Research Article Fault Tolerant Control for Uncertain Time-Delay Systems with a Trajectory Tracking Approach

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1 Mahemaical Problems in Engineering Volume 215 Aricle ID pages hp://dx.doi.org/1.1155/215/ Research Aricle Faul Toleran Conrol for Uncerain Time-Delay Sysems wih a Trajecory Tracking Approach ShiLei Zhao Hong Guo and YuPeng Liu School of SofWare Harbin Universiy of Science and Technology Harbin 1536 China Correspondence should be addressed o ShiLei Zhao; zhaosl121@126.com Received 13 Sepember 214; Revised 31 December 214; Acceped 2 January 215 Academic Edior: Vincen Cocquempo Copyrigh 215 ShiLei Zhao e al. This is an open access aricle disribued under he Creaive Commons Aribuion License which permis unresriced use disribuion and reproducion in any medium provided he original work is properly cied. This paper sudies he problem of faul oleran conrol by rajecory racking for a class of linear consan ime-delay sysems. The aim is o design a conrol law by considering he faul deeced by he observer o make he fauly sysem rack he reference model even if fauls occur. By considering wo kinds of acuaor fauls one consan and anoher ime-varying he corresponding proporional inegral observers and acive FTC conrol laws are designed respecively. Sae racking error sae esimaion error oupu esimaion error and faul esimaion error are combined ino a descripor sysem. Based on Lyapunov-Krasovskii funcional approach sabiliy problems of he descripor sysem are easily solved in erms of he Linear Marix Inequaliies (LMI). Finally a numerical example is considered o prove he effeciveness in boh cases. 1. Inroducion Over he pas few decades problems of faul oleran conrol wellknownasftcindynamicsysemshavearacedlosof aenion [1 2]. FTC has been developed o preserve he sysem sabiliy and mainain accepable performances in case of fauls occurring. The exising FTC sraegies can be divided ino wo caegories. The firs one named as he passive FTC reas he faul as uncerainy; herefore i involves no faul deecion and esimaion (see [3 5]).Thesecondonehe acive FTC differs from he passive FTC in ha i requires a faul deecion and isolaion (FDI) block o deec isolae andesimaefaulswhichareusedocompensaehefaul and ensure an accepable sysem performance (e.g. [6 8]). AsheobainedfaulinformaionisusedheaciveFTCis more reliable. On he oher hand ime-delay is anoher facor ha can degrade sysem performance; i is a buil-in feaure in many engineering sysems. The presence of ime-delay ogeher wih fauls could cause sysem o be insable easily. Therefore researching on FTC design of ime-delay sysem has grea pracical and heoreical significance [9]; his challenging opic has ignied he ineres of some auhors. For example [1 11] provide a kind of faul deecion mehod based on an ieraive learning observer for nonlinear consan sae delay sysems. Reference [12] designs H faul deecion filers for muliple ime-delay discree-ime sysems. Based on a swiched descripor observer approach [13] deals wih sensor faul esimaion and compensaion problems of ime-delay swiched sysems. In [14] for boh addiive and muliplicaive fauls a robus faul deecion and isolaion scheme is proposed for uncerain coninuous linear sysems wih discree sae delays. In [15] a faul deecion filer is invesigaed for a class of discree-ime swiched linear sysems wih imevarying delays so ha he differen esimaion errors are minimized. In [16] some adapive faul diagnosis observers (AFDO) are designed o deal wih fas faul esimaion and