Unknown Input High Gain Observer for Fault Detection and Isolation of Uncertain Systems

Transcription

1 Engneerng Leer, 7:, EL_7 08 Unknown Inp Hgh Gan Observer or Fal Deecon and Isolaon o Unceran Sysems Sharddn Mondal*, G Chakrabory and K Bhaacharyya Absrac An nknown np hgh gan observer (UIHGO) based componen al deecon and solaon (FDI) echnqe s presened Frs, a redced order UIHGO s derved or a lnear sysem whose parameers are nceran o some exen he observer gan s deermned by solvng he well-known algebrac Rcca eqaon (ARE) hen, sng a bank o sch observers, a FDI algorhm s devsed o deec and solae he componen al (e, paramerc al) o an nceran sysem he FDI algorhm consss o wo seps In he rs sep, he deecon o al and he solaon o aly regon are accomplshed and n he nex sep, he aly parameer s solaed rom he aly regon Eecveness o he proposed observer as well as he FDI echnqe s shown wh he help o a nmercal example Index erms Unknown np hgh gan observer; componen al; al deecon and solaon; nceran sysem; parameer esmaon I INRODUCION Wh he rsng demands o hgh relably and saey o advanced processes lke avoncs, nclear power saons, aomobles ec have led o ncreasng reqremens o developng new mehods o spervson and monorng as a par o overall process conrol scheme Deren al deecon and solaon (FDI) schemes have been developed or avodng alre o he plans Model based al deecon echnqes (lke Kalman ler or observer based) have receved ncreasng aenon ollowng he poneerng work o Beard [] he FDI concep sng observers or Kalman lers s devsed based on he assmpon ha he mahemacal model o a sysem s perecly known In realy, however hs assmpon does no hold becase he parameers o a process are n general nceran or me varyng Agan he characerscs o dsrbances or nose are no compleely known; hence hey canno be perecly modeled here s always a msmach beween he acal process model and s mahemacal model (even here s no al n he process), whch somemes prodces alse alarms corrpng he Manscrp receved Aprl 5, 009 S Mondal s wh he Robocs and Bo-mecharoncs Laboraory, Deparmen o Mechancal Engneerng, Pohang Unversy o Scence and echnology (Posech), Pohang, , Soh Korea, el No: /84, Fax: , e-mal: sharm@posechackr G Chakrabory and K Bhaacharyya are wh he Deparmen o Mechancal Engneerng, Indan Inse o echnology, Kharagpr, Pn- 70, Inda, e-mals: goam@mechkgpernen, kng@mechkgpernen * Correspondng ahor perormance o he FDI echnqe o avod alse alarms, he FDI mehod shold be robs e, nsensve o modelng nceranes B he algorhm shold no be oo robs o gnore he al e, a sgncanly large varaon o he parameer vales Over he years, varos knds o robs al deecon and solaon echnqes have been developed o dagnose deren ypes o als lke sensors, acaors or componens [-4, 7, 9, 0,, 6-0] Frank [8], n a srvey paper, descrbed deren ypes o observer based robs al dagnoss echnqes Paon and Chen [5] dscssed varos robsness sses relaed observer based al dagnoss echnqes Lnear marx neqaly (LMI) based robs al deecon echnqes or nceran sysems have been developed n [9, 0] he dencaon based FDI echnqes have been sed by many researchers [,, ] o deec parameer als Daley and Wang [5] sed a hgh gan observer, whch was developed by Peersen and Hollo [4], as a ool or sensor al deecon In he presen work, an nknown np hgh gan observer (UIHGO) based componen (e, parameer) al deecon and solaon echnqe s derved Frs, an nknown np hgh gan observer s developed or a lnear nceran sysem Sch ype o nknown np observers has wde applcaons n modern conrol sysems he nceranes (modelng or paramerc or boh) are navodable Nex, sng a bank o sch observers, a parameer al deecon and solaon echnqe s devsed or a paramercally nceran sysem on he assmpons ha sensors and acaors