SUPERVISORY FAULT TOLERANT CONTROL BASED ON DWELL-TIME CONDITIONS

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1 SUPERVISORY FAUOERAT COTRO BASED O DWE-TIME CODITIOS Den Efmov, Jerome Cela, Dav enry Unverty of Boreaux, IMS-lab, Automatc control group 351 cour e la lbératon, Talence, France {Den.Efmov; Jerome.Cela; Dav.enry}@m-boreaux.fr Abtract: The problem of the actve fault tolerant control (FTC) wth reconfguraton mechanm for lnear ytem wth external turbance aree wth applcaton of the upervory control approach. Startng from the well nown n FTC lterature conton for nepenent egn of fault etecton, olaton an fault compenaton ytem we propoe new et of unte conton an the computatonal proceure provng mutual performance of the ytem. The effcency of the approach emontrate on a flght ytem benchmar example. 1. ITRODUCTIO The contnuouly ncreang requrement on afety an relablty of control algorthm lea to the egn of reconfgurable fault tolerant control ytem (ee (Atröm et al., 2000; Blane et al., 2003; Boovc an Mehra, 2002; Chen an Patton, 1999) an the reference theren). Such ytem have ablte to manage aequately faulty tuaton recoverng control capablte qucly n the preence of a fault. The man objectve of FTC to mantan the pecfe performance of a ytem n the preence of fault. Epecal attenton to the FTC egn problem pa n flght an aeronautc applcaton (Cela et al., 2008; enry an Zolghar, 2005). Two approache can be tnguhe n th area: the pave an the actve metho. In the pave approach, a unque control algorthm egne to acheve the gven objectve n healthy a well a n faulty tuaton (the robut control). Unfortunately, guarantee robutne to certan fault contract to qualty preervaton n the nomnal moe. The actve approache react to fault event by ung a reconfguraton mechanm enurng the nomnal performance n fault-free tuaton an ome amble level of performance n faulty tuaton (Zhang an Jang 2002; 2003). The great beneft of the actve FTC approache that the fault tolerance oe not egrae the performance level n normal (fault-free) operatng moe. The actve FTC characterze by on-lne fault etecton an olaton wth poteror fault compenaton va a control reconfguraton mechanm. A ubcla of actve FTC approache bae on precompute control law epenng on the fault whch have been entfe by fault etecton an olaton unt (ee (Starowec an Berjag, 2009) for ntance), that we wll coner n th wor. In the reconfgurable FTC ytem bae on actve approache, the appearance of the control reconfguraton mechanm naturally lea to upervory framewor applcaton to FTC egn problem. To explan the feature of th approach coner the famly of lnear ytem: x = Ax+ Bu+Δ + G, y = Cx, = 1,. (1) n m q where x R, u R, y R an R are tate, control, output an turbance repectvely. It aume that n (1) all matrce A, B, G, C an vector Δ, = 1, are nown, they preent the nomnal fault moel (the fference between the nomnal fault hft Δ an a real one Δ () t can be hen n ). The generc form of (1) allow for coneraton the fferent type of fault, n actuator an n capactor, for ntance. For ome I = {1,..., } the famly (1) contan the fault-free moel. Ang the wtchng gnal : R + I, that etermne the current value of the nex n (1) for all t 0, we obtan the lnear wtche ytem. Then the FTC egn problem can be formulate a the tanar problem of the wtche ytem (1) tablzaton (more precely t neceary to egn a tablzng control u R m for the ytem (1) enurng the turbance