Unified Optimal Linear Estimation Fusion Part I: Unified Models and Fusion Rules

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1 Unified Optimal Linear Estimation usion Part I: Unified Models usion Rules Rong Li Dept of Electrical Engineering University of New Orleans New Orleans, LA Phone: , ax: -30 xliunoedu unmin Zhu Dept of Mathematics Sichuan University Chengdu, Sichuan , P R China ymzhuscueducn Chongzhao Han Electronic & Information Engr i an Jiaotong University i an, Shaanxi 71004, P R China czhanxjtueducn Abstract This paper deals with data fusion for the purpose of estimation Three fusion architectures are considered: centralized, distributed, hybrid A unified linear model general framework for these three architectures are established Optimal fusion rules in the sense of best linear unbiased estimation (BLUE), weighted least squares (WLS), their generalized versions are presented for cases with either complete, incomplete, or no prior information These rules are much more general flexible than previous results or example, they are in a unified form that are optimal for all the three fusion architectures with arbitrary correlation of local estimates or observation noises across sensors or across time They are also in explicit forms convenient for implementation The relationships among these rules are also presented Keywords: Estimation fusion, track fusion, fusion rules, BLUE, least squares, MMSE estimation 1 Introduction Estimation fusion, or data fusion for estimation, is the problem of how to best utilize useful information contained in multiple sets of data for the purpose of estimating an unknown quantity a parameter or process These data sets are usually obtained from multiple sources (eg, multiple sensors) Evidently, estimation fusion has wide-spread applications since many practical problems involve data from multiple sources One of the most important application areas of estimation fusion is target tracking using multiple (homogeneous or heterogeneous) sensors, where it is more commonly referred to as track fusion or track-to-track fusion The multiple sets of data could alsobe collected from a single source over a time period With this wider view of estimation fusion, the conventional problems of filtering, prediction, particularly smoothing of an unknown process are special estimation fusion problems In fact, some of the results presented in this paper have been applied to the development of optimal linear smoothers for linear Supported in part by ONR via grant N , NS via Grant ECS-73428, LES via Grant (16 )-RD-A-32 Supported in part by The National Key Project Natural Science oundation of China systems with arbitrarily colored (not necessarily Markov) cross-correlated process measurement noise [12] There are two basic fusion architectures: centralized decentralized (or distributed) (referred to as measurement fusion estimate fusion in target tracking, respectively), depending on whether raw data are sent to the fusion center or not They have pros cons in terms of optimality, channel requirements, reliability, survivability, information sharing, etc Centralized fusion is nothing but a conventional estimation problem with distributed data Distributed fusion is more challenging has been a focal point of most fusion research for many years Estimation/track fusion has been investigated for more than two decades Many results have been obtained (see, eg, [,, 14]) While these results do represent major progress, to the authors knowledge, they are still rather limited in several aspects or example, it appears that a systematic approach to the development of distributed fusion rules is still lacking Most of the existing optimal fusion formulas for distributed fusion were obtained by ingenious but ad hoc manipulations of formulas of local estimates so that they are equivalent to the corresponding centralized fusion formulas (see, eg, [, 4, 1]) Many of these results rely on the celebrated Kalman filters for the centralized fusion architecture An obvious drawback of this ad hoc approach is that the fusion rules thus developed cannot be expected to have a wider applicability than that of the Kalman filter The following are examples of the limitations of these fusion rules Sensor observation errors have to be uncorrelated This is a real limitation in many practical situations or example, many sensor errors are dependent on the common rom estimatee (the quantity to be estimated) thus are correlated Another example is that the estimatee is observed by multiple sensors in a common noisy environment, such as during noise jamming generated by a target The local dynamic models have to be identical In reality, local estimates may have been obtained based on different dynamic models of the estimatee to account for sensor-specific situations or requirements or example, different sensors may use different