are al ree Snce he hgh gan observer [5, 4] s robs agans parameer nceranes o some exen, he FDI echnqe s also robs agans he nceranes he FDI algorhm works n wo seps In sep-, he deecon o al and solaon o aly zone s accomplshed sng a bank o UIHGOs In he nex sep, aly parameer s solaed by parameer solaon mehod In he presen work, a par o he sysem parameers (e, he parameers o he aly sbsysem) s esmaed n sep- and only when a al occrs n he sysem In hs respec he complexy o al solaon s drascally redced n comparson wh sandard parameer dencaon echnqe [,, ] all he parameers o a sysem are esmaed a every me sep rrespecve o he occrrence o any al and he esmaed vales are compared wh her nomnal vales A nmercal example s presened o demonsrae he eecveness o he proposed observer as well as he FDI echnqe he basc mehodology o desgnng he nknown np hgh gan observer or an nceran sysem s dscssed n (Advance onlne pblcaon: May 009)

2 Engneerng Leer, 7:, EL_7 08 secon II he al deecon and solaon algorhm s explaned n secon III In secon IV, a nmercal example s presened o demonsrae he perormances o he proposed mehods he concldng remarks are nclded n secon V II UNKNOWN INPU HIGH GAIN OBSERVER In hs secon, an nknown np hgh gan observer s developed or a lnear nceran sysem he scen condons or he exsence o he observer are provded Consder a lnear me-nvaran sysem wh nknown nps x() = ( A+ A) x() + ( B+ B) () + Ed() () y() = Cx () () n x() R - he sae vecor, m () R - he p measrable np vecor, y() R - he op vecor and q d() R - he nknown np vecor he marces ABC,, and E o sable dmensons are known he marces A and B are he nceranes o he sysem and np marces respecvely hese may be consan or me varyng dependng on he sysem I s assmed ha ( A+ A) s always asympocally sable or all A I hs condon s no sased hen rs a conroller s o be desgned o sablze he sysem I s also assmed ha he sysem sases he rank condon: rank( CE) = rank( E) hs s a basc assmpon or desgnng any nknown np observer Now, sng a sae ransormaon marx, he saes are redened as z() = x () sch ha φ( n r) q E = E E s r q dmensonal marx wh rank( E) = rank( E ) and φ s a nll marx he sysem and op eqaons can be recas as ollows z A A z A A z B = + + z A A z A A z B () B φ( n r) q + + d B E z y() = C C (4) z Now s assmed ha he measremen sgnals are sch ha he ollowng rank condon s sased: rank( C) > rank( E) hs s a necessary condon or desgnng hs observer as he exra measremen sgnals are sed o desgn he redced order observer aer decoplng he nknown nps hs condon allows he rearrangemen o he op eqaon n he ollowng orm wh he help o a ransormaon y = Vy, V s a nonsnglar marx, y as y = C φ z y C C z (5) Now he eqaons () and (5) can be wren n expanded orm as ollows z = Az+ Az+ Az+ Az+ B + B (6) z = Az+ Az + Az+ Az + B+ B+ Ed (7) y C z (8) = y = Cz + Cz (9) Elmnang z rom he eqaon (6) sng he eqaon (9), one ges z = Az + AC ( y Cz) + B + Ed (0) Ed = Az + Az + B wh E - known marx and - nknown sgnal d I can be seen ha C shold be ll rank marx, whch wll be always so as rank( CE) = rank( E) Now, he eqaon (0) can be re-wren n smpled orm as z = Az + B+ Ed, () s s s = A A A C C, Bs = B AC and = y For desgnng an observer, he sysem shold sasy he observably condon: rank( O( AC, )) = n Now one can desgn an observer or he sysems () and (8) o esmae he sae z ˆ ˆ ˆ = s + s + ( () yˆ ˆ = Cz () he observer gan marx K s deermned by solvng he ollowng algebrac Rcca eqaon (ARE) [5, 4] q EE PEE P AP s + PAs + Q+ PC CP + = 0 (4) σ q σ wh K = PC, (5) Q s a pre-chosen posve dene marx and he consans q & σ are speced nmbers I was shown n [4] ha or any σ > 0, here exss q sch ha gan obaned rom he above eqaons wll lead o C( jwi As + KC) E < σ or w R w s he reqency hs condon mples ha he eec o nknown sgnal becomes very small n error dynamcs d or an approprae vale o σ Now he sae ẑ s esmaed rom eqaon (9) sng he esmaed sae ẑ as zˆ = C ( y C z ˆ ) (6) Fnally sng ˆ = { ˆ ˆ } ond o as complees z z z, he esmaed saes ˆx are xˆ = zˆ Wh hs, observer desgn process (Advance onlne pblcaon: May 009)

