PIO SUSCEPTIBILITY IN FLY-BY-WIRE SYSTEMS

Transcription

1 Hrlandson Cardoso d Moura*, Gabril Stim Porto Algr*, Jorg Hnriqu Bidinotto*, Eduardo Morgado Blo* * Univrsity of São Paulo, Enginring School of São Carlos, Brazil. Av. João Dagnon, , phon Kywords: PIO, Fly-By-Wir, Human pilot modls, Modl rfrnc, ROVER Abstract Th prsnt work proposs to invstigat th rlationship btwn th longitudinal dynamics of Fly-by-Wir aircrafts and th rsultant suscptibility to th Pilot Inducd Oscillations phnomna. Diffrnt valus of th stability drivativs will b simulatd in softwar implmntations with multipl thortical pilot modls. Th aircraft movmnt is considrd happn only along th longitudinal axis. It is xpctd from ths trials a rang of valus of th drivativs that must b avoidd in futur aircraft dsigns to minimiz th PIO occurrnc and which drivativs hav influnc on PIO occurrnc. 1 Introduction In th last fw dcads, th automation of aircrafts has bn growing fast. Considring it, th dvloping procss must concrn about safty and flying qualitis of such systms, spcially on th calld Fly-By-Wir (FBW) which is applid nowadays in larg scal. It is xtrmly hlpful in th projct dvlopmnt to avoid an undsird rspons, lik Pilot Inducd Oscillations (PIO), prior to flight tsts of th aircraft. Th PIO phnomna can b dfind as sustaind or uncontrollabl oscillations rsulting from th fforts of th pilot to control th airplan [1]. It has bn studid sinc th 1960s in manual flight control airplans, as sn in [2-4], that invstigat th causs of ths vnts. Anothr works prformd, such as [5,6], studid xprimntally human pilot modls to prdict th pronnss to PIO in th simulatd systms. Algorithms to dtctd th occurrnc of th phnomna in ral tim hav also bn studid, lik Th Ral Tim Oscillation Vrifir (ROVER), which was dfind by [7] and implmntd in [8]. Dspit ths phnomna hav bn studid for 50 yars thr ar fw rsarchs focusd on FBW systms. Thos issus hav motivatd th prsnt work, which aims to look for conditions that PIO suscptibility is prsnt for aircrafts with FBW systms, in ordr to incras th flight safty and quality on aircraft providd with this kind of command. For FBW systms, th stability drivativs can b implmntd via softwar. Consquntly, th rsultant rspons of such aircrafts can b altrd prior to flight opration. Considring this scnario, th prsnt work proposs to invstigat th rlation btwn th longitudinal dynamic charactristics of FBW airplans and its suscptibility to PIO. This pronnss is invstigatd with computr simulations, using th Matlab platform and also an intgration of this softwar with commrcial flight simulators that can provid a fast visual mthod to vrify th rsultant systm rspons. Th rquird dynamics of th simulatd aircraft is implmntd by th us of a Modl Rfrnc Adaptiv Control (MRAC), which can altr th original systm dynamics to follow th modl rspons. This papr is organizd as follows: th Aircrafts Dynamics Modlling is dscribd in sction 2, followd by th controllr framwork dscription in Sction 3. This chaptr also includs a brifly xplanation of pilot modls in sction 3.1 and th ROVER algorithm in Sction 3.2. Nxt, th xprimntal stup is dscribd in Sction 4. Exprimntal rsults ar thn prsntd in Sction 5 and a discussion of ths 1

