Consensus of the Distributed Varying Scale Wireless Sensor Networks

Abstract

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.

1. Introduction

Consensus is of great importance in many applications of wireless sensor networks (WSNs) [1], such as the clock synchronization [2], data fusion [3, 4], and formation control [5]. To achieve the consensus, it requires that all nodes of the networks achieve the special value, while they may begin with the different initial values.

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 [6], where the consensus problems are investigated for the fixed scale networks. The most further researches focus on optimizing the algorithms and the communicating topology, such as in [7–10], and the optimal algorithms are taken to solve the consensus. Other researchers devote to optimize the topology, such as Lin and Jia solve the average consensus problems of multiagents with time delay in the jointly connected topology [11]; Atrianfar and Haeri deal with the average consensus problems in the weak connected topology [12]; consensus problems are studied under the sleeping-awaking method [13, 14].

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 [15, 16].

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 [17]. Based on the connected network, consensus regions are investigated in the multiagent dynamical systems [18, 19].

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 2, some basic concepts and notations are introduced. In Section 3, we provided the main results. For each certain topology, the consensus criterions and the consensus regions are investigated in detail. For the varying topology in which some nodes may be added while some other nodes may be removed, the consensus criterions and the consensus regions are also provided. In Section 5, a numeral example shows the reliability of the proposed results. In Section 6, several conclusions are obtained.

2. Preliminaries

2.1. Graph Theory and Notations

Let ℵ={0,1,2,…,𝜅,𝜅+1,…} be all possible topologies of WSN, where the relations of the elements can be expressed by the partial sequence 0⪯1⪯2⪯⋯⪯𝜅⪯⋯, and 0 is the initial topology.

Let 𝜃∶𝑇→ℵ be the discernible function among the topologies, 𝜃(𝑘)=𝜅∈ℵ, where (𝑚^∼_𝜅−1+1)𝑙≤𝑘≤𝑚^∼_𝜅𝑙, 𝑙∈𝑅⁺, 𝑘=𝑧𝑙, 𝑧∈𝑁, 𝑇={0,𝑙,2𝑙,…,𝑚^∼₀𝑙,(𝑚^∼₀+1)𝑙,…,𝑚^∼₁𝑙,…,𝑚^∼_𝜅𝑙,…}, 𝑚^∼_𝜅=𝑚₀+𝑚₁+⋯+𝑚_𝜅, {𝑚₀,𝑚₁,…,𝑚_𝜅,…}⊆𝑍⁺, and 𝑚_𝜅 is called the dwell time.

According to the function 𝜃, the varying scale WSN (VSWSN) can be denoted by a varying graph 𝐺(𝜃(𝑚^∼_𝜅𝑙))=𝐺(𝜅)={𝑉(𝜅),𝐴(𝜅),𝐸(𝜅)}, where 𝑉(𝜅)=𝑉₁(𝜅)∪𝑉₂(𝜅), 𝑉₁(𝜅)={1_𝜅,2_𝜅,…,𝑖_𝜅,…,𝑢_𝜅} is the valid node set that all elements come from the former topology 𝜅−1, 𝑉₂(𝜅)={1^′_𝜅,2^′_𝜅,…,𝑣^′_𝜅} is the newly joined node set, and 𝐸(𝜅)={(𝑖_𝜅,𝑗_𝜅)∣𝑖_𝜅≠𝑗_𝜅,𝑖_𝜅,𝑗_𝜅∈𝑉(𝜅)} is the edge set. 𝐴(𝜅)=(𝑎_𝑖𝑗(𝜅))_𝑤𝜅∈𝑅^{𝑤𝜅×𝑤𝜅} is the weighted symmetric matrix, where 𝑤_𝜅=𝑢_𝜅+𝑣_𝜅, if 𝑎_𝑖𝑗(𝜅)=1, then there exists information flow between 𝑖_𝜅 and 𝑗_𝜅, if 𝑎_𝑖𝑗(𝜅)=0, then there is no communication between 𝑖_𝜅 and 𝑗_𝜅. 𝑁_𝑖𝜅={𝑗_𝜅∣𝑖_𝜅≠𝑗_𝜅,𝑖_𝜅,𝑗_𝜅∈𝑉(𝜅)} refers to the neighbor set of node 𝑖_𝜅 in the topology 𝜅.

𝐿(𝑤_𝜅)=(𝑙_𝑖𝑗(𝜅))_{𝑤𝜅×𝑤𝜅} is a Laplacian matrix, and 𝑙_{𝑖𝜅𝑗𝜅} is defined as 𝑙_𝑖𝑗(𝜅)=∑_𝑗𝜅𝑎_𝑖𝑗(𝜅),𝑖_𝜅=𝑗_𝜅,𝑗_𝜅∈𝑁_𝑖𝜅(𝑘),−𝑎_𝑖𝑗(𝜅),𝑖_𝜅≠𝑗_𝜅,𝑗_𝜅∈𝑁_𝑖𝜅(𝑘). (1)

Similarly, a component which the connected subgraph of 𝐺(𝜅) can be denoted by 𝐶^′(𝜅)={𝑉^′(𝜅),𝐴^′(𝜅),𝐸^′(𝜅)}, 𝑉^′(𝜅)⊂𝑉(𝜅),𝐸^′(𝜅)⊂𝐸(𝜅). The relation between 𝐺(𝜅) and 𝐶^′(𝜅) is denoted by 𝐶^′(𝜅)⊂𝐺(𝜅).

Definition 1. For all 𝑖_𝜅∈𝑉(𝜅), if there exists a component 𝐶^′(𝜅) of 𝐺(𝜅) such that 𝑖_𝜅∈𝑉^′(𝜅), then 𝐶^′(𝜅), is said to be the node attached component of 𝑖_𝜅, if there exists the sequence 𝐶^′(𝜅),𝐶^′(𝜅+1),…, such that 𝑖_𝜅∈𝑉^′(𝜅),𝑖_𝜅∈𝑉^′(𝜅+1)…, then the sequence 𝐶^′(𝜅),𝐶^′(𝜅+1),… is said to be the node attached component sequence of node 𝑖_𝜅.