accommodaion problems for ime-varying delay sysems. Recenly here is also an acive FTC approach based on rajecory racking developed o solve he FTC problem. Thisschemeiscomposedoffaulysysemreferencemodel observer and conroller and is aim is o design a conrol law by considering acuaor fauls deeced by observers and o make he fauly sysem saes rack he reference model saes which are no effeced by fauls [17 19]. This paper is abou o develop a sraegy for linear consan ime-delay sysems based on rajecory racking. The moivaion of his paper mainly sems from wo facs:

2 2 Mahemaical Problems in Engineering (1) some FTC schemes of ime-delay sysems are obained [2 21] bu less work which sudies on FTC problems employs he descripor redundancy propery and solves he faul isolaion esimaion and FTC problems ogeher; (2) here is some work addressing FTC designs based on rajecory racking which focused on linear ime invarian (LTI) sysem wihou ime-delay bu few work is focused on FTC of ime-delay sysems. Our work will exend earlier resuls of faul esimaion using rajecory racking o he ime-delay sysems. In his paper our purpose is o sudy he FTC design problem for linear sae ime-delay sysems subjeced o consan or ime-varying fauls. The main idea is o design an acive FTC conroller and PI observer and o use he virual dynamic [22 24] in boh acive FTC law and oupu esimaion error expression o urn he problem under sudy ino a descripor sysem. By using he Lyapunov-Krasovskii funcional approach he sabiliy of he descripor sysem has been proved. The advanages of he proposed mehod isalsobasedonheabovewofacs:(1) he inroducion of rajecory racking can ensure he racking of fauly sysems o reference models which could guaranee an accepable sysem performance even if fauls occur; (2) he descripor redundancy propery can avoid crossed erms in he LMI andhendecreasehenumberoflmicondiionsandconsequenly relax he conservaism [17]. This paper is organized as follows. In he nex secion he sysem under sudy and he acive FTC scheme based on rajecory racking are presened. In Secion 3 FTC design for linear sae ime-delay sysems affeced by consan faul wihou uncerainies is esablished. Then some FTC design for linear sae ime-delay sysems affeced by ime-varying fauls wih uncerainies is given. In he las secion a numerical example for consan fauls wihou uncerainies and ime-varying fauls wih uncerainies is considered o illusrae he applicabiliy and effeciveness of he proposed approaches. Noaions. In a block marix he noaion sands for he erms induced by symmery. The superscrip T denoes marix ranspose sym(a) denoes A+A T anddiag{ } sands for a block-diagonal marix. The following lemma is needed o provide LMI condiions. Lemma 1. For any marices X Y andσ() wih appropriae dimensions and Σ T ()Σ() < I and for any posiive real number ifollowsha X T Σ T () Y+Y T Σ () X X T X+ 1 Y T Y. (1) 2. Problem Formulaion Consider he following sysem wihou fauls corresponding o a reference model: x () =Ax() +Bu() +A d x () y=cx() +C d x () (2) where x() R n is he sae vecor u() R m is he inpu vecor and y() R p is he oupu vecor. A B A d CandC d are known consan real marices of appropriae dimensions. h is he sae delay and h R + is a consan real number. Consider he fauly sysem given by x f () = Ax f () + A d x f () + Bu f () + Bf () y f () = Cx f () + C d x f () where x f () R n u f () R m y f () R p andf() R m are he fauly sae vecor he faul oleran conrol vecor he fauly oupu vecor and he faul vecor affecing he sysem behavior. And he uncerainies of