State Dependent Difference Riccati Equation ... - Semantic Scholar
State Dependent Difference Riccati Equation based Estimation for Maneuvering Target Liat Peled-Eitan and Ilan Rusnak RAFAEL, P.O.Box 2250, Haifa 31021, Israel Abstract— Estimation of evading target maneuvers with unknown turning rate is considered. The modeling of the target's equations of motion takes into account the rotation of the velocity and acceleration vectors as the target maneuvers. Coordinated turn and barrel roll target evasive maneuver are dealt with. The inclusion of the more detailed kinematic behavior of the maneuvering target creates nonlinear equations of motion. The state - position, velocity and acceleration, and the angular rate of the velocity vector are estimated. This is done without inclusion of the angular rate into the state vector, but in separate equation. As the equations of motion are nonlinear the State Dependent Differential-Difference Riccati Equation based estimator (SDDRE) is implemented and compared to the Kalman Filter based on the constant-step acceleration target maneuver model. It is demonstrated via simulations for constant turning rate and barrel roll evading target's maneuvers that the detailed modeling of the maneuvering target based filter-estimator have improved performance with respect to the Kalman Filter based on the constant-step acceleration target maneuver model. Keywords- maneuver estimation; SDRE; SDDRE; Constant turning rate, barrel roll.
I.
INTRODUCTION
The issue of estimating a maneuvering target is widely treated subject. A comprehensive survey of models and estimators is presented in [1-5]. The simplest approach is to implement three independent Constant-Step acceleration filters (CA) or Exponentially Correlated Acceleration (ECA) filters [6], one filter for each coordinate. However these filters may not achieve the required performance for coordinated turn target maneuvers (CT) or barrel roll evasive maneuver (BREM) as they are not matched to these maneuvers, i.e. steady state errors are created. For more advanced estimators it has been understood that incorporating detailed information on the target dynamics and kinematics into the estimator's equations has the potential to increase the quality of estimation. However, the inclusion of more detailed target maneuver model and the related constraints lead to nonlinear models. Thus the Kalman Filter is not directly applicable. The most common approach to deal with nonlinear systems is the Extended Kalman Filter (EKF). In [7] the issue of pseudo measurements had been introduced and the Extended Kalman Filter was applied. In [7] it was pointed out that inclusion of a constraint is usually difficult to incorporate into the dynamic equation and it is much easier to incorporate them into the measurement equations.
1689
In this paper the coordinated turn evading maneuver (CT) and barrel roll evasive maneuver (BREM) with unknown turning-barrel roll rate are considered. One of the main and important issues is the estimation of the angular turning rate. This is needed for achieving better matching of the estimator and for the derivation of high performance guidance law for this type of evading maneuver [8]. In [9, chapter 10] it is shown that the algorithms that include the turning rate as part of the state are problematic, to say the least. The current approaches to estimation of nonlinear systems include many methods. A comprehensive survey of such methods applied to maneuvering target estimation is presented in [1-5]. For example: in [10] a multiple model approach is applied; in [11] algebraic constraint is incorporated in the state equations; and in [12] the IMM approach is applied; to estimate maneuvering target. Two approaches to modeling the maneuvering target equations of motion are dealt with here: i) Rotating velocity vector based approach (velocity based jerk equations - VJ); and ii) Rotating acceleration vector based approach (acceleration based jerk equations – AJ). The later approach is introduced as it requires less computational effort. The rotating velocity based approach has been used and EKF was applied in [22]. In this paper the State Dependent Differential-Difference Riccati Equation (SDDRE) based estimator is applied to the nonlinear equations of motion. The authors are unaware of any publication estimating a maneuvering target by the SDDRE method. The SDDRE approach is very intuitive, although it is not optimal as shown for the State Dependent Algebraic Riccati Equation (SDARE) approach in [13-16]. The optimal filter requires additional terms for optimality [14-17]. Albeit the sub-optimality of the direct SDDRE approach it is known that the SDDRE based estimator is BIBO stable [18]. The novelty in this paper is: Application of the SDDRE to estimation of a maneuvering target state and turning rate; Introduction of rotating acceleration vector kinematics equations of motion of a maneuvering target; Comparison of rotating velocity vector and rotating acceleration vector kinematics equations of motion application with the SDDRE based estimator on common basis. The mathematical derivations are presented in the three dimensional space, however for simplicity the simulations are presented in two dimensions, i.e. planar target's maneuver. Simulations show that estimator based on the rotating velocity
and acceleration vector kinematics equations of motion give better performance than the Kalman Filter based on constant acceleration target model. For coordinated turn target's evading maneuver the difference between the rotating velocity and acceleration vector based approaches is small. However, for barrel roll target's evading maneuver (higher turning rate) the rotating velocity vector based approach gives faster convergence of the state and turning rate estimation error. II.
PROBLEM STATEMENT AND APPROACH TO THE SOLUTION
f x(t ) *w(t ), x(to )
xo ,
g x(t ) v(t )
z (t )
where x(t) is the state vector, z(t) is the measurement, w(t), v(t) are the white Gaussian stochastic processes representing the system driving noise and the measurement noise, respectively, x(to) is a Gaussian random vector, and E[w(t)w(W)T ]= WG(t-W),E[v(t)v(W)T ]=VG(t-W),
T
@
4 ! 0.
0
(2.9)
As xˆ(t) Q(t ) Q(t ) AsT xˆ(t) *W* T Q(t )C T V 1CQ(t )
This is a suboptimal estimator. The optimal estimator has additional terms as detailed in [14,15,16]. Here the discrete version of the following SDDRE based state estimator [18] is implemented. (2.10)
Q(to )
All vectors and matrices are of appropriate dimensions. The problem being considered here is finding the optimal estimate xˆ (t ) as a functional of {z(t), to≤t≤tf} that minimizes the quadratic criterion:
>
(2.8)
The SDRE/SDARE based estimator is [13]
(2.2)
T
E >x(t ) xˆ (t )@ 4>x(t ) xˆ (t )@ ;
Cx (t ) v(t )
K xˆ(t), t Q(t )CV 1 Q (t ) As xˆ(t) Q(t ) Q(t ) AsT xˆ(t) *W*T Q(t )C TV 1CQ(t ),
E[(xo -E(xo ))(xo -E(xo )T ]=Qo .
J
z (t )
xo ,
xˆ(t) = As xˆ(t) xˆ(t)+K xˆ(t), t >z (t ) Cxˆ(t)@, xˆ(to ) = xo ,
E[w(t)v(W)T ]=0, E[w(t)xo ]=0, E[v(t)xo ]=0, T
As x(t ) x(t ) *w(t ), x(to )
K xˆ(t) Q(t )CV 1
(2.1)
E[xo ]=xo , E[w(t)]=0, E[v(t)]=0,
x (t )
xˆ(t) = As xˆ(t) xˆ(t)+K xˆ(t) >z (t ) Cxˆ(t)@, xˆ(t o ) = xo ,
The problem considered here is the state estimation of the nonlinear stochastic system x (t )
measurement of the state, i.e. g x(t ) Cx(t ) , The state equation are represented as
Qo .
The difference between the SDARE as applied in [14,15,16] , eq. (2.9), and eq. (2.10), the one implemented in this paper, is that in this paper the differential-difference Riccati equation is used for computation of the gains and not the algebraic Riccati equation (2.9) that is solved each sampling interval. III.
