Sensor Selection Based on Generalized Information Gain for Target Tracking in Large Sensor Networks
aa r X i v : . [ m a t h . O C ] F e b Sensor Selection Based on Generalized Information Gain for TargetTracking in Large Sensor Networks
Xiaojing Shen, Member, IEEE, Pramod K. Varshney, Fellow, IEEE ∗† May 9, 2018
Abstract
In this paper, sensor selection problems for target tracking in large sensor networks with linear equalityor inequality constraints are considered. First, we derive an equivalent Kalman filter for sensor selection,i.e., generalized information filter. Then, under a regularity condition, we prove that the multistage look-ahead policy that minimizes either the final or the average estimation error covariances of next multipletime steps is equivalent to a myopic sensor selection policy that maximizes the trace of the generalizedinformation gain at each time step. Moreover, when the measurement noises are uncorrelated betweensensors, the optimal solution can be obtained analytically for sensor selection when constraints are tem-porally separable. When constraints are temporally inseparable, sensor selections can be obtained byapproximately solving a linear programming problem so that the sensor selection problem for a largesensor network can be dealt with quickly. Although there is no guarantee that the gap between the per-formance of the chosen subset and the performance bound is always small, numerical examples suggestthat the algorithm is near-optimal in many cases. Finally, when the measurement noises are correlatedbetween sensors, the sensor selection problem with temporally inseparable constraints can be relaxed toa Boolean quadratic programming problem which can be efficiently solved by a Gaussian randomizationprocedure along with solving a semi-definite programming problem. Numerical examples show that theproposed method is much better than the method that ignores dependence of noises. keywords:
Sensor selection; generalized information gain; sensor networks, target tracking ∗ This work was supported in part by U.S. Air Force Office of Scientific Research (AFOSR) under Grants FA9550-10-1-0263 andFA9550-10-1-0458 and in part by the NNSF of China † Xiaojing Shen (corresponding author, [email protected]) and Pramod K. Varshney ([email protected]) are with the Depart-ment of Electrical Engineering and Computer Science, Syracuse University, NY, 13244, USA. Xiaojing Shen ([email protected])is on leave from Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China. Introduction
Over the past twenty years, advances in sensor technologies have led to the emergence of large numbers oflow-cost sensing devices with a fair amount of computing and communication capabilities. Large sensornetworks have attracted much attention both from theoretical and practical standpoints and have become afast-growing research area. To efficiently manage large sensor networks, one typically designs a policy fordetermining the optimal sensor network performance and resource utilization at each time, within logical orbudget constraints. The most comprehensive recent survey on sensor management is provided in the book[1]. Discussion on more advances in this area is available in the recent survey paper [2] and referencestherein. In this paper, we concentrate on sensor selection problems in which a subset of sensors are selectedat each time instant while tracking a target that provides optimal performance–resource usage tradeoffs.The sensor selection problem arises in various applications, including target tracking, e.g., [3, 4], robotics[5], and wireless networks [6]. Sensor selection for the target tracking problem will be considered here. Inthe literature, the sensor selection problem has been formulated for different dynamic systems. In [3], thestate model was assumed to be deterministic without noise. A convex optimization procedure was developedbased on a heuristic to solve the problem of selecting k sensors from a set of m sensors. Although nooptimality guarantees could be provided for the solution, numerical experiments showed that it performedwell. Another important contribution comes from the work reported in [4] where the state model was assumedrandom with noise and a general objective function of the sensor selection problem was transformed to aquadratic form by introducing the gain matrix as an additional decision variable. However, the resultingoptimization problem cannot efficiently take advantage of the structure of the covariance of measurementnoise such as it being a diagonal matrix in the uncorrelated case. In this paper, the sensor selection problemformulated by the use of the Moore-Penrose generalized inverse only relies on Boolean decision variableswithout introducing additional decision variables. The resulting optimization problem can efficiently takeadvantage of the structure of the measurement noise and obtain the optimal solution analytically. Manyother excellent results on sensor selection for state estimation in different situations can be found in, e.g.,[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and references therein.Sensor management problems are often considered with different criteria and objectives. Representa-tive approaches for sensor management include optimization of estimation error covariance [3, 4], Fisherinformation [9, 10], and entropy or mutual information [5, 11, 12, 13, 14, 15, 18, 19]. Various functions ofthe estimation error covariance and Fisher information matrix, including their determinant and trace, havebeen used as reward functions for optimal sensor management. Several popular measures, including R ´ e nyientropy, Kullback-Leibler (KL) divergence, and Hellinger-Battacharya distance, have been used for the cal-culation of information gain between two densities. In this paper, based on the Moore-Penrose generalizedinverse, we will derive a closed-form expression of information gain for sensor selection called generalizedinformation gain whose trace function is taken as the reward function for optimal sensor selection. When themeasurement noises are assumed independent, the notion of information measure based on the information2ain has been discussed in the literature, see [19].In this paper, we consider the problem of state estimation for a linear dynamic system being monitoredby multiple sensors. For sensor selection, we first derive an equivalent Kalman filter for sensor selection,i.e., generalized information filter. Then, under a regularity condition, we prove that the multistage look-ahead policy that minimizes either the final or the average estimation error covariance of next N time steps isequivalent to a myopic sensor selection policy that maximizes the trace of the generalized information gain ateach time step. Thus, trace of the generalized information gain is defined as a measure of information that theselected sensors provide at each time step. Moreover, when the measurement noises are uncorrelated betweensensors, the optimal solution can be obtained analytically when the constraints are temporally separable.When the constraints are temporally inseparable, the solution of the sensor selection problem can be obtainedby approximately solving a linear program (LP) so that sensor selections for a large sensor network can beperformed quickly. Although there is no guarantee that the gap between the performance of the chosensubset and the performance bound is always small, numerical examples suggest that the algorithm is near-optimal in many cases. Finally, when the measurement noises are correlated between sensors, the sensorselection problem when the constraints are temporally inseparable can be relaxed to a Boolean quadraticprogramming (BQP) which can be efficiently solved by a Gaussian randomization procedure along withsolving a semi-definite programming (SDP) problem which can be solved by interior-point methods [20].Numerical examples show that the proposed method yields solutions that are much better than the methodthat ignores dependence.The rest of the paper is organized as follows. Preliminaries are given in Section 2, where the generalizedinformation filter for sensor selection and multistage sensor selection problems that minimize either the finalor the average estimation error covariances over the next N time steps are formulated. In Section 3, undera regularity condition, we prove that multistage look-ahead policies are equivalent to the myopic sensorselection policy that maximizes the trace of the generalized information gain at each time step. In Section4, the case of uncorrelated measurement noises is considered. The optimal solution is derived analyticallyfor sensor selection when the constraints are temporally separable. When the constraints are temporallyinseparable, the sensor selection scheme is obtained by approximately solving an LP. In Section 4, the caseof correlated measurement noises is considered. The sensor selection problem is relaxed to a BQP which canbe efficiently solved by a Gaussian randomization procedure along with solving an SDP problem. In Section5, numerical examples are given and discussed. In Section 6, concluding remarks are provided. We consider a surveillance region of interest (ROI) that is being monitored by a sensor field for potentialtargets crossing the ROI. The fusion center tracks the target by optimally selecting a fixed number of sensors3rom a large sensor network under some logical or budget constraints. Specifically, we consider a L -sensorlinear dynamic system x k +1 = F k x k + w k , (1) y ik = H ik x k + v ik , i = 1 , , . . . , L, (2) z ik = γ ik H ik x k + γ ik v ik , i = 1 , , . . . , L, (3)where x k ∈ R r , F k ∈ R r × r is an invertible matrix ; y ik ∈ R n i , H ik ∈ R n i × r , { w k } and { v ik } are bothtemporally uncorrelated with zero means and invertible covariances Q k and R ik respectively. The