A Backstepping Simple Adaptive Control Application to Flexible Space Structures

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1 Chins Journal of Aronauics 5 (01) Conns liss availabl a cincdirc Chins Journal of Aronauics journal hompag: A Backspping impl Adapiv Conrol Applicaion o Flxibl pac rucurs LIU Min, XU hiji, HAN Chao chool of Asronauics, Bihang Univrsiy, Bijing , China Rcivd Novmbr 011; rvisd 4 January 01; accpd Fbruary 01 Absrac Alhough h simpl adapiv conrol (AC) is widly sudid boh in hory and applicaion in flxibl spac srucur conrol and ohr conrol problms, i is rsricd by h almos sricly posiiv ral (APR) condiions. In mos pracical conrol problms, h APR condiions ar no saisfid. hrfor, basd on h AC hory, his papr proposs a backspping simpl adapiv conrol algorihm which suis h sysm wih arbirary rlaiv dgr wih no nd of paralll fdforward compnsaor. h proposd conrol algorihm consiss of dcomposiion of h arbirary rlaiv dgr sysm ino a known subsysm and an unknown APR subsysm which ar conncd in cascad, dsign of consan oupu fdback conrollr for h known subsysm, and implmnaion of backspping mhod and AC of h unknown APR subsysm. Inhriing h characrisics of h AC, his mhod can b adapiv onlin for h paramr uncrainis. hn, h applicaion of h proposd conrollr o larg flxibl spac srucur wih collocad snsors and acuaors is sudid, and h simulaion rsuls valida h proposd conrollr. I is a nw sragy o apply h classical AC o high rlaiv dgr plans. Kywords: simpl adapiv conrol; almos sricly posiiv ral; backspping; flxibl srucur; inrmdia conrol law 1. Inroducion 1 h simpl adapiv conrol (AC) mhodology was firs inroducd by obl, al. in 1979 [1] and furhr dvlopd by Bar-Kana, al. []. I has also bn dvlopd by Balas, al. o infini-dimnsional sysms [3], discr sysms [4] and sysms wih unknown dlays and prsisn disurbancs [5]. I givs a low-ordr conrollr wih no nd of h ordr knowldg of h conrolld plan, so i can sabiliz larg-scal sysm wih small numbr of adjusabl paramrs. h simpliciy and robusnss of AC hav ld o succssful implmnaions in such divrs applicaions as flxibl spac srucurs [,6-8], fligh conrol [9-10], powr sysms [11], roboics [1-13] and wind Corrsponding auhor. l.: addrss: sarsjxu@yahoo.com.cn Foundaion im: Naional Naural cinc Foundaion of China ( ) Elsvir Ld. Opn accss undr CC BY-NC-ND licns. doi: / (11) urbins [14-16]. In ordr o apply a simpl adapiv conrollr, h sysm should comply wih som rquirmns. On of h rquirmns is ha h plan should b almos sricly posiiv ral (APR) [8], ha is, hr xiss a consan oupu fdback so ha h rsuling closd-loop sysm is sricly posiiv ral. I has bn shown ha plans wih minimum-phas ransfr funcions of rlaiv dgr 1 in h singl-inpu singl-oupu (IO) cas, or of rlaiv dgr m in h mulipl-inpu mulipl-oupu (MIMO) mulivariabl cas, ar APR [13]. Howvr, mos pracical plans ar no APR. aking h flxibl spac srucur as an xampl, i has a rlaiv dgr of, so i is no an APR sysm. In ordr o apply h AC o h flxibl spac srucur sysm, Mhil and Balas [17-18], Mufi [7] and Bar-Kana, al. [,8,19] hav don a lo of jobs o solv h APR problm. Bu hos soluions ar no suiabl for many ohr conrol problms, spcially h sysm wih highr rlaiv dgr. On nds a dsign mhod which can xpand h AC o high rlaiv dgr sysms.

