Event-Based Control for Average Consensus of Wireless Sensor Networks with Stochastic Communication Noises

Abstract

This paper focuses on the average consensus problem for the wireless sensor networks (WSNs) with fixed and Markovian switching, undirected and connected network topologies in the noise environment. Event-based protocol is applied to each sensor node to reach the consensus. An event triggering strategy is designed based on a Lyapunov function. Under the event trigger condition, some sufficient conditions for average consensus in mean square are obtained. Finally, some numerical simulations are given to illustrate the effectiveness of the results derived in this paper.

1. Introduction

Wireless sensor network (WSN) has attracted significant attention as an emerging communication architecture. It has many practical applications in such areas as robotics, surveillance and environment monitoring, and information collection.

A WSN can be viewed as a multiagent system (MAS) from a network-theoretic perspective. Each node represents a sensor and each edge performs information exchange between sensors. In some cases, the agreement is a common value which may be the average of the initial states of the system, is often called average consensus, and has wide application background in the areas such as formation control [1], distributed filtering [2], and distributed computation [3]. It means to achieve the accordance of the states of MAS. In [4], Olfati-Saber and Murray consider the average consensus control for the directed and undirected networks with fixed and switching topologies. In [5], Kingston and Beard extend the results of [4] to the discrete-time models and weakened the condition of instantaneous strong connectivity. In [6], Xiao and Boyd consider the distributed averaging consensus of the networks with fixed and undirected topologies. In [7], Q. Zhang and J. Zhang design a distributed consensus protocol to analyze the multiagent systems in uncertain communication environments including the communication noises and Markov topology switches. In [8], Wang et al. investigate the H∞ consensus control problem for a class of discrete time-varying multiagent systems with both missing measurements and parameter uncertainties. Also, the distributed estimation problems over sensor networks have been widely discussed in [9–11].

For the node of the WSNs with limited energy, control over networks with limited resources is a challenging task. Consequently, the most important problem in WSN is the energy consumption, which is directly proportional to the transmit power of the information exchange between the sensors. The event-based control can facilitate the efficient usage of the resources. Based on the event-based control mechanism, it can reduce the useless communication between neighboring agents along with the energy consumption. In [12], Wang and Lemmon discuss the event-triggered data transmission in distributed networked control systems with packet loss and transmission delays. In [13], Mazo and Tabuada focus on reducing the number of messages from sensors to controllers and from controllers to actuators by the use of the decentralized event-triggered mechanism. In [14], Seyboth et al. propose a novel control strategy for multiagent coordination with event-based broadcasting. The proposed control strategy guarantees either asymptotic convergence to average consensus or convergence to a ball centered at the average consensus. In [15], Meng and Chen study an average consensus problem for multiagent systems by event-based control, which is used on each agent to drive the state to their initial average eventually.

The main contribution of this paper is studying average consensus problem for the WSN with fixed topology and Markovian switching topology in the noise environment; the event-based average consensus control protocol is designed for the WSN based on a Lyapunov function. Under the event trigger condition, sufficient conditions for average consensus in mean square are obtained.

The remainder of the paper is organized as follows. In Section 2, some concepts in graph theory are described, and the problem to be investigated is formulated. In Section 3, the main results are presented. In Section 4, some numerical examples show the reliability of the main results. In Section 5, some conclusions are given.

2. Problem Formulation and Preliminaries

2.1. Concepts in Graph Theory

Let 𝒢={𝒱,ℰ,𝒜} be an undirected graph, where 𝒱={1,2,…,𝑛} is the set of nodes, node 𝑖 represents the 𝑖th sensor node, ℰ is the set of edges, and an edge in 𝒢 is denoted by an ordered pair (𝑗,𝑖). (𝑗,𝑖)∈ℰ if and only if the 𝑗th sensor node can send information to the 𝑖th sensor node directly. The neighborhood of the 𝑖th sensor node is denoted by 𝑁_𝑖={𝑗∈𝒱∣(𝑗,𝑖)∈ℰ}.

𝒜=[𝑎_𝑖𝑗]∈ℝ^𝑛×𝑛 is called the adjacency matrix of 𝒢. For any 𝑖,𝑗∈𝒱,𝑎_𝑖𝑗≥0, and 𝑎_𝑖𝑗>0⇔𝑗∈𝑁_𝑖.  deg_in(𝑖)=∑^𝑛_𝑗=1𝑎_𝑖𝑗 is called the in-degree of 𝑖; deg_out(𝑖)=∑^𝑛_𝑗=1𝑎_𝑖𝑗 is called the out-degree of 𝑖; 𝐿=𝒟−𝒜 is called the Laplacian matrix of 𝒢, where 𝒟=diag(deg_in(1),…,deg_in(𝑛)). Its eigenvalues are real and can be ordered as 𝜆₁≤𝜆₂≤⋯≤𝜆_𝑛 (1) with 𝜆₁=0 and 𝜆₂ being the smallest nonzero eigenvalue for connected graphs.

