Fuzzy Reliable Tracking Control for Flexible Air-breathing Hypersonic Vehicles

Transcription

1 Inernaonal Journal of Fuzzy Sysems, Vol. 3, No. 4, December 33 Fuzzy Relable rackng Conrol for Flexble Ar-breahng Hypersonc Vehcles Xaoxang Hu, Hujun Gao, Hamd Reza arm, Lgang Wu, and Changhua Hu Absrac In hs paper, we presen a fuzzy relable rackng conrol desgn mehod for flexble ar-breahng hypersonc vehcles (FAHVs) subjec o dsurbances and possble sensor/acuaor falures. hs problem s challengng due o he srong couplng effecs, varable operang condons and possble falures n FAHVs. Frs, akag-sugeno (-S) fuzzy model s used o represen he longudnal dynamcs model of FAHVs. hen, by consderng he dsurbances and he fauls, he fuzzy relable rackng problem s proposed, and he rackng conrol problem s ransformed no a sablzaon problem. A fuzzy relable sae-feedback conroller s desgned o guaranee he asympoc sably of he closed-sysem. By he Lyapunov approach, he exsence condons for such a conroller are esablshed n erms of lnear marx nequales. Wh he desgned conrollers, he reference command can be racked n spe of acuaor/sensor fauls. Smulaon has demonsraed he proposed desgn scheme. eywords: Flexble ar-breahng hypersonc flgh vehcles (FAHVs), nonlnear dynamc sysem, akag-sugeno (-S) fuzzy modelng, fuzzy relable conrol, acuaor/sensor fauls.. Inroducon he scramje-powered ar-breahng hypersonc vehcles (AHVs) presens a more cos effcen way o make access o space roune, or even make he space ravel roune and nerconnenal ravel as easy as nercy ravel [, ]. Compared o radonal flgh vehcles, AHVs have rreplaceable advanage. Beng dfferen from rocke engne, scramje s capable of obanng oxygen drecly from amosphere, so he AHVs can carry more payloads. For he urboje engne, he maxmum speed s usually lmed o a Mach number of abou 3.5 Correspondng Auhor: Hujun Gao s wh Research Insue of Inellgen Conrol and Sysems, Harbn Insue of echnology, Harbn, 5, P.R. Chna. E-mal: hjgao@h.edu.cn Manuscrp receved 8 Jun. ; revsed 6 Aug. ; acceped 8 Nov.. by he allowable urbne blade emperaure, whle he AHVs can fly a a hgh Mach number (greaer han 5) whou carryng oxdzer. Jus as every con has wo sdes, here are dsadvanage for he applcaon of scramje. AHVs use he echnology of arframe negraed wh scramje engne confguraon [3,4], whch makes he neracons beween he elasc arframe, he propulson sysem, and he srucural dynamcs very srong [5]. Modelng and flgh conrol of such vehcles has been an acve subjec of research n recen years. For he modelng and flgh conrol of AHVs, modelng s very mporan. he man modelng ssue s how o clearly and accuraely descrbe he propulsve forces and vehcle moons under he srong couplng among aerodynamcs, propulson sysem wh scramje, and flexbly of he arcraf. Due o he dynamcs' enormous complexy, only he longudnal dynamcs models of AHVs were suded and he modelng and conrol problems of hypersonc arcrafs were dscussed. Recenly, n [6,7], a longudnal nonlnear model suable for conrol desgn was derved and could descrbe he complex dynamcs wh he flexble couplng effecs n a scramje-powered vehcle. Based on he nonlnear flexble ar-breahng hypersonc vehcles (FAHVs) model, several sudes on he flgh conrol and navgaon were conduced. In [8], a conrol-orened model was derved for he FAHVs usng curve fs calculaed drecly from he forces and momens ncluded n he ruh model, and hen an approxmae feedback lberalzaon example of conrol desgn was gven o derve a nonlnear conroller. In [], he auhors presened wo oupu feedback conrol desgn mehods for he FAHVs models, and adapve conrol echnques were also consdered n []. In [], dynamc oupu feedback echnque was used o provde robus reference velocy and alude rackng conrol n he presence of model unceranes and varyng flgh condons, and n [,3], lnear conrollers wh npu consrans usng on-lne opmzaon and an-wndup echnques were also proposed. More recenly, a nonlnear robus adapve conrol desgn mehod was presened n [4], and n [5], he auhors consdered he modelng of aerohermoelasc effecs and gave a Lyapunov-based rackng conroller. hough a lo of works have been done, he robus conrol for he hgh nonlnear dynamcs of FAHVs s sll a penden problem, especally when possble sensor/acuaor fauls exs. FSA

