DESIGN OF LONGITUDINAL CONTROL SYSTEM FOR A NONLINEAR F-16 FIGHTER USING MSS METHOD. Eric H.K. Fung*, Y.K. Wong**, Hugh H.T. Liu + and Y.C.

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1 DESIGN OF LONGITUDINAL CONTROL SYSTEM FOR A NONLINEAR F- FIGHTER USING MSS METHOD Eric H.K. Fng*, Y.K. Wong**, Hgh H.T. Li + and Y.C. Li* *Dpartmnt of Mchanical Enginring, **Dpartmnt of Elctrical Enginring, Th Hong Kong Polytchnic Univrity, Hng Hom, Hong Kong + Intitt for Aropac Stdi, Univrity of Toronto Toronto, Ontario, Canada, M3H 5T Abtract: In atomatic flight control ytm or atopilot, mltipl pcification critria ar ndd to b atifid concrrntly, ch a good holding (mall tatic altitd holding rror), fat rpon, mooth tranition (l ocillation, ovrhoot). So how to dign th MSS (Mltipl Simltano Spcification) controllr ffctivly and practically i a vry ignificant and challnging job. Li propod a MSS controllr dign mthod (Li and Mill, ). In thi papr, w frthr apply th mthod togthr with th fin-tning tchniq to th DoF nonlinar F- fightr longitdinal control channl. Simlation rlt how it applicability to nonlinar flight control ytm. Copyright 5 IFAC Kyword: MSS control, flight control, longitdinal control channl, pitch attitd control, pd control.. INTRODUCTION Aircraft manfactrr hav rachd a high lvl of xprti and xprinc in flight control. Th crrnt dign and analyi tchniq applid in indtry nabl flight control nginr to addr virtally any ralitic dign challng. Howvr, th dign and implmntation of flight control law i till a vry complx tak and th many dign problm that hav to b conidrd mak it a cotly and lngthy proc. For xampl, in atomatic flight control ytm or atopilot, mltipl pcification critria ar ndd to b ati fid concrrntly, ch a good holding (mall tatic altitd holding rror), fat rpon, mooth tranition (l ocillation, ovrhoot). So how to dign th MSS (Mltipl Simltano Spcification) controllr ffctivly and practically i a vry ignificant and challnging job. Li propod a MSS controllr dign approach for th abov problm (Li and Mill, ). In thi papr, w apply th mthod frthr to a mor practical nvironmnt-dof nonlinar F- fightr longitdinal control channl. In th longitdinal control channl of th F- fightr, th pitch control loop and pd control loop ar conidrd for th flight control intgration (Etkin, B.,98). Th pitch attitd control channl i th baic longitdinal atopilot channl; it control th pitch angl by applying appropriat dflction of th lvator if th actal pitch angl diffr from th dird rfrnc val. Th pd control channl i alo an atopilot channl; it maintain a contant pd or Mach nmbr throgh coordinatd control of throttl and lvator. For th longitdinal control loop, w nd to dign propr controllr to atify mltipl objctiv. Hr w apply th MSS controllr dign mthod togthr with th fin-

2 tning tchniq to obtain th final controllr for th DoF nonlinar F- fightr. Th rt of th papr i organizd a follow. In Sction w rviw th ncary thortical backgrond of th MSS controllr dign mthod. And in Sction 3, w giv th DoF F- fightr nonlinar modl and th linar modl at th trimmd opration point. Sction 4 giv th dign implmntation of th individal control channl and th intgratd control of th longitdinal channl of th F- linar modl, and th imlation rlt of linar modl and non-linar modl. Th conclion and on-going/ftr rarch ar givn in Sction 5.. MSS CONTROLLER DESIGN METHOD A gnral framwork for control ytm incld th plant rprntd by a tranfr matrix P, an xogno inpt w and actator inpt, a controllr rprntd by a tranfr matrix K, and a rglatd otpt z and nor otpt y, a hown in figr (Boyd and Barratt, 99). w z P K Fig.. Control ytm fram work W partition th plant tranfr matrix P a Pzw Pz P () Pyw Py Hnc z Pzww + Pz () y P w + P whr ij yw P i th tranfr matrix from j to i, i z, y; j, w. Now ppo th controllr i oprating, o that w hav Ky (3) W can olv for z in trm of w to gt z ( Pzw + PzK( I PyK) Pyw) w (4) that i, th clod-loop tranfr matrix H can b rprntd a H Pzw + PzK( I PyK) Pyw (5) Many control dign pcification ar convx fnction with rpct to th clod-loop tranfr matric H (Boyd and Barratt, 99), that i, all prformanc pcification can b conidrd imltanoly a fnction in trm of H, which ar valatd ndr vry diffrnt controllr K. If thr ar n convx pcification rqird to b atifid imltanoly, dnotd a y y whr i φ ( H ) α φ ( H ) α L () φn ( H ) α n α ( i,,..., n ) dnot th xpctd pcification val, thn a MSS control problm can b formalizd a: Dign a controllr K ch that all th pcification hold imltanoly. W call ch a controllr a atifactory controllr. Li propod th convx combination mthod (Li, ) to obtain th MSS controllr. 