2 HERLANDSON C. MOURA, GABRIEL S. PORTO ALEGRE, JORGE H. BIDINOTTO, EDUARDO M. BELO rsults is don in Sction 6. Finally concluding rmarks ar discussd in th last sction. 2 Aircrafts Modlling: Stability and Control drivativs An aircraft modlld as a rigid body hav six dgr of frdom, thr of translation (x,y and z) and thr of rotational motion angls (roll-φ, pitch-θ and yaw-ψ). Commonly, in th flight dynamics formulation an airfram s stability axs O B locatd at th aircraft s cntr of gravity (CG) and moving with it is dfind as prsntd in Fig. (1), which is rfrncd to a rfrnc fram O E. Onc this airfram is dfind, th rsultant arodynamic forc can b dscribd by thr componnts [X,Y,Z] acting along ach corrspondd axis. Similarly, th rsultant momnt vctor has th componnts [L,M,N] rprsnting th rolling, pitching and yawing momnt rspctivly. Th vlocity of th aircraft s CG also has th componnts [u,v,w] and angular vlocity th componnts [p,q,r], which rprsnts th rat of roll, pitch and yaw corrspondingly. Th linarizd quations of aircrafts motion in th longitudinal plan can thn b drivd considring som simplifying assumptions: th aircraft is tratd as a singl rigid body with six dgrs of frdom; th gyroscopic ffcts of th spinning rotors ar nglctd; th wind vlocity is considrd zro; and finally th longitudinal motion is dcoupld of th latral on. Considring ths simplifications and th Etkin[9] formulation, which applis th small-disturbanc thory, th linar quations of motion can b dfind as a stat spac modl. In th prsnt notation, th variabls rfrnc valus ar rprsntd by a subscript zro and th small prturbations from th rfrnc condition ar dnotd by th prfix Δ. Th longitudinal aircraft dynamics can b thn rprsntd by th quations: x = Ax + Bu (1) y = Cx + Du (2) whr th stats x is rprsntd by th vctor [ u w q θ] T and for a constant throttl control condition th input is dfind as th lvator angl position δ. Figur 1: Aircraft Modlling Th xpandd xprssion of Eq. (1) is prsntd in Eq. (3). Th trms X u, X w, Z u, Z w, Z q, Z w, M u, M q and M w rprsnt th longitudinal dimnsional stability drivativs and th trms X δ, Z δ and M δ rprsnt th longitudinal control drivativs. Ths paramtrs can also b xprssd in a nondimnsional way, which will b xprssd in th prsnt work by th variabls C xu, C xα, C zu, C zα, C zq, C zα, C mu, C mα, C mq, rspctivly for th stability drivativs and C, C xδ z and C δ m δ for th control drivativs. Th stability and control drivativs will b valuatd with diffrnt valus in th prsnt work, aiming to find th rlation btwn thir valus and th rsultant pronnss to th PIO phnomna. Th matrixs C and D ar dfind dpnding of th output considrd. At th prsnt work C is considrd an idntity matrix and D is a zro matrix, which nabls th output to b th stats dfind, as shown in Eq. (4). 2

3 u w [ ] = q θ X u m X w m 0 g cos θ o Z u Z w Z q + mu o mg sin θ o m Z w m Z w m Z w m Z w [ 1 [M I u + M w Z u y (m Z w ) ] 1 [M I w + M w Z u y (m Z w ) ] 1 [M I q + M w (Z q + mu o ) ] M w mg sin θ o y (m Z w ) I y (m Z w ) [ ] + M δ I y [ X δ m Z δ m Z w + M w Z δ I y (m Z w ) 0 ] δ u w ] q θ (3) u w y = [ ] [ ] q θ (4) In ordr to simulat a mor ralistic aircraft a simpl actuator srvo-hydraulic modl of th lvator control surfac was incorporatd to th systm. This modl nabls th modling of th rat and position saturation of th actuator systm and can b dscribd by th diagram showd in Fig. (2). In th normal condition of opration, which occurs whn th commands ar of small amplitud, th actuator dynamics can b dscribd by Eq. (5). G ac (s) = 1 τs + 1 Figur 2: Actuator dynamics (5) 3 Control systm framwork Th control systm structur is showd in Fig. (3). Th rfrnc signal, commonly namd as synthtic task (Syntask), consists in a squnc of pitch angls to b followd by th simulatd aircraft. Th