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

Authors: Chuan Ji; Wuneng Zhou; Huashan Liu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## 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.


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## Body

## 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 Section2, 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

Let𝒢={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 Ni={j∈𝒱∣(j,i)∈ℰ}.𝒜
=
[
a
i
j
]
∈
ℝ
n
×
n is called the adjacency matrix of 𝒢. For any i,j∈𝒱,aij≥0, and aij>0⇔j∈Ni.  degin(i)=∑j=1naij is called the in-degree of i; degout(i)=∑j=1naij is called the out-degree of i; L=𝒟-𝒜 is called the Laplacian matrix of 𝒢, where 𝒟=diag(degin(1),…,degin(n)). Its eigenvalues are real and can be ordered as
(1)λ1≤λ2≤⋯≤λn
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(2)x˙i(t)=ui(t),
where xi(t)∈ℝ is the state of the ith sensor and ui(t)∈ℝ is the control input. The initial state xi(0) is deterministic.Each sensor includes a digital microprocessor and dynamics. The microprocessor of sensori monitors its own measurement value xi(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
(3)x^i(t)=xi(tki),tki≤t<tk+1i,
where t0i,t1i,… is sequence of event-times of sensor i. We can see that the discrete-time signal xi(tki) is converted into the continuous-time signal x^i(t).Theith sensor can receive information from its neighbors
(4)y^ji(t)=x^j(t)+σjiωji(t),
where yji(t) denotes the measurement of the jth sensor’s state xj(t) by the ith sensors, {ωji(t)∣i,j=1,2,…,n} are the communication noises, and σji is the noise intensity function.For the dynamic network(𝒢,X), we give the event-based consensus protocl. 								(5)ui(t)=∑j∈Niaij(y^ji(t)-x^i(t)).The dynamics of agenti for t∈[tki+lh,tki+lh+h) is given by
(6)x˙i(t)=∑j∈Ni(t)‍aij[xj(tk′j)-xi(tki)+σjiωji(t)]=∑j∈Ni(t)‍aij[xj(tki+lh)-xi(tki+lh)]+∑j∈Ni(t)‍aij[xi(tki+lh)-xi(tki)]+∑j∈Ni(t)‍aij[xj(tk′j)-xj(tki+lh)]+∑j∈Ni(t)‍aijσjiωji(t)=∑j∈Ni(t)‍aij[xj(tki+lh)-xi(tki+lh)]+∑j∈Ni(t)‍aij[ej(tki+lh)-ei(tki+lh)]+∑j∈Ni(t)‍aijσjiωji(t),
where tk′j is defined as
(7)tk′j=max{t∣t∈{tkj,k=0,1,…},t≤tki+lh}.Substituting the protocol (5) into the system (2) leads to
(8)dx(t)=-[L(x(kh)+e(kh))]dt+GdW(t),
where L is the Laplacian matrix of 𝒢, G is the noise intensity matrix, and W(t)=(W1(t),…,Wn(t))T is an n-dimensional Brownian motion.The event condition for agenti has the following form:
(9)λn∥xi(tki+lh)∥2≤λ2∥ei(tki+lh)∥2+2∥G∥2,
where
(10)ei(tki+lh)=xi(tki)-xi(tki+lh).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:
(11)dδ(t)=-L[δ(kh)+(I-J)e(kh)]dt+(I-J)GdW(t),
where δ(t)=x(t)-Jx(t) and  J=(1/n)1n1nT.If we define(12)f(δ,t)=-L[δ(kh)+(I-J)e(kh)],g(δ,t)=(I-J)G,
then (11) becomes
(13)dδ(t)=f(δ,t)dt+g(δ,t)dW(t).In WSNs, each sensor node communicates with other sensor nodes through the unreliable networks. If the communication channel between sensorsi and j is (j,i)∈ℰf, and ℰf 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=(γij)N×N given by
(14)P{r(t+Δ)=j∣r(t)=i}={γijΔ+o(Δ),ifi≠j,1+γiiΔ+o(Δ),ifi=j,
where Δ>0 and γij≥0 is the transition rate from i to j if i≠j, while
(15)γii=-∑‍j≠iγij
and r(0)=r0.We denote the undirected communication graph by𝒢={𝒢(1),𝒢(2),…,𝒢(N)}, where 𝒢(k)={𝒱,ℰ𝒢(k),𝒜𝒢(k)} is the undirected graph. Denote the topology graph by 𝒢t at moment t(t≥0), so r(t)=k if and only if 𝒢t=𝒢(k).Under Markovian switching topology, we have(16)dx(t)=-[L(r(t))(x(kh)+e(kh))]dt+G(r(t))dW(t),(17)dδ(t)=-L(r(t))[δ(kh)+(I-J)e(kh)]dt+(I-J)G(r(t))dW(t).So, under this condition, the event condition for agenti has the following form:
(18)λn(L(r(t)))∥xi(tki+lh)∥2≤λ2(L(r(t)))∥ei(tki+lh)∥2+2∥G∥2,
where
(19)ei(tki+lh)=xi(tki)-xi(tki+lh).