(2.3)
THE CONSTANT ACCELERATION TARGET MANEUVER MODEL
z (t ) Cx (t ) v(t )
For comparison in this paper we present the performance of a most common estimator of target motion. This estimator is based on the target Constant Acceleration (CA) Step target maneuver. The following dynamic model of the target [1, 2, 6] is assumed, for each inertial coordinate, the CA target maneuver model is
the solution of the preceding problem is the Kalman filter [19,20]. (2.5) xˆ(t) = Axˆ(t)+K(t)[ z (t ) Cxˆ(t)], xˆ(to ) = xo ,
ª x (t ) º d « x (t ) » « » dt «¬aT (t )»¼
A. Estimator for Linear System For the linear system x (t )
Ax(t ) *w(t ), x(to )
xo ,
(2.4)
K (t ) Q(t )CV 1 Q (t ) AQ(t ) Q(t ) AT *W*T Q(t )C T V 1CQ (t ), Q(to ) Qo . B. Estimators for Nonlinear System For nonlinear systems there are several approaches. Here the SDRE [13-16] approach is considered. The SDRE approach is based on the dual of the SDRE based nonlinear control [16]. This approach parameterizes the state equation (2.1) into the linear structure called State Dependent Coefficient Form, and is also called Extended Linearization. This approach includes the State Dependent Algebraic Riccati Equation (SDARE) based estimation and the State Dependent Differential-Difference Riccati Equation (SDDRE) based estimation. Then for linear
1690
ª0 1 0 º ª x ( t ) º ª0 º «0 0 1» « x (t ) » «0» w (t ), « »« » « » T «¬0 0 0»¼ «¬aT (t )»¼ «¬1»¼ ª x (t ) º z (t ) >1 0 0@« x (t ) » v (t ) « » «¬aT (t )»¼ where x - target-missile separation distance [m]
x
aT wT (t ) v(t )
z
(3.1)
- target-missile separation velocity [m/s] - target acceleration [m/s2] - target process driving noise (jerk) [m/s3] - target-missile separation measurement noise [m] - measured target distance[m]
The preceding assumes that the target performs an
&
ZT - angular rate of the target velocity direction (turning rate of the target) [rad/s]
evasive maneuver (a stochastic process), that is, a step acceleration maneuver of amplitude aT 0 whose initiation instant is uniformly distributed in the interval [to,tf]. Essentially this model assumes a piecewise constant acceleration target maneuver in the inertial space. The continuous shaping filter [1,2,6] of this process is
d aT dt
2) Jerk equations – VJ Second order equations of motion based on the targets velocity (4.1) are presented. As the target is maneuvering the velocity vector is rotating. The jerk (derivative of acceleration) is the given by [21,22]. This is called here Velocity based Jerk (VJ) equations of motion. The target's jerk is [21,22] (4.4)
(3.2)
wT (t ),
where the spectral density of the target maneuver (the process noise), wT (t ) , is
aT2 0 2 [ m / s 3 / Hz] [m 2 / s 5 ] tm where 2 aT 0 - target step maneuver value [m/s ], tm WT
& jT
(3.3)
t f to is the
& jT
Comprehensive survey of modeling the behavior of a maneuvering target can be found in [1-5]. Here one specific case is considered. A redundant set of assumptions with respect to maneuvering target behavior is: i. Maneuver in one plane, i.e. angular rate vector & & constant direction - ZT u ZT 0 ; Constant absolute value of target velocity, & vT constant ;
iii.
Aerodynamically controlled aircraft (velocity perpendicular to acceleration); Coordinated turn (no loss of target energy); No roll rate.
iv. v.
vT 1v
& aT
(4.1)
ZT
(4.6)
1) Jerk equations - AJ First order equations of motion based on the target's acceleration (4.1) are derived. As the target is maneuvering the acceleration vector is rotating. The jerk (derivative of acceleration) is the given by [21]. This is called Acceleration based Jerk (AJ) equations of motion. The target's jerk is given by
1) Acceleration equations First order equations of motion based on the target's velocity (4.1) are presented. As the target is maneuvering the velocity vector is rotating. The target's acceleration is given by [21]
&
aT 1a
1a -unit vector in the target acceleration direction (perpendicular to 1v) then it is possible to derive the following kinematic relations.
vT - absolute value of the target's velocity[m/s] 1v - unit vector in the target velocity direction
& & & d vT vT 1v ZT u vT dt & & vT u aT & 2 ; vT
(4.5)
where & 2 aT - target acceleration vector [m/s ] 2 aT - absolute value of the target's acceleration [m/s ]
where & vT - target velocity vector [m/s]
& aT
& & & & d vT & & & vT 1v ZT u vT 2ZT u ZT u ZT u vT dt & & & & & & & vT 1v ZT u vT 2ZT u aT ZT u ZT u vT
B. The kinematics as a function of acceleration In effort to reduce the computational effort with the SDDRE based on the Velocity based Jerk (VJ) equations of motion, as can be seen in section 6, here an acceleration based equations are derived. For aerodynamically controlled aircraft, under the aforementioned assumptions (the acceleration and velocity are perpendicular) the target's velocity's angular rate is equal to the target's acceleration angular rate. If the target's acceleration is expressed as
A. The kinematics as a function of velocity We assume that the target's velocity is expressed as
& vT
& & & & & & vT 1v ZT u vT 2 ZT u vT 1v ZT u ZT u vT
Substituting vT 1v from (4.2) gives [22]
KINEMATIC EQUATIONS OF MANEUVERING TARGET
ii.
& d 2 vT dt 2
& 3 jT - target's jerk vector [m/s ]
time interval in which the target is expected to take an evasive maneuver. IV.
& d aT dt
& d aT dt
& jT
(4.2)
where & vT
(4.3)
(4.7)
- target velocity vector [m/s]
& jT
- target jerk vector [m/s3]
ZT
- turning rate (4.3) of the target [rad/s]
&
& 2 aT - target acceleration vector [m/s ]
& & aT 1a ZT u aT
The VJ and AJ models assume piecewise constant turning
1691
ªxº « x » « » « x» « » y d « » « y » dt « » « y» «z» « » « z » « z» ¬ ¼
rate target maneuver. This is essentially piecewise constant target acceleration in the target's coordinates. THE VARIANCE AND SPECTRUM OF THE GLINT NOISE
V.
For simplicity it is assumed that the only measurement noise, v(t), is the glint noise. The standard deviation of the glint noise, Vg, for uniformly distributed reflectors, is [23] 1 2 2 (5.1) D ; D L V g2 12 S where D - the effective linear dimension of the target perpendicular to the target-missile LOS [m]. L - the linear dimension of the target perpendicular to the [m]. LOS target-missileWhen frequency agility is applied at rate of fs=1/Ts [Hz], the spectral density of (stair type random stochastic process - i.i.d. sequence) the glint is given by [24].
V go
&
ZT
(5.2)
ª xº « x » « » d « x» « » dt « y » « y » « » ¬« y¼»
INCORPORATION OF THE KINEMATIC CONSTRAINT
A. Velocity based jerk kinematic equation With velocity based jerk equations (4.4) it is assumed that & Z 0 . Thus from (4.4b) the kinematics of maneuvering target is modeled as
wT
& & & & & & d aT 2ZT u aT ZT u ZT u vT wT dt & & vT 1v Z T u vT
0 0 2Z z 0 1 0 0 0 2Z x
0 0 0 0 0 Z xZ z 0 0 0 0 0 Z zZ y 0 1 0 0 2 0 Z y Z z2
º ª x º ª0 »« » « » « x » «0 2Z y » « x» «1 » 0 » «« y »» ««0 0 » « y » «0 »« » « 2Z x » « y» «0 0 » « z » «0 »« » « 1 » « z » «0 0 »¼ «¬ z»¼ «¬0 0 0
0 0 0 0 0 1 0 0 0
0º 0» » 0» » 0» ª w x º 0» «« w y »» » 0» «¬ w z »¼ 0» » 0» 1»¼
(6.2)
ª y z z yº 1 « z x x z »; » x 2 y 2 z 2 « «¬ x y y x»¼
& & vT u aT & 2 vT
(6.3)
and in two dimensions the equations of motion reduce to
The kinematic equations in section IV are constraints that can be incorporated into the estimator equations. It is possible to incorporate the kinematic constraint into the state equation or measurement equation and the "unknown" quantities are interpreted as either a measurement noise or system driving noise. The approach here has the advantage that the kinematic constraint is incorporated into the system equations, the unknown is the system driving noise and the measurements are linear. The derivation here is in three dimensions although the simulations are performed in two dimensions.
& jT
2Z z 0 0 2Z y
0 0 0 0 0 Z xZ y 0 1 0 0 0 Z x2 Z z2 0 0 0 0 0 Z yZ z
where
where Ts is the sampling interval of the frequency agile radar. Therefore, we assume that the spectral density of the measurement noise, v(t), is Vgo >m 2 Hz@. VI.
