GAME THEORY SOLUTIONS USING NEURAL NETWORKS FOR MISSILE GUIDANCE
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1 GAME THEORY SOLUTIONS USING NEURAL NETWORKS FOR MISSILE GUIDANCE Vnkat Durbha * and S.N. Balakrishnan ** vdkc7@umr.du, bala@umr.du Dartmnt of Mchanical and Arosac Enginring Univrsity of Missouri-Rolla, Rolla, MO INTRODUCTION Elmntary gam thory is concrnd with otimization of crtain ay off btwn two contsting artis. Pursuit-Evasion gams ar a class of roblms with conflicting intrsts of layrs. In such roblms, th ay-off can b th tim of intrct or th final miss distanc. Whil th Pursur tris to minimiz th afor-mntiond ay-off, th vadr tris to maximiz it. This is analogous to situation involving an attacking Missil/Airlan and an Airlan/Missil trying to avoid intrction. This can b considrd as otimal control roblm with th ay-off as th cost function. Nural ntworks hav bn succssfully usd to solv various otimal control roblms in rcnt yars. In th rsnt study, Adativ-Critic basd nural ntworks ar mloyd to solv a linar ursuitvasion roblm. This is achivd by succssivly adating thr ntworks, an action ntwork ach for th ursur and th vadr and a critic ntwork common for both th ursur and vadr. Th final miss distanc is takn as th valu to b otimizd. Problms with nonlinar dynamics ar in rogrss. This ar is dividd into fiv sctions. In Sction 2, thr is a brif dscrition of th mathmatical modl for a linar ursuit-vasion gam. This is basd on basic two rson gam dscribd in Bryson[1]. Th third sction givs an ovrviw of th alications of Adativ Critic basd nural ntworks for solving both linar and nonlinar otimal control roblms. Th fourth sction xlains th sts involvd in adating th nural ntwork for a finit tim roblm such as this. Sction fiv discusss th rsults obtaind and th furthr work bing don. * Graduat studnt ** Profssor, Associat fllow, AIAA, Contact Author
2 2. LINEAR PURSUIT-EVASION GAME Considr a scalar linar ursuit-vasion gam dscribd in [1]. Th dynamics of Pursur and Evadr ar givn by x =Ax +bu x =Ax +bv (1) whr rrsnts th ursur and rrsnts th vadr. x is th osition of th ursur and x is th osition of th vadr. u and v ar th control availabl to th ursur and th vadr rsctivly. A and b rlat th vlocity of th ursur with its osition and control rsctivly. A and b rlat th vlocity of th vadr with its osition and control. Th final miss by th ursur is takn as th cost to b minimizd. This is givn by th wighd quadratic form Eq (2) ( ) ( ) 2 T x t x t (2) f f M M vadr. whr T M M is th wight alid to trminal distanc btwn ursur and. It is to b notd that th objctiv of th vadr is to maximiz this cost function. In ral lif situations, th ursur is rquird to com as clos to th vadr within a fixd final tim. Th control variabls of both th ursur and th vadr ar limitd by th intgral quadratic constraint (3), t f to t f to 2 u dt E R 2 v dt E R (3) whr E and E ar th maximum control availabl to ursur and vadr rsctivly. control. R is th wight on th ursur control and R is th wight on vadr 2
3 Both th ursur and th vadr would us maximum control availabl to achiv thir dsird objctiv. Howvr, it is assumd, that th control availabl to th ursur is gratr than that availabl to th vadr. This condition is ssntial for any dgr of succss in intrction. Th constraints on th availabl control ar addd to th cost function. Th augmntd cost function with th abov constraints is givn by Eq (4), t f ( ) ( ) T ( ) 1 1 J = x tf x tf + u v dt R R 2 M M 2 (4) As th vadr tris to maximiz th J valu its constraint is subtractd. For th uros of