Observability Analysis on Radome Aberration Estimation

Transcription

1 Pepints of the 9th Wold Congess he Intenational Fedeation of Automatic Contol Cape own, South Afica. August 4-9, 4 Obsevability Analysis on adome Abeation Estimation in-guk Seo*. in-jea ahk** Chang-Kyung yoo*** *Division of Aeospace Engineeing, KAIS, Daejeon, Koea, ( mgseo@fdcl.kaist.ac.k). ** Division of Aeospace Engineeing, KAIS, Daejeon, Koea, ( mjtahk@fdcl.kaist.ac.k). *** Depatment of Aeospace Engineeing, Inha Univesity, Incheon, Koea, ( ckyoo@inha.ac.k). Abstact: he obsevability of adome abeation estimation with line-of-sight (LOS) angle-only measuement is studied in this pape. When missile seeke can only measue LOS angle of taget, taget state vaiables ae equied to be estimated with this measuement fo accuate missile guidance. Since LOS angle measuement contains eo, caused by adome attached at missile nose and called adome abeation angle, the state estimation pefomance is degeneated. hus, the adome abeation effects should be estimated and compensated fo accuate taget state vaiable estimation. he adome abeation angle is modeled as the poduct of adome slope and gimbal angle. he obsevability of state vaiables and adome slope estimation is analyzed fo the elative dynamics with stationay taget model in pola coodinate system. Simulations ae conducted to veify the obsevability conditions obtained in this pape.. INODUCION o accomplish high homing missile guidance pefomance, it is essential to figue out taget state vaiables accuately. Active homing missiles obtain taget state infomation fom seeke measuement. Howeve, when the taget is equipped with anti-electonic Counte easues (anti-ec), the seeke measuement is inteupted. Especially fo missile with micowave seeke case, the seeke cannot measue the elative distance of taget fom missile, and only measues the elative diection of the taget, line-of-sight (LOS) angle. hus, taget state vaiables ae equied to be estimated fom LOS angle-only measuement. adome is a stuctue attached at the nose of active homing missile to potect missile seeke and educe aeodynamic dag foce acting on missile. Since missile seeke is placed inside of adome, the micowave eflected by taget should pass adome to each missile seeke. When the shape of adome is hemisphee, the micowave is not efacted by the adome, but the aeodynamic dag foce is the maximum. his means that actual adome shape cannot be pefect hemisphee, so the micowave is efacted by adome. As a esult, LOS angle measuement contains eo, called adome abeation angle, and the taget state estimation becomes inaccuate with this eoneous measuement. hus, it is equied to estimate and compensate the adome effect fo high taget state estimation pefomance. Seveal adome abeation estimation methods have been developed in pevious studies. Zachan (999) suggested a simple method using dithe signal added to missile guidance command. Howeve, due to this additional signal, homing guidance pefomance is able to be degaded. Neual netwok algoithm is intoduced by Lin () fo adome abeation estimation. But, since its leaning pocess is pefomed on the gound and the chaacteistics of adome changes with envionment and flight conditions, it is difficult with this algoithm to accuately estimate adome abeation duing actual flight. Yueh (983, 985), Lin (995), and Song (4) applied ultiple odel Filte (F) o Inteacting ultiple odel (I) Filte to adome abeation estimation poblem. Howeve, those methods equie a lot of models fo accuate estimation. adome slope, which is paamete epesenting the efaction chaacteistics of a cetain adome, is defined as a state vaiable of filte algoithm by Gufil (4), but this algoithm equies lage numbe of state vaiables fo dynamic system modelling. ost of those algoithms, except fo Zachan (999), have the same shotage that they utilize elative ange between missile and taget as a seeke measuement. In this pape, the obsevability of adome abeation estimation with LOS angle-only measuement is analysed. he adome abeation angle is modelled with adome slope, and it is consideed as a state vaiable. he local obsevability analysis method fo nonlinea dynamic system based on Lie deivative and gadient opeato, intoduced by Hwang (97), Batosiewicz (995), and a (), is utilized to deive the obsevable condition of the taget state vaiables and adome abeation estimation system. his pape is composed as follows. he obsevability analysis theoy studied in the pevious liteatues is explained in Section. he system dynamics and measuement ae modelled in Section 3. Section 4 deals with the analysis on the obsevability of this dynamic system. Simulation esults ae pesented in Section 5.he conclusion of this pape is addessed in Section 6. Copyight 4 IFAC 396

