Consensus Analysis for High-Order Multi-Agent Systems without or with Delays

Abstract

This paper studies the consensus problem for a high-order multi-agent systems without or with delays. Consensus protocols, which only depend on the own partial information of agents and partial relative information with its neighbors, are proposed for consensus and quasi-consensus, respectively. Firstly, some lemmas are presented, and then a necessary and sufficient condition for guaranteeing the consensus is established under the consensus protocol without delays. Furthermore, communication delays are considered. Some necessary and sufficient conditions for solving quasi-consensus problem with delays are obtained. Finally, some simulations are given to verify the theoretical results.

1. Introduction

In recent years, the consensus problem of multi-agent systems has received a great deal of attention due to its broad applications in formation control of unmanned air vehicles, the design of sensor networks, the cooperative control of mobile robots, and automated highway system. In a word, we say all agents reach the consensus upon certain quantities of interest if all agents agree on a common state eventually; the precise definition will be introduced in the following sections. Many researches have been devoted to the consensus problem of multi-agent systems. In 1995, Vicsek et al. [1] proposed a discrete-time model and concluded that all the headings of the agents converged to a common value by simulations. In 2003, Jadbabaie et al. [2] provided a theoretical explanation for Vicsek’s simulation results by graph theory. Gradually, the consensus problem of multi-agent systems received a growing attention. For example, Olfati-Saber and Murray [3] discussed consensus problems for networks of dynamic agents with fixed and switching topologies under three cases. Ren and Beard [4] considered the problem of information consensus under dynamically changing interaction topologies and weighting factors. Recently, the second-order consensus problem of multi-agent systems has attracted more and more researchers’ attention [5–14]. Different from [5–7, 10], Yu et al. [9], Song et al. [11], and Wen et al. [14] all studied the second-order consensus problem of multi-agent systems with nonlinear dynamics. Compared with synchronization in complex networks which are of nonlinear dynamics, a generalized consensus was investigated. From a unified viewpoint, [15] provided a novel framework for the consensus of multi-agent systems and the synchronization of complex networks. Viewed from the leader-following consensus, both Song et al. [11] and Zhu and Cheng [12] considered the consensus problem with a leader. In general, the second-order consensus problem pays attention to whether the relative position and velocity of each agent will converge to common state.

At the same time, the high-order consensus problem of multi-agent systems is also worth being studied. Generally speaking, the first-order consensus problem of multi-agent systems can be considered as a special case of the synchronization of complex dynamic networks. Moreover, second-order consensus problem of multi-agent system means that the relative position and velocity of each agent will be converged to common. However, the birds achieving the consensus should be on the acceleration besides the relative position and velocity for the flocks of birds [16], which means that all the position, velocity, and acceleration will be converged to common. Recently, there are more and more studies on the high-order consensus or swarming problem (see [16–20] and the references therein).

Motivated by the previously mentioned results, consensus of high-order multi-agent systems is investigated in this paper. A new consensus protocol which only depends on the own partial information of agents and partial relative information with its neighbors is proposed. In other words, the consensus protocol in this paper does not need the global information of agents and the global relative information. It is convenient to be designed when the global information states cannot be available. Furthermore, we consider the quasi-consensus problem of high-order multi-agent systems with a time-delay protocol. The delay is nonnegligible in the information exchanges due to the limited transmission speed or memory effect. Therefore, there are many papers focusing on the consensus problem with delay [21–25]. It is worth noting that Yu et al. [25] considered the quasi-consensus problem of second-order multi-agent systems with delay under the protocol 𝑢_𝑖(𝑡)=𝛼^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−𝛼^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑡−𝜏)−𝑥_𝑖(𝑡−𝜏). (1) This delay is caused by the fact that the agent needs some memory to store the outdated information from its neighboring agents. It found that this delay can induce quasi-consensus; that is, all agents have the same velocity but keep a distance from each other eventually. Therefore, the delay in this sense looks very interesting. This paper considers quasi-consensus of high-order multi-agent systems with the delay in the sense of memory effect. Based on the Nyquist stability criterion, some results on the quasi-consensus of high-order multi-agent systems are obtained. It is worth noting that the results on the consensus of high-order multi-agent systems with a protocol of communication delay hold similarly. Furthermore, some simulations are given to verify the theoretical results.

Notations. Let ℝ^𝑛 denote the 𝑛-dimensional Euclidian space. Let 0_𝑁 and 𝐼_𝑁 be the 𝑁×𝑁 zero matrix and identity matrix, respectively. If 𝑀 is a matrix, then rank 𝑀 and 𝜎(𝑀) denote the rank and the spectrum of 𝑀, respectively. Let 𝟏 and 𝟎 denote the 𝑁×1 column vectors of all ones and all zeros, respectively. The symbol ⊗ denotes the Kronecker product.

2. Preliminaries and Model

Consider a multi-agent systems of 𝑁 agents. The dynamics of each agent is governed by the following general ℓth-order integrator: 𝜉̇_𝑖⁽⁰⁾=𝜉_𝑖⁽¹⁾,⋮𝜉̇_𝑖^(ℓ−2)=𝜉_𝑖^(ℓ−1),𝜉̇_𝑖^(ℓ−1)=𝑢_𝑖, (2) where 𝑖=1,2,…,𝑁,  ℓ≥1 is a positive integer and denotes the order of the differential equations, 𝜉_𝑖^(𝑘)∈ℝ^𝑚,𝑘=0,1,…,ℓ−1, are the states of agent 𝑖, 𝑢_𝑖∈ℝ^𝑚 is the consensus protocol, and 𝜉_𝑖^(𝑘) denotes the 𝑘th-order derivative of 𝜉_𝑖. The initial state denotes 𝜉_𝑖0=[𝜉_𝑖(0),𝜉_𝑖⁽¹⁾(0),…,𝜉_𝑖^(ℓ−1)(0)].