2.2. Average Consensus for WSNs

In this paper, we study the average consensus control for a WSN with dynamics 𝑥̇_𝑖(𝑡)=𝑢_𝑖(𝑡), (2) where 𝑥_𝑖(𝑡)∈ℝ is the state of the 𝑖th sensor and 𝑢_𝑖(𝑡)∈ℝ is the control input. The initial state 𝑥_𝑖(0) is deterministic.

Each sensor includes a digital microprocessor and dynamics. The microprocessor of sensor 𝑖 monitors its own measurement value 𝑥_𝑖(𝑡) continuously and decides when to communicate with the neighboring sensors by broadcasting the actual measurement value. Therefore the latest broadcasted value of sensor 𝑖 can be described by the piecewise constant function 𝑥_𝑖(𝑡)=𝑥_𝑖𝑡^𝑖_𝑘,𝑡^𝑖_𝑘≤𝑡<𝑡^𝑖_𝑘+1, (3) where 𝑡^𝑖₀,𝑡^𝑖₁,… is sequence of event-times of sensor 𝑖. We can see that the discrete-time signal 𝑥_𝑖(𝑡^𝑖_𝑘) is converted into the continuous-time signal 𝑥_𝑖(𝑡).

The 𝑖th sensor can receive information from its neighbors 𝑦_𝑗𝑖(𝑡)=𝑥_𝑗(𝑡)+𝜎_𝑗𝑖𝜔_𝑗𝑖(𝑡), (4) where 𝑦_𝑗𝑖(𝑡) denotes the measurement of the 𝑗th sensor’s state 𝑥_𝑗(𝑡) by the 𝑖th sensors, {𝜔_𝑗𝑖(𝑡)∣𝑖,𝑗=1,2,…,𝑛} are the communication noises, and 𝜎_𝑗𝑖 is the noise intensity function.

For the dynamic network (𝒢,𝑋), we give the event-based consensus protocl. 𝑢_𝑖(𝑡)=_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑦_𝑗𝑖(𝑡)−𝑥_𝑖(𝑡). (5)

The dynamics of agent 𝑖 for 𝑡∈[𝑡^𝑖_𝑘+𝑙ℎ,𝑡^𝑖_𝑘+𝑙ℎ+ℎ) is given by 𝑥̇_𝑖(𝑡)=_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝑥_𝑗𝑡^𝑗_𝑘′−𝑥_𝑖𝑡^𝑖_𝑘+𝜎_𝑗𝑖𝜔_𝑗𝑖(𝑡=_{𝑗∈𝑁𝑖(𝑡)})𝑎_𝑖𝑗𝑥_𝑗𝑡^𝑖_𝑘+𝑙ℎ−𝑥_𝑖𝑡^𝑖_𝑘+𝑙ℎ+_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝑥_𝑖𝑡^𝑖_𝑘+𝑙ℎ−𝑥_𝑖𝑡^𝑖_𝑘+_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝑥_𝑗𝑡^𝑗_𝑘′−𝑥_𝑗𝑡^𝑖_𝑘+𝑙ℎ+_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝜎_𝑗𝑖𝜔_𝑗𝑖(𝑡)=_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝑥_𝑗𝑡^𝑖_𝑘+𝑙ℎ−𝑥_𝑖𝑡^𝑖_𝑘+𝑙ℎ+_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝑒_𝑗𝑡^𝑖_𝑘+𝑙ℎ−𝑒_𝑖𝑡^𝑖_𝑘+𝑙ℎ+_{𝑗∈𝑁𝑖(𝑡)}𝑎_𝑖𝑗𝜎_𝑗𝑖𝜔_𝑗𝑖(𝑡), (6) where 𝑡^𝑗_𝑘′ is defined as 𝑡^𝑗_𝑘′=max𝑡∣𝑡∈𝑡^𝑗_𝑘,𝑘=0,1,…,𝑡≤𝑡^𝑖_𝑘+𝑙ℎ. (7)

Substituting the protocol (5) into the system (2) leads to 𝑑𝑥(𝑡)=−[𝐿(𝑥(𝑘ℎ)+𝑒(𝑘ℎ))]𝑑𝑡+𝐺𝑑𝑊(𝑡), (8) where 𝐿 is the Laplacian matrix of 𝒢, 𝐺 is the noise intensity matrix, and 𝑊(𝑡)=(𝑊₁(𝑡),…,𝑊_𝑛(𝑡))^𝑇 is an 𝑛-dimensional Brownian motion.

