Adaptive Consensus of Distributed Varying Scale Wireless Sensor Networks under Tolerable Jamming Attacks

Abstract

Consensus problem is investigated for a varying scale wireless sensor network (VSWSN) under tolerable jamming attacks, where the scale of the network is increasing or decreasing due to the newly joined nodes or the removed nodes, respectively; the tolerable jamming attack means that the attack strength is limited. It supposes that during the communications, all nodes may encounter with the tolerable jamming attacks; when the attack power is larger than the given value, the attacked nodes fall asleep, or otherwise the nodes are awakened. Under the sleep method, based on the Lyapunov method, it shows that if the communicating graph is the global limited intersectional connection (GLI connection) and the system has the enough dwell time in the intersectional topology, then under the designed consensus protocol, all nodes achieve the global average consensus.

1. Introduction

In the past decades, distributed coordination of wireless sensor network (WSN) has been widely investigated, such as formation control, target-tracking, and environmental monitoring [1, 2]. For the distributed coordination, consensus is the fundamental requirement in which all states of sensors achieve a common value, such as the average consensus [3], and sample data-based consensus [4, 5]. The characteristics of WSN including the unreliable links and the limited energy supply render the challenges of developing algorithms and optimizing topology to achieve the consensus control; therefore, many topology optimization and algorithm development problems have been studied.

The early consensus work can be found in [6], where the general methods of consensus control are proposed. In recent years, consensus problems coupling with optimizing topology have been investigated. For instance, under a leader-following framework, the consensus problems were studied [7, 8]. More details can be found in [9–12]. Based on the sleeping-awaking method, consensus problem of the Markovian switching WSN with multiple time delays was studied [13]. Based on the stochastic matrices, the consensus algorithm was proposed in [14]; more results are proposed in [15, 16].

Recently, adaptive consensus problem has attracted much attention. For instance, a distributed consensus protocol with an adaptive law was proposed by adjusting the coupling weights [17]. According to iterative learning method, an adaptive consensus protocol was designed for all follower agents to track a leader [11]. More results are shown in [18, 19].

Notice that most of the above results on the consensus are associated with the fixed node set. However, in the real applications, the scale of WSN often is varying due to the node removal or the new nodes joining the network, where the node removal means that some nodes quit from the network because the energy is exhausted.

In recent years, the related consensus problems of the varying scale networks (VSNs) have risen researchers attentions, such as consensus of the scale-free network (SFN), where degree distribution follows a power law, at least asymptotically [20, 21]. In the literature [22], consensus problem of varying scale wireless sensor network (VSWSN) was investigated, where the varying topology of VSWSN is expressed by the node attached component sequences. As a result, the global limited intersectional connection (GLI connection) is the necessary condition of system achieving the global consensus.

2. Related Work

In fact, networks often encounter with some attacks, such as jamming attacks, tampering attacks, and exhaustion attacks. Under some attacks, the networks may be broken down, the coordinative behavior cannot be kept, and all nodes cannot achieve the consensus. In the literature [23], the synchronization against the removed nodes of complex dynamical networks was studied, where the communications are based on the switching topology. In recent years, consensus problem with the attacks has attracted some researchers attention. In [24], Wang et al. studied consensus problem of networks under the recoverable attacks, where after being attacked, the system becomes paralyzed; in the next period, the system recovers and achieves the consensus, and the relation between the state of system and the attack signal is not considered.

However, in real applications, the system often is influenced by the attack signals whatever the topology is optimized. Namely, the dynamic state of the system is impacted by the attack signals, and up to now, consensus problem of VSWSNs with the tolerable jamming attack has not attracted much attention.

The main contribution of this paper is to investigate the consensus problem of VSWSN under the tolerable jamming attacks. It begins with the introduction on communicating graph. Namely, all nodes communicate information among the components; the communications among all different node attached components are based on the intersectional topologies. Then the states of all components can be described by the different stochastic equations (DSEs); the consensus can be regarded that all trivial solutions of DSEs converge to the same value. The aim of this paper is to establish some criteria of VSWSN under the tolerable jamming attacks.

It should point out that the introduced topology in this paper is different from the previous results. In most literatures, such as the node set of network is fixed, system switches among the different spinning trees, or system communicates information in the union connected topology, and the related results cannot be applied for VSWSN because of the fixed scale. In fact, whatever VSWSN runs sleeping algorithm in any surroundings, for example, system encounters with attacks, the topology can be expressed by the node attached sequences, the connectivity of the network can be shown by local limited intersectional connection (LLI connection) or GLI connection, and the general connection is the special case of GLI connected.

The outline of this paper is listed as follows. In Section 3, some basic concepts, notations, and problem formulation are introduced. In Section 4, the main results are proposed. In Section 5, a numeral example shows the reliability of the proposed results. In Section 6, several conclusions are obtained.

3. Preliminaries

3.1. Notations and Some Conceptions

Notations. Throughout this paper, ℵ={0,1,2,…,𝜅,…} denotes the topology set of the varying scale wireless sensor network (VSWSN); the elements of the set satisfy the following partial sequence: 0⪯1⪯2⪯⋯⪯𝜅₁⪯𝜅₂⪯⋯, (1) where the listed topologies will appear in succession, 0 is the initial topology, and 𝜅₁, 𝜅₂ are called the adjacent topologies. Accordingly, [𝑡_𝜅1,𝑡_𝜅2) denotes the dwell time interval of topology 𝜅₁; if 𝜅₁≠𝜅₂, then 𝜅₁≺𝜅₂ denotes the relation between 𝜅₁ and 𝜅₂.