0 1 0 0 0
ª xº « x » « » « x» « » ª1 0 0 0 0 0 0 0 0 º « y » ª v x º «0 0 0 1 0 0 0 0 0» « y » «v » « »« » « y » «¬0 0 0 0 0 0 1 0 0»¼ « y» «¬ v z »¼ «z» « » « z » « z» ¬ ¼
ª xº « y» « » «¬ z »¼ msr
2
ª § ZTs · º ¸» « sin ¨ 2 ¹» ªm2 º ªm2 º 2 V g Ts , « »; V g (Z ) V go « © , « § ZTs · » «¬ Hz »¼ ¬ Hz ¼ ¸ » « ¨ ¬ © 2 ¹ ¼
1 ª0 «0 0 « «0 Z y2 Z z2 « 0 «0 «0 0 « «0 Z x Z y «0 0 « 0 «0 «0 Z Z x z ¬
ª xº « y» ¬ ¼m
Zz
ª0 1 «0 0 « «0 Z z2 « «0 0 «0 0 « ¬«0 0
0 1 0 0 0 2Z z
0 0 0 0 0 0 0 1 0 0 0 Z z2
0 º ª x º ª0 0 »» «« x »» «0 « 2Z z » « x» «1 »« » « 0 » « y » «0 1 » « y » «0 »« » « 0 ¼» ¬« y¼» ¬«0
(6.4)
ª xº « x » « » ª1 0 0 0 0 0º « x» ªvx º «0 0 0 1 0 0 » « y » « v » ¬ ¼« » ¬ y ¼ « y » « » ¬« y¼»
x y y x . x y 2
(6.5)
2
B. Acceleration based jerk kinematic equation - AJ With acceleration based jerk equations it is not & imperative to assume that Z 0 . Therefore from (4.7) we model this as
& d aT & & ZT u aT wT dt wT aT 1a & j
(6.1)
0º 0»» 0» ª wx º »« » 0 » ¬ wy ¼ 0» » 1¼»
(6.6)
where wT represents the deviation of the actual behavior of the target from constant absolute value of the velocity assumption. We have
where wT represents the deviation of the actual behavior of the target from the constant angular turning rate and constant absolute value of the velocity assumptions. We have the state space representation in the State Dependent Coefficient Form in three dimensions
Z z Z y º ª xº ª wx º ª xº ª 0 (6.7) d « » « » y « Z z 0 Z x » « y» « wy » « » « » dt « » «¬ z»¼ «¬ Z y Z x 0 »¼ «¬ z»¼ «¬ wz »¼ The state space representation in the State Dependent Coefficient Form is in three dimensions
1692
ª xº « x » « » « x» « » y d « » « y » dt « » « y» «z» « » « z » « z» ¬ ¼
ª xº « y» « » «¬ z »¼ msr
ªZ x º «Z » « y» ¬«Z z ¼»
ª0 «0 « «0 « «0 «0 « «0 «0 « «0 «0 ¬
1 0 0 1 0 0 0 0 0 0 0 Zz 0 0 0 0 0 Zy
0 0 0 0 0 0 0 0 0
0 0 0 0 0 Zz 1 0 0 1 0 0 0 0 0 0 0 Zx
0 0 0 0 0 0 0 0 0
0 0 º ª x º ª0 0 0 » « x » «0 »« » « 0 Z y » « x» «1 » 0 0 » «« y »» ««0 0 0 » « y » «0 »« » « 0 Z x » « y» «0 1 0 » « z » «0 »« » « 0 1 » « z » «0 0 0 »¼ «¬ z»¼ «¬0
0 0 0 0 0 1 0 0 0
0º 0» » 0» » 0» ª w x º 0» «« w y »» » 0» «¬ w z »¼ 0» » 0» 1»¼
ª xº « x » « » « x» « » 1 0 0 0 0 0 0 0 0 ª º « y » ªv x º «0 0 0 1 0 0 0 0 0» « y » «v » « »« » « y » «¬0 0 0 0 0 0 1 0 0»¼ « y» «¬ v z »¼ «z» « » « z » « z» ¬ ¼
C. The trajectory The trajectory of coordinated turn and barrel roll is modeled as in [1,9,25]. It is
ª xº « y» ¬ ¼m
0 0 º ª x º ª0 0 0 » « x » «0 »« » « 0 Z z » « x» «1 »« » « 1 0 » « y » «0 0 1 » « y » «0 »« » « 0 0 ¼ ¬ y¼ ¬0 ª xº « x » « » ª1 0 0 0 0 0º « x» ªv x º « 0 0 0 1 0 0 » « y » «v » ¬ ¼« » ¬ y ¼ « y » « » ¬« y¼» 1 0 0 1 0 0 0 0 0 0 0 Zz
0 0 0 0 0 0
zT
0
aT cos Z z t M
xT
xT 0 xT 0t
yT
yT 0 yT 0t
zT
zT 0 zT 0t ,
(6.8)
(7.1)
aT
sin Z z t M ,
Z z2 aT
Z z2
(7.2)
cos Z z t M ,
where (x,y,z) are the target's coordinates and the rest of the variables are self evident. VIII. THE SDRE BASED DISCRETE ESTIMATOR
ª yz zyº 1 « zx xz» » x 2 y 2 z 2 « ¬« xy yx¼»
ª0 «0 « «0 « «0 «0 « ¬0
aT sin Z z t M
then
We consider the discrete nonlinear stochastic system in the State Dependent Coefficient Form
(6.9)
x(t 1)
0º 0» » 0 » ª wx º »« » 0» ¬ w y ¼ 0» » 1¼
A x(t ) x(t ) *wd (t ), x(to )
xo ,
(8.1)
z (t ) Cx (t ) vd (t )
Note by comparing Eqns' (6.2, 6.4) to (6.8, 6.10) that the computational effort of AJ based estimator is smaller than the computational effort of VJ based estimator. In two dimensions the equations of motion are ª xº « x » « » d « x» « » dt « y » « y » « » ¬ y¼
xT yT
where x(t) is the state vector, z(t) is the measurement, u(t) is the input, w(t), v(t) are the white Gaussian stochastic sequences representing the system driving noise and the measurement noise, respectively, xo is a Gaussian random vector, and
E[xo ]=xo , E[wd (t)]=0, E[v(t)]=0,
(6.10)
E[wd (t)wd (W)T ]= Wd G tW ,E[vd (t)vd (W)T ]=Vd G tW ,
(8.2)
E[wd (t)vd (W)T ]=0, E[wd (t)xo ]=0, E[vd (t)xo ]=0, T
T
E[(xo-E(xo ))(xo-E(xo )T ]=Qo . All vectors and matrices are of appropriate dimensions. The State Dependent Difference Riccati Equation (SDDRE) based estimator is, k=0,1,2,…k-1, is adaptation of the State Dependent Differential Riccati Equation (SDDRE) (8.3)
xˆ(k 1 ) = A xˆ(k) xˆ(k) +K(k)[ z (k ) Cxˆ(k)], xˆ(to ) = xo , 1 ~ ~ K (k ) Q (k )C T CQ (k )C T Vd ~ T Q (k 1) A xˆ(k) Q(k ) A xˆ(k) *Wd *T , Q(ko ) Qo . / predictor prior / ~ Q(k ) >I K (k )C @Q (k ) / corrector posterior /
VII. TARGET MANEUVER MODEL A. Coordinated Turn Target Evasive Maneuver The target moves with constant velocity. At certain moment it starts a coordinated turn (constant acceleration and constant speed (absolute value of the velocity) at constant altitude, i.e. in the (x-y) plane only.
IX.
SIMULATION RESULTS
In this section the performance of the constant acceleration (CA) (3.1), the velocity based jerk (VJ) equation based filters (4.5, 6.4) and the acceleration based jerk (AJ) equation based filters (4.7, 6.11) is compared via simulations. In effort to perform the comparison on common basis all simulation are performed for system driving noise (process noise) with power spectral density of W process 100 [ m / s 3 2 / Hz];
B. Barrel Roll Target Evasive Maneuver It is assumed that the target moves in the z direction and the barrel roll trajectory is created by accelerations in the (xy) plane.
1693
for sampling interval Ts 10 m sec , and for measurement noise level of Vg =3m. Further
ZT
aT ; VT VT
Zz
xT2 yT2
constant
Table 9.1 presents the parameters of the examples that are presented in this paper. The signal-to-noise ratio is the "power" of the target position to the noise power (half of the squared ratio of the radius of the maneuvering trajectory to the measurement noise standard deviation) S N
1 § aT · ¨ ¸ 2 ¨© ZT2 ¸¹
2
Figure 9.1.2: Deterministic Tracking error of coordinated turn target evasive maneuver.
;
V g2 and the "bandwidth" of the filter can be approximately estimated by
Zo
6
W Vgo
6
W process .
V g2Ts
Table 9.1: Parameters of the examples.
CT BREM
aT
VT
[m/s2]
[m/s]
Z7
Vg
[rad/s]
[m]
9 8
250 40
0.36 2
3 3
Zo
WT [(m/s3) 2 /Hz]
S/N [dB]
[rad/s]
100
42.5 13.5
3.2 3.2
Figure 9.1.3: Deterministic turning rate estimation error of coordinated turn target evasive maneuver.
A. Constant Turn Evading Maneuver - CT Simulation results in this section are presented for constant-coordinated turn target evasive maneuver for the parameters as presented in table 9.1. Fig. 9.1.1 presents the trajectory of the coordinated turn target maneuver. The target moves at constant velocity and at t=1sec starts a coordinated turn.
Fig. 9.1.4 presents the measurement noise norm (Msr) and the ensemble average of the tracking error based on Monte-Carlo method (based on 100 runs) for the different filters. One can see that the CA filter develops higher tracking error than the VJ and AJ filters thus demonstrating the improved performance of the VJ and AJ based filter.
