Flight Midcourse Guidance Control Based On Genetic Algorithm

Transcription

1 Flght Mdcourse Gudance Control Based On Genetc Algorthm Zhao-hua Yang Bejng Unversty of Aeronautcs and Astronautcs The school of Instrumentaton Scence & Optoelectroncs Engneerng, BUAA Jan-cheng Fang Bejng Unversty of Aeronautcs and Astronautcs The school of Instrumentaton Scence & Optoelectroncs Engneerng, BUAA Zhen-qang Q Bejng Aerospace Automatc Control Insttute qzq@ht.edu.cn ABSTRACT An advanced flght mdcourse gudance law based on genetc algorthm (GA) s proposed. The proposed mdcourse gudance formulaton mnmzes the flght tme and maxmzes the termnal energy subject to a termnal ntercept condton. GA s used to search the optmal attack angle for the flght trajectory. By combnng GA and sngular perturbaton technque (SPT), the optmal flght gudance law s obtaned consequently. SPT s appled to approxmate the termnal flght tme. Meanwhle, the paper completely elmnates the need for solvng two-pont boundary-value problems (TBPVP), whch s too complex for dervaton and mplementaton. The smulaton results show that the resultng gudance law s near-optmal and the proposed method s vald. Especally, the GA gudance law can apply to ntercept the maneuver s successfully. Categores & Subject Descrptors: G.1.6 Optmzaton; I..8 Problem Solvng, Control Methods and Search; J. Physcal Scence and Engneerng. General Terms: Desgn, Algorthms Keywords: Genetc algorthm; Flght mdcourse gudance control; Sngular perturbaton technque 1. INTRODUCTION Flght gudance control problems n general and ntercepton of arborne s n partcular, have been an mportant area of research on technologes supportng ant-ballstc mssle defense n the last decade. Flght gudance control schemes of early generatons appled proportonal navgaton gudance (PNG) or ts modfcatons. Later, modern gudance laws were developed Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. GECCO 5, June 5 9, 5, Washngton, DC, USA. Copyrght 5 ACM /5/6 $5.. based on a lnearzed knematcal model and a lnear quadratc optmal formulaton (Zarchan,199, Cottrel, 1971). Recently, varous solutons such as feedback lnearzaton, 1 sldng model control, dfferental games, 3 predctve control 4 and neural networks 5 etc. are ntroduced nto ths feld. However, a straghtforward soluton for the nonlnear optmal gudance law usng the conventonal methods may lead to a two-pont boundary-value problem (TPBVP) whch nvolves teratons of forward ntegraton of the state equaton and backward ntegraton of the adjont equaton untl the soluton converges. Not only does the TPBVP nvolve substantal computaton, but ts convergence propertes can be poor; furthermore, t s very senstve to the ntal condtons and nose dsturbances. The great potental of genetc algorthm (GA) has long been recognzed n combnaton optmzaton and nonlnear control. GA uses search procedures based on the mechansm of natural genetcs, whch combnes a Darwnan survval-of-the-ftness strategy to elmnate unft characterstcs and uses random nformaton exchange. In addton, the GA optmzaton course nvolves lttle exteror nformaton, and the ftness functon s taken as ts only optmzaton crtera. The sngular perturbaton technque (SPT) has been shown to produce good approxmatons to the optmal control for a varety of problems. The chef advantage of ths approach s that t nvolves only the solutons to a set of coupled nonlnear equatons and the nonteratve ntegraton of quadratures. Ths approach can be appled to the problem of generatng near-optmal solutons. To the best knowledge of the authors, combnng GA wth SPT to applyng to flght gudance control has not yet been performed. The man work ncluded n ths paper begns wth the nonlnear model of the flght gudance control, so the errors caused by lnearzaton are avoded. In ths paper, the flght trajectory of s frstly dvded nto N elements to transform the optmal gudance problem to N flght trajectory optmzaton problems, and a lnear combnaton of flght tme and termnal specfc energy s taken as the performance ndex of the optmzaton problems. Then, an advanced mdcourse gudance law based on genetc algorthm (GA) s developed. The proposed mdcourse gudance formulaton mnmzes flght tme and maxmzes the resdual energy subject to a termnal ntercept condton. GA s used to search the optmal attack angle for the flght trajectory and the SPT s appled to approxmate the termnal flght tme of 151

