Robust Fuzzy Fault Detection for Non-Linear Stochastic Dynamic Systems

Transcription

1 he Oe Automato ad Cotrol Systems Joural, 009,, Oe Access Robust Fuzzy Fault Detecto for No-Lear Stochastc Dyamc Systems Magdy G. El-ghatwary,* ad Steve X. Dg Idustral Electroc ad Automatc Cotrol Deartmet, Faculty of Electrcal Egeerg, Meoufa Uversty, Meouf 395, Meofa, Egyt Isttute for Automatc Cotrol ad Comlex System, Uversty of Dusburg-Esse, Germay Abstract: Oe of the dffcults for fault detecto techques for o-lear stochastc systems va model-based methods s the desg of resdual geerato. I ths aer, a ew fault detecto (FD aroche for o-lear stochastc systems s roosed. he o-lear system s rereseted by a dscrte akag-sugeo (S fuzzy model. he use of (S theory allows to rereset o-lear systems as a set of lear systems, whch rereset the local system behavor aroud dfferet oeratg ots. he global system behavour s descrbed by a fuzzy fuso of all systems. he FD system for each local sub system s desged by solvg the corresodg Dscrte Algebra Recat Equato (DARE. Otmzato algorthm based o mmzg the resdual covarace matrx s used to obta a robust FD for global system behavor. he observer ga matrces are solved usg a set of Lear matrx Iequaltes (LMIs.. INRODUCION Over the ast two decades, fault detecto (FD system has made a sgfcat rogress ad recved cosderable atteso both research ad alcato doma. It leads to robust FD system for techcal rocesses that ca be modelled as lear tme varet (LI systems. If LI system cotas measuremet ose, kalma flter s used to desg FD system [, ]. Recetly akag-sugeo (S fuzzy model was develoed successfully to vestgate o-lear system [3]. he robust fuzzy observer was dscussed for S fuzzy system wth arameter ucertates []. For systems wth measuremet dsturbaces, ew descrtor observer aroach was develoed [5-7]. Exteded kalma flter desg method based o basc of adatve fuzzy logc s show [8]. Robust fault estmato aroach for vehcle lateral dyamc model s show [9]. I ths aer, aother fault detecto aroach s used. he roosed aroach use fuzzy logc bascs to desg kalma flter for each fuzzy subsystem by solvg the corresodg Dscrte Algebra Recat Equato (DARE. Otmzato algorthm based o mmzg the resdual covarace matrx s used to obta a robust FD for global system. he geerated flter s robust agast stochastc ose ad sestve wth resect to faults. hs aer s orgazed as follows: Secto shows some relmares about akag-sugeo (S fuzzy model, fault geerato ad fault evaluato, the roosed aroache s reseted secto 3; a alcato examle s foud secto ; the coclusos are gve secto 5.. PRELIMINARIES Some cocets relevat to ths work are revewed. Frst, the S fuzzy model for o-lear system s reset. he *Address corresodece to ths author at the Idustral Electroc ad Automatc Cotrol Deartmet, Faculty of Electrcal Egeerg, Meoufa Uversty, Meouf 395, Meofa, Egyt; el: & Fax: ; E-mal: magdyelghatwary@yahoo.com resdual geerato ad resdual evaluato cocets are brefly descrbed... S Fuzzy Model Costructo he fuzzy model roosed by akag ad Sugeo s descrbed by fuzzy IF-HEN rules, whch rereset local lear ut-outut relatos of a o-lear system [3]. I order to cosder stochastc oses ad faults the dscrete S fuzzy systems, we roose the dscrete S fuzzy system whch the th rule s formulated the