Dynamic Programming for Trajectory Optimization of Engine-out Transportation Aircraft

Transcription

1 Dynmic Progrmming for Trjectory Optimiztion of Engine-out Trnporttion Aircrft Hongying Wu, Nyibe Chio Cho, Hkim Boudi *, Lunlong Zhong *, Felix Mor-Cmino * *LARA, ENAC, Touloue 355,Frnce E-mil: whyhgh@hotmil.com, hkimboudi@yhoo.fr, lunlong.zhong@enc.fr, morcmino@hotmil.fr # Progrm de Ingenierí Mectrónic, UnB, Bucrmng, Colombi nchio@unb.edu.co Abtrct: The purpoe of thi communiction i to contribute to the development of new trjectory mngement cpbility for n engine-out trnporttion ircrft. Engine-out i drmtic itution for flight fety nd thi tudy focue on the deign of mngement ytem for emergency trjectorie t thi pecil itution. Firt the gliding chrcteritic nd flying qulitie of trnport ircrft with totl engine filure re nlyzed while gliding rnge etimtion i conidered. Then new repreenttion of the flight dynmic of n engine-out ircrft i propoed where the pce vrible i choen independent prmeter inted of the time vrible. Thi llow to propoe new formultion of the correponding trjectory optimiztion problem nd to develop revere dynmic progrmming olution technique. Simultion reult re diplyed nd new development perpective re dicued. Key Word: Flight Sfety, Trjectory Optimiztion, Qui Stedy Glide, Revere Dynmic Progrmming INTRODUCTION The filure of engine i drmtic event for ir flight fety nd mny incident nd ccident re reulting from engine filure. Here n undeirble nd very pecil ce, ll engine out t given point of the flight, i conidered. Thi itution my led to crh unle flyble decent trjectory towrd fe lnding plce i performed. There re mny different reon for engine-out while it pper tht in thi itution ny wrong deciion mde by the pilot my led to cttrophic conequence. So it look quite deirble to develop n emergency guidnce mode for thi itution. Thi new functionlity could be integrted in Flight Mngement Sytem which hould be ble to elect proper lnding ite nd propoe feible trjectory towrd thi ite. To chieve thi purpoe there re mjor tep which hould be performed: etblih nd nlyze the flight dynmic of n ir trnporttion ircrft with totl engine filure (power off, tudy the gliding chrcteritic nd flying qulitie of trnporttion ircrft, develop method to etblih fe rechble re from given itution nd finlly develop method to optimize gliding trjectory towrd poible fe lnding plce. In thi tudy, it i uppoed tht engine out occur once the ircrft h lredy gined ome peed nd ltitude fter tke-off.. Only glide of engine-out irplne in the verticl plne i conidered trt //$6. c IEEE 98 ENGINE OUT FLIGHT DYNAMICS To etblih nd nlyze the flight dynmic of n ir trnporttion ircrft with totl engine filure (power off, the clicl eqution of flight hould be lightly dpted to thi prticulr ce. The erodynmic force (drg D, lift L, nd ide force Y re defined in term of dynmic preure, reference re nd dimenionle erodynmic coefficient []: D / V S C D,, M (- e / V S C L, M (- L, Y / V S CY, r, p, r, M (-3 Here V i the irpeed, i the ir denity (kg/m 3, i the ngle of ttck, e i the elevtor deflection, p i the ircrft roll rte, r i the yw rte nd M i the current Mch number, C D, CL nd CY re dimenionle erodynmic coefficient. C D nd C L re uppoed relted by the polr model CD CD ' K CL where K i contnt. It i conidered tht ome hydrulic power remin vilble to ctivte the elevtor, ileron nd rudder erodynmic urfce, o tht dynmic tbility well ttitude control cn till be performed by the flight control ytem. Indeed, mny trnport ircrft re equipped with n deployble uxiliry turbine (RAM which llow inuring in the control chnnel of the ircrft the vilbility of reidul hydrulic power. While the dditionl drg generted by the RAM remin minor [], the extinction of the ircrft engine reult in noticeble incree of the drg, while lift nd ide force remin quite the me. The drg coefficient i now given by: e

