Optimal Sensor Collaboration for Parameter Tracking Using Energy Harvesting Sensors

1 Optimal Sensor Collaboration for P arameter T racking Using Ener gy Harv esting Sensors Shan Zhang, Sijia Liu, Member , IEEE, V inod Sharma, Senior Member , IEEE, and Pramod K. V arshney , Life F ellow , IEEE Abstract —In this paper , we design an optimal sensor collabo- ration strategy among neighboring nodes while tracking a time- varying parameter using wireless sensor networks in the presence of imperfect communication channels. The sensor network is assumed to be self-power ed, where sensors are equipped with energy harvesters that replenish energy from the en vironment. In order to minimize the mean square estimation error of parameter tracking, we pr opose an online sensor collaborati on policy subject to real-time energy harvesting constraints. The proposed energy allocation strategy is computationally light and only relies on the second-order statistics of the system parameters. For this, we ﬁrst consider an ofﬂine non-con v ex optimization problem, which is solved exactly using semideﬁnite programming. Based on the ofﬂine solution, we design an online power allocation policy that requir es minimal online computation and satisﬁes the dynamics of ener gy ﬂow at each sensor . W e prov e that the proposed online policy is asymptotically equi valent to the optimal ofﬂine solution and show its conver gence rate and rob ustness. W e empirically show that the estimation perf ormance of the proposed online scheme is better than that of the online scheme when channel state information about the dynamical system is available in the low SNR regime. Numerical results are conducted to demonstrate the effectiveness of our appr oach. Index T erms —Wir eless sensor netw orks, parameter tracking, node collaboration, energy harv esting, semideﬁnite program- ming. I . I N T RO D U C T I O N Recent advances in wireless communications and electron- ics ha ve enabled the development of low-cost, low-po wer , multifunctional sensor nodes that are deployed as networks for man y applications such as en vironment monitoring, source localization and target tracking [1]–[3]. These sensor nodes sense and measure some attributes of the targets of inter- est, and are able to exchange their information through in- network communications [4]. In this paper , we study the problem of parameter tracking in the presence of inter -sensor communications that is referred to as sensor collaboration. Here sensors are equipped with ener gy harv esters, which can replenish energy from the en vironment, such as sun light and wind. The act of sensor collaboration allows sensors to update Copyright (c) 2015 IEEE. Personal use of this material is permitted. Howe ver , permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. S. Zhang and P . K. V arshney are with the Department of EECS, Syracuse Univ ersity , Syracuse, NY 13244, USA. Email: { szhang60,v arshney } @syr .edu. S. Liu is with the MIT -IBM W atson AI Lab, IBM Research, Cambridge, MA 02142, USA. Email: sijia.liu@ibm.com. V . Sharma is with the Department of ECE, Indian Institute of Science, Bangalore 560012, India. Email: vinod@ece.iisc.ernet.in. This work was supported in part by AR O grant number W911NF-14-1-0339 and AFOSR grant number F A9550-16-1-0077. their measurements by taking a linear combination of the measurements of those they interact with, prior to transmission to a fusion center (FC). As suggested in [5], collaboration smooths out the observation noise, thereby improving the quality of the signal transmitted to the FC and eventual estimation performance. In the absence of sensor collaboration, the proposed estima- tion architecture reduces to a classical distributed estimation system that uses an amplify-and-forward transmission strategy [6]–[8]. The power allocation problem for distributed esti- mation under an orthogonal multiple access channel (MA C) model was studied in [6], where the optimal po wer amplifying factors were found by minimizing the estimation distortion subject to energy constraints. In [7], the same problem was studied under a different communication model, coher ent MA C, where sensors coherently form a beam into a common channel receiv ed at the FC. In [8], the design of optimal power allocation strategies was addressed over orthogonal fading wireless channels for source estimation in a network of energy-harv esting sensors. In [9], [10], the problem of power allocation for parameter tracking was studied where the (scalar) parameter of interest was modeled as a Gauss-Markov process. In the aforementioned literature [6]–[10], inter-sensor communication was not considered. Moreov er , the work in [6]–[8] only focused on one-snapshot estimation, while the work [9], [10] only studied the conv entional sensor network with battery-limited sensors. In contrast, here we seek an optimal sensor collabor ation scheme for a dynamic parameter trac king using ener gy-harvesting sensors. The problem of distributed estimation with sensor collabo- ration has attracted recent attention [5], [11]–[15]. In [11], the network topology was assumed to be fully connected, where all the sensors are allowed to share their observations. It was shown that the optimal strategy is to transmit the processed signal (after collaboration) over the best av ailable channels with po wer le vels consistent with the channel qualities. In [12], the optimal collaboration strategy was designed under star , branch and linear network topologies. In [5], [13], [14], the problem of collaboration network topology design was studied by incorporating the cost of sensor collaboration. It was shown that a partially connected network can yield estimation perfor- mance close to that of a fully connected network. Ho we ver , the aforementioned literature on collaborativ e estimation focused only on static networks, and the parameter to be estimated is temporally in variant. In [15], the sensor collaboration strategy was designed for the estimation of temporally correlated pa- rameters. Howe ver , in [15] a batch-based linear estimator was 2 used, where all the historical measurements have to be stored for decision making. Different from [15], a Kalman ﬁlter- like estimator is adopted in this paper , and a computationally inexpensi ve b ut efﬁcient real-time sensor collaboration scheme is developed in conjunction with parameter tracking. In the existing work on power allocation for collaborativ e estimation [5], [11]–[15], sensor networks were composed of (con ventional) battery-limited sensors. Motiv ated by the rapid advances in the emerging energy-harvesting de vice technology [16], it is attractive to perform collaborativ e estimation using energy harvesting sensors. In this context, each sensor can replenish energy from the en vironment without the need of battery replacement. The effect of energy harvesting on parameter estimation was considered recently in [17]–[20]. In [17], a single sensor equipped with an energy harvester was used for parameter estimation, and the optimal power allocation scheme was designed under causal and non-causal side information of energy harvesting. In [18], a deterministic energy harvesting model was proposed for parameter estimation over an or- thogonal MAC. In [19], the optimal communication (sensors to the FC) strategy was obtained for remote estimation with energy harvesting sensors. In [17]–[19], the power allocation policy w as designed in the absence of sensor collaboration. In [20], the problem of energy allocation and storage control for collaborativ e estimation was addressed by solving noncon ve x optimization problems through con ve x relaxations. Ho wev er , the strategy dev eloped in [20] only provides an ofﬂine solution for one snapshot estimation. In contrast, we propose a sensor collaboration policy that obeys the real-time energy harvesting constraints. Another difference between our work and the previous ones is that the previous studies [5], [11]–[15], [20] hinge on the noise-free inter-sensor communication model, while the noiseless assumption is relaxed in this paper . Moreover , sensors communicate with neighbors using point to point orthogonal channels in [5], [11]–[15], [20], while sensors exchange their information with neighbors through a coherent MA C in this paper . Multiple-access channel communication introduces collaboration noise, which has to be taken into account while designing the optimal sensor collaboration scheme. In a preliminary version of this paper [21], we studied the sensor collaboration problem for static parameter estimation in the absence of collaboration noise. In this paper , we focus on tracking of a scalar dynamic parameter with noise-corrupted sensor collaboration. The assumption of scalar random process enables us to obtain e xplicit expressions for the estimation dis- tortion and the transmission cost, and to e xplore the asymptotic behavior of the estimation error with respect to the energy constraint. The assumption of scalar random process was also used in other power allocation problems [5], [6], [10], [11]. Howe v er , different from [5], [6], [10], [11], we consider a different problem in which a scalar random process is tracked in the presence of noise-corrupted inter-sensor collaboration and energy-harvesting sensors. More speciﬁcally , we have the following new contributions in this paper • W e consider tracking of a