The event condition for agent 𝑖 has the following form: 𝜆_𝑛𝑥_𝑖(𝑡^𝑖_𝑘+𝑙ℎ)²≤𝜆₂𝑒_𝑖(𝑡^𝑖_𝑘+𝑙ℎ)²+2‖𝐺‖², (9) where 𝑒_𝑖𝑡^𝑖_𝑘+𝑙ℎ=𝑥_𝑖𝑡^𝑖_𝑘−𝑥_𝑖𝑡^𝑖_𝑘+𝑙ℎ. (10)

Remark 1. Each sensor broadcasts its state information to the neighbors and also receives state information from its neighbors for event detection at each sampling instant. The event detector in (9) guarantees that it reduces the sensor energy consumption and network bandwidth usage because it only checks the event condition at discrete sampling instants.

Next, we consider the average consensus control protocol for the system (8) as follows: (𝑡)[𝛿()+()𝑒(+()(𝑡),𝑑𝛿=−𝐿𝑘ℎ𝐼−𝐽𝑘ℎ)]𝑑𝑡𝐼−𝐽𝐺𝑑𝑊 (11) where 𝛿(𝑡)=𝑥(𝑡)−𝐽𝑥(𝑡) and 𝐽=(1/𝑛)1_𝑛1^𝑇_𝑛.

If we define 𝑓()[𝛿()+()𝑒(,𝑔()=()𝛿,𝑡=−𝐿𝑘ℎ𝐼−𝐽𝑘ℎ)]𝛿,𝑡𝐼−𝐽𝐺, (12) then (11) becomes 𝑑𝛿(𝑡)=𝑓(𝛿,𝑡)𝑑𝑡+𝑔(𝛿,𝑡)𝑑𝑊(𝑡). (13)

In WSNs, each sensor node communicates with other sensor nodes through the unreliable networks. If the communication channel between sensors 𝑖 and 𝑗 is (𝑗,𝑖)∈ℰ_𝑓, and ℰ_𝑓 is the set of the communication channels which probably lost the signal, then the time-varying topologies under link failure or creation can be described by the Markov switching topology.

Let {𝑟(𝑡)_𝑡≥0} be a right-continuous Markov chain on the probability space taking values in a finite state set 𝑆={1,2,…,𝑁} with generator Γ=(𝛾_𝑖𝑗)_𝑁×𝑁 given by 𝑃𝑟()(𝑡)=𝛾𝑡+Δ=𝑗∣𝑟=𝑖^𝑖𝑗Δ+𝑜(Δ),if𝑖≠𝑗,1+𝛾_𝑖𝑖Δ+𝑜(Δ),if𝑖=𝑗, (14) where Δ>0 and 𝛾_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗, while 𝛾_𝑖𝑖=−_𝑗≠𝑖𝛾_𝑖𝑗 (15) and 𝑟(0)=𝑟₀.

We denote the undirected communication graph by 𝒢={𝒢(1),𝒢(2),…,𝒢(𝑁)}, where 𝒢(𝑘)={𝒱,ℰ_𝒢(𝑘),𝒜_𝒢(𝑘)} is the undirected graph. Denote the topology graph by 𝒢_𝑡 at moment 𝑡(𝑡≥0), so 𝑟(𝑡)=𝑘 if and only if 𝒢_𝑡=𝒢(𝑘).

Under Markovian switching topology, we have (𝑡)[𝐿(𝑟(𝑡𝑥()((𝑟(𝑡(𝑡),(𝑡)(𝑟(𝑡𝛿()+()𝑒(+()𝐺(𝑟(𝑡(𝑡).𝑑𝑥=−))(𝑘ℎ+𝑒𝑘ℎ))]𝑑𝑡+𝐺))𝑑𝑊𝑑𝛿=−𝐿))[𝑘ℎ𝐼−𝐽𝑘ℎ)]𝑑𝑡𝐼−𝐽))𝑑𝑊 (16)(17)

So, under this condition, the event condition for agent 𝑖 has the following form: 𝜆_𝑛(𝐿(𝑟(𝑡)))𝑥_𝑖(𝑡^𝑖_𝑘+𝑙ℎ)²≤𝜆₂(𝐿(𝑟(𝑡)))𝑒_𝑖(𝑡^𝑖_𝑘+𝑙ℎ)²+2‖𝐺‖², (18) where 𝑒_𝑖𝑡^𝑖_𝑘+𝑙ℎ=𝑥_𝑖𝑡^𝑖_𝑘−𝑥_𝑖𝑡^𝑖_𝑘+𝑙ℎ. (19)

3. Main Results

In this section, we consider the average consensus for the WSNs with fixed topology and Markovian switching topology by event-based control.