2 multiple models that are most suitable to take advantage of the merits of each local sensor The network structure information patterns of the distributed systems have to be simple When a distributed system is too complicated, even a genius cannot device a fusion rule by manipulating the local estimates such that it is equivalent to the corresponding centralized fusion rule It has been realized for many years following the original work of [2] that local estimates (tracks) have correlated errors How to counter this cross correlation has been a central topic in distributed fusion Several ingenious techniques, such as the tracklets technique [11, 10] those based on the information graph [8], have been developed Their main underlying idea is to reduce the correlation by smart processing /or management at either the sensor or central level It is, however, virtually impossible to annihilate this cross correlation completely in practical situations Satisfactory approaches to distributed fusion in the presence of this cross correlation is still not available In addition, these techniques cannot overcome the other limitations mentioned above In this paper, a general systematic approach to estimation/track fusion is presented It is much more general flexible than the above-mentioned approach based on the Kalman filter A variety of fusion rules in unified forms that are optimal in the linear class for centralized, distributed, hybrid fusion architectures are presented, along with their relationships The key to this development is simple: The local estimates are nothing but observations of the estimatee These rules are optimal for an arbitrary number of sensors in the presence of the above-mentioned cross correlation in the sense of either the weighted least-squares (WLS) or best linear unbiased estimation (BLUE) ie, linear minimum variance (LMS) or mean-square error (LMMSE) [4] The results presented do not suffer from the drawbacks mentioned above are very easy to be implemented The remainder of this paper is organized as follows Section 2 formulates the fusion problems Section 3 introduces a unified data model for centralized, distributed, hybrid architecture Various optimal fusion rules are presented in Section 4 Section clarifies the relationships of these rules Section 6 provides a summary 2 ormulation of Estimation usion or simplicity, consider a distributed system with a fusion center sensors (local stations), each connected to the fusion center Denote by the set of observations ( the ariances of its associated noises) of the th sensor of the estimatee (ie, the quantity to be estimated) Denote by the th sensor s local estimate of, the ariance of its associated estimation error Let be the estimate of the ariance of its associated error at the fusion center, respectively The problem considered in this paper is the following: Obtain using all available information at fusion center Estimation fusion can be classified into three categories, depending on what information is available at the fusion center: If all unprocessed 1 data (or observations) are made available directly at the fusion center, that is, if, then it is known as a centralized fusion, central-level fusion, or measurement fusion problem If only data-processed observations are available at the fusion center, that is, if for non-trivial mappings, then the problem is known as decentralized fusion, distributed fusion, sensor-level fusion, or autonomous fusion In the most commonly used distributed fusion systems, ; that is, only local estimates (based on ) are available at the fusion center Such an architecture is known as estimate fusion in target tracking will be referred toas the stard distributed fusion for convenience