#### Abstract

This paper considers a distributed sampled-data filtering problem in sensor networks with stochastically switching topologies. It is assumed that the topology switching is triggered by a Markov chain. The output measurement at each sensor is first sampled and then transmitted to the corresponding filters via a communication network. Considering the effect of a transmission delay, a distributed filter structure for each sensor is given based on the sampled data from itself and its neighbor sensor nodes. As a consequence, the distributed sampled-data filtering in sensor networks under Markovian switching topologies is transformed into mean-square stability problem of a Markovian jump error system with an interval time-varying delay. By using Lyapunov Krasovskii functional and reciprocally convex approach, a new bounded real lemma (BRL) is derived, which guarantees the mean-square stability of the error system with a desired performance. Based on this BRL, the topology-dependent sampled-data filters are obtained. An illustrative example is given to demonstrate the effectiveness of the proposed method.

#### 1. Introduction

A sensor network is composed of a large number of sensor nodes which are usually distributed in a spatial region. These sensor nodes are capable of cooperatively achieving some special tasks by communicating with neighbour nodes. In recent years, lots of attentions have been paid to sensor networks due to their wide applications such as environment monitoring and forecasting, object tracking, infrastructure safety, intelligent traffic system, and space exploration. An important problem in sensor networks is to observe the states of a system via the information exchange among sensor nodes. Since sensor nodes are spatially assigned in a large scale domain, it is practical to design a distributed algorithm for state estimation or filtering in sensor networks. Recently, the distributed filtering or state estimation over sensor networks has drawn considerable interests of many researchers. For example, a distributed filtering for sensor network was addressed in [1], where a consensus algorithm was introduced to make the estimate of each sensor asymptotically converge to the average estimation of these sensors. The distributed Kalman filtering algorithm in [1] was further improved in [2]. In [3], pining observers were designed under the condition that sensor can only observe partial states of the target. In [4], a novel distributed estimation scheme was proposed based on local Luenberger-like observers with a consensus strategy, where network-induced delays and package dropouts were considered.

In addition, filtering has been widely investigated in the past two decades due to its capability of minimizing the highest energy gain of the estimation error for all initial conditions and noise; see [5–7]. The aim of filtering is to design a stable filter by using the measurements outputs to estimate the system states or a combination of them. Such an approach has been recently applied to the distributed filtering for sensor networks. To mention some, in [8], the distributed filtering problem for sensor networks with multiple missing measurements was investigated, where the concept of consensus performance was defined to quantify the consensus degree over a finite horizon. By using the vector dissipativity, a novel approach to the design of distributed robust consensus filters was given in [9]. The filtering problem was investigated in [10] for class of nonlinear systems with randomly occurring incomplete information including both sensor saturations and missing measurements. The distributed filtering problem in sensor networks for discrete-time systems with missing measurements and communication link failure was studied in [11].

With the rapid development of digital technologies, the filtering based on sampled data has been exploited in recent years; see, for example, [12–16]. It is well known that sampled-data control can bring some advantages for multiagent systems such as flexibility, robustness, low cost, and energy saving. Due to the limited energy power of sensor networks, it is therefore more practical and significant to design distributed filters by using the sampled data of each sensor. In [17], a stochastic sampled-data approach to the analysis and design of distributed filters for sensor networks was proposed. It is worth pointing out that the sampled-data filtering in sensor networks has not yet been fully investigated. In addition, the communication topologies of sensor network may randomly change duo to the effect of link failures, packet dropouts, external disturbances, channel fading, task execution alteration, and so forth. In [18], the consensus-based distributed filtering problem for a discrete-time linear system was solved where the communication topology was time-varying with a stochastic process. Considering lossy sensor networks, the distributed finite-horizon filtering problem for a class of time-varying systems was investigated in [19]. In [20], a distributed robust estimation over sensor network with Markovian randomly varying topology was addressed, where the sufficient conditions were given to guarantee a suboptimal level of disagreement of estimates. Notice that these results with stochastically switching topologies are mainly concerned with the distributed filtering in pure continuous-time or discrete-time setting; however, there is little related work done for the distributed filtering in a sampled-data setting over sensor network with Markovian switching topologies, which motivates the current study.

In this paper, we aim to investigate the distributed sampled-data filtering in sensor networks, where the communication topologies are switched by a Markov chain. A new filter structure for each sensor node is given with the sampled data of the output measurements from itself and its neighbour sensor nodes, where the communication delay is taken into consideration. Based on this filter structure, the sampled-data filtering problem is transformed into the control problem of Markovian jump systems with a time-varying delay. Then, a new BRL for the system is obtained by employing Lyapunov-Krasovskii functional and reciprocally convex approach. Correspondingly, based on this the appropriate topology-dependent sampled-data filters are derived by solving a set of linear matrix inequalities, whose effectiveness is illustrated by a numerical example.