In order to express the varying topology, a discernible function 𝜃∶𝑇→ℵ is introduced; 𝜃(𝑡)=𝜅₁∈ℵ, where 𝑡∈[𝑡_𝜅1,𝑡_𝜅2). According to 𝜃(𝑡), the varying topology of VSWSN can be denoted by a varying graph 𝐺(𝜃(𝑡))=𝐺(𝜅₁)=(𝑉(𝜅₁),𝐸(𝜅₁),𝐴(𝜅₁)), where 𝑉(𝜅₁)=𝑉₁(𝜅₁)∪𝑉₂(𝜅₁) denotes the varying node set, 𝑉₁(𝜅₁)={1_𝜅1,2_𝜅1,…,𝑖_𝜅1,…,𝛼_𝜅1} refers to the valid node set of which all elements inherit from the former topology, 𝑉₂(𝜅₁)={1^′_𝜅1,2^′_𝜅1,…,𝛽^′_𝜅1}is the newly joined node set, and 𝐸(𝜅₁)={(𝑖_𝜅1,𝑗_𝜅1)∣𝑖_𝜅1≠𝑗_𝜅1,𝑖_𝜅1,𝑗_𝜅1∈𝑉(𝜅₁)} stands for the edge set.

𝑁_𝑖(𝜅)={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. 𝐴(𝜅)=(𝑎_𝑖𝑗(𝜅))_𝑤𝜅∈𝑅^{𝑤𝜅×𝑤𝜅} stands for the weighted symmetric matrix, where 𝑤_𝜅=𝛼_𝜅+𝛽_𝜅, 𝑎_𝑖𝑗(𝜅) takes value in 0 or 1. ∀𝑗_𝜅∈𝑁_𝑖(𝜅), 𝑎_𝑖𝑗(𝜅)=1 means that there exists information flow between the awaking nodes 𝑗_𝜅 and 𝑖_𝜅; if one of 𝑗_𝜅 or 𝑖_𝜅 is asleep, then 𝑎_𝑖𝑗(𝜅)=0; if 𝑗_𝜅∉𝑁_𝑖(𝜅), then 𝑎_𝑖𝑗(𝜅)≡0.

𝐿(𝑤_𝜅)=(𝑙_𝑖𝑗(𝜅))_{𝑤𝜅×𝑤𝜅} is the Laplacian matrix, and 𝑙_{𝑖𝜅𝑗𝜅} is defined by 𝑙_𝑖𝑗(𝜅)=^𝑗_𝜅𝑎_𝑖𝑗(𝜅),𝑖_𝜅=𝑗_𝜅,𝑗_𝜅∈𝑁_𝑖(𝜅),−𝑎_𝑖𝑗(𝜅),𝑖_𝜅≠𝑗_𝜅,𝑗_𝜅∈𝑁_𝑖(𝜅). (2)

The following conceptions are used frequently [22].

Definition 1. ∀𝑖_𝜅∈𝑉(𝜅); if there exists a component 𝐶^′_𝑖(𝜅) of 𝐺(𝜅) such that 𝑖_𝜅∈𝑉^′_𝑖(𝜅), then 𝐶^′_𝑖(𝜅) is said to be the node attached component of 𝑖_𝜅; if there exists the sequence 𝐶^′_𝑖(𝜅₁),𝐶^′_𝑖(𝜅₂),… such that 𝑖_𝜅∈𝑉^′_𝑖(𝜅₁), 𝑖_𝜅∈𝑉^′_𝑖(𝜅₂)…, then sequence 𝐶^′_𝑖(𝜅₁),𝐶^′_𝑖(𝜅₂),… is said to be the node attached component sequence of node 𝑖_𝜅.

𝐶^′_𝑖(𝜅) denotes a component of 𝐺(𝜅), where 𝐶^′_𝑖(𝜅)={𝑉^′_𝑖(𝜅),𝐴^′_𝑖(𝜅),𝐸^′_𝑖(𝜅)}, 𝑖_𝜅∈𝑉^′_𝑖(𝜅)⊂𝑉(𝜅),𝐸^′_𝑖(𝜅)⊂𝐸(𝜅). 𝑉^′_𝑖(𝜅)=𝑉^′_𝑖¹(𝜅)∪𝑉^′_𝑖²(𝜅), where 𝑉^′_𝑖¹(𝜅) refers to the valid node set of which all elements inherit from the former attached component of 𝑖_𝜅, 𝑉^′_𝑖²(𝜅) is the newly joined node set.

Definition 2. ∀𝑖_𝜅,𝑗_𝜅∈𝑉(𝜅); if there exists two related attached component sequences 𝐶^′_𝑖(𝜅₁), 𝐶^′_𝑖(𝜅₂),…,𝐶^′_𝑖(𝜅_𝑟),… and 𝐶^′′_𝑗(𝜅₁),𝐶^′′_𝑗(𝜅₂),…,𝐶^′′_𝑗(𝜅_𝑟),…, respectively, and if there exists 𝜅₀∈ℵ and ℵ^′={𝜅_𝑟1,𝜅_𝑟2,…}⊂ℵ, such that 𝐶^′_𝑖(𝜅_𝑟)=𝐶^′′_𝑗(𝜅_𝑟), where 𝜅₀⪯⋯⪯𝜅_𝑟⪯⋯, then the communicating graph is said to be the global limited intersectional connection (GLI connection), and 𝜅_𝑟 is called the intersectional topology.

Assumption 3. After the intersectional topology, the node set may be varied. For example, let 𝜅₁ be the intersectional topology, and 𝜅₂ is the next topology of it; in [𝑡_𝜅1,𝑡_𝜅2), all nodes will not be removed, but at time 𝑡_𝜅2, some new nodes will be removed and some nodes will be added.

Remark 4. Under Assumption 3, it follows that between every two adjacent intersectional topologies 𝜅_𝑟 and 𝜅^′_𝑟, each node 𝑖_𝜅 will appear in 𝜅^′_𝑟, where 𝜅_𝑟≺𝜅⪯𝜅^′_𝑟.
𝑖_𝜅1 and 𝑖_𝜅2 refer to a node in the different topology, where 𝜅₁≠𝜅₂.

3.2. Problem Statement

In many applications, the communication topology of VSWSN is based on the multiple components. In this paper, the communication is the component-based.