Note, however, that distributed fusion in general does not need to have If the information available to the fusion center includes unprocessed data from some sensors processed data from other sensors (ie, if ), then it is referred to as a hybrid fusion problem The results presented in this paper are valid for many other systems or architectures, such as those in which there is no fusion center, but there is communication among some or all of the local sensors 3 Unified Linear usion Model 31 The Unified Model Let! be any (multi-dimensional) observation of the th sensor; that is,! is a generic observation in It is said that! is a linear observation if it is an affine function of ; that is, if it satisfies the following linear equation! " # $ (1) where " is not a function of $ is the observation noise The key to the development of the unified linear observation model of this paper for centralized, distributed, hybrid fusion is the following: A local estimate can be viewed as an observation of the estimatee using the following identity # # (2) 1 By unprocessed it is meant that no data processing has been done, although observation data are usually obtained after signal processing In general, data are obtained by processing certain signals

3 + + ) + M L + + where the estimation error % '& ( acts as the observation noise It should be emphasized that this observation model of the distributed estimation fusion is valid for every case without exception it does not rely on any assumption or example, it is valid for every linear or nonlinear estimator ') ( every linear or nonlinear observation * ( This model shall be referred to as the data model for stard distributed fusion Clearly, the above two equations can be rewritten in a unified form More generally, however, consider also distributed systems in which* ( is processed linearly first then sent to the fusion center; that is,, ( - ( / * ( 0-1 ( 2 3 ( * ( - 3 ( 4 ( ( (3) is available at the fusion center, where1 ( 3 ( are known are not functions of observations in 7 ( This model shall be referred to as the linearly-processed data model for distributed fusion Clearly, the centralized fusion model (1) is a special case of this model with3 ( ( - 6 ( Also, for a linear estimator ) ( with linear observation * (, (2) is a special case of (3) with 3 ( 4 ( ( - % & ( However, (2) holds universally, not limited to linear case Note also that the data, ( sent to the fusion center in a distributed system could also be a nonlinear function of * ( However, this is not considered in this paper, which deals only with linear estimation fusion It is now ready to describe the unified model CL - centralized with linear observation - stard distributed - distributed with linearly-processed data <= >?, ( - :; * ( CL (, ( A ( - :; 4 ( CL 8 3 ( 4 (, GH, B - CDE, I A B? - CDE A A B ( - Let :; 6 ( CL % & ( 6 ( GH I? B - CDE B Then, a unified linear model of the data available to the fusion center is, ( - A ( 2 ( (4) The corresponding batch form for all data is, B - A 2 B () Clearly, this model is valid not only for centralized distributed fusion, but also for hybrid fusion It is the foundation for the unified treatment of linear estimation fusion presented in this paper The key to this unified model is to view each piece of data (in particular, ) ( ) available at the fusion center as an observation of 32 Correlation in the Unified Model In general, noise components? J J J? B in the unified model are correlated; that is, the following matrix is not GH I necessarily block diagonal: K - / B 0 - :; L CL wherel M - / 6? J J J? 6 B 0 is the observation noise ariance; - / &? J J + J? & B 0 is the joint error ariance of local estimates; L - / 6+? J J J? 6 B 0 is the ariance of equivalent observation noise It should be emphasized M that the framework of the unified model requires that be the a priori ariance before any data is made available L is almost always assumed block-diagonal in the literature Unfortunately, this is not true for many important practical situations irst, as shown in a forthcoming paper, the observation noises of a discrete-time multisensor system obtained by sampling a continuous-time multisensor system are correlated Second, 6? J J J? 6 B are in general correlated if is observed in a common noisy environment, such