2 34 Inernaonal Journal of Fuzzy Sysems, Vol. 3, No. 4, December In order o develop effcen conrol approaches o address he rackng conrol ask of FAHVs subjec o complex nonlnear and couplng, s necessary o mplemen he akag-sugeno (-S) fuzzy conrol scheme. -S modelng echnque s an effecve approach of nonlnear sysems, whch could approxmae any smooh nonlnear funcon o any specfed accuracy whn any compac se [6]. he -S fuzzy models s represened by a se of lnear models by fuzzy IF-HEN rules, so s possble for he exsng radonal lnear sysems resuls o be appled o analyss and synhess of nonlnear sysems based on he parallel-dsrbued compensaon (PDC) scheme [7]. here have been several resuls n leraures on he conrol of nonlnear sysems based on -S fuzzy echnque [8-]. Smlar o all oher arplanes and space vehcles, possble sensor/acuaor falures are no avodable n FAHVs. Due o he complexy of FAHVs, s hard o compleely avod he fauls exsng n he sensors or acuaors, whch have much o do wh he safey and accuracy of he rendezvous. Hence, relable conrol agans he possble fauls s also a major challenge n he flgh conrol of FAHVs. Bu o our knowledge, hs problem has no been well dscussed. Relable conrol has araced many researchers and a number of resuls have been repored. In [3-5], relable conroller desgn mehods for lnear sysems are presened, whch sably and he gven performances are ensured n spe of some admssble conrol componen ouages. Fuzzy relable conrol problem has also been suded n [6], bu n mos of he sudes, he fauls are assumed o be zero when fauls occur. hs modelng mehod can smplfy he conroller desgn mehod. However, s sgnfcan o adop a more general model o descrbe he fauls wh scalng facors wh upper and lower bounds n pracce. Besdes he possble fauls, varous dsurbances always exs n he realsc envronmen, whch can resul and can have srong adverse effecs on he performance of FAHVs conrol sysems, so he dsurbance aenuaon problem mus be consdered n he conrol desgn. In recen years, he dsurbance aenuaon conrol problem has been wldly suded n [7-3], bu here are few resuls on he relable H conrol of FAHVs subjec o possble sensor/acuaor falures. he complex and challengng problem of robus dsurbance aenuaon conrol desgn for he longudnal model of FAHVs has no been fully nvesgaed and many mporan ssues reman unsolved, whch movaes he presen sudy. Movaed by he above dscussons, n hs paper, we sudy he fuzzy relable rackng conrol problem for FAHVs wh dsurbance. Based on he -S fuzzy modelng echnology, a -S fuzzy model, wh possble acuaor fauls and sensor fauls, s consruced o represened he complex nonlnear longudnal model of FAHVs. By consderng he wo cases (acuaor fauls case and sensor fauls case) respecvely, he reference command rackng problem s ransformed no a sablzaon problem. hen, he fuzzy relable sae-feedback conroller desgn mehod s developed by a Lyapunov approach. he exsence condons for he admssble relable conrollers, n spe of he sensor/acuaor falures, are formulaed n he form of lner marx nequales. Afer geng he fuzzy sae-feedback conroller, an llusrave example s provded o show he effecveness and advanage of he proposed conrol desgn mehod. he res of hs paper s organzed as follows. In Secon, he dynamc model of FAHVs s esablshed, and hen, a -S model of FAHVs s consruced. Based on he esablshed -S model, he fuzzy relable conroller desgn problem s formulaed. Secon 3 presens he fuzzy conroller desgn mehod. hen, an example s gven o llusrae he applcably of he proposed approach n Secon 4. Fnally, we conclude he paper n Secon 5. Noaon: he noaons used hroughou he paper are farly sandard. hroughou hs paper, he superscrp sands for marx ransposon; and R n denoes he n-dmensonal Eucldean space and R n m denoes he se of all n m real marces; dag{ } sands for a block-dagonal marx, and sym{a} s defned as A+A ; I and denoe he deny marx and zero marx wh compable dmensons. In symmerc block marces or complex marx expressons, we use an * o represen a erm ha s nduced by symmery. Marces are assumed o be compable for algebrac operaons f her dmensons are no explcly saed.. Problem Formulaon In hs secon, a akag-sugeno fuzzy model of FAHVs s esablshed wh consderng dsurbances. he fuzzy model s descrbed by fuzzy IF-HEN rules and s employed here o deal wh he conrol desgn problem for he nonlnear longudnal dynamcs of FAHVs. wo knds of possble fauls, acuaor fauls and sensor fauls, are modeled. hen he fuzzy relable rackng conrol problem s proposed. A. Model Descrpon he hypersonc vehcle model consdered n hs paper was developed by Bolender and Doman [6, 7]. Flexbly effecs are ncluded n he model. A longudnal skech of he vehcle s gven n Fgure. he nonlnear equaons are descrbed as follows:

3 X. Hu e al: Fuzzy Relable rackng Conrol for Flexble Ar-breahng Hypersonc Vehcles 35 ( θ α ) h = V sn V = ( cosα D) g sn( θ α ) m g α = snα + + cos θ α mv V θ = Q M Q = I yy η= ς ωη ωη + N η = ς ω η ω η + N ( L) Q ( ) () able. Ls of fuzzy rules. Rule NO. Premse varables V α S S S M 3 S B 4 M S 5 M M 6 M B 7 B S 8 B M B B Fgure. Geomery of he flexble hypersonc vehcle. h and V represen he flgh alude and velocy of FAHVs, respecvely; α s he angle of aack of he vehcle and θ s he flgh pch angle wh Q represens he pch rae. η denoes he h generalzed elasc coordnae., L, D and N are he hrus, lf, drag, and generalzed elasc forces, respecvely; M s he pchng momen. hs nonlnear model s composed of fve rgd-body sae varables [ hv,, α, θ, Q] and four flexble saes [ η, η, η, η]. he conrol npus: fuel-o-ar rao Φ and elevaor deflecon δ e, affec he forces and momen n hese equaons n a complex nonlnear way, he deals can be found n [8]. Snce he nonlnear dynamc of FAHVs s hghly complex, desgnng a nonlnear conroller drecly s dffcul. In hs paper, we wll ulze he well-known -S fuzzy model echnology o approach he nonlnear sysems (). In -S fuzzy model consrucng, V and α are chose as he premse varables and he modelng echnque expressed n [3] s employed. Each premse varable s supposed o have hree levels: a lower bound, a upper bound and a equlbrum pon, whch named as small (S), bg (B), and mddle (M), respecvely. Consder he sysem wh dsurbances, he nonlnear model () can hen be represened by a -S fuzzy model composed of (3²) fuzzy rules, as lsed n able, S, M, and B represen small, mddle, and bg, respecvely. An example of he fuzzy IF-HEN rules correspondng o able s explaned as follow: Rule ) If V s V and α s α, hen x () = Ax () + Bu () + D f () y () = Cx () x( ) = [ h, V, α, θ, Q, η, η, η, η], f() s he unceran exraneous dsurbance or he nonlneary, D s he gan marx of f(), C = C = = C = C, and C=. he fuzzy membershp funcons of V and α are defned as follows: hs ( V) = f V > VM, hm( V ) = hb( V ) 4 hb( V) = exp( 3.5 V( ) VB ) 4 hs( V) = exp( 3.5 V( ) VS ) f V < VM, hm( V ) = hb( V ) hb ( V) = hs ( α) = f α > αm, hm( α) = hb( α) 4 hb( α) = exp(.6 α( ) αb ) 4 hs( α) = exp(.6 α( ) αs ) f α < αm, hm( α) = hb( α) hb ( α) =. hus, he -S fuzzy model whch represens he nonlnear hypersonc vehcle model () can be descrbed by x () = h() Ax () + Bu () + D f () () y () = Cx (),

4 36 Inernaonal Journal of Fuzzy Sysems, Vol. 3, No. 4, December h () s correspondng o able, wh h (), =,,, and h () =. he conrol objecve s o rack a command velocy and alude vecor y com () = [ V com (), h com ()] whou seady-sae rackng error, ha s, lm e ( ) = lm y ( ) y ( ) =. ( ) In order o elmnae he seady-sae rackng error, we nroduce he error negral acon n he conroller. Defne com τ τ τ com τ, d () = e( ) d = ( y( ) y ()) d hen d () = e () = y () ycom (). In order o oban a robus rackng conroller wh sae-feedback, he followng augmened sae-space descrpon s nroduced: Rule ) If V s V and α s α, hen ζ() = Aζ() + Bu () + Gw () y () = Cζ () A B D A =, B, G C = = I x () f() ζ () =, w(), C [ C. ] d () = = ycom () For descrpon brevy, he above equaon s wren as ζ() = Ahζ() + Bhu() + Ghw() (3) y () = Cζ () A = h() A, B = h() B, G = h() G. h h h Based on he parallel dsrbued compensaon (PDC) concep, he fuzzy sae-feedback conroller for fuzzy model () s consruced as u () = h() ζ() = ζ() (4) h h = h(). he augmened close-loop sysem can be wren as ς() = A + B ζ() + G w ( h h h) h () h() hj() ( A B j) ζ () Gw () = + + (5) hen he oupu rackng conroller desgn problem can be ransformed no a sablzaon problem