analysis th distanc btwn th ursur and vadr is mad as stat variabl dnotd by z() t as givn in Eq (5). whr ( tf, t) to ( ) () Φ ( ) () () Φ ( ) () () ˆ () ˆ () z t M x t x t xˆ t t, t x t f xˆ t t, t x t f Φ rrsnts th fundamntal matrix of th fundamntal matrix of A. A and ( tf, t) Thrfor th nw stat quation and th cost function can b simlifid to, Φ rrsnts (5) whr, () = ρ() ε() z t t u t v (6) t f ( f ) ( R ) R 1 1 J = z t + u v dt 2 2 (7) to () ( f, ) ρ t = MΦ t t b (8) () ( f, ) ε t = MΦ t t b (9) Th Hamiltonian for th modifid stat quation is givn by 1 T T T H = ( u Ru v Rv) + λ ( ρ() t u ε() t v) (10) 2 whr, λ is th Lagrang multilir. 3
4 Th co-stat quations and th otimality conditions ar as follows ; ( t ) ( ) f z tf λ = 0 λ = (11) u = R ρ λ ; 1 T v= R ε λ (12) 1 T 3. NEURAL NETWORKS FOR CONTROL It is wll known that th dynamic rogramming formulation offrs th most comrhnsiv solution to nonlinar otimal control; howvr, a hug amount of comutational and storag rquirmnts ar ndd to solv th associatd Hamilton- Jacobi-Bllman (HJB) quation [Bryson and Ho, 1975] (also known as th Bllman quation). Wrbos [1992] roosd a mans to gt around this numrical comlxity by using aroximat dynamic rogramming (ADP) formulations. His mthods aroximat th original roblm with a discrt formulation. Th solution to th ADP formulation is obtaind through th two-nural ntwork adativ critic aroach. In on vrsion of th adativ critic aroach calld th dual huristic rogramming (DHP) on ntwork calld th action ntwork rrsnts th maing btwn th stat variabls of a dynamic systm and control and th scond ntwork, calld th critic, oututs th costats with th stat variabls as its inuts. This ADP rocss, through th nonlinar function aroximation caabilitis of nural ntworks, ovrcoms th comutational comlxity that lagud th dynamic rogramming formulation of otimal control roblms. Mor imortant, this solution can b imlmntd on-lin, sinc th control comutation rquirs a fw multilications of th ntwork wights which ar traind off-lin. This tchniqu was alid by Balakrishnan and Biga[1996] to various linar and non-linar roblms including an aircraft control roblm. Not that thr ar various tys of adativ critic dsigns availabl in litratur. An intrstd radr can rfr to [Prokhorov and Wunsch, 1997] for mor dtails on ADP and DHP. Svral authors hav usd nural ntworks to "otimally" solv nonlinar control roblms [4,5,6]. For xaml, Kim and Calis[7] hav roosd a nural ntwork basd control corrction basd on Lyaunov thory. A major diffrnc btwn thir aroach and this study is that th dvlomnt of guidanc law/control is basd on otimal control; hnc, it is stabilizing and at th sam tim minimizing a cost. 4
5 In this study, a cascad of dual nural ntworks is usd as a framwork for th solutions of linar as wll as nonlinar, finit-tim otimal control roblms with a scial alication to th ursur-vadr roblms using diffrntial gam thory. In a tyical adativ critic dsign, th controllr outut dos not dnd th currnt tim but only th currnt stats; in this roblm, by contrast, th controllr outut has to b diffrnt for th sam valus of th stat sinc th tim lft to comlt th task lays a rol in how much control has to b usd. Hnc, a cascad of controllrs is synthsizd by indxing th indndnt variabl. Han and Balakrishnan [1999,2002] furthr alid this mthod to an agil missil control roblm. Th contribution of this work is that w xtnd this ida to solv roblms cast in a diffrntial gams stting whr no nural ntwork aroach has bn usd bfor. In th currnt roblm of intrst, two action ntworks hav bn usd at ach stag. On action ntwork is traind to gnrat th control for th ursur and th othr, for th vadr. Thr is a singl critic ntwork to gnrat th costat at ach stag. Th rlativ distanc btwn th ursur and vadr is th inut for both th action and th critic ntworks. Th roblm is formulatd as a finit tim roblm. Thrfor th training rocds backwards. 