2 the mdcourse phase. Through these methods, the paper completely elmnates the need to solve two-pont boundaryvalue problems whch s too complex for dervaton and mplementaton. The smulaton results show that the resultng gudance law s near-optmal and the proposed method s vald and near-optmal. Ths paper s organzed as follows. Secton gves the detaled representaton of the mathematcal model of the BVR mssle ncludng the force analyss and moton analyss, secton 3 develops the optmal gudance law based on the sngular perturbaton technque and genetc algorthm, secton 4 verfes the performance of the proposed optmal gudance law through smulaton on a BVR mssle, and the fnal secton presents conclusons and future work to be done.. MATHEMATICAL MODEL The mdcourse phase of the flght s defned as the perod followng launch untl seeker lock-on s realzed. The man purpose of the mdcourse gudance s to navgate a mssle so that s may operate n optmal condtons n regard to mssle performance and relatve geometry aganst a when seeker lock-on s acheved. Fg. 1 shows the force analyss of a beyond-vson -range (BVR) mssle and the symbols used n ths paper. D Y L Z mg T v X Fg. 1 The force analyss of the mssle For smplcty, the motons of the BVR mssles are constraned wthn X-Z plane. As for the 3-D moton of the BVR mssle, we can also study them n the -D space through proper coordnate transformatons. 6 Durng the mdcourse flght, the BVR mssle s modeled as a pont mass. The state varables are the velocty of the BVR mssles v, the flght path angle, and the poston ( x, h) of the BVR mssle n the nertal space, where, h denotes the flght alttude. Then the state equatons of the BVR mssle are expressed as follows. ( LT sn ) gcos (1) mv v ( T cos D) v gsn () m x v cos (3) h vsn (4) L 1/vsC C L L CL ( ) (5) D 1/vsC C D D CD KCL The aerodynamc dervatves CL, CD and k are gven as functons of the Mach number M, whch are gven n Table1. Table 1 Fundamental data for BVR mssle 7 M CD k CL The mass m and the thrust T of the BVR mssle are gven as functons of tme t, whch are shown n Fg.. (kg) 18 (N) 33 thrust tme/s Fg. Tme hstory of mssle mass and thrust In our study, the attack angle of the BVR mssles s treated as control varable, whch should satsfy the nequalty constrant: (6) mn max Durng the mdcourse gudance, tactcal consderaton dctates short flght tme as the basc performance ndex. In addton, the energy of the BVR mssles must be conserved durng the mdcourse phase so that suffcent energy s avalable for termnal engagement of ntellgent s. So, the optmal performance ndex s represent as t f J E( t ) (1 ) dt (7) f mass where, t f and Et ( f ) denote the fnal tme and the termnal energy of the mdcourse phase respectvely. The weghtng factor enables the tradeoff between flght tme and termnal energy. Ths factor s constraned as tme/s 1 (8) 3. OPTIMAL GUIDANCE LAW BASED ON GENETIC ALGORITHM 3.1 Representaton of Problem In ths paper, after the trajectory of the s dvded nto N elements, the flght gudance control problem s changed nto N 15