2 No.3 LIU Min al. / Chins Journal of Aronauics 5(01) I has bn subsqunly shown how various forms of paralll fdforward compnsaors (PFCs) [10,13,0] can b usd o saisfy h APR condiions. Howvr, wih h addiion of h fdforward compnsaor, h oupu racking rror of h conrolld sysm is no longr guarand o approach zro asympoically, bu rmains boundd. Bsids, h dsign of an appropria PFC accura paramrs of h original plan. omims his may b unfasibl, bcaus usually h AC mhod is adopd in plans wih paramr uncrainis. In his papr, basd on backspping conrol algorihm [1], a nonlinar backspping simpl adapiv conrollr is drivd for plans wih arbirary rlaiv dgr. On of h advanags is ha h proposd conrol algorihm is fr of PFCs. h basic dsign procdur consiss of hr sps. Firsly, h arbirary rlaiv dgr unknown plan is dcomposd ino a known subsysm and an unknown nonlinar subsysm wih h linar par bing APR sysm and h nonlinar par bing Lipchiz coninuous. hn, h oupu of h unknown subsysm is slcd as h virual conrol vcor and a consan fdback conrol is dsignd o sabiliz h known subsysm. Finally, via adoping backspping conrol mhod, h unknown subsysm is considrd and h backspping simpl adapiv conrollr is drivd. h sabiliy of h proposd conrollr is invsigad by Lyapunov sabl hory and posiiv ral hory. o valida h fficincy of h conrollr, is applicaion o larg flxibl spac srucur is sudid by dcomposing h larg flxibl spac srucur ino h kinmaics subsysm and h dynamics subsysm. h APR propry of h dynamics subsysm is confirmd, and hn h backspping simpl adapiv conrollr for flxibl srucurs wih collocad snsors and acuaors is givn. Finally, a numrical xampl is prsnd.. Problm amn In many pracical nginring problms, h conrolld sysm can b dscribd by subsysms in sris. h mos common cas is ha h conrolld sysm is formd by a dynamics subsysm which dscribs h voluion of h vlociis as im progrsss and a known kinmaics subsysm which dscribs h displacmn rsponss, and hs subsysms ar conncd by vlociis. For xampl, h flxibl spac srucurs can b dscribd by a dynamics subsysm and a simpl known kinmaics subsysm in cascad. Evn in a gnral cas, on can us h mhod dscribd in Rf. [] o dcompos h conrolld sysm ino subsysms. o w assum ha h coninuous nonlinar im invarian conrolld sysm wih rlaiv dgr n can b dcomposd ino h wo subsysms conncd in cascad as follows: x () Ax() Bu() f( x,) : (1) y() Cx() p x p() Apxp() Bp y() : yp() Cpxp() () whr h subsysm is an unknown nonlinar sysm wih rlaiv dgr m. h plan sa x () is a n 1 vcor, u () h m 1 conrol inpu, y () h m 1 sysm oupu vcor, and A, B and C ar unknown marics whil h plan {A, B, C } is APR. W assum hroughou his papr ha f (x, ) is Lipchiz coninuous. h subsysm p is a known coninuous linar im invarian sysm wih rlaiv dgr nm, and h subsysm p can b sabilizd by an oupu fdback conrollr. x p () is h n p 1 plan sa vcor, y () h m 1 conrol vcor of subsysm p, y p () h m 1 oupu vcor, and A p, B p and C p ar known marics wih appropria dimnsions. h objciv of conrol in his papr is o dsign h conrol inpu u () o caus h oupu y p () of h subsysm p o rack h oupu of h coninuous