State Estimation in TPN and PPN Guidance Laws by Using Unscented and Extended Kalman Filters

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1 Stte Estmton n PN nd PPN Gudnce Lws by Usng Unscented nd Extended Klmn Flters S.H. oospour*, S. oospour**, mostf.sdollh*** Fculty of Electrcl nd Computer Engneerng, Unversty of brz, brz, Irn, *s.h.moospour@gml.com ** s.moospour@scu.c.r ***mostf.sdollh@gml.com Abstrct: In ths pper two mportnt strteges of proportonl nvgton gudnce lw,.e. PN nd PPN, consderng hghly nonlner model of mssle nd trget enggement re smulted for tctcl homng mssle (r to r). It s supposed tht trget hs n unnown constnt ccelerton wth rndom strtng tme. Wth consderng process nose nd mesurement nose for the system, both gudnce lws PN nd PPN re smulted by pplyng Unscented Klmn Flter (). We consder flght control system dynmcs n the gudnce system nd model them s sngle-lg nd then both gudnce lws re smulted by pplyng. At the end, obtned results from pplyng re compred to the well nown Extended Klmn Flter (). Our Smulton hs shown tht, n comprson wth, hs much better performnce n stte estmton nd reducng the effect of nose on the mssle commnd ccelerton. Keywords: gudnce, PN, PPN, estmton, unscented lmn flter.. Introducton the proportonl nvgton gudnce hs been wdely recognzed s n effcent gudnce scheme for homng mssles. In ths scheme, the ccelerton commnd gven to the mssle s proportonl to the ngulr rte of the Lne-Of-Sght (LOS) between the mssle nd the trget. he proportonl gudnce lw s usully clssfed nto two types: rue Proportonl Nvgton Gudnce (PNG) nd Pure Proportonl Nvgton Gudnce (PPNG). In [] PNG nd PPNG re compred extensvely from the respect of mplementton nd pursut behvor. he equtons of mssle nd trget enggement model re hghly nonlner. Over the lst 2-3 yers, the extended Klmn flter () hs become the lgorthm of choce n numerous nonlner estmtons. hs ncludes estmtng the stte of nonlner dynmc system. he pples the stndrd lner Klmn flter methodology to lnerzton of the true nonlner system. hs pproch s sub-optml, nd cn esly led to dvergence. Juler et l. [2] proposed the Unscented Klmn Flter () s dervtve-free lterntve to the extended Klmn flter n the frmewor of stte-estmton. he mn dfference to the s tht the pproxmtes the Gussn probblty dstrbuton by set of smple ponts wheres the lnerses the nonlner model equtons. hs leds to results whch re usully both more ccurte, becuse the orgnl equtons re used, nd less costly, becuse no Jcobn mtrces need to be clculted [2]. In [2] ws presented nd n [3] ws used for stte estmton n nonlner systems. In [4] the gudnce lgorthm of n nt-mssle-mssle (AA) system, whch ws bsed on the proportonl nvgton nd combned wth n ws studed. [5], [6] hs used other nds of extended lmn flter n PNG. In [7] stte estmton n the two-dmensonl ntercept problem, bsed on the ssumpton tht certn trgets execute evsve mneuvers orthogonl to ther velocty vectors, were consdered. he objectve ws to estmte the enggement sttes n the presence of unnown trget ccelerton nd gude the nterceptor to ht the trget bsed on these stte estmtes. In ths pper we study for stte estmton n PNG nd PPNG. We compre nd performnces on reducng the effect of nose n sttes estmton nd ther effects on the control sgnl n mssle gudnce system. he present rtcle s orgnzed s follows: In Secton 2, we expln the equtons of the mssle nd the trget enggement model. We lso expln the two gudnce lws,.e. PNG nd PPNG. In Secton 3, we present n overvew of the Unscented rnsform (U) used to compute the frst two moments of rndom vrble undergong rndom vrble trnsformton. We then derve the, usng the U to del wth the nonlnertes. he smulton results re n secton 4. Fnlly, the results of the study re brefly summrzed n secton wo Clsses of PN Lws

