Maneuvering Target Tracking Method based on Unknown but Bounded Uncertainties

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1 Preprints of the 8th IFAC World Congress Milano (Ital) ust 8 - Septemer, Maneuvering arget racing Method ased on Unnown ut Bounded Uncertainties Hodjat Rahmati*, Hamid Khaloozadeh**, Moosa Aati*** * Facult of Electrical and Computer Engineering, Sstems and Control Department, K. N. oosi Universit of echnolog, ehran, Iran, (h_rahmati@ee.ntu.ac.ir). ** Facult of Electrical and Computer Engineering, Sstems and Control Department, K. N. oosi Universit of echnolog, ehran, Iran, ( h_haloozadeh@ntu.ac.ir) *** Facult of Electrical and Computer Engineering, Sstems and Control Department, K. N. oosi Universit of echnolog, ehran, Iran, ( aati@dena.ntu.ac.ir) Astract: In this paper a new maneuvering target tracing method ased on a model with unnown ut ounded uncertainties is proposed. Firstl, a model with unnown ut ounded uncertainties is proposed. hen, an algorithm for tracing the target in oth maneuvering and non-maneuvering cases is generated. he proposed method can e used for a large diversit of input tpes. Also, this method does not require maneuver detection and covariance resetting that are necessar in previous maneuvering target tracing methods. herefore, it doesn t consume an time for maneuver detection and covariance resetting. Also simulation results are provided to confirm the theoretical development. Kewords: target tracing, unnown ut ounded uncertainties, input estimation, Kalman filtering. INRODUCION arget tracing is estimation of states of the target that include position and velocit of contemporar and mostl its future from the contemporar nois oservations. racing of the maneuvering targets is one of the most important prolems in air trafficing, navigation and guidance sstems. Since the radar cannot measure the target accelerations (maneuver terms) directl, it ecome more complicated (Mcintre et al, 998). One of the most important classes of maneuvering target tracing (M) methods is input detection and estimation (IDE) method. he major idea in the IDE method is that at first unnown inputs ( u ) should e estimated, then the states will e estimated using the estimated inputs. he asic IDE methods need additional attempt for detecting and estimating the maneuver start time, thus maneuver detection dela is inevitale. A lot of literature tried to decrease the maneuver detection dela. For instance, in recentl proposed modified input estimation () method (Khaloozadeh et al, 9), states and unnown accelerations (maneuvers) are augmented with each other. B this modification original maneuvering model is transformed into non-maneuvering model then the original states and maneuvers are estimated simultaneousl with a standard Kalman filter (KF). Although the method has good performance in low and medium maneuvering case, it has two essential drawacs; firstl, in this method for having state and maneuver estimations with suitale speed, it needs covariance resetting whenever a change in maneuver happens. Practicall the covariance resetting cannot e done on time since the time of starting maneuver is undetermined. herefore, in high maneuver case or when maneuver changed constantl the method does not have acceptale performance. In order to solve this prolem (Bahari et al. (9a, ) and Beheshtipour et al. (9)) have used Fuzz forgetting factor or a Fuzz fading memor to improve tracing accurac for high maneuvering targets. However, the Fuzz reasoning rules depend on some prior nowledge of the maneuvering targets, demands a high computational cost, and mostl leads to a poor real-time performance. In ang et al. () a new strong tracing algorithm (S) has een proposed using the idea of strong tracing filter (Zhou et al, 996), and the formula of multiple fading factor matrices are derived ased on the residuals. However, all mentioned methods are relativel low speed estimators in high maneuver case. Another prolem with the method that none of mentioned papers solved is this fact that, the method onl can trac a limited class of maneuvers. For eample if maneuver term ehaves lie a ramp function, the method will fail tracing. Moreover, with the rapid development of modern navigation technolog, the maneuverailit of aircrafts are growing stronger and stronger, so having the ailit of tracing different maneuvers is an important feature that a good tracing method must have it. In this paper for solving aove mentioned prolems, a novel target tracing model (called UBB model) and algorithm (called UBBM filter) are proposed. In UBB model process and measurement noises are considered as unnown ut ounded (UBB) noises (Schweppel, 973), that never have een used for M purpose. M methods with Baesian Copright the International Federation of Automatic Control (IFAC) 9

