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DINS: A Distributed Scheme for Sensor Fusion over Fading Channels*
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th9-14, World Congress Control The International Federation Toulouse, France, July 2017 The International Federation of of Automatic Automatic Control Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 14284–14289
DINS: A Distributed Scheme for Sensor DINS: A Distributed Scheme for Sensor DINS: A Distributed Scheme for DINS: A Distributed Scheme for Sensor Sensor Fusion over Fading Channels Fusion over Fading Channels Fusion over Fading Channels Fusion over Fading Channels ∗ ∗∗ ∗
Jinming Xu ∗∗ Cailian Chen ∗ Xinping Guan ∗∗ ∗∗ Cailian Jinming Xu Chen Jinming Xu Chen ∗ Xinping Xinping Guan Guan ∗ Binqiang Xue ∗∗∗ ∗∗∗Chen ∗ Xinping Guan ∗ Jinming Xu ∗∗ Cailian Cailian ∗∗∗ Binqiang Xue Binqiang Binqiang Xue Xue ∗∗∗ ∗ Department of Automation, Shanghai Jiao Tong University, and Key ∗ ∗ Department of Automation, Shanghai Jiao Tong University, and Key Department of Automation, Shanghai Jiao University, and Laboratory of System Control and Information Processing, Ministry of ∗ Department of Automation, Shanghai Jiao Tong Tong University, and Key Key Laboratory of System Control and Information Processing, Ministry of Laboratory of System Control and Information Processing, Ministry Education of China, Shanghai 200240, China (e-mails:{shyzhu, Laboratory of System Control and 200240, Information Processing, Ministry of of Education of China, Shanghai China (e-mails:{shyzhu, Education of Shanghai 200240, cailianchen, xpguan}@sjtu.edu.cn). Education of China, China, Shanghai 200240, China China (e-mails:{shyzhu, (e-mails:{shyzhu, cailianchen, xpguan}@sjtu.edu.cn). ∗∗ cailianchen, andxpguan}@sjtu.edu.cn). Electronic Engineering, Nanyang ∗∗ School of Electrical cailianchen, xpguan}@sjtu.edu.cn). ∗∗ School of Electrical and Electronic Engineering, Nanyang of Electrical and Engineering, Nanyang Technological University, Singapore 639798 (e-mail: ∗∗ School School of Electrical and Electronic Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: Technological University, Singapore 639798 (e-mail: [email protected]). Technological [email protected]). University, Singapore 639798 (e-mail: ∗∗∗ [email protected]). and Electrical Engineering, Qingdao ∗∗∗ College of Automation [email protected]). ∗∗∗ College of Automation and Electrical Engineering, of Automation and Electrical Engineering, Qingdao University, Qingdao China (e-mail: Qingdao ∗∗∗ College CollegeUniversity, of Automation and 266071, Electrical Engineering, Qingdao Qingdao 266071, China (e-mail: University, Qingdao 266071, China (e-mail: [email protected]). University, Qingdao 266071, China (e-mail: [email protected]). [email protected]). [email protected]). Shanying Shanying Shanying Shanying
Zhu ∗ ∗ Zhu Zhu Zhu ∗
Abstract: In this paper, we consider the problem of distributed sensor fusion in relayAbstract: In this paper, consider the problem of distributed sensor fusion in relayAbstract: In this we consider problem of fusion in assisted sensor networks withwe time-varying asymmetric communications over fading channels. A Abstract: In networks this paper, paper, we consider the the problem communications of distributed distributed sensor sensor fusion in relayrelayassisted sensor with time-varying asymmetric over fading channels. A assisted sensor networks with time-varying asymmetric communications over fading channels. A Distributed INnovation based Scheme (DINS) is proposed by fusing both the local measurements assisted sensor networksbased with time-varying asymmetric communications over fading channels. A Distributed INnovation Scheme (DINS) is proposed fusing both the local measurements Distributed INnovation based (DINS) by fusing the measurements and the information received from neighboring nodes. Weby present a hybrid TDMA scheme for Distributed INnovation based Scheme Scheme (DINS) is is proposed proposed bypresent fusing both both the local local measurements and the information received from neighboring nodes. We a hybrid TDMA scheme and the information received from neighboring nodes. We present a hybrid TDMA scheme for efficient implementation of DINS and the design of parameters of DINS to bring the effectsfor of and the information received fromand neighboring nodes. We present a hybrid TDMA scheme for efficient implementation of DINS the design of parameters of DINS to bring the effects of efficient implementation of DINS and the design of parameters of DINS to bring the effects of fading under a satisfactory level. With an appropriate design, it is shown that our algorithm is efficient implementation of level. DINS With and the design of parameters of DINS to bring the effects of fading under a satisfactory an appropriate design, it is shown that our algorithm is fading under a satisfactory level. With an appropriate design, it is shown that our algorithm is able to produce estimates with desired asymptotic properties of unbiasedness and consistency. fading under a satisfactory level.desired With an appropriate design, of it is shown that and our algorithm is able to produce estimates with asymptotic unbiasedness consistency. able to estimates with asymptotic properties of and Simulation results are provided to demonstrate the properties performance of DINS. able to produce produce estimates with desired desired asymptotic properties of unbiasedness unbiasedness and consistency. consistency. Simulation results are provided to demonstrate the performance of DINS. Simulation results are to the of DINS. Simulation are provided provided to demonstrate demonstrate the performance performance DINS.Ltd. All rights reserved. © 2017, IFACresults (International Federation of Automatic Control) Hosting byofElsevier Keywords: Relay-assisted network; analog fading channel; unbiasedness; consistency. Keywords: Keywords: Relay-assisted Relay-assisted network; network; analog analog fading fading channel; channel; unbiasedness; unbiasedness; consistency. consistency. Keywords: Relay-assisted network; analog fading channel; unbiasedness; consistency. 1. INTRODUCTION communications only. In order to achieve energy efficiency 1. communications only. In order to efficiency 1. INTRODUCTION INTRODUCTION communications only. In achieve energy efficiency for large-scale field monitoring, theachieve notionenergy of relay-assisted 1. INTRODUCTION communications only. In order order to to achieve energy efficiency for large-scale field monitoring, the notion of relay-assisted for large-scale field monitoring, the notion of relay-assisted networks has been proposed in Zhu et al. (2015), Consider a typical problem of sensor fusion over sensor sensor for large-scale field monitoring, the notion of relay-assisted sensor networks has been proposed in Zhu et al. (2015), Consider aanodes typical problem of sensor fusion over sensor sensor networks has been proposed in Zhu et al. (2015), where a number of relay nodes (RNs) comprising of wireConsider typical problem of sensor fusion over sensor networks, measure an unknown vector of interest, sensor anetworks has beennodes proposed incomprising Zhu et al. of (2015), Consider anodes typical problem of sensor fusion over sensor where number of relay (RNs) wirenetworks, measure an unknown vector of interest, where a number of relay nodes (RNs) comprising of less transceivers but no sensors are deployed in the network networks, nodes measure an unknown vector of interest, obtaining noisy versions of the vector, and then report the where a number but of relay nodesare (RNs) comprising of wirewirenetworks, nodesversions measureofan unknown vector ofreport interest, less transceivers no sensors deployed in the network obtaining noisy the vector, and then the less transceivers but no sensors are deployed the network assist the data exchange between sensorin nodes (SNs), obtaining noisy versions of the the vector, and then processing report the the to samples tonoisy a remote fusion center forand further less transceivers but no sensors are deployed in the network obtaining versions of vector, then report to assist the dataOne exchange sensor nodes (SNs), samples to remote fusion for processing to assist the exchange between sensor nodes (SNs), the meters. majorbetween advantage of such of samples to a remote (Xiao fusionetcenter center for further further processing and decision making al. (2006)). For instance, in i.e., to assist the data dataOne exchange between sensor nodeskind (SNs), samples to aa making remote fusion center for further processing i.e., the meters. major advantage of such kind of and decision (Xiao et al. (2006)). For instance, in i.e., the meters. One major advantage of such kind of networks is that long distance communications between and decision making (Xiao et al. (2006)). For instance, in a power system, we need to monitor system components i.e., the meters. One major advantage of such kind of and decision making (Xiaotoetmonitor al. (2006)). For components instance, in networks is that long distance communications between aausing power system, we need system networks is that long distance communications between meters can be avoided with only a few RNs, while certain power system, we need to monitor system components meters and estimate the state vector of the entire networks is that long distance communications between a power system, we need to monitor system ofcomponents can be avoided can with only aa guaranteed. few RNs, while certain using estimate state vector meters can be with only few network specifications using meters and estimatetothe the statemeter vectormeasurements of the the entire entire meters power meters systemand according these meters can be avoided avoided can withstill onlybe a guaranteed. few RNs, RNs, while while certain certain using meters and estimate the state vector of the entire network specifications still be power system according to these meter measurements network specifications can still be guaranteed. power system according to al.these these meter measurements measurements network specifications can still be guaranteed. (Huangsystem et al. (2012); Xie et (2012)) power according to meter Some preliminary results regarding distributed sensor fu(Huang (2012); Xie et al. (Huang et et yal. al.