Distributed fusion in wireless sensor networks based on a novel event-triggered strategy

doi:10.1016/j.jfranklin.2018.04.021

Journal of the Franklin Institute

Volume 356, Issue 17, November 2019, Pages 10315-10334

https://doi.org/10.1016/j.jfranklin.2018.04.021 Get rights and content

Highlights

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A novel event-triggered-strategy that can relate the dynamic threshold with the communication rate intuitively is presented.
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By adopting the novel event-triggered-strategy, a novel distributed multisensor data fusion algorithm is proposed.
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By fusing the measurements in the neighborhood only, the energy consumption is saved in the proposed algorithm.

Abstract

In this paper, the event-triggered distributed multi-sensor data fusion algorithm is presented for wireless sensor networks (WSNs) based on a new event-triggered strategy. The threshold of the event is set according to the chi-square distribution that is constructed by the difference of the measurement of the current time and the measurement of the last sampled moment. When the event-triggered decision variable value is larger than the threshold, the event is triggered and the observation is sampled for state estimation. In designing the dynamic event-triggered strategy, we relate the threshold with the quantity in the chi-square distribution table. Therefore, compared to the existed event-triggered algorithms, this novel event-triggered strategy can give the specific sampling/communication rate directly and intuitively. In addition, for the presented distributed fusion in wireless sensor networks, only the measurements in the neighborhood (i.e., the neighbor nodes and the neighbor’s neighbor nodes) of the fusion center are fused so that it can obtain the optimal state estimation under limited energy consumption. A numerical example is used to illustrate the effectiveness of the presented algorithm.

Introduction

Due to the constraints on network bandwidth and resources in the wireless sensor networks (WSNs), in order to reduce the unnecessary waste, it is necessary to design a better communication mechanism for the system. The event-triggered mechanism is determined by a preset event-triggered constraint to determine whether the information shall be transmitted, which to some extent can save the network bandwidth, as well as the sensor and state estimator energy consumption. The optimization design of the event-triggered sampling/transmission strategy has gradually turned into a research hotspot at home and abroad.

In the early 1980s, Ho et al. first propose an event-triggered sampling strategy for discrete systems in [1]. In 1999, a comparison between periodic and event-based sampling for first order stochastic systems is given by Ãström and Bernhardsson in [2]. Arzén et al. propose a variable sampling event-triggered system based on the deviation signal and apply it to water tank level control in [3]. Based on the residual triggering condition, the event-triggered state estimator is deduced by mining the information behind the event in [4], [5]. Paper [6] addresses the state estimation based on the send-on-delta (SOD) triggered strategy. In [7], the distributed filtering problem is investigated for a class of discrete time-varying systems in wireless sensor networks with an event-based communication mechanism. Shi et al. take advantage of the “set-valued” filtering architecture to implement event-triggered state estimation in [8]. The event-triggered state estimation problem of the hidden Markov model is studied in [9], it also compares and analyzes the reliable channel and the unreliable channel with packet loss. An open-loop and a closed-loop stochastic event-triggered sensor schedule for remote state estimation are proposed in [10]. In [11], the problem of event-triggered state estimation under mixed delay and nonlinear interference is studied. By the convex optimization method, the theoretical analysis of the algorithm and the upper bound of the estimation error are given. Through combining quantized control with event-triggered control, [12] studies the consensus problem of multi-agent system with external interference, which reduces the communication burden and overcomes the network constraints. In [13], the problem of event-triggered synchronization of heterogeneous complex networks is investigated, and the influence of transmission delays on event-triggered synchronization is considered. In order to decrease communication load of integrant, the problem of distributed event-triggered pinning control for practical consensus of multi-agent systems with quantized communication based on a directed graph is investigated in [14]. The event-triggered strategies mentioned in the above literatures are effective, but couldn’t provide the actual communication rate directly.