Definition 2. For all 𝑖_𝜅,𝑗_𝜅∈𝑉(𝜅), if there exists two related attached component sequences 𝐶^′(𝜅₁), 𝐶^′(𝜅₂),…,𝐶^′(𝜅_𝑟),…, and 𝐶^′′(𝜅₁),𝐶^′′(𝜅₂),…,𝐶^′′(𝜅_𝑟),…, respectively, and if there exist 𝜅₀∈ℵ and ℵ^′={𝜅_𝑟1,𝜅_𝑟1+1,…,𝜅_𝑟1+𝑚1,𝜅_𝑟2+1,…,𝜅_𝑟2+𝑚2,…,𝜅_𝑟𝑞+1,𝜅_{𝑟𝑞+𝑚𝑞},…}⊂ℵ, such that 𝐶^′𝜅_𝑟1+𝑚1=𝐶^′′𝜅_𝑟1+𝑚1,𝐶^′𝜅_𝑟2+𝑚2=𝐶^′′𝜅_𝑟2+𝑚2𝐶,…,_′𝜅_{𝑟𝑞+𝑚𝑞}=𝐶^′′𝜅_{𝑟𝑞+𝑚𝑞},…, (2) where 𝑟₁,𝑚₁,…,𝑟_𝑞,𝑚_𝑞∈𝑍⁺, 𝜅₀⪯𝜅_𝑟1⪯𝜅_𝑟1+1⪯⋯⪯𝜅_{𝑟𝑞+𝑚𝑞}, then the communicating graph is said to be the local limited intersection connected (LLI connected), and 𝜅_𝑟1+𝑚1,𝜅_𝑟2+𝑚2,…,𝜅_{𝑟𝑞+𝑚𝑞} are called the intersection topologies.

Definition 3. For all 𝑖_𝜅,𝑗_𝜅∈𝑉(𝜅), if there exists two related attached component sequences 𝐶^′(𝜅₁), 𝐶^′(𝜅₁),…,𝐶^′(𝜅_𝑟),… and 𝐶^′′(𝜅₁),𝐶^′′(𝜅₁),…,𝐶^′′(𝜅_𝑟),…, respectively, and if there exist 𝜅₀∈ℵ and ℵ^′={𝜅_𝑟,𝜅_𝑟+1,…}⊂ℵ, such that for all 𝜅₀⪯𝜅_𝑟, 𝐶^′(𝜅_𝑟)=𝐶^′′(𝜅_𝑟), then the communicating graph is said to be the global limited intersection connected (GLI connected).

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

2.2. Problem Statement

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 𝑖_𝜅∈𝑉(𝜅), let 𝑥_𝑖(𝑘,𝜅) be the state of sensor or node 𝑖_𝜅, where 𝑥_𝑖(𝑘,𝜅)∈𝑅^𝑑,𝑑∈𝑍⁺. Suppose the state of 𝑖_𝜅 is given by 𝑥_𝑖(𝑘+𝑙,𝜅)=𝑏𝑥_𝑖(𝑘,𝜅)+𝑢_𝑖𝜅(𝑘), (3) where 𝑏 is a constant, 𝑢_𝑖𝜅(𝑘) is the consensus protocol and is given by 𝑢_𝑖𝜅(𝑘)=𝜖_{𝑖𝜅𝑗𝜅∈𝑁𝑖𝜅(𝑘)}𝑎_𝑖𝑗(𝜅)𝑥_𝑗(𝑘,𝜅)−𝑎^∼_𝑖𝑗(𝜅)𝑥_𝑖(𝑘,𝜅), (4) where 𝑎^∼_𝑖𝑗(𝜅) is a positive constant.

For all 𝐶^′(𝜅)⊂𝐺(𝜅), under the dynamic (3) and protocol (4), the state of component 𝐶^′(𝜅) which attaches on node 𝑖_𝜅 can be displayed as follows: 𝑋𝑛_𝜅(𝑘+𝑙)=𝐵𝑛_𝜅−𝜖𝑛_𝜅𝐿𝑛_𝜅𝑋𝑛_𝜅(𝑘), (5) where 𝑚^∼_𝜅−1+1𝑙≤𝑘≤𝑚^∼_𝜅𝑙,𝑛_𝜅=𝑝_𝜅+𝑞^′_𝜅≤𝑤_𝜅,𝐵𝑛_𝜅{},𝜖𝑛=diag𝑏,…,𝑏,…,𝑏_𝜅=diag𝜖_1𝜅,…,𝜖_𝑖𝜅,…,𝜖𝑛_𝜅. (6)𝐿𝑛_𝜅=(𝑙_{𝑖𝜅𝑗𝜅})𝑛_𝜅^×𝑛_𝜅, and 𝑙_{𝑖𝜅𝑗𝜅} is defined as 𝑙_{𝑖𝜅𝑗𝜅}=∑𝑎_𝑗𝜅^∼_𝑖𝑗(𝜅),𝑖_𝜅=𝑗_𝜅,𝑗_𝜅∈𝑁_𝑖𝜅(𝑘),−𝑎_𝑖𝑗(𝜅),𝑖_𝜅≠𝑗_𝜅,𝑗_𝜅∈𝑁_𝑖𝜅(𝑘),𝑋𝑛_𝜅(𝑘)=𝑋_𝑝𝜅(𝑘)^𝑇,𝑋_𝑞′𝜅(𝑘)^𝑇𝑇, (7)𝑋_𝑝𝜅(𝑘) is the vector state of the valid nodes that come from the former topology 𝜅−1, 𝑋_𝑞′𝜅(𝑘) is the vector state of the newly joined nodes, 𝑝_𝜅≤𝑢_𝜅, 𝑞^′_𝜅≤𝑣_𝜅, 𝑋_𝑝𝜅(𝑘)=𝑥_1𝜅(𝑘,𝜅),…,𝑥_𝑖(𝑘,𝜅),…,𝑥_𝑝(𝑘,𝜅)^𝑇,𝑖_𝜅∈𝑉₁(𝜅),𝑋_𝑞′𝜅(𝑘)=𝑥_1′(𝑘,𝜅),…,𝑥_𝑖′(𝑘,𝜅),…,𝑥_𝑞′(𝑘,𝜅)^𝑇,𝑖^′_𝜅∈𝑉₂(𝜅). (8)