We use the weighted undirected graph 𝒢 to model the interaction topology of the multi-agent systems; that is, each vertex represents an agent of systems; each arc (edge) 𝑒_𝑗𝑖 represents that there is a communication link from agent 𝑗 to agent 𝑖. 𝑎_𝑖𝑗>0 is the weight of the communication link 𝑒_𝑗𝑖. One can see [26] for more details. Now, we propose the following consensus protocol: 𝑢_𝑖=−_𝑘∈𝑀1𝑐_𝑘𝜉_𝑖^(𝑘)−𝛼^𝑁_𝑗=1𝑎_{𝑖𝑗𝑘∈𝑀2}𝛾_𝑘𝜉_𝑖^(𝑘)−𝜉_𝑗^(𝑘), (3) where 𝑀={0,1,2,…,ℓ−1}. Let 𝑀₁,𝑀₂⊂𝑀 be two nonempty subsets such that 𝑀₁∪𝑀₂=𝑀,𝑀₁∩𝑀₂=∅, 𝛼 is the coupling strength and 𝛾_𝑘>0 is the feedback gain of relative information. 𝑐_𝑘≠0⇔𝑘∈𝑀₁, otherwise, 𝑐_𝑘=0 and let 𝑐₀=0; 𝛾_𝑘≠0⇔𝑘∈𝑀₂, otherwise, 𝛾_𝑘=0 and let 𝛾₀≠0. Let 𝑛₀=min{𝑘∶𝑘∈𝑀₁}.

Let 𝜉⁽⁰⁾=[𝜉₁^𝑇(0),𝜉₂^𝑇(0),…,𝜉_𝑁^𝑇(0)]^𝑇 and 𝜉^(𝑛+1)=𝜉̇^(𝑛), 𝑛=0,1,…,ℓ−2. By employing the consensus protocol (3), the dynamics equations of all agents can be written as follows: 𝜉̇⁽⁰⁾𝜉̇⁽¹⁾⋮𝜉̇^(ℓ−1)=Φ⊗𝐼_𝑚𝜉⁽⁰⁾𝜉⁽¹⁾⋮𝜉^(ℓ−1), (4) where Φ=𝐴⊗𝐼_𝑁−𝛼𝐵⊗𝐿,𝐴=010⋯0001⋯0⋮⋮⋮⋮⋮000⋯10−𝑐₁−𝑐₂⋯−𝑐_{ℓ−1ℓ×ℓ},𝛾𝐵=00⋯000⋯0⋮⋮⋮⋮00⋯0₀𝛾₁⋯𝛾_{ℓ−1ℓ×ℓ}, (5)(6)𝐿=[ℓ_𝑖𝑗] is the associated Laplacian of 𝒢; that is ℓ_𝑖𝑖=∑_𝑗≠𝑖𝑎_𝑖𝑗, ℓ_𝑖𝑗=−𝑎_𝑖𝑗. It is easy to note that 0Φ=^𝑁𝐼_𝑁0_𝑁⋯0_𝑁0_𝑁0_𝑁𝐼_𝑁⋯0_𝑁0⋮⋮⋮⋮⋮_𝑁0_𝑁0_𝑁⋯𝐼_𝑁−𝛼𝛾₀𝐿−Δ₁−Δ₂⋯−Δ_ℓ−1, (7) where Δ_𝑘=𝑐_𝑘𝐼_𝑁+𝛼𝛾_𝑘𝐿.

Definition 1. The multi-agent systems governed by (4) are said to achieve consensus if lim_𝑡→∞‖𝜉_𝑖^(𝑘)(𝑡)−𝜉_𝑗^(𝑘)(𝑡)‖=0,  𝑘=0,1,2,…,ℓ−1, 𝑖,𝑗=1,2,…,𝑁,𝑖≠𝑗, for any initial values.

Remark 2. For convenience, we assume 𝑚=1 in the following discussion; however, all the results hereafter remain valid for 𝑚>1 by the Kronecker product.

3. Consensus Analysis

In this section, we consider the consensus problem defined in the Section 2 for the multi-agent systems (4) with fixed topology. To this end, we establish the following lemmas which are needed for the main results.

Denote Γ=𝐼_𝑁⊗𝐴−𝛼𝐿⊗𝐵. One has the following lemmas.

Lemma 3. Φ is similar to Γ.

Proof. See the appendix.

Lemma 4. Assume that Φ is defined in (5) and 𝜆₁,𝜆₂,…,𝜆_𝑁∈𝜎(𝐿); then the characteristic polynomial of Φ is det𝜇𝐼_𝑁ℓ−Φ=^𝑁_𝑖=1𝜇^ℓ+𝛿^𝑖_ℓ−1𝜇^ℓ−1+⋯+𝛿^𝑖₁𝜇+𝛼𝜆_𝑖𝛾₀, (8) where 𝛿^𝑖_𝑘=𝑐_𝑘+𝛼𝜆_𝑖𝛾_𝑘,𝑘=1,2,…,ℓ−1,𝑖=1,2,…,𝑁.

Proof. See the appendix.

Lemma 5. Φ has at least 𝑛₀ zero eigenvalues. It has exactly 𝑛₀ zero eigenvalues if and only if 𝐿 has a simple zero eigenvalue. If 𝐿 has a simple zero eigenvalue, then the geometric multiplicity of the zero eigenvalue of Φ equals one. Moreover, the left and right eigenvector of Φ associated with eigenvalue zero are as follows: 𝑣_𝑙=1𝑐_𝑛0𝑐₁,…,𝑐_ℓ−1,1⊗𝟏^𝑇,𝑤_𝑟=1√𝑁𝑒₁⊗𝟏. (9)

Proof. See the appendix.

Based on the previous lemmas, here we give the main result.

Theorem 6. Under the consensus protocol (3), the multi-agent systems (4) achieve the consensus if and only if Φ has exactly 𝑛₀ zero eigenvalues and all of other eigenvalues are negative.