*Notation*. represents the set of natural numbers. is the identity matrix. and represent vectors whose entries are ones and zeros, respectively. denotes the Euclidean norm. is a diagonal matrix with diagonal entries . means that matrix is symmetric positive definite. The symbol denotes the symmetric terms in a symmetric matrix.

#### 2. Problem Statement

In this paper, we consider that the sensor network with sensor nodes is spatially distributed, whose topology is represented by a directed weighted graph of order , where and are the set of nodes and edges, represents a weighted adjacency matrix with nonnegative adjacency elements . An edge defined as implies that node can receive information from node . Node is considered as a neighbor of node if . For all , , and if ; otherwise, .

Consider continuous-time systems as follows: where and are the state vector and the output signal to be estimated, respectively; is exogenous disturbance belonging to ; , , and are known constant matrices of appropriate dimensions. The state of the system (1) is observed by a network of sensors, where each sensor is given as where is the measured output received by the sensor from the plant, is output measurement noise belonging to , and and are known constant matrices of appropriate dimensions.

Due to the possible occurrence of random events in sensor networks, we consider a group of directed graph , where is a continuous-time Markov process with values in a finite set . The transition probabilities are defined as where , as , and is the transition rate from mode to mode , which satisfies for and for . Then, it is easily known that .

*Remark 1. *In many practical applications, for example, the marine oil pervasion monitoring systems deployed in an oceanic area, see in [21], the topologies of the mobile sensor network may vary (switch) with the coverage area and the spatial distribution of the pervading oil, which is affected by some random factors such as wind, sea wave, and temperature. In this case, it is suitable to model the randomly switching network topologies as a Markov process [21]. Another example for Markovian switching topology can be seen in [22]. Hence, in this paper we focus on the filtering problem in the sensor network with Markovian switching topologies.

In this paper, we investigate the filter design issue for sensor networks in a sampled-data setting. Denote as the estimate of the plant’s state at sensor node and define the output estimation error for sensor node as It is assumed that the sampler is time-driven and the output estimation error of sensor is first sampled at each sampling instant , and then the sampled output estimation error will be sent to the ZOHs (zero order hold) of itself and its neighbors for the filter design through a communication network. Then, the ZOH of sensor is used to collect the sampled data from itself and its neighbor sensor and keep them constant until a new sampled data arrives. All the collected sampled data of the ZOH is then sent to filter for the estimate of the state . Considering the negative effects of network uncertainty, all the sampled-data transmitted via communication network is assumed to suffer an expected communication delay , where is constant and larger than zero. Also, we assume that there exists a constant such that . Based on the above analysis, the filter to be designed for sensor is given as follows: where , is the estimate of at sensor node , and matrix are parameters of filter to be designed later.

*Remark 2. *In fact, due to the introduction of a communication network, sampled-data information during transmission may suffer the network uncertainty such as communication delay, data packet dropouts, and disorders. In order to make the analysis easier, we only take the effect of communication delay into account. Considering the effect of communication delays, data packet dropouts and disorders simultaneously will be investigated in the future work.

Let and be the local estimate error and the local filtering error at sensor . From (1), (2), and (5), we have the following filtering error system for sensor node :

Defining and , we have where , , , , , , and for each fixed .

Define an “artificial delay’’ as , . Apparently, is piecewise-linear with the derivative at and is discontinuous at . Then, it is clear that , . Thus, the system (7) can be written as The initial condition of state is supplemented as , , with and , where denotes the Banach space of absolutely continuous functions with square-integrable derivatives and the norm .

Next, we need to introduce the following definition.

*Definition 3. *The system (8) with is said to be exponentially mean-square stable if there exist constants and such that

*The distributed ** sampled-data filtering problem* under consideration in the paper is to determine the parameters of the filter (5) such that the following requirements are simultaneously satisfied:(i)the filtering error system (8) with is exponentially mean-square stable;(ii)under the zero-initial condition, for a prescribed performance level , the filtering error satisfies
for any nonzero .

Before ending this section, the following lemmas are very useful for the proofs of the main results.

Lemma 4 (Schur complement [23]). *Let be a symmetric real matrix represented by , where is square and nonsingular. Then if and only if and .*

Lemma 5 (see [24]). *For any constant matrix , scalar , and vector function such that the following integration is well defined, then
*

Lemma 6 (see [25]). *For given positive integers , , a scalar in the interval , and a given matrix , two matrices and , define the function for all vector in as
**
If there exists a matrix such that , then the following inequality holds:
*

#### 3. Main Results

##### 3.1. Filtering Analysis

In this subsection, we will derive a BRL for the filtering error system (8). For this purpose, we first choose a Lyapunov-Krasovskii functional as where with , , , , , . For the sake of simplicity, let , and be a block entry matrix with . For example, and .

