Robust State and fault Estimation of Linear Discrete Time Systems with Unknown Disturbances

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1 Robut Stte nd fult Etition of Liner Dicrete ie Ste with Unnown Diturbnce Beoudi lel, Ben Hid Fçl Electricl Engineering Deprtent of ESS, Reerch Unit C3S, uni Univerit, 5 Avenue h Huein, BP 56, 8 uni, unii beouditlel@hoo.fr fcl.benhid@ett.rnu.tn Abtrct hi pper preent new robut fult nd tte etition bed on recurive let qure filter for liner tochtic te with unnown diturbnce. he novel eleent of the lgorith re : iple, eil ipleentble, qure root ethod which i hown to olve the nuericl proble ffecting the unnown input filter lgorith nd relted infortion filter nd oothing lgorith; n itertive frewor, where infortion nd covrince filter nd oothing re equentill run in order to etite the tte nd fult. hi ethod provide direct etite of the tte nd fult in ingle bloc with iple forultion. A nuericl eple i given in order to illutrte the perfornce of the propoed filter. Keword Kln filtering, unbied iniu-vrince, tte nd fult etition, unnown diturbnce, qure root, liner dicret- tie te. I. INRODUCION In the pt few er, the proble of filtering in the preence of unnown input h ttrcted big ttention, due to it ppliction in environent. he unnown input filtering proble h been treted in the literture b different pproche. he firt pproch ue tht the odel for dnicl evolution of the unnown input i vilble. When the propertie of the unnown input re nown, the ugented tte Kln filter (ASKF i olution. o reduce coputtion cot of the ASKF [] propoed the two tge Kln filter where the etition of the tte nd unnown input re decoupled. he econd pproch tret the ce when we not hve prior nowledge bout the dnicl evolution for unnown input. Kitnidi [] w the firt to olve the proble uing the liner unbied iniu-vrince. An etend Kitnidi filter uing prterizing technique to obtin n optil filter (OEF hve been propoed b Drouch et l [3]. Heih [4] h been developed robut-two tge Kln filter (RSKF equivlent to Kitnidi filter. An (OMVF reported b C.S Hieh [5] hve been ued in order to developed n optil iniu vrince filter (OMVF to olve degrdtion proble encountered in (OEF. Gillijn nd De Moor [6] h treted the proble of etiting the tte in the preence of unnown input which ffect the te odel. he hve been developed recurive filter which i optil in the ene of iniu-vrince. hi filter h been etended b the e uthor for joint input nd tte etition to liner dicrete-tie te with direct feedthrough where the tte nd the unnown input etition re interconnected. hi filter i clled recurive three tep filter (RSF [7] nd i liited to direct feedthrough tri with full rn. Cheng et l, [8] propoed recurive optil filter with globl optilit in the ene of unbied iniu-vrince. hi filter i liited to etite the tte. he ce of n rbitrr rn h been propoed b Hieh in [9] the deigned optil filter Known ERSF (Etend RSF. Recentl, nother technique uing let qure ethod hve been propoed b lel et l, [] to etite the tte nd unnown input. he Fult Detection nd Ioltion (FDI proble for liner te with unnown diturbnce i generll tudied, ee e.g. Niouhh [], Keller, [], Chen nd Ptton, [3,4], Ben Hid et l, [5]. According to [], robut fult detection nd ioltion in continou-tie i developed uing the error innovtion technique to generte n unbied white reidul ignl. he fult i dignoed b ttiticl teting. A new ethod i developed in [6] to detect nd iolte ultiple fult ppering iultneouel or equentill in liner tie-invrint tochtic dicrete-tie te with unnown input []. heir ethod conit of generting directionl