FLIGHT TRAJECTORIES OPTIMIZATION

Transcription

1 Collee o Aeronaucal Enneern, PAF Acadey, Rsalpur, Naonal Unversy o Scence and echnoloy, Pasan ICAS CONGRESS Key Words: Flh dynacs, Opzaon, Opal rajecores, rajecory Opzaon Absrac Syse opzaon s a process o ranslan he dynacs o a syse and s desred objecves no he aheacal lanuae, whch ve rse o wha s called a conrol proble and hen o nd he soluon o hs proble Such a soluon s called opal conrol and he pah ollows o acheve he desred objecves s called opal rajecory rajecory opzaon s an opal ranser proble For any speced end condon and perorance nde, he proble o deernn he opal rajecory n powered lh o an arcra n aospherc condons, subjec o ceran physcal consrans, s very cople proble In eneral canno be solved whou usn nuercal copuaon based on a speced odel o he aosphere and arcra aerodynac and enne characerscs In he pas an nensve research has been carred ou n he area o syse opzaon and opal rajecores In he wor presened n hs paper, ephass s ade on eneralzaon o he opal rajecores o arcra, he basc nredens o he opzaon proble and orulan he precse saeen o he opzaon proble he denon o opal conrol proble, orulaon o a conrol proble and soluon o such a conrol proble are presened Deren opzaon echnques are dscussed and copared o show her ers and laons he opal rajecores n deren phases o lh, e rajecores n horzonal plane, vercal clb rajecores, are analyzed usn he Ponryan s Mau Prncple Conrol Proble In a eneral way a conrol proble s conrolln o a dynac syse Opzaon o a syse s a process o ranslan he syse dynacs and s desred objecves no he absrac lanuae o aheacs, whch ve rse o wha s called a conrol proble, and hen o nd he soluon o hs proble Such a soluon s also called opal conrol, and he pah ollows o reach he desred oal s called opal rajecory he aheacal odel, whch represens he physcal syse, consss o a se o relaons beween sae o he syse and npu o he syse Consrans are ncorporaed n hs se o relaons, whch are usually epressed n he or o equaons called he sae equaons he requreen o obann a ceran oupu s replaced wh he requreen o hn a ceran are se n he sae space o he syse he physcal resrcons or consrans upon he se o npus lead o a se o adssble npus or conrols he desred objecves can be aaned by any adssble npus, each o whch resuls n a deren response So s requred o evaluae each response and possble pc up he bes one hs requres he use o perorance creron, whch s a easure o perorance or cos o conrol Such a perorance creron s called perorance nde or cosuncon (or unconal) he soluon o a conrol proble s o deerne he adssble npus whch enerae he desred oupu and whch, n don so, nze (opze) he cosunconal

2 he syse s overned by a se o derenal equaons or he sae equaons & (, () where s sae vecor, u s conrol vecor and s e I s assocaed wh a perorance nde o eneral or: J ( u ) () K [ ( ), ] + L [ ( ), u ( ), ] d where K [ ), ] s he ernal cos and L [,, ] s cos beween he e nerval he opu conrol proble s o nd [, ] u ( ha nzes he perorance nde alon he opal rajecory Based on how o nd u (, an opal conrol proble can be handled n wo possble ways: Paraeer or Sac Opzaon Process or Dynac Opzaon Paraeer Opzaon he ehod where he conrol vecor u ( can be ound by a se o paraeers a, a, a 3, a n, s called paraeer opzaon and he perorance nde, whch s o be nzed, becoes a uncon o hese paraeers Such a perorance nde s also called cosuncon hs ehod s appled o rajecory opzaon proble where he rajecory s epressed n ers o a ne nuber o paraeers and a se o paraeers s opzed o e he opu perorance I s used o aze or nze a cosuncon J K ( ), ) + L (, u, ) d (3) As rajecory depends on u (, hereore, J depend on u ( I can be epressed n ers o a ne nuber o paraeers a, a, a n, hen n he nal analyss, J can be wren as a uncon o ne nuber o paraeers: J a, a, a ) (4) ( n he laon o paraeer opzaon s ha becoes dcul o solve he proble he nuber o paraeers ncreases or he copley o he proble ncreases where he perorance creron s a cosunconal, e s uncon o derenuncons he donan opzaon echnques used n paraeer opzaon are: he heory o Maa and Mna, he Drec Mehods o opzaon, such as Sple, Rosenbroc-Powel Mehod, Mehod o Gradens or Seepes Descen, Conjuae Graden Mehod, ec 3 Process Opzaon he process opzaon s he eneralzaon o paraeer opzaon proble and s concerned wh ndn a au or nu or a quany (perorance nde) whch depends upon n ndependenuncons (, (, n ( hs s a eneralzaon o he concep o a uncon and s called unconal For ore coplcaed conrol probles, where he perorance creron s cosunconal, or where ore accurae soluon s requred, process opzaon s used, e n case o rajecores n unseady lh condons he cosunconal s o he ype: J K[ ), ] + L[,, ] d (5) J depends on ( ( ), u ( )), bu ( ) also depends on u ( ) hereore, we can say ha n nal analyss, he perorance nde J s a unconal o deren conrols and we can wre J J (6) (u ) he aous opzaon echnques used n process opzaon are: he Calculus o Varaons, he Belan s Dynac Proran, he Ponryan s Mau Prncple, ec he an heorecal approaches, whch had a ajor pac upon research n opzaon, are based on he basc heory o ordnary Ma and Mna