## 2.1. Concepts in Graph Theory

Let𝒢={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 Ni={j∈𝒱∣(j,i)∈ℰ}.𝒜
=
[
a
i
j
]
∈
ℝ
n
×
n is called the adjacency matrix of 𝒢. For any i,j∈𝒱,aij≥0, and aij>0⇔j∈Ni.  degin(i)=∑j=1naij is called the in-degree of i; degout(i)=∑j=1naij is called the out-degree of i; L=𝒟-𝒜 is called the Laplacian matrix of 𝒢, where 𝒟=diag(degin(1),…,degin(n)). Its eigenvalues are real and can be ordered as
(1)λ1≤λ2≤⋯≤λn
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(2)x˙i(t)=ui(t),
where xi(t)∈ℝ is the state of the ith sensor and ui(t)∈ℝ is the control input. The initial state xi(0) is deterministic.Each sensor includes a digital microprocessor and dynamics. The microprocessor of sensori monitors its own measurement value xi(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
(3)x^i(t)=xi(tki),tki≤t<tk+1i,
where t0i,t1i,… is sequence of event-times of sensor i. We can see that the discrete-time signal xi(tki) is converted into the continuous-time signal x^i(t).Theith sensor can receive information from its neighbors
(4)y^ji(t)=x^j(t)+σjiωji(t),
where yji(t) denotes the measurement of the jth sensor’s state xj(t) by the ith sensors, {ωji(t)∣i,j=1,2,…,n} are the communication noises, and σji is the noise intensity function.For the dynamic network(𝒢,X), we give the event-based consensus protocl. 								(5)ui(t)=∑j∈Niaij(y^ji(t)-x^i(t)).The dynamics of agenti for t∈[tki+lh,tki+lh+h) is given by
(6)x˙i(t)=∑j∈Ni(t)‍aij[xj(tk′j)-xi(tki)+σjiωji(t)]=∑j∈Ni(t)‍aij[xj(tki+lh)-xi(tki+lh)]+∑j∈Ni(t)‍aij[xi(tki+lh)-xi(tki)]+∑j∈Ni(t)‍aij[xj(tk′j)-xj(tki+lh)]+∑j∈Ni(t)‍aijσjiωji(t)=∑j∈Ni(t)‍aij[xj(tki+lh)-xi(tki+lh)]+∑j∈Ni(t)‍aij[ej(tki+lh)-ei(tki+lh)]+∑j∈Ni(t)‍aijσjiωji(t),
where tk′j is defined as
(7)tk′j=max{t∣t∈{tkj,k=0,1,…},t≤tki+lh}.Substituting the protocol (5) into the system (2) leads to
(8)dx(t)=-[L(x(kh)+e(kh))]dt+GdW(t),
where L is the Laplacian matrix of 𝒢, G is the noise intensity matrix, and W(t)=(W1(t),…,Wn(t))T is an n-dimensional Brownian motion.The event condition for agenti has the following form:
(9)λn∥xi(tki+lh)∥2≤λ2∥ei(tki+lh)∥2+2∥G∥2,
where
(10)ei(tki+lh)=xi(tki)-xi(tki+lh).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:
(11)dδ(t)=-L[δ(kh)+(I-J)e(kh)]dt+(I-J)GdW(t),
where δ(t)=x(t)-Jx(t) and  J=(1/n)1n1nT.If we define(12)f(δ,t)=-L[δ(kh)+(I-J)e(kh)],g(δ,t)=(I-J)G,
then (11) becomes
(13)dδ(t)=f(δ,t)dt+g(δ,t)dW(t).In WSNs, each sensor node communicates with other sensor nodes through the unreliable networks. If the communication channel between sensorsi and j is (j,i)∈ℰf, and ℰf 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=(γij)N×N given by
(14)P{r(t+Δ)=j∣r(t)=i}={γijΔ+o(Δ),ifi≠j,1+γiiΔ+o(Δ),ifi=j,
where Δ>0 and γij≥0 is the transition rate from i to j if i≠j, while
(15)γii=-∑‍j≠iγij
and r(0)=r0.We denote the undirected communication graph by𝒢={𝒢(1),𝒢(2),…,𝒢(N)}, where 𝒢(k)={𝒱,ℰ𝒢(k),𝒜𝒢(k)} is the undirected graph. Denote the topology graph by 𝒢t at moment t(t≥0), so r(t)=k if and only if 𝒢t=𝒢(k).Under Markovian switching topology, we have(16)dx(t)=-[L(r(t))(x(kh)+e(kh))]dt+G(r(t))dW(t),(17)dδ(t)=-L(r(t))[δ(kh)+(I-J)e(kh)]dt+(I-J)G(r(t))dW(t).So, under this condition, the event condition for agenti has the following form:
(18)λn(L(r(t)))∥xi(tki+lh)∥2≤λ2(L(r(t)))∥ei(tki+lh)∥2+2∥G∥2,
where
(19)ei(tki+lh)=xi(tki)-xi(tki+lh).


## 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≤1/2λn.Proof.
Define the functionV:Rn×R+→R+ by
(20)V(δ,t)=12δT(t)δ(t).
Computing dV(δ,t) along the trajectory generated by the system (11) for any t∈[kh,(k+1)h), we have
(21)dV(δ,t)=[δT(t)f(δ,t)+trace[gT(δ,t)g(δ,t)]]dt+δT(t)(I-J)GdW(t)=[-δT(t)L[δ(kh)+(I-J)e(kh)]+∥(I-J)G∥2[-δT(t)L[δ(kh)+(I-J)e(kh)]]dt+δT(t)(I-J)GdW(t)=[(t-kh)[δ(kh)+(I-J)e(kh)]T×LTL[δ(kh)+(I-J)e(kh)]-δT(kh)L[δ(kh)+(I-J)e(kh)]+∥(I-J)G∥2]dt-(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)GdW(t)≤(hλn[δ(kh)+(I-J)e(kh)]T×L[δ(kh)+(I-J)e(kh)]-δT(kh)L[δ(kh)+(I-J)e(kh)]+∥(I-J)G∥2)dt-(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)GdW(t)=((hλn-1)δT(kh)Lδ(kh)+(2hλn-1)×δT(kh)L(I-J)e(kh)+hλn∥I-J∥2eT(kh)Le(kh)+∥(I-J)G∥2((hλn-1)δT(kh)Lδ(kh)+(2hλn-1))dt-(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)GdW(t).
Applying the inequality(22)δT(kh)L(I-J)e(kh)≤12∥L∥2δT(kh)δ(kh)+12∥I-J∥2eT(kh)e(kh),
it can be derived that(23)dV(δ(t),t)≤(-12δT(kh)Lδ(kh)+12∥I-J∥2eT(kh)Le(kh)+∥(I-J)G∥2(-12δT(kh)Lδ(kh))dt-(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)GdW(t)
with 2hλn≤1.