reidul uing n ioltion filter. In [3] the optil filtering nd robut fult dignoi proble h been tudied for tochtic te with unnown diturbnce. he output etition error with diturbnce decoupling i ued reidul ignl. After tht, ttiticl teting procedure i pplie to eine the reidul nd to dignoe fult. Netherle, the iultneou ctutor nd enor fult nd tte proble i not treted in [3, 4]. Recentl, [7] preent new optil recurive filter for tte nd fult etition of liner tochtic te with unnown diturbnce. hi ethod i bed on the uption tht no prior nowledge bout the dnicl evolution of the unnown diturbnce i vilble. he filter h two dvntge: it conider n rbitrr direct feedthrough tri of the fult nd it perit ultiple fult etition. In order the overcoe thi proble, we introduce the qure root pproch into recurive let qure filter. We ue tht the unnown diturbnce ffect onl the tte eqution, while, the fult ffect both the tte nd the output eqution where the direct feedthrough tri h n rbitrr rn nd under the pecific condition tht the proce nd eureent noie re correlted. hi pper i orgnized follow. In Section, the proble of fult detection i pecificll tted for tochtic te. In ection 3, we develop robut filter. hen, the perfornce of the deigned filter i deontrted through

2 iulted eple in ection 4, followed b few concluding rer in ection 5. II. PROBLEM FORMULAION Conider the liner tochtic dicrete-tie te in the following for: A Bu F f Gd w C F f v n R i the tte vector, r q ( + + ( Where R i the p eureent vector, u R i the nown input, R i the dditive fult vector nd d R i the unnown diturbnce vector. he proce noie w nd the eureent noie v re correlted white noie equence of zero-en with joint covrince tri v R S ε v w δ w S Q (3 With R >, where δ i the unit pule. he trice A, B, F, G, C nd F re nown nd hve pproprite dienion. We ue tht ( C, A i obervble, p+ q nd the initil tte i uncorrelted with the white noie w nd v. he initil tte i guin rndo vrible with ε [ ] ˆ nd ε ( ˆ( ˆ P where ε [.] denote the epecttion opertor. he i of thi pper i to deign n unbied iniu-vrince liner etitor of the tte nd fult f without n infortion concerning the fult. Firt we repreent the proce nd eureent noie in the following for: / v R v v / with w, X w (, I+ n (4 Q w he trice ϒ X nd, Q tif f / X SR, Q, Q SR S (5 o repreent the tte infortion n eqution fort, we introduce n uilir rndo vrible / with en zero nd covrince tri I n, tht i / (, I n. Since we ued the covrince tri P / to be ei-poitive / definite, we cn copute it qure root P / uch tht: / / P / P / P / (6 hi qure root cn be choen to be upper or lower tringulr / With /, ˆ /, d nd P / o defined, the vrible cn be odeled through the following tri eqution : / ˆ / Gd P/ / (7 hi eqution i clled generlized covrince repreenttion / ˆ / + P / / + G d (8 Fro eqution (, ( nd (9, we obtin the following et of contrint eqution on the unnown, f nd + : / ˆ I / / n G P / d / C F f R + v d + Bu / A G F I n + X w Q, Let thi et of eqution be denoted copctl b ( F( ( + G( d( + L( μ( ( he weighted let-qure proble for the derivtion of the qure-root filter lgorith b: in μ( μ( ubject to ( F( ( + G( d( + L( μ( he gol of the nli of the weighted let-qure proble i the derivte of qure root olution for the filtred nd one tep hed predicted tte nd fult etition. herefore, we will ddre the nuericl trnfortion involved in olving ( in two conecutive prt. We trt with the derivtion of the qure-root lgorith for coputing the filtered tte nd fult etition in ection 3. nd the derivtion for the coputtion of the one-tep hed prediction i preented in ection 3.. We define the