rror btwn th rfrnc and actual position of th pitch angl is thn snt to a pilot modl, which aims to simulat th bhavior of a ral human pilot. This modl so calculats a control signal which is passd to a rfrnc stat spac modl. This modl consists of a stat spac quation, as dfind in Eq. (1) and (2), which modls th longitudinal dynamics of an aircraft to a givn st of valus for th stability and control drivativs. Furthrmor, this rfrnc modl oprats in a closd-loop schm considring for that th Linar Quadratic Rgulator (LQR) mthodology. Th output of th modl is thn usd to calculat an rror signal btwn its output and th actual output of th controlld aircraft, which in th prsnt work is rprsntd by a flight simulator modl. Ths st of signals is so usd in th implmntation of th M-MRAC (Modifid Modl Rfrnc Adaptiv Control), 3

4 HERLANDSON C. MOURA, GABRIEL S. PORTO ALEGRE, JORGE H. BIDINOTTO, EDUARDO M. BELO which guarants that th controlld aircraft follows th modl rfrnc dynamics. Lastly, th ROVER algorithm dtcts th PIO phnomna in ral tim in th controlld aircraft output. Th ROVER output will b thn usd in th valuation of th implmntd modls pronnss to th phnomna. Th nxt sctions dscrib ach of th citd control systm lmnts. δ p (s) (s) = K p 0.3s 0.2s + 1 δ p (s) (s) = K (6s + 1) 0.3s p (3s + 1) G NM(s) (7) (8) G NM (s) 1 = ( (0.1s + 1) ( s ) 20 s + 1) Figur 3: Control systm structur 3.1 Pilot Modlling Du to th ability to adapt its bhavior, th fild of modlling th human rspons consists in a challnging and difficult issu. Nvrthlss, som transfr functions can simulat th human pilot bhavior in simpl tasks as th on considrd in th prsnt work, whr a rfrnc signal of th pitch angl must b trackd. For this purpos, thr modls will b considrd: th Tustin[10] pilot modl, th crossovr[11] modl and th prcision[12] pilot modl dscribd by th Eqs. (6), (7) and (8) rspctivly. Th input for ths modls is considrd th rror (s) for th tracking task considrd and th output is dfind as th lvator angl command δ p (s). 3.2 Rfrnc Modl: LQR Th rfrnc modl prsntd in Fig. (3) is dfind in closd loop schm using th LQR control stratgy. This approach nabls th possibility of modification in th original rspons of th modl. Bsids, th stability of this systm is assurd with th LQR control systm. For its implmntation is aimd, for a givn MIMO (Multipl-Input-Multipl-Output) systm x = Ax + Bu(x R n and u R m ), a control law u = Kx which minimizs th quadratic cost function: J = (x T Qx + u T Ru)dt 0 (9) whr Q 0 and R > 0 ar symmtric, positiv smi-dfinit matrics with appropriatd dimnsions. Th solution for this LQR problm givs th gain K of th form: K = R 1 B T P (10) to a P R n n positiv dfinit, symmtric matrix which satisfis th quation: δ p (s) (s) = K p(5s + 1) 0.3s s (6) PA + A T P PBR 1 B T P + Q = 0 (11) 4

5 Howvr, th LQR problm prsntd is basd on th minimization of a st of signals, commonly known as controlld outputs, in th shortst amount of tim. Thrfor, som modifications ar ncssary in ordr to nabl this mthod to b applid in systm which aims to track a rfrnc signal. On of ths mthods, as statd in [13], adds to th systm a nw stat z of th rror btwn th rfrnc r and th systm output, as follows: z = r C lqr x (12) whr C lqr dnots a matrix of appropriatd dimnsions. Th systm is thn altrd in a augmntd form with th inclusion of th nw stat: [ x A 0 n m ] = [ z C lqr 0 ] [ x m m z ] B + [ ] u 0 m m + [ 0 n m I m m ] r (13) whr I rprsnts an idntity matrix. Onc th augmntd systm is dfind, a control law which minimizs th quadratic cost function prsntd in Eq. (9) is sarchd in th form u = [K x K z ] [ x z ]. 