followg form: Rule : IF z s M ad... ad z s M he x(k + = A x(k + B u(k + E, (k + E f, f(k ( y(k = Cx(k + D u(k + (k + F f, f(k where M j ( =,,...,, j =,, are fuzzy sets, z =[z,..., z ] are remse varables, x(k R s state vector, u(k R ad y(k R q are the ut ad measured outut vectors resectvely, (k R m vector of zero mea whte Gaussa oses wth ostve defte covarace matrx, f(k R l s the fault to be detected. A, B, E,, E f,,, D,, F f, are kow matrces wth arorate dmeso. he defuzzfed outut of S fuzzy system ( s rereseted as: x(k + = y(k = where (z(k = [A x(k + B u(k + E, (k + F f, f (k] [ x(k + D u(k + (k + F f, f (k] ( h (z(k, h (z(k = M j (z j (k. j= h (z(k M j (z j (k > 0 s the grade of membersh of z j (k M j. Assume that M j (z j (k 0. We have j= 87-3/ Betham Oe

2 6 he Oe Automato ad Cotrol Systems Joural, 009, Volume El-ghatwary ad Dg k h (z(k = For smlfyg otato (z(k s relaced by, whch wll also be adoted the sequet sectos f o cofuso s caused. Uder the assumto that the curret error s deedet of the curret ose, we rovde E[e k ] = E[ k e ] = 0, for all,k... Problem Formulato Fault detecto roblem ca be formulated as desg fault detecto system whch s robust wth resect stochastc oses ad sestve wth resect to faults..3. Resdual Geerato he frst ste to acheve robust FD system s to geerate resdual sgal whch s decouled from the ut sgal u(k. I ths case, we cosder the so-called S fuzzy flter whch s descrbed as follows: Rule : If z s M ad... ad z s M he ˆx(k + = A ˆx(k + B u(k + (L * + L [ y(k ŷ(k] (3 ŷ(k = ˆx(k + D u(k where L * s the flter ga matrx for sub-model obtaed from solvg DARE for each local system, L s cremet ga matrces obtaed from reducg covarace matrx of resdual sgal. he fuzzy flter based resdual geerator s ferred as the weghted sum * ˆx(k + = [A ˆx(k + B u(k + (L + L ( y(k ŷ(k] ŷ(k = [ ˆx(k + D u(k], where s the same weght fucto used S model (. o aalyze the covergece of the flter, the state error vector e(k = x(k ˆx(k s gve by the followg dfferece equato. e(k + = j= [( A (L * + L e(k +(E, (L * + L F, j (k + (E f, (L * + L F f, j f (k] r(k = μ [ e(k + (k + F f, f (k], where r(k s resdual sgal. Eq. (5 ca be rereseted as: e(k + = j= r(k = [( A j L e(k + (E,j L F, j (k +(E f,j L F f, j f (k] [ e(k + (k + F f, f (k], where A j = A L *, E,j = E, L * F, j ad E f,j = E f, L * F f, j. Eq. (6 ca be more smlfed ad rereseted as ( (5 (6 e(k + = r(k = j= [( A j e(k + E,j (k + E f,j f (k] [ e(k + (k + F f, f (k], where Aj = A j L, E,j = E,j L F,j ad E f,j = E f,j L F f,j... Resdual Evaluato After the desg of the resdual geerator, the remag mortat task for robust fault detecto s the evaluato of the geerated resdual. Based o [0], threshold value J th > 0 ca be calculated. Usg the followg logc relatosh for fault detecto: r(k,n < J th o fault (8 r(k,n < J th fault, where the so-called resdual evaluato r(k,n s determed by N r(k,n = r (kr(k, (9 k=0 wth N s legth of the evaluated wdow. Sce a evaluato of the sgal over the whole tme rage s mractcal, t s desred that the fault wll be detected as easy as ossble. Based o (7, we have r(k,n = r (k + r f (k,n where r (k ad r f (k are defed as: r (k = r(k f(k =0, r f (k = r(k =0. Moreover, the fault-free case resdual evaluato fucto s r(k,n r (k,n J th,, where J th, = su L r (k,n. We choose the threshold J th as J th = J th,. Where J th s costat ad ca be evaluated off-le. 3. ROBUS FAUL DEECION DESIGN Robust fault detecto desg s show the followg sub-sectos. 