as when a target state is observed in the presence of countermeasures (eg, noise jamming) or atmospheric noise This is particularly true when the sensors are on the same platform Third, observation errors of many practical sensors are coupled through their dependence on or example, the observation noise statistics is dependent on how far the target is because of difference in such quantities as cross-sections or SNR M In general, is almost never block-diagonal even if L is This is easily understable is analogous to the fact that the estimation errors of a Kalman filter at different time instants are generally correlated even if the observation noise is white A more detailed discussion is given in Part II [13] Moreover, the noise the estimatee (if is rom) could be correlated in general: /? B 0 - N? /? ( 0 - N ( (6)? O P or example, for the stard distributed system with optimal linear unbiased local estimates with prior mean R, one has N ( - /? ( 0 - S/ & ( 2 ) ( % R 0? % & ( T - % / & ( 0 % U S ) ( & V( T 2 R U S & V( T - % W ( (7) where the last equation above follows from the orthogonality principle In summary, (4) () is indeed a unified data model for all linear centralized, distributed, hybrid fusion problems However, the price paid is that the observation noises may be correlated with each other with the estimatee As a consequence, any optimal fusion rule based on this unified model must be valid for such correlation 33 An Alternative Distributed usion Architecture It is clear that the stard distributed fusion model (2) is universally valid However, it leads to correlation among the equivalent noises of different sensors between the noise the estimatee These two types of correlation may be avoided by using instead the linearly-processed data

4 y y y Š model (3) for distributed fusion This model is, however, not universally valid It works only for linear observations with the uncorrelated actual observation noises under some assumed structure of the local estimators Assume that Z is given in the following form Z [ \ Z ] ^ Z _` Z a b Z \ Z c [ _d a ^ Z b Z c \ Z ] ^ Z ` Z with linear observations ` Z [ b Z ] e Z Note that this holds true for all linear unbiased estimators (not necessarily optimal in the linear class) with linear observations Then f Z [ Z a _d a ^ Z b Z c \ Z [ ^ Z b Z ] ^ Z e Z [ g Z b Z ] h Z (8) is a linearly processed observation Note that the ariance of the corresponding noise h is i [ j h k [ _i Zl c [ _^ Z m Zl ^ ln c [ ^ m ^ n () wherem [ j e k ^ [ diagj ^ o p qqqp ^ r k i Thus, is block-diagonal provided m is block-diagonal This avoids the difficulty associated with the unified model based on the stard i distributed model, which has a non-blockdiagonal In this model, in addition to the data f Z Z a [ _d a ^ Z b Z c \ Z, gain^ Z needs to be sent to or computed at the fusion center This distributed architecture will be referred to as the simple non-stard distributed fusion 4 Unified Optimal usion Rules Z is linear unbiased Only linear unbiased estimation fusion will be considered; that is, it is assumed in this paper that estimators of only linear unbiased fusion rules will be considered Consider two most commonly used types of linear estimation methods: optimal (linear) weighted least squares (WLS) best linear unbiased estimation (BLUE) The latter is also known as linear minimum mean-square error (LMMSE), linear minimum variance (LMS), or linear unbiased minimum variance (LUMS) estimation They are defined by, for the unified model (), BLUE WLS [ s t u z { v } wx~ _j a k j a k n ƒ c (10) [ s t u v zwx j f r a k ni o j f r a k (11) where g do not depend on data ƒ They minimize mean-square error fitting error, respectively When multiple solutions of the minimization problem v zwx j f r a k ni o j f r a k exist, it is assumed that the one with the minimum norm (ie, smallest length ) is always chosen since this solution is the one that has smallest error ariance _j a k j a k nc is unique As a consequence, there is no need to assume that ni o is nonsingular However, it is normally assumed that i o is positive definite ( thus nonsingular) for least squares fusion, otherwise the least squares approach is very questionable zero fitting error does not imply zero estimation error This assumption will not be stated every time