基于數(shù)據(jù)驅(qū)動(dòng)的KPI系統(tǒng)最優(yōu)濾波器設(shè)計(jì)

姚智剛　 彭開香

(北京科技大學(xué)自動(dòng)化學(xué)院,北京100083)

基于數(shù)據(jù)驅(qū)動(dòng)的KPI系統(tǒng)最優(yōu)濾波器設(shè)計(jì)

姚智剛 彭開香

(北京科技大學(xué)自動(dòng)化學(xué)院,北京100083)

摘要:為了優(yōu)化基于數(shù)據(jù)驅(qū)動(dòng)的嵌入式故障診斷濾波器設(shè)計(jì),采用KPI(key performance index)思想設(shè)計(jì)FDI(fault detection and isolation)殘差,研究基于mDOs觀測(cè)器的閉環(huán)Kalman濾波器設(shè)計(jì)方法,實(shí)現(xiàn)故障診斷和系統(tǒng)狀態(tài)的有效觀測(cè).首先,基于采樣數(shù)據(jù)得到大型復(fù)雜系統(tǒng)的KPI子空間模型,定義了跟蹤誤差,得到了閉環(huán)濾波器;其次,將殘差序列表示為Hankel模型,通過(guò)定義正交投影補(bǔ)矩陣并選擇恰當(dāng)?shù)臄?shù)據(jù)列,構(gòu)建出新的閾值矩陣;最后,得到Kalman濾波增益的計(jì)算方法,并給出了最優(yōu)Kalman濾波器的設(shè)計(jì)步驟.結(jié)果表明,優(yōu)化后殘差的幅值降低至優(yōu)化前的1/2.基于數(shù)據(jù)驅(qū)動(dòng)的KPI系統(tǒng)最優(yōu)濾波器設(shè)計(jì),可提高對(duì)弱小故障監(jiān)測(cè)的靈敏度,實(shí)現(xiàn)系統(tǒng)狀態(tài)估計(jì)和故障診斷的性能優(yōu)化.

關(guān)鍵詞:KPI;殘差;觀測(cè)器;濾波器

近幾十年來(lái),基于解析模型的故障診斷理論和方法得到了深入發(fā)展,其基本原理是通過(guò)比較實(shí)際測(cè)量的信息與解析模型的測(cè)量估計(jì)來(lái)構(gòu)建殘差,進(jìn)一步對(duì)殘差進(jìn)行優(yōu)化,并與閾值進(jìn)行比較以判斷是否有故障發(fā)生[1-7].但是對(duì)于大型復(fù)雜系統(tǒng),存在模型龐大、模型不確定、建模困難等實(shí)際問(wèn)題.采用數(shù)據(jù)驅(qū)動(dòng)故障診斷方法,尤其是子空間輔助的數(shù)據(jù)驅(qū)動(dòng)FDI(fault detection and isolation)系統(tǒng),可擺脫對(duì)模型的依賴,通過(guò)構(gòu)建并分析殘差,達(dá)到故障診斷的目的.大型復(fù)雜系統(tǒng)數(shù)據(jù)的維數(shù)和數(shù)量過(guò)于龐大,基于數(shù)據(jù)驅(qū)動(dòng)的設(shè)計(jì)方法可能導(dǎo)致FDI系統(tǒng)的階數(shù)過(guò)高,從而降低殘差數(shù)據(jù)對(duì)某些重要參數(shù)和信號(hào)的靈敏度,造成漏報(bào)現(xiàn)象.KPI(key performance index)思想最早源于經(jīng)濟(jì)學(xué),并被廣泛應(yīng)用于汽車系統(tǒng)設(shè)計(jì)與管控中.將KPI思想引入FDI系統(tǒng)殘差的設(shè)計(jì)中,基于KPI選擇關(guān)鍵參數(shù)后FDI系統(tǒng)的階次降低,可以減小FDI設(shè)計(jì)的難度和工作量,提高殘差信號(hào)對(duì)系統(tǒng)重要參數(shù)和信號(hào)的靈敏度,從而構(gòu)建出數(shù)據(jù)驅(qū)動(dòng)KPI-FDI系統(tǒng).

嵌入式診斷濾波器備受關(guān)注.基于故障診斷系統(tǒng)與被控對(duì)象的對(duì)應(yīng)關(guān)系,在實(shí)現(xiàn)對(duì)被控對(duì)象故障異常監(jiān)視與診斷的同時(shí),還可達(dá)到對(duì)系統(tǒng)狀態(tài)的有效觀測(cè).Steven等[8]在數(shù)據(jù)驅(qū)動(dòng)的故障診斷系統(tǒng)設(shè)計(jì)基礎(chǔ)上,進(jìn)一步研究了系統(tǒng)狀態(tài)估計(jì)問(wèn)題,提出了一種基于數(shù)據(jù)驅(qū)動(dòng)的觀測(cè)器設(shè)計(jì)方法,并指出其對(duì)于基于數(shù)據(jù)驅(qū)動(dòng)的控制器設(shè)計(jì)的重要意義;但是, 設(shè)計(jì)中無(wú)差拍觀測(cè)器在處理隨機(jī)擾動(dòng)時(shí)的性能有待于進(jìn)一步提高.