For 𝑖_𝜅∈𝑉(𝜅), let 𝑥_𝑖(𝑡,𝜅) be the state of sensor 𝑖_𝜅, where 𝑥_𝑖(𝑡,𝜅)∈𝑅. Suppose the state of 𝑖_𝜅 is given by 𝑥̇_𝑖(𝑡,𝜅)=𝑢_𝑖𝜅(t), (3) where 𝑢_𝑖𝜅(𝑡) is the consensus protocol, and it is given by 𝑢_𝑖𝜅(𝑡)=𝜀𝑛_{𝜅𝑗𝜅∈𝑁𝑖(𝜅)}𝑎_𝑖𝑗(𝜅)𝑦_𝑗(𝑡,𝜅)−𝑥_𝑖(𝑡,𝜅),𝑡∈𝑡_𝜅,𝑡_𝜅′, (4)𝜅^′ is the next topology of 𝜅, 𝑦_𝑗(𝑡,𝜅) is the state of 𝑗_𝜅 that is measured by 𝑖_𝜅, 𝑦_𝑗(𝑡,𝜅)=𝑥_𝑗(𝑡,𝜅)+𝑓_𝑖𝑗(𝑡)+𝑤_𝑖𝑗(𝑡), 𝑓_𝑖𝑗(𝑡) is the measured attack signal, 𝑓_𝑖𝑗(𝑡)=𝑓_𝑗𝑖(𝑡), and 𝑤_𝑖𝑗(𝑡) is the white noise.

Consider ∀𝐶^′_𝑖(𝜅)⊂𝐺(𝜅); based on the dynamic (3) and protocol (4), the dynamic state of component 𝐶^′_𝑖(𝜅) which attaches on node 𝑖_𝜅 is described by 𝑋̇^𝑖𝑛_𝜅(𝑡)=−𝜀𝑛_𝜅𝐿𝑛^𝑖_𝜅𝑋𝑛^𝑖_𝜅(𝑡)+𝜀𝑛_𝜅Γ^𝑖𝐹𝑛^𝑖_𝜅(𝑡)+𝑊𝑛^𝑖_𝜅(𝑡,𝑖_𝜅)∈𝑉^𝑖₁(𝜅),𝑡∈𝑡_𝜅,𝑡_𝜅′, (5) where 𝐿𝑛^𝑖_𝜅=(𝑙_{𝑖𝜅𝑗𝜅})𝑛_𝜅^×𝑛_𝜅, 𝑋𝑛^𝑖_𝜅(𝑘)=[𝑋^𝑖_𝑝𝜅(𝑘)^𝑇,𝑋^𝑖_𝑞′𝜅(𝑘)^𝑇]^𝑇, 𝑛_𝜅=𝑝_𝜅+𝑞_𝜅, 𝑋_𝑝𝜅(𝑡) refers to the state vector of node set 𝑉^𝑖₁(𝜅)={1_𝜅,2_𝜅,…,𝑖_𝜅,…,𝑝_𝜅}⊆𝑉₁(𝜅), 𝑝_𝜅≤𝛼_𝜅, 𝑋^𝑖_𝑞′𝜅(𝑘) is the state vector of the newly joined node set 𝑉^𝑖₂(𝜅)={1^′_𝜅,2^′_𝜅,…,𝑖^′_𝜅,…,𝑞^′_𝜅}, 𝑞_𝜅≤𝛽_𝜅, and 𝑋^𝑖_𝑝𝜅(𝑡)=𝑥_1𝜅(𝑡,𝜅),…,𝑥_𝑖(𝑡,𝜅),…,𝑥_𝑝(𝑡,𝜅)^𝑇,𝑖_𝜅∈𝑉^𝑖₁(𝜅),𝑋^𝑖_𝑞′𝜅(𝑡)=𝑥_1′(𝑡,𝜅),…,𝑥_𝑖′(𝑡,𝜅),…,𝑥_𝑞′(𝑡,𝜅)^𝑇,𝑖^′_𝜅∈𝑉^𝑖₂(𝜅),Γ^𝑖=diag𝛽^𝑇₁,𝛽^𝑇₂,…,𝛽^𝑇_𝑖,…,𝛽𝑛^𝑇_𝜅,𝛽^𝑇_𝑖=𝑎_𝑖1(𝜅),𝑎_𝑖2(𝜅),…,𝑎_𝑖𝑛_𝜅(𝜅),𝑊^𝑖(𝑡)=diag𝑊¹_𝑡,𝑊²_𝑡,…,𝑊^𝑖_𝑡𝑛,…,𝑊^𝜅_𝑡,𝑊^𝑖_𝑡=𝑤_𝑖1(𝑡),𝑤_𝑖2(𝑡),…,𝑤_𝑖𝑗(𝑡),𝑤_𝑖𝑛_𝜅(𝑡)^𝑇,𝐸𝑤_𝑖𝑗(𝑡)=0,𝐸𝑤_𝑖𝑗(𝑡)𝑤_𝑖𝑗(𝑡)^𝑇𝐹=1,_𝑖𝑛_𝜅(𝑡)=diag𝐹¹(𝑡),𝐹²(𝑡),…,𝐹^𝑖(𝑡),…,𝐹𝑛_𝜅(𝑡,𝐹_𝑖)(𝑡)=𝑓_𝑖1(𝑡),𝑓_𝑖2(𝑡),…,𝑓_𝑖𝑗(𝑡),𝑓_𝑖𝑛_𝜅(𝑡)^𝑇. (6)

Remark 5. Analogously, if 𝑛_𝜅 is substituted by 𝑤_𝜅, then system (5) refers to the whole system. Namely, 𝑋̇_𝑤𝜅(𝑡)=−𝜀_𝑤𝜅𝐿_𝑤𝜅𝑋_𝑤𝜅(𝑡)+𝜀_𝑤𝜅Γ𝐹_𝑤𝜅(𝑡)+𝑊_𝑤𝜅(𝑡,𝑖_𝜅)∈𝑉₁(𝜅),𝑡∈𝑡_𝜅,𝑡_𝜅′. (7)