We can see that(24)d[e-tV(δ(t),t)]=e-t[-V(δ(t),t)dt+dV(δ(t),t)]≤e-t[-V(δ(t),t)-12δT(kh)Lδ(kh)+12∥I-J∥2eT(kh)Le(kh)+∥(I-J)G∥2[-V(δ(t),t)-12δT(kh)Lδ(kh)]dt-e-t(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+e-tδT(t)(I-J)GdW(t).
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
(25)e-t𝔼V(δ(t),t)≤∫kht-e-t𝔼V(δ(t),t)dt+12e-khδT(kh)δ(kh)-(12δT(kh)Lδ(kh)-12∥I-J∥2eT(kh)Le(kh)-∥(I-J)G∥2(12δT(kh)Lδ(kh))(e-t-e-kh)≤∫kht-e-t𝔼V(δ(t),t)dt+12e-khδT(kh)δ(kh)+(12δT(kh)Lδ(kh)-12∥I-J∥2eT(kh)Le(kh)-∥(I-J)G∥2(12δT(kh)Lδ(kh))(e-kh-e-(k+1)h)=∫kht-e-t𝔼V(δ(t),t)dt+12e-khδT(kh)δ(kh)+(12∥I-J∥2xT(kh)Lx(kh)-12∥I-J∥2eT(kh)Le(kh)-∥(I-J)G∥2(12∥I-J∥2xT(kh)Lx(kh))(e-kh-e-(k+1)h)≤∫kht-e-t𝔼V(δ(t),t)dt+12e-khδT(kh)δ(kh).
By using Gronwall inequality, we have
(26)𝔼V(δ(t),t)≤12e-kh∥δ(kh)∥2e-(t-kh)=12∥δ(kh)∥2e-t.
Letting t→∞(k→∞), we obtain
(27)limt→∞𝔼∥δ(t)∥2=0.
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≤1/2λn(r(t)).Proof.
Define the functionV:Rn×R+×S→R+ by
(28)V(δ,t,r(t))=12δT(t)δ(t).
Computing dV(δ,t,r(t)) along the trajectory generated by the system (11) for any t∈[kh,(k+1)h), we have
(29)dV(δ,t,r(t))=[+trace[gT(δ,t,r(t))g(δ,t,r(t))]]δT(t)f(δ,t,r(t))+trace[gT(δ,t,r(t))g(δ,t,r(t))]]dt+δT(t)(I-J)G(r(t))dW(t)=[+∥(I-J)G∥2+∑j=1N‍γijV(δ,t,r(t))]-δT(t)L(r(t))[δ(kh)+(I-J)e(kh)]+∥(I-J)G∥2+∑j=1NγijV(δ,t,r(t))]dt+δT(t)(I-J)G(r(t))dW(t)≤((hλn-1)δT(kh)L(r(t))δ(kh)+(2hλn-1)δT(kh)L(r(t))(I-J)e(kh)+hλn∥I-J∥2eT(kh)L(r(t))e(kh)+∥(I-J)G(r(t))∥2+γ-V(δ,t,r(t)))dt-(WT(t)-WT(kh))G(r(t))(I-J)×L(r(t))[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)G(r(t))dW(t)≤(-12δT(kh)L(r(t))δ(kh)+12∥I-J∥2eT(kh)L(r(t))e(kh)+∥(I-J)G(r(t))∥2+γ-V(δ,t,r(t))(-12δT(kh)L(r(t))δ(kh))dt-(WT(t)-WT(kh))G(r(t))(I-J)×L(r(t))[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)G(r(t))dW(t)
with 2hλn≤1.
Forγ->0,γ-=max{|γii:i∈S|}, compute
(30)d[e-2γ-tV(δ(t),t,r(t))]=e-2γ-t[-2γ-V(δ(t),t,r(t))dt+dV(δ(t),t,r(t))]≤e-2γ-t[∥(I-J)G(r(t))∥2+γ-V(δ,t,r(t))]-2γ-V(δ(t),t,r(t))-12δT(kh)L(r(t))δ(kh)+12∥I-J∥2eT(kh)L(r(t))e(kh)+∥(I-J)G(r(t))∥2+γ-V(δ,t,r(t))]dt-e-2γ-t(WT(t)-WT(kh))G(I-J)×L(r(t))[δ(kh)+(I-J)e(kh)]+e-2γ-tδT(t)(I-J)G(r(t))dW(t).