following trnfortion l C I In I n then uing l,reulting trnfored et of contrint eqution i: l ( l F( ( + l G( d( + l L( μ( hen we hve: / / Cˆ / / F CG CP R / ( / ˆ d / In f G + P / v d + Bu / A G F I n + X w Q, So fro ( we forulte the proble (LS follow: C ˆ/ F in ˆ ˆ / I f f, ˆ, ˆ+ Bu A F I n + where W i the weighting tri choen follow: / / / / CP / R CP / R / / P/ P/ / / X Q, X Q, W (9 ( W (3 he objectif of the net ection i to deign n unbied iniu vrince liner etitor of the tte nd the fult f without n infortion concerning the fult f.. III. FILER DESIGN o olve the proble (, we propoe to decopoe it into two prt: firt prt to etite n unbied iniu vrince of the tte nd fult nd econd prt to the tie updte of the filter. A Meureent updte he eureent updte i derived fro ( b etrcting the row tht depend onl on nd f. hi ield,

3 ˆ C/ F in fˆ, ˆ ˆ / I f W, (4 Where W, denote the weighting which we give tochtic interprettion b chooing / / / / CP / R CP / R, / / P/ P/ W (5 he proble i to deterine liner etite f ˆ, ˆ of on the given dt nd ˆ / which hve the following for fˆ ˆ C/ M ˆ ˆ (6 / n ( + n With M R, uch tht both etite re iniu-vrince unbied etite tht i etite with the propertie: ε( ε( ˆ, (7 ε( f ε( f f ˆ, (8 nd the epreion below re inil: ε ( ˆ( ˆ, (9 ( ˆ ( ˆ ε f f f f, (. Unbied etition o obtin n unbied etition of tte nd fult, the tri M ut tif the following two lgebric contrint: We prtition the tri F I n M I In n CG M G M M M M M ( follow : M M F In M In M F hence { M, M F I n, M I n, M F } ( ( in the contrint (3 (4 p n With M R, M R. On ubtituting the contrint eqution ( it cn be given follow M M CG M G M (5 M CG M G + M CG M CG M G + M CG G he etitor f ˆ nd ˆ re unbied if M nd tif the following contrint: M E (6 M (7 M E Γ (8 where E F CG, I p nd Γ [ G ] (9 he innovtion error h the following for Cˆ / F f + CGd+ e (3 where e C / + v (3 / A + F f+ w (3 F p ; the necer nd ufficient Le: Let rn ( condition o tht the etitor ˆ nd fˆ tri i full colu rn, tht i, E ( K ( rn E rn F C G p + q. re unbied M In the net ubection, we propoe to deterine the gin nd M b tifing the unbiedne contrint (7 nd (8. b. fult etition Eqution (3 will be written f E + e (33 d Sine e not hve unit vrince nd doe not tif the uption of the Gu-Mrov theore [7], the let qure olution do not hve iniu-vrinve. Netherle, the covrince tri of e h the following for H ε ee CP / C + R, P/ ε / / where ( (34 For tht, f ˆ cn be obtined b weighted let qure (WLS etition with weighting tri H. heore : Let / be unbied ; the tri H i poitive definite nd the tri E on i full colun rn, then to hve UMV fult etition, the tri gin M i given b ( M E where ( E E H E E H (35 Proof: Under tht H i poitive definite nd n invertible tri L R verifie LL H, o we cn rewrite (3 follow: f L L E + L e (36 d If the tri E i full colun rn, tht i, rn ( E p + q, then the tri ( E H E i invertible. Solving (36 b n LS etition i equivlent to olve (33 b WLS olution: ( ˆ f E H E E H (37

4 uppoe tht ( ( M E H E E H L e In thi w, we conider tht (38 h unit vrince nd (36 cn tif the uption of the Gu- Mrov theore. Hence, (37 i the UMV etite of f. he fult etition error i given b: ( (39 f ˆ f f IM F f M CGdM e hen, the fult error etition i rewritten follow: ( (4 f M e fro eqution (4 we cn clculte f : f Uing (34, the covrince P tri i given b f P ε f f M H M EH E ( ( ( ( (4 c. tte etition In thi prt, we propoe to obtin to obtin n unbied iniu vrince tte etitor to clculte the gin tri M wich will iniie the trce of covrince tri P under the unbiedne contrint (8. héoree: Let E H E be noningulr, then the tte gin tri M