Consider ∀𝑖_𝜅∈𝑉^𝑖₁(𝜅); let 𝑒_𝑗(𝑡,𝜅)=𝑥_𝑗(𝑡,𝜅)−𝑥⁰_𝑖(𝑡,𝜅), 𝑒(𝑡,𝜅)=𝑋(𝑡,𝜅)−𝑥⁰_𝑖(𝑡,𝜅)⊗1𝑛_𝜅, it obtains the systematic error of (5) as follows: 𝑒^𝑖(𝑡,𝜅)=𝑋𝑛^𝑖_𝜅(𝑡)−𝟏𝑛_𝜅⊗𝑥⁰_𝑖(𝑡,𝜅)=Φ_𝜅𝑋^𝑖𝑛_𝜅(𝑡), (8) where 𝑥⁰_𝑖(𝑡,𝜅)=1𝑛_{𝜅𝑖𝜅∈𝑉𝑖(𝜅)}𝑥_𝑖(𝑡,𝜅),𝑒^𝑖(𝑡,𝜅)=𝑒₁(𝑡,𝜅),…,𝑒_𝑖(𝑡,𝜅),…,𝑒𝑛(𝑡,𝜅)^𝑇,Φ=𝐼𝑛_𝜅−1𝑛_𝜅𝑛×𝟏_𝜅×𝑛_𝜅, (9) where 𝟏𝑛_𝜅^×𝑛_𝜅 is the 𝑛_𝜅×𝑛_𝜅 matrix in which each entry is 1.

From (8), one gets 𝑒^𝑖̇(𝑡,𝜅)=Φ^𝑖_𝜅𝑋̇𝑛^𝑖_𝜅(𝑡)=Φ^𝑖_𝜅−𝜀^𝑖_𝜅𝐿𝑛_𝜅𝑋𝑛^𝑖_𝜅(𝑡)+Φ_𝜅𝜀^𝑖𝑛_𝜅Γ^𝑖𝐹𝑛^𝑖_𝜅(𝑡)+𝑊𝑛^𝑖_𝜅(𝑡). (10)

Assumption 6. Suppose that each node can sense the strength of the attack signal in its perceivable areas; in terms of carrier sense of ASCENT algorithm [25], every sensor is awake or asleep according to the attacks, namely, 𝑎_𝑖𝑗(𝜅)=1,𝑓_𝑖𝑗(𝑡≤𝛼()𝜅)𝑥^𝑖(𝑡,𝜅)𝑑^𝑖_𝜅,0,otherwise, (11) where 𝛼(𝜅) is the constant and 𝑑^𝑖_𝜅 is the maximal degree of component which is attached by node 𝑖_𝜅.

Assumption 7. Under criterion (11), the topology of VSWSN is GLI connection.

Definition 8. For VSWSN (7), the jamming attacks are said to be the tolerable if VSWSN (7) satisfies Assumption 7.

Definition 9. Consider ∀𝐶^′_𝑖(𝜅)⊂𝐺(𝜅) and ∀𝑖_𝜅,𝑞_𝜅∈𝑉_𝑖(𝜅); if lim_𝑡→∞𝐸𝑥_𝑖(𝑡,𝜅)−𝑥_𝑞(𝑡,𝜅)²=0, (12) then VSWSN (5) is said to achieve the component consensus. In addition, if 𝐶^′_𝑖(𝜅)=𝐺(𝜅), and (12) holds, then VSWSN (5) is said to achieve the global consensus.
If lim_𝑡→∞𝐸(‖𝑥_𝑖(𝑡,𝜅)−𝑥⁰_𝑖(𝑡,𝜅)‖²)=0, then VSWSN (5) is said to achieve the component average consensus. In addition, if 𝐶^′_𝑖(𝜅)=𝐺(𝜅) and (12) holds, then VSWSN (5) is said to achieve the global average consensus.

In the following section, the consensus problem under the tolerable attacks is investigated via the error system (10).

Remark 10. According to Assumption 3, if VSWSN is GLI-connected, then each intersectional topology is connected, and it holds that ∑_{𝑗𝜅∈𝑁𝑖𝜅}𝑎_𝑖𝑗(𝜅)>0. According to the literature [26], all eigenvalues of −𝐿𝑛^𝑖_𝜅 satisfy 0=𝜆¹(𝜅)>𝜆²(𝜅)≥⋯≥𝜆𝑛_𝜅(𝜅)≥−2Δ(𝜅), (13) where Δ(𝜅)=max_{𝑗𝜅∈𝑁𝑖𝜅}𝑎_𝑖𝑗(𝜅)∣𝑖_𝜅∈𝑉(𝜅). (14)
For convenience, 𝜆^𝑖𝑗(𝜅) denotes the second largest eigenvalue of −𝐿(𝑤_𝜅), 𝑖_𝜅,𝑗_𝜅∈𝑉(𝜅), 𝜆^𝑖(𝜅) refers to the second largest eigenvalue of −𝐿(𝑛_𝜅), 𝑖_𝜅∈𝑉(𝜅), and 𝑗_𝜅∉𝑉(𝜅).

Remark 11. If a sensor leaves from its neighbors and becomes an isolated node, its state may not keep in coordination with other nodes temporarily. Note that if the communicating graph is GLI-connected, the node has a chance to communicate with other nodes and keep coordination with other nodes.

Remark 12. In model (5), the state of each node may be influenced by the attack signal function 𝐹^𝑖(𝑡).

Remark 13. System (5) achieves the component average consensus refers that to the fact that the norm of 𝑒^𝑖(𝑡,𝜅) converges to zero. Similarly, if 𝐶^′_𝑖(𝜅)=𝐺(𝜅), then the global average consensus means that the norm of 𝑒(𝑡,𝜅) converges to zero.