covarianceof the noise v k , (( v k ) ′ , . . . , ( v Lk ) ′ ) ′ is denoted by R k , Cov( v k ) which is assumed invertible, R ijk , Cov( v ik , v jk ) so that R iik = R ik . If the i -th sensor is selected, we let γ ik = 1 , otherwise γ ik = 0 (see, e.g., [4]); γ k , ( γ k , . . . , γ Lk ) ′ . We shall focus on Equations (1) and (3) for sensor selection. The stacked measurementequation is written as z k = ˜ H k x k + ˜ v k , (4)where z k , (( z k ) ′ , . . . , ( z Lk ) ′ ) ′ , (5) ˜ v k , (( γ k v k ) ′ , . . . , ( γ Lk v Lk ) ′ ) ′ , (6) ˜ H k , (( γ k H k ) ′ , . . . , ( γ Lk H Lk ) ′ ) ′ . (7)The covariance of the noise ˜ v k is denoted by ˜ R k , Cov(˜ v k ) , ˜ R ijk , Cov( γ ik v ik , γ jk v jk ) = γ ik γ jk R ijk . (8)Moreover, we denote by z k , ( z ′ , . . . , z ′ k ) ′ , x k | k , E [ x k | z k ] , P k | k , E [( x k | k − x k )( x k | k − x k ) ′ ] .At time t k , the fusion center has γ ik , i = 1 , . . . , L , x k | k and P k | k (or measurements z k ). The fusioncenter is to design the sensor selection scheme for the next N time steps. At time t k + n , m k + n sensors will beselected from L sensors, for n = 1 , . . . , N . They will send their measurements, compressed measurementsor local estimates to the fusion center. The fusion center makes the final estimates for the state at times t k + n , n = 1 , . . . , N . The problem is how to select m k + n sensors from L sensors (i.e. determine the Booleandecision variables γ ik + n , i = 1 , . . . , L ), n = 1 , . . . , N that minimize the objective function which is • either the final estimation error covariance f , P k + N | k + N , (9) The invertibility of the transition matrix can be guaranteed in tracking problems, see [21]. or the average estimation error covariance f , N N X n =1 P k + n | k + n . (10)The constraint that m k + n sensors are selected from L sensors, n = 1 , . . . , N induces a constraint thatis temporally separable. Moreover, we shall also consider constraints that are temporally inseparable, forexample, energy constraints.Note that the objective functions (9)–(10) are matrices. Matrix optimization considered here is in thesense that if x ∗ is an optimal solution, then for an arbitrary feasible solution x , P ( x ) (cid:23) P ( x ∗ ) , i.e., P ( x ) − P ( x ∗ ) is a positive semi-definite matrix (see, e.g., [20]). It is well known that the Kalman filter provides the globally optimal solution if the noises are assumedGaussian, otherwise it provides the best linear unbiased estimate. It is recursive no matter whether thecovariances of noises are invertible or not and is given as follows (see, e.g., [22]), x k +1 | k +1 = x k +1 | k + K k +1 ( z k +1 − ˜ H k +1 x k +1 | k ) , (11) P k +1 | k +1 = ( I − K k +1 ˜ H k +1 ) P k +1 | k , (12)where I is an identity matrix with compatible dimensions, x k +1 | k = F k x k | k , (13) K k +1 = P k +1 | k ˜ H ′ k +1 ( ˜ H k +1 P k +1 | k ˜ H ′ k +1 + ˜ R k +1 ) + , (14) P k +1 | k = F k P k | k F ′ k + Q k . (15)The superscript “ + ” means Moore-Penrose generalized inverse (see, e.g., [23]) . If P k +1 | k , P k +1 | k +1 and ˜ R k +1 are invertible (for example, the case that all L sensors are selected), then we have the followingequivalent Kalman filter x k +1 | k +1 = P k +1 | k +1 ( P − k +1 | k x k +1 | k + ˜ H ′ k +1 ˜ R − k +1 z k +1 ) , (16) P k +1 | k +1 = ( P − k +1 | k + ˜ H ′ k +1 ˜ R − k +1 ˜ H k +1 ) − , (17)which is usually called the information filter and ˜ H ′ k +1 ˜ R − k +1 ˜ H k +1 is called the information gain (see e.g.,[24]). Once the sensors are selected, the covariance of the noise vector of the selected sensors is invertible. Here, the Moore-Penrose generalized inverse is used since the ( ˜ H k +1 P k +1 | k ˜ H ′ k +1 + ˜ R k +1 ) may not be invertible. The reasonis that ˜ H k +1 and ˜ R k +1 defined by (7) and (8) for the sensor selection problem include the decision variables { γ k +1 , . . . , γ Lk +1 } which have L − m k +1 number of zeros. Q k , k = 1 , , . . . , are invertible, and it is easy to check that P k +1 | k and P k +1 | k +1 which are updated by the Kalman Filter based on the selected sensors are also invertible. Here, however, ˜ R k +1 are not invertible, since there are L − m k +1 number of zeros in the decision variables { γ k +1 , . . . , γ Lk +1 } .Thus, we first prove that, for the dynamic system (1) and (4) defined under sensor selection where ˜ R k +1 arenot invertible, there still exists an equivalent Kalman filter similar to (16)–(17). Theorem 2.1.
For the dynamic system defined by (1) and (4) under sensor selection, we have the followingequivalent Kalman filter (generalized information filter) x k +1 | k +1 = P k +1 | k +1 ( P − k +1 | k x k +1 | k + ˜ H ′ k +1 ˜ R + k +1 z k +1 ) , (18) P k +1 | k +1 = ( P − k +1 | k + ˜ H ′ k +1 ˜ R + k +1 ˜ H k +1 ) − . (19) Proof.
See appendix.The key difference in Theorem 2.1 is that ˜ R − k +1 where ˜ R k +1 is invertible in (16)–(17) has been replacedby ˜ R + k +1 in (18)–(19). Due to this difference, ˜ H ′ k +1 ˜ R + k +1 ˜ H k +1 will be called generalized information gain .Notice that P k +1 | k +1 is a function of γ k +1 , . . . , γ Lk +1 , since ˜ H k +1 and ˜ R k +1 are functions of γ k +1 , . . . , γ Lk +1 .Thus, it is denoted by P k +1 | k +1 ( γ k +1 , . . . , γ Lk +1 ) . Similarly, P k + n | k + n ( γ k +1 , . . . , γ Lk +1 , . . . , γ k + n , . . . , γ Lk + n ) is a function of γ k +1 , . . . , γ Lk +1 , . . . , γ k + n , . . . , γ Lk + n , for n = 1 , . . . , N .It is the generalized information filter based on Moore-Penrose generalized inverse that helps us decou-ple the multistage look-ahead policies to an equivalent myopic sensor selection policy that maximizes thegeneralized information gain with a lower computational complexity in Section 3. Another advantage isthat the sensor selection problem formulated by the use of the Moore-Penrose generalized inverse only re-lies on Boolean decision variables without introducing additional decision variables and can efficiently takeadvantage of the structure of the measurement noise to obtain the optimal solution and efficient algorithms. Thus, by using Theorem 2.1, the two sensor selection problems can be stated as min γ ik + n P k + N | k + N ( γ k +1 , . . . , γ Lk +1 , . . . , γ k + N , . . . , γ Lk + N )= ( P − k + N | k + N − + ˜ H ′ k + N ˜ R + k + N ˜ H k + N ) − (20)subject to L X i =1 γ ik + n = m k + n , n = 1 , . . . , N, (21) γ ik + n ∈ { , } , i = 1 , , . . . , L, n = 1 , . . . , N, (22)6nd min γ ik + n N X n =1 P k + n | k + n ( γ k +1 , . . . , γ Lk +1 , . . . , γ k + N , . . . , γ Lk + N )= N X n =1 ( P − k + n | k + n − + ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) − (23)subject to L X i =1 γ ik + n = m k + n , n = 1 , . . . , N,γ ik + n ∈ { , } , i = 1 , , . . . , L, n = 1 , . . . , N. In this section, we consider some properties of the optimization problems presented in Section 2.3 that willsimplify their solution. We will show that if the primal sensor selection problem (20) has an optimal solution,then both the problem (20) and the problem (23) can be transformed to equivalent optimization problems thatmaximize an information measure at each time step.
Lemma 3.1.
Consider two optimization problems: ( A ) max x ∈S M ( x ) , (24) ( A ) max x ∈S tr( M ( x )) , (25) where M ( x ) is a matrix for an arbitrary x ∈ S ; S specifies the constraint on the decision variable x . If theproblem ( A ) has an optimal solution, then the problem ( A ) is equivalent to ( A ). Proof.
See appendix.
Lemma 3.2.
Consider two optimization problems: ( B ) min x i ∈S i ,i =1 ,...,n M n ( x , . . . , x n ) , f or n = 1 , . . . , N, (26) ( B ) min x n ∈S n ,n =1 ,...,N, N X n =1 M n ( x , . . . , x n ) (27) where M n ( x , . . . , x n ) is a function of decision variables x , . . . , x n , for n = 1 , . . . , N . If the optimal solu-tion that minimizes M n ( x , . . . , x n ) , ( x ∗ , . . . , x ∗ n ) , is the same as the one that minimizes M n +1 ( x , . . . , x n +1 ) ,for n = 1 , . . . , N − , then the optimal solution that minimizes M N ( x , . . . , x N ) (( B ) with n = N ) is thesame as that for ( B ). roof. See appendix.Based on Lemma 3.2, the solution to both the problem (20) and the problem (23) can be simplified andobtained by solving N optimization problems separately. Lemma 3.3.
If the primal sensor selection problem (20) has an optimal solution, then both the problem (20)and the problem (23) can be transformed to the equivalent problem that solves N optimization problems thatmaximize ˜ H ′ k + n ˜ R + k + n ˜ H k + n , n = 1 , . . . , N respectively, i.e., max γ ik + n ˜ H ′ k + n ˜ R + k + n ˜ H k + n for n = 1 , . . . , N, (28) subject to L X i =1 γ ik + n = m k + n ,γ ik + n ∈ { , } , i = 1 , , . . . , L, where ˜ H k + n and ˜ R k + n are defined in Equations (7) and (8) respectively; the superscript “ + ” indicatesMoore-Penrose generalized inverse [23]. That is, the problems (20), (23) and (28) have the same optimalsolution. Proof.
See appendix.
Remark . Lemma 3.3 shows that multistage look-ahead policies, i.e., the problem (20) and the problem(23), are equivalent to a myopic sensor selection policy that maximizes the generalized information gain witha lower computational complexity. Why do the problem (20) and the problem (23) with different objectiveshave the same optimal solution? The main reason is that the objectives and constraints are temporally sepa-rable. For example, consider m k + n = 1 , i.e., select one sensor at each time step, if there is a sensor whichhas the smallest noise and provides the most information at each time step, then the selection of the sensor ateach time step is the optimal sensor selection scheme no matter whether the objective is the final estimationerror covariance or the average estimation error covariance.Moreover, based on Lemmas 3.1 and 3.3, we have the following theorem. Theorem 3.5.