3 4 heory o Maa and Mna he heory o ordnary aa and na s concerned wh he proble o ndn he values o each o n ndependen varables a, a, a n a whch soe uncon o varable (a, a, a n ) reaches eher a au or a nu he esence o a soluon o such a proble s uaraneed by he heore o Weersrass as lon as he uncon s connuous 4 Necessary condons or aa or na he locaon o local erea or a real-valued uncon (), whch s dened and connuous on he closed nerval [a, b] n he neror o a reon R ay be deerned by he wo necessary condons, e s an ereu hen ( ) () Such a pon s also called saonary pon or crcal pon I here s a pon where ( ) does no ess bu s pecewse connuous, e ( ) and ( + ) ess or all bu no necessarly () he necessary condons n such a case s l ( ) and l ( ) or l + + ( ) (local nu) (8) and l ( ) (local au) (9) or he end pons a, b (local nu) l a + ( ) or l ( ) (local au) () b We ay conclude ha he poenal canddaes or he absolue ereu o () are he saonary pons, he pons where one or ore rs paral dervaves o () are dsconnuous, and he end pons However, neher saonary pon nor dsconnues have o be erea In order o deerne a local ereu o all he poenal canddaes or ereu o a uncon, he sucen condons are: 4 Funcon o one ndependen varable In case o a uncon () o one ndependen varable, s a pon a whch () s zero and he dervave () chanes s sn ro posve o neave (or neave o posve) when passes hrouh zero, hen s au (or nu) o () hs s he necessary condon or ereu o a uncon o one ndependen varable, whch ay be phrased as: s neror pon a whch () and > or < () ( ) hen s local ereu s local nu () s posve, e () and s local au () s neave, e < > (3) 43 Funcon o wo ndependen varables For a uncon o wo ndependen varables (, ), ereu s relave nu > and ( ) > (4) and s relave au < and ( ) > (5) 44 Funcon o n ndependen varables he correspondn necessary condons or local erea o a uncon o n ndependen varables (,, n ) ay be epressed as: a saonary pon wll be local nu D > (, 3, 5, ) and D (, 4, 6, ) (6) < 3

