31 1 2014 1 DOI: 10.7641/CTA.2014.30119 Control Theory & Applications Vol. 31 No. 1 Jan. 2014,,, (, 102206) : (principal component analysis, PCA),. (multivariate exponentially weighted moving average principal component analysis, MEWMA PCA) PCA. MEWMA PCA EWMA, MEWMA PCA. MEWMA PCA λ,. MEWMA PCA EWMA PCA,,. : ; ; ; : TP277 : A Incipient fault detection of multivariate exponentially weighted moving average principal component analysis QIU Tian, BAI Xiao-jing, ZHENG Xi-yu, ZHU Xiang (State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China) Abstract: The principal component analysis (PCA) is a useful tool for data analysis and has been widely used in fault detection and process monitoring. MEWMA PCA (multivariate exponentially weighted moving average principal component analysis) is used to solve the problem where PCA cannot detect incipient faults properly. This paper further investigates the mechanism of the effect of EWMA on the fault detection of PCA in MEWMA PCA. The reason that MEWMA PCA can detect incipient faults is analyzed. The relationship among the forgetting factor, the detectable amplitude of a single sensor and the delay time introduced by EWMA is derived. Both numerical simulation results and historical data simulation result of a coal grinding unit in a power plant validate the mechanism of the improvement of fault detection by MEWMA PCA. An example is given for showing the detection of an incipient fault within a specified time range satisfying the practice requirement, by setting appropriate forgetting factor. Key words: incipient faults; principal component analysis (PCA); exponentially weighted moving average (EWMA); fault detection 1 (Introduction),.,,,. [1],., (principal component analysis, PCA),. [2 3], PCA, PCA., PCA. (exponentially weighted moving average, EWMA), [4 5]., Dunia [6] EWMA PCA : 2013 02 17; : 2013 09 17.. E-mail: qiutian@ncepu.edu.cn; Tel.: +86 13810697170. : 973 (2012CB215203); (51036002);.
20 31, Qin [7 8],, EWMA PCA. EWMA PCA, Chen [9] EWMA PCA T 2 SPE. [10] EWMA EWMA, PCA, MEWMA PCA(multivariate EWMA PCA), EWMA PCA. MEWMA PCA., λ. EWMA, EWMA.,,,. L(, ) λ. MEWMA PCA EWMA PCA,, EWMA, SPE T 2 PCA., MEWMA PCA. MEWMA PCA, λ.. 2 MEWMA--PCA (MEWMA--PCA method) X = T P + E, (1) : X n m, n, m ; T n k,, k ; P k m, ; E n m,. X, R, R = UDU, (2) : U m m, γ i R, D = diag{γ 1,, γ m }. u i m, U U = [u 1 u m ], k, P = [u 1 u k ], D k = diag{γ 1,, γ k }. T 2 SPE [11]. T 2 T 2 = D 1 2 k P x 2 T 2 α, (3) : Tα 2 T 2, α, F α,k,n k α F, k, n k. Tα 2 T 2 k(n + 1)(n 1) α = F n 2 α,k,n k. (4) nk SPE SPE = Cx 2 SPE α, (5) : C = I P P, SPE α SPE, α. SPE α = θ 1 [ c α 2θ2 h 2 0 + 1 + θ 2h 0 (h 0 1) ] 1 θ 1 θ1 2 h 0, (6) : θ i = m j=k+1 γj, i i = 1, 2, 3, h 0 = 1 2θ 1θ 3. 3θ 2 2 EWMA,,, i x i, i x i : x i = (1 λ) x i 1 + λx i, (7) : λ, λ,. X, X = [x 1 x 2 x 3 x n], : n, x i m, x 0 = 0. x i, x i = (1 λ) x i 1 + λx i = λ i (1 λ) i j x j. EWMA X = [ x 1 x n] (8) R R [9] : R = 1 n 1 X λ X = R. (9) 2 λ EWMA, P. T 2 SPE : T 2 λ 1 = ( 2 λ D) 2 P x 2. (10) t (0 < t n) T 2 [9 10] T t 2 = x t P λ ( 2 λ D) 1 P x t = λ t λ (1 λ) t j x j P ( 2 λ D) 1
