Guaranteed Cost Control for a Class of Uncertain Delay Systems with Actuator Failures Based on Switching Method

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1 49 Inrnaonal Journal of Conrol, Ru Wang Auomaon, and Jun and Zhao Sysms, vol. 5, no. 5, pp. 49-5, Ocobr 7 Guarand Cos Conrol for a Class of Uncran Dlay Sysms wh Acuaor Falurs Basd on Swchng Mhod Ru Wang and Jun Zhao Absrac: hs papr focuss on h problm of guarand cos conrol for a class of uncran lnar dlay sysms wh acuaor falurs. Whn acuaors suffr srous falur h nvr fald acuaors can no sablz h sysm, basd on swchng sragy of avrag dwll m mhod, undr h condon ha acvaon m rao bwn h sysm whou acuaor falur and h sysm wh acuaor falurs s no lss han a spcfd consan, a suffcn condon for xponnal sably and wghd guarand cos prformanc ar dvlopd n rms of lnar marx nquals (LMIs). Fnally, as an xampl, a rvr polluon conrol problm llusras h ffcvnss of h proposd approach. Kywor: Acuaor falurs, avrag dwll m, guarand cos conrol, lnar marx nquals (LMIs), swchd dlay sysm.. INRODUCION m dlay s a common phnomnon ncounrd n ngnrng conrol. Also, w noc ha m dlay s frqunly a sourc of nsably and ofn droras sysm prformanc. Rcn yars hav wnssd an normous growh of nrs n sably analyss [-5] and conrollr synhss [6-8]. On h ohr hand, whn conrollng a ral plan, s always rabl o gn a conrol sysm whch s no only asympocally sabl bu also guarans an adqua lvl of prformanc. On way o addrss h robus prformanc problm s o consdr a lnar quadrac cos funcon. hs approach s h so-calld guarand cos conrol [9]. Snc h wor of Chang and Png, hs ssu has bn addrssd xnsvly [-3]. Owng o h growng dman of sysm rlably n arospac and ndusral procss, h sudy of rlabl conrol has rcnly aracd consdrabl anon. hrfor, h problm of rlabl guarand cos conrol aracs mor and mor rsarch nrss n rcn yars [4-6]. Howvr, hs rlabl conrol gn mho ar all basd on a basc assumpon ha h nvr fald acuaors can sablz a gvn sysm. For h cas Manuscrp rcvd March 3, 6; rvsd Novmbr 8 6; accpd July 6, 7. Rcommndd by Edoral Board mmbr Km A. Slson undr h drcon of Edor Ja Won Cho. hs wor was suppord by h NSF of Chna undr grans Ru Wang and Jun Zhao ar wh h School of Informaon Scnc and Engnrng, Norhasrn Unvrsy, Shnyang, 4, P. R. Chna (-mals: ruwang@6.com, Corrspondng auhor. whr acuaors suffr srous falur h nvr fald acuaors can no sablz h sysm, h xsng gn mho of rlabl conrol do no wor. In our rcn wor [7], swchng chnqu s nroducd o dal wh hs cas and a gn mhod s oband by usng mulpl Lyapunov funcon mhod. Bu for h cas whr acuaors suffr srous falur n dlay sysms, no rsuls hav bn avalabl up o now. In hs papr, w nroduc swchng sragy o solv h rlabl guarand cos conrol problm for dlay sysms wh srous fald acuaors. h avrag dwll m mhod, whch has bn shown an ffcv ool n h sudy of sably analyss for hybrd or swchd sysms [8-], s adopd o gn conrollrs such ha h closd-loop sysm sasfs guarand cos conrol n prsnc of srously fald acuaors. Fnally, a numrcal xampl s gvn o show h ffcvnss of h proposd mhod. Alhough h da ha swchng chnqu s nroducd o dal wh