4 a saonary pon wll be local au D < (, 3, 5, ) and D > (, 4, 6, ) () where D 45 Soluons subjec o consrans I he varables,, n are subjec o ceran relaons called consrans, o he or: (,, n ), (,, n ), (,, n ) (8) wh < n, he nuber o ndependen varables s reduced o n- he proble s o nd an ereu o (,, n ) or one or ore ndependen varables subjec o consran equaons here are deren ehods o nd he soluon o hs proble subjec o consrans, such as Subsuon Mehod, Larane Mulpler Mehod, Calculus o Varaons ec 46 Larane Mulpler Mehod Consder a uncon o wo ndependen varables (, ) wh consran, ) (9) ( (, ) (, he dervave ) can be wren [ 3] as: also called he Jacoban (, ) () (, ) de he necessary condon or he esence o saonary pon s also a pon where he Jacoban deernan us vansh, e (, ) (, ) or () hs equaon ay be rewren as: / / () / / λ where λ s a consan called Larane ulpler Equaon () can also be wren as: + λ + and λ (3) Equaon (3) can be consdered as he necessary condons or he esence o a saonary pon o he uncon + λ whou he consrans hereore, he necessary condon or a saonary pon o (, ) wh consran (, ) s ound by orn he auened uncon + λ, and rean he proble as one whou consrans hs resul can be eended o he eneral case o a uncon o n varables wh consran equaons, wh he nroducon o a se o consans, Larane ulplers, λ, λ, λ he use o auened uncon allows a proble wh consrans o be replaced by a proble whou consrans hs new proble can be solved by any echnque used or solvn probles whou consrans 5 he Calculus o Varaons he calculus o varaons s ha branch o calculus n whch ereal probles are nvesaed under ore eneral condons han hose consdered n he ordnary heory o aa and na More speccally, he calculus o varaons s appled o process opzaon probles, where he aa or na o unconal epressons are nvesaed he os eneral probles o he calculus o varaons n one ndependen varable are he proble o Bolza, Mayer, and Larane hese probles are heorecally equvalen and any one o he can be 4

5 ransored no anoher by chane o coordnaes 5 Proble o Bolza In order o orulae he proble o Bolza, consder a class o uncons ( ), (,, n) sasyn he consrans (, &, whch nvolve n- deree o reedo Assun ha hese uncons us be conssen wh he end condons: ψ (, ) ( r,, q ) r ψ r (, ) ( r q+, q+, s n+ ) (4) where (n+) are he oal boundary condons, q are he nal condons, and (s-q) are he nal condons Fnd ha specal se ha nzes he unconal or J K [ ), ] + L(, &, d (5) he above orulaed proble can be reaed n a sple anner a se o varables, called Larane ulples, λ, λ, λ s nroduced and he ollown epresson, called he auened uncon, s ored: F L + λ (6) j j j or F L + λ + λ + + λ he ereal arc us sasy consrans and he Euler-Larane equaon s F F (,, n ) () d d & A aheacal consequence o he Euler- Larane equaons s he derenal relaonshp d d F + n F & & F + (8) For he probles where he auened uncon s ndependen o, he ollown neral, called he rs neral s vald: n F & F + & C where C s an neraon consan (9) 5 he Proble o Mayer he proble o Mayer s ha parcular case o Bolza n whch he neral par o cos unconal J s zero, e when F (3) As he consran equaon s (, & j,, hereore, he auened uncon n Mayer proble s equal o zero F L λ (3) As F + j, hereore, he cosunconal s J K ( ), ] (3) [ 53 he Proble o Larane he proble o Larane s ha parcular case o he proble o Bolza n whch he cos unconal J s epressed n he neral or only, e K (33) j J L (, &, ) d (34) he ransversaly condon s spled o Cd + n F & d j (35) 54 Probles Involvn Inequales In any physcal probles, here are varous nequaly consrans on he conrol vecor or eaple, he au hrole sen, au conrol delecn, ec So n a proble wh nzn a cosunconal J L ( &,, ) d (36) wh equaly consrans o he or: ( &,, ) (3) and nequaly consrans o he or: Γ Γ( &,, Γ (38) n ) a In order or a rajecory o be adssble, us sasy boh he equaly and he nequaly consrans he slac-varable ehod, conver he nequaly consran equaon (38) o an 5

