Constrained Optimal Controller Design of Aerial Robotics Based on Invariant Sets

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1 JOURNAL OF SOFWARE, VOL. 6, NO. 2, FEBRUARY 2 93 Costrad Optmal Cotrollr Dsg of Aral Robotcs Basd o varat Sts Jaqag L Collg of Computr Scc ad Softwar Egrg, Shzh Uvrsty, Shzh 586, Cha Emal: ljq@ szu.du.c Yahu Lu Collg of Computr Scc ad Softwar Egrg, Shzh Uvrsty, Shzh 586, Cha. Emal: luyahu@63.com Zh J Collg of Computr Scc ad Softwar Egrg, Shzh Uvrsty, Shzh 586, Cha. Emal: jzh@szu.du.c Halog P Dpartmt of Automato, South Cha Uvrsty of chology, Guagzhou 564, Cha. Emal: auhlp@scut.du.c Abstract Costrad optmal cotrol problm of umad aral vhcl (UAV) whch s also calld aral robotcs s studd ths papr. h olar systm of small umad hlcoptr wth boudd dsturbac s abstractd ad modld by PWA hybrd systms modl. As th complty o-l computato for a class of larg hybrd systms, a plct optmal cotrollr for hybrd systms basd o mult-paramtrc quadratc programmg (mp-qp) s proposd. h fasbl doma whch s th mamal cotrolld varat sts for hybrd systms s parttod backward dyamc programmg by mp-qp mthod. At ach stp, o stp rachabl sts ar computd, optmal cotrol laws ar costructd to th corrspodg rgos, ad th plct optmal cotrollr s obtad. Fally smulato rsults vrfy th ffctvss of th proposd cotrol mthod. d rms aral robotcs, hybrd systms, varat sts, mp-qp, plct cotrollr. NRODUCON Aral Robotcs (Umad hlcoptr) whch s also calld umad aral vhcls (UAV), plays mportat rols th applcato of motor, rscu ad aral photograph. hr has b a grat dal of trsts th study of umad hlcoptr th last dcad []. As th UAV modl s a compl hgh ordr olar systm ad th varty of flyg vromt, th cotrollr s dffcult to dsg. t s commo to dsg doubl closd loops cotrollr for autoomous hlcoptr, hs work was supportd by th Natoal Natural Scc foudato of Cha (66738, 6973, 6934). Corrspodg author Ja-qag L : ljq@szu.du.c r loop for th atttud ad outr loop for th trajctory []. A lar quadratc rgulator s dsg th study of Hu ad Yu [2-5]. Aftr modld by hybrd systm, L dsgd a modl prdctv cotrollr of UAV [6]. Howvr, th study of optmal prformac ad th ral tm cotrol ar stll th mportat pots for th UAV. ths papr, basd o th study bfor, th olar umad hlcoptr s modld by hybrd systms o th stats of hovr ad forward flyg. th flyg vlop of th UAV systm, mamal varat st s computd for th atttud as th fasbl doma. th scurty ara, a plct optmal algorthm s dsgd for th UAV cotrollr by back-stp dyamc program, th prformac of ral tm s guaratd. th optmal cotrol, o stp mult-paramtrc program mthod s tdd to hybrd systm. Optmal cotrol problm s rsolvd by mult ffct domas ad sub-systms mult-paramtrc program, o back-stp rachabl sts s computd, ad th plct rlato of optmal cotrol ad systms stats s costructd. Dffrt from th papr[7], boudd dsturbac s cosdrd th varat st computg ad cotrollr dsg. h rst of ths papr s orgazd as follow: th hybrd modl of umad hlcoptr wth boudd dsturbac s dscrbd Scto 2, th th varat st s troducd Scto3, th optmal cotrollr s dsgd Scto 4, followd by som smulato th t Scto. Coclusos ad furthr dscusso ar gv Scto 6.. HYBRD SYSEMS MODEL OF UAV A. Nolar modl of UAV 2 ACADEMY PUBLSHER do:.434/jsw

2 94 JOURNAL OF SOFWARE, VOL. 6, NO. 2, FEBRUARY 2 Fgur. Umad hlcoptr systm h systm of umad hlcoptr s show Fg., whch cotas stats ad 4 cotrol puts, th olar dyamc s gv as follow[2]: = f(, u, t) + () = [ u v w p q r φ θ ψ a b] u = [ col lo lat r ], quato (), s th boudd dsturbac of, th colum vctor s th stats of th small umad hlcoptr, uvw,, dot th vlocty wth rspct to th body-coordat fram, φ, θψ, dot roll, ptch ad yaw, ad p, qrar, thr rats. h logtudal ad a, b ;th lattudal swg agl of rotary wg ar colum vctor u s th cotrol vctor composd by th collctv ptch agl col, th ma rotor blads logtudal ad lattudal ptch agls ( lo, lat ), ad th tal blad ptch agl r. f s a olar fucto whch dscrbs th rlatoshp of umad hlcoptr wth put cotrol ad tral dsturbac, t ca b gv as follow: u = vr wq g sθ + ( X mr + X fus) m v = wp ur + g sφ cosθ + ( Ymr + Yfus + Ytr + Yvf ) m w = uq vp + g cosφ cosθ + ( Zmr + Z fus + Zht ) m p = qr( yy zz ) + ( Lmr + Lvf + Ltr ) q = pr( zz ) yy + ( M mr + M ht ) yy r = pq( yy) zz + ( Q + Nvf + Ntr ) zz φ = p + q sφ taθ + r cosφ taθ θ = q cosφ r sφ ψ = q sφ cosθ + r cosφ cosθ a A a u uw a w ww lo a = q + ( + ) + lo µ ΩR µ z ΩR b b v v B w lat b = p + lat µ v ΩR (2) quato ( 2 ), m s th mass of UAV, X,Y,Z, L,M,N dot th forcs ad torqus actg o th hlcoptr,, yy, zz ar momts of rta of th hlcoptr, Q s torqu of motor, th rotatoal spd of ma motor s Ω, rot s th rotato rta of ma rotor blad, A lo ad ar logtudal ad B lat lattudal gas, th subscrpt mr stads for ma blad, fus stads for hlcoptr, tr stads for tal rotor, vf stads for vrtcally tal, ht stads for lvl tal. h olar modl dtal of small umad hlcoptr ca b foud [2]. B. rmmg calculato of UAV t s dffcult to dsg th cotrollr of UAV, as hgh ordr olar modl of umad hlcoptr. At a quvalt pot, th olar modl s appromat larzd, cotrollr s dsg basd o th lar modl. hs papr maly studs th stats of hovr ad th forward spd of 3m/s. h quvalt pots s trmmg calculatd frst, th cotrol put ad atttuds ar computd at trm stat as gv th fly codto. h costat lar flght s dscussd, so th forc ad torqu ar at balac codto. Acclrato, agular vlocty, ad agular acclrato ar qual to zro, th th trmmg quatos ca b costructd, th trmmg valu also ca b calculatd. C. Hybrd systms modl For UAV, svral mod ca b costructd as Fg.2, stats of hovr ad 3m/s forward spd s cosdrd ths papr. At th quvalt pots of hovr ad 3m/s forward spd, th olar modl s larzd frst. Fgur 2. Mult-mod of UAV For th olar systm (), th output s gv as (3): y = [ ] (3) l outputs ca b slctd for th olar systm, ad th corrspodg quvalt pots ar calculatd. By Jacob larzato, lar modl ca b obtad at th quvalt pots [2]. At th quvalt pots (, u, y ), 2 ACADEMY PUBLSHER

3 JOURNAL OF SOFWARE, VOL. 6, NO. 2, FEBRUARY 2 95 th form of lar modl σ ca b dscrbd as follow [8]: f f = u + = + u = u= u u= u. t. y ε y y + ε (4) f f Suppos A =, B = =, rgd body u= u u = u= u assumpto s cosdrd to rsolv th problm that th swg agl ca ot b masurd. h dffrtal quato of swg agl s rplacd by stabl rlato. As th horzotal pla of umad hlcoptr s studd ths papr, th s dgrs of frdom ca abstractd to thr dgrs of frdom modl. h logtudal varabls of modl ar st to zro [4]: w =, p =, q =, φ =, θ =, th lar modl ca b obtad by rducd ordr: = A () t + Bu () t + (5) h colum vctor s th systm stats of abstract modl, cotag 9 stats: two dmsos spd, th agl ad atttud of yaw: = [ u v r ψ ] (6) [ ] u = col lo lat r (7) y = [ ] (8) h olar systm ca b dscrbd by th subsystm of hybrd systms (5). h trasto codto btw th mods should b dtrmd accordg to th systm codto. varat sts s computd as safty scto. NVARAN SES A (postv) varat st of a dyamc systm s a subst of th stat spac that oc th stat trs ths st t wll rma t for all futur tms t [5].. () X ( t) X for all t>. Whr () t s th subst of th dyamc at tm of t ad X s a subst of th stat spac. Cosdr th cotuous dyamc systm () t = f( ()) t (9) Whr f : s a cotuous Lpschtz fucto. A suffct ad cssary for X s th varat st s that th dffrtal quato s drctd to th st at ach pot a th boudary: X. Dfto 2 (Boulgad, 932).Lt a closd st κ. h tagt co to κ s th st dst( + hz, κ ) κ = z = h :lmf h () Fgur 3. h tagt cos h dfto s du to Boulgad as Fg. 3. Not that κ, th tror of κ,th κ =,f f t{ } t{ κ}, th κ = φ ad κ, th tagt co s a co whch cotas all th vctors whch drct from to th st κ. horm (Nagumo, 942) Assum th systm (9) admts a uqu soluto for.h closd st κ s postvly st for th systm (9) f ad oly f for all κ. f( ) κ () From th thorm, a cssary ad suffct codto for systm (9) s vry pot o th boudary κ s drctd to th st. hs ca b prssd as blow: κ ( ) f( ) κ (2) Whr κ ( ) dots a ormal to κ at. h varat st s dscrbd by a qualty κ = { V( ) b} (3) V( ) whch dfs th varat st s a fucto of. hr ar two mportat famls of varat sts. hs ar th classs of llpsodal sts ad polyhdral sts. Mod trasto dyamc systm or cotuous systms hav ths typs of varat sts. A Ellpsodal sts Ellpsodal sts ar usd wdly as varat sts cotuous systm. From th stc of a quadratc Lyapuov fucto for such systm ad that lvl sts of Lyapuov fuctos ar varat sts [5]. A corollary ca b dducd from t: Corollary A systm = A ( ),, A, f A has all o-postv ral-part gvalus, th th systm has llpsodal varat st. Ellpsodal sts ar popular varat sts. A llpsodal varat st ca b prssd as follow: 2 ACADEMY PUBLSHER

4 96 JOURNAL OF SOFWARE, VOL. 6, NO. 2, FEBRUARY 2 = { P } (4) Or = { ( ) ( ) } (5) P a P s symmtry matr, ad a s th ctr of th llpsodal varat. h varat sts ca b computd as cov optmzato problms as follows. P Mmz log dt Subjct to P LM(P,q) (6) P P R Whr = ad P s postv dft. hs s cov optmzato problm ad ca b solvd by LM tools [6]. As a llpsodal varat problm, ths st ca b computd as follows: Mmz log dt P Sub. to AP+ PA, P vpv (7) Gv a st of tal stats, th codto ca b formulatd as a lar matr qualty usg th so calld S-procdur[4]. t ca b dscrbd as follow: >, P P P P P a, (8) P ad ar th varabls of th lar matr qualty. h last p qualts () ca b substtutd. A cov optmzato problm for computg th smallst varat sts that cotas. h LM s chag as follow by a slght varato. C N Cd Q( ) C N Cd, (9) C s th psudo vrs of C, th colums of C N s a bass of th ullspac of C, ad Q( ) s th lft sd of th LM (9) [7]. For th systm = A + Bu wth cotrol, cosdr th follow quato for th varat st s cotractv. ( A+ BK) P+ P( A+ BK) (2) As P ad K ar blar th quato, thus t s ot asy to hadl. But th both sd of th -quato ca b multply by Q= P, ad paramtrzd K as K = YP to achv th lar matr qualty as follow. QA + AQ + Y B + BY, Q (2) h lar matr qualty s asy to hadl. t ca b computd by LM toolbo of MALAB, ad s cotractv wth lar cotrol as follow form: ut () = γ BPt (), γ > (22) h abov proprty ca b asly tdd to th cas of a ucrta par (A, B). B Polyhdral sts fact, polyhdral sts ar oft atural prssos of physcal costrats o stats ad cotrol varabl to th varat sts. Howvr, th shap of th polyhdral sts s mor flbl tha that of th llpsod, ths lads to bttr appromato to th varat sts ad doma of dyamc systms. hs flbl proprty maks polyhdral sts mor rprstato th computato. A polyhdral st ca b rprstd th followg form[5]: = { : F } (23) whr F s a r matr, ad of th form [ ] r R = dots a vctor h computato of polyhdral varat sts s dffcult tha th computato of a llpsodal sts. hr ar maly two mthods to costructv th polyhdral varat sts: tratv mthods whch us a backward procdur to comput th varat sts [5] ad gstructur aalyss/assgmt mthods. varat st ca b computd by th tratv mthods as follow algorthm. Algorthm talzato: k= ad Rpat k, = k = { : u( ) U : A + Bu( ) + Ew w W} = = Utl = th = At ach stp, th algorthm comput th st of stats for whch all soluto of th systm stay th.h squc s th th subst of thos stats for whch f a trasto s possbl, th stat aftr th trasto s also th. Egstructur aalyss/assgmt mthod s aothr ffct mthod. Som cotrbutos show how to dtrm varat sts cludd polyhdra of th form ( G, ρ) { X : ρ G ρ} (23) A stablzg cotrol law u = Ks assgd [7]. ths papr, polyhdral varat sts s cosdrd. 2 ACADEMY PUBLSHER

5 JOURNAL OF SOFWARE, VOL. 6, NO. 2, FEBRUARY 2 97 varat sts for hybrd systms ca also b computd by tratv mthod. V. MP-QP OF HYBRD SYSEMS A Optmal cotrol of hybrd systms Cosdr pcws aff hybrd systms (5), whr t () Ω ut (), th ft-tm optmal cotrol problm of costrat hybrd systms ca b dfd as follow: * J ( ()) = m J( (), U ) (24a) U Subj. to: ' L () t + Eu () t W, f [ (), t u()] t Ω ar computd accordg to cotrol put by multparamtrc quadratc program mthod. () s slctd t ( + ) = fpwa( u, ) = At ( ) + But ( ) + (24b) (), to provd u ( ()) as follow, ut ( ( )) ca also b f provd. J( (), U ) = P( ) + Q( t) 2 2+ Ru( t) Proof: Gv talzato stat () ad 2 t= corrspodg U = [ u()', u( k )'], o th (25) optmal cotrol problm (25), f dfs th fucto of stats ad cotrol srs, k ( ) χ, lt trmal costrats, s th optmal tm, m U = [ u'(),, u'( ) ]' R s th optmal cotrol, Q 2 = ' Q ad R = R', Q= Q', P = P'. { Ω } s = s th sub-systm doma of hybrd systms. As modl prdctv cotrol mthod [9], th optmal cotrol vctor U * ( ()) s dtrmd by () (24). Oc () s chagd, th cotrol put should b calculatd aga, so t s a o-l optmal cotrol problm. Cosdr problm (24) as a mult-paramtrc quadratc problm, () s cosdrd as paramtrc vctor, th plct fucto of U * ( ()) ad () s bult. h optmal cotrol ca b obtad ral-tm cotrol as th stats plct fucto s rplacd by th currtly stats. h problm of mult tm o-l computg problm ca b solvd by ths mthod. A mthod of mult-paramtrc quadratc program mthod s proposd ths papr, th plct pc aff fucto of U ad s costructd by stat doma parttos ad optmal cotrol dsg. B Eplct cotrollr dsg Stat polyhdro parttos ar obtad by multparamtrc program [3], th pc-aff proprty of cotrol law ad valu fucto s also provd. hs mthod s tdd to hybrd systms ths scto, ad plct cotrollr of hybrd systms s dsgd by dyamc program. horm 2 : Cosdr th hybrd systms optmal cotrol problm (24), th optmal cotrol vctor ca b dscrbd as pc-aff cotrol law (26): * t t u ( ( t)) = F ( t) + (26) h prformac d s dfd as follow: * k t t J ( ( t)) = '( t) Q ( t) + L ( t) + C, (27) (26) ad (27), t t t t ( k) CR { : t ( )' Lt ( ) + Mt ( ) N}, (28) t =, CR ar cov parttos of k k D whch s fasblty st ( k ). thorm 2, optmal cotrol vctor ad valu fucto s + s th amout of parttos whr (), ( ) may b, support v, =,, s + ar trasto srs of tm, v s th k lmt of v, th: k k v = j f ( k) χ (29) Slct v ad ts trasto, th optmal problm (24) ca b trasform to (3): J ( ()) = m J( (), U ) (3a) * v U Subj.to ' L t + Eu t t W, f [ ( t), u( t)] Ω t ( + ) = At ( ) + But ( ) + ( k) χ k, f k =,, (3 b) v (), f f thr s a problm that th costrats optmal cotrol (3) s tm varyg, pcws aff fdback cotrol law ca b obtad by mult- paramtrc quadratc mthod. As th quvalc btw md logc dyamc (MLD) modl ad pcws aff modl for th optmal cotrol problm (3), mult- paramtrc md tgr quadratc mthod basd o MLD modl s usd []. ordr to mprov computato ffccy, o stp back mult- paramtrc dyamc program [7] s tdd to hybrd systms optmal cotrol. h back rachabl st s computd to dsg optmal cotrollr. Cosdr th optmal cotrol problm (24) by dyamc program: j 2 ACADEMY PUBLSHER

6 98 JOURNAL OF SOFWARE, VOL. 6, NO. 2, FEBRUARY 2 J ( ) = m( Q() t + Ru() t + J ( ) * * j j 2 2 j+ j+ u j (3a) Subj. to: ' L + Eu W, f [, u ] Ω j+ = fpwa( j, uj) χ j+ j =, j j j j h boudary s: f, J * ( ) = P. 2 (3b) h problm of dyamc program (3) ca b solvd by back dyamc for mult- paramtrc md tgr quadratc mthod. Cosdr th frst stp dyamc program: * * J ( ) = m( Q + Ru + J ( ) 2 2 u Subj.to = f (, u ) χ PWA f (32) h optmal problm (32) for PWA hybrd systm ca b dvdd to s problms for o stp mult- paramtrc program (s s th amout of th sub- systm). thorm, χ s cov polyhdro st, u s pcws aff fdback cotrol law, J * ( ) s pcws r aff quadratc fucto for, N s th stat parttos amout o χ. By frst stp program, th plct pcws lar stat fdback fucto btw u ad χ ca b obtad. From stp j = 2 to fal stp j =,s problm of o stp MP-QP ca b costructd vry stat partto of χ j +, so th amout of s o stp MP-QP r problm s N s, ad th stat partto covrs χ j. MP-QP mthod, thr may st that o cov polyhdro blogs to svral stat parttos or svral * goal fucto valus J j+ ( j+ ). h cotrol sgal ad * goal fucto ca b dcdd by J j+ ( j+ ). Calculatg stp j for dyamc program, th plct optmal cotrol ca b obtad ach stp. Wth th mamal cotrollabl varat st [], mamal st tat st s st as fasbl st. h plct cotrollr ca b obtad th followg stps:..h mamal cotrollabl varat st s computd by trat mthod; 2. St th varat st as th fasbl st, solv th optmal problm (24) by mp-qp mthod, varat st s dvdd to ffct doma, ad th back stp rachabl st, pcws cotrol ad valu fucto ar computd to th corrspodg parttos. 3. Gv a tal stats, th optmal ca b obtad accordg to th parttos Cosdr th boudd dsturbac th plct cotrollr dsg, th robust cotrollr ca b computd by Potryag dffrc calculato for stats st ad dsturbac [2]. V. SMULAON Accordg to th plct cotrollr dsg mthod scto 3, th optmal cotrollr s dsgd for systm (5) basd o th PWA modl o quvalt pots hovr ad 3m/s. Optmal cotrollr dsg ca b cludd as follow:. Costruct th PWA modl o UAV; 2. Mamal cotrollabl st s computd as fasbl st flght vlop. 3. h plct fucto btw cotrol put ad stats s computd basd o mq-qp mthod. 4. Comput th optmal rsoluto o th stats of UAV. h UAV X-Cll 6SE s slctd as modl mach. Aftr trmmg calculato, ordr rducd ad dscrtzato, th dyamc o quvalt pots ca b dscrbd as follow: = A u+ + B (33) σ = : y.5; A =, B =, σ = : y.5; A =, B = St systm costrats as follow: u, (34).. (35) 2 ACADEMY PUBLSHER

7 JOURNAL OF SOFWARE, VOL. 6, NO. 2, FEBRUARY 2 99 h wgh matr s dsgd for optmal cotrollr aftr prmts: P s zro matr, Q= dag(.. ), R= dag( 4 4 3). Aftr trato calculato mamal cotrollabl varat sts, fasbl parttos ar dvdd to 75 domas as Fg.4, whr s tagt pla 4 = as th systm s 4 dmsos systm. Fgur 6. rajctory of systm output Fgur 4. Cotrol parttos 4 = fasbl doma, th stat ca rtur stabl stats = [;;;] from th tal stat = [2.5;-.449;;] by ffct puts. Smulato as follow: Fgur 7. rajctory of systm put Fgur 5. rajctory of systm stats Fgur 8. Dsturbacs of systm From Fg.5 ad Fg.6, h systm stats ad output rtur to qulbrato pot although wth boudd dsturbac. Fg. 7 ad Fg.8 show th put cotrol ad 2 ACADEMY PUBLSHER

8 2 JOURNAL OF SOFWARE, VOL. 6, NO. 2, FEBRUARY 2 systm dsturbac. Cosdr boudd dsturbac, st th mamal robust cotrollabl varat sts as fasbl doma, th optmal cotrollr ca b obtad by Potryag dffrc calculato for rachabl st ad dsturbac st. V.CONCLUSON Basd o th mthod of papr [7], boudd dsturbac s cosdrd ths papr. Costrad optmal cotrol problm for umad hlcoptr s studd ths papr. UAV Systm s abstractd ad modld by PWA hybrd systms frst, mult- paramtrc quadratc program mthod s proposd to off-l computd optmal cotrol law for ral tm cotrol. th fasbl doma whch s th mamal cotrollabl varat st, optmal cotrol s computd ach stp by back dyamc program. h optmal cotrollr ca b obtad by Potryag dffrc calculato for rachabl st ad dsturbac st. h smulato rsult provs th valdty of ths mthod at th d of ths papr. ACKNOWLEDGMEN h authors wsh to thak Bmporad, Kvasca. h work s partally supportd by th Natoal Natural Scc Foudato of Cha (Grat o , 6973 ad 6934) ad th Natoal Hgh-ch Rsarch ad Dvlopmt Program of Cha (Grat No.27AA4Z35). REFERENCES [] M. E. Drr. troducto to Hlcoptr ad ltrotor Flght Smulato [M]. Amrca sttut of Aroautcs ad Astroautcs, 26. [2] Y. F. Hu. Modl ad cotrol of aral robotc [D]. Graduat thss, Guagdog provc, Cha: South Cha Uvrsty of tchology, [3] H. L. P, Y. F Hu ad Y. Wu. Ga Schdulg Cotrol of A Small Umad Hlcoptr [A]. 27 EEE tratoal Cofrc o Cotrol ad Automato: [4] W. Yu. Lus Rodrgus ad Brado Gordo. Pcws- Aff Cotrol of a hr DOF Hlcoptr [A]. Amrca Cotrol Cofrc 26. [5] M. Bsgaard. Modllg, Estmato ad Cotrol of Hlcoptr Slug Load Systm [D]. Ph.D.thss, Dpartmt of Elctroc Egrg Scto of Automato ad Cotrol, Aalborg Uvrsty, Dmark, 27. [6] J. Q. L, H. L. P. Hybrd systms Modl ad Cotrol of a Small Umad Hlcoptr. 27th Chs Cotrol Cofrc, 28: [7] J. Q. L, J. H. Y. Rsarch ad Dsg o Optmal Cotrollr of Aral Robotcs [A]. EEE tratoal Cofrc o Robotcs ad Bommtcs 29 (ROBO 29). [8] J. Zhag, R. X. Qu. Costrat Lar Optmal Multparamtrc Programmg Approach for Lar Costrad Quadratc Optmal Cotrol Problms [J]. Joural of sg Uvrsty, 27, 47(S2): [9] X Yu-Gg, Wag Fa. Nolar mult-modl prdctv cotrol [J]. Acta Automato Sca, 996, 22(4): [] J. Q. L, H.L. P, H. P. Wag. Md tgr Programmg Cotrol Basd o Hybrd Systms ad rmal Costrat by varat Sts[J]. Acta Automato Sca, 28, 34(8): [] A. Bmporad, F. Borrll, ad M. Morar. O th Optmal Cotrol Law for Lar Dscrt m Hybrd Systms [A]. HSCC 22: 5-9. [2] S. Rakovc, P. Grdr, M. Kvasca, D. Q. May, ad M. Morar. Computato of varat Sts for Pcws Aff Dscrt Systms Subjcts to Boudd Dsturbacs [A]. EEE Cofrc o Dcso ad Cotrol 24: [3] E. C. Krrga. Roubust Costrat Satsfacto: varat Sts ad Prdctv Cotrol [D]. Ph. D. thss, Cotrol Group Dpartmt of Egrg Uvrsty of Cambrdg, 2. [4] A. Bmporad, M. Morar, V. Dua, ad E. N. Pstkopoulos. h Eplct Lar Quadratc Rgulator for Costrad Systms [J]. Automatca, 22, 38: 3-2. [5] M. Jrstrad. varat Sts for a Class of Hybrd Systms [A]. EEE Cofrc o Dcso ad Cotrol 998: [6] S. Boyd. Lar Matr qualts Systm ad Cotrol hory [M]. SAM, 994. [7] F. Blach. St varac Cotrol [J]. Automatca, 999, 35: L Jaqag (98-), bor Guagdog Cha. H s currtly a lcturr wth th Collg of Computr Scc ad Softwar Egrg, Shzh Uvrsty, Cha. H rcvd hs B.S dgr from th Dpartmt of Automato, South Cha Uvrsty of chology, Guagzhou, 23 ad th Ph.D dgr from Dpartmt of Automato, South Cha Uvrsty of chology, Guagzhou 28. Hs rsarch trsts clud: hybrd systms, robotcs, ad optmal cotrol. Lu Yahu (976-), bor Shag Cha. H s currtly a lcturr wth th Collg of Computr Scc ad Softwar Egrg, Shzh Uvrsty, Cha. H rcvd hs B.S dgr from th Dpartmt of Automato, Najg uvrsty, Guagzhou, 996 ad th Ph.D dgr from Dpartmt of computr, sghua Uvrsty, Bjg 28.Hs rsarch trsts clud: modlg, robotcs, ad optmal computg. J Zh (973-), bor Jagshu Cha. H s currtly a profssor wth th Collg of Computr Scc ad Softwar Egrg, Shzh Uvrsty, Shzh Cha. H rcvd hs Ph.D dgr from Dpartmt of Commucato ad formato Systms, X'a Jaotog Uvrsty, X'a 999. P Halog (965-), bor Ha Cha. H s currtly a profssor wth th Dpartmt of Automato, South Cha Uvrsty of chology, Guagzhou Cha. H rcvd Ph.D dgr from Dpartmt of Automato, South Cha Uvrsty of chology, Guagzhou ACADEMY PUBLSHER