it is needed In the case where i is block-diagonal ( ˆ [ ), both batch recursive forms of these two estimators are simple well-known To be valid for the general case, in particular for the stard distributed fusion, it is most desirable to develop BLUE WLS fusers that are valid for observation noises that are correlated with each other with the estimatee 41 BLUE usion with Complete Prior Knowledge or convenience, it is said that a BLUE fusion problem has complete prior knowledge if both the prior mean \ the prior ariance Š of the estimatee (as well as its correlation ˆ with the observation error h ) are known, because the only prior information of the estimatee used by a BLUE estimator is its first two moments The following theorem presents the BLUE fusion rule in the case of complete prior knowledge Theorem 1 Using data from (), the unique, unified, optimal, BLUE fuser (10) of with prior knowledge \ [ _ c, Š [ Œ Ž j k, ˆ [ Œ Ž j p h r k is [ \ ] ^ _f r a \ c (12) [ j a ƒ k [ Š a ^ ^ n (13) where [ Š n ] i ] ˆ ] j ˆ k n (14) ^ [ j Š n ] ˆ k o (1) The error ariance the gain matrices have the following alternative forms Š [ Š n ] ^ i ^ n a ˆ ^ n a _ ˆ ^ ncn (16) ^ [ j Š n ] ˆ k j i ] ˆ k o (17) where the inverse is assumed to exist [ d a ^ In the case where is singular, Theorem 1 is still valid with o replaced by }, the Moore-Penrose pseudoinverse of Note, however, that (17) is no longer valid in this case This BLUE fuser reduces to the familiar BLUE estimator with uncorrelated errors if ˆ [ 42 BLUE usion Without Prior Knowledge The BLUE fusion rule of Theorem 1 is not valid if either there is no prior knowledge about the estimatee or the knowledge is incomplete (eg, the prior ariance is not known or does not exist) In either case, the popular BLUE estimation formulas are not applicable However, the BLUE fusion rules as defined by (10) for these cases still exist They are presented as Theorems 2 3 in this the next subsections Theorem 2 The unique, unified, optimal, BLUE fuser (10) of using data from () without prior knowledge is [ ^ f r [ ^ f r (18) Š [ ^ i ^ n [ ^ i ^ n (1) ^ [ } _d a i j i k } c (20)

5 ³ ¼ Ø Ø ¹ Ø where the optimal gain matrix is given uniquely by š œ ž œ ž Ÿ (21) š œ ž ž œ ž Ÿ ž (22) if only if œ has full row rank, where ž is a full-rank square-root matrix of ž š œ is any square-root matrix of œ Otherwise, ž (23) where is any matrix of compatible dimension satisfying œ ž the superscript sts for the unique Moore-Penrose pseudoinverse Note that an alternative form of is ª œ (24) if only if œ is nonsingular This is the same as (17) with «or the stard distributed BLUE fusion without prior information, one has a clearer expression for each element of, where š š thus Denote by the ± ² Ÿ th ³ ³ submatrix of š œ ž œ ž Ÿ Then is the average of over the² th column thus the estimate follows µ (2) ¹ º º ª œ (26) 43 BLUE usion With Incomplete Prior Knowledge In practice, it is sometimes more desirable to define a singular ª» thus the corresponding ª» does not exist 2 This would be the case if prior information about some but not all components of ¹ ¼ were not available or example, when tracking an aircraft just taking off from an airport, it is easy to determine the prior velocity vector with certain ariance, but not the prior position vector A practical means of specifying such knowledge is to set the corresponding elements (or eigenvalues) of ª» to infinity, or more appropriately, set the corresponding elements (or eigenvalues) of ª» to zero Theorem 3 Given the following as the incomplete prior knowledge of ¹ : ¹ ¼ ½ ¹ a positive semi-definite symmetric but singular matrix ª», ¹ ¾ Ÿ «, the unique, unified, optimal, BLUE fuser (10) for data model () is given by ¹ ¹ ¼ Ÿ º (27) ª œ (28) where the optimal gain matrix is as given in Theorem 2 with œ replaced by œ : À š Á  Á à œ À Ä ««Ÿ œ à (2) 2Å Æ Ç È is just a convenient symbol here, not the inverse any matrix since it is singular, although it is meant to be the inverse ofå È É is the orthogonal matrix that diagonalizes ª» : ª» Ê ËÌ Í Ä Ÿ such that Ä diag Î Î Á Ÿ Ï Ð rank ª» Ÿ 44 Optimal Weighted LS usion Theorem 4 The unique, unified, optimal, WLS fuser (11) having minimum norm using data from () is ¹ º (30) ª œ œ (31) where the gain matrix is given by ª œ (32) This minimum-norm fuser