Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 610169, 8 pages

http://dx.doi.org/10.1155/2013/610169

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

College of Information Sciences and Technology, Donghua University, Shanghai 200051, China

Received 1 November 2013; Accepted 20 November 2013

Academic Editor: Guoliang Wei

Copyright © 2013 Chuan Ji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 be an undirected graph, where is the set of nodes, node represents the th sensor node, is the set of edges, and an edge in is denoted by an ordered pair . if and only if the th sensor node can send information to the th sensor node directly. The neighborhood of the th sensor node is denoted by .

is called the adjacency matrix of . For any , and . is called the in-degree of ; is called the out-degree of ; is called the Laplacian matrix of , where . Its eigenvalues are real and can be ordered as with and 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 where is the state of the th sensor and is the control input. The initial state is deterministic.

Each sensor includes a digital microprocessor and dynamics. The microprocessor of sensor monitors its own measurement value continuously and decides when to communicate with the neighboring sensors by broadcasting the actual measurement value. Therefore the latest broadcasted value of sensor can be described by the piecewise constant function where is sequence of event-times of sensor . We can see that the discrete-time signal is converted into the continuous-time signal .

The th sensor can receive information from its neighbors where denotes the measurement of the th sensor’s state by the th sensors, are the communication noises, and is the noise intensity function.

For the dynamic network , we give the event-based consensus protocl.

The dynamics of agent for is given by where is defined as

Substituting the protocol (5) into the system (2) leads to where is the Laplacian matrix of , is the noise intensity matrix, and is an -dimensional Brownian motion.

The event condition for agent has the following form: where

*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: where and .

If we define then (11) becomes

In WSNs, each sensor node communicates with other sensor nodes through the unreliable networks. If the communication channel between sensors and is , and 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 be a right-continuous Markov chain on the probability space taking values in a finite state set with generator given by where and is the transition rate from to if , while and .

We denote the undirected communication graph by , where is the undirected graph. Denote the topology graph by at moment , so if and only if .

Under Markovian switching topology, we have

So, under this condition, the event condition for agent has the following form: where

#### 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 .*

*Proof. *Define the function by
Computing along the trajectory generated by the system (11) for any , we have

Applying the inequality

it can be derived that
with .

We can see that
Integrating both sides of the previous inequality from to (), making use of (9), and taking the expectation, one can obtain that
By using Gronwall inequality, we have
Letting , we obtain
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 .*

*Proof. *Define the function by
Computing along the trajectory generated by the system (11) for any , we have
with .

For , compute

Integrating both sides of the above inequality from to (), making use of (18), and taking the expectation, one can obtain that

By using Gronwall inequality, we have
Letting , we obtain that

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 , where (see the topology in Figure 1, which is used in Dimarogonas [16]) and the initial state is .

The adjacent matrix is

The related degree matrix is

The related Laplacian matrix is

The nonzero largest and smallest eigenvalues are and , respectively.

Let the noise intensity matrix be

The sampling period for all sensor nodes is chosen as which satisfies . 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.

*Example 2. *The switching topology of the four sensors is determined by the Markov chain whose state space is . The related topology graph is (see the topologies in Figure 4).

The adjacent matrices are

The related degree matrices are

The related Laplacian matrices are

The nonzero largest and smallest eigenvalue are and , respectively.

Let the noise intensity matrices be

The sampling period for all sensor nodes is chosen as which satisfies . 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.

#### 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).

#### References

- J. A. Fax and R. M. Murray, “Information flow and cooperative control of vehicle formations,”
*IEEE Transactions on Automatic Control*, vol. 49, no. 9, pp. 1465–1476, 2004. View at Publisher · View at Google Scholar · View at Scopus - R. Olfati-Saber, “Distributed Kalman filter with embedded consensus filters,” in
*Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC '05)*, pp. 8179–8184, Seville, Spain, December 2005. View at Publisher · View at Google Scholar · View at Scopus - N. Lynch,
*Distributed Algorithms*, Morgan Kaufmann, San Matero, Calif, USA, 1996. - R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,”
*IEEE Transactions on Automatic Control*, vol. 49, no. 9, pp. 1520–1533, 2004. View at Publisher · View at Google Scholar · View at Scopus - D. B. Kingston and R. W. Beard, “Discrete-time average-consensus under switching network topologies,” in
*Proceedings of the American Control Conference*, pp. 3551–3556, Minneapolis, Minn, USA, June 2006. View at Scopus - L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging,”
*Systems and Control Letters*, vol. 53, no. 1, pp. 65–78, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Q. Zhang and J. Zhang, “Distributed consensus of continuous-time multi-agent systems with markovian switching topologies and stochastic communication noises,”
*Journal of Systems Science and Mathematical Sciences*, vol. 31, no. 9, pp. 1097–1110, 2011. View at Google Scholar · View at Zentralblatt MATH - Z. Wang, D. Ding, H. Dong, and H. Shu, “H∞ consensus control for multi-agent systems with missing measurements: the finite-horizon case,”
*Systems and Control Letters*, vol. 62, no. 10, pp. 827–836, 2013. View at Publisher · View at Google Scholar - D. Ding, Z. Wang, H. Dong, and H. Shu, “Distributed H∞ state estimation with stochastic parameters and nonlinearities through sensor networks: the finite-horizon case,”
*Automatica*, vol. 48, no. 8, pp. 1575–1585, 2012. View at Publisher · View at Google Scholar - H. Dong, Z. Wang, and H. Gao, “Distributed H∞ filtering for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks,”
*IEEE Transactions on Industrial Electronics*, vol. 60, no. 10, pp. 4665–4672, 2013. View at Publisher · View at Google Scholar - H. Dong, Z. Wang, and H. Gao, “Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts,”
*IEEE Transactions on Signal Processing*, vol. 60, no. 6, pp. 3164–3173, 2012. View at Publisher · View at Google Scholar - X. Wang and M. D. Lemmon, “Event-triggering in distributed networked control systems,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 3, pp. 586–601, 2011. View at Publisher · View at Google Scholar · View at Scopus - M. Mazo and P. Tabuada, “Decentralized event-triggered control over wireless sensor/actuator networks,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 10, pp. 586–601, 2011. View at Google Scholar - G. Seyboth, D. Dimarogonas, and K. Johansson, “Event-based broadcasting for multi-agent average consensus,”
*Automatica*, vol. 49, no. 1, pp. 245–252, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Meng and T. Chen, “Event based agreement protocols for multi-agent networks,”
*Automatica*, vol. 49, no. 7, pp. 2125–2132, 2013. View at Publisher · View at Google Scholar - D. V. Dimarogonas, E. Frazzoli, and K. H. Johansson, “Distributed event-triggered control for multi-agent systems,”
*IEEE Transactions on Automatic Control*, vol. 57, no. 5, pp. 1291–1297, 2012. View at Publisher · View at Google Scholar · View at Scopus