Robust control for aircraft anti-skid braking system based on dynamic tire/road friction force model

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1 Pocdings of th nd Intnational Confnc on Coput Scinc and Elctonics Engining (ICCSEE 3 Robust contol fo aicaft anti-skid baking syst basd on dynaic ti/oad fiction foc odl Li Fngyu School of Autoation Scinc and Elctical Engining Bijing Univsity of Aonautics and Astonautics Bijing, China alxand.l983@gail.co Jiao Zongxia School of Autoation Scinc and Elctical Engining Bijing Univsity of Aonautics and Astonautics Bijing, China zxjiao@buaa.du.cn Abstact Th aicaft anti-skid baking syst plays an ipotant ol fo th safty duing taxiing. Aicaft ti/oad fiction foc is ssntial infoation fo odl basd contol law. In this pap, a obust contol statgy with an obsvbasd dynaic aicaft ti/oad odl is givn to ipov th bak quality. Fistly, th nonlina fuslag and whl odls a divd. Th ti/oad fiction foc is divd with LuG dynaic fiction odl considing vlocity colation, in which th intnal fiction stat is stiatd by dual-obsv. In cas of xtnal distubanc and odling o, a obust contoll is ployd to tack th onlin calculatd dsid slip at. Th siulation sults show that th wll-dsignd contoll has stabl bak pfoanc. Kywods-aicaft baking contol; LuG dynaic fiction odl; obust contol; stat stiation; nonlina contol I. INTRODUCTION Th liability of th aicaft anti-skid baking syst plays an ipotant ol fo th safty of passngs and aipots, which can stop th aicaft soothly and ffctivly. Whn applying pssu on aicaft anti-skid baking syst (ABS duing taking off, landing and taxing, high fficincy contol statgy is ndd to nsu th aicaft gound safty. Th aicaft baking syst is a coplx nonlina syst with unctaintis. Th nonlinaitis of syst a constuctd by th nonlina lationship btwn aicaft ti and unny (o oad adhsion cofficint, adhsion cofficint and slip at, and bak toqu and pssu []. It is ath difficult to gt an accuat baking syst and ti/oad fiction odls and thi paats. Thus, nonlina contol thoy is intoducd to solv th pobl ntiond abov. In th past, any kinds of contol statgis w utilizd in siulations, hadwa-in-th-loop (HIL xpints and aicaft tsts. Most of th fo sach was not odl basd. Th pssu bias contol law is studid by Tang in [], which shows liabl pfoanc in siulation and tsts. And th vaiabl stuctu contol and slip at optial contol w also dsignd in th litatus [3, 4]. Th ti/oad adhsion foc odl is ssntial fo odl basd contol laws. Ealy sach [5] focusd on th adhsion foc psntation, which is wll-known as Magic Foula, givs a nic appoxiation to xpintal sults of th lationship btwn fiction cofficint and longitudinal slip. Rcnt sach bings on kind of dynaic ti/oad fiction foc odl into ABS contol, which has a potntial advantag to dscib closly so of th physical phnona and to dpnd on paats dictly latd with phnona, calld th LuG Modl. This odl concns th any-to-on apping fo vlocity and oth vaiabls to adhsion cofficint [6]. Many ABS contol statgis a alizd using th LuG dynaic fiction odl und vaious oad conditions. Espcially, th paatic unctaintis du to oad condition o/and tpatu changing was considd in [7], in which on stat obsv was constuctd to stiat intnal fiction stat and longitudinal vlocity. By utilizing LuG dynaic fiction foc odl, th dsid axiu slip at is stiatd onlin. Thus, an obsv-basd baking contoll is dsignd to tack th dsid slip at. Ths odls hav bn usd succssfully to idntify and copnsat