4. Main Results

This section will investigate the consensus problem while the system encounters with the jamming attacks. The aim of this section is to establish some consensus criteria of VSWSN.

Let 𝛿^𝑖𝑗(𝑡,𝜅)=12𝑒^𝑖𝑗(𝑡,𝜅)^𝑇𝑒^𝑖𝑗(𝑡,𝜅,𝛿_𝑖)(𝑡,𝜅)=12𝑒^𝑖(𝑡,𝜅)^𝑇𝑒^𝑖(𝑡,𝜅),𝛿^𝑗(𝑡,𝜅)=12𝑒^𝑗(𝑡,𝜅)^𝑇𝑒^𝑗(𝑡,𝜅), (15) where 𝑒^𝑖𝑗(𝑡,𝜅)=(𝑒₁(𝑡,𝜅)^𝑇,…,𝑒_𝑖(𝑡,𝜅)^𝑇,…,𝑒_𝑗(𝑡,𝜅)^𝑇,…,𝑒𝑛(𝑡,𝜅)^𝑇), then the update laws of 𝜀𝑛_𝜅(𝑛=0,1,2,…) are provided by 𝜀𝑛_𝜅=𝑐𝐸𝛿^𝑗𝑡_𝜅𝑚−1,𝜅_𝑚−1+𝐸𝛿^𝑗𝑡_𝜅𝑚,𝜅_𝑚, (16) where 𝑚≥1 and 𝑐 is a positive constant, and one proposition is obtained as follows.

Proposition 14. 𝛿^𝑖𝑗(𝑡,𝜅) satisfies 𝐸𝛿̇^𝑖𝑗(𝑡,𝜅)≤𝜀𝑛_𝜅𝜆^𝑖𝑗_𝜅𝐸𝛿^𝑖𝑗(𝑡,𝜅)+Δ𝛿^𝑖, (17) where Δ𝛿^𝑖=1/2𝐸[𝐹^𝑖(𝑡)^𝑇Γ^𝑇Φ^𝑇_𝜅Φ_𝜅𝑋𝑛_𝜅(𝑡)+𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅Φ_𝜅Γ𝐹^𝑖(𝑡)].

Proof. Note that 𝛿̇^𝑖𝑗(𝑡,𝜅)=12𝑒̇^𝑖𝑗(𝑡,𝜅)^𝑇𝑒^𝑖𝑗(𝑡,𝜅)+𝑒^𝑖𝑗(𝑡,𝜅)^𝑇𝑒̇^𝑖𝑗(𝑡,𝜅≤1)2𝑋𝑛_𝜅(𝑡)^𝑇−𝜀𝑛_𝜅𝐿𝑛_𝜅^𝑇Φ^𝑇_𝜅+𝐹^𝑖(𝑡)+𝑊(𝑡)^𝑇Γ^𝑇𝜀𝑛^𝑇_𝜅Φ^𝑇_𝜅×Φ_𝜅𝑋𝑛_𝜅(𝑡)+12𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅×Φ_𝜅−𝜀𝑛_𝜅𝐿𝑛_𝜅𝑋𝑛_𝜅(𝑡)+Φ_𝜅𝜀𝑛_𝜅Γ𝐹^𝑖(𝑡)(𝑡=1+𝑊)2𝑋𝑛_𝜅(𝑡)^𝑇−𝜀𝑛_𝜅𝐿𝑛_𝜅^𝑇Φ^𝑇_𝜅Φ_𝜅𝑋𝑛_𝜅(𝑡)+𝐹^𝑖(𝑡)+𝑊(𝑡)^𝑇×Γ^𝑇𝜀𝑛^𝑇_𝜅Φ^𝑇_𝜅Φ_𝜅𝑋𝑛_𝜅(𝑡)+𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅Φ_𝜅−𝜀𝑛_𝜅𝐿𝑛_𝜅𝑋𝑛_𝜅(𝑡)+𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅Φ_𝜅𝜀𝑛_𝜅Γ𝐹^𝑖(𝑡)+𝑊(𝑡); (18) then it holds that 𝐸𝛿̇^𝑖𝑗(𝑡,𝜅1)≤𝐸2𝑋𝑛_𝜅(𝑡)^𝑇−𝜀𝑛_𝜅𝐿𝑛_𝜅^𝑇Φ_𝜅^𝑇Φ_𝜅𝑋𝑛_𝜅(𝑡+1)2𝐸𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅Φ_𝜅−𝜀𝑛_𝜅𝐿𝑛_𝜅𝑋𝑛_𝜅(𝑡)+𝐹^𝑖(𝑡)^𝑇Γ^𝑇𝜀𝑛^𝑇_𝜅Φ^𝑇_𝜅Φ_𝜅𝑋𝑛_𝜅(𝑡)+𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅Φ_𝜅𝜀𝑛_𝜅Γ𝐹^𝑖(𝑡𝑛)≤𝜀_𝜅𝜆^𝑖𝑗_𝜅𝐸𝑒^𝑖𝑗(𝑡,𝜅)^𝑇𝑒^𝑖𝑗(𝑡,𝜅)+Δ𝛿^𝑖=𝜀𝑛_𝜅𝜆^𝑖𝑗_𝜅𝐸𝛿^𝑖𝑗(𝑡,𝜅)+Δ𝛿^𝑖. (19) This completes the proof.

Let 𝛿^𝑖(𝑡,𝜅)=𝛿^𝑖₁(𝑡,𝜅)+𝛿^𝑖₂(𝑡,𝜅), (20) where 𝛿^𝑖₁(𝑡,𝜅)=12𝑒^𝑖₁(𝑡,𝜅)^𝑇𝑒^𝑖₁(𝑡,𝜅),𝛿^𝑖₂(𝑡,𝜅)=12𝑒^𝑖₂(𝑡,𝜅)^𝑇𝑒^𝑖₂(𝑡,𝜅),𝑒^𝑖₁(𝑡,𝜅)=𝑒₁(𝑡,𝜅)^𝑇,…,𝑒_𝑖(𝑡,𝜅)^𝑇,…,𝑒_𝑝(𝑡,𝜅)^𝑇,𝑒^𝑖₂(𝑡,𝜅)=𝑒_1′(𝑡,𝜅)^𝑇,…,𝑒_𝑖′(𝑡,𝜅)^𝑇,…,𝑒_𝑞′(𝑡,𝜅)^𝑇; (21) under Assumption 6, it holds the following proposition.

