Adaptive Extended Kalman Filter for Ballistic Missile Tracking

Transcription

1 World Academy of Science, Engineering and echnology Vol:, No:4, 7 Adaptive Etended Kalman Filter for Ballitic Miile racing Gaurav Kumar, Dharmbir Praad, Rudra Pratap Singh International Science Inde, Electrical and Computer Engineering Vol:, No:4, 7 waet.org/publication/748 Abtract In the current wor, adaptive etended Kalman filter (AEKF) i preented for olution of ground radar baed ballitic miile (BM) tracing problem in re-entry phae with unnown ballitic coefficient. he etimation of trajectory of any BM in reentry phae i etremely difficult, becaue of highly non-linear motion of BM. he etimation accuracy of AEKF ha been teted for a typical tet target tracing problem adopted from literature. Further, the approach of AEKF i compared with etended Kalman filter (EKF). he imulation reult indicate the uperiority of the AEKF in olving joint parameter and tate etimation problem. Keyword Adaptive, AEKF, ballitic miile, EKF, re-entry phae, target tracing. I. INRODUCION HE BM tracing i quite crucial for ecurity of any nation againt any BM attac from enemy country. In India, a multilayered BM defence ha been developed to guard againt uch attac []. racing of BM can be done with the help of ground baed radar but traced parameter contain certain uncertainty and error. hi error can be minimized by Kalman filter [] which ue a et of equation and conecutive data input (from radar) to etimate the accurate value. However, thi filter i only uitable for linear ytem model and hence, it cannot be ued to trace the motion of BM in re-entry phae which i highly nonlinear and unpredictable becaue of the parameter lie drag, drift, ballitic coefficient and atmopheric condition. In cae of non-linear proce, etended Kalman filter ( EKF) i ued which i non-linear verion of Kalman filter. EKF adopt a technique called aylor erie epanion to linearize a non-linear tate model. EKF etimate the non-linear tate by uing a priori gue of the proce and meaurement noie. A the target enter in the endo-atmopheric reaon, the etimation become difficult ince the phyical parameter lie ballitic coefficient, drag, lift etc. are not nown accurately. Further, it really become difficult to etimate the poition, when the priori etimated value i too far from it real value. In uch ituation the etimate offered by EKF may diverge. A modified non-linear tate etimator named a adaptive etended Kalman filter (AEKF) ha been propoed in [] for GPS integration. he algorithm update the proce and meaurement noie at every meaurement to find it proper value and hence reduce the chance of divergence of the etimation. Gaurav Kumar i with Aanol Engineering College, Wet Bengal, India. India (phone: : gr@gmail.com). Dharmbir Praad and Dr. Rudra Pratap Singh are Aitant Profeor, Electrical Engineering Department, Aanol Engineering College, Wet Bengal, India ( armbirpraad@yahoo.com, rpingh.aec@gmail.com). In thi paper, an attempt ha been made to trac the BM having unnown ballitic coefficient uing EKF and AEKF algorithm. he reult obtained have been compared in term of etimation error and covariance. In view of the dicuion carried out in the above ection, thi paper i organized a follow: Section II how the trajectory a well a the reference ytem of BM and tate model for unnown ballitic coefficient i covered. Moreover, the meaurement model of BM i alo covered. Ballitic coefficient i different for every flying object. he Jacobian matri of unnown ballitic coefficient i alo hown. Section III cover the EKF and AEKF filter which are ued in order to etimate the poition of BM. he theory a well a algorithm of both filer i covered. he theory a well a algorithm of both filer i covered. Section IV preent imulation reult and dicuion. Section V preent the concluion. II. PROBLEM FORMULAION A. rajectory he force acting on the BM are gravity, drag and other force lie centrifugal acceleration, Corioli acceleration, wind acceleration, lift force. Spinning motion i ignored becaue of mall effect on it trajectory. he trajectory of BM from launch to impact i mainly divided into three phae boot phae, ballitic flight and re-entry a hown in Fig.. Fig. Different phae of BM rajectory Boot phae i the period during which the rocet of a ballitic miile operate to attain it pea velocity. hi thrut may vary which mae the trajectory etimation more comple. he total acceleration in boot phae i given by a a ad ag () International Scholarly and Scientific Reearch & Innovation (4) cholar.waet.org/7-689/748