Figure 9.1.4: Stochastic RMS tracking error of coordinated turn target evasive maneuver.
Figure 9.1.1: Trajectory of coordinated turn target evasive maneuver.
Fig. 9.1.5 presents the stochastic estimation error of the turning rate of the maneuver. Again the CA filter develops higher tracking error than the VJ and AJ based filters. Throughout the figures one can see the transient at t=1sec, when the target initiates the coordinated turn. Fig. 9.1.6 presents the stochastic turning rate mean estimation error and ±1 one standard deviation of the estimated turning rate thus demonstrating the estimation quality.
Fig. 9.1.2 presents the deterministic tracking error (i.e. without the presence of the measurement noise). One can see that the CA filter develops a constant tracking error while the VJ and AJ filters have zero steady state error. Fig. 9.1.3 presents the deterministic estimation error of the turning rate of the maneuver. Again the CA filter develops a constant estimation error while the VJ and AJ filters have zero steady state estimation error.
1694
constant estimation error while the VJ and AJ filters have zero steady state estimation error.
Figure 9.1.5: Stochastic turning rate mean estimation error for coordinated turn target evasive maneuver. Figure 9.2.2: Deterministic Tracking error of barrel roll target evasive maneuver.
Figure 9.1.6: Stochastic turning rate mean estimation error and ±1 one standard deviation of the estimated turning rate of coordinated turn target evasive maneuver.
Figure 9.2.3: Deterministic turning rate estimation error of barrel roll target evasive maneuver.
B. Barrel Roll Evading Maneuver - BREM Simulation results in this section are presented for barrel roll evasive maneuver. It is assumed that the target moves in the z direction and the barrel roll trajectory is created by accelerations in the (x-y) plane. The simulations are
Fig. 9.2.4 presents the stochastic tracking error. One can see that the CA filter develops higher tracking error than the VJ and AJ filters thus demonstrating the improved performance of the VJ and AJ based filter.
presented for the parameters as presented in table 9.1 . Fig. 9.2.1 presents the trajectory of the coordinated turn target maneuver. The target moves at constant velocity and at t=1sec starts a coordinated turn. Fig. 9.2.2 presents the deterministic tracking error. One can see that the CA filter develops a constant tracking error while the VJ and AJ filters have zero steady state error.
Figure 9.2.4: Stochastic RMS tracking error of barrel roll target evasive maneuver.
Figure 9.2.1: Trajectory of barrel roll target evasive maneuver. Fig. 9.2.3 presents the deterministic estimation error of the turning rate of the maneuver. Again the CA filter develops a
1695
Fig. 9.2.5 presents the stochastic estimation error of the turning rate of the maneuver. Again the CA filter develops higher tracking error than the VJ and AJ based filters. Throughout the figures one can see the transient at t=1sec, when the target initiates the coordinated turn. Fig. 9.2.6 presents the stochastic turning rate mean estimation error and ±1 one standard deviation of the estimated turning rate thus demonstrating the estimation quality. From Figs. 9.2.3-9.2.6 one can see that the VJ based estimation demonstrates faster convergence relative to the AJ based estimator.
[5]
[6]
[7]
[8]
Figure 9.2.5: Stochastic turning rate mean estimation error of barrel roll target evasive maneuver.
[9]
[10]
[11]
[12]
[13]
Figure 9.2.6: Stochastic turning rate mean estimation error and ±1 one standard deviation of the estimated turning rate of barrel roll target evasive maneuver.
[14]
X.
CONCLUSIONS
Estimation of evading target maneuvers with unknown turning rate is considered. The equations of motion take into account the rotation of the velocity and acceleration vectors as the target maneuvers. As the equations of motion are nonlinear the State Dependent Differential-Difference Riccati Equation based estimator (SDDRE) is implemented and compared to the Kalman Filter based on the constant-step acceleration target maneuver model. It is demonstrated via simulations on turning rate and barrel roll evading maneuvers that the detailed modeling of the maneuvering target based filters-estimators have improved performance with respect to the Kalman Filter based on the constant-step acceleration target maneuver model. The velocity based jerk (VJ) estimator has faster convergence rate relative to the acceleration based jerk (AJ) estimator.
[2]
[3]
[4]
[16]
[17]
[18]
[19] [20] [21]
REFERENCES [1]
[15]
Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part I: Dynamic Models, IEEE Trans. on AES, Vol. 39, No. 4, Oct. 2003, pp. 1333-1364. Also: In Proceedings of the 2000 SPIE Conference on Signal and Data Processing of Small Targets, Vol. 4048, pp. 212-234. Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part II: Motion Models of Ballistic and Space Targets, IEEE Trans. on AES, Vol. 46, No. 1, Jan. 2010, pp. 96-119. Also: In Proceedings of the 2001 SPIE Conference on Signal and Data Processing of Small Targets, Vol. 4473. Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part III: Measurement Models, In Proceedings of the 2001 SPIE Conference on Signal and Data Processing of Small Targets, Vol. 4473, pp. 423-446. Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part IV: Decision –Based Methods, In Proceedings of the 2002 SPIE
1696
[22]
[23] [24] [25]
Conference on Signal and Data Processing of Small Targets, Vol. 4728, pp. 511-534. Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part V: Multiple-Models Methods, IEEE Trans. on AES, Vol. 41, No. 4, Oct. 2005, pp. 1255-1321. Fitzgerald, R. J.: "Simple tracking filters: Closed-form solutions", IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-17, November 1981, pp. 781-785. Tahk, M. and Speyer J.L.: Target Tracking Problems Subject to Kinematic Constraints, IEEE Trans. on AC, Vol. 35, No. 3, March 1990, pp. 324-326. Aggarwal, R.K.: Optimal Missile Guidance for Weaving Targets, Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, December 1997, pp. 2775-2779. Zarchan, P. and Musoff, H.: Fundamentals of Kalman Filtering: a Practical Approach, 2nd Edition, Progress in Astronautics and Aeronautics, Vol. 208, AIAA, 2005, Chapter 10. Marks, G.M.: Multiple Models Adaptive Estimation (MMAE) for Improving Guidance Performance against Weaving Targets, AIAA Guidance, Navigation and Control Conference, 21-24 August 2006, Keystone, CO, AIAA 2006-6697 1Chen, R.H., Speyer, J.L. and Lianos, D.: Homing Missile Guidance and Estimation under Agile Target Acceleration, Journal of Guidance, Control and Dynamics, Vol. 30, No. 6, November-December 2007, pp. 1577-1589. DOI:10.25141/1.30107 Jean Dezert and Benjamin Pannetier: A PCR-BIMM filter for maneuvering target tracking, Fusion 2010, 13th International Conference on Information Fusion, 26-29 July 2010, EICC, Edinburgh, UK, Mracek,C.P., Cloutier, J.R., and D'Souza, C.A.: A New technique for Nonlinear Estimation, Proceedings of the 1996 IEEE International Conference on Control Applications, Dearborn, MI, September 15-18, 1996. Shue S.P. and Agarwal, R. K.: Extension of Kalman Filter Theory to Nonlinear Systems with Application to Wing Rock Motion, Asian Journal of Control, Vol. 2, No. 1, pp. 42-49, March 2000. Xin, M. and Balakrishnan, S.N.: A New Filtering Technique for a Class of Nonlinear Systems, Proceedings of the 41st IEEE Conference on Decision and Control, Las-Vegas, Nevada USA, December 2002. Lam, Q. M., Anderson, B. and Xin, M.: T-D Filter vs. Nonlinear Filtering Techniques in Nonlinear Target Tracking Applications, AIAA 2010 InfoTech Conference @Aerospace 2010, 20 - 22 April, 2010, Atlanta, Georgia, AIAA 2010-3521. Xin, M., Balakrishnan, S.N. and Ohlmeyer, E.J.: "Integrated Guidance and Control of Missile with T - D Method," IEEE Trans. On Control Systems Technology, Vol. 14, No. 6, November 2006, pp. 981-992. Haessig, D.A. and Friedland, B.: State Dependent Differential Riccati Equation for Nonlinear Estimation and Control, 2002 IFAC, 15th Triennial World Congress, Barcelona, Spain. Bryson, A.E. and Ho, Y.C.: Applied Optimal Control, Hemisphere, New-York, 1975. Kwakernaak, H. and Sivan, R.:" Linear Optimal Control Systems, Wiley-Interscience, 1972. Blakelock, J.H.: Automatic Control of Aircraft and Missiles, John Wiley & Sons, Inc., 1991. Asseo, S.J. and Ardila, R.J.: Sensor-Independent Target State Estimator Design and Evaluation, Proc. National Aerospace and Electronics Conference, NAECON 1982, pp. 916-924. Barton, D. K. and Ward, H. R.: "Handbook of Radar Measurement, Prentice-Hall, 1969. Papoulis, A.: Probability, Random Variables, and Stochastic Processes, McGraw-Hill, Inc., 1991. Imado, F. and Miwa, S.: Missile Guidance Algorithm against High-g Barrel Roll maneuvers, Journal of Guidance, Control and Dynamics, Vol. 17, No. 1, January-February 1994, pp. 123-128.
