Module C1: Adaptive Charting Techniques for Feedback-Controlled and Dynamic Systems

Transcription

1 Module C: Adapive Charing Techniques for Feedback-Conrolled and Dynamic Sysems Prof. Fugee Tsung Hong Kong Universiy of Science and Technology Qualiy Lab: hp://qlab.ielm.us.hk

2 Ouline I. Inroducion II. III. IV. The adapive T char Monioring feedback-conrolled processes Monioring inerial processes V. Conclusions Dr. Fugee Tsung, HKUST

3 I. Inroducion

4 SPC (Saisical Process Conrol) and variaion reducion Process mean Observaions..34 UCL=.34 Process mean.. Measuremens _ X= Time period Toal Variaion = 7 4 Common Cause Variaion Time period Assignable + Cause Variaion LCL=-.34 Dr. Fugee Tsung, HKUST 4

5 A basic model for SPC Noise + Failure signal = Observaion a + δ f = x a :whienoise δ : shif magniude f : shif paern Tsung and Apley (), JQT Dr. Fugee Tsung, HKUST 5

6 Examples of failure signals A consan shif Dynamic shifs...5. Process mean Time sequence Time period Oupu Time period Wang and Tsung (7b), IJPR Wang and Tsung (7a), JQT Dr. Fugee Tsung, HKUST 6

7 The developmen of SPC δ, f δ, f f = f = a + δ f = x Dr. Fugee Tsung, HKUST 7

8 Anoher view of CUSUM chars The sopping ime of a univariae CUSUM char δ TC = inf : max δ xi h k > i= k+ Expressed as likelihood raio x H i ~ N( μσ, ) : μ = vs. H : μ = δ f( x H ) TC = > h k inf : max ln ' k i= k+ f( xk H) Dr. Fugee Tsung, HKUST 8

9 A graphical illusraion of CUSUM chars Expressed as likelihood raio of vecors X x x x x N T k = [ k+, k+,..., ],where i ~ ( μ, σ ) H : E[ X ] = vs. H : E[ X ] = δ k f( X H ) TC = > h k inf : max ln ' k f( Xk H) k - f( X H) ln f( X H) Dr. Fugee Tsung, HKUST 9

10 Cuscore and RF-Cuscore Chars Cuscore (Luceno and Box, 999) r T r x h i SO = inf : max i i Cuscore k > i= k+ RF-Cuscore (Han and Tsung (5), JASA) x T x x h i RFC = inf : max i i > RFCuscore k i= k+ Dr. Fugee Tsung, HKUST

11 A unified framework for SPC Process variables Xk = x, L x, p L x k+, L x k+, p T T x x k+ Shif paern H : E( Xk ) = T T H: E( Xk) = [ d,..., d k+ ] A unified form T inf { : max ( ) } CA = Lk > h k K Key elemens dimension: p shif paern: d covariance: Σ search window: K Dr. Fugee Tsung, HKUST

12 δf : known (= R ) Σ = I, k = δf :known (=δ ) Σ = I, k f : known, δ : unknown (replaced by is MLE) Σ = I, k CUSCORE (Luceno (999)) CUSUM for Univariae i.i.d. process GLRT (Apley and Shi (999)) RF-CUSCORE (Han and Tsung (5)) d : unknown (replaced by X ) Σ = I, k = The unified form Shif paern: d = δf Covariance: Σ Change-poin: k K p = p > Key elemens dimension: p shif paern: d covariance: Σ search window: K T δf : unknown (replaced by X ) k = K + (no searching) Dynamic (Tsung and Apley ()) Adapive T d : unknown (replaced by predicion) k = K + (no searching) Dr. Fugee Tsung, HKUST

13 II. The adapive T char Wang, K., and Tsung, F., Journal of Qualiy Technology, 7

14 Direcionally invarian char for mulivariae processes Process variables x = [ x, x ] T ARL conour plo 3 Hypohesis H = vs. : μ H : μ μ Hoelling s T Char T = x Σ x > T h μ Dr. Fugee Tsung, HKUST 4

15 Direcionally varian char A known shif, d Hypohesis H = vs. : μ H : μ = d T T Λ = = exp d Σ x d Σ d. f( x H) Charing saisic (log-likelihood raio): T f( x H ) = d Σ x d Σ d > h T T Dr. Fugee Tsung, HKUST 5

16 Properies of direcionally varian chars Simplified forma: T = d Σ x d Σ d > h T T T = d Σ x > h' T Propery: he char is sensiive o all shifs in he same direcion H : = H H μ : μ : μ = d = δd Dr. Fugee Tsung, HKUST 6