b M P/ C H I EE +Γ E (4 ( ( Proof: According to eqution (6 nd fter (4, we cn deduce tht ˆ/ M ˆ/ + M ( Cˆ/ (43 Fro (4, we now tht M In then we hve: ˆ/ ˆ/ + M ( Cˆ/ (44 Uing (44 the tte etition error, given b ˆ (45 ( IM C / M F f( M CGG dm v Conidering (8 nd (45, we deterine P follow: ( / ( ( P IM C P I M C + M R M M H M P/ H M P/ ( ( + (46 So, the optiiztion proble cn be olved uing Lgrnge ultiplier (47 ( / ( / ( trce MC M P C M + P trce M EΓ Λ where Λ i the tri of Lgrnge ultiplier. Setting the derivte of (47 with repect to ( M we obtin: H M CP/ E Eqution (8 nd (48 for the liner te of eqution Λ (48 H E( M C P / E Γ Λ So, if ( E H E (49 i non ingulr, (49 will hve unique olution. B. he filter tie updte For the tie updte, we etrct fro ( the eqution tht depend on + nd ubtitute nd f for their LS etite ˆ / nd fˆ / obtined during the eureent updte. hi ield, Aˆ / + F f + Bu + ( A / + F f + w (5 he correponding LS proble i given b in + Aˆ / F f Bu (5 + W3, where W3, denote the weighting tri which we chooe ε ( A / + F f + w( A / + F f + w 3, W (5 Fro eqution (5, we hve ˆ ˆ + / A ˆ/ + F f + Bu (53 Fro eqution (3, the prior covrince P / h the following for: f P P A P / A F f f t + Q (54 P P ( F f P/ ε f f P/ I ( M C P/ C ( M + M R( M Where i clculted b uing (6 (55 4. EXENDED FILER In thi ection, we ee to etend thi filter to conider the ce where < rng ( F p. o olve thi proble, we ue the e pproch developed b []. If we introduce (3 et (3 in (39, then we will be ble to write the fult error etition in the following for : f I M F f M C G d M C + v (56 ( ( / M CF f M CA ( In M F f n + M CG dm Cw M v ε we define the following nottion : f Φ M F IpΣ, E M CF (57 Auing tht [ ] + where I ( F ( F d E M CG Σ (58 Uing the e technique preented in [9] the epecttion vlue of the f i given b:

5 f f f f ε f Σ f E Σ f+ E ( EΣ f+ + ( E E f d f d f f d E Σ f E d + E E d + + E E E d (59 ( ( f We ue tht Ei Σ i nd E i for i,,, then we obtin: ε f Σ f (6 o obtin n unbied etition of the fult, the gin tri M F Φ, M CF Σ, M hould repect the following contrint : he eqution (6 cn be writen M E d M C G (6 (6 where Φ [ ], E F CF CG Σ (63 Uing (63, we cn deterie the gin tri M follow: M E + E E H E E H where ( (64 he tte etition error i given in the following for : ( / ( ( ( ( + ( I M C w M v (65 I M C M F f M CG G d M v I M C A / + IM C FfM F f M CG G d o obtin n unbied etite of the tte, the gin hould be tif the following contrint: M F, M CF F M CG G M Σ Σ (66 (67 Fro (66-(67 we obtin: M E Γ,where Γ F G Σ (68 Refer to (65, the error tte covrince tri i given in following for ( / ( P IM C P I M C + M C M (74 + (75 / / M P M C M P C M P he gin tri i deterin b iniizing the trce of the covrince tri P uch (67. ( / ( M P C H I EE +Γ E (76 Updting the filter i given b the eqution (53 - (54 5. AN ILLUSRAIVE EXAMPLE We conider the e nuericl eple ued in [4]. he linerized odel of iplified longitudinl flight control te i the following: ( A +Δ A + ( B +Δ B u + F f + w + C + F f + v where the tte vrible re pitch ngle δ z, pitch rte w z nd norl velocit η, the control input i elevtor control ignl. F nd F re the trice ditribution of the ctutor fult f nd enor fult f. he preented te eqution cn be rewritten follow: + A + Bu + F f + Gd + w C + F f + v Where F nd F re the trice injection of the fult vector in the e nd eureent eqution. F F, F F he ter Gd repreent the preter perturbtion in trice A nd B. Gd Δ A +Δ Bu he te preter trice re: A , B.8, C I 3 3, δz wz η, R. ee( 4 Q dig {.,.,. } We inject iultneoul two fult in the te, f 4u( 4u( 6 f u( 3 + u( 7 where u i the unit -tep function. he firt fult f occu in the ctutor nd the econd fult f occu in the enor δz he unnown diturbnce i given b: Δ Δ Δ3 Δb Gd G + u 3 b Δ Δ Δ Δ where Δij nd Δ bij ( i, ; j,, 3 re perturbtion in erodnic nd control coefficient. he trice injection of the fult nd unnown diturbnce re ten follow:.45 G, F.8, F,.83 In thi iultion, the erodnic coefficient re perturbed b ± 5%, i.e Δ ij.5ij nd Δ b ij.5b ij. In ddition, we et u, [ ], P. ee( réel etié

