Fault Diagnosis and Fault Identification for Fault-Tolerant Control of Chemical Processes

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1 Fault Dagno and Fault Identcaton or Fault-olerant Control o Checal Procee Kap-Kyun Noh * and En Sup Yoon School o Checal Engneerng, Seoul Natonal Unverty, Seoul Korea (*Eal: noh@plab.nu.ac.r) Abtract: Fault-tolerant control (FC) o nonlnear yte preented wthn an adaptve control raewor. FC can be accoplhed by three ubta; aultdagno, ault dentcaton and adaptve nonlnear control. In oer to dagnoe a ault at a te, a et o redual generator or ault dagno degned by ean o unnown nput oberver. When dturbance et, dturbance-decoupled odel could be derved or relable dagno. Fault dentcaton ollowng ault dagno an analogue to control ta; the dagnoed ault regaed a a control nput and ound out uch that the redual ro redual generator ncorporatng dentcaton ta drven to reerence zero. And, eedbac lnearzng control led wth ault dagno and ault dentcaton to copenate or a ault to the proce. A three-tan yte taen a an eaple or deontraton o the preented FC. Keywo: ault dagno, ault dentcaton, ault-tolerant control, nonlnear yte.. INRODUCION Autoated checal proce ha yeld to hgh qualty and hgh ecency o noral operaton, but ha becoe ore coplcated and ore vulnerable to ault becaue procee have been ntegrated nto wder operaton plator and operaton algorth ay be another ault ource. Furtherore, a ple ault could be apled by the control yte and developed nto aluncton o the loop, even nto a alure at the plant level. h requre advanced ault dagno and upervon to prove relablty and aety. A cot-eectve way to acheve the goal by ean o a ault-tolerant control (FC) (Blan et al., 200). he FC could be acheved by ergng the ault noraton obtaned ro ault dagno and ault dentcaton nto the control yte or ault accoodaton. Fault dagno chee ha to ecently detect and denty a ault even when the proce under cloed-loop control and vare over wde range, and ollowng ault dagno, ault dentcaton etate te-varyng behavor o the dagnoed ault whch then relected nto the control law to accoodate the ault. Nonlnear oberver-baed ault dagno the reearch that ha been ade around a lnear yte ha been lately etended to nonlnear yte (Fran and Dng, 997) preented. he odel ued or oberver degn could be decoupled ro dturbance and/or a dedcated ault by ean o tate tranoraton, and o reultng oberver unaected by dturbance and poe tructured entvte to the ault. he tate tranoraton baed on the concept o ault (dturbance) detectablty nde and o t contan output and ther dervatve. A ban o redual generator provdng generalzed redual et and decon uncton uch a ed threhold are needed to dagnoe a ault. he degn ethod and condton o uch nonlnear oberver wll be preented. When a ault detected and localzed, t agntude and te -evolvng behavor hould be dented to tae a countereaure eepng proce perorance. he ae odel and the concept o ault relatve oer a thoe o ault dagno are utlzed to obtan degn odel whch allow to recontruct the ault and on whch the redual generator degned. Fault