nonlinar im invarian rfrnc modl asympoically which can also b dcomposd ino h wo subsysms conncd in cascad as follows: x m1() Am1xm1() Bm1um1() g( xm1,) m1 : ym1() Cm1xm1() m x m() Apxm() Bp ym1() : ym() Cpxm() (3) (4) whr h rfrnc subsysm m1 sa x m1 () is an n m1 1 vcor wih m 1 oupu y m1 (), and g(x m1,) h nonlinar par of h m1. h subsysm m1 is dsignd o m som dsird prformanc propris and has h sam numbr of oupu as h subsysm, bu x m1 () and x () nd no o hav h sam dimnsions, and i will b prmissibl o hav n =dim( x ) dim( x ) n (5) p m1 whr h opraor dim dnos h dimnsion of h vcor. u m1 () is m 1 conrol vcor of rfrnc modl and can b prsnd as h oupu of an command gnraing sysm of h form v m() Av vm() (6) um1() Cvvm () whr v m () is h command sa vcor. h marics A v and C v ar unknown, and only masurmns of h inpu u m1 () ar prmid. inc subsysm p is a known sysm, h subsysm m is slcd h sam as p. x m () is h n p 1 plan sa vcor, y m1 () h m 1 conrol vcor of subsysm p, and y m () h m 1 oupu vcor. 3. Backspping Adapiv Conrollr According o h backspping conrol algorihm,

3 448 LIU Min al. / Chins Journal of Aronauics 5(01) No.3 y () is slcd as h virual conrol vcor o dsign an inrmdia conrol law for subsysm p. Firs, dfin h oupu rror of h subsysm as () y () y () (7) p m hn, l h sa rror of h sysm p b dfind as () x () x () (8) p p m o, h rror quaions of p ar p () App () Bp ( y() ym1()) () Cpp() (9) lc h inrmdia conrol law as () y () k () (10) m1 1 whr k 1 is a posiiv dfini marix. If y ()= (), h nx quaion holds. () ( A B k C ) () A () (11) p p p 1 p p c p h posiiv dfini Lyapunov funcion is slcd as V () P () (1) 1 p p whr P = P > 0. aking h drivaiv of h Lyapunov quaion V 1 and subsiuing Eq. (11), hr is V ()( A PPA ) () (13) 1 p c c p inc p can b mad sabl by oupu fdback, hr is a posiiv dfini marix Q which saisfis AcPPAc Q (14) ubsiuing Eq. (14) ino Eq. (13) rsuls in V 1 p() Qp() 0 (15) I is obvious ha V 1 0 if and only if p ()=0, namly h s E={ p p ()=0} is h largs invarian s conaind in V 1 0. hn according o h Laall invarianc principl of diffrnial quaion, whn y ()= (), undr h inrmdia conrol law dscribd in Eq. (10), h oupu of subsysm p racks y m () asympoically, ha is p ()0 and ()0 as. hn, backing a sp, considring h subsysm, backspping conrol algorihm is adopd o g h adapiv conrollr for h conrolld sysm. Whn prfc oupu acking occurs (i.., whn y () = y m1 () for ), h corrsponding sa and conrol rajcoris ar dfind o b h idal sa and idal conrol rajcoris, rspcivly. hs idal rajcoris ar dnod by x () and u (), whr h suprscrip dnos h idal racking condiion. By dfiniion, hs idal rajcoris saisfy Eq. (1), hrfor, () () () (,) (16) x Ax Bu f x : y() Cx() ym1() Assuming ha h idal rajcoris ar a linar combinaion of h modl rfrnc sysm, a linar ransformaion can b wrin as x() 11x m1() + 1um1() (17) u () K x () + K u () whr 11, 1, x m1 u m1 K x and Ku ar consan bu unknown marics. aking h drivaiv of h linar ransformaion Eq. (17) and subsiuing Eq. (3) and Eq. (6) ino i rsuls in h following quaions which ar calld h maching condiions. A11 