Nonlinear Minimum Variance Estimation For Fault Detection Systems

Transcription

1 Nonlinear Minimum Variane Etimation For Fault Detetion Sytem Abtrat A novel model-baed algorithm or ault detetion (FD) in tohati non-linear ytem i propoed. The Nonlinear Minimum Variane (NMV) etimation tehnique i ued to generate a reidual ignal whih i then ued to detet atuator and enor ault in the ytem. The main advantage o the approah i the impliity o the nonlinear etimator theory and the traightorward truture o the reulting olution. Simulation example are preented to illutrate the deign proedure and the type o reult obtained. The reult demontrate that both atuator and enor ault an be deteted ueully.. Introdution The need or high perormane, eiieny, aety and reliability in modern engineering ytem ha oued interet in the Fault Detetion and Iolation (FDI) problem. A ault i deined a an unexpeted hange in a ytem with omponent maluntion or variation in operating ondition. Some ault, i not promptly and properly deteted, ould turn into unreoverable ailure, auing eriou damage and even lo o human live []. In the literature ault an be aume to take plae in dierent part o a ytem, and are laiied a atuator ault or enor ault [2]. Atuator ault an repreent partial or omplete lo o ontrol ation. A total atuator ault an our a a reult o a breakage, ut or burned wiring, hort-iruit or the preene o oreign body in the atuator [2]. Senor ault in inorret output rom the enor. They an alo be ubdivided into partial and total ault. Fault Detetion (FD) method an be laiied into two major ategorie; model-baed and data-driven approahe [3]. The model-baed Fault Detetion Iolation (FDI) approahe inlude parity pae, parameter etimation and oberver baed approahe. The oberverbaed FDI method i one o the mot eetive and ha reeived igniiant interet rom indutry [4]. Model baed approahe typially rely on two tep: reidual generation; the proedure o extrating ault ymptom rom the proe, and reidual evaluation; the proedure o deiion making [5]. The reidual are oten generated uing either an oberver; or determiniti model, or an optimal ilter or tohati model. Oberver baed FD method ue meaurement o the atual ignal and etimate o the ignal to generate the reidual. The reidual hould be deined to beome large when a ault our, to avoid ale alarm [6], but remain a mall a poible due to other unertaintie uh a unknown diturbane and modelling error.

2 generation approahe have been developed ueully or linear ytem. However, muh le work ha been done or nonlinear ytem. Thi i primarily due to the omplexity o nonlinear ytem. The area o FDI or nonlinear ytem i not overed ompletely yet, o it i worthy o tudy [7]. There i ome exiting literature on the ue o a nonlinear etimator or ault detetion and iolation. The mot popular etimator or nonlinear proee i known to be extended Kalman ilter (EKF) [8]. Although widely ued, EKF have ome deiienie, inluding the requirement o dierentiability o the tate dynami a well a ueptibility to bia and divergene in the tate etimate. The unented Kalman ilter (UKF), on the ontrary, ue the nonlinear model diretly intead o linearizing it [9] and hene doe not need to alulate the Jaobian and an ahieve higher order auray. Partile ilter (PF) or Sequential Monte Carlo Method are onidered a general numerial tool to approximate the a poteriori denity in nonlinear and non-gauian iltering problem. The main drawbak with the partile ilter i that it i very demanding omputationally[]. In thi tudy, the Nonlinear Minimum Variane (NMV) etimator i ued or the irt time to generate a reidual ignal or ault detetion appliation. The trong point o thi tehnique i that a general nonlinear operator i ued to repreent the nonlinearity o the hannel or o the meaurement enor. Thi might involve a et o nonlinear equation or even inlude a look-up table or be a model obtained rom a neural or uzzy-neural network. The main advantage o propoed etimator i that no on-line linearization i required, a in the extended Kalman ilter, and implementation i eay. The ot-untion to be minimized i the variane o the etimation error and a relatively imple optimization proedure and olution reult []. The roadmap or thi tudy i a ollow. The derivation o NMV etimation method i given in etion 2. NMV baed reidual generation or ault detetion i deribed in etion 3. The perormane o the propoed ault detetion method i illutrated by a ae tudy in etion 4. Finally the onluion are ummaried in etion Nonlinear Minimum Variane Etimation The theory o Nonlinear Minimum Variane Etimation (NMVE) wa intrıdued by Grimble [] uing polynomial ytem model [2, 3] and later tate-equation baed model [4, 5]. The NMVE tehnique involve the etimation o a ignal that pae through a ommuniation hannel having nonlinearitie and ommuniation/tranport delay [3]. The meaurement are aumed to be orrupted by a noie ignal, whih i orrelated with the ignal to be etimated. Signal and noie model are aumed to be linear and time-invariant. The Nonlinear Minimum Variane (NMV) etimator derivation i baed on the minimization o the error variane riterion. Conider the ytem hown in Fig., whih inlude the nonlinear ignal hannel model and linear meaurement noie and ignal model. 2