for (5). B. Fuzzy Relable Conroller Desgn Problem he faul model consdered n hs paper s supposed o depend on wo dfferen sources of possble fauls, ha s, acuaor faul and sensor faul. In wha follows, he wo cases wll be consdered respecvely. Acuaor faul case he ype of acuaor faul consdered hs paper s loss F of acuaor effecveness, here we use u () o descrbe he acuaor sgnal and F u () = h() F () ζ() = F () ζ() (6) a a a a ah s he acuaor fauls-oleran feedback conroller whch needs o be deermned, = h(), ah a he faul marx Fa() = dag{ faφ (), fa δ ()}. f () e a, = { Φ, δe} s an unknown funcon, whch means he acuaor reducon coeffcen. Assumpon : fa () s supposed o sasfy fal fa () fau, (7) f al and f au are known consans whch represen he lower and upper bounds of fa (), respecvely, and fal fau <. Wh Assumpon, f f al = f au =, he correspondng h acuaor u () s complee broken down. If f al = f au =, he h acuaor u () s n he faul-free case. Oherwse, f < f al < f au, and fa (), here exss he paral faul n he correspondng acuaor. Sensor faul case Smlarly o acuaor faul, he sensor faul consdered n hs paper can be defned n he followng form: u S ( ) = h() F () ζ() = F ( ) ζ( ), (8) s s s sh s s he sensor fauls-oleran feedback conroller o be deermned, = h(), he sensor sh s falure marx Fs () = { fsh(), fsv (), fs α(), fs θ (), f ( ), f ( ), f ( ), f ( ), f ( ), f ( ), f ( ) sq sη s η sη s η sdh sd. V { α θ η η η, η } s h V f ( ), h, V,,, Q,,,, d, d s an unknown funcon, whch sands for he sensor reducon coeffcen. }

5 X. Hu e al: Fuzzy Relable rackng Conrol for Flexble Ar-breahng Hypersonc Vehcles 37 Assumpon : f () s supposed o sasfy s fsl fs () fsu () f sl and f su are known consans whch represen he lower and upper bounds of fs (), respecvely, and fsl fsu <. Here, f sl = f su =, and f sl = f su =, denoe he cases of complee sgnal loss and no faul n he correspondng sensor, respecvely, and oher values of fs () represen paral faul of he sensor. hen he close-loop sysem wh sae-feedback conroller can be wren as ς() = A + B ζ() + G w () ( h h h) h ( ) = F () for he acuaor fauls case, or h a ah h = Fs() for he sensor fauls case. Because of he exsence of w (), he dsurbance aenuaon mus be consdered when desgnng he sae-feedback conroller. o hs end, dsurbance aenuaon performance s se as follow: y () Ωy() d ρ w () w() d () w () L[, ), Ω s a posve defne weghng marx, ρ s a prescrbed aenuaon level. Consder FAHVs wh possble sensor/acuaor fauls, he fuzzy relable conroller desgn problem s o fnd a sae-feedback conroller, such ha: he closed-loop sysem s robusly sable; he oupu of he sysem can rack a command vecor y [, ] com = Vcom hcom whou seady-error; In he even of possble sensor/acuaor fauls, he sably and he rackng performance of he sysem can be guaraneed. 3. Man Resuls In hs secon, based on he parallel dsrbued compensaon (PDC) scheme, he fuzzy sae-feedback relable conroller desgn problem wll be nvesgaed. Before proceedng, he followng lemmas are gven. Lemma [33]: Le E, F and Σ are marces of approprae dmensons wh Σ. hen, for any scalar ε>, EΣ F + F Σ E ε EE + εf F. Lemma [34]: For a me-varyng dagonal marx Φ () = dag { σ(), σ(),, σ p () } and marces R and S wh approprae dmensons, f Φ() V, { σ σ σ p } Φ () = dag (), (),, () and V> s a known dagonal marx, hen for any scalar ε>, RΦ S + S Φ R εrvr + ε S VS. Lemma 3 [35]: he parameerzed lnear marx nequales, n n μμ M j j <, s fulflled, f he followng condon holds: M < ; M + ( Mj + M j ) <, j k. k A. Acuaor Fauls-Relable conroller We nroduce Fa = dag{ faφ, fa δ } e La = dag{ laφ, la δ } e J = dag{ j, j } a aφ aδ e fa = ( fal + fau ), la = [ fa ( ) fa] fa and ja = ( fau fal ) ( fau + fal ) wh = { Φ, δe}. hen, we have F ( ) a = Fa I + L and a LaLa Ja Ja I. A suffcen condon for he sably of he fuzzy sysem () o solve he problem of acuaor fauls-relable conroller desgn for FAHVs s gven as follows. heorem : For he -S fuzzy sysem () assocaed wh acuaor fauls descrbed n (6), f here exs marx P>, Yj ( j =,,, ), and a scalar ε a >, sasfyng Θ <, =,,,. () Θ + ( Θ j +Θ j ) <, j, k sym{ AX + BY j} + ε abj ab Yj Gh XC ε aja (3) Θ j = ρ I Ω hen here exss a proper acuaor fauls-oleran conroller such ha he close-loop sysem n () s asympocally sable n spe of he acuaor fauls, and he dsurbance aenuaon performance n () s guaraneed for a prescrbed performance ndex ρ. he desred sae-feedback conrol gan s gven by a F = a YX. (4) Proof: For sysem (), defne he followng Lyapunov funcon V () = ζ () Pζ (), hen ake me dervave of V (), we have