sysem (3) are defined by (3) X () =X+ΔX() X {AA d BCC d } (4) where ΔA ΔA d ΔB ΔC and ΔC d are ime-varying unknown marices describing he bounded model uncerainies defined by ΔX () =M x F () N x (X x) ={(A a) (A d ad)(b b) (C c) (C d cd)} where M a M ad M b M c M cd N a N ad N b N c andn cd are knownconsanrealmariceswihappropriaedimensions and he marix funcion F() is bounded by (5) F () F () T I. (6) In order o esimae he faul vecor f() which is required by he FTC scheme and he fauly sysem saes x f () we consider he PI observer as follows: x f () =A x f () +A d x f () +Bu f () +B f () +H 1 (y f () y f ()) f () =H 2 (y f () y f ()) y f () =C x f () +C d x f () where H 1 R n p and H 2 R r p are he observer s gain marices o be deermined. 3. Faul Toleran Conroller Design In his secion wo cases are considered according o he characerisics of fauls. Firs we assume ha he faul is a consan one and here are no uncerainies in fauly sysem (6). Second assume ha he faul is a ime-varying one and here are uncerainies in he fauly sysem. The FTC design scheme is illusraed in Figure 1. The objeciveofhisworkisoensureherackingofhefauly sysem o he nominal one. In oher words he scheme is o design FTC law and observer gain marices o minimize he differences beween he fauly saes of (3) and he reference saes given by model (2) he fauly sysem saes and he observer saes he fauly sysem esimaion oupu and (7)

3 Mahemaical Problems in Engineering 3 u() u f () Sysem f() y f () The faul esimaion error dynamics e f () is expressed as follows: e f () = f () f () = H 2 e y (). (12) u c () Observer f() Conroller Reference model x f () x f ( h) + x() x( h) Figure 1: The conroller scheme. he reference model oupu and he nominal inpu and he FTC inpu plus he faul. From he FTC scheme of Figure 1hefollowingFTClaw srucureisproposed[17]: u f () =u() +K 1 (x () x f ()) f () +K 2 (x () x f ()) where K 1 K 2 R m n arehesaefeedbackgainmariceso guaranee he sabiliy of he fauly sysem even if he faul occursandminimizehedifferencebeweenhefaulysysem and he reference one Firs Case: Consan Faul wihou Uncerainies. I is here considered ha he faul which affecs he sysem acuaor is aconsanbias.obviouslyiisaspecialcasehahefaul saisfies f () =. (9) In he following par o ensure he racking we firs give sae error faul esimaion error oupu error and racking error and he difference beween nominal inpu and FTC inpu plus he faul respecively by e p () =x() x f () e s () =x f () x f () e f () =f() f () e y () =y f () y f () e u () =u() (u f () +f()). (8) (1) By using formulae (2) (3) (7)and(1)hedynamicsof e p () and e s () are given by The oupu esimaion error e y () can be wrien in he form of e y () =Ce s () +C d e s (). (13) In order o organize he above equaions ino he form of descripor sysems we can inroduce a virual dynamics in he oupu esimaion error; his laer can be rewrien as e y () =Ce s () +C d e s () e y (). (14) By adding and subracing K 1 x f () K 2 x f ( h)andf() in (8) and using (1)onecanobain e u () =K 1 e p () +K 1 e s () +K 2 e p () +K 2 e s () +e f () +e u (). (15) The combinaions of (11) (12) (14) and(15) yield a descripor sysem expressed as follows: where E=diag{I 2n+m m+p } Ee () = Ae () + A d e () (16) e () =(e T p () et s () et f () et u () et y ())T A B A B H 1 A = H 2 [ K 1 K 1 I I ] [ C I ] A d A d A d =. [ K 2 K 2 ] [ C d ] The main proposed resul can now be esablished [25]. (17) Theorem 2. The racking error e p ()hesaeerrore s ()and faul esimaion error e f () asympoically converge o zero if here exis some marices P 1 > P 2 > P 3 >andmarices Y 2 and Y wih appropriae dimensions such ha he following inequaliies hold: e p () =Ae p () +Be u () +A d e p () e s () =Ae s () +Be f () H 1 e y () +A d e s (). (11) Ω Y [ Y T P ] 3 < (18) [ h ]