3 Engneerng Leer, 7:, EL_7 08 III FAUL DEECION AND ISOLAION ALGORIHM In hs secon, a componen al deecon and solaon echnqe or a lnear nceran sysem s derved I consss o wo seps In he rs sep, a se o resdals s generaed wh he help o a bank o nknown np hgh gan observers (UIHGOs) o deec he al and solae he aly regon In he second sep, aly parameer s solaed rom he aly regon Consder a lnear me nvaran sysem as x() = ( A+ A) x() + ( B+ B) () (7) he sgncance o he marces and vecors are same as descrbed n secon II Sppose a al occrs n a componen o he plan he deecon and solaon o he al are carred o n wo seps as ollows Sep-: Deecon and paral solaon o al he aly sysem s wren as x() = ( A+ A + A) x() + ( B+ B + B) (), (8) A and B are he aly pars o he marces A and B respecvely I can be emphaszed ha he magnde o als (e, A and B ) shold be sgncanly larger compared o he magnde o nceranes (e, A and B ) he sae eqaon (8) can now be rearranged as x() = ( A+ A) x() + ( B+ B) () + Ed(), (9) q E s a known marx and d() R s he nknown np sasyng he relaon Ed() = A x() + B () (0) Now he sysem s dvded no N nmbers o sbsysems wh each characerzed by a ew parameers he choce o sbsysems s qe arbrary b here shold no have any common elemens beween hem In a physcal sysem, he sbsysems are chosen based on he physcal proxmy o deren parameers Assme ha he al has been occrred n he -h sbsysem he sysem eqaon consderng he al n he -h sbsysem s wren as x () = ( A+ A) x () + ( B+ B) () + E d () () () () () () he sbscrp () ndcaes ha he al has been consdered n he -h sbsysem he op eqaon or hs sysem s wren as y () = C x (), () () () () he eqaons () and () are smlar o he eqaons () and () Now, ollowng he procedre dscssed n secon II, an nknown np hgh gan observer s desgned o esmae he saes xˆ () Once he saes () () xˆ () calclaed as r () = y () yˆ () = y () C x ˆ () () () () () () () () are esmaed, he resdals are Now, an nknown np observer, properly desgned, can esmae he saes rrespecve o nknown nps So he resdal r, calclaed rom he eqaon (), () () converges whn a bonded vale known as hreshold vale he al occrs n he -h sbsysem or here s no al n he sysem as he eec o possble als n -h sbsysem s consdered as nknown nps In deal case, e, n he absence o nose and parameer nceranes, he resdals shold converge o zero (hogh a small hreshold vale s always se o ake care o errors de o he nmercal lmaons) as n he presen case he convergence akes place whn a hreshold vale, whch agan depends on he amon o nceranes and np sgnal appled o he sysem In hs way, one can deec a al and solae he aly sbsysem sng N nmbers o resdals calclaed wh he help o ha nmbers o UIHGOs However ( N ) sch observers wll be scen o solae a aly sbsysem when N > becase once ( N ) sbsysems are ond al ree, he remanng sbsysem s aomacally dened as he aly one A decson able (as shown n able ) s drawn o solae he aly sbsysem rom observaon o ( N ) resdals able : Decson able or solaon o aly sbsysem Observa on cases For gven resdals ( r() ) and hreshold vales ( ε ): r > ε () () () () () hen se and r ε hen se 0 r r r () () () r( N ) Decson Case No al Case- 0 Fal: SS Case- 0 Fal: SS Case- 0 Fal: SS Case- (N-) 0 Fal: SS (N-) Case-N Fal: SS (N) Sep-: oal solaon o al Once he aly sbsysem s solaed, he aly parameer n he aly sbsysem s dened n hs sep Frs, he eec o he aly sbsysem s smlaed as an nknown np sgnal, say F he relaonshp beween, he () F () parameers o he aly sbsysem, say s, and he saes x() are known and can be wren as F () = ( s, x ()), (4) he ncon s lnear or a lnear sysem he sysem eqaon or hs case becomes x() = ( A+ A) x() + ( B+ B) () + Ed (), (5) (Advance onlne pblcaon: May 009)