3 nonlnear trajectory optmzaton problems. As for the th problem, we can take use of GA to search the optmal trajectory, and the sngular perturbaton technque s appled to approxmate the fnal tme t ( 1,, N) of each problem. f The flght trajectory of the s guaranteed by the ground support system through flterng and dentfcaton and transmtted to the mssle durng the mdcourse gudance. A gven trajectory s dvded nto N elements whch are represented as EE 1 ( 1,,, N) (as shown n Fg. 3). As for the poston E, we can compute the orentaton angle of the and the slant range R between the and the mssle through the Eqs.9 and 1. ht hm arctan( ) (9) x x t m t m t m R ( h h ) ( x x ) (1) Accordng to and R, we can get the approxmate expresson of the fnal tme from the mssle poston P to the poston E, where t s the course angle of the, v t s the velocty, the subscrpts m and t denote mssle and respectvely. The fnal tme t f s derved from zero-order sngular perturbaton technque. 8 We assume the fles at constant speed n the nterval from E to E 1, ( 1,,..., N). So, there must be a shortest course for ntercepton of the, and the fnal tme s the tme used by the nterceptor. The dervaton process of t f s as follows. The symbols employed n ths secton are shown n Fg.3 P R E v t v m By projectng R and v t to PE 1, we can obtan the expresson of the fnal flght tme, R cos( ) t f (11) vm vt cos( t ) In order to nsure the valdty of the homng gudance, we must conserve suffcent energy for termnal engagement of ntellgent s. In other words, the energy consumed n the mdcourse t E 1 Fg. 3 Geometry of estmatng fnal tme t f phase should be mnmal. We represent the consumed energy by the terms of the power of mssle E, ( T cos( ) D) vm E (1) mg where, D s the drag force on the mssle, whch s represented n Eq D vm scl (13) Then, we can transfer the performance ndex functon n Eq. 7 to the followng expresson, mn J t f J E (1 ) dt (14) where, the factor s constraned by Eq GA Optmzaton 3..1 Codng scheme of chromosome Bnary codng scheme s usually adopted n normal GA for ts smplcty and good feasblty. But the defect of Hammng clff lmts ts applcaton. In order to acheve satsfactory nonlnear optmzaton effect, we adopt length-varable codng scheme Constructon of ftness functon In GA, the ftness functon s the only motvaton for the nonlnear optmzaton. The value of the ftness functon should not be negatve. Accordng to the performance ndex functon J n Eq. 14, we construct the ftness functon as shown n Eq. 17 f () x C E dt E dt C, tf tf max (1 ), (1 ) max F M T F where, denotes the th trajectory optmzaton problem, C max s t f the maxmum of E (1 ) dt. So the mnmum of performance J s equvalent to the maxmum of the ftness functon f ( x ) Genetc operatons In ths paper, the attack angle of the mssle s treated as the control varable, whch s constraned by Eq. 6. Usng GA, the ntal populaton of the attack angle s prepared at random, whch contans m ndvduals. The generaton s updated n the followng process. 1) Generaton of parents: m parent ndvduals are ntally selected at random. ) Calculaton of ftness: As for each ndvdual of v m, the varable and are ganed through Eqs.1 and. Then accordng 15) 153

4 to Eq.16, the ftness functon of each ndvdual n the populaton s calculated. 3) Selecton: Accordng to ftness of each ndvdual, the fnest chromosomes are selected from the roulette selecton whch determnes a fne chromosome at random accordng to the specfed probablty based on the rankng. 4) Crossover and mutaton operaton: n order to nsure the dversty of the populaton, the crossover and mutaton operaton are performed accordng to the specfed probablty. 5) Through step 3) and 4), the new generaton s determned consequently. The ftness of each ndvdual n the new generaton s calculated. If the fnest soluton does not converge, return to the step 3). 6) If the fnest soluton converges or the maxmum tmes of teraton arrves, we beleve the optmal attack angle s ganed, the optmal v m and are ganed, too. Meanwhle, the poston of the mssle at next tme nstant E 1 ( xm, 1, hm, 1) s calculated through Eqs.3 and 4. To enhance ntercept performance aganst maneuverng s, we repeat the prevous optmzaton process on each element of the trajectory, usng the latest avalable nformaton about the and mssles states. Fnally, we can gan the optmal gudance law for the mdcourse phase. In ths paper, the varables between nodal values are nterpolated usng a thrd order splne functon, and the 4th order Runge-Kutta algorthm s used as the ntegraton scheme. The flow chart of the whole optmzaton process s shown n Fg SIMULATION RESULTS The proposed optmal gudance law based on GA s tested usng a BVR mssle smulaton. The mass m and the thrust T of the BVR mssle are shown n Fg.. About the flght trajectory of the, we consder two types of condtons. One s that the fles at constant speed followng the horzontal trajectory, whch s descrbed by (a) and (b). Another s that the possesses the horzontal or vertcal maneuverng ablty, whch s descrbed by (c) and (d). The latter can avod the attack from frepower. The flght trajectores used n ths paper are as the followng. (a) The ntrudes at speed of m/s from the dstance of 4km at the heght of 8km. (b) The escapes at speed of m/s from the dstance of km at the heght of 8km. (c) The ntrudes at speed of m/s from the dstance of 4km at the heght of 8km. When the dstance between the and mssle decreases to km, the begns to maneuver wth the vertcal overloadng of 1.5g. (d) The ntrudes at speed of m/s from the dstance of 4km at the heght of 8km. Meanwhle, the maneuvers wth the horzontal overloadng of 1.5g. We apply the proposed gudance law to ntercept or pursut the above-mentoned s respectvely. In GA, the populaton sze m s, the probablty of crossover, P c, s set to.45, and the probablty of mutaton, P m, s set to.6. =+1 Dvde the trajectory nto N elements = 1 Parameters ntalzaton for mssle and ( ncludng v,, x, h ) Calculate and R, Parameters ntalzaton for GA ( ncludng C max, *, m ) Generaton of the ntal populaton Codng Decodng Calculate v, Estmate t f, Y Calculate x +1, h +1 N > N $ Y Determnate the optmal gudance law and the optmal trajectory End E Calculate ftness functon Check stop condton? Selecton, crossover and mutaton Fg. 4 The flowchart of the proposed scheme The optmal flght trajectores of the mssle are shown n Fg. 5, where the fgures (a-d) descrbe the above- mentoned four knds of s. The mss dstance and fnal tme of the mssle correspondng to the four knds of s are gven n table. As shown n Fg. 5 and table, the mss dstance meets the requrement of performance ndex, and the s ntercepted by the mssle successfully, whch s guded by GA gudance law. N 154

