Influence of a Piston Propeller-driven Engine Model on the Design of a Cruise Autopilot

Size: px

Start display at page:

Download "Influence of a Piston Propeller-driven Engine Model on the Design of a Cruise Autopilot"

August Smith
5 years ago
Views:

AAA Aviation -7 Jun 6, Washington, D.. AAA Atmosphric Flight Mchanics onfrnc AAA 6-58 nflunc of a Piston Propllr-drivn Engin Modl on th Dsign of a ruis Autopilot Pdro J.

Boschtti Univrsidad Simón Bolívar, Naiguatá, Vargas, 6, Vnzula and Downloadd by NS E DE AERONAUA (A) on Jun, 6 http://arc.aiaa.org DO:.54/6.

1 AAA Aviation -7 Jun 6, Washington, D.. AAA Atmosphric Flight Mchanics onfrnc AAA 6-58 nflunc of a Piston Propllr-drivn Engin Modl on th Dsign of a ruis Autopilot Pdro J. Gonzálz nstituto cnológico d Aronáutica, São José dos ampos, SP, 8-9, Brazil Pdro J. Boschtti Univrsidad Simón Bolívar, Naiguatá, Vargas, 6, Vnzula and Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ A Jn A Pn A ηn L D P d H J k n P av P brak t V W Flávio J. Silvstr nstituto cnológico d Aronáutica, São José dos ampos, SP, 8-9, Brazil Piston propllr-drivn ngins ar usually modld assuming constant brak powr (simpl modl) to obtain powr availabl. A mor ralistic modl can b usd stimating th powr availabl using a complx procdur. A complx modl is proposd to obtain th powr availabl. ruis flight autopilots rquir accurat rprsntations of th aircraft. h aim of this papr is to valuat th influnc of a piston propllr-drivn ngin modl on th dsign of a cruis autopilot for a mdium rang unmannd arial vhicl. A cruis flight autopilot was dsignd with th capacity to track flight path angl and vlocity, and both propulsion modls ar usd for comparison. A sris of simulations was prformd to track ach rfrnc sparatly and togthr. h simpl and complx propulsions modls for th autopilots wr capabl of tracking th rfrnc signals with a low rror; howvr, for th simulations, th throttl of th simpl modl was lowr than that for th complx modl. t can b obsrvd that th simpl modl is gnrating mor thrust availabl for lowr throttl positions, which rduc th accuracy of th modl. Nomnclatur = cofficints of th advanc paramtr polynomial curv = cofficints of powr availabl polynomial curv = cofficints of th propllr fficincy polynomial curv = lift cofficint = drag cofficint = powr cofficint = propllr diamtr = hight = advanc paramtr = inducd drag factor = propllr rvolutions pr scond = powr availabl = brak powr = thrust = tim = vlocity = wight of th aircraft PhD andidat, Arospac Enginring Division, São José dos ampos, Mmbr AAA. Associat Profssor, Dpartmnt of ndustrial chnology, amurí Grand Vally, Snior Mmbr AAA. Assistant Profssor, Arospac Enginring Division, São José dos ampos, Mmbr AAA. Amrican nstitut of Aronautics and Astronautics opyright 6 by Pdro J. Gonzálz, Pdro J. Boschtti, Flávio J. Silvstr. Publishd by th Amrican nstitut of Aronautics and Astronautics, nc., with prmission.