2 9th IFAC Wold Congess Cape own, South Afica. August 4-9, 4. OBSEVABLE CONDIION OF NONLINEA DYNAIC SYSE he local obsevability analysis method is explained in this section. An abitay nonlinea dynamic system with single measuement is consideed as below. x f z h, x x u n x is state vaiable vecto and measuement. () z is single wo diffeent points, x x and x x, ae said to be distinguishable when thee exists a contol input u d which makes the following condition to be satisfied. z x x z x x () he dynamic system given in () is locally obsevable at x if thee exists an neighbouhood of x x such that all the points included in this neighbouhood, except fo x x, ae distinguishable fom x x. o figue out the local obsevability condition, the Lie deivative of the given dynamic system, l xu,, is calculated as below. x x h h l xu, ( n) h x Fom the Lie deivative vecto shown in (3), the gadient O x u, is defined as opeato of the dynamic system,, d O x, u d xx (3) l xu, d (4) x A Y I V Fig.. Engagement Kinematics V and ae velocity and flight-path angle of missile, and V is assumed to be constant. he missile acceleation A is assumed to be pependicula to the missile velocity. stands fo the elative distance between taget and missile, and is line-of-sight(los) angle. With the stationay taget model, the engagement kinematics is modelled as follows. V sin V cos A V 3. Seeke easuement odel he elative geomety between missile seeke system with adome effect and taget is shown in Fig.. he efeence axis is defined in -dimensional inetial coodinate system. Y I Appaent aget aget X I (6) If the nonlinea dynamic system in () is locally obsevable at an abitay point x, Ox, u d is full-ank. his condition can be ewitten in tems of the deteminant. Body Axis issile Velocity det O x, u (5) d issile X I 3. SYSE ODELING 3. Engagement Kinematics odel he elative engagement kinematics between missile and taget is defined in -dimensional inetial pola coodinate system as shown in Fig.. Fig.. Seeke Geomety and ae angle-of-attack and pitch angle of missile. is adome abeation angle. Since the micowave eflected by taget is efacted by adome befoe it eaches seeke, appaent taget is detected by the seeke. his means that the LOS angle of the appaent taget is measued, athe than the LOS angle of the eal taget. hus, the LOS angle measuement E is modelled as below. (7) E 396

3 9th IFAC Wold Congess Cape own, South Afica. August 4-9, 4 he adome abeation angle model suggested by Gufil (4) is defined as below. (8) is adome slope. With the assumption that the angle-of-attack of missile is negligible, fom Fig., the following elationship holds. (9) Fom (7), (8), and (9), the LOS angle measuement can be ewitten as E () he adome slope is consideed to be constant. () 4. OBSEVABILIY OF ADOE ABEAION ESIAION 4. Nonlinea System Definition he obsevable conditions of adome abeation and state vaiables estimation with LOS angle-only measuement is handled in this pape. his implies that the adome slope is needed to be defined as a state vaiable of the system. Also, the obsevability analysis method explained in Section equies the measuement model not to include the contol input. he flight path angle of missile is selected as a state vaiable to satisfy this measuement model equiement. hus, fom (6), (), and (), the nonlinea system to analyse is defined as follows. x f z h, x x u () x u A (3) 4. Obsevability Analysis f V sin V cos xu, A (4) V h x (5) o analyse the obsevability of the dynamic system given in (), the Lie deivative of this system is obtained fom (), (4) and (5). l xu, h h h h x x x x (6) 3 dh d h d h h x x x x hx h x (7) 3 dt dt dt hus, the gadient opeato Oxu, is deived fom (6) as O xu, x x x x h h h h hx hx hx hx h x h x hx hx hx hx hx hx (8) Afte some algebaic calculations, the deteminant of O xu, is deived as follows. det O xu, V 3 V sin 6V 6V sin cos 3 V V V sin cos 3 6V 8V 3 cos sin lead angle is defined as (9) () Duing teminal homing guidance stage, the diection of missile velocity vecto is almost aligned towads the taget, and the magnitude of missile manoeuve equied to educe the miss distance is small. hus, it is easonable to assume that is small enough duing the teminal stage, esulting in the following appoximations. sin () cos Fom (9) and (), obsevability condition deteminant is appoximated as 396