I.
INTRODUCTION
The issue of estimating a maneuvering target is widely treated subject. A comprehensive survey of models and estimators is presented in [1-5]. The simplest approach is to implement three independent Constant-Step acceleration filters (CA) or Exponentially Correlated Acceleration (ECA) filters [6], one filter for each coordinate. However these filters may not achieve the required performance for coordinated turn target maneuvers (CT) or barrel roll evasive maneuver (BREM) as they are not matched to these maneuvers, i.e. steady state errors are created. For more advanced estimators it has been understood that incorporating detailed information on the target dynamics and kinematics into the estimator's equations has the potential to increase the quality of estimation. However, the inclusion of more detailed target maneuver model and the related constraints lead to nonlinear models. Thus the Kalman Filter is not directly applicable. The most common approach to deal with nonlinear systems is the Extended Kalman Filter (EKF). In [7] the issue of pseudo measurements had been introduced and the Extended Kalman Filter was applied. In [7] it was pointed out that inclusion of a constraint is usually difficult to incorporate into the dynamic equation and it is much easier to incorporate them into the measurement equations.
1689
In this paper the coordinated turn evading maneuver (CT) and barrel roll evasive maneuver (BREM) with unknown turning-barrel roll rate are considered. One of the main and important issues is the estimation of the angular turning rate. This is needed for achieving better matching of the estimator and for the derivation of high performance guidance law for this type of evading maneuver [8]. In [9, chapter 10] it is shown that the algorithms that include the turning rate as part of the state are problematic, to say the least. The current approaches to estimation of nonlinear systems include many methods. A comprehensive survey of such methods applied to maneuvering target estimation is presented in [1-5]. For example: in [10] a multiple model approach is applied; in [11] algebraic constraint is incorporated in the state equations; and in [12] the IMM approach is applied; to estimate maneuvering target. Two approaches to modeling the maneuvering target equations of motion are dealt with here: i) Rotating velocity vector based approach (velocity based jerk equations - VJ); and ii) Rotating acceleration vector based approach (acceleration based jerk equations – AJ). The later approach is introduced as it requires less computational effort. The rotating velocity based approach has been used and EKF was applied in [22]. In this paper the State Dependent Differential-Difference Riccati Equation (SDDRE) based estimator is applied to the nonlinear equations of motion. The authors are unaware of any publication estimating a maneuvering target by the SDDRE method. The SDDRE approach is very intuitive, although it is not optimal as shown for the State Dependent Algebraic Riccati Equation (SDARE) approach in [13-16]. The optimal filter requires additional terms for optimality [14-17]. Albeit the sub-optimality of the direct SDDRE approach it is known that the SDDRE based estimator is BIBO stable [18]. The novelty in this paper is: Application of the SDDRE to estimation of a maneuvering target state and turning rate; Introduction of rotating acceleration vector kinematics equations of motion of a maneuvering target; Comparison of rotating velocity vector and rotating acceleration vector kinematics equations of motion application with the SDDRE based estimator on common basis. The mathematical derivations are presented in the three dimensional space, however for simplicity the simulations are presented in two dimensions, i.e. planar target's maneuver. Simulations show that estimator based on the rotating velocity
and acceleration vector kinematics equations of motion give better performance than the Kalman Filter based on constant acceleration target model. For coordinated turn target's evading maneuver the difference between the rotating velocity and acceleration vector based approaches is small. However, for barrel roll target's evading maneuver (higher turning rate) the rotating velocity vector based approach gives faster convergence of the state and turning rate estimation error. II.
PROBLEM STATEMENT AND APPROACH TO THE SOLUTION
f x(t ) *w(t ), x(to )
xo ,
g x(t ) v(t )
z (t )
where x(t) is the state vector, z(t) is the measurement, w(t), v(t) are the white Gaussian stochastic processes representing the system driving noise and the measurement noise, respectively, x(to) is a Gaussian random vector, and E[w(t)w(W)T ]= WG(t-W),E[v(t)v(W)T ]=VG(t-W),
T
@
4 ! 0.
0
(2.9)
As xˆ(t) Q(t ) Q(t ) AsT xˆ(t) *W* T Q(t )C T V 1CQ(t )
This is a suboptimal estimator. The optimal estimator has additional terms as detailed in [14,15,16]. Here the discrete version of the following SDDRE based state estimator [18] is implemented. (2.10)
Q(to )
All vectors and matrices are of appropriate dimensions. The problem being considered here is finding the optimal estimate xˆ (t ) as a functional of {z(t), to≤t≤tf} that minimizes the quadratic criterion:
>
(2.8)
The SDRE/SDARE based estimator is [13]
(2.2)
T
E >x(t ) xˆ (t )@ 4>x(t ) xˆ (t )@ ;
Cx (t ) v(t )
K xˆ(t), t Q(t )CV 1 Q (t ) As xˆ(t) Q(t ) Q(t ) AsT xˆ(t) *W*T Q(t )C TV 1CQ(t ),
E[(xo -E(xo ))(xo -E(xo )T ]=Qo .
J
z (t )
xo ,
xˆ(t) = As xˆ(t) xˆ(t)+K xˆ(t), t >z (t ) Cxˆ(t)@, xˆ(to ) = xo ,
E[w(t)v(W)T ]=0, E[w(t)xo ]=0, E[v(t)xo ]=0, T
As x(t ) x(t ) *w(t ), x(to )
K xˆ(t) Q(t )CV 1
(2.1)
E[xo ]=xo , E[w(t)]=0, E[v(t)]=0,
x (t )
xˆ(t) = As xˆ(t) xˆ(t)+K xˆ(t) >z (t ) Cxˆ(t)@, xˆ(t o ) = xo ,
The problem considered here is the state estimation of the nonlinear stochastic system x (t )
measurement of the state, i.e. g x(t ) Cx(t ) , The state equation are represented as
Qo .
The difference between the SDARE as applied in [14,15,16] , eq. (2.9), and eq. (2.10), the one implemented in this paper, is that in this paper the differential-difference Riccati equation is used for computation of the gains and not the algebraic Riccati equation (2.9) that is solved each sampling interval. III.
(2.3)
THE CONSTANT ACCELERATION TARGET MANEUVER MODEL
z (t ) Cx (t ) v(t )
For comparison in this paper we present the performance of a most common estimator of target motion. This estimator is based on the target Constant Acceleration (CA) Step target maneuver. The following dynamic model of the target [1, 2, 6] is assumed, for each inertial coordinate, the CA target maneuver model is
the solution of the preceding problem is the Kalman filter [19,20]. (2.5) xˆ(t) = Axˆ(t)+K(t)[ z (t ) Cxˆ(t)], xˆ(to ) = xo ,
ª x (t ) º d « x (t ) » « » dt «¬aT (t )»¼
A. Estimator for Linear System For the linear system x (t )
Ax(t ) *w(t ), x(to )
xo ,
(2.4)
K (t ) Q(t )CV 1 Q (t ) AQ(t ) Q(t ) AT *W*T Q(t )C T V 1CQ (t ), Q(to ) Qo . B. Estimators for Nonlinear System For nonlinear systems there are several approaches. Here the SDRE [13-16] approach is considered. The SDRE approach is based on the dual of the SDRE based nonlinear control [16]. This approach parameterizes the state equation (2.1) into the linear structure called State Dependent Coefficient Form, and is also called Extended Linearization. This approach includes the State Dependent Algebraic Riccati Equation (SDARE) based estimation and the State Dependent Differential-Difference Riccati Equation (SDDRE) based estimation. Then for linear
1690
ª0 1 0 º ª x ( t ) º ª0 º «0 0 1» « x (t ) » «0» w (t ), « »« » « » T «¬0 0 0»¼ «¬aT (t )»¼ «¬1»¼ ª x (t ) º z (t ) >1 0 0@« x (t ) » v (t ) « » «¬aT (t )»¼ where x - target-missile separation distance [m]
x
aT wT (t ) v(t )
z
(3.1)
- target-missile separation velocity [m/s] - target acceleration [m/s2] - target process driving noise (jerk) [m/s3] - target-missile separation measurement noise [m] - measured target distance[m]
The preceding assumes that the target performs an
&
ZT - angular rate of the target velocity direction (turning rate of the target) [rad/s]
evasive maneuver (a stochastic process), that is, a step acceleration maneuver of amplitude aT 0 whose initiation instant is uniformly distributed in the interval [to,tf]. Essentially this model assumes a piecewise constant acceleration target maneuver in the inertial space. The continuous shaping filter [1,2,6] of this process is
d aT dt
2) Jerk equations – VJ Second order equations of motion based on the targets velocity (4.1) are presented. As the target is maneuvering the velocity vector is rotating. The jerk (derivative of acceleration) is the given by [21,22]. This is called here Velocity based Jerk (VJ) equations of motion. The target's jerk is [21,22] (4.4)
(3.2)
wT (t ),
where the spectral density of the target maneuver (the process noise), wT (t ) , is
aT2 0 2 [ m / s 3 / Hz] [m 2 / s 5 ] tm where 2 aT 0 - target step maneuver value [m/s ], tm WT
& jT
(3.3)
t f to is the
& jT
Comprehensive survey of modeling the behavior of a maneuvering target can be found in [1-5]. Here one specific case is considered. A redundant set of assumptions with respect to maneuvering target behavior is: i. Maneuver in one plane, i.e. angular rate vector & & constant direction - ZT u ZT 0 ; Constant absolute value of target velocity, & vT constant ;
iii.