BKx 11Am1 ( A1 BKu) Cv 11Bm1Cv 1CvAv f( x, ) 11g( xm1, ) (18) C11 Cm1 C1Cv 0 If h maching condiions ar saisfid and y () y (), on gs m1 y () () (19) Dfin h oupu rror of subsysm as () y () () (0) Whn y () diffrs from () a =, asympoic racking is achivabl providd sabilizing oupu fdback is includd in h conrol law. h conrol law u () K x () K u () K () (1) x m1 u m1 rsuls in y () ()y m1 () and ()0, whr K is a sabilizing oupu fdback gain, and K always xiss bcaus h subsysm saisfis h APR condiions. Whn asympoic racking occurs, h corrsponding sa and conrol inpu ar dfind o b h asympoic sabl sa rajcoris and conrol rajcoris. hs rajcoris ar dnod by x ' () and u ' () rspcivly, and h suprscrip ' dnos h asympoic racking condiion. o h nx quaions saisfy: ' ' ' ' ' x () Ax() Bu() f( x,) : () ' ' y() Cx() and h rror ' () y () () (3) vanishs asympoically. h sa racking rror is dfind as

4 No.3 LIU Min al. / Chins Journal of Aronauics 5(01) ' () x () x () (4) From Eq. (0) and Eq. (3), w hav () C () () (5) Wih h rsul abov, h backspping adapiv law is chosn o hav a similar form o Eq. (1) and is dscribd as u() K() r() Kx () xm1() Ku() um1() K() () (6) whr K() [ Kx() Ku() K ()] (7) r() [ x () u () ()] (8) m1 h gains K x (), K u () and K () ar adapd as follows: K() () r (), K(0) K0 K x() () xm1() x, K x(0) K x0 K u() () um1() u, Ku(0) Ku0 K () () (), K(0) K0 m1 (9) whr, x, u and ar posiiv dfini adapiv paramr marics ha drmin h ra of adapaion, K 0, K x0, K u0 and K 0 ar iniial valus. 4. Closd-loop abiliy Analysis aking h drivaiv of h sa racking rror () and subsiuing Eq. (1) and Eq. () ino i rsuls in () x () x () A ()+ ' x x m1 u u m1 B ( K K ) x ( )+ B ( K K ) u ( )+ B ( K K ) () B K () B K ()+ W form ' f( x, ) f( x, ) K x Kx Kx K u Ku Ku K K K K [ K x K u K ] hn, Eq. (30) can b dscribd as () A ()+ B Kr ()+ B K () BK () f ' (30) (31) (3) whr f f( x, ) f( x, ). ubsiuing h dfiniion of () ino Eq. (3) givs () A ()+ B Kr () f (33) c1 whr Ac1 A BKC. Equaions (5) and (33) ar combind o obain h racking rror sysm: () Ac1()+ BKr () f (34) () C() () h sabiliy of h nonlinar sysm dscribd as Eq. (34) can b analyzd by Lyapunov hory. h posiiv dfini Lyapunov funcion is slcd as V V1V V1() P() 1 r( K K ) 0 (35) whr P is a posiiv symmric marix. V 1 quals o h Lyapunov funcion dfind in viw of h subsysm p and dscribd as Eq. (1). inc 1 ()0, h drivaiv of V 1 is no Eq. (15) any mor. ubsiuing y ()=()+ () ino Eq. (1) rsuls in p () Ac p () Bp() (36) V () Q () () PB () (37) 1 p p p p aking h drivaiv of h Lyapunov quaion V along h rajcoris of Eq. (34) and using Eq. (9) yild and V ()( ) () Ac1P PAc1 ( ) P B Kr ( ) r( K K) 1 ( ) P f 1 1 r( K K ) r( ( ) r ( ) K ) r( () () ) () () (38) r K Kr (39) ubsiuing Eq. (39) ino Eq. (38) rsuls in V ()( ) () Ac1P PAc1 () P B Kr () () Kr () ( ) P f (40) inc subsysm saisfis APR condiions, hr is a posiiv dfini marix Q which saisfis h Kalman-Yacubovic condiions: and A P P A Q PB C c1 c1 ubsiuing Eq. (5) ino Eq. (40) rsuls in V () Q () () Kr () ( ) P f V () Q () () Q () p p ( ()) Kr () () PB () p p p () p () () (41) (4) PB C P f (43) inc () vanishs asympoically as im nds o