Remark 4. Analogously, if 𝑛_𝜅 is substituted by 𝑤_𝜅, then system (5) expresses the state of the whole VSWSN.
For the node attached component 𝐶^′(𝜅), sensors 1^′_𝜅,…,𝑞^′_𝜅 are the newly joined nodes, and the state of each newly joined node at time 𝑘 is equal to the state at time 𝑘−𝑙 in the topology 𝜅−1.

Definition 5. For all 𝐶^′(𝜅)⊂𝐺(𝜅), if _{𝑖𝜅,𝑗𝜅∈𝑉′(𝜅)}𝑥_𝑖(𝑚^∼_𝜅𝑙,𝜅)−𝑥_𝑗(𝑚^∼_𝜅𝑙,𝜅)²<𝑐𝑑^𝑚𝜅∼_{𝑖𝜅𝑖𝜅∈𝑉′(𝜅)}𝑥𝑚_𝑖^∼_𝜅−1+1𝑙,𝜅², (9) where 0<𝑐<1, 𝑑^∼_𝑖𝜅 refers to the maximal degree node of 𝐶^′(𝜅), then VSWSN (5) is said to be the component potential consensus. In addition, if 𝐶^′(𝜅)=𝐺(𝜅) and (9) holds, then VSWSN (5) is said to be the global potential consensus.

Definition 6. For all 𝐶^′(𝜅)⊂𝐺(𝜅) and for all 𝑖_𝜅,𝑞_𝜅∈𝑉^′(𝜅), if lim_𝑘→∞𝑥_𝑖(𝑘,𝜅)−𝑥_𝑞(𝑘,𝜅)=0, (10) then VSWSN (5) is said to be achieved the component consensus. In addition, if 𝐶^′(𝜅)=𝐺(𝜅) and (10) holds, then VSWSN (5) is said to be achieved the global consensus.
Let 𝑥⁰_𝑖(𝑘,𝜅)=(1/𝑛_𝜅)∑_{𝑖𝜅∈𝑉′(𝜅)}𝑥_𝑖(𝑘,𝜅), if lim_𝑘→∞‖𝑥_𝑖(𝑘,𝜅)−𝑥⁰_𝑖(𝑘,𝜅)‖=0, then VSWSN (5) is said to be achieved the component average consensus. In addition, if 𝐶^′(𝜅)=𝐺(𝜅) and (10) holds, then VSWSN (5) is said to be achieved the global average consensus.

Theorem 7. For all 𝐶^′(𝜅)⊂𝐺(𝜅), if there exists a constant 𝜂, such that 𝐵𝑛_𝜅−𝜖𝑛_𝜅𝐿𝑛_𝜅^𝑇𝑚𝜅𝐿𝑛_𝜅𝐵𝑛_𝜅−𝜖𝑛_𝜅𝐿𝑛_𝜅^𝑚𝜅<𝑑^∼_𝑖𝜅𝜂^2𝑚𝜅𝐼𝑛_𝜅, (11) where 0<𝜂<1, then VSWSN (5) is the component potential consensus. In addition, if 𝐶^′(𝜅)=𝐺(𝜅), then VSWSN (5) is the global potential consensus.

Proof. Note that ∑_{𝑖𝜅,𝑞′𝜅∈𝑉′(𝜅)}‖𝑥_𝑖(𝑚^∼_𝜅𝑙,𝜅)−𝑥_𝑗(𝑚^∼_𝜅𝑙,𝜅)‖²=𝑋^𝑇𝑛_𝜅(𝑚^∼_𝜅𝑙)𝐿(𝑛_𝜅)𝑋𝑛_𝜅(𝑚^∼_𝜅𝑙)=𝑋𝑛^𝑇_𝜅((𝑚^∼_𝜅−1+1)𝑙)[(𝐵𝑛_𝜅−𝜖𝑛_𝜅𝐿𝑛_𝜅)^𝑇]^𝑚𝜅𝐿(𝑛_𝜅)(𝐵𝑛_𝜅−𝜖𝑛_𝜅𝐿𝑛_𝜅)^𝑚𝜅𝑋𝑛_𝜅((𝑚^∼_𝜅−1+1)𝑙), if [(𝐵𝑛_𝜅−𝜖𝑛_𝜅𝐿𝑛_𝜅)^𝑇]^𝑚𝜅𝐿(𝑛_𝜅)(𝐵𝑛_𝜅−𝜖𝑛_𝜅𝐿𝑛_𝜅)^𝑚𝜅<𝑑^∼_𝑖𝜅𝜂^2𝑚𝜅𝐼𝑛_𝜅, then _{𝑖𝜅,𝑞′𝜅∈𝑉′(𝜅)}𝑥_𝑖(𝑚^∼_𝜅𝑙,𝜅)−𝑥_𝑗(𝑚^∼_𝜅𝑙,𝜅)²<𝑑^∼_𝑖𝜅𝜂^2𝑚𝜅𝑋𝑛_𝜅((𝑚^∼_𝜅−1+1)𝑙)². (12)
Similarly, we can get the result when 𝐶^′(𝜅)=𝐺(𝜅).