2 Preprints of the 8th IFAC World Congress Milano (Ital) ust 8 - Septemer, model require some nowledge on the statistical characteristics of the noise. Baesian model are not useful most of the times, since these statistical characteristics of the noise are sometimes not availale, time varing or incorrect, or impossile to estimate ecause of lac of adequate data. Also, in M prolems most of the time it is etter to consider unnown accelerations as uncertain stochastic processes ut not white noise since white noise does not have time structure. Modeling uncertainties with UBB stochastic process (instead of white noise) is more appropriate since most uncertainties in phsical sstems have time structure and ounded values. In this wor ased on UBB model a UBBM filter is proposed that can estimate the position, velocit, and acceleration of a target with unnown maneuver. UBBM filter does not require covariance resetting and it could handle all maneuver tpes. he numerical simulations of the proposed UBBM filter shows that it can quicl trac ever maneuvering target with small tracing errors. his paper also compares the proposed filter with the method in Khaloozadeh et al. (9) to show performance of MBB filter.. NON-MANEUVERING UBB MODEL FORMULAION In this paper a linear discrete time model for nonmaneuvering targets under the UBB uncertainties is considered as follows ( + ) = F( ) ( ) + G( ) w ( ) () z( ) = H( ) ( ) + v ( ) () m p Where the vectors R, u R, and z R are the unnown state and unnown input to e estimated and measurale sstem output respectivel; F () R, m G () R p and H () R are state, process noise and m p output matrices respectivel. w () R, and v () R are ounded noise vectors with unnown statistical characteristics that ma include modeling inaccuracies, discretization errors or computer round-off errors. v () represents the measurement noise and w () is unnown ut ounded process noise. he onl information aout v () and w () are as follows w( ) Ωw ( ) w ( ) Q ( ) w ( ), n N (3) v( ) Ωv ( ) v ( ) R ( ) v ( ), n N () And also for initial condition following constraint is considered () Ω () () Ψ (), n N () Where Ω (), Ω () and Ω () are ellipsoidal sets whose size w v and shape can var with time. Also, Q(), R () and Ψ () are positive definite matrices. he center of Ω (), Ω () and Ω () are and the orientations of Ω (), Ω () and Ω () w v are respectivel determined the eigenvectors of Q(), R () w v and Ψ (), also the lengths of semimajor aes of Ω (), Ω () w v and Ω () are determined respectivel the lengths of the eigenvalues of Q (), R () and Ψ (). It is assumed that the target moves in a plane, which is a two dimensional case. herefore state vector ecomes as follows ( ) = ( ) v ( ) ( ) ( ) v (6) Where (), v() and (), v() are respectivel the target positions and velocities in and directions in Cartesian coordinate. Also F(), G () and H () are defined as follows F() =, () =, () = G H (7) 3. SANDARD UBBM FILER For the non-maneuvering target motion model () and (), asic UBBM filter formulas are as follows (Schweppel, 973) - { Ω ( + + ) = : - ( + + ) Σ ( + + ) - ( + + ) (8) ( + + ) = F( ) ( ) + K ( + ) { z( + ) H( + ) F( ) ( )} (9) K( + ) =Σ ( + + ) H ( + ) R ( + ) () Σ ( + + ) = δ ( + ) Σ ( + + ) () Σ ( + + ) =Σ ( + )- Σ ( + ) H ( + ) ( ) ( ) ( ) ( ) ( ) R + + H + Σ + H + H + Σ ( + ) () Σ ( + ) = F()( ) () () () () - β( ) Σ F + G Q G (3) Σ ( + ) = ( ) - ρ( + ) Σ + Q() = Q(), () () β() R = R ρ } () δ ( + ) = ( + )- ( + ) ( ) ( ) z H F ( ) ( ) ( ) + Σ + + ( + )- ( + ) ( ) ( ) H H z H F () For an ρ( + ), β( ) ρ( + ), β( ) ( ) = Σ ( ) = Ψ Where centre of the ounding ellipsoid estimate set Ω is, and the size and shape of this set is determined Σ. 9

3 Preprints of the 8th IFAC World Congress Milano (Ital) ust 8 - Septemer, Although, UBBM filter ields an estimate which is a set rather than a single vector, it is quite reasonale to consider eing the est estimation. is a min-ma estimate in a prolem minimizing the maimum error etween and the true ut unnown value.. PROPOSED UBBM FILER he target motion model in maneuvering case can e considered as ( + ) = F() () + C()() u + G() w() (6) z() = H() () + v () (7) m Where C() R m is the input matri and u () R is the completel unnown maneuver (input). he are defined elow C( n) =, u( n) = [ u ( n) u ( n)] he standard UBBM filter cannot e used for maneuvering targets ecause it does not consider maneuver. his section aims to design an estimation algorithm for sstem (6) and (7) such that a) a set that contains all possile value of the true state vector is quantified at each time step. ) the output error vector z()- H() () is acceptale, i.e., it remains in the interior of the