3. NONLINEAR F- MODEL AND LINEARIZATION Now w apply th propod MSS controllr dign mthod to a DoF F- fightr. In th following, w giv th non-linar ordinary diffrntial qation (ODE) dcribing th motion of a F- fightr (Lat, T. Ngyn, t al., 979): q T & rv qw g in + CX, t + m m q v& pw r + g co inϕ + CY, t m q w& q pv + g co coϕ + CZ, t m I I qb (7) p& Z XZ qr + ( r& + pq) + Cl, t I X I X IX I I I qc q& Z X pr XZ + ( r p ) + Cm, t Hr IX I XZ qb r& pq + ( p& qr) + Cn, t Hq IZ IZ IZ whr, v, w and p, q, r ar th body-ax componnt of linar vlociti and rotational vlociti, rpctivly; yaw angl ψ, pitch angl, and roll angl ϕ, that i, th Elr angl dnot th attitd of th aircraft with rpct to th Earth; g i acclration d to gravity, m i airplan ma; q i th fr-tram dynamic prr; dnot wing ara, b i wing pan, c i wing man arodynamic chord.t i th ngin thrt, H i th ngin anglar momntm; I X,, IZ, IXY, IXZ, Z ar inrtia tnor ; th cofficint C X, t, CY, t, CZ, t, Cl, t, Cmt,, Cn, t, ar th total arodynamic cofficint, which wr drivd from low-pd tatic and dynamic wind-tnnl tt condctd with bcal modl of th F- in windtnnl faciliti at th NASA Am and Langly Rarch Cntr fightr (Lat, T. Ngyn, t al., 979). Th motion qation and th blow kinmatic qation togthr mak p th indpndnt ODE, which i th F- nonlinar ODE modl.

3 q inϕ + rcoϕ ψ& co & q coϕ rinϕ ϕ& p + ψ& in x& { co + ( v inϕ + wcoϕ)}coψ (8) ( v coϕ winϕ)inψ y& { co + ( vinϕ + wcoϕ)}inψ ( v coϕ winϕ)coψ z& in + ( vinϕ + wcoϕ)co whr th Elr angl ψ,, and ϕ dnot th attitd of th aircraft with rpct to th arth; x, y, z dnot th poition of th aircraft with rpct to th arth-fixd rfrnc fram. In ordr to dign th MSS controllr for th nonlinar F- fightr, firt, w nd to obtain th linarizd modl, and ynthiz th linar controllr bad on linar ytm thory. Thn, with th nonlinar fightr modl, w apply and fin-tn th linar controllr to driv th propr controllr. In thi papr, w apply th MSS controllr dign mthod to dign th initial controllr for th individal loop and thn intgrat th MSS controllr ing th opn-loop combination mthod (Li, ). According to th imlation rlt of th nonlinar modl, th trialand-rror mthod i d to add th drivativ control trm to th initial controllr, and th propr controllr for th nonlinar ytm i finally obtaind. At th bginning, w nd to find th tady-tat flight condition which can b d a oprating point for th linarization, and a initial condition for imlation. In thi papr, w conidr th tady wing-lvl flight of F- fightr at an altitd of 5m and Mach nmbr i.. Thn w compt th tady-tat flight condition and linariz th ODE nonlinar modl by mall-prtrbation mthod. Th linarizd F- ytm dcribd by th tatpac matric A, B, C, D, can b dnotd by th tandard Matlab LTI ytm for tdy convninc. W conidr two inpt two otpt bytm, who inpt ar dflction of lvator δ, dflction of ngin thrtδ t, th otpt ar pitch attitd, airpd along x-ax and th tat vctor x i [, v, w, p, q, r, ψ,, ϕ, x, y, z].bca th matrix A i a matrix, th dnominator of th tranfr fnction i of ordr. It i tr that ndr th condition of mall prtrbation from tadytat, wing-lvl, non-idlipping flight, th rigidaircraft qation of motion cold b plit into two ncopld t. Th ar th longitdinal qation that involv, w, q, and th latral-dirctional qation that involv v,ϕ, p, r. It i poibl to xtract implifid b-matrix ALo from A by pcifying a vctor with th lmnt nmbr of th rqird tat variabl. Similarly, th bmatric B Lo, C Lo, DLo can b obtaind from B, C, D, rpctivly. Thn th drivd imp lifid longitdinal ytm ha th following xprion: & w& w δt A + Lo B q& Lo q δ & (9) w δ t CLo + D Lo q δ whr A Lo B Lo ; C -. Lo DLo Th tranfr fnction of th F- longitdinal control ytm ar thn drivd at lat. G δ t G δ t G δ.748( -.33) ( ) ( )( ).54( +.4) ( )( ).3988 ( + 48.) ( ) ( ) ( ) G δ ( -. ( ) ( +.) ) ( ) 4. INTEGRATED PITCH/SPEED AUTOPILOT DESIGN AND SIMULATION Now w conidr th intgratd longitdinal control ytm of aircraft (Li, ), hown in figr. c c K K δ δ t G ϑδ G δ G ϑδt G δ t Fig.. Intgratd longitdinal control ytm Am that th ovrall mltipl prformanc rqirmnt ar: th pitch attitd and pd control both hav good dign critria in trm of tracking (mall tady tat rror and fat tting tim) and afty (accptabl ovrhoot). Th cro ffct i rprntd by th imlation top tim val ndr th cro tp command.