Remark 8. If VSWSN (5) is the component potential consensus or the global potential consensus, then the errors among the adjacent nodes are decreased, but it does not mean that system achieves consensus, in any case, the consensus is constrained by the dwell time 𝑚_𝜅.

Assumption 1. We assume that for each intersection topology 𝜅, their exists 𝑚>0, such that 𝑚_𝜅>𝑚.

Remark 9. 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 𝑖_𝜅∈𝑉^′(𝜅), let 𝑒^𝑖_𝜅(𝑘)=𝑥_𝑖(𝑘,𝜅)−𝑥⁰_𝑖(𝑘,𝜅); we get the systematic error of (5) as follows: 𝑒_𝜅(𝑘)=𝑋𝑛_𝜅(𝑘)−𝟏𝑛_𝜅⊗𝑥⁰_𝑖(𝑘,𝜅)=Γ𝑋𝑛_𝜅(𝑘)=Γ𝐵𝑛_𝜅−𝐿𝑛_𝜅𝑋𝑛_𝜅(𝑘−𝑙), (13) where 𝑒_𝜅(𝑘)=(𝑒¹_𝜅(𝑘),𝑒²_𝜅(𝑘)…,𝑒𝑛_𝜅(𝑘))^𝑇, Γ=𝐼𝑛_𝜅−(1/𝑛_𝜅)×𝟏𝑛_𝜅×𝑛_𝜅, and 𝟏𝑛_𝜅^×𝑛_𝜅 is the 𝑛_𝜅×𝑛_𝜅 matrix in which each entry is 1.

The primary purpose of this paper is to provide the basic consensus analysis of (5) for the aforementioned scenarios. Because the scale of the network is a variable, different from the traditional literatures which investigate the consensus problems are based on the global structure, to study consensus problems, we depend on the component sequences. In the meanwhile, the consensus regions are gotten in the following sections.

3. Main Results

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.

3.1. Consensus Investigation for WSN in a Certain Topology

In this subsection, we study the potential consensus problems of the certain topology. For convenience, we focus on the component 𝐶^′(𝜅), its topology index is dropped and denoted by 𝐶^′, and its state is denoted by 𝑋𝑛(𝑘). The error of 𝑋𝑛(𝑘) is denoted by 𝑒(𝑘)=𝑋𝑛(𝑘)−𝟏𝑛_×1⊗𝑥⁰_𝑖(𝑘). To express the state of 𝑒(𝑘), we depend on the eigenvalues. Let eigen(𝐵𝑛−𝜖𝐿𝑛) be the eigenvalue set of the matrix (𝐵𝑛−𝜖𝐿𝑛), let 𝜆_𝑖 be the 𝑖th eigenvalue of 𝐵𝑛−𝜖𝐿𝑛, and 𝜆=max{|𝜆_𝑖|∣𝑖∈𝑉^′(𝜅)}.

To achieve the potential consensus, it requires that the error 𝑒(𝑘)=𝑋𝑛(𝑘)−𝟏𝑛_×1⊗𝑥⁰_𝑖(𝑘)𝑛(𝑘)𝐵𝑛𝑛𝑋𝑛()=Γ𝑋=Γ−𝜖𝐿𝑘−𝑙 (14) is decreasing. To keep the decreasing error, it requires that 𝜆 takes in some values, and the following discussion focuses on how to take the value of parameter 𝜖_𝑖 to get the needed 𝜆.

3.1.1. Potential Consensus under 𝑎^∼_𝑖𝑗=𝑎_𝑖𝑗

For 𝑎^∼_𝑖𝑗=𝑎_𝑖𝑗, we get 𝐿(𝑛_𝜅)=𝐿𝑛_𝜅, and the simplified protocol of (4) can be shown by 𝑢_𝑖=𝜖_{𝑖𝑗∈𝑁𝑖(𝑘)}𝑎_𝑖𝑗𝑥_𝑗(𝑘)−𝑥_𝑖(𝑘). (15)

Theorem 10. If 0≤𝑏<1 and 𝜖_𝑖 satisfies 0<𝜖_𝑖<1+𝑏2𝑑_𝑖, (16) where 𝑑_𝑖=∑_{𝑗∈𝑁𝑖}𝑎_𝑗𝑖, then under (15), each component of VSWSN (5) is the component potential consensus.

Proof. If 0≤𝑏<1, and 𝜖_𝑖 satisfies (16), based on the Gersgorin disc theorem, it holds that −1<𝑏−2𝑑_𝑖𝜖_𝑖<𝜆_𝑖<𝑏<1, (17) hence |𝜆|<1. We have [(𝐵𝑛−𝜖𝐿𝑛)^𝑇]^𝑚𝐿(𝑛_𝜅)(𝐵𝑛−𝜖𝐿𝑛)^𝑚<𝜆^∼2𝑚𝑑_𝑖𝐼𝑛. Following Theorem 7, subsystem (5) is the potential consensus. This completes the proof.

Theorem 11. If −1≤𝑏<0 and 𝜖_𝑖 satisfies −1+𝑏2𝑑_𝑖<𝜖_𝑖<0, (18) then under (15), subsystem (5) is the component potential consensus.

The proof is similar to that of Theorem 10, and it is omitted.

Remark 12. If 𝑏=1 and 𝑉(𝜅) is the fixed node set, then the related consensus problems discussed here are the same as in literatures [6, 20]. However, when |𝑏|>1, the protocol (15) and the protocol listed in the literatures [6, 20] cannot insure subsystem (5) to achieve the consensus, and the following section will solve the problems.
In (15), if 𝜖_𝑖 is equal to a certain value and 𝑉(𝜅) is the fixed node set, then system converges to one equilibrium point. In order to get the wide consensus regions, the value of 𝜖_𝑖 takes in (0,(1+𝑏)/2𝑑_𝑖), and the similarly reasons are taken by the Theorems 13 and 14 as follows.

