Design of Closed Loop Optimal Guidance Law Using Neural Networks

V ol . 15　 N o . 2 　　　　　　　　　　　　 CHIN ESE JO U RN A L O F A ER O N A U T IC S　　　　　　　　　　　　　　M ay 2002

Design of Closed Loop Optimal Guidance Law Using Neural Networks ZHOU Rui ( Dep artment of A utomat ic Control, Beij ing Uni versity of A er onaut ics and A stronauti cs, Beij i ng 100083, China ) Abstract: 　It is gener ally impo ssible to obtain the analytic o pt imal g uidance law fo r complex no nlinear g uidance sy st ems o f hom ing missiles, and the o pen lo op o ptimal guidance law is oft en o btained by numer ical methods , w hich can not be used dir ectly in pr actice. T he neural netw or ks ar e t rained off-line using the optimal tr ajecto ry of the missile pro duced by the num erical open loo p o ptimal guidance law , and t hen, the conver g ed neur al netw or ks ar e used on-line as t he feedback optimal guidance law in r eal -time . T he r esear ch sho ws that differ ent selectio ns o f the neur al netw or ks inputs , such as the system stat e var iables o r the rat e of L O S ( line of sight ) , may hav e g reat effect o n the per for mances o f the guidance systems for homing missiles. T he ro bustness fo r sever al g uidance la ws is inv estig ated by simulatio ns, and the mo dular neural netw o rks ar chitectur es ar e used to incr ease the appro ximating and gener alizing abilit ies in the larg e state space. Some useful conclusio ns ar e obtained by simulatio n result s . Key words: 　neural netw or ks; missile guidance; o ptimal guidance law ; pr opor tional nav ig atio n g uidance 神 经网络最优闭环制导律设计. 周　锐. 中国航空学报( 英文版 ) , 2002, 15( 2) : 98- 102. 摘 　要: 对于 复杂的非 线性导 弹制导 系统, 很难求 得其解 析的最 优制导 律, 只 能求 得开 环的数 字 解 , 不能适用于具有时变不确定性的导弹制导系统。利用神经网络的学习和推广能力, 对开环的数 字 最优制导律进行离线的学习, 作为闭环的神经 最优制导律在线应用 。研究分别选择系统 状态变 量 和视线角速 率等不同的 神经网络 输入对制导 系统性能 的影响, 以及各种 制导律的鲁 棒性问题, 并 采用模块化神经网络结构提高神经网络的学习和推广能力, 仿真结果得到一些有益的结论。 关 键词: 神经网络; 导弹制导; 最优制导律; 比例导引 文 章编号: 1000-9361( 2002) 02-0098-05　　 　中图分类号: V 249. 1　　　文献标识码: A

　　T he proport ional navig at ion guidance ( PNG)

diff icult t o solv e and o ft en obtained of f -line by nu-

is used w idely in practice because of it s simple

m erical methods, thus can be only used as open

st ruct ure and easily implement ing, but its per for -

l oop OGL . Mo reo ver , the OGL is sensit iv e t o t he

mances ar e limited . Opt imal guidance law s ( OGL )

initial condit ions and noises in t he missile guidance system , w hich lim it s it s applicat io ns in pract ice .

hav e theoret ically solv ed t he problem ex ist ing in PNG , and pro ved that the P NG is t he o pt imal so -

Neural net w orks ar e w idely used in synt hesis

lut ion under t he co nditions of zero miss dist ance

o f vario us feedback OGL using t he opt imal open

and minim izing t he sum o f t he squares o f t he missile accel erat ion , assuming no t arg et m aneuver ing

l oop t raject ory of m issiles

[ 1]

and no delay in missile dy namics . U nf ort unat ely , solv ing t he OGL w il l result in a complex tw o po int boundary value pro blem ( T PBVP) , w hich is Received date: 2001-10-08; Revision recei ved dat e: 2002-03-10 Foundati on it em: N at ional S cience Found at ion of China ( 69904002) A rt icl e U R L: ht t p: / / ww w . hk xb. net. cn/ cja/ 2002/ 02/ 0098/

