Fully distributed clock synchronization in wireless sensor networks under exponential delays

doi:10.1016/j.sigpro.2016.02.007

Signal Processing

Volume 125, August 2016, Pages 261-273

https://doi.org/10.1016/j.sigpro.2016.02.007 Get rights and content

Highlights

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Network-wide clock synchronization problem under exponential message delays is solved.
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The problem is cast into a linear programming problem.
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ADMM is exploited to obtain a fully-distributed algorithm.
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Closed-from expressions are available for each update step in the algorithm.
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The proposed algorithm is guaranteed to converge to the global optimal solution.

Abstract

In this paper, we study the global clock synchronization problem for wireless sensor networks under unknown exponential delays in the two-way message exchange mechanism. The joint maximum likelihood estimator of clock offsets, clock skews and fixed delays for the network is formulated as a linear programming (LP) problem. Then, based on the alternating direction method of multipliers (ADMM), we propose a fully distributed synchronization algorithm which has low communication overhead and computation cost. Simulation results show that the proposed algorithm achieves better accuracy than consensus algorithm and the distributed least squares algorithm, and can always converge to the centralized optimal solution.

Introduction

A variety of applications in wireless sensor networks (WSNs), such as object tracking, machine health monitoring, home appliance control, etc. [1], [2], require cooperative execution of distributed tasks amongst a set of sensor nodes. To execute these tasks, nodes are usually required to run on a common time frame. However, this is far from trivial due to many factors, including imperfection of oscillators, environmental changes, and unknown delays in message delivery. Therefore, global clock synchronization constitutes an important prerequisite in many services of WSNs.

One critical component of clock synchronization is the modeling of the random network delays that perturb the message-exchange process. Existing probabilistic models for random delays in WSNs include Gaussian, exponential, Gamma, Weibull, and log-normal [3], [4], [5]. Not a single model dominates, because different models are justified in different applications. For example, it was observed in [6] that when the point-to-point hypothesis reference connection topology is of interest, the cumulative link delay in WSN is approximately represented as a single server M/M/1 queue, and the random delays should be modeled as exponential random variables. Furthermore, the minimum link delay algorithm [6], which was later derived in [7] based on the assumption of exponentially distributed network delays, shows good performance in practice. In addition, among all distributions with support in $[0, + \infty)$ and a fixed mean, the exponential distribution has the maximum entropy, and thus least informative. Hence the assumption of exponential delay distribution is worth investigating.

The other crucial challenge in clock synchronization is to design algorithms enabling global clock calibration among all the nodes in the network. Earlier protocols were mainly based on pairwise synchronization. More specifically, before synchronization, a particular network structure (spanning tree or cluster) must be built, and then layer-by-layer pairwise synchronization was executed starting from a reference node. As an important component in this approach, pairwise synchronization under exponential delay distribution has been extensively studied in the past decade. For example, the maximum likelihood estimator (MLE) is studied in [7], [8], while the linear unbiased estimator using order statistics (BLUE-OS) and the minimum variance unbiased estimator (MVUE) are derived in [9] under asymmetric exponential distributed delays in uplink and downlink. Recently, max-product algorithm is used in [10], [11] to derive a message passing method for the clock offset tracking in the presence of exponential distributed random delays. Unfortunately, the clock skew is not considered in these works, resulting in potentially frequent re-synchronization. To avoid frequent offset resetting, skew compensation is adopted for a number of synchronization algorithms. The MLE for joint clock offset and skew estimation has been derived in [12], [13], while a low-complexity MLE has been proposed in [14], [15]. Furthermore, an approximate MLE for joint offset and skew estimation has been reported in [16].

However, in such pairwise-based synchronization approach, not only large communications overhead are required to maintain the specific network structure, synchronization errors also accumulate rapidly as distance from reference node increases. This hinders the scalability of the synchronization protocol and reduces its robustness to changes in the network topology. To relieve these problems, more recent synchronization algorithms were designed to work in a fully distributed way. Specifically, each node calibrates its own clock by exchanging messages with its direct neighbors and performing computations locally, with no special network structure maintained. Existing fully distributed synchronization schemes can be categorized into pulse-coupled based and packet-coupled based. For the pulse-coupled based approach, physical pulses are transmitted from different nodes to achieve a unified clock ticking rhythm [17], [18]. On the other hand, for packet-coupled based synchronization, clock reading at each node is encoded in time stamps and transmitted as data packets to its neighbors. Within the packet-coupled based category, two major ideas have been frequently used: consensus principle [19], [20], [21], [22] and factor graph based statistical inference [23], [24], [25]. Unfortunately, large mean-square-error (MSE) occurs in estimates from consensus based algorithms since the fixed message delays are not explicitly considered. Furthermore, for existing factor graph based algorithms, either only clock offset is considered [23], or the algorithms are derived explicitly for Gaussian delays [24], [25].