correlated parameter sequence. • W e incorporate a recursiv e linear minimum mean squared error estimator (R-LMMSE) to track the parameter in the presence of multiplicativ e noise. • W e consider noise-corrupted sensor collaboration accom- plished through a coherent MA C in a network of energy harvesting sensors for parameter tracking. • W e propose an optimal ofﬂine sensor collaboration scheme under the assumption that the energy constraint is on an a verage basis, and show that although the resulting optimization problem is non-con vex, a globally optimal solution can be found via semideﬁnite programming. Also, we prove the stability of this scheme. • W e propose an online sensor collaboration policy that satisﬁes the real-time energy harvesting constraints with minimal online computational comple xity , and prove that its resulting long-term estimation performance is the same as that of the optimal of ﬂine solution. Moreover , we show the con v ergence rate and robustness of the proposed optimal online scheme. • W e demonstrate the efﬁcienc y of our methodology by comparing it with a greedy optimal online policy when the channel state information (CSI) regarding the dynam- ical system is assumed to be known causally at the FC. The main generalization compared to [21] is that the pa- rameter to be estimated in this paper is a correlated sequence compared to the independent and identically distrib uted (i.i.d.) sequence in [21]. Thus in [21], the estimation procedure required estimating the parameter via only the data obtained at that time. In this paper, we incorporate an R-LMMSE to track and estimate the parameter because of the correlation in time. Moreov er , it becomes a distributed R-LMMSE with online constraints (due to energy harv esting) and multiplication of the system parameters with random coefﬁcients (due to channel gains). Thus, technically it is much more complicated and different from [21]. The optimization problem solved also is much more complicated than in [21]. The rest of the paper is organized as follows. In Section II, we introduce the collaborati ve tracking system, and present the general statement of the sensor collaboration problem. In Section III, we formulate the ofﬂine sensor collaboration problem for parameter tracking, where only partial knowledge of the second-order statistics of system parameters is required. In Section IV, we solve the non-con ve x ofﬂine sensor collab- oration problem and obtain its globally optimal solution via semideﬁnite programming. W e also design an online sensor collaboration policy . In Section V, for comparison, we study a sensor collaboration problem with known channel state information. In Section VI, we demonstrate the effecti veness of our online solution obtained in Section IV through numer- ical examples and compare with the greedy optimal solution obtained in Section V. Finally , in Section VII we summarize our work and discuss future research directions. I I . P RO B L E M S TA T E M E N T In this section, we formally state the problem of sen- sor collaboration for parameter tracking using a network of 3 energy harvesting sensors. In the wireless sensor network under consideration, each sensor is equipped with an energy harvesting device which replenishes itself from a renew able energy source. In the overall inference system, sensors ﬁrst acquire raw measurements of the phenomenon of interest via a linear sensing model, and then perform spatial collaboration via a coherent MA C. A coherent MAC channel can be realized through transmit beamforming [22], where sensor nodes simultaneously transmit a common message and the phases of their transmissions are controlled so that the signals constructiv ely combine at the FC. This requires that each of the sensor nodes kno ws its channel gains. It has been sho wn in [9] that the coherent MAC can be realized by the distributed synchronization schemes described under moderate amounts of phase error [23], [24]. An adv antage of a coherent MA C is that all the sensors use the same channel to transmit their observations. W e also remark that the coherent MAC model has been widely used in the study of the po wer allocation problem for distributed estimation [5], [7], [10], [15], [21]. After collaboration, the signals are transmitted to the FC, which tracks the random process. W e show the collaborative estimation system to be studied in Fig. 1. A list of the system parameters is provided in T able I. Ambient energy random process n oisy sensor collaboration using MAC {   } LMMSE estimate fading coherent - MAC   , 1   ,    , 2   , 1   , 2   ,                  { 󰆹   } Kalman filter FC Sensor with energy harvest er Communication Fig. 1: Collaborative tracking system. A. Sensor collaboration via coher ent MAC Let { θ k ∈ R } be the discrete-time random process to be estimated, which evolv es as a ﬁrst-order Gauss-Markov process θ k = αθ k − 1 + τ k , k = 1 , 2 , . . . , (1) where θ k denotes the random parameter at time k , α is the known state transition coefﬁcient, and τ k is i.i.d with zero mean and variance σ 2 τ . W e also assume that θ k has zero mean, and | α | < 1 . Then, θ k is an ergodic Markov chain with a unique stationary distribution. In addition, if τ k has a positiv e density on whole of real line and E [ | τ k | ] < ∞ , then θ k is geometrically ergodic [25]. Many practically relev ant distributions satisfy these conditions: Gaussian, mixtures of Gaussians and some heavy tailed stable distributions modeling electromagnetic interference. W e consider a linear measurement model x k = h k θ k +  k , k = 1 , 2 , . . . , (2) where x k ∈ R N is the v ector of measurements acquired by N sensors at time k , h k ∈ R N is the vector of observation gains with known second-order statistics, E [ h k ] = h and cov ( h k ) = Σ h ;  k ∈ R N is the vector of measurement noises with zero mean and cov ariance E [  k 1  T k 2 ] = Σ  I ( k 1 − k 2 ) . W e also assume that { h k } , { θ k } and {  k } are independent of each other . After linear sensing, each sensor may pass its observation to other sensors for collaboration, prior to transmission to the FC. The sharing of measurements is accomplished via a coherent MA C. Collaboration among sensors is represented by a ﬁxed topology matrix A with binary entries, namely , A mn ∈ { 0 , 1 } for m ∈ [ M ] and n ∈ [ N ] , where for ease of notation, [ M ] denotes { 1 , 2 , . . . , M } . Here A mn = 1 signiﬁes that there exists a communication link from the n th sensor to the m th sensor; otherwise, A mn = 0 . The neighborhood structure of the sensor network can be determined by the communication range of each sensor in the network. Note that A is essentially a truncated adjacency matrix with a self loop because each sensor can collaborate with itself, namely , A mm = 1 . W e note that if A mn = 0 for all m 6 = n , the resulting sensor collaboration strategy reduces to the amplify- and-forward transmission strategy considered in [7], [10]. W ith a relabeling of sensors, we assume that the ﬁrst M sensors (out of a total of N sensor nodes) communicate with the FC. The condition M ≤ N is motiv ated by the need to reduce possible high transmission energy expense. The prior knowledge about the network topology A is commonly determined based on the quality of observation and channel gains, and the pairwise sensing distance [5]. Giv en the network topology , the collaborativ e signal re- ceiv ed at the i th sensor at time k is modeled as z k,i = W k ( i, i ) x k,i +   X j ∈N i W k ( i, j ) x k,j + κ k,i   , N i = { j | A ij = 1 , j ∈ [ N ] and j 6 = i } , (3) where x k,i is the i th entry of the observ ation vector x k , N i is the set of immediate neighbors of the i th sensor , W k ( i, i ) is a power amplifying factor used for transmitting the measurement of the i th sensor to the FC, W k ( i, j ) ( i 6 = j ) is the power amplifying factor used for transmitting the measurement of the j th sensor to the i th sensor , and { κ k,i , k ≥ 0 } denotes the communication noise associated with a coherent MA C for inter-sensor collaboration, which is an independent, identically distributed (i.i.d) sequence, independent of other system pa- rameters, with zero mean and variance σ 2 κ . The sensor collaboration model (3) can also be written in a compact form z k = W k x k + κ k , W k ◦ ( 1 M 1 T N − A ) = 0 , (4) where z k = [ z k, 1 , . . . , z k,M ] T ∈ R M denotes the collaborati ve signals at time k , W k ∈ R M × N is the collaboration matrix that contains collaboration weights (based on the allocated energy) used to combine sensor measurements, W k ( i, j ) is the ( i, j ) th entry of W k , κ k = [ κ k, 1 , . . . , κ k,M ] T is the vector of receiv er communication noises for sensor collaboration, ◦ denotes the Hadamard product, 1 N is the N × 1 vector of all ones, and 0 is the M × N matrix of all zeros. In what 4 T ABLE I: List of System Parameters Parameter Symbol Mean Cov ariance Parameter of interest θ k 0 s k State transition coefﬁcient α - - Parameter model noise τ k 0 σ 2 τ Observation gain v ector h k h Σ h Measurement noise vector  k 0 Σ  Collaboration noise at sensor i κ k,i 0 σ 2 κ Channel g ain vector g k g Σ g T ransmission noise ς k 0 σ 2 ς Rayleigh f ading channel parameter ξ ξ q π 2 4 − π 2 ξ 2 follows, while referring to vectors of all ones and all zeros, their dimensions will be omitted for simplicity but can be inferred from the context. In (4), the second identity