3.1. WSNs with the Fixed Topology

Theorem 2. Consider the system (8) over a connected communication graph driven by event condition in (9). Then the system (11) is asymptotically stable in mean square; it means that the system (8) reaches average consensus in mean square if 0≤ℎ≤1/2𝜆_𝑛.

Proof. Define the function 𝑉∶𝑅^𝑛×𝑅₊→𝑅⁺ by 𝑉()=1𝛿,𝑡2𝛿^𝑇(𝑡)𝛿(𝑡). (20) Computing 𝑑𝑉(𝛿,𝑡) along the trajectory generated by the system (11) for any 𝑡∈[𝑘ℎ,(𝑘+1)ℎ), we have 𝑑𝑉(𝛿,𝑡)=𝛿^𝑇(𝑡)𝑓(𝛿,𝑡)+trace𝑔^𝑇(𝛿,𝑡)𝑔(𝛿,𝑡)𝑑𝑡+𝛿^𝑇(𝑡)(𝑡)=)(𝐼−𝐽𝐺𝑑𝑊−𝛿^𝑇(𝑡)𝐿[𝛿()+()𝑒(+‖‖𝑘ℎ𝐼−𝐽𝑘ℎ)](𝐼−𝐽)𝐺²𝑑𝑡+𝛿^𝑇(𝑡)(𝑡)=𝛿()+()𝑒()(𝐼−𝐽𝐺𝑑𝑊(𝑡−𝑘ℎ)[𝑘ℎ𝐼−𝐽𝑘ℎ)]^𝑇×𝐿^𝑇𝐿[𝛿(𝑘ℎ)+(𝐼−𝐽)𝑒(𝑘ℎ)]−𝛿^𝑇()𝐿[𝛿()+()𝑒(+‖‖𝑘ℎ𝑘ℎ𝐼−𝐽𝑘ℎ)](𝐼−𝐽)𝐺²−𝑊𝑑𝑡_𝑇(𝑡)−𝑊^𝑇()𝐺()[𝛿()+()𝑒(𝑘ℎ𝐼−𝐽×𝐿𝑘ℎ𝐼−𝐽𝑘ℎ)]+𝛿^𝑇(𝑡)(𝑡)≤)(𝐼−𝐽𝐺𝑑𝑊ℎ𝜆_𝑛[𝛿(𝑘ℎ)+(𝐼−𝐽)𝑒(𝑘ℎ)]^𝑇×𝐿[𝛿(𝑘ℎ)+(𝐼−𝐽)𝑒(𝑘ℎ)]−𝛿^𝑇()𝐿[𝛿()+()𝑒(+‖‖𝑘ℎ𝑘ℎ𝐼−𝐽𝑘ℎ)](𝐼−𝐽)𝐺²−𝑊𝑑𝑡_𝑇(𝑡)−𝑊^𝑇()𝐺()[𝛿()+()𝑒(𝑘ℎ𝐼−𝐽×𝐿𝑘ℎ𝐼−𝐽𝑘ℎ)]+𝛿^𝑇(𝑡)(𝑡)=)(𝐼−𝐽𝐺𝑑𝑊ℎ𝜆_𝑛−1𝛿^𝑇(𝑘ℎ)𝐿𝛿(𝑘ℎ)+2ℎ𝜆_𝑛−1×𝛿^𝑇(𝑘ℎ)𝐿(𝐼−𝐽)𝑒(𝑘ℎ)+ℎ𝜆_𝑛‖𝐼−𝐽‖²𝑒^𝑇()()+)𝐺‖𝑘ℎ𝐿𝑒𝑘ℎ‖(𝐼−𝐽²−𝑊𝑑𝑡_𝑇(𝑡)−𝑊^𝑇()𝐺()[𝛿()+()𝑒(𝑘ℎ𝐼−𝐽×𝐿𝑘ℎ𝐼−𝐽𝑘ℎ)]+𝛿^𝑇(𝑡)(𝐼−𝐽)𝐺𝑑𝑊(𝑡). (21)
Applying the inequality 𝛿^𝑇()𝐿()𝑒()≤1𝑘ℎ𝐼−𝐽𝑘ℎ2‖𝐿‖²𝛿^𝑇()𝛿()+1𝑘ℎ𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇(𝑘ℎ)𝑒(𝑘ℎ), (22)