6 38 Inernaonal Journal of Fuzzy Sysems, Vol. 3, No. 4, December ( h h h) ( h h h) V () = ζ() P A B A B P ζ() + ζ() PG w + w G Pζ(). h () () h (5) he close-loop fuzzy sysem () s sable wh dsurbance aenuaon performance ρ f he followng nequaly holds: V () + y() Ωy () ρ w wd< (6) ( ) ( ) By Lemma ζ() PGhw() + w () GhPζ() ρ ζ() PGhGhPζ() + ρ w () w() from (5) and (6), we can ge ha (6) lead o sym PAh + PBF h a + PBF h al a ah + ρ PG G P + C Ω C <. h L a, can be easly ob- X = P, For he srucure of F a and aned ha FaLa LaFa Yh Fa X on o above nequaly by h =, by defnng (7) (8) = and performng a congruence ransforma- P, we have sym AhX + BhYh + BhLaYh + ρ GhGh + XC Ω CX < By Lemma, for a scalar ε a >, we have BhLY a h + Yh LB a h εa BJ h abh + ε ayh JY a h hen (8) can be rewren as: sym A X + B Y + ε B J B { } h h h a h a h ayh JaYh ρ GhGh XC CX + ε + + Ω < By Schur complemen, he above nequaly equals o Ψaj Yj Gh XC ε aja hh () j() < ρ I Ω Ψ = sym A X + BY + ε B J B, ha s, { } aj j a a h( ) h ( ) Θ <,,,,. j hen by Lemma 3, nequales () and (3) can easly be obaned. he proof s compleed. B. Sensor Fauls-Relable conroller Smlarly o he acuaor faul case, we nroduce he followng marces: Fs= dag{ fsh, fsv, fs α, fs θ, fsq, fs η, fs η, fs η, fs η, fsd, f }, h sdv Ls = dag{ lsh, lsv, ls α, ls θ, lsq, l, l, l, l, l, l }, sη s η sη s η sd sd h V Js = dag{ jsh, jsv, js α, js θ, jsq, js η, j,,,, }, s η j sη j s η j sd j h sdv fs = ( fsl + fsu ), ls = [ fs( ) fs] fs, and wh { hv,, αθ,, Q, η, η, ηη,, dh, dv} js = ( fsu fsl ) ( fsu + fsl) =. hus, we have ( ) Fs = Fs I+ Ls and LL s s Js Js I. heorem : For he -S fuzzy sysem () assocaed wh sensor fauls descrbed n (8), f here exs marx X>, Y j, V j, ( j =,,, ), and a scalar ε s >, sasfyng Θ <, =,,,. () Θ + ( Θ j +Θ j ) <, j, k sym{ AX + BY j} + εs BV jb X Gh XC ε sjs () Θ j = ρ I Ω X () Js Vj Yj hj () < j= Yj X (3) hen here exss a proper sensor fauls-oleran conroller such ha he close-loop sysem n () s asympocally sable n spe of he sensor fauls, and he dsurbance aenuaon performance n () s guaraneed for a prescrbed performance ndex ρ. he desred sae-feedback conrol gan can be gven by = YX Fs. (4) Proof: Smlarly o he proof of heorem, for sensor fauls case, he close-loop sysem n () s asympocally sably f (5) holds. Defnng X = P and performng a congruence ransformaon o (5) by P, we have { ( ) sym A } h + Bh shfs I + Ls X + XC Ω CX + ρ GhGh < By defnng Yh = Fs X, we have sym( AhX + BhYh) + Bh shfs LsX (5) + XLsFs Bh + XC Ω CX + ρ GhGh < By Lemma, for any scalar ε s >, B h shmslx s + XLM s sbh εs B h shfsjf s sbh + ε s XJX s. From () and (3), we have

7 X. Hu e al: Fuzzy Relable rackng Conrol for Flexble Ar-breahng Hypersonc Vehcles 3 ε sb h shfsjf s sbh = ε sbyx h h JsX Yh Bh εsbhyx h YhBh εsbvb h h h and hus, B h shmslx s + XLM s sbh (6) εs BVB h h h, + ε sxjsx By Schur complemen, he above nequaly equals o Ψs j X Gh XC ε sjs hh () j() < ρ I Ω Ψ = sym A X + BY + ε BV B, ha s, { } s j j s j h() h () Θ <, =,,,. j hen by Lemma 3, nequales ()-(3) hold. he proof s compleed. 4. Smulaon Resuls In hs secon, a numercal example s provde o llusrae he effecveness and advanages of he fuzzy relable conroller desgn mehods proposed n hs paper. For he consruc of -S fuzzy model, he lower and he upper bounds of V and α are chosen as: V = f/ s, VS = 64 f/ s, and α B = 5deg, α S = deg, oher saes are chosen accordng o he flgh envelop. he membershp funcons of he fuzzy model re shown n Fgures -3, Fgure s he membershp funcon of V and Fgure 3 s ha of α. he hypersonc vehcle model parameer values are borrowed from [8]. By usng he fuzzy modelng mehod descrbed n Secon, he -S fuzzy rackng model of hypersonc vehcle can be esablshed. he conrol objecve s o rack a se sep wh respec o a rm condon, whch are reasonable requremens for FAHVs. he npu reference commands