h problm of srous falur coms from [7], hr ar hr faurs n hs papr. Frs, dlay ffc s consdrd whl s nglcd n [7]; scond, h condons of xponnal sably and guarand cos conrol ar dvlopd whl asympocal sably and H prformanc ar consdrd n [7]; h ool usd n hs papr s h avrag dwll m mhod whl mulpl Lyapunov mhod s adopd n [7]. In hs papr, x() dnos h usually -norm and x = sup x( θ). dnos h symmrc h θ bloc n on symmrc marx. λ max ( S) and λ mn ( S) dno h maxmum and mnmum

2 Guarand Cos Conrol for a Class of Uncran Dlay Sysms wh Acuaor Falurs Basd on Swchng 493 gnvalus of marx S, rspcvly.. PROBLEM FORMULAION W consdr h followng uncran lnar sysm x () = ( AΔ A) x() Ex( h) Bu(), x ( θ) = ϕ( θ), θ [ h,], n () whr x R s h sa, u R s h conrol npu, A, BE, ar consan marcs of appropra dmnsons, ϕ( θ ) s a dffrnabl vcor-valud nal funcon on [ h,], h > dnos h sa dlay, Δ A s a ral-valud marx rprsnng mvaryng paramr uncrans sasfyng Δ A = DF() N for som nown consan marcs D, N, F () s an unnown marx funcon sasfyng F () F() I. Acuaor falurs ar assumd o occur whn a prscrbd subs of conrol channls. W classfy acuaors of h sysm () no wo groups. On s a s of acuaors suscpbl o falurs, whch s dnod by Ω {,,, q}, hs acuaors may occasonally fal. h ohr s a s of acuaors robus o falur, whch s dnod by Ω {,,, q} Ω. Usng hs noaons w nroduc h dcomposon B = B Ω B Ω, () whr B Ω, B Ω ar formd from B by zrong ou columns corrspondng Ω, Ω rspcvly. L Ωcorrspond o a parcular subs of suscpbl acuaors ha acually xprnc falurs. Now, nroduc h dcomposon smlar o (): B = B B, whr B, B ar formd from B by zrong ou columns corrspondng, rspcvly. hus h followng nquals can b asly oband, BΩBΩ BB, BB BΩBΩ. (3) Rmar : In h sudy of rlabl conrol, hr s a usual assumpon ha ( A, B ) s conrollabl [4-6] snc s asr o gn a conrollr n hs cas. Hr rmov hs assumpon o covr mor gnral suaons ncludng boh cass of ( A, B ) bng q conrollabl and bng unconrollabl. Whou loss of gnraly, w consdr h cas of ( A, B ) bng unconrollabl n hs papr. Suppos ha h fauly acuaors can b rcovrd hrough a m nrval. hn, h sa of h sysm s domnad by h followng pcws dffrnal quaon: ( A A) x Ex( h) Bu x Δ = ( A Δ Ax ) Ex ( h) B u. (4) W gn wo n of sa fdbac conrollrs: on s for h sysm whou acuaor falur; h ohr s for h sysm wh acuaor falurs. hrfor, h sysm (4) can b rwrn no h followng swchd dlay sysm x () = ( AΔ Ax ) () Ex ( h) B() u(), x ( θ) = ϕ( θ), θ [ h,], (5) whr ():[, ) M = {, }, B = B, B = B. W gn sa fdbac conrollrs for swchd sysm (5) n h followng sa fdbac form: u = K x, (6) whr K ( =,) ar conrollr gans. Rmar : For h slf-rparng conrol sysms and faul olran conrol sysms, monorng h sysm and dcng h nsanc of acuaor falur (.., dnfy whch cr sa ) can b ralzd by h adapv dcon obsrvr or sldng mod obsrvrs, and so on (s for xampl, [3,4]). Dfnon : h sysm (5) s sad o b xponnally sabl undr swchng sgnal f h soluon x( ) of h sysm (5) sasfs γ ( ) x() Γ x for consans Γ and γ >. Movad by h da of wghd urbanc anuaon n [], h wghd cos funcon assocad wh h sysm (5) s gvn by () () J = [ x( ) Qx( ) u Ru ] d, (7) whr λ s a posv consan, Q and R ar posv dfn wghd marcs. Now, h wghd guarand cos conrol problm for h swchd sysm (5) s sad as follows: Dfnon : Consdr h sysm (5). If hr xs a conrol law u for ach subsysm and a swchng