implies that W k preserves the sparsity structure of the topology matrix A . After sensor collaboration, the collaborativ e signal z k is transmitted through a coherent MA C so that the recei ved signal y k at the FC is a coherent sum y k = g T k z k + ς k , k = 1 , 2 , . . . , (5) where g k ∈ R M is the vector of channel gains with known second-order statistics E [ g k ] = g and co v ( g k ) = Σ g , and { ς k } is an i.i.d noise sequence independent of { g k } and { z k } with zero mean and variance σ 2 ς . W e assume that the FC kno ws the second-order statistics of the observ ation gain, channel gain, and additive noises. Our goal is to design the optimal energy allocation scheme W k such that the estimation distortion of tracking θ k is optimized under real-time energy harvesting constraints. Prior to formally stating our problem, we next introduce the transmission costs, energy-harvesting sensors, and the estimation approach. B. T r ansmission cost In the entire estimation system, sensor energy is consumed for both inter-sensor communication and sensor-to-FC com- munication. Based on (3), the cost of energy consumed at sensor j ∈ [ N ] and time k for spatial collaboration is T (1) k,j ( W k ) : = X i ∈N j ( W k ( i, j ) x k,j ) 2 , (6) and its expectation over stochastic parameters is given by E [ T (1) k,j ( W k )] ( a ) =  e T j E [ x k x T k ] e j  M X i =1 ,i 6 = j W 2 k ( i, j ) =  e T j E [ x k x T k ] e j  [ e T j ( W k ◦ ˜ I ) T ( W k ◦ ˜ I ) e j ] , (7) where E [ x k x T k ] = Σ  + σ 2 θ ( hh T + Σ h ) , the equality (a) holds since W k ( i, j ) = 0 if i / ∈ N j and i 6 = j , ◦ denotes the Hadamard product, ˜ I : = 11 T − [ I M , 0 M × ( N − M ) ] , and e j is a basis vector with 1 at the j th coordinate and 0 s elsewhere. Moreov er from (4), the cost of energy consumed at sensor i ∈ [ M ] and time k for sensor-to-FC communication becomes T (2) k,i ( W k ) = z 2 k,i (8) and its expectation is given by E [ T (2) k,i ( W k )] = e T i W k E [ x k x T k ] W T k e i + σ 2 κ . (9) The total transmission cost can now be represented as T k,i ( W k ) = ( T (1) k,i ( W k ) + T (2) k,i ( W k ) , i ≤ M , T (1) k,i ( W k ) , M < i ≤ N , (10) where we recall that only the ﬁrst M sensors are used to communicate with the FC, and thus T (2) k,i ( W k ) = 0 if i > M . C. Ener gy ﬂow of energy harvesting sensors Let H k,i denote the harvested energy of the i th sensor at time k . Let H k = { H k,i , i = 1 , 2 , . . . , N } . Follo wing [26], [27], we assume that { H k,i , k ≥ 0 } satisfy the strong law of large numbers (SLLN) with the limiting mean E [ H k,i ] = µ i : lim 1 n P n k =1 H k,i = µ i almost surely . An asymptotically stationary and er godic process { H k } satisﬁes this assumption. This energy generation process is general enough to cover the assumptions usually made in the literature [28]–[32] and is quite realistic. In particular, this allows correlation in time of the energy harvested at a sensor as well as spatial correlation in the energy harvested at neighboring sensors. Moreov er , we assume that the energy harv ested at the end of each time step, namely , H k,i is not av ailable until the be ginning of time k + 1 . The energy storage buf fer of each sensor is considered to be large enough (here assumed with inﬁnite capacity). And each sensor is supposed to know the energy it has in its buf fer at a giv en time. W e denote by S k,i the energy av ailable in the i th sensor at time k , which ev olves as S k +1 ,i = ( S k,i − T k,i ) + + H k,i , i ∈ [ N ] , (11) where for notational simplicity , we use T k,i to denote the transmission cost giv en by (10), and ( x ) + = max { 0 , x } . D. Pr oblem F ormulation In this setup, the goal is to solve the follo wing problem: At each time k , ﬁnd an energy allocation scheme W k such that P ∞ k =1 E [( θ k − ˆ θ k ) 2 ] is minimized with the real-time energy constraints T k,i ≤ S k,i , ∀ i = 1 , 2 , . . . , N and ∀ k ≥ 1 . In the abov e problem, ˆ θ k is the estimate of θ k . Howe v er , this problem may not remain ﬁnite for all choices of W k . Therefore, we consider the minimization of lim sup N →∞ 1 N N X k =1 E [( θ k − ˆ θ k ) 2 ] . (12) This is close to the standard av erage cost Markov decision problem (MDP) [33] except that we ha ve real-time constraints, which are not considered by the usual constrained MDP [34]. Thus, we may consider a greedy approach: At each time k , ﬁnd W k that minimizes E [( θ k − ˆ θ k ) 2 ] such that T k,i ≤ S k,i for each i . This will imply that we should take T k,i = S k,i most of the time. But, this may not be a good strategy from a long term point of view because this would then mean T k,i = H k,i most of the time and hence whenev er H k,i is lo w , we may get high estimation error . Thus, a more sustainable strategy , as we will see later , will be T k,i ≤ min ( S k,i , µ k,i − η ) , where η is a small positiv e constant. Also, for computational reasons, because we need to perform the computations at each time k , we consider a linear estimator . Furthermore, ours is a distributed setup: the energy constraint S k,i will only be 5 known to sensor i at time k and is randomly changing with time. Also, we would like the FC to compute W k because it has the maximum computational power and is assumed to be connected to a regular power grid. Howe ver , sending the needed information h k , g k and ς k at each time k will incur too much communication cost. Also, after computation, the FC will need to transmit W k to all the sensors for them to decide on the collaboration strategy . T aking the above considerations into account, in the follow- ing, we ﬁrst formulate an ofﬂine optimization problem that can be solved using only the second-order statistics of h k , g k , H k , α and the variances of the receiver noises at different sensor nodes. This solution W ∗ will not necessarily satisfy the real- time energy constraint at the sensor nodes. The solution W ∗ is provided to all the sensor nodes and the FC before hand. Based on this solution, each sensor can obtain a real-time solution W k using only its o wn ener gy information S k,i at time k . W e will show that asymptotically the mean squared error (MSE) of the real-time solution matches that obtained when using W ∗ . Furthermore, at the end, via simulations, we will also show that the performance of this scheme is close to that of an optimal linear estimator computed at the FC at a far more computational cost and using all the current information of h k and g k . Remark 1: The second-order statistics can be estimated us- ing robust mean and v ariance estimators [35]. Most commonly used rob ust estimators, e.g., M-estimators and S-estimators, to estimate the cov ariance matrix hav e been shown to have very low estimation error with a reasonal number of samples [35]. I I I . O FFL I N E O P T I M I Z A T I O N P RO B L E M Based on (1) – (5), we consider the state space model at the FC, θ k = αθ k − 1 + τ k , y k = g T k W k h k θ k + v k , (13) where v k = g T k W k  k + g T k κ k + ς k . Here v k can be regarded as the measurement noise with statistics E [ v k ] = 0 and co v( v k ) = tr( W k Σ  W T k E [ g k g T k ]) + σ 2 κ tr( E [ g k g T k ]) + σ 2 ς = tr( W k Σ  W T k Λ g ) + σ 2 κ tr( Λ g ) + σ 2 ς , (14) where Λ g = E [ g k g T k ] = gg T + Σ g from (5). W e will consider W k = W in this section and ﬁnd W that minimizes (12). Since we are considering a linear estimator , we can formulate it as a Kalman ﬁlter . Howe ver , even with W k = W , the ﬁlter for the state space model given in (13) is not a standard Kalman ﬁlter because of the multiplicative noise g T k Wh k in the measurement model. A recursive linear minimum mean squared error estimator (R-LMMSE) in the presence of multiplicativ e noise has been considered in [36], [37]. Motiv ated by the results in [36], [37] and since we hav e already assumed that | α | < 1 , the following Lemma shows that (13) satisﬁes the assumption in [37] and hence that the R-LMMSE to be deriv ed is stable. In the following, we denote u k := g T k Wh k . Lemma 1: Giv en the state space model in (13), we hav e the following results: u k is uncorrelated with v k . Moreov er , { v k } is a zero mean, uncorrelated sequence. Proof : See Appendix A.  