6 Fig. Actul tte nd etited ˆ Fig. Actul fult f nd etited f ˆ Fig3. rce of the covrince tri P Figure nd preent the ctul tte nd fult vector nd their etited vlue obtined b the propoed filter.convergence of the trce of the tte covrince tri nd fult covrince tri re hown repectivel in Fig 3. CONCLUSION In thi pper, the robut filter i developed to obtin n effective tte nd fult etition of liner tochtic te in preence of unnown input. he dvntge of thi filter re epecill iportnt in the ce when we do not hve n prior infortion bout the unnown diturbnce nd fult. An ppliction nd the robutne of the propoed filter h been hown b n illutrtive eple. REFERENCES [] B. Friedlnd: retent of bi in recurive filtering. IEEE rnction Control, vol. 4, pp , 969. [] P. K. Kitnidi: Unbied iniu vrince liner tte etition. Autotic, vol.3, no. 6, pp [3] M. Drouch, M. Zdzini, nd M. Bouteb: Etenion of iniu vrince etition for te with unnown input. Autotic, vol.39, no.5 pp , 3. [4] C.S. Hieh: Robut twi-tge Kln filter for te with unnown input. IEEE rnction on Autotic Control, vol. 45, no., pp [5] C.S. Hieh: Optil iniu vrince filtering for te with unnown input. In procceding of the 6th World Congre on Intelligent Control nd Autotic (WCICA 6, vol. Dlin, Ghin, pp , 6. [6] S. Gillijn nd B. Moor: Unbied iniuvrince input nd tte etition for liner icret-tie te with direct feedthrough. Autotic, vol.43, no.5, pp , 7. [7] S. Gillijn nd B. Moor: Unbied iniuvrince input nd tte etition for liner icret-tie te. Autotic,vol.43, no., pp [8] Y. Cheng, H. Ye, Y. Wng, nd D. Zhou: Unbied iniu-vrince tte etition for liner dicrettie te with unnown input. Autotic, vol.45,no., pp [9] C.S. Hieh: Etenion of unbied iniuvrince input nd tte etition for te with unnown input. Autotic, vol.45, no pp , (9. [] B. lel nd B.H. Fçl : Recurive Let Squre Etition for the joint input-tte etition of liner dicrete tie te with unnown input, 8 th Interntionl Multi- Conference on Ste, Signl & device, Het unii,; [] Niouhh, R. (994. Innovtion genertion in the preence of unnown input: Appliction to robut filure detection, Autotic 3(: [] Keller, J. (998. Fult ioltion filter deign for liner Stochtic te with unnown input, 37th IEEEConference on Deciion nd Control, p, FL, USA, pp [3] Chen, J. nd Ptton, R. (996. Optil filtering nd robut fult dignoi of tochtic te with unnown diturbnce, IEE Proceeding: Control heor Appliction 43(: [4] Chen, J. nd Ptton, R. (999. Robut Modelbed Fult Dignoi for Dnic Ste, Kluwer Acdeic Publiher, Norwell, MA. [5] Ben Hid, F., Khéiri, K., Rgot, J. nd Go, M. (. Unbied iniu-vrince filter for tte nd fult etition of liner tie-vring te with unnown diturbnce, Mtheticl Proble in Engineering, Vol., Article ID , 7 pge. [6] Jouli, H., Keller, J. nd Suter, D. (3. Fult ioltion filter with unnown input in tochtic te, Proceeding of Sfeproce, Whington, DC, USA, pp [7] K.hhéri, F;Ben hid Novel optil recurive filter for tte nd fult etition of liner tochtic te with unnown diturbnce. Int. J. Appl. Mth. Coput. Sci.,, Vol., No. 4, [8] Golub nd Vn Lon, G. Golub & C. vn Lon Mtri coputtion. hird edition. London: he John Hopin Univerit Pre, 996