4 9th IFAC Wold Congess Cape own, South Afica. August 4-9, 4 det O xu, V V 6V 3V () his appoximated deteminant in () becomes zeo when one o moe of the following conditions ae satisfied. he value of det, as t t t (3) O xu fo each case in (3) is obtained Ox u t Ox u t Ox u det, det, det, t Ox u 6 V 8 4 det, sin 7 (4) hus, fom (5) and (4), the dynamic system defined in () is locally obsevable at an abitay point x when all of the following obsevability conditions ae satisfied. a) b) c) t t As shown fom (5) and (4), t (5) is not a necessay condition fo the local obsevability of the system given in () at an abitay point x. Howeve, since (9) becomes when t and () equals to fo t case, the state estimation pefomance with the dynamic model in () is expected to be low fo non-maneuveing missile, t, in small lead angle case. 5. SIULAION In ode to veify the local obsevability conditions deived in Section 4, state vaiables and adome abeation estimation simulations ae pefomed. he estimatio is designed by applying Extended Kalman Filte (EKF) algoithm to the dynamic system in (). he taget is consideed to be stationay. he initial position and velocity of missile and taget ae given in inetial Catesian coodinate system as X Y m 6 / V V X m sec Y X Y m (6) Fom (6), the actual initial state vecto is obtained as below. x m t (7) he initial guess of state vecto and covaiance matix ae defined to be xˆ m m (8) P ˆ diag.5 deg 5.5 deg.5 (9) he actual adome slope is selected as. (3) he tansfe function fom missile acceleation A to missile pitch angle is expessed with tuning ate time constant,. s s,. sec A s V (3) he missile autopilot system is modelled as st -ode lag system by intoducing time constant as below. A s,.sec (3) A s s cmd A cmd is missile acceleation command. wo diffeent guidance algoithms, Popotional Navigation Guidance (PNG) law and Adaptive Intemittent aneuve (AI), ae utilized to geneate missile guidance command. AI is designed by Lee () in ode to maintain LOS angle ate to be lage than a cetain level duing flight. he design paametes of guidance laws ae defined as below. NPNG 4 N AI k.5 U L (33) N is the navigation atio of each guidance algoithm. he simulation is pefomed as an open-loop test, meaning that the guidance command is geneated by the tue state vaiable values, not the estimated ones. he following esults ae accomplished by utilizing onte-calo method with times of simulation uns fo each guidance law case. 3963

5 9th IFAC Wold Congess Cape own, South Afica. August 4-9, PNG AI aget ajectoy.5 Lead Angle ate 7.5 Y (m) ate (/sec) (a) issile ajectoy X (m) (e) Lead Angle ate Histoy ime (s) Zeo Effot iss.5 Eo in LOS Angle Estimates Zeo Effot iss (m) Eo () ime (s) ime (s) (b) Zeo-Effot-iss Distance (f) LOS Angle Estimation Eo.9 LOS Angle ate 3 Eo in ange Estimates ate (/sec) ime (s) Eo (m) ime (s) (c) LOS Angle ate Histoy (g) elative ange Estimation Eo 3.5 issile Flight Path Angle ate.5 Eo in Flight Path Angle Estimates 3.5 ate (/sec).5 Eo () ime (s) ime (s) (d) Flight Path Angle ate Histoy (h) issile Flight Path Angle Estimation Eo 3964