3.3 M-MRAC Th control systms namd MRAC (Modl Rfrnc Adaptiv Control) ar a class of algorithms which th dsird rspons of th controlld systm is spcifid by a modl. Th paramtrs of th controllr ar thn adjustd basd upon th rror btwn th output of th controlld and th modl systms. Th controllr paramtrs so convrg asymptotically to a st of idal valus, which can nabl th controlld systm to tracks a rfrnc following th dynamics of th modl systm. For a MIMO systm, x = Ax + Bu(x R n and u R m ) with x(0) = x 0, a MRAC controllr can b dfind using th invrs Lyapunov analysis, as statd in [14]. Th main purpos of th this MRAC systm is to dsign a stat fdback adaptiv control law so th systm stat x globally uniformly asymptotically tracks th stat x rf R n of th rfrnc modl: x rf(t) = A rf x rf (t) + B rf r(t) x rf (0) = x 0 (14) whr A rf R n n is a Hurwitz matrix, B rf R n m and r(t) R m is an xtrnal boundd command vctor. Th control input u thus nds to b slctd so th tracking rror prsntd in Eq. (15) globally uniformly asymptotically tnds to zro. (t) = x(t) x rf (t) (15) Through th application of th invrs Lyapunov analysis th control law can b computd as statd in Eq. (16), whr K x R n X m K r R m X m ar control gains (or adaptiv laws) which ar gnratd onlin with th controllr. u = K xt x + K rt r (16) If th adaptiv laws ar slctd as statd in Eq. (17) a Lyapunov function V can b dfind with its tim drivativ globally ngativ smidfinit, in th form V = T Q 0 (for som Q = Q T > 0 ), and with its scond drivativ boundd in th form V = 2 T Q. Consquntly, th Lyapunov analysis stablishs that th stat tracking rror (t) dfind in Eq. (15) tnds to th origin globally, uniformly and asymptotically. Th trms Γ x = Γ T x > 0 and Γ r = Γ T r > 0 ar usr slctd matrics of rat of adaption and P = P T > 0 satisfis th algbraic Lyapunov quation PA rf + A T rf P = Q for som Q = Q T > 0. K x = Γ x x T PB K r = Γ r r(t) T PB (17) Nvrthlss, th classical MRAC systm dscribd abov can hav an oscillatory transint bhavior which dtriorat th controllr rspons and limits its applicability, 5

6 HERLANDSON C. MOURA, GABRIEL S. PORTO ALEGRE, JORGE H. BIDINOTTO, EDUARDO M. BELO spcially in a PIO scnario, whr an oscillatory rspons alrady xists. To ovrcom this problm, a simpl modification in th rfrnc modl can b don by fding back th tracking rror signal. This approach is calld modifid rfrnc modl MRAC or M-MRAC and is dfind in [15]. In th M-MRAC framwork, th tracking rror signal m (t) = x(t) x m (t) is computd using th modifid modl rfrnc: x m(t) = A m x m (t) + B m r(t) + λ m (t) x m (0) = x 0 (18) whr λ > 0 is a dsign paramtr. On can notic that for λ = 0 th convntional MRAC dsign is achivd again, with (t) = m (t). Th application of th Lyapunov stability thory can also prov that th modifid stat tracking rror m (t) tnds to th origin globally, uniformly and asymptotically. Th paramtr λ, according to [16] mthodology, can b calculatd in th trms of th adaption rat matrix Γ r and th matrix P ( P = P T > 0 satisfis th algbraic Lyapunov quation PA rf + A T rf P = Q for som Q = Q T > 0) as follows: λ = 2αΓ r λ máx (B m T PB m ) (19) whr α = x sup t r(t) 2 and λ máx is a adjustabl paramtr. 