(t) (2012); Xie et(t), al. (2012)) (2012)) preliminary results regarding distributed sensor fu= Hi θXie +w i = 1, 2, . . . , m, (1) Some Some preliminary results regarding distributed sensor fu(Huang et al. et sion in relay-assisted networks can be found in Zhu et al. i (2012); i al. (2012)) Some preliminary results regarding distributed sensor fuy = H + w ii = 1, 2, .. .. .. ,, m, (1) sion in relay-assisted networks can be found in Zhu et al. i (t) iθ i (t), y (t) = H θ + w (t), = 1, 2, m, (1) sion in relay-assisted networks can be found in Zhu et al. i i i (2015, 2016). For sensor networks without RNs, i.e., only where a linearized DC model is adopted in deriving above yi (t) = H θ + w (t), i = 1, 2, . . . , m, (1) sion in 2016). relay-assisted networks canwithout be foundRNs, in Zhu etonly al. i i (2015, For sensor networks i.e., where a linearized DC model is adopted in deriving above J (2015, 2016). For sensor networks without RNs, i.e., only SNs are present, a number of distributed solutions have where a is adopted in deriving above model, θlinearized ∈ RJ is DC the model state vector including bus voltage (2015, 2016). For sensor networks without RNs, i.e., only where a linearized DC model is adopted in deriving above SNs are present, a number of distributed solutions have J is the state vector model, θ ∈ magnitudes, R including voltage SNs are present, number of solutions been proposed in aathe literature. For example, Refs. have Das model, θ R including bus voltage angles Hi vector ∈ RJJii ×J is the bus observation are present, number of distributed distributed solutions have ×J model, and θ ∈ ∈ magnitudes, RJ is is the the state state vector including bus voltage SNs been proposed in the literature. For example, Refs. Das Ji ×J angles and H R is the observation been proposed in the literature. For example, Refs. Das i ∈ and Mesbahi (2009); Liu et al. (2013); Schizas et al. (2008); angles and magnitudes, H ∈ R is the observation matrix with J ≤ J, and w (t) is the observation noise i i∈ RJi ×J is the observation i been proposed in theLiu literature. For Schizas example, Refs. Das angles and magnitudes, Hw and Mesbahi Mesbahi (2009); et al. al. (2013); et al. al. (2008); i i (t) is the observation noise matrix with J ≤ J, and and (2009); Liu et (2013); Schizas et (2008); i Xiao et al. (2005) proposed to use consensus algorithms to matrix with J ≤ J, and w (t) is the observation noise following a Gaussian distribution with zero mean. i i and Mesbahi (2009); Liu et al. (2013); Schizas et al. (2008); matrix with J ≤ J, and w (t) is the observation noise Xiao et al. (2005) proposed to use consensus algorithms to i i following a Gaussian distribution with zero mean. Xiao et al. (2005) proposed to use consensus algorithms to do the sensor fusion. In this case, it is assumed that nodes following a Gaussian distribution with zero etsensor al. (2005) proposed to use itconsensus algorithms to following zero mean. mean. do the fusion. In this case, is assumed that nodes Recently, aa Gaussian number ofdistribution distributedwith solutions to the above Xiao do the sensor fusion. In this case, it is assumed that nodes first take a snapshot of measurements and then adopt Recently, a number of distributed solutions to the above do the sensor fusion. In this case, it is assumed that nodes take algorithms aa snapshot of measurements and thenOn adopt Recently, a of the above sensor fusion problem have been solutions reported,to name a first first of measurements and adopt the data iteratively. the Recently, a number number of distributed distributed solutions toto the above sensor fusion problem have been reported, to name aa consensus first take take algorithms a snapshot snapshot to of fuse measurements and then thenOn adopt consensus to fuse the data iteratively. the sensor fusion problem have been reported, to name few, Das and Mesbahi (2009); Kar et al. (2012); Xie consensus algorithms to fuse the data iteratively. On other hand, if the measurements of the unknown vector sensorDas fusion problem have beenKar reported, to nameXie a consensus algorithms to fuse the data iteratively. On the few, and Mesbahi (2009); et al. (2012); the other hand, if the measurements of the unknown vector few, Das and Mesbahi (2009); Kar et al. (2012); Xie et al. (2012); Schizas et al. (2008); Xiao et al. (2005); other if measurements of unknown are hand, streamingly at SNs, a natural wayvector is to few, Das and Schizas Mesbahiet (2009); Kar Xiao et al.et(2012); Xie θ et al. (2012); al. (2008); al. (2005); other hand, if the the available measurements of the the unknown vector θ are streamingly available at SNs, a natural way is to et al. (2012); Schizas et al. (2008); Xiao et al. (2005); Xu et al. (2015), where the nodes collaboratively estimate θ are available at aa natural is the consensus step and innovation stepway together et al. (2012); Schizas et al. (2008); Xiao et al.estimate (2005); combine Xu et al. (2015), where the nodes collaboratively θ are streamingly streamingly available at SNs, SNs, natural way is to to combine the consensus step and innovation step together Xu et al. (2015), where the nodes collaboratively estimate the unknown vector by means of local computations and combine the consensus and innovation step in one update, e.g., Karstep et al. (2012); Zhang andtogether Zhang Xu et al. (2015), where the nodes collaboratively estimate the unknown vector by means of local computations and combine the consensus step and innovation step together in one update, e.g., Kar et al. (2012); Zhang and Zhang the unknown vector by of local computations and in Kar (2012); Zhang and Zhu et e.g., al. (2015). Most of the aforementioned This the unknown by means means local computations and (2012); researchvector is partially fundedof by National Key Research in one one update, update, e.g., Kar et et al. al. (2012); Zhang and Zhang Zhang (2012); Zhu et al. (2015). Most of the aforementioned (2012); Zhu et al. (2015). Most of the aforementioned works assume that the communication topology is fixed This research is partially funded by National Key Research and Development of China (No. 2016YFB0901903, research is Program partially funded by National Key Research (2012); Zhu etthat al. the (2015). Most of thetopology aforementioned This works assume communication is fixed This research is partially funded by National Key Research and Program of works assume that communication topology is with perfect is not practical in realistic 2016YFB0901901), of China under(No. the 2016YFB0901903, grants 61633017, and Development Development NSF Program of China China (No. 2016YFB0901903, works assumechannels, that the the which communication topology is fixed fixed with perfect channels, which is not practical in realistic and Development NSF Program of China (No. 2016YFB0901903, 2016YFB0901901), of China under the grants 61633017, with perfect channels, which is not practical in realistic applications. Actually, communication links among nodes U1405251, 61603251, 61521063, 61603205, and Shandong 2016YFB0901901), NSF of Chinaand under the grants 61633017, with perfect channels, which is not practical in realistic 2016YFB0901901), NSF of China under the grants 61633017, applications. Actually, communication links among nodes U1405251, 61603251, 61521063, and Shandong applications. Actually, among nodes are usually unidirectional since nodes links broadcast difProvincial Foundation of Chinaand under grant U1405251, Natural 61603251,Science 61521063, and 61603205, 61603205, and Shandong applications. Actually, communication communication links amongat nodes U1405251, 61603251, 61521063, and 61603205, and Shandong are usually unidirectional since nodes broadcast at difProvincial ZR2015FQ012 are usually unidirectional since nodes broadcast at Provincial Natural Natural Science Science Foundation Foundation of of China China under under grant grant are usually unidirectional since nodes broadcast at difdifProvincial Natural Science Foundation of China under grant
ZR2015FQ012 ZR2015FQ012 ZR2015FQ012 Copyright © 2017, 2017 IFAC 14849 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 14849 Copyright ©under 2017 responsibility IFAC 14849 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 14849 10.1016/j.ifacol.2017.08.1847
Proceedings of the 20th IFAC World Congress Shanying Zhu et al. / IFAC PapersOnLine 50-1 (2017) 14284–14289 Toulouse, France, July 9-14, 2017
ferent power levels and may have different interference patterns. Moreover, communications links between nodes generally suffer from noise and errors and are subject to channel fading. These negative effects make the extension highly non trivial in terms of both the algorithm design and theoretical analysis. In our previous work Zhu et al. (2016), we proposed a distributed algorithm to solve the problem of sensor fusion if the unknown variable is scalar. In this paper, we follow the same line and aim to extend it to the more general vector case (1). The paper is organized as follows. In Section 2, we formulate our problem. In Section 3, we propose a distributed solution to the problem of sensor fusion in relay-assisted networks, called DINS, followed by the performance analysis in Section 4. Section 5 presents simulation results demonstrating the effectiveness and efficiency of DINS. 2. PROBLEM FORMULATION Consider a relay-assisted network consisting of SNs and RNs that monitors a field of interest. There are n local nodes observing the unknown vector θ ∈ RJ with the measurements given by (1). For notational simplicity, we label the SNs from 1 to m, and RNs from m + 1 to n. We assume that all wi (t) are independent of t, but may be correlated across nodes, and supt≥0 E{ w(t) 2 } < ∞, where w(t) = [w1 (t)T , . . . , wm (t)T ]T . Generally, Ji ≤ J, for each i, this means that each SN only have limited measurement and θ is not locally observable at each SN. As such, cooperation betweens the nodes is then necessary to solve such sensor fusion problem. We assume half-duplex operation of nodes, i.e., nodes can not simultaneously transmit and receive. The nodes use the analog-and-forward scheme for the information exchange, where data are scaled and then transmitted without any coding (Liu et al. (2007)). It is convenient to model the communication network as a dynamic directed graph G(t) = (V, E(t)), where V = IS ∪ IR , IS and IR are the sets of SNs and RNs, respectively, E(t) ⊂ V ×V collects all communication links between nodes at time instant t. The directed edge (i, j) ∈ E(t) means that node i can transmit data to node j directly at time instant t. As such, each node i has a local neighborhood Ni (t), which is the set of nodes from which it can receive data at time instant t. The SN and RN neighbors of node i are denoted as NiS (t) = Ni (t) ∩ IS and NiR (t) = Ni (t) ∩ IR , respectively.
We consider the scenario where transmissions suffer from fading and additive noise. With the analog-and-forward scheme, each node i amplifies its data with αi (t) first and then broadcasts to its neighbors. At the receiver side, the data received by node i at time instant t is zi (t) = αj (t)hij (t)xj (t) + vij (t) , (2) j∈NiS (t)∪NiR (t)
where xi (t) is the estimate of θ at node i, hij (t) is the random fading coefficient of link (j, i) with bounded variance, vij (t) is the receiver noise in the transmission from node j to node i, which is assumed to be of zero mean and bounded variance. Eq. (2) describes the signal reception from the neighbors to each node. In the subsequent analysis, we consider the case where all hij (t) are independent across receivers and across time instants, and the receiver
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noise vij (t) are uncorrelated from one time instant to the next and uncorrelated with hij (t). 3. DISTRIBUTED SENSOR FUSION SCHEME OVER FADING CHANNELS In this section, we provide a distributed sensor fusion scheme for relay-assisted sensor networks subject to fading. In the scheme, each node only relies on local information, which is confined to one-hop communications. 3.1 Development of DINS Our scheme is motivated by the observation that both the own measurement (1) and the local information (2) received from neighboring nodes can be regarded as some kind of measurements. Actually, the local information (2) depends on θ through the local estimates xj (t), which follows an indirect relation. In this way, (2) can be treated as virtual measurement. This motivates us to propose the following Distributed INnovation based Scheme (DINS) by fusing these two kinds of measurements: • At the end of one round of transmissions, each SN i ∈ IS updates its estimate xi (t) as xi (t + 1) = xi (t) + ρ(t)HTi (yi (t) − Hi xi (t)) + ρ(t)(zi (t) − bi (t)xi (t)),
(3)
where ρ(t) > 0 is the decaying weight for attenuating the noise, and bi (t) is the factor introduced to compensate for the effect of channel fading. • As for RNs, we denote the reception at each RN i ∈ IR as its estimate, i.e., xi (t) = zi (t).