attenuaton). The problem of upervory FTC egn ha been wely aree n the lterature (Blane et al., 1997; Blane et al., 2003; Boovc an Mehra, 2002), an many approache have been apple for nepenent optmzaton of the fault etecton, olaton an compenaton ytem. The conton of the wtche ytem (1) tablty equppe wth the fault etecton an compenaton bloc are analyze n the wor (Yang et al., 2009a; 2009b). The both wor coner the cheme for multaneou fault etecton, olaton an compenaton for nonlnear ytem applyng upervory control approach uner aumpton that there no external turbance n the ytem ( = 0 ). The wor (Yang et al., 2009b) nvetgate the cae when the full tate vector x avalable for meaurement, then the proceure of fault etecton an olaton realze wthout atonal flter by rect falfcaton of a canate control after t ubttuton n the cloe loop wth plant. The nex of the plant uppoe to be contant. Uner mlar aumpton n the paper (Yang et al., 2009a) the cae of partal meaurement y x conere, then the oberver bae fault etecton cheme egne. Applcaton of the upervory FTC framewor to nonlnear plant lea to rather complex tablty con- Copyrght by the Internatonal Feeraton of Automatc Control (IFAC) 13717

2 ton obtane n (Yang et al., 2009a; 2009b). The man contrbuton of th paper cont n the approach evelopment that orente on the multaneou egn of the fault etecton, olaton an compenaton ytem for (1). Contrarly the conventonal approache (Blane et al., 1997; Blane et al., 2003; Boovc an Mehra, 2002; enry an Zolghar, 2005) the propoe proceure bae on overall optmzaton of the FTC ytem tablty properte regarle a performance lot n each partcular ubytem. Such a trategy lea to qualty mprovement for the whole ytem. The more etale problem tatement an the ytem ecrpton are gven n the ecton below. 2. SUPERVISORY FTC SYSTEM STRUCTURE To eal wth the poe problem we are gong to apply the approach for upervory control egn propoe n (epanha et al., 2002; 2003; epanha an More, 1995). In accorance wth (epanha et al., 2002; 2003; epanha an More, 1995) the upervory control ytem ha to nclue the mult etmator of the ytem (1) tate (the bloc of the fault etecton an olaton), the control algorthm for each = 1, (the bloc of the fault compenaton) an the wtchng logc (the reconfguraton mechanm) that orchetrate the control actvaton. et u ecrbe all thee bloc conequently. A. Mult etmator Th bloc for the ytem (1) cont n uenberger-type oberver of the followng form: where z = A z + Bu+Δ + ( y Cz ), = 1,, (2) n z R the tate x etmaton for the nex = 1, n (1), are the oberver gan. Aumpton 1. The matrce A C, = 1, are urwtz. Defne the etmaton error e = x z, then the choce of the oberver gan n accorance wth aumpton 1 enure for the matche cae (when nexe n the plant (1) an n the oberver (2) are the ame) e = ( A C) e + G, = 1,, (3) an for the unmatche cae e j = ( Aj jc) ej + ( A Aj) x+ ( B Bj) u+ (4) + ( Δ Δ j) + G, I, j = 1,, j. The ytem (3) aymptotcally table for the cae = 0 an ha boune oluton for any boune turbance, the properte of the error e j are har to etermne from (4) nce they epen on x an u (that may be unboune for a wrong control choce). The property, that the matche etmaton error e tay boune or converge to zero, can be ue for etecton of the nex value n (1). B. Fault tolerant control The FTC bloc equaton can be wrtten a follow: ξ = Rξ + K y, u = S ξ + M y w, = 1,, (5) where ξ R the tate of the ytem (5), the matrce R, K, S, M an the