3 ε Signal oure W z ( ) Nonlinear noie deign parallel hannel W ( z ) Signal y + + W n n ( z ) W ( ) z Linear ub-ytem or main hannel z Λ z Λ Channel delay n d d ( ) z ( z ) Nonlinear main hannel + + n z ( ) z Nonlinear etimator Etimated ŝ Meage ignal: = Weighted meage: Wy q = W q ε Coloured Noie Meage ignal: = W Figure : Signal and Noie Model and Communiation Channel Dynami 3

4 The ignal hannel model inlude the nonlinearitie that may involve both linear and nonlinear dynami. The ignal hannel dynami with a delay an be expreed a: where z Λ Λ = k I Λ ( )( ) = ( )( ) hannel t z W t () denote a diagonal matrix o the k tep delay element in the ignal path and. The parallel path dynami hown in Fig., by a dotted line, an be expreed a: ( z ) = ( z ) z Λ (2) Thi i a ititiou hannel, added to provide deign tuning option, that an be ued to repreent unertaintie in hannel knowledge, whih provide additional deign reedom. The r ombined ignal oure and noie ignal (t) R i given a: (t) = y(t) + n(t) (3) Conider the nonlinear ytem or the optimal etimation problem illutrated in Fig.. The input and noie generating proee have an innovation ignal model with white noie ignal r input: ε () t R and it may be aumed to be zero-mean with ovariane matrix: ov[ ε( t), ετ ( )] = Iδt τ where δ tτ denote the Kroneker delta-untion. The ignal hown in the loed-loop ytem model o Fig. may be lited a: Noie: nt ( ) Wε ( t) Input ignal: y( t) Wε ( t) = (4) n = (5) Channel input: ( t) = yt ( ) + nt ( ) (6) Linear hannel ubytem: () t ( W )( t) = (7) Weighted hannel intererene: n () t = ( ε )() t (8) Nonlinear hannel ubytem: () t ( )( t) Λ Nonlinear hannel input: ( t) z () t ( t k) d = (9) d = = () Obervation ignal: z( t) n ( t) ( t) Meage ignal to be etimated: t W yt WWε ( t) = + () () = () = (2) Weighted meage ignal: Etimation error ignal: () t = WW y() t (3) q q ( t t ) = ( t) ˆ ( t t ) (4) 4

5 where t ˆ( t ) denote the etimate o the ignal (t) at time t, given obervation z(t) up to time t-. Value o may be poitive or negative aording to the ollowing ondition: =, or etimation; >, or predition and <, or ixed-lag moothing. The riterion or the nonlinear minimum variane etimator i given below: T J= traeewt { { ( t )( Wt ( t )) }} (5) q where E {.} denote the expetation operator and W q [6] denote a linear tritly minimumphae dynami ot-untion weighting untion matrix whih i aumed to be tritly minimum phae, quare and invertible. The etimate t ˆ( t ) i aumed to be generated rom a nonlinear etimator o the orm: q where t ˆ( t ) = H (, t z ) zt ( ) (6) (, tz ) = W H( + W Y) (7) q where (, tz ) denote a minimal realiation o the optimal nonlinear etimator. Sine an ininite-time ( t = ) problem i o interet thereore no initial ondition term i required. The blok diagram repreentation o (, tz ) will be a hown in Fig. 2. Nonlinear etimator Obervation + - m(t) p(t) Etimate Tranmiion Figure 2: Implementation o the Nonlinear Etimator The term H,A and Y ued in equation (8) an be alulated uing the onept o power * * petrum or the ombined linear model uing: φ = (W + W )(W + W ), and where the notation or the adjoint o W implie: n n * T W (z ) = W (z), and in thi ae the z denote z- 5