If the primal sensor selection problem (20) has an optimal solution, both the problem (20)and the problem (23) can be transformed to the equivalent problem requiring the solution of N optimizationproblems that maximize tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) , n = 1 , . . . , N respectively, i.e., max γ ik + n tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) for n = 1 , . . . , N, (29) subject to L X i =1 γ ik + n = m k + n ,γ ik + n ∈ { , } , i = 1 , , . . . , L, here ˜ H k + n and ˜ R k + n are defined in Equations (7) and (8) respectively. That is, the problems (20), (23)and (29) have the same optimal solution.Remark . Theorem 3.5 shows that both the minimization of the final estimation error covariance andminimization of the average estimation error covariance are equivalent to maximization of the trace of thegeneralized information gain of each time step. Thus, the objective function tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) (30)of the problem (28), i.e., trace of the generalized information gain, is defined as a measure of informationthat the selected sensors provide at ( k + n ) -th time step. Determinant of the generalized information gain canbe similarly defined as a measure of information if ˜ H ′ k + n ˜ R + k + n ˜ H k + n is a positive definite matrix. When themeasurement noises are assumed independent, more information measures based on information gain can beformulated, see e.g., [19].Furthermore, the information measure (30) has an advantage that it does not depend on pdfs of the statesand measurements, but only relies on covariances of noises and the measurement matrices. Maximizingthis measure can be employed as an alternative criterion for sensor selection, which will be used in sensorselection problems when the constraints are temporally inseparable in the following sections. When pdfsare known, it is better to try to use the information criteria based on pdfs such as optimization of Fisherinformation, entropy or mutual information for sensor selection (see, e.g., [2]). When sensor measurement noises are uncorrelated and the constraints are temporally separable, we have thefollowing result that defines the optimal sensor selection scheme.
Theorem 4.1.
Let the information measure corresponding to the i -th sensor at ( k + n ) -th time be de-noted as a ik + n , tr(( H ik + n ) ′ ( R ik + n ) − H ik + n ) , i = 1 , . . . , L . Let { a i k + n , . . . , a i L k + n } denote { a k + n , . . . , a Lk + n } rearranged in descending order. If the problem (20) has an optimal solution, then the optimalsensor selection scheme for both the problem (20) and the problem (23) is γ i k + n = 1 , . . . , γ i mk + n k + n = 1 , γ i mk + n +1 k + n = 0 , . . . , γ i L k + n = 0 , for n = 1 , . . . , N . The optimality of sensor selection scheme is in the senseof either the minimization of the covariance of the final estimation error (9) or the average estimation error(10) or maximization of the information measure (30). If the problem (20) does not have an optimal solution,the optimality of the sensor selection scheme is only in the sense of maximization of the information measure(30). Proof.
If the measurement noises are uncorrelated between sensors, then ˜ R k + n is a block diagonal matrix9ith ˜ R ijk + n = 0 , i = j . Thus, ˜ R + k + n = (diag( γ k + n R k + n , . . . , γ Lk + n R Lk + n )) + = diag( γ k + n ( R k + n ) − , . . . , γ Lk + n ( R Lk + n ) − ) (31)which follows from the definition of Moore-Penrose generalized inverse. Moreover, by Theorem 3.5, theproblem (20) is equivalent to max γ ik + n tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) for n = 1 , . . . , N, = L X i =1 γ ik + n tr(( H ik + n ) ′ ( R ik + n ) − H ik + n ) subject to L X i =1 γ ik + n = m k + n ,γ ik + n ∈ { , } , i = 1 , , . . . , L. If we define a ik + n , tr(( H ik + n ) ′ ( R ik + n ) − H ik + n ) , i = 1 , . . . , L and { a i k + n , . . . , a i L k + n } denotes { a k + n , . . . , a Lk + n } rearranged in descending order, then the optimal solution is γ i k + n = 1 , . . . , γ i mk + n k + n = 1 , γ i mk + n +1 k + n =0 , . . . , γ i L k + n = 0 . Many constraints on sensor selection can be represented as linear equalities or inequalities such as logicalconstraints and budget constraints (see, e.g., [3, 4]). Let us denote linear equalities or inequalities as follows a ′ p γ D b p , p = 1 , . . . , P, (32)where γ , ( γ ′ k +1 , . . . , γ ′ k + N ) ′ , γ k + n , ( γ k + n , . . . , γ Lk + n ) ′ , for n = 1 , . . . , N ; (33) a p is a vector with a compatible dimension and b p is a scalar; “ D ” can represent either “ ≥ ” “ ≤ ” or “ = ”for each n . In general, these constraints are temporally inseparable, which makes the optimization problemswith objectives (20) and (23) not separable and highly nonlinear in variables γ ik + n . The correspondingoptimization problems are very hard to solve.However, from Remark 3.6, the trace of generalized information , tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) , can be definedas the measure of information that selected sensors provide. Thus, we can try to maximize the availableinformation gain from time k + 1 to k + N by optimizing the selection of sensors so that better estimation10erformance can be expected. We shall consider the following objective (i.e, sum of the weighted informationmeasure) f , N X n =1 ω n tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) , (34)where ω n , n = 1 , . . . , N are weights which place different importance on different time steps. For example,if the state estimation at the final time is more important, a larger weight ω N can be used. If the stateestimation of each time step is equally important, an equal weight structure ω = · · · = ω N can be used.Therefore, we consider the following optimization problem for sensor selection: max γ ik + n N X n =1 ω n tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) (35)subject to a ′ p γ D b p , p = 1 , . . . , P,γ ik + n ∈ { , } , i = 1 , , . . . , L, n = 1 , . . . , N. Since sensor measurement noises are assumed uncorrelated in this section, by Equation (31), the problem(35) is equivalent to max γ ik + n N X n =1 ω n L X i =1 γ ik + n tr(( H ik + n ) ′ ( R ik + n ) − H ik + n ) (36)subject to a ′ p γ D b p , p = 1 , . . . , P,γ ik + n ∈ { , } , i = 1 , , . . . , L, n = 1 , . . . , N, which is a Boolean linear programming (BLP) problem and the optimal objective function value is denotedby f ∗ BLP . It can be relaxed by replacing the nonconvex constraints γ ik + n ∈ { , } with the convex constraints ≤ γ ik + n ≤ , i = 1 , , . . . , L, n = 1 , . . . , N . Thus, we have the following LP problem: max γ ik + n tr(Γ D ′ ) (37)subject to a ′ p γ D b p , p = 1 , . . . , P, ≤ γ ik + n ≤ , i = 1 , , . . . , L, n = 1 , . . . , N, where γ is defined by (33); Γ is a L × N matrix with i -th row and n -th column element being γ ik + n , i.e, Γ , γ k +1 γ k +2 · · · γ k + N γ k +1 γ k +2 · · · γ k + N ... ... . . . ... γ Lk +1 γ Lk +2 · · · γ Lk + N (38)and D is a L × N matrix D , ( d in ) L × N , d in , ω n tr(( H ik + n ) ′ ( R ik + n ) − H ik + n ) . (39)11t is well known that LP problems can be solved efficiently. The solution of the problem (37) is denotedby ( γ ik + n ) ∗ LP , i = 1 , . . . , L, n = 1 , . . . , N . The corresponding objective function is denoted by f ∗ LP . Notethat the feasible solution set of the problem (37) contains that of the problem (36) so that f ∗ BLP ≤ f ∗ LP .Based on ( γ ik + n ) ∗ LP , we can generate a suboptimal feasible solution of the problem (36) denoted by ˆ γ ik + n , i = 1 , . . . , L, n = 1 , . . . , N . The corresponding objective function is denoted by ˆ f BLP and ˆ f BLP ≤ f ∗ BLP .The difference g = f ∗ LP − ˆ f BLP is called the gap in [3]. The gap is useful in evaluating the performance ofthe suboptimal solution ˆ γ ik + n . We can say ˆ γ ik + n is no more than g -suboptimal.Note that the procedure of generating a feasible solution of the problem (36) from ( γ ik + n ) ∗ LP is problemdependent, i.e., relying on the equalities or inequalities (32) and the Boolean constraint. As an illustration,let us consider a representative example. Besides the temporally separable constraints (21) and (22), weconsider an energy constraint which is temporally inseparable as follows N X n =1 γ ik + n ≤ m ik , i = 1 , . . . , L, (40)which means that the i -th sensor can only be selected m ik times from time k + 1 to time k + N ( m ik < N ),for i = 1 , . . . , L . Thus, the specific form of the optimization problem (37) with the constraints (21) and (40)can be represented to max γ ik + n tr(Γ D ′ ) (41)subject to a ′ p γ D b p , p = 1 , . . . , P, ≤ γ ik + n ≤ , i = 1 , , . . . , L, n = 1 , . . . , N, where Γ and γ are defined by (38) and (33) respectively; P = N + L , a p , ( c ′ ,p , . . . , c ′ N,p ) ′ , c n,p , ( , p = n , , p = n , n = 1 . . . , N,b p , m k + p , “ D ” means “ = ” , (42)for p = 1 , . . . , N, ( corresponds to the constraints (21) ); a p , ( ′ i , . . . , ′ i ) ′ ,b p , m ik , “ D ” means “ ≤ ” , (43)for p = N + i, i = 1 , . . . , L, ( corresponds to the constraints (40) ) , where and denote L -dimensional vectors with 1 entries and 0 entries respectively and i means an L -dimensional vector whose i -th entry is 1 others are 0s.The sensor selection scheme with the energy constraint for uncorrelated sensors is described by thefollowing algorithm. 12 lgorithm 4.2 (Sensor selection scheme with the energy constraint for uncorrelated sensors) . • Step 1: Given an optimal solution of (41) ( γ ik + n ) ∗ LP , obtain the optimal objective function f ∗ LP . • Step 2: Generate a feasible solution of the problem (36) with the constraints (21), (22) and (40) from ( γ ik + n ) ∗ LP as follows.We generate the feasible solution based on the importance (weight) of the information of each time step.Without loss of generality, assume that ω ≤ ω ≤ . . . ≤ ω N . Thus, we generate the selection schemefrom the N -th time step to the first time step. Set the index set of candidate sensors i , { , . . . , L } . – Iteratively generate ˆ γ ik + N − n for the ( N − n ) -th time step, n = 0 , . . . , ( N − as follows:for n = 0 : ( N −
1) ˆ γ ik + N − n , ( , if i ∈ i , , if i ∈ i , , f or i = 1 , . . . , L, (44) where i is the index set of the first m k + N − n maximum entries of ( ( γ k + N − n ) ∗ LP , . . . , ( γ Lk + N − n ) ∗ LP )in the index set of candidate sensors i and denote i = i − i . Set m ik := m ik − , for i ∈ i .Update the index set of candidate sensors i , { i : m ik > , i = 1 , . . . , L } .end • Step 3: Output g -suboptimal solution ˆ γ ik + n , i = 1 , . . . , L, n = 1 , . . . , N and the corresponding objec-tive ˆ f BLP , where g = f ∗ LP − ˆ f BLP is the gap.