Integrating both sides of the above inequality fromkh to t (t∈[kh,(k+1)h)), making use of (18), and taking the expectation, one can obtain that
(31)e-2γ-t𝔼V(δ(t),t,r(t))≤∫kht-γ-e-2γ-t𝔼V(δ(t),t,r(t))dt+12e-2γ-khδT(kh)δ(kh)-(12δT(kh)L(r(t))δ(kh)-12∥I-J∥2eT(kh)L(r(t))e(kh)-∥(I-J)G(r(t))∥2(12δT(kh)L(r(t))δ(kh))(e-2γ-t-e-2γ-kh)≤∫kht-γ-e-2γ-t𝔼V(δ(t),t,r(t))dt+12e-2γ-khδT(kh)δ(kh)+(12δT(kh)L(r(t))δ(kh)-12∥I-J∥2eT(kh)L(r(t))e(kh)-∥(I-J)G(r(t))∥2(12δT(kh)L(r(t))δ(kh))×(e-2γ-kh-e-2γ-(k+1)h)=∫kht-γ-e-2γ-t𝔼V(δ(t),t,r(t))dt+12e-2γ-khδT(kh)δ(kh)+(12∥I-J∥2xT(kh)L(r(t))x(kh)-12∥I-J∥2eT(kh)L(r(t))×e(kh)-∥(I-J)G(r(t))∥2(12∥I-J∥2xT(kh)L(r(t))x(kh))×(e-2γ-kh-e-2γ-(k+1)h)≤∫kht-γ-e-2γ-t𝔼V(δ(t),t,r(t))dt+12e-2γ-khδT(kh)δ(kh).
By using Gronwall inequality, we have(32)𝔼V(δ(t),t,r(t))≤12e-2γ-kh∥δ(kh)∥2e-γ-(t-kh)=12e-γ-(t+kh)∥δ(kh)∥2.
Letting t→∞(k→∞), we obtain that
(33)limt→∞𝔼∥δ(t)∥2=0.
So the system (16) reaches average consensus in mean square.


## 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≤1/2λn.Proof.
Define the functionV:Rn×R+→R+ by
(20)V(δ,t)=12δT(t)δ(t).
Computing dV(δ,t) along the trajectory generated by the system (11) for any t∈[kh,(k+1)h), we have
(21)dV(δ,t)=[δT(t)f(δ,t)+trace[gT(δ,t)g(δ,t)]]dt+δT(t)(I-J)GdW(t)=[-δT(t)L[δ(kh)+(I-J)e(kh)]+∥(I-J)G∥2[-δT(t)L[δ(kh)+(I-J)e(kh)]]dt+δT(t)(I-J)GdW(t)=[(t-kh)[δ(kh)+(I-J)e(kh)]T×LTL[δ(kh)+(I-J)e(kh)]-δT(kh)L[δ(kh)+(I-J)e(kh)]+∥(I-J)G∥2]dt-(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)GdW(t)≤(hλn[δ(kh)+(I-J)e(kh)]T×L[δ(kh)+(I-J)e(kh)]-δT(kh)L[δ(kh)+(I-J)e(kh)]+∥(I-J)G∥2)dt-(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)GdW(t)=((hλn-1)δT(kh)Lδ(kh)+(2hλn-1)×δT(kh)L(I-J)e(kh)+hλn∥I-J∥2eT(kh)Le(kh)+∥(I-J)G∥2((hλn-1)δT(kh)Lδ(kh)+(2hλn-1))dt-(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)GdW(t).
Applying the inequality(22)δT(kh)L(I-J)e(kh)≤12∥L∥2δT(kh)δ(kh)+12∥I-J∥2eT(kh)e(kh),
it can be derived that(23)dV(δ(t),t)≤(-12δT(kh)Lδ(kh)+12∥I-J∥2eT(kh)Le(kh)+∥(I-J)G∥2(-12δT(kh)Lδ(kh))dt-(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)GdW(t)
with 2hλn≤1.
