Optimal control of multi-missile system based on analytical method

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1 een Adnes on Elerosene nd Compuers Opml onrol o mul-mssle ssem bsed on nll mehod Xng Lu, Yongj Wng, Shu Dong, Le Lu Absr The mnmum-me snhronous onrol problem o mul-mssle ssem s onsdered n hs pper. B desgnng he opml onroller lke bng-bng sruure whh hs sngle swh dependen on me prmeer, he ses npus n be epressed mhemll nd nlll. Wh he spe epressons o he ses npus, he orgnl me opml problem s rnsormed no sndrd nonlner progrmmng problem. Aer opmzng ll he prmeers opmzon ools, he mnmum-me problem s soled. Compred wh he numerl mehods whh should dsree he ses nd onrol npus, he proposed sreg perorm less ompuonl burden nd more preson. Smulon resuls demonsre he een nd ld o he proposed mehod. Kewords mu-ehle, nll mehod, snhronzon, mnmum-me opml onrol, enrlzed onrol O I. INTODUCTION VE he ps ew ers, he reserh o oopere onrol o mulple ehles hs red more nd more enons nd eors[1-3]. Some mure nd ehnl mehods he been emploed n he ormon o spers[4-5], mulple unmnned erl ehles (UAVs)[6-8] nd mssles[9-1]. Compred wh one sngle ehle, mu-ehle n hee more omple nd derse sks whn ommon ommunon nework. Sne he nresng mlr demnds or he k nd deense ssems, group o well-orgnzed nd low-os mssles s onronng more dnges hn sngle one. And he oopere ghng mnner wll denel be he mn pern n he uure bleeld. Thereore, s sgnn o sud on he reserh o oopere onrol o mul-mssle ssem (MMS). The oopere k o MMS hs wo ommon mnners: oopere gme nd suron k. The onep o oopere gme s proposed n reen ers nd he sud on hs beome er populr re or reserh. Shm[11] Ths work ws suppored n pr b he Nonl Nure Sene Foundon o Chn (No nd ), Doorl Fund o Mnsr o Eduon o Chn (No ), Fundmenl eserh Funds or he Cenrl Unerses o HUST (NO ), nd Preson Mnuurng Tehnolog nd Equpmen or Mel Prs (No. 212DFG764). Correspondng uhor, Xng Lu s wh Ke Lboror o Mnsr o Eduon or Imge Proessng nd Inellgen Conrol,Shool o Auomon,Huzhong Uners o Sene nd Tehnolog, Wuhn, Chn (e-ml: glun@gml.om). YongJ Wng (Ined-Dmro) s proessor he Shool o Auomon, Huzhong Uners o Sene nd Tehnolog, Wuhn, Chn. (e-ml: wngjh@ml.hus.edu.n). Shu Dong s wh he Shool o Auomon, Huzhong Uners o Sene nd Tehnolog, Wuhn, Chn. (e-ml: husds@hus.edu.n). nesged he opml oopere pursu nd eson sreges gns homng mssle bsed on he opml onrol heor. ubnsk[12] reserhed he hree-pler onl b derenl gme. And Andre[13] proposed he oopere derenl gmes sreges or e rr proeon rom homng mssle. I s noed h he sreg o oopere gme requres he MMS should onl ke he mnner o enrlzed onrol. Howeer, eher he enrlzed onrol[14] or deenrlzed onrol[15-19] n hee suron k, whh s he ommon nd useul w o oopere enggemen. An MMS n desro nd pere he deense ssem b kng he rge smulneousl. Jeon[14] proposed homng gudne lw lled oopere proporonl ngon (CPN) or oopere k o mulple mssles bsed on enrlzed onrol. Zho[16] proposed oopere gudne sheme where oordnon lgorhms nd lol gudne lws re ombned ogeher. Zhng[19] onsdered he oopere nerepon o mong rge b mulple ehles wh olerne o uor or nework lures. Ths pper sudes he MMS oopere onrol h ms smulneous k on s rge nd mnl onsdered he mnmum-me snhronous onrol problem (MTSCP) o n MMS. The purpose s o obn he mnmum me when ll he ses o mssles re snhronous b opml onrol npus, whh n redue he me o oopere k. Sne he deenrlzed onrol s no opml due o lk o globl normon, he enrlzed onrol s ppled here o sole he opml onrol problem. Furhermore, he onsrn o oerlod s onsdered owng o he resron o he ehle s