(4)
Recursions (3) and (4) constitute our distributed sensor fusion scheme. It is noted that with this scheme each node only need local information to update its estimate. Of special note is the formulation of (3), where we divide it into two parts, one is the innovation yi (t) − Hi xi (t) w.r.t. its own data and the other is the innovation zi (t) − bi (t)xi (t) w.r.t. received data from neighboring nodes. Similar forms can also be found in Kar et al. (2012); Stankovi´c et al. (2011); Zhang and Zhang (2012). However, the superposition in the multiple access scheme (2) would introduce an undesired non-vanishing noise term contributed by the channel fading, which makes the aforementioned existing schemes not applicable in our case. This is why we introduce the compensation factor bi (t) in (3). To make the presentation clearer, we consider the special case that each SN can directly transmit data to other SNs or via at most one-hop RN. Let −αj (t)hij (t), j ∈ NiS (t), 0, j ∈ NiS (t), , i ∈ IS , pij (t) = pij (t), j = i, − j=i
stack pij (t), xi (t) into vectors pi (t) = [pi1 , pi2 , . . . , pim ]T and x(t) = [x1 (t)T , x2 (t)T , . . . , xm (t)T ]T ∈ RmJ , respectively, one then obtains
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zi (t) − bi (t)xi (t) = γi (t)xi (t) − (pi (t)T ⊗ IJ )x(t) + αj (t)hij (t)(pj (t)T ⊗ IJ )x(t) j∈NiR (t)
(5) + ρ(t)vi (t), where γi (t) = j∈N R (t) αj (t)hij (t) k∈N S (t) αk (t)hjk (t)+ i j j∈NiS (t) αj (t)hij (t) − bi (t) and vi (t) = j∈Ni (t) vij (t) + α (t)h (t) v (t). With the above noR j ij jk j∈Ni (t) k∈Nj (t) tations, we rewrite the proposed DINS in a compact form x(t + 1) = ImJ − ρ(t)(HT H + (L(t) − Γ(t)) ⊗ IJ ) x(t) + ρ(t) HT y(t) + v(t) , (6)
where y(t) = [y1 (t)T , y2 (t)T , . . . , ym (t)T ]T , v(t) = [v1 (t)T , v2 (t)T , . . . , vm (t)T ]T , H = diag{H1 , H2 , . . . , Hm }, Γ(t) = diag {γ1 (t), γ2 (t), . . . , γm (t)}, and p1 (t)T + αj (t)h1j (t)pj (t)T j∈N1R (t) .. . L(t) = . T pm (t)T + αj (t)hmj (t)pj (t) R (t) j∈Nm
In summary, the total number of time slots for a relayassisted network is m + 2 − m , where m is the number of neighbors of all RNs. In this way, n − m − 2 + m slots can be saved compared with the traditional node-based TDMA scheme in Das and Mesbahi (2009); Ergen and Varaiya (2010). 3.3 Design of algorithm parameter We design the algorithm parameters αj (t) and bi (t) by bringing the effect of fading on the algorithm under an unbiased level. To this end, we require that the channel ¯ ij E{hij (t)} > 0, is available statistics, i.e., the mean h at the receiver i. The next result gives a choice of such parameters with desired properties. The next result provides a choice of such parameters, which shows that the fading only leads to an unbiased effect. Proposition 1. For each node i = 1, 2, . . . , n, let αi (t) = √ 1/ m, and for each SN i ∈ IS we design bi (t) as 1 ¯ ij + 1 ¯ ij ¯ jk , h h h bi (t) = √ m m S R S j∈Ni (t)
j∈Ni (t)
k∈Nj (t)
3.2 Implementation of DINS
then E{Γ(t)} = 0.
In this section, we outline how the proposed DINS may be implemented over wireless networks. First, note that DINS is a synchronous algorithm, that is, the nodes updates their states synchronously at the end of one round of transmissions. One popular protocol suitable for such algorithms is the TDMA scheme, e.g., Das and Mesbahi (2009); Ergen and Varaiya (2010). In the traditional TDMA scheme, the time axis is divided into slots with each slot assigned to one user, and users are permitted to transmit only during their assigned slots. One critical aspect is the low scalability, where the number of slots is n, growing linearly with the number of nodes. This would jeopardize the efficiency of TDMA scheme in practical scenarios, especially for largescale networks.
Proof. The proof is straightforward, so we omit the details, see also Zhu et al. (2016).
In this section, we adopt a hybrid TDMA scheme to implement DINS by exploiting both the broadcast property and the superposition property of wireless channels. Our key insight is that each node does not need to know the exact data from its neighboring nodes. Rather it only needs to know the sum. That is the interference property of the multiple access scheme (2) may be exploited to our advantage. Similar idea has already been used in Nazer et al. (2011). The time slots of proposed hybrid TDMA scheme are assigned according to the following rules: 1) All the neighbors of RNs share the first slot. They broadcast simultaneously to the neighbors. According to (2), the received data at RNs are αj (t)hij (t)xj (t) + vij (t) . zR i (t) = j∈NiS (t)
2) All the RNs share the ensuing one slot. Accordingly, the received data at SN neighbors of RNs are αj (t)hij (t)xj (t) + vij (t) . zSi (t) = j∈NiR (t)
3) Each of the remainder SNs consumes one slot to transmit its data to the neighbors.
4. PERFORMANCE ANALYSIS OF DINS In this section, we will examine the convergence properties of DINS in the presence of fading transmissions. First, let us define the estimation error ei (t) = xi (t) − θ for each SN i ∈ IS . Stack them in the column vector e(t) = [e1 (t)T , . . . , em (t)T ]T , and define the matrix A(t) = ImJ − ρ(t)(HT H + L(t) ⊗ IJ ). Consequently, subtracting θ m = 1m ⊗ θ from both sides of (6), one obtains e(t + 1) = A(t) + ρ(t)Γ(t) ⊗ IJ e(t) + ρ(t)Γ(t)θ m + ρ(t)(HT w(t) + v(t)), (7) where we use the property that L(t)1m = 0 in terms of the definition of L(t).
It is noted that the coefficient matrix A(t) + ρ(t)Γ(t) ⊗ IJ of (7) is generally nonsymmetric, which makes the approaches for undirected graphs in Xie et al. (2012); Kar et al. (2012); Zhang and Zhang (2012) not applicable here. Actually, the coexistence of SNs and RNs in the network leads to a nonsymmetric L(t) even if all communication links between the nodes are bidirectional. On the other hand, the eigenvalues of A(t) + ρ(t)Γ(t) ⊗ IJ can be sufficiently close to 1 with large t, if we use decaying weight limt→∞ ρ(t) = 0. These two aspects present a challenge of using the classical results of linear systems (see Rugh (1996)) to investigate the convergence property of (10). Before moving onto our main results, we need two additional concepts. Definition 2. The unknown parameter θ is collaboratively m observable in one time-step, if i=1 HTi Hi is of full rank. Definition 3. Graph G(t) is said to be uniformly strongly connected (USC), if there exists a constant B > 0 such
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(k+1)B−1 that the union graph G[kB,(k+1)B) = V, kB E(t)
is strongly connected 1 for all nonnegative integers k.
Definition 2 means that θ can be collaboratively recovered by all the nodes, even if it is not locally observable at each node. Definition 3 is introduced to ensure the repeated influence of the nodes on each other. 4.1 Fixed network In this case, the communication network is fixed, i.e., G(t) = G, ∀t. We will introduce a transformation utilizing the spectral properties of graph G, based on which we obtain the following convergence result. Lemma 4. For a strongly connected graph G, the left ¯ eigenvector √ ξ of L w.r.t. the eigenvalue 0 is positive. Denote E = diag{ξ1 , ξ2 , . . . , ξm }, then F = 2HT H + √ √ −1 √ −1 T √ ¯ E + E L ¯ E) ⊗ IJ is positive definite if θ ( EL ¯ = E{L(t)}. is collaboratively observable, where L Proof. The proof is a direct consequence of Theorem 2 and Lemma 2 in Zhu et al. (2015). Theorem 5. For a strongly connected graph G, if θ is collaboratively observable, and ∞ ρ(t) = ∞, (8) t=0
then the estimate sequence {xi (t)}t≥0 generated DINS at each SN i ∈ IS is asymptotically unbiased. lim E{xi (t)} = θ, ∀i = 1, 2, . . . , m. t→∞
If further,
∞ t=0
ρ(t)2 < ∞,
(9)
t→∞
Proof. Asymptotic Unbiasedness: Taking expectations of both sides of (7) and using Proposition 1, we can obtain ¯ e(t), ¯(t + 1) = A(t)¯ e (10) ¯ ¯(t) = E{e(t)} √ where e and A(t) = E{A(t)}. By Lemma 4, we know that D = Ξ ⊗ IJ is nonsingular. Thus utilizing the transformation D¯ e(t), we have D¯ e(t + 1) = −1 ¯ DA(t)D D¯ e(t). Based on Lemma 4, and noting that (8) is satisfied, we use the same line of arguments in Zhu et al. (2015) to show that limt→∞ D¯ e(t) = 0, from which the asymptotic unbiasedness follows, since D is nonsingular. Strong Consistency: Letting V (t) = ˆ e(t)2 = De(t)2 , given the historical information {V (s) : s ≤ t} we obtain from (7) that E{V (t + 1)|V (s) : s ≤ t} = E D(A(t) + ρ(t)Γ(t) ⊗ IJ )e(t)2 |V (s) : s ≤ t + ρ(t)2 E{DΓ(t)θ m 2 } + ρ(t)2 E{DHT w(t)2 } + E{Dv(t))2 }
− 2ρ(t)2 e(t)T E{((L(t) − Γ(t)) ⊗ IJ )T DΓ(t)}θ m , (11)
1
where we use the facts that {wi (t)}, {vi (t)}, {hij (t)} are independent from e(t), and E{Γ(t)} = 0 by Proposition 1. In lieu of the independence assumption on {wi (t)}, {vi (t)} and {hij (t)}, one has E D(A(t) + ρ(t)Γ(t) ⊗ IJ )e(t)2 |V (s) : s ≤ t ˆ(t)T ImJ − ρ(t)E{B(t)} + ρ(t)2 E{C(t)} e ˆ(t), =e
where B(t) = 2HT H + D(L(t) ⊗ IJ )D−1 + D−1 (L(t)T ⊗ IJ )D and C(t) = D−1 (HT H + L(t)T ⊗ IJ )D2 (HT H + L(t) ⊗ IJ )D−1 . It is easy to see that E{B(t)} = F and E{C(t)} is bounded by recalling that {wi (t)}, {vi (t)} and {hij (t)} all have bounded variances. It thus follows that there is a t∗ > 0 such that ∀t ≥ t∗ , ˆ(t)T E{B(t)}ˆ e(t) ≤ 1 − ρ(t)λmin (F) + ρ(t)2 O(1) V (t), e where we use the fact that the minimum eigenvalue λmin (F) of F is positive by Lemma 4. On the other hand, it follows from Proposition 1 and the fact that {wi (t)}, {vi (t)} and {hij (t)} all have bounded variances that E{DΓ(t)θ m 2 }, E{DHT w(t)2 } and E{Dv(t))2 } are bounded, i.e., O(1). This along with the inequality 2zT1 z2 ≤ z1 2 + z2 2 for any two vectors z1 , z2 , and the H¨older inequality yields e(t)T E{((L(t) − Γ(t)) ⊗ IJ )T DΓ(t)}θ m ≤ O(1)V (t). Substituting the above relations into (11) implies for all t ≥ t∗ , E{V (t + 1)|V (s) : s ≤ t} ≤ 1 + ρ(t)2 O(1) V (t)
− ρ(t)λmin (F)V (t) + O(1)ρ(t)2 . It is noted that t=t∗ ρ(t)2 < ∞ by (9). Hence, by the Robbins-Siegmund theorem (see Polyak (1987)) we know that there is a random variable v ≥ 0 such that ∞ ρ(t)V (t) < ∞ = 1. P lim V (t) = v ∞
t→∞
∞
then {xi (t)}t≥0 is also strongly consistent, P lim xi (t) = θ = 1, ∀i = 1, 2, . . . , m.