1 : 21 P λ t (1 λ) t j x j. (11) T 2 k n, T 2 α = T 2 α. (12) t (0 < t n) SPE SPE t = x t C x t = λ t t (1 λ) t j x j Cλ (1 λ) t j x j. (13) (9) R γ i = λ/(2 λ)γ i, i = 1,, m. SPE α [9] SPE α = λ 2 λ SPE α. (14) i (0 < i < t) l, d, ξ = [0 0 1 0 0], t l T 2 [9] T 2 t = x t P D 1 P x t + 2x t P D 1 P ξ d + ξdp D 1 ξ d. (15) EWMA MEWMA PCA T 2 T 2 t = λ t λ (1 λ) k j x j P ( 2 λ D) 1 P λ t (1 λ) t j x j + 2λ t λ (1 λ) t j x j P ( 2 λ D) 1 P λ t (1 λ) t j ξ d + λ t λ (1 λ) t j ξdp ( 2 λ D) 1 P λ t (1 λ) t j ξ d. (16) i (0 < i < t) l d, t SPE MEWMA PCA SPE : SPE t = x t Cx t + 2x t CP ξ d + ξd Cξ d, (17) SPE t = λ t t (1 λ) t j x j Cλ (1 λ) t j x j + 2λ t t (1 λ) t j x j Cλ (1 λ) t j ξ d + λ t (1 λ) t j ξd Cλ t (1 λ) t j ξ d. (18) 3 (EWMA s effect to PCA) 3.1 EWMA PCA (EWMA s effect to the mean value of PCA indices) (12)(14), EWMA, T 2, SPE λ/(2 λ)., EWMA PCA. EWMA, T 2 SPE. PCA T 2 [12] E(T 2 ) = n Tj 2 /n, (19) n. MEWMA PCA T 2 E( T 2 ) = n T j 2 /n. (20), x i x j (i j), x i x j = 0(i j). M = P λ ( 2 λ D) 1 P, T t 2 = λ 2 t (1 λ) 2(t j) x j Mx j, t = 1, 2,, n, n = λ 2 [ n ( j (1 λ) 2(j i) )x j Mx j] = T 2 j i=1 λ 2 [ n 1 (1 λ) 2(n j+1) 1 (1 λ) 2 x j Mx j] = λ 2 1 (1 λ) [n2 λ 2 λ E(T 2 ) S], (21) S = n (1 λ) 2(n j+1) x j Mx j. x max Mx max = max(x 1 Mx 1,, x n Mx n), S = n (1 λ) 2(n j+1) x max Mx max = (1 λ) 2 (1 (1 λ) 2n ) 1 (1 λ) 2 x max Mx max. (22) n, S /n=0, n, S/n=0, n E( T 2 ) = 1. (23) E(T 2 ), n SPE j =
22 31 λ 2 [ n ( j (1 λ) 2(j i) )x Cx j j ] = i=1 C( x + λ t (1 λ) t j ξd) λ 2 [ n 1 (1 λ) 2(n j+1) x 1 (1 λ) Cx 2 j j ] = C x Cλ t (1 λ) t j ξd δ. (29) λ 2 1 (1 λ) [ne(spe) 2 S ],, x = 0, (0, ) (24) d d δ S = n (1 λ) 2(n j+1) x Cx j j. Θ( Cξ)(1 (1 λ) t i+1 ), (30) Θ( ). (, ), d n, E(SPE) E(SPE) = λ d = 2d. (31) 2 λ. (25) (30) (31) SPE EWMA T 2 SPE. [9], EWMA T 2 SPE,, EWMA, T 2 (2 λ)/λ, SPE. EWMA PCA. (23)(25),, EWMA, T 2, SPE, λ/(2 λ). (16), PCA EWMA,,, T 2 (2 λ)/λ,. T 2. (18),, SPE,, SPE. 3.2 EWMA PCA (EWMA s effect to the detectable amplitude of PCA) EWMA PCA, PCA, EWMA PCA, EWMA PCA., EWMA, λ, EWMA., λ, λ. SPE k = C x k 2 δ 2, (26) δ 2 = SPE α., (0, ),, C x k δ. (27) x k x ξd, C( x + ξd) δ. (28) 2δ ln(1 dθ( L SPE = Cξ) ) 1. (32) ln(1 λ), T 2 2χ d. λ Θ(P ( 2 λ D) 1 P ξ)(1 (1 λ) t i+1 ) (33) L T 2 2χ ln(1 ) λ dθ(p ( L T 2 = 2 λ D) 1 ξ) 1, (34) ln(1 λ) χ 2 = T α 2. SPE T 2., T 2, SPE. SPE T 2., λ (32). (30)(33), λ PCA, λ,,, T 2 λ/(2 λ). (14), EWMA, δ λ/(2 λ),, SPE λ/(2 λ). EWMA,,, λ. (32)(34) L,, λ, L,, PCA, λ,, L,