2 α γ θ η ρ ρ o σ φ ω = angl of attack = flight-path angl = pitch angl = propllr fficincy = air dnsity at flight altitud = air dnsity at sa lvl = dnsity ratio = powr-altitud factor = angular vlocity of th motor shaft Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ ntroduction HE dsign of autopilot systms rquir an accurat mathmatical modl of th airplan. his modl consists of th arodynamic modl, th inrtial-mass modl, and th propulsion modl. Accuracy of th plant lads to a bttr bhavior in closd loop. xtbooks as Stvns and Lwis and Zipfl prsnt also modlling tchniqus for th propulsion of th airplan, and vn missils, must b. Zipfl discuss on arodynamic modlling of airplans and missils for rockts, turbojts and turbofans, but not for piston propllr-drivn ngins, but dos not prsnt modls for piston propllr-drivn ngins. Usually, th piston propllr-drivn ngins ar modld assuming constant powr and constant propllr fficincy, and an altitud corrction for piston ngins is applid. avcar 4 assums a constant brak powr of th ngin (for piston ngin or turboprop) and usd a polynomial function to stimat th propllr fficincy. Smtana 5 proposs to modl th brak powr output by a polynomial function and assumd th propllr fficincy as a constant valu. Miss 6 xplains th procss to dtrmin th powr availabl for a piston propllr-drivn ngin and suggsts a spcific quation to stimat th powr-altitud factor to considr th variations of output brak powr with altitud. hn, Smtana 5 rcommnds a modifid formula for th sam aim. Boschtti t al 7 modl a piston propllr-drivn ngin using an algorithm to comput th powr availabl. his modl simulats th brak powr by a polynomial function, which is corrctd at ach tim stp basd on th Miss modl for th powraltitud factor, and th propllr fficincy is calculatd using two polynomial functions. o simplify, hr th modl dscribd by Andrson is calld th simpl modl and th on cratd by Boschtti t al., 7 th complx modl. h complx modl is mor difficult to crat than th simpl modl. h first on nds many xprimntal data from th ngin and th propllr, and it rquirs a complx algorithm to rprsnt it. h simpl modl is rprsntd by a singl quation, nding only th maximum brak powr at sa lvl, and th propllr fficincy is constant. Qustions aris on how th ngin modl would influnc th dsign of an autopilot systm. h main objctiv of this papr is to valuat th influnc of a piston propllr-drivn ngin modl into th dsign of cruis autopilot for a mdium rang unmannd arial vhicl.. Propulsion modls h piston propllr drivn ngin simpl modl is dscribd by avcar 4 and Andrson as, P V () av P P () av brak whr th brak powr (P brak) and th propllr fficincy (η) ar constant, for th sam hight th powr availabl is constant (P av), and th thrust () is function of flight vlocity (V). h complx modl of a piston propllr-drivn ngin basd on Rf. [6] and prsntd by Boschtti t al 7 uss as input data th ngin charts providd by th manufacturr, propllr charts obtaind xprimntally, th propllr diamtr, and an atmosphric modl for dnsity according to th air tmpratur and altitud. h procdur to calculat th availabl powr prsntd in Rf. [7] it is xplaind hrin. Basd on th nginpropllr configuration, th brak powr and th propllr rvolutions pr scond of th ngin ar arrangd in two vctors, rspctivly, and th powr cofficint ( P) vctor for th spcific altitud can b computd by, Amrican nstitut of Aronautics and Astronautics

3 p i Pbrak i ; i,, N 5 n d i () whr φ is powr altitud factor, ρ is th air dnsity at flight altitud, d is th propllr diamtr, and n ar propllr rvolutions pr scond. h powr-altitud factor is computd at ach tim stp considring that it is a function of hight through th air dnsity ratio (σ= ρ/ρ sa lvl), H (4) Using th information of propllr charts, polynomial quations ar usd to xprss th advanc paramtr (J) as a function of powr cofficint ( p), and th propllr fficincy as a function of advanc paramtr, rspctivly, Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ J P AJ P AJ P AJ P AJ (5) J A J A J A J A (6) h advancd paramtr and th propllr fficincy ar xprssd in vctor form: f J (7) f P J (8) h powr availabl and th corrsponding vlocity is obtaind as a function of brak powr, fficincy, and powr altitud factor φ as: P P ; i,, N av i brak i (9) V J n d; i,, N i i i i () According to Smtana, 5 th powr availabl for a spcific altitud may b xprssd as a polynomial quation, P av A V A V A V A () P P P P Using th mthod of last squars, th cofficints of th polynomial ar computd, basd in th valus of powr availabl and flight vlocity obtaind using Eqs. (9) and ().. Rigid body quations of motion n this sction, th quations of motion of a rigid body aircraft ar shown. h flight dynamics of a gnric rigid aircraft is drivd and th kinmatic quations ar prsntd. h body rfrnc systm is usd; th quation of forcs and momnt, and finally th body angular vlocitis in trms of th Eulr angls and Eulr rats ar givn. A. Dfinition of forcs and momnts for a rigid aircraft h quations of motion ar obtaind from Nwton s scond law; th summation of th xtrnal forcs acting on th aircraft is qual to th tim rat of chang of th momntum of th body. h xtrnal momnts acting on th aircraft ar qual to th tim rat of chang of th angular momntum; Eqs. ()-() xprss ths statmnts, 9 d F ( mv ) b mv () dt Amrican nstitut of Aronautics and Astronautics