th fiction in chanical syst. In [8], a dualobsv was divd to captu diffnt nonlinaitis of fiction stat intoducd by LuG dynaic odl, which can b usd to ipov th dsid slip at tacking pfoanc. In cas of possibility of nting nonobustnss, an adaptiv obust contoll is dsignd to guaant th stability of stiation [9, ]. Howv, ths statgis a dvlopd without obust considation. In this study, th obust analysis is intoducd to ABS contol syst. Fistly, th nonlina fuslag and whl odls a stablishd. Th ti/oad fiction foc is odld by LuG dynaic fiction odl considing vlocity colation, in which th unasuabl intnal fiction stat is stiatd by dual-obsv. Th contol tagt, dsid slip at, is calculatd onlin with th hlp of psudo-static distibutd LuG odl. At last a obust contoll fo aicaft ABS is dsignd considing xtnal distubanc lik vtical load changing and odling o. Siulations a iplntd to vify th thotical dsign. II. SYSTEM DYNAMICS A. Aicaft fuslag and whl dynaics odling Th aicaft gound dynaics odl is shown in figu, fo which w can s th a lift (L, ngin thust (T, aodynaic dag (D, gavity (G and landing ga focs acting on th fuslag. Th landing ga focs a tansfd fo intaction btwn aicaft whls and Publishd by Atlantis Pss, Pais, Fanc. th authos 69

2 Pocdings of th nd Intnational Confnc on Coput Scinc and Elctonics Engining (ICCSEE 3 unny, including longitudinal and noal nos landing ga foc (F ln and F nn and lft and ight ain landing ga focs (F ll, F l, F nl and F n. Whn only considing th longitudinal focs, w hav v = T D Fln Fl Fll ( wh v is aicaft longitudinal spd. ω R v u τ T F G F ( F ln ll l F nn Fnl ( Fn L D Figu. Focs on th fuslag Fo th sak of sipl, th a so assuptions blow: Assuption : Th a sytic bak pssu signals btwn lft and ight baking syst and sa woking conditions (unway stat and tpatu, tc., thus, w hav Fx Fll = Fl ; ( Th a no latal focs o acclation, which ans that th vtical load has unifo distibution, thus, w hav th sa noal foc Fn Fnl = Fn ; (3 3 Th fuslag gavity cnt spd v and aicaft whl spd ω can b dtctd with snsos; 4 Longitudinal nos landing ga foc (F ln, ngin thust and lift a nglctd account fo thi sall agnitud copad with ain landing ga longitudinal foc (F x and gavity. Rak : It is obviously that th syst is idally odld and so conditions a ignod, which ans that th ust b odl isatch du to xtnal distubanc and odling o. And this t is dfind as Δ a which will b takn into consid lat in th dsign of obust contol. Thus, basd on Assuptions, w hav v =F x K a v. (4 wh K a is th aodynaic dag cofficint and D = K a v. Th aicaft whl dynaics odl is shown in figu, fo which w can s th bak toqu (u τ is applid on th olling whl. Th bak toqu agnitud has a coplicatd nonlina lationship with bak valv od signal (P b. But h w siplify it into lina lationship du to its high fquncy spons and wak ffct on th contol pfoanc. Th psntation is u = K b P b, and K b = n p μ p n h A p R b, which consists th infoations of th fiction facs of fiction pats nub (n p, fiction cofficint of fiction pats (μ p, bak piston nub (n h, piston aa (A p and fiction adius (R b. F x Figu. Dynaics of aicaft whl Thus, th aicaft ti odl daftd in figu can b dscibd by J ω = Fx KbPb (5 wh J is otational intia of aicaft whl and is th whl olling adius. B. Aicaft ti/oad fiction odl Th aicaft ti/oad fiction cofficint is dfind by th atio of baking foc and th noal foc, which can b xpssd as: Fx μ =, (6 Fn wh th fiction cofficint μ is a coplx function