā)生函數(shù)理論的基礎上,探究一類適用于復雜系統(tǒng)可靠性建模、分析與評估的程式化解算方法,用以解決由于多態(tài)性能觀測數(shù)據(jù)有限或模型簡化處理,可能導致的解算精度下降、解算結(jié)果失真等技術(shù)難題,進而為發(fā)展多狀態(tài)復雜系統(tǒng)可靠性分析理論和解算方法提供技術(shù)借鑒。

1 模糊多狀態(tài)系統(tǒng)

1.1 常規(guī)多狀態(tài)系統(tǒng)

首先,引入多狀態(tài)系統(tǒng)基本理念。不失一般性,假設任意多狀態(tài)系統(tǒng)S,由n個不同功能元件j(j=1,2,…,n)組成,且第j個元件具有kj種狀態(tài)性能,相關(guān)狀態(tài)向量gj由式(1)確定

gj=[gj1,gj2,…,gjkj]

(1)

進一步,假設元件j在任意時刻t的性能狀態(tài)為Gj(t),則有Gj(t)∈gj。這里,Gj(t)為隨機變量,相關(guān)狀態(tài)取值概率向量pj,計算式為

pj=[pj1(t),pj2(t),…,pjkj(t)]

(2)

綜上,如果多狀態(tài)系統(tǒng)S具有K種輸出性能狀態(tài)qi(i=1,2,…,K),則有式(3)成立。

(3)

式中,f(·)為由系統(tǒng)相應物理特性確定的構(gòu)型函數(shù)。

1.2 模糊多狀態(tài)系統(tǒng)

如前論述可知,多狀態(tài)系統(tǒng)S的輸出性能qi及其概率分布,可由系統(tǒng)組成元件j的狀態(tài)向量gj、概率向量pj以及系統(tǒng)構(gòu)型函數(shù)f(·)唯一確定。然而,鑒于觀測時間、觀測手段有限,獲取gj、pj兩類向量的精確值往往較難,而多采用區(qū)間值或模糊值代替。由此,為更加有效地基于現(xiàn)有觀測手段、觀測數(shù)據(jù),分析獲取所研多狀態(tài)系統(tǒng)的性能分布及可靠性參數(shù),模糊多狀態(tài)理論被引入其中。后續(xù)內(nèi)容,統(tǒng)一使用“～”上標符號,標識模糊數(shù)學量。

(4)

式中,模糊數(shù)的表達方式有多種,這里選用滿足kaufumann-gupta排序規(guī)則的三角模糊數(shù)M,數(shù)學定義為

M=[a,b,c]

(5)

式中,模糊數(shù)間的排序運算,參照如下次序?qū)嵤?/p>

(6)

(7)

(8)

SRi={ri∈Ri|ri≥0}

(9)

由此,可將式(7)左側(cè)表達式改寫為

(10)

(11)

2 發(fā)生函數(shù)法

多狀態(tài)建模理論,與傳統(tǒng)的二元建模理論相比,關(guān)于系統(tǒng)運行歷程的刻畫,更為貼近真實工況,能夠較為準確地反映系統(tǒng)運行特性。但鑒于建模狀態(tài)增多,導致解算工作趨于復雜,對于引入模糊多狀態(tài)建模理論的復雜系統(tǒng)而言,更是如此。由此,為提升復雜系統(tǒng)可靠性建模的程式化程度,有效釋放解算資源,基于生成序列的發(fā)生函數(shù)法,被應用于可靠性分析、評估領(lǐng)域,并發(fā)揮重要作用。

2.1 通用發(fā)生函數(shù)

首先,給出發(fā)生函數(shù)的基本定義。不失一般性,考慮n個獨立隨機變量Xj,以及與其相關(guān)的隨機變量Y=f(X1,X2,…,Xn),其中,變量Xj存有kj種狀態(tài),且滿足概率分布

pxj=Pr{Xj=xj}

則可定義變量Xj的z變換函數(shù)為

進一步,通過引入通用發(fā)生算子Ωf,則可確定關(guān)聯(lián)變量Y的z變換函數(shù)為

(12)

注意,若將變量Y視為系統(tǒng)輸出,Xj視為組成元件的性能輸入,f(·)視為系統(tǒng)構(gòu)型函數(shù),且按規(guī)則

(xj,pj)=[(xj1,pj1),…,(xjkj,pjkj)]

量化變量Xj的概率分布,則式(12)可進一步寫為

(13)

2.2 模糊通用發(fā)生函數(shù)

引入模糊多狀態(tài)理論,仍針對由n個元件j組成的多狀態(tài)系統(tǒng)S,沿用式(4)中符號定義,則有元件j的模糊z變換函數(shù)uj(z)為

(14)

(15)