2 dentcaton orulated nto control ta proble uch that the redual ro the redual generator drven by dented ault orced to zero. he ethod baed on the condton o nput obervablty (Hou and Patton, 998; Kabore and Wang, 999); the redual repond to only a pecc ault and recontruct the actual ault. Fault-tolerant control acheved by cobnng nput-output eedbac lnearzng control law baed on the ault-paraeterzed odel wth ault noraton obtaned ro ault dagno ault dentcaton, whch reduced to the adaptve lnearzng control (Satry and Idor, 989). 2. ON-LINE FAUL DIAGNOSIS Conder nonlnear yte ane n control, dturbance and ault ode; n d n g0( ) + g( u ) () t + D( d ) () t + e( ) ( t) (a) y h ( ) (b) n p Γ R, u Ω R, y R are the tate u vector, the control nput vector, and the output vector. d n Ω d d R the dturbance vector ncludng odel error, and n Ω R the ault ode vector or coponent ault and actuator ault, and both vector cont o unnown te -varyng uncton. g( ), 0,,.., ; D( ),,.., n and e(),,.., n d are ooth vector eld, and h (),,.., q are ooth calar eld, repectvely, on Γ and they are nown. Here, Γ a phycally eable and bounded et, and Ω wth ubcrpt are bounded et or correpondng nput. FDI Strategy: When the proce ubect to dturbance, odel-baed FDI trategy ay gve leadng analytcal redundance. hu, to prove the FDI perorance, a ean o creatng analytcal redundancy not aected by dturbance hould be deved a n the ollowng tep; S: Obtan a reduced odel decoupled ro dturbance, but tll aected by ault. S2: For the detecton o ault reaned at tep S, produce a redual generator baed on dturbance-decoupled odel obtaned at tep S. S3: For ault olaton aong the ault at tep S, partton ault nto olable ault ubet and generate a ban o redual generator n whch each one dedcated to each ault ubet. It baed on a ault-added dturbance-decoupled odel and provde generalzed redual et gvng derent entvte to derent ault ubet. h eature enable to unquely olate a ault ubet by checng the value o all redual. Fault Detectablty: A ault ad to be detectable a ault aect at leat one o obervable output. Fro th pont o vew, the detectablty o a ault can be characterzed by ung the concept o relatve oer nown a a ault detectablty nde n ault dagno eld (Lu and S, 997), whch dened a the allet nteger, r e g0 r, uch that L L h( ) 0 [,.., p], Γ (2) e() a ault vector eld o a ault, and h ( ) a calar uncton o the output, y. I ault relatve oer, r, le than or equal to n, the ault detectable. Otherwe, the ault not detectable. Dturbance relatve oer,, or the dturbance, r d d, can be dened n the ae way. Dturbance-Decoupled Model: Redual generator wll be obtaned through the degn o oberver baed on dturbance-decoupled or a ault-added dturbance-decoupled odel. he condton that guarantee a tate tranoraton nducng dturbance-decoupled nonlnear odel are baed on the concept o well-dened dturbance relatve oer and are analoge to thoe o eedbac lnearzaton n nonlnear control theory (Idor, 989). Conder the yte wth only dturbance; n d g ( ) D ( d ) () t 0 + (3a) y h () (3b) For above yte, condton below are et, C. he relatve oer, r d, o the output, y,,.., l (l < p), wth repect to dturbance vector, d, well dened.

3 C2. he charactertc atr, C (), ored at the th te dervatve o each output, y, beore the dturbance vector, d, ha ull row ran. r d LD L g h 0 LD L n g h 0 d C ( ) ( ) D r nd d nd L L h L L h D g0 nd Dn g d 0 nd C3. he dtrbuton, () pan{ D (),.., D ()}, dented and panned by dturbance vector eld ha contant ran, q( q) and t nvolutve,.e., Le bracet o any par o vector eld belongng to () belong to () agan. then, there et a tate tranoraton y h ξ zφ () () () [,.., p] (5) ( r ) r d η y Lg h 0 η η l ξ ξ,.., ξ, ( ),..,,.., ξ ξ ξ r y y d, nr and d η R wth r cont o calar d nd eld, η, uch that L η ( ) 0 and ae the tate D tranoraton locally nvertble (Idor, 989); ( ξ η ) Φ z Φ l ( ), [,.., ] (6) he yte n the tranored tate r d ξ r () () d g0 + g g0 g g0 + D g0 Dn d g0 D n d (4) Lh L L h L L h u L L h L L h d LL e ( ) g h 0 Le L n g h (7a) + 0 Φ ( ξη, ) ξ ξ Lh g Lg h A + Bξ r+ () u d ξ 2 2 n n ξ LL g g h 0 LL g g h 0 Lh e Le h n (7b) + ( ) Φ ( ξη, ) 2 2 n n LL e g h 0 Le L n g h 0 η ηξη (,, u, ) (7c) A the ( r ) ( r ) atr and B the ( r ) vector and A, ) n a canoncal or. ( B he tranored yte dvded nto two ubyte accong to eplct dependence on the dturbance. Dturbance-decoupled odel cont o lower ubyte (7b)(7c) and t wll be utlzed or the degn o nonlnear oberver that are robut to dturbance but entve to ault. Rear: he odel drven by the ault and a well ξ a new nput, not drectly avalable. It etaton to ue a derentator lter. But, nce each lter drven by a eaured output, the eect o the ault are relected nto etated output and t ucceve dervatve, and the dturbance decoupled odel actually ueul or FDI lted to the (7c). But, ther etate ro lter wll be ued or provon o unavalable tate wth (7c). h ean that the ault whoe relatve oer are ore than two cannot be eparable through the reultng decoupled odel unle all tate are avalable. Fault Detecton: he degn o a nonlnear oberver or FD (Fault Detecton) perored on the odel (7c) and varou degn ethod o an oberver n the lterature can be condered. I tate etate ro derentator lter are ucently accurate, the ba odel becoe; (,, u, ) η ηηξ (8a) Ye Cη (8b) Under oe aupton below, A Etra output, Y, not nvolved n dturbance e decouplng are avalable and lnear n the tate. A2 he ba odel obervable ro etra output. A3 he ba odel can be put nto a te -varyng lnear yte wth a Lpchtz nonlnear perturbaton; (, ) (,, ) (,, ) η Q ξ u η + N ηξ u + E ηξ (9) N ηξ,, u N ηξ,, u ηo ηη (0) ( ) ( ) η a contant. o A4 Contant atr, K, can be choen uch that ( ( ) ) ( ( ) ) λ Q ξ, u KC + Qξ, u KC <α ξ, u λ[ t; ] () denote the egenvalue o te-varyng atr at te t and α > 0 a contant. A canddate oberver can be taen a

4 η Q ξ u η + Nηξ u + KC ηη (2) (, ) (,, ) ( ) and then the error dynac wth a redual e Q u KC e+ N u N u + E ( ( ξ, ) ) ( ηξ,, ) ( ηξ,, ) ( ηξ,, ) ( ecce) /2 (3a) r (3b) e ηη. Aup ton A3 can be ealy et nce the ba odel reduced and a nonlnear perturbaton ter allowed, and when the odel dened over bounded doan. Aupton A4 ae the error dynac table the degn atr, K, can be choen uch that or α> 2η t ated (Slotne and L, 0 984). In the abence o any ault ( 0 ), the error dynac are ade table around the equlbru, e 0 and o the redual decayng to zero, ndcatng no ault. But, n the preence o any ault ( 0), the error no longer tay at zero due to nonlnear nonvanhng eect by a ault and o reultng nonzero redual ndcate the occurrence o a ault. Fault I olaton: Due to the allowance or dturbance decouplng, the generalzed oberver chee (GOS) (Chen and Patton, 2000) provdng a generalzed redual et or ault olaton adopted. o pleent the GOS, the odel decoupled ro dturbance and a well a ault ubet needed, and t can be obtaned n the ae way a beore or augented dturbance. Robut ault olaton between two ault n ene o the GOS can be checed by; η ran ( e( ), e( ) ) ran( e( ), e( ) ) 2 (4) e () and e () are vector eld o condered ault. h condton ae ure that two ault are not only relected nto dturbance-decoupled odel but alo not decoupled at a te. he obervablty o a ault-added dturbance-decoupled odel aued. he degn o redual generator or ault olaton baed on a degnated ault and dturbance decoupled odel and ther degn proceed a beore. When one ault occur at a te, the rght ault olaton can be done by checng value o all redual. 3. FAUL IDENIFICAION When the ocu ade on the degn o ault tolerant control, n addton to ault detecton and olaton, ault dentcaton dentyng the ze o the ault and t te varyng behavor ha to be olved. Fault to be dented are lted to the ault olated by ault dagno. Fault dentcaton proble can be reorulated nto control ta proble. In th approach, ault gnal regaed a a control nput to the yte and a eedbac control ound uch that the redual trac a zero reerence. Snce the redual uually gven a the derence between eaured output and etated output, the ault orce the etated output ro redual generator to trac eaured output. When there no ault, the eedbac control wll decay to zero, ndcatng no ault, whle n the preence o a ault, the eedbac control wll provde a control nput whch actually an etate o the ault. Conder the yte wth ngle ault vector; n g0 ( ) + g( u ) () t + e( ) () t (5a) y h ( ) (5b) one eleent o ault vector,, and decrpton o varable, yu,,, and uncton, g, e, h, are the ae a thoe o the yte (). A the ault detectablty nde,, or the ault vec- tor,, uch that e g0 LL h ( ) 0 {,.., n }, {,.., p}, Γ (6) well dened. A2. the atr, n Le L 0 ( ) g h E, ( ) n E 2 2, ( ) L 2 e L 0 2( ) g h E ( ) Ep, ( ) p n p Le L ( ) g h 0 p a ull colun ran or all Γ. (7) hen, the ollowng tate coonate can be taen

5 ξ ξ h ( ) ξ L h ( ) 2 g 0,,.., p ξ ( ) L h g 0 (8) and the yte tranored nto ollowng or, whch the ae a (7); A + BG 0, (, ) + G, (, ) (, ) u + E ξ ξ ξς ξς ξς ς G ( ξς, ) G ( ξς, ) u + y C ξ ξ (9a) (9b) ξ ( ξ,.., ξ ), p ξ ( ξ,.., ξ ) Φ () R wth p, 2 n ς Φ ( ) R uch that η ( ξς, ) nvertble. G, 0,.., Lg L ( ), 0,,.., g h 0 p n, and ( A E 0,.., LL e ( ) g h 0, C ) n a oberv- able canoncal or and ( A, B ) n a controllable canoncal or. Baed on the tructure o the above yte, a canddate redual generator ncorporatng ault dentcaton taen a (Kabore and Wang, 999): ξ Aξ + BG ( ξς, ) + G (,) ξς u + E( ξς, ) KC ( ξ y) 0,, (20a) ς 2 2 G ( ξς, ) G ( ξ, ς) u (20b) 0 + r Cξ y (20c) G0, ( ξ, ς) + G, ( ξς, ) u G 0, ( ξ, ς) G 2, ( ξς, ) u + 2 E ( ξς, ) v G0, ( ξ, ς) + G, (, ) p ξ ς u p v a proper new nput and taen a ( ) v y ( 2 ) v2 y2 v v ( p) p yp and oberver gan, (2) (22), K, choen uch that each obervable or ay be table and ae the error eponentally converge to zero. 4. FAUL-OLERAN CONROL o accoodate dented ault, the lnearzng control law can be lned wth ault dentcaton, whch reduced to adaptve lnearzng control (Satry and Idor, 989; eel et. al., 99; Hu, 999). When the reultng control law appled to the aulty proce, qua-lnear yte reult n. e Ae+ W ( e, η, u ) Θ (23) c a A c a ( r+ ) ( r + ) Hurwtz atr, u a approately lnearzng control law and e error coonate. Θ the ault error and the atr, W ( ) nonlnear uncton beore the ault error. A ault error decayng to zero, the qua-lnear yte becoe ayptotcally lnearzed. he tablty o the perturbed yte enured the perturbng ter bounded over the doan and the pole o the Hurwtz lnear yte are placed ucently deep nto the let hal o -plane (Za, 990). 