3.. Ga Matrx Desg Based o DARE he ga matrx for each local sub-system s obtaed. he comutato of covarace of resdual sgal geerated by kalma flter-based resdual geerator ad fault detecto flter s show. Cosder system (7 wth ΔL = 0, the followg system s obtaed. e(k + = r(k = j= j= [( A j e(k + (E,j (k + E f,j f (k] [ e(k + (k + F f, f (k] r(k = r (k + r f (k, (7 (0 Based o [], the followg theorem rovdes a soluto to obta L *, the roof for lear system s gve []. heorem. Each sub-system s stable ad satsfy H orm f

3 Robust Fuzzy Fault Detecto he Oe Automato ad Cotrol Systems Joural, 009, Volume 7 L * = (S + A P ( P + R, ( where Q = E, E,, R = F,, S = E, ad P 0 s the covarace of the estmato error, t s gve as a soluto of the followg DARE P = A PA + Q (S + A PC ( PC + R (S + A PC, ( 3.. Covarace of Resdual Geerated by Kalma Flter For FD of the dyamc system wth oly stochastc ose, the steady-state oe-ste redctve kalma flter s ofte used as resdual geerator [3, ]. I ths case, the geerated resdual s a zero-mea whte Gaussa sgal wth mmal covarace the fault-free case, ad the resdual covarace ca be easly calculated. Based o the statstcal characterstc of resdual sgal, the covarace matrx r ( of resdual r(k s equal to the covarace matrx of ose duced resdual sgal r (k, therefore r (l = r (l = CP + R l = 0 (3 r (l = r (l = 0, l 0, Sce the resdual vector r ks,k the evaluated wdow s defed as r ks,k =[r (k s,..., r (k], thus the covarace matrx of resdual vector r ks,k s = E{r ks,k r ks,k } r (0 r ( r (S = r ( r (0 0 r (S 0 r (0 (s+(s+ PC + R 0 0 = r ( 0 r (s 0 PC + R (s+(s+ ( Sce the geerated resdual r(k s u-correlated, t ca be foud from above exresso, the covarace matrx of resdual sgal r ks,s s a block dagram matrx, there fore a statstcal resdual for the resdual vector r ks,k ca be easly carred out based o ths roerty Icremet Ga Matrx Desg Based o LMI Icremeted ga matrx ΔL show (7 s desged. For ths urose, the resdual covarace wll be frstly aalyzed. If the resdual dyamc s stable, the uque stablzg soluto of followg DARE deoted by s the covarace of estmated error + lm E{e(k k= +e (k +} = = {( A L ( A L +(E, L (E, L } < j {( A j L L j ( A j L L j +(E,j L F, j + E, j L j (E,j L F, j + E, j L j } eq. (5 cotd.. (5 Sce the estmated error e(k + s deedet of (k, the covarace of resdual sgal r s lm k= E{r(kr (k} = r = { C + F, } herefore, + < j {( ( +( + F, j ( + F, j } tr( r = tr C + < + < ( ( ( + F, j ( + F, j = tr + tr μ F, + tr μ μ (C + C (C + C j j j < < + tr μ μ (F + F (F + F j,, j,, j where (6 (7 tr μ + tr μ μ (F + F j,, j (F, + F, j s < oly decded by ose ad s a ostve scalar costat. As tr(ab = tr(ba the, tr( r = tr( C + + < j < j ( ( F, ( + F, j ( + F, j (8 Based o the above results, the otmzato of FD desg ca be exressed as: Fd ΔL such that, the resdual dyamc (7 s stable ad tr μ C + tr μ μ (C + C j j ( m < j