Aerodynamically controlled aircraft (velocity perpendicular to acceleration); Coordinated turn (no loss of target energy); No roll rate.
iv. v.
vT 1v
& aT
(4.1)
ZT
(4.6)
1) Jerk equations - AJ First order equations of motion based on the target's acceleration (4.1) are derived. As the target is maneuvering the acceleration vector is rotating. The jerk (derivative of acceleration) is the given by [21]. This is called Acceleration based Jerk (AJ) equations of motion. The target's jerk is given by
1) Acceleration equations First order equations of motion based on the target's velocity (4.1) are presented. As the target is maneuvering the velocity vector is rotating. The target's acceleration is given by [21]
&
aT 1a
1a -unit vector in the target acceleration direction (perpendicular to 1v) then it is possible to derive the following kinematic relations.
vT - absolute value of the target's velocity[m/s] 1v - unit vector in the target velocity direction
& & & d vT vT 1v ZT u vT dt & & vT u aT & 2 ; vT
(4.5)
where & 2 aT - target acceleration vector [m/s ] 2 aT - absolute value of the target's acceleration [m/s ]
where & vT - target velocity vector [m/s]
& aT
& & & & d vT & & & vT 1v ZT u vT 2ZT u ZT u ZT u vT dt & & & & & & & vT 1v ZT u vT 2ZT u aT ZT u ZT u vT
B. The kinematics as a function of acceleration In effort to reduce the computational effort with the SDDRE based on the Velocity based Jerk (VJ) equations of motion, as can be seen in section 6, here an acceleration based equations are derived. For aerodynamically controlled aircraft, under the aforementioned assumptions (the acceleration and velocity are perpendicular) the target's velocity's angular rate is equal to the target's acceleration angular rate. If the target's acceleration is expressed as
A. The kinematics as a function of velocity We assume that the target's velocity is expressed as
& vT
& & & & & & vT 1v ZT u vT 2 ZT u vT 1v ZT u ZT u vT
Substituting vT 1v from (4.2) gives [22]
KINEMATIC EQUATIONS OF MANEUVERING TARGET
ii.
& d 2 vT dt 2
& 3 jT - target's jerk vector [m/s ]
time interval in which the target is expected to take an evasive maneuver. IV.
& d aT dt
& d aT dt
& jT
(4.2)
where & vT
(4.3)
(4.7)
- target velocity vector [m/s]
& jT
- target jerk vector [m/s3]
ZT
- turning rate (4.3) of the target [rad/s]
&
& 2 aT - target acceleration vector [m/s ]
& & aT 1a ZT u aT
The VJ and AJ models assume piecewise constant turning
1691
ªxº « x » « » « x» « » y d « » « y » dt « » « y» «z» « » « z » « z» ¬ ¼
rate target maneuver. This is essentially piecewise constant target acceleration in the target's coordinates. THE VARIANCE AND SPECTRUM OF THE GLINT NOISE
V.
For simplicity it is assumed that the only measurement noise, v(t), is the glint noise. The standard deviation of the glint noise, Vg, for uniformly distributed reflectors, is [23] 1 2 2 (5.1) D ; D L V g2 12 S where D - the effective linear dimension of the target perpendicular to the target-missile LOS [m]. L - the linear dimension of the target perpendicular to the [m]. LOS target-missileWhen frequency agility is applied at rate of fs=1/Ts [Hz], the spectral density of (stair type random stochastic process - i.i.d. sequence) the glint is given by [24].
V go
&
ZT
(5.2)
ª xº « x » « » d « x» « » dt « y » « y » « » ¬« y¼»
INCORPORATION OF THE KINEMATIC CONSTRAINT
A. Velocity based jerk kinematic equation With velocity based jerk equations (4.4) it is assumed that & Z 0 . Thus from (4.4b) the kinematics of maneuvering target is modeled as
wT
& & & & & & d aT 2ZT u aT ZT u ZT u vT wT dt & & vT 1v Z T u vT
0 0 2Z z 0 1 0 0 0 2Z x
0 0 0 0 0 Z xZ z 0 0 0 0 0 Z zZ y 0 1 0 0 2 0 Z y Z z2
º ª x º ª0 »« » « » « x » «0 2Z y » « x» «1 » 0 » «« y »» ««0 0 » « y » «0 »« » « 2Z x » « y» «0 0 » « z » «0 »« » « 1 » « z » «0 0 »¼ «¬ z»¼ «¬0 0 0
0 0 0 0 0 1 0 0 0
0º 0» » 0» » 0» ª w x º 0» «« w y »» » 0» «¬ w z »¼ 0» » 0» 1»¼
(6.2)
ª y z z yº 1 « z x x z »; » x 2 y 2 z 2 « «¬ x y y x»¼
& & vT u aT & 2 vT
(6.3)
and in two dimensions the equations of motion reduce to
The kinematic equations in section IV are constraints that can be incorporated into the estimator equations. It is possible to incorporate the kinematic constraint into the state equation or measurement equation and the "unknown" quantities are interpreted as either a measurement noise or system driving noise. The approach here has the advantage that the kinematic constraint is incorporated into the system equations, the unknown is the system driving noise and the measurements are linear. The derivation here is in three dimensions although the simulations are performed in two dimensions.
& jT
2Z z 0 0 2Z y
0 0 0 0 0 Z xZ y 0 1 0 0 0 Z x2 Z z2 0 0 0 0 0 Z yZ z
where
where Ts is the sampling interval of the frequency agile radar. Therefore, we assume that the spectral density of the measurement noise, v(t), is Vgo >m 2 Hz@. VI.