6 equaly consran by nroducn new sae varables z, sasyn he equaon ( Γ Γn )( Γa Γ) z (39) he proble under consderaon becoes dencal wh handn he ereal arc, n class, z(, whch sasy he equaly consrans, and Larane ulpler can now be used o adjon equaons (36 o 39) and used he usual necessary condons are he applcable hereore, a proble wh wo-sded nequaly consran s replaced by he equaly consran n he new proble Slarly, a proble wh one-sded nequaly consran, Γ ( &,, ) Γ a s replaced by he equaly consran ( Γ Γ ) z a 55 Forulaon o Opal Conrol Proble n Calculus o Varaons he orulaon o he opal conrol proble usn he Halonan uncon s se o he ollown epressons: Syse dynacs & (, u, ) ed (4) Perorance nde J K [ ( ), ] + L (, u, ) d (4) Fnal sae consran Halonan Sae equaon Ψ ( ( ), ) (4) H (, u, ) L (, u, ) + λ (, u, ) H λ (43) & Cosae equaon H λ Saonary condon u (, u, ) (44) & L (45) L u + u ( H (46) λ Boundary condons ) ven ( K + Ψ µ λ ) d ( ) (4) + ( K + Ψ µ + H ) d λ hereore, he opzaon depends on he soluon o a wo-pon boundary value proble Norally he nal sae ) s nown and he nal cosae λ ( ) s deerned by equaon (45) I s enerally very dcul o solve he wo-pon boundary value probles he os serous laon o he Calculus o Varaons s ha here are consrans on he conrol npu as n alos all he praccal probles, s no on o wor, because he necessary condons or he ereu would no apply 6 Dynac Proran Dynac proran s based on he Bellan s prncple o opaly, can be vewed as an ourow o he Halon-Jacob approach o varaonal probles he Dynac proran approach o opal conrol proble s bascally eanor dscree-e syses However, can be used or connuouse syses Consder a syse wh plan dynacs & (, and assocaed wh a perorance nde J K ( ), ] + L (, u, ) d [ (48) We are neresed n deernn a connuous opal conrol u ( on a ven nerval [, ] ha nzes J and drves a ven nal sae ) o a nal sae sasyn consran ( Ψ [ ( ), ] Suppose s he curren e and + s a uure e close o hen he cosunconal can be wren as: + J(, K[ ), ] + L(, τ ) dτ + L(, τ) dτ (49) + Coparn equaon (48) and (49) + J(, n L(, τ) dτ + J( +, + τ) < τ < + where + s he sae a e +, hereore, and (, u, ) (5) 6

7 Hence, + J (, n L(, τ ) dτ + J τ ) < τ < + ( +, + (5) he aylor seres epanson o J ( +, + abou, and an an approaon o he neral n equaon (5), we can wre o rs order J (, where J n τ ) < τ < + J (, ) J J ( L + J (, + J + J and J J (, ) + J + n u ( τ ) < τ < + ( L + J ) or J n ( L + J ) Len J u ( τ ) < τ < + n u ( ) J L + (5) (53) (54) (55) hs paral derenal equaon or he opal cos J (, s called he Halon-Jacob- Bellan (HJB) equaon I s solved bacward n e ro, by sen n equaon (55), s boundary condons seen o be (56) J ( ( ), ) K ( ( ), ) on hyper surace Ψ ( ( ), ) I we dene he Halonan uncon as: H (, λ, L(, + λ (, and J λ hen he HJB equaon can be wren as: J (5) n u ( H (, u, J, ) ) he HJB equaon provdes he soluon o he opal conrol proble or eneral nonlnear e-varyn syses; however, n os cases s possble o solve analycally When can be solved, provdes an opal soluon he aheacal odeln o a lh rajecory opzaon usn he Ponryan s Mau Prncple can be done wh he assupon ha he opal conrol ess and s unque Suppose ha he opal conrol u ( ) ess, s unque, and ha ( ) s he eneraed opal rajecory hen, correspondn o u ( ) and ( ), here es a cosae vecor p ( ) such ha he ollown relaons hold: Plan Dynacs & s n-densonal (58) ( [,, ] Conrol Consrans [, ] u s -densonal (59) u ( Ω Cos Funconal J u) K [ ), ] + L[,, ] d ( (6) he Necessary Condons or Opal Conrol d ( ) sae equaons (6) d dp ( ) d p ( ) ( ) o cosae equaons (6) where eans ha he paral dervave us be evaluaed a u ( ), ( ) and p ( ) Boundary Condons Inal Condons ( ) (norally ven) (63) ernal Condons ψ ψ s -densonal (64) [ ), ] ransversaly Condons p ( ) K µ s -densonal he Halonan ψ + µ (65) he Ponryan s Mau Prncple H H[, p(,, ] p L[,, ] + p [,, ] (66)