sruure. Then he boundr o onrol npu dened n hs pper s ed. Wh he desgned onroller lke bng-bng sruure whh hs sngle swh h s dependen on me prmeer, he nll epressons o se npus re obned. Aordng o he epressons, we n ge regon n he se spe. And s worh nong h he me o snhronzon s shores when he boundr o eh mssle s se regon hs ommon pon wh he ohers. Bsed on he boe prnple, he orgnl me opml problem n be rnsormed no sndrd nonlner progrmmng problem. Fnll, b opmzng ll he prmeers esng n he nll epressons, he mnmum me nd opml onrol npus o ll he mssons re obned. The proposed mehod n be ppled o hee he suron k nd sequene k he sme me. The sequene k demnds h he mssles k he rge wh ISBN:

2 een Adnes on Elerosene nd Compuers ed me sequene. Thereore, he ke o he opml onrol sreg s o gurnee he snhronzon o ll he msses on ed me sequene. The mhem desrpons o he MTSCP bou boh suron k nd sequene k re presenng n hs pper. The orgnzon o hs pper s s ollows. Se. II nrodues he dnm equon o mssle gen nd hen presens he desrpon o he MTSCP. Se. III proposes sreg h rnsormng he me opml problem o nonlner progrmmng problem. In Se. IV n emple s gen o demonsre he eeeness o he proposed pproh. Fnll, Se. V summrzes some onlusons o hs rle. II. POBLEM STATEMENT The oopere gudne o n smlr mssles whh re denoed s 1, 2. n s onsdered n he rle. Assume h he mssle gen n onl hnge he dreon o he elo nd he speeds o ll he mssles re onsn. And hen deompose he rjeor o mssles no moons on longudnl plne nd lerl plne, nd desgn ndependen onrollers. When he mssle s lose o s rge, lmos remns sgnn upon he lerl plne. Whou loss o generl, we ssume ll mssles re loed nd moe n sme longudnl plne, nd gnore he ollson h ours mong he group o mssles [2]. The rele moon o mssle s shown n Fg.1, nd he XOY oordnes re esblshed, whh n obn he orrespondng equon:,, where [, ] T s he poson eor, nd [, ] T s he elo eor, [, ] T s he eleron eor, s he dsne slr beween he mssle nd he rge nd s he eleron slr. Dene nd s he nlokwse ngles rom he OX s o he eor [, ] T nd[, ] T respeel, nd here ess,,, (1) (2) Beuse he gudne lw s dul o desgn bsed on he model (1) nd he sensor ssembled on he mssle n onl mesure he ses o, nd, he model below s doped o desgn he onroller os( ) sn( ) (3) Due o he resron o eh ehle s sruure he lue o ehle s oerlod hs boundr: n.then ordng o (3) here s n sn( ) sn( ) n sn( ) (4) sn( ) Sne he lue o s n h n be gnored n ompred wh, here ess n sn( ) n (5) Assume r, nd, u sn / r /,so he equon (3) n be rewren s r os u u u where u n /. n, nd u re ll onsns. Consder he oopere gudne o n smlr mssles nd he mssle hs he ollowng dnm equon r os u (7) u u u The nl ses o mssle re r() r nd().i s obous h here es, / 2 (6) nd r ordng o he denons o he ses. Furhermore, we ssume ll he onrol npus he he sme boundr u, so here hs u u. Ths pper ms o desgn enrlzed onrol or smulneous k on s rge, so he desrpon o he MTSCP s mn u [ u, u ] s.. (1) r ( ) r ( ) r ( ) 1 2 n (2) ( ) ( ) ( ) 1 2 n where s he me o snhronzon, whh s he objee o he MTSCP. B desgnng he opml onroller u, he MTSCP n be rnsormed nd soled nlll. (8) ISBN:

3 een Adnes on Elerosene nd Compuers Y mssle he boundr o ( T), r( T).And he hree-dmensonl rjeor showed n Fg.3 presens he boundr o ( ), r( ) deren me r(t) 25 O( rge) Fg.1 rele moon beween he mssle nd he rge III. OPTIMAL CONTOLLE GENEATED Aordng o (7), he ses he me re epressed s ollowed ( ) u ( ) d r ( ) r os ( ) d Assume he onroller u o mssle s desgned s s u u () (1) s u T, T nd s { 1,1}.I s eden h he Where desgned onroller s smlr o he bng-bng sruure nd he swh o s he.subsung (1) no (9), one n he s u () 2s u s u T And he epresson o r s X (9) (11) 1 r sn( s u ) su r () (12) 1 r [2sn( s u ) sn( ) T su sn( 2 s u s u )] I s obous h he se () hnges lnerl wh he me. When he res rom o T, eh lue o s orrespondng o nl se ( T), where ( T) ( ). So T he me T, ll he lues o ( T) orm he rnge o.beuse o he ondon, / 2 ( T) nd (9), s obousl nong h r s monoonll nresng unon o.thereore, he rnge o rt ( ) n be obned. Fnll, he boundr o ( T), r( T) n he se spe n be deped b he nll epressons (11) nd (12). For sngle mssle, ssume he nl ses re /3 nd r 3.The lue o T s 1 nd u s.5. Fg.2 shows r() (T)(rd) Fg.2 he boundr o ( T), r( T) when T 1s (s) ()(rd) Fg.3 he boundr o ( ), r( ) deren me From Fg. 2 nd Fg. 3, he onluson s esl obned h he me o snhronzon s shores when boundr n he se spe o eh mssle s showed n Fg.3 hs ommon pon wh he ohers, whh mens h he ses o ll he mssles re equl on he boundres. Thereore, he epressons (11) nd (12) n be ppled o sole he opml onrol problem (8) b rnsormng no nonlner progrmmng one. Then some opmzon ools n MTLAB re emploed here o opmze he prmeers, nd s.fnll he opml nd onroller u re obned. In hs work, he proposed mehod n hee suron k nd lso sequene k. Eh problem s desrped s ollows. The suron k mens ll he mssles moe rom he sme me nd hee snhronzon he sme momen. So he MTSCP semen o suron k s mn, s, s.. 1) r ( ) r ( ) r ( ) n 2) ( ) ( ) ( ) 1 2 n (13) The sequene k requres ll he mssles reh ISBN:

4 een Adnes on Elerosene nd Compuers onsensus on ed me sequene. Done s he me nerl o mssle, whh s onsn. So he MTSCP semen o sequene k s mn, s, s.. 1) r ( ) r ( ) r ( ) (14) n 1 n1 2) ( ) ( ) ( ) n 1 n1 Fg.4 lerl demonsres he MTSCP o sequene k. For he onenene o solng he MTSCP, he boe semen n be rnsormed no noher one h he mssles begn o ondu he oopere gudne rom he deren me bu hee snhronzon he sme me, whh llusres n Fg.5. Thereore, he MTSCP n be esl soled b nll epressons (11) nd (12). u () 1, u () 2 nd u 3 ().I s lerl o noe h he opml onrol npus re smlr o he bng-bng sruure nd he swh s dependen on.trjeores o he se npus r () nd () o ll he hree mssles re shown n Fg.6 nd Fg.7 respeel. In Fg.9 he ures represen he boundres o he se npus he me. (rd) mssle3 mssle2 mssle1.8 4 () 1 ().7.6 * 2 () 3 () Fg.4 he MTSCP o sequene k rom he sme me (s) Fg.6 he rjeores o () mssle3 mssle2 mssle1 4 () () ( ) ( ) 17 r * 3 () 2 () 2( ) 3( ) Fg.5 he MTSCP o sequenl k rom he deren me IV. NONLINEA SIMULATION In hs seon, n MTSCP o sequene k s proded or smulon o demonsre he eeeness o he heorel resuls n hs rle. The oopere k o hree mssles s onsdered n hs emple. The lues o me nerls re 1 1, 2 5 nd he onroller boundr s u.5.assumng h he nl ses re r 1 18, 1 / 5.5, r2 21, 2 /3, r 3 2, 3 /4. Aordng o he boe ondons nd he proposed mehod, he resuls o smulon re obned. And he opml me o hee snhronzon s s.The lues o oher prmeers re s, s 3 1, s, s 2 1, s nd s 1 1.Thereore, he spe epressons o ll he hree onrollers re obned. Fg.8 shows he rjeores o he hree opml onrollers u (s) Fg.7 he rjeores o r () (s) Fg.8 he rjeores o u () mssle3 mssle2 mssle1 ISBN:

5 een Adnes on Elerosene nd Compuers r mssle3 mssle2 mssle (rd) Fg.9 he boundres o he hree mssles when The rjeores o r () nd () n Fg.6 nd Fg.7 llusre h he ses o he hree mssles re snhronous or he rs me he me, whh proes he ronl o he resuls. Fg.9 shows h he se boundres o hree mssles he ommon pon el he me nll mehod s sgnn nd een. V. CONCLUSION nd demonsres hs The oopere gudne problem wh oerlod onsrns bsed on enrlzed onrol s suded n hs pper. The proposed sreg s ppled o mke he MMS hee snhronzon whn mnmum me. B desgnng opml onroller lke bng-bng sruure whh hs sngle swh dependen on me prmeer, he nll epressons o se npus re obned. So he orgnl me opml problem n be rnsormed no sndrd nonlner progrmmng one. Fnll, he MTSCP s soled b opmzng ll he prmeers. Ths nll mehod s pble o omplee he suron k nd sequenl k. Moreoer, he resuls o smulon demonsre he less ompuonl burden nd more preson ompred wh numerl soluons. In uure sud seerl ors suh s he deren boundr o eh mssle s oerlod nd mp ngle onsrn wll be onsdered or prl mplemenons. Furhermore, he proposed mehod wll be epnded nd ompleed heorell o hee he opml oopere gudne. [6] Ollero, A. nd I. Mz, Mulple Heerogeneous Unmnned Aerl Vehles, Berln, Sprnger-Verlg, 27 [7] Ddek, Z.T., A.M. Annswm nd E. Lresk, Adpe onguron onrol o mulple UAVs, Conrol Engneerng Pre, Vol.21, No.8, 213, pp [8] Moon, S., E. Oh nd D.H. Shm, An Inegrl Frmework o Tsk Assgnmen nd Ph Plnnng or Mulple Unmnned Aerl Vehles n Dnm Enronmens, Journl o Inellgen & obo Ssems, Vol.7, No.1, 213, pp [9] We, C., e l., Opml ormon keepng onrol n mssle oopere enggemen, Arr Engneerng And Aerospe Tehnolog, Vol.84, No.6, 212, pp [1] Cu, N., e l., Sud on mssle ormon reonguron opmzed rjeor generon nd onrol, Journl o Applled Mhns, Vol.77, No.5, 21, pp [11] Shm, T., Opml oopere pursu nd eson sreges gns homng mssle,journl o Gudne Conrol And Dnms, Vol. 34, No.2, 211, pp [12] ubnsk, S. nd S. Gumn, Three-pler pursu nd eson onl, Journl o Gudne Conrol And Dnms, Vol. 37, No.1,214, pp [13] Perelmn, A., T. Shm nd I. usnk, Coopere derenl gmes sreges or e rr proeon rom homng mssle, Journl o Gudne Conrol And Dnms, Vol.34, No.3, 211, pp [14] Jeon, I., J. Lee nd M. Thk, Homng gudne lw or oopere k o mulple mssles, Journl o Gudne Conrol And Dnms, Vol.33, No.1, 21, pp [15] Wez, L.A., J.E. Hurdo nd A.J. Snlr, Deenrlzed oopere-onrol desgn or mulehle ormons, Journl o Gudne Conrol And Dnms, Vol.31, No.4, 28, pp [16] Zho, S. nd. Zhou, Coopere Gudne or Mul-mssle Slo Ak, Chnese Journl o Aeronus, Vol.21, No.6, 28, pp [17] Mln, T.W. nd.w. Berd, Coordnon rbles, oordnon unons, nd oopere-mng mssons, Journl o Gudne Conrol And Dnms, Vol.28, No.1, 25, pp [18] Zheng, Y., Y. Zhu nd L. Wng, Consensus o heerogeneous mul-gen ssems, IET Conrol Theor nd Applons, Vol.16, No.5, 211, pp [19] Zhng, P., e l., Ful olerne o oopere nerepon usng mulple lgh ehles, Journl o he Frnkln Insue, Vol. 35, No.9, 213, pp [2] Y. Wng, S. Dong, L. OU, nd L. Lu, Coopere onrol o mul-mssle ssems, IET Conrol Theor & Applon, o be publshed. BIOGAPHY: Yongj Wng he reeed hs PhD n Power Pln Engneerng rom Huzhong Uners o Sene nd Tehnolog, Chn n 199. He s urrenl Proessor n Shool o Auomon Huzhong Uners o Sene nd Tehnolog. Hs reserh neress nlude neurl nework, ssem denon nd onrol, lgh ehle onrol. EFEENCES [1] noo, A. nd T. Shm, Formon-lng gudne or oopere rdr deepon, Journl o Gudne Conrol And Dnms, Vol.35, No.6, 212, pp [2] Cmp, G., e l., Desgn nd lgh-esng o non-lner ormon onrol lws, Conrol Engneerng Pre, Vol.15, No.9, 27, pp [3] Ousngsw, J. nd M.E. Cmpbell, Opml oopere reonnssne usng mulple ehles, Journl o Gudne Conrol And Dnms, Vol.3, No.1, 27, pp [4] Bndo, M. nd A. Ihkw, Ae ormon lng long n ellp orb,journl o Gudne Conrol And Dnms,Vol.36, No.1,213, pp [5] Mssr, M., F. Bernell-Zzzer nd S. Cnes, Nonlner onrol o ormon lng wh se onsrns,journl o Gudne Conrol And Dnms, Vol.35, No.6, 212, pp ISBN:

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