2 World Academy of Science, Engineering and echnology Vol:, No:4, 7 International Science Inde, Electrical and Computer Engineering Vol:, No:4, 7 waet.org/publication/748 where a i total acceleration, a i acceleration due to thrut, ad i acceleration due to drag, ag i acceleration due to gravity. In thi cae, the earth rotation can be omitted becaue of it mall duration hence the Corioli acceleration and centrifugal force i neglected. Ballitic flight i the travelling of BM between boot phae and re-entry phae. Re-entry phae i when BM re-enter in atmophere. he force act on BM in re-entry phae i given by a a d a g () he hypothei of flat earth i conidered here o the orthogonal co-ordinate ytem depicted in Fig. can be ued with following variable a in [4]: i the abcia, y i the ordinate, and y are target co-ordinate at time t, v i the velocity module, and i the angle between the horizontal ai and the direction of motion. Fig. Co-ordinate reference ytem of BM B. State Model of BM for Unnown Ballitic Coefficient State model of BM during it trajectory in dicrete time non- linear dynamic equation i given by [5] g ( ) G w () where tate vector, y y [5], here i poition along -ai, i velocity along -ai, y i poition along y-ai, y i velocity along y ai, 5 i unnown ballitic coefficient. Gf (4) G (5) (6) where i ampling time. Drag force i the force that oppoe the motion of the body. It can be conidered a aerodynamic reitance. Here the drag force will act againt the motion of BM and given by [6] D g v (7) where β i ballitic co-efficient, g i gravitational force 9.8m / ec ), i air denity, v i the velocity module. Air denity ( ) i the function of height (eponentially decaying function of height) and it decreae with increae in altitude. It alo change with the variation in temperature and humidity, however in cae of tracing BM it i avoided a it hardly affect BM motion. Here air denity can be given by where, c y c e c.7g / m, c.49 c.7554g / m, c.49 (8) 4 m 4 - m - for y 944m for y 944m Drag force can be written in term of tate vector component which can be given by, f g 4 By eploiting the following identitie y in arctg y co arctg Equation (9) can be implified a co arctg in arctg y y y 4 4 (9) () () g 4 f 5 () 4 he proce noie i aumed a Gauian and it can change dynamically during filter operation and in thi cae Q i given by Q where q * q * () International Scholarly and Scientific Reearch & Innovation (4) cholar.waet.org/7-689/748

3 World Academy of Science, Engineering and echnology Vol:, No:4, 7 International Science Inde, Electrical and Computer Engineering Vol:, No:4, 7 waet.org/publication/748 5 where q [7] in m / and q in g m are tuning parameter which are elected by deigner to get the proce noie in the target dynamic, i matri with element equal to zero. he better filter operation and reult can be achieved by tuning the filter parameter Q and R and it value may change in each time of meaurement. he trajectory of BM depend on the gravity of Earth, aerodynamic force and aerodynamic moment. he major uncertainty in etimating the trajectory of a ballitic target i due to the aerodynamic force and moment acting on it. he initial information matri J i the invere of initial covariance matri and given by he uitable epreion for P / P d d / J P / (4) d d h h i h h (5) i the variance of unnown ballitic coefficient baed on the prior value or aumed nowledge of ma of the BM. C.Meaurement Model he radar i conidered to be located at ground level i.e., y R with range r and elevation. he error R tandard deviation of thee meaurement are denoted a r (for range) and e (for elevation). he meaurement equation i given by where z H v (6) z d h, H he meaurement of the radar i in polar coordinate wherea the etimation & prediction equation are in the rectangular coordinate, thi require tranformation in one direction. he relationhip between the target poition in rectangular coordinate and meaurement in polar coordinate i given by polar coordinate tranformation d= r co and h= r in to mae it linear. v i the noie meaured Carteian coordinate and it i independent of proce noie w. Proce noie i modeled a a zero-mean white Gauian proce with covariance matri d h R [8] with element r co ( ) r in ( ) r in( ) r co ( ) ( r r )in( )co( ) (7) An eample of target trajectory i hown with the following parameter are taen into conideration, β 4 g.m m,γ -.,q 9 m.,, y,v 9m/. 8m, he tate model of BM i imulated in MALAB with above parameter to get the trajectory a well a velocity and acceleration graph. Fig. how the trajectory of BM in -y plane, Fig. 4 and 5 how the target velocity and target of BM during it trajectory repectively. Fig. rajectory of BM Fig. 4 Velocity of BM here i trong deceleration becaue of the drag force which increae with increae in velocity of the object which can be een in Fig. 5. D.Jacobian Matri for Unnown Ballitic Coefficient Ballitic coefficient decribe how air reitance low down projectile in it flight [9]. It generally depend on ma, body hape and cro-ectional area. Accurate value of International Scholarly and Scientific Reearch & Innovation (4) cholar.waet.org/7-689/748