2 ' C D CD, e, M ncde, M ( where C DE i the dditionl drg of hut down engine, nd n i the number of engine of the ircrft. The flight eqution cn be written : V ( D( V,, m g in m (3- ( L( V,, m g co mv (3- x V co (3-3 z V in (3-4 where i the pth ngle, i the pitch ngle. Once fuel dumping h been performed, the m m of the ircrft i conidered to remin contnt. Here x nd z re repectively the current longitudinl nd verticl poition of the ircrft center of grvity. Then, it height bove Erth i given by: h z H (x (4 where H (x i ground level t poition x. where z i the initil ltitude. Now, conidering the bove expreion of D, we get: min mg / ( z S K / C' V ( z (9 D Then the ir peed decree during the qui ttic glide decent. For wide body ircrft, from cruie level, bout 8 m/ re lot for qui tedy initil decent of m. A tll contrint cn be conidered to check the feibility of the glide mneuver: V ( z Vtll ( z mg / ( z S C L mx (- or K / C' D / C (- L mx Thi condition i generl erodynmic condition for gliding feibility of given ircrft under pecific erodynmic itution. Alo, the expreion of D min how tht dynmic preure remin contnt during the qui ttic glide. Figure. diply the irpeed nd tll peed during tedy decent. 3 ESTIMATION OF GLIDING RANGE A firt etimtion of gliding rnge cn be obtined by conidering tht the ircrft remin in qui tedy gliding condition where ir peed nd pth ngle chnge tedily ccording to current ir denity during the whole decent. Fig. Airpeed during tedy gliding Fig. Aircrft force for qui tedy decent In thi itution the pth ngle i uch [3]: rcin (/ f ( V / g (5 or ccording to [4]: det / f mg / where ET mg z mv (6 dz i the ircrft totl energy. Here f =L/D i the lift to drg rtio. According to eqution (5, minimum glide ngle,, mx i chieved with mximum lift to drg rtio. Since L mg, thi correpond to minimum drg. Then it cn be hown tht: / ' K (7- mx C D ' Dmin ( z V (CD Q (C' D (7- A firt pproximtion to the mximum rnge to e level i then given by: R (8 z K C' D Then reltion between the qui ttic glide pth ngle nd the ltitude cn be introduced nd the minimum glide pth ngle i given by: mx tn P C' D K Q g /( R T z/ T ( where ir denity in tndrd tmophere cn be expreed g z RT ( z e with T ( z T z, T 88. 5K, K / m, R m 3 87 / K,.5kg / m. Then thi ngle incree while the ltitude i decreing during the qui tedy glide, hown in Figure 3. The mximum flight rnge R i then more ccurtely determined by: R dz z K C' D z tn mg T R z SC' D P g R T g R ( Thi i illutrted in Figure 4. If the ircrft loe engine power t higher ltitude, it cn glide over n increed 4th Chinee Control nd Deciion Conference (CCDC 99

3 rnge. In the ce of the ccident occurred on 4/8/, the A33 ircrft glided for km. With thi informtion, the rechble lnding ite cn be determined ccording to ome flight plnner [5], [6]. Fig 3. tn during qui tedy gliding Fig. 4 Rechble rnge for qui tedy glide. 4 GLIDE TRAJECTORY OPTIMIZATION FOR SAFETY In thi ection the problem of mnging the trjectory of trnporttion ircrft gliding from given initil flight itution i conidered. Contrrily to the clicl mx rnge gliding problem, by the end of the gliding mneuver, the ircrft mut be in condition (peed nd ttitude to perform fe touch down t lnding. In thi ce, the flight guidnce eqution written in the ircrft wind xi re given by eqution (3. Oberve the eqution tht the only independent input prmeter which i vilble here i the pitch ngle,, which cn, even in n engine-out itution, be controlled by the pilot either through the hydrulic power provided by the RAT or the uxiliry power unit-apu, or through the trim control chnnel. Here the initil flight