1 : 23,,, PCA,. (35) λ 2δ λ = 1 (1 dθ( Cξ) ) 1 L SPE +1. (35) 4 (Simulation) 4.1 (Numerical simulation) [13], X = As + E, A 4, (36), s = [s 1 s 2 s 3 s 4 ] 4 n, s 1, s 2, s 3, s 4 0, 1, E 0, 0.2. 0.6266 0.3935 0.1911 0.2075 0.0397 0.3559 0.4510 0.0722 0.6209 0.3288 0.2797 0.1234 A = 0.1381 0.4254 0.0618 0.6010 0.3020 0.0518 0.7609 0.0280. 0.2399 0.1791 0.0567 0.5458 0.1735 0.4960 0.1127 0.5117 0.1494 0.3855 0.2884 0.1216 (36) 500, N. 4, EWMA, R λ/(2 λ), EWMA. (21)(24) T 2 SPE, 1 2. 1 T 2 Table 1 Forgetting factor s effect to the mean value of T 2 N 500 2500 4500 6500 8500 λ = 0.2 3.7151 3.9607 3.9412 3.8904 3.8626 λ = 0.4 3.9114 3.9844 3.9694 3.9474 3.9134 λ = 0.6 3.9893 4.0052 3.9852 3.9682 3.9393 λ = 0.8 4.0277 4.0160 3.9921 3.9742 3.9511 λ = 1.0 4.0414 4.0127 3.9879 3.9689 3.9514 2 SPE Table 2 Forgetting factor s effect to the mean value of SPE N 500 2500 4500 6500 8500 λ = 0.2 0.0346 0.0360 0.0360 0.0363 0.0365 λ = 0.4 0.0776 0.0812 0.0810 0.0816 0.0818 λ = 0.6 0.1338 0.1397 0.1391 0.1400 0.1400 λ = 0.8 0.2095 0.2180 0.2166 0.2180 0.2176 λ = 1.0 0.3161 0.3276 0.3248 0.3273 0.3260 1 2 λ T 2 SPE.,, EWMA, T 2, SPE λ/ (2 λ)., T 2 SPE. λ PCA, 3 4 2 λ. 3 T 2 Table 3 The detectable amplitudes of T 2 statistic corresponding to different forgetting factors and delays 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 λ = 0.2 10.1 7.3 6.3 5.7 5.4 5.2 5.1 5.1 5.0 5.0 5.0 5.0 5.0 4.9 4.9 λ = 0.4 9.4 8.0 7.6 7.5 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 λ = 0.6 10.4 9.8 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 λ = 0.8 12.2 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 λ = 1.0 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 14.8 4 SPE Table 4 The detectable amplitudes of SPE statistic corresponding to different forgetting factors and delays 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 λ = 0.2 1.97 1.43 1.21 1.11 1.05 1.01 0.99 0.98 0.97 0.97 0.96 0.96 0.96 0.96 0.96 λ = 0.4 1.84 1.56 1.48 1.45 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 λ = 0.6 2.01 1.90 1.89 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 λ = 0.8 2.37 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 λ = 1.0 2.88 2.88 2.88 2.88 2.88 2.88 2.88 2.88 2.88 2.88 2.88 2.88 2.88 2.88 2.88
24 31, λ, PCA,. EWMA,, λ,. λ,,,, ;,.,, λ,. PCA, 2 100 300 1, PCA λ = 0.2 MEWMA PCA λ = 0.5 MEWMA PCA, 1 3. 1 2, PCA. EWMA,,, 106. 1 PCA Fig. 1 Fault detection results of PCA without EWMA 2 λ 0.2 PCA Fig. 2 Fault detection results of MEWMA PCA with λ = 0.2 3 λ 0.5 PCA Fig. 3 Fault detection results of MEWMA PCA with λ = 0.5 3, λ 0.5, 103, 19%., EWMA PCA, PCA. λ,,., EWMA λ,, λ,, λ,. EWMA PCA, λ. λ,, PCA ; λ,,., (35) λ. 4.2 (Example simulation) A ( 1, 2, 1, 2, 3, 3,, A), 1 min, 8,. 2. 300 min,, 235 min. λ 3.5, 10 min, ( ) 3.5, 10 min. EWMA, 5. 5, PCA,