4 M d ( H b ) b Hb () dt whr F is th vctor of forcs, m is th mass of th airplan, ω is th angular vlocity vctor of th body systm rspct to th arth but xprssd in th body systm and H b=ω, is th mass momnts of inrtia of th aircraft on ach axis. Onc ths xprssions ar dvlopd, th scalar quations ar, F x m u qw rv F m v ru pw y F m w pv qu z (4) Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ assuming x to th frontal dirction of th aircraft, y to th right wing, and z to th ground. h products of inrtia acting on th plans yz and xy ar zro, th angular acclrations can b writtn as p q r L L N M ( N ( ( yy yy ) pr ( r p ) yy ) pq ( ) pq ( yy yy ) qr ) qr L, M and N ar th roll, pitch and yaw momnts, rspctivly. About to th position of th aircraft, th rlationship btwn th angular vlocitis in th body fram (p,q,r) and th Eulr rats is dtrmind by th nxt st of quations; p q r cos sin sin cos sin cos cos whr ϕ is th bank angl, θ is th pitch angl, and ψ is th yaw angl. his quation could also b xprssd in function of th body angular vlocitis to calculat th Eulr rats, sin tan cos sin sc cos tan p sin q cos sc r (5) (6) (7) B. ontribution of forcs and momnts on th aircraft h quations prsntd prviously can b linarizd using th small disturbanc thory. h motion of th aircraft is basd on small dviations from th stady flight condition. 9 h forcs and momnts acting on th complt aircraft ar dfind in dimnsionlss arodynamics cofficints. h arodynamic cofficints ar sparatd in longitudinal cofficints acting on th lift and drag axs and th pitching momnt primarily dpndnt on α and on latral-dirctional cofficints acting on th sidslip forc and on th β dpndnt roll and yaw momnts. Equation (8) dfins th angls of incidncs and Eq. (9) th tru/ral vlocity and th dynamic prssur. Equation () shows th tmporal drivativs of ths angls and th tru/ral spd. v rprsnts th latral vlocity and V th tru vlocity 4 Amrican nstitut of Aronautics and Astronautics

5 w v arctan, arcsin (8) u V V, q (/ ) V u v w (9) u w w u v V vv uu v v w w V,, u w V u w V () h atmosphr dnsity and th tmpratur ar dpndnt on th gomtric hight; th intrnational standard atmosphr SA stats that: Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ ,56 ( ah / ), p ( ah / ) / R () whr =88.5 K, a= 6.5x km, R=87. m k s and p = Nm Adding arodynamic, thrust, and gravitational forcs and momnts acting on th aircraft, it is possibl to obtain Eq. () and Eq. (). D F cos F g sin b F L w Yw m g sin cos () l F sin F g cos cos bl M qs cm bn Fz F F \ Fr () h thrust forc is not only in function of th angl α F btwn th lin of traction and th X b axis. h thrust of a piston fixd proplld aircraft is a function of throttl δ t. raditionally, it is modld with Eqs. ()-(), and Eq. (4) includs th throttl prcntag. P av t (4) V n this papr, it is proposd to includ th stimation of th powr availabl with Eq. (). h magnitud of th arodynamic cofficints dpnds on th arodynamic proprtis of th vhicl; ths cofficints could b stimatd analytically, with numrical mthods and simulations, wind tunnl tsting or flight tsts.,. Stady flight and trim Stady flight is a flight condition whr th total of th forcs and momnts acting on th aircraft ar qual to zro. n ordr to hold a flight condition, th aircraft rquirs modifying th position of th control surfacs and adjusting th throttl. h arodynamic forcs and momnt cofficints ar a function of th dflction of th control surfacs: lvator δ, ailron δ a, and ruddr δ r.,9, h stat-spac modling tchniqus wr applid in ordr to simulat th aircraft in flight; this is applicabl just around linarizd and stady flight condition. h flight quations may b writtn around an quilibrium point. Equation (5) rprsnts th trim flight condition, q S q S q Sc M D D cos sin L L sin cos F cos mg sin( ) F sin F F mg cos( ) (5) 5 Amrican nstitut of Aronautics and Astronautics