[ 2, 3]

, and provide an ef-

f ect ive approach t o desig n t he opt imal co nt rol ler s f or co mpl ex opt im al problem s, because o f it s abilit ies t o com put ing par al lell y , learning , generalizing , and f aul t t olerance . But how t o select t he

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Design of Cl osed Loop O pt imal G uidance Law U s ing N eural N et w orks

st ruct ures and tr aining sam ples of t he neural net w orks guidance ( NNG ) may hav e an impo rt ant ef fect on t he per for mances of m issile guidance systems. T his paper discusses t he select io n of neur al net w orks st ruct ures in NNG , and anal yzes t he per fo rmances and ro bust ness t o noises o f NNG w it h dif ferent input s.

1　Generation of Opt imal Guidance L aw T he t w o dimensional eng agement bet w een t he missile and t arget is show n in Fig . 1. T he relat ive range o f t he m issile and t arget in t he direct io n o f X and Y is denot ed as x and y . Assuming no t ar get maneuvering, t he relative mot ion equat ions are

2　Select ion of N NG St ruct ures F or t he com plex nonlinear opt imal guidance f or hom ing m issiles, o nly o pen lo op numerical OGL can be obt ained of f -l ine, w hich can not be used in t he time varying and uncert ain guidance systems. But the opt imal open loo p t raject ory of t he missile based o n T PBVP can be used t o t rain t he neur al net wo rks of f -l ine , and then , the converg ed neural net wo rks are used as an o pt imal f eedback guidance o n-line. F or long range int ercept ion w it h larg e st at es space , a neural netw o rk m ay hav e g ood learning and generalizing abilit ies fo r so me st at es subspace, but may hav e po or o nes fo r anot her st at es subspace. So a m odular connect io nist archit ect ure show n in Fig . 2 can be used to all ocat e diff erent

F ig . 1　M issile and tar g et inter ceptio n geo metr y

given by

[ 2]

x = v t cos - v m cos , x ( 0) = x 0 y = v t sin - v m sin , y ( 0) = y 0 an = , ( 0) = vm

( 1)

0

Fig. 2　 M o dular neura l netw or ks a rchitectur e

w here an is t he normal acceler at ion of t he missil e, and assuming t he tr ansfer funct ion o f t he missile aut opilot is w it hout del ay . A simple f orm of OGL is defined as

netw o rks t o learn t raining pat t erns f rom diff erent [ 4] r eg io ns o f t he input space . Each netw or k can it self be a local or global appro ximat or for a part icuth

t

1 a2n dt ( 2) 2 0 and t he t erminal condit ions, x ( t f ) = 0, y ( t f ) = 0 are sat isf ied, where tf is t he fr ee t erminal t ime. J = min a n

∫ f

It is w ell know n that solv ing Eq. ( 2) under Eq. ( 1) w ill r esul t in a complex nonlinear T PBVP, w hich can be so lved num erically of f -line as o pen loop OGL . T he P NG of t he abo ve problem is given as an = N ′ v cq ( 3) w here N ′is t he navig ation rat io; v c and q are separat ely relativ e v elo cit y alo ng t he l ine of sight ( L OS) and ang le of L OS.

lar r eg io n of t he space. T he k o ut put unit of t he g ating net w ork , deno ted by g k , is esk gk = ( 4) m ∑i= 1e si w here sk is t he unit k′ s out put s o f t he gat ing net w ork and m is t he number of exper t netw o rks. T he o ut put of t he ent ire archit ect ure is act ually t he m issile guidance co mmand, and g iven as m

an =

∑g

k

( 5)

ank

k= 1

th

w here a nk denot es t he out put of the k ex pert net w ork. During t raining , t he weight s of t he ex pert and gat ing net w orks are adjusted sim ult aneously

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ZHO U Rui

using t he backpropagat ion alg orit hm so as t o max imize t he f unct ion m

lnL = ln ∑g k e

-

k= 1

1 ‖a* - a ‖2 n nk 22

( 6)

k

* n

w here a denot es t he ideal missile guidance com mand pro duced by t he o pen loop opt im al t raject ory based on T PBVP under Eq. ( 1) and Eq. ( 2) ; 2i deth

no tes a scaling param et er associat ed w ith t he i ex pert netw o rk.