2 2. Equtons Of ssle-rget Enggement In Polr Coordnte We consder mssle-trget enggement n polr coordnte system nd ssume them to be mss pont. he trget moves wth the velocty V nd mneuvers wth constnt norml ccelerton A. he mssle pursues the trget wth the velocty V nd the norml ccelerton A. he lne connectng the mssle nd the trget s nown s the lne of sght. he lne of sght mes n ngle of wth respect to the fxed reference, nd the length of the lne of sght (nstntneous seprton between mssle nd trget) s rnge denoted R. he closng velocty V C s defned s the negtve rte of chnge of the dstnce from the mssle to the trget. he mssle nd the trget velocty vectors me ngles of nd respectvely, wth respect to the fxed reference. Snce the trget ccelerton A s perpendculr to the trget velocty vector, the ngulr velocty of the trget cn be expressed s A / V. he dfferentl equtons of moton re obtned by resolvng the mssle nd the trget veloctes long nd norml to the LOS nd performng certn lgebrc opertons [9], s R V Cos( ) V Cos( ) V C V Sn( ) V Sn( ) R () Where s the tme dervtve of the lne of sght ngle or the LOS rte. 2.2 rue nd pure proportonl nvgton gudnce lws heoretclly, the true proportonl nvgton (PN) gudnce lw ssues ccelerton commnds, perpendculr to the LOS, whch re proportonl to the LOS rte nd closng velocty. he PN gudnce lw cn be stted s A NR Where N s untless desgner chosen gn nown s the effectve nvgton rto. he mssle ccelerton component long the LOS drecton s V A. Sn( ). he ngulr velocty of the mssle s ( A / V ). Cos( ). In PPN gudnce lw the ccelerton commnd lw s perpendculr to the mssle velocty vector nd expresses s A N V. he ngulr velocty of the mssle s obtned s A / V. In ths study we model the flght control system dynmcs s sngle lg,.e. N A L (2) S AU Where N L s the cheved mssle ccelerton nd AU s the flght control system constnt. he bove equton cn be wrtten s N ( A N ) / (3) L L AU We consder the nput nose nd mesurement nose for the system. We ssume the LOS rte mesurements re corrupted wth nose (whte Gussn nose). We lso ssume tht trget mneuvers wth constnt mgntude nd unformly dstrbuted strtng tme. It cn be shown mthemtclly tht trget ccelerton s equl to whte Gussn nose whch s pssed. 3. he Unscented Klmn Flter Unscented Klmn flter ws frst proposed by Juler et l to ddress nonlner stte estmton n the context of control theory. he lgorthm uses set of crefully chosen smple ponts to cpture men nd covrnce of the system. he smples re propgted through true nonlner equtons, the lnerzton of whch s unnecessry t ll. hey cn cpture the sttes of the stte up to d order, nd hs the sme order of computton complexty wth Extended Klmn flter. It s superor to both n theory nd n mny prctcl stutons. he Unscented rnsform (U) forms the core of the lgorthm. he U s method to effcently compute the frst two moments of rndom vrble undergong n rbtrry non-lner trnsformton. It s bsed on the de tht t s eser to pproxmte Gussn dstrbuton thn t s to pproxmte n rbtrry non-lner functon. 