4 8 he Oe Automato ad Cotrol Systems Joural, 009, Volume El-ghatwary ad Dg Based o [5], the followg lemma s obtaed. Lemma. Assume that the matrces L stablzes the resdual dyamcs (7 the tr(v = tr μ C + tr μ μ (C + C j j ( where < j V = (E, L (E, L + < j (E,j L F, j + E, j L j (E,j L + E, j L j, ad > 0 s the uque stable soluto of DARE = [( A L ( A L + ] + < j [A j L L j ] [A j L L j ]+ ( (. (9 (0 Proof: From the soluto of DARE (5 ad (6, we kow that tr C + tr ( ( < j = tr μ ( A L l V ( A L l l=0 + tr μ μ j ( A j L L j l < j V ( A j L L j l ( ( = tr( μ ( A L l C l ( A L l=0 + tr μ μ j ( A j L L j l < j ( ( ( A j L L j l V = trv, ( Based o lemma, the followg theorem s used to obta the chage ga matrx ΔL. heorem. Assume that ΔL s gve ad there exsts a symmetrc matrx X > 0 ad >0 such that:. I + L XL > 0. I (L + L j X(L + L j > 0 3. X + A X A + + A X L ( I L XL L X A < 0. X + ( ( A j + A j X ( A j + A j + ( ( + ( A j + A j X(L + L j ( I (L + L j X(L + L j (L + L j X ( A j + A j < 0, are satsfed for I,j. he: system (7 s stable < X cosequetly tr( μ C + tr( μ μ (C j ( < tr(xv. < j Proof: he results ( follows drectly from the bouded real lemma [6]. For the results (, based o the schur comlemets lemma, codto (3 s equvalet to the followg LMI: I + L X L A X L L X A X + A X A + < 0, ( for hold f X > A XA + C. Ad codto ( s equvalet to the followg LMI: l + (L + L j X (L + L j ( A j + A j X (L + L j (L + L j X ( A j + A j X + ( A j + A j X ( A j + A j +( ( (3 for < < hold f X > ( A j + A j X ( A j + A j + ( (. Comerg ( ad (3 wth DARE (, based o the mootocty of the DARE [7], we kow X. Ad based o Lemma, we have that tr( + tr( ( < j ( = tr(v tr( XV Based o heorem, the revous roblem ca be reformulated as: for a gve > 0 ad symmetrc X > 0, fd ΔL such that: X + A X A + + A X L ( I L X L L XA < 0, ad tr(xv m for X + ( A j + A j X ( A j + A j + ( ( +( A j + A j X (L + L j ( I (L + L j X (L + L j (L + L j X ( A j + A j < 0, ( (5

5 Robust Fuzzy Fault Detecto he Oe Automato ad Cotrol Systems Joural, 009, Volume 9 ad tr(xv m. for < j. It s kow that tr( XV = tr (E, L X (E, L, for ad tr( XV = tr ([E,j L F, j ]+[ E, j L j ] X ([E,j L F, j ]+[E, j L j ], for < j. herefore the mmzato of tr(xv, ca be realzed wth the followg method, ( E, for, ad j. L X[E, L <, ([E,j L F, j ]+[E, j L j ] X ([E,j L F, j ]+[E, j L j hs formulato ca be rereseted as LMI as: S(E, X F, Y * X > 0 (6 for, ad S(E,j X F, j Y + E, j X F, Y j * X > 0 (7 for < j, where S =, Y = XΔL. herefore, ths roblem ca be reformulated as the followg otmzato roblem: For a gve > 0, fd symmetrc matrces X > 0 ad matrx Y, so that the followg LMIs: X XA Y Y 0 * X 0 < 0, (8 * * l 0 * * * l S(E, X F, Y * X > 0 (9 for, ad X M(j Y + Y j 0 * X 0 ( * * l 0 * * * l S(E,j X F, j Y + E, j * X for < j where X F, Y j < 0, (30 > 0, (3 M(j = XA j Y + XA j Y j. he the roblem ca be solved ad the soluto for ΔL s ΔL = X Y.. LAERAL VEHICLE DYNAMIC MODEL I recet years may research have bee doe the feld of vehcle dyamcs, may achevemets have bee fulflled [8-0]. Ad may alcatos dfferet vehcle dyamc models have bee acheved. he dervato of the vehcle dyamc model s based o the hyscal moto equatos, therefore the dfferet models ca be classfed accordg to the qualty of model s freedom. he geeral used oe-track model (or bcycle model s a 3 DOF model [], for the vehcle s smlfed as a whole mass wth the ceter of gravty o the groud, whch ca oly move x axs, y axs, ad yaw aroud z axs. he coordate system s show Fg. (. Coordate system of vehcle model.