condition re written : x ( x, h( h, V ( V, ( (3 while the finl lnding condition re uch : h ( t f hg ( x( t f, V ( t f V, ( t f (4 where V nd hould llow fe lnding t ltitude h G ( x( t f where function h G i repreenttive of the ground topogrphy under the conidered flight re. Since finl time i unknown nd i only chrcterized by the tifction of the finl condition, the replcement of independent prmeter t by the pce vrible x llow to diminih the complexity of the problem ince now finl x f i known once the lnding ite h been choen. Moreover, thi pproch hould fcilitte the conidertion of ground eprtion contrint nd could mke eier the conidertion of the effect of wind over the glide trjectory. From eqution (3 with : dt / dx /( V co (5 we get: z tg V ( D( V,, m g in mv co ( L( V,, m g co mv co (6 (6 (6 3 where repreent the derivtive with repect to the longitudinl poition x of the ircrft. The dditionl intnt contrint re: V x V ( z x [ x, x ], z [ z, z ] (7- mx ( min f, ( x ( x min, ( x min min (7- mx mx z x h ( x x [ x, x ] (7-3 ( G f Contrint (7- nd (7- prevent from tlling nd contrint (7-3 from ome flight into terrin-fit itution t n intermediry point of the glide. Then, different formultion of n optimiztion problem [7] cn be conidered to deign fe glide trjectory. For exmple the following criterion could be minimized with repect to the ucceive vlue of long the glide: min ( h( x f hg ( x f (8 under finl contrint V v V ( x f V ( (9- ( min vmx ( gmin ( x f ( g mx (9- where vmin, vmx, g min nd g mx re poitive mrgin nd with tte eqution (6, flight contrint (7 nd initil condition (3. The olution of thi non liner, trongly contrined trjectory optimiztion problem i difficult from the numericl point of view nd direct on line computtion of it olution doe not pper to be feible. For intnce, n pproch bed on the minimum principle [8] hould reult in very difficult two point boundry problem ince the reulting Hmiltonin h not n ffine tructure with f 4th Chinee Control nd Deciion Conference (CCDC

4 repect to the input prmeter. Mny other complex technique hve been developed for trjectory genertion [9], [], [] while Dynmic Progrmming [] pper to provide ome good perpective [3],[4]. To pply effectively Dynmic Progrmming olution trtegy, dicretiztion of thi problem pper necery nd the choice of the pce vrible x independent vrible for the flight eqution pper mot convenient. 5 THE PROPOSED SOLUTION STRATEGY Here dynmic progrmming i ued to generte feible glide trjectory towrd fe lnding plce. To inure the tifction of the finl lnding configurtion given by the qulity contrint (4, which i more criticl condition, revere pproch i dopted. Then the gliding trjectory i computed bckwrd from thee finl condition through the feible glide et defined by contrint (7 nd the pce dicretized tte eqution (6. With the objective of getting mooth flyble trjectory which void wting unnecerily the remining hydrulic energy ued to control the erodynmic ctutor (elevtor, THS, flp nd ero brke long the engine-out glide trjectory, new optimiztion criterion i dopted here. Thi urrogte criteri llow penlizing lrge vrition on pitch ttitude ngle, decent pth ngle, peed nd flight level, o tht it i evlution long feible pth Pk leding to tte i t tge k i given by formul uch : Here, i C k ( E ET ( T i P nd k ET re poitive weight whoe vlue chnge with the ditnce to the lnding ite. Dynmic progrmming, either direct or revere, conider t ech tge different feible tte nd elect for ech of them the bet pth leding to them from the initil tte t the firt tge of the erch proce. Under given vlue of input prmeter i t tge, bckwrd integrtion i ued to e the dditionl cot involved in going from tte (,i to new feible tte t the next tge of the erch proce. However, whtever