In the WSNs, data fusion could be accomplished by distributed or centralized fusion architecture. Centralized fusion does not applicable to WSNs because there are large quantities of sensors in the WSNs, so it requires lots of calculation to augment all the sensors, and it is highly possible to be a failure at the center node. In contrast, the distributed fusion architecture embodies a great advantage, for every sensor performs not only as a sensor but also as an estimator with both sensing and computational capabilities, so each sensor may be used as a fusion center. In addition, the estimation is generated by the measurements which is collected only from its neighborhood, so distributed fusion architecture has higher robustness and less communication burden. In [15], an event-triggered distributed data fusion algorithm is proposed in which the event-triggered mechanism is based on the estimated error covariance, and each sensor is triggered individually. In literature [16], the problem of distributed event-triggered state estimation is studied in the background of multi-agent coordination, and the triggering conditions of the event are designed according to the value of the residual. The problem of event-triggered distributed state estimation for a class of discrete nonlinear stochastic systems with time-varying delays, randomly occurring uncertainties and randomly occurring nonlinearities is concerned in [17]. In [18], the distributed filtering problem of event-triggered wireless sensor networks with bandwidth and energy constraints is studied. Considering the random measurement fading phenomenon, an adaptive algorithm for determining the triggering threshold is given, and the desired average transmission rate can be obtained under limited wireless channel resources. The event-triggered distributed H∞ filtering problem is considered in [19]. Dong et al. design a set of event-triggered time-varying distributed state estimators that allow dynamic state estimation errors to satisfy the average H∞ performance constraints, which to some extent can save constrained computational resources and network bandwidth [20]. The problem of distributed event-triggered H∞ filtering over sensor networks for a class of discrete-time systems modeled by a set of linear Takagi-Sugeno (T-S) fuzzy models is studied in [21]. In [22], the distributed estimation problem for networked sensing system with event-triggered communication schedules on both sensor-to-estimator channel and estimator-to-estimator channel is studied.

As mentioned above, there are many event-triggered distributed fusion algorithms. These existed event-triggered transmission/sampling strategies can be divided into the following categories: (1) triggering an event based on the observation residual; (2) based on the estimation error covariance; (3) based on the observations, including SOD or measurement-based triggering (MBT). The first two categories need to compute the state estimation at the sensor side, which require large calculation; the third category based on the measurements only, it is therefore more efficient. It is well known that event-triggered strategy is different from time-triggered strategy, in that not all the measurements shall be transmitted to the fusion center, therefore it has a communication rate that related to the event-triggered threshold. However, the existing event-triggered methods can not give the specific communication rate directly and intuitively. To solve this problem, a new event-triggered strategy is given, and based on which a novel event-triggered distributed data fusion algorithm for wireless sensor networks is presented in this paper. We improve SOD method by setting the threshold according to the chi-square distribution table where the difference of the measurement of the current time and measurement of the last sampled moment is concerned. When the chi-square distribution value exceeds the preset threshold, the event is triggered and the observation is sampled for estimation. Based on the chi-square distribution table, one can find the sampling/communication rate of the presented algorithm directly, that is, the threshold of the novel dynamic event-triggered strategy is corresponded to the relevant value in the chi-square distribution table. Since the energy of the wireless sensor networks is limited, based on the network topology of sensors, we fuse the neighbor nodes and the neighbor’s neighbor nodes of the fusion center in the distributed fusion algorithm, which is more practical and energy efficient and the fusion result is optimal. The algorithm proposed in this paper can also be generalized to the state estimation of nonlinear systems through referring to [23], [24], [25].

This paper is organized as follows. In Section 2, the problem is formulated. Section 3 presents the event-triggered multisensor state estimation algorithm. Section 4 is the simulation and Section 5 draws the conclusion.

Section snippets

System model characterization

Consider the following linear dynamic system $\begin{matrix} x (k + 1) = A (k) x (k) + w (k), k = 0, 1, \dots \end{matrix}$ $\begin{matrix} z_{i} (k) = C_{i} (k) x (k) + v_{i} (k), i = 1, 2, \dots, N \end{matrix}$ where x(k) ∈ Rⁿ is the system state, A(k) ∈ R^n × n is the state transition matrix, w(k) is the system noise assumed to be white Gaussian distributed with zero mean and variance Q(k), $z_{i} (k) \in R^{m_{i}}$ is the measurement of sensor i at time k, and $C_{i} (k) \in R^{m_{i} \times n}$ is the measurement matrix. The measurement noise v_i(k) is assumed to be white Gaussian with zero mean and $E {v_{i} (k) v_{j}^{T} (l)} = R_{i} (k) δ_{k l} δ_{i j},$ where δ_kl