6 50 he Oe Automato ad Cotrol Systems Joural, 009, Volume El-ghatwary ad Dg Fg. (, whch s fxed to the CG. For the urose of studyg the roll moto of the vehcle, the CG s ot assumed o the groud. Comarg wth oe track model, the roll moto aroud the xaxs s troduced, so t s called a DOF model. For a more recse descrto of the vehcle dyamc, the vehcle s modeled as a mult-body system. Some large DOF models have bee costructed, such as the vehcle smulato software rucksm whch cludes a DOF model. But such kd of model s too comlcated to be used for the o-le alcato, oly sutable for some offle or smulato alcato. I IFAIS roject [], order to establsh a desg framework of model based motorg system for vehcle lateral dyamcs cotrol systems, the DOF model ad oetrack model have bee studed. I ths aer, oe-track model s used. Because of S fuzzy model ca be used wth tme varg systems so S fuzzy model ca be obtaed for Vehcle Lateral Dyamc model... Smulato Results Sesor fault for Lateral vehcle dyamc model s studed. Lateral accelerato sesor fault ad Yaw rate sesor fault wth stochastc oses are detected. I ths case, dscrete S fuzzy model s used. After the dscretzato of each sub system, usg 0 mllsecods as samlg tme, the vehcle lateral dyamc model s rereseted ba the followg: x(k + = x(k = 0 V (k = ay (k 0 r (k where A = A = A 3 = A = [A x(k + B ( L * (k + L (k + E f, f (t] [ x(k + D L * (k + v(k + F f, f (k] , B = E f,, B = E f,, B = E 3 f,3, B = E f, (3.. Resdual Geerator Desg As troduced above, the resdual for olear system s rereseted by S fuzzy flter of the form lke ˆx(k + = [A ˆx(k + B u(k + (L * + L ( y(k ŷ(k] ŷ(k = [ ˆx(k + D u(k], (33 where L ad L are defed as (7. he followg are the detals of the sub models ad corresodg flter-based resdual geerators. he Frst Sub Model I ths case, the steerg agle s take as ut sgal, ad lateral accelerato as outut sgal. he resdual geerated s r = a y â y (3 he ga matrces obtaed from solvg the DARE ( are L * = L * 3 = , L * =, L * = he cremet ga matrces are obtaed by solvg (8 -(3 L = L 3 = ,, L = ,, L = he covarace matrces for each sub-system based o ( are, = =, =,3 =, he Secod Sub Model I ths case, the steerg agle s adoted as ut sgal, yaw rate as outut sgal, the resdual geerated s r = r ˆr (35

7 Robust Fuzzy Fault Detecto he Oe Automato ad Cotrol Systems Joural, 009, Volume 5 Fg. (. Robust fault detecto for lateral accelerato wth stochastc oses. he ga matrces obtaed from solvg the DARE ( are L * = L * 3 = , L * =, L * = he cremet ga matrces are obtaed by solvg (8 -(3 L = L 3 = ,, L = ,, L = he covarace matrces for each sub-system based o ( are = = = 3 = Resdual Evaluato After the desg of the resdual geerator, the remag mortat task for robust fault detecto s the resdual evaluator. he resdual evaluato cossts of evaluato fucto ad threshold value. Usg L orm as evaluato fucto wth the legth of evaluato wdow N = 0. he threshold value s calculated at fault free case. he Frst Sub Model he kow ut (steerg agle wth ose s show Fg. (a. he data wth a offset sesor fault of 5m/s occured at t = 8 secod s used to valdate the desged robust FD system. he threshold value ths case s J th = I Fg. (b, from t = 8 secod the evaluated sgal has exceeded the threshold value. he Secod Sub Model he kow ut (steerg agle wth ose s show Fg. (3a. he data wth a offset sesor fault of 5m/s occured at t = secod are used to valdate the desged