5 450 LIU Min al. / Chins Journal of Aronauics 5(01) No.3 b infiniy, h hird and h fourh ims on h righ hand of Eq. (43) can b omid sinc hs ims hav no ffc on h sabiliy of h conrolld sysm. On can prov his judgmn by h modifid invarianc principl and h daild vidncs can b found in Rf. [3]. hrfor, V p () Qp () () Q() () PB C () () P f (44) p p Using h Cauchy-chwarz inqualiy on p ( ) P B p C () if nonlinariy f(x,) is Lipchiz, hn ' ' L f ( x, ) f ( x, ) x x (45) or f L (), whr L is a posiiv consan cofficin. ubsiuing Eq. (45) ino Eq. (44) givs whr V () Q () () Q () p p 1 PBpC ( p () () ) () P L () p ( ) ( ) (46) 1 min ( Q) sup PBpC 0 1 min ( Q ) sup PBpC sup( P L) 0 (47) and min (Q) dnos h smalls ignvalu of Q. o V ( ) ( ) 0 (48) p If and only if ()=0 and p ()=0, V 0 holds. W can us Laall invarianc principl of diffrnial quaion o mak h sysm oupu rack h rfrnc oupu asympoically, i.. h oupu racking rror () and () vanish asympoically as im nds o b infiniy whil h adapiv gains rmain boundd. 5. Applicaion o Flxibl pac rucurs In his scion, backspping simpl adapiv conrol algorihm is usd o dsign conrollr for larg spac srucur in h collocad cas. h larg flxibl spac srucur modl usd in h subsqun simulaions can b dfind by h following marics [8] : 0 A I p 1/ B whr B is h inpu marix, p 0 B (49) (50) 1 n 1 diag(,,, ), (51) n dnos h nh modal frquncy, and is h damping cofficin. h vlociy is slcd as oupu, hn Cp [ 0 C] (5) whr C is h oupu marix. If h snsors and h acuaors ar collocad, i.., if B C (53) hn, 0 CB p p [ 0 C] CC 0 (54) B his shows ha h flxibl srucur wih collocad snsors and acuaors is minimum-phas and is produc C p B p >0. In ohr words, h dynamics subsysm of h flxibl srucur is APR [8]. hr xiss an unknown consan oupu fdback conrol gain K b and posiiv dfini symmric marics P 1, Q 1, making h Kalman-Yacubovic quaions P1 ( Ap BKC p b p) ( Ap BKC p b p ) P1 Q1 PB 1 p Cp (55) saisfid. Whn h snsors and h acuaors ar collocad, h sysm is APR by slcing vlociy as h oupu. Bu in spac missions, dsignrs car abou posiion as wll as vlociy. hrfor, in Rf. [19], vlociy plus scald posiion fdback is usd. In his papr, h backspping simpl adapiv mhod proposd abov is usd o dsign conrollrs for larg spac srucur by aking h posiion ino considraion. According o h backspping algorihm, h flxibl spac srucur modl is dfind as x () Apx() Bpu() 1 : (56) y() Cp x() : ( ) y( ) (57) whr x() [ () ()], y() () is slcd as h virual conrol vcor, and () h sysm oupu. o a rfrnc modl can b dfind as 1m x m() Amxm() Bmum() : ym() Cmxm() m m m n (58) : ( ) y ( ) (59) whr A m, B m and C m ar known marics, m () is h rfrnc oupu, and h rfrnc modl has idal closd-loop rsponss. o h backspping simpl adapiv conrollr applid o larg spac srucur wih collocad snsors and acuaors is