式中,sup(·)為上確界函數(shù);min(·)為最小值函數(shù)。

3 模糊復雜系統(tǒng)可靠性評估建模

3.1 通用模型

工程復雜系統(tǒng)一般多具有元件構(gòu)成多、邏輯關(guān)聯(lián)結(jié)構(gòu)多變等特征,就可靠性建模分析而言,存有串聯(lián)、并聯(lián)、串并混聯(lián)、橋聯(lián)等多種結(jié)構(gòu)可能。鑒于串聯(lián)、并聯(lián)結(jié)構(gòu)均為串并混聯(lián)結(jié)構(gòu)的一類特殊情形,橋聯(lián)結(jié)構(gòu)亦可利用數(shù)學手段轉(zhuǎn)換為串并混聯(lián)結(jié)構(gòu),考慮到模型確立的普適性,這里選用典型串并混聯(lián)結(jié)構(gòu)模型,作為文中模糊多狀態(tài)復雜系統(tǒng)可靠性分析的通用模型,如圖1所示。

圖1 典型串并混聯(lián)結(jié)構(gòu)Fig.1 Classic series parallel hybrid structure

圖1中,變量m,n,s,p均取自然數(shù),uij(z)為相關(guān)元件的z變換函數(shù)。

3.2 模糊多狀態(tài)系統(tǒng)可用度評估

(16)

(17)

式中

4 典型算例

4.1 初始假設

以某型船用汽輪發(fā)電系統(tǒng)為例,如圖2所示,由供汽、主發(fā)電、備用發(fā)電3個子單元構(gòu)成。其中,供汽單元包括兩臺蒸汽鍋爐,負責聯(lián)合提供蒸汽工質(zhì);發(fā)電單元由汽輪機和發(fā)電機構(gòu)成,主、輔發(fā)電單元互為備用,擇一提供電力載荷。鑒于船用汽輪發(fā)電系統(tǒng),組成結(jié)構(gòu)復雜,海上運行環(huán)境惡劣,運行工況多樣,且性能狀態(tài)精確觀測不易,可將其視為模糊多狀態(tài)復雜系統(tǒng)處理,并選用三角模糊數(shù),量化描述系統(tǒng)模糊狀態(tài)。

進一步,假定利用期望功率輸出比η,判定系統(tǒng)及其各級組成單元的性能狀態(tài)等級。對于圖2所示系統(tǒng),參照文獻[28]中穩(wěn)態(tài)計算結(jié)果,并綜合行業(yè)專家經(jīng)驗,有如下初始條件,見表1。

圖2 船用汽輪發(fā)電系統(tǒng)組成框架Fig.2 Frame structure of marine boiler turbine generator system

j1(鍋爐1)2(鍋爐2)pj1(0．00007，0．0001，0．00012)(0．00007，0．0001，0．00012)pj2(0．0061，0．0066，0．0070)(0．0061，0．0066，0．0070)pj3(0．9927，0．9933，0．9938)(0．9927，0．9933，0．9938)gj1(0，0，0)(0，0，0)gj2(0．5，0．6，0．7)(0．5，0．6，0．7)gj3(0．9，0．9，0．9)(0．9，0．9，0．9)j3(汽輪機1)4(發(fā)電機1)pj1(0．00015，0．0002，0．00025)(0．0025，0．003，0．0034)pj2(0．0160，0．0164，0．0169)(0．9968，0．997，0．9972)pj3(0．9829，0．9834，0．9838)-gj1(0，0，0)(0，0，0)gj2(0．73，0．8，0．88)(1．0，1．0，1．0)gj3(1．0，1．0，1．0)-j5(汽輪機2)6(發(fā)電機2)pj1(0．008，0．0083，0．0088)(0．0024，0．003，0．0035)pj2(0．9910，0．9917，0．9920)(0．9965，0．997，0．9976)pj3--gj1(0，0，0)(0，0，0)gj2(0．80，0．85，0．90)(1．0，1．0，1．0)gj3--

由表1可知,汽輪發(fā)電系統(tǒng)中供汽鍋爐運行性能存有3種狀態(tài),即完全失效狀態(tài)、部分失效狀態(tài)和充分工作狀態(tài),且2臺供汽鍋爐運行性能完全一致;類似地,主發(fā)電單元汽輪機運行性能也存有3種狀態(tài),完全失效狀態(tài)、部分失效狀態(tài)和完全工作狀態(tài);與主發(fā)電單元不同,輔發(fā)電單元汽輪機運行性能要求較寬松,僅存有2種狀態(tài),完全失效狀態(tài)和充分工作狀態(tài);最后,主輔發(fā)電機均性能穩(wěn)定,較難發(fā)生失效,僅存有完全失效和完全工作2種狀態(tài)。

4.2 系統(tǒng)模糊發(fā)生函數(shù)