Based on Lemma 1, R-LMMSE [37] is employed to track the random parameter θ k , which is shown in Algorithm 1. Algorithm 1 R-LMMSE with known second-order statistics of the system. Inputs: ˆ θ 0 = 0 , P 0 assumed known 1) Prediction step: ˆ θ k | k − 1 = α ˆ θ k − 1 , (15) with the predicted estimation error P k | k − 1 = α 2 P k − 1 + σ 2 τ . (16) 2) Updating step: ˆ θ k = ˆ θ k | k − 1 + d k ( y k − E [ u k ] ˆ θ k | k − 1 ) , (17) with the updated estimation error P k = (1 − d k E [ u k ]) P k | k − 1 , (18) where d k is the estimator gain d k = P k | k − 1 E [ u k ] ( E [ u k ]) 2 P k | k − 1 + cov( v k ) + cov( u k ) s k , (19) and s k : = co v( θ k ) , s k = α 2 s k − 1 + σ 2 τ (20) with known initial condition s 0 , E [ u k ] = g T Wh and E [ u k u j ] =  tr( WΛ h W T Λ g ) k = j, 0 k 6 = j, (21) where Λ g has been deﬁned in (14), and Λ h = E [ h k h T k ] = hh T + Σ h . Also, cov( v k ) has been deﬁned in (14) with W k = W and co v( u k ) = E [ u 2 k ] − ( E [ u k ]) 2 . Remark 2: It is clear from (15) – (18) that if the mul- tiplicativ e noise u k is deterministic, namely , cov( u k ) = 0 , R-LMMSE reduces to the commonly used Kalman ﬁltering algorithm. From [37], we obtain that the estimation error E [( θ k − ˆ θ k ) 2 ] = P k → P ( ∞ ) and cov( θ k ) = s k → s ∞ . From (20), we obtain s k = α 2 k s 0 + σ 2 τ k X i =1 α 2 i − 2 = α 2 k s 0 + σ 2 τ 1 − α 2 k 1 − α 2 . (22) Using the fact that α 2 < 1 , we note that s ∞ = σ 2 τ 1 − α 2 . Moreov er , P ∞ satisﬁes the Riccati equation P ∞ = α 2 P ∞ ¯ d ¯ c 2 P ∞ + ¯ d + σ 2 τ , (23) where ¯ c = g T Wh , and ¯ d = tr( WΣ  W T Λ g ) + σ 2 κ tr( Λ g ) + σ 2 ς + tr( WΛ h W T Λ g ) s ∞ − ( g T Wh ) 2 s ∞ . Also, then (12) equals P ( ∞ ) . Thus, we would like to obtain the scheme W that minimizes P ( ∞ ) which at the same time also satisﬁes the energy constraint in the mean, i.e., E [ T i ( W )] ≤ µ i − η , for all i and a small positive η , where T i ( · ) denotes the total transmission cost as k → ∞ . It is known from [9, Lemma 1] that P ∞ is a decreasing function of ¯ c 2 / ¯ d . Thus, the optimal W can be obtained by solving the following optimization problem: maximize W f ( W ) subject to W ◦ ( 11 T − A ) = 0 , E [ T i ( W )] ≤ µ i − η , i ∈ [ N ] , (24) 6 T ABLE II: Ke y Equations Equation Number Problem Statement (24) Ofﬂine optimization problem (36) Constructed online energy allocation policy (38) Statistics based optimization problem (42) Full kno wledge based online policy where T i ( W ) is the transmission cost giv en by (10), the expectation is with respect to the stationary distribution of θ k , µ i is the mean of the harvested energy at sensor i , η > 0 is a giv en small constant and f ( W ) = ( g T Wh ) 2 tr( W [ Λ h s ∞ + Σ  ] W T Λ g ) + σ 2 κ tr( Λ g ) + σ 2 ς , (25) where s ∞ = σ 2 τ 1 − α 2 . In the of ﬂine optimization problem (24), the last inequality constraint implies that the energy cost cannot exceed the amount of harvested energy on an averag e , where η can be regarded as the remaining av erage energy that is deliberately kept in the ener gy storage b uffer for further usage [21]. W e observe that { θ k } is a geometrically ergodic Markov chain and the regeneration epochs of { θ k } and { ( T k ( W ) , θ k ) } are the same. Therefore, { ( T k ( W ) , θ k ) } is also a geometrically ergodic Markov chain. In particular , T k ( W ) satisﬁes SLLN. Remark 3: Note that the stated optimization problem in (24) is an ofﬂine sensor collaboration scheme. Therefore, it may not satisfy real-time energy constraints. This optimal of ﬂine sensor collaboration scheme needs to be determined only once, and thus has signiﬁcant computational merit. In the next section, we use the ofﬂine scheme to derive an online solution that asymptotically achie ves the optimal P ∞ subject to the real-time energy constraints. Some ke y equations are summarized in T able II. I V . O P T I M A L S E N S O R C O L L A B O R A T I O N : F RO M O FFL I N E T O O N L I N E S O L U T I O N In this section to solve the of ﬂine problem (24), we be- gin by concatenating the nonzero entries of a collaboration matrix into a collaboration vector . There are two beneﬁts of using matrix vectorization: a) the topology constraint can be eliminated without loss of performance, which results in a less complex problem; b) the structure of noncon ve xity is easily characterized via such a reformulation. W e then deriv e the optimal solution of ofﬂine problem (24) with the aid of semideﬁnite programming. Given the optimal ofﬂine sensor collaboration scheme, we provide an online scheme which is asymptotically equiv alent to the optimal ofﬂine solution. In the of ﬂine problem (24), the only optimization variables are the nonzero entries of the collaboration matrix. T o solve this problem, we begin by concatenating the nonzero entries of W (columnwise) into the collaboration vector w = [ w 1 , w 2 , . . . , w L ] T ∈ R L , (26) where L is the total number of nonzero entries in the topol- ogy matrix A . W e also note that there exists a one-to-one correspondence between the elements of w and W , that is, there exists a row index m l and a column index n l such that w l = [ W ] m l n l , where [ X ] ij (or X ij ) denotes the ( i, j ) th entry of a matrix X . Another important observation from the ofﬂine problem (24) is that both the objectiv e function and the energy constraint in volv e quadratic matrix functions 1 . Inspired by this fact, we explore the relationship between W and w in Proposition 1. Proposition 1: Giv en a matrix W ∈ R M × N and its columnwise vector w ∈ R L that only contains the nonzero elements of W , the expressions of b T W and tr( CWDW T ) can be equiv alently expressed as functions of w , b T W = w T B , (27) tr( CWD W T ) = w T Ew , (28) where B ∈ R L × N and E ∈ R L × L are given by B ln =  b m l , n = n l 0 , otherwise , (29) E kl = [ C ] m k m l [ D ] n k n l , (30) for n ∈ [ N ] , k ∈ [ L ] and l ∈ [ L ] , where the indices m l and n l satisﬁes w l = W m l n l for l ∈ [ L ] . Proof: See Appendix F.  Based on Proposition 1, we can simplify the of ﬂine problem (24) by: a) eliminating the netw ork topology constraint, and b) expressing the objective function f ( W ) and the transmission cost E [ T i ( W )] in the forms that only inv olve quadratic v ector functions. Consequently , the ofﬂine problem (24) becomes maximize w w T Ω N w w T Ω D w + σ 2 κ tr( Λ g ) + σ 2 ς , subject to w T Ω T ,i w + σ 2 κ ≤ µ i − η , i ≤ M , w T Ω C ,i w ≤ µ i − η , M < i ≤ N , (31) where Ω N , Ω D , Ω T ,i and Ω C ,i are all positive semideﬁnite matrices. W e elaborate on the expressions of the aforemen- tioned coefﬁcient matrices in Appendix B. W e remark that problem (31) is a noncon ve x optimization problem as it requires the maximization of a ratio of conv ex quadratic functions. Howe ver , we will show that semideﬁnite programming (SDP) can be used to ﬁnd the globally optimal solution of problem (31). By introducing a new variable t , together with a new constraint 1 /t 2 = w T Ω D w + σ 2 κ g T g + σ 2 ς , problem (31) can be rewritten as maximize w ,t t 2 w T Ω N w , subject to t 2 w T Ω T ,i w + t 2 σ 2 κ ≤ t 2 ( µ i − η ) , i ≤ M , t 2 w T Ω C ,i w ≤ t 2 ( µ i − η ) , M < i ≤ N , t 2 w T Ω D w + t 2 σ 2 κ g T g + t 2 σ 2 ς = 1 , (32) where t and w are optimization variables. Upon deﬁning ¯ w = [ t w T , t ] T ∈ R L +1 , problem (32) can be rewritten as maximize ¯ w ¯ w T Q 0 ¯ w , subject to ¯ w T Q i, 1 ¯ w ≤ 0 , i ≤ M , ¯ w T Q i, 2 ¯ w ≤ 0 , M < i ≤ N , (33) 1 A quadratic matrix function is a function f : R n × r → R of the form f ( X ) = tr( X T AX ) + 2 tr( B T X ) + c for certain coef ﬁcient matrices A , B and c . 7 ¯ w T Q N +1 ¯ w ≤ 1 , where ¯ w is the optimization variable, and Q 0 =  Ω N 0 0 T 0  , Q i, 1 =  Ω T ,i 0 0 T σ 2 κ − ( µ i − η )  , Q i, 2 =  Ω C ,i 0 0 T − ( µ i − η )  , Q N +1 =  Ω D 0 0 T σ 2 κ g T g + σ 2 ς  . In problem (33), we have replaced ¯ w T Q N +1 ¯ w = 1 with its inequality counterpart without loss of performance since the inequality constraint is satisﬁed with equality at the solution while maximizing a conv ex quadratic function. W e also note that the solution of (31) can be obtained from the solution of problem (33) via w = [ ¯ w ] 1: L / [ ¯ w ] L +1 , where [ ¯ w ] 1: L denotes the vector that consists of the ﬁrst L entries of ¯ w , and [ ¯ w ] L +1 is the ( L + 1) th entry of w . Problem (33) no w contains a special noncon ve x structure: maximization of a con ve x homogeneous quadratic function (i.e., no linear term with respect to w is in volv ed) subject to homogeneous quadratic constraints. Although problem (33) is not con vex, its globally optimal solution can be obtained via SDP [10], [38]. W e provide the solution of problem (33) in Proposition 2. In the following Proposition, [ ¯ W ∗ ] L = s ∗ ( s ∗ ) T denotes the submatrix of ¯ W ∗ formed by deleting its ( L + 1) st row and column. It is rank 1 . Proposition 2: Let ¯ W ∗ be the solution of the following SDP problem, maximize ¯ W tr ( Q 0 ¯ W ) , subject to tr ( Q i, 1 ¯ W ) ≤ 0 , i ≤ M , tr ( Q i, 2 ¯ W ) ≤ 0 , M < i ≤ N , tr ( Q N +1 ¯ W ) ≤ 1 , ¯ W  0 , (34) where ¯ W ∈ R ( L +1) × ( L +1) is the optimization variable. The solution of problem (31) is then given by w ∗ = s ∗ / q [ ¯ W ∗ ] L +1 ,L +1 , (35) where [ ¯ W ∗ ] L +1 ,L +1 is the ( L + 1) th diagonal element of ¯ W ∗ . Proof: See Appendix C.  The optimal ofﬂine sensor collaboration matrix, namely , the solution of the ofﬂine problem (24), can no w be found by using Proposition 2 and the sparsity pattern of the network topology matrix A . In what follo ws, we will utilize the optimal ofﬂine sensor collaboration scheme to design an online sensor collaboration scheme W k that satisﬁes the real-time energy constraint T k,i ( W k ) ≤ S k,i for e very time k and ev ery sensor i . Here we recall that at time k , each sensor has access only to its stored energy S k,n giv en by (11). W e construct the following online energy allocation policy: W k = W ∗ , if T k,i ( W ∗ ) ≤ S k,i , ∀ i ∈ [ N ] , W k = β k ◦ W ∗ , otherwise , (36) where W ∗ denotes the optimal of ﬂine sensor collaboration matrix, ◦ denotes the Hadamard product, β k = min i { β k,i } , and β k,i = min { 1 , p S k,i /T k,i ( W ∗ ) } . Once W k is obtained from (36), the transmission cost at time k is then given by T k,i = T k,i ( W k ) . Subsequently , the status of the energy storage buf fer at the next time step is updated as S k +1 ,i according to (11). Remark 4: Note that the computation of β k requires the global knowledge β k,i , i ∈ [ N ] . Howe ver , we can use the max-consensus protocol [39] such that the computation β k = max i {− β k,i } can be performed in a decentralized setting via inter-sensor message passing. In Theorem 1, we show that the proposed online sensor collaboration policy is asymptotically consistent with the op- timal ofﬂine solution in the mean-squared-error sense under the framework of R-LMMSE. T o show this asymptotic con- sistency , we ﬁrst present Lemma 2. Lemma 2: As k → ∞ , the following results hold: 1) θ k d → θ ∞ , E [ θ γ k ] → E [ θ γ ∞ ] for γ = 1 , 2 , where d → denotes conv ergence in distribution. 