4 φ( φovrhoot (max ( t) ) t, c ( t) φ( H ) φttl ( T) α; φ3( φovrhoot (max ( t) ) t, c ( t) φ4( H ) φttl ( T) α4; φ5( φcro ( T) c ( t) α5; φ( φcro ( T ) c ( t) α α; α3; () whr T i th imlation nd tim. Th dird pcification val of th F- fightr longitdinal control ytm ar dfind by: α.3; α.5; α 3.3; α 4.; α 5 ; α. In thi papr, w apply th opn-loop combination mthod (Li, ) to dign th propr intgratd controllr. Firt, dign th individal controllr by MSS controllr dign mthod (Li and Mill, ) and thn intgrat th individal controllr to mt th total pcification. For th pd control loop, w nd to atify th pcification φ and φ. Uing th MSS controllr dign mthod (Li and Mill, ), w dign th ampl controllr [Eq.()] to atify on pcification at on tim (Goodwin, t al., ), 5 J 5 + J + () thn φ ( H ).7597 ; φ ( H ).557 φ ( H ).35498; φ ( H ).5 Figr 3 i th imlation rlt of th controllr; it atifi th two pcification i.. φ.7353; φ.433. For pitch attitd control loop, w alo th am mthod. W dign two ampl controllr [Eq.(4)] to atify on pcification at a tim (Goodwin, t al., ), 5 J 5 J 5 (4) Thn w can gt th pcification matrix, i.. φ3( φ3( (5) φ4( φ4( Sinc φ3( H ) φ3( H ) α 3, φ4( φ4( H ) α 4 w hav λ.793, λ Thn th MSS controllr () i givn (Li and Mill, ): : -.9 ( +.55) ( ) () J ( ) Th imlation rlt ar hown in Figr 4. It how that th control objctiv can b atifid ccflly. From th MSS controllr dign mthod (Li and Mill, ), w nd to olv th inqality [Eq.()] φ ( φ( λ α ( ) ( ) φ H φ H λ α λ, λ + λ i () Uing th linar programming optimization rotin in Matlab, λ. 784, λ. 979 wr fond. Thn th final MSS controllr [Eq.(3)] wa drivd (Li and Mill, ):.459( )( +.585)( +.3) J (3) ( )( +.473) Fig.3. Spd control loop imlation Fig.4. Pitch control loop From th abov imlation rlt, it i obvio that th MSS controllr atifi th rqird objctiv of th rpctiv loop. Now w intgrat th individal loop (Li, ) throgh opn-loop combination mthod to form th intgratd control ytm whr th total pcification ar alo valatd. According to th opn-loop combination mthod (Li, ), th intgratd controllr K λ J ; K λ J (7) whr λ and λ ar contant cofficint. W manag to lct propr cofficint to achiv or pcification. Comparing diffrnt imlation rlt with diffrnt cofficint, λ λ ar fond to b

5 th propr cofficint for controllr [Eq.(7)]. Th imlation rlt of th intgratd pitch/pd atopilot ar prntd in Figr 5. It can b fond from Figr 5(a) that th prformanc ndr th intgratd control ytm i φ.783, φ.3, rpctivly. It man that ovrhoot of th intgratd ytm bcom largr than individal loop bt it till at th atifactory lvl. At th am tim, th ttling tim val bcom mallr than bfor probably d to th inflnc of th othr channl. Th cro ffct i vry mall, φ.45. Similar conclion may b drawn from Figr 5(b), in which th following prformanc 8 i obtaind: φ 3., φ , th cro 5 ffct φ (a) and of ( φ.88, φ ; φ.95 ) c (a) and of (b) and of c Fig.5. Longitdinal intgratd control of F- linar modl It i obvio that th MSS controllr can b applid to th linar ytm of th F- fightr. In ordr to valat th ffctivn of th MSS controllr in a practical aircraft longitdinal control channl, w prform th nonlinar imlation of F- fightr flying at th altitd of 5 mtr with Mach nmbr val.. Th rlt ar hown in Figr. c (b) and of c 3 ( φ 3.84, φ.88, φ ) Fig.. Longitdinal intgratd control of DoF F- nonlinar modl Th imlation rlt of th nonlinar F- fightr how that th MSS controllr i not atifactory inc om prformanc pcification cannot b atifid. So w nd to fin-tn th MSS controllr to achiv an accptabl control ffct. Not that th drivativ control will hav th ffct of incraing th tability of th ytm, rdcing th ovrhoot, and improving th tranint rpon (Goodwin, t al., ). Now w add a drivativ control to th intgratd controllr K [Eq.