0 1 0 0 0
ª xº « x » « » « x» « » ª1 0 0 0 0 0 0 0 0 º « y » ª v x º «0 0 0 1 0 0 0 0 0» « y » «v » « »« » « y » «¬0 0 0 0 0 0 1 0 0»¼ « y» «¬ v z »¼ «z» « » « z » « z» ¬ ¼
ª xº « y» « » «¬ z »¼ msr
2
ª § ZTs · º ¸» « sin ¨ 2 ¹» ªm2 º ªm2 º 2 V g Ts , « »; V g (Z ) V go « © , « § ZTs · » «¬ Hz »¼ ¬ Hz ¼ ¸ » « ¨ ¬ © 2 ¹ ¼
1 ª0 «0 0 « «0 Z y2 Z z2 « 0 «0 «0 0 « «0 Z x Z y «0 0 « 0 «0 «0 Z Z x z ¬
ª xº « y» ¬ ¼m
Zz
ª0 1 «0 0 « «0 Z z2 « «0 0 «0 0 « ¬«0 0
0 1 0 0 0 2Z z
0 0 0 0 0 0 0 1 0 0 0 Z z2
0 º ª x º ª0 0 »» «« x »» «0 « 2Z z » « x» «1 »« » « 0 » « y » «0 1 » « y » «0 »« » « 0 ¼» ¬« y¼» ¬«0
(6.4)
ª xº « x » « » ª1 0 0 0 0 0º « x» ªvx º «0 0 0 1 0 0 » « y » « v » ¬ ¼« » ¬ y ¼ « y » « » ¬« y¼»
x y y x . x y 2
(6.5)
2
B. Acceleration based jerk kinematic equation - AJ With acceleration based jerk equations it is not & imperative to assume that Z 0 . Therefore from (4.7) we model this as
& d aT & & ZT u aT wT dt wT aT 1a & j
(6.1)
0º 0»» 0» ª wx º »« » 0 » ¬ wy ¼ 0» » 1¼»
(6.6)
where wT represents the deviation of the actual behavior of the target from constant absolute value of the velocity assumption. We have
where wT represents the deviation of the actual behavior of the target from the constant angular turning rate and constant absolute value of the velocity assumptions. We have the state space representation in the State Dependent Coefficient Form in three dimensions
Z z Z y º ª xº ª wx º ª xº ª 0 (6.7) d « » « » y « Z z 0 Z x » « y» « wy » « » « » dt « » «¬ z»¼ «¬ Z y Z x 0 »¼ «¬ z»¼ «¬ wz »¼ The state space representation in the State Dependent Coefficient Form is in three dimensions
1692
ª xº « x » « » « x» « » y d « » « y » dt « » « y» «z» « » « z » « z» ¬ ¼
ª xº « y» « » «¬ z »¼ msr
ªZ x º «Z » « y» ¬«Z z ¼»
ª0 «0 « «0 « «0 «0 « «0 «0 « «0 «0 ¬
1 0 0 1 0 0 0 0 0 0 0 Zz 0 0 0 0 0 Zy
0 0 0 0 0 0 0 0 0
0 0 0 0 0 Zz 1 0 0 1 0 0 0 0 0 0 0 Zx
0 0 0 0 0 0 0 0 0
0 0 º ª x º ª0 0 0 » « x » «0 »« » « 0 Z y » « x» «1 » 0 0 » «« y »» ««0 0 0 » « y » «0 »« » « 0 Z x » « y» «0 1 0 » « z » «0 »« » « 0 1 » « z » «0 0 0 »¼ «¬ z»¼ «¬0
0 0 0 0 0 1 0 0 0
0º 0» » 0» » 0» ª w x º 0» «« w y »» » 0» «¬ w z »¼ 0» » 0» 1»¼
ª xº « x » « » « x» « » 1 0 0 0 0 0 0 0 0 ª º « y » ªv x º «0 0 0 1 0 0 0 0 0» « y » «v » « »« » « y » «¬0 0 0 0 0 0 1 0 0»¼ « y» «¬ v z »¼ «z» « » « z » « z» ¬ ¼
C. The trajectory The trajectory of coordinated turn and barrel roll is modeled as in [1,9,25]. It is
ª xº « y» ¬ ¼m
0 0 º ª x º ª0 0 0 » « x » «0 »« » « 0 Z z » « x» «1 »« » « 1 0 » « y » «0 0 1 » « y » «0 »« » « 0 0 ¼ ¬ y¼ ¬0 ª xº « x » « » ª1 0 0 0 0 0º « x» ªv x º « 0 0 0 1 0 0 » « y » «v » ¬ ¼« » ¬ y ¼ « y » « » ¬« y¼» 1 0 0 1 0 0 0 0 0 0 0 Zz
0 0 0 0 0 0
zT
0
aT cos Z z t M
xT
xT 0 xT 0t
yT
yT 0 yT 0t
zT
zT 0 zT 0t ,
(6.8)
(7.1)
aT
sin Z z t M ,
Z z2 aT
Z z2
(7.2)
cos Z z t M ,
where (x,y,z) are the target's coordinates and the rest of the variables are self evident. VIII. THE SDRE BASED DISCRETE ESTIMATOR
ª yz zyº 1 « zx xz» » x 2 y 2 z 2 « ¬« xy yx¼»
ª0 «0 « «0 « «0 «0 « ¬0
aT sin Z z t M
then
We consider the discrete nonlinear stochastic system in the State Dependent Coefficient Form
(6.9)
x(t 1)
0º 0» » 0 » ª wx º »« » 0» ¬ w y ¼ 0» » 1¼
A x(t ) x(t ) *wd (t ), x(to )
xo ,
(8.1)
z (t ) Cx (t ) vd (t )
Note by comparing Eqns' (6.2, 6.4) to (6.8, 6.10) that the computational effort of AJ based estimator is smaller than the computational effort of VJ based estimator. In two dimensions the equations of motion are ª xº « x » « » d « x» « » dt « y » « y » « » ¬ y¼
xT yT
where x(t) is the state vector, z(t) is the measurement, u(t) is the input, w(t), v(t) are the white Gaussian stochastic sequences representing the system driving noise and the measurement noise, respectively, xo is a Gaussian random vector, and
E[xo ]=xo , E[wd (t)]=0, E[v(t)]=0,
(6.10)
E[wd (t)wd (W)T ]= Wd G tW ,E[vd (t)vd (W)T ]=Vd G tW ,
(8.2)
E[wd (t)vd (W)T ]=0, E[wd (t)xo ]=0, E[vd (t)xo ]=0, T
T
E[(xo-E(xo ))(xo-E(xo )T ]=Qo . All vectors and matrices are of appropriate dimensions. The State Dependent Difference Riccati Equation (SDDRE) based estimator is, k=0,1,2,…k-1, is adaptation of the State Dependent Differential Riccati Equation (SDDRE) (8.3)
xˆ(k 1 ) = A xˆ(k) xˆ(k) +K(k)[ z (k ) Cxˆ(k)], xˆ(to ) = xo , 1 ~ ~ K (k ) Q (k )C T CQ (k )C T Vd ~ T Q (k 1) A xˆ(k) Q(k ) A xˆ(k) *Wd *T , Q(ko ) Qo . / predictor prior / ~ Q(k ) >I K (k )C @Q (k ) / corrector posterior /
VII. TARGET MANEUVER MODEL A. Coordinated Turn Target Evasive Maneuver The target moves with constant velocity. At certain moment it starts a coordinated turn (constant acceleration and constant speed (absolute value of the velocity) at constant altitude, i.e. in the (x-y) plane only.
IX.
SIMULATION RESULTS
In this section the performance of the constant acceleration (CA) (3.1), the velocity based jerk (VJ) equation based filters (4.5, 6.4) and the acceleration based jerk (AJ) equation based filters (4.7, 6.11) is compared via simulations. In effort to perform the comparison on common basis all simulation are performed for system driving noise (process noise) with power spectral density of W process 100 [ m / s 3 2 / Hz];
B. Barrel Roll Target Evasive Maneuver It is assumed that the target moves in the z direction and the barrel roll trajectory is created by accelerations in the (xy) plane.
1693
for sampling interval Ts 10 m sec , and for measurement noise level of Vg =3m. Further
ZT
aT ; VT VT
Zz
xT2 yT2
constant
Table 9.1 presents the parameters of the examples that are presented in this paper. The signal-to-noise ratio is the "power" of the target position to the noise power (half of the squared ratio of the radius of the maneuvering trajectory to the measurement noise standard deviation) S N
1 § aT · ¨ ¸ 2 ¨© ZT2 ¸¹
2
Figure 9.1.2: Deterministic Tracking error of coordinated turn target evasive maneuver.
;
V g2 and the "bandwidth" of the filter can be approximately estimated by
Zo
6
W Vgo
6
W process .
V g2Ts
Table 9.1: Parameters of the examples.
CT BREM
aT
VT
[m/s2]
[m/s]
Z7
Vg
[rad/s]
[m]
9 8
250 40
0.36 2
3 3
Zo
WT [(m/s3) 2 /Hz]
S/N [dB]
[rad/s]
100
42.5 13.5
3.2 3.2
Figure 9.1.3: Deterministic turning rate estimation error of coordinated turn target evasive maneuver.
A. Constant Turn Evading Maneuver - CT Simulation results in this section are presented for constant-coordinated turn target evasive maneuver for the parameters as presented in table 9.1. Fig. 9.1.1 presents the trajectory of the coordinated turn target maneuver. The target moves at constant velocity and at t=1sec starts a coordinated turn.
Fig. 9.1.4 presents the measurement noise norm (Msr) and the ensemble average of the tracking error based on Monte-Carlo method (based on 100 runs) for the different filters. One can see that the CA filter develops higher tracking error than the VJ and AJ filters thus demonstrating the improved performance of the VJ and AJ based filter.
Figure 9.1.4: Stochastic RMS tracking error of coordinated turn target evasive maneuver.
Figure 9.1.1: Trajectory of coordinated turn target evasive maneuver.
Fig. 9.1.5 presents the stochastic estimation error of the turning rate of the maneuver. Again the CA filter develops higher tracking error than the VJ and AJ based filters. Throughout the figures one can see the transient at t=1sec, when the target initiates the coordinated turn. Fig. 9.1.6 presents the stochastic turning rate mean estimation error and ±1 one standard deviation of the estimated turning rate thus demonstrating the estimation quality.