3.1.2. Potential Consensus under 𝑎^∼_𝑖𝑗=𝑏𝜉/𝑑_𝑖𝜖_𝑖+𝑎_𝑖𝑗

When |𝑏|≥1, for each node 𝑖, changes its weight appropriately, the related consensus problems can be transformed to the problems of |𝑏|<1. If we choose 𝑎^∼_𝑖𝑗=𝑏𝜉/𝑑_𝑖𝜖_𝑖+𝑎_𝑖𝑗, then the protocol (4) can be expressed by 𝑢_𝑖=𝜖_{𝑖𝑗∈𝑁𝑖(𝑘)}𝑎_𝑖𝑗𝑥_𝑗(𝑘)−𝑥_𝑖(𝑘)−𝑏𝜉𝑥_𝑖(𝑘). (19) In this situation, the dynamic of each component in VSWSN can be shown by 𝑋𝑛(𝑘+𝑙)=𝐵𝑛−𝜖𝐿𝑛𝑋𝑛(𝑘), (20) where 𝐵𝑛=diag{𝑏(1−𝜉),𝑏(1−𝜉),…,𝑏(1−𝜉)}, 12𝑛𝜖=diag{𝜖,𝜖,…,𝜖}.

Based on Theorems 7 and 10, we get the following theorem.

Theorem 13. When 𝑏≥1, if 𝜖_𝑖 and 𝜉 satisfy 0<𝜖_𝑖<1+𝑏(1−𝜉)2𝑑_𝑖1,1≤𝜉<𝑏+1, (21) then, under (19), subsystem (20) is the component potential consensus.

Proof. According to Gersgorin disc theorem, we obtain |𝜆_𝑖−𝜖_𝑖𝑑_𝑖|≤|𝜖_𝑖|𝑑_𝑖. From (21), we get −1<𝑏(1−𝜉)−2𝜖_𝑖𝑑_𝑖<𝜆_𝜅<𝑏(1−𝜉)<1. Similarly to the proof of Theorem 10, based on Theorem 7, it is known that subsystem (20) is the component potential consensus. This completes the proof.

Similarly to Theorem 13, we obtain theorem as follows.

Theorem 14. When 𝑏≤−1, if 𝜖_𝑖 and 𝜉 satisfy −1+𝑏(1−𝜉)2𝑑_𝑖<𝜖_𝑖1<0,1<𝜉<−𝑏+1, (22) then under protocol (19), subsystem (20) is the component potential consensus.

3.2. Consensus Investigation for WSN in the Varying Topology

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 (14), we get 𝑒𝑚_𝜅^∼_𝜅𝑙=Γ𝑋𝑛𝑚_𝜅^∼_𝜅𝑙. (23) Let 𝜆_𝜅=max{|𝜆^∼|∣𝜆^∼∈eigen(𝐵𝑛_𝜅−𝜖𝑛_𝜅𝐿𝑛_𝜅)}, 𝑦_𝜅(𝑘)=norm²(𝑋_𝑞′𝜅(𝑘,𝜅)); it holds the following proposition.

Proposition 15. For all 𝐶^′(𝜅)⊂𝐺(𝜅), where 𝐶^′(𝜅) attaches on node 𝑖_𝜅, it can be shown that 𝑒𝑚_{𝑖𝜅,𝑗𝜅∈𝑉′(𝜅)}^𝑖_𝜅^∼_𝜅𝑙−𝑒𝑚^𝑗_𝜅^∼_𝜅𝑙²≤𝜆^2𝑚𝜅_𝜅⋯𝜆𝑑^2𝑚0′_0′^∼_𝑖𝜅𝑋_0′(0)²+Λ(𝜅), (24) where 0^′,1^′,2^′,…,𝜅 are the different topologies that 𝑖_𝜅 will appear successively as follows: Λ(𝜅)=𝑑^∼_𝑖𝜅𝜆^2𝑚𝜅_𝜅𝑦𝑚_𝜅^∼_𝜅𝑙+⋯+𝜆^2𝑚𝜅_𝜅⋯𝜆^2𝑚3′_3′𝜆^2𝑚2′_2′𝑦𝑚_2′^∼₂𝑙+𝜆^2𝑚𝜅_𝜅⋯𝜆^2𝑚1′_1′𝑦𝑚_1′^∼₁𝑙. (25)