Proof (Sufficiency). If Φ has exactly 𝑛₀ zero eigenvalues, then Φ has the Jordan canonical form 𝐽₁𝐽₂, where 𝐽₁=010⋯0001⋯0⋮⋮⋮⋮⋮000⋯1000⋯0_𝑛0×𝑛0; (10)𝐽₂ is the upper diagonal block matrix with diagonal entries being the nonzero eigenvalues of Φ. Moreover, one has Φ=𝑤₁,𝑤₂,…,𝑤_ℓ𝑁×𝐽₁0_{𝑛0×(𝑁ℓ−𝑛0)}0_{(𝑁ℓ−𝑛0)×𝑛0}𝐽₂×𝑣^𝑇₁𝑣^𝑇₂⋮𝑣^𝑇_ℓ𝑁, (11) where 𝑤_𝑖∈ℝ^ℓ𝑁,𝑖=1,2,…,ℓ𝑁, can be chosen to be the right eigenvectors or generalized eigenvectors of Φ, 𝑣_𝑖∈ℝ^ℓ𝑁, 𝑖=1,2,…,ℓ𝑁, can be chosen to be the left eigenvectors or generalized eigenvectors of Φ. Without of loss generality, assume that 𝑀₂={0,1,…,𝑛₀−1} and 𝑀₁={𝑛₀,𝑛₀+1,…,ℓ−1}. Let 𝑤₁=[𝟏^𝑇,𝟎^𝑇,…,𝟎^𝑇]^𝑇, 𝑤₂=[𝟎^𝑇,𝟏^𝑇,…,𝟎^𝑇]^𝑇,…,𝑤_𝑛0=[𝟎^𝑇,𝟎^𝑇,…,𝟏^𝑇_𝑛0,𝟎^𝑇,…,𝟎^𝑇_ℓ−𝑛0]^𝑇; one can verify that 𝑤₁,𝑤₂,…,𝑤_𝑛0 are a right eigenvector and 𝑛₀−1 generalized right eigenvectors of Φ associated with zero eigenvalue, respectively. Since Φ has exactly 𝑛₀ zero eigenvalues, without loss of generality, one can denote them by 𝜇₁=𝜇₂=⋯=𝜇_𝑛0=0. By Lemma 5, we know that 𝐿 has a simple zero eigenvalue, which implies that there is a nonnegative vector 𝑝=(1/𝑁)𝟏 such that 𝑝^𝑇𝐿=𝟎, 𝑝^𝑇𝟏=1 by [16]. Then we choose, 𝑣₁=(1/𝑐_𝑛0)[𝑐_𝑛0𝑝^𝑇,…,𝑐_ℓ−1𝑝^𝑇,𝑝^𝑇,𝟎^𝑇,…,𝟎^𝑇_𝑛0−1]^𝑇,…, 𝑣₂=(1/𝑐_𝑛0)[𝟎^𝑇,𝑐_𝑛0𝑝^𝑇,…,𝑐_ℓ−1𝑝^𝑇,𝑝^𝑇,𝟎^𝑇,…,𝟎^𝑇_𝑛0−2]^𝑇,…,  𝑣_𝑛0=(1/𝑐_𝑛0)[𝟎^𝑇,…,𝟎^𝑇_𝑛0−1,𝑐_𝑛0𝑝^𝑇,…,𝑐_ℓ−1𝑝^𝑇,𝑝^𝑇]^𝑇. It follows from Lemma 5 that they are 𝑛₀−1 generalized left eigenvectors and a left eigenvector of Φ associated with eigenvalue zero, respectively. Moreover, 𝑣^𝑇_𝑖𝑤_𝑖=1, 𝑖=1,2,…,𝑛₀. It is noted that 𝑒^Φ𝑡=𝑤₁,𝑤₂,…,𝑤_ℓ𝑁×𝑒^𝐽1𝑡0_{𝑛0×(𝑁ℓ−𝑛0)}0_{(𝑁ℓ−𝑛0)×𝑛0}𝑒^𝐽2𝑡𝑣^𝑇₁𝑣^𝑇₂⋮𝑣^𝑇_ℓ𝑁, (12) where lim_𝑡→∞𝑒^𝐽2𝑡=0_{𝑁ℓ−𝑛0} since the nonzero eigenvalues have negative real parts and 𝑒^𝐽1𝑡=𝑡1𝑡²⋯𝑡2!𝑛^𝑛0−1₀!𝑡−101𝑡⋯^𝑛0−2𝑛₀−2!⋮⋮⋮⋮⋮000⋯𝑡000⋯1_𝑛0×𝑛0. (13) Therefore, for enough large 𝑡, the dominant terms of 𝑒^Φ𝑡 denoted by Φ_𝑑 is𝟏𝑝^𝑇𝑐_𝑛0+1𝑐_𝑛0+𝑡𝟏𝑝^𝑇⋯^𝑛0−1_𝑘=0𝑐_𝑛0+𝑘𝑡^{𝑛0−𝑘−1}𝑐_𝑛0𝑛₀−𝑘−1!𝟏𝑝^𝑇𝑛0_𝑘=1𝑐_𝑛0+𝑘𝑡^𝑛0−𝑘𝑐_𝑛0𝑛₀−𝑘!𝟏𝑝^𝑇⋯𝑡^𝑛0−1𝟏𝑝^𝑇𝑐_𝑛0𝑛₀!0−1_𝑁𝟏𝑝^𝑇⋯^𝑛0−2_𝑘=0𝑐_𝑛0+𝑘𝑡^{𝑛0−𝑘−2}𝑐_𝑛0𝑛₀−𝑘−2!𝟏𝑝^{𝑇𝑛0−1}_𝑘=1𝑐_𝑛0+𝑘𝑡^{𝑛0−𝑘−1}𝑐_𝑛0𝑛₀−𝑘−1!𝟏𝑝^𝑇⋯𝑡^𝑛0−2𝟏𝑝^𝑇𝑐_𝑛0𝑛₀!0−2⋮⋮⋮⋮⋮⋮⋮_𝑁0_𝑁⋯𝟏𝑝^𝑇𝑐_𝑛0+1𝑐_𝑛0𝟏𝑝^𝑇⋯𝟏𝑝^𝑇𝑐_𝑛00_𝑁0_𝑁⋯0_𝑁0_𝑁⋯0_𝑁0⋮⋮⋮⋮⋮⋮⋮_𝑁0_𝑁⋯0_𝑁0_𝑁⋯0_𝑁ℓ×ℓ. (14)Thus, we write Φ_𝑑 by a compact form 𝟏𝑝^𝑇𝑟₁₂(𝑡)𝟏𝑝^𝑇⋯𝑟_1𝑛0(𝑡)𝟏𝑝^𝑇⋯𝑟_1ℓ(𝑡)𝟏𝑝^𝑇0_𝑁𝟏𝑝^𝑇⋯𝑟_2𝑛0(𝑡)𝟏𝑝^𝑇⋯𝑟_2ℓ(𝑡)𝟏𝑝^𝑇0⋮⋮⋮⋮_𝑁0_𝑁⋯𝟏𝑝^𝑇⋯𝑟_𝑛0ℓ(𝑡)𝟏𝑝^𝑇0_𝑁0_𝑁⋯0_𝑁⋯0_𝑁0⋮⋮⋮⋮_𝑁0_𝑁⋯0_𝑁⋯0_𝑁, (15) where 𝑟_𝑖𝑗(𝑡),𝑗>𝑖 is a polynomial of 𝑡 with degree 𝑗−𝑖. It follows from (4) that 𝜉⁽⁰⁾(𝑡)𝜉⁽¹⁾(𝑡)⋮𝜉^(ℓ−1)(𝑡)=𝑒^Φ𝑡𝜉⁽⁰⁾(0)𝜉⁽¹⁾(0)⋮𝜉^(ℓ−1)(0), (16) that is, for enough large 𝑡, 𝜉^(𝑘)(𝑡)=[𝜉₁^(𝑘)(𝑡),𝜉₂^(𝑘)(𝑡),…,𝜉_𝑁^(𝑘)(𝑡)]^𝑇=𝟏𝑝^𝑇𝜉^(𝑘)(0)+𝑟_{𝑘+1,𝑘+2}(𝑡)𝟏𝑝^𝑇𝜉^(𝑘+1)(0)+⋯+𝑟_𝑘+1,ℓ(𝑡)𝟏𝑝^𝑇𝜉^(ℓ−1)(0),𝑘≤𝑛₀−1, (17) which implies that 𝜉_𝑖^(𝑘)(𝑡)→𝜉_𝑗^(𝑘)(𝑡) as 𝑡→∞ for 𝑘≤𝑛₀−1.
If 𝑘≥𝑛₀, then 𝜉^(𝑘)(𝑡)→𝟎 as 𝑡→∞. Therefore, 𝜉_𝑖^(𝑘)(𝑡)→𝜉_𝑗^(𝑘)(𝑡) as 𝑡→∞, for all 𝑖≠𝑗, 𝑘=0,1,…,ℓ−1. This implies that the multi-agent systems achieve the consensus.
(Necessity). By reduction to absurdity, suppose that the sufficient condition dose not hold; then there are two cases may be hold.
Case  1. Φ has more than 𝑛₀ zero eigenvalues.
Case  2. Φ has exactly 𝑛₀ zero eigenvalues, but it has at least one eigenvalue with nonnegative real part.
If Case  1 holds, then rank(lim_𝑡→∞𝑒^Φ𝑡)≥rank(lim_𝑡→∞𝑒^𝐽1𝑡)>𝑛₀.
If Case  2 holds, then lim_𝑡→∞𝑒^𝐽2𝑡≠0_{𝑁ℓ−𝑛0}, and rank(lim_𝑡→∞𝑒^Φ𝑡)≥rank(lim_𝑡→∞𝑒^𝐽1𝑡)+rank(lim_𝑡→∞𝑒^𝐽2𝑡)>𝑛₀. However, it follows from Φ_𝑑 (the dominant terms of 𝑒^Φ𝑡) that if the consensus is achieved, then rank(lim_𝑡→∞𝑒^Φ𝑡)≤𝑛₀. This is a contradiction.