of th aicaft longitudinal slip and oth factos, such as ti and unny conditions. Th longitudinal slip is dfind by: v ω λ =. (7 v Th dynaic ti/oad contact fiction odl is dscibd by th functions blow [6] : dz = z, (8 g( dz fx( t = ( z+ + Fn, (9 with / / vs = μc + ( μs μc wh, z th intnal wightd an fiction stat, v = (v ω is th lativ vlocity, f x is th ti/oad fiction foc odl; is th noalizd ubb longitudinal lupd stiffnss, is th noalizd ubb longitudinal lupd daping, is th noalizd viscous lativ daping, μ c is th noalizd Coulob fiction, μ s is th noalizd static fiction (μ c μ s, [, ], v s is th Stibck lativ vlocity. Rak : z psnts th whl bistl dflction which an ssntial stat vaiabl fo th fiction foc odl [6]. And th valu of z is unknown and unasuabl, so obsvs a ployd to stiat th agnitud of z, which will b studid in pat III. It is shown in [6] that th odl has following finit bistl dflction popty: P. If z( μs /, thn zt ( μs /, t, Publishd by Atlantis Pss, Pais, Fanc. th authos 63

3 Pocdings of th nd Intnational Confnc on Coput Scinc and Elctonics Engining (ICCSEE 3 P. > μs μ C >,. Rak 3: P is physically intuitiv and usd in th subsqunt contoll dsign fo a guaantd obust pfoanc. And it nsus that th intnal fiction stats a boundd and that its upp bound is givn by th static fiction paat and bistl stiffnss. Considing th odling o of nti syst and LuG odl copad with th actual ti/oad fiction foc Δ = Fx fx +Δ a, ( wh Δ a is th syst odling o dfind in Rak. Fo (4 and (5, th whol syst can b wittn as: v =fx Kav + Δ, ( J ω = f u +Δ. ( III. x ROBUST CONTROLLER DESIGN A. Contol tagt Th contol law dsignd in this pap is in th pupos of aking full us of th aicaft ti/oad adhsion foc/cofficint, which can shotn th baking distanc. In th oth wods, a axiu dclation is achivd with th sa tagt. Th dsid slip at λ d is cosponding to th axiu adhsion foc F xax and cofficint μ ax and v d is th dsid lativ spd cosponding to λ d though th dfinition in (7. Thus, w can gt th dfinition of baking fficincy by λ H μλ ( λl η =, (3 μax( λh λl wh, λ H and λ L a slip at psnt th fist slip occunc valu and last bak lasing occunc valu. Accoding to (3, it is known that high baking fficincy can b obtaind by holding baking syst woking in th aa aound λ d. By solving th distibutd LuG ti/oad fiction odl and obtain th following psudo-static lationship btwn μ and λ, w hav Lη g( μ = g( + γ +, (4 L η with λ η η =, γ =, λ ωg( v wh L is lngth of th ti/oad contact patch. Thus, assuing th paats in quivalnt psudostatic odl Equation (4 a known, th axiu slip at λ d cosponding to μ ax can b calculatd by solving it nuically. Thn, v d is obtaind fo its dfinition. Thus, w hav τ d dv dω =. (5 Though linaly paatizing th syst odl by substituting Equations (8, (9, (, ( to (5 givs dv v g = Pb gz+ g z g(, (6 gk a g( + v +Δ wh, g = J/(K b, g = F n (J/+ /(K b, Δ = Δ (J/+ /(K b + Δ n. Noting that, g and g a positiv; Δ n is th unctain nonlinaity, so Δ psnts th odling o and unctain nonlinaity. Assuption : Th odling o and unctain nonlinaity a boundd by a known positiv constant Δ δ, (7 wh δ is th known bound of Δ. Obviously, th physical ti/oad fiction can sv liitd adhsion foc fo th syst, which can b tatd as th axiu upp bound fo δ. B. Obsvs dsign Basd on th aicaft ti/oad fiction foc dsciption in (8 and (9, w can gt th syst dynaics psntd by ( and (. Considing th dsciption of pat A in this sction, th contol tagt is to tack th dsid lvant spd v d