Proof. Noting that 𝑒𝑚_{𝑖𝜅,𝑗𝜅}^𝑖_𝜅^∼_𝜅𝑙−𝑒𝑚^𝑗_𝜅^∼_𝜅𝑙²=𝑒𝑚^𝑇_𝜅^∼_𝜅𝑙𝐿𝑛_𝜅𝑒𝑚_𝜅^∼_𝜅𝑙, (26) from (23), it results in 𝑒𝑚_{𝑖𝜅,𝑗𝜅}^𝑖_𝜅^∼_𝜅𝑙−𝑒𝑚^𝑗_𝜅^∼_𝜅𝑙²=𝑋𝑛𝑚_𝜅^∼_𝜅𝑙^𝑇Γ^𝑇𝐿𝑛_𝜅Γ𝑋𝑛𝑚_𝜅^∼_𝜅𝑙=𝑋𝑛𝑚_𝜅^∼_𝜅−1+1𝑙^𝑇𝐵𝑛_𝜅−𝜖_𝜅𝐿𝑛_𝜅^𝑚𝜅Γ^𝑇𝐿𝑛_𝜅Γ×𝐵𝑛_𝜅−𝜖_𝜅𝐿𝑛_𝜅^𝑚𝜅𝑋𝑛𝑚_𝜅^∼_𝜅−1+1𝑙. (27) Since the absolute of each eigenvalue of Γ is no more than 1, the absolute of each eigenvalue of 𝐿(𝑛_𝜅) is no more than 𝑑^∼_𝑖𝜅, and the absolute of each eigenvalue of 𝐵𝑛_𝜅−𝜖𝐿𝑛_𝜅 is no more than 𝜆_𝜅. The following inequality is obtained: 𝑒𝑚^𝑇_𝜅^∼_𝜅𝑙𝐿𝑛_𝜅𝑒𝑚_𝜅^∼_𝜅𝑙≤𝜆𝑑^2𝑚𝜅_𝜅^∼_𝑖𝜅𝑋𝑛𝑚_𝜅^∼_𝜅−1+1𝑙^𝑇𝑋𝑛𝑚_𝜅^∼_𝜅−1+1𝑙. (28) On the other hand, 𝑋𝑛𝑚_𝜅^∼_𝜅−1+1𝑙^𝑇𝑋𝑛𝑚_𝜅^∼_𝜅−1+1𝑙≤𝑋𝑚_𝑝𝜅^∼_𝜅−1+1𝑙^𝑇𝑋𝑚_𝑝𝜅^∼_𝜅−1+1𝑙+𝑋𝑚_𝑞′𝜅^∼_𝜅−1+1𝑙^𝑇𝑋𝑚_𝑞′𝜅^∼_𝜅−1+1𝑙≤𝑋𝑚_𝑝𝜅^∼_𝜅−1+1𝑙^𝑇𝑋𝑚_𝑝𝜅^∼_𝜅−1+1𝑙+𝑦𝑚_𝜅^∼_𝜅−1𝑙, (29) where 𝑋_𝑞′𝜅(𝑘) refers to the state vector of all newly jointed nodes in component 𝐶^′(𝜅); 𝑋_𝑝𝜅(𝑘) means the state vector of the component which attaches on node 𝑖_𝜅 and all related valid nodes that come from the former topology 𝜅−1.
Considering some nodes in topology 𝜅−1 may be invalid at time 𝑚^∼_𝜅𝑙, the node attached component of 𝑖_𝜅 in the topology 𝜅−1 contains 1_𝜅,…,𝑝_𝜅 and may contain other nodes. In addition, the values of 1_𝜅,…,𝑝_𝜅 at time 𝑘 are equal to the 1_𝜅−1,…,𝑝_𝜅−1 at time 𝑘−𝑙, then it holds 𝑋_𝑝𝜅(𝑘)^𝑇𝑋_𝑝𝜅(𝑘)≤𝑋𝑛_𝜅−1(𝑘−𝑙)^𝑇𝑋𝑛_𝑝𝜅−1(𝑘−𝑙). (30) Following (29), (30), it can be shown that 𝑋𝑛𝑚_𝜅^∼_𝜅𝑙^𝑇𝐿𝑛_𝜅𝑋𝑛𝑚_𝜅^∼_𝜅𝑙≤𝜆𝑑^2𝑚𝜅_𝜅^∼_𝑖𝜅𝑋𝑛𝑚_𝜅−1^∼_𝜅−1+1𝑙^𝑇𝑋𝑛𝑚_𝜅−1^∼_𝜅−1+1𝑙+𝑦𝑚_𝜅^∼_𝜅−1𝑙. (31) Similaring to (31), we have 𝑋𝑛𝑚_𝜅−1^∼_𝜅−1+1𝑙^𝑇𝑋𝑛𝑚_𝜅−1^∼_𝜅−1+1𝑙≤𝜆^{2(𝑚𝜅−1)}_𝜅−1𝑋𝑚_𝑝𝜅−1^∼_𝜅−2+1𝑙^𝑇𝑋𝑚_𝑝𝜅−1^∼_𝜅−2+1𝑙+𝑦𝑚_𝜅−1^∼_𝜅−2𝑙. (32) Combining (32) and (31), we get 𝑋𝑚_𝜅^∼_𝜅𝑙^𝑇𝐿𝑛_𝜅𝑋𝑛𝑚_𝜅^∼_𝜅𝑙≤𝑑^∼_𝑖𝜅𝜆^2𝑚𝜅_𝜅𝜆^{2𝑚𝜅−1}_𝜅−1𝑋𝑚_𝑝𝜅−1^∼_𝜅−2+1𝑙^𝑇𝑋𝑚_𝑝𝜅−1^∼_𝜅−2+1𝑙+𝜆^2𝑚𝜅_𝜅𝜆^{2𝑚𝜅−1}_𝜅−1𝑦_𝜅−1𝑚_𝜅−1𝑙+𝜆^2𝑚𝜅_𝜅𝑦_𝜅𝑚_𝜅𝑙. (33) Based on (30), (31), (32), and (33), by the limited iterative, we get (24).

Theorem 16. If each component (5) of WSN is LLI connected, then the following statements hold. (i) If 0≤𝑏<1, 𝜖_𝑖𝜅 satisfies (16), and (34) holds, then under protocol (15), each component (5) achieves the component consensus.
(ii) If −1<𝑏≤0, 𝜖_𝑖𝜅 satisfies (18), and (34) holds, then under protocol (15), each component (5) achieves the component consensus, where lim_𝑚→∞Λ(𝜅)=0. (34)
In addition, if subsystem (5) is GLI connected, then under protocol (15), subsystem (5) achieves the global consensus.