are chosen as sep npus, so each command wll pass hrough a prefler as ωn H() s = (7) s + ζω s+ ω ζ denoes dampng rao, ω n sands for naural frequency. he sgnal ou of he prefler s defned as reference command whch s amed o be racked. In smulaon, o llusrae he effecveness of he proposed conroller, we wll use he orgnal nonlnear model (no he consruced fuzzy model) o es he performance of n n B he conrol sysem. Accordng o [36], he dsurbance are assumed o be bounded, whch can be regarded as a gus of wnd n aerospace. In order o compare he rackng performances of FAHVs wh fuzzy relable conroller and he nomnal conroller, a nomnal fuzzy sae-feedback conroller u () = ς () s nroduced whch s desgned whou consderng he possble fauls. h() h() h B (V) h M (V) h S (V) V(f/s) Fgure. Membershp funcons of V. h B (α) h M (α) h S (α) α(deg) Fgure 3. Membershp funcons of α. Here, we consder a clmbng maneuver wh longudnal acceleraon usng separae reference commands for alude and velocy. In hs smulaon, he reference commands for alude and velocy are chosen as follow: f/s for velocy and f for alude, respecvely. he parameers of (7), ζ and ω n are chosen as. and.rad/s, respecvely, whch can make he clmbng of velocy and alude fnsh n abou 5s. In he followng, we wll dscuss he acuaor fauls and sensor fauls, respecvely.

8 33 Inernaonal Journal of Fuzzy Sysems, Vol. 3, No. 4, December A. Acuaor Fauls Case In realsc envronmen, boh of he wo acuaors may lose s effecveness. So for acuaor fauls case, suppose ha f al =.6, f au =, = { Φ, δe}, hen Fa = dag{.8,.8}. hen, our purpose s o desgn a fuzzy relable sae-feedback conroller = h(), ah a such ha FAHVs can rack he reference command n spe of he acuaor fauls and exernal dsurbances. 4 Leng γ =, Ω= I, and by heorem, we can oban he conroller gans a, (,,...,), hen he fuzzy relable sae-feedback conroller can be consruced. o llusrae he advanage of he fuzzy fauls-oleran conroller desgn mehod, we compare he effecveness of he conroller ah whch s desgned by heorem and he nomnal conroller nom. he effecveness and rackng performance for FAHVs wh nom and ah are depced n Fgure 4, he sold lne s he reference command, dashed lne s for fuzzy relable conroller ah and dash-doed lne s for nomnal conroller. I s observed from Fgure 4 ha, he proposed nom fuzzy relable conroller and he nomnal conroller all acheves good performance for he rackng problem. Compared o he nomnal conroller, under he same fauls, he fuzzy relable conrol sraegy ensures beer rackng performance. hs confrms ha he proposed fuzzy relable conrol sraegy can realze excellen performance wh hghly nonlnear sysem n (). Velocy Change, f/s Velocy rackng Error, f/s Reference command Alude Change, f Alude rackng Error, f x Reference command Fgure 4. rackng performance of acuaor fauls. Fgures 5 and 6 show he oher mporan saes and he npus of FAHVs, respecvely. More specfcally, he angle of aack, flgh pah angle and he npus of he plan: fuel-o-ar rao Φ and elevaor deflecon δ e are shown. he angle of aack s of grea mporance snce represens he vehcle's aude. If he amplude of α becomes oo large, he vehcle may no funcon. From Fgure 5, α remans whn abou.8deg from he rm condon and he pch angle θ s kep reasonably small n he above wo cases. Fgure 6 gves he npus of he plan. he npus are all smooh and bounded. In summary, he smulaons resuls demonsrae ha he proposed fuzzy relable conroller s effecve n presence of acuaor fauls and exernal dsurbances. Angle of Aack, deg Pch Angle, deg Ideal Fuel-o-Ar Rao Fgure 5. Angle of aack, flgh pah angle of acuaor fauls. Ideal Elevaor Deflecon, deg Fgure 6. Inpu of acuaor fauls. B. Sensor Fauls Case For sensor fauls case, suppose ha m sl =.4, m su =, { h, V, αθ,, Q, η, η, ηη,, dh, dv}, hen we have Ms = dag{.7,.7,.7,.7,.7,.7,.7,.7,.7,.7,.7}. Assume ha he fauls of he sensors occur smulaneously, hen, our purpose s o desgn a fuzzy relable sae-feedback conroller sh, such ha he FAHVs can rack he reference command n spe of he sensor fauls and exernal dsurbances. Leng γ =.5, 6 Ω= I, and by heorem, we can oban he conroller gans, (,,...,). In he smulaon, we s