6 domain omplex number. The generalized petral-ator: YY * Y may be omputed uing: = φ,where Y = A D = DA. The ytem model are aumed uh that D i tritly Shur polinomial matrix [7, 8] atiying: D D = (C + C )(C + C ) (8) * * * n n The right-omprime polynomial matrix model an be deined a: C D A = WWW q Y (9) The polynomial operator H now may be optained rom the minimal degree olution (H,F ), with repet to F, o the ollowing Diophantine equation: k FA + Gz = C (2) The etimation error an be penalied in a partiular requeny range by uing a dynami aymptotially table weighting untion W A Ω = Ω BΩ, where A Ω and BΩare polynomial matrie. The weighted error involve a linear path at the optimum. In the linear ae the modiied ot untion will have the orm (Pareval theorem doe not apply in the nonlinear ae): T * J = trae{ E(WΩe( t t ))(WΩe( t t )) } = trae / ( 2π j ) (WΩΦeeW Ω )dz / z (2) z= 3. NMVE Baed Fault Detetion In nonlinear minimum-variane etimation, the nonlinearitie are aumed to be in the ignal hannel or poibly in a noie hannel repreenting the unertainty. The imple olution that ollow arie beaue o the aumption o linearity or the ignal generating model and the reult obtained here involve only a leat-quare type o analyi [9]. The Fault detetion tehnique are oten baed on the generation o appropriate reidual ignal whih have to be enitive to ault themelve but independent o diturbane. 6

7 Model-baed FD method are baed on omparing the behavior o the atual ignal and an etimated ignal o the ytem. Typially, it i hown that in the abene o a ault, the oberver reidual approahe zero. When a ault exit, thi reidual will be non-zero, and it may thereore erve a a ault indiator. The blok diagram o the propoed nonlinear minimum variane etimator, taking =, baed on reidual generation or ault detetion, i hown in Fig. 3. n Weighted hannel intererene Meaured ignal + + NMV Etimator ŝ W ŷ = ˆ W (z ) Linear ub-ytem (z ) nonlinear main hannel ˆ Generation r, a: Fig. 3: NMVE Baed Generation Sheme The reidual ignal an be generated by uing meaured ignal ( t) and it etimate ŝ ( t) ( ) ( ) ˆ ( ) r t = t t (22) The NMV algorithm etimate the ignal ŝ o ŝ might be deined in term o ŝ ignal by uing eqn(6), eqn(7), eqn(9) and eqn() a ollow ( ) ˆ( ) ˆyt = W t (23) ˆ t = ˆyt = W ˆt (24) ( ) ( ) ( ) ˆ ( ) ˆ ( ) t = W t (25) Then inally reidual ignal an be alulated ubtituting eqn(28) into eqn (23): ( ) ( ) ˆ( ) rt = t WW t (26) Thi reidual ignal r i going to be heked with a reaonable threhold to detet that a ault ha ourred in the ytem. 7

8 When there i a ault at the ignal etimation point, the reidual beome rt = t W W t ˆ + φ (27) () () () I the plant i linear thi impliie a: = + (28) W ˆ W () t WW () t φ ( ˆ ) () () () rt = W W W t t + φ (29) ( ) = W W W (t t ) + φ (3) Whereφ i a ault and where φ i the output ariing rom the ignal ault. However, it an be only deteted i term i large ompared with etimation error and the ignal noie ε ( t). 3.. Threhold omputation To ahieve a ueul ault detetion baed on the available reidual ignal, urther eort i needed. evaluation and threhold etting are ued to ditinguih the ault rom the diturbane and unertaintie. A deiion on the poible ourrene o a ault will then be made by mean o a imple omparion between the reidual eature and the threhold, a hown in Fig. 4. Threhold Computation + Evaluation _ > No Ye Fault Fault-ree Fig. 4: evaluation In pratie, the o-alled limit monitoring and trend analyi are, due to their impliity, widely ued or the purpoe o ault detetion. For a given ignal r, the primary orm o limit monitoring i i r < T or r > T then, Alarm, ault i deteted min min max max i T r T then, No Alarm, ault-ree 8