17 ARL performance of direcional chars Invarian char Varian char μ 5 - μ μ μ 3 Dr. Fugee Tsung, HKUST 7

18 ARL performance comparison d =[,] d =[,] μ - 5 μ μ μ Conclusion: a direcionally varian char is more sensiive o designaed shifs! Dr. Fugee Tsung, HKUST 8

19 Exended direcional chars - Combine direcionally varian & invarian chars Shif diagram Shifs along General shifs μ, μ T = x Σ x > h T = μσ x > h T = μσ x > h T T T Zhou, Jin, Jin (5), IIE Trans. Dr. Fugee Tsung, HKUST 9

20 Exended direcional chars -. Variable E X.5 V = x e U d U -char : = [, ] = dσ V > h U d U -char : = [, ] = d Σ V > h Jiang (4), IIE Trans. Dr. Fugee Tsung, HKUST

21 Sequenial opimal ess. Variable E X U -Char U j -Char U -Char d d j d Dr. Fugee Tsung, HKUST

22 The adapive T procedure Known shif, Hypohesis vs. H : μ = H : μ = d f( x H ) Λ = = T T exp d Σ x d Σ d. f( x H) Charing saisic (log-likelihood raio): T d = d Σ x d Σ d > h T T AT AT Dr. Fugee Tsung, HKUST

23 Philosophy of he Adapive T Procedure racking and learning (a) general T chars (b) direcional T chars (c) adapive T chars Dr. Fugee Tsung, HKUST 3

24 Shif direcion forecasing using EWMA Adapive T Char d = λx + ( λ) d T = d Σ x d Σ d > h T T AT AT λ = λ = T = x Σ x T Hoelling s T Char T = d Σ x T Direcional T Char Dr. Fugee Tsung, HKUST 4

25 ARL properies of he AT char Noncenraliy parameer (NCP): T = μσ μ The ARL performance of he AT char is deermined by NCP. Design guidelines c Dimension In-conrol ARL Ineresed NCP Design Opimal uning parameer Conrol limi Dr. Fugee Tsung, HKUST 5

26 The choice of an opimal λ I increases wih: Shif magniude, NCP I decreases wih: In-conrol ARL Dimension, p I is usually less han.5 Opimal λ Noncenraliy parameer Dr. Fugee Tsung, HKUST 6

27 The choice of a conrol limi Nonlinear conrol limis Conrol Limis Soluion: fi a regression funcion CL = f ( λ, ICARL, p) λ Dr. Fugee Tsung, HKUST 7

28 Performance sudy A bivariae process Ideniy covariance Σ =. ARL 5 5 AT λ=. AT λ=.5 GT δ Dr. Fugee Tsung, HKUST 8

29 III. Monioring feedback-conrolled processes

30 A feedback-conrolled process Sysem model: e = μ + x + d d = φd + ε θε PI-conroller x = k e + k e P I k k = MMSE-conroller A feedback-conrolled process d y e x x = φx + ( θ φ) e Dr. Fugee Tsung, HKUST 3

31 Differen zones of PI-conrolled processes θ e = μ + x + d < μ = δσ d. Zone Zone 4 3 Zone 4 Behavior of oupu Ee [ ] = δσ d Ee [ ] = ( + kp + ki) δσ d lim Ee [ ]. Behavior of inpu Ex [ ] = ( kp + ki) δσ d Ex k k k k lim Ex [ ] δσ d. δσ [ ] = ( P + I + P + I) d Zone 3. φ.5. Dr. Fugee Tsung, HKUST 3

32 Trajecories of PI-conrolled processes..5 Variable E X Zone Zone..5 Variable E X (a) Zone : φ =.8, θ =.7, k =.5 ; k = P I (b) Zone : φ =.8, θ =.5, k =.5, k = P I..5 Variable E X Zone 3 Zone 4..5 Variable E X (c) Zone 3: φ =.7 ; θ =., k P =.4, k I =.645 k P =.5, k I =.47 Dr. Fugee Tsung, HKUST (d) Zone 4: φ =.8, θ =.3, 4 6 8

33 Trajecories of MMSE-conrolled processes. Variable E X. Variable E X (a) Zone : φ =.5, θ =.9 (b) Zone : φ =.7, θ =...75 Variable E X.5.5. ϕ (c) Zone 3: φ =., θ = φ Tsung and Tsui (3), IIE Trans. Dr. Fugee Tsung, HKUST 33

