IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL AC-27, NO. 6, DECEMBER 1982. E;'B,,*F-'= YF-1 + E;'Z*. ( 16) with YF- I strictly proper. Then VI = W, + ...
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IEEE TRANSACTIONS
+ E;‘Z* with YF- I strictly proper. Then VI = W, + E ; ’ Z ,,where E;‘B,,*F-’=
YF-1
w~=Y~A-’-E(PA~+QB~)-~QA-~
( 16)
(17)
and similarly V2 = W2+ E ; ’ Z , where
W~=YF-I-E(PA~+QB~)-’RF-~.
( 18)
The cost can then be given the form p
(Vo)
+ (( WlC - W2G + V3S + V4),(
( 19)
Y=R.
Relation (7) gives M,Dl + N T B l = I . But NT = E-IR due to (6) and (12). so that M T = E - IT where T is a polynomial matrix such that TD, + R B I = E .
(20)
Further, (16) implies via the second relation of (19) that
E,R
+Z,F=
B1,9.
Postmultiplying this equation by [ I B,] and employing (20). results in (13). Concerning the second part of the proof, the cost (10) will converge iff the sequences @ u and 9 e both converge. Since Fo is a stable polynomial matrix by assumption, it follows from (9) that u and e converge for every C , G. and S iff the rational matrices K,,N , , N,, and L , , M I , M T l are all stable. Their stability is, in turn, equivalent to the existence of a (stable) spectral factor E , as can be concluded from (4), (6) and (12). ( 5 ) and (20). (7). The spectral factor itself exists iff every HurwiL? common right factor of @ A l and 9 B l is stable; see Kukra [9]. The stability of M I .X , then implies the asymptotic internal stability of the tracking system.
AC-27,NO. 6, DECEMBER 1982
Finally. let us note that if Fo is unimodular (i.e.. F is a right divisor of A ) , then ( I 3) can be replaced by (20). which is a significant simplification of the design procedure. The polynomial approach offers an alternative to the standard statespace design methods. It is particularly attractive for systems with inaccessible states, for it obviates the design of observers for the plant and for the reference. Instead,the optimal controller is found as a single djnamical system arhose realization is left to the designer. This realization is limited only by the requirement of internal stability. i.e., the controller must be realized without unstable hidden modes.
WlC - W2G + V3S + V4))
where V4 = E ; ’ ( Z I , C + Z , G ) . Clearly, W , , W,, and V3 depend on the controller, whereas Vo and V4 depend just on the given data. A detailed analysis shows that p attainsits minimum, mith the controller being independent of C, G , and S , iff Wl = 0. W2 = 0. and V3 = I. Relation (15) then immediately implies (12). To obtain (13). first observe that ( 17) and (1 8) yield Yl=Q,
ON AUTOMATIC CONTROL, VOL
REFERENCES K. J. ktr6m. Inrroduction ro Stochastic Conrrol Theof?‘. New York: Academic, 1970. G . Bengtsson. “Output regulation and internal models-A frequency domain approach.” Auromanca. vol. 13, pp. 333-345. 1977. B. A. Francis. ”The multivariable servomechanism problem from the input-output viewpoint.” IEEE Tram. Auromar. Conrr.. vol. AC-22. pp. 322-328. 1977. B. A. Francis and W . M. Wonham. “The internal model principle for linear multivariable regulators.” Appl. Math. Oprrmir., vol. 2, pp. 170- 194, 1975. T. Kailath. Linear Sysrem. E n g l e w d Cliffs,NJ: Prentice-Hall. 1980. V. Kukra, Discrere finearConrrol-ThePo/vnonlial Equanon Approach. New York: Wile?. 1979. - “Discrete stochastic regulation and tracking,” Kvbernerrka, vol. 16, pp. 263-272. 1980. “A dead-beat servo problem,” Inr. J . Conrr., vol. 32. pp. 107- 113. 1980. - “New results in state estimation and regulation,” AuIomarzca. vol. 17. pp. 745-748.1981. H. H.Rosenbrock. Srale-Spoceand Mulrrrulriable Theon.. New York: Wile).. 1970. M. Sebek. “Multivariable deadbeat servo problem,” K~bernerika.vol. 16. pp. 442-453. 1980. “Optimal tracking via polynomial matrix equations.” I n t . J . Sysr. Sci., vol. 12, pp. 357-369, 1981. M Sebek and V . Ku&a “Matrix equations arising in regulator problems,” Kybernerrh. vol. 17. pp. 128-139. 1981. L. N. Volgin. The Fundamenralr of rhe Theov of Controlling Machrnes (in Russian). Moscow. U.S.S.R.: Soviet Radio. 1%2. W. A. Wolovich. &near Mulrroariable S~srem. New York: Springer-Verlag. 1974. W. A. Wolovich and P. Ferreira “Output regulation and tracking in linear mdtivariable wstems.” IEEE Tram. Auromar. Conrr.. vol. AC-24, pp 460-465, 1979. W. MY Wonham. Linear Mulriwriable Conrrol- A Geomernc Approach. New York: Springer-Verlag 1974.