Here, to construct the feasible solution satisfying the constraints (21), (22) and (40), we employ theequation (44). The main computation complexity is in Step 1 where an LP problem needs to be solved.Illustrative examples will be presented in Section 6.
In this section, for correlated sensor measurement noises, we again determine the sensor selection scheme bymaximizing the weighted information measure: max γ ik + n N X n =1 ω n tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) (45)subject to a ′ p γ D b p , p = 1 , . . . , P,γ ik + n ∈ { , } , i = 1 , , . . . , L. where the linear constraints are defined in (32) that may include both the temporally separable and inseparableconstraints. Since sensor measurement noises are correlated, to obtain the optimal solution, an exhaustive13earch is necessary since ˜ R k + n has no special structure such as it being a diagonal matrix. For the simplestcase of the temporally separable constraint (21) and N = 1 , there are a total of Lm k + n ! feasible solutions.For large L and m k + n , such an exhaustive search may not be feasible in real time. Thus, to make the solutioncomputationally more efficient, the problem (45) is approximately solved by replacing ˜ R + k + n by R − k + n . Thisapproximation is lossless for the case of uncorrelated sensor noises and temporally separable constraint (i.e.,does not change the optimal solution in Theorem 4.1). More discussion on approximation loss for differentdependences will be given in Section 6. Thus, we consider the approximate problem max γ ik + n N X n =1 ω n tr( ˜ H ′ k + n R − k + n ˜ H k + n ) (46)subject to a ′ p γ D b p , p = 1 , . . . , P,γ ik + n ∈ { , } , i = 1 , , . . . , L. Moreover, from the definition of ˜ H k + n (7), we have tr( ˜ H ′ k + n R − k + n ˜ H k + n )= tr( L X i =1 L X s =1 γ ik + n γ sk + n ( H ik + n ) ′ T isk + n H sk + n )= L X i =1 L X s =1 γ ik + n γ sk + n tr(( H ik + n ) ′ T isk + n H sk + n )= − γ ′ k + n B k + n γ k + n , (47)where T isk + n is the i -th row block and s -th column block of the matrix T k + n , T k + n , R − k + n ; the i -throw and s -th column of B k + n is − tr(( H ik + n ) ′ T isk + n H sk + n ) . Thus, the problem is equivalent to solving thefollowing Boolean quadratic programming (BQP) problem min γ ik + n N X n =1 ω n γ ′ k + n B k + n γ k + n (48)subject to a ′ p γ D b p , p = 1 , . . . , P,γ ik + n ∈ { , } , i = 1 , , . . . , L, n = 1 , . . . , N. For this problem, however, it is still hard to obtain an optimal solution, since the nonconvex Boolean con-straints and B k + n may not be a positive semi-definite matrix. It is known to belong to the class of NP-hardproblems. Fortunately, this class of problems can be solved by a recently developed computationally efficientapproximation technique (see, e.g., [25]). We apply it to the problem (48) as follows.14y semidefinite relaxation (SDR) technique (see, e.g., [20, 25]), the problem (48) can be relaxed to min X ∈ S ( NL +1) , X (cid:23) tr( CX ) (49)subject to tr( E p X ) D (4 b p − ′ diag( a p ) ) , p = 1 , . . . , P, tr( F s X ) = 1 , s = 1 , . . . , ( N L + 1) , where F s is a matrix with s -row and s -column F s ( s, s ) = 1 , others are equal 0, for s = 1 , . . . , ( N L + 1) , E p , diag( a p ) ′ ! , C , B B ′ B ! , B , diag( ω B k +1 , . . . , ω N B k + N ) , where B k + n is defined by (47) for n = 1 , . . . , N ; I and are an identity matrix and a “1” vector withcompatible dimensions respectively. The problem (49) is an SDP problem. The derivation of the problem(49) is given in Appendix.Note that the procedure for generating a feasible solution of the problem (45) from the solution of theproblem (49) is also problem dependent, i.e., relying on the equalities or inequalities (32) and the Booleanconstraint. As an illustration, let us again consider the representative constraints (21), (22) and (40). Thus,the specific expressions of a p and b p in the optimization problem (49) are given by (42) and (43).Based on the SDP (49), a typical Gaussian randomization procedure is used to construct an approximatesolution to the problem (45) here (see [25]). Thus, we have the following algorithm. Algorithm 5.1 (Sensor selection scheme with the energy constraint for correlated sensors) . • Step 1: Given an optimal solution of the SDP (49) X ∗ ∈ S ( NL +1) , and a number of randomizations S. • Step 2: Generate S feasible solutions by Gaussian randomization procedure based on X ∗ :for s=1:S1. Generate a vector ξ s ∼ N (0 , X ∗ ) . Set η s , ξ s (1 : N L ) which means the first NL entries of ξ s .2. Without loss of generality, assume that ω ≤ ω ≤ . . . ≤ ω N . We generate the selectionscheme from the N -th time step to the first time step. Set the index set of candidate sensors i , { , . . . , L } . ∗ Iteratively generate γ ik + N − n,s for the ( N − n ) -th time step, n = 0 , . . . , ( N − for n = 0 : ( N − γ ik + N − n,s , ( , if i ∈ i , , if i ∈ i , , f or i = 1 , . . . , L, (50)15 here i is the index set of the first m k + N − n maximum entries of η s ( J ( L − n −
1) + 1 : J ( L − n )) in the index set of candidate sensors i and i = i − i . Set m ik := m ik − , for i ∈ i . Update the index set of candidate sensors i , { i : m ik > , i = 1 , . . . , L } .end3. Denote γ k + N − n,s , ( γ k + N − n,s , . . . , γ Lk + N − n,s ) and γ s , ( γ k +1 ,s , . . . , γ k + N,s ) .end • Step 3: Determine s ∗ = argmax s =1 ,...,S f ( γ s ) where f ( · ) may be the objective funcrions f , f or f defined in (9), (10) and (34) respectively. • Step 4: Output ˆ γ = γ s ∗ as the sensor selections of the problem (45). Note that specific design of the randomization procedure technique is problem dependent. Here, toconstruct the feasible solution satisfying the constraints (21), (22) and (40), we employ Equation (50). Thechoice of S will be discussed in Section 6. Based on simulations, the randomized solution can often achievea good performance with a small S , which is similar to that in [25]. The main computational complexityof the algorithm is in Step 1 where an SDP problem needs to be solved. The SDP problem can be solvedefficiently by using interior-point methods (see, e.g., [20]). In this section, we present a number of illustrative examples. Both uncorrelated and correlated sensor mea-surement noise cases are considered.
We first compare the performance of the approach given in Theorem 4.1 with the one in Joshi an Boyd [3]and the one in Mo et al. [4].
Example 6.1.
Let us consider a dynamic system with L = 40 sensors which are uniformly distributed overa square of size 100 m . The parameter matrices and noise covariances for the dynamic system (1)–(4) are F k = ! , Q k = ! , (51) H ik = ! , R ik = r i r i ! , i = 1 , . . . , L, (52) where r i and r i are randomly sampled from the uniform distribution in [5, 7] and [10 12] respectively. Weconsider a constraint, i.e., select m k + n = [1 , , , , sensors from 40 sensors at the next time steprespectively.