3.4 ROVER Th ROVER algorithm considrd in th prsnt work can dtct in ral tim th occurrnc of th PIO phnomna basd on th monitoring of four paramtrs: th amplitud and frquncy of th pitch rat aircraft rspons, th amplitud of th pilot command and th phas angl diffrnc btwn ths signals. If on of ths monitord paramtrs xcd th valus prsntd in Tabl (1) th valu 1 is st to a corrspondd flag. Whn th four flags ar st with 1 a svr PIO condition occurs. Th implmntd ROVER algorithm givs thn a signal which in th PIO condition (four flags st) has th valu on, othrwis it has a zro valu. Tabl 1. ROVER paramtrs Paramtr Thrshold valu Pitch rat magnitud 12º /s Pitch rat frquncy 1 10 rad/s Pilot command 1.2 pak-to-pak (60% of maximum dflction) Phas diffrnc 180º 4 Mthodology In ordr to invstigat th rlation btwn th longitudinal stability /control drivativs and th rsultant pronnss of th implmntd aircraft dynamics to th PIO, simulations of th dscribd systm ar prformd in th softwar Matlab. Th aircrafts modls implmntd in th simulations ar basd on th flight data of th modl Boing , drivd from [9], in a cruising and horizontal flight condition at a fixd altitud of ft at a Mach Numbr of 0,8. Th original stability and control drivativs of this aircraft modl is shown in Tabl (2). Tabl 2. Boing Stability/ Control Drivativs Drivativ Valu Drivativ Valu C xu C mα C xα C mq C zu C zα C xδ C zq C zδ C zα C mu x In th first stag of th prsnt work, th longitudinal stability /control drivativs drivd from this flight condition data ar thn variatd individually to diffrnt valus and a stat spac rfrnc modl dynamic is thn drivd from ths altrd valus. Simulations ar prformd 6

7 considring just th rfrnc modl oprating in a closd loop condition with th LQR systm architctur dscribd in Sction 3 for diffrnt pilot modls. Th rfrnc modls implmntd in this stag also includ th actuator dynamics dscribd in Sction 2. Th pronnss of ach rfrnc modl obtaind to th PIO is thn valuatd basd on th output of th ROVER algorithm, which is xprssd as a prcntag lvl of activation of th algorithm in th tim intrval considrd. It is xpctd of ths simulations a rang of valus for ach drivativ which must b avoidd in ordr not to occur th phnomna. In th scond stag of th work, sts of combination of ths drivativs obtaind from th last stag ar drivd, with low and high pronnss to th PIO. Rfrnc modls ar so calculatd from ths valus, and th complt control architctur with th M-MRAC systm dscribd in Sction 3 is simulatd. Th Boing modl implmntd in th flight simulator softwar FlightGar is considrd as th controlld systm in this stag. Th rsultant pronnss to th PIO of th controlld aircraft is thn valuatd basd on th ROVER activation again. Ths simulations aim to confirm th influnc of som of th drivativs in th rsulting rspons of th controlld systm. Furthrmor, th flight simulator can also provid a fast visual mthod to vrify th occurrnc of oscillations in th rsultant systm rspons. 5 Rsults In th first stp, simulations ar prformd using th stat spac modls in ordr to obtain a rang of drivativs, with high pronnss to PIO, to b avoidd for ach pilot modl considrd. Ths rangs obtaind, for th stability and control drivativs, as rsult of this procss ar shown in Tabls (3),(4) and (5) for Tustin, Crossovr and Prcision pilot modls rspctivly. In ths tabls th ROVER rang obtaind, with its minimum and maximum valus, for ach corrspondd drivativ is also prsntd. Th valus xprssd in parnthss stablishs th corrspondd drivativ valu for th PIO lvl stablishd. Stability / Control Drivativ Tabl 3. Rang with high pronnss to Tustin Pilot Modl Non-dimnsional rang C mq C zδ Stability / Control Drivativ ROVER Rang Lvl (%) 8.40 (-7.25) (-4.07) 6.19 (-3.30) 6.34 (45.94) (-95.51) (-94.07) (1.37) (1.36) Tabl 