becomes the unique optimal WLS fuser if only if œ is nonsingular Although new for fusion, this is a well-known result for estimation or the stard distributed WLS fusion, the following theorem holds for each submatrix of the gain matrix Theorem Each element Á of of the optimal WLS rule for the stard distributed fusion is given explicitly in terms of œ alone by Á ÑÒ Ó œ Ô Õ Ö where œ Ô Õ is the ± ² Ÿ th ³ ³ submatrix of œ œ Ô Õ Á (33) This theorem is attractive mainly in a theoretical sense It makes the dependence of the weights of the optimal fuser on œ explicit Note that these weights are uniquely determined by œ alone for the stard distributed fusion It can be shown that many existing fusion formulas, such as the stard distributed fusion with feedback of [7] the two-sensor track fusion formula of [3], are special cases of this theorem The fuser of this theorem is, however, computationally expensive 4 Optimal Generalized Weighted LS usion Consider now a generalized WLS problem of using the (non-bayesian) least-squares method to provide an optimal fused estimate of a rom variable ¹ that has prior mean ¹ arianceª», is correlated with the data errors, with the cross-ariance given by ¹ ¾ Ÿ «Consider the following model º» ¹ ¾» ¹ ¼ (34) with ¾» Ÿ ¹ ¼ ¹ Ÿ ª» Let Then, º À ¼ º Ã Ø À š à ¾ À ¾» ¾ à œ Ø À ª» ««œ à º Ø ¹ ¾ Ø (3)

6 å ö õ ì å Ü Þ Û Ü Ý Þ Ù Ü æ ç è é í êë î àú ã ï ð Ú Ùå ñ òóú ô á î àú ã ï ð Ú Ùå ñ Ü õ àú ã (36) Ü ø î Ù ï Ùå ñ î Ù ï Ùå ñ ò ùûú ú Ü õ óú õ ò Ü øð Ú òóú ô á ð Ú úû (37) Ü ö ð Ú òóú ô á (38) the corresponding optimal WLS estimator of Ù using data set Ú Ù ß à á ß â â â ß à ã ä is which follows from the optimal WLS fuser above by re- ó placing à ã ð ó,, with àú ã ð, Ú, Ú Relationships Among usion Rules 1 Equivalence of Optimal WLS BLUE usion Theorem 6 Assume that there is no prior knowledge about the estimatee Ù ó is positive definite Then the unique BLUE fuser the unique minimum-norm optimal WLS fuser are identical (ie, they coincide) ð Remark 1: In the case where òó ô á ð is singular, the optimal WLS problem amounts to the underdetermined case of a system of linear equations, å thus has infinitely many solutions; any of these solutionsù yields zero fitting error: î à ã ï ð Ùå ñ òó ô á î à ã ï ð Ùå ñ Ü ü Consequently, an optimal WLS fuser does not exist in a strict sense since there is no way of selecting one out of these solutions without using an additional criterion The solution given by Theorem 4 is the one having smallest norm is unique This theorem states that this minimum-norm solution turns out to have the smallest error ariance Remark 2: The above ó equivalence is roughly the best one can expect since is always assumed positive definite whenever the optimal WLS fusion is considered ó If is not positive definite, the WLS ö fuser Ü õ ó of õ Theorem 4 with the following replacement ò õ Ü øð òó û ð ú û ð òó û does not coincide with the BLUE fuser, in fact, is often a meaningless fuser Remark 3: The optimal WLS fusers using data of uncorrelated errors are statistically consistent in the sense the estimation error approaches zero as more more data are used When the data have correlated errors, these fusers are in general no longer consistent By this theorem, the BLUE fuser for data of correlated errors could also be inconsistent Conversely, an optimal WLS fusion problem (ie, without prior knowledge about Ù ) can be treated as a BLUE fusion problem as follows Use the first piece (or block) of data à á to obtain the optimal WLS (or equivalently, the BLUE) fused estimate of Ù, which is given by ö á Ü øð áò ó áô á ð á ú û å, Ù á Ü ö á ð áò ó áô á à á Then use the remaining å data to compute the BLUE fused estimates by treatingù á ö á Þ as Ù ö ý in the usual formulation of BLUE fusion Note that if the data errors are correlated, the errors of the remaining data are in general correlated with Ùþ Ü Ù ï ÙÞ Ü Ù ï Ùå á Theorem 6 above alsoguarantees that the optimal gener- ó ÿ ü alized WLS fuser coincides with the BLUE fuser if Ú 2 Relationships of Various usion Rules Of all fusion rules presented above, the BLUE fusion rule with incomplete prior knowledge is the most general one It clearly reduces to the BLUE fusion rule without prior knowledge with complete prior knowledge in the case ö ý ô á Ü ü of ùö ýô á ù Ü ü, respectively Somewhat