it can be derived that (𝛿(𝑡))≤−1𝑑𝑉,𝑡2𝛿^𝑇()()+1𝑘ℎ𝐿𝛿𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()()+‖‖𝑘ℎ𝐿𝑒𝑘ℎ(𝐼−𝐽)𝐺²−𝑊𝑑𝑡_𝑇(𝑡)−𝑊^𝑇()𝐺()[𝛿()+()𝑒(𝑘ℎ𝐼−𝐽×𝐿𝑘ℎ𝐼−𝐽𝑘ℎ)]+𝛿^𝑇(𝑡)(𝐼−𝐽)𝐺𝑑𝑊(𝑡) (23) with 2ℎ𝜆_𝑛≤1.
We can see that 𝑑𝑒^−𝑡𝑉(𝛿(𝑡),𝑡)=𝑒^−𝑡[−𝑉(𝛿(𝑡),𝑡)𝑑𝑡+𝑑𝑉(𝛿(𝑡),𝑡)]≤𝑒^−𝑡(𝛿(𝑡))−1−𝑉,𝑡2𝛿^𝑇()()+1𝑘ℎ𝐿𝛿𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()()+‖‖𝑘ℎ𝐿𝑒𝑘ℎ(𝐼−𝐽)𝐺²𝑑𝑡−𝑒^−𝑡𝑊^𝑇(𝑡)−𝑊^𝑇(𝐺()[𝛿()+()𝑒(𝑘ℎ)𝐼−𝐽×𝐿𝑘ℎ𝐼−𝐽𝑘ℎ)]+𝑒^−𝑡𝛿^𝑇(𝑡)(𝐼−𝐽)𝐺𝑑𝑊(𝑡). (24) Integrating both sides of the previous inequality from 𝑘ℎ to 𝑡 (𝑡∈[𝑘ℎ,(𝑘+1)ℎ)), making use of (9), and taking the expectation, one can obtain that 𝑒^−𝑡𝔼𝑉(𝛿(𝑡),𝑡)≤^𝑡_𝑘ℎ−𝑒^−𝑡(𝛿(𝑡))+1𝔼𝑉,𝑡𝑑𝑡2𝑒^−𝑘ℎ𝛿^𝑇()𝛿()−1𝑘ℎ𝑘ℎ2𝛿^𝑇()()−1𝑘ℎ𝐿𝛿𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()()−)𝐺‖𝑘ℎ𝐿𝑒𝑘ℎ‖(𝐼−𝐽²𝑒^−𝑡−𝑒^−𝑘ℎ≤^𝑡_𝑘ℎ−𝑒^−𝑡(𝛿(𝑡))+1𝔼𝑉,𝑡𝑑𝑡2𝑒^−𝑘ℎ𝛿^𝑇()𝛿()+1𝑘ℎ𝑘ℎ2𝛿^𝑇()()−1𝑘ℎ𝐿𝛿𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()()−)𝐺‖𝑘ℎ𝐿𝑒𝑘ℎ‖(𝐼−𝐽²𝑒^−𝑘ℎ−𝑒^{−(𝑘+1)ℎ}=^𝑡_𝑘ℎ−𝑒^−𝑡(𝛿(𝑡))+1𝔼𝑉,𝑡𝑑𝑡2𝑒^−𝑘ℎ𝛿^𝑇()𝛿()+1𝑘ℎ𝑘ℎ2‖𝐼−𝐽‖²𝑥^𝑇()()−1𝑘ℎ𝐿𝑥𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()()−‖‖𝑘ℎ𝐿𝑒𝑘ℎ(𝐼−𝐽)𝐺²𝑒^−𝑘ℎ−𝑒^{−(𝑘+1)ℎ}≤^𝑡_𝑘ℎ−𝑒^−𝑡(𝛿(𝑡))+1𝔼𝑉,𝑡𝑑𝑡2𝑒^−𝑘ℎ𝛿^𝑇(𝑘ℎ)𝛿(𝑘ℎ). (25) By using Gronwall inequality, we have (𝛿(𝑡))≤1𝔼𝑉,𝑡2𝑒^−𝑘ℎ‖𝛿(𝑘ℎ)‖²𝑒^{−(𝑡−𝑘ℎ)}=12‖𝛿(𝑘ℎ)‖²𝑒^−𝑡. (26) Letting 𝑡→∞(𝑘→∞), we obtain lim_𝑡→∞𝔼‖𝛿(𝑡)‖²=0. (27) So the system (8) reaches average consensus in mean square.

