Dynamic Average Consensus and Consensusability of General Linear Multiagent Systems with Random Packet Dropout

Abstract

This paper is concerned with the consensus problem of general linear discrete-time multiagent systems (MASs) with random packet dropout that happens during information exchange between agents. The packet dropout phenomenon is characterized as being a Bernoulli random process. A distributed consensus protocol with weighted graph is proposed to address the packet dropout phenomenon. Through introducing a new disagreement vector, a new framework is established to solve the consensus problem. Based on the control theory, the perturbation argument, and the matrix theory, the necessary and sufficient condition for MASs to reach mean-square consensus is derived in terms of stability of an array of low-dimensional matrices. Moreover, mean-square consensusable conditions with regard to network topology and agent dynamic structure are also provided. Finally, the effectiveness of the theoretical results is demonstrated through an illustrative example.

1. Introduction

Multiagent systems (MASs) are kinds of networked systems in which each agent updates its states based on the information exchanges over communication networks. Due to their broad applications in many areas such as sensor networks [1], distributed computation [2], swarms and flocks [3], and formation control [4], the consensus problem of MASs has attracted increasing attention in recent years, and considerable interesting results have been obtained on this problem.

At the beginning, the consensus problem is investigated mainly for first-order and second-order MASs. Olfati-Saber and Murray [5] studied the consensus problem for a network of first-order integrators under various topology conditions, including directed or undirected, fixed or switching, and with or without communication time-delays; Hatano and Mesbahi [6] studied the asymptotic agreement of first-order continuous MASs over random information network based on stochastic stability analysis; Zhang and Tian [7] investigated the MASs consisting of discrete-time second-order agents under stochastic switching topology and proved that MASs can achieve mean-square consensus if and only if the union graphs in the topology set are connected. Sun et al. [8] investigated the finite-time consensus for first- and second-order leader-following Multiagent systems. In [9], Lin et al. further studied the consensus problems of a class of high-order Multiagent systems with dynamically changing topologies and time-delays and proved that the communication time-delays do not affect the stability of the Multiagent systems.

Recently, some researches on MASs with general linear dynamics have been conducted. Li et al. [10] proved that MASs with a communication topology having a spanning tree can reach consensus under an observe-type protocol if and only if each agent is stabilizable and detectable. Xu et al. [11] investigated the information structure for the consensusability of MASs under both fixed and switching topologies with dynamic output feedback control. In [12], Xu et al. further studied the leader-following consensus problem of discrete-time Multiagent systems with switching and undirected topology by applying two kinds of distributed observer-based consensus protocols to each agent. Su and Huang [13] proved that if the system matrix is marginally stable and the dynamic graph is jointly connected, both the leaderless and the leader-following consensus problems for linear discrete-time Multiagent systems under switching network topology can be solved.

Note that most of the existing results on the consensus problem of MASs are derived from the assumption that the network communication environment is perfect. However, communication constraints which may deteriorate the performance of MASs are inevitable in practice. There are numerous studies focusing on the consensus problem for MASs with either communication time delay or presence of perturbation or both of them, such as [14–18], but as one of the main reasons for performance deterioration, packet dropout, which takes place frequently during the information exchange between agents, is not fully investigated. Therefore, it is of great significance to study the effect of packet dropout in MASs. Ren and Beard [19] studied the consensus problem for first-order continuous-time agents communicating via unreliable network through dynamically changing interaction topology method and proved that asymptotic consensus can be achieved if the union of the directed graph had a spanning tree frequently enough. Sun et al. [20] investigated the convergence and convergence speed for the second-order and the high-order discrete-time Multiagent systems with random networks and arbitrary weights. Zhang and Tian [21] investigated the mean-square consensus problem for continuous time second-order MASs disturbed by noise, variable delays, and occasional packet dropout and discussed the necessary and sufficient condition for mean-square robust consensusability. And in [22], Zhang and Tian further studied the consensus seeking problem for linear MASs, where each agent communicates via a weighted random lossy network and derived the maximum allowable loss probability.