9 X. Hu e al: Fuzzy Relable rackng Conrol for Flexble Ar-breahng Hypersonc Vehcles 33 compare he conrol performance of he fuzzy fauls-oleran conroller sh and he nomnal conroller nom. he rackng performance for FAHVs wh sh and nom s gven n Fgures 7-. Ideal Fuel-o-Ar Rao x Velocy Change, f/s Velocy rackng Error, f/s Reference command Alude Change, f Alude rackng Error, f Reference command Ideal Elevaor Deflecon, deg Fgure. Inpu of sensor fauls. 5. Concluson Angle of Aack, deg Pch Angle, deg Fgure 7. rackng performance of sensor fauls Fgure 8. Angle of aack, flgh pah angle of sensor fauls. From Fgure 7, we can see ha, he proposed fuzzy relable conroller can sll ge a good rackng performance n presence of sensor fauls, bu under he same fauls, he rackng error of he nomnal conroller s more bgger han he fuzzy relable conroller. hs confrms ha he proposed fuzzy relable conrol sraegy ensures beer rackng performance for FAHVs n spe of he sensor fauls. he oher mporan saes and he npus are shown n Fgures 8 and, respecvely. From he smulaons resuls we can see ha, he proposed fuzzy relable conroller can ge a good performance n presence of sensor fauls and exernal dsurbances. In hs paper, a fuzzy relable conrol sraegy has been presened for he rackng problem of he longudnal dynamcs of FAHVs model wh acuaor or sensor fauls and exernal dsurbance. Based on he -S fuzzy modelng echnology, a -S fuzzy model has been consruced o represen he nonlnear dynamcs of he FAHVs. By defnng an augmened sysem and consderng he wo cases (acuaor fauls and sensor fauls) respecvely, he rackng problem has been ransformed no a dsurbance aenuaon problem of he close-loop problem. hen, based on PDC scheme, fuzzy relable conroller desgn problems have been suded for he menoned wo faul cases by usng Lyapunov mehod, respecvely. Suffcen condons for desgnng such a conroller have been proposed n erms of LMIs. Illusrave examples have shown he effecveness of he proposed conroller desgn mehod. Acknowledgmen hs work was parally suppored by Naonal Naural Scence Foundaon of Chna (6746, 65, 654 & 67366), Avaon Scence Fund of Chna (ZA77), and Naonal Naural Scence Foundaon of Helongjang Provnce of Chna (F). References [] J. Bern and R. Cummng, Ffy years of hypersoncs: we've been, we're gong, Progress n Aerospace Scences, vol. 3, no. 6-7, pp , 3. [] J. Bern, J. Peraux, and J. Ballmann, Advances n

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11 X. Hu e al: Fuzzy Relable rackng Conrol for Flexble Ar-breahng Hypersonc Vehcles 333 [6] B. Chen and X. Lu, Relable conrol desgn of fuzzy dynamc sysems wh me-varyng delay, Fuzzy Ses and Sysems, vol. 46, no. 3, pp , 4. [7] Y. Zhao, J. Wu, and P. Sh, H conrol of non-lnear dynamc sysems: a new fuzzy delay paronng approach, IE Conrol heory & Applcaons, vol. 3, no. 7, pp. 7-8,. [8] F. Yang, Z. Wang, Y. Hung, and M. Gan, H conrol for neworked sysems wh random communcaon delays, IEEE rans. on Auomac Conrol, vol. 5, no. 3, pp. 5-58, 6. [] Z. D. Wang, F. Yang, D. W. C. Ho, and X. Lu, Robus H conrol for neworked sysems wh random packe losses, IEEE rans. on Sysems, Man and Cyberne cs, Par B, vol. 37, no. 4, pp. 6-4, 7. [3] Z. Wang, D. Ho, Y. Lu, and X. Lu, Robus H conrol for a class of nonlnear dscree me-delay sochasc sysems wh mssng measuremens, Auomaca, vol. 45, no. 3, pp ,. [3] H. Dong, Z. Wang, D. Ho, and H. Gao. Robus H Fuzzy oupu-feedback conrol wh mulple probablsc delays and mulple mssng measuremens, IEEE rans. on Fuzzy Sysems, vol. 8, no. 4, pp. 7-75,. [3] M. exera and S. Zak, Sablzang conroller desgn for unceran nonlnear sysems usng fuzzy models, IEEE rans. on Fuzzy sysems, vol. 7, no., pp. 33-4,. [33] I. Peersen, A sablzaon algorhm for a class of unceran lnear sysems, Sysems and Conrol Leers, vol. 4, no. 8, pp , 87. [34] Y. Wang, L. Xe, and C. de Souza. Robus conrol of a class of unceran nonlnear sysems, Sysems and Conrol Leers, vol.. no., pp. 3-4, [35] H. uan, P. Apkaran,. Narkyo, and Y. Yamamoo, Parameerzed lnear marx nequaly echnques n fuzzy conrol sysem desgn, IEEE rans. on Fuzzy sysems, vol., no., pp ,. [36]. Gbson, L. Crespo, and A. Annaswamy, Adapve conrol of hypersonc vehcles n he presence of modelng unceranes, Proceedng of he Amercan Conrol