Proposition 15. Functions 𝛿^𝑖𝑗(𝑡,𝜅), 𝛿^𝑖(𝑡,𝜅), and 𝛿^𝑗(𝑡,𝜅) satisfy the following inequality: 𝐸𝛿^𝑖𝑗𝑡,𝜅_𝑚≤exp𝜀𝑛_𝜅𝑚𝜆^𝑖𝑗_𝜅𝑚𝑇_𝜅𝑚+^𝜅𝑚−1_{𝑠𝑖=𝜅0}𝜆^𝑖_𝑠𝑇_𝑠×𝐸𝛿^𝑖𝑡₀^𝑖,0^𝑖+exp𝜀𝑛_𝜅𝑚𝜆^𝑖𝑗_𝜅𝑚𝑇_𝜅𝑚+^𝜅𝑚−1_{𝑠𝑗=𝜅0}𝜆^𝑗_𝑠𝑇_𝑠×𝐸𝛿^𝑗𝑡₀^𝑗,0^𝑗+𝑓𝑡,𝜅_𝑚+𝑔𝑡,𝜅_𝑚, (22) where 𝜆^𝑖𝑗_𝜅=max{𝛼(𝜅)+𝜆^𝑖𝑗_𝜅}, 𝜆^𝑖_𝜅=max{𝛼(𝜅)+𝜆^𝑖_𝜅}, 𝑇_𝑠=𝑡_𝜅𝑠−𝑡_𝜅𝑠−1, and 𝑔^𝑖𝑡,𝜅_𝑚=exp𝜀𝑛_𝜅𝑚−1𝜆^𝑖_𝜅𝑚−1𝑡−𝑡_𝜅𝑚−2𝛿^𝑖₂𝑡_𝜅𝑚−2,𝜅_𝑚−2𝑇_𝜅𝑚−2+exp𝜀𝑛_𝜅𝑚−1𝜆^𝑖_𝜅𝑚−1𝑇_𝜅𝑚−1+𝜀𝑛_𝜅𝑚−1𝜆^𝑖_𝜅𝑚−2𝑇_𝜅𝑚−2×𝛿^𝑖₂𝑡_𝜅𝑚−3,𝜅_𝑚−3𝜀𝑛+⋯+exp_𝜅𝑚−1𝜆^𝑖_𝜅𝑚−1𝑇_𝜅𝑚−1+⋯+𝜀𝑛_𝜅1𝜆^𝑖_𝜅1𝑇_𝜅1×𝛿^𝑖₂𝑡_𝜅0,𝜅₀,𝑔^𝑗𝑡,𝜅_𝑚=exp𝜀𝑛_𝜅𝑚−1𝜆^𝑗_𝜅𝑚−1𝑡−𝑡_𝜅𝑚−2𝛿^𝑗₂𝑡_𝜅𝑚−2,𝜅_𝑚−2𝑇_𝜅𝑚−2+exp𝜀𝑛_𝜅𝑚−1𝜆^𝑗_𝜅𝑚−1𝑇_𝜅𝑚−1+𝜀𝑛_𝜅𝑚−1𝜆^𝑗_𝜅𝑚−2𝑇_𝜅𝑚−2×𝛿^𝑗₂𝑡_𝜅𝑚−3,𝜅_𝑚−3𝜀𝑛+⋯+exp_𝜅𝑚−1𝜆^𝑗_𝜅𝑚−1𝑇_𝜅𝑚−1+⋯+𝜀𝑛_𝜅′1𝜆^𝑗_𝜅′1𝑇_𝜅′1×𝛿^𝑗₂𝑡_𝜅0′,𝜅_0′, (23) where 𝜅₀ and 𝜅^′₀ are the initial topologies of 𝑖_𝜅,𝑗_𝜅, respectively.