Remark 7. From Theorem 6, if (4) achieves the consensus, then Φ has exactly 𝑛₀ zero eigenvalues. Therefore, 𝐿 has a simple zero eigenvalue, which implies that the information exchange topology 𝒢 is connected.

Remark 8. Compared to the consensus protocols proposed in [16–19], protocol (3) only depends on the own partial information of agents and the partial relative states with its neighbors, which does not need the global information of agents or the global relative information. It is convenient to design when global information states cannot be available.

To illustrate the previous result, we consider the following example.

Example 9. Considering the following third-order system of 8 agents, where the dynamics of each agent are governed by 𝑥̇_𝑖=𝑣_𝑖,𝑣̇_𝑖=𝑎_𝑖,𝑎̇_𝑖=𝑢_𝑖, (18) where 𝑥_𝑖, 𝑣_𝑖, 𝑎_𝑖, 𝑖=1,2,…,8, denote the position, velocity, and acceleration of agent 𝑖, respectively, let Laplacian matrix 𝐿 be 10−10000002−100−100−1−14−100−1000−12−1000000−110000−100010000−100100000000−11. (19) It follows from the 𝐿 that the graph is connected. Moreover, the eigenvalues of 𝐿 are 0, 0.382, 0.6086, 1, 1, 2.2271, 2.618, and 5.1642. If we choose 𝛼=1, 𝛾₀=1, 𝛾₁=2, and 𝑐₂=2, then 𝑛₀=2. Furthermore, Φ has 2 zero eigenvalues and the real parts of all other eigenvalues are negative. Let the initial values be the 𝑥(0)=[10,20,30,5,−10,−20,−30,−5], 𝑣(0)=[2,−8,3,−3,−2,5,8,−5], 𝑎(0)=[−2,2,3,−3,5,−5,1,−1]. By Theorem 6, consensus in dynamical multi-agent systems (18) with network topology (19) can be achieved. Figures 1, 2, and 3 show the plots of 𝑥_𝑖, 𝑣_𝑖, 𝑎_𝑖, 𝑖=1,2,…,8, respectively.

Figure 1 

The trajectories of 𝑥𝑖, 𝑖=1,2,…,8.

Figure 2 

The trajectories of 𝑣𝑖, 𝑖=1,2,…,8.

Figure 3 

The trajectories of 𝑎𝑖, 𝑖=1,2,…,8.

4. Quasi-Consensus Problem with Delays

Because of the limited transmission speed or memory effect, there exist nonnegligible delays in the communication process. Therefore, it is necessary to consider the influence of delays on the consensus. In [25], the authors considered the delays in the sense of memory store by the fact that the agent uses the memory to store the outdated information of its neighbors. Such delays are so interesting that can deduce quasi-consensus. In this section, we consider the consensus problem with the protocol of delays in the sense of memory store as follows: 𝑢_𝑖=−_𝑘∈𝑀1𝑐_𝑘𝜉_𝑖^(𝑘)+𝛼^𝑁_𝑗=1𝑎_{𝑖𝑗𝑘∈𝑀2}𝛾_𝑘𝜉_𝑗^(𝑘)(𝑡)−𝜉_𝑖^(𝑘)(𝑡)+_𝑘∈𝑀1𝑐_𝑘𝜉_𝑖^(𝑘)(𝑡−𝜏)−𝛼^𝑁_𝑗=1𝑎_{𝑖𝑗𝑘∈𝑀2}𝛾_𝑘𝜉_𝑗^(𝑘)(𝑡−𝜏)−𝜉_𝑖^(𝑘)(𝑡−𝜏), (20) where 𝜏>0. Let 𝜂_𝑖=[𝜉_𝑖⁽⁰⁾,𝜉_𝑖⁽¹⁾,…,𝜉_𝑖^(ℓ−1)], 𝜂=[𝜂^𝑇₁,𝜂^𝑇₂,…,𝜂^𝑇_𝑁]^𝑇.