Proof. (i) When 0≤𝑏<1 and 𝜖_𝑖𝜅 satisfies (16), if WSN is LLI connected and (34) holds, then for all 𝑖_𝜅′,𝑗_𝜅′∈𝑉(𝜅^′), there exist two related node attached components 𝐶^′ and 𝐶^′′ of 𝑖_𝜅′ and 𝑗_𝜅′, respectively, such that in the jointed topology 𝜅₀, (𝜅^′⪯𝜅₀) Proposition 15 holds.
Based on Theorem 10, Assumption 1, subsystem (5) is the component potential consensus, then lim_𝑚→∞𝜆^2𝑚𝜅_𝜅⋯𝜆𝑑^2𝑚0′₀^∼_𝑖𝜅𝑋_0′(0)²=0. (35)
On the other hand, if (34) holds, then lim_𝑚→∞(∑_{𝑖𝜅,𝑗𝜅∈𝑉′(𝜅)}‖[𝑒^𝑖_𝜅(𝑚^∼_𝜅𝑙)−𝑒^𝑗_𝜅(𝑚^∼_𝜅𝑙)]‖²)=0.
So it holds lim_𝑚→∞𝑥_𝑖(𝑘,𝜅)=𝑥_𝑗(𝑘,𝜅).
(ii) Similarly, we can get the result when −1<𝑏≤0; we also can show that VSWSN is the global consensus when WSN is GLI connected. This completes the proof.

Theorem 17. If component (5) is LLI connected, then the following statements hold. (i) If 𝑏≥1, 𝜖_𝑖𝜅,𝜉 satisfy (21), and if (34) holds, then under (19), subsystem (20) is the component consensus.
(ii) If −1≥𝑏, 𝜖_𝑖𝜅,𝜉 satisfy (22), and if (34) holds, then under (19), subsystem (20) is the component consensus.
In addition, if VSWSN (5) is GLI connected, then under protocol (19), VSWSN (20) is the global consensus.

Proof. Similar to the proof of Theorem 16, it is omitted.

4. The Consensus Regions

In fact, the former sections discuss consensus problems that require the state of each node keeps in coordination with the average value 𝑥⁰(𝑘,𝜅), and the value of 𝑥⁰(𝑘,𝜅) is decided by the parameters 𝜖_𝑖𝜅 and 𝜉. The regions of 𝜖_𝑖𝜅 and 𝜉 mean that the changeable value of 𝑥⁰(𝑘,𝜅) has a widely range. Hence, in this section, we investigate the consensus regions.

Definition 18. Under protocol (4) or (19), the region of 𝜖_𝑖𝜅 and 𝜉, such that VSWSN (5) is the component consensus or the global consensus, is called the consensus region.

In the varying topology, denote 𝑑_𝑖𝜅=∑_{𝑗𝜅∈𝑁𝑖𝜅}𝑎_𝑖𝑗(𝜅), Ψ¹_𝑖𝜅=−1+𝑏2𝑑_𝑖𝜅<𝜖_𝑖𝜅,Ψ<0_2𝑖𝜅=0<𝜖_𝑖𝜅<1+𝑏2𝑑_𝑖𝜅,Υ¹_𝑖𝜅=−1+𝑏(1−𝜉)2𝑑_𝑖𝜅<𝜖_𝑖𝜅,Υ<0_2𝑖𝜅=0<𝜖_𝑖𝜅<1+𝑏(1−𝜉)2𝑑_𝑖𝜅,Ξ¹_𝜅=11<𝜉<−𝑏+1,Ξ²_𝜅=11<𝜉<𝑏,Ψ+1_1𝜅=_{𝑖∈𝑉′(𝜅)}Ψ¹_𝑖𝜅,Ψ²_𝜅=_{𝑖∈𝑉′(𝜅)}Ψ²_𝑖𝜅,Υ¹_𝜅=_{𝑖∈𝑉′(𝜅)}Υ¹_𝑖𝜅,Υ²_𝜅=_{𝑖∈𝑉′(𝜅)}Υ²_𝑖𝜅, (36) and following Theorems 7, 10, and 13, we get the following corollary.

Corollary 19. Under protocol (4) or (19), if 𝐺(𝜅) is LLI connected or GLI connected, then the following statements hold.(1)If 0≤𝑏<1, then for all 𝜖_𝜅∈Ψ²_𝜅, system (5) achieves the component potential consensus or the global potential consensus, respectively.(2)If −1<𝑏<0, then for all 𝜖_𝜅∈Ψ¹_𝜅, system (5) achieves the component potential consensus or the global potential consensus, respectively.(3)If 𝑏≥1, then for all 𝜖_𝜅×𝜉∈Υ²_𝜅×Ξ²_𝜅, system (20) achieves the component potential consensus or the global potential consensus, respectively.(4)If 𝑏≤−1, then for all 𝜖_𝜅×𝜉∈Υ¹_𝜅×Ξ¹_𝜅, system (20) achieves the component potential consensus or the global potential consensus, respectively.

Following Theorem 16 and Corollary 19, it holds the following results.

Corollary 20. Under protocol (4) or (19), if 𝐺(𝜅) is LLI connected or GLI connected and (34) holds, then the following statements hold.(1)If 0≤𝑏<1, then for all 𝜖_𝜅∈Ψ²_𝜅, system (5) achieves the component consensus or the global consensus, respectively.(2)If −1<𝑏<0, then for all 𝜖_𝜅∈Ψ¹_𝜅, system (5) achieves the component consensus or the global consensus, respectively.(3)If 𝑏≥1, then for all 𝜖_𝜅×𝜉∈Υ²_𝜅×Ξ²_𝜅, system (20) achieves the component consensus or the global consensus, respectively.(4)If 𝑏≤−1, then for all 𝜖_𝜅×𝜉∈Υ¹_𝜅×Ξ¹_𝜅, system (20) achieves the component consensus or the global consensus, respectively.

Proof. (1) If 𝐺(𝜅) is LLI connected, based on Corollary 19, when 0≤𝑏<1, for all 𝜖_𝜅∈Ψ²_𝑖𝜅, under (4), WSN is the potential consensus. Following Theorem 16, if (34) holds, then subsystem (5) is the component consensus. Analogously, we can proof (2), (3), and (4).