A graph is said to be strongly connected if there is a sequence of directed edges such that any vertex can be reached from another one.
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Recalling that t=t∗ ρ(t) = ∞ by (8), it thus follows that v = 0, from which we have P{limt→∞ V (t) = 0} = 1. Therefore, P{limt→∞ De(t) = 0} = 1. Remember that D is nonsingular, we complete the proof. 4.2 Time-varying network
For the sake of brevity, we assume that HTi Hi is diagonal and rank deficient for each SN i ∈ IS . This is saying that each SN i ∈ IS only observes part of the components of θ.
Since the network G(t) is time-varying and directed, a common Lyapunov function for (7) is generally not available in this case. Hence, the method developed for fixed networks in Section 4.1 fails. To address this issue, we consider an appropriate reordered space, built on which we can deal with asymmetry in a tractable framework.
Consider the error dynamics (10), for each component l = 1, 2, . . . , J, let reorder the state {¯ e1,l (t), e¯2,l (t), . . . , e¯m,l (t)} in such a way that r1,l (t) ≥ r2,l (t) ≥ · · · ≥ rm,l (t), where ri,l (t) = e¯it ,l (t) and {1t , 2t , . . . , mt } is a permutation of {1, 2, . . . , m} at time t. The next result shows that the ordered states will converge to consensus for each component l. Proposition 6. Suppose that the graph G(t) is USC, then for each l = 1, 2, . . . , J, we have lim (ri,l (t) − rj,l (t)) = 0, ∀i, j = 1, 2, . . . , m.
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Proof. It is easy to see that there is a permutation matrix ¯ ⊗ IJ )PT = IJ ⊗ L(t) ¯ P such that P(L(t) and PHT HPT = diag{Λ1 , Λ2 , . . . , ΛJ }, where Λi is a diagonal matrix for each i = 1, 2, . . . , J, since HT H is diagonal. Let us denote r(t) = P¯ e(t) and divide it 1
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Consider a sensor network with n = 50 nodes using the random geometric graph model and two nodes are linked by two unidirectional links if their distance is less than log n/n. We assign 5 critical nodes without which the network will separate into groups as RNs. Afterwards, we randomly delete 20% of unidirectional links to obtain an asymmetric network. The time-varying nature of G(t) is simulated by letting 10% of links fail at each time instant.
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V˜ (t) = E{Γ(t)θ m 2 } + E{HT w(t)2 } + E{v(t))2 } − ¯ 2e(t)T E{((L(t) − Γ(t)) ⊗ IJ )T Γ(t)}θ m . Since 1T L(t) = ¯ T ⊗ ¯ ¯ L(t)1 = 0, it can shown that L(t) ⊗ IJ + L(t) IJ is positive semidefinite, which similar to Lemma 4 ˜ implies that B(t) is positive definite. Furthermore, we ˜ find that E{C(t)} and V˜ (t) are all bounded following the similar arguments in the proof of Theorem 5. Applying the Robbins-Siegmund theorem (cf. Polyak (1987)) again leads to the conclusion P{limt→∞ e(t) = 0} = 1, from which the theorem follows.
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Fig. 1. Behaviors of the ordered states r1 (t), r3 (t) for a ring graph SN 1→SN 2→RN 1→SN 3→SN 1. into r(t) = [r1 (t)T , r2 (t)T . . . , rJ (t)T ]T , where rl (t) ∈ Rm , ∀l = 1, 2, . . . , J, then we derive from (10) that ¯ rl (t). (12) rl (t + 1) = Im − ρ(t)(Λl + L(t)) In lieu of (12), we can show that either r1,l (t) is nonin∞ creasing and converges to a nonnegative value r1,l ≥ 0 or rm,l (t) is nondecreasing and converges to a nonpositive ∞ value rm,l ≤ 0. The proof is similar to that of Zhu et al. (2016), whose intuitive explanation is illustrated in Fig. 1. Next, it suffices to show that ri,l (t) converges to the same value for all i = 1, 2, . . . , m. By careful examination of (12) and following similar arguments as those for the scalar case in Zhu et al. (2016), we can show that ri,l (t), ∀i, converges to the same value under the assumption of USC. The proposition thus follows. Theorem 7. Suppose that the graph G(t) is USC and ¯ ¯ 1T L(t) = L(t)1 = 0. If θ is collaboratively observable and (8), (9) are satisfied, then the estimate sequence {xi (t)}t≥0 generated by DINS at each SN i ∈ IS is asymptotically unbiased and strongly consistent. Proof. Asymptotic Unbiasedness: By applying a wellknown convergence result on recurrent inequalities (Polyak, 1987, p.45), we can show that limt→∞ rl (t) = 0, and thus ¯(t) = 0 in view of the fact that P is nonsingular. limt→∞ e We omit the details due to page limitation. Strong Consistency: Using V (t) = e(t)2 as a Lyapunov candidate, given the historical information {V (s) : s ≤ t} we obtain from (7) that E{V (t + 1)|V (s) : s ≤ t} ˜ = e(t)T ImJ + ρ(t)2 E{C(t)} e(t) ˜ − ρ(t)λmin (B(t))V (t) + ρ(t)2 V˜ (t), (13) T T ˜ ¯ ¯ where B(t) = 2H H + L(t) ⊗ IJ + L(t) ⊗ IJ and ˜ C(t) = (HT H + L(t)T ⊗ IJ )(HT H + L(t) ⊗ IJ ) and
The state of the field is represented by the vector θ ∈ R10 . Each component corresponds to a random field sample, which is generated by a zero-mean Gaussian random variable of variance 20. Moreover, each node is associated with only one component of θ, i.e., Hi = [0, . . . , 1, . . . , 0], where the only nonzero 1 is the (i mod 10) + 1-th entry, for all i. Obviously, θ ∈ R10 is collectively observable, m since i=1 HTi Hi = 4I10 + diag{0, 1, 1, 1, 1, 1, 0, . . . , 0} is nonsingular. The measurement noise wi (t) is i.i.d. with unit variance. We consider the Rayleigh channel fading between nodes which is assumed to be i.i.d. with parameter σr = 1. The receiver noise are drawn i.i.d. from a Gaussian distribution with variance σ 2 = 0.5. Parameters αi (t) and bi (t) are designed according to Proposition 1. We use the decaying weight ρ(t) = 2(t + 1)−0.7 . It can be shown that ρ(t) satisfies the conditions of Theorem 5. DINS is implemented with random initial values xi (0) ∈ [θ − 1, θ + 1], ∀i.
The first simulation is to demonstrate the performance of unbiasedness of DINS. In Fig. 2(a), we plot the ensemble average of the estimate xi (t) for SNs with minimum and maximum neighbors, respectively, where each curve is averaged over 10 independent runs. The theoretical limit, i.e., the true value θ is also plotted for comparison purpose. As we can notice from the figure, each ensemble average closely approaches θ, although θ is not locally observable at each SN. This corroborates the theoretical result obtained in Theorem 7, i.e., DINS is capable of enabling all the nodes to collaborate. To quantify the performance of the proposed algorithm DINS, we use the the mean square error averaged over all 45 1 2 SNs emse (t) = 45×10 i=1 xi (t) − θ as a measure. Simulations are presented by averaging over 10 independent runs. The average mean square error is shown in Fig. 2(b), from which we note that emse (t) decreases gradually to 0, thus conforming the theoretical result of Theorem 7. We also plot the results of mean square error of DINS for four different choices of ρ(t) = 2(t + 1)η , i.e., η = −0.7, −1, −2, in Fig.2(c). For η = −1 and η = −0.7,
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Fig. 2. (a) Ensemble average of estimates for SNs with maximum and minimum neighbors, respectively; (b) Average mean square error of DINS; (c) Average mean square error of DINS for different choices of ρ(t) = 2(t + 1)−0.7 , 2(t + 1)−1 , 2(t + 1)−2 . both of them satisfy the conditions of Theorem 7, and the convergence is observed. It is worth noting that the convergence for η = −1 is quite slow. However, for the case η = −2, the decay is too fast to obtain a small error and a large residual occurs. 6. CONCLUSIONS We considered the problem of distributed estimation of an unknown vector in relay-assisted sensor networks. Particularly, we focused on the effect of time-varying asymmetric topology due to link failure and the fading channels. We proposed a distributed sensor fusion scheme DINS based on innovation techniques with a hybrid TDMA scheme enabling its implementation. Theoretically, we showed that DINS can effectively solve the sensor fusion problem in noisy environments with guaranteed properties of asymptotic unbiasedness and consistency for both fixed and time-varying topologies. We also presented simulations to validate its effectiveness and efficiency. REFERENCES Das, A.K. and Mesbahi, M. (2009). Distributed linear parameter estimation over wireless sensor networks. IEEE Transactions on Aerospace and Electronic Systems, 45(4), 1293–1306. Ergen, S.C. and Varaiya, P. (2010). TDMA scheduling algorithms for wireless sensor networks. Wireless Networks, 16(4), 985–997. Huang, Y.-F., Werner, S., Huang, J., Kashyap, N., Gupta, V. (2012). State estimation in electric power grids. IEEE Signal Processing Magazine, 29(4), 33–43. Kar, S., Moura, J.M.F., and Ramanan, K. (2012). Distributed parameter estimation in sensor networks: Nonlinear observation models and imperfect communication. IEEE Transactions on Information Theory, 58(6), 3575–3605. Liu, K., Gamal, H.E., and Sayeed, A. (2007). Decentralized inference over mutiple-access channels. IEEE Transactions on Signal Processing, 55(7), 3445–3455. Liu, S., Li, T., Xie, L., Fu, M., and Zhang, J. (2013). Continuous-time and sampled-data based average consensus with logarithmic quantizers. Automatica, 49(11), 3329–3336.
Nazer, B., Dimakis, A.G., and Gastpar, M. (2011). Local interference can accelerate gossip algorithms. IEEE Journal of Selected Topics in Signal Processing, 5(4), 876–887. Polyak, B.T. (1987). Introduction to Optimization. Optimization Software, New York. Rugh, W.J. (1996). Linear System Theory. Prentice Hall, Upper Saddle River, NJ, second edition. Schizas, I.D., Ribeiro, A., and Giannakis, G.B. (2008). Consensus in ad hoc WSNs with noisy links-Part I: Distributed estimation of deterministic signals. IEEE Transactions on Signal Processing, 56(1), 342–356. Stankovi´c, S.S., Stankovi´c, M.S., and Stipanovi´c, D.M. (2011). Decentralized parameter estimation by consensus based stochastic approximation. IEEE Transactions on Automatic Control, 56(31), 531–543. Xiao, J., Ribeiro, A., Luo, Z.Q., and Giannakis, G.B. (2006). Distributed compression-estimation using wireless sensor networks. IEEE Signal Processing Magazine, 23(4), 27–41. Xiao, L., Boyd, S., and Lall, S. (2005). A scheme for robust distributed sensor fusion based on average consensus. In Proc. Int. Conf. Information Process. Sensor Networks(IPSN), 63–70. Los Angeles, USA. Xie, L., Choi, D.-H., Kar, S., Poor, H. V. (2012). Fully distributed state estimation for wide-area monitoring systems. IEEE Transactions on Smart Grid 3(3), 1154– 1169. Xu, J., Zhu, S., Soh, Y.C., and Xie, L. (2015). Augmented distributed gradient methods for multi-agent optimization under uncoordinated constant stepsizes. In Proc. 54th IEEE Conf. Decision and Control, 2055– 2060. Osaka, Japan. Zhang, Q. and Zhang, J. (2012). Distributed parameter estimation over unreliable networks with Markovian switching topologies. IEEE Transactions on Automatic Control, 57(10), 2545–2560. Zhu, S., Chen, C., Ma, X., Yang, B., and Guan, X. (2015). Consensus based estimation over relay assisted sensor networks for situation monitoring. IEEE Journal of Selected Topics in Signal Processing, 9(2), 278–291. Zhu, S., Soh, Y.C., and Xie, L. (2016). Distributed inference for relay-assisted sensor networks with intermittent measurements over fading channels. IEEE Transactions on Signal Processing, 64(3), 742–756.