2) E [ T k,i ( W ∗ )] → E [ T i ( W ∗ )] for each i ∈ [ N ] . Proof : See Appendix D.  Theorem 1: Under the R-LMMSE framework (15)–(18), let ˆ θ k | k and ˆ θ 0 k | k denote the estimates when using the online sensor collaboration scheme (36) and the ofﬂine policy W ∗ . Then lim k →∞ E [( ˆ θ k | k − ˆ θ 0 k | k ) 2 ] = 0 . (37) Proof : See Appendix E.  In practice, the system parameters α, σ 2 τ , σ 2 ς , σ 2 κ , Σ  , Λ g , Λ h needed for obtaining the online scheme W k in (36), will need to be estimated. This will incur estimation error . The following proposition shows that the MSE P ∞ of the ofﬂine scheme W ∗ is robust with respect to the estimation error . The same can be shown for P k . Proposition 3: The asymptotic MSE P ∞ is a continuous function of the system statistics parameters α, σ 2 τ , σ 2 ς , σ 2 κ , Σ  , Λ g , Λ h . Proof : See Appendix G.  It is also of interest to get the rate of conv ergence of E [( ˆ θ k | k − θ k ) 2 ] → P ( ∞ ) . W e know from [37] that the rate of con ver gence for the ofﬂine scheme W ∗ is exponential. Howe v er , for the online ﬁlter, the rate will depend on the distributions of the channel gains and τ k . The follo wing lemma provides information for such rates. Lemma 3: The rate of con ver gence of E [( θ k − ˆ θ k | k ) 2 ] to P ∞ depends on the rate P ( 1 k P k j =1 ( H j,i − T j,i ( W ∗ )) ≤ $ ) and P ( T k,i ( W ∗ ) k > $ ) to go to 0 as k tends to inﬁnity , where $ is a small positiv e constant less than η . Proof : See Appendix H.  Now we provide some conditions on the distributions of the system parameters to obtain some rates of con ver gence of E [( ˆ θ k | k − θ k ) 2 ] → P ( ∞ ) . W e can show that θ ∞ has an exponential moment if τ k has a positi ve density everywhere and has ﬁnite moment generating function in a neighborhood of 0 . Then if h k,i , θ ∞ and  k hav e light tails, T k,i has light tail. If h k,i is ﬁnite valued, then h k,i θ ∞ is light tailed. Otherwise, this condition can get violated ev en if h k,i is light tailed. If H k is also strongly aperiodic, geometrically er godic Marko v chain, we can, using concentration inequalities for Mark ov processes [40] and the above Lemma, show that we obtain exponential rate of conv ergence of E [( ˆ θ k | k − θ k ) 2 ] → P ( ∞ ) . Otherwise, one may only get a polynomial rate of conv ergence. 8 W e emphasize that the proposed online policy has a sig- niﬁcant computational merit in that it is only required to solve the sensor collaboration ofﬂine problem (24) once and then minimal extra computation (36) is required at each time. Theorem 1 relates the performance of the online strategy with that of the optimal ofﬂine strategy . Note that we can also directly obtain a real-time optimal online sensor collaboration scheme by minimizing P k , the estimation error when using R-LMMSE, subject to the ex- pected transmission cost and scaling its solution using (36). The corresponding optimization problem is giv en as minimize W k P k subject to W k ◦ ( 11 T − A ) = 0 , E [ T k,i ( W k )] ≤ µ i − η , i ∈ [ N ] , (38) where P k = ( g T W k h ) 2 tr( W k [ Λ h s k + Σ  ] W T k Λ g ) + σ 2 κ tr( Λ g ) + σ 2 ς . By conv erting the collaboration matrix to the collaboration vector , statistics based problem (38) can be solved using the optimization approach proposed in Section IV. Howe ver , the computational complexity for this ﬁlter is very high since we need to solve the optimization problem at each time instant. W e do not know the stability of this ﬁlter either . Thus, unlike the optimal online ﬁlter obtained abov e, this ﬁlter is not asymptotically optimal, but rather a greedy optimal scheme. W e will see in Section VI that this ﬁlter does not perform better than the above proposed online ﬁlter . T o demonstrate the efﬁcacy of the proposed online policy , in Section VI we will also present a comparison between our proposed sensor collaboration scheme with the online solution based on the full knowledge about the dynamical system, i.e., the realizations of observation and channels gains at each time. V . C O M PA R I S O N W I T H C S I B A S E D S E N S O R C O L L A B O R A T I O N In this section, for the purpose of comparison, we assume that the channel state information, i.e., h k and g k , of the state- space model (13) is av ailable at the FC at time k . In this ideal case, R-LMMSE simpliﬁes to the standard Kalman ﬁlter shown in Algorithm 2. Algorithm 2 R-LMMSE with CSI. Inputs: ˆ θ 0 = 0 , P 0 assumed known 1) Prediction: (15) and (16) 2) Update: ˆ θ k = ˆ θ k | k − 1 + q k ( y k − g T k W k h k ˆ θ k | k − 1 ) , (39) with the updated estimation error P k = (1 − q k g T k W k h k ) P k | k − 1 , (40) where q k is the Kalman ﬁlter gain q k = P k | k − 1 g T k W k h k P k | k − 1 ( g T k W k h k ) 2 + E [ v 2 k ] , (41) where E [ v 2 k ] = g T k W k Σ  W T k g k + σ 2 κ g T k g k + σ 2 ς . The sensor collaboration problem given the CSI about the estimation system is cast as maximize W k ( g T k W k h k ) 2 g T k W k Σ  W T k g k + σ 2 κ g T k g k + σ 2 ς subject to E [ T k,i ( W k )] ≤ µ i − η , i ∈ [ N ] , W k ◦ ( 1 M 1 T N − A ) = 0 . (42) Similarly , by con verting the collaboration matrix to the collab- oration vector , the CSI based problem (42) can be solved using the optimization approach proposed in Section IV. Howe v er , unlike the Kalman ﬁlter in Section IV with W k = W , the stability of the Kalman ﬁlter for the CSI based case has not been established in the literature. Let W ∗ k be the solution of the CSI based problem (42) at time k . W e then use (36) to construct the real-time sensor collaboration scheme W k . This Kalman ﬁlter is a greedy scheme and not necessarily ev en asymptotically optimal. This is because in this setup we cannot use our approach used for the ofﬂine ﬁlter to optimize P ∞ since stability of this Kalman ﬁlter is not kno wn. Another possible approach to obtain an optimal ﬁlter could be via Markov decision theory . But this would be computationally infeasible even for a small number of sensors. Also, such CSI at the FC is not av ailable in practice. Instead, we usually hav e access only to the second-order statistics of the system parameters for parameter inference as in Section IV. In the next section, we will empirically sho w that in terms of estimation accuracy , the sensor collaboration policy proposed in Section IV performs better than the CSI based approach at low signal-to-noise ratios (SNRs), where the SNR refers to SNRs in terms of measurement noise and collaboration noise. Moreov er , the former is computationally light, while the latter in addition to requiring much more information, also requires the solution of the CSI based problem (42) at e very time step, which thus introduces high computational complexity . V I . N U M E R I C A L R E S U L T S In this section, we demonstrate the efﬁcacy of our proposed sensor collaboration schemes through numerical examples. The collaborative tracking system is sho wn in Fig. 1, where the vector of observation gains h k and channel gains g k are chosen randomly and independently from Rayleigh( ξ ) distri- bution with ξ = 1 . Therefore, we obtain h = g = p π / 2 1 and Σ h = Σ g = (4 − π ) / 2 I . The observation noise  k and the channel noise ς k are drawn from zero-mean Gaussian distribu- tions with variances Σ  = 0 . 5( I + 11 T ) and σ 2 ς = 1 . Similarly , the v ector of collaboration noises κ k at time k is assumed to be Gaussian with cov ariance Σ κ = σ 2 κ I , where unless speciﬁed otherwise, we set σ 2 κ = 0 . 5 . The parameter θ k ev olves as a Gauss-Markov process with Gaussian noise τ k of variance σ 2 τ = 0 . 5 . The state transition parameter is set as α = 0 . 8 . The initialized variance of θ k denoted by s 0 is 100. W e assume that the spatial placement and neighborhood structure of the sensor network is modeled as a random geometric graph, RGG( N , r ) [5], [41], where sensors are uniformly distributed over a unit square meter area and bidirectional communication links are possible only for pairwise distances at most r meters, where r refers to collaboration radius. Namely , A mn = 1 if the distance between the m th sensor and the n th sensor is less 9 than r , otherwise it is 0 . W ithout loss of generality , we assume that N = 10 sensors are uniformly distributed ov er the region of interest and unless otherwise speciﬁed, we will assume M = 10 and r = 0 . 6 m . At each sensor, the harvested energy H k,i is modeled as an Exponential random variable with mean µ i = 10 for i ∈ [ N ] . In the considered sensor collaboration problems (24) and (42), we choose η ∈ { 0 . 