In this paper, based on the two-way message exchange mechanism, a network-wide joint estimator of clock offsets, clock skews and fixed delays is derived under exponential delay model. The joint maximum likelihood estimation problem is first cast into a linear programming (LP) problem, and then a distributed solution is derived based on alternating direction method of multipliers (ADMM) [26], [27]. Exploiting the special structure of the derived solution, further improvement in terms of computation efficiency is proposed. Both theoretical analysis and simulation results show that the proposed distributed algorithm approaches the MSE performance of the centralized LP solution.

The rest of the paper is organized as follows. In Section 2, the system model is first described, and then the MLE for global clock synchronization problem is cast as a LP-problem. In Section 3, a distributed synchronization algorithm based on ADMM is derived. Simulation results are presented in Section 4. Finally, conclusions are drawn in Section 5.

Notation: The operator $Tr {A}$ takes the trace of matrix $A$ , and $1_{N}$ denotes the vector of N ones, while $I_{N}$ represents an $N \times N$ identity matrix. $diag (\cdot)$ denotes the block diagonal concatenation of input arguments. Superscript ${(\cdot)}^{T}$ denotes the transpose operator. $N_{i}$ denotes the set of neighbors of node S_i and $| N_{i} |$ indicates the number of elements in set $N_{i}$ . Finally, $∥ x ∥_{2}$ represents the Euclidean norm of vector $x$ .

Section snippets

System model and maximum likelihood estimator

Consider a network with N sensor nodes ${S_{1}, S_{2}, \dots, S_{N}}$ . These sensors are randomly distributed in the field and can be self-organized into a network¹ by establishing connections between neighbor nodes lying within each other׳s communication range. An example of 25 sensor nodes is shown in Fig. 1, where each edge represents the ability to transmit and receive packets between the

Distributed clock synchronization algorithm

First, we transform the inequality constraints in (7) into equality constraints by introducing nonnegative slack variables [27], [31]. In particular, with the definitions $x_{i} = {[α_{i} γ_{i}]}^{T}$ , $a_{i} = {[\sum_{j \in N_{i}} (T_{s}^{{i, j}} + T_{r}^{{j, i}}) 0]}^{T}$ , $B^{{i, j}} = [\begin{matrix} T_{1, 1}^{{i, j}} & - T_{4, 1}^{{i, j}} \\ - 1 & 1 \end{matrix} \dots {\begin{matrix} T_{1, K}^{{i, j}} & - T_{4, K}^{{i, j}} \\ - 1 & 1 \end{matrix}]}^{T}, E^{{i, j}} = [\begin{matrix} - T_{2, 1}^{{i, j}} & T_{3, 1}^{{i, j}} \\ 1 & - 1 \end{matrix} \dots {\begin{matrix} - T_{2, K}^{{i, j}} & T_{3, K}^{{i, j}} \\ 1 & - 1 \end{matrix}]}^{T},$ (7) can be written into a LP form as $\min_{{x_{i}}_{i = 1}^{N}, {d_{ij}, w_{ij}}_{i = 1, j \in N_{i}}^{N}} \sum_{i = 2}^{N} a_{i}^{T} x_{i} + \sum_{i = 1}^{N} \sum_{j \in N_{i}} (- 2 K) d_{ij} + \sum_{i = 1}^{N} \sum_{j \in N_{i}} I_{D} (d_{ij}) + \sum_{i = 1}^{N} \sum_{j \in N_{i}} I_{W} (w_{ij}) s . t . B^{{i, j}} x_{i} + E^{{i, j}} x_{j} + d_{ij} 1_{2 K} + w_{ij} = 0, (i, j) \in L,$

Simulation results and discussions

In this section, numerical simulations will be presented to assess the performance of the proposed distributed clock synchronization algorithm. The measure of parameter estimation fidelity at iteration m is the average mean squared error (AMSE) of clock offsets and clock skews over the whole network: $AMSE (ν (m)) = \frac{1}{N - 1} \sum_{i = 2}^{N} {({\hat{ν}}_{i} (m) - ν_{i})}^{2},$ where $ν \in {θ, β}$ . In all figures except Fig. 7, network of 25 nodes are randomly deployed in an area $5 m \times 5 m$ , with each node having communication radius 1.5 m (an example

Conclusions and future works

Global clock synchronization for WSN in the presence of unknown exponential delays was investigated under the two-way message exchange mechanism. The MLE for the joint estimation of clock skews, clock offsets and fixed delays was cast into a linear programming problem. A distributed message-passing synchronization algorithm based on ADMM was derived. The proposed algorithm requires communications only between neighboring sensors and the communication overhead was shown to be low. Simulation