With the consensus protocol (20), the closed loop of multi-agent systems (2) can be written as 𝜂̇_𝑖=𝐴𝜂_𝑖−𝐴𝜂_𝑖(𝑡−𝜏)−𝛼^𝑁_𝑗=1𝑙_𝑖𝑗𝐵𝜂_𝑗+𝛼^𝑁_𝑗=1𝑙_𝑖𝑗𝐵𝜂_𝑗(𝑡−𝜏), (21) where 𝐴, 𝐵 are defined as (5) and 𝐴=00⋯000⋯0⋮⋮⋮⋮00⋯00−𝑐₁⋯−𝑐_{ℓ−1ℓ×ℓ}. (22)

Therefore, 𝜂̇=𝐼_𝑁⊗𝐴𝜂−𝐼_𝑁⊗𝐴𝜂()()()𝜂().𝑡−𝜏−𝛼𝐿⊗𝐵𝜂+𝛼𝐿⊗𝐵𝑡−𝜏 (23) Let 𝐽 be the Jordan matrix associate with the 𝐿. Therefore, there exists orthogonal matrix 𝑄 such that 𝑄^𝑇𝐿𝑄=𝐽=diag{𝜆₁,𝜆₂,…,𝜆_𝑁}. Then, 𝑄^𝑇⊗𝐼_ℓ𝜂̇(𝑡)=𝑄^𝑇⊗𝐼_ℓ𝐼_𝑁(𝜂(𝑡)−𝑄⊗𝐴−𝛼𝐿⊗𝐵)_𝑇⊗𝐼_ℓ𝐼_𝑁⊗𝐴(𝜂()=𝐼−𝛼𝐿⊗𝐵)𝑡−𝜏_𝑁⊗𝐴−𝛼(𝐽⊗𝐵)𝑄^𝑇⊗𝐼_ℓ𝜂(𝑡)−𝐼_𝑁⊗𝐴−𝛼(𝐽⊗𝐵)𝑄^𝑇⊗𝐼_ℓ𝜂(𝑡−𝜏). (24) Let 𝜃(𝑡)=(𝑄^𝑇⊗𝐼_ℓ)𝜂(𝑡)=[𝜃^𝑇₁(𝑡),𝜃^𝑇₂(𝑡),…,𝜃^𝑇_𝑁(𝑡)]^𝑇. Then, it follows from (24) that 𝜃̇(𝑡)=𝐼_𝑁(𝜃(𝑡)−𝐼⊗𝐴−𝛼𝐽⊗𝐵)_𝑁⊗𝐴−𝛼(𝐽⊗𝐵)𝜃(𝑡−𝜏); (25) that is 𝜃̇_𝑖(𝑡)=𝐴−𝛼𝜆_𝑖𝐵𝜃_𝑖(𝑡)−𝐴−𝛼𝜆_𝑖𝐵𝜃_𝑖(𝑡−𝜏). (26) The characteristic equation of the multi-agent system (26) is 𝜆^ℓ+𝑐_ℓ−1+𝛼𝜆_𝑖𝛾_ℓ−11−𝑒^−𝜆𝜏𝜆^ℓ−1+⋯+𝑐₁+𝛼𝜆_𝑖𝛾₁1−𝑒^−𝜆𝜏𝜆+𝛼𝜆_𝑖𝛾₀1−𝑒^−𝜆𝜏=0,𝑖=1,2,…,𝑁. (27)

Definition 10. The multi-agent systems (2) are said to achieve quasi-consensus if lim_𝑡→∞𝜉_𝑖⁽⁰⁾(𝑡)−𝜉_𝑗⁽⁰⁾(𝑡)=𝑐_𝑖𝑗,lim_𝑡→∞𝜉_𝑖^(𝑘)(𝑡)−𝜉_𝑗^(𝑘)(𝑡)=0, (28) for any initial conditions, where 𝑐_𝑖𝑗 are constants, for all 𝑘=1,2,…,ℓ−1 and 𝑖,𝑗=1,2,…,𝑁.

Theorem 11. Assume that 𝒢 is connected. Quasi-consensus of multi-agent systems (2) with protocol (20) can be achieved if and only if each equation of (27) (𝑖=2,3,…,𝑁) has only simple zero root and the real parts of all other roots are negative.