01 , 1 } . For clarity , we summarize the empirically studied sensor collaboration schemes as follows. • The proposed ofﬂine sensor collaboration scheme W ∗ is the solution of problem (24). • The proposed online sensor collaboration scheme W k is deﬁned in (36). • The online sensor collaboration scheme based on temporally-dependent statistics of system parameters W 1 k is the solution of problem (38) that satisﬁes (36). • The CSI based online greedy sensor collaboration scheme W 2 k is the solution of problem (42) that satisﬁes (36). W e remark that compared to our proposed sensor collabora- tion schemes W ∗ and W k , the schemes W 1 k and W 2 k require the solution of the optimization problems at ev ery time step. This naturally leads to high computational cost. The tracking performance for the above four schemes is measured through the empirical mean squared error (MSE), which is computed by simulating over 500 samples and taking av erage. 0 2 4 6 8 10 0 2 4 6 8 10 12 Empirical MSE Time Proposed online scheme W k Statistics based online scheme W k 1 Proposed offline scheme W * CSI based online scheme W k 2 9 9.05 9.1 0.28 0.3 0.32 0.34 Fig. 2: Empirical MSE under η = 0 . 01 . In Fig. 2, we present the empirical MSE when using the aforementioned sensor collaboration schemes as a function of time k with η = 0 . 01 . As we can see, the empirical MSE decreases as k increases, and it con v erges to a steady- state value by k = 10 . Note that the average variance of the parameter is decreasing as k increases and eventually con ver ges to a constant which implies that the av erage SNR is decreasing as k increases; see (22). As we can see, our proposed sensor collaboration schemes W ∗ and W k yield MSE that is better than the greedy CSI based scheme W 2 k at low SNR. This indicates that our proposed scheme W k is efﬁcient not only in computation but also in estimation performance at lo w SNR. The empirical MSE against SNRs with respect to measurement noise and collaboration noise will 0 2 4 6 8 10 12 14 16 18 20 −50 0 50 100 150 200 250 Time Confidence interval Fig. 3: 95% conﬁdence intervals of ˆ θ k for the proposed optimal online scheme under η = 0 . 01 . 0 2 4 6 8 10 10 −1 10 0 10 1 10 2 10 3 Time Empirical MSE Proposed online scheme W k , η = 0.01 Proposed online scheme W k , η = 1 Fig. 4: Empirical MSE using the proposed online scheme W k for η ∈ { 0 . 01 , 1 } . 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 x 10 −3 Time E [( ˆ θ k | k − ˆ θ 0 k | k ) 2 ] Fig. 5: Empirical v eriﬁcation of Theorem 1: Asymptotic consistency of the proposed online scheme and the optimal ofﬂine scheme under η = 0 . 01 . be explored in the following. W e also remark that the ofﬂine scheme W ∗ performs as well as the online schemes. But the 10 0 5 10 15 20 10 −2 10 0 10 2 10 4 10 6 Time Energy flow at sensor 1 S k,1 , stored energy under scheme W k T k,1 , consumed energy under scheme W k S k,1 , stored energy under scheme W * T k,1 , consumed energy under scheme W * Not satisfying real−time energy constraint Fig. 6: Real-time energy ﬂow . 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 SNR dB Empirical MSE Proposed online scheme W k Statistics based online scheme W k 1 CSI based online scheme W k 2 Fig. 7: Empirical MSE as a function of SNR in terms of measurement noise, η = 0 . 01 . 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Radius Normalized empirical MSE Proposed online scheme W k No collaboration Fully connected network Fig. 8: Normalized empirical MSE as a function of collaboration radius. former might not obey the real-time energy constraint. In Fig. 3, we present the 95% conﬁdence interv als of ˆ θ k for the proposed optimal online scheme under η = 0 . 01 using two tailed t-test with a 1000 number of samples. As we can see, our proposed online scheme performs very well as k increases. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Empirical MSE α Proposed online scheme W k Statistics based online scheme W k 1 CSI based online scheme W k 2 Fig. 9: Empirical MSE as a function of α , η = 0 . 01 at low SNR in terms of measurement noise. 2 3 4 5 6 7 8 9 10 0.3 0.4 0.5 0.6 0.7 0.8 Empirical MSE N Proposed online scheme W k Statistics based online scheme W k 1 CSI based online scheme W k 2 Fig. 10: Empirical MSE as a function of N , η = 0 . 01 at lo w SNR in terms of measurement noise. 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 3.5 Empirical MSE SNR dB Proposed online scheme W k Statistics based Online scheme W k 1 CSI based Online scheme W k 2 Fig. 11: Empirical MSE as a function of SNR in terms of collaboration noise, η = 0 . 01 . In Fig. 4, we show the impact of the choice of η on the tracking performance for the proposed online scheme. W e recall that the parameter η is the remaining a verage ener gy that 11 0 2 4 6 8 10 0.25 0.3 0.35 0.4 0.45 0.5 M Empirical MSE Proposed online scheme W k Fig. 12: Empirical MSE as a function of M using the proposed online scheme W k , η = 0 . 01 . is deliberately kept in the energy storage buf fer . As we can see, the smaller the η is, the better the tracking performance we obtain since more energy is av ailable to be used in sensor collaboration. As a result, a larger η leads to a less effecti ve energy allocation scheme. In Fig. 5, we show the asymptotic performance of the proposed online scheme and the optimal ofﬂine scheme under η = 0 . 01 , namely , when k is large enough, the estimation performance of the proposed online scheme is the same as the ofﬂine scheme. As expected, the empirical MSE between W k and W ∗ con ver ges to 0 as k increases which is consistent with our theoretical result in Theorem 1. In Fig. 6, we present the real-time stored, and consumed energy at the ﬁrst sensor for the proposed online schemes W k and the proposed ofﬂine scheme W ∗ with η = 0 . 01 . Here the stored energy and the consumed energy at sensor 1 and time k are denoted by S k, 1 and T k, 1 , respectively . As we can see, the consumed energy T k, 1 is always less than or equal to the stored energy S k, 1 for the proposed efﬁcient online scheme. Such dynamics of the energy ﬂo w is consistent with (11). W e hav e veriﬁed that the online schemes W 1 k and W 2 k also follow such energy dynamics. Moreover , we can see that the ofﬂine scheme violates real-time energy constraints. In Fig. 7, we show the empirical MSE as a function of the av erage SNR in terms of measurement noise with η = 0 . 01 and k = 20 . This k is large enough to provide the conv erged estimation error . As we can see, the empirical MSE decreases when the a verage signal to measurement noise ratio increases. This is e xpected. By ﬁxing the a verage signal to measurement noise ratio, we can see that the proposed online scheme W k nev er performs worse than W 1 k which requires much more computational effort. Also, it performs better than the full information ﬁlter W 2 k for signal to measurement noise ratio less than 15 dB. In Fig. 8, we present the normalized empirical MSE as a function of the collaboration radius r at time step k = 20 with η = 0 . 01 . As we can see, the normalized empirical MSE decreases signiﬁcantly as r increases, since more collaboration links are established. Moreov er , the normalized empirical MSE ceases to signiﬁcantly decrease when r > 0 . 7 . This indicates that a large part of the performance improvement is achieved only through partial collaboration. In Fig. 9, we show the empirical MSE as a function of α with η = 0 . 01 , k = 20 for signal to measurement noise ratio less than 3 dB. The empirical MSE increases when α increases. This is expected because the estimation error is a monotonic function of α that is the state transition coefﬁcient of the parameter model; see (18) - (20). By ﬁxing α , we can see that the proposed online scheme W k performs close to or even better than the statistics based online scheme W 1 k . Moreov er , the proposed online scheme W k performs better than the full knowledge based online scheme W 2 k . In Fig. 10, we present the empirical MSE as a function of the number of sensors N under η = 0 . 01 , k = 20 for signal to measurement noise ratio less than 3 dB. As we can see, the empirical MSE decreases when N increases since more information is transmitted to the FC. Moreover , we can see the proposed online scheme W k performs better than the full knowledge based online scheme W 2 k . In Fig. 11, we sho w the empirical MSE as a function of the av erage SNR in terms of collaboration noise with η = 0 . 01 and k = 20 . As we can see, the empirical MSE decreases when the average signal to collaboration noise increases. This is expected. By ﬁxing the average signal to collaboration noise, we can see that the proposed online scheme W k nev er performs worse than statistics based online scheme W 1 k which requires much more computational effort. Also, it performs better than the full information ﬁlter W 2 k up to 5 dB, which is the region of interest for energy harvesting sensor nodes [26]. In Fig. 12, we show the empirical MSE as a function of M , the number of sensors that communicate with the FC, for the proposed online scheme at η = 0 . 