We can see that(24)d[e-tV(δ(t),t)]=e-t[-V(δ(t),t)dt+dV(δ(t),t)]≤e-t[-V(δ(t),t)-12δT(kh)Lδ(kh)+12∥I-J∥2eT(kh)Le(kh)+∥(I-J)G∥2[-V(δ(t),t)-12δT(kh)Lδ(kh)]dt-e-t(WT(t)-WT(kh))G(I-J)×L[δ(kh)+(I-J)e(kh)]+e-tδT(t)(I-J)GdW(t).
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
(25)e-t𝔼V(δ(t),t)≤∫kht-e-t𝔼V(δ(t),t)dt+12e-khδT(kh)δ(kh)-(12δT(kh)Lδ(kh)-12∥I-J∥2eT(kh)Le(kh)-∥(I-J)G∥2(12δT(kh)Lδ(kh))(e-t-e-kh)≤∫kht-e-t𝔼V(δ(t),t)dt+12e-khδT(kh)δ(kh)+(12δT(kh)Lδ(kh)-12∥I-J∥2eT(kh)Le(kh)-∥(I-J)G∥2(12δT(kh)Lδ(kh))(e-kh-e-(k+1)h)=∫kht-e-t𝔼V(δ(t),t)dt+12e-khδT(kh)δ(kh)+(12∥I-J∥2xT(kh)Lx(kh)-12∥I-J∥2eT(kh)Le(kh)-∥(I-J)G∥2(12∥I-J∥2xT(kh)Lx(kh))(e-kh-e-(k+1)h)≤∫kht-e-t𝔼V(δ(t),t)dt+12e-khδT(kh)δ(kh).
By using Gronwall inequality, we have
(26)𝔼V(δ(t),t)≤12e-kh∥δ(kh)∥2e-(t-kh)=12∥δ(kh)∥2e-t.
Letting t→∞(k→∞), we obtain
(27)limt→∞𝔼∥δ(t)∥2=0.
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≤1/2λn(r(t)).Proof.
Define the functionV:Rn×R+×S→R+ by
(28)V(δ,t,r(t))=12δT(t)δ(t).
Computing dV(δ,t,r(t)) along the trajectory generated by the system (11) for any t∈[kh,(k+1)h), we have
(29)dV(δ,t,r(t))=[+trace[gT(δ,t,r(t))g(δ,t,r(t))]]δT(t)f(δ,t,r(t))+trace[gT(δ,t,r(t))g(δ,t,r(t))]]dt+δT(t)(I-J)G(r(t))dW(t)=[+∥(I-J)G∥2+∑j=1N‍γijV(δ,t,r(t))]-δT(t)L(r(t))[δ(kh)+(I-J)e(kh)]+∥(I-J)G∥2+∑j=1NγijV(δ,t,r(t))]dt+δT(t)(I-J)G(r(t))dW(t)≤((hλn-1)δT(kh)L(r(t))δ(kh)+(2hλn-1)δT(kh)L(r(t))(I-J)e(kh)+hλn∥I-J∥2eT(kh)L(r(t))e(kh)+∥(I-J)G(r(t))∥2+γ-V(δ,t,r(t)))dt-(WT(t)-WT(kh))G(r(t))(I-J)×L(r(t))[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)G(r(t))dW(t)≤(-12δT(kh)L(r(t))δ(kh)+12∥I-J∥2eT(kh)L(r(t))e(kh)+∥(I-J)G(r(t))∥2+γ-V(δ,t,r(t))(-12δT(kh)L(r(t))δ(kh))dt-(WT(t)-WT(kh))G(r(t))(I-J)×L(r(t))[δ(kh)+(I-J)e(kh)]+δT(t)(I-J)G(r(t))dW(t)
with 2hλn≤1.