the ize of the dicrete tep dopted to perform thi revere erch proce, from one tge to nother, lrge number of new tte hould be generted to gurntee the ccurcy of the reulting olution. Thi led to n exploive number of olution to be conidered when the tge order incree. So the exploion of the point mut be voided to inure the computer procee the problem. After ech bckwrd integrting, mny point hould be cut by uing the dynmic progrmming principle. Here, to llevite thi foreeeble computtionl burden, heuritic melting procedure i developed where cloer tte to centrl tte of the current tge in the erch proce re deleted while thi centrl tte i mintined. ij The ditnce between two tte i nd j of tge which h been dopted to generte thee cluter within one tge i given by: i j V Vmx i j z z zmx i j mx ij ( V ( ( ( V z Here two level weighting h been dopted: Vmx, z mx nd mx re cling prmeter nd V, z nd with V z re poitive reltive weighting. The bove pproch which h been developed i biclly n open loop pproch nd require very lrge computtionl effort which i unlikely to be performed on bord n ircrft which i lredy in criticl engine-out itution. Our propol here, which hould be developed in the ner future i to tke profit of the mount of dt generted by the revere dynmic progrmming erch proce, conidering different itution nd prmeter uch ircrft initil flight level, ltitude nd m, to trin neurl network devie deigned to generte pitch ngle directive t ech point long the decent o tht the glide trjectory led fely to the lnding itution. Here the computtionl burden ocited with revere dynmic progrmming i tken into profit to generte the trining dt be for the neurl network [5]. The generted pitch ngle directive cn be either ent to the utopilot when it i till operting or to flight director. In tht lt ce thi will llow thi mneuver to be performed efficiently in mnul mode by the pilot. Oberve tht long the glide trjectory, ech new olicittion of the neurl network will generte new piloting directive in ccordnce with the current itution of the ircrft which i lo the reult of externl perturbtion uch wind. 6 SIMULATION RESULTS A imultion tudy h been performed uing the RCAM wide body trnporttion ircrft model [6]. Then conidering the ce in which n engine filure occur 5km wy from poible lnding ite, different glide trjectorie obtined by revere dynmic progrmming re diplyed on Figure 5. nd on Figure 6 ccording to different initil itution. It pper tht if the ircrft h lrge initil totl energy, which men high peed nd/or high ltitude, the reulting glide trjectory i not be very mooth: the peed nd ltitude re ubject to lrge nd rpid chnge o tht the ircrft loe energy in exce ufficiently quickly to rrive to the lnding ite with cceptble flying prmeter. When initil totl energy i not too much exceive, the reulting glide trjectorie reult to be moother. For exmple, for initil condition with n Figure 7. nd Figure 8. diply n optimized glide trjectory in the ce in which initil ltitude i km (FL33 nd initil irpeed i m/ (bout 4 knot. Figure 9. nd Figure. diply the lnding rnge which cn be reched fely by n ircrft whoe initil glide condition re n ltitude of km nd n irpeed of 4th Chinee Control nd Deciion Conference (CCDC

5 8m/ (bout 36 knot. The lrget obtined glide rnge i bout 37 km while the hortet obtined glide rnge i 6 km. Then in tht ce, the gliding ircrft cn rech fely lnding ite locted between 6km nd 37km wy. Oberve on thee figure tht, the horter the rnge, the rougher i the trjectory. Compring figure 7. nd Figure 9., it pper lo tht with higher initil irpeed the gliding ircrft rnge i lo higher. Thee numericl reult indicte tht revere dynmic progrmming cn be ued to olve the glide trjectory genertion problem nd contribute to the deign of glide trjectory genertor either off line or on line. Fig 7. A mooth optiml glide trjectory in 3-D Fig 8. A mooth optiml glide trjectory in verticl plne Fig 5. Optiml glide trjectorie with different initil peed 3-dimenionl repreenttion Fig 9. Trjectorie of ircrft in verticl plne with different lnding rnge, 3D view Fig. 6 Optiml glide trjectorie in verticl plne with different initil ltitude Fig. Trjectorie of ircrft in verticl plne with different lnding rnge 4th Chinee Control nd Deciion Conference (CCDC