3 494 Ru Wang and Jun Zhao law ( ), and a posv scalar J such ha for all admssbl uncrans, h closd-loop sysm s asympocally sabl and h valu of h cos funcon (7) sasfs J J, hn h sysm (5) s sad o sasfy wghd guarand cos conrol, J s sad o b a wghd guarand cos uppr bound. Dfnon 3 [8,9]: For any swchng sgnal and any τ, l N ( τ, ) dno h numbr of connus of on an nrval ( τ, ). If τ N (,) τ N (8) τ a hol for gvn N, τ a >, hn h consan τ a s calld h avrag dwll-m and N s h char bound. As commonly usd n h lraur, for convnnc, w choos N = n hs papr. L ( ) (rsp., ( ) ) dno h oal acvaon m of h sysm wh acuaor falurs (rsp. h sysm whou acuaor falurs,) durng [, ). For any gvn λ (, λ ), w choos an arbrary λ ( λ, λ ). Movad by h da n [], w propos h followng swchng law: (S) Drmn h swchng sgnal ( ) such ha h nqualy () λ λ () λ λ (9) hol for any gvn nal m, whr λ s a posv numbr o b chosn lar. Rmar 3: h da of swchng condon (S) s n fac o consran acvaon m of h sysm wh acuaor falurs (, ) rlavly small compard wh ha of h sysm whou acuaors falur. 3. MAIN RESULS In hs scon, w frs consdr h non-swchd dlay sysm (). Choos h Lyapunov funconal candda of h form h λ ( s) V( x ) = x Px x () s Zx() s, () whr P, Z ar posv dfn marcs o b chosn lar. Lmma : Gvn consan dlay h and posv consans λ, ε, f hr xs posv dfn marcs P and Z such ha h followng marx nqualy Θ PE h < E P Z hold, whr Θ= Q Z PA A P εpdd P ε N N λ P PBR B P, hn, undr h sa fdbac conrol u K = R B P, w hav λ( ) λ ( s) V( x ) V( x ) () = Kx wh [ x ( s) Qx( s) u ( s) Ru( s)]. Proof: S h Appndx. Whn acuaors falur occurs, h sysm () bcoms h form x () = ( AΔ A) x() Ex( h) B u, x ( θ) = ϕ( θ), θ [ h,]. By gnng sa fdbac conrol u () = Kx wh K = R B P, w hav h followng rsul. Lmma : Gvn consan dlay h and posv consan λ, ε, assum ha hr xs posv dfn marcs P and Z such ha h followng marx nqualy Π PE h < E P Z hold, whr Π= Q Z PA A P εpdd P ε N N hn, w hav Ω λ PPB R B P, Ω λ( ) λ ( s) V( x ) V( x ) (3) [ x ( s) Qx( s) u ( s) Ru( s)]. Proof: S h Appndx. Rmar 4: Lmma gvs som dcay sma for Lyapunov funconal candda V( x ) n (), whl Lmma gvs som sma of xponnal growh for V( x ). hs smas wll b usd o dvlop h man rsul. horm : If hr xs a s of posv scalars ε, λ and posv dfn marcs P, Z ( =,) such ha h marx nquals

4 Guarand Cos Conrol for a Class of Uncran Dlay Sysms wh Acuaor Falurs Basd on Swchng 495 Ξ PE, h < E P Z Ξ PE, h < E P Z hold, whr (4) (5) Q PA A P εpdd P ε N N Z λppbr B P, Q PA A P εpdd P ε N N Z λp PBΩR BΩP. Ξ = Ξ = hn, h sysm (5) undr h sa fdbac conrollrs (6) wh K = R B P s xponnally sabl for any swchng sgnal sasfyng h condon (S) and h avrag dwll m ln μ τa τa =, (6) whr μ sasfs P μp, Z μz,, j M. (7) j j Morovr, a wghd guarand cos uppr bound s gvn by λ [ λ s J = x P( ) x x ( s ) Z ( ) x ( s ) ]. h λ (8) Proof: Dfn a pcws Lyaponov funconal candda for sysm (5) as follows V( x ) = V ( x ) () λ ( s) = () h () x P x x () s Z x() s, (9) whr P, Z ( =,) sasfyng (4) and (5). Accordng o (7) and h dfnon of V () ( x ) n (9), w can asly oban V μv,, j M. () j For any gvn >, w l = < < < = N (, ) dno h swchng m nsans of ovr h nrval (, ). Usng (4), (5), Lmma and Lmma, w hav V( x ) = V ( x ) = V ( x ) () λ( ) λ( s) V( x ) ( ),, s = ( ) λ λ( s) V( x ) ( ),, s = () whr () s = x () s Qx() s u () s Ru(). s Combnng () and () la o λ (,) λ (,) ( ) λ ( s, ) λ ( s, ) ( ) λ (, ) λ (, ) μ ( ) ( ) λ ( s,) λ ( s,) ( )( s) λ (, ) λ (, ) μv ( ) x λ (, ) λ (, ) V( x ) V ( x ) V x μ μ λ ( s,) λ ( s,) ( ) λ ( s,) λ ( s,) ( )( ) ( ) ( s) () s s N (, ) λ (, ) λ (, ) V ( ) ( x) N (,) s λ (,) s (,) s μ λ (, ) λ (, ) N (, )ln μ ( ) ( ) λ (,) s λ (,) s N ( s,)ln μ = V x () s () s. () Frs, w gv h proof of h xponnal sably for swchd dlay sysm (5). Accordng o (8) and (6), w hav N (, )ln μ λ, >. (3) hrfor, follows from () and (3) ha λ (, ) λ (, ) N (, )ln μ ( ) λ (, ) λ (, ) ( λ λ) ( )( ) ( λ λ) ( ) ( ) ( ) ( ). V( x ) V ( x ) V x V x = V x From h Lyapunov funconal n (9), w hav (4) ax () Vx ( ) bx, (5) whr

5 496 Ru Wang and Jun Zhao a = mn λ ( P), M mn b= max λ ( P) hmax λ ( Z ). M max M Usng (4) and (5), w g max b x () V( x ) x. a a hrfor, b λ x() x, (6) a whch mpls h sysm (5) s xponnally sabl. In h followng, w show ha h closd-loop sysm sasfs h prformanc uppr bound. N (, )ln Mulplyng boh s of () by μ rsuls n N (, )ln μ V( x ) λ (, ) λ (, ) V ( )( x) λ (,) s λ (,) s N (,)ln s μ V ( )( x) λ (,) s λ (,) s N (,)ln s μ From (3), (7) can b wrn as ( ) ( ) ( ) λ ( s,) λ ( s,) s V x V x No ha V( x ), w now λ ( s, ) λ ( s, ) s V ( )( x). hrfor, () s () s. λ ( s) s λ ( s) s V( ) x (7) () s. () ( ).(8) Now, ngrang (8) from = o yl λ ( s) s ( s) () s d s λ ( s) = () s d s s = () s. I obvously hol ha s λ () s λ V ( ) x d V ( ) x ( ) = ( ). hus, w hav s () s λ V ( )( x) λ λ [ s = x P( ) x x ( s) Z ( ) x( s) ]. h λ hs s h nd of proof. Rmar 5: h conrollr gnd s swchng conrollr, n whch swchng law mus sasfy wo condons. On s h condon (S), whch consrans h srous falur m no oo larg; h ohr condon s abou avrag dwll m, whch s a consran of srous falur frquncy on acuaors. hrfor, horm ndcas ha h sysm () can sasfy guarand cos conrol on condon ha srous falur m s corrspondngly shorr and srous falur frquncy s also corrspondngly lowr. Rmar 6: Whn μ =, namly, τ a =, whch mpls ha swchng sgnals can b arbrary ons and a common Lypunov funconal s formd. In hs cas, h swchd sysm () sasfs guarand cos conrol undr arbrary swchng. Morovr, sng λ =, whch mans no swchng bwn subsysms, h wghd guarand cos conrol dgnras no a rgular guarand cos conrol problm whou wghng λτ for a sngl subsysm. horm prov a suffcn condon for h soluon o h wghd guarand cos conrol problm. Howvr, nquals (4) and (5) ar no asy o solv snc hy ar no LMIs. h followng rmar shows how o urn (4) and (5) no LMIs, whch can b asly solvd by Malab. Rmar 7: Pr- and pos-mulplyng boh s of nquals (4) and (5) by dag{ X, X } and dag{ X, X }, rspcvly, whr X,), w oban X QX AX X A ε P = ( = DD ε X N NX EX XZX X BR B h XZX <, (9)