4 World Academy of Science, Engineering and echnology Vol:, No:4, 7 Ballitic coefficient i very neceary in order to now the true nature and trajectory of BM and in cae of target tracing, ballitic coefficient i unnown to u. So in thi paper we have conidered it a unnown for all calculation purpoe and alo etimated the unnown ballitic coefficient. Jacobian matri i the matri of all the firt order partial derivative of a vector-valued function. Here Jacobian can be written a [4] GF (8) International Science Inde, Electrical and Computer Engineering Vol:, No:4, 7 waet.org/publication/748 he element of the matri are a follow F are a: f F (9) F,, F,, g 4 F,, 5 4 F, F, 4, g F,. c 4, 5 g F,. c 4 4, 5 g. 4 F, 4, 5 4 g 4 F, 4, 5 4 g () F, 5 4, 5 g F, III. FILER A. EKF EKF i widely ued for the poition etimation of non-linear ytem. o etimate the proce and meaurement noie covariance EKF ue a fied priori etimate during it whole proce. he ummary for EKF algorithm i given a []: ) ime update equation (a) he tate Projection (b) he error covariance projection: P ψ G g () P Q () Fig. 5 Acceleration of BM ) Meaurement update equation (a) Kalman Gain update Here HP H R K P H () (b) he error covariance update P P I K H (4) (c) he etimate update with meaurement K z H (5) i the predicted etimate of tate at th tep,,i a poteriori etimate of tate at th tate, P i the error covariance, i the tate tranition matri, K i the Kalman Gain matri, Q and R repreent the proce and meaurement noie covariance repectively. o implement EKF we require priori nowledge of proce and meaurement noie. However, there i a chance of high etimation error when the priori noie covariance i preumed with improper value. o overcome thi demerit of EKF, AEKF i propoed which adjut the proce and meaurement noie at every meaurement to find proper noie covariance at each tep. B. AEKF AEKF help to reduce the influence of prior Q and R on the etimation by updating it in every meaurement. he main two approache for Q and R etimation are correlation of the innovation equence and the other one i correlation of the output innovation equence. Here the correlation of innovation equence i ued to determine Q and R and further the maimum-lielihood etimation for the multivariate normal ditribution approach i ued to mae the actual value of the covariance conitent with it theoretical value []. he innovation equence can be written a International Scholarly and Scientific Reearch & Innovation (4) cholar.waet.org/7-689/748

5 World Academy of Science, Engineering and echnology Vol:, No:4, 7 e ) H ( v (6) International Science Inde, Electrical and Computer Engineering Vol:, No:4, 7 waet.org/publication/748 he covariance can be obtained by taing the variance on both ide of (6) he covariance of HP e H R i written a (7) E[( e )( e ) ] (8) According to the maimum-lielihood etimation for the multivariate normal ditribution approach, the tatitical ample variance i given a: ( e )( e ) (9) i From ( 7), the etimate of the meaurement noie covariance i a R HP H () Proce noie covariance Q i the uncertainty in the nonlinear dynamic equation meaured during meaurement update and it i applied to the ytem in order to get the proper etimate. It can be written a P P ( () i Q ) he proce and meaurement noie covariance are modified adaptively by uing () and (). So, in EKF, it i mandatory that the initial value Q and R hould be et correctly otherwie there i high chance of divergence from true trajectory wherea AEKF update the proce and meaurement noie at every meaurement to find noie covariance proper value and hence reduce the chance of divergence of the etimation. IV. SIMULAION RESULS AND DISCUSSIONS In thi paper, the two non-linear filter namely, EKF and AEKF are ued for olution of tracing of a typical ballitic target a conidered in [4]. he filtering algorithm are implemented uing MALAB Ra on a computer with.ghz Intel Core I-66U proceor with 4 GB RAM. ypical BM tracing problem ha been olved and the tet run for each cae ha been performed time and bet reult of etimated ballitic coefficient and etimation error ha been hown in Fig. 8, 9 and, repectively. he comparative reult for etimation error covariance (a conidered in [ ]) for both EKF and AEKF are preented in able I. he mean value for run i taen into conideration which i hown in able I. ime Update (a) he State Projection: (b) Error Covariance Projection: Meaurement Update (a) Kalman Gain Update: (b) Error Covariance Update: (c) Etimate update with meaurement: Q and R Fig. 6 Bloc Diagram for Adaptive Kalman filter ABLE I COVARIANCE VALUE FOR UNKNOWN BALLISIC COEFFICIEN Mean Etimation Covariance Percentage (%) State EKF( -5 ) AEKF( -5 ) Improvement y y [5] With the % improvement in able I, the AEKF offer a mean etimation error covariance of.95-5,.57-5,.88-5,. -5 and.8-5 repectively for the five tate under conideration with an improvement.4-5,.46-5,. -5 and.6-5 and. -5 of repectively, compared to EKF. hu, AEKF offer more accurate etimation of target trajectory. he comparative reult correponding to AEKF (repreented by dotted line) and EKF ( repreented by dahed line) for typical trajectory i preented in Fig. 7. he International Scholarly and Scientific Reearch & Innovation (4) cholar.waet.org/7-689/748