Kalman filter with event-triggered mechanism

Theorem 1 The Kalman filter (KF) with event-triggered mechanism

For system (1), (2), the estimation of x(k) by the Kalman filter is given by $\begin{matrix} {\begin{matrix} {\hat{x}}_{i} (k | k - 1) & = & A (k - 1) {\hat{x}}_{i} (k - 1 | k - 1) \\ P_{i} (k | k - 1) & = & A (k - 1) P_{i} (k - 1 | k - 1) A^{T} (k - 1) + Q (k - 1) \\ {\hat{x}}_{i} (k | k) & = & {\hat{x}}_{i} (k | k - 1) + K_{i} (k) {\tilde{z}}_{i} (k | k - 1) \end{matrix} \end{matrix}$ $\begin{matrix} P_{i} (k | k) & = {\begin{matrix} (I - K_{i} (k) C_{i} (k)) P_{i} (k | k - 1) {(I - K_{i} (k) C_{i} (k))}^{T} + K_{i} (k) R_{i} (l_{i}) K_{i}^{T} (k) \\ + K_{i} (k) {[C_{i} (k) - C_{i} (l_{i}) \prod_{p = l_{i}}^{k - 1} A^{- 1} (p)] Ξ (k) {[C_{i} (k) - C_{i} (l_{i}) \prod_{n = l_{i}}^{k - 1} A^{- 1} (n)]}^{T} \\ + C_{i} (l_{i}) (\sum_{m = l_{i}}^{k - 1} \prod_{p = l_{i}}^{m} A^{- 1} (p) Q (m) \prod_{n = m + 1}^{k - 1} A^{T} (n)) {[C_{i} (k) - C_{i} (l_{i}) \prod_{p = l_{i}}^{k - 1} A^{- 1} (p)]}^{T} \\ + C_{i} (k) (\sum_{m = l_{i}}^{k - 1} \prod_{n = m + 1}^{k - 1} A (n) Q (m) \prod_{p = l_{i}}^{m} A^{- 1, T} (p)) C_{i}^{T} (l_{i})} K_{i}^{T} (k) - (I - K_{i} (k) C_{i} (k)) \\ \cdot {P_{i} (k | k - 1) (C_{i} (l_{i}) \prod_{p = l_{i}}^{k - 1} A^{- 1} (p) - \end{matrix} \end{matrix}$

Numerical example

We use an example to illustrate the effectiveness of the presented algorithm in this section. A wireless sensor network with 12 sensor nodes is deployed to monitor the target, and the topology of the WSNs is shown in Fig. 1 [26], [27]. $\begin{matrix} x (k + 1) & = [\begin{matrix} 1 & T_{s} \\ 0 & 1 \end{matrix}] x (k) + Γ (k) ξ (k) \end{matrix}$ $\begin{matrix} z_{i} (k) & = C_{i} x (k) + v_{i} (k), k = 1, 2, \dots, L \end{matrix}$ where $L = 1000$ is the length of the signal x to be estimated. $T_{s} = 0.01$ is the sampling period. The state $x (k) = {[s (k) \dot{s} (k)]}^{T},$ where s(k) and $\dot{s} (k)$ are the position and velocity of the target at time kT_s,

Conclusions

This article studies the distributed multi-sensor event-triggered data fusion problem with a new event-triggered strategy in wireless sensor networks. The threshold of the event-trigger is set according to the chi-square distribution which is constructed by the difference of the measurement of the current time and the measurement of the last sampled moment. This novel trigger method can easily give the specific communication rate based on the chi-square distribution table. In the distributed

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^☆: This work was supported by the National Natural Science Foundation Projects under Grants 61773063, 61773334 and 61720106010, the Beijing Natural Science Foundation under Grant 4161001, and by Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant 61621063.

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Distributed fusion in wireless sensor networks based on a novel event-triggered strategy☆

Highlights

Abstract

Introduction

Section snippets

System model characterization

Kalman filter with event-triggered mechanism

Numerical example

Conclusions

Automatica

Automatica

IEEE Trans. Circuit Syst. I: Reg.

IEEE Trans. Syst. Man Cybern.: Syst.

J. Frankl. Inst.

Inf. Sci.

Inf. Fus.

J. Frankl. Inst.

J. Frankl. Inst.

Automatica

Comparison of periodic and event based sampling for first-order stochastic systems

Proceedings of the Fourteenth IFAC World Congress

A simple event-based PID controller

Proceedings of the Fourteenth IFAC World Congress

Stochastic event-triggered sensor scheduling for remote state estimation

Proceedings of the 52nd IEEE Conf. Decision and Control