Forγ->0,γ-=max{|γii:i∈S|}, compute
(30)d[e-2γ-tV(δ(t),t,r(t))]=e-2γ-t[-2γ-V(δ(t),t,r(t))dt+dV(δ(t),t,r(t))]≤e-2γ-t[∥(I-J)G(r(t))∥2+γ-V(δ,t,r(t))]-2γ-V(δ(t),t,r(t))-12δT(kh)L(r(t))δ(kh)+12∥I-J∥2eT(kh)L(r(t))e(kh)+∥(I-J)G(r(t))∥2+γ-V(δ,t,r(t))]dt-e-2γ-t(WT(t)-WT(kh))G(I-J)×L(r(t))[δ(kh)+(I-J)e(kh)]+e-2γ-tδT(t)(I-J)G(r(t))dW(t).
Integrating both sides of the above inequality fromkh to t (t∈[kh,(k+1)h)), making use of (18), and taking the expectation, one can obtain that
(31)e-2γ-t𝔼V(δ(t),t,r(t))≤∫kht-γ-e-2γ-t𝔼V(δ(t),t,r(t))dt+12e-2γ-khδT(kh)δ(kh)-(12δT(kh)L(r(t))δ(kh)-12∥I-J∥2eT(kh)L(r(t))e(kh)-∥(I-J)G(r(t))∥2(12δT(kh)L(r(t))δ(kh))(e-2γ-t-e-2γ-kh)≤∫kht-γ-e-2γ-t𝔼V(δ(t),t,r(t))dt+12e-2γ-khδT(kh)δ(kh)+(12δT(kh)L(r(t))δ(kh)-12∥I-J∥2eT(kh)L(r(t))e(kh)-∥(I-J)G(r(t))∥2(12δT(kh)L(r(t))δ(kh))×(e-2γ-kh-e-2γ-(k+1)h)=∫kht-γ-e-2γ-t𝔼V(δ(t),t,r(t))dt+12e-2γ-khδT(kh)δ(kh)+(12∥I-J∥2xT(kh)L(r(t))x(kh)-12∥I-J∥2eT(kh)L(r(t))×e(kh)-∥(I-J)G(r(t))∥2(12∥I-J∥2xT(kh)L(r(t))x(kh))×(e-2γ-kh-e-2γ-(k+1)h)≤∫kht-γ-e-2γ-t𝔼V(δ(t),t,r(t))dt+12e-2γ-khδT(kh)δ(kh).
By using Gronwall inequality, we have(32)𝔼V(δ(t),t,r(t))≤12e-2γ-kh∥δ(kh)∥2e-γ-(t-kh)=12e-γ-(t+kh)∥δ(kh)∥2.
Letting t→∞(k→∞), we obtain that
(33)limt→∞𝔼∥δ(t)∥2=0.
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 isx˙i=ui, 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(34)A=(0110101011010010)The related degree matrix is(35)D=(2000020000300001).The related Laplacian matrix is(36)L=(2-1-10-12-10-1-13-100-11).The nonzero largest and smallest eigenvalues areλn=4 and λ2=1, respectively.Let the noise intensity matrixG be
(37)G=(0.00100000.00200000.00300000.001).The sampling period for all sensor nodes is chosen ash=0.003 which satisfies 0≤h≤1/2λn. 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 vectorsx1(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 chainr(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).The topologies of the four sensors in states 1, 2, and 3.
(a)
(b)
(c)The adjacent matrices are(38)A(1)=(0110100010010010),A(2)=(0100100100010110),A(3)=(0010000110010110).The related degree matrices are(39)D(1)=(2000010000200001),D(2)=(1000020000100002),D(3)=(1000010000200002).The related Laplacian matrices are(40)L(1)=(2-1-10-1100-102-100-11),L(2)=(1-100-120-1001-10-1-12),L(3)=(10-10010-1-102-10-1-12).The nonzero largest and smallest eigenvalue areλn=3.4142 and λ2=0.5858, respectively.Let the noise intensity matricesG be
(41)G(1)=(0.00100000.00200000.00300000.001),G(2)=(0.00200000.00300000.00300000.001),G(3)=(0.00300000.00200000.00100000.001).The sampling period for all sensor nodes is chosen ash=0.002 which satisfies 0≤h≤1/2λn(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 vectorsx1(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.



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*Source: 610169-2013-12-26.xml*