01 and k = 20 . As we can see, the empirical MSE decreases when M increases since more sensors are used for communication to the FC and, therefore the FC can acquire more information. V I I . C O N C L U S I O N In this paper , we studied the problem of sensor collaboration for dynamic parameter tracking in energy harvesting sensor networks. W e incorporated the effect of sensor collaboration noise while designing the online collaboration scheme. Based on the second-order statistics of the system parameters, we dev eloped an online sensor collaboration policy that is built on the solution of the ofﬂine sensor collaboration problem. Although the latter is a complex non-conv ex optimization problem, we showed that it can be solved exactly via semidef- inite programming. W e also prov ed that the proposed online policy is asymptotically consistent with the optimal ofﬂine solution ov er an inﬁnite time horizon. W e hav e also shown via simulations that at low SNRs, in the region of interest in sensor networks, our online scheme actually outperforms a greedy optimal online policy which has full information at the fusion center about the channel g ains and energy at the sensors at any giv en time. Numerical results have illustrated the efﬁcienc y of our approach and the impact of energy harvesting and collaboration noise on the performance of collaborativ e estimation. 12 In future work, it would be desirable to consider a more practical sensor collaboration model by taking into account the impact of pairwise distances between sensors on their channel gain. Also, it is interesting to take into account the energy consumption and delay due to the estimation of second- order statistics. Moreov er , the scalar random parameter can be generalized to a vector of random parameters. Also, one can study quantization-based inter -sensor collaboration. Lastly , the design of network topologies could be taken into account under the current framework. A P P E N D I X A P RO O F O F L E M M A 1 Since u k = g T k Wh k and v k = g T k W  k + g T k κ k + ς k , we hav e E [ u k v k ] = E [ g T k Wh k ( g T k W  k + g T k κ k + ς k )] (43) = E [ g T k Wh k g T k W  k ] + E [ g T k Wh k g T k κ k ] + E [ g T k Wh k ς k ] . The ﬁrst term on the right hand side (RHS) of (43) yields E [ g T k Wh k g T k W  k ] = E [ g T k Wh k g T k W ] × E [  k ] = 0 . Applying similar analysis for other terms on the RHS of (43), we can conclude that u k is uncorrelated with v k . Like wise, we can sho w that { v k } is an uncorrelated sequence.  A P P E N D I X B Q UA D R A T I C V E C T O R F U N C T I ON S Based on Proposition 1, the ofﬂine problem (24) can be simpliﬁed such that it only contains quadratic vector functions with respect to w . Using the identities (27) and (28), we can rewrite the quadratic matrix functions in f ( W ) as ( g T Wh ) 2 = w T Ω N w , (44) tr( Λ g W [ Λ h s ∞ + Σ  ] W T ) = w T Ω D w , (45) where Ω N = Ghh T G T , G is deﬁned as in (29) that satisﬁes g T W = w T G , the ( k , l ) th entry of Ω D is deﬁned as in (30), namely , [ Ω D ] kl = [ Λ g ] m k m l [ Λ h s ∞ + Σ  ] n k n l . Recall the expected transmission cost for sensor-to-FC in (9), E [ T (2) i ( W )] = e T i W [ Λ h s ∞ + Σ  ] W T e i + σ 2 κ (46) = tr ( e i e T i W [ Λ h s ∞ + Σ  ] W T ) = w T Ω T(2) i w , where Ω T(2) i is also obtained according to (30), [ Ω T(2) i ] kl = [ e i e T i ] m k m l [ Λ h s ∞ + Σ  ] n k n l . According to (7), the expected transmission cost for inter- sensor communication is a function of W ◦ ˜ I , where ˜ I = 11 T − [ I M , 0 M × ( N − M ) ] . W e further deﬁne ˜ W = W ◦ ( 11 T − [ I M , 0 M × ( N − M ) ]) , and one can view the matrix ˜ W as the collaboration matrix W with zero diagonals. Therefore, E [ T (1) i ( W )] = tr ( e i e T i [ Λ h s ∞ + Σ  ]) tr ( ˜ We i e T i ˜ W T ) . (47) Similar to w , let ˜ w be the vector that consists of the nonzero entries (columnwise) of ˜ W , namely , ˜ w l = W m l n l for certain indices m l ∈ [ M ] , n l ∈ [ N ] and m l 6 = n l . Clearly , ˜ w is a vector of ( L − M ) dimension, and it can be readily expressed as a linear function of w , namely , ˜ w = Fw with F ∈ R ( L − M ) × L , given by F ij =  1 , K ( i ) = j, j ∈ [ L ] , i ∈ [ L − M ] , 0 , otherwise , (48) where K = { l | w l = W m l n l , m l 6 = n l } . W e elaborate this linear transformation through the follow- ing example. A 1 =   1 0 1 0 1 1 1 0 1   , w =         W 11 W 31 W 22 W 13 W 23 W 33         , ˜ w =   W 31 W 13 W 23   (49) Giv en A 1 , we can obtain that K = { 2 , 4 , 5 } . Therefore, F 12 = 1 , F 24 = 1 and F 35 = 1 , and otherwise F ij = 0 . W e hav e F 1 =   0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0   . By concatenating ˜ W to ˜ w and using the identity (28) as well as the linear transformation ˜ w = Fw , we can re write the quadratic matrix function in (47) as tr ( e i e T i [ Λ h s ∞ + Σ  ]) tr ( ˜ We i e T i ˜ W T ) = ˜ w T Ω T(0) i ˜ w , (50) = w T Ω T(1) i w , (51) where the ( ˜ k , ˜ l ) th entry of Ω T(0) i is also obtained according to (30), namely , [ Ω T(0) i ] ˜ k ˜ l = tr ( e i e T i [ Λ h s ∞ + Σ  ])[ e i e T i ] n ˜ k n ˜ l , here ˜ k , ˜ l ∈ [ L − M ] and Ω T(1) i = F T Ω T(0) i F . Upon deﬁning Ω C ,i = Ω T(1) i as the characterization of expected transmission cost consumed for inter-sensor commu- nication, and Ω T ,i = Ω T(1) i + Ω T(2) i as the characterization of both the e xpected transmission cost consumed for inter-sensor communication and sensor-to-FC, we ev entually obtain the problem (31). In what follows, we summarize the coefﬁcients precisely correspond to the problem (31) Ω N = Ghh T G T , Ω D , Ω C ,i = Ω T(1) i , Ω T ,i = Ω T(1) i + Ω T(2) i , where [ Ω D ] kl = [ Λ g ] m k m l [ Λ h s ∞ + Σ  ] n k n l , Ω T(1) i = F T Ω T(0) i F , [ Ω T(0) i ] ˜ k ˜ l = tr ( e i e T i [ Λ h s ∞ + Σ  ])[ e i e T i ] n ˜ k n ˜ l and [ Ω T(2) i ] kl = [ e i e T i ] m k m l [ Λ h s ∞ + Σ  ] n k n l , for k , l ∈ [ L ] , ˜ k , ˜ l ∈ [ L − M ] . W e note that the matrix Ω N is positi ve semideﬁnite and of rank one. Since the denominator of f ( W ) in (24) must be positive when the channel noise and collaboration noise are ignored (namely , σ 2 ς = 0 and σ 2 κ = 0 ), the matrix Ω D is positiv e deﬁnite. The matrices Ω C ,i and Ω T ,i are also positiv e deﬁnite according to the deﬁnition of transmission cost.  A P P E N D I X C P RO O F O F P R O PO S I T I O N 2 By introducing a new variable ¯ W ∈ R ( L +1) × ( L +1) together with the constraint ¯ W = ¯ w ¯ w T , problem (33) can be refor- 13 mulated as maximize ¯ W tr ( Q 0 ¯ W ) , subject to tr ( Q 1 i ¯ W ) ≤ 0 , i ≤ M , tr ( Q 2 i ¯ W ) ≤ 0 , M < i ≤ N , tr ( Q N +1 ¯ W ) ≤ 1 , rank ( ¯ W ) = 1 , ¯ W  0 , (52) where we have used the fact that the constraint rank ( ¯ W ) = 1 together with ¯ W  0 is equiv alent to the constraint ¯ W = ¯ w ¯ w T , and ¯ W  0 indicates that ¯ W is positive semideﬁnite. After dropping the (nonconv ex) rank-one constraint, problem (52) is relaxed to the semideﬁnite program (SDP) (34). It is clear from (33)-(34) that the solution of (31) is achiev able based on the solution of problem (34) if the latter renders a rank-one solution. Similar to [10, Theorem 1] and [38, Theorem 1.4], the rank-one property of ¯ W ∗ can be explored by studying the KKT conditions of problem (34). Details of the proof are omitted here for the sake of brevity . Since ¯ W ∗ is of rank one, we can immediately construct (35) as the solution of problem (31).  A P P E N D I X D P RO O F O F L E M M A 2 1) Since | α | < 1 , θ k d → θ ∞ where θ ∞ is an a.s. ﬁnite random variable [42]. Also, then θ ∞ d = αθ ∞ + τ , where θ ∞ and τ are independent, and τ has the distribution of τ 1 and d = denotes equality in distribution. Therefore, E [ θ ∞ ] = α E [ θ ∞ ] + E [ τ ] and hence E [ θ ∞ ] = E [ τ ] / (1 − α ) . Also, E [ θ 2 ∞ ] = α 2 E [ θ 2 ∞ ] + E [ τ 2 ] , because E [ τ ] = 0 . Therefore, E [ θ 2 ∞ ] = E [ τ 2 ] / (1 − α 2 ) . From the dynamical equation (1), θ k = α k θ 0 + τ k + ατ k − 1 + α 2 τ k − 2 + . . . + α k − 1 τ 1 . Therefore, E [ θ k ] = α k E [ θ 0 ] + (1 + α + . . . + α k − 1 ) E [ τ 1 ] → E [ τ 1 ] / (1 − α ) , as k → ∞ and E [ θ 2 k ] = E [ τ 2 1 ](1 + α + . . . + α k − 1 ) → E [ τ 2 1 ] / (1 − α 2 ) . 2) From (7) and (9), we have E [ T (1) k,i ( W ∗ )] =  e T i ( E [ θ 2 k ]( hh T + Σ h ) + Σ  ) e i  × [ e T i ( W ∗ ◦ ˜ I ) T ( W ∗ ◦ ˜ I ) e i ] , E [ T (2) k,i ( W ∗ )] = e T i W ∗ ( E [ θ 2 k ]( hh T + Σ h ) + Σ  ) × W ∗ T e i + σ 2 κ . From the total transmission cost in (10), we obtain E [ T k,i ( W ∗ )] → E [ T i ( W ∗ )] since from 1) E [ θ 2 k ] → E [ θ 2 ∞ ] .  A P P E N D I X E P RO O F O F T H E O R E M 1 In the following, we consider a space H of all real valued random v ariables on the common underlying probability space, with E [ | x | 2 ] < ∞ . This is a Hilbert space with inner product E [ xy ] and k x k 2 = E [ | x | 2 ] . Let { y k } and { y 0 k } denote the observations receiv ed at the FC by using the online sensor collaboration scheme (36) and the ofﬂine policy , respectiv ely . The estimates of θ k based on { y k } and { y 0 k } are gi ven by the projection operator P o in the Hilbert space H : ˆ θ k | k = P o ( θ k | y 1 , . . . , y k , ˆ θ 0 ) , (53) ˆ θ 0 k | k = P o ( θ k | y 0 1 , . . . , y 0 k , ˆ θ 0 ) . (54) Consider the equation S 0 k +1 ,i = ( S 0 k,i − T k,i ( W ∗ )) + + H k,i for i ∈ [ N ] , with S 0 0 ,i = S 0 ,i . W e hav e { H k,i , k ≥ 0 } and { T k,i ( W ∗ ) , k ≥ 0 } , independent, each satisfying SLLN. Also, from Lemma 2, E [ T k,i ( W ∗ )] → E [ T i ( W ∗ )] . From ofﬂine problem (24), we ha ve E [ T i ( W ∗ )] ≤ µ i − η . Therefore, since S 0 k +1 ,i ≥ P k l =1 ( H l,i − T l,i ( W ∗ )) , S 0 k +1 ,i → ∞ almost surely (a.s.) for all i as k → ∞ . Since the actual energy consumption for sensor i at time k is T k,i ( W k ) ≤ T k,i ( W ∗ ) , we also have S k +1 ,i → ∞ a.s. for all i . Therefore, from (36), we obtain T k,i ( W k ) → T i ( W ∗ ) a.s. for all i . Accordingly , we obtain that W k → W ∗ a.s. as k → ∞ . W e will use k x k to denote p E [ x T x ] for a random vector x . Since W k ≤ W ∗ , we also obtain k W k − W ∗ k → 0 as k → ∞ . From (13), we