16n Figure 1, the traces of the estimation error covariance are plotted for the sensor selection method givenin Theorem 4.1, the one in Joshi an Boyd [3] and the one in Mo et al. [4] respectively. The CPU time isplotted in Figure 2 for the three algorithms respectively. Figure 1 shows that the three methods obtained veryclose and similar estimation performance for the numerical example, while Figure 2 shows that the CPU timeof the method in Theorem 4.1 is much smaller than that of the one in Joshi an Boyd [3] and the one in Mo etal. [4]. The reason is that the method in Theorem 4.1 is an analytical solution. In addition, the computationtime of the three methods is not an increasing function of the number of selected sensors. The reason isthat when the number of selected sensors increases, the number of the decision variables does not increaseand the structure of the optimization does not change; only some parameters of the equality constraints arechanged.Moreover, we consider a representative target tracking dynamic system with energy constraints. Weassume that each target will be tracked in a Cartesian frame. The four state variables include position and ve-locity ( x, ˙ x, y, ˙ y ) respectively (see e.g., [24]). The parameter matrices and noise covariances for the dynamicsystem (1)–(4) are F k = T T , Q k = T / T / T / T T / T /
20 0 T / T , (53) H ik = ! , R ik , i = 1 , . . . , L, (54)where T = 1 s is the sampling interval; F k , Q k , H ik are the same in the following examples. The differenceis the noise covariance of measurements, R ik , in the following examples. Since the algorithm in Joshi anBoyd [3] that requires the measurement matrix is full-column rank when the measurement matrices of eachsensor are the same and the one in Mo et al. [4] does not present how to threshold the approximate solutionto generate a feasible solution satisfying the energy constraints, we evaluate the performance of Algorithm4.2 by comparing with the exhaustive method for a monitoring system which has a small number of sensorsand using the gap given in the Step 3 of Algorithm 4.2 for a monitoring system which has a large number ofsensors in the following examples respectively. Example 6.2.
First, to compare with the exhaustive method, let us consider a relatively small monitoringsystem with L = 9 sensors which are uniformly distributed over a square of size 100 m . The parametermatrices of the dynamic system are given in (53)–(54) where R ik = r i r i ! , (55) r i and r i are randomly sampled from the uniform distribution in [5, 10]. We consider the optimizationproblem (37) with temporally inseparable constraint (40) and the constraints (21), (22) where N = 3 , k + n = 2 , n = 1 , . . . , N and m ik = 2 , i.e., select 2 sensors from 9 sensors at each time step and select eachsensor less than twice in 3 time steps. In Figure 3, the traces of the final estimation error covariance f based on the three methods are plottedrespectively, where r i and r i are randomly sampled 50 times. The three methods are 1) the exhaustivemethod that minimizes the final estimation error covariance f , 2) Algorithm 4.2 that maximizes the weightedinformation measure f with weights [1 / , / , / and 3) Algorithm 4.2 that maximizes the weightedinformation measure f with weights [0 , , respectively. Similarly, the traces of the average estimationerror covariance f are plotted in Figure 4. In Figure 5, the sum of information measures of N time steps P Nn =1 tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) is plotted for the two sensor selection schemes respectively. They are obtainedfrom 2) and 3) respectively.From Figures 3–5, we have following observations: • Figure 3 shows that the trace of the final estimation error covariances obtained from Algorithm 4.2 withtwo different weights that maximizes the weighted information measure f are very close to that ofthe exhaustive method. Similarly, Figure 4 shows that the trace of average estimation error covarianceobtained from Algorithm 4.2 with weights [1 / , / , / that maximizes the weighted informa-tion measure f is very close to that of the exhaustive method. These indicate that maximization ofthe weighted information measure f is a good alternative criterion for minimizing final or average estimation error covariance for sensor selection. • Moreover, in Figure 3, when the objective is minimization of the final estimation error covariance f ,both Algorithm 4.2 with weights [0 , , and Algorithm 4.2 with weights [1 / , / , / are nearoptimal for sensor selection. However, Figure 5 shows that the sum of information measures of N timesteps for Algorithm 4.2 with the weights [1 / , / , / is larger than that of Algorithm 4.2 with theweights [0 , , . Thus, it is better to choose the weights [0 , , , since a larger sum of informationmeasures of N time steps implies that more good sensors are used. • Finally, Figure 4 shows that when the objective is to minimize the average estimation error covari-ance f , it is better to choose the weights [1 / , / , / since Algorithm 4.2 with the weights [1 / , / , / is near optimal. Example 6.3.
Next, let us consider a large monitoring system with L = 20 ×
20 = 400 sensors which areuniformly distributed in a square of size 100 m . We consider the optimization problem (37) with temporallyinseparable constraint (40) and the constraints (21), (22) where N = 5 , m k + n = 10 , n = 1 , . . . , N and m ik = 2 , i.e., select 10 sensors at each time step from 400 sensors and select each sensor less than twice in 5time steps. Moreover, we consider the performance of Algorithm 4.2 for different cases of m k + n from 10 to100. Obviously, the exhaustive method is infeasible. In Figure 6, the upper bound and lower bound of the objective function of the optimization problem (37)are plotted based on 50 Monte Carlo runs. The corresponding gaps, i.e, the upper bound minus the lower18ound are plotted in Figure 7. Figures 6 and 7 show that the gaps are very small and Algorithm 4.2 canobtain the optimal solution in the sense of maximizing the weighted information measure f in many cases,although the sensor network is large where the Boolean decision variables are more than 2000. Figure 8presents the gaps as a function of m k + n from 10 to 100. It shows that the gaps are increasing as the numberof selected sensors. In this subsection, we will compare Algorithm 5.1 with the exhaustive method for a simple problem so thatthe approximation loss can be computed. For this, we assume that only the sensor selection scheme for thenext step is to be designed, i.e., N = 1 . For N = 5 , we will compare Algorithm 5.1 with Algorithm 4.2 thatignores dependence. In this case, the exhaustive method is infeasible, since we have to enumerate . × cases. At the end, an example that compares the root mean square error (RMSE) of state estimation based onsensor selection is presented. Example 6.4.
Let us consider L = 25 for the sensor network shown in Figure 14. Assume that there is ajammer signal v k with a covariance R k at the position (550 m, m ) , besides the natural noises v ik , i =1 , . . . , L which are independent of v k . The jamming signal introduces dependence among measurementnoises. Thus, the noises at the i -th sensor is given as follows ˘ v ik = v ik + P αd ni, v k = v ik + β i v k (56) where β i , P αd ni, ; d i, is the distance between the jammer and the i -th sensor; the signal decay exponent n = 2 , the scaling parameter α = 1 and different values for the signal power P = [1 , , , , , , × are used in simulations respectively. Thus, noises of sensors are correlated and the i -th block and j -thblock of the noise covariance ˘ R k can be computed by (56) to be ˘ R ijk = Cov (˘ v ik , ˘ v jk ) = R ijk + β i β j R k , (57) where R ijk =
10 00 10 ! , if i = j ! , i = j . , R k = ! , are used in simulations. Note that the corresponding Pearson’s correlation coefficients between sensors areapproximately equal to [0 . , . , . , . , . , . , . corresponding to P = [1 , , , , , , × respectively. We consider the optimization problem (45) with temporally separable constraints (21), (22)where N = 1 , m k +1 = 2 , i.e., select 2 sensors from 25 sensors at the next time step.
19n Figure 9, comparisons of the objective function tr( ˜ H ′ k +1 ˜ R + k +1 ˜ H k +1 ) of the optimization problem(45) (i.e., the information measure of the ( k + 1) -th time step) based on the exhaustive method, Algorithm5.1 and Theorem 4.1 that ignores dependence are plotted for different jammer signal powers respectively.We present the performance of Algorithm 5.1 with small numbers of randomizations S = 20 , S = 100 .Similarly, comparisons of the traces of the estimation error covariance of ( k + 1) -th time step are plotted inFigure 10.From Figures 9–10, we have the following observations: • For all the three methods, the larger is the signal power of jammer, the smaller is the informationmeasure of the ( k + 1) -th time step obtained from the selected sensors and the larger is the trace of theestimation error covariance at the ( k + 1) -th time step. • Figures 9–10 also show that the exhaustive method yields better results than Algorithm 5.1 with small S and the latter is better than the method of Theorem 4.1 that ignores dependence, especially in thecase of strong dependence (i.e., strong signal power of the jammer). • Figures 9–10 indicate that larger the value of S is , the closer is the performance of Algorithm 5.1 tothat of the exhaustive method, i.e., the smaller is the approximation loss of Algorithm 5.1. Example 6.5.
Next, let us consider a monitoring system with a large N = 5 and L = 5 × sensorswhich are uniformly distributed in a square of size 100 m . We consider the optimization problem (37) withtemporally inseparable constraint (40) and the constraints (21), (22) where m k + n = 2 , n = 1 , . . . , N and m ik = 2 , i.e., select 2 sensors at each time step from 25 sensors and select each sensor less than twice in next5 time steps. In Figure 11, comparisons of the objective function P Nn =1 ω n tr( ˜ H ′ k + n ˜ R + k + n ˜ H k + n ) of the optimizationproblem (45) (i.e., the sum of the weighted information measure of N time steps, f defined in (34)) basedon approaches of Algorithm 5.1 and Algorithm 4.2 that ignores dependence are plotted for different jammersignal powers respectively. We examine the performance of Algorithm 5.1 as a function of the numberof randomizations S = 20 , S = 100 , S = 2000 which are small, compared with the exhaustive number . × . Similarly, comparisons of the traces of the average estimation error covariances of N time stepsare plotted in Figure 12.Figures 11–12 show that Algorithm 5.1 with a small value of S is better than Algorithm 4.2 that ignoresdependence, especially in the case of strong dependence (i.e., strong signal power of the jammer). In addition,Figures 11–12 indicate that larger the value of S is, the better is the performance of Algorithm 5.1 than thatof Algorithm 4.2 that ignores dependence. Example 6.6.