Fig. 9.1.2 presents the deterministic tracking error (i.e. without the presence of the measurement noise). One can see that the CA filter develops a constant tracking error while the VJ and AJ filters have zero steady state error. Fig. 9.1.3 presents the deterministic estimation error of the turning rate of the maneuver. Again the CA filter develops a constant estimation error while the VJ and AJ filters have zero steady state estimation error.
1694
constant estimation error while the VJ and AJ filters have zero steady state estimation error.
Figure 9.1.5: Stochastic turning rate mean estimation error for coordinated turn target evasive maneuver. Figure 9.2.2: Deterministic Tracking error of barrel roll target evasive maneuver.
Figure 9.1.6: Stochastic turning rate mean estimation error and ±1 one standard deviation of the estimated turning rate of coordinated turn target evasive maneuver.
Figure 9.2.3: Deterministic turning rate estimation error of barrel roll target evasive maneuver.
B. Barrel Roll Evading Maneuver - BREM Simulation results in this section are presented for barrel roll evasive maneuver. It is assumed that the target moves in the z direction and the barrel roll trajectory is created by accelerations in the (x-y) plane. The simulations are
Fig. 9.2.4 presents the stochastic tracking error. One can see that the CA filter develops higher tracking error than the VJ and AJ filters thus demonstrating the improved performance of the VJ and AJ based filter.
presented for the parameters as presented in table 9.1 . Fig. 9.2.1 presents the trajectory of the coordinated turn target maneuver. The target moves at constant velocity and at t=1sec starts a coordinated turn. Fig. 9.2.2 presents the deterministic tracking error. One can see that the CA filter develops a constant tracking error while the VJ and AJ filters have zero steady state error.
Figure 9.2.4: Stochastic RMS tracking error of barrel roll target evasive maneuver.
Figure 9.2.1: Trajectory of barrel roll target evasive maneuver. Fig. 9.2.3 presents the deterministic estimation error of the turning rate of the maneuver. Again the CA filter develops a
1695
Fig. 9.2.5 presents the stochastic estimation error of the turning rate of the maneuver. Again the CA filter develops higher tracking error than the VJ and AJ based filters. Throughout the figures one can see the transient at t=1sec, when the target initiates the coordinated turn. Fig. 9.2.6 presents the stochastic turning rate mean estimation error and ±1 one standard deviation of the estimated turning rate thus demonstrating the estimation quality. From Figs. 9.2.3-9.2.6 one can see that the VJ based estimation demonstrates faster convergence relative to the AJ based estimator.
[5]
[6]
[7]
[8]
Figure 9.2.5: Stochastic turning rate mean estimation error of barrel roll target evasive maneuver.
[9]
[10]
[11]
[12]
[13]
Figure 9.2.6: Stochastic turning rate mean estimation error and ±1 one standard deviation of the estimated turning rate of barrel roll target evasive maneuver.
[14]
X.
CONCLUSIONS
Estimation of evading target maneuvers with unknown turning rate is considered. The equations of motion take into account the rotation of the velocity and acceleration vectors as the target maneuvers. As the equations of motion are nonlinear the State Dependent Differential-Difference Riccati Equation based estimator (SDDRE) is implemented and compared to the Kalman Filter based on the constant-step acceleration target maneuver model. It is demonstrated via simulations on turning rate and barrel roll evading maneuvers that the detailed modeling of the maneuvering target based filters-estimators have improved performance with respect to the Kalman Filter based on the constant-step acceleration target maneuver model. The velocity based jerk (VJ) estimator has faster convergence rate relative to the acceleration based jerk (AJ) estimator.
[2]
[3]
[4]
[16]
[17]
[18]
[19] [20] [21]
REFERENCES [1]
[15]
Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part I: Dynamic Models, IEEE Trans. on AES, Vol. 39, No. 4, Oct. 2003, pp. 1333-1364. Also: In Proceedings of the 2000 SPIE Conference on Signal and Data Processing of Small Targets, Vol. 4048, pp. 212-234. Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part II: Motion Models of Ballistic and Space Targets, IEEE Trans. on AES, Vol. 46, No. 1, Jan. 2010, pp. 96-119. Also: In Proceedings of the 2001 SPIE Conference on Signal and Data Processing of Small Targets, Vol. 4473. Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part III: Measurement Models, In Proceedings of the 2001 SPIE Conference on Signal and Data Processing of Small Targets, Vol. 4473, pp. 423-446. Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part IV: Decision –Based Methods, In Proceedings of the 2002 SPIE
1696
[22]
[23] [24] [25]
Conference on Signal and Data Processing of Small Targets, Vol. 4728, pp. 511-534. Li, X.R. and Jilkov, V.P.: Survey of Maneuvering Target Tracking. Part V: Multiple-Models Methods, IEEE Trans. on AES, Vol. 41, No. 4, Oct. 2005, pp. 1255-1321. Fitzgerald, R. J.: "Simple tracking filters: Closed-form solutions", IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-17, November 1981, pp. 781-785. Tahk, M. and Speyer J.L.: Target Tracking Problems Subject to Kinematic Constraints, IEEE Trans. on AC, Vol. 35, No. 3, March 1990, pp. 324-326. Aggarwal, R.K.: Optimal Missile Guidance for Weaving Targets, Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, December 1997, pp. 2775-2779. Zarchan, P. and Musoff, H.: Fundamentals of Kalman Filtering: a Practical Approach, 2nd Edition, Progress in Astronautics and Aeronautics, Vol. 208, AIAA, 2005, Chapter 10. Marks, G.M.: Multiple Models Adaptive Estimation (MMAE) for Improving Guidance Performance against Weaving Targets, AIAA Guidance, Navigation and Control Conference, 21-24 August 2006, Keystone, CO, AIAA 2006-6697 1Chen, R.H., Speyer, J.L. and Lianos, D.: Homing Missile Guidance and Estimation under Agile Target Acceleration, Journal of Guidance, Control and Dynamics, Vol. 30, No. 6, November-December 2007, pp. 1577-1589. DOI:10.25141/1.30107 Jean Dezert and Benjamin Pannetier: A PCR-BIMM filter for maneuvering target tracking, Fusion 2010, 13th International Conference on Information Fusion, 26-29 July 2010, EICC, Edinburgh, UK, Mracek,C.P., Cloutier, J.R., and D'Souza, C.A.: A New technique for Nonlinear Estimation, Proceedings of the 1996 IEEE International Conference on Control Applications, Dearborn, MI, September 15-18, 1996. Shue S.P. and Agarwal, R. K.: Extension of Kalman Filter Theory to Nonlinear Systems with Application to Wing Rock Motion, Asian Journal of Control, Vol. 2, No. 1, pp. 42-49, March 2000. Xin, M. and Balakrishnan, S.N.: A New Filtering Technique for a Class of Nonlinear Systems, Proceedings of the 41st IEEE Conference on Decision and Control, Las-Vegas, Nevada USA, December 2002. Lam, Q. M., Anderson, B. and Xin, M.: T-D Filter vs. Nonlinear Filtering Techniques in Nonlinear Target Tracking Applications, AIAA 2010 InfoTech Conference @Aerospace 2010, 20 - 22 April, 2010, Atlanta, Georgia, AIAA 2010-3521. Xin, M., Balakrishnan, S.N. and Ohlmeyer, E.J.: "Integrated Guidance and Control of Missile with T - D Method," IEEE Trans. On Control Systems Technology, Vol. 14, No. 6, November 2006, pp. 981-992. Haessig, D.A. and Friedland, B.: State Dependent Differential Riccati Equation for Nonlinear Estimation and Control, 2002 IFAC, 15th Triennial World Congress, Barcelona, Spain. Bryson, A.E. and Ho, Y.C.: Applied Optimal Control, Hemisphere, New-York, 1975. Kwakernaak, H. and Sivan, R.:" Linear Optimal Control Systems, Wiley-Interscience, 1972. Blakelock, J.H.: Automatic Control of Aircraft and Missiles, John Wiley & Sons, Inc., 1991. Asseo, S.J. and Ardila, R.J.: Sensor-Independent Target State Estimator Design and Evaluation, Proc. National Aerospace and Electronics Conference, NAECON 1982, pp. 916-924. Barton, D. K. and Ward, H. R.: "Handbook of Radar Measurement, Prentice-Hall, 1969. Papoulis, A.: Probability, Random Variables, and Stochastic Processes, McGraw-Hill, Inc., 1991. Imado, F. and Miwa, S.: Missile Guidance Algorithm against High-g Barrel Roll maneuvers, Journal of Guidance, Control and Dynamics, Vol. 17, No. 1, January-February 1994, pp. 123-128.