34 Window of Opporuniy (a) (b) Tsung, Shi, and Wu (999, JQT) Dr. Fugee Tsung, HKUST 34

35 Lieraure review Monior process oupu Monior conrol acion Falin, e al. (99), Box and Kramer (99), Capilla, e al. (999) Monior boh inpu and oupu Tsung e al. (998), Tsung (999a), Tsung (999b), Tsung, e al. (999), Tsung (), Tsung (), Tsung and Apley (), Tsung and Tsui (3), Shi and Tsung (3), Jiang (4), Tsung e al. (6) Jiang e al. (6), Runger e al. (6), V = [ e, x ] T T = V Σ V > T h Dr. Fugee Tsung, HKUST 35

36 On he efficiency and robusness of discree proporional-inegral conrol schemes (Tsung e al. (998), Technomerics) Process models Absolue efficiency (AE) e = Y + D Y = c+ δy + g( δ) X X = k kpe ki e B φb ARMA(,) : D = a θ B φb ARIMA(,,) : D = a ( θ B)( B) a AE = σ σ e σ : variance of whie noise; σ a e : variance of oupu Dr. Fugee Tsung, HKUST 36

37 AE of he opimal P, I, PI schemes under ARMA(,) disurbance Dr. Fugee Tsung, HKUST 37

38 AE of he opimal I and PI schemes under ARIMA(,,) disurbance Dr. Fugee Tsung, HKUST 38

39 Join monioring of PID-conrolled processes (Tsung e al. (999), JQT) Process models and he PID conrol schemes e = Y + D = X + D D = φd + a θa X = k e k e k ( e e ) P I j D j= Dr. Fugee Tsung, HKUST 39

40 PID design maps for ARMA(,) disurbance processes Dr. Fugee Tsung, HKUST 4

41 Join monioring schemes Bonferroni s approach CL CL =± L σ e e e =± L σ X X X Le = LX = z( α / 4) Hoelling s mulivariae conrol char χ = x Σ x T σe σ e, X Σ = σex, σ X Dr. Fugee Tsung, HKUST 4

42 The dynamic T char for monioring feedback-conrolled processes (Tsung and Apley (), IIE Trans.; IIET Bes Paper Award) Challenges Window of Opporuniy Dynamics and auocorrelaion Charing saisics L T DT = X Σ X X = [ y, u, y, u,..., y L, u L] T Dr. Fugee Tsung, HKUST 4

43 Dynamic T char: Pracical issues and soluions Marix inversion Generalized inverse DT = Σ = i Choice of L X Σ X L T ( L+ ) i= i λ ee i λ T i AR( ) expansion and check coefficiens Dr. Fugee Tsung, HKUST 43

44 Dynamic T char: Performance Sudy Mean shif Disurbance parameer shif Dr. Fugee Tsung, HKUST 44

45 PID Chars for Process Monioring (Jiang e al. (), Technomerics) PID conroller U = k e k e k ( B) e B P I D PID predicor ˆ D = ke k e k ( Be ) B P I D PID char e = D Dˆ = ( k ) e k ( B) e I P D( ) ( ) k B e + D D e > Lσ e Dr. Fugee Tsung, HKUST 45

46 Capabiliy indices C C T S μ = σ e μ CT = if ki = = σ e( + kp) + kp if ki > Dr. Fugee Tsung, HKUST 46

47 Design guidelines Specify ineresed shif level Maximize C T by adjusing PID parameers If Max C T >5, choose PID, hen sop; Max C S by varying P&D parameers Based on C S, chose opimal parameers Dr. Fugee Tsung, HKUST 47

48 Improved Design of PID Chars (Tsung e al. (6), JQT) Problems wih he exising design mehod C T, C S (ransien and seady-sae capabiliy) canno represen ARL disribuions Maximizaion of C and Cs does no consider ARL Dr. Fugee Tsung, HKUST 48

49 Mean shif paerns Dr. Fugee Tsung, HKUST 49

50 Guidelines for PID char design Specify ARL, auocorrelaion coefficien, and mean shif Use Table of Tsung e al. (6), find PID charing parameers Evaluae he enire ARL performance Dr. Fugee Tsung, HKUST 5

51 The AT char for feedbackconrolled processes (Wang and Tsung, IJPR, 7) Join monioring of oupu and inpu V [, ] T = e x T T = d Σ V d Σ d > T AT AT h EWMA forecasing d = λv + ( λ) d Problems EWMA and Oscillaions? Time sequence Dr. Fugee Tsung, HKUST 5