Finally, let us consider the L -sensor noise covariance R k , k = 1 , , . . . which depends onthe state x k . A frequently made assumption is that larger is the distance between the sensor and the target,larger is the noise covariance. However, when we design the sensor selection scheme of next N time steps at ime k , we do not know the state x k + n so that we replace it by the state prediction x k + n | k which is used tocompute the R k + n , n = 1 , . . . , N . Specifically, the noise covariance R k + n is R k + n = ¯ R k + n + ˘ R k + n , (58) where ˘ R k + n is the noise covariance from the jammer signal v k defined in (57) and the signal power ofjammer P = 1 . × ; ¯ R k + n is a diagonal matrix with the i -th diagonal block defined as follows ¯ R iik + n = α d i,n α d i,n ! , i = 1 , . . . , L, , n = 1 , . . . , N, (59) where α = 0 . is a scaling parameter; d i,n is the distance between the target prediction x k + n | k andthe i -th sensor. We consider the optimization problem (37) with temporally separable constraint (40) andthe constraints (21), (22) where N = 5 , m k + n = 2 , n = 1 , . . . , N and m ik = 2 , i.e., select 2 sensorsat each time step and select each sensor less than twice in 5 time steps. The initial state of the target is (600 m, − m/s, m, m/s ) . In Figure 13, RMSE of the state estimates based on 200 Monte Carlo runs is given. We compare Algo-rithm 4.2 that ignores dependence with Algorithm 5.1 with S=20, S=100 and S=2000 respectively.As far as the RMSE is considered, Figure 13 also shows that Algorithm 5.1 with a small value of S is better than Algorithm 4.2 that ignores dependence and that larger the value of S is, the better is theperformance of Algorithm 5.1 than that of Algorithm 4.2 that ignores dependence. In this paper, we have proposed a generalized information filter for target tracking in wireless sensor networkswhere measurements from a subset of sensors are employed at each time step. Then, under a regularity condi-tion, we proved that the multistage look-ahead policy that minimizes either the final or the average estimationerror covariances of next N time steps is equivalent to the myopic sensor selection policy that maximizes thetrace of the generalized information gain at each time step. When the measurement noises are uncorrelated,the optimal solution has been derived analytically for sensor selection with temporally separable constraints.For temporally inseparable constraints, the sensor selection scheme can be obtained by approximately solv-ing an LP problem. Although there is no guarantee that the gap between the performance of the chosensubset and the performance bound is always small, numerical examples showed that the algorithm is near-optimal in many cases and the selection scheme for a large sensor network with more than 2000 Booleandecision variables can be dealt with quickly. Finally, when the noises of measurements are correlated, thesensor selection problem with temporally inseparable constraints was relaxed to a BQP problem which canbe efficiently solved by a Gaussian randomization procedure by solving an SDP problem which can be solvedby interior-point methods and related software tools. Numerical examples showed that the proposed methodis much better than the method that ignores dependence.21uture work will involve the generalization from the linear dynamic systems to nonlinear dynamic sys-tems. The equivalence between multistage look-ahead optimization policy for sensor management and themyopic sensor optimization policy and the corresponding sensor management schemes will be investigated.In addition, it can be considered for various applications such as robotics, sensor placement for structuresand different types of wireless networks. Appendix
The proof of Theorem 2.1.Proof.
Notice that there are m k +1 number of γ ik +1 = 1 and L − m k +1 number of γ ik +1 = 0 so that thereexists a permutation matrix P such that P ˜ H k +1 = H k +1 (1 : m k +1 ) ! (60)and P ˜ R k +1 P ′ = R k +1 (1 : m k +1 )
00 0 ! , (61)where H k +1 (1 : m k +1 ) and R k +1 (1 : m k +1 ) are the stacked measurement matrices and the covariance ofnoises of the m k +1 selected sensor respectively; is a zero matrix with compatible dimensions. From theproperty of the permutation matrix P × P ′ = P ′ × P = I and the definition of Moore-Penrose inverse, wehave P ˜ R + k +1 P ′ = ( P ˜ R k +1 P ′ ) + = R k +1 (1 : m k +1 )
00 0 ! + = ( R k +1 (1 : m k +1 )) −
00 0 ! (62)22oreover, by equations (60)–(62) and repeatedly using the definition of Moore-Penrose generalized inverseand the property of the permutation matrix P × P ′ = P ′ × P = I , we have the following derivation (cid:16) ˜ R k +1 + ˜ H k +1 P k +1 | k ˜ H ′ k +1 (cid:17) + = P ′ R k +1 (1 : m k +1 )
00 0 ! P + P ′ ˜ H k +1 (1 : m k +1 ) P k +1 | k ˜ H k +1 (1 : m k +1 ) ′ )
00 0 ! P ! + = P ′ R k +1 (1 : m k +1 ) + ˜ H k +1 (1 : m k +1 ) P k +1 | k ˜ H k +1 (1 : m k +1 ) ′ )
00 0 ! P ! + = P ′ ( R k +1 (1 : m k +1 ) + ˜ H k +1 (1 : m k +1 ) P k +1 | k ˜ H k +1 (1 : m k +1 ) ′ )
00 0 ! + P = P ′ ( R k +1 (1 : m k +1 ) + ˜ H k +1 (1 : m k +1 ) P k +1 | k ˜ H k +1 (1 : m k +1 ) ′ ) −
00 0 ! P = P ′ ( R k +1 (1 : m k +1 )) − ( I + ˜ H k +1 (1 : m k +1 ) · P k +1 | k ˜ H k +1 (1 : m k +1 ) ′ ( R k +1 (1 : m k +1 )) − ) − P = P ′ ( R k +1 (1 : m k +1 )) −
00 0 ! · ( I + ˜ H k +1 (1 : m k +1 ) P k +1 | k ˜ H k +1 (1 : m k +1 ) ′ ( R k +1 (1 : m k +1 )) − ) − I ! P P ′ ( R k +1 (1 : m k +1 )) −
00 0 ! · I + ˜ H k +1 (1 : m k +1 ) P k +1 | k ˜ H k +1 (1 : m k +1 ) ′
00 0 ! · ( R k +1 (1 : m k +1 )) −
00 0 !! − P = P ′ ( R k +1 (1 : m k +1 )) −
00 0 ! · P ′ + P ′ ˜ H k +1 (1 : m k +1 ) P k +1 | k ˜ H k +1 (1 : m k +1 ) ′
00 0 ! · ( R k +1 (1 : m k +1 )) −
00 0 !! − = P ′ ( R k +1 (1 : m k +1 )) −
00 0 ! P ′ + P ′ ˜ H k +1 (1 : m k +1 ) P k +1 | k ˜ H k +1 (1 : m k +1 ) ′
00 0 ! P ˜ R + k +1 P ′ ! − = P ′ ( R k +1 (1 : m k +1 )) −
00 0 ! ( P ′ + P ′ P ˜ H k +1 P k +1 | k ˜ H ′ k +1 P ′ P ˜ R + k +1 P ′ ) − = P ′ ( R k +1 (1 : m k +1 )) −
00 0 ! ( P ′ + ˜ H k +1 P k +1 | k ˜ H ′ k +1 ˜ R + k +1 P ′ ) − = P ′ ( R k +1 (1 : m k +1 )) −
00 0 ! P ( I + ˜ H k +1 P k +1 | k ˜ H ′ k +1 ˜ R + k +1 ) − = ˜ R + k +1 ( I + ˜ H k +1 P k +1 | k ˜ H ′ k +1 ˜ R + k +1 ) − . (63)From Equations (14) and (63), we have K k +1 = P k +1 | k H ′ k +1 (cid:16) ˜ R k +1 + ˜ H k +1 P k +1 | k ˜ H ′ k +1 (cid:17) + = P k +1 | k H ′ k +1 ˜ R + k +1 (cid:16) I + ˜ H k +1 P k +1 | k ˜ H ′ k +1 ˜ R + k +1 (cid:17) − , K k +1 (cid:16) I + ˜ H k +1 P k +1 | k ˜ H ′ k +1 ˜ R + k +1 (cid:17) = P k +1 | k H ′ k +1 ˜ R + k +1 , so that K k +1 I + K k +1 ˜ H k +1 P k +1 | k ˜ H ′ k +1 ˜ R + k +1 = P k +1 | k H ′ k +1 ˜ R + k +1 . Moreover, we have K k +1 = ( I − K k +1 ˜ H k +1 ) P k +1 | k H ′ k +1 ˜ R + k +1 = P k +1 | k +1 H ′ k +1 ˜ R + k +1 , (64)where the derivation of (64) is based on (12).From Equations (12) and (64), we have P − k +1 | k = P − k +1 | k +1 ( I − K k +1 ˜ H k +1 ) , = P − k +1 | k +1 − P − k +1 | k +1 K k +1 ˜ H k +1 , = P − k +1 | k +1 − H ′ k +1 ˜ R + k +1 ˜ H k +1 . Thus, we have P − k +1 | k +1 = P − k +1 | k + H ′ k +1 ˜ R + k +1 ˜ H k +1 .By (11), (12) and (64), x k +1 | k +1 = ( I − K k +1 ˜ H k +1 ) x k +1 | k + K k +1 z k +1 , = ( I − K k +1 ˜ H k +1 ) x k +1 | k + P k +1 | k +1 H ′ k +1 ˜ R + k +1 z k +1 , = P k +1 | k +1 P − k +1 | k x k +1 | k + P k +1 | k +1 H ′ k +1 ˜ R + k +1 z k +1 , = P k +1 | k +1 { P − k +1 | k x k +1 | k + H ′ k +1 ˜ R + k +1 z k +1 } , Thus, we have x k +1 | k +1 = P k +1 | k +1 { P − k +1 | k x k +1 | k + ˜ H ′ k +1 ˜ R + k +1 z k +1 } . The proof of Lemma 3.1.Proof.