4. Rang with high pronnss to Crossovr Pilot Modl Non-dimnsional rang C mq C zδ Stability / Control Drivativ ROVER Rang Lvl (%) (28.30) (-7.25) (23.09) (9.76) (-96.44) (-97.56) (1.47) (1.39) Tabl 5. Rang with high pronnss to Prcision Pilot Modl Non-dimnsional rang C mq C zδ ROVER Rang Lvl (%) (27.20) (-21.10) (44.06) (-4.12) ( ) (-93.68) (1.53) (1.22) Considring th valus obtaind for ach stability/control drivativ, gnral rangs could b drivd for an indpndnt pilot modl systm. Ths rangs ar prsntd in Tabl (6). 7

8 HERLANDSON C. MOURA, GABRIEL S. PORTO ALEGRE, JORGE H. BIDINOTTO, EDUARDO M. BELO Furthrmor, th proposd mthod can nabl on to find also a rang of valus with low pronnss to th phnomna. Ths valus ar prsntd in Tabl (7). Tabl 6. Gnral rang with high pronnss Stability / Non-dimnsional rang Control Drivativ C mq C zδ rangs foundd with low and high pronnss. Th original modl was also simulatd in ordr to compar th PIO lvl obtaind. Th high pronnss modl was st with C mq = 6. 92, = , C zδ = and = Th low pronnss modl was st with C zq = , C zα = and = Th othrs stability/control drivativs of ths modls wr maintaind with its original valus. Th Figs. (4), (5) and (6) prsnt th plot rspons of th simulations prformd. In ths figurs th PIO lvl obtaind to ach modl considrd is also shown, which was multiplid by a factor of tn to facilitat th visualization. Tabl 7. Gnral rang with low pronnss Stability / Non-dimnsional rang Control Drivativ C zq x 10 3 C zα C mα C mq C zδ x x In th scond stp simulations wr prformd with th M-MRAC systm intgratd with th flight simulator FlightGar dscribd in sction (3). Th rfrnc modls wr built applying th Figur 4: M-MRAC+ Flight Simulator: Tustin Pilot Modl 8

9 Figur 5: M-MRAC+ Flight Simulator: Crossovr Pilot Modl 6 Discussions In th first stp of th prsnt work simulations wr prformd with th purpos of obtaining rangs for th stability/control drivativs with high pronnss to PIO. In this stag, th original drivativs valus (prsntd in Tabl (2)) wr varid and th rsultant pronnss of th systm was valuatd. For this intnt, a closd loop systm was simulatd consisting in th stat spac modl of th aircraft, a LQR controllr and th ROVER algorithm. Th systm also includd a pilot modl dscribd function in ordr to simulat a mor ralistic scnario. In this stag, to obtain th high pronnss drivativs, th systm was first simulatd with its original drivativs valus with th pilot gain adjustd on th onst of a PIO condition, basd on th ROVER signal. Figur 6: M-MRAC+ Flight Simulator: Prcision Pilot Modl Th PIO lvl obtaind in this condition is thn usd as a rfrnc to b compard with th valus obtaind with th variation of th drivativs. Th rsultant stat spac modl, obtaind from th drivativs variation, is thn considrd as a high pronnss if lads to a rspons with a gratr ROVER lvl activation. For th Tustin pilot modl, a gain of K p = ld th original stat spac systm to a PIO lvl of activation of 0% (prcntag of th tim duration of th simulation). As prsntd in Tabl (3) th most snsitiv drivativs ar C mq,, C zδ and. This rsult was partially xpctd, sinc ths drivativs appars dirctly in th xprssions of q and θ 0 in th stat spac modlling prsntd in Sction (2). Th variation of th othr drivativs ld th modl to unsatisfactory rsponss, and thn ar not 9