5. Numerical Example

Example 21. Consider two node attached components 𝐶^′(𝜅) and 𝐶^′′(𝜅) of 1_𝜅,3_𝜅 (as shown in Figure 1), where 0,1, and 2 are the part topologies in sequence, 𝑎_𝑖𝑗(𝜅) takes value in 0 or 1, and 𝐶^′(1)=𝐶^′′(1) is the intersection topology. For 𝐶^′(𝜅) and 𝐶^′′(𝜅), the related node set sequences are shown as follows: 𝑉^′0=1₀,2₀,6₀,𝑉^′1=1₁,2₁,3₁,4₁,5₁,𝑉^′2=1₂,2₂,…,𝑉^′′0=3₀,4₀,𝑉^′′1=1₁,2₁,3₁,4₁,5₁,𝑉^′′2=3₂,5₂,…. (37)
Suppose that the state of 𝑖_𝜅 is shown as in (3), and consider the following three questions:(i)when 𝑏=0.6 or 𝑏=−0.6, in topology 0, under protocol (15), determine whether system (5) achieves the component potential consensus or not;(ii)when 𝑏=5 or 𝑏=−5, in topology 0, under protocol (19), determine whether system (20) achieves the component potential consensus or not;(iii)suppose that WSN is LLI connected, the partial node attached components of 1, 3 are in topology sequence 0,1, and 2 (as shown in Figure 1), and 𝑏=0.6, under protocol (15), determine whether system (5) achieves the component consensus or not. When WSN is GLI connected, two node attached components of 1, 3 are in topology sequence 0,1 (as shown in Figure 1), under protocol (15), determine whether system (5) achieves the global consensus or not.

Figure 1 

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

For the above questions, under the related protocol, we can get (34) holds firstly. Then we propose the following analysis.

For the first question, when 𝑏=0.6, from (16), we get the consensus region of 0 as follows: Ψ²₀=_{𝑖∈𝑉′(0)}Ψ²_𝑖0,Ψ²_𝑖0=𝜖_𝑖0∣0<𝜖_𝑖0<0.8𝑑_𝑖0,𝑖₀∈{1,2,3,4,6}. (38) Choosing (0.5,0.3,0.5,0.7,0.6)^𝑇∈Ψ²₀, we get 𝜖₀=diag{0.5,0.3,0.5,0.7,0.6}; when 𝑙=0.2, 𝑚₀=10, the simulation result is shown in Figure 2.

Figure 2 

In topology 0, the node attached components of 1 and 3 achieve the potential consensus, respectively.

When 𝑏=−0.6, from (16), we get the consensus region of topology 0 as follows: Ψ¹₀=_{𝑖∈𝑉′(0)}Ψ¹_𝑖0,Ψ¹_𝑖0=𝜖_𝑖0∣−0.8𝑑_𝑖0<𝜖_𝑖0,𝑖<0₀∈{1,2,3,4,6}. (39) Choosing (−0.55,−0.3,−0.5,−0.6,−0.4)^𝑇∈Ψ¹₀, we get 𝜖₀=diag{−0.55,−0.3,−0.5,−0.6,−0.4}; when 𝑙=0.2, 𝑚₀=10, the simulation result is shown in Figure 3.

Figure 3 

In topology 0, node attached components of 1 and 3 keep the component potential consensus, respectively.

For the second question, when 𝑏=5, from (21), we get the consensus region Υ²₀=∏_{𝑖0∈𝑉′(0)}Υ²_𝑖0, where Υ²₀=0<𝜖_𝑖0<1+5(1−𝜉)2𝑑_𝑖,Ξ²₀=61<𝜉<5,𝑖₀∈{1,2,3,4,6}. (40) Choosing 0.9×(0.5,0.3,0.5,0.7,0.6)^𝑇∈Ξ²×Υ²₀, we get 𝜖₀=diag{0.5,0.3,0.6,0.7,0.65}; when 𝑙=0.2, 𝑚₀=10, the simulation result is shown as Figure 4.

Figure 4 

In topology 0, node attached components of 1 and 3 keep the component potential consensus, respectively.

When 𝑏=−5, from (21), we get the consensus region Υ¹₀=∏_{𝑖0∈𝑉′(0)}Υ¹_𝑖0, where Υ¹_𝑖0=−1+5(1−𝜉)2𝑑_𝑖0<𝜖_𝑖0,Ξ<0₁₀=61<𝜉<5,𝑖₀∈{1,2,3,4,6}. (41) Choosing 0.9×(0.5,0.3,0.5,0.7,0.6)^𝑇∈Ξ¹×Υ¹₀, we have 𝜖₀=diag{−0.6,−0.2,−0.5,−0.5,−0.3}; when 𝑙=0.2, 𝑚₀=10, the simulation result is shown in Figure 5.

Figure 5 

In topology 0, node attached components of 1 and 3 keep the component potential consensus, respectively.

For the last question, when 𝑏=0.6, from (21) we get the consensus region of 𝜅 below, where 𝜅∈{0,1,2}, Ψ²_𝜅=_{𝑖∈𝑉′(𝜅)}Ψ²_𝑖𝜅,Ψ²_𝑖𝜅=0<𝜖_𝑖𝜅<1+5(1−𝜉)2𝑑_𝑖𝜅. (42) Similarly, we choose 𝜖₀=diag{0.5,0.3,0.6,0.7,0.65},??𝜖_𝑖1=𝜖_𝑖2=diag{0.5,0.3,0.6,0.7,0.65}. When 𝑙=0.2, if VSWSN is LLI connected, the part simulation results are shown in Figures 2, 6, and 7. And if VSWSN is GLI connected, the simulation results are shown in Figures 2 and 6.

Figure 6 

In topology 1, node attached components of 1 and 3 keep the component potential consensus, respectively.

Figure 7 

In topology 2, node attached components of 1 and 3 keep the component potential consensus, respectively.

6. Conclusions

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.

Acknowledgments

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).