Proof (Sufficiency). If 𝒢 is connected, then 𝐿 has only a simple zero eigenvalue with associated right eigenvector 𝑞₁=(1/√𝑁)𝟏, where 𝑄^𝑇𝐿𝑄=𝐽, 𝑄=[𝑞₁,𝑞₂,…,𝑞_𝑁], 𝑞_𝑖=[𝑞_𝑖1,𝑞_𝑖2,…,𝑞_𝑖𝑁]^𝑇, 𝑖=1,2,…,𝑁. If each equation of (27) (𝑖=2,3,…,𝑁) has only simple zero root and the real parts of all other roots are negative, then the states in (26) converge to some constants. It follows from the 𝜃(𝑡)=(𝑄^𝑇⊗𝐼_ℓ)𝜂(𝑡) that 𝜃_𝑖(𝑡)=(𝑞^𝑇_𝑖⊗𝐼_ℓ)𝜂(𝑡)=∑^𝑁_𝑗=1𝑞_𝑖𝑗𝜂_𝑗(𝑡). Denote 𝜃_𝑖(𝑡)=[𝜃_𝑖0(𝑡),𝜃_𝑖1(𝑡),…,𝜃_𝑖ℓ−1(𝑡)]^𝑇; then one has 𝜃_𝑖𝑘(𝑡)=^𝑁_𝑗=1𝑞_𝑖𝑗𝜉_𝑗^(𝑘)(𝑡),𝑘=0,1,…,ℓ−1. (29) Noting that 𝜃̇_𝑖𝑘(𝑡)=𝜃_𝑖𝑘+1(𝑡), if one of the 𝜃_𝑖1(𝑡),𝜃_𝑖2(𝑡),…,𝜃_𝑖ℓ−1(𝑡) does not converge to zero, then lim_𝑡→∞‖𝜃_𝑖0(𝑡)‖=∞. Therefore, lim_𝑡→∞‖𝜃_𝑖𝑘(𝑡)‖=0, 𝑖=2,3,…,𝑁, 𝑘=1,…,ℓ−1. Moreover, one has 𝜂_𝑖(𝑡)=∑^𝑁_𝑗=1𝑞_𝑗𝑖𝜃_𝑗(𝑡); therefore, 𝜉_𝑖^(𝑘)(𝑡)=^𝑁_𝑗=1𝑞_𝑗𝑖𝜃_𝑗𝑘(𝑡),𝑘=0,1,…,ℓ−1. (30) Thus, lim_𝑡→∞𝜉^(𝑘)(𝑡)−1√𝑁𝜃_1𝑘(𝑡),𝜃_1𝑘(𝑡),…,𝜃_1𝑘(𝑡)^𝑇=0, (31) where 𝑘=1,2,…,ℓ−1,  𝜃̇₁(𝑡)=𝐴𝜃₁(𝑡). Therefore, the quasi-consensus is reached.
(Necessity). If the quasi-consensus can be reached, then there exists 𝜎(𝑡)=[𝜎₀(𝑡),𝜎₁(𝑡),…,𝜎_ℓ−1(𝑡)]^𝑇 such that lim_𝑡→∞‖𝜉^(𝑘)(𝑡)−1_𝑁⊗𝜎_𝑘(𝑡)‖=0. Since 𝐺 is connected, 𝐿 has only one zero eigenvalue with associated eigenvector 𝑞₁=(1/√𝑁)𝟏. Thus, 𝟎=𝑄^𝑇𝐿𝟏=𝐽𝑄^𝑇𝟏=[𝜆₁𝑞^𝑇₁𝟏,𝜆₂𝑞^𝑇₂𝟏,…,𝜆_𝑁𝑞^𝑇_𝑁𝟏]^𝑇; that is, 𝑞^𝑇_𝑖𝟏=0,𝑖=2,3,…,𝑁. Therefore, it follows from 𝜃_𝑖(𝑡)=𝑞_𝑖1𝜂₁(𝑡)+𝑞_𝑖2𝜂₂(𝑡)+⋯+𝑞_𝑖𝑁𝜂_𝑁(𝑡)=(𝑞^𝑇_𝑖⊗𝐼_ℓ)𝜂(𝑡) that lim_𝑡→∞𝜃_𝑖𝑘(𝑡)=lim_𝑡→∞𝑞^𝑇_𝑖𝟏⊗𝜎_𝑘(𝑡)=0, (32)𝑖=2,3,…,𝑁, 𝑘=1,2,…,ℓ−1, 𝜃̇₁(𝑡)=𝐴𝜃₁(𝑡). By way of contradiction, assume that the sufficient condition does not hold; then there are two cases.
Case  1. Each of 𝑁−1 equations (27) (𝑖=2,3,…,𝑁) has more than one zero roots.
Case  2. Each of 𝑁−1 equations (27) (𝑖=2,3,…,𝑁) has simple zero root; however, there exists nonnegative real part.
If one of the previous two cases holds, then there exists a 1≤𝑘≤ℓ−1 such that lim_𝑡→∞‖𝜃_𝑖𝑘(𝑡)‖≠0 for 𝑖=2,3,…,𝑁, which contradicts formula (32).

𝑒^−𝜆𝜏 can be expressed by Taylor expansion; therefore, (27) can be rewritten as follows: 𝜆^ℓ+𝜆𝜏−^∞_𝑘=2(−𝜏)^𝑘𝜆^𝑘−1𝑘!^ℓ−1_𝑘=0𝛿^𝑖_𝑘𝜆^𝑘=0. (33)

Since 𝜏>0 and 𝛿^𝑖₀=𝛼𝜆_𝑖𝛾₀>0, 𝑖=2,3,…,𝑁, (33) has only simple zero root for 𝑖=2,3,…,𝑁. When 𝑠≠0, let 𝐹_𝑖(𝜆)=1+𝐺_𝑖(𝜆), where 𝐺_𝑖(𝜆)=∑^ℓ−1_𝑘=0𝛿^𝑖_𝑘𝜆^𝑘𝜆^ℓ1−𝑒^−𝜆𝜏,𝑖=2,3,…,𝑁. (34) Based on the Nyquist stability criterion, therefore, the real parts of nonzero roots of (27) are negative if and only if 𝐺_𝑖(𝑗𝜔) does not enclose the point (−1,𝑗0) for 𝜔∈ℝ, where 𝑗 is imaginary unit.

Without loss of generality, assume that ℓ is an odd number. Denote 𝑆₁={0,2,…,ℓ−1} and 𝑆₂={1,3,…,ℓ−2}. By some calculations, one can obtain 𝐺_𝑖=∑𝑗𝜔^{ℓ−1𝑘=0}𝛿^𝑖_𝑘(𝑗𝜔)^𝑘(𝑗𝜔)^ℓ1−𝑒^{−𝑗𝜔𝜏}=∑^ℓ−1_𝑘=0𝛿^𝑖_𝑘𝜔^𝑘𝑗^𝑘+ℓ(−𝜔)^ℓ2sin𝜔𝜏2𝑒^{𝑗((𝜋−𝜔𝜏)/2)}=𝑗_𝑘∈𝑆1𝛿^𝑖_𝑘𝜔^𝑘(−1)^{(𝑘+ℓ−1)/2}+_𝑘∈𝑆2𝛿^𝑖_𝑘𝜔^𝑘(−1)^(𝑘+ℓ)/2×−𝜔^ℓ−12sin𝜔𝜏2𝑒^{𝑗((𝜋−𝜔𝜏)/2)}. (35)

Let 𝑎_𝑖=∑_𝑘∈𝑆2𝛿^𝑖_𝑘𝜔^𝑘(−1)^(𝑘+ℓ)/2,  𝑏_𝑖=∑_𝑘∈𝑆1𝛿^𝑖_𝑘𝜔^𝑘(−1)^{(𝑘+ℓ−1)/2}.

Therefore, 𝐺_𝑖=𝑎𝑗𝜔^𝑖+𝑏_𝑖𝑗−𝜔^ℓ2sin𝜔𝜏2𝑒^{𝑗((𝜋−𝜔𝜏)/2)}=2sin(𝜔𝜏/2)𝑎²_𝑖+𝑏²_𝑖𝜔^ℓ𝑒^{𝑗(((−𝜋−𝜔𝜏)/2)+arctan(𝑏𝑖/𝑎𝑖))}. (36)

Therefore, Theorem 11 is equivalent to the following theorem.

Theorem 12. Assume that 𝒢 is connected. Quasi-consensus of multi-agent systems (2) with protocol (20) can be achieved if and only if 𝐺_𝑖=()𝑗𝜔2sin𝜔𝜏/2𝑎²_𝑖+𝑏²_𝑖𝜔^ℓ𝑒^{𝑗(((−𝜋−𝜔𝜏)/2)+arctan(𝑏𝑖/𝑎𝑖))} (37) does not enclose the point (−1,𝑗0) for 𝑖∈{2,3,…,𝑁} and 𝜔∈ℝ, where  𝑎_𝑖=∑_𝑘∈𝑆2𝛿^𝑖_𝑘𝜔^𝑘(−1)^(𝑘+ℓ)/2, 𝑏_𝑖=∑_𝑘∈𝑆1𝛿^𝑖_𝑘𝜔^𝑘(−1)^{(𝑘+ℓ−1)/2}.