(7)] to bcom K λj + τ (8) whr τ i th drivativ control gain. Uing th trial- τ and-rror tchniq, w lct th gain 5. Thn th final controllr [Eq.(9)] i givn a:.459( )( +.585)( +.3) K ( )( +.473) (9) - 5 (+.58)( +.775)( ) K ( )

6 Th imlation rlt of th nonlinar F- fightr with th final intgratd controllr ar hown in Figr 7. controllr and imlation rlt a rportd arlir in th papr. Th imlation rlt dmontrat that th MSS dign mthod (Li and Mill, ) with controllr fin-tning can b applid to th nonlinar F- fightr longitdinal control ytm. 5. CONCLUSIONS (a) and of c In thi papr, w apply th MSS controllr dign mthod (Li and Mill, ) to th nonlinar F- fightr imlation. Fin-tning i prformd by adding a drivativ control action to th MSS loop controllr. Th imlation rlt vrify th ffctivn of th abov dign mthod in nonlinar F- longitdinal control ytm. In th ftr work, w will contin to tdy th robtn of MSS controllr dign mthod and it application in th mor complicatd aircraft atopilot. ACKNOWLEDGMENT Th athor wold lik to thank Th Hong Kong Polytchnic Univrity for th financial pport (Projct No. A-PE77) toward thi work. REFERENCES (b) and of c Fig.7. Longitdinal intgratd control of DoF F- nonlinar modl with th final controllr It can b n that th rlting fin-tn control ytm atifi th dign pcification imltanoly in pit of th diffrnc btwn th rlt of th linar and th nonlinar F- fightr modl. Rmark: Sinc thr ar diffrnc btwn th linar modl of th ytm and it original nonlinar modl, th initial linar controllr applid to th nonlinar ytm may not atify th prformanc pcification that wr fond achivabl in th linar modl. So w nd to adjt th linar controllr by adding a drivativ control whn applid to th nonlinar modl. In fact, in thi papr, w can alo carry ot th fintning at th MSS controllr dign tag of th individal loop. Whn w hav drivd a MSS controllr, w apply it to th nonlinar modl of th individal loop. If th MSS controllr i fond to b natifactory, th MSS controllr hold b adjtd (or fin-tnd) ntil an accptabl on i obtaind. Uing thi approach, w can obtain th imilar Etkin, B. and L.D. Rid (99). Dynamic of Flight Stability and Control, Wily, Nw York, USA. Goodwin, G.C., S.F. Grab and M.E. Salgado (). Control Sytm Dign, Prntic Hall, Nw Jry, USA. Li, H.H.T. and J.K. Mill (). Mltipl pcification dign in flight control ytm, In Procding of Amrican Control Confrnc, Chicago, Vol., pp Li, H.H.T. (). Dign combination in intgratd flight control, In Procding of Amrican Control Confrnc, Arlington, Vo l., pp Li, H.H.T. (). Mlti-objctiv dign for an intgratd flight control ytm: a combination with modl rdction approach, In Procding of IEEE Intrnational Sympoim on Comptr Aidd Control Sytm Dign, Glagow, pp.-. Lat, T.N., M.E. Ogbrn and P.G. William (979). Simlator Stdy of Stall/Pot-Stall Charactritic of a Fightr Airplan With Rlaxd Longitdinal Static Stability, NASA Tchniq papr 538. Boyd, S.P. and C.H. Barratt (99). Linar Controllr Dign: Limit of Prformanc, Prntic Hall, Nw Jry, USA.