On a decoupled multitarget tracking algorithm

SL+l=AS; where
as initial estimations and covariances of the decoupled filters. If & / k = { E ; j ( k ) ) ~ , ~ , E j j ( k{e;j(k)}2,2 = ) is the covariance matrix of the Kalman filter of (1) and Pk/k = diag(Pl(k),PZ(k), ...,P N ( ~ ) ) that of the is decoupled filters of (6), a relation between these two matrices was given as
Manuscript received May 12, 1989. IEEE Log No. 36741.
0018-9251/90/0700-01$1.00 @ 1990 IEEE
In some tracking applications, one may be confronted with the problem of tracking simultaneously several maneuvering targets with correlated measurements and maneuvers. The correlated maneuvers may be produced by a formation of aircraft or during air combat and the correlated measurements may be the result of a sensor itself when this single sensor simultaneously takes measurements on the targets. If the dynamic model of each target has dimension n, to track N targets simultaneously, the direct use of a Kalman filter will require handling a dynamic model with dimension n N . In view of the computational burden and the reliability of the tracking system, it is not suitable for real tracking application, to simultaneously track several targets with a single filter. There are two types of completely different ways to deal with the problem, i.e., group tracking filtering scheme [ l ]and decoupled tracking filtering approach. A group tracking filter tracks multiple targets as a group and is not interested in each individual target specially. Conversely, by the decoupled filtering approach, N targets are simultaneously tracked by N independent filters, each with the dimension identical to that of a single target. Under this investigation, we consider the decoupled tracking filtering approach. We notice that the correlation in target maneuvers and the correlation in sensor measurements reflect non-zero off-diagonal terms, respectively, in corresponding covariance matrices Q and R of normal description of N target dynamics. Based on the technique of simultaneous diagonalization of Q and R [2], decoupled system model has been obtained a and a decoupled tracking filtering approach has been proposed in [3].It is shown below that because of internal correlation in the original system the Kalman filter of the decoupled system is not, in general, decoupled and hence, the decoupled filters proposed in [3]are suboptimal and, in fact, not Kalman filters. However, it is believed that this decoupled filtering scheme is much better than a simple decoupling approach which ignores the correlations in maneuvers and measurements. We show below that the decoupled filters proposed in [3]will converge asymptotically to the stable version of the Kalman filter, if the Kalman filter is uniformly asymptotically stable. At the end of Section 111, some conclusions on the asymptotical stability of the decoupled filters are presented.
CORRESPONDENCE
1
11.
PROBLEM FORMULATION
For as in [3], we focus Our investigation on one space dimension. Let the dynamics of all targets be modeled as
+ BwL Y ; + ~ CSL+l + vL+], =
Transactions on Aerospace and Elecfmnic Systems to deal with
the problem of tracking multiple targets with correlated
measurements and maneuvers. It is proved that the decoupled filters are, in general, suboptimal and are not, in fact, Kalnian filters. However, it is shown also that if the standard Kalman filter is asymptotically stable the decoupled filters will converge asymptdically to the stable version of the standard Kalman
I.
INTRODUCTION
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ROBERT J. KELLY Allied-Signal Aerospace Company Bemlix Communications Division 1300 East Joppa Road Towson, MD 21204
REFERENCES [l] Kelly, R. J. (1990) Reducing geometric dilution of precision using ridge regression. IEEE Eansactwns on Aerospace and EIectronic Systems, AES-26, 1 (Jan. 1990), 154-168. Hoerl, A., and Kennard, R. (1970) Ridge regression and bias estimation for nonorthogonal problems. Echnmetrics, 12 (Feb. 1970). Agee, W. S., and nmer, R. H. The use of Ridge regression in trajectory estimation. Proceedings of the Twenty-Shh Conference on the Design of Eqeriments in Army Research, Development and Testing, ARO Report 81-2. Kelly, R. J. (1990) GDOP, Ridge regression and the Kalman filter. The Joumal of the Institute of Navigation UK., (Sept. 43 19) 90.
All references to “least mean squares (LMS)” should read “least squares (LS).” In the footnote 1, page 154, change all references to eigenvalue (s) to singular value (5). There i a calculation error in (14). s Use instead (32) with A$ = 0. Change the equation above (18) to read ‘ ‘ ~ = a2/a: for i = 1,p.”The first i n and second terms i (29) should read “pTP,TG;P~p” and “2PTP;fG;HTAB.” Delete the 2 in (33). Change “eigenvalue” in the paragraph below (30) to read “singular yalue.” Change (36) to read “e WS(0,R).” Change “PK” to “PK”in the line below (43).