6 whr γ is th flight-path angl in th quilibrium stat (6) h arodynamic cofficints ar a function of δ. hy could b stimatd for a flight condition. h angl of trim α, th dflction of th lvator to hold it, and th amount of throttl to sustain th spd ar th rsults of th trim calculation. o rprsnt in stat-spac variabl quations, th slctd stats and obsrvabl variabls ar prsntd in Eqs. (7)-(8). Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ h control vctor is, X u v w p q r x x x (7) Y V p q r a a a h (8) h final form of th invariant linar systm is: h linar quation for th outputs of th systm is, tc c ac x rc N y E z D U (9) X AX BU () Y X DU () Basd on this st of quation, a sris of scripts wr writtn in Matlab, whr th dynamics of th rigid motion of th aircraft is calculatd and th stats ndd for th computation of this dynamics ar obsrvabl. Onc th modl of an aircraft is dfind, th stady flight condition may b dtrmind around a fixd spd; th final output of th trim calculation offrs th initial conditions to run th simulations; from this start point, th flight dynamics valuation of th aircraft may b compltd and th controllr dsignd. Longitudinal Automatic Pilot h aim is to dsign a longitudinal flight control systm that will b capabl of holding th flight spd and th flight-path. his is don by rgulating som stats of th aircraft to zro whil th dsirabl clos-loop rspons is obtaind. h control systm computs multipl inputs and multipl outputs and thn it drivs a matrix quation that solvs th control gains., h control tchniqu usd was LQR (linar quadratic rgulator), which consists in minimizing a quadratic cost or prformanc indx J. J x Qx u Ru dt () For a systm of th form, x Ax Bu y x u Kx () h rsult of th closd loop systm is 6 Amrican nstitut of Aronautics and Astronautics

7 x ( A BK) x A x (4) h output fdback gain K ncssary to minimiz J is obtaind by th algbraic Ricatti quation c A P PA Q PBR B P (5) h wighting matrix Q=H *H. hs H matrics dfin th trackd output. R is an idntity matrix of th siz of th control inputs. hn, th Kalman gain is calculatd as K R B P (6) Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ Assuming that th clos loop systm is stabl, th prformanc indx bcoms J x () Px() (7) Figur shows th structur of th control systm proposd for this work. h outr loop is basd on classical flight control outr loops for flight path angl and vlocity. h lvator controls th flight path angl and th ngin th changs in vlocity. -z r v ompnsator -L Actuator Aircraft as Study o valuat diffrnt ngin modls in th cruis flight autopilot, th Unmannd Aircraft for Ecological onsrvation (ANE) was slctd. h ANE is a rconnaissanc-unmannd aircraft for th dtction of oil lakags. h aircraft was slctd bcaus nough information of th airplan is availabl. abl shows th charactristics of th ANE. 7 h modld ngin is th Simonini Victor Plus with a propllr , lark Y sction, two blads with blad angl of dg and a diamtr of.947 m. 4 abl shows th charactristic of th ngin, and Fig. prsnts th propllr charts. abl prsnts th arodynamic modl of th ANE. h flight condition was linarizd and th stady stat flight arrangmnt of th aircraft was dtrmind for a spd V to=54 m/s and a hight of 48 m. h simulations will start from th stady stat condition prsntd in tabl 4. h automatic control systm for an aircraft must hav ths functionalitis: a fdback output to incras th stability of th intrnal systm, which could b idntifid as a stability augmntation systm, and a dynamic controllr for tracking outr loop variabls; this is carrid out with a proportional intgral lad-lag compnsator for th flight path angl and a lad-lag for th vlocity. h structur of th compnsators is dpndnt on th dynamics of th actuator; thy ar indpndnt paramtrs of ach projct. h transfr function for th clos loop systm for th flight-path angl and th tru vlocity compnsator ar prsntd as Eq. (8). From a Bod diagram, th phas margins of th closd loop systm wr -7.4 db and 6.8 db for th flight path angl trackr; th clos loop is stabl. -H -K Figur. Structur of th control systm u y x x 7 Amrican nstitut of Aronautics and Astronautics

8 .56s 5.s.9.s f g, f V (8) s.s s 5 abl. haractristics of th ANE Dimnsions Longitud Hight Span hord m.6 m 5.87 m.64 m Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ Surfac.9 m Wight nrtia Matrix Prformanc ruis spd ak-off distanc Engin Propllr abl. haractristics of th Simonini Victor Plus ngin. ω, rv/min P brak, hp kg 4 54 m/s 7 m kg m 4 Simonini Victor Plus , lark Y sction, two blads with blad angl of dg Figur. urvs of propllr , lark Y sction, two blads with blad angl of dg. 7,4 8 Amrican nstitut of Aronautics and Astronautics