52 Oscillaed EWMA Specifically for oscillaed signals Calculaion Weighs EWMA m = ( λ) m + λe Δ = e m p = ( λ) p + λδ μ = m + p. Weighs Lags Oscillaed EWMA -. 5 Lags 5 Dr. Fugee Tsung, HKUST 5

53 Shif predicion in oscillaed processes..5 Variable E X EWMA_E EWMA_X OEWMA_E OEWMA_X Dr. Fugee Tsung, HKUST 53

54 Performance sudy The AT-OE char ouperforms T for small shifs ouperforms MEWMA for large shifs gives overall good performance AT-OE T MEWMA δ λ =. λ =.5 λ =. λ = Dr. Fugee Tsung, HKUST 54

55 IV. Monioring general dynamic processes Wang and Tsung (7a), JQT.

56 Modeling dynamic processes Process model y = δ y + g( δ ) x + a N x = φn + ε θε = μ + N x, A ank example Dr. Fugee Tsung, HKUST 56

57 Modeling dynamic processes Shif model - Ea ( ) < = M ( δσ ) y, Ey [ ] = M( δ ) σ Mσ y y i= δ i [ Mσ y ( δ),] T [ y, x] : T [ Mσ y,] T Shif model - < μx = Mσ x, Ey [ ] = g( δ ) Mσ δ Mgσ Dr. Fugee Tsung, HKUST 57 x x i= [ Mgσx( δ), Mσx] T [ Mgσx, Mσx] i T

58 Dynamic changes y Oupu Concenraion 4 3 Oupu Concenraion Time Period Time Period x Inpu Concenraion - Inpu Concenraion Time Period Time Period Dr. Fugee Tsung, HKUST 58

59 The AT char for monioring dynamic processes Process variables V = [ y, y,..., y, x, x,..., x ] T L + L + Charing saisics T = D Σ V D Σ D > h T T AT Tsung and Apley (), IIE Trans. Dr. Fugee Tsung, HKUST 59

60 The choice of L and L Expansion as an AR series ACF plo ( ) k = θ φ θ k + x k = x x e Auocorrelaion Lag Dr. Fugee Tsung, HKUST 6

61 Shif predicion One-sep-ahead predicion of imeseries model Z = [ y, x ] T b = φx θ ( x b ) x, x, b = δ y + g( δ ) b y, x, EWMA predicion Z d = λz + ( λ) d D d = [ y, x ] T = [ d,..., d, y, y, L +,..., d ] x, x, L + T D b = [ b,..., b, y, y, L +,..., b ] x, x, L + T Dr. Fugee Tsung, HKUST 6

62 ARL disribuion under model uncerainies AT-E, samples AT-OSA, samples ARL 5 ARL Mean Shif Mean Shif AT-E, samples AT-OSA, samples ARL 5 ARL Mean Shif Mean Shif Dr. Fugee Tsung, HKUST 6

63 The choice of sample sizes Variance decreases when sample size increases Decrease sharply for small sample sizes The use of a leas 5 samples is recommended Iner-quarile Iner-quarile plo of he in-conrol ARLs 5 5 Sample size Dr. Fugee Tsung, HKUST 63

64 V. Conclusions

65 Conclusions Criical poins in modeling dynamic processes Idenify process dynamics Modeling process dynamics Monioring Adapive T char Modify for special process dynamics Dr. Fugee Tsung, HKUST 65

66 References Box, G. E. P., and Kramer, T. (99). "Saisical Process Monioring and Feedback Adjusmen - A Discussion." Technomerics, 34(3), Capilla, C., Ferrer, A., Romero, R., and Hualda, A. (999). "Inegraion of saisical and engineering process conrol in a coninuous polymerizaion process." Technomerics, 4(), 4-8. Falin, F. W., and Woodall, W. H. (99). "Some Saisical Process-Conrol Mehods for Auocorrelaed Daa - Discussion." Journal of Qualiy Technology, 3(3), Fan, S.-K., Lo, L.-C., Chang, Y.-J., Lin, C.-J. (), " Predicion of imevarying merology delay for dewma and RLS-LT conrollers, " Journal of Process Conrol,, Han, D., and Tsung, F. (6). "A reference-free Cuscore char for dynamic mean change deecion and a unified framework for charing performance comparison." Journal of he American Saisical Associaion,, Han, D. and Tsung, F. (9), " Opimal Sopping Time for Deecing Changes in Discree Time Markov Processes," Sequenial Analysis, 8, Han, D. and Tsung, F. (), " Deecion of Changes in a Random Financial Sequence wih a Sable Disribuion," Journal of Applied Saisics, 37, 89-. Han, D., Tsung, F., Li, Y. and Wang, K. (), " A Nonlinear Filer Conrol Char for Deecing Dynamic Changes," Saisical Sinica,, Dr. Fugee Tsung, HKUST 66