1 : 25,, PCA ;,. 6, 6.05, PCA,,., (35) λ = 0.5, EWMA,, 3.5. 4 5 PCA EWMA PCA MEWMA PCA. 5 PCA Table 5 The detectable amplitudes of PCA without filtering x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 T 2 27.93 43.05 23.79 19.80 17.52 8.39 25.62 36.16 SPE 3.79 3.64 3.84 3.91 3.98 6.05 3.80 3.75 4 EWMA PCA Fig. 4 Fault detection results of PCA without EWMA,,, 30 min, 6, 3.5, 30 min,,. 5, MEWMA PCA. λ, MEWMA PCA. 5 (Conclusion). MEWMA PCA EWMA PCA,, EWMA, EWMA T 2 SPE., λ, MEWMA PCA, EWMA PCA., MEWMA PCA EWMA. λ, MEWMA PCA. (References): [1],,,. [J]., 2012, 29(12): 1517 1529. (LI Juan, ZHOU Donghua, SI Xiaosheng, et al. Review of incipient fault diagnosis methods [J]. Control Theory & Applications, 2012, 29(12): 1517 1529.) 5 λ = 0.5 MEWMA PCA Fig. 5 Fault detection results of MEWMA PCA with λ = 0.5 4, PCA, 235, [2] WANG H, ZHOU H, HANG B. Number selection of principal components with optimized process monitoring performance [C] //Proceedings of Decision and Control. Hangzhou, China: Zhejiang University, 2004: 4726 4731. [3],,. [J]., 2002, 23(3): 232 235. (WANG Haiqing, SONG Zhihuan, LI Ping. Study on the fault de-
26 31 tectability of principal component analysis [J]. Chinese Journal of Scientific Instrument, 2002, 23(3): 232 235.) [4] AKHILESH J, RAJEEV U, SUMANA C. Exponentially weighted moving average scaled PCA for on-line monitoring of Tennessee Eastman challenge process [J]. International Journal of Systems, Algorithms & Applications, 2012, 2(12): 183 186. [5] ZHANG G, LI N, LI S. A modified multivariate EWMA control chart for monitoring process small shifts [C] //Proceedings of Modelling, Identification and Control. Shanghai: Shanghai JiaoTong University, 2011: 75 80. [6] DUNIA R, QIN S J, EDGAR T F, et al. Identification of faulty sensors using principal component analysis [J]. AIChE Journal, 2004, 42(10): 2797 2812. [7] QIN S J, YUE H, DUNIA R. Self-validating inferential sensors with application to air emission monitoring [J]. Industrial & Engineering Chemistry Research, 1997, 36(5): 1675 1685. [8] QIN S J, YUE H, DUNIA R. A self-validating inferential sensor for emission monitoring [C] //Proceedings of American Control. Austin, TX, USA: Texas University, 1997: 473 477. [9] CHEN J, LIAO C M, LIN F R J, et al. Principle component analysis based control charts with memory effect for process monitoring [J]. Industrial & Engineering Chemistry Research, 2001, 40(6): 1516 1527. [10],,. MEWMA PCA [J]., 2007, 36(5): 650 656. (GE Zhiqiang, YANG Chunjie, SONG Zhihuan. Research and application of small shifts detection method based on MEWMA PCA [J]. Information and Control, 2007, 36(5): 650 656.) [11] JACKSON J E. A User s Guide to Principal Components [M]. New York: John Wiley & Sons, 1991. [12],,. [M]. :, 2001. (SHENG Zhou, XIE Shiqian, PAN Chengyi. Probability Theory and Mathematical Statistics [M]. Beijing: Higher Education Press, 2001.) [13] QIN S J. Statistical process monitoring: basics and beyond [J]. Journal of Chemometrics, 2003, 17(8): 480 502. : (1976 ),,,,, E-mail: qiutian@ncepu.edu.cn; E-mail: xiaoj bai@163.com; (1987 ),,,, (1989 ),,,, E-mail: xiyuer.zheng@gmail.com; (1988 ),,,, E-mail: xuchunhonghappy03@163.com.