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Measurement Correlation for Multiple Sensor Tracking in a Dense Target Environment
c. B. C
W G AND L.c. YOUENS
CONCLUDING &-MARKS A new solution of the quadratic optimal tracking problem in linear
multivariable discrete-time systems has been presented. The approach is based on polynomial input-output models and the design procedure is reduced to spectral factorization and the solution of two linear equations in polynomial matrices. There are efficient algorithms to implement all steps of the design procedure. These algorithms are described in [6]. The optimal controller has been shown to consist of a reference feedfomard and an output feedback. Thus the reference and the plant output are each processed in a different way; the classical error-actuated controllers are obtained as a special case when forcing Q = R. Moreover. the feedback part of the controller turns out to be independent of the reference, and thus represents the optimal regulator of the effect of the plant’s initial state. This part is specified by (12) which can be interpreted as the assignment of desired pole locations given by the spectral factor. The feedfomard part is specified by (13) which can be thought of as adjusting the zeros of the optimal tracking system. The theoqand solution of these matrix polynomial equations is treated in [13]. The assumption that Fo be stable is necessary only for a positive definite @.If 0 is merely nonnegative definite, the tracking problem may well have a solution for an unstable F,. The related problems of tracking, output regulation, and internal stability were studied from the general point of view in [17], [4], [3], [2], and [I61 using the internal model principle. The particular results pertinent to the quadratic tracking are currently under investigation and will be reported in a futurepaper.
Abstract-In this note, we describe an algorithm for correlating measurements from several sensors. This is a problem area in multiple sensor tracking in a dense target environment. It is shown that the correlation problem is similar to the assignment problem in operation research with assignment penalties being equal to the sufficient statistic of the generalized likelihood ratio test.
I. INTRODUCTION In many target tracking applications. one is confronted with the problem of multiple sensors simultaneously tracking a number of targets. This situation brings up at least three problem areas: 1) the identification of a time sequence of measurements associated with the same target, 2) the identification of measurements from several sensors associated with the same target, and 3) efficient algorithms for processing multiple sensor data after both 1) and 2) have been completed. The first problem above is often referred to as the problem of multiobject tracking and has received considerable attention in recent years (see, for example, [ I ] and [2]). The second problem is the subject of this note. Manuscript recei\*ed February 9, 1981; revised September 30. 1981. November 2, 1981, February 1982. and May 6, 1982. This work was supported by the Department of the Army. The United States Government assumes no responsibility for the information presented The authors are with Lincoln Laboraton.. Massachusetts Institute of Technology, Lexington. MA 02173.
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0018-9286/82/1200-1250$00.75 81982 IEEE
IEEE TRANSACTIONS ON AUTOMATIC C O ~ O L VOL. ,
AC-27,NO. 6, DECEMBER 1251 1982
Problem 3) above was the subject of[3] where Kalman filter configurations for multiple s e m r data processing were studied in detail. In this note, we show that the sensor-to-sensor measurement correlation problem is the same as the classical assignment problem in operations research. The correlation algorithm is presented in the next section. An example is given in Section 111.
therefore use this fact to prescreen / , j for rejecting very unlikely correlations. Consider the case where one would like to decide wrhich y, is correlated with a particular zJ. For a fixed j , the generalized multiple hypothesis testing procedure will select they, satisfymg ,=I
11. CORRELATION
A. The Generalized Likelihood Ratio Test
Let yij denote the j t h measurement of the ith sensor, assuming that there are M sensors taking measurements on the same N targets. The problem where the target sets observed by each sensor are not identical will be discussed later. We note that the term “measurement” should be interpreted in a more general sense. For example, y f j may represent a state estimate of the j t h target from the ith sensor. In another example, a subset of yz, may be actual measurements and the remaining subset are predicted measurements from target state estimators. This second case is the same as the track maintenance problem of multiple sensor tracking in a dense target environment. We assume that all yij’s are in the same coordinate system. For a specific application, the required transformation for satisfying this assumption can be defined. Let Hk denote the hypothesis that the same target has generated measurements ( yij,; i = 1;. . , M } . There is therefore a total of N M possible hypotheses. Let x denote the true measurement vector corresponding to Hk; then the generalized likelihood ratio test requires the following decision function:
where the above decision function is obtained by using the maximum likelihood estimate of x, ie., the i satisfying
in the density p ( X,,; i = 1; . .,M / x ) . Equations (2.1) and (2.2) form the sufficient statistic for correlating measurements from multiple sensors. If the measurement noise vector is independent between sensors and follows a Gaussian density function with zero mean and covariance P f j , then (2.1) becomes I
I
min ... I
.
for a givenj.
jq
ALGORITHM
M
\
r=l
I
where
If one simply repeats the above procedure for all j , then ambiguous situations may arise. This is because the same measurement of a given sensor may be selected as correlated with several measurements of the other sensor, while some measurements may be declared uncorrelated completely. This situation maybe possible when sensors do not have perfect resolution, e.g., a sensor may report only one measurement for several closely spaced targets while the second sensor has already resolved these targets. Situations like this can be treated by considering the properties of the chi-square distribution. For the case that unambiguous matches between y; and zJ must be made (i.e., the completely resolved case), this gives rise to the classical assignment problem.’