8 5 he Oe Automato ad Cotrol Systems Joural, 009, Volume El-ghatwary ad Dg Fg. (3. Robust fault detecto for yaw rate wth stochastc ose. robust FD system. he threshold value ths case s J th = I Fg. (3b, from t = sec ad the evaluated sgal has exceeded the threshold value. 5. CONCLUSIONS I ths aer, robust FD aroach for o-lear system wth measuremet oses has bee develoed. he olear system s rereseted by S fuzzy model. he geerated algorthm cossts of two arts, the frst art, the fault detecto for each fuzzy subsystem s obtaed by solvg DARE, the secod art, the cremeted fault detecto s obtaed from reducg covarace matrx of resdual sgal. he geerated FD system s robust agast stochastc oses ad sestve to the fault. he desg rocedure has bee rovded term of LMIs. REFERENCES [] S.X. Dg, P. Zhag, P.M. Frak ad M. Sader, Multobjectve desg of fault detecto flters, ECC 03, echcal Sesso, Cambrdge, UK, 003. [] D. Smo, Kalma flter for fuzzy dscrete tme dyamc systems, Aled Soft Comutg, vol. 3,. 9-07, 003. [3]. akag ad M. Sugeo, Fuzzy detfcato of systems ad ts alcatos to modellg ad cotrol, IEEE raacto o System Ma ad Cyberetcs, vol. 5, o., 66-3, 985. [] M.G. El-ghatwary, S.X. Dg ad Z. Gao, Robust fault detecto for ucerta akag-sugeo fuzzy systems wth arametrc ucertaty ad rocess dsturbaces, Proceedg of IFAC Symosum SAFE PROCESS, Bejg, Cha, 006. [5] Z. Gao, X. Sh ad S.X. Dg, Observer desg for -S fuzzy systems wth measurmet outut oses, IFAC World Cotrol Cogress, Prague, 005. [6] Z. Gao,. Cha ad H. Wag, A robust fault detecto flter for stochastc system va descrtor estmator ad arametrc ga desg, IE Proceedgs Cotrol heory Alcatos, vol., o. 5, , 007. [7] Z. Gao ad S.X. Dg, Sesor fault recostructo ad comesato for a class of olear state-sace systems va descrtor system aroach, IE Proceedgs Cotrol heory Alcatos, vol., o. 3, , 007. [8] N. Zhag, C. Doald ad I. Wusch, A exteded Kalma flter (EKF aroach o fuzzy system otmzato roblem, IEEE Iteratoal Coferace of Fuzzy Systems, 003, [9] Z. Gao, S.X. Dg ad Y. Ma, Robust fault estmato aroach ad ts alcato vehcle lateral dyamc systems, Otmal cotrol Alcatos ad Methods, vol. 8, o. 3,. 3-56, 007. [0] S.X. Dg, P. Zhag ad P.M. Frak, hreshold calculato usg lm-techque ad ts tegrato the desg of fault detecto systems, Proceedgs of the d IEEE Coferece O Decso ad Cotrol, 003, [] Y. Ma, Itegrated desg of observer-based fault dagoss systems ad ts alcato to vehcle lateral dyamc cotrol systems AKS, Dusburg, Germa, 006. [] M. Zhog, S.X. Dg, B. ag ad J. Lam, A otmzato aroach to fdf desg for ucerta dscrete-tme systems, Proceedgs 5 th IFAC World Cogress, 00. [3] M. Bassevlle ad I.V. Nkforov, Detecto of abrut chages: theorem ad alcato, Pretce Hall, Ic., 993. [] S.X. Dg, P. Zhag, B. Huag, E.L. Dg ad P.M. Frak, A aroach to orm ad statstcal methods based resdual evaluato, 0 th Iteratoal Coferece o Methods ad Models Automato ad Robotcs, 00. [5] Y. Ma ad S.X. Dg, Itegrated desg of fault detecto system wth mult-objectve otmzato, Proceedgs of IFAC Symosum, SAFEPROCESS, Bejg, Cha, 006. [6] K. Zhou, Essetal of robust cotrol, Pretce-Hall, Ic, 998. [7] S.X. Dg, P.M. Frak, E.L. Dg ad. Jesch, A ufed aroach to the otmzato of fault detecto systems, Iteratoal Joural of Adatve Cotrol ad Sgal Processg, vol., , 988.

9 Robust Fuzzy Fault Detecto he Oe Automato ad Cotrol Systems Joural, 009, Volume 53 [8].D. Gllese, Fudametals of vehcle dyamcs, Socety of Automatc Egeers, Warredale, USA, 99. [9] M. Borer ad R. Iserma, Adarve oe-track model for crtcal lateral drvg stuatos, I Iteratoal Syosum o Adavaced Vehcle Cotrol (AVEC, Jaa, 00. [0] Y. Fukada, Sl-Agle estmato for vehcle stablty cotrol, Vehcle System Dyamcs, vol. 3, o,, , 999. [] D. Art, IFAIS Vehcle Models of WP8, echcal Rerot, IFAIS, 00. Receved: December 05, 008 Revsed: December 5, 008 Acceted: December 6, 008 El-ghatwary ad Dg; Lcesee Betham Oe. hs s a oe access artcle lcesed uder the terms of the Creatve Commos Attrbuto No-Commercal Lcese (htt://creatvecommos.org/lceses/byc/3.0/, whch ermts urestrcted, o-commercal use, dstrbuto ad reroducto ay medum, rovded the work s roerly cted.