9 where T min, T max denote the minimum and maximum value o T in the ault-ree ae. They are the threhold value. 4. Deign and Simulation Reult The omputation o the etimator i relatively traightorward. The polynomial matrix equation an be olved uing the Matlab polynomial toolbox PolyX. Given thee matrie the etimator may be implemented very neatly, a hown in Fig. 2. The eletion o the unertainty tuning untion i a dual problem to the eletion o optimal ontrol ot untion weighting [6]. The requirement or the nonlinear operator i that it hould have a table invere. A imple tarting point i thereore to aume the unertainty model i a ontant and o a mall magnitude. Thi orrepond to the ituation where the unertainty i imply white noie added at the output o the ommuniation hannel beore it enter the etimator. Unertainly i o oure oten aoiated with high requeny behavior and hene a imple linear lead term might be ued to repreent the requeny repone o a in the example whih ollow. To validate the eetivene o the NMV ilter baed ault detetion ytem, nonlinear SISO ytem i ued a an example. The NMV ilter i omputed below or the example and a imulation i ued to veriy the reult.we onider a ytem having the ollowing ignal and noie model; W =.. 99z, 6. W n =.z and let weighting W q =, W.. z The linear hannel harateriti are deined a harateriti o the ytem i given in Fig. 5. = 5 5 and Channel delay = W (. z ) Λ = =. z,o that k = 5. The tati nonlinear The d-gain and hange in the ut-o requeny o the weighting ilter inluene the auray o etimation. The tuning untion, whih i optimized or thi example, ha the ollowing repreentation:.4z =.z 9

10 5 ytem harateriti output input Fig. 5. Nonlinear Behavior o the Output Sub-Sytem The overall ytem and imulink model o NMV ilter or ault detetion i a hown in Fig. 6. Wn Noie Delay z^-l NF Nonlinear F Nonlinear etimator Random Number W Signal oure W Delay NW Meaured Signal z _hat inv(w) nonlinear W Linear NW Nonlinear W3 ACTUATOR FAULT INPUT Linear z^-l CHANNEL Nonlinear W2 SENSOR FAULT INPUT etimated Etimated Signal meaured Fault Fault Detetion Fault indiator Fig.6: Simulink Model o NMV Baed Fault Detetion Sytem Under normal operation ondition (ault-ree) meaured ignal and etimated ignal are illutrated in Fig.7. The minimum variane or the NMV etimator i.4e-2. Tuning ilter repone i hown in Fig.8. Calulated reidiual ignal and onidene level threhold are dediated in Fig.9. A hown in Fig. 9, reidual ignal i under the threhold. It mean ytem i under normal operation.

11 .8.6 Meaured Signal Etimated Signal Amplitude Time (Se) Fig.7. Meaured and Etimated Signal (no ault) 2 Frequeny Repone Value o Signal, Noie (dahed), Linear Channel (dotted) 5 5 Singular Value (db) Frequeny (rad/) Fig.8. Tuning Filter Frequeny Repone

12 2.5 Threhold Time (Se) Fig.9. Signal with Threhold (no ault) Two type o ault are applied to validate the eetive o the propoed NMV etimator in ault detetion implementation. 4.. Senor Fault For the enor ault; the ignal hown in Fig. i applied to the enor ault input o the ytem a illutrated in Fig. 6 imulink model. The ault i onidered a a drit on the meaurement enor. Ater applied enor ault, atual ignal and etimated ignal are illutrated in Fig.. Calulated reidual ignal and onidene level threhold are dediated in Fig. 2. Fault ha been deteted ueully with aurate time a hown in Fig. 2. 2

13 2.5 Applied Fault Signal 2 Amplitude Time (Se) Fig.. Applied Fault Signal Meaured Signal Etimated Signal.5 Amplitude Time (Se) Fig.. Atual and Etimated Signal (aulty) 3

14 2.5 2 Threhold Time (Se) Fig.2. Signal with Threhold (aulty) 4.2. Atuator Fault For the atuator ault; the ignal hown in Fig. 3 i applied to the atuator ault input o the ytem a illutrated in Fig. 6 imulink model. The ault i onidered a a lot ontat o the atuator input or a while. Ater the atuator ault i applied, atual ignal and etimated ignal are a illutrated in Fig. 4. Calulated reidual ignal and onidene level threhold are dediated in Fig. 5. Fault ha been deteted ueully with aurate time a hown in Fig. 5. Applied Fault Signal -.5 Amplitude Time (Se) Fig.3. Applied Fault Signal 4