have k y k − y 0 k k ≤ (55) k g T k W k h k θ k + g T k W k  k − g T k W ∗ h k θ k − g T k W ∗  k k ≤ k g k kk W k − W ∗ kk  k k + k g k kk W k − W ∗ kk h k kk θ k k . Since k g k k , k  k k , k h k k and E [ θ 2 ∞ ] (namely , s ∞ ) are bounded and k W k − W ∗ k → 0 and k θ k − θ ∞ k → 0 , as k → ∞ , we obtain k y k − y 0 k k → 0 , for k → ∞ . (56) In what follows, we will show that k P o ( θ k | y 1 , . . . , y k , ˆ θ 0 ) − P o ( θ k | y 0 1 , . . . , y 0 k , ˆ θ 0 ) k → 0 , (57) for k → ∞ . From (1), we hav e θ k = α N θ k − N + τ k + ατ k − 1 + . . . + α N − 1 τ k − N +1 . Therefore, we get P o ( θ k | y 1 , . . . , y k , ˆ θ 0 ) = P o ( α N θ k − N + τ k + ατ k − 1 (58) + . . . + α N − 1 τ k − N +1 | y 1 , . . . , y k , ˆ θ 0 ) . Since { τ k , . . . , τ k − N +1 } are zero mean and independent of { y k − N , . . . , y 1 , θ 0 } , from (58), we obtain that P o ( θ k | y 1 , . . . , y k , ˆ θ 0 ) = α N P o ( θ k − N | y 1 , . . . , y k , ˆ θ 0 )+ (59) P o ( τ k + ατ k − 1 + . . . + α N − 1 τ k − N +1 | y k , . . . , y k − N +1 ) . For the ﬁrst term in the RHS of (59), we note that α N P o ( θ k − N | y 1 , . . . , y k , ˆ θ 0 ) ≤ α N E [ θ 2 k − N ] , (60) where we ha ve used the fact that k P o ( x | z ) k ≤ k x k . Moreover , from Lemma 2, we obtain that sup k E [ θ 2 k − N ] < ∞ . Therefore, RHS of (60) can be taken arbitrarily small by taking N large enough. In what follows, we keep N ﬁxed. Now , let us focus on the second term in (59). For ease of notation, we deﬁne ¯ τ k : = τ k + ατ k − 1 + . . . + α N − 1 τ k − N +1 , (61) ¯ y k : = [ y k , . . . , y k − N +1 ] T , ¯ y 0 k : = [ y 0 k , . . . , y 0 k − N +1 ] T . From (56), we obtain that k ¯ y k − ¯ y 0 k k → 0 , for k → ∞ . (62) 14 Then, we hav e k P o ( ¯ τ k | ¯ y k ) − P o ( ¯ τ k | ¯ y 0 k ) k =     E [ ¯ τ T k ¯ y k ] ¯ y k k ¯ y k k 2 − E [ ¯ τ T k ¯ y 0 k ] ¯ y 0 k k ¯ y 0 k k 2     = 1 k ¯ y k k 2 k E [ ¯ τ T k ¯ y k ] ¯ y k − E [ ¯ τ T k ¯ y 0 k ] ¯ y 0 k ( k ¯ y k k 2 / k ¯ y 0 k k 2 ) k . (63) In (63), we note that k ¯ y k k ≥ co v( v k ) > 0 due to (13). This also holds for k ¯ y 0 k k . Moreover , we hav e sup k k ¯ y k k < ∞ and sup k k ¯ y 0 k k < ∞ as k → ∞ due to sup k k θ k k < ∞ . Also, we hav e ( k ¯ y k k 2 / k ¯ y 0 k k 2 ) → 1 as k → ∞ since |k ¯ y k k − k ¯ y 0 k k| ≤ k ¯ y k − ¯ y 0 k k → 0 as k → ∞ . Therefore, we obtain k E [ ¯ τ T k ¯ y k ] ¯ y k − E [ ¯ τ T k ¯ y 0 k ] ¯ y 0 k ( k ¯ y k k 2 / k ¯ y 0 k k 2 ) k (64) ≤ k E [ ¯ τ T k ¯ y k ] ¯ y k − E [ ¯ τ T k ¯ y k ] ¯ y 0 k k + k E [ ¯ τ T k ¯ y k ] ¯ y 0 k − E [ ¯ τ T k ¯ y 0 k ] ¯ y 0 k ( k ¯ y k k 2 / k ¯ y 0 k k 2 ) k , where k E [ ¯ τ T k ¯ y k ] ¯ y k − E [ ¯ τ T k ¯ y k ] ¯ y 0 k k ≤ k E [ ¯ τ T k ¯ y k ] kk ¯ y k − ¯ y 0 k k ≤ k ¯ τ k kk ¯ y k ] kk ¯ y k − ¯ y 0 k k → 0 , as k → ∞ and k E [ ¯ τ T k ¯ y k ] ¯ y 0 k − E [ ¯ τ T k ¯ y 0 k ] ¯ y 0 k ( k ¯ y k k 2 / k ¯ y 0 k k 2 ) k ≤ k ¯ y 0 k kk E [ ¯ τ T k ¯ y k ] − E [ ¯ τ T k ¯ y 0 k ( k ¯ y k k 2 / k ¯ y 0 k k 2 )] k ≤ k ¯ y 0 k kk ¯ τ k kk ¯ y k − ¯ y 0 k ( k ¯ y k k 2 / k ¯ y 0 k k 2 ) k → 0 , as k → ∞ . As a result, RHS of (63) con ver ges to 0 as k → ∞ . T ogether with (60), we ha ve completed the proof of (57). Therefore, we obtain the result in (37).  A P P E N D I X F P RO O F O F P R O PO S I T I O N 1 Let w ∈ R L be the v ector of column-wise nonzer o entries of W ∈ R M × N . The one-to-one mapping between w and W en- sures that for any w l , we have a certain pair of indices ( m l , n l ) such that w l = W m l n l , where m l ∈ [ M ] , n l ∈ [ N ] , l ∈ [ L ] . Giv en b ∈ R M , we obtain b T W = [ b T W · 1 · · · b T W · N ] , (65) where W · j is the j th column of W and b T W · j = P M i =1 b i W ij , for j ∈ [ N ] . Similarly , giv en B ∈ R L × N , we hav e w T B = [ w T B · 1 · · · w T B · N ] , (66) where B · j is the j th column of B and w T B · j = P L l =1 W m l n l B lj , for j ∈ [ N ] . Consider the j th entry of w T B , we obtain [ w T B ] j = L X l =1 W m l n l B lj = L X l =1 ,n l = j b m l W m l j (67) = M X m l =1 b m l W m l j = [ b T W ] j , where we have used the facts that B lj = b m l if j = n l , and 0 otherwise. Based on (67), we can conclude that w T B = b T W . W e next show identity (28). Giv en C ∈ R M × M and D ∈ R N × N , we hav e tr( CWD W T ) = X i =1 e T i CWD W T e i . (68) Let c i := e T i C . By applying the identity (27), we hav e e T i CW = w T ˜ C i and W T e i = ˜ E T i w , where h ˜ C i i ln =  [ c i ] m l , n = n l , 0 , otherwise , and h ˜ E i i ln =  [ e i ] m l , n = n l , 0 , otherwise , for l ∈ [ L ] and n ∈ [ N ] . Therefore, tr( CWD W T ) = X i =1 w T ˜ C i D ˜ E T i w = w T ( X i =1 ˜ C i D ˜ E T i ) w . (69) Upon deﬁning d n := e T n D and h ˜ D i i nk =  [ d n ] n k , m k = i, 0 , otherwise , for n ∈ [ N ] , l ∈ [ L ] , we obtain X i =1 ˜ C i D ˜ E T i = X i =1 ˜ C i ˜ D i = E , (70) where we hav e used the fact that E lk = [ C ] m l m k [ D ] n l n k for n ∈ [ N ] , k ∈ [ L ] and l ∈ [ L ] . Based on (70) and (69), we can conclude that tr( CWD W T ) = w T Ew .  A P P E N D I X G P RO O F O F P R O PO S I T I O N 3 From Proposition 2, we have the optimal solution ¯ W ∗ that is of rank one. According to [43, Theorem 2.4], we can obtain that ¯ W ∗ is unique. Moreover , in Proposition 2, we ha ve shown that the optimal solution of problem (31) is obtained by decomposing the submatrix of ¯ W ∗ formed by deleting its ( L + 1) st row and column, namely [ ¯ W ∗ ] L = s ∗ ( s ∗ ) T . Then, if we choose | s ∗ | , we obtain a unique globally optimal solution W ∗ for the ofﬂine problem. Since the optimizing function of problem (34) is continuous and the feasibility set is compact for a given parameter set, by [44, Theorem 9.14], the unique optimal solution ¯ W ∗ of problem (34) is a continuous function of the parameters. Hence, the optimal solution W ∗ of problem (31) is also a continuous function. Next consider the asymptotic mean squared error (MSE) P ∞ in (23) of the ofﬂine sensor collaboration scheme (and hence also of the online sensor collaboration scheme). Equation (23) is quadratic in P ∞ and hence P ∞ is a continuous function of its coefﬁcients which are continuous functions of W ∗ and the other parameters. Therefore, P ∞ is a continuous function of the system parameters.  A P P E N D I X H P RO O F O F L E M M A 3 In the follo wing, we sho w the con v ergence rate of    ˆ θ k | k − θ k    → P ∞ , where ˆ θ k | k denotes the estimate using the proposed optimal online sensor collaboration scheme W k and θ k is the true value. According to triangle inequality , we have    ˆ θ k | k − θ k    ≤    ˆ θ k | k − ˆ θ 0 k | k    +    ˆ θ 0 k | k − θ k    , (71) where ˆ θ 0 k | k denote the estimate when using the ofﬂine policy W ∗ . 15 From [37, Theorem 2],    ˆ θ 0 k | k − θ k    con ver ges to P ∞ expo- nentially . Now , we consider    ˆ θ k | k − ˆ θ 0 k | k    . From (65) in proof of Theorem 1, for any N > 0 , we have    ˆ θ k | k − ˆ θ 0 k | k    ≤ α N E [ θ 2 k − N ] + k P o ( ¯ τ k | ¯ y k ) − P o ( ¯ τ k | ¯ y 0 k ) k . where ¯ τ k : = τ k + ατ k − 1 + . . . + α N − 1 τ k − N +1 , ¯ y k : = [ y k , . . . , y k − N +1 ] T , ¯ y 0 k : = [ y 0 k , . . . , y 0 k − N +1 ] T and P o is the projection operator which determines the estimate of θ k . By taking N large enough, the ﬁrst term on RHS can be taken arbitrarily small. This also happens at an exponential rate. From (69) and the inequality below it, we hav e k P o ( ¯ τ k / ¯ y k ) − P o ( ¯ τ k / ¯ y 0 k ) k ≤ c k ¯ y k − ¯ y 0 k k for c is an appropriately large constant. Using triangle inequal- ity and Cauchy-Schwarz inequality , we obtain k ¯ y k − ¯ y 0 k k ≤ k X j = k − N +1   y j − y 0 j   , (72) and   y j − y 0 j   2 = E [(( g T j W ∗ h j θ j + v j ) − ( g T j β j ◦ W ∗ h j θ j + v j )) 2 ] (73) ≤ k g j k 2 k h j k 2 k θ j k 2 k W ∗ − β j ◦ W ∗ k 2 . Also, k W ∗ − β k ◦ W ∗ k ≤ E [(1 − β k ) 2 ] k W ∗ k . Therefore, from (72), we have k ¯ y k − ¯ y 0 k k ≤ k g k k h k k W ∗ k k X j = k − N +1 k θ j k E [(1 − β j ) 2 ] . Since sup j k θ j k < ∞ and N is taken a ﬁnite ﬁxed number , we then need to show rate of con vergence of E [(1 − β k ) 2 ] . For δ k > 0 , E [(1 − β k ) 2 ] = E [(1 − β k ) 2 | β k ≤ δ k ] P ( β k ≤ δ k ) (74) + E [(1 − β k ) 2 | β k > δ k ] P ( β k > δ k ) ≤ P ( β k ≤ δ k ) + (1 − δ k ) 2 . T ake δ k = 1 − 1 δ k for a small δ > 0 . Then second term (1 − δ k ) 2 = 1 δ 2 k and decays exponentially . Now we consider P ( β k ≤ δ k ) . P ( β k ≤ δ k ) = P ( β k ≤ 1 − 1 δ k ) = P ( S k,i ≤ (1 − 1 δ k ) 2 T k,i ( W ∗ ) , i = 1 , . . . , N ) ≤ N X i =1 P  S k,i ≤ (1 − 1 δ k ) 2 T k,i ( W ∗ )  ≤ N X i =1 P   k X j =1 H j,i − k X j =1 T j,i ( W ∗ ) ≤ (1 − 1 δ k ) 2 T k,i ( W ∗ )   ≤ N X i =1 P   k X j =1 ( H j,i − T j,i ( W ∗ )) ≤ T k,i ( W ∗ )   = N X i =1 P   1 k k X j =1 ( H j,i − T j,i ( W ∗ )) ≤ T k,i ( W ∗ ) k ; T k,i ( W ∗ ) k ≤ $   + N X i =1 P   1 k k X j =1 ( H j,i − T j,i ( W ∗ )) ≤ T k,i ( W ∗ ) k ; T k,i ( W ∗ ) k > $   ≤ N X i =1 P   1 k k X j =1 ( H j,i − T j,i ( W ∗ )) ≤ $ ) + P ( T k,i ( W ∗ ) k > $ )   , where we choose $ < η . Since E [ T i ( W ∗ )] ≤ E [ H i ] − η , then E [ H i ] − E [ T i ( W ∗ )] ≥ η .  R E F E R E N C E S [1] Jennifer Y ick, Biswanath Mukherjee, and Dipak Ghosal, “Wireless sensor network survey , ” Computer networks , v ol. 52, no. 12, pp. 2292– 2330, 2008. 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Optimal Sensor Collaboration for Parameter Tracking Using Energy Harvesting Sensors

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