As the symmetric property of the Nyquist plot, we only need to consider the case of 𝜔>0. When 𝜔∈(0,+∞), |𝐺_𝑖(𝑗𝜔)|≤(2∑^ℓ−1_𝑘=0𝛿^𝑖_𝑘𝜔^𝑘)/𝜔^ℓ is monotonously decreasing; furthermore, arg𝐺_𝑖=𝑏𝑗𝜔arctan^𝑖𝑎_𝑖−𝜋2−𝜔𝜏2,2𝑛𝜋≤𝜔𝜏2𝑏<2𝑛𝜋+𝜋;arctan^𝑖𝑎_𝑖+𝜋2−𝜔𝜏2,2𝑛𝜋+𝜋≤𝜔𝜏2<2𝑛𝜋+2𝜋, (38) where 𝑛≥0 is the nonnegative integer.

When 𝜔∈(0,+∞), 𝐺_𝑖(𝑗𝜔) crosses the negative real axis for the first time at 𝜔^𝑖₁, one has 𝜔^𝑖₁𝜏2𝑏=arctan^1𝑖𝑎¹_𝑖+𝜋2,𝐺_𝑖𝑗𝜔^𝑖₁≤2∑_𝑘∈𝑆2𝛿^𝑖_𝑘(𝜔^𝑖₁)^𝑘(𝜔^𝑖₁)^ℓ, (39) where 𝑎¹_𝑖=∑_𝑘∈𝑆2𝛿^𝑖_𝑘(𝜔^𝑖₁)^𝑘(−1)^(𝑘+ℓ)/2, 𝑏¹_𝑖=∑_𝑘∈𝑆1𝛿^𝑖_𝑘(𝜔^𝑖₁)^𝑘(−1)^{(𝑘+ℓ−1)/2}.

Therefore, one has the following result equivalently.

Theorem 13. Assume that 𝒢 is connected. Quasi-consensus of multi-agent systems (2) with protocol (20) can be achieved if and only if 2_𝑘∈𝑆2𝛿^𝑖_𝑘𝜔^𝑖₁^𝑘<𝜔^𝑖₁^ℓ,𝑓𝑜𝑟𝑖=2,3,…,𝑁, (40) where 𝜔^𝑖₁ is defined by 𝜔^𝑖₁𝜏/2=arctan(𝑏¹_𝑖/𝑎¹_𝑖)+𝜋/2.

Denote 𝛿_max=max_𝑖,𝑘𝛿^𝑖_𝑘. If all 𝜔^𝑖₁ satisfy 2_𝑘∈𝑆2𝛿_max𝜔^𝑖₁^𝑘≤𝜔^𝑖₁^ℓ,for𝑖=2,3,…,𝑁, (41) then Theorem 13 holds.

Remark 14. If one considers consensus protocol with communication delays as follows: 𝑢_𝑖=−_𝑘∈𝑀1𝑐_𝑘𝜉_𝑖^(𝑘)+𝛼^𝑁_𝑗=1𝑎_{𝑖𝑗𝑘∈𝑀2}𝛾_𝑘𝜉_𝑗^(𝑘)(𝑡)−𝜉_𝑖^(𝑘)(𝑡)−_𝑘∈𝑀1𝑐_𝑘𝜉_𝑖^(𝑘)(𝑡−𝜏)+𝛼^𝑁_𝑗=1𝑎_{𝑖𝑗𝑘∈𝑀2}𝛾_𝑘𝜉_𝑗^(𝑘)(𝑡−𝜏)−𝜉_𝑖^(𝑘)(𝑡−𝜏), (42) then the similar results can be obtained.

Example 15. Considering the multi-agent systems which is similar to example 1, here we take (20) as the quasi-consensus protocol. We take the same parameters; that is, the Laplacian matrix 𝐿 is also 10−10000002−100−100−1−14−100−1000−12−1000000−110000−100010000−100100000000−11, (43)𝛼=1, 𝛾₀=1, 𝛾₁=2 and 𝑐₂=2 and 𝑛₀=2. Then, eigenvalues of 𝐿 are 𝜆₁=0, 𝜆₂=0.382, 𝜆₃=0.6086, 𝜆₄=𝜆₅=1, 𝜆₆=2.2271, 𝜆₇=2.618, and 𝜆₈=5.1642. By simulation, 𝜏_max=1.3224. Moreover, we choose 𝜏=0.5. By Theorem 13, one knows that the quasi-consensus can be reached. Figures 4, 5 and 6 show that the quasi-consensus is reached.

Figure 4 

The trajectories of 𝑥𝑖, 𝑖=1,2,…,8.

Figure 5 

The trajectories of 𝑣𝑖, 𝑖=1,2,…,8.

Figure 6 

The trajectories of 𝑎𝑖, 𝑖=1,2,…,8.

5. Conclusions

When the global states of agents and global relative information are unavailable, this paper designs a consensus protocol which only depends on the own partial information of agents and partial relative information with its neighbors. Under the consensus protocol, a necessary and sufficient condition for guaranteeing the consensus for a high-order integrator multi-agent systems is established. Moreover, the consensus protocol shows that the consensus can be reached when the own states of agents are complementary to the relative information with its neighbors; the consensus may not be reached when there are only a few information exchanges. Moreover, the paper also considers the quasi-consensus problem under the protocol with the delays. It is interesting that multi-agent systems can reach consensus without delays and quasi-consensus with delays.

Appendix

Proof of Lemma 3. In fact, let 𝑥=(𝜉⁽⁰⁾)^𝑇,(𝜉⁽¹⁾)^𝑇,…,(𝜉^(ℓ−1))^𝑇𝑇, (A.1) where 𝜉=[𝜉₁,𝜉₂,…,𝜉_𝑁]^𝑇. Then 𝑥̇=Φ𝑥. On the other hand, let 𝜂_𝑖=[𝜉_𝑖⁽⁰⁾,𝜉_𝑖⁽¹⁾,…,𝜉_𝑖^(ℓ−1)]^𝑇 and 𝑦=[𝜂^𝑇₁,𝜂^𝑇₂,…,𝜂^𝑇_𝑁]^𝑇; then 𝑦̇=Γ𝑦. (A.2) Noting that 𝑦 is a rearrange of 𝑥, there exists a nonsingular matrix 𝑃 in which every row and every column has only simple one such that 𝑦=𝑃𝑥, which together with (A.2) 𝑥̇=𝑃⁻¹Γ𝑃𝑥; (A.3) that is, Φ=𝑃⁻¹Γ𝑃. Therefore, Φ is similar to Γ. Moreover, Γ and Φ have the same characteristic polynomial and eigenvalues.