本文以某復(fù)雜系統(tǒng)為例,引入KPI思想,研究了基于數(shù)據(jù)驅(qū)動(dòng)的故障診斷濾波器優(yōu)化設(shè)計(jì)問(wèn)題.

1KPI系統(tǒng)子空間模型

穩(wěn)態(tài)下的大型復(fù)雜系統(tǒng)可以表示為線性時(shí)不變系統(tǒng),其模型描述如下:

(1)

式中,E()為方差；Q，R，S為噪聲矩陣.

為了解決復(fù)雜系統(tǒng)輸出變量過(guò)于龐大的問(wèn)題,定義如下的KPI變量:

(2)

式中,Wy為權(quán)重函數(shù).

進(jìn)一步給出KPI模型為

(3)

假設(shè)系統(tǒng)矩陣A,B,C，D,系統(tǒng)階數(shù)n和矩陣Q，R，S均未知.對(duì)采集的數(shù)據(jù)進(jìn)行處理,構(gòu)建如下的Hankel數(shù)據(jù)結(jié)構(gòu):

X(i)∈Rn×N,U(i)∈Rl×N,Y(i)∈Rm×N

W(i)∈Rl×N,V(i)∈Rm×N,Usp-1∈Rspl×N

Ysp-1∈Rspm×N, Usf-1∈Rspl×N, Ysf-1∈Rspm×N

Zsp-1∈Rsp(l+m)×N,Zsf-1∈Rsf(l+m)×N

Wsf-1,u∈Rspl×N,Vsf-1,f∈Rsfm×N

式中,N為測(cè)試數(shù)據(jù)長(zhǎng)度;sp,sf為整數(shù),且大于n.

參考文獻(xiàn)[9],將式(3)改寫為 References)

(4)

式中,K表示Kalman增益;e表示白噪聲.

基于式(4),構(gòu)建KPI模型的Hankel等式:

Ysf-1=Γsf-1X(i)+Hsf-1,uUsf-1+Hsf-1,wWsf-1,u+

Vsf-1,f=Γsf-1X(i)+Hsf-1,uUsf-1+Hsf-1,eEsf-1

(5)

式(5)即為大型復(fù)雜系統(tǒng)的KPI子空間模型.

利用Zsp-1,Usf-1和Ysf-1構(gòu)建如下的QR分解:

(6)

求解齊次方程組

(7)

(8)

2KPI系統(tǒng)多通道殘差生成器設(shè)計(jì)

本文考慮KPI離散線性時(shí)不變系統(tǒng),基于文獻(xiàn)[8]的研究結(jié)果,可得多通道殘差生成器mDOs的緊湊形式為

(9)

式中

基于文獻(xiàn)[8],Luenberger方程中的矩陣T滿足條件rank(T)=n,由此可構(gòu)建全狀態(tài)觀測(cè)器為

(10)

式中

Ax=T-AzT∈Rn×n,Bx=T-Bz∈Rn×1

Lx=T-L∈Rn×m,Cx=G-1CzT∈Rm×n

Dx=G-1Dz∈Rm×l

3基于mDOs觀測(cè)器的Kalman濾波器設(shè)計(jì)

mDOs觀測(cè)器設(shè)計(jì)沒有充分考慮系統(tǒng)噪聲的影響,而是根據(jù)等價(jià)空間設(shè)計(jì)了無(wú)差拍觀測(cè)器.當(dāng)系統(tǒng)中存在未知噪聲時(shí),無(wú)差拍觀測(cè)器的估計(jì)精度將受到影響.除了狀態(tài)估計(jì)外,殘差信號(hào)同樣受到噪聲的影響,如果設(shè)計(jì)的殘差不考慮噪聲,則故障診斷系統(tǒng)性能將有所下降.假定系統(tǒng)噪聲為高斯白噪聲, Kalman濾波器被公認(rèn)為該情況下的最優(yōu)觀測(cè)器.本節(jié)主要研究在mDOs觀測(cè)器的基礎(chǔ)上,如何基于數(shù)據(jù)進(jìn)一步設(shè)計(jì)Kalman濾波器,實(shí)現(xiàn)系統(tǒng)狀態(tài)估計(jì)與故障診斷的性能優(yōu)化.