6 7 CONCLUSION The purpoe of thi communiction h been to preent the firt reult of tudy turned towrd the deign of n emergency mngement ytem ble to cope with n engine-out itution for trnporttion ircrft. The min contribution of thi communiction re: - review of the qui tedy glide rnge for trnport ircrft; - the propol of new repreenttion of the flight dynmic of gliding ircrft with the introduction of ptil dimenion independent vrible; - the development of olution trtegy bed on bckwrd integrtion nd revere dynmic progrmming whoe feibility i upported by the diplyed imultion reult. Thi work hould be completed by the integrtion of lterl mneuver nd the conidertion of the effect of wind over the glide trjectory. Thi lt point could be tckled by the development of n dptive pproch bed on the online etimtion of wind nd the ue of neurl mchine to generte control directive on rective bi. Thi remin for further tudie. REFERENCES [] Robert C. Nelon, Flight Stbility nd Automtic Control, McGrw-Hill Book Compny, US 989. [] Zhiyou Liu, Minjie Hou, Gng Wen, Experimentl Determintion of Aero-Engine Windmilling Drg, Journl of Aeropce Power, Vol., No., 6. [3] Nguyen X. Vinh, Flight Mechnic of High-Performnce Aircrft, Cmbridge Univerity Pre, UK, 993. [4] Hongying Wu, Mngement of Emergency Trjectory for Trnport Aircrft, MSc Repport, ENAC, Touloue, September. [5] Ell M. Atkin, Igor Alono Portillo, Mtthew J. Strube, Emergency Flight Plnning Applied to Totl Lo of Thrut. Journl of Aircrft, Vol.43, No. 4, 5-6, 6. [6] Ryn Rpetti, Srigul-Klijn, A 3-phe fe trjectory hping for ditreed ircrft, IEEE Aeropce Conference, USA,. [7] John T. Bett, Survey of Numericl Method for Trjectory Optimiztion, Journl of Guidnce, Control, nd Dynmic,Vol., No., 93-7, 998. [8] S. Khn, Flight Trjectory Optimiztion, Toronto, ICAS. [9] Nerin Srigul-Klijn,, R. Rpetti, A. Jordn, I. Lopez, M. Srigul-Klijn, P. Nepec, Intelligent Flight-Trjectory Genertion to Mximize Sfe-Outcome Probbility After Ditre Event. Journl of Aircrft, Vol.47, No., 55-67,. [] B. Outtr, F. Mor-Cmino, Trjectory Genertion for Reltive Guidnce of Merging Aircrft, in Optimiztion nd Coopertive Control Strtegie, M. Hirch editor, Lecture Note in Control nd Informtion Science Serie, Springer Berlin, 8. [] F.J. Vormer, M. Mulder, M.M. vn Pen, J.A. Mulder, Optimiztion of Flexible Approch Trjectorie Uing Genetic Algorithm, Journl of Aircrft, Vol.43, No.4, 94-95, 6. [] D. Bertek, Dynmic Progrmming nd Optiml Control, Athen Scientific, 7. [3] P. Hgeluer nd F. Mor-Cmino, Evlution of Prcticl Solution for Onbord Aircrft Four Dimenionl Guidnce, AIAA Journl of Guidnce, Control nd Dynmic (5,5-54, 997. [4] P. Hgeluer nd F. Mor-Cmino, A Soft Progrmming Approch for On-line Aircrft 4-D Trjectory Optimiztion, Europen Journl of Opertion Reerch, 87-95, 998. [5] Frncelin, R.A., F.A.C. Gomide, A Neurl Network for n-tge Optiml Control Problem, Proc. IEEE Int. Conf. Neurl Network - ICNN 94, Vol. 7, pp , Orlndo-FL, USA (994. [6] P. Lmbrecht, S. Bennni, G. Looye nd D. Moormnn, The RCAM Deign Chllenge Problem Decription in Robut Flight Control, J.F. Mgni, S. Bennni nd J. Terlouw editor, Lecture Note in Control nd Informtion Science 4, Springer, th Chinee Control nd Deciion Conference (CCDC 3