10 HERLANDSON C. MOURA, GABRIEL S. PORTO ALEGRE, JORGE H. BIDINOTTO, EDUARDO M. BELO considrd in th prsnt work. In this cas, for instanc, th adjustmnt of th drivativ C mq to a valu of C mq = 4.07 lads th systm to a PIO condition with a ROVER lvl of %. For th Crossovr Pilot modl a gain of K p = mak th original systm to oprat in a PIO lvl condition of 2%. Th most propns drivativ, in this cas, was C zδ = which rsults in a systm with a ROVER lvl of %. In th last pilot dscrib function considrd, th Prcision modl, a gain of K p = 4 mak th original systm to oprat in a PIO lvl condition of 1.93%. Th most propns drivativ was thn C zδ = 93.68, which lads to a systm with a ROVER lvl of %. Onc ths rangs, with high pronnss, wr obtaind, th analysis of th Tabls (3), (4) and (5) can lad on to dtrminat a gnral rang, for ach drivativ considrd, with high pronnss to th phnomna, which is indpndnt of th pilot modl considrd. Ths rang valus ar shown in Tabl (6). Th rlvanc of th mthod dvlopd in th prsnt work rsids in th fact that it is indpndnt of th control systm architctur considrd. Thrfor, it can b applid to any control systm considrd, just by varying th drivativs valus and valuating th rsultant pronnss to PIO. Bsids, th mthod can nabl on also to find rangs for th drivativs with low pronnss to th phnomna. For this cas, th pilot gain of th modls is adjustd with highr valus in ordr to obtain a svr PIO condition. Th drivativs ar thn variatd, and thir corrspondd valus ar considrd to b part of a low pronnss modl if lad th systm to a lowr ROVER activation lvl. Th rangs obtaind for this cas ar prsnt in Tabl (7). As can b noticd by comparing th Tabls (6) and (7), th sam drivativs of th high pronnss cas can lad th systm to a low pronnss condition. Th diffrnc rsids just in th rangs considrd. In th scond stag of th work, simulations with commrcial flight simulator softwar wr prformd in ordr to confirm th ffctivnss of th proposd mthod. In this stp, th M-MRAC control systm architctur dscribd in Sction (3) was applid. Th valus obtaind in Tabl (6) for th drivativs wr combind, and a high pronnss modl was stablishd. In a similar procdur, a low pronnss modl was obtaind from th rsults of Tabl (7). Th original stat spac systm and ths nw systms obtaind ar thn usd as rfrnc modls to th M-MRAC systm. Each rsultant systm was thn simulatd with th sam pilot modls considrd in th first stp with high gains to incit a svr PIO condition. Th controlld plant, in this cas, is considrd th aircraft dynamics modl of th flight simulator. In fact, as can b noticd from th rsults illustratd in Figs. (4), (5) and (6), th high pronnss modl ld th systm to a mor svr PIO condition for th thr pilot modls considrd, sinc th ROVER lvl was gratr for this cas. On th othr hand, th low pronnss modl built mad th systm to stop th PIO occurrnc, which confirm th fficacy of th mthod. 7 Conclusions Th mthod proposd in th prsnt work can nabl on to find th rangs for th stability/control drivativs mor propns to th PIO phnomna. Th mthod can b mployd to any control systm architctur applid to crat a closd loop systm with th stat spac modl considrd, sinc it is basd on th outputs of th rsultant systm. Th proposd systm can also b applid to dtrmin low pronnss modls and to any pilot modl considrd. Furthrmor, its ffctivnss can b confirmd with th proposd M-MRAC systm intgratd with commrcial flight simulators, which can improv th analysis of th rsultant systm by th us of th visualization of th aircraft rspons in ral tim. Bsids, considring th systm is alrady intgratd with a flight-simulator, futur works can b prformd with human pilots using th visual fdback providd by a full flight simulator, including th architcturs intgratd with movabl platforms. 10