Motivated by the above analysis, this paper is concerned with consensus problem of general linear discrete-time MASs with random packet dropout that happens during the information exchange between agents, which, to the best of our knowledge, has not been fully investigated until now. The contribution of this paper lies in the following: first, a distributed consensus protocol with weighted graph is proposed to address the packet dropout phenomenon; second, through introducing a new disagreement vector, the dynamic average consensus problem is studied based on the established new framework; third, the necessary and sufficient condition for MASs to reach mean-square consensus is derived in terms of stability of an array of low-dimensional matrices, which is easier to be used in reality; moreover, mean-square consensusable conditions with regard to network topology and agent dynamic structure are also provided.

The rest of this paper is organized as follows. In Section 2, modeling construction of Multiagent systems with packet dropout is proposed, and some useful lemmas are introduced. In Section 3, a new framework is derived to address the consensus problem of MASs. The consensusable conditions with regard to network topology and agent dynamic structure are presented in Section 4. In Section 5, an illustrative example is provided to demonstrate the effectiveness of our results. Concluding remarks are drawn in Section 6.

Notation.  ℝ^𝑛 and ℝ^𝑚×𝑛 denote the 𝑛-dimensional Euclidean space and the set of 𝑚×𝑛 real matrix, respectively, and 𝐼_𝑁 represents an 𝑁-dimensional identity matrix. diag{⋅} refers to a diagonal matrix, or a block diagonal matrix. For matrices 𝐴 and 𝐵, 𝐴>𝐵 implies that 𝐴 and 𝐵 are symmetric matrices and that 𝐴−𝐵 is positive definite. The superscript 𝑇 and −1 mean transpose and inverse of a real matrix separately, and symbol ∗ in a symmetric matrix implies a block that can be induced by symmetry. 𝜆_min(⋅), 𝜆_max(⋅) and 𝜌(⋅), denote, respectively, the smallest eigenvalue, the largest eigenvalue and the spectral radius of a matrix. Prob{⋅} stands for the occurrence probability of a stochastic event and Prob{⋅∣⋅} the conditional probability. 𝐸(⋅) is the mathematical expectation operator. ‖⋅‖ and ‖⋅‖₁ stand the 2-norm and 1-norm respectively.

2. Problem Formulation and Preliminaries

Consider a system of 𝑁 agents with identical general linear dynamics. The dynamic equation of agent 𝑖 is described by 𝑥_𝑖(𝑘+1)=𝐴𝑥_𝑖(𝑘)+𝐵𝑢_𝑖(𝑘),𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖=[𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛]^𝑇∈𝑅^𝑛 is the state variable, 𝑢_𝑖∈ℝ^𝑚 is the control input, and 𝐴∈ℝ^𝑛×𝑛 and 𝐵∈ℝ^𝑛×𝑚 are known constant matrices.

Suppose that there is a predesigned undirected fixed topology 𝐺 which guarantees the effective communication between distinct nodes. This assumption is quite reasonable, since it is impossible to reach consensus under an unconnected topology. Denote 𝑁_𝑖 as the neighbor set of the node 𝑖 and 𝐴_𝐺=[𝑎_𝑖𝑗]∈𝑅^𝑁×𝑁 as the weighted adjacency matrix of undirected graph 𝐺 with 𝑎_𝑖𝑖=0, and for any 𝑖 belongs to set 𝑁_𝑖, 𝑎_𝑖𝑗=𝑎_𝑗𝑖>0, otherwise, 𝑎_𝑖𝑗=𝑎_𝑗𝑖=0.

Under ideal communication environment, the commonly used linear distributed consensus protocol in MASs is as follows: 𝑢_𝑖(𝑘)=𝐾_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑥_𝑗(𝑘)−𝑥_𝑖(𝑘),𝑖=1,2,…,𝑁, (2) where 𝐾∈ℝ^𝑚×𝑛 is the gain matrix.

As mentioned in Section 1, packet dropout is inevitable in practical environment. In this paper, Bernoulli distributed white sequence {𝜃_𝑖𝑗(𝑘)} is introduced to model the unreliability of the link between agent 𝑖 and agent 𝑗. At any given time instant 𝑘, for any 𝑗∈𝑁_𝑖, 𝜃_𝑖𝑗(𝑘) takes value in the set {0,1}. Suppose that for any 𝑖,𝑗∈{1,…,𝑁}, 𝜃_𝑖𝑗(𝑘) is mutually independent; that is, the packet dropout probability between each pair of agents is independent. When the packet is transmitted successfully between agent