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DINS: A Distributed Scheme for Sensor DINS: A Distributed Scheme for Sensor DINS: A Distributed Scheme for DINS: A Distributed Scheme for Sensor Sensor Fusion over Fading Channels Fusion over Fading Channels Fusion over Fading Channels Fusion over Fading Channels ∗ ∗∗ ∗
Jinming Xu ∗∗ Cailian Chen ∗ Xinping Guan ∗∗ ∗∗ Cailian Jinming Xu Chen Jinming Xu Chen ∗ Xinping Xinping Guan Guan ∗ Binqiang Xue ∗∗∗ ∗∗∗Chen ∗ Xinping Guan ∗ Jinming Xu ∗∗ Cailian Cailian ∗∗∗ Binqiang Xue Binqiang Binqiang Xue Xue ∗∗∗ ∗ Department of Automation, Shanghai Jiao Tong University, and Key ∗ ∗ Department of Automation, Shanghai Jiao Tong University, and Key Department of Automation, Shanghai Jiao University, and Laboratory of System Control and Information Processing, Ministry of ∗ Department of Automation, Shanghai Jiao Tong Tong University, and Key Key Laboratory of System Control and Information Processing, Ministry of Laboratory of System Control and Information Processing, Ministry Education of China, Shanghai 200240, China (e-mails:{shyzhu, Laboratory of System Control and 200240, Information Processing, Ministry of of Education of China, Shanghai China (e-mails:{shyzhu, Education of Shanghai 200240, cailianchen, xpguan}@sjtu.edu.cn). Education of China, China, Shanghai 200240, China China (e-mails:{shyzhu, (e-mails:{shyzhu, cailianchen, xpguan}@sjtu.edu.cn). ∗∗ cailianchen, andxpguan}@sjtu.edu.cn). Electronic Engineering, Nanyang ∗∗ School of Electrical cailianchen, xpguan}@sjtu.edu.cn). ∗∗ School of Electrical and Electronic Engineering, Nanyang of Electrical and Engineering, Nanyang Technological University, Singapore 639798 (e-mail: ∗∗ School School of Electrical and Electronic Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: Technological University, Singapore 639798 (e-mail: [email protected]). Technological [email protected]). University, Singapore 639798 (e-mail: ∗∗∗ [email protected]). and Electrical Engineering, Qingdao ∗∗∗ College of Automation [email protected]). ∗∗∗ College of Automation and Electrical Engineering, of Automation and Electrical Engineering, Qingdao University, Qingdao China (e-mail: Qingdao ∗∗∗ College CollegeUniversity, of Automation and 266071, Electrical Engineering, Qingdao Qingdao 266071, China (e-mail: University, Qingdao 266071, China (e-mail: [email protected]). University, Qingdao 266071, China (e-mail: [email protected]). [email protected]). [email protected]). Shanying Shanying Shanying Shanying
Zhu ∗ ∗ Zhu Zhu Zhu ∗
Abstract: In this paper, we consider the problem of distributed sensor fusion in relayAbstract: In this paper, consider the problem of distributed sensor fusion in relayAbstract: In this we consider problem of fusion in assisted sensor networks withwe time-varying asymmetric communications over fading channels. A Abstract: In networks this paper, paper, we consider the the problem communications of distributed distributed sensor sensor fusion in relayrelayassisted sensor with time-varying asymmetric over fading channels. A assisted sensor networks with time-varying asymmetric communications over fading channels. A Distributed INnovation based Scheme (DINS) is proposed by fusing both the local measurements assisted sensor networksbased with time-varying asymmetric communications over fading channels. A Distributed INnovation Scheme (DINS) is proposed fusing both the local measurements Distributed INnovation based (DINS) by fusing the measurements and the information received from neighboring nodes. Weby present a hybrid TDMA scheme for Distributed INnovation based Scheme Scheme (DINS) is is proposed proposed bypresent fusing both both the local local measurements and the information received from neighboring nodes. We a hybrid TDMA scheme and the information received from neighboring nodes. We present a hybrid TDMA scheme for efficient implementation of DINS and the design of parameters of DINS to bring the effectsfor of and the information received fromand neighboring nodes. We present a hybrid TDMA scheme for efficient implementation of DINS the design of parameters of DINS to bring the effects of efficient implementation of DINS and the design of parameters of DINS to bring the effects of fading under a satisfactory level. With an appropriate design, it is shown that our algorithm is efficient implementation of level. DINS With and the design of parameters of DINS to bring the effects of fading under a satisfactory an appropriate design, it is shown that our algorithm is fading under a satisfactory level. With an appropriate design, it is shown that our algorithm is able to produce estimates with desired asymptotic properties of unbiasedness and consistency. fading under a satisfactory level.desired With an appropriate design, of it is shown that and our algorithm is able to produce estimates with asymptotic unbiasedness consistency. able to estimates with asymptotic properties of and Simulation results are provided to demonstrate the properties performance of DINS. able to produce produce estimates with desired desired asymptotic properties of unbiasedness unbiasedness and consistency. consistency. Simulation results are provided to demonstrate the performance of DINS. Simulation results are to the of DINS. Simulation are provided provided to demonstrate demonstrate the performance performance DINS.Ltd. All rights reserved. © 2017, IFACresults (International Federation of Automatic Control) Hosting byofElsevier Keywords: Relay-assisted network; analog fading channel; unbiasedness; consistency. Keywords: Keywords: Relay-assisted Relay-assisted network; network; analog analog fading fading channel; channel; unbiasedness; unbiasedness; consistency. consistency. Keywords: Relay-assisted network; analog fading channel; unbiasedness; consistency. 1. INTRODUCTION communications only. In order to achieve energy efficiency 1. communications only. In order to efficiency 1. INTRODUCTION INTRODUCTION communications only. In achieve energy efficiency for large-scale field monitoring, theachieve notionenergy of relay-assisted 1. INTRODUCTION communications only. In order order to to achieve energy efficiency for large-scale field monitoring, the notion of relay-assisted for large-scale field monitoring, the notion of relay-assisted networks has been proposed in Zhu et al. (2015), Consider a typical problem of sensor fusion over sensor sensor for large-scale field monitoring, the notion of relay-assisted sensor networks has been proposed in Zhu et al. (2015), Consider aanodes typical problem of sensor fusion over sensor sensor networks has been proposed in Zhu et al. (2015), where a number of relay nodes (RNs) comprising of wireConsider typical problem of sensor fusion over sensor networks, measure an unknown vector of interest, sensor anetworks has beennodes proposed incomprising Zhu et al. of (2015), Consider anodes typical problem of sensor fusion over sensor where number of relay (RNs) wirenetworks, measure an unknown vector of interest, where a number of relay nodes (RNs) comprising of less transceivers but no sensors are deployed in the network networks, nodes measure an unknown vector of interest, obtaining noisy versions of the vector, and then report the where a number but of relay nodesare (RNs) comprising of wirewirenetworks, nodesversions measureofan unknown vector ofreport interest, less transceivers no sensors deployed in the network obtaining noisy the vector, and then the less transceivers but no sensors are deployed the network assist the data exchange between sensorin nodes (SNs), obtaining noisy versions of the the vector, and then processing report the the to samples tonoisy a remote fusion center forand further less transceivers but no sensors are deployed in the network obtaining versions of vector, then report to assist the dataOne exchange sensor nodes (SNs), samples to remote fusion for processing to assist the exchange between sensor nodes (SNs), the meters. majorbetween advantage of such of samples to a remote (Xiao fusionetcenter center for further further processing and decision making al. (2006)). For instance, in i.e., to assist the data dataOne exchange between sensor nodeskind (SNs), samples to aa making remote fusion center for further processing i.e., the meters. major advantage of such kind of and decision (Xiao et al. (2006)). For instance, in i.e., the meters. One major advantage of such kind of networks is that long distance communications between and decision making (Xiao et al. (2006)). For instance, in a power system, we need to monitor system components i.e., the meters. One major advantage of such kind of and decision making (Xiaotoetmonitor al. (2006)). For components instance, in networks is that long distance communications between aausing power system, we need system networks is that long distance communications between meters can be avoided with only a few RNs, while certain power system, we need to monitor system components meters and estimate the state vector of the entire networks is that long distance communications between a power system, we need to monitor system ofcomponents can be avoided can with only aa guaranteed. few RNs, while certain using estimate state vector meters can be with only few network specifications using meters and estimatetothe the statemeter vectormeasurements of the the entire entire meters power meters systemand according these meters can be avoided avoided can withstill onlybe a guaranteed. few RNs, RNs, while while certain certain using meters and estimate the state vector of the entire network specifications still be power system according to these meter measurements network specifications can still be guaranteed. power system according to al.these these meter measurements measurements network specifications can still be guaranteed. (Huangsystem et al. (2012); Xie et (2012)) power according to meter Some preliminary results regarding distributed sensor fu(Huang (2012); Xie et al. (Huang et et yal. al.