3. he Unscented rnsformton Suppose tht x R d s rndom vrble wth men x nd covrnce P xx, nd tht y f ( x ), wth f n rbtrry non-lner functon. he problem of developng consstent, effcent nd unbsed trnsformton procedure cn be exmned by consderng the ylor seres expnson of equton y f ( x ) bout x. hs seres cn be expressed s f ( x) f ( x x) f ( x) f x f x f x... 3! 2 2 Where x s zero men Gussn vrble wth n n covrnce P xx, nd f x s the pproprte nth order term n the multdmensonl ylor Seres. ng expecttons, t cn be shown tht the trnsformed men nd covrnce re (4) y f ( x) fpxx fe x (5) Pyy fpxx ( f ) f ( E x E x P xx 2 4! E Pxx x Pxx ) f... If the moments nd dervtves cn be evluted correctly up to the nth order, the men s correct up to the nth order s well. Smlr comments hold for the (6)

3 covrnce equton s well, lthough the structure of ech term s more complcted. Extended Klmn Flter ssumes tht the second nd hgher order terms of x n Equton bove cn be neglected. Under ths ssumpton, y f ( x) P fp ( f ) yy xx he U computes set of weghted sgmponts { w, } n tht exctly cpture the men nd covrnce of x. hese sgm-ponts re chosen s follows: x W n x ( ( n ) Pxx ), W 2( n ) n x ( ( n ) Pxx ) W n 2( n ) Where ( ( n ) Pxx ) denotes the -th row or column of the mtrx squre root of ( n ) P xx, nd R. W s the weght ssocted wth the -th pont. he prmeter provdes n extr degree of freedom tht llows the fne tunng of hgher order moments of the pproxmton. hs choce for the sgm-ponts ensures tht the men nd covrnce of both x nd y re computed ccurtely up to the second order. he chosen sgm-ponts exctly cpture the men nd covrnce of x, for exmple we hve for covrnce: P W ( x)( x) W ( n )( P )( P ) P xx xx xx he men nd covrnce of y cn now be computed by trnsformng ech of the sgm-ponts ccordng to Y f ( ) nd settng: y W Y P W ( Y y)( Y y) yy (7) (8) (9) () 3.2 Unscented Flterng In ths secton the s revewed. he flter presented n [8] s derved for dscrete-tme nonlner equtons, where the system model s gven by x f ( x, u, v, ) y h( x, u, w, ) () Where x s the n stte vector nd y s the m mesurement vector. Note tht contnuous-tme model cn lwys be expressed n the form of the Eq. () through n pproprte numercl ntegrton scheme. We ssume tht the process nose w nd mesurement nose v re zero-men Gussn nose processes wth covrnces gven by Q nd R, respectvely. he equtons re gven n Algorthm:. We crete the ugmented stte vector s follows: x x v, E w (2) 2. he set of sgm ponts re creted by pplyng Equton (8) to the ugmented stte vector: x x ( n q r ) P x ( n q r ) P (3) 3. he trnsformed set s gven by nstnttng ech pont through the process model, f (, u, ) (4) x 4. he predcted men s computed s x W, (5) 5. And the predcted covrnce s computed s (6),,,, P W x x 6. Instntte ech of the predcton ponts through the observton model, Y h( x, u, w, ) (7) Y, 7. he predcted observton s clculted by WY (8), 8. Snce the observton nose s ddtve nd ndependent, the nnovton covrnce s yy,,, (9) P R W Y Y Y Y 9. Fnlly the cross correlton mtrx s determned by,, xy,, (2) P W x Y Y 4. SIULAION results We smulte both PN nd PPN gudnce lws wth nd wthout the presence of nose (process nose nd mesurement nose) for both systems wth nd wthout consderton of flght system dynmc. It s supposed the