vector w have approprate menon, Bw, = 1,. =Δ Aumpton 2. The matrce A + BMC BS = KC R are urwtz for all = 1,. The matrce ecrbe ynamc of the ytem (1), (5) n the matche cae for all = 1,. The choce of the matrce, = 1, can be alo performe n a way provng ere qualty of turbance attenuaton n the cloe loop ytem (1), (5). For the unmatche cae connecton of the plant (1) wth the nex an the control (5) wth the nex j may reult n unboune repone. C. Supervor m q n The wtchng logc a map : R R R I that generate the wtchng gnal σ ( t ) = ( u, y, z1,..., z ) (6) an agn the current control algorthm from (5), whch ha to be actvate n cloe loop wth the plant (1). In the eal cae σ( t) (the control nex match the plant one). The upervor ha to enure rght contnuty of the gnal σ () t (epanha et al., 2002; 2003; epanha an More, 1999) (the gnal ha to be pecewe contnuou an between any two jump a tme elay houl ext). The egn of the map ffer epenng on operaton conton an the bloc (2), (5) properte. D. Problem tatement The ytem (2) reponble for the fault etecton an olaton, then next the ytem (5) realze the fault compenaton. The aumpton 1 an 2 ecrbe both mentone properte of the ytem (2) an (5). Uner aumpton 1 there ext a convergng oberver n (2), that may olve the etecton problem. The aumpton 2 tate that beng etecte there ext a controller n (5) compenatng the fault. Typcally n FTC theory thee bloc are egne nepenently optmzng ome performance functonal (enry an Zolghar, 2005; Zhang an Jang, 2008). The oberver (2) are egne to maxmze ther entvty to a partcular fault an robutne agant turbance. The control n (5) are calculate to enure 2 / performance. A t well nown, the optmalty of the ubytem oe not mply the ame property for the whole ytem. In our cae the optmal properte are crtcally epenent on the upervor (6). In th wor we are gong to preent an approach to the y

3 tem (1), (2), (5), (6) egn orente on the mutual performance optmzaton of th wtche ytem. For th purpoe we have to chooe a charactertc of the hybr ytem (1), (2), (5), (6) to be optmze n parallel wth the conventonal one ue for the oberver (2) an control (5) egn. For th purpoe n th wor we tae the mnmal amble tme between wtche. It well nown fact (berzon, 2003) that wtchng among table lnear ytem oe not lea to ntablty f the elay between wtche are bg enough (the mnmum elay between wtche calle well-tme (berzon, 2003)), th why the trategy orente on thee elay ncreang frequently apple n practce to enure tablty n wtche ytem. owever, for FTC ytem uch approach not amble, nce t reult n the tme of fault etecton an olaton ncreang. Atonally, t may lea to more long pero of a wrong control actvaton for the faulty plant. The both hortage are namble for FTC ytem from practcal pont of vew. Then the mnmzaton of the well-tme value for the upervory FTC ytem (1), (2), (5), (6) loo reaonable. The conton of the upervory FTC ytem (1), (2), (5), (6) tablty for the cae = 0 are gven n ecton 3. Ther evelopment for the cae 0 preente n ecton 4. The nfluence of the well-tme value on the ytem performance alo evaluate n ecton 3 an 4. ext, bae on thee reult new computaton proceure for the conton verfcaton an the FTC ytem (1), (2), (5), (6) ynthe formulate n ecton 5. Applcaton to a flght afety control ytem conere n ecton TE CASE WITOUT DISTURBACES In th ecton we aume that ( t ) = 0 for all t 0. In th cae for each fxe plant nex I accorng to (3) there t ext C > 0 an