(t) (2012); Xie et(t), al. (2012)) (2012)) preliminary results regarding distributed sensor fu= Hi θXie +w i = 1, 2, . . . , m, (1) Some Some preliminary results regarding distributed sensor fu(Huang et al. et sion in relay-assisted networks can be found in Zhu et al. i (2012); i al. (2012)) Some preliminary results regarding distributed sensor fuy = H + w ii = 1, 2, .. .. .. ,, m, (1) sion in relay-assisted networks can be found in Zhu et al. i (t) iθ i (t), y (t) = H θ + w (t), = 1, 2, m, (1) sion in relay-assisted networks can be found in Zhu et al. i i i (2015, 2016). For sensor networks without RNs, i.e., only where a linearized DC model is adopted in deriving above yi (t) = H θ + w (t), i = 1, 2, . . . , m, (1) sion in 2016). relay-assisted networks canwithout be foundRNs, in Zhu etonly al. i i (2015, For sensor networks i.e., where a linearized DC model is adopted in deriving above J (2015, 2016). For sensor networks without RNs, i.e., only SNs are present, a number of distributed solutions have where a is adopted in deriving above model, θlinearized ∈ RJ is DC the model state vector including bus voltage (2015, 2016). For sensor networks without RNs, i.e., only where a linearized DC model is adopted in deriving above SNs are present, a number of distributed solutions have J is the state vector model, θ ∈ magnitudes, R including voltage SNs are present, number of solutions been proposed in aathe literature. For example, Refs. have Das model, θ R including bus voltage angles Hi vector ∈ RJJii ×J is the bus observation are present, number of distributed distributed solutions have ×J model, and θ ∈ ∈ magnitudes, RJ is is the the state state vector including bus voltage SNs been proposed in the literature. For example, Refs. Das Ji ×J angles and H R is the observation been proposed in the literature. For example, Refs. Das i ∈ and Mesbahi (2009); Liu et al. (2013); Schizas et al. (2008); angles and magnitudes, H ∈ R is the observation matrix with J ≤ J, and w (t) is the observation noise i i∈ RJi ×J is the observation i been proposed in theLiu literature. For Schizas example, Refs. Das angles and magnitudes, Hw and Mesbahi Mesbahi (2009); et al. al. (2013); et al. al. (2008); i i (t) is the observation noise matrix with J ≤ J, and and (2009); Liu et (2013); Schizas et (2008); i Xiao et al. (2005) proposed to use consensus algorithms to matrix with J ≤ J, and w (t) is the observation noise following a Gaussian distribution with zero mean. i i and Mesbahi (2009); Liu et al. (2013); Schizas et al. (2008); matrix with J ≤ J, and w (t) is the observation noise Xiao et al. (2005) proposed to use consensus algorithms to i i following a Gaussian distribution with zero mean. Xiao et al. (2005) proposed to use consensus algorithms to do the sensor fusion. In this case, it is assumed that nodes following a Gaussian distribution with zero etsensor al. (2005) proposed to use itconsensus algorithms to following zero mean. mean. do the fusion. In this case, is assumed that nodes Recently, aa Gaussian number ofdistribution distributedwith solutions to the above Xiao do the sensor fusion. In this case, it is assumed that nodes first take a snapshot of measurements and then adopt Recently, a number of distributed solutions to the above do the sensor fusion. In this case, it is assumed that nodes take algorithms aa snapshot of measurements and thenOn adopt Recently, a of the above sensor fusion problem have been solutions reported,to name a first first of measurements and adopt the data iteratively. the Recently, a number number of distributed distributed solutions toto the above sensor fusion problem have been reported, to name aa consensus first take take algorithms a snapshot snapshot to of fuse measurements and then thenOn adopt consensus to fuse the data iteratively. the sensor fusion problem have been reported, to name few, Das and Mesbahi (2009); Kar et al. (2012); Xie consensus algorithms to fuse the data iteratively. On other hand, if the measurements of the unknown vector sensorDas fusion problem have beenKar reported, to nameXie a consensus algorithms to fuse the data iteratively. On the few, and Mesbahi (2009); et al. (2012); the other hand, if the measurements of the unknown vector few, Das and Mesbahi (2009); Kar et al. (2012); Xie et al. (2012); Schizas et al. (2008); Xiao et al. (2005); other if measurements of unknown are hand, streamingly at SNs, a natural wayvector is to few, Das and Schizas Mesbahiet (2009); Kar Xiao et al.et(2012); Xie θ et al. (2012); al. (2008); al. (2005); other hand, if the the available measurements of the the unknown vector θ are streamingly available at SNs, a natural way is to et al. (2012); Schizas et al. (2008); Xiao et al. (2005); Xu et al. (2015), where the nodes collaboratively estimate θ are available at aa natural is the consensus step and innovation stepway together et al. (2012); Schizas et al. (2008); Xiao et al.estimate (2005); combine Xu et al. (2015), where the nodes collaboratively θ are streamingly streamingly available at SNs, SNs, natural way is to to combine the consensus step and innovation step together Xu et al. (2015), where the nodes collaboratively estimate the unknown vector by means of local computations and combine the consensus and innovation step in one update, e.g., Karstep et al. (2012); Zhang andtogether Zhang Xu et al. (2015), where the nodes collaboratively estimate the unknown vector by means of local computations and combine the consensus step and innovation step together in one update, e.g., Kar et al. (2012); Zhang and Zhang the unknown vector by of local computations and in Kar (2012); Zhang and Zhu et e.g., al. (2015). Most of the aforementioned This the unknown by means means local computations and (2012); researchvector is partially fundedof by National Key Research in one one update, update, e.g., Kar et et al. al. (2012); Zhang and Zhang Zhang (2012); Zhu et al. (2015). Most of the aforementioned (2012); Zhu et al. (2015). Most of the aforementioned works assume that the communication topology is fixed This research is partially funded by National Key Research and Development of China (No. 2016YFB0901903, research is Program partially funded by National Key Research (2012); Zhu etthat al. the (2015). Most of thetopology aforementioned This works assume communication is fixed This research is partially funded by National Key Research and Program of works assume that communication topology is with perfect is not practical in realistic 2016YFB0901901), of China under(No. the 2016YFB0901903, grants 61633017, and Development Development NSF Program of China China (No. 2016YFB0901903, works assumechannels, that the the which communication topology is fixed fixed with perfect channels, which is not practical in realistic and Development NSF Program of China (No. 2016YFB0901903, 2016YFB0901901), of China under the grants 61633017, with perfect channels, which is not practical in realistic applications. Actually, communication links among nodes U1405251, 61603251, 61521063, 61603205, and Shandong 2016YFB0901901), NSF of Chinaand under the grants 61633017, with perfect channels, which is not practical in realistic 2016YFB0901901), NSF of China under the grants 61633017, applications. Actually, communication links among nodes U1405251, 61603251, 61521063, and Shandong applications. Actually, among nodes are usually unidirectional since nodes links broadcast difProvincial Foundation of Chinaand under grant U1405251, Natural 61603251,Science 61521063, and 61603205, 61603205, and Shandong applications. Actually, communication communication links amongat nodes U1405251, 61603251, 61521063, and 61603205, and Shandong are usually unidirectional since nodes broadcast at difProvincial ZR2015FQ012 are usually unidirectional since nodes broadcast at Provincial Natural Natural Science Science Foundation Foundation of of China China under under grant grant are usually unidirectional since nodes broadcast at difdifProvincial Natural Science Foundation of China under grant
ZR2015FQ012 ZR2015FQ012 ZR2015FQ012 Copyright © 2017, 2017 IFAC 14849 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 14849 Copyright ©under 2017 responsibility IFAC 14849 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 14849 10.1016/j.ifacol.2017.08.1847
Proceedings of the 20th IFAC World Congress Shanying Zhu et al. / IFAC PapersOnLine 50-1 (2017) 14284–14289 Toulouse, France, July 9-14, 2017
ferent power levels and may have different interference patterns. Moreover, communications links between nodes generally suffer from noise and errors and are subject to channel fading. These negative effects make the extension highly non trivial in terms of both the algorithm design and theoretical analysis. In our previous work Zhu et al. (2016), we proposed a distributed algorithm to solve the problem of sensor fusion if the unknown variable is scalar. In this paper, we follow the same line and aim to extend it to the more general vector case (1). The paper is organized as follows. In Section 2, we formulate our problem. In Section 3, we propose a distributed solution to the problem of sensor fusion in relay-assisted networks, called DINS, followed by the performance analysis in Section 4. Section 5 presents simulation results demonstrating the effectiveness and efficiency of DINS. 2. PROBLEM FORMULATION Consider a relay-assisted network consisting of SNs and RNs that monitors a field of interest. There are n local nodes observing the unknown vector θ ∈ RJ with the measurements given by (1). For notational simplicity, we label the SNs from 1 to m, and RNs from m + 1 to n. We assume that all wi (t) are independent of t, but may be correlated across nodes, and supt≥0 E{ w(t) 2 } < ∞, where w(t) = [w1 (t)T , . . . , wm (t)T ]T . Generally, Ji ≤ J, for each i, this means that each SN only have limited measurement and θ is not locally observable at each SN. As such, cooperation betweens the nodes is then necessary to solve such sensor fusion problem. We assume half-duplex operation of nodes, i.e., nodes can not simultaneously transmit and receive. The nodes use the analog-and-forward scheme for the information exchange, where data are scaled and then transmitted without any coding (Liu et al. (2007)). It is convenient to model the communication network as a dynamic directed graph G(t) = (V, E(t)), where V = IS ∪ IR , IS and IR are the sets of SNs and RNs, respectively, E(t) ⊂ V ×V collects all communication links between nodes at time instant t. The directed edge (i, j) ∈ E(t) means that node i can transmit data to node j directly at time instant t. As such, each node i has a local neighborhood Ni (t), which is the set of nodes from which it can receive data at time instant t. The SN and RN neighbors of node i are denoted as NiS (t) = Ni (t) ∩ IS and NiR (t) = Ni (t) ∩ IR , respectively.
We consider the scenario where transmissions suffer from fading and additive noise. With the analog-and-forward scheme, each node i amplifies its data with αi (t) first and then broadcasts to its neighbors. At the receiver side, the data received by node i at time instant t is zi (t) = αj (t)hij (t)xj (t) + vij (t) , (2) j∈NiS (t)∪NiR (t)
where xi (t) is the estimate of θ at node i, hij (t) is the random fading coefficient of link (j, i) with bounded variance, vij (t) is the receiver noise in the transmission from node j to node i, which is assumed to be of zero mean and bounded variance. Eq. (2) describes the signal reception from the neighbors to each node. In the subsequent analysis, we consider the case where all hij (t) are independent across receivers and across time instants, and the receiver
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noise vij (t) are uncorrelated from one time instant to the next and uncorrelated with hij (t). 3. DISTRIBUTED SENSOR FUSION SCHEME OVER FADING CHANNELS In this section, we provide a distributed sensor fusion scheme for relay-assisted sensor networks subject to fading. In the scheme, each node only relies on local information, which is confined to one-hop communications. 3.1 Development of DINS Our scheme is motivated by the observation that both the own measurement (1) and the local information (2) received from neighboring nodes can be regarded as some kind of measurements. Actually, the local information (2) depends on θ through the local estimates xj (t), which follows an indirect relation. In this way, (2) can be treated as virtual measurement. This motivates us to propose the following Distributed INnovation based Scheme (DINS) by fusing these two kinds of measurements: • At the end of one round of transmissions, each SN i ∈ IS updates its estimate xi (t) as xi (t + 1) = xi (t) + ρ(t)HTi (yi (t) − Hi xi (t)) + ρ(t)(zi (t) − bi (t)xi (t)),
(3)
where ρ(t) > 0 is the decaying weight for attenuating the noise, and bi (t) is the factor introduced to compensate for the effect of channel fading. • As for RNs, we denote the reception at each RN i ∈ IR as its estimate, i.e., xi (t) = zi (t).