ellipsoid () enclosing all proale values of the vector v () : - ()- () () () z H R z()- H() (), n N c) u () could e calculated. d) to have a good estimation for state vector as quic as possile. he following algorithm for UBBM filter is proposed here v ( )- ( ) ( )- ( ) + v v v + u( n) =, u( n) = Where v( ) and v( ) are estimation of target velocities in and direction respectivel. So, following equations are used for UBBM filter - { Ω ( + + ) = : - ( + + ) Σ ( + + ) - ( + + ) (8) ( + ) = F() ( ) + C()( u ) (9) v ( )- v ( ) u ( ) + v ( + )- v ( ) u ( + ) = () } ( + ) = F() ( ) + C()( u + ) () ( + + ) = ( + ) + K( + ) z( + )- H( + ) ( + ) () { } K H R (3) - ( + ) =Σ ( + + ) ( + ) ( + ) ( ) Σ + + = - δ ( ) + Σ ( + + ) () Σ ( + + ) = Σ ( + )- Σ ( + ) H ( + ) R ( + ) + H( + ) Σ ( + ) H ( + ) ( ) H + Σ ( + ) () Σ ( + ) = F()( ) () () () () - β( ) Σ F + G Q G Σ ( + ) = ( ) - ρ( + ) Σ + Q () = Q() β() R () = R() ρ (6) (7) δ ( + ) = ( + )- ( + ) ( + ) z H ( ) + Σ ( + ) ( + ) H H ( + )- ( + ) ( + ) z H (8) For an ρ( + ),β() ρ( + ), β( ) ( ) = Σ ( ) = Ψ. SIMULAION RESULS In this section the theoretical developments in M are verified numerical simulations. he effectiveness of the proposed UBBM filter is compared with the method that has een proposed Khaloozadeh et al. (9). In their approach, the maneuver is augmented with the state space vector in a new single state vector. In this scheme, the maneuvering model is changed into non maneuvering model. hen, the original state and maneuver vectors are estimated simultaneousl with a standard Kalman Filter. he algorithm given in Khaloozadeh et al. (9) is as follows K ( + ) = [ P ( + + ) H ( + ) + G R + (9) - () ()] ( ) P ( + + ) = P ( + )- P ( + ) H ( + ) [ R ( + ) + H ( + ) P ( + ) H ( + )] H ( + ) P ( + ) (3) P ( + ) = F ( ) P ( ) F ( ) + G ( ) Q ( ) G ( ) (3) ( + + ) = F ( ) ( ) + K ( + ) [ Z ( + )- H ( + ) F ( ) ( )] (3) where - 9

4 Preprints of the 8th IFAC World Congress Milano (Ital) ust 8 - Septemer, ( ) ( ) ( ) ( ), ( ) F C = F = u( ) I ( ) ( ) G G =, w ( ) = w ( ) z () = H () () + v () H () = [ H()() Φ H()()] C v () = H() G() w() + v( + ) { } { } Q () = E w () w () = Q () R () = E v () v () = H() G() Q() G () H () + R() { } () = E w () v () = Q() G () H () In the simulations in order to have a fair comparison etween method and the proposed UBBM filter, process and measurement noises are considered as zero mean white processes. Since the method is usale for white noise processes. In the eamples, = second, g = 98. m / s, Q =, and R =. Initial conditions are ii ii () = m m / s m - m / s, ( ) = [ ] and parameters ρ = 8., β = 8.. ( ) = I and P( )= I where I is the identit matri with appropriate dimensions. he actual value and the estimated position in -direction(km) Position Error in -direction(km) he actual value and the estimated position in Y-direction(Km) 3 -. ime(sec).. -. Position Error in Y-direction(Km) ime(sec) Fig.. he actual and estimated positions, and their relevant estimation errors for the constant acceleration case he actual value and the estimated velocit in -direction(km/s) he actual value and the estimated velocit in Y-direction(Km/s) 8 6 Eample : in this eample, a constant acceleration model is considered such that for t s; [ ] u = m / s. he target egins to maneuver at th second with high maneuver u = [ ] m / s Velocit Error in -direction(km/s).. -. Velocit Error in Y-direction(Km/s) Figs. - compare the estimated positions, velocities and accelerations of the target estimated the method and the proposed method in the first eample. he first row of Figs. - depicts respectivel the actual positions, velocities and accelerations in and directions in dash-dot line, the estimated positions, velocities and accelerations the method in solid line and estimated values the proposed UBBM filter in dashed line. he second row depicts respectivel the positions, velocities and accelerations estimation errors in and directions method in solid line and the proposed UBBM filter in dashed line. For t < s (non-maneuvering stage) oth of the methods could trac the states of the target correctl without stead state error ut when target egins to maneuver, the method tracing performance decreases and it tries to estimate the states ver slow such that until t = s it has ig estimation errors. Figures show that the proposed UBBM filter could trac the target ver quicl without stead state error ime(sec) ime(sec) Fig.. he actual and estimated velocities, and their relevant estimation errors for the constant acceleration case he actual value and the estimated acceleration in -direction(g) Acceleration Error in -direction(g) ime(sec) Fig. 3. he actual and estimated accelerations, and their relevant estimation errors for the constant acceleration case in -direction 93