4 4 Mahemaical Problems in Engineering where he expression of Ω is shown as follows: Ω=W p P 2 W T p +hw p1e T P 3 EW T p1 W php 2 W T ph + sym [W p P 1 W T p1 +YEWT P +YEWT ph Proof. Choose a Lyapunov-Krasovskii funcional candidae as V () =V 1 () +V 2 () +V 3 () (22) V 1 () =e T () E T P 1 e () +(Y 2 A HC 1 XC 2 )W T p +(Y 2 A d KC 3 )W T ph ] W p =[I 2n+2m+p ] W ph =[ I 2n+2m+p ] W p1 =[ I 2n+2m+p ] Y 2 =[ Y T 2 Y T 2 Y T 2 ]T X=[ X T X T X T ] T V 2 () = h +V e T (ω) E T P 3 Ee (ω) dω dv V 3 () = e T (ω) P 2 e (ω) dω where P 1 saisfies E T P 1 =P 1 E. The ime derivaives of V() are given by V=2e T () P 1 Ee () +he T () E T P 3 Ee () (23) X =[ X T 41 ]T K = [ K T K T K T ] T K =[ K T 42 ]T H=[ H T H T H T ] T H =[ H T 21 HT 32 ]T C 1 =[ I] C 2 =[I I ] C 3 =C 2 A B A d A B A d A = A d = [ I I ] [ ] [ C I] [ C d ] Y 11 Y 12 Y 2 = Y 13 [ Y 16 Y 14 Y 18 ] [ Y 17 Y 15 ] (19) where Y 12 Y 13 andy 14 are inverible marices and Y 11 Y 15 Y 16 Y 17 andy 18 are slack marices. The observer and conroller gains are hen compued by K 1 =(Y 14 ) 1 X 41 K 2 =(Y 14 ) 1 K 42 H 1 =(Y 12 ) 1 H 21 H 2 =(Y 13 ) 1 H 32. (2) Remark 3. I should be menioned ha he free weighing marix Y 2 has he srucure of Y 2 =[ Y T 2 Y T 2 Y T 2 ]T.Buhe conservaism would increase; in order o obain a racable marix condiion we can adop he mehod by defining α 1 I Y 2 Y 2 = [ α 2 I ] [ Y ] 2 (21) [ α 3 I] [ Y 2 ] where α 1 α 2 andα 3 are he real number. By choosing hese scalars appropriaely he conservaism canno increase much. This mehod has been used by [23]. e T (ω) E T P 3 Ee (ω) dω + e T () P 2 e () e T () P 2 e (). (24) According o he Newon-Lebniz formula and closedloop sysem equaion he following equaions are rue: Π 1 =ξ T () YE [e () e() e (ω) dω]= (25) Π 2 =ξ T () Y 2 [ Ae () A d e () Ee ()] = where Y Y 2 are appropriae dimensioned marices and ξ() = [e T () e T ( h) (Ee()) T ] T. Adding 2Π 1 2Π 2 andhξ T ()YP 1 3 YT ξ() and subracing ξt ()YP 1 3 YT ξ()dωrespecivelyin(24) yield (26) which is shown as follows: V () =2e T () E T P 1 e () +he T () E T P 3 Ee () e T (ω) E T P 3 Ee (ω) dω + e T () P 2 e () e T () P 2 e () +2ξ T () YE [e () e() e (ω) dω] +2ξ T () Y 2 [ Ae () + A d e () Ee ()] =2e T () P 1 Ee T () +he T () E T P 3 Ee () e T (ω) E T P 3 Ee (ω) dω + e T () P 2 e () e T () P 2 e () +2ξ T () YE [e () e() e (ω) dω] +2ξ T () Y 2 [ Ae () + A d e () Ee ()] +hξ T () YP 1 3 YT ξ () ξ T () YP 1 3 YT ξ () dω

5 Mahemaical Problems in Engineering 5 =ξ T () Ωξ () +hξ T () YP 1 3 YT ξ () [ξ T () Y+ e T (ω) E T P 3 ] P 1 3 [Y T ξ () +P 3 Ee (ω)]dω. (26) Then we know from (26) ha Ω+hYP 1 3 YT < which guaranees V() is nonposiiveness for nonzero ξ(). One can always find a sufficienly small ε > such ha V() ε e() 2 and he asympoic sabiliy of he sysem is proved Second Case: Time-Varying Faul wih Uncerainies. In his par we consider sysem (3) wih he uncerainies and he faul being ime-varying one. Here we modify PI observer slighly as follows: x f () =A x f () +A d x f () +Bu f () +B f () +H 1 (y f () y f ()) f () =H 2 (y f () y f ()) f () y f () =C x f () +C d x f (). The dynamics of e p () e s ()ande f () are given by e p () =Ae p () +ΔAe p () +Be u () +ΔBe u () +A d e p () ΔAx() +ΔA d e p () ΔA d x () ΔBu() e s () = ΔAe p () +Ae s () +Be f () ΔBe u () ΔA d e p () +A d e s () +ΔAx() H 1 e y () +ΔBu() +ΔA d x () e f () = f () f () = e f () H 2 e y +f() + The oupu esimaion error e y () is given by e y () = ΔCe p () +Ce s () ΔC d e p () +C d e s () +ΔCx() +ΔC d x (). The subsiuion of e u () in (8) implies e u () = K 1 e p () K 1 e s () K 2 e p () K 2 e s () e f (). (27) f (). (28) (29) (3) Equaion (29) and (3) can be rewrien as e u () =K 1 e p () +K 1 e s () +K 2 e p () +K 2 e s () +e u () +e f () e y () = ΔCe p () +Ce s () e y () ΔC d e p () +C d e s () +ΔC d x () +ΔCx(). (31) The combinaion of (28) and (31) leads o he following descripor sysem: Ee () =( A+Δ A) e () +( A d +Δ A d )e() +( B+Δ B) V () (32) where E A d and e() have been given above and he expressions of A Δ A B Δ B Δ A d andv() are shown as follows: A B A B H 1 A = I H 2 [ K 1 K 1 I I ] [ C I ] ΔA ΔB ΔA ΔB Δ A = [ ] [ ΔC ] B = I I [ ] [ ] ΔA ΔA d ΔB ΔA ΔA d ΔB Δ B = [ ] [ ΔC ΔC d ] ΔA d ΔA d Δ A d = [ ] [ ΔC d ] V () =[x T () x T ()u T () f T () f T ()] T. (33) The condiion ensuring he sabiliy of he descripor sysem (32) and he aenuaion level γ > from he perurbaion-like erm V() o he error dynamic e() are provided in he following heorem. Theorem 4. Sysem (32) describing he differen errors is sable and he gain from V() o e() is bounded by γ>if here exis some marices P 1 > P 2 >andp 3 >and marices Y 2 Y

6 6 Mahemaical Problems in Engineering wih appropriae dimensions and posiive scalars such ha he marix inequaliy (34) holds. Ω Y 2 B Y Y 2 M ea Y 2 M eb Y 2 M ead γ 2 I+ 2 N T eb N eb P 3 h < [ 1 I 2 I ] [ 3 I ] (34) and he observer and conroller gains are hen compued by +2ξ T () YE [e () e() e (ω) dω] +2ξ T () Y 2 [ Ae () + A d e () + BV () Ee ()] + sym (ξ T () Y 2 M ea F 5 () N ea e () +ξ T () Y 2 M ead F 5 () N ead e () +ξ T () Y 2 M eb F 5 () N eb V ()). By applying Lemma 1 (38) yields (38) K 1 =(Y 14 ) 1 X 41 K 2 =(Y 14 ) 1 X 42 H 1 =(Y 11 ) 1 H 21 H 2 =(Y 12 ) 1 H 32. Proof. Le us consider he weighed L 2 consrain given by (35) e T (σ) Qe (σ) dσ < γ 2 V T (σ) V (σ) dσ (36) where γ is he aenuaion level from he perurbaion-like erm V() o he error e() in (32) and Q is a symmeric semiposiive-definie weighed marix. I is well known ha he consrain is saisfied if here exiss a Lyapunov-Krasovskii funcion such ha V () ξ T () Ωξ () ξt () Y 2 M ea M T ea YT 2 ξ () ξt () Y 2 M ead M T ead YT 2 ξ () ξt () Y 2 M eb M T eb YT 2 ξ () +2ξ T () Y 2 BV () + 2 V T () N T eb N ebv () wherehe Ω is expressed as follows: Ω =W p QW T p + 1W p N T ea N eaw T p + 3 W ph N T ead N eadw T ph +W pp 2 W T p (39) V () +e T () Qe () γ 2 V T () V () <. (37) Choose Lyapunov-Krasovskii funcional candidae as V() = V 1 () + V 2 () + V 3 () and V 1 () V 2 () V 3 () are defined in (22). Adding 2Π 1 2Π 2 andhξ T ()YP 1 3 YT ξ() and subracing ξt ()YP 1 3 YT ξ()dωrespecivelyin(37) yield (38) which is shown as follows: V () 2e T () E T P 1 e () +he T () E T P 3 Ee () e T (ω) E T P 3 Ee (ω) dω + e T () P 2 e () e T () P 2 e () +2ξ T () YE [e () e() e (ω) dω] +2ξ T () Y 2 [( A+Δ A) e () +( A d +Δ A d )e() +( B+Δ B) V () Ee ()] =2e T () P 1 Ee () +he T () E T P 3 Ee () e T (ω) E T P 3 Ee (ω) dω + e T () P 2 e () e T () P 2 e () +hw p1 E T P 3 EW T p1 W php 2 W T ph + sym (W p P 1 W T p1 +YEWT P +YEWT ph +(Y 2 A HC 1 XC 2 )W T p +(Y 2 A d KC 3 )W T ph ). (4) W p W p1 W ph A A d Y 2 X K H C 1 C 2 andc 3 are defined by Theorem 2 and M ea M eb M ead N ea N eb andn ead are shown as follows: M a M b M a M b M ea = [ ] [ ] N a N a N ea = N b [ N b ] [ N c M c ] M ad M ad M ead = [ ] [ M cd ]

