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Consensus problems are investigated for a type of the distributed varying scale wireless sensor network (VSWSN), where the scale of the network is increasing or decreasing due to the new nodes joining in or the invalid nodes quitting from WSN, respectively. In order to demonstrate the communicating behaviors more realistic, we offer the node attached component sequence for each valid node under the general sleep algorithm. Based on the component sequences, several concepts, such as the local limited intersection connection (LLI connection) and the global limited intersection connection (GLI connection), are provided to display the connectivity of the varying topology. Under certain conditions, the designed protocol ensures that all sensors arrive at the component consensus or the global consensus if the communicating graph is LLI connected or GLI connected, respectively. By the basic theoretical analysis, several consensus criterions are obtained. In the meanwhile, the consensus regions are investigated. The numerical example shows the reliability of the proposed results.

Consensus is of great importance in many applications of wireless sensor networks (WSNs) [

In the past few years, consensus problems have been widely investigated, and many exciting results have been obtained. The early work of consensus problems can be found in [

In recent years, some consensus results of the scale-free networks, where the scale of the networks may be increasing, are obtained in the exciting literatures [

And so far, consensus regions of the networks have attracted much attention by some researchers, such as Li et al. who propose an observer-type protocol to solve the consensus problem, and meanwhile an unbounded consensus region was provided [

It should point out that in the traditional consensus literatures, the scale of the network is fixed or increasing. Namely, the newly joined nodes and the invalid removed nodes have not been considered. In fact, the varying scale wireless sensor network (VSWSN) exists in many applications. For example, after the limited power exhausted, the nodes become invalid, or when the new nodes join the network, the scale of WSN is changed, and the related consensus regions are indefinite. However, up to now, consensus problems of VSWSN with the consensus regions have not caused much attention.

The main purpose of this paper is to investigate the consensus problems of VSWSN and its consensus regions. In the traditional literatures, the connected topology, the weak connected topology, or the jointly connected topology is the basic condition for the network communication. Different from the traditional literatures, in VSWSN, while some new nodes join in the network and some nodes quit from the network, VSWSN may neither connected, nor jointly connected, nor weak connected, and the communicating behaviors among those nodes are indefinite and random. The communicating graph of VSWSN which we proposed in this paper is displayed by the node attached component sequence. It turns out that if every two nodes have chance to connect, namely, the topology is the local limited intersection connected (LLI connected) or the global limited intersection connected (GLI connected), and the intersection topology has the enough dwell time, then under the designed protocol, VSWSN achieves the component consensus or the global consensus.

Moreover, the consensus regions are different from the traditional literatures. In the traditional literatures, the consensus value is determined by the state of the fixed node set. In VSWSN, the consensus value is varying according to the varying node set. So the considered systems in this paper are more general than that of network with the fixed node set, which the topology is the special case of the global limited intersection connection.

The outline of this paper is listed as follows. In Section

Let

Let

According to the function

Similarly, a component which the connected subgraph of

For all

For all

For all

Under LLI connected and GLI connected, we are going to discuss the consensus problems of VSWSN in the following sections.

In many applications, the communication of VSWSN is based on the multiple components. In this paper, the consensus problems which we considered are the component-based communication.

For

For all

Analogously, if

For the node attached component

For all

For all

Let

For all

Note that

Similarly, we can get the result when

If VSWSN (

We assume that for each intersection topology

If a sensor leaves from its neighbors and becomes an isolated node, and its state may not keep in coordination with other nodes. Note that if the communicating graph is LLI connected or GLI connected, the node has chance to keep in coordinating with other nodes.

For all

The primary purpose of this paper is to provide the basic consensus analysis of (

In this section, we are going to discuss the component potential consensus and the global potential consensus firstly. Then the relations between the component potential consensus and the component consensus, the global potential consensus and the global consensus are studied.

In this subsection, we study the potential consensus problems of the certain topology. For convenience, we focus on the component

To achieve the potential consensus, it requires that the error

For

If

If

If

The proof is similar to that of Theorem

If

In (

When

Based on Theorems

When

According to Geřsgorin disc theorem, we obtain

Similarly to Theorem

When

Based on the discussion of the potential consensus, in this subsection, we investigate the consensus problems of the node attached component in VSWSN.

From analog equation (

For all

Noting that

Considering some nodes in topology

If each component (

(ii) If

In addition, if subsystem (

(i) When

Based on Theorem

On the other hand, if (

So it holds

(ii) Similarly, we can get the result when

If component (

(ii) If

In addition, if VSWSN (

Similar to the proof of Theorem

In fact, the former sections discuss consensus problems that require the state of each node keeps in coordination with the average value

Under protocol (

In the varying topology, denote

Under protocol (

If

If

If

If

Following Theorem

Under protocol (

If

If

If

If

Consider two node attached components

Suppose that the state of

when

when

suppose that WSN is LLI connected, the partial node attached components of 1, 3 are in topology sequence

For nodes 1 and 3, there are two related node attached components.

For the above questions, under the related protocol, we can get (

For the first question, when

In topology

When

In topology

For the second question, when

In topology

When

In topology

For the last question, when

In topology

In topology

This paper has investigated the consensus problems of VSWSN. According to the local limited intersection connected and the global limited intersection connected, it has provided several criterions of the component consensus and the global consensus. All gotten results have been based on the structures of the node attached component sequences. Based on the structure, the results cannot be influenced by the increasing-decreasing node set. In the meanwhile, it has given the consensus regions.

This work is supported by the National Natural Science Foundation of China (61075060), the innovation program of Shanghai Municipal Commission (12zz064), and the Open Project of State Key Laboratory of Industrial Control Technology (ICT1231).