Proof. From criterion (11), it is straightforward that ‖Γ𝐹^𝑖(𝑡)‖≤‖𝑋𝑛_𝜅(𝑡)‖; then Δ𝛿^𝑖=12𝐸𝐹^𝑖(𝑡)^𝑇Γ^𝑇Φ^𝑇_𝜅Φ_𝜅𝑋𝑛_𝜅(𝑡)+𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅Φ_𝜅Γ𝐹^𝑖(𝑡≤𝛼()𝜅)2𝐸𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅Φ_𝜅𝑋𝑛_𝜅(𝑡)+𝑋𝑛_𝜅(𝑡)^𝑇Φ^𝑇_𝜅Φ_𝜅𝑋𝑛_𝜅(𝑡()≤𝛼𝜅)𝐸𝛿^𝑖(𝑡,𝜅). (24) Combine (24) with (19); it holds that 𝐸𝛿̇^𝑖𝑗(𝑡,𝜅)≤𝜀𝑛_𝜅𝛼(𝜅)+𝜆^𝑖𝑗_𝜅𝐸𝛿^𝑖𝑗(𝑡,𝜅𝑛)=𝜀_𝜅𝜆^𝑖𝑗_𝜅𝐸𝛿^𝑖𝑗(𝑡,𝜅); (25) then ∀𝑡∈[𝑡_𝜅𝑚+1,𝑡_𝜅𝑚), 𝐸𝛿^𝑖𝑗𝑡,𝜅_𝑚+1≤𝐸𝛿^𝑖𝑗𝑡_𝜅𝑚,𝜅_𝑚exp^𝑡_𝑡𝜅𝑚𝜀_𝜅𝑚𝜆^𝑖𝑗_𝜅𝑚,𝛿𝑑𝑠≤𝐸^𝑖𝑗𝑡_𝜅𝑚,𝜅_𝑚exp𝜀_𝜅𝑚𝜆^𝑖𝑗_𝜅𝑚𝑡−𝑡_𝜅𝑚. (26) Notice that 𝐸𝛿^𝑖𝑗𝑡_𝜅𝑚,𝜅_𝑚≤𝐸𝛿^𝑖𝑡_𝜅𝑚,𝜅_𝑚+𝐸𝛿^𝑗𝑡_𝜅𝑚,𝜅_𝑚,𝐸𝛿^𝑖𝑡_𝜅𝑚,𝜅_𝑚≤exp𝜀_𝜅𝑚𝜆^𝑖_𝜅𝑚𝑡_𝜅𝑚−𝑡_𝜅𝑚−1×𝐸𝛿_𝑖𝑡_𝜅𝑚−1,𝜅_𝑚−1𝜀≤⋯≤exp_𝜅𝑚𝜆^𝑖_𝜅𝑚𝑡_𝜅𝑚−𝑡_𝜅𝑚−1+𝜀_𝜅𝑚−1𝜆^𝑖_𝜅𝑚−1𝑡_𝜅𝑚−1−𝑡_𝜅𝑚−2+⋯+𝜀_𝜅1𝜆^𝑖_𝜅1𝑡_𝜅1−𝑡_𝜅0𝐸𝛿^𝑖𝑡_𝜅0,𝜅₀+𝑔^𝑖𝑡,𝜅_𝑚,=exp^𝜅𝑚−1_𝑠=𝜅0𝜀_𝑠𝜆^𝑖_𝑠𝑇_𝑠𝐸𝛿^𝑖𝑡_𝜅0,𝜅₀+𝑔^𝑖𝑡,𝜅_𝑚,𝐸𝛿^𝑗𝑡_𝜅𝑚,𝜅_𝑚≤exp𝜀_𝜅𝑚𝜆^𝑗_𝜅𝑚𝑡_𝜅𝑚−𝑡_𝜅𝑚−1×𝐸𝛿_𝑗𝑡_𝜅𝑚−1,𝜅_𝑚−1𝜀≤⋯≤exp_𝜅𝑚𝜆^𝑗_𝜅𝑚𝑡_𝜅𝑚−𝑡_𝜅𝑚−1+𝜀_𝜅𝑚−1𝜆^𝑗_𝜅𝑚−1𝑡_𝜅𝑚−1−𝑡_𝜅𝑚−2+⋯+𝜀_𝜅1𝜆^𝑗_𝜅1𝑡_𝜅1−𝑡_𝜅0𝐸𝛿^𝑗𝑡_𝜅0,𝜅₀+𝑔^𝑗𝑡,𝜅_𝑚,=exp^𝜅𝑚_𝑠=𝜅0𝜀_𝑠𝜆^𝑗_𝑠𝑇_𝑠𝐸𝛿^𝑗𝑡_𝜅0,𝜅₀+𝑔^𝑗𝑡,𝜅_𝑚; (27)(28) from inequalities (26), (27), and (28), it holds the inequality (22). This completes the proof.

Next, the consensus criterion is proposed as follows.

Theorem 16. If subsystem (7) is the GLI-connected and the each attack signal satisfies Assumption 6, then under (4), VSWSN (7) achieves the global average consensus if there is the enough dwell time in the interactional topology 𝜅_𝑚, namely, 𝑇_𝜅𝑚1>max𝜆^𝑖𝑗_𝜅𝑚−^𝜅𝑚−1_𝑠=𝜅0𝜆^𝑖_𝑠𝑇_𝑠,1𝜆^𝑖𝑗_𝜅𝑚−^𝜅𝑚−1_{𝑠=𝜅0′}𝜆^𝑗_𝑠𝑇_𝑠, (29) and 𝜀𝑛_𝜅 satisfies (16).

Proof. For inequality (29), it follows that 𝜆^𝑖𝑗_𝜅𝑚<0, 𝜆^𝑖_𝜅𝑚≤0, based on Proposition 15, and if each attack signal satisfies Assumption 6, then lim_{𝑇𝜅𝑚→∞}𝐸𝛿^𝑖𝑗(𝑡,𝜅)=0; (30) this completes the proof.

Remark 17. In 𝐸[𝛿^𝑖𝑗(𝑡,𝜅_𝑚+1)]≤𝐸[𝛿^𝑖𝑗(𝑡_𝜅𝑚,𝜅_𝑚)]exp[𝜀_𝜅𝑚𝜆^𝑖𝑗_𝜅(𝑡−𝑡_𝜅𝑚)], even though 𝜆^𝑖𝑗_𝜅<0, it cannot ensure that the system achieves the average consensus; see simulation results of Example 2. In addition, if adaptive parameter (16) is utilized, whatever 𝐸[𝛿^𝑖𝑗(𝑡_𝜅𝑚,𝜅_𝑚)] tends to infinite, the system achieves the consensus, and this result is different from the literature [7].

Remark 18. Theorem 16 shows that under the tolerable jamming attacks, if VSWSN is the GLI-connected and the value of 𝜀𝑛_𝜅 is chosen appropriately, then VSWSN achieves the consensus. The numerical example of the following section shows the reliability.