4 Engneerng Leer, 7:, EL_7 08 d() = F Wh a measremen marx, an () C observer s hen desgned o esmae he saes Knowng he saes, he nknown np sgnal s esmaed rom he sae eqaons neglecng he nceranes sng he nomnal vales o he parameers o he oher non-aly sbsysems he esmaed sgnal Fˆ () s now sed o esmae he parameers s 's rom he relaon (4), whch s rewren as Fˆ ˆ () = (, s x ()) (6) All he elemens o eqaon (6) excepng he parameers s are known Deren parameer esmaon echnqes can be sed o esmae s rom eqaon (6) However, a very smple logcal approach s appled n he presen work n order o solae he aly elemen Le s consder he k-h parameer as he aly one From he above relaon, s k can be esmaed sng nomnal vales o res o he parameers Mahemacally, sˆ ˆ ˆ k = gs (, s,, sk, sk+, sl, x ( ), F( )) (7) l s he nmber o parameers o he aly sbsysem and g s a nconal I s observed ha n seady sae, he esmaed vales lcae very less he assmpon s correc he movng averages echnqe can be sed o smoohen he lcaon o he esmaed vales de o nceranes I he assmpon s wrong, he esmaed vales vary sgncanly large Now, as he sngle al case s beng consdered, here wll be only one case when he esmaed parameer wll vary less he parclar parameer or whch happens s he aly one In hs way, he aly parameer s solaed In he same way, any paramerc al o any s k sbsysem can be deeced and solaed ollowng he above wo seps he FDI echnqe can be smmarzed n a block dagram as shown gre IV NUMERICAL EXAMPLE Consder a mechancal sysem (as shown n gre ) ha consss o wo mass elemens and hree ses o sprngs and dampers he sae space model o he sysem can be wren as ollows: x() = ( A+ A) x() + ( B+ B) (), , ( K+ K) K ( C+ C) C A= M M M M K ( K + K) C ( C + C) M M M M 000(/ ) = M, x = X X X X, () = F(), B [ ] { } X and mass elemen X are he dsplacemen and velocy o he M respecvely, K - he sness elemen j and C - he dampng coecen (=, and j=,, ) he j marces A and B are he nceranes o he sysem and np marces respecvely Fgre : Mechancal sysem havng wo masses and hree ses o sprng-damper Fgre : Srcre o he FDI algorhm he nmercal vales o he sysem parameers are M = 870 kg, M = 550 kg, K, = N / m K = N / m, K = N / m, C = 500 Ns / m, C = 000 Ns / m and C = 5675 Ns / m he sysem response wh arbrary nal condons s smlaed sng MALAB-SIMULINK oolbox he paramerc nceranes are smlaed n sch a way ha he elemens o he marces A and B der maxmm o ± 5 % rom her nomnal vales A al s now nrodced n he sprng o sness K a he new vale o = 50sec K s se o N/m Now sng he FDI algorhm, (Advance onlne pblcaon: May 009)

5 Engneerng Leer, 7:, EL_7 08 dscssed n secon III, he al s deeced and he aly elemen (here K ) s solaed as ollows Sep-: Deecon and paral solaon o he al Frs, he sysem s dvded no hree sbsysems as ollows: SS : K, & C M ; : SS K & and C SS : K, C & M he nceranes are nrodced as ollows: A= MΣ N and B = MΣ N wh, M = In N = 005 A, N and = 005 B Σ = Σ0sn( w ) Σ 0 = 05 I and w rad/s he marces and = 005 A B are same as A and B excepng he elemens conanng consan erms are replaced wh zeros he snsodal varaon n sysem parameers s nrodced n smlaon he ollowng np sgnal s appled or hs case: = sn( w) wh 0 0 = 00 N and w = rad/s As he sysem s dvded no hree sbsysems, so wo UIHGOs are scen as a par o sep- he observers are desgned or SS and SS he nknown np marces E s and nknown np sgnals d s or hose observers are gven below E [ ], () = d() = A() x() + B() () E () = [ ], d () = A() x() + B() () he op marces are C () = and C() = 0 0 Now applyng he mehod dscssed n secon-ii, wo E s are ond o as E () [ 0 0 ] = and E () [ 0 0 ] = he vales o he nng parameers σ and q are consdered as σ () = 005, σ () = 005, q () = 5 and q he vale o Q s chosen as or boh () = 5 Q = 5I n he observers he gan marces are calclaed by solvng he eqaons (4) and (5) he vales o he observer gans or he above observers are K () = [ ] and K () = [ ] respecvely wo hgh gan observers are hen desgned or he above sysems Fnally he resdals are calclaed and ploed n gre and gre 4 In deal saon (e, al ree and n he absence o parameer nceranes), he resdals shold be zero However n he presen case hese wll no be zero de o presence o parameer nceranes Hence wo small 5 0 ε() = and consan hreshold vales { } 9 { } 0 ns are chosen as he smlaon ε() = 0 0 s carred o applyng xed np sgnals In real saon, adapve hreshold vales [5] shold be chosen as np sgnals vary dependng on operang condons Here he hreshold vales are calclaed n normal operang condon e, when here s no al n he sysem Fgre : Componens o he resdal r () () Fgre 4: Componens o he resdal r () () (Advance onlne pblcaon: May 009)

6 Engneerng Leer, 7:, EL_7 08 As boh he resdals cross he hreshold vales, he exsence o a al s conrmed o solae he aly sbsysem a decson able (able ) s consrced as shown below able : Decson able or solaon o aly sbsysem Observaon Is r () > () r () () ε (se ) or ε (se 0 )? r r () () Decson Case Fal: SS From able-, s seen ha he al s n sbsysem So he nex sep (e, sep-) s carred o o solae he aly parameer Sep-: oal solaon o he al Here he aly sbsysem (SS) s rs replaced wh an nknown orce F () as F () = K( X X) + C( X X ) hen he sysem s remodeled as ollows x() = ( A+ A) x() + ( B+ B) () + Ed() E = 0 0 / M / M and d() = F () I can [ ] be noced ha he parameer nceranes or hs sysem are only n rd and 4 h rows o A and B marces, whch ndcae A and B have non-zero elemens n rd and 4 h rows only Agan he marx E conans non-zero elemens n he same rows For hs smlary here he sysem eqaon s remodeled combnng he nknown nps and nceranes as x() = Ax() + B() + E d () and A, 4x+ B,+ F / M Ec = dc = 0 A4, 4x+ B4, F / M 0 hs s a specal case, whch may no appear or all sysems Now a ll order nknown np observer [6] s desgned 0 0 wh op marx C = o esmae he saes xˆ( ) Usng he esmaed saes xˆ( ) and he nomnal vales o he parameers o sbsysem-, he nknown orce F () s esmaed rom he ollowng relaonshp F ˆ () = M x ˆ + K x ˆ + C x ˆ, ˆ x s calclaed akng he dervave o wh ˆx respec o me Fnally he aly parameers are esmaed sng he ollowng relaon Fˆ () ˆ ˆ ˆ ˆ = K ( x ) ( ) x + C x4 x Frs, he al s assmed o resde n he sness elemen and sng he nomnal vale o C =000 Ns/m, K c c K s esmaed he movng averages are aken o redce he eec o nceranes and nmercal errors o he esmaed vales he nal movng average s aken wh a me wndow o 5 (5-0 seconds o daa) sec Inal daa are no aken o redce he ranson eec ha comes de o nal condons Now wh an ncremen o sec, he movng averages are calclaed po 0 sec o me span and he esmaed vales are ploed n gre 5 he plo shows ha esmaed vales vary very less (maxmm varaon o 5 % rom s mean vale) Now s assmed o be aly and sng he nomnal vale o K =70000 N/m, s esmaed he esmaed vales C aer akng movng averages n smlar manner as n case o are ploed n gre 6 he plo shows ha he esmaed K vales o C C vary wdely (as hgh as 75 % rom s mean vale), whch s becase o he wrong assmpon hs Fgre 5: Esmaed sness Fgre 6: Esmaed dampng coecen (Advance onlne pblcaon: May 009)