鑒于工程實際應用中,往往重點關(guān)注所研系統(tǒng)的穩(wěn)態(tài)輸出性能。因此,文中以下各類量化計算,均以此為基本前提,不再額外說明。

首先,計算船用汽輪發(fā)電系統(tǒng)各組成單元的模糊z變換函數(shù)Uj(z),這里以汽輪機1和發(fā)電機1組成的主發(fā)電單元U2(z)為例,解算說明如下,其他相關(guān)單元不再贅述。

(18)

(0.002 6,0.003 2,0.003 7)z0+

(0.015 9,0.016 4,0.016 9)z(0.73,0.80,0.88)+

(0.979 8,0.980 4,0.981 0)z1

(19)

類似地,可求供汽單元和輔發(fā)電單元模糊z變換函數(shù)U1(z)、U3(z)分別為

U1(z)=10-7(0.049,0.1,0.144)z0+

10-5(0.085 4,0.132,0.168)z(0.25,0.3,0.35)+

10-3(0.139,0.198 7,0.238 5)z0.45+

10-4(0.372 1,0.435 6,0.49)z(0.5,0.6,0.7)+

(0.012 1,0.013 1,0.013 9)z(0.7,0.75,0.80)+

(0.985 5,0.986 6,0.987 6)z0.9

(20)

U3(z)=(0.010 4,0.011 3,0.012 3)z0+

(0.987 5,0.988 7,0.989 6)z(0.80,0.85,0.90)

(21)

10-4(0.269 8,0.361 7,0.456 1)z0+

10-5(0.085 1,0.132,0.168 6)z(0.25,0.30,0.35)+

10-3(0.138 5,0.198 7,0.239 3)z0.45+

10-4(0.370 7,0.435 6,0.491 7)z(0.5,0.6,0.7)+

(0.012 1,0.013 1,0.013 9)z(0.7,0.75,0.8)+

10-3(0.163,0.182 8,0.205 3)z(0.73,0.8,0.88)+

(0.018,0.019 1,0.020 1)z(0.80,0.85,0.9)+

(22)

分析式(22)可知:①船用汽輪發(fā)電系統(tǒng)作為一類模糊多狀態(tài)復雜裝備系統(tǒng),具有8種不同運行狀態(tài),相關(guān)狀態(tài)性能及其概率分布模糊特征明顯,較難實現(xiàn)量化精確描述,這里采用三角模糊數(shù)方法,實現(xiàn)模糊特征的量化描述;②若8類運行狀態(tài),按照量化性能數(shù)值,依次從小到大排序(0→1,(0.25,0.30,0.35)→2,…,0.9→8),則比照不同狀態(tài)概率分布量值可知,汽輪發(fā)電系統(tǒng)運行處于第5、7、8類狀態(tài)的幾率較大,即系統(tǒng)常態(tài)運行功率長期保持于設計功率的70%以上;③已知汽輪發(fā)電系統(tǒng)各構(gòu)成單元發(fā)生函數(shù)Ui(z)的前提下,全系統(tǒng)的多類模糊運行狀態(tài),可通過引入模糊發(fā)生算子Ω,嵌套迭代解算,運算過程僅涉及數(shù)乘、求和等簡單代數(shù)運算,計算資源依賴程度低、程式化大、解算速度快。

4.3 系統(tǒng)可用度評估

(23)

表2 模糊集勢函數(shù)計算結(jié)果

10-3(0.163,0.182 8,0.205 3)0.32+

(0.018,0.019 1,0.020 1)0.791 7+

(0.963 6,0.967 3,0.970 7)=

(0.977 9,0.982 5,0.986 7)

(24)

5 結(jié) 論

以串并混聯(lián)船用汽輪發(fā)電結(jié)構(gòu)為對象,開展了模糊多狀態(tài)復雜系統(tǒng)可靠性通用評估方法研究。結(jié)合通用發(fā)生函數(shù),給出了模糊多狀態(tài)可靠性建模、分析、評估方法,并針對典型案例實施了模糊需求下多狀態(tài)復雜系統(tǒng)的可用度解算和評估。解算與評估表明,文中建模、分析、評估方法合理,能真實反映多狀態(tài)復雜系統(tǒng)的可靠性特征;克服觀測數(shù)據(jù)不確定性的模糊數(shù)學處理方法有效,可靠性評估結(jié)論更真實可靠;回避大自由度微分運算的可靠性單元通用發(fā)生函數(shù)描述方法,解算優(yōu)勢明顯、適用范圍更廣。

文中建模、解算方法,豐富、發(fā)展了復雜系統(tǒng)多狀態(tài)可靠性評估理論,可為有關(guān)模型實體的維修決策提供理論支撐和技術(shù)借鑒。下一步擬拓展文中研究結(jié)果,結(jié)合元啟發(fā)式計算技術(shù),探討多狀態(tài)復雜系統(tǒng)的可靠性設計及部組件配置優(yōu)化方法。

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