9 abl. Arodynamic modl of th ANE Forcs qc /( ) L V D L Y r.865a.48 pb /(V ).465rb /(V ) l F ( P av / V o Momnts ) t pb /(V ).58rb /(V ).749 a 497 r Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ n M qc /(V ) pb /(V ).564rb /(V ).98 a abl 4. Stady flight condition for th aircraft. 789 Vlocity 54 m/s Hight 48 m Pitch angl.754 dg hrottl simpl modl 8.5% hrottl complx modl 47.4% Elvator dflction -.8 dg Ailron dflction dg Ruddr dflction dg Having ths spcifications, th linar modl of th aircraft is dfind and th modl could b built in Simulink to mak clos-loop simulations. h xognous rfrncs ar prsntd as tracking commands, to hold spd and flight-path. h gain vctor was tund with Matlab. h routin fmincon was usd to obtain th optimal gain. his routin finds a constraind minimum of a scalar function of svral variabls starting at an initial stimat. his is gnrally rfrrd to as constraind nonlinar optimization or nonlinar programming minimum of constraind nonlinar multivariabl function. h control systm was stimatd with th complx modl. hn, th simulation was prformd with both modls and compard. Figur and 4 ar th rsult of a s simulation tracking an incras of vlocity from 54 m/s to 58 m/s. h masurd outputs (Fig. ) ar th vlocity, th flight path angl, th hight, pitch rat, angl of attack, and pitch angl. Figur 4 shows th variation of lvator and throttl commands. t is possibl to obsrv th diffrnc btwn th rspons of both controllr to attain commandd spd. n fact, during th s of simulation th simpl modl controllr dos not attain th commandd and has a final rror of prcnt. h commandd flight path angl is satisfactorily kpt in both cass. Pitch rat and hight sms to b changing similarly but at diffrnt rats. Angl of attack and pitch angl ar diffrnt during th ntir simulation. Whn ngin modls ar compard, it is obsrvd how th complx modl holds full throttl for mor than 4 s whil th aircraft is acclrating. hr is a diffrnc of 7prcnt in th final throttl obtaind for both ngins; th simpl modl appars to b producing mor thrust at lowr throttl to hold th commandd spd. h saturation limits applid to th complx modl ar not compatibl with th simpl modl. t is producing mor thrust in th initial sction of th manuvr. A scond valuation was prformd to track a flight path angl of two dgrs. Figur 5 shows th variation of th flight path angl until it attains th command; th incras of hight associatd with this manuvr is also apprciabl. hr is a diffrnc of.7 m/s btwn th vlocitis of both modls, which rprsnt a diffrnc lss that prcnt on stationary stat rror. Manwhil, th rst of th monitord variabls bhav similarly in both cass. Figur 6 prsnts th commandd inputs during th manuvr whil th lvator sms to hav a vry similar dflction for both cass. h throttl shows a diffrnc of prcnt btwn th two modls. r 9 Amrican nstitut of Aronautics and Astronautics

10 Vt, m/s Simpl modl omplx modl 54.. q, rad/s Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ Hight, m, dg , dg, dg Figur. Masurd Outputs for vlocity tracking, dg Simpl modl omplx modl t.5. Figur 4. Elvator and throttl commands Amrican nstitut of Aronautics and Astronautics

11 h third cas consists in an incras of vlocity of m/s and a commandd flight path angl of dg. Fig. 7 shows how th ANE tracks th commandd flight path angl and starts to climb whil th vlocity incrass. Onc again, Fig. 8 shows that thr is a discrpancy btwn th throttl of th simpl modl, which is providing with mor thrust at lowr throttl positions than th complx modl. h diffrnc is prcnt for th stationary condition. t appars that th simpl modl is gnrating mor powr for th sam throttl position than th complx modl. h tracking commands wr achivd using both modls, although th rror with th simpl modl was about 5 prcnt on avrag. Onc again th dynamics of th simpl modl dos not rspct th sam boundaris applid to th complx modl which caus and unfasibl situation for th first scond of th manuvr. n this flight conditions, th utilization of both modls sms not to influnc drastically in th rsult of th manuvrs, xcpt in th first cas th variation from th trim condition was compltly rsponsibility of th ngin. Howvr, thr xists consistnt inaccuracy btwn th throttl lvl rquird for ach manuvr. Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ Vt, m/s Hight, m , dg Simpl modl omplx modl q, rad/s, dg, dg Figur 5. Masurd Outputs for flight path angl tracking Amrican nstitut of Aronautics and Astronautics