7 Engneerng Leer, 7:, EL_7 08 conrms ha aly elemen s K and hereby he al solaon process complees hs s seen ha he FDI scheme works well or he occrrence o a al n sbsysem I can be shown easly ha he mehod works wh eqal ease or he occrrence o any parameer al n any oher sbsysem V CONCLUSIONS An nknown np hgh gan observer (UIHGO) based componen al deecon and solaon (FDI) scheme s presened Frs an UIHGO or a lnear nceran sysem s derved hese ypes o observers have wde applcaons n robs conrol and al dagnoss hen, sng a bank o sch observers, a FDI echnqe s devsed he advanage o he FDI algorhm s ha s capable o esmang als even he parameers are copled n he sysem marx I also redces he complexy o esmang all he parameers a every me nsan nlke exsng dencaon based parameer al dagnoss echnqes he same FDI echnqe can also be sed o deec a al n a nosy sysem or a nonlnear sysem provded oher ypes o nknown np esmaors capable o handlng nose or nonlneary shold be sed Proc 44h IEEE Con on Decson and Conrol, and he Eropean Conrol Con 005, Sevlle, Span, pp 88-84, 005 [4] Peersen, I R and Hollo, C V, Hgh-gan observer approach o dsrbance aenaon sng measremen eedback, In J o Conrol, Vol 48, No 6, pp , 988 [5] Paon, R J and Chen, J, Observer-based al deecon and solaon: robsness and applcaons, Conrol Engneerng Pracce, Vol 5, No 5, pp 67-68, 997 [6] Pg, V, Qevdo, J, Escobe, and Sanc, A, Robs Fal Deecon sng Lnear Inerval Observers, In Proc 5h IFAC Symposm on Fal Deecon, Spervson and Saey o echncal Processes, Washngon DC, pp , 00 [7] Sa M and Gan, Y, A New Approach o Robs Fal Deecon and Idencaon, IEEE rans Aerospace and Elecronc Sysems, Vol 9, No, pp , 99 [8] Shen, L C and Hs, P L, Robs Desgn o Fal Isolaon Observers, Aomaca, Vol 4, No, pp 4-49, 998 [9] Wang, H B, Wang, J L and Lam, J, Robs Fal Deecon Observer Desgn: Ierave LMI Approaches, J o Dynamc Sysems, Measremen, and Conrol, Vol 9, pp 77-8, 007 [0] Zhong, M, Dng, S X, Lam, J and Wang, H, An LMI approach o desgn robs al deecon ler or nceran LI sysems, Aomaca, Vol 9, No, pp , 00 REFERENCES [] Beard, R V, Falre Accommodaon n Lnear Sysems hrogh Sel-Reorganzaon, Ph D dsseraon, Deparmen o Aeronacs and Asronacs, MI, Cambrdge, Massachses, 97 [] Bloch, G, Oladsne, M and homas, P, On-lne al dagnoss o dynamc sysems va robs parameer esmaon, Conrol Eng Pracce, Vol, No, pp , 995 [] Chen, J, Paon, R J and Zhang, H-Y, Desgn o nknown np observers and robs al deecon lers, In J o Conrol, Vol 6, No, pp 85-05, 996 [4] Chow, E Y and Wllsky, A S, Analycal Redndancy and he desgn o robs alre deecon, IEEE rans Aomac Conrol, Vol 9, No 7, pp 60-64, 984 [5] Daley, S and Wang, H, Applcaon o a Hgh Gan Observer o Fal Deecon, In Proc nd IEEE Con on Conrol Applcaons, Vancover, B C, Sepember -6, pp 6-6, 99 [6] Daroach, M, Zasadznsk, M and X, S J, Fll order observers or lnear sysems wh nknown nps, IEEE rans Aomac Conrol, Vol 9, No, pp , 994 [7] Dan, G R and Paon, R J, Robs Fal Deecon n Lnear Sysems sng Lenberger Observers, In Proc UKACC In Con on CONROL 98, Wales Swansea, UK, pp , 998 [8] Frank, P M, Enhancemen o robsness n observer-based al deecon, In J o Conrol, Vol 59, No 4, pp , 994 [9] Ge, W and Fang, C Z, Deecon o aly componens va robs observaon, In J o Conrol, Vol 47, No, pp , 988 [0] Jang, B, Wang, J L and Soh, Y C, Robs Fal Dagnoss or a Class o Lnear Sysems wh Uncerany, In Proc he Amercan Conrol Con, San Dego, Calorna USA, pp , 999 [] Jang,, Khorasan, K and aazol, S, Parameer Esmaon- Based Fal Deecon, Isolaon and Recovery or Nonlnear Saelle Models, IEEE rans Conrol Sysems echnology, Vol 6, No 4, pp , 008 [] Mondal, S, Chakrabory, G and Bhaacharyya, K, An nknown np Kalman ler based componen FDI algorhm and s applcaon n aomoble, In J o Vehcle Aonomos Sysems, Vol 5, No /4, pp 74-87, 007 [] Palma, L B, Coo, F V and da Slva, R N, Dagnoss o Paramerc Fals Based on dencaon and Sascal Mehods, In (Advance onlne pblcaon: May 009)