Conference, San Lous, Mssour, pp , June. Xaoxang Hu receved hs BE degree n Conrol Scence and Engneerng and MS degree n Gudance Navgaon and Conrol from X an Research Insue of Hgh-ech, X'an, P.R. Chna, n 5 and 8, respecvely. He s currenly workng owards a PhD degree n X an Research Insue of Hgh-ech. Hs research neress nclude sldng mode conrol, nonlnear conrol, and -S fuzzy conrol. Hujun Gao receved he Ph.D. degree n conrol scence and engneerng from Harbn Insue of echnology, Chna n 5. He was a Research Assocae wh he Deparmen of Mechancal Engneerng, he Unversy of Hong ong, from November 3 o Augus 4.From Ocober 5 o Ocober 7, he carred ou hs pos docoral research wh he Deparmen of Elecrcal and Compuer Engneerng, Unversy of Albera, Canada. Snce November 4, he has been wh Harbn Insue of echnology, he s currenly a Professor and drecor of he Research Insue of Inellgen Conrol and Sysems. Dr. Gao s research neress nclude nework-based conrol, robus conrol/fler heory, me-delay sysems and her engneerng applcaons. He s an Assocae Edor for Auomaca, IEEE ransacons on Indusral Elecroncs, IEEE ransacons on Sysems Man and Cybernecs Par B: Cybernecs, IEEE ransacons on Fuzzy Sysems, IEEE ransacons on Crcus and Sysems I, IEEE ransacons on Conrol Sysems echnology ec. Hamd Reza arm receved he B.Sc. degree n power sysems engneerng and he M.Sc. and Ph.D. degrees boh n conrol sysems engneerng from Sharf Unversy of echnology, ehran, Iran, n 8,, and 5, respecvely. He s currenly a Professor of Conrol Sysems a he Faculy of echnology and Scence of he Unversy of Agder, Grmsad, Norway. He has publshed more han papers n referred journals and ransacons, book chapers and conference proceedngs. Hs research neress are n he areas of nonlnear sysems, neworked conrol sysems, robus conrol/fler desgn, me-delay sysems, waveles and vbraon conrol of flexble srucures wh an emphass on applcaons n engneerng. Dr. arm was he recpen of he Juan de la Cerva Research Award n 8, he Alexander-von-Humbold-Sfung Research Fellowshp n 6, he German Academc Exchange Servce (DAAD) Research Fellowshp n 3, he Naonal Presdency Prze for Dsngushed Ph.D. suden of Elecrcal Engneerng n 5, and he Naonal Sudens Book Agency s Award for Dsngushed Research hess n 7. He serves as Charman of he IEEE Chaper on Conrol Sysems IEEE Norway Secon. He s also servng as an edoral board member for some nernaonal journals, such as he Journal of Mecharon-

12 334 Inernaonal Journal of Fuzzy Sysems, Vol. 3, No. 4, December cs-elsever, he Journal of Mecharoncs and Applcaons, he Inernaonal Journal of Conrol heory and Applcaons, he Inernaonal Journal of Arfcal Inellgence, ec. He s a member of he IEEE echncal Commee on Sysems wh Uncerany, IFAC echncal Commee on Robus Conrol and IFAC echncal Commee on Auomove Conrol. Lgang Wu receved hs BE degree n Auomaon from Harbn Unversy of Scence and echnology, Harbn, Chna, n, and hs MS and PhD degrees n Conrol Scence and Conrol Engneerng from Harbn Insue of echnology, Harbn, n 3 and 6, respecvely. From January 6 o Aprl 7, he was a Research Assocae n he Deparmen of Mechancal Engneerng, Unversy of Hong ong, Hong ong. From Sepember 7 o June 8, he was a Senor Research Assocae n he Deparmen of Mahemacs, Cy Unversy of Hong ong, owloon, Hong ong. In March 8, he joned Harbn Insue of echnology, he s currenly an Assocae Professor. He was seleced as a member of New Cenury Excellen alens n Unversy of he Chnese Mnsry of Educaon. He s an Assocae Edor of Crcus, Sysems, and Sgnal Processng. Hs curren research neress nclude robus conrol/fler heory, me-delay sysems, muldmensonal sysems and model reducon. Changhua Hu receved he B.Eng. and M.Eng. degrees from he Hgh-ech Insue of X an, X an, Chna, n 87 and, respecvely, and he Ph.D. degree from he Norhwesern Polyechnc Unversy, X an, n 6. He s currenly a Professor wh he Hgh-ech Insue of X an. Durng Sepember 8 December 8, he was a Vsng Scholar wh he Unversy of Dusburg, Dusburg, Germany. Hs curren research was suppored by he Naonal Scence Foundaon of Chna. He has auhored or coauhored wo books and abou arcles. Hs curren research neress nclude faul dagnoss and predcon, lfe prognoss, and faul-oleran conrol.