First, we prove that the optimal solution of ( A ) is also the optimal solution of ( A ). If x is theoptimal solution of ( A ), then, for arbitrary x ∈ S , M ( x ) (cid:22) M ( x ) which yields tr( M ( x )) ≤ tr( M ( x )) .Thus, x is also the optimal solution of ( A ).On the other hand, if x is the optimal solution of ( A ), then, for arbitrary x ∈ S , tr( M ( x )) ≤ tr( M ( x )) which implies tr( M ( x )) ≤ tr( M ( x )) . Notice that the problem ( A ) has an optimal solution x which yields tr( M ( x )) ≤ tr( M ( x )) . Thus, tr( M ( x )) = tr( M ( x )) so that tr( M ( x ) − M ( x )) = 0 .By tr( M ( x ) − M ( x )) = 0 and M ( x ) − M ( x ) (cid:23) , we have M ( x ) = M ( x ) . Therefore, x is alsothe optimal solution of ( A ). 25 he proof of Lemma 3.2.Proof. In one direction: if x ∗ , . . . , x ∗ N is the optimal solution of (( B ) with n = N ), then, for n = 1 , . . . , N , M n ( x ∗ , . . . , x ∗ n ) (cid:22) M n ( x , . . . , x n ) for arbitrary x i ∈ S i , i = 1 , . . . , n, since the optimal solution that minimizes M n ( x , . . . , x n ) is the same as that minimizes M n +1 ( x , . . . , x n +1 ) ,for n = 1 , . . . , N . Thus, N X n =1 M n ( x ∗ , . . . , x ∗ n ) (cid:22) N X n =1 M n ( x , . . . , x n ) for arbitrary x i ∈ S i , i = 1 , . . . , n, which yields x ∗ , . . . , x ∗ N is also the optimal solution of ( B ).On the other hand, assume that x ∗ , . . . , x ∗ N is the optimal solution of ( B ). If x ∗ , . . . , x ∗ N is not theoptimal solution of ( B ), then there exists an optimal solution x , . . . , x N which has a smaller objectivefunction value than that of x ∗ , . . . , x ∗ N . Since the optimal solution that minimizes M n ( x , . . . , x n ) is thesame as that minimizes M n +1 ( x , . . . , x n +1 ) , x , . . . , x n is the optimal solution of M n ( x , . . . , x n ) for n = 1 , . . . , N . Thus, M n ( x ∗ , . . . , x ∗ n ) (cid:23) M n ( x , . . . , x n ) , n = 1 , . . . , N, so that N X n =1 M n ( x ∗ , . . . , x ∗ n ) (cid:23) N X n =1 M n ( x , . . . , x n ) , which yields a contradiction. Thus, x ∗ , . . . , x ∗ N is the optimal solution of ( B ). The proof of Lemma 3.3.Proof.
If the problem (20) has an optimal solution, from the fact that any positive definite matrix A (cid:23) A implies A − (cid:22) A − , then we have that the problem (20) is equivalent to solve max γ ik + n ( P k + N | k + N ( γ k +1 , . . . , γ Lk +1 , . . . , γ k + N , . . . , γ Lk + N )) − = P − k + N | k + N − + ˜ H ′ k + N ˜ R + k + N ˜ H k + N subject to L X i =1 γ ik + n = m k + n , n = 1 , . . . , N,γ ik + n ∈ { , } , i = 1 , , . . . , L, n = 1 , . . . , N, F k , it is equivalent to solve the following two problems max γ ik + n ˜ H ′ k + N ˜ R + k + N ˜ H k + N subject to L X i =1 γ ik + n = m k + n , n = N,γ ik + n ∈ { , } , i = 1 , , . . . , L, n = N, and min γ ik + n P k + N − | k + N − ( γ k +1 , . . . , γ Lk +1 , . . . , γ k + N − , . . . , γ Lk + N − )= ( P − k + N − | k + N − + ˜ H ′ k + N − ˜ R + k + N − ˜ H k + N − ) − subject to L X i =1 γ ik + n = m k + n , n = 1 , . . . , N, − ,γ ik + n ∈ { , } , i = 1 , , . . . , L, n = 1 , . . . , N, − . Both of them have an optimal solution respectively.After N -step recursive decomposition, the problem (20) is equivalent to solve the following N optimiza-tion problems max γ ik + n ˜ H ′ k + n ˜ R + k + n ˜ H k + n for n = 1 , . . . , N, subject to L X i =1 γ ik + n = m k + n ,γ ik + n ∈ { , } , i = 1 , , . . . , L. All of N optimization problems have an optimal solution respectively.Moreover, for the problem (20), we consider minimizing P k + n | k + n and P k + n +1 | k + n +1 respectively.Both of them have a recursive decomposition similar to that of minimizing P k + N | k + N for the problem (20).Thus, we have min γ ik + s P k + n | k + n ( γ k +1 , . . . , γ Lk +1 , . . . , γ k + n , . . . , γ Lk + n ) (65)subject to L X i =1 γ ik + s = m k + s , s = 1 , . . . , n,γ ik + s ∈ { , } , i = 1 , , . . . , L, s = 1 , . . . , n, min γ ik + s P k + n +1 | k + n +1 ( γ k +1 , . . . , γ Lk +1 , . . . , γ k + n +1 , . . . , γ Lk + n +1 ) (66)subject to L X i =1 γ ik + s = m k + s , s = 1 , . . . , n + 1 ,γ ik + s ∈ { , } , i = 1 , , . . . , L, s = 1 , . . . , n + 1 , have the same optimal solutions ( γ ik + s ) ∗ , i = 1 , . . . , L, s = 1 , . . . , n respectively. By Lemma 3.2, theproblem (20) is also equivalent to solving the problem (23). Therefore, if the primal sensor selection problem(20) has an optimal solutions, both the problem (20) and the problem (23) can be equivalently transformedto solve the problem (28). The derivation of the problem (49 )By ( γ ik + n ) = γ ik + n , the problem (48) is equivalent to min γ ik + n γ ′ B γ subject to γ ′ diag( a p ) γ D b p , p = 1 , . . . , P,γ ik + n ∈ { , } , i = 1 , , . . . , L, n = 1 , . . . , N, where γ is defined in (33); diag( a p ) and B = diag( ω B k +1 , . . . , ω N B k + N ) are diagonal matrix and diag-onal block matrix respectively. If we let τ ik + n = 2 γ ik + n − and denote by τ k + n , ( τ k + n , . . . , τ Lk + n ) ′ and τ , ( τ ′ k +1 , . . . , τ ′ k + N ) ′ , then the problem is equivalent to min τ ik + n
14 ( τ + ) ′ B ( τ + ) (67)subject to
14 ( τ + ) ′ diag( a p )( τ + ) D b p , p = 1 , . . . , P, ( τ ik + n ) = 1 , i = 1 , , . . . , L, n = 1 , . . . , N, where is a 1 vector with compatible dimensions. Moreover, it is equivalent to min τ ik + n , t ( τ + ) ′ B ( τ + ) (68) = ( τ ′ k + n t ) B B ′ B ! τ k + n t ! subject to ( τ ′ t ) diag( a p ) ′ ! τt ! D b p − ′ diag( a p ) , p = 1 , . . . , P, ( τ ik + n ) = 1 , i = 1 , , . . . , L, n = 1 , . . . , N,t = 1 . ( τ ∗ , t ∗ ) is the optimal solution to (68), then τ ∗ (respectively − τ ∗ ) is an optimal solution to (67) when t ∗ = 1 (respectively t ∗ = − ). Moreover, the problem is equivalentto min τ, t ( τ ′ t ) C τt ! (69)subject to ( τ ′ t ) E τt ! D (4 b p − ′ diag( a p ) ) , p = 1 , . . . , P, ( τ ′ t ) E s τt ! = 1 , s = 1 , . . . , ( N L + 1) , where C = B B ′ B ! , E = diag( a p ) ′ ! ; E s is a matrix with s -row and s -column E s ( s, s ) = 1 , others equal 0, for s = 1 , . . . , ( N L + 1) . Byintroducing a new variable X = ( τ ′ t ) ′ ( τ ′ t ) and removing the constraint rank ( X ) = 1 , the problem (69)can be relaxed to the problem (49). Acknowledgment
We would like to thank Yunmin Zhu for his helpful suggestions that greatly improved the quality of thispaper.