假定在KPI白噪聲模型中白噪聲e的方差為Σe,穩(wěn)態(tài)卡爾曼增益為Kz.根據(jù)mDOs觀測(cè)器,可得到如下的閉環(huán)濾波器:

(11)

式中

Kz,u=[Kz,u,1Kz,u,2…Kz,u,su]∈Rsu×m

u=1,2,…,m; Kz,u,j∈Rm; j=1,2,…,su

(12)

由式(12)可知, 如果存在Kz使得

TK-L=KzG

(13)

則式(11)所表示的閉環(huán)濾波器即為Kalman濾波器.

mDOs與白噪聲模型的誤差方程為

(14)

將式(14)進(jìn)行擴(kuò)展可得

(15)

式中

Hsf-1,k=

Ξsp-1rsp-1(k-sp+1)≈Ξsp-1rsp-1(k-sp+1)

(16)

式中

綜合式(15)和(16),殘差序列rsf-1可通過(guò)esf-1(k)和rsp-1(k-sp+1)直接表述為

rs(k)=Hsf-1,rrsp-1(k-sp+1)+Hsf-1,kesf-1(k)

(17)

式中,Hsf-1,r=Γsf-1Ξsp-1.與Yf和Yp相似,將式(17)表示為如下的Hankel模型:

Rsf-1(k)=Hsf-1,rRsp-1(k-sp+1)+

Hsf-1,kEsf-1(k)

(18)

假設(shè)殘差序列為某一靜態(tài)零均值隨機(jī)噪聲,當(dāng)s≥n時(shí),Rsf-1(k)與Rsp-1(k-sp+1)不相關(guān),則有

(19)

定義Rsp-1(k-sp+1)的正交投影補(bǔ)矩陣為

Rsp-1(k-sp+1)

(20)

且式(20)滿足

(21)

由此可得數(shù)據(jù)矩陣Σr為

(22)

根據(jù)式(20)～(22)可得

[Hsf-1,rRsp-1(k-sp+1)+Hsf-1,kEsf-1(k)]T=

(23)

(24)

由此可得

(25)

(26)

4仿真分析

圖1為某型火控系統(tǒng)的原理框圖[10].該系統(tǒng)由多個(gè)傳感器、執(zhí)行器和子系統(tǒng)組成,共計(jì)30余個(gè)單體設(shè)備,其中火控計(jì)算機(jī)為系統(tǒng)的核心,單體之間的連接和數(shù)據(jù)通信以總線為主.其工作過(guò)程概略為:在獲取目標(biāo)位置的基礎(chǔ)上,采集各傳感器的信息,包括裝備所處陣地的氣象信息、火藥溫度、裝備所處的地理坐標(biāo)、車體平面的姿態(tài)量和身管的空間指向等數(shù)據(jù);其次,以火控計(jì)算機(jī)為主動(dòng)方,向隨動(dòng)系統(tǒng)、發(fā)射控制系統(tǒng)、操作顯示系統(tǒng)和通訊管理系統(tǒng)等發(fā)送命令,并進(jìn)行數(shù)據(jù)交互,在火控計(jì)算機(jī)內(nèi)部完成射擊所需的控制量計(jì)算;最后,各執(zhí)行器在接收到命令數(shù)據(jù)后,迅速調(diào)整車體平臺(tái)姿態(tài),并控制身管迅速轉(zhuǎn)向預(yù)定的空間指向.為了確保射擊精度,要求車體平臺(tái)的姿態(tài)量和身管空間指向的輸出誤差限制在10個(gè)密位以內(nèi).

圖1　某型火控系統(tǒng)原理框圖

本文采用數(shù)據(jù)驅(qū)動(dòng)的方法,從若干輸出量中將對(duì)系統(tǒng)性能影響最大的5個(gè)輸出量(平臺(tái)北向角y1、縱傾角y2、橫傾角y3、身管方位角y4、身管高低角y5)選擇為KPI參量,基于數(shù)據(jù)構(gòu)建KPI-FDI系統(tǒng).

Az=diag(Az,1,Az,2,Az,3,Az,4，Az,5)

Bz,1={-0.07,0.14,0.02,-0.07,-0.24,

-0.05,0.17,0.03,0.10,-0.34,0,

0.12,0,0.04,-0.33,-0.03,0.15,

0.03,0.12,-0.35,-0.04,0.11,0.08,

0.11,-0.23,-0.03,0.03,0.05,-0.09,

-0.10,0,0.03,0.02,0.03,-0.03,

-0.05,0,0.07,-0.13,-0.05,-0.02,

0,0.08,-0.01,-0.10,-0.03,0.13,

0.03,0.07,-0.27}

其余Bz,i∈R50與此類似.