(4)
Recursions (3) and (4) constitute our distributed sensor fusion scheme. It is noted that with this scheme each node only need local information to update its estimate. Of special note is the formulation of (3), where we divide it into two parts, one is the innovation yi (t) − Hi xi (t) w.r.t. its own data and the other is the innovation zi (t) − bi (t)xi (t) w.r.t. received data from neighboring nodes. Similar forms can also be found in Kar et al. (2012); Stankovi´c et al. (2011); Zhang and Zhang (2012). However, the superposition in the multiple access scheme (2) would introduce an undesired non-vanishing noise term contributed by the channel fading, which makes the aforementioned existing schemes not applicable in our case. This is why we introduce the compensation factor bi (t) in (3). To make the presentation clearer, we consider the special case that each SN can directly transmit data to other SNs or via at most one-hop RN. Let −αj (t)hij (t), j ∈ NiS (t), 0, j ∈ NiS (t), , i ∈ IS , pij (t) = pij (t), j = i, − j=i
stack pij (t), xi (t) into vectors pi (t) = [pi1 , pi2 , . . . , pim ]T and x(t) = [x1 (t)T , x2 (t)T , . . . , xm (t)T ]T ∈ RmJ , respectively, one then obtains
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zi (t) − bi (t)xi (t) = γi (t)xi (t) − (pi (t)T ⊗ IJ )x(t) + αj (t)hij (t)(pj (t)T ⊗ IJ )x(t) j∈NiR (t)
(5) + ρ(t)vi (t), where γi (t) = j∈N R (t) αj (t)hij (t) k∈N S (t) αk (t)hjk (t)+ i j j∈NiS (t) αj (t)hij (t) − bi (t) and vi (t) = j∈Ni (t) vij (t) + α (t)h (t) v (t). With the above noR j ij jk j∈Ni (t) k∈Nj (t) tations, we rewrite the proposed DINS in a compact form x(t + 1) = ImJ − ρ(t)(HT H + (L(t) − Γ(t)) ⊗ IJ ) x(t) + ρ(t) HT y(t) + v(t) , (6)
where y(t) = [y1 (t)T , y2 (t)T , . . . , ym (t)T ]T , v(t) = [v1 (t)T , v2 (t)T , . . . , vm (t)T ]T , H = diag{H1 , H2 , . . . , Hm }, Γ(t) = diag {γ1 (t), γ2 (t), . . . , γm (t)}, and p1 (t)T + αj (t)h1j (t)pj (t)T j∈N1R (t) .. . L(t) = . T pm (t)T + αj (t)hmj (t)pj (t) R (t) j∈Nm
In summary, the total number of time slots for a relayassisted network is m + 2 − m , where m is the number of neighbors of all RNs. In this way, n − m − 2 + m slots can be saved compared with the traditional node-based TDMA scheme in Das and Mesbahi (2009); Ergen and Varaiya (2010). 3.3 Design of algorithm parameter We design the algorithm parameters αj (t) and bi (t) by bringing the effect of fading on the algorithm under an unbiased level. To this end, we require that the channel ¯ ij E{hij (t)} > 0, is available statistics, i.e., the mean h at the receiver i. The next result gives a choice of such parameters with desired properties. The next result provides a choice of such parameters, which shows that the fading only leads to an unbiased effect. Proposition 1. For each node i = 1, 2, . . . , n, let αi (t) = √ 1/ m, and for each SN i ∈ IS we design bi (t) as 1 ¯ ij + 1 ¯ ij ¯ jk , h h h bi (t) = √ m m S R S j∈Ni (t)
j∈Ni (t)
k∈Nj (t)
3.2 Implementation of DINS
then E{Γ(t)} = 0.
In this section, we outline how the proposed DINS may be implemented over wireless networks. First, note that DINS is a synchronous algorithm, that is, the nodes updates their states synchronously at the end of one round of transmissions. One popular protocol suitable for such algorithms is the TDMA scheme, e.g., Das and Mesbahi (2009); Ergen and Varaiya (2010). In the traditional TDMA scheme, the time axis is divided into slots with each slot assigned to one user, and users are permitted to transmit only during their assigned slots. One critical aspect is the low scalability, where the number of slots is n, growing linearly with the number of nodes. This would jeopardize the efficiency of TDMA scheme in practical scenarios, especially for largescale networks.
Proof. The proof is straightforward, so we omit the details, see also Zhu et al. (2016).
In this section, we adopt a hybrid TDMA scheme to implement DINS by exploiting both the broadcast property and the superposition property of wireless channels. Our key insight is that each node does not need to know the exact data from its neighboring nodes. Rather it only needs to know the sum. That is the interference property of the multiple access scheme (2) may be exploited to our advantage. Similar idea has already been used in Nazer et al. (2011). The time slots of proposed hybrid TDMA scheme are assigned according to the following rules: 1) All the neighbors of RNs share the first slot. They broadcast simultaneously to the neighbors. According to (2), the received data at RNs are αj (t)hij (t)xj (t) + vij (t) . zR i (t) = j∈NiS (t)
2) All the RNs share the ensuing one slot. Accordingly, the received data at SN neighbors of RNs are αj (t)hij (t)xj (t) + vij (t) . zSi (t) = j∈NiR (t)
3) Each of the remainder SNs consumes one slot to transmit its data to the neighbors.
4. PERFORMANCE ANALYSIS OF DINS In this section, we will examine the convergence properties of DINS in the presence of fading transmissions. First, let us define the estimation error ei (t) = xi (t) − θ for each SN i ∈ IS . Stack them in the column vector e(t) = [e1 (t)T , . . . , em (t)T ]T , and define the matrix A(t) = ImJ − ρ(t)(HT H + L(t) ⊗ IJ ). Consequently, subtracting θ m = 1m ⊗ θ from both sides of (6), one obtains e(t + 1) = A(t) + ρ(t)Γ(t) ⊗ IJ e(t) + ρ(t)Γ(t)θ m + ρ(t)(HT w(t) + v(t)), (7) where we use the property that L(t)1m = 0 in terms of the definition of L(t).
It is noted that the coefficient matrix A(t) + ρ(t)Γ(t) ⊗ IJ of (7) is generally nonsymmetric, which makes the approaches for undirected graphs in Xie et al. (2012); Kar et al. (2012); Zhang and Zhang (2012) not applicable here. Actually, the coexistence of SNs and RNs in the network leads to a nonsymmetric L(t) even if all communication links between the nodes are bidirectional. On the other hand, the eigenvalues of A(t) + ρ(t)Γ(t) ⊗ IJ can be sufficiently close to 1 with large t, if we use decaying weight limt→∞ ρ(t) = 0. These two aspects present a challenge of using the classical results of linear systems (see Rugh (1996)) to investigate the convergence property of (10). Before moving onto our main results, we need two additional concepts. Definition 2. The unknown parameter θ is collaboratively m observable in one time-step, if i=1 HTi Hi is of full rank. Definition 3. Graph G(t) is said to be uniformly strongly connected (USC), if there exists a constant B > 0 such
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(k+1)B−1 that the union graph G[kB,(k+1)B) = V, kB E(t)
is strongly connected 1 for all nonnegative integers k.
Definition 2 means that θ can be collaboratively recovered by all the nodes, even if it is not locally observable at each node. Definition 3 is introduced to ensure the repeated influence of the nodes on each other. 4.1 Fixed network In this case, the communication network is fixed, i.e., G(t) = G, ∀t. We will introduce a transformation utilizing the spectral properties of graph G, based on which we obtain the following convergence result. Lemma 4. For a strongly connected graph G, the left ¯ eigenvector √ ξ of L w.r.t. the eigenvalue 0 is positive. Denote E = diag{ξ1 , ξ2 , . . . , ξm }, then F = 2HT H + √ √ −1 √ −1 T √ ¯ E + E L ¯ E) ⊗ IJ is positive definite if θ ( EL ¯ = E{L(t)}. is collaboratively observable, where L Proof. The proof is a direct consequence of Theorem 2 and Lemma 2 in Zhu et al. (2015). Theorem 5. For a strongly connected graph G, if θ is collaboratively observable, and ∞ ρ(t) = ∞, (8) t=0
then the estimate sequence {xi (t)}t≥0 generated DINS at each SN i ∈ IS is asymptotically unbiased. lim E{xi (t)} = θ, ∀i = 1, 2, . . . , m. t→∞
If further,
∞ t=0
ρ(t)2 < ∞,
(9)
t→∞
Proof. Asymptotic Unbiasedness: Taking expectations of both sides of (7) and using Proposition 1, we can obtain ¯ e(t), ¯(t + 1) = A(t)¯ e (10) ¯ ¯(t) = E{e(t)} √ where e and A(t) = E{A(t)}. By Lemma 4, we know that D = Ξ ⊗ IJ is nonsingular. Thus utilizing the transformation D¯ e(t), we have D¯ e(t + 1) = −1 ¯ DA(t)D D¯ e(t). Based on Lemma 4, and noting that (8) is satisfied, we use the same line of arguments in Zhu et al. (2015) to show that limt→∞ D¯ e(t) = 0, from which the asymptotic unbiasedness follows, since D is nonsingular. Strong Consistency: Letting V (t) = ˆ e(t)2 = De(t)2 , given the historical information {V (s) : s ≤ t} we obtain from (7) that E{V (t + 1)|V (s) : s ≤ t} = E D(A(t) + ρ(t)Γ(t) ⊗ IJ )e(t)2 |V (s) : s ≤ t + ρ(t)2 E{DΓ(t)θ m 2 } + ρ(t)2 E{DHT w(t)2 } + E{Dv(t))2 }
− 2ρ(t)2 e(t)T E{((L(t) − Γ(t)) ⊗ IJ )T DΓ(t)}θ m , (11)
1
where we use the facts that {wi (t)}, {vi (t)}, {hij (t)} are independent from e(t), and E{Γ(t)} = 0 by Proposition 1. In lieu of the independence assumption on {wi (t)}, {vi (t)} and {hij (t)}, one has E D(A(t) + ρ(t)Γ(t) ⊗ IJ )e(t)2 |V (s) : s ≤ t ˆ(t)T ImJ − ρ(t)E{B(t)} + ρ(t)2 E{C(t)} e ˆ(t), =e
where B(t) = 2HT H + D(L(t) ⊗ IJ )D−1 + D−1 (L(t)T ⊗ IJ )D and C(t) = D−1 (HT H + L(t)T ⊗ IJ )D2 (HT H + L(t) ⊗ IJ )D−1 . It is easy to see that E{B(t)} = F and E{C(t)} is bounded by recalling that {wi (t)}, {vi (t)} and {hij (t)} all have bounded variances. It thus follows that there is a t∗ > 0 such that ∀t ≥ t∗ , ˆ(t)T E{B(t)}ˆ e(t) ≤ 1 − ρ(t)λmin (F) + ρ(t)2 O(1) V (t), e where we use the fact that the minimum eigenvalue λmin (F) of F is positive by Lemma 4. On the other hand, it follows from Proposition 1 and the fact that {wi (t)}, {vi (t)} and {hij (t)} all have bounded variances that E{DΓ(t)θ m 2 }, E{DHT w(t)2 } and E{Dv(t))2 } are bounded, i.e., O(1). This along with the inequality 2zT1 z2 ≤ z1 2 + z2 2 for any two vectors z1 , z2 , and the H¨older inequality yields e(t)T E{((L(t) − Γ(t)) ⊗ IJ )T DΓ(t)}θ m ≤ O(1)V (t). Substituting the above relations into (11) implies for all t ≥ t∗ , E{V (t + 1)|V (s) : s ≤ t} ≤ 1 + ρ(t)2 O(1) V (t)
− ρ(t)λmin (F)V (t) + O(1)ρ(t)2 . It is noted that t=t∗ ρ(t)2 < ∞ by (9). Hence, by the Robbins-Siegmund theorem (see Polyak (1987)) we know that there is a random variable v ≥ 0 such that ∞ ρ(t)V (t) < ∞ = 1. P lim V (t) = v ∞
t→∞
∞
then {xi (t)}t≥0 is also strongly consistent, P lim xi (t) = θ = 1, ∀i = 1, 2, . . . , m.