5 The valdty of the proposed scheme s confrmed. Moreover, the proposed GA gudance law can ntercept or pursut the maneuver, whch s superor to the conventonal proportonal gudance law. (a) h /m mssle x /m x Table Smulaton results on dfferent s Index Target 1 Target Target 3 Target 4 Mss Dstance /m Fnal tme /s The energy consumpton and the fnal tme varaton of the ntrudng wth constant speed are showed n Fg. 6. It s found that the mssle can capture the n shorter tme and fewer energy consumpton. That s to say, GA gudance law can ntercept the s quckly and save more energy for the homng gudance. So the proposed gudance law s a knd of near optmal control law. Above all, the GA gudance law can avod solvng the two-pont boundary-value problem. 9 x 17 8 (b) h /m mssle x /m x energy consumng / J tme / s 1 1 (c) h / m mssle fnal tme / s tme / s (d) h /m x / m x mssle x /m x 1 4 Fg. 5 The optmal flght trajectory of the mssle and the Fg. 6 The energy consumpton and the fnal tme of the ntrudng wth constant speed 5. CONCLUSION In ths paper, we transform the mdcourse gudance problem to N nonlnear trajectory optmzaton problems and develop a mdcourse gudance law based on genetc algorthm. Smulaton results show that the proposed optmal gudance law s vald and near optmal. Above all, t s not necessary to solve the two-pont boundary-value problem whch s rather complex to solve. As for the onlne mplementaton, we wll adopt the table lookup method based on the results of ths paper. Another method wll be developed to mprove the convergence speed of GA by betterng the generaton method of ntal populaton accordng to mmune prncple. 1 The above-mentoned two suggestons are our future work. 155

6 6. REFERENCES [1] Sanguk Lee, J. E. Cochran Jr., Orbtal maneuvers va feedback lnearzaton and bang-bang control, Journal of Gudance, Control and Dynamcs, Vol., No. 1, pp , [] Brerley, S.D., Longchamp, R., Applcaton of sldng-mode control to ar-ar ntercepton problem, IEEE transactons on aerospace and electronc systems, v 6, n, pp , 199. [3] Vladmr Turetskey, Josef Shnar, Mssle gudance laws based on pursut-evason game formulatons, Automatca, Vol. 39, pp , 3. [4] Wen-Hua Chen, Donald J. Balance, Peter J Gawthrop, Optmal control of nonlnear systems: a predctve control approach, Automatca, Vol. 39, pp , 3. [5] Ru Zhou, Dfferental game controllers desgn usng neural networks, Control and Decson, Vol. 18, No. 1, pp , 3. [6] Nguyen X. Vnh, Peree T. Kabamaba, Tetsuya Takehtra, Acta Astronautca, Vol. 48, No. 1, pp. 1-19, 1. [7] Eun-Jung Song, Mn-Jea Tahk, Real-tme mdcourse gudance wth ntercept pont predcton, Control Engneerng Practce, Vol. 8, pp , [8] Chang-Me Xao, Optmal fuzzy gudance law for nterceptng maneuverng evader, PhD. of Harbn Insttute of Technology, [9] Goldberg D E, Dev K, Kob B, Don t worry be messy, Proc. of ICGA 91, pp [1] Lcheng Jao, Le Wang. A Novel Genetc Algorthm Based on Immunty. IEEE Transactons on Systems, Man, And Cybernetcs Part A: Systems and Humans, vol. 3, No. 5, pp