6 No.3 LIU Min al. / Chins Journal of Aronauics 5(01) f() ym() k( () m()) f() y() f() u() Kf() rf() Kf x () xm() Kf () m() f () f() u u K K f() () rf () f (60) whr h firs quaion dnos h inrmdia conrol law, k is a posiiv dfini marix, K f () = [K fx () K fu () K f ()] h adapiv conrol marix and r f () m m f [ x ( ) u ( ) ( )], f h posiiv dfini adapiv paramr marix. 6. Illusraiv Exampl Fig. Closd-loop im rspons o an impuls disurbanc (k=1 and c=0.3). In his scion, a wo-mass spring sysm which is wll known as a ypical xampl of flxibl spaccraf srucur is considrd [4] (s Fig. 1). Fig. 1 wo-mass spring sysm. In Fig. 1, x 1 and x ar displacmns of h masss, and U is h conrol inpu of h sysm. I can b assumd ha boh bodis hav uni mass, i.., m 1 =m = 1 kg, and h wo masss ar conncd by a spring wih a spring cofficin k and a dashpo wih a damping cofficin of c. h spring consan k has a nominal valu of 1 and is assumd o b uncrain (0.5k). o h flxibl mod frquncy can vary bwn 1 rad/s and rad/s. h dashpo damping cofficin c has a nominal valu of 0.3 and is assumd o b unknown oo. Whn x 1 is masurd, h sysm is a flxibl srucur wih collocad snsor and acuaor. o h sysm saisfis APR condiions. h backspping adapiv conrollr can b usd dircly. In h collocad cas, h backspping adapiv conrollr is Eq. (60). h aim of h conrol is o mak x 1 sabl, so, h rfrnc sa x m ()=0 is slcd, h rfrnc inpu u m ()=0, and h rfrnc oupu m ()=0. Hr w choos k=1, =0.5. h iniial condiion is ha hr is an impuls disurbanc wih iniial valu of x 1 (0)=1. imulaion rsuls wih k and c bing nominal valu ar shown in Fig.. h rsuls wih k= and c=0 ar shown in Fig. 3. h simulaion rsuls show ha h conrol prformanc of h conrollr dvlopd in his papr is good, vn whn h sysm has paramr uncrainis, and h conrollr has srong robusnss o h sysm paramr uncrainis. Fig. 3 Closd-loop im rspons o an impuls disurbanc (k= and c=0). 7. Conclusions 1) In his papr, a backspping simpl adapiv conrol mhod is dvlopd for high rlaiv dgr nonlinar sysms which can b dcomposd ino subsysms mniond abov. h proposd conrollr is simpl, low-dimnsiond, wih no nd of full sa fdback or sa obsrvr. I dos no dpnd on h accura modl of conrolld sysm, and has srong robusnss o sysm paramr uncrainis. ) h proposd conrollr is applid o flxibl spac srucur wih collocad snsors and acuaors and h simulaion rsuls valida h fficincy of h conrollr. 3) h conrol mhod prsnd in his papr is a nw approach o xpand h applicaion of simpl adapiv conrol o high rlaiv dgr sysms wih no nd of paralll fdforward configuraions. Rfrncs [1] obl K, Kaufman H, Mablius L. Modl rfrnc oupu adapiv conrol sysm wihou paramr idnificaion. 18h IEEE Confrnc on Dcision and Con-

7 45 LIU Min al. / Chins Journal of Aronauics 5(01) No.3 rol including h ymposium. 1979; [] Bar-Kana I, Kaufman H, Balas M. Modl rfrnc adapiv conrol of larg srucural sysms. AIAA Journal of Guidanc 1983; 6(): [3] Balas M J. Fini-dimnsional dirc adapiv conrol for discr-im infini-dimnsional linar sysms. Journal of Mahmaical Analysis and Applicaion 1995; 196: [4] Balas M J, Funs R, Mhil E. Adapiv conrol and rjcion of prsisn disurbancs: coninuous and discr-im. AIAA , 003. [5] Balas M J. Adapiv conrol of nonlinar sysms wih unknown dlays and prsisn disurbancs. AIAA , 010. [6] Bar-Kana I, Kaufman H. om applicaions of dirc adapiv conrol o larg srucural sysms. Journal of Guidanc 1984; 7(6): [7] Mufi I H. Modl rfrnc adapiv conrol for larg srucural sysms. Journal of Guidanc 1987; 10(5): [8] Barkana I, Bn-Ashr J Z. A simpl adapiv conrol applicaion o larg flxibl srucurs. 