12 , dg Simpl modl omplx modl Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ Vt, m/s, dg Simpl modl omplx modl t Figur 6. Elvator and throttl commands q, rad/s, dg Hight, m , dg.5 4 Amrican nstitut of Aronautics and Astronautics.5 Figur 7. Masurd outputs for vlocity and flight path angl tracking

13 , dg Simpl modl omplx modl -5.5 Downloadd by NS E DE AERONAUA (A) on Jun, 6 DO:.54/ t.5 Figur 8. Elvator and throttl commands Summary A control systm for longitudinal cruis flight was projctd to valuat th influnc of two diffrnt piston fixd propllr ngin modls. Both ngin modls wr dscribd as wll as th aircraft rigid modl and th control structur. hr cass of simulation wr carrid out using th ANE. h complx modl is capabl of providing a mor accurat information of th powr availabl for th ngin and as rsult of th providd thrust. h simpl modl sms to dvlop mor thrust at lowr throttl position than th complx modl in all simulation cass, which could b an inaccuracy of th modl. n flight conditions that prcis mor powr availabl th discrpancy btwn ngin modls rally affcts th prformanc of th autopilot and commands an unfasibl situation. h autopilot sms to b capabl of tracking all commandd rfrncs with both modls with small rrors. h simpl modl dos not rprsnt accuratly nough th prformanc of th ngin what affct dirctly on th xpctd prformanc of th aircraft. Rfrncs Stvns, B. L., and Lwis, F. L. Aircraft ontrol and Simulation, nd d., John Wily & Sons, nc., Hobokn, NJ,. Zipfl, P. H. hr-dgrs-of-frdom Simulation, Modling and Simulation of Arospac Vhicl Dynamics, nd d., AAA Education Sris, AAA, Rston, VA, 7. Andrson, J. D., Aircraft Prformanc and Dsign, WB/McGraw-Hill, Boston, 999, haps avcar, A., limb Prformanc of Piston Propllr Airplan with ambrd Wing and Variabl Propllr Efficincy, Journal of Aircraft, Vol. 48, No. 5,, pp Smtana, F. O., Flight Vhicl Prformanc, Flight Vhicl Prformanc and Arodynamic ontrol, AAA Education Sris, AAA, Rston, VA,, pp. 45-5, 6-65, 7-8, Miss (von), R., h gnral prformanc problm, hory of Flight, Dovr Publications, Nw York, 959, pp Boschtti, P. J., Gonzálz, P. J., and árdnas, E. M. Program to alculat th Prformanc of Airplans Drivn by a Fixd-Pitch Propllr, AAA papr 5-5, 5. 8 Miss (von), R., h airplan ngin, hory of Flight, Dovr Publications, Nw York, 959, pp. 54,65. 9 Nlson, R.., Longitudinal Motion, Flight Stability and Automatic ontrol, nd d., McGraw Hill, Boston, MA, 998 Etkin, B., Dynamics of Atmosphric Flight, John Wily & Sons, nc., Nw York ity, NY, 97. MALAB, h MathWorks, nc., Softwar Packag, Vr (Ra),. McLan, D, Automatic Flight ontrol Systms, Prntic Hall, nc, Southampton, Hampshir, UK, 99. Simonini Flying, VOR PLUS [onlin databas], URL: [citd Sptmbr ]. 4 Hartman, E. P., and Birmann, D., h arodynamic charactristics of full-scal propllrs having,, and 4 blads of lark Y and R.A.F. 6 airfoil sctions, NAA R-64, 98. Amrican nstitut of Aronautics and Astronautics

Rotor Stationary Control Analysis Based on Coupling KdV Equation Finite Steady Analysis Liu Dalong1,a, Xu Lijuan2,a

204 Intrnational Confrnc on Computr Scinc and Elctronic Tchnology (ICCSET 204) Rotor Stationary Control Analysis Basd on Coupling KdV Equation Finit Stady Analysis Liu Dalong,a, Xu Lijuan2,a Dpartmnt of