15 .5 Meaured Signal Etimated Signal -.5 Amplitude Time (Se) Fig.4. Atual and Etimated Signal (aulty) Threhold Time (Se) Fig.5. Signal with Threhold (aulty) 5

16 5. CONCLUSIONS A NMV etimator baed ault detetion ytem or nonlinear ytem ha been developed. The NMV etimator i ued to generate the reidual ignal whih indiate poible ault ondition in the ytem. The NMV etimator ha ome beneit relative to ome other nonlinear etimator in three repet i.e. it require le omputational ot, eay to implement and to tune. The algorithm i illutrated uing the imulation o a nonlinear proe ontrol example. The imulation reult how that the method ha a good perormane in deteting ault at either input or output. 6. REFERENCES [] Alkaya A, Eker İ. Variane enitive adaptive threhold-baed PCA method or ault detetion with experimental appliation. ISA Tranation ISA Tranation 5 (2) [2] Iermann R. Fault-diagnoi ytem: an introdution rom ault detetion to ault tolerane. Berlin: Springer; 26. [3] Venkataubramanian VR, Rengawamy KY, Kavuri SN. A review o proe ault detetion and diagnoi, part I: quantitative model baed method. Computer & Chemial Engineering 23;27: [4] Venkataubramanian VR, Rengawamy KY, Kavuri SN. A review o proe ault detetion and diagnoi, part II: qualitative model and earh trategie. Computer & Chemial Engineering 23;2: [5] Chow, E. Y., and Willky, A. S., 984. Analytial redundany and the deign o robut ailure detetion ytem. IEEE Tranation on Automati Control, vol. 29(7): [6] Hur, S.H., Katebi, R., Taylor, A., Model-baed ault monitoring o a plati ilm extruion proe. IET Control Theory Appl., 2, Vol. 5, I. 8, pp [7] Alrowaie, F., Gopaluni, R.B., Kwok, K. E., Fault detetion and iolation in tohati nonlinear tate-pae model uing partile ilter. Control Engineering Pratie, 2, (22), 6 32 [8] Geraimo G. R., A Derivative-Free Kalman Filtering Approah to State Etimation-Baed Control o Nonlinear Sytem. IEEE Tranatıon On Indutrial Eletroni, Vol. 59, No., 22, [9] Mirzaee, A., Salahhoor, K., Fault diagnoi and aommodation o nonlinear ytem baed on multiple-model adaptive unented Kalman ilter and withed MPC and H-ininity loop-haping ontroller. Journal o Proe Control 22 (22) [] Shamher, A. N., Linear and nonlinear polynomial baed etimator, PhD thei, Indutrial Control enter, Strathlyde Univerity,29, p(63). 6

17 [] Grimble, M.J.,NMV optimal etimation or nonlinear direte-time multi-hannel ytem, 46th IEEE Con. on Deiion and Control,New Orlean, 2 4 Deember 27, pp [2] Grimble, M.J.,Time-varying polynomial ytem approah to multihannel optimal linear iltering. Int. Con. o Aouti, Speeh and Signal Proeing, Detroit, MI, 9 2 May 995, vol. 2, pp [3] Grimble, M.J., Robut indutrial ontrol ytem: optimal deign approah or polynomial ytem, John Wiley, Chiheter, 26. [4] Grimble, M.J, 2, 'Nonlinear Minimum Variane State Baed Etimation or Direte- Time Multi-Channel Sytem', IET Journal on Signal Proeing, Volume 5, Iue 4, July, Volume: 5, Iue 4, Page(): Digital Objet Identiier:.49/iet-pr [5] Grimble M J, 22, 'Nonlinear Minimum Variane Etimation or State-Dependent Direte-Time Sytem', IET Journal on Signal Proeing, Volume 6, Iue 4, June, p , DOI:.49/iet-pr.2.22, Print ISSN , Online ISSN [6] Grimble M. J., Nonlinear generalied minimum variane eedbak, eedorward and traking ontrol. Automatia, 25, 4, [7] Kuera V., Direte Linear Control, Wiley, Chiheter,979. [8] Kuera V., Stohati multivariable ontrol:a polynomial equation approah, IEEE Tran.,98,AC-25,(5),pp [9] Grimble M.J., Shamher A. N., Optimal minimum variane etimation or non-linear direte-time multihannel ytem. IET Journal on Signal Proeing, 2, Vol. 4, I. 6, pp ISSN: , INSPEC Aeion Number: