FAULT DETECTION AND IDENTIFICATION BASED ON FULLY-DECOUPLED PARITY EQUATION

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1 Control 4, Unversty of Bath, UK, September 4 FAUL DEECION AND IDENIFICAION BASED ON FULLY-DECOUPLED PARIY EQUAION C. W. Chan, Hua Song, and Hong-Yue Zhang he Unversty of Hong Kong, Hong Kong, Chna, Emal: mehan@hku.hk, Fax: (8)89796 Bejng Unversty of Aeronauts and Astronauts, Bejng, Chna Keywords: Party equaton, fault deteton and dentfaton, reursve least squares method Abstrat A multple fault deteton and dentfaton method based on the fully-deoupled party equaton for dynam systems wth known lnear and unknown non-lnear terms s presented. It s shown that the resduals generated from the fully-deoupled party equatons derved here are senstve only to spef atuator or sensor faults, but are deoupled from other faults and the unknown nonlnear term. he ondtons for the exstene of these equatons are also gven. From the resduals generated from the fully-deoupled party equaton, faults are estmated usng the reursve least squares method. he applaton of the proposed method to detet, solate and dentfy faults n a smulated DC motor s also presented. Introduton Fault deteton, solaton and dentfaton are now beng ntegrated nto pratal ontrol systems to mprove the safety and relablty of these systems. Party equaton s a popular approah to detet faults. Frst, resduals are generated from the party equatons. he resduals are large, f a fault ours; but small otherwse. Further, f the resduals are senstve only to a spef fault, then t an be solated and dentfed. Party equatons for systems wth modelng errors arsng from hanges n the operatng pont are derved n [], and optmal robust party equatons maxmally senstve to faults and mnmally senstve to modellng errors are gven n [4], whh are further refned n [9]. he optmal party vetor method for solatng faults s dsussed n [3], but the dentfaton of faults was not dsussed. A popular fault dagnoss tehnque for systems wth unknown nonlnearty s the unknown nput observer (UIO) method nvolvng dsountng frst the unknown nonlnearty usng ts estmate [6]. Observers are proposed to deouple the effet of the unknown nputs []. However, fault dentfaton has not been onsdered n these works. In ths paper, a new approah not only to detet and solate, but also to dentfy faults for systems wth know lnear and unknown nonlnear terms s presented. A fullydeoupled party equaton s derved for generatng resduals senstve only to spef sensor or atuator faults, but deoupled from the system state, the unknown nonlnearty and other faults. Faults an now be dentfed from the resduals by the reursve least squares method. he performane of the proposed method s llustrated usng a smulated DC motor wth a shunt feld rut. From the smulaton results, t s shown that the proposed method s able to dentfy, solate and dentfy the faults. Fully-deoupled equaton Consder the followng dsrete nonlnear system, k + ) = A + Bu( + Fw( x, u, () y( = C + Gψ ( x, u, where, R n s the state, u( R p s the nput, y( R q s the sensor output, w(x, u, R r, ψ(x, u, R r are the unknown nonlnear dynam of the system, A, B, C, F and G are known matres wth approprate dmensons. Defne w( = w(x, u, and ψ( = ψ (x, u,. For s >, the measurement equaton of () s gven by [3], Y ( = H + H U ( + H W ( + Hψ Ψ( () where Y( s the sensor output under normal ondtons, s s the order of the party spae, Y( = [y (k s) K y (], U( = [u (k s), K u (], W( = w (k s ) K w (], Ψ( = [ψ (k s) K ψ (], and H, H, H w and H ψ are: H = CAB s B CA L CF H w = CAF CF s F CA F L C G CA ; G H = H =. ψ s G w L L, L CAB L L L CAF CF Defnton :he party spae V s defned as [6, 8], V = { v v H = } (3) where ν R p s the party vetor, H and s are gven n (). Defnton :he party equaton at tme k s, = v [ Z( HU ( ] (4) where s the resdual, whh may ontan the fault. U ( = [u (k s) K u (],and Z( = [Z(k s) K Z(]. U ( s the normal nput to the atuator, and Z(,

2 Control 4, Unversty of Bath, UK, September 4 s the sensor output that may ontan faults. Under normal operatng ondton, U ( = U( and Z( = Y(. A fully deoupled party equaton s derved here for solatng and dentfyng faults, suh that the resduals generated by (4) are senstve only to spef faults. H = CAB B CA s B L L CAB where B s B wth the th olumn orrespondng to the th atuator beng removed. Sne the party vetor senstve to the th atuator must satsfy: v [ H H Hψ H ] = () Lemma :he resduals generated by the fullydeoupled party equaton (4) an be used to detet, solate and dentfy fault, f there exsts a fully-deoupled party vetor v, suh that s: w () deoupled from the system state, () deoupled from the unknown nonlnear part W( and Ψ(, whh may ontan the unknown nputs and unertantes, (3) senstve only to spef atuator faults or sensor faults, but deoupled from eah other. hene the party equaton onstruted by the party vetor that satsfes () s referred as the fully-deoupled party equaton. he ondton for the exstene of nonzero solutons of () s gven n heorem. heorem Let [H H w H ψ H ] onsst of n x ndependent olumns, the neessary and suffent ondton for () to have nonzero solutons s: n s > x () q Proof: Condton () an be obtaned from (). For a system operatng normally, obtaned from () s, = v [ Y ( H U ( = v [ H + H ww ( + Hψ Ψ( ] From (3), s deoupled from the system state, when the system s normal. If the unknown nonlnear terms W( and Ψ( are dentally zero, s zero only f the system s operatng normally. If v [H w W( + H ψ Ψ(] s dentally zero, then ondton () follows, and hene s deoupled from W( and Ψ(. If ondtons () and () are satsfed, s non-zero only f the system s faulty. herefore, faults an be deteted from, as follows, rh faulty (6) < rh normal where r h s the threshold obtaned from a statstal table. he resdual generated from (4) an be used to detet, but not solate fault, sne s a funton of atuator or sensor faults. However, f ondtons (3) s satsfed, s senstve only to spef faults, and nsenstve to other faults. Consequently, faults an be solated and dentfed from, usng a fault model to be desrbed later. Condtons () and (3) are the key results n dervng the fully deoupled party equaton. 3 Isolaton of faults 3. Isolaton of Atuator Faults o solate atuator faults assumng that the sensors are operatng normally, the party vetor must satsfy (3) to ensure the resduals are deoupled from the system state. he resduals are deoupled from the unknown nonlnear terms W( and Ψ(, f [3], v =, v = (7) H w he party vetor must satsfy the followng equaton for the resduals generated from the fully-deoupled party equaton to be senstve only to a spef atuator fault: v H = =,, L p (8) H ψ, where p s the number of atuators n the system, v s the () and the suffent ondton s: n + r s >, q > ( r + p) () q + ( r + p) where n s the dmenson of the system state, q and p are the number of sensors and atuators, r and r are the dmenson of the unknown nonlnear terms n (). Proof: he left null spae and hene the nonzero soluton of () exsts, f and only f the number of ndependent rows of [H H w H ψ H ] s greater than n x. From (), and the dmensons of H, H w, H ψ and H, whh are (s + )q n, (s + )q (s + ) r, (s + )q r and (s + )q (s + ) (p - ), t follows that the dmenson of [H H w H ψ H ] s (s + )q [n + r + (s + )(r + p )]. herefore, the neessary and suffent ondton for the exstene of nonzero solutons s: (s + )q > n x, or s > n x /q. Further, f the number of ndependent rows of [H H w H ψ H ] s greater than the number of ndependent olumns, then nonzero solutons of () exst, gvng the suffent ondton: (s + )qn + r + (s + ) (r + p ). From (), an approprate wndow wth sze s an be seleted to obtan the fully-deoupled party vetor to onstrut H, H w, H ψ and H. If q > (r + p). If q > n + r + (r + p), then s =, mplyng the hardware redundany of the system s enough to solate faults and the spatal redundany s unneessary. 3. Isolaton of sensor faults In solatng sensor faults, t s assumed the atuators are operatng normally. he fully-deoupled party equaton s onstruted suh that resduals generated from t are senstve only to spef sensor faults, but nsenstve to other sensor faults. he fully-deoupled party equaton (4) for eah sensor s obtaned from H, H, H w and H ψ and wth C and G replaed by C and G obtaned from C and G by settng the th row orrespondng to the th sensor to. If the sensors are operatng normally, then party vetor senstve to the th atuator, H s gven by, r ( v [ H + HwW ( + H Ψ( ] (9) = ψ (3)

3 Control 4, Unversty of Bath, UK, September 4 For r ( to be deoupled from the system state, v must From (), v H =, v HwW ( =, and satsfy: v Hψ W ( =, then v H = (4) he party vetor v r ( = v [ HU ( HU ( should also satsfy the followng ] () ondton f t s deoupled from W( and Ψ(: Also, v H =, where H s obtaned by removng the v Hw =, v H ψ = () th olumn of B n H. Sne r ( obtaned from () s It follows that f the party vetor s nsenstve to the th senstve only to the th atuator, and nsenstve to other sensor, the followng ondton holds, atuators, then () beomes, U ( = η ( U( + λ ( E (3) v [ H H w Hψ ] = (6) where E = [ L ] s a (s + )p dmenson vetor. For v to be nsenstve to the th sensor, the elements n Rewrtng () gves, H, H, H w and H ψ orrespondng to the th sensor must be. he party equaton onstruted by v satsfyng (6) r ( = v [ H ( η ( U( + λ ( E) HU ( ] s referred to as the fully-deoupled party equaton. he = v [( η ( ) HU ( + Hλ ( E] (4) ondton for the exstene of v s gven n heorem. = φ ( θ ( heorem Let n be the number of ndependent rows n After takng nto aount modelng and measurement [ H Hw Hψ ]. he neessary and suffent ondton nose n(, (4) beomes, for the exstene of nonzero solutons of (6) s: r ( k ) = φ ( k ) θ ( k ) + n ( k ) () n s > x (7) where φ ( = [ v HU (, v HE], θ( = [(η ( ), q λ (]. Assumng n( s a zero mean whte nose wth a and the suffent ondton s: ovarane matrx of R(, then from (), θˆ ( k ) an be n + r s >, q > r + (8) obtaned by the reursve least squares method, H H w q r H ψ where,, and are obtaned from H, H w and H ψ by removng the th row (a vetor) of C, D and G. he proof of ths theorem s smlar to that of theorem. From (7) and (8), H, H, H w and H ψi are onstruted, one s s seleted. 4 Identfaton of faults 4. Identfaton of Atuator Faults Atuator faults arsng from a hange n the salng fator and/or the onstant bas an be desrbed by the followng eqaton [6]: z = η y + λ (9) where z s the fault, y s the output wthout fault, η the salng fator, and λ the bas. If an atuator s operatng normally, t s lear from (9) that η = and λ =. If there are a onstant output fault, then η =, and λ s a non-zero onstant. For salng fator faults, η and λ =, and for onstant bas faults, η = and λ. If the th atuator s faulty, then, (9) beomes, u ( = η ( u ( + λ ( () where, u ( s the nput of the th atuator, u ( s the normal nput, η ( s the salng fator, and λ ( s the bas. After the resduals are generated from the fullydeoupled party equatons, the parameters of the fault model (9) an be estmated from the resduals usng the reursve least squares method. From () and (4), the resduals senstve to the th atuator are gven by: r ( = v [( H + HU ( + HwW ( + Hψ Ψ( ) HU ( ] () K( = P( φ ( [ I + φ( P( φ ( ] (6) ˆ θ ( = ˆ( θ + K( [ φ( ˆ( θ ] (7) P( = P( K( φ ( P( (8) where K( and P( are the gan matrx and the error ovarane matrx respetvely, and I s the unt matrx. From (), the fully-deoupled party vetor must satsfy the followng ondton to ensure the fault n the th atuator appears n the resduals: v H (9) hs result s summarzed n heorem 3. heorem 3 he fully-deoupled party equaton (4) an detet, solate and dentfy atuator faults, f the fullydeoupled party vetor satsfes () and (9). 4. Sensor Faults Identfaton he fault model of the th sensor an be desrbed by the followng model [7]: z ( k ) = y ( k ) f ( k ) (3) + where, z ( s the atual output of the th sensor and y ( s the normal output. In matrx form, z ( = y( + f ( (3) where z( = [z ( z ( z q (], y( = [y ( y ( y q (], f( = [f ( f ( f q (]. hen Z( = Y ( + I f ( (3) where Z( = [z (k s) z (k s + ) z (] s the atual sensor output, I = [I I L I ] s a matrx wth dmenson (s + )q x q and I s a q x q dentty matrx. he resdual nsenstve to the th sensor s,

4 Control 4, Unversty of Bath, UK, September 4 r ( = v [ Z ( H = v [ Y ( + I U ( ] f ( H U ( ] (33) where y and f are replaed by n Y ( and f (. When the atuators are operatng normally, the resduals nsenstve to the th sensor an be obtaned from (3) and (33), r ( = v [ H + H U( + H ww ( + H Ψ( + I f ( H U( ] ψ w = (34) From (6), v H =, v H W (, and v Hψ W ( =, hene r ( = v I f ( (3) = vi f ( = φ( θ( (36) where = [r ( r ( r q (], ν = [ν ν ν q ]. For systems wth measurement nose, n(, (36) beomes r ( = φ ( θ ( + n( (37) where θ( = f(, φ( = νi, and n( are defned n (). From (3), sensor faults an be estmated by the reursve least squares algorthms as dsussed prevously. Example he proposed method s appled to detet and dentfy faults of the DC motor wth a shunt feld rut desrbed by the nonlnear ordnary dfferental equatons [9] & a R f & f = f + V, f () = f L f L f Ra = a f ωr + V, a () = a D & ωr = f a ωr, ωr () = ωr J J (38) where, f s the feld urrent, a the armature urrent and ω r the speed of revoluton, V the nput voltage, R f and L f the feld resstane and ndutane, R a and L a the armature resstane and ndutane, the mutual ndutane between L f and L a, D and J the vsous resstane and the moment of nerta of the load. Let x = [ f a ω r ] be the state, and y = [ f a f + a ω r ] be the measurement output, and u = V, (38) an be rewrtten n dsrete tme as [],.88 k + ) = (39) u( + w( y( = (4) where w s the unknown nonlnear term n () n dsrete form. From (39) and (4), there are one atuator and four sensors n ths system. Assumng the ovarane matrx R( s a x dagonal matrx wth. along ts dagonal. he party vetors an be obtaned from () and (6), and resduals senstve to spef faults an be omputed by (4). he fault models an be estmated from () and (37) by the reursve least squares method. () Fault dentfaton of atuator he resduals senstve before any fault ours are shown n Fg.. When a fault gven by the fault model (9) wth η = and λ = ours at seonds, the resduals are omputed and are shown n Fg.. he dentfaton of the fault under normal operatng ondton s shown n Fg. 3, and under faulty ondton n Fg. 4. Clearly, the resduals generated from the fully-deoupled party equaton s senstve only to the spef atuator fault, and s ndependent of the system state and the unknown nonlnear term, ndatng that the atuator fault an be deteted, solated and dentfed. () Fault dentfaton of sensors A onstant off-set of unts ours n sensors and, and. unts n sensors 3 and 4 after seonds. he estmated parameters of the fault model are shown n Fg., whh are lose to the atual fault. Next, onsder the ase when sensors and 4 are operatng normally, whlst sensors and 3 have developed a fault wth a onstant off-set of 3. and unts respetvely after seonds. he estmated faults are shown n Fg. 6, showng that the estmated faults are lose to the atual ones, llustratng that the proposed method s able to detet, solate and dentfy suessfully sensor faults. 6 Conlusons hs paper presents an approah for fault dagnoss n the system wth unknown nonlnearty based on the fullydeoupled party equaton. he resduals generated from the fully-deoupled party equaton s senstve only to spef faults and s deoupled from the system state, the unknown nonlnearty and other faults. he parameters of the fault models an be estmated from the resduals by the reursve least squares method. he performane of the proposed method s llustrated by usng t to detet and dentfy suessfully multple atuator or sensor faults n a smulated DC motor. Aknowledgements hs projet s supported by the HKSAR RGC Grant (HKU 7/E) and the Natonal Natural Sene Foundaton of Chna (634) Referenes [] E. Y. Chow and A. S. Wllsky. Analytal redundany and the desgn of robust falure deteton systems, IEEE rans. Automat Control, AC-9(7), pp , (984). []. Darouah,. Zasadznsk and S. J. Xu. Fullorder observer for lnear systems wth unknown

5 Control 4, Unversty of Bath, UK, September 4 nputs, IEEE rans. Automat Control, 39, pp , (994). [3] H. Jn and H. Y. Zhang. FDI of slow-grown faults of dynam systems [J]., IEEE ransatons on Aerospae and Eletron Systems, 3(4), pp. - 8, (999). [4] X. C. Lou, A. S. Wllsky and G. C. Verghese. Optmally robust redundany relatons for falure deteton n unertan systems, Automata, (3), pp , (986). [] H. Song and, H. Y. Zhang. An approah to sensor fault dagnoss based on fully-deoupled party equaton and parameter estmate, Proeedngs of the 4 th World Congress on Intellgent Control and Automaton, pp. 7-74, (). [6] Jay F. u and J. L., Sten. odel error ompensaton for observer desgn, Internatonal Journal of Control, 69, pp , (998). [7] Y. Wang. On-lne fault dagnoss of nonlnear dynamal systems usng reurrent neural networks, Ph.D. hess, he Unversty of Hong Kong, (). [8] X. Wen, H. Y. Zhang and I. Zhou. Fault dagnoss and fault tolerant ontrol for ontrol system, ahne Industry Press, (998). [9] H. Y. Zhang and R. J. Patton. Optmal desgn of robust analytal redundany for unertan systems, IEEE Regon Conferene on Computer, Communaton, Control and Power Engneerng, (993).. sale tma/s bas/v 3 4 tme/s Fg. 4 Estmated parameters of atuator fault model (faulty) f f f f tme/s Fg. Estmaton parameters of Sensor faults ourrng at f =. s, [ ] sale tme/s Fg. Resduals for atuator (normal) tme/s Fg. Resduals for atuator (Faulty) 3 4 tme/s bas/v 3 4 tme/s Fg. 3 Estmated parameters of atuator fault model (normal) f f f3 f tme/s Fg. 6 Estmaton of sensor faults ourrng ats, f = [ 3. ]