7 Mahemaical Problems in Engineering 7 N ad N ad N ead = [ ] [ ] M a M b M ad M a M b M ad M eb = [ ] [ M c M cd ] N a N b N eb = N ad. [ N c ] [ N cd ] (41) Faul esimaion Faul Figure 2: Faul and is esimaion. Following he similar seps of previous proofs and applying Schur complemen on (39) he sufficien LMI condiions proposed in Theorem 4 follow. Remark 5. I should be menioned ha our sudy o his paperismainlymoivaedbyheworkof[17]. There are hree differences beween he work given in [17] and ours.firs he sysem in [17] are Takagi-sugeno fuzzy models and in our sudy he linear sysem is invesigaed; however he resuls of he linear sysem heories can be applied for he design of Takagi-sugeno fuzzy models. Second he Takagi-sugeno models of he work [17] aken are combinaions of linear ime invarian sysems. Our sudy is he exend of linear sysems o linear ime-delay sysems bu i is no aken ino accoun in [17]. Third problems of he sabiliy and he H conrol for delayed sysems use he Lyapunov-Krasovskii approach oher han he Lyapunov-Razumikhin approach and he resuls of using Lyapunov-Krasovskii approach are usually less conservaive han hose using Lyapunov-Razumikhin approach [25]. 4. Numerical Example In his secion we will provide a numerical example o demonsrae he effeciveness of he design mehods proposed in he previous secion. This example is aken from [23] and he fauly sysem is defined by A=[ ] B=[ ] A d =[.25 ] C=[ ] C d =[.1.1 ]. (42) We assume ha he ime delay is given by h =.1.Leing he nominal inpu signal u() = sin() and he consan faul f=1affecing he sysem behavior a 3 s 1s one can obain he following soluion by solving he condiions in he Theorem 2: H 1 =[ ] H 2 =[ ] K 1 =[ ] K 2 =[ ]. (43) The simulaion resuls are shown in Figures 2 3 4and5. In Figure 2 he real faul and is esimae are depiced. Figures 3 and 4 compare he sae variables of he reference model he observerandhe fauly sysem wih FTC.Figure 5 shows he comparison of he nominal conrol inpu and FTC signal. In he nex we will consider he ime-varying faul wih sysem uncerainies; he uncerainies are defined by M ad =M b =M a = diag {.1.1} M c =M cd = diag {.1.1} N a =N ad =N c =N cd = diag {.1.1} N b = [.1.1] T. (44) The sysem uncerainies are given by F() =.5 sin(). The definiion of he ime delay h and he nominal inpu signal u() is given by h =.1 and u() = sin(). Theimevarying faul f=sin(.5 8) cos() affecs he sysem behavior a 3 s 1s. When choosing he free weighing marix Y 2 = [ Y T 2 Y T 2 ]T and solving he condiions in Theorem 4 wih

8 8 Mahemaical Problems in Engineering Sae of fauly sysem Sae of observer Sae of reference sysem Nominal inpu FTC Figure 3: Comparison of reference mode sae x 1 fauly sysem sae x f1 and he observer sae x f1. Figure 5: Nominal inpu u() and FTC u f () Sae of fauly sysem Sae of observer Sae of reference sysem Figure 4: Comparison of reference mode sae x 2 fauly sysem sae x f2 and he observer sae x f Faul esimaion Faul Figure 6: Faul and is esimaion..15 Q=diag{I 2 I 2 I 1 I 1 I 2 }onecanobainheaenuaionvalue γ = wih H 1 =[ ].1.5 H 2 =[ ] 1 3 K 1 =[ ] (45).5.1 K 2 =[ ]. InorderoshowheeffeciveinfluenceofheFTCon he sysem rajecory we make he comparison beween he sysems wih and wihou FTC when fauls occur. The simulaion resuls are shown in Figures and11. Figure 6 illusraes he simulaion resul of faul esimaion. Figures 7 and 8 show he sae variables of he reference Sae of fauly sysem Sae of observer Sae of reference sysem Figure 7: Comparison of reference mode sae x 1 fauly sysem sae x f1 and he observer sae x f1 wih FTC.