E . The Assignment Problem
In the classical assignment problem of operations research, the l i j may represent the penalty (or payoff) of assigning the ith person to work on the jth job. The optimum assignment is selected as these entries giving min
i. j ;,
I;,
with the constraint that each column/row can be assigned to a row/column only once. A trivial approach to the above optimization problem is to enumerate all possible combinations and choose the one with minimum sum. This results in N! trials. A classical method called the Hungarian algorithm which requires substantially fewer operations has been known for many years in the field of combinatorial programming. A particular method of implementing the Hungarian algorithm given by Munkre [4] requires at most (1 1 N3 + 12N2 +31N)/6 operations representing a substantial savli ing for large N ’ s . The details of the Hungarianflunkre algorithm w not be repeated here. The above discussions illustrate the problem of correlating measurements from mo sensors. For more than two sensors, the computational load increases rapidly although the problem is conceptually straightforward. In a multiple sensor problem, one first computes the sufficient statistic using (2.1), (2.2). Let it be denoted as / j l , , 2 , ; 3 ...., where A4 is the as an entry of a total number of sensors. One cannow visualize the “super matrix” with dimension ( N X N X N X . . . X N) where there are MN’s. The correlation solution is given by the combination of mutually independent entries achieving the minimum sum. It is interesting to note that a corresponding Hungarian/Munkre algorithm for the aforementioned problem does not seem to be available yet. An exhaustive search method will require enumerating (N!)”possibilities and this number looks prohibitive even for a modest range of N and M .
’
Notice that the 2 above is the maximum likelihood estimate of x assuming that all yij,’s are correlated. When this is true, it is the compressed measurement discussed in [3]. Equations (2.3),(2.4)give the sufficient statistic for the Gaussian measurement noise case. Consider a simple case where there are two sensors tracking with Gaussian measurement noise. Letting yz and z j denote measurements of the first and second sensors with covariances P, and X,, respectively, one can obtain the sufficient statistic for the two sensor case as [5]
I t .l . = ( y . I- t .
,>[ 0.9 is rather small. REFERENCE$ [ I ] Y . Bar-Shalom.“Tracking methodsin amulti-targetenvironment.” IEEE Tram. Auromor. Conrr.. vol. AC-23, pp. 618-626. Aug. 1978. [2] D Reid. “ A n algorithmfor uacking multipletargets.” I E E E Tmm. Auromur. Conrr.. vo1. AC-24. pp. 843-854. Dec. 1979. [3] D . Willner. C . B. Chang. and K. P. Dunn.“Kalman filter algorithms for a multi-sensor system.” in Proc. 1976 I E E E Con/. Decisron Conrr.. Dec. 1976. [4] J. Munkres. “Atgorilhms fortheassignmentandtransportationproblems.’‘ S I A M 1.. r.01. 5 , pp. 32-38. Mar. 1957 [5] C. B. Chang and L. C. Y o u m . “Measurement correlation for multiple sensor tracking [6]
inadensetargetenvironment.”Lincoln Lab, M.I.T.. Cambridge. MA. Tech. Rep. 549. Jan. 1981. C.B. Chang.“Ballistictralecton.estimation with angle-onlymeasurements.” IEEE Trum. Auromur. Conrr.. vol. AC-25. pp. 474-480. June 1980.
IterativeLeast-SquaresParameterEstimationfor Pulse Response and Output Disturbance
ARMA
P. LOHNBERG AND G. H. J. WISSELINK Absiracf --% parameters of an ARMA system and of additive ARMA output disturbance VAII mutually different poles wereestimated from a record of the disturbed output on a single pulseinput. Alternating the steps of iterative inverse filtering with iterations according to generalized least squaresappearedveryeconomical for this system. This procedure converged for second-ordersimulations. Statistics of theestimates were evaluated from these simulation results for two signal-to-noise ratios. Manuscript received September 24, 1981: revised March 29, 1982 and May 27. 1982. P Lbhnberg is withtheDepartment of ElectricalEngineering. Tuente University of Technologg. 7500 A E Enschede. The Netherlands. G. H. J. Wisselink is with Vitatron Medical B.V.. 6950 A B Dieren. The Netherlands.
0018-9286/82/1200-1252S00.75 91982 IEEE