3　Desig n of T raining Sam ples for NN G Out put s of t he t raining sam ples for NNG are

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3. 2　Rate of LOS as NNG inputs Similar t o P NG , q , t he rat e o f L OS , and v c , t he closing velocit y betw een t he m issile and t arget , are select ed as input s o f NNG . Based o n t he m issile and t arget intercept io n geomet ry in Fig . 1, o ne can obt ain q=

x ( v t sin - v m sin ) - y ( v t cos - v mcos ) x2 + y2

vc =

x ( v t cos - v m cos ) + y ( v t sin - v m sin ) x 2 + y2

q can be used direct ly as input , and v c is norm alized as v c = v c/ ( v m + v t)

* n

a show n in Eq. ( 6) . T o assure t he st abilit y and co nverg ence of neural net wo rks, a *n can be norm al ized as [ 2] a*n x 0 a n = 0. 3 + 3v 2m

4　Simulation Result s of Feedback NN G ( 7)

T he init ial condit io ns of t he syst em are select [ 2]

　　Input s of t he t raining samples f or NNG can be dif ferent according to the desig ner' s pref erence,

ed as

t hus t he perf orm ances of t he guidance syst em may

t ain an open loo p OGL and opt imal t raject ory ,

be diff er ent . Considering t he P NG and OGL have

based on which, tw o diff erent inputs in t raining samples, ( x, y , -) and ( v c , q ) can be developed,

distinct ly diff erent inputs, t w o cases are invest igat ed in Fig . 3, in w hich the inputs of NNG are se-

x 0 = 7500m, v m = 500m/ s, v t = 300m / s,

110° , 0 = 80° . Chebyshev t echnique

[ 5]

=

is used t o ob-

lect ed respectively as x , y and , o r q and v c ; t he

and the co rresponding out put s ar e all t he same, a-n. T he num ber of ex pert net w orks is f our. De-

st ruct ure of NNG is show n as F ig. 2. In al l cases,

pending on t he dif ferent input select ion, t he st ruc-

it is assumed t hat there is neit her targ et m aneuver ing nor delay in t he missile aut opilot .

t ure of each expert net w or k is select ed as 3-5-1 or 2-5-1, and t he st ruct ure of t he gat ing net w or k is select ed as 3-5-4 o r 2-5-4, respect iv ely . T he mo dular net w orks can be used as f eedback g uidance on l ine af ter they are co nverg ed . T he simulatio n r esult s under no minal condit ions are show n in Figs. 4～6, and compared w it h t he ones of PNG in the sam e coor dinat e. T o invest igat e t he robust ness of each missil e guidance law , t he t raject ories of t he missile and t ar get under t he

ig . 3　Schem e of N N G system w it h two different inputs

3. 1　System states variables as NNG i nputs Since t he f ull st at es variables rev eal t he int erior behavior s of t he g uidance syst em , x , y , and in Eq. ( 1) are first investig at ed as NN G input s . In order t o assure the st abil it y and co nverg ence o f t he neural net wo rks, input stat es can be norm alized as x = x / x 0, y = y / x 0, - = 2 / ! ( 8)

condit ions of measuring noises in missil e pat h ang le are sho w n in F ig. 7. T he st reng t h of measuring noises in , deno ted by n , is g iv en by n = 0. 02sin5t ( r ad) 　　 T he simulation r esult s show that t he perf orm ances of N NG using x , y , and as its input s are consist ent w it h t hose of nominal OGL , and have t he best r obustness in all cases. But t he m issile w it h NNG using q and v c as it s input s f ails t o kill

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Design of Cl osed Loop O pt imal G uidance Law U s ing N eural N et w orks

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t he t arget under measuring noises in . T he o pen loop OGL has been of f it s optim alit y , and is very sensit ive t o t he measuring noise. T he missile w it h PNG has paid m uch mor e cont rol ef for ts w it h a large curvat ur e, but it s miss dist ance f or P NG is sm all in all cases .