5 Preprints of the 8th IFAC World Congress Milano (Ital) ust 8 - Septemer, he actual value and the estimated acceleration in -direction(g) 8 he actual value and the estimated velocit in -direction(km/s) he actual value and the estimated velocit in Y-direction(Km/s) Acceleration Error in Y-direction(g). Velocit Error in -direction(km/s). Velocit Error in Y-direction(Km/s) ime(sec) Fig.. he actual and estimated accelerations, and their relevant estimation errors for the constant acceleration case in -direction Eample : in this eample, a non-constant acceleration is considered. Before th there is no maneuver and after th the target egins to maneuver with u = [ ] (t -)m / s. Figs. -8 compare the results of the method and the UBBM method for the second eample. he first row of Figs. -8 depicts respectivel the actual positions, velocities and accelerations in and directions in dash-dot line and theestimated positions, velocities and accelerations method in solid line and the proposed method in dash line. he second row depicts respectivel the positions, velocities and accelerations estimation errors in and directions method in solid line and for the proposed UBBM filter in dashed line. For t < s (non-maneuvering stage) oth of the methods could trac states of the target correctl without stead state error ut when target egins to maneuver, the method could not trac the target and the estimation errors increase graduall. Also, it can e seen that the proposed filter tracs the target fast with no stead state error. he actual value and the estimated position in -direction(km) he actual value and the estimated position in Y-direction(Km) Position Error in -direction(m) Position Error in Y-direction(m) ime(sec) ime(sec) Fig. 6. he actual and estimated velocities, and their relevant estimation errors for the ramp acceleration case he actual value and the estimated acceleration in -direction(g) Acceleration Error in -direction(g) ime(sec) Fig. 7. he actual and estimated accelerations, and their relevant estimation errors for the ramp acceleration case in -direction he actual value and the estimated acceleration in -direction(g) Acceleration Error in Y-direction(g) ime(sec) ime(sec) Fig.. he actual and estimated positions, and their relevant estimation errors for the ramp acceleration case ime(sec) Fig. 8. he actual and estimated accelerations, and their relevant estimation errors for the ramp acceleration case in -direction 9

6 Preprints of the 8th IFAC World Congress Milano (Ital) ust 8 - Septemer, 6. CONCLUSIONS In this paper a new M method ased on UBB model has een proposed. Firstl a UBB model for target motion is developed and then a new M algorithm called UBBM filter is given. B UBBM filter target tracing in oth maneuvering and non-maneuvering cases is possile. It can e seen that using the proposed filter the target positions, velocities and accelerations were estimated ver well without stead state errors. Also, the UBBM filter does not have an limitation on the tpe of the unnown maneuver and it can e used for a large diversit of input tpes. Moreover, it does not require maneuver detection and covariance resetting that are necessar in previous M methods. herefore, UBBM filter reduces the tracing time. Simulation results are provided to confirm the theoretical development and to compare the filter with method. Results show the high performance of the UBBM filter for oth maneuvering and non-maneuvering targets with different inputs. 7.REFERENCES Bahari, M. H., Naghii, M. B., and Pariz, N. (9). Intelligent fading memor for high maneuvering target tracing. Int. J. Phs. Sci, vol.(),pp. 8-. Bahari, M. H., and Pariz, N. (9). High maneuvering target tracing using an input estimation technique associated with Fuzz forgetting factor. Sci. Res. Essas, vol.(), pp Beheshtipour, Z., and Khaloozadeh, H. (9). An Innovative Fuzz Covariance Presetting For High Maneuvering arget racing Prolems. Control and Decision Conference, pp Khaloozadeh, H., and Karsaz, A. (9). Modified input estimation technique for tracing maneuvering targets. IE Radar, Sonar, Navigation, vol. 3, no., pp. 3-. Mcintre, G. A., and Hintz, K. J. (998). Comparison of several maneuver tracing models. SPIE Conf. Signal Processing, Sensor Fusion and arget Recognition VII, vol. 337, pp Schweppe, F.C. (973). Uncertain dnamic sstems. Prentice-Hall, Inc., Englewood Cliffs. Yang, J. L., and Ji, H. B. (). High maneuvering targettracing ased on strong tracing modified input estimation. Sci. Res. Essas, vol. (3), pp Zhou, D. H., and Fran, P. M. (996). Strong tracing filtering of nonlinear time-varing stochastic sstems with colored noise: application to parameter estimation and empirical roustness analsis. Int. J. Control, vol. 6():pp