觀測(cè)增益矩陣表示為

Lz,1={0,0.01,-0.03,-0.02,0.01,0,0.03,

0.01,0,0.15,0.02,-0.06,0.03,-0.16,

-0.14,0.02,0.01,0.09,-0.02,0.03,

-0.06,-0.11,-0.11,-0.09,-0.03}

其余Lz,i∈R25與此類似.

其他矩陣表述如下:

Cz=diag(Cz,1，Cz,2,Cz,3,Cz,4,Cz,5)

在辨識(shí)等價(jià)空間及其向量和構(gòu)建mDOs觀測(cè)器后,對(duì)基于mDOs觀測(cè)器的Kalman濾波器增益設(shè)計(jì)方法進(jìn)行驗(yàn)證,y1,y2,y3,y4,y5的KPI輸出誤差曲線如圖2所示(Ey1,Ey2,Ey3,Ey4,Ey5為誤差幅值),對(duì)應(yīng)的Kalman增益為

圖3給出了優(yōu)化之后的Kalman濾波器的KPI輸出估計(jì)誤差曲線.由圖可知,優(yōu)化后殘差的幅值降低至優(yōu)化前的1/2,從而驗(yàn)證了數(shù)據(jù)驅(qū)動(dòng)設(shè)計(jì)針對(duì)mDOs觀測(cè)器的Kalman濾波器的有效性.

5結(jié)語(yǔ)

本文針對(duì)故障診斷濾波器的最優(yōu)化問(wèn)題進(jìn)行了研究,提出了一種數(shù)據(jù)驅(qū)動(dòng)的基于mDOs觀測(cè)器的Kalman濾波器設(shè)計(jì)方法及實(shí)施步驟,并利用某型火控系統(tǒng)進(jìn)行了驗(yàn)證.結(jié)果表明,優(yōu)化后殘差的幅值降低至優(yōu)化前的1/2,從而故障診斷系統(tǒng)可對(duì)更小的故障進(jìn)行監(jiān)測(cè).但是在所提算法中,采用數(shù)據(jù)驅(qū)動(dòng)方式設(shè)計(jì)并使用mDOs觀測(cè)器狀態(tài),往往存在因階數(shù)過(guò)大而求逆困難的問(wèn)題,可通過(guò)對(duì)觀測(cè)器的重新設(shè)計(jì)解決這一問(wèn)題.

(a)y1輸出誤差曲線

(b) y2輸出誤差曲線

(c) y3輸出誤差曲線

(d) y4輸出誤差曲線

(e) y5輸出誤差曲線

(a) y1輸出誤差曲線

(b) y2輸出誤差曲線

(c) y3輸出誤差曲線

(d) y4輸出誤差曲線

(e) y5輸出誤差曲線

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Data-driven based optimal filter design for KPI system

Yao Zhigang Peng Kaixiang

(School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China)

Abstract:In order to optimize the data-driven design of embedded fault diagnosis filters, a FDI (fault detection and isolation) residual generator is presented based on KPI (key performance index), and a design method for the closed-loop Kalman filter based on the multi-diagnostic observers (mDOs) is studied to realize fault diagnosis and effective observation for system status. First, the KPI subspace model for the large complex system is obtained based on the sampling data. The tracking error is defined, and the closed-loop filter is realized. Then, the residual sequence is expressed as the Hankel mode. A new threshold matrix is constructed by defining an orthogonal projection complement matrix as well as selecting appropriate data columns. Finally, the calculation method for the Kalman filter gain is obtained, and the design steps for the optimal Kalman filter are described. The results show that the amplitude of the optimized residual is half that of the pre-optimized residual. The data-driven based optimal filter design for the KPI system can improve the monitoring sensitivity for tiny fault and optimize both status estimating and fault diagnosis.

Key words:KPI(key performance index); residual; observer; filter

DOI:10.3969/j.issn.1001-0505.2016.02.004

收稿日期:2015-08-04.

作者簡(jiǎn)介:姚智剛(1980—)，男，博士，講師，xyhk_yzg@163.com.

基金項(xiàng)目:國(guó)家自然科學(xué)基金資助項(xiàng)目(61473033).

中圖分類號(hào):TP29

文獻(xiàn)標(biāo)志碼:A

文章編號(hào):1001-0505(2016)02-0249-06

引用本文: 姚智剛,彭開香.基于數(shù)據(jù)驅(qū)動(dòng)的KPI系統(tǒng)最優(yōu)濾波器設(shè)計(jì)[J].東南大學(xué)學(xué)報(bào)(自然科學(xué)版),2016,46(2):249-254. DOI:10.3969/j.issn.1001-0505.2016.02.004.