3.2. WSNs with Markovian Switching Topology

Theorem 3. Consider the system (16) over a number of connected switching communication graphs driven by event condition in (18). Then the system (17) is asymptotically stable in mean square; it means that the system (16) reaches average consensus in mean square if 0≤ℎ≤1/2𝜆_𝑛(𝑟(𝑡)).

Proof. Define the function 𝑉∶𝑅^𝑛×𝑅₊×𝑆→𝑅⁺ by 𝑉((𝑡=1𝛿,𝑡,𝑟))2𝛿^𝑇(𝑡)𝛿(𝑡). (28) Computing 𝑑𝑉(𝛿,𝑡,𝑟(𝑡)) along the trajectory generated by the system (11) for any 𝑡∈[𝑘ℎ,(𝑘+1)ℎ), we have 𝑑𝑉(𝛿,𝑡,𝑟(𝑡))=𝛿^𝑇(𝑡)𝑓((𝑡𝑔𝛿,𝑡,𝑟))+trace^𝑇(𝛿,𝑡,𝑟(𝑡))𝑔(𝛿,𝑡,𝑟(𝑡))𝑑𝑡+𝛿^𝑇(𝑡)𝐺(𝑟(𝑡(𝑡)=)(𝐼−𝐽))𝑑𝑊−𝛿^𝑇(𝑡)𝐿(𝑟(𝑡𝛿()+()𝑒(+)𝐺‖))[𝑘ℎ𝐼−𝐽𝑘ℎ)]‖(𝐼−𝐽²+^𝑁_𝑗=1𝛾_𝑖𝑗𝑉(𝛿,𝑡,𝑟(𝑡))𝑑𝑡+𝛿^𝑇(𝑡)𝐺(𝑟(𝑡(𝑡)≤)(𝐼−𝐽))𝑑𝑊ℎ𝜆_𝑛−1𝛿^𝑇()𝐿(𝑟(𝑡𝛿()+𝑘ℎ))𝑘ℎ2ℎ𝜆_𝑛−1𝛿^𝑇(𝑘ℎ)𝐿(𝑟(𝑡))(𝐼−𝐽)𝑒(𝑘ℎ)+ℎ𝜆_𝑛‖𝐼−𝐽‖²𝑒^𝑇()𝐿(𝑟(𝑡𝑒()+)𝐺(𝑟(𝑡𝑘ℎ))𝑘ℎ‖(𝐼−𝐽))‖²+((𝑡−𝑊𝛾𝑉𝛿,𝑡,𝑟))𝑑𝑡_𝑇(𝑡)−𝑊^𝑇()𝐺(𝑟(𝑡)(𝑟(𝑡𝛿()+()𝑒(𝑘ℎ))(𝐼−𝐽×𝐿))[𝑘ℎ𝐼−𝐽𝑘ℎ)]+𝛿^𝑇(𝑡)𝐺(𝑟(𝑡(𝑡)≤−1)(𝐼−𝐽))𝑑𝑊2𝛿^𝑇()𝐿(𝑟(𝑡𝛿()+1𝑘ℎ))𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()𝐿(𝑟(𝑡𝑒()+‖‖𝑘ℎ))𝑘ℎ(𝐼−𝐽)𝐺(𝑟(𝑡))²+((𝑡−𝑊𝛾𝑉𝛿,𝑡,𝑟))𝑑𝑡_𝑇(𝑡)−𝑊^𝑇(𝐺(𝑟(𝑡)(𝑟(𝑡𝛿()+()𝑒(𝑘ℎ)))(𝐼−𝐽×𝐿))[𝑘ℎ𝐼−𝐽𝑘ℎ)]+𝛿^𝑇(𝑡)(𝐼−𝐽)𝐺(𝑟(𝑡))𝑑𝑊(𝑡) (29) with 2ℎ𝜆_𝑛≤1.