15h IEEE Mdirranan Elcrochnical Confrnc. 010; [9] Mooij E. Modl rfrnc adapiv guidanc for rnry rajcory racking. AIAA , 004. [10] Barkana I. Classical and simpl adapiv conrol for nonminimum phas auopilo dsign. Journal of Guidanc, Conrol, and Dynamics 005; 8(4): [11] Barkana I, Fischl R. A simpl adapiv nhancr of volag sabiliy for gnraor xciaion conrol. Procdings of h Amrican Conrol Confrnc. 199; [1] Ulrich, asiadk J Z, Barkana I. Modling and dirc adapiv conrol of a flxibl-join manipulaor. Journal of Guidanc, Conrol, and Dynamics 01; 35(1): [13] Kaufman H, Barkana I, obl K. Dirc adapiv conrol algorihms. nd d. Nw York: pringr, [14] Johnson K E, Fingrsh L J, Pao L Y, al. Adapiv orqu conrol of variabl spd wind urbins for incrasd rgion nrgy capur. AIAA , 005. [15] Balas M J, Li Q, Prman R. Adapiv disurbanc racking conrol for larg horizonal axis wind urbins in variabl spd rgion II opraion. AIAA , 010. [16] Fros A, Balas M J, Wrigh A D. Modifid adapiv conrol for rgion 3 opraion in h prsnc of wind urbin srucural mods. AIAA , 010. [17] Mhil E A, Balas M J. om condiions for sric posi- iv ralnss of scond ordr and almos scond ordr sysms. AIAA , 005. [18] Mhil E A, Balas M J. Drmining sricly posiiv ralnss from sysm modal characrisics. Journal of Guidanc, Conrol, and Dynamics 007; 30(5): [19] Bar-Kana I, Fischl R, Kalaa P. Dirc posiion plus vlociy fdback conrol of larg flxibl spac srucurs. IEEE ransacions on Auomaic Conrol 1991; 36(10): [0] Iwai Z, Mizumoo I, Dng M. A paralll fdforward compnsaor virually ralizing almos sricly posiiv ral plan. Procdings of h 33rd IEEE Confrnc on Dcision and Conrol. 1994; [1] Zhng M J. paccraf aiud conrol sysm dsign via backspping conrol mhod. PhD hsis, chool of Aronauics, Bihang Univrsiy, 007. [in Chins] [] Dosho, Ohmori H, ano A. Nw dsign mhod of simpl adapiv conrol for sysms wih arbirary rlaiv dgr. Procdings of h 36h IEEE Confrnc on Dcision and Conrol. 1997; [3] Barkana I. On gain condiion and convrgnc of simpl adapiv conrol. AIAA , 003. [4] Wi B, Byun K W. Nw gnralizd srucural filring concp for aciv vibraion conrol synhsis. Journal of Guidanc 1989; 1(): Biographis: LIU Min rcivd B.. dgr from Bihang Univrsiy in 007 and is now a Ph.D. sudn in h chool of Asronauics, Bihang Univrsiy. His main rsarch inrss ar flxibl spaccraf aiud conrol and simpl adapiv conrol. liumin@sa.buaa.du.cn XU hiji is a profssor and Ph.D. suprvisor a h chool of Asronauics, Bihang Univrsiy. H rcivd h Ph.D. dgr from Hnri Poincaré Univrsiy in His currn rsarch inrss ar guidanc, navigaion, dynamics and conrol of spaccraf. sarsjxu@yahoo.com.cn HAN Chao is a profssor and Ph.D. suprvisor a h chool of Asronauics, Bihang Univrsiy. H rcivd h Ph.D. dgr from h sam univrsiy in His currn rsarch inrss ar guidanc, navigaion, dynamics and conrol of spaccraf. hanchao@buaa.du.cn