11 Rfrncs [1] DEPARTMENT OF DEFENSE INTERFACE STANDARD. Mil Standard, MIL1797A: Flying Qualitis of Pilotd Airplans. Washington D. C., [2] ASHKENAS, I. L.; JEX, H. R.; MCRUER, D. T. Pilot-inducd oscillations: thir caus and analysis. SYSTEMS TECHNOLOGY INC INGLEWOODCA, [3] SMITH, J. W.; BERRY, D. T. Analysis of longitudinal pilot-inducd oscillation tndncis of YF-12 aircraft [4] ANDERSON, M. R. Pilot-Inducd Oscillations Involving Multipl Non-linaritis. Journal of Guidanc Control and Dynamics, v. 21, n. 5, p , [5] TOADER, A.; URSU, I. Pilot Modling Basd on Tim-Dlay Synthsis. Procdings of th Institution of Mchanical Enginring Part G - Journal of Arospac Enginring, v. 228, n. 5, p , [6] ACOSTA, D. M.; YILDIRAY, Y.; CRAUN, R. W.; BEARD, S. D.; LEONARD, M. W.; HARDY, G. H.; WEINSTEIN, M. Pilot Evaluation of a Control Allocation Tchniqu to Rcovr from Pilot-Inducd Oscillations. Journal of Aircraft, v. 52, n. 1, p , [7] MITCHELL, D. G.; ARENCIBIA, A. J.; MUNOZ, S. Ral-tim dtction of pilot-inducd oscillations. In: AIAA Atmosphric Flight Mchanics Confrnc and Exhibit, Providnc, Rhod Island, p. 4700, [8] JOHNSON, D. A. Supprssion of pilot-inducd oscillation (PIO). AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOOL OF ENGINEERING AND MANAGEMENT, [9] ETKIN, B.; REID, L. D. Dynamics of flight stability and control. 3rd Edition. Nw York: Nw York Wily, [10] TUSTIN, A. Th natur of th oprator's rspons in manual control, and its implications for controllr dsign. Journal of th Institution of Elctrical Enginrs-Part IIA: Automatic Rgulators and Srvo Mchanisms, v. 94, n. 2, p , [11] MCRUER, D. T.; KRENDEL, E. S. Mathmatical modls of human pilot bhavior. Advisory Group For Arospac Rsarch And Dvlopmnt. Nuilly-Sur- Sin (Franc), [12] MCRUER, D. T., Graham, D., Krndl, E., and Risnr, W., Human Pilot Dynamics in Compnsatory Systms. Air Forc Flight Dynamics. Lab. AFFDL [13] GHAFFAR, Alia Abdul; RICHARDSON, Tom. Modl rfrnc adaptiv control and LQR control for quadrotor with paramtric uncrtaintis. World Acadmy of Scinc, Enginring and Tchnology, Intrnational Journal of Mchanical, Arospac, Industrial, Mchatronic and Manufacturing Enginring, v. 9, n. 2, p , [14] EUGENE, Lavrtsky; KEVIN, Wis. Robust and adaptiv control with arospac applications. 1st Edition. London, Springr, [15] STEPANYAN, Vahram; KRISHNAKUMAR, Kalmanj. MRAC rvisitd: guarantd prformanc with rfrnc modl modification. In: Amrican Control Confrnc (ACC), IEEE, p [16] STEPANYAN, Vahram; KRISHNAKUMAR, Kalmanj. On th Robustnss Proprtis of M-MRAC. In: Infotch@ Arospac Acknowldgmnts This work is supportd by São Paulo Rsarch Foundation (FAPESP), Grant 2016/ , and by th National Council for Scintific and Tchnological Dvlopmnt (CNPq). Contact Author Addrss Hrlandson C. d Moura: hrlandsonc@usp.br Gabril S. Porto Algr: gabril.algr@usp.br Jorg H. Bidinotto: jhbidi@sc.usp.br Eduardo M. Blo: blo@sc.usp.br Copyright Statmnt Th authors confirm that thy, and/or thir company or organization, hold copyright on all of th original matrial includd in this papr. Th authors also confirm that thy hav obtaind prmission, from th copyright holdr of any third party matrial includd in this papr, to publish it as part of thir papr. Th authors confirm that thy giv prmission, or hav obtaind prmission from th copyright holdr of this papr, for th publication and distribution of this papr as part of th ICAS procdings or as individual off-prints from th procdings. 11