η> 0 uch that e( t) C e (0) e η for all t 0. Recall, that the gnal Ce j ( t ), j = 1, are the only one avalable for egn purpoe. Defne the wtchng logc a follow: t+ 1 = argnf t t { ( )( ) ( ), +τ Ce D σ t t > Ce j t (7) j = 1,, j σ ( t ) }, 0 ; t 0 = 0 ; σ ( t ) = argmn 1 j Ce j( t ), 0 ; (8) σ ( t) =σ ( t ) for all t t < t + 1, 0, (9) where t, 0 are ntant of wtche, τ D > 0 welltme contant. For t 0 = 0 the wtchng gnal ntalze a σ (0) = argmn 1 j Ce j(0), the ame rule (8) ue for all tme ntant t. The tme ntant of wtch t + 1 calculate n (7) a the frt tme ntant after t +τ D when the output etmaton error of an oberver become maller than the current one ue for control. The well-tme τ D enure tme elay between any two wtche an abence of chatterng. The wtchng logc (7) (9) mlar to the hytere one ue n (epanha et al., 2002; 2003). Theorem 1. et aumpton 1, 2 hol, ( t ) = cont an () t = 0for all t 0. Then there ext τ D > 0 uch that for ( 1) n any ψ (0) R + +, ψ = [ ξ x z1... z ] the oluton of the ytem (1), (2), (5), (7) (9) poe the etmate / ( ) t ψ t ν D e μ τ ψ(0) +υ δ [0, t) +ϖmax 1 for all t 0, for ome ν > 0, μ > 0, ϖ > 0 an υ > 0, where C[ e( t) eσ( t )( t)] f t [ t, ) t +τd δ( t) = Ce( t) < Ce σ( t )( t); 0 otherwe. All proof are exclue ue to pace lmtaton. The proof ea bae on the obervaton that the wtche ytem ynamc (1), (2), (5) can be preente n the new coornate T ζ = [ z ξ x z1... z ] (the lat part z 1... z oe not T contan z ) n the form: ζ = W, ζ + V, Ce + ι, + G, (10) where V, = [( BM+ ) K ( BM+ ) ( B1M + 1)......( B M + ) ], ι, = [00( Δ Bw ) ( Δ1 Bw 1 )...( Δ Bw ) ], G = [00G 0...0] an the matrx W, left bloc trangular (all bloc above the man agonal are zero) an the bloc on the man agonal are, A C, A1 1C,, A C. Snce all bloc on the man agonal are urwtz, the matrx W, ha the ame property. To calculate τ D note that there ext permutaton tranformaton matrce T j an E j, j = 1, provng ψ = Tj ζ j an ej = Ejψ where ψ = [ ξ x z1... z ] (the lat part z 1... z contan all term), then for any actve control I from (10) we have: ψ = TW, T ψ + TV, Ce + Tι, + TG = = T[ W, T + V, CE] ψ + Tι, + TG for all t [ t, t + 1 ). Owng the tanar reult on welltme wtche ytem tablty (berzon, 2003; More, 1995; Xe et al., 2001; Efmov et al., 2008) the value of τ D houl be taen to atfy τ D = max 1 j { αj, ln( λβ j, )}, (11) where 0<λ< 1 a egn contant, α j, the mnmal n norm real part of the matrx W j, egenvalue, an β j, = up t 0 exp( TW j j, T j t ) (the norm of a matrx 13719

4 compute a t maxmum ngular value). The reult mean that f the ytem not etectable wth repect to the output Ce σ( t ) ( t ), then only practcal aymptotc tablty can be enure n general cae. The gnal δ the output fault etecton error (th gnal not avalable for meaurement), the ampltue of δ follow by the multetmator (2) properte an etectablty of the plant (1). Of coure, the theorem preent the wort cae etmate. ow let u rop the aumpton that () t = cont for all t 0. Suppoe, that t () = T ( r ), t [ Tr, T r + 1 ) an T ( r ) I for all r 0,.e. the true moel of the plant (1) can be fferent on fferent nterval [ Tr, T r + 1 ) an t ( ) a pecewe contant gnal, then retrctng the rate of t ( ) varaton we can ubtantate the overall ytem tablty. Further we wll aume that there ext alo Τ D > 0 uch that Tr+ 1 Tr Τ D for all r 0, when Τ D efne the amble rate of fault n the ytem Corollary 1. et aumpton 1, 2 hol, Tr+ 1 Tr Τ D for all r 0 an () t = 0 for all t 0. Then there ext ( 1) n Τ D > 0 an τ D > 0 uch that for any ψ (0) R + + the oluton of the ytem (1), (2), (5), (7) (9) poe the etmate t / ( t) e μ ψ ν Τ D ψ(0) +υ δ [0, t) +ϖ max 1 for all t 0, for ome ν > 0, μ > 0, ϖ> 0 an υ> 0. An example of the well-tme Τ D choce a follow: Τ D = max 1 j { τdμj ln( λν j )}, where 0<λ< 1 a egn contant an τ D > 0, μ j > 0, ν j>0, j = 1, come from Theorem 1. Thu, uner conton of corollary 1 the ytem (2), (5), (7) (9) realze a reconfgurable fault tolerant control algorthm for the plant (1). The accuracy of the fault tolerant control realze by (2), (5), (7) (9) epen on ablty of the mult-etmator (2) to etect the correct current moe of the ytem (1) (.e. on the ampltue of the error δ ). 4. TE CASE WIT DISTURBACES The prevou ecton reult can be ealy extene to the cae () t 0, t 0. We wll aume that,.e. e up t 0{ ( t ) } <+. The man obtacle n th cae that for any plant nex I accorng to (3) there ext C 1 > 0, C 2 > 0 an η> 0 uch that ηt e( t) C1 e(0) e + C2 for all t 0. Thu the etmaton error even for the matche oberver n (2) oe not converge to zero. Theorem 2. et aumpton 1, 2 hol an ( t ) = cont for all t 0. Then there ext τ D > 0 uch that for any ( 1) n ψ (0) R + + an the oluton of the ytem (1), (2), (5), (7) (9) poe the etmate / ( ) t ψ t ν D e μ τ ψ(0) +υ { δ [0, t) + + [0, t) } +ϖmax 1 for all t 0, for ome ν > 0, μ > 0, ϖ > 0 an υ > 0, where δ ( t ) efne n Theorem 1. Corollary 2. et aumpton 1, 2 hol an Tr+ 1 Tr Τ D for all r 0. Then there ext Τ D > 0 an ( 1) n τ D > 0 uch that for any ψ (0) R + +, the oluton of (1), (2), (5), (7) (9) poe the etmate t / ( t) e μ ψ ν Τ D ψ( 0) +υ { δ [0, t ) + + [0, t) } +ϖ max 1 for all t 0, for ome ν > 0, μ > 0, ϖ> 0 an υ> 0. Therefore, the appearance of the turbance oe not change the properte of the well-tme upervor (7) (9). The ytem (1), (2), (5) n th cae emontrate proportonal evaton from the unperturbe behavor. Secton 3 an 4 preent the tablty conton an the expreon for well-tme computaton, n other wor they are evote to the FTC ytem analy, the ynthe phae preente n the next ecton. 5. FTC DESIG In th ecton we are gong to propoe the computaton proceure for the upervory FTC ytem (1), (2), (5), (6) ynthe fnng a trae-off between well-tme value optmzaton an the etmator (2) or the FTC (5) entvty/robutne. The aumpton 1 fxe the tablty property of the etmator. In practcal applcaton atonal requrement are mpoe on the matrce to ncreae entvty to fault an robutne wth repect to turbance. Typcally (enry an Zolghar, 2005), the matrce are erve a oluton of the followng / 2 optmzaton problem: = arg mn W (, ) / W (0, ) (12) for mn{re[ λ ( A + C ) ]} < 0, = 1,, where W (, ) = ( I A + C) G for = the tranfer functon for the etmaton error e from the nput, W (, ) = ( I A + C ) correpon for = 0 to aymptotc gan between the error an atve fault n (3), λ( A ) the vector of egenvalue for a matrx A, the norm n (12) unertoo n or 2 ene. Then the numera