For 𝛾>0,𝛾=max{|𝛾_𝑖𝑖∶𝑖∈𝑆|}, compute 𝑑𝑒^−2𝛾𝑡𝑉(𝛿(𝑡),𝑡,𝑟(𝑡))=𝑒^−2𝛾𝑡−2𝛾𝑉(𝛿(𝑡),𝑡,𝑟(𝑡))𝑑𝑡+𝑑𝑉(𝛿(𝑡),𝑡,𝑟(𝑡))≤𝑒^−2𝛾𝑡−2(𝛿(𝑡)(𝑡−1𝛾𝑉,𝑡,𝑟))2𝛿^𝑇()𝐿(𝑟(𝑡𝛿()+1𝑘ℎ))𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()𝐿(𝑟(𝑡𝑒()+‖‖𝑘ℎ))𝑘ℎ(𝐼−𝐽)𝐺(𝑟(𝑡))²+𝛾𝑉(𝛿,𝑡,𝑟(𝑡))𝑑𝑡−𝑒^−2𝛾𝑡𝑊^𝑇(𝑡)−𝑊^𝑇()𝐺()(𝑟(𝑡𝛿()+()𝑒(𝑘ℎ𝐼−𝐽×𝐿))[𝑘ℎ𝐼−𝐽𝑘ℎ)]+𝑒^−2𝛾𝑡𝛿^𝑇(𝑡)(𝐼−𝐽)𝐺(𝑟(𝑡))𝑑𝑊(𝑡). (30)
Integrating both sides of the above inequality from 𝑘ℎ to 𝑡 (𝑡∈[𝑘ℎ,(𝑘+1)ℎ)), making use of (18), and taking the expectation, one can obtain that 𝑒^−2𝛾𝑡(𝛿(𝑡)(𝑡≤𝔼𝑉,𝑡,𝑟))_𝑡𝑘ℎ−𝛾𝑒^−2𝛾𝑡(𝛿(𝑡)(𝑡+1𝔼𝑉,𝑡,𝑟))𝑑𝑡2𝑒^{−2𝛾𝑘ℎ}𝛿^𝑇()𝛿()−1𝑘ℎ𝑘ℎ2𝛿^𝑇()𝐿(𝑟(𝑡𝛿()−1𝑘ℎ))𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()𝐿(𝑟(𝑡𝑒()−)𝐺(𝑟(𝑡𝑘ℎ))𝑘ℎ‖(𝐼−𝐽))‖²𝑒^−2𝛾𝑡−𝑒^{−2𝛾𝑘ℎ}≤^𝑡_𝑘ℎ−𝛾𝑒^−2𝛾𝑡(𝛿(𝑡)(𝑡+1𝔼𝑉,𝑡,𝑟))𝑑𝑡2𝑒^{−2𝛾𝑘ℎ}𝛿^𝑇()𝛿()+1𝑘ℎ𝑘ℎ2𝛿^𝑇()𝐿(𝑟(𝑡𝛿()−1𝑘ℎ))𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()𝐿(𝑟(𝑡𝑒()−‖‖𝑘ℎ))𝑘ℎ(𝐼−𝐽)𝐺(𝑟(𝑡))²×𝑒^{−2𝛾𝑘ℎ}−𝑒^{−2𝛾(𝑘+1)ℎ}=^𝑡_𝑘ℎ−𝛾𝑒^−2𝛾𝑡(𝛿(𝑡)(𝑡+1𝔼𝑉,𝑡,𝑟))𝑑𝑡2𝑒^{−2𝛾𝑘ℎ}𝛿^𝑇()𝛿()+1𝑘ℎ𝑘ℎ2‖𝐼−𝐽‖²𝑥^𝑇()𝐿(𝑟(𝑡𝑥()−1𝑘ℎ))𝑘ℎ2‖𝐼−𝐽‖²𝑒^𝑇()𝐿(𝑟(𝑡()−)𝐺(𝑟(𝑡𝑘ℎ))×𝑒𝑘ℎ‖(𝐼−𝐽))‖²×𝑒^{−2𝛾𝑘ℎ}−𝑒^{−2𝛾(𝑘+1)ℎ}≤^𝑡_𝑘ℎ−𝛾𝑒^−2𝛾𝑡(𝛿(𝑡)(𝑡+1𝔼𝑉,𝑡,𝑟))𝑑𝑡2𝑒^{−2𝛾𝑘ℎ}𝛿^𝑇(𝑘ℎ)𝛿(𝑘ℎ). (31)
By using Gronwall inequality, we have (𝛿(𝑡)(𝑡≤1𝔼𝑉,𝑡,𝑟))2𝑒^{−2𝛾𝑘ℎ}‖𝛿(𝑘ℎ)‖²𝑒^{−𝛾(𝑡−𝑘ℎ)}=12𝑒^{−𝛾(𝑡+𝑘ℎ)}‖𝛿(𝑘ℎ)‖². (32) Letting 𝑡→∞(𝑘→∞), we obtain that lim_𝑡→∞𝔼‖𝛿(𝑡)‖²=0. (33)
So the system (16) reaches average consensus in mean square.