6 Guarand Cos Conrol for a Class of Uncran Dlay Sysms wh Acuaor Falurs Basd on Swchng 497 XQX AX XA εdd ε X N NX EX <. XZX X BΩR BΩ h XZX (3) W dfn XZX = Y, XZX = Y. Accordng o Schur complmn Lmma, marx nquals (9) and (3) ar quvaln o h followng LMIs AX XA ε DD EX X XN Y XBR B h Y, < Q ε I (3) AX X A Y εdd X EX X X N BΩR BΩ <. h Y Q ε I (3) Rmar 8: No ha n LMIs (3) and (3)ε can b rgardd as a varabl. In addon, n ordr o g a lowr guarand bound, w nd a largr μ and a smallr λ, whch can b ralzd by paramr rav mhod. Bu hs rsuls n largr avrag dwll m n (6), whch s of cours unrabl. hus, w nd o slc h paramrs μ and λ accordng o praccal rqurmn. 4. EXAMPLE In hs scon, w apply h proposd gn mhod o llusra h rvr polluon conrol problm. L z( ) and q ( ) dno h concnraons pr un volum of bochmcal oxygn dmand (BOD) and solvd oxygn (DO), rspcvly, a m, n a rach of a pollud rvr. L z and q, corrspondng o som masur of war qualy sandar, dno h rd sady valus of z and q, rspcvly. Dfn = = () = ( () () ). x () z() z, x () q() q, x x x hn h dynamc quaon for x s crbd by [5,6]. x () = ( AΔ A) x() ( EΔE) x( h) ( E ΔE) x( h) Bu w, whr η η A =, 3 η η Δ() η Δ A () =, B=, 3() () Δ Δ βη ηδβ E =, Δ E =, βη ηδβ ( β) η E =, ( β) η η Δβ Δ E =, η Δβ v() Δ() z w =, s v() Δ3() z Δ()( q q) ( ) (33) u () = u() u() s h conrol varabls of rvr polluon. h physcal manng of hs paramrs can b found n [5]. Whn w=, h = h, (33) can b rwrn as h followng uncran lnar dlay sysm x () = ( AΔ A) x() Ex( h) Bu, (34) η whr E =. η In hs smulaon, w choos =.8, =, =.5, η =.4, η =.6, Δ ( ) =.sn, Δ 3 3 =Δ ( ) =.4sn, h =.. hus ()..4 A=, B=, Ω= {}, BΩ =, E =,.6 h paramr uncrans Δ A( ) = DF( ) N,.. whr F ( ) = sn, D=, N = Suppos ha h fauly acuaors can b rcovrd hrough a m nrval, h guarand cos conrol problm of sysm (34) can b solvd by usng horm Choosng λ =, ε =, w g posv dfnon marcs P, Z ( =,) by solvng LMIs (3), (3)

7 498 Ru Wang and Jun Zhao P =, P =, Z =, Z = Choosng μ =.4, λ =.5, λ =, from (6), w ln μ gτ a = = Accordng o horm, f h acvaon m rao bwn h sysm whou acuaor falur and h sysm wh acuaor falurs s no lss han () λ λ = 9, () λ λ xponnal sably s achvd. Morovr, from (6), w hav.5( ) x ( ) 4.78 x. L h nal sa of sysm (34) b x () = (.5) for.. From (8), h wghd guarand cos uppr bound s λ [ λ s J = xpx x ( s ) Zx ( s ) λ ] h = CONCLUSIONS In hs papr, w hav nvsgad h problm of guarand cos conrol for a class of lnar dlay sysms for h cas whr acuaors suffr falurs. W focusd on h cas ha h nvr fald acuaors ar nadqua o sablz h sysms by a sngl sa fdbac. Suppos ha h fauly acuaors can b slf-rpard hrough a m nrval, h nr sysm can b rgardd as a swchd sysm Basd on avrag dwll m schm, w hav gnd h swchng sa fdbac conrollrs n rms of LMIs such ha h consdrd dlay sysms s xponnally sabl and a wghd guarand cos uppr s drvd. APPENDIX Proof of Lmma : h drvav of V( x ) along h rajcory of h dlay sysm () s gvn by ( ) ( ) ( ) ( ) ( ) h λ x = x Px x Zx x ( h) s λ ( ) λ h [ ] Zx( h) x ( szxs ) ( ) = x ( ) P ( AΔ A) x( ) Ex( h) Bu h x ( ) Zx( ) x ( h) Zx( h) λ ( s) λ x ( s) Zx( s) h x ( PA A P εpdd P ε N N Z PBR B P) x x ( ) PEx( h) ( ) ( ) h x h E Px x ( h) Zx( h) hrfor, V λ ( s) h λ x ( s) Zx( s). x ( PA A P εpdd P ε N N Z P PBR B P) x x ( ) PEx( h) λ h x ( h) E Px( ) x ( h) Zx( h) ΘQ x () PE x () = PBR B P. x( h) x( h) h E P Z From (), w hav λ V x ( Q PBR B P) x λ V ( x Qx u Ru). (35) By usng h dffrnal hory and (35) for (), w hav λ( ) λ ( s) V( x ) V( x ) [ x ( s) Qx( s) u ( s) Ru( s)]. Proof of Lmma : Smlarly o h proof of Lmma, dffrnang V( x ) along h rajcory of sysm () rsuls n [ ] ( x ) = x P ( AΔ A) x Ex( h) BKx h x ( ) Zx( ) x ( h) Zx( h) λ ( s) h λ x ( s) Zx( s). No ha h oupus of fauly acuaors corrspondng o any Ω ar assumd o b zro,.., h conrol npu u (x) may b appld o h plan only hrough normal acuaors, w hav

8 Guarand Cos Conrol for a Class of Uncran Dlay Sysms wh Acuaor Falurs Basd on Swchng 499 BK = B R B P. From (3), w hav BΩR BΩ B R B. (36) hrfor, [ ] ( x ) = x P ( AΔ A) x Ex( h) B u h x ( ) Zx( ) x ( h) Zx( h) λ ( s) λ x ( s) Zx( s) h x ( PA A P εpdd P ε N N Z PB R B P) x x ( ) PEx( h) h x ( h) E Px( ) x ( h) Z λ ( s ) h x ( h) x () s Zx() s. Accordng o (36) and (3), w hav V x ( PA A P εpdd P ε N N Z P PB R B P) x x ( ) PEx( h) λ Ω Ω h x ( h) E Px( ) x ( h) Zx( h) λ ( s) 4 λ x ( s) Zx( s) h x ( PA A P εpdd P ε N N Z P PBΩR BΩP) x x ( ) PEx( h) x ( h) E ( ) h Px x ( h) Zx( h) ΠQ x () PE x () = PBΩR BΩP x( h) x( h) h E P Z. hrfor, λ V ( x Qx u Ru), whch n urn gvs λ( ) λ ( s) V( x ) V( x ) [ x ( s) Qx( s) u ( s) Ru( s)]. REFERENCES [] Q. L. Han and K. Q. Gu, On robus sably of m-dlay sysms wh norm-boundd uncrany, IEEE rans. on Auomac Conrol, vol. 46, no. 9, pp ,. [] Y. H, M. Wu, J. H. Sh, and G. P. Lu, Dlaydpndn robus sably crra for uncran nural sysms wh mxd dlays, Sysms & Conrol Lrs, vol. 5, no., pp , 4. [3] Y. H, M. Wu, J. H. Sh, and G. P. Lu, Paramr-dpndn Lyapunov funconal for sably of m-dlay sysms wh polyopcyp uncrans, IEEE rans. on Auomac Conrol, vol. 49, no. 5, pp , 4. [4] M. Wu, Y. H, J. H. Sh, and G. P. Lu, Dlaydpndn crra for robus sably of mvaryng dlay sysms, Auomaca, vol. 4, no. 8, pp , 4. [5] Y. S. Moon, P. Par, W. H. Kwon, and Y. S. L, Dlay-dpndn robus sablzaon of uncran sa-dlayd sysms, Inrnaonal Journal of Conrol, vol. 74, no. 4, pp ,. [6] H. J. Gao and C. H. Wang, Dlay-dpndn robus and flrng for a class of uncran nonlnar m-dlay sysms, IEEE rans. on Auomac Conrol, vol. 48, no. 9, pp , 3. [7] B. Chn, J. Lam, and S. Y. Xu, Mmory sa fdbac guarand cos conrol for nural dlay sysms, In. J. Innovav Compung, Informaon and Conrol, vol., no., pp , 6. [8] W. H. Chn, Z. H. Guan, and X. M. Lu, Dlaydpndn oupu fdbac guarand cos conrol for uncran m-dlay sysms, Auomaca, vol. 4, no. 7, pp , 4. [9] S. S. L. Chang and. K. C. Png, Adapv guarand cos conrol of sysms wh uncran paramrs, IEEE rans. on Auomac Conrol, vol. 7, no. 4, pp , 97. [] L. Yu and F. Gao, Opmal guarand cos conrol of cr-m uncran sysms wh boh sa and npu dlays, Journal of Franln Insu, vol. 338, no., pp. -,. [] I. R. Prsn and D. C. McFarlan, Opmal