6 9th IFAC Wold Congess Cape own, South Afica. August 4-9, 4 Eo (/) Eo in adome Slope Estimates ime (s) (i) adome Slope Estimation Eo Fig. 3. State Vaiables and adome Abeation Estimation As shown in Fig. 3-(f) and (h), the estimation eos of LOS angle and missile flight path angle in both guidance law cases convege to zeo as missile appoaches taget. Howeve, it is given in Fig. 3-(g) and (i) that the estimation eos of elative ange and adome slope in PNG law case ae significantly lage than those in AI case. When missile guidance command is geneated fom PNG law, LOS angle ate monotonically conveges to as missile appoaches taget. Also, missile acceleation command is popotional to LOS angle ate in this case, and missile flight path angle ate is popotional to missile acceleation, as shown in (6). hus, missile flight path angle ate also diminishes to as LOS angle ate gets smalle. Lead angle ate is a diffeence between missile flight path angle ate and LOS angle ate, so its magnitude also deceases. hose chaacteistics ae addessed in Fig. 3-(c), (d), and (e). hus, it is shown fom simulation esults in Fig. 3-(c), (d), and (e) that PNG law dives the system to violate the local obsevability conditions defined in (5), esulting in low estimation pefomance as shown in Fig. 3-(g) and (i). On the othe hand, in AI case, the magnitude of LOS angle ate is maintained to be lage than cetain cuve. issile flight path angle ate has non-zeo value when guidance command is geneated, except fo the time when missile does not maneuve. As a esult, lead angle ate does not convege to. hose tends ae also shown in Fig. 3-(c), (d), and (e), and these esults mean that the guidance command calculated fom AI make the system to satisfy the local obsevability conditions in (5). he simulation esults in Fig. 3-(f), (g), (h), and (i), the estimation pefomance in the AI case is shown to be supeio to that of the PNG law case, suppot this agument. Howeve, since the estimation eos in AI case is lage at the ealy stage, the time inteval when missile does not maneuves and flight path angle ate is, a guidance algoithm to enhance the obsevability duing this peiod is equied to be developed. 6. CONCLUSION he obsevability analysis of adome abeation and taget state estimation with LOS angle-only measuement is handled in this pape. he engagement kinematics is defined in -dimensional pola coodinate system and the adome abeation angle is modelled with the paamete called adome slope. he local obsevability conditions ae deived by applying the obsevability analysis method based on the ful-ank condition of gadient opeato deived fom Lie deivatives. Simulation esults imply that the obtained obsevability conditions on state vaiables ae elated to the adome abeation and taget state estimation pefomance, and it is shown that time histoies of state vaiables ae consideably affected by guidance laws. hus, it is necessay to intoduce a guidance law which can guaantee the obsevability of the adome slope estimation. EFEENCES Zachan, P., and Gatt, H. (999). Adaptive adome Compensation using Dithe. Jounal of Guidance, Contol, and Dynamics, (), Lin, C. L. (). Stability Analysis of adome Eo and Calibation Using Neual Netwoks. IEEE ansactions on Aeospace and Electonic Systems, 37(4), Yueh, W.. (983). Adaptive Estimation Scheme fo adome Eo Calibation. In Decision and Contol, 983. he nd IEEE Confeence on,, Yueh, W.., and Lin, C. F. (985). Guidance Pefomance Analysis with In-Flight adome Eo Calibation. Jounal of Guidance, Contol, and Dynamics, 8(5), Lin, J.., and Chau, Y. F. (995). adome Slope Compensation using ultiple-odel Kal-man Filtes. Jounal of Guidance, Contol, and Dynamics, 8(3), Song,. L., Lee, D. G., and Shin, S. J. (5). Active homing pefomance enhancement with multiple model adome slope estimation. Poceedings of the Institution of echanical Enginees, Pat G: Jounal of Aeospace Engineeing, 9(3), 7-4. Gufil, P. and Kasdin, N. J. (4). Impoving missile guidance pefomance by in-flight two-step nonlinea estimation of adome abeation. IEEE ansactions on Contol System echnology, (4), Hwang,. and Seinfeld, J. H. (97). Obsevability of nonlinea systems. Jounal of Optimization heoy and Applications, (), Batosiewicz, Z. (995). Local obsevability of nonlinea systems. Systems & Contol Lettes, 5(4), a, Q. (). Stuctual Conditions on Obsevability of Nonlinea Systems. Intenational Jounal of Infomation echnology and Compute Science, 3(4), 6-. Lee, H. I., ahk,. J., and Sun, B. C. (). Pactical dualcontol guidance using adaptive intemittent maneuve stategy. Jounal of Guidance, Contol, and Dynamics, 4(5),