Now, we state and establish the following result.

Theorem 7. *For given scalars , , and filter parameters , the filtering error system (8) with is exponentially mean-square stable if there exist real matrices , , , , and some matrices of appropriate dimensions such that
**
for , where
*

*Proof. *Define the weak infinitesimal operator of along the trajectory (8) with respect to as
Then, along with (14), we have
where
with and . Applying Lemma 5, we have
Notice that
Using Lemma 6, we obtain

Taking the mathematical expectation on both sides of (20) and from (20)–(24), we have for
where . It is clear that if , then there exists a small enough constant such that , which means that the system (8) is mean-square stable.

Define a new function , where is a scalar to be determined. Then, we have

Integrating both sides of (26) from 0 to and taking the expectation, one can obtain that

Using the similar method in [26], we can know that there exists a scalar such that for

Due to the fact that , where ,
Therefore, it can be concluded from Definition 3 that the error system (8) is exponentially mean-square stable. Applying Lemma 4 to , one can arrive at (17). The proof is completed.

Next, we are in a position to obtain a sufficient condition that guarantees the performance in (11) for the filtering error system (8).

Theorem 8. *For given scalars , , and filter parameters , the filtering error system (8) is exponentially mean-square stable with a prescribed performance level if there exist real matrices , , , , and some matrices of appropriate dimensions such that
**
for , where
*

*Proof. *First, it is easily known that if the inequality (31) holds, then the inequality (17) is satisfied, which ensures that the filtering error system (8) with is exponentially mean-square stable. The remaining proof is to guarantee that under zero initial conditions, the filtering error system (8) satisfies the performance (11). Similar to the proof of Theorem 7, we can calculate
where . It is clear from Lemma 4 that the inequality (31) leads to . Under the zero initial condition, it follows from (33) that the inequality (11) holds. The proof of Theorem 8 is completed.

##### 3.2. Filter Design

Based on Theorem 8, we will derive a sufficient condition on the existence of the topology dependent filter parameters as follows.

Theorem 9. *For given scalars , , , the distributed filtering problem is solvable by the filters (5) if there exist real matrices , , , , , and some matrices of appropriate dimensions such that
**
for , where
**
Moreover, the filter gain is given by
*

*Proof. *First, for any real symmetric matrix and a given scalar , we have that , which implies that
Let for the conditions (17) in Theorem 7, and define , . Pre- and postmultiplying both sides of (31) by , respectively, and using Lemma 4, one can obtain (35), where the term is dealt with by the inequality (38). The proof of Theorem 9 is completed.

*Remark 10. *It should be pointed out that, to simplify the analysis, we assume that the communication delay is constant and larger than zero. In fact, the design method proposed in this paper can be extended to the more general case that the communication delay is time-varying with an upper bound.

#### 4. A Numerical Example

In this section, a numerical example is given to illustrate the effectiveness of the proposed method.

*Example 1. *Consider a continuous linear system as
with the initial state and the exogenous disturbance . The state of plant (39) is estimated by a sensor network with 4 sensor nodes, where the parameters of sensor are given by
and output measurement noise is given by . The network topologies of the sensor network are given by two directed graphs, whose adjacency matrices are given by
The topology switching is trigged by a Markov chain, whose transition rate is given by
Then, the switching rule of Markov chain is shown in Figure 1.

By Theorem 9, we design the filter parameters . Set the sampling period
Then, it is easily known that the maximum sampling period . It is assumed that all the sampled data is subject to a communication delay in the process of transmission. By choosing the performance level and the parameter , we employ Theorem 9 to obtain the suitable filer gains as follows
Based on the derived filter gains, we depict the filtering errors of node in Figure 2 and the system output and its estimates from filter in Figure 3. It follows from all the figures that the method proposed in the paper can effectively solve the distributed sampled-data filtering problem in sensor networks with a desired performance level .

#### 5. Conclusion

We have proposed a distributed sampled-data filtering in sensor networks with Markovian switching topologies. Each sensor node can receive sampled data of output measurement from itself and its neighbouring nodes for filter design. In this scheme, the distributed sampled-data filtering problem in sensor networks can be converted into the mean-square stable problem of a Markovian jump system with an interval time-varying delay. By using Lyapunov-Krasovskii functional approach, we have given a new BRL to guarantee the mean-square stability of the transformed system with a desired performance index . Based on this BRL, we have derived the filter gains corresponding to the network topology switching by solving a set of LMIs. Finally, the effectiveness of the proposed method has been illustrated by a numerical example.

In the future research, we will investigate network-based filtering for sensor networks like [27]. Another possible research direction is to extend the proposed method to sensor networks under energy constraints [28].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was supported by the Natural Science Foundation of China under Grant no. 61263003.