9 Mahemaical Problems in Engineering Sae of fauly sysem Sae of observer Sae of reference sysem Figure 8: Comparison of reference mode sae x 2 fauly sysem sae x f2 and he observer sae x f2 wih FTC Sae of fauly sysem Sae of observer Sae of reference sysem Figure 11: Comparison of reference mode sae x 2 faulysysem sae x f2 and he observer sae x f2 wihou FTC Nominal inpu FTC Figure 9: Nominal inpu u() and FTC u f (). model he observer and he fauly sysem wih FTC. The comparison of he nominal conrol and FTC signal is shown in Figure 9. Figures1 and 11 show he sae variables of he reference model he observer and he fauly sysem wihou FTC. Two differen cases are considered including consan faul wihou uncerainies and ime-varying faul wih uncerainies. From he above simulaion resuls one can see ha he synhesized observers and FTC conrollers showed heir effeciveness since he faul is esimaed (Figures 2 and 6) and he sae variables of he fauly sysem wih FTC are closed o reference model (Figures and8) and he racking beween he fauly sysem and he reference model is ensured. On he oher hand if FTC is no employed in he fauly sysem (Figures 1 and 11) he sae rajecories of fauly sysem deviae from he sae rajecories of reference model and he racking is no achieved Sae of fauly sysem Sae of observer Sae of reference sysem Figure 1: Comparison of reference mode sae x 1 faulysysem sae x f1 and he observer sae x f1 wihou FTC. 5. Conclusion In his paper he problem of acive FTC design for linear ime-delay sysems wih and wihou uncerainies is reaed. The aim of he FTC law and observer design is o ensure he rajecory racking of fauly sysem. By considering he descripor redundancy of closed-loop sysems and using Lyapunov-Krasovskii funcional approach he proposed FTC scheme has been easily formulaed in LMI erms. Two kinds of fauls have been considered. The firs one deals wih he consan fauls wihou uncerainies as a special case and he oher deals wih he ime-varying fauls wih sysem uncerainies. Finally one example has been considered o illusrae he efficiency of he proposed scheme in boh cases. In addiion i is ineresing o develop he FTC conrol law by aking ino accoun modeling muliplicaive fauls and some exernal perurbaions and considering how o deal wih sensor fauls and how o apply his scheme o T-S models.

10 1 Mahemaical Problems in Engineering Conflic of Ineress The auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmens The auhors would like o hank he reviewers for heir consrucive commens and suggesions which have helped o improve he presenaion of he paper. This work was suppored by he Naional Naural Science Foundaion of China (no ). References [1] J.ChenandR.J.PaonRobus Model-Based Faul Diagnosis for Dynamic Sysems Kluwer Academic Publishers Boson Mass USA [2] M. Blanke M. Kinnaer and J. Lunze Diagnosis and Faul- Toleran Conrol Springer Berlin Germany 26. [3] H.NiemannandJ.Sousrup Passivefauloleranconrolofa double invered pendulum a case sudy Conrol Engineering Pracicevol.13no.8pp [4] H. Wang and S. Daley Acuaor faul diagnosis: an adapive observer-based echnique IEEE Transacions on Auomaic Conrolvol.41no.7pp [5]B.JiangM.SaroswieckiandV.Cocquempo Faulidenificaion for a class of ime-delay sysems in Proceedings of he American Conrol Conference pp Anchorage Alaska USA May 22. [6] C. Edwards S. K. Spurgeon and R. J. Paon Sliding mode observers for faul deecion and isolaion Auomaica vol. 36 no. 4 pp [7] N. E. Wu Y. Zhang and K. Zhou Deecion esimaion and accommodaion of loss of