4 A (Ft/s 2 ) DLANDA (rd/s) A (Ft/s 2 ) A (Ft/s 2 ) A (Ft/s 2 ) dstnce between mssle nd trget s 4 ft. he ntl vlues of flght ngle of trget nd LOS ngle re 8 nd zero degree, respectvely, the nvgton constnt rto s 4, tme constnt of flght system dynmc s.5 second nd trget ccelerton s 3g. We lso suppose tht ntl veloctes of mssle nd trget re ft V ; 3 ft s V s. Fgures -2 show the smulton results. We drw trget ccelerton, lne of sght rte nd mssle ccelerton commnd for both gudnce lws nd both cses wth nd wthout flght system dynmc. It s cler when we use for both gudnce lws nd both cses, the mssle ccelerton hs better performnce nd LOS rte s estmted better thn. But there s no dfference between estmted trget ccelerton n both flters nd tht s becuse of tht n the stte equtons of system, the ccelerton trget stte vrble hs no nonlner components nd we now the dvntge of over the s n nonlner systems. 5. CONCLUSIONS he consstently performs better thn the well nown, wth the dded beneft of ese of mplementton n tht no nlytcl dervtves (Jcobns or Hessns) need to be clculted. For stteestmton, the nd hve equl complexty. In ths pper, we use n the mssle gudnce hghly nonlner system for dfferent cses. Our smulton hs shown tht the hs better performnce n stte estmton nd reducng the effect of nose on the mssle commnd ccelerton. 2 5 Actul Actul DLANDA -.2 fg. 3: Lne of sght rte for PPNG wthout consderton of tme dely tme (sec) Fg. 4. ssle ccelerton commnd for PNG wthout consderton of tme dely ACUAL ACUAL tme (sec) Fg. : ssle ccelerton commnd for PPNG wthout consderton of tme dely Actul -2 tme (sec) fg. 2: rget ccelerton component for PPNG wthout consderton of tme dely -2 tme (sec) Fg. 5: rget ccelerton component for PNG wthout consderton of tme dely ACUAL -.2 tme (sec) Fg. 6: Lne of sght rte for PNG wthout consderton of tme dely

5 DLANDA (rd/s) A (Ft/s 2 ) A (Ft/s 2 ) 2 5 ACUAL -8-9 ACUAL tme (sec) Fg. 7: ssle ccelerton commnd for PPNG wth consderton of tme dely A ACUAL Fg. 8: rget ccelerton component for PPNG wth consderton of tme dely ACUAL DLANDA -.2 Fg. 9: Lne of sght rte for PPNG wth consderton of tme dely ACUAL A tme (sec) Fg. : rget ccelerton component for PNG wth consderton of tme dely ACUAL -.2 tme (sec) Fg. 2: Lne of sght rte for PNG wth consderton of tme dely References [] U. S. Shul nd P. R. hptr, he proportonl nvgton dlemm-pure or true?, IEEE rnsctons on Aerospce nd Electronc Systems, pp , 99. [2] S. J. Juler nd J. K. Uhlmnn, A New Extenson of the Klmn Flter to Nonlner Systems, n Proc. 997 AeroSense: he th Int. Symp. on Aerospce/Defence Sensng, Smulton nd Controls. [3] E. A. Wn, R. vn der erwe, he unscented lmn flter for nonlner estmton, n Proc of Symposum 2 on Adptve Systems for Sgnl Processng, Comuncton, nd Control. pp [4] F. Imdo, A study on ssle gudnce system gnst rndomely mneuverng r-to-surfce mssle, n proc of the 24 IEEE, Interntonl Conference on Control Aplctons, pp [5] G. E. Hssoun, C. C. lm, Advnced gudnce control system desgn for homng mssles wth berngs-only mesurements, 994 IEEE Interntonl Conference on Industrl echnology, Gungzhou, Chn. [6] G. E. Hssoun, C. C. lm, performnce of the extended lmn flter under proportonl nvgton, echncl Report CRL 93-, he unversty of Adelde, Dpt. of Electrcl nd Electronc Engneerng, rch [7] R. H. Chen, J. L. Speyer, D. Lnos, Homng ssle Gudnce nd Estmton Under Agle rget Accelerton, Journl Of Gudnce Control And Dynmcs, vol. 3, pp , 27. [8] P. Zrchn, ctcl nd Strtegc ssle Gudnce, AIAA,fourth Ed., 22 [9] A. Green, J. Shnr, nd. Guelmn, Gme optml Gudnce Lw Synthess for Short Rnge ssles, Journl Of Gudnce, Control nd Dynmcs. vol. 5, no., pp. 9-97, 992. Fg. : ssle ccelerton commnd for PNG wth consderton of tme dely