5. Numerical Example

Example 1. Suppose that VSWSN (7) is composed of the following two node attached components: 𝐶¹_𝜅=𝑉¹(𝜅),𝐸¹(𝜅),𝐴¹(𝜅,𝐶_3𝜅)=𝑉³(𝜅),𝐸³(𝜅),𝐴³(𝜅), (31) where 𝑇₀(𝑡)={1,2,3}, the figure of 𝑇₀(𝑡) is shown in Figure 1, 𝑎, 𝑏, 𝑐 refer to the attack signals under the different topologies, and the circles refer to the sensible regions. For convenience, the topology indexes of nodes are dropped in the following. Suppose that 𝑉¹1={1,2},𝑉³2={},𝑉1,2,5₁3={},𝑉1,2,3,4,5₃1={3},𝑉³2={},𝑉3,4₃3={1,2,3,4,5}. (32)
The related matrices are listed as follows: 𝐿¹1=,𝐿1−1−11₁2=,𝐿1−10−12−10−11₁3=,𝐿1−1000−1200−1001−1000−1100−1001₃2=,𝐿1−1−11₃3=𝐿¹3. (33)
Under protocol (4), the dynamic state of the node attached component is given by (5); the related parameters are listed below.
In topology 1, the second largest eigenvalue of −𝐿¹(1) and −𝐿³(1) is −2; the jamming attack signal is 𝑓_𝑎(𝑡)=2‖𝑥¹(𝑡,1)‖, 𝛼(1)=1, the measured signals are 𝑓₁₃(𝑡)=𝑓₂₃(𝑡)=𝑓_𝑎(𝑡), and Assumption 6 cannot be satisfied; node 3 falls in sleeping (see topology 1 of Figure 1). Taking 𝑐=0.1, suppose that 𝜀_𝜅 satisfies (16); the simulation results are shown in Figure 2.
In topology 2, according to carrier sense, node 3 is awakened, node 5 joined the node attached component of 1, and node 4 joined the node attached component of 3 (see topology 2 of Figure 1). For this topology, the second largest eigenvalue of −𝐿¹(2) and −𝐿³(2) is −1. Suppose that the jamming attack signal is 𝑓_𝑏(𝑡)=‖𝑥²(𝑡,2)‖, 𝛼(1)=0.2, the measured signals are 𝑓₁₃(𝑡)=𝑓₂₃(𝑡)=0.01𝑓_𝑏(𝑡), and Assumption 6 is satisfied; taking 𝑐=0.1, suppose that 𝜀_𝜅 satisfies (16); the simulation results are shown in Figure 3.
In topology 3, two node attached components are merged (see topology 3 of Figure 1). For this topology, the second largest eigenvalue of −𝐿¹(3) is −1. Suppose that the jamming attack signal is 𝑓_𝑐(𝑡)=‖𝑥³(𝑡,3)‖, 𝛼(1)=0.3, the measured signals are 𝑓₁₃(𝑡)=𝑓₃₁(𝑡)=0.01×[(4/25)𝑓_𝑐(𝑡)], and Assumption 6 is satisfied; taking 𝑐=0.1, suppose that 𝜀_𝜅 satisfies (16); the simulation results are shown in Figure 4.

(a)

(b)

(c)

Figure 1 

Two node attached components of nodes 1 and 3 in VSWSN.

(a) The dynamic states of all sensors

(b) The errors among the average values and states of all nodes, where 𝑒𝑖(𝑡)=𝑦𝑖(𝑡),𝑖=1,2,3

(c) State of the attack signal, where 𝑓3(𝑡)=𝑓𝑎(𝑡)

(d) The update law of 𝜀𝜅1, where 𝑓0(𝑡)=𝜀𝜅1

Figure 2 

The dynamic states of 𝑋𝑘, 𝑌𝑘, jamming attack signal and the update law of 𝜀𝜅1 in topology 1.

(a) The dynamic states of all sensors

(b) The errors among the average values and states of all nodes, where 𝑒𝑖(𝑡)=𝑦𝑖(𝑡),𝑖=1,2,3,4,5

(c) 𝑠1(𝑡) is the attack signal, 𝑠2(𝑡)=𝜀𝜅2

(d) 𝑓1(𝑡),𝑓2(𝑡),𝑓5(𝑡) stand the measured attack signals by nodes 1, 2, and 5, respectively

Figure 3 

The dynamic states of 𝑋𝑘, 𝑌𝑘, attack signals, and the update law of 𝜀𝜅2 in topology 2.

(a) The dynamic states of all sensors

(b) The errors among the average values and states of all nodes, where 𝑒𝑖(𝑡)=𝑦𝑖(𝑡),𝑖=1,2,3,4,5

(c) The state of the jamming attack signal, where 𝑠(𝑡)=𝑓𝑐(𝑡)

(d) The update law of 𝜀𝜅3, where 𝑠(𝑡)=𝜀𝜅3

Figure 4 

The dynamic states of 𝑋𝑘, 𝑌𝑘, jamming attack signal, and the update law of 𝜀𝜅3 in topology 3.

Example 2. Following Example 1, if 𝜀_𝜅 is replaced by the gain function 𝑎(𝑡)=log(𝑡+2)/(𝑡+2) studied in [7], the system with topology 3 cannot achieve the consensus; the simulation results are shown in Figure 5.

(a) The dynamic states of all sensors

(b) The errors among the average values and states of all nodes, where 𝑒𝑖(𝑡)=𝑦𝑖(𝑡),𝑖=1,2,3,4,5

(c) Curves of the gain function 𝑠(𝑡)=log(𝑡+2)/(𝑡+2)

Figure 5 

The dynamic states of 𝑋𝑘, 𝑌𝑘, and gain function 𝑠(𝑡) of Example 2.

6. Conclusion

This paper has investigated the consensus problem of VSWSN under the tolerable jamming attacks. It has disclosed the relations among the attack power, initial values of the newly joined nodes, dwell time, and GLI-connected topology. According to the errors of the node attached components, the adaptive parameters were provided; then the adaptive consensus protocol was obtained, and the designed protocol ensures that the system achieves the consensus whatever the values of the newly joined nodes. The obtained results in this paper have extended some existing results which are associated with the fixed node set system. In fact, according to the attack power, this paper has provided a sleep method of VSWSN when the system encounters with the jamming attacks. Finally, simulation results have shown the effectiveness of the obtained results.

For the future research, relations among the time delays of multiple hop-relays, accumulated errors, and the consensus will be considered.

Acknowledgment

This work is supported by Cultivation Fund of Taizhou University (2013PY09).