against many people s intuition, the BLUE fusion rule with complete prior knowledge is a special case of the BLUE fusion rule without prior knowledge since the prior mean the ariance can be treated as the initial data the associated error ariance rom Theorem 6, it is clear that the optimal WLS fusion rule is a special case of the BLUE fusion rule without prior knowledge It is also easy to see that the optimal generalized WLS fusion rule is a special case of the BLUE fusion rule with complete prior knowledge Note that almost all optimal linear fusion formulas presented before in the literature are special cases of the optimal WLS fusion rules of this paper 6 Summary Centralized, distributed, hybrid estimation fusion architectures have been considered in a general framework A unified linear data model for these three architectures have been presented The best linear unbiased estimation (BLUE), the optimal weighted least squares (WLS), the optimal generalized WLS fusion rules for an arbitrary number of sensors, with complete, incomplete, or no prior information about the estimatee, have been presented in explicit forms These rules have unified forms for the above three fusion architectures are extremely convenient for implementation They are much more general flexible than previously available results or example, they remain to be optimal for the cases where observation errors are arbitrarily correlated across sensors, over time, /or arbitrarily correlated with the estimatee They are also optimal for cases with complete, incomplete, or no prior information The fusion rules for the stard distributed architecture are extremely general in the sense that they virtually do not rely on any assumption on the local estimates or example, they are optimal given the cross ariance of local estimates no matter what these estimates are how they were obtained Relationships among the various fusion rules presented have also been provided A Appendix: Proofs A1 Proof of Theorem 1 The unique BLUE estimator of Ù Û Ü Ý using à á ß â â â ß à ã ä is given by (see, eg, [6]) Ù ã Ü øù ú î Ùþ ß àþ ã ñ û î àþ ã ñ î à ã ï øà ã ú ñ Ù õ øà ã ï ð ÙÞ ú (3) ö ã Ü ö ý ï î Ùþ ß àþ ã ñ û î àþ ã ñ î àþ ã Ü ö ý ï õ õ ß Ùþ ñ ò (40)

7 where (41) (42) (1) follows from The estimation error is (16) thus follows from the property of the BLUE estimator that Its equivalence to can also be easily shown by algebraic manipulations (17) follows from below:! " # $ " " A2 Proof of Theorem 2 Let % % where% do not depend on! & & & # When there is no prior knowledge about, the unbiasedness of requires that be unbiased for all possible ( unknown), that is, ' This is true iff % ( $ & This also agrees with the fact that in order for the BLUE estimator $ to be unbiased for all possible, it is required that with satisfying $ Consequently, the BLUE estimator is given by, where satisfies $, or equivalently $, minimizes the corresponding error ariance since $ In other words,, where ) * + /, (43) Note that this is a weighted least squares problem with linear constraint $ It is well known that the general solution of $ is 4 (44) where is any matrix of suitable dimension $ 4 $ (since 4 4 ) It is straightforward to show from the basic properties of the matrix pseudoinverse that Substituting the last two equations above into the quadratic form in (43) yields! 4 #! 4 # " Clearly, minimizing the quadratic objective function in (43) thus amounts to solving ( for It can be shown that Hence, the solution is 4 4 ' : 4 ( It thus follows from (44) that the general solution of (43) is $ (4) where is any matrix of suitable dimension satisfying 4 ( (23) thus follows To show necessary sufficient condition for the uniqueness of, note first that only of (4) with 4 ( can be a unique solution, otherwise of (4) with ; 4 for any real number ; < ( would be a distinct solution since ; 4 ( When 4 (, (4) gives a unique solution due to the uniqueness of the M-P pseu- doinverse Thus is unique iff 4 ( A necessary sufficient condition for 4 ( when 4 ( is that the vector 4 is in the row space of Since 4 $ is a projector onto the orthogonal complement of the row space (ie, subspace spanned by the row vectors) of, the above necessary sufficient condition holds iff the row space of is the orthogonal complement of the row space of, which is equivalent to having full row rank The second equation in (21) follows from the first one because it can be shown that Note that 4 = ( where Thus, 4 is unique Its error ariance is