6 World Academy of Science, Engineering and echnology Vol:, No:4, 7 etimation error for AEKF and EKF correponding to poition and velocity along the and y direction are given in Fig. 9. he trajectory offered by AEKF i found to be cloer to the true trajectory of BM under conideration. or other more comple etimation problem. Zoomed view International Science Inde, Electrical and Computer Engineering Vol:, No:4, 7 waet.org/publication/748 Beta [g/ m ec ] Fig. 7 Etimated rajectory of BM ime [ec] Fig.8 Etimated Ballitic coefficient (beta) of BM Fig. 8 how etimated ballitic coefficient (beta) compared with true value of ballitic coefficient (olid line). Here alo AEKF able to etimate the unnown beta very cloe to it true value a compared with EKF. Fig. how the error of etimated beta a compared to true value and it alo prove that etimation error i le in AEKF when compared with EKF. V. CONCLUSION rue EKF AEKF BM tracing problem in re-entry phae with unnown ballitic coefficient i olved uing AEKF. In thi wor, the performance of AEKF i teted uing typical target problem and further it reult i compared with EKF counterpart. he imulation reult obtained uing MALAB how that AEKF tand uperior to EKF in term of accuracy. Further, it may be oberved that, the AEKF not only uperede EKF in tate etimation but alo in parameter etimation too. hu, the AEKF may be recommended a a promiing tool for olving imilar etimation problem with unnown ballitic coefficient beta error [g /m ec ] Fig. 9 Error of EKF (dahed line) and AEKF (dotted-line) ime [ec] Fig. Error of EKF (dahed line) and AEKF (dotted-line) for beta etimation REFERENCES EKF AEKF [] Raju G. C. homa India nuclear and pace program: Defene or Development? Cambridge Univerity Pre, vol. 8, iue, pp. 5-4 Jan 986. [] Greg Welch and Gary Bihop, An Introduction to the Kalman Filter Dept. of Computer Science, Univ. of Nort Carolina, R , July, 6. [] Weidong Ding, Jingling Wang and Chri Rizo, Improving Adaptive Kalman Etimation in GPS/INS Integration. he Journal of Navigation, vol 6, pp , 7. [4] A. Farina, B, Ritic, D. Benvenuti, racing a ballitic target Comparion of Several Filter. IEEE ran on AES, vol 8, iue no, pp , July. [5] K. Vami Krihna Reddy. Anand M. apa, R. Venatram Reddy, N. Prabhaar, State etimation of ballitic miile for homing phae of Eo-atmopheric Interceptor, Int. Conf. on Advance in Control and Optimization of Dynamical Sytem, 7. [6] Zarchan, P. (997) actical and Strategic Miile Guidance ( rd ed.). AIAA, Progre in Atronautic and Aeronautic, vol.76, 997. [7] Farina, A. Studer, F.A. Radar Data Proceing, Introduction and racing, Vol..England: Reearche Studie Pre, May, 985. [8] Michael Courtney, Amy Courtney he ruth about the ballitic coefficient Popular Phyic, Cornell Univerity Library, May 7. [9] Bar-Shalom, Y., Li, X-R Etimation and tracing principle echnique and oftware, MA: Artech Houe, 99. [] Dongyun Kim, Jihyun Ha and Kwanhi You Adaptive Etended Kalman Filter Baed Geolocation Uing DOA/FDOA Int, J.Vol-4 No. June,. [] Karl G. Joreog, Structural analyi of covariance and correlation Matrice, Pychometria, vol., 4,no. 4, Dec International Scholarly and Scientific Reearch & Innovation (4) 7 48 cholar.waet.org/7-689/748