4. Numerical Examples

In this section, we give two examples to examine the average consensus of the systems (8) and (16).

Example 1. Consider a WSN composed by four sensors in which each dynamic state of the sensor is 𝑥̇_𝑖=𝑢_𝑖, where 𝑖=1,2,3,4 (see the topology in Figure 1, which is used in Dimarogonas [16]) and the initial state is 𝑋(0)=(4−1,5,2)^𝑇.

Figure 1 

The topology of the four sensors.

The adjacent matrix is 𝐴=0110101011010010 (34)

The related degree matrix is 𝐷=2000020000300001. (35)

The related Laplacian matrix is 𝐿=2−1−10−12−10−1−13−100−11. (36)

The nonzero largest and smallest eigenvalues are 𝜆_𝑛=4 and 𝜆₂=1, respectively.

Let the noise intensity matrix 𝐺 be 𝐺=0.00100000.00200000.00300000.001. (37)

The sampling period for all sensor nodes is chosen as ℎ=0.003 which satisfies 0≤ℎ≤1/2𝜆_𝑛. Using the event condition in (9), we can draw the dynamic curve of the states of the sensors by Matlab as Figure 2. It shows us that the four sensor nodes reach the average consensus in mean square with fixed topology. The control signal when the events occur for each sensor is shown in Figure 3. We can see that the number of sensor control updates that only occur at the sampling instant under the event trigger condition is rapidly reduced to reach average consensus.

Figure 2 

The trajectories of the state vectors 𝑥1(𝑡), 𝑥2(𝑡), 𝑥3(𝑡), and 𝑥4(𝑡).

Figure 3 

Control inputs for the sensors.

Example 2. The switching topology of the four sensors is determined by the Markov chain 𝑟(𝑡) whose state space is 𝑆={1,2,3}. The related topology graph is 𝒢(𝑖)={𝒱(𝑖),ℰ(𝑖),𝒜(𝑖)} (see the topologies in Figure 4).

(a)

(b)

(c)

Figure 4 

The topologies of the four sensors in states 1, 2, and 3.

The adjacent matrices are 𝐴(1)=(2)=,𝐴(3)=.0110100010010010,𝐴01001001000101100010000110010110 (38)

The related degree matrices are 𝐷(1)=(2)=,𝐷(3)=.2000010000200001,𝐷10000200001000021000010000200002 (39)

The related Laplacian matrices are 𝐿(1)=,𝐿(2)=,𝐿(3)=.2−1−10−1100−102−100−111−100−120−1001−10−1−1210−10010−1−102−10−1−12 (40)

The nonzero largest and smallest eigenvalue are 𝜆_𝑛=3.4142 and 𝜆₂=0.5858, respectively.

Let the noise intensity matrices 𝐺 be 𝐺(1)=,𝐺(2)=,𝐺(3)=.0.00100000.00200000.00300000.0010.00200000.00300000.00300000.0010.00300000.00200000.00100000.001 (41)

The sampling period for all sensor nodes is chosen as ℎ=0.002 which satisfies 0≤ℎ≤1/2𝜆_𝑛(𝑖)(𝑖=1,2,3). Using the event condition in (18), we can draw the dynamic curve of the states of the sensors by Matlab as Figure 5. It shows us that the four sensor nodes reach the average consensus in mean square with Markovian switching topologies.

Figure 5 

The trajectories of the state vectors 𝑥1(𝑡), 𝑥2(𝑡), 𝑥3(𝑡), and 𝑥4(𝑡).

5. Conclusions

In this paper, we have dealt with the problem of average consensus in mean square of the WSNs. By using the event-based control mechanism, we have obtained several sufficient conditions to ensure the average consensus in mean square for WSNs with fixed and Markovian switching topologies. There are many other topics worth investigating, such as the packet loss and time-delay cases of the WSN’s consensus problem under event condition.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61075060, 61203337), the Innovation Program of Shanghai Municipal Education Commission (12zz064), the Specialized Research Fund for the Doctoral Program of Higher Education under Grant no. 20120075120009, and the Natural Science Foundation of Shanghai (12ZR1440200).