8 T T Z T ST A Z A3 Proof of Theorem 3 Since any symmetric matrix can be diagonalized by orthogonal transformation, one has > A? B C diagd E? E G H C I, where C is an orthogonal matrix (C? B C I ), E B diagd J K K K J L H M N O B rankd > A? H E G B N Let P G R B P B C B C C I GI R S P T G R B B C IST P B C C I GI R where T are the subvectors of T that correspond to E in the sense that D U H B E Note that T E G B N is practically equivalent 3 to no prior knowledge about G, thus the information about S contained in S > A?, D S V W H B are equivalent to that of contained in T T, E, D V W H B C I Treating as an observation A of leads to the following data model A B T B Z D T U H B [\ N ] Z V The original data model () of S can be converted to a model of observing : W B ^ S Z V W B D ^ C H Z V W Combining these two model yields P T P W B W R B [\ ^ N ] P T W C R U V W R B ^ W V W (46) ` _ The ariance B D _ V W H is given by ` P T P B a U V W R b B E U C I T U D C I H I ` R Once is treated as an observation, there is no prior information about at all This means that Theorem 2 is applicable now Therefore, all formulas in this theorem follows from Theorem 2 for the data model (46) the relationship S B C Sc B C c A4 Proof of Theorem > B C D d _ U c W H C I In the case of stard ` distributed fusion,^ I `? ^ is nonsingular since is always assumed to have full rank for the optimal WLS fusion ^ B [\ K K K \ ]I for the stard distributed fusion Hence, the theorem follows from direct manipulations 3e f g h is not rigorously equivalent to no prior knowledge abouti f in theory In reality, however, j mk l is chosen in such a way it represents the fact that there is no prior knowledge about i f in the first place This equivalence is thus rigorous provided j mk l is chosen this way nonrigorously A rigorous approach here will never be exactly equivalent to this real fact thus leads to inferior results than this theorem for this practical problem Note also that ni f o p q r g s ftu contains no useful information for BLUE estimation at all since there is no prior knowledge about i f anyway References [1] A T Alouani, T R Rice, R E Helmick, On Sensor Track usion, in Proc 14 American Control Conf, (Baltimore, MD), pp , June 14 [2] Bar-Shalom, On the Track-to-Track Correlation Problem, IEEE Trans Automatic Control, vol AC- 26, pp 71 72, Apr 181 [3] Bar-Shalom L Campo, The Effect of the Common Process Noise on the Two-Sensor used- Track Covariance, IEEE Trans Aerospace Electronic Systems, vol AES-22, pp , Nov 186 [4] Bar-Shalom R Li, Estimation Tracking: Principles, Techniques, Software Boston, MA: Artech House, 13 (Reprinted by BS Publishing, 18) [] Bar-Shalom R Li, Multitarget-Multisensor Tracking: Principles Techniques Storrs, CT: BS Publishing, 1 [6] H Chen, Recursive Estimation Control for Stochastic Systems New ork: Wiley, 18 [7] C Chong, Hierarchical Estimation, in Proc Second MIT/ONR Workshop on C3, (Monterey, CA), July 17 [8] C Chong, S Mori, W H Barker, K C Chang, Architectures Algorithms for Track Association usion, IEEE AES Systems Magazine, pp 13, Jan 2000 [] C Chong, S Mori, K C Chang, Distributed Multitarget Multisensor Tracking, in Multitarget-Multisensor Tracking: Advanced Applications, ( Bar-Shalom, ed), pp 247 2, Norwood, MA: Artech House, 10 [10] O E Drummond, A Hybrid Sensor usion Algorithm Architecture Tracklets, in Proc 17 SPIE Conf on Signal Data Processing of Small Targets, vol 3163, pp 12 24, 17 [11] O E Drummond, Tracklets a Hybrid usion With Process Noise, in Proc 17 SPIE Conf on Signal Data Processing of Small Targets, vol 3163, pp 12 24, 17 [12] R Li, C Z Han, J Wang, Recursive Smoothing Prediction for Linear Discrete-Time Systems, in Proc 3th IEEE Conf on Decision Control (submitted), (Sydney, Australia), Dec 2000 [13] R Li J Wang, Unified Optimal Linear Estimation usion Part II: Discussions Examples, in Proc 2000 International Conf on Information usion, (Paris, rance), July 2000 [14] M Liggins, C Chong, I Kadar, M G Alford, V Vannicola, S Thomopoulos, Distributed usion Architectures Algorithms for Target Tracking, Proceedings of the IEEE, vol 8, pp 107, Jan 17