conrol effeciveness Inernaional Adapive Conrol and Signal Processingvol.14no.7 pp [8] B. Marx D. Koenig and D. Georges Robus faul-oleran conrol for descripor sysems IEEE Transacions on Auomaic Conrolvol.49no.1pp [9] M. Wang B. Chen and S. Tong Adapive fuzzy racking conrol for sric-feedback nonlinear sysems wih unknown ime delays Inernaional Innovaive Compuing Informaion and Conrolvol.4no.4pp [1] W. Chen and M. Saif An ieraive learning observer for faul deecion and accommodaion in nonlinear ime-delay sysems Inernaional Robus and Nonlinear Conrol vol.16no.1pp [11] W. Chen and M. Saif Faul deecion and accommodaion in nonlinear ime-dealy sysems in Proceedings of he American Conrol Conferencepp June23. [12] B. Jiang M. Saroswiecki and V. Cocquempo H faul deecion filer design for linear discreeime sysems wih muliple ime delays Inernaional Sysems Science vol. 34 no. 5 pp [13] D. Du B. Jiang and P. Shi Sensor faul esimaion and compensaion for ime-delay swiched sysems Inernaional Sysems Sciencevol.43no.4pp [14] C. Jiang D. H. Zhou and F. Gao Robus faul deecion and isolaion for uncerain linear rearded sysems Asian Journal of Conrolvol.8no.2pp [15] D. Zhang L. Yu and W.-A. Zhang Delay-dependen faul deecion for swiched linear sysems wih ime-varying delayshe average dwell ime approach Signal Processing vol. 91 no. 4 pp [16] B. Jiang K. Zhang and P. Shi Less conservaive crieria for faul accommodaion of ime-varying delay sysems using adapive faul diagnosis observer Inernaional Adapive Conrol and Signal Processingvol.24no.4pp [17] T. Bouarar B. Marx D. Maquin and J. Rago Faul-oleran conrol design for uncerain Takagi-Sugeno sysems by rajecory racking: a descripor approach IET Conrol Theory & Applicaionsvol.7no.14pp [18] D. Ichalal B. Marx J. Rago and D. Maquin Observer based acuaor faul oleran conrol for nonlinear Takagi- Sugeno sysems: an LMI approach in Proceedings of he 18h Medierranean Conference on Conrol and Auomaion (MED 1) pp Marrakech Morocco June 21. [19] T. Bouarar B. Marx D. Maquin and J. Rago Trajecory racking faul oleran conroller design for Takagi-Sugeno sysems subjec o acuaor fauls in Proceedings of he Inernaional Conference on Communicaions Compuing and Conrol Applicaions (CCCA 11) pp. 1 6 Hammame Tunisia March 211. [2] K. Guelon T. Bouarar and N. Manamanni Robus dynamic oupu feedback fuzzy Lyapunov sabilizaion of Takagi-Sugeno sysems a descripor redundancy approach Fuzzy Ses and Sysemsvol.16no.19pp [21] T. Bouarar K. Guelon and N. Manamanni Saic oupu feedback conroller design for Takagi-Sugeno sysems a fuzzy Lyapunov LMI approach in Proceedings of he 48h IEEE Conference on Decision and Conrol Held Joinly wih 28h Chinese Conrol Conference (CDC/CCC 9) pp IEEE Shanghai China December 29. [22] T. Bouarar K. Guelon and N. Manamanni Robus fuzzy Lyapunov sabilizaion for uncerain and disurbed Takagi- Sugeno descripors ISA Transacions vol. 49 no. 4 pp [23] Y.ZhaoJ.WuandP.Shi H conrol of non-linear dynamic sysems: a new fuzzy delay pariioning approach IET Conrol Theory & Applicaionsvol.3no.7pp [24] X. F. Jiang and Q.-L. Han Robus H conrol for uncerain Takagi-Sugeno fuzzy sysems wih inerval ime-varying delay IEEE Transacions on Fuzzy Sysems vol.15no.2pp [25] G. J. Zhang C. S. Han and L. G. Wu Admissibiliy and H performance analysis of T-S fuzzy descripor sysems wih imedelay in Proceedings of he 2nd Inernaional Conference on Inelligen Conrol and Informaion Processing (ICICIP 11) pp July 211.

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