5. APPLICAION Fgure how a cheatc o a three-tan yte. Ung the a balance and a low by orrcell law, the yte can be decrbed a; ( u a gn( ) 2 g )/ A ( u a gn( ) 2g a 2 g )/ A ( a gn( ) 2g a gn( ) 2 g )/ A the tate,, the level o each tan and u, 0 u are a nlow. 20 A the cro-ecton o tan and 3,, 32 are the cro ecton o ntercon- 20 nected and outlet ppe, repectvely. a calng contant and g the gravty contant. denote ault caued by varou reaon uch a lea, cloggng and pup alure. Level are avalable. Fault dtrbuton atr, E( ), to the ault,, gven a; 2g u gn( )2g A A A 2g gn( )2 2 u g g 3 E () A A A A 2g gn( 3 3)2g3 gn( 3 2)2g A A A

6 Unnown dturbance not condered. And, all odeled ault have the ault relatve oer o one. Output carry wth eaureent noe. Redual generator or ault detecton wll be degned baed on the whole odel (Fgure2) and olable ault ubet or ault olaton are a ollow; {, }, S {,, }, S { }, S { }, S { } S Generalzed redual et, { r, r, r, r, r } , (6.7) 5 7, generated ro redual generator va oberver baed on the odel n new tate, η uch that Leη 0 e all ault vector eld belongng to a ault ubet, S (Fgure 3). A or ault dentcaton, only one ault at a te etated and t ault relatve oer one. he tate o ς avalable ro etra output and the rt dervatve o the output, y, obtaned by a der- entator lter. he reult are hown n Fgure4. Perorance o lnearzng control lned wth ault dentcaton copared wth ple lnearzng control a hown n Fgure5. ACKNOWLEDGEMEN We acnowledge the nancal ad or th reearch provded by the Bran Korea 2 Progra upported by the Mntry o Educaton. In addton, we would le to than the Autoaton & Syte Reearch Inttute and Inttute o Checal Engneerng o Seoul Natonal Unverty. REFERENCES Blane, M., Starowe, M. and Wu, N.E. (200) Concept and Method n Fault-olerant Control. Proc. Aercan Control Conerence, Arlngton, June, Fran, P. M. and Dng, X. (997) Survey o Robut Redual Generaton and Evaluaton Method n Oberver-Baed Fault Detecton Syte, J. Proc. Contr. 7(6), Hou, M. and Patton, R.J. (998) Input Obervablty and Input Recontructon. Autoatca. 34(6), Kabore, P. and Wang, H. (999) On the Degn o Fault Dagno Flter and Fault olerant Control. Aercan Control Conerence, San Dego, Calorna, June. Lu, B. and S, J. (997) Fault Iolaton Flter Degn or Lnear e-invarant Syte, IEEE ran. Auto. Control, 42, Idor, A. (989) Nonlnear Control Syte, 2nd, Sprnger-Verlag. Satry, S. and Idor, A. (989) "Adaptve Control o Lnearzable Syte," IEEE ran. Auto. Control, 34, Chen, J. and Patton, R. J. (2000) Robut Model-baed Fault Dagno or Dynac Syte. Kluwer Acadec Publher, Boton/Dorecht/London. Slotne, J-J. E. and L, W. (99) Appled Nonlnear Control, Prentce Hall, New Jerey. Hu, Q. and Rangaah, G.P. (999), "Adaptve Internal Model Control o Nonlnear Procee," Che. Engng. Sc., Za, S.H. (990) On the Stablzaton and Obervaton o Nonlnear/Uncertan Dynac Syte. IEEE ran. Autoatc Control. 35, 604. r0 r r2 r3 r4 r Fgure. Scheatc o a three-tan yte Fault Fault 2 Fault 3 Fault 4 Fault 5 Fault 6 Fault 7 Fault 8 Fgure2. Redual or ault detecton Fgure3. Generalzed redual et ( r,,..,5 ) or ault olaton Fault Mode Level Level 2 Level (a) (b) (c) (d) (e) Fault Fault 2 Fault 3 Fault 4 Fault 5 Fault 6 Fault 7 Fault 8 Fgure4. Fault dentcaton o ault ode by FC by FC by FC Fgure5. Perorance by FC and ple nonlnear control