A graph is said to be strongly connected if there is a sequence of directed edges such that any vertex can be reached from another one.
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t=t∗
Recalling that t=t∗ ρ(t) = ∞ by (8), it thus follows that v = 0, from which we have P{limt→∞ V (t) = 0} = 1. Therefore, P{limt→∞ De(t) = 0} = 1. Remember that D is nonsingular, we complete the proof. 4.2 Time-varying network
For the sake of brevity, we assume that HTi Hi is diagonal and rank deficient for each SN i ∈ IS . This is saying that each SN i ∈ IS only observes part of the components of θ.
Since the network G(t) is time-varying and directed, a common Lyapunov function for (7) is generally not available in this case. Hence, the method developed for fixed networks in Section 4.1 fails. To address this issue, we consider an appropriate reordered space, built on which we can deal with asymmetry in a tractable framework.
Consider the error dynamics (10), for each component l = 1, 2, . . . , J, let reorder the state {¯ e1,l (t), e¯2,l (t), . . . , e¯m,l (t)} in such a way that r1,l (t) ≥ r2,l (t) ≥ · · · ≥ rm,l (t), where ri,l (t) = e¯it ,l (t) and {1t , 2t , . . . , mt } is a permutation of {1, 2, . . . , m} at time t. The next result shows that the ordered states will converge to consensus for each component l. Proposition 6. Suppose that the graph G(t) is USC, then for each l = 1, 2, . . . , J, we have lim (ri,l (t) − rj,l (t)) = 0, ∀i, j = 1, 2, . . . , m.
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Proof. It is easy to see that there is a permutation matrix ¯ ⊗ IJ )PT = IJ ⊗ L(t) ¯ P such that P(L(t) and PHT HPT = diag{Λ1 , Λ2 , . . . , ΛJ }, where Λi is a diagonal matrix for each i = 1, 2, . . . , J, since HT H is diagonal. Let us denote r(t) = P¯ e(t) and divide it 1
1
r
1
0.5
r1
0.8
r3
r3
0.6
0 −0.5
3
1
0.2
−1 −1.5 0
r (t) is nonincreasing
0.4
r (t) is nondecreasing
200 400 Iteration index
600
0 0
(a) r1 (t0 ) ≤ 0
200 400 Iteration index
r1
0.5
5. SIMULATION RESULTS
600
(b) r3 (t0 ) ≥ 0 2.5
1
r3
Consider a sensor network with n = 50 nodes using the random geometric graph model and two nodes are linked by two unidirectional links if their distance is less than log n/n. We assign 5 critical nodes without which the network will separate into groups as RNs. Afterwards, we randomly delete 20% of unidirectional links to obtain an asymmetric network. The time-varying nature of G(t) is simulated by letting 10% of links fail at each time instant.
r1
2
r3
1.5
0 −0.5
1
r (t) is nondecreasing 3
−1 −1.5 0
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V˜ (t) = E{Γ(t)θ m 2 } + E{HT w(t)2 } + E{v(t))2 } − ¯ 2e(t)T E{((L(t) − Γ(t)) ⊗ IJ )T Γ(t)}θ m . Since 1T L(t) = ¯ T ⊗ ¯ ¯ L(t)1 = 0, it can shown that L(t) ⊗ IJ + L(t) IJ is positive semidefinite, which similar to Lemma 4 ˜ implies that B(t) is positive definite. Furthermore, we ˜ find that E{C(t)} and V˜ (t) are all bounded following the similar arguments in the proof of Theorem 5. Applying the Robbins-Siegmund theorem (cf. Polyak (1987)) again leads to the conclusion P{limt→∞ e(t) = 0} = 1, from which the theorem follows.
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Fig. 1. Behaviors of the ordered states r1 (t), r3 (t) for a ring graph SN 1→SN 2→RN 1→SN 3→SN 1. into r(t) = [r1 (t)T , r2 (t)T . . . , rJ (t)T ]T , where rl (t) ∈ Rm , ∀l = 1, 2, . . . , J, then we derive from (10) that ¯ rl (t). (12) rl (t + 1) = Im − ρ(t)(Λl + L(t)) In lieu of (12), we can show that either r1,l (t) is nonin∞ creasing and converges to a nonnegative value r1,l ≥ 0 or rm,l (t) is nondecreasing and converges to a nonpositive ∞ value rm,l ≤ 0. The proof is similar to that of Zhu et al. (2016), whose intuitive explanation is illustrated in Fig. 1. Next, it suffices to show that ri,l (t) converges to the same value for all i = 1, 2, . . . , m. By careful examination of (12) and following similar arguments as those for the scalar case in Zhu et al. (2016), we can show that ri,l (t), ∀i, converges to the same value under the assumption of USC. The proposition thus follows. Theorem 7. Suppose that the graph G(t) is USC and ¯ ¯ 1T L(t) = L(t)1 = 0. If θ is collaboratively observable and (8), (9) are satisfied, then the estimate sequence {xi (t)}t≥0 generated by DINS at each SN i ∈ IS is asymptotically unbiased and strongly consistent. Proof. Asymptotic Unbiasedness: By applying a wellknown convergence result on recurrent inequalities (Polyak, 1987, p.45), we can show that limt→∞ rl (t) = 0, and thus ¯(t) = 0 in view of the fact that P is nonsingular. limt→∞ e We omit the details due to page limitation. Strong Consistency: Using V (t) = e(t)2 as a Lyapunov candidate, given the historical information {V (s) : s ≤ t} we obtain from (7) that E{V (t + 1)|V (s) : s ≤ t} ˜ = e(t)T ImJ + ρ(t)2 E{C(t)} e(t) ˜ − ρ(t)λmin (B(t))V (t) + ρ(t)2 V˜ (t), (13) T T ˜ ¯ ¯ where B(t) = 2H H + L(t) ⊗ IJ + L(t) ⊗ IJ and ˜ C(t) = (HT H + L(t)T ⊗ IJ )(HT H + L(t) ⊗ IJ ) and
The state of the field is represented by the vector θ ∈ R10 . Each component corresponds to a random field sample, which is generated by a zero-mean Gaussian random variable of variance 20. Moreover, each node is associated with only one component of θ, i.e., Hi = [0, . . . , 1, . . . , 0], where the only nonzero 1 is the (i mod 10) + 1-th entry, for all i. Obviously, θ ∈ R10 is collectively observable, m since i=1 HTi Hi = 4I10 + diag{0, 1, 1, 1, 1, 1, 0, . . . , 0} is nonsingular. The measurement noise wi (t) is i.i.d. with unit variance. We consider the Rayleigh channel fading between nodes which is assumed to be i.i.d. with parameter σr = 1. The receiver noise are drawn i.i.d. from a Gaussian distribution with variance σ 2 = 0.5. Parameters αi (t) and bi (t) are designed according to Proposition 1. We use the decaying weight ρ(t) = 2(t + 1)−0.7 . It can be shown that ρ(t) satisfies the conditions of Theorem 5. DINS is implemented with random initial values xi (0) ∈ [θ − 1, θ + 1], ∀i.
The first simulation is to demonstrate the performance of unbiasedness of DINS. In Fig. 2(a), we plot the ensemble average of the estimate xi (t) for SNs with minimum and maximum neighbors, respectively, where each curve is averaged over 10 independent runs. The theoretical limit, i.e., the true value θ is also plotted for comparison purpose. As we can notice from the figure, each ensemble average closely approaches θ, although θ is not locally observable at each SN. This corroborates the theoretical result obtained in Theorem 7, i.e., DINS is capable of enabling all the nodes to collaborate. To quantify the performance of the proposed algorithm DINS, we use the the mean square error averaged over all 45 1 2 SNs emse (t) = 45×10 i=1 xi (t) − θ as a measure. Simulations are presented by averaging over 10 independent runs. The average mean square error is shown in Fig. 2(b), from which we note that emse (t) decreases gradually to 0, thus conforming the theoretical result of Theorem 7. We also plot the results of mean square error of DINS for four different choices of ρ(t) = 2(t + 1)η , i.e., η = −0.7, −1, −2, in Fig.2(c). For η = −1 and η = −0.7,
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Fig. 2. (a) Ensemble average of estimates for SNs with maximum and minimum neighbors, respectively; (b) Average mean square error of DINS; (c) Average mean square error of DINS for different choices of ρ(t) = 2(t + 1)−0.7 , 2(t + 1)−1 , 2(t + 1)−2 . both of them satisfy the conditions of Theorem 7, and the convergence is observed. It is worth noting that the convergence for η = −1 is quite slow. However, for the case η = −2, the decay is too fast to obtain a small error and a large residual occurs. 6. CONCLUSIONS We considered the problem of distributed estimation of an unknown vector in relay-assisted sensor networks. Particularly, we focused on the effect of time-varying asymmetric topology due to link failure and the fading channels. We proposed a distributed sensor fusion scheme DINS based on innovation techniques with a hybrid TDMA scheme enabling its implementation. Theoretically, we showed that DINS can effectively solve the sensor fusion problem in noisy environments with guaranteed properties of asymptotic unbiasedness and consistency for both fixed and time-varying topologies. We also presented simulations to validate its effectiveness and efficiency. REFERENCES Das, A.K. and Mesbahi, M. (2009). Distributed linear parameter estimation over wireless sensor networks. IEEE Transactions on Aerospace and Electronic Systems, 45(4), 1293–1306. Ergen, S.C. and Varaiya, P. (2010). TDMA scheduling algorithms for wireless sensor networks. Wireless Networks, 16(4), 985–997. Huang, Y.-F., Werner, S., Huang, J., Kashyap, N., Gupta, V. (2012). State estimation in electric power grids. IEEE Signal Processing Magazine, 29(4), 33–43. Kar, S., Moura, J.M.F., and Ramanan, K. (2012). Distributed parameter estimation in sensor networks: Nonlinear observation models and imperfect communication. IEEE Transactions on Information Theory, 58(6), 3575–3605. Liu, K., Gamal, H.E., and Sayeed, A. (2007). Decentralized inference over mutiple-access channels. IEEE Transactions on Signal Processing, 55(7), 3445–3455. Liu, S., Li, T., Xie, L., Fu, M., and Zhang, J. (2013). Continuous-time and sampled-data based average consensus with logarithmic quantizers. Automatica, 49(11), 3329–3336.
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