5 tor n (12) W (, ) evaluate the ytem robutne an enomnator W (0, ) etmate the enblty to fault. The FTC (5) egn (enry an Zolghar, 2005) typcally performe applyng QR approach or followng mlar / 2 optmzaton: ( R, K, S, M ) argmn W (, ) (13) = for mn{re[ λ ( ) ]} < 0, = 1,, 1 where W (, ) = ( I ) G, G = [ G 0] for = the ytem (1), (5) tranfer functon from the nput T to the tate [ x ξ ]. The contrane optmzaton problem (12), (13) prove the nepenent oluton for the mult-etmator (2) an FTC (5) egn. The prncpal novelty of the preent paper cont n propoton of mutual reegn of (2), (5), (6) to enure global performance an tablty of the reulte wtche ytem tang nto account properte of the upervor. A a global performance crtera t wa propoe to ue the value of the well-tme τ D (that etermne the fatet amble wtche pero among the fault tolerant control law (5)) an Τ D (that efne the amble rate of fault n the ytem). By amble we mean that wtchng wth thee well-tme o not etroy the ytem tablty. The value Τ D erve a a complementary charactertc of the ytem, but the value τ D volaton may lea to the ytem (1), (2), (5), (6) performance egraaton or/an tablty lo. The well-tme value τ D gven n (11), accorng to α j,, β j, efnton τ D =τd( 1,..., ; 1,..., ), then the propoe optmzaton problem to be olve can be formulze a follow (by the matrx efnton t epen on R, K, S, M, = 1, ) (, R, K, S, M ;...;, R, K, S, M ) = (14) arg mn (,..., ;,..., ), =,..., ;,..., J mn{re[ λ ( A + C ) ]} < 0, mn{re[ λ ( ) ]} < 0, = 1,, (15) J (,..., ;,..., ) = τ (,..., ;,..., ) D 1 1 2max 1 { W (, ) / W ( 0, ) } 3max 1 { W (, ) }, where (14) efne the optmzaton crtera an (15) gve the contrant, 0, = 1, 2, 3 are the egn parameter. > Another varant of th problem formulaton bae on the max maxmum amble value of well-tme τ D ntroucton. max It aume that f τ D > τ D, then the ytem reacton tme (the mnmal tme between wtche an the maxmum tme of a wrong controller actvty) not acceptable from the ytem performance pecfcaton. Thu, (, R, K, S, M ;...;, R, K, S, M ) = (16) arg mn (,..., ;,..., ), =,..., ;,..., J mn{re[ λ ( A + C ) ]} < 0, mn{re[ λ ( ) ]} < 0, = 1,, (17) τ max D ( 1,..., ; 1,..., ) τd, (18) J (,..., ;,..., ) = = max { (, ) / ( 0, ) } W W 2 max 1 { W (, ) }, where (16) efne the optmzaton crtera an (17), (18) tate for the contrant to be atfe, 1 an 2 are potve egn parameter. et u tre, that (14), (15) an (16) (18) belong to the cla of nonlnear optmzaton problem. Depte the conere ytem (1) an the oberver (2) wth the control (5) are lnear, the cloe by the upervor (6) ytem wtche an, hence, nonlnear. Conequently, any optmzaton problem orente on mutual (2), (5), (6) egn an global performance optmzaton become a nonlnear one. 6. APPICATIOS A fourth orer F-8 arcraft moel (Zhang an Jang, 2008) ue to emontrate the avantage of the propoe approach wth the tate pace vector x = [ prβφ] T ( p, r, β, φ repreent the roll rate, the yaw rate, the elp an the ban angle repectvely) an the control u = [ δ1 δ2] T ( δ 1 an δ 2 are the two aleron eflecton on the wng). FTC ytem egn for the cae of tuc actuator aree. The ytem matrce for the fault free cae n (1) have form: A 1 = , B 1 = , C = , Δ 1 = The fault correpon to cenaro wth tuc actuator, thu ; B 3 = , Δ 3 = α, B 2 =, Δ 2 = α2 where α 2 = π /6, α 3 = π /6 are the angle of tuc actuator (t aume that the actuator can be tuc n the maxmum evaton poton), the matrce A2 = A3 = A 1. It requre 13721