67 References Jiang, W. (4). "A join monioring scheme for auomaically conrolled processes." IIE Transacions, 36(), -. Jiang, W., Shu, L. J., and Tsung, F. G. (6). "A comparaive sudy of join monioring schemes for APC processes." Qualiy and Reliabiliy Engineering Inernaional, (8), Jiang, W., Wu, H. Q., Tsung, F., Nair, V. N., and Tsui, K. L. (). "Proporional inegral derivaive chars for process monioring." Technomerics, 44(3), 5-4. Li, Y., Pu, X., and Tsung, F. (9), " Adapive Charing Schemes Based On Double Sequenial Probabiliy Raio Tess," Qualiy and Reliabiliy Engineering Inernaional, 5, -39. Luceno, A., and Gonzalez, F. J. (999). "Effecs of dynamics on he properies of feedback adjusmen schemes wih dead band." Technomerics, 4(), 4-5. Runger, G. C., Tesik, M. C., and Tsung, F. (6). "Relaionships among conrol chars used wih feedback conrol." Qualiy and Reliabiliy Engineering Inernaional,, Shi, D. F., and Tsung, F. (3). "Modelling and diagnosis of feedbackconrolled processes using dynamic PCA and neural neworks." Inernaional Journal of Producion Research, 4(), Dr. Fugee Tsung, HKUST 67

68 References Shu, L. J., Apley, D. W., and Tsung, F. (). "Auocorrelaed process monioring using riggered Cuscore chars." Qualiy and Reliabiliy Engineering Inernaional, 8, 4-4. Tsung, F. (999a). "Improving auomaic-conrolled process qualiy using adapive principal componen monioring." Qualiy and Reliabiliy Engineering Inernaional, 5(), Tsung, F. (999b). "On hree-erm adjusmen schemes for saisical process conrol." Inernaional Journal of Indusrial Engineering, 6, 6-7. Tsung, F. (). "Saisical monioring and diagnosis of auomaic conrolled processes using dynamic PCA." Inernaional Journal of Producion Research, 38(3), Tsung, F. (). "A noe on saisical monioring of engineering conrolled processes." Inernaional Journal of Reliabiliy,. Qualiy and Safey Engineering, 8, -4. Tsung, F., and Apley, D. W. (). "The dynamic T- char for monioring feedback-conrolled processes." IIE Transacions, 34(), Tsung, F., Shi, J., and Wu, C. F. J. (999). "Join monioring of PIDconrolled processes." Journal of Qualiy Technology, 3(3), Tsung, F., and Shi, J. J. (999). "Inegraed design of run-o-run PID conroller and SPC monioring for process disurbance rejecion." IIE Transacions, 3(6), Dr. Fugee Tsung, HKUST 68

69 References Tsung, F., and Tsui, K. L. (3). "A mean-shif paern sudy on inegraion of SPC and APC for process monioring." IIE Transacions, 35(3), 3-4. Tsung, F., Wu, H. Q., and Nair, V. N. (998). "On he efficiency and robusness of discree proporional-inegral conrol schemes." Technomerics, 4(3), 4-. Tsung, F. G., Zhao, Y., Xiang, L. M., and Jiang, W. (6). "Improved design of proporional inegral derivaive chars." Journal of Qualiy Technology, 38(), Wang, K., and Tsung, F. (7). "Monioring feedback-conrolled processes using adapive T schemes." Inernaional Journal of Producion Research, 45, Wang, K., and Tsung, F. (8). "An Adapive T Char for Monioring Dynamic Sysems." Journal of Qualiy Technology, 4, 9-3. Wang, K. and Tsung, F. (9), " An Adapive Dimension Reducion Scheme for Monioring Feedback-Conrolled Processes," Qualiy and Reliabiliy Engineering Inernaional, 5, Wang, K. and Tsung, F. (), " Recursive parameer esimaion for caegorical process conrol," Inernaional Journal of Producion Research, 48, Zhou, S., Jin, N., and Jin, J. (5). "Cycle-based signal monioring using a direcionally varian mulivariae conrol char sysem." IIE Transacions, 37(), Dr. Fugee Tsung, HKUST 69