Proof of Lemma 4. Let 𝑄 be a nonsingular matrix such that 𝑄⁻¹𝐿𝑄=𝐽, where 𝐽 is the Jordan matrix associated to 𝐿. Thus det𝜇𝐼_𝑁ℓ−Φ=det𝜇𝐼_𝑁ℓ−Γ=det𝜇𝐼_𝑁ℓ−𝑄⁻¹⊗𝐼_ℓΓ𝑄⊗𝐼_ℓ=det𝜇𝐼_𝑁ℓ−𝐼_𝑁=⊗𝐴+𝛼𝐽⊗𝐵_𝑁𝑖=1det𝜇𝐼_ℓ−𝐴+𝛼𝜆_𝑖𝐵=^𝑁_𝑖=1𝜇^ℓ+𝛿^𝑖_ℓ−1𝜇^ℓ−1+⋯+𝛿^𝑖₁𝜇+𝛼𝜆_𝑖𝛾₀. (A.4)

Proof of Lemma 5. It follows from (8) that there are 𝑛₀ zero eigenvalues of Φ that correspond to each zero eigenvalue of 𝐿. In fact, for every zero eigenvalue 𝜆_𝑖 of 𝐿; that is, 𝜆_𝑖=0, 𝜇^ℓ+𝛿^𝑖_ℓ−1𝜇^ℓ−1+⋯+𝛿^𝑖₁𝜇+𝜆_𝑖𝛾₀=𝜇^ℓ+𝛿^𝑖_ℓ−1𝜇^ℓ−1+⋯+𝛿^𝑖_𝑛0+1𝜇^𝑛0+1+𝑐_𝑛0𝜇^𝑛0=𝜇^𝑛0𝜇^ℓ−𝑛0+𝛿_{ℓ−𝑛0−1}𝜇^{ℓ−𝑛0−1}+⋯+𝛿_𝑛0+1𝜇+𝑐_𝑛0, (A.5) which implies that every zero eigenvalue of 𝐿 corresponds to 𝑛₀ zero eigenvalues of Φ. It is directly to see that 𝐿 has at least one zero eigenvalue with an associated eigenvector 𝟏, since 𝐿𝟏=0. Therefore, Φ has at least 𝑛₀ zero eigenvalues. Moreover, It follows from (8) again that Φ has exactly 𝑛₀ eigenvalues if and only if 𝐿 has exactly a simple zero eigenvalues.
Next, by a similar method from [16], let 𝑠 be an eigenvalue of Φ with associated eigenvector 𝑞=[𝑝^𝑇₁,𝑝^𝑇₂,…,𝑝^𝑇_ℓ]^𝑇, where 𝑝^𝑇₁,𝑝^𝑇₂,…,𝑝^𝑇_ℓ are 𝑁×1 column vectors. Therefore, 0Φ𝑞=^𝑁𝐼_𝑁0_𝑁⋯0_𝑁0_𝑁0_𝑁𝐼_𝑁⋯0_𝑁0⋮⋮⋮⋮⋮_𝑁0_𝑁0_𝑁⋯𝐼_𝑁−𝛼𝛾₀𝐿−Δ₁−Δ₂⋯−Δ_ℓ−1×𝑝₁𝑝₂⋮𝑝_ℓ−1𝑝_ℓ𝑝=𝑠¹𝑝₂⋮𝑝_ℓ−1𝑝_ℓ, (A.6) which implies that 𝑝₂=𝑠𝑝₁, 𝑝₃=𝑠𝑝₂,…,𝑝_ℓ=𝑠𝑝_ℓ−1,−𝛼𝛾₀𝐿𝑝₁−𝑐₁𝑝₂−𝛼𝛾₁𝐿𝑝₂−⋯−𝑐_ℓ−1𝑝_ℓ−𝛼𝛾_ℓ−1𝐿𝑝_ℓ=𝑠𝑝_ℓ; moreover, −𝛼∑_𝑘∈𝑀2𝛾_𝑘𝑠^𝑘𝐿𝑝₁−∑_𝑗∈𝑀1𝑐_𝑗𝑠^𝑗𝑝₁=𝑠^ℓ𝑝₁. Let 𝜅=∑_𝑗∈𝑀1𝑐_𝑗𝑠^𝑗+𝑠^ℓ/−𝛼∑_𝑘∈𝑀2𝛾_𝑘𝑠^𝑘; that is, 𝜅 is an eigenvalue of 𝐿 with associated eigenvector 𝑝₁. If 𝐿 has a simple zero eigenvalue, without loss of generality, suppose that 𝜅=0, then Φ has a eigenvalue 𝑠=0 (the algebraic multiplicity is 𝑛₀), so 𝑝₂=𝑝₃=⋯=𝑝_ℓ=0 and 𝑞=[𝑝^𝑇₁,0^𝑇_𝑁,…0^𝑇_𝑁]^𝑇, which is only one linearly independent eigenvector for Φ associated with eigenvalue zero; that is, the geometric multiplicity of zero eigenvalue of Φ is one.
It is easy to note that Φ1√𝑁𝑒₁1⊗𝟏=√𝑁𝐴𝑒₁⊗𝐼_𝑁𝟏−1√𝑁𝐵𝑒₁⊗()1𝐿𝟏=0,𝑐_𝑛0𝑐₁,…,𝑐_ℓ−1,1⊗𝟏^𝑇Φ=1𝑐_𝑛0𝑐₁,…,𝑐_ℓ−1,1𝐴⊗𝟏^𝑇𝐼_𝑁−1𝑐_𝑛0𝑐₁,…,𝑐_ℓ−1,1𝐵⊗𝟏^𝑇𝐿=0. (A.7) Therefore, we obtain 𝑤_𝑟=1√𝑁[1,0,…,0]^𝑇_ℓ×1𝑣⊗𝟏,_𝑙=1𝑐_𝑛0𝑐₁,…,𝑐_ℓ−1,1⊗𝟏^𝑇. (A.8)

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant no. 61304169, the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant no. 13KJB110022, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK20130857 and the Scientific Research Foundation of Nanjing University of Posts and Telecommunications under Grant no. NY213052.