6 to egn FTC ytem (2), (5), (6). Three control law an oberver have to be egne for the mult-etmator (2) an for the ban of control (5). The cae of nepenent egn performe n accorance wth (12), (13) eparately. The reulte ytem ha τ D = 2.01 [ec] an Τ D = 32.3 [ec], that clearly not acceptable for a flght ytem (the crah can happen for 2 ec wth tuc actuator). Applcaton of the preente egn proceure gve the et of control an oberver wth τ D = 0.4 [ec] an Τ D = 1.14 [ec]. The reult of the ytem mulaton are hown n Fg. 1. In Fg. 1,a the ban angle φ trajectory wth the t reference φ r are plotte, the gnal t () an σ () t are preente n Fg. 1,b, the output of the etmator ε = Ce, = 1, 2,3 are hown n Fg. 1,c. 1.2 a. b. 1 ϕ c. ϕr ε3 ε ε 2 Fg. 1. Smulaton reult. For the mulaton the oberver pole have been choen to mnmze etecton elay an the actual etecton tme are proportonal to τ D (ee Fg. 1,b). owever, ue to trct ntablty of the plant ubjecte by tuc fault, even uch mall etecton tme reult n gnfcant evaton of the regulate varable from t reference (Fg. 1,a). 7. COCUSIO The problem of the actve fault tolerant control for lnear ytem wth external turbance olve wth applcaton of the upervory control approach. Startng from the well nown nepenent egn of fault etecton, olaton an compenaton ytem we propoe the new egn proceure provng overall performance of the ytem. The effcency of the approach emontrate on F-8 flght ytem benchmar example. REFERECES Atröm K., Alberto P., Blane M., Ior A., Schaufelberger W., Sanz R. (2000). Control of Complex Sytem. σ Sprnger Verlag. Blane M., Knnaert M., unze M. et Starowec M. (2003). Dagno an fault tolerant control. Sprnger, ew Yor. Blane M., Iza-Zamanaba R., Bogh S.A., unau C.P. (1997). Fault-tolerant control ytem a holtc vew. Control Eng. Practce, 5(5), pp Boovc J.D., Mehra R.K. (2002). Falure Detecton, Ientfcaton an Reconfguraton n Flght Control. Fault Dagno an Fault Tolerance for Mechatronc Sytem, Sprnger, ew Yor. Chen J., Patton R.J.(1999). Robut Moel-Bae Fault Dagno for Dynamc Sytem. Kluwer Acaemc Publher, orwell, MA. Cela J., enry D., Zolghar A. an Goupl P. (2008). Development of an Actve Fault Tolerant Flght Control Strategy. AIAA Journal of Guance, Control an Dynamc, 31(1), pp Efmov D.V., Panteley E., ora A. (2008). On Input-to- Output Stablty of Swtche onlnear Sytem. Proc. 17th IFAC WC, Seoul, Korea. enry D., Zolghar A. (2005). Degn an analy of robut reual generator for ytem uner feebac control, Automatca, 41(2), pp epanha J.P., More A.S. (1999). Certanty equvalence mple etectablty. Sytem Control ett., 36, pp. 13. epanha J.P., berzon D., More A.S. (2002). Supervon of Integral-Input-to-State Stablzng Controller, Automatca, pp epanha J.P., berzon D., More A.S. (2003). ytere- Bae Supervory Control of Uncertan near Sytem, Automatca, pp berzon D. (2003). Swtchng n Sytem an Control. Brhäuer, Boton. More A.S. (1995). Control ung logc-bae wtchng. In: Tren n control (A. Iory (E.)), Sprnger-Verlag, pp Starowec M., Berjag D. (2009). Pave/actve fault tolerant control for TI ytem wth actuator outage. Proc. European Control Conference, Buapet. Xe W., Wen C., Z. (2001). Input-to-tate tablzaton of wtche nonlnear ytem. IEEE Tran. Automat. Control, 46, pp Yang., Jang B., Cocquempot V. (2009a). A fault tolerant control framewor for peroc wtche non-lnear ytem. Int. J. Control, 82(1), pp Yang., Jang B., Starowec M. (2009b). Supervory fault tolerant control for a cla of uncertan nonlnear ytem. Automatca, 45, pp Zhang Y., Jang J. (2002). Graceful performance egraaton n actve fault tolerant ytem. Proc. IFAC congre 2002, Barcelona. Zhang Y., Jang J. (2003). Fault tolerant control ytem egn wth explct coneraton of performance egraaton. IEEE Tran. Aerop. Electron. Syt., 39(3), pp Zhang Y.M., Jang J. (2008). Bblographcal Revew on Reconfgurable Fault-Tolerant Control Sytem. IFAC Annual Revew n Control