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Most existing clock synchronization algorithms in wireless sensor networks require full timestamp information to estimate clock skew and offset so as to maintain the same time scale between nodes. However, in many scenarios, with the purpose of decreasing energy consumption, reducing communication overhead, or improving security, timestamps are not exchanged in synchronization process, resulting in insufficient access to all time information by nodes to be synchronized. Hence, how to jointly estimate clock skew and offset with limited observation is a significant challenge. In order to solve this issue, this paper proposes a partial timestamp synchronization (PTS) scheme, which can implement complete synchronization function only according to the local timestamps and the timestamp relationships obtained by the basic synchronization protocol. Then, based on PTS, a joint Maximum Likelihood Estimator is developed to estimate clock skew and offset when stochastic delays follow Gaussian distribution. In addition, a Best Linear Unbiased Estimator for PTS with exponential random delays is presented, which can also estimate clock skew and offset simultaneously. Finally, simulations are conducted to illustrate the effectiveness of the proposed algorithms.
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Thus far, there have been considerable amount of algorithms provided in the literature for the time synchronization problem in WSNs. Distributed time synchronization, as an important category of time synchronization algorithms, has received great concern, due to its improved robustness to node failures and dynamic network topology (Barooah & Hespanha, 2007; Freris, Borkar, & Kumar, 2009; Freris, Graham, & Kumar, 2011; Giridhar & Kumar, 2006; Luo, Cheng, & Wu, 2016; Simeone, Spagnolini, Bar-Ness, & Strogatz, 2008; Stanković, Stanković, & Johansson, 2018). In particular, inspired by the consensus theory in multi-agent systems, one promising area in distributed synchronization for WSNs, called consensus-based time synchronization (CBTS), has also attracted much attention over the years (He, Cheng, Shi, Chen, & Sun, 2014; Schenato & Fiorentin, 2011; Wang, Xiong, Chen, & Wang, 2018; Xiong & Kishore, 2008).
Consensus-based time synchronization, which is built on multi-agent consensus technique, becomes an important synchronization strategy to achieve accurate time synchronization for wireless sensor networks in a fully distributed way. In practical network scenarios, random communication delays are inevitable and have a significant influence on the information exchange procedure. Without dealing with the delays, many synchronization algorithms based on consensus will diverge and the nodes in the network will fail to synchronize with each other. A feasible approach addressing this problem is to design powerful and effective relative skew estimator to suppress the influence caused by delays. In this paper, we propose a sequential least squares algorithm to estimate the skew under delays, which improves the estimation accuracy while reducing the memory requirement. Based on the estimation algorithm and considering the case of the varying skews, we also devise an improved estimation scheme by incorporating a forgetting factor so as to track the skew evolution. By applying the proposed estimation method to average consensus protocol, each node can achieve accurate synchronization in the presence of delays. We theoretically prove the convergence of the whole scheme by using Lyapunov theory and considering strongly rooted topology condition, and provide simulations to show that our scheme outperforms existing consensus-based synchronization schemes under random delays.
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View all citing articles on Scopus

^☆: This work was presented in part at the 14th International Conference on Communication Systems 2014, Macau, November 2014 [38].

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Fully distributed clock synchronization in wireless sensor networks under exponential delays☆

Highlights

Abstract

Introduction

Section snippets

System model and maximum likelihood estimator

Distributed clock synchronization algorithm

Simulation results and discussions

Conclusions and future works

Comput. Netw.

Wireless Sensor Networks: A Systems Perspective

Clock synchronization of wireless sensor networks

IEEE Signal Process. Mag.

Analysis of synchronization algorithms with time-out control over networks with exponentially symmetric delays

IEEE Trans. Commun.

On maximum-likelihood estimation of clock offset

IEEE Trans. Commun.

Energy-efficient estimation of clock offset for inactive nodes in wireless sensor networks

IEEE Trans. Inf. Theory

On minimum variance unbiased estimation of clock offset in a two-way message exchange mechanism

IEEE Trans. Inf. Theory

A factor graph approach to clock offset estimation in wireless sensor networks

IEEE Trans. Inf. Theory

On maximum likelihood estimation of clock offset and skew in networks with exponential delays

IEEE Trans. Signal Process

Maximum likelihood estimators of clock offset and skew under exponential delays

Appl. Stoch. Models Bus. Ind.

Low complexity maximum likelihood estimators for clock synchronization of wireless sensor nodes under exponential delays

IEEE Trans. Signal Process

Novel clock phase offset and skew estimation using two-way timing message exchanges for wireless sensor networks

IEEE Trans. Commun.

A scalable synchronization protocol for large scale sensor networks and its applications

IEEE J. Sel. Areas Commun.

On the scalability of cooperative time synchronization in pulse-connected network

IEEE Trans. Inf. Theory

Global clock synchronization in sensor networks

IEEE Trans. Comput.