References [1] A. O. Hero, D. A. Casta ˜ n ´ o n, D. Cochran, and K. Kastella, Foundations and applications of sensormanagement . New York: Springer, 2008.[2] A. O. Hero and D. Cochran, “Sensor management: Past, present, and future,”
IEEE Sensors Journal ,vol. 11, pp. 3064–3075, December 2011.[3] S. Joshi and S. Boyd, “Sensor selection via convex optimization,”
IEEE Transactions on Signal Pro-cessing , vol. 57, pp. 451–462, February 2009. 294] Y. Mo, R. Ambrosino, and B. Sinopoli, “Sensor selection strategies for state estimation in energy con-strained wireless sensor networks,”
Automatica , vol. 47, pp. 1330–1338, 2011.[5] G. Zhang, S. Ferrari, and M. Qian, “An information roadmap method for robotic sensor path planning,”
Journal of Intelligent & Robotic Systems , vol. 56, no. 1, pp. 69–98, 2009.[6] F. Zhao and L. Guibas,
Wireless Sensor Networks: An Information Processing Approach . San Francisco,CA: Morgan Kaufmann, 2004.[7] V. Gupta, T. H. Chung, B. Hassibi, and R. M. Murray, “On a stochastic sensor selection algorithm withapplications in sensor scheduling and sensor coverage,”
Automatica , vol. 42, no. 2, pp. 251–260, 2006.[8] S. Jiang, R. Kumar, and H. E. Garcia, “Optimal sensor selection for discrete-event systems with partialobservation,”
IEEE Transactions on Automatic Control , vol. 48, no. 3, pp. 369–381, 2003.[9] M. L. Hernandez, T. Kirubarajan, and Y. Bar-Shalom, “Multisensor resource deployment using pos-terior cramer-rao bounds,”
IEEE Transactions on Aerospace and Electronic Systems , vol. 40, no. 2,pp. 399–416, 2004.[10] R. Tharmarasa, T. Kirubarajan, M. L. Hernandez, and A. Sinha, “PCRLB-based multisensor arraymanagement for multitarget tracking,”
IEEE Transaction on Aerospace and Electronic Systems , vol. 43,no. 2, pp. 539–555, 2007.[11] T. Zhao and A. Nehorai, “Information-driven distributed maximum likelihood estimation based ongauss-newton method in wireless sensor networks,”
IEEE Transactions on Signal Processing , vol. 55,pp. 4669–4682, September 2007.[12] J. Denzler and C. M. Brown, “Information theoretic sensor data selection for active object recognitionand state estimation,”
IEEE Transactions on Pattern Analysis and Machine Intelligence , vol. 24, no. 2,pp. 145–157, 2002.[13] I. Mark P. Kolba, Student Member and L. M. Collins, “Information-based sensor management in thepresence of uncertainty,”
IEEE Transactions on Signal Processing , vol. 55, pp. 2731–2735, June 2007.[14] C. M. Kreucher, A. O. Hero, K. D. Kastella, and M. R. Morelande, “An information-based approach tosensor management in large dynamic networks,”
Proceedings of the IEEE , vol. 95, pp. 978–999, May2007.[15] C. Kreucher, K. Kastella, and A. O.Hero, “Sensor management using an active sensing approach,”
Signal Processing , vol. 85, no. 3, pp. 607–624, 2005.[16] E. Masazade, R. Niu, and P. K. Varshney, “Dynamic bit allocation for object tracking in wireless sensornetworks,”
IEEE Transactions on Signal Processing , vol. 60, pp. 5048–5063, October 2012.3017] J. L. Williams, J. W. Fisher, and A. S. Willsky, “Approximate dynamic programming forcommunication-constrained sensor network management,”
IEEE Transactions on Signal Processing ,vol. 55, no. 8, pp. 4300–4311, 2007.[18] G. M. Hoffmann and C. J. Tomlin, “Mobile sensor network control using mutual information methodsand particle filters,”
IEEE Transactions on Automatic Control , vol. 55, pp. 32–47, January 2010.[19] N. Xiong and P. Svensson, “Multi-sensor management for information fusion: issues and approaches,”
Information Fusion , vol. 3, no. 2, pp. 163–186, 2002.[20] S. Boyd and L. Vandenberghe,
Convex Optimization . Cambridge University Press, 2004.[21] Y. Bar-Shalom, H. Chen, and M. Mallick, “One-step solution for the multistep out-of-sequence mea-surement problem in tracking,”
IEEE Transactions on Aerospace and Electronic Systems , vol. 40,pp. 27–37, January 2004.[22] Y. Zhu, J. Zhou, X. Shen, E. Song, and Y. Luo,
Networked Multisensor Decision and Estimation Fusion:Based on Advanced Mathematical Methods . CRC Press, 2012.[23] A. Ben-Israel and T. N. E. Greville,
Generalized inverses: theory and applications . New York: JohnWiley, second ed., 2003.[24] Y. Bar-Shalom and X. Li,
Multitarget-Multisensor Tracking: Principles and Techniques . Storrs, CT:YBS Publishing, 1995.[25] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic opti-mization problems,”
IEEE Signal Processing Magazine , vol. 27, pp. 20–34, May 2010.31 T h e t r ace o f e s ti m a ti on e rr o r c ov a r i a n ce Mo et al’s algorithmJoshi et al’s algorithmOur algorithm in Theorem 4.1
Figure 1: The traces of the estimation error covariance are plotted as a function of number of selected sensors. −2 Number of selected sensors T i m e ( s e ond s ) Mo et al’s algorithmJoshi et al’s algorithmOur algorithm in Theorem 4.1
Figure 2: The cpu times are plotted as a function of number of selected sensors.32 … ,50 T h e t r ace o f f Exhaustive methodAlogrithm 4.2 with weights=[1/3 1/3 1/3]Alogrithm 4.2 with weights=[0 0 1]
Figure 3: The traces of the final estimation error covariance for 50 Monte Carlo runs. … ,50 T h e t r ace o f f Exhaustive methodAlogrithm 4.2 with weights=[1/3 1/3 1/3]Alogrithm 4.2 with weights=[0 0 1]
Figure 4: The traces of the average estimation error covariance in 50 Monte Carlo runs.33 … ,50 T h e s u m o f i n f o r m a ti on m ea s u r e o f N s t e p s Alogrithm 4.2 with weights=[1/3 1/3 1/3]Alogrithm 4.2 with weights=[0 0 1]
Figure 5: The sum of information measures of N time steps for 50 Monte Carlo runs … ,50. upper bounds of the objectivelower bounds of the objective Figure 6: The upper bound and lower bound of the objective function of the optimization problem (37) for50 Monte Carlo runs. 34 … ,50. G a p s Figure 7: The gaps, i.e. the upper bounds minus the lower bounds shown in Figure 6.
10 20 30 40 50 60 70 80 90 10000.020.040.060.080.10.120.14 Number of selected sensors G a p s Figure 8: The average gaps based on 50 Monte Carlo runs are plotted as a function of number of selectedsensors. 35 T h e i n f o r m a ti on m ea s u r e o f t h e ( k + ) − t h ti m e s t e p Exhaustive methodIgnore dependenceAlgorithm 5.1 with S=20Algorithm 5.1 with S=100
Figure 9: Information measure based on the selected sensors at the ( k + 1) -th time step from weak to strongsignal power of the jammer (from weak to strong correlation between sensors). T h e t r ace o f P k + | k + Exhaustive methodIgnore dependenceAlgorithm 5.1 with S=20Algorithm 5.1 with S=100
Figure 10: The trace of estimation error covariance at the ( k + 1) -th time step from weak to strong signalpower of the jammer (from weak to strong correlation between sensors).36 T h e s u m o f t h e w e i gh t e d i n f o r m a ti on m ea s u r e o f N ti m e s t e p s Algorithm 5.1 with S=2000Algorithm 5.1 with S=100Algorithm 5.1 with S=20Ignore dependence
Figure 11: The sum of the weighted information measures of N time steps from weak to strong signal powerof the jammer (from weak to strong correlation between sensors). T h e t r ace o f t h e a v e r a g e e s ti m a ti on e rr o r c ov a r i a n ce Algorithm 5.1 with S=20Algorithm 5.1 with S=100Algorithm 5.1 with S=2000Ignore dependence
Figure 12: The trace of the average estimation error covariance of N time steps from weak to strong signalpower of the jammer (from weak to strong correlation between sensors).37 R M S E Ignore dependenceAlgorithm 5.1 wih S=20Algorithm 5.1 wih S=100Algorithm 5.1 wih S=2000
Figure 13: RMSE of the state estimates based on 200 Monte Carlo runs. −100 0 100 200 300 400 500 600 700−1000100200300400500 i=1i=2i=3i=4i=5 i=6i=7i=8i=9i=10 i=11i=12i=13i=14i=15 i=16i=17i=18i=19i=20 i=21i=22i=23i=24i=25Fusion center JammerThe initial positionof the targetThe initial velocity(−20 m/s, 0m/s)−100 0 100 200 300 400 500 600 700−1000100200300400500 i=1i=2i=3i=4i=5 i=6i=7i=8i=9i=10 i=11i=12i=13i=14i=15 i=16i=17i=18i=19i=20 i=21i=22i=23i=24i=25Fusion center JammerThe initial positionof the targetThe initial velocity(−20 m/s, 0m/s)