8 Mnzaon o he Halonan H [ ( ), u ( ), p ( ), ] H [ ( ), u ( ), p ( ), ] or every, o u (, e u ( ) Ω (6), and all adssble values Selecon o sae equaons In case o an arcra, wh he assupon ha here s no wnd, no sdeslp and all he oens are n equlbru, he eneral equaons o oon over sphercal and non roan earh are he sae equaons Usually, he opal rajecory probles are souh under hese condons; oherwse, he proble becoes coplcaed and can be solved hereore, he sae equaons are epressed as: dv P D & v& ( cosα snγ d dγ v & & γ ( ( Psnα + L) cosϕ cosγ d v v R + h dχ & & 3 χ( ( Psnα + L) snϕ d vcosγ d & 4 & ( v cosγ cos χ d dy & y& 5 ( v cosγ sn χ d dh & 6 h& ( v snγ d d & & ( q s d (68) 3 Selecon o Conrol varables We ay chose a se o varables as conrol varable, whch wll deerne he rajecory, such as hrole sen η, anle o aac α, anle o roll ϕ ec Dependn on he analyss o a desred rajecory any one or cobnaon o he conrol varables can be seleced, bu hese conrols are subjeced o physcal laons or consrans as enoned below: Conrol Consrans [, ] u s -densonal (69) η n η η a α n α α a 3 CL C n L CL a 4 n L n L ( M ) per 5 ϕ ϕ per 6 M M per 4 Dervaon o Cosae Equaons Dependn on he nuber o sae varables ( v, γ, χ,, y, h, ec), we us have equal nuber o cosae varables [ p (, p (, p 3 (, p 4 (, p 5 (, p 6 (, p ( ec] hus, we have he cosae vecor as: p ( ) p ( ) p ( ) p ( ) () hereore, he Halonan, dened by equaon (66) can be epressed as: H pl[,, ] + p ( Pcosα ρv SCD snγ v + p( Psnα + ρv SCL cosϕ cos v v R h γ + snϕ + p3 ( Psnα + ρv SCL + p4( vcosγ cosχ vcosγ + p ( vcosγ snχ + p ( vsnγ p ( q 5 6 () Where p o (consan scalar and norally s zero) he cosae equaons, as dened by equaon (6), can be derved by an he paral dervave o he Halonan () wh respec o each sae varable as enoned below: p& p& p&, ( ) v( ) ( ) γ ( ) ( ) ( ) () 5 Cos Funconal or Objecve Funcon he eneral or o he cosunconal o a syse s ven by he relaon dened by equaon (6), bu he decson abou he cos unconal or objecve uncon depends on he as o he syse hereore, he decson abou he cosunconal or objecon uncon depends upon he ype o he arcra and he requreen 8

9 o he parcular sson o be accoplshed by he arcra In eneral, a cosunconal can be a unconal o derenuncons or varables he scaln aon hese uncons or varables ay no be sae due o derence n densons; or ay be desred o ae he cosunconal ore sensve o one varable as copared o ohers In such case, wehn (or scaln) acors are nroduced n he cosunconal o pleen he requred scaln or desred eecs o he deren varables on he cosunconal For eaple, n case o an arcra, consdern boh e and he uel consupon as he easure o perorance, he cosunconal can be epressed as: or where J ( u ) d + W ( ) J ( u ) d + W ( ) and (3) s he ass o he arcra a o he he nal e and he nal e rajecory respecvely, and W s he wehn acor 6 Mnzaon o he Halonan and Soluon o he Derenal Equaons Once all he aerodynac and enne daa s avalable, hen he derenal equaons are solved o e he sae and cosae varables A each e nerval, he Halonan s calculaed or all perssble values o he conrol paraeer and he opal value s seleced hs opal value s used n he soluon o he derenal equaons or ha parcular e he equaons are solved a each e nerval, ro he nal e o he ernal e o he rajecory o be opzed he values o he cosae varables evaluaed a he ernal e are copared wh he desred values o he cosaes, whch are obaned usn ransversaly condons (65) as ollown: p j K j ψ + j ψ (4) j µ µ ( j,, 3, n ) where K s he ernal cos, ψ s -denson ernal condons, and µ s also -denson unnown paraeers analoous o Larane ulplers, he values o hese paraeers are chosen as a uess whle assnn he nal uess values o he cosae varables hereore, he desred nal values are uncon o hese nal uess values o µ he coparson beween he values o he cosae varables evaluaed a he ernal e and he desred values o he cosaes wll deerne wheher or no he boundary condons are sased I no, he nal uess value o he cosaes and µ are chaned usn opzaon subroune Sple and he process s repeaed unl he boundary condons are e he nal values o cosae and µ, ha resul n een he ernal condons, are used alon wh he oher ernal condons o evaluae he opu rajecory Eaples o very he approach, a anoeuvrable arcra C (an eperenal arcra [6] was seleced or he evaluaon o opal rajecores n horzonal lh and n vercal lh and he resuls are shown n he ures hrouh () 8 6 X () V(/s) 5 V X 4 6 (sec) Fure Mnu uel sae rajecores (Case II) 9