4. DEVELOPMENT OF NEURAL NETWORK SOLUTIONS Dvlomnt of th nural ntwork solution is outlind in this sction. Th finit tim is slit into N sts. Not that th solution rrsnts a backward sw rocss. Last Ntwork: 1. For random valus of final miss distanc Z N, calculat λ N from th final costat condition. 2. Us otimality condition H u=0; H v=0, Eq.(12), to solv for aroriat U N-1 and V N From th costat roagation quation, calculat λ N Train thr nural ntworks: Th U N-1 ntwork oututs U N-1, and th V N-1 ntwork oututs V N-1 for diffrnt valus of Z N 1. Th λ N-1 ntwork oututs λ N-1 for diffrnt valus of Z N-1.W hav otimal U N-1, V N-1 and λ N-1 now. 4.2 Othr Ntworks: 5. Assum diffrnt valus of Z N-2 and us random nural ntworks (or initializd with U N-1 ntwork and V N-1 ntwork) calld U N-2,V N-2 ntworks to outut U N-2 and V N-2. Us 5
6 Z N-2, U N-2 and V N-2 to obtain Z N-1. Inut Z N-1 to λ N-1 ntwork to gt λ N-1. Us Z N-2, λ N-1 in H =0; H =0 to solv for U N-2 and V N-2. Us this U N-2 and V N-2 to corrct th ntworks. u v Continu this rocss until both th ntworks convrg. Ths ntworks yild otimal U N-2 and V N Using random Z N-2 into U N-2 ntwork obtains otimal U N-2. Similarly obtain otimal V N-2. Us Z N-2,U N-2 and V N-2 to obtain Z N-1 and inut to λ N-1 ntwork to gnrat λ N-1. Us Z N-2, U N-2, V N-2 and λ N-1 in costat quation to obtain otimal λ N-2. Train λ N-2 ntwork with Z N-2 as inut. W hav λ N-2 ntwork that yilds otimal λ N Rat sts 5 and 6 with K=N-1, N-2,..., 0, until w gt U 0,V 0. A schmatic of th ntwork dvlomnt is rsntd in Fig.1 Figur.1 Schmatic of Adativ-Critic training 6
7 5. RESULTS AND DISCUSSION Prliminary rsults hav bn obtaind for th roblm using th standard rocdurs. Furthr analysis will b don using th adativ critic aroach. Th currnt rsults will hl in validating thos obtaind from adativ critic. Th ursuit-vasion roblm will b studid for diffrnt and mor comlx stratgis of vadr and that of th ursur. Ths rsults would b rsntd at th confrnc. Figur 2. Distanc btwn Pursur and Evadr with Tim 7
8 Figur 3. Control of Pursur and Evadr Vs tim 8
9 Figur 4. Control of Pursur and Evadr Vs Rlativ Distanc 6. REFERENCES [1] A. E. Bryson and Y. Ho, Alid Otimal Control, Hmishr Publishing Co., 1975, [2] P. J. Wrbos(1992), Aroximat Dynamic rogramming for Ral-Tim Control and Nural Modling, in D.A. Whit and D. Sofg(Eds.) Handbook of Intllignt Control, Multiscinc Prss. [3] Prokhorov, D.V., and Wunsch D.C., II, Adativ Critic Dsigns,. IEEE Transactions on Nural Ntworks, Vol. 8, No. 5, (1997), [4] D. Han and S.N.Balakrishnan, Robust Adativ Critic Basd Nural Ntwork for Control-Constraind Agil Missil Control. Amrican Control Confrnc, Jun 1999, San digo, CA [5] D. Han and S.N.Balakrishnan, Adativ Critic Basd Nural Ntwork for Agil Missil Control. AIAA Journal of Guidanc Control and Dynamics, Vol. 25, No.2, Mar
10 [6] D. Han and S.N.Balakrishnan, Robust Adativ Critic Basd Nural Ntwork for Sd-constraind Agil Missil Control. IEEE Transactions on Control Systm Tchnology, Vol. 10, No. 4, July 2002, [7] B.S.Kim and A.J.Calis. Nonlinar Flight Control Using Nural Ntworks. AIAA Journal of Guidanc, Control, and Dynamics, Vol.20,No.1, ,
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