guarand cos conrol and flrng for uncran lnar sysms, IEEE rans. on Auomac Conrol, vol. 39, no. 9, pp , 994. [] L. Yu, Z. H. Guan, and M. X. Sun, Opmal guarand cos conrol of lnar uncran sysms wh npu consrans, Inrnaonal Journal of Conrol, Auomaon, and Sysms, vol. 3, no. 3, pp , 5. [3] J. H. Par and K. Cho, Guarand cos conrol of uncran nonlnar nural sysms va mmory sa fdbac, Chaos, Solons and Fracals, vol. 4, no., pp. 83-9, 5. [4] J. S. Y, F. H. Yang, and J. L. Wang, Rsln guarand cos conrol o olra acuaor fauls

9 5 Ru Wang and Jun Zhao for cr-m uncran lnar sysms, IEE Procdngs-Conrol hory and Applcaons, vol. 47, no. 3, pp ,. [5] G. H. Yang, J. L. Wang, and C. S. Yng, Rlabl guarand cos conrol for uncran nonlnar sysms, IEEE rans. on Auomac Conrol, vol. 45, no., pp. 88-9,. [6] L. Yu, An LMI approach o rlabl guarand cos conrol of cr-m sysms wh acuaor falur, Appld Mahmacs and Compuaon, vol. 6, no. 3, pp , 5. [7] R. Wang and J. Zhao, Rlabl H conrol for a class of swchd nonlnar sysms wh acuaor falurs, Nonlnar Analyss: Hybrd Sysms, vol., no. 3, pp , 7. [8] J. P. Hspanha and A. S. Mors, Sably of swchd sysms wh avrag dwll-m, Proc. of h 38h IEEE Confrnc on Dcson and Conrol, pp , Dc [9] D. Lbrzon, Swchng n Sysms and Conrol, Brhausr, Boson, 3. [] G. S. Zha, B. Hu, K. Yasuda, and A. N. Mchl, Dsurbanc anuaon proprs of mconrolld swchd sysms, Journal of h Franln Insu, vol. 338, no. 7, pp ,. [] X. M. Sun, J. Zhao, and D. J. Hll, Sably and L -gan analyss for swchd dlay sysms: A dlay-dpndn mhod, Auomaca, vol. 4, no., pp , 6. [] G. S. Zha and H. Ln, Conrollr falur m analyss for symmrc H conrol sysms, Inrnaonal Journal of Conrol, vol. 77, no. 6, pp , 4. [3] M. A. Dmrou, Adapv rorganzaon of swchd sysms wh fauly acuaors, Proc. of h 4h IEEE confrnc on Dcson and Conrol, pp ,. [4] C. Edwar and C. P. an, Faul olran conrol usng sldng mod obsrvrs, Proc. of h 43rd IEEE confrnc on Dcson and Conrol, pp , 4. [5] C. S. L and G. Lmann, Connuous fdbac guaranng unform ulma bounddnss for uncran for uncran lnar dlay sysms: An applcaon o rvr polluon conrol, Compurs and Mahmacs wh Applcaons, vol. 6, no. -, pp , 988. [6] F. Zhng, Q. G. Wang, and. H. L, Adapvrobus conrol of uncran m dlay sysms, Auomaca, vol. 4, no. 8, pp , 5. conrol. Ru Wang rcvd h B.S. and M.S dgrs n Mahmacs n and 4, rspcvly, boh from Boha Unvrsy, Chna. Sh compld h Ph.D. dgr n Conrol hory and Applcaons n 7 a Norhasrn Unvrsy, Chna. Hr man rsarch nrss nclud swchd sysms, dlay sysms, and faul-olran Jun Zhao rcvd h B.S. and M.S. dgrs n Mahmacs n 98 and 984, rspcvly, boh from Laonng Unvrsy, Chna. H compld h Ph.D. dgr n Conrol hory and Applcaons n 99 a Norhasrn Unvrsy, Chna. From 99 o 993, h was a Posdocoral Fllow a h sam unvrsy. Snc 994 h has bn wh h School of Informaon Scnc and Engnrng, Norhasrn Unvrsy, Chna, whr h s currnly a Profssor. From Fbruary 998 o Fbruary 999, h was a Vsng Scholar a h Coordnad Scnc Laboraory, Unvrsy of Illnos a Urbana-Champagn. Hs man rsarch nrss nclud hybrd sysms, nonlnar sysms, gomrc conrol hory, and robus conrol.