10 X 6 4 () 6 X () 4 V (/s) V Alude () 8 h 5 4 Fuel Consupon ( ) Fure Mnu e and nu uel sae rajecores (Case III) 4 3 Fure (de) 6 8 h () (sec) Coparson aon he hree lh cases 6 8 X () V (/s) Mnu e case (I) Fure 4 Opal sae rajecores o lh n vercal plane V Mnu e and nu uel case (III) Mnu uel case (II) 4 6 e (sec) X (sec) h Horzonal dsance () Fure 5 Flh pah n vercal plane 8 Analyss o resuls he horzonal lh rajecores o he above eaples shown n ures,, and 3 reveal he desred resuls he resuls o lh n vercal plane as shown n ure 4 and 5 reveal he desred lh pah conrol However, was observed ha as he ernal condons can be se n wo possble ways, or eaple: ψ ( ( ), ) req or req Boh should yeld he sae resuls, bu praccally was observed he nspe o converence n boh he cases he resuls were deren In nu e case I o eaple n he horzonal lh rajecory, or he ernal condon req, he resul s local, e resuls n au hrole conrol and nu e Whle or he ernal condon req, resuls n nu hrole sen and aes au e, alhouh ees he end condons hs shows ha he Ponryan s nu prncple provdes only he necessary condons and no he sucen condons or opaly he soluon ay or ay no be opal In hs case boh he resuls yeld he ereal rajecores, bu only one s opal I was also observed ha he chane n boundary condons or he sae varables resuls n chane o he opal rajecores, however, he chane n he nal

11 uess o he cosae varables does no eec he resuls 8 CONCLUSION he ollown conclusons are drawn ro he analyss o he resuls: Flh rajecory opzaon probles can be solved by any o he opzaon echnques: he Calculus o Varaons, he Belan s Dynac Proran, he Ponryan s Mau Prncple, ec dependn on he copley and naure o he consrans, however, he Ponryan s Mau Prncple s appled as a es case I has been proved by he resuls ha Ponryan s au prncple provdes only he necessary condons and no he sucen condons or opaly, bu sll can be successully appled or opzaon o rajecores o any arcraor any lh phase Only has o be ensured o analyze all he ereal rajecores and nd he opal rajecory by her coparson [3] JV Breawell, he Opzaon o rajecores, J SIAM, Vol, pp5-4, June 959 [4] C A Desoer, Ponryan s Mau Prncple and he prncple o Opaly, Franln Ins, Vol, pp 36-36, 96 [5] EM Maashov, GM Kubs, DG Belyh Uned Syse o Eperens based on es Characerscs and as o Flh rajecory Opzaon MAI Press, Moscow, 989 [6] R E Kopp, Ponryan s Mau Prncple, n G Leann (ed) Opzaon echnques, Acadec Press, New Yor, 96 [] N X Vnh, Opal rajecores n Aospherc Flh, Elsever, Aserda, 98 [8], Arcra Flyn Quales Analyss and Flh rajecores Opzaon Ph D heses, Bejn Unversy o Aeronaucs and Asronaucs, Chna February, 999 [9], Xo Ylun, Flh rjecores Oprzaon Usn Ponryan s Mau Prncple he 3 rd Asan-Pacc Conerence on Aerospace echnoloy and Scence (APCAS-), Kunn he chane n boundary condons or he sae varables resuls n chane o he opal rajecores; however, he chane n he nal uess o he cosae varables does no eec he resuls he Ponryan s au prncple ay be appled n slar way or he opzaon o rajecores o he spacecra, and he sae aospherc odel can be used or he launch or re-enry phase o he spacecra Reerences [] M Ahans, Modern Conrol heory, Deernsc Opal Conrol, Lecure Noes, MI, 94 [] M Ahans, he Saus o Opal Conrol heory and Applcaon or Deernsc Syses IEEE, vol AC-, No 3, pp58-596, July 966