obtaind onlin by th calculation of dsid slip at λ d. Thus, th nti syst dynaics can b xpssd in (6. Du to th unasuabl intnal fiction stat z, th obsv should b ployd to obtain it. Considing th diffnt nonlina chaactistics of fiction stat z in th two ts g z and g v z/g(v of (6, a dual-obsv stuctu is ployd h which is poposd by Tan t al in [8]. This anipulation taks o flxibility to dal with diffnt nonlinaitis in th dynaics. Tho : In th cas of th absnc of unctain nonlinaitis and odling o, i.., Δ =, suppos that th aicaft ti/oad fiction foc is dscibd by th odl in (8 and (9, and consid th aicaft baking syst quation in ( and (, th globally asyptotic tacking of onlin calculatd dsid slip at λ, which is cosponding to μ ax tajctoy, is achivd by th following nonlina contol law: Pbn = k+ gzˆ g zˆ. (7 gk a + g ( + v + gv + v d wh psnts th dsid lativ vlocity tacking o and and psnt th stiats of fiction stat. Th stiats of intnal fiction stat z a obtaind by th obsvs blow to handl th diffnt nonlinaitis: Publishd by Atlantis Pss, Pais, Fanc. th authos 63

4 Pocdings of th nd Intnational Confnc on Coput Scinc and Elctonics Engining (ICCSEE 3 v v zˆ, (8 = γ g( v ˆ g g( v = z+γ (, (9 wh γ and γ a positiv constants to b dsignd. Poof: Bas on th fdback linaization pincipls. With th choic of P bn in (7, w hav d v g =k g z + g z, ( g( wh th odling o and unctain nonlinaity Δ =. Dfin th following positiv-si-dfinit (p.s.d. candidat Lyapunov function V = n g z + g z g γ + γ. ( It is woth announcing that, th obsvation os in ( can b dscibd by (this will b utilizd in th lat poof dz = z +γ g( dz = zγ g( z = zzˆ z = zzˆ, (, (3 wh, and z is th unasuabl al valu of fiction stat. Th divativ of V n along tajctois in (8, (9 and ( is g v g v V = k z z, (4 n γ g( γ g( With P, th al valu (z, stiation valu (, and stiation os ( z, z of fiction stat a boundd. Basd on th dfinition of v d, physical liitation of v and v, th bounddnss of tacking o is obvious. Du to th psntations of g and g(v, th constants, and, th bounddnss of contoll is appant fo th stuctu in (7. Th bounddnss of all intnal vaiabls and constants a povd. Thus, th divativ of th lativ vlocity tacking o is also boundd. By (4, w hav L. Cobining th fact that L and ė L and utilizing th Babalat s la, w conclud that,, as t. Thus, by th dfinition of and λ in (7, w hav λ λ d, as t. C. Robust contoll dsign To avoid nting th unstabl stiation of th unasuabl fiction stats z, th following obust obsvs with pojction-typ odifications which a basd on th obsvs dsign in pat B = Pojz ˆ z γ g( = Pojz v ˆ z+ γ g(, (5. (6 Th pojction apping Poj ζ (* has siila stuctu with Sasty in [], if ˆ ζ = ζ and > Pog ( if ˆ ζ = ζ = ζin and <, (7 othwis wh ζ is a sybol that can b placd by z and z. P. Th pojction apping has th following poptis: v ax z Pojz v ˆ z γ, (8 zˆ γ g( z Pojz v ˆ z+ γ. (9 zˆ + γ g( La : Considing th aicaft baking syst dscibd in (, ( and ti/oad fiction foc in th fo of (8 and (9 with unctain nonlinaitis, i.., Δ, w can find th nonlina obust contol law in th following stuctu Pb = Pbn + Pb, (3 wh P bn is shown in (7. Th xists a nonlina obust contol t P b, in a fo which satisfis th following lationships: s= Pb gz + g z +Δ ε, (3 wh P b is synthsizd to attnuat th ffct of th unctain nonlinaitis and odling o, and ε is a constant to b dsignd. And P. (3 b Poof: Basd on last pat of th poof fo Tho, th intnal fiction stat and paats hav th poptis of bounddnss. Thus, th obust contol t P b h always xists. Using th siila tchniqu in Yao [], w choos sooth bounding function h satisfying: Publishd by Atlantis Pss, Pais, Fanc. th authos 63