Fig. 7　T r aject or ies of missile and t arg et under no ises

5　Conclus ions U nder nom inal condit ions, OGL is superior

F ig . 4　N ominal tr ajecto ries o f missile and tar g et

g reatly t o PNG on all sides. But f or nonlinear g uidance syst em s, onl y open l oop numerical OGL can be obt ained, which is very sensitive t o no ises and uncert aint ies in t he g uidance syst em , and t hus, can not be used direct ly in pract ice . But t he o pt imal t raject ory obtained fr om t he open OGL can be used t o t rain a f eed f orw ard m ult ilayer neural netw o rk off -line. T he netw o rk can be t hen used o n-line as feedback NNG. T he feedback NNG using f ul l sy st em st at es as it input s present s goo d perfo rmances and st rong r obustness w ith respect t o noises and uncert aint ies. Mo reo ver, t he NNG only needs t o st ore a set

Fig. 5　N om inal ang ular rat e of L O S

o f w eig ht s w it h small com put at ion and sto rage, t hus, it is v ery suit able t o use on -line in real t ime . Since only t he rate of L OS and closing v elo cit y betw een t he missile and targ et can not uniquely det ermine t he sy st em st at es , t hey can not r eveal t he int erio r dynamic behav io r in t he guidance syst em . U nder no minal condit ions, the feedback NNG using t he rate of L OS and clo sing velocity as it s input s has similar perfo rmances t o the OGL , but t he m issile f ails t o kill t he t arget under measuring noises. It is show n t hat the OGL may be a more complex f unctio n of t he rat e o f L OS and closing ve-

F ig . 6　M issile no minal nor mal acceleration

lo cit y bet ween t he m issile and t arget . It should be po int ed out t hat a sing le neural netw o rk ar chitecture can be select ed f or t hose

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shor t rang e engag em ent bet w een t he missil e and t arget , and maintains goo d guidance per for mances, so as t o simplif y t he NNG. References [ 1] 　 Bryson A E , Ho Y C. A ppl ied opt imal cont rol [ M ] . W ashington D C: Hals ted Pres s, 1975. [ 2] 　 Rahbar N , Bahr ami M . S ynt hes is of opt imal f eedback guidance law f or hom ing mis sil es us ing neural net w orks [ J ] . O ptimal Con trol A ppl icat ions & M et hods, 2000, 21: 137- 142. [ 3] 　 Cot tr el R G , V incent T L. M inimiz ing in tercept or si ze using neu ral net work s f or t er minal guid ance l aw s yst em [ J] . Journ al of G uidance , Cont rol and D ynam ics, 1996, 19( 3) : 557- 562. [ 4] 　 Jacobs R A , J or dan M I . Learning piecew ise cont rol s tr at egies in a modular neur al netw ork archit ect ure [ J ] . IEEE Tr ans acti on s on S yst ems , M an and Cybernet ics, 1993, 23

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( 2) : 337- 345. [ 5] 　 V l as senbroeck J. A Chebyshev t echnique f or s ol ving nonl inear opt imal cont rol probl ems [ J] . IEEE T rans on A ut omat ic Cont rol , 1988, 33( 4) : 333- 340.

Biography: ZHOU Rui 　 He wa s bo rn in 1968, and r eceived his docto ral deg ree fro m Ha rbin Institut e o f T echno lo gy in 1997, and then became a teacher in Depart ment of A uto matic Contr ol, Beijing U niver sity of A er onautics and A stro nautics, w here he became an asso ciate pro fesso r in 1999. His resear ch inter ests are flig ht co ntro l, missile guidance, and intelligent co ntr ol. T el: ( 010) 82317318, E-mail: zhr @ . dept 3. buaa. edu. cn