5 Pocdings of th nd Intnational Confnc on Coput Scinc and Elctonics Engining (ICCSEE 3 v h g z + + δ, (33 g ( wh z = z ax z in (also xists z = /ax - /in by pojction apping in (5 and (6. And w choos th obust contol t P b in a fo as Pb 3 h 4ε Thus, it is asily to hav fo (3 that =. (34 v δ. (35 s Pb g z g z Substituting P b into (35, w hav s 3g z 3 ε g z ε ε 3 ε 3 3δ ε + ε ε ε 3 Thus th inquation in (3 is povd. Accoding to th dfinition of and P b, it is asily to hav that (3 is satisfid. Tho : Considing th aicaft baking syst dscibd in (, ( and ti/oad fiction foc in th fo of (8 and (9 with unctain nonlinaitis, i.., Δ, w can find th nonlina obust contol law in th sa stuctu as in (3, which is wittn h as following, Pb = Pbn + Pb, wh P bn is givn in (7, th aicaft ti/oad intnal fiction stat obsvs a givn in (5 and (6; P b psnts th obust contol t in th fo of (34, w hav, All th signals psnt in syst a boundd. Futho, th candidat Lyapunov function in (39 satisfis: k k t t g ε g g V( t V( + (, (37 k If aft a finit ti t, in absnc of any unctaintis, i.., Δ =, by utilizing th poposd contoll in (3, th onlin calculatd dsid slip at λ d, which is cosponding to th μ ax, can b achivd. Poof: Applying P b synthsizd in th fo of (3 into th poposd syst, w hav d g = k gz + g z + Δ+ Pb, (38 g( Considing th following p.s.d. candidat Lyapunov function,. (36 V = g. (39 Th divativ of V along tajctois in (5, (6 and (38 is V v = k + ( g z + g z +Δ+ P, g( (4 Considing (3 povd in La and th stuctu of V, w hav k. (4 V k ε V g + + ε Utilizing copaison la, w hav th condition in (37. Thus, V (t xponntially dcays and is ultiatly boundd. With th hlp of Assuption, th lativ vlocity v tacking o is boundd. Account fo th P, th intnal fiction stats has th poptis of bounddnss. Though th pojction apping in (5 and (6, th stiats of stat and a boundd by μ s /. Thus th baking pssu input P b is boundd. Dfin th sa p.s.d. candidat Lyapunov function V n as in (. By substituting ( and (3, th divativ of V n along tajctois in (5, (6 and (38 is zˆ V = k + P g z ( + + n b γ zˆ g z g z g z ( + + γ γ γ g z v = + ˆ ˆ γ g z v zˆ v zˆ + ( k Pb z z γ + γ γ g b g z v g z z + zˆ ˆ + z + z γ γ. (4 Substituting th dfinition of ż in (8, (8 and (9 in P and (3 in La, cobining th positiv paats in V n, w hav g v g V = n k z z k γ γ. (4 Accoding to (4, w hav L. Cobining th fact that L and ė L and utilizing th Babalat s la, w conclud that,, as t. Thus, by th dfinition of and λ in (7, w hav λ λ d, as t, which ans th μ ax is achivd cospondingly. Publishd by Atlantis Pss, Pais, Fanc. th authos 633

6 Pocdings of th nd Intnational Confnc on Coput Scinc and Elctonics Engining (ICCSEE 3 IV. SIMULATION RESULTS Siulations hav bn pfod basd on th aicaft syst dscibd by aicaft baking syst in (8, (9 and aicaft ti/oad fiction foc in (, (. Th fiction paats and syst paats a givn in TABLE, TABLE I. SIMULATION PARAMETERS Na Valu Units Na Valu Units / s/ 37 Kg.38 s/ J.63 Kg μ C.5 - g 9.8 /s μ S.9 - v 5.94 /s v s.5 /s z.8 N/ Th initial aicaft spd and whl lina spd a th sa (5.94 /s. Sinc th s, a baking pssu is applid by th baking actuato on fiction pats. Th dsid slip at is calculatd onlin, and cospondingly th lativ vlocity is givn to th obust contoll as th tacking signal. Noting that, a band-liitd whit nois signal with pow of. is injctd into th dsid slip at signal fo s to 7s to siulat xtnal distubanc. Th initial valus of all intgatos a. Th siulation sults show that, th pfoancs of two obsvs a ffctiv and th syst spons (slip at tc. convg to th al/dsid valus fast and thn cov to thi noal valus aft distubanc. Figu 3 shows th al valu captud fo objct odl and th stiation valus of th dual-obsv and stiation os a shown in figu 4. Figu 5 shows th baking pssu od snt to th baking actuato. Figu 6 shows th slip at calculatd by th aicaft and whl vlocity (shown in figu 7 and th dsid valu calculatd onlin with xtnal distubanc. Onc th tagt of tacking th onlin calculatd dsid slip at is achivd, th axiu adhsion foc and dclation a achivd as wll. So w can conclud that, accoding to th sult in figu 6, th dsid bak pfoanc is achivd by applying bak pssu signal in figu 5. Figu. 4 Estiation os of dual-obsv Figu. 5 Baking pssu signal applid to th actuato. Figu. 6 Aicaft syst siulatd and dsid slip at valus. Figu. 3 Estiatd and al intnal fiction stat valus. Figu. 7 Aicaft and lina whl vlocitis. Publishd by Atlantis Pss, Pais, Fanc. th authos 634

7 Pocdings of th nd Intnational Confnc on Coput Scinc and Elctonics Engining (ICCSEE 3 V. CONCLUSION In this study, a nonlina obust contoll is poposd basd on th LuG dynaic fiction foc odl. Th baking pssu contoll is dtind by th stiats of unasuabl intnal fiction stats. A dual-obsv is ployd to stiat unasuabl stats with th pojction apping and a sious of bounddnss assuptions in od not to obtain unstabl stiats. Th siulation sults show that th aicaft can b stoppd ffctivly with a na axiu slip at o dclation by applying th poposd contoll. Th asyptotic convgnc of th stiatd stats is also povd thotically and by siulation sults. REFERENCES [] Wang Ji sn. Nonlina contol thoy and application to aicaft antiskid syst. Xi an: Nothwstn Poly tchnical Univsity, (in Chinss. [] Tang Chuany. Siulation studis on aicaft anti-skid baking syst. Xi an: Nothwstn Poly tchnical Univsity, 7(in Chinss. [3] Li Yun, Ma Ruiqing, Xu Jing, Xi Lili. Ipoving iabl stuctu contol of aicaft. Jounal of Nothwstn Poly tchnical Univsity, 8,6(6:75-754(in Chinss. [4] Tian Guanlai, Xi Lili, Yu Kaixian, Chang Shunkong. Study on optial contol thod of an aicaft anti-skid baking syst basd on slip-atio. Acta Aonautica t Astonautica Sinica, 5, 6(4:46-464(in Chinss. [5] Bakk, E., Nybog, L. and Pacjka, H.B., Ty odlling fo us in vhicl dynaic studis. SAE Pap. No. 874, 987. [6] C. Canudas d Wit, M. L. Ptsn, A. Shiiav. A nw obsv fo ti/oad distibutd contact fiction. Confnc on Dcision and Contol, Hawaii, USA, 3. [7] J. Yi, L. Alvaz, X. Clays, R. Howitz. Adaptiv gncy baking contol in autoatd highway syst using a dynaic ti/oad fiction odl. Poccdings of 39th IEEE Confnc of Dcision Contol, Sydny, Austalia,, pp [8] Yaolong Tan, Ioannis Kanllakopoulos. Adaptiv nonlina fiction copnsation with paatic unctaintis. Pocding of th Aican Contol Confnc. San Digo: 999,pp [9] P. Toi, Robust adaptiv fiction copnsation fo tacking contol of obot anipulatos, IEEE Tan. on Auto. Cont., 45, pp 64 69,. [] B. Yao, M. Toizuka, Adaptiv obust contol of SISO non-lina systs in a si-stict fdback fo, Autoatica, 33, pp , 997. [] Sasty S, Bodson M. Adaptiv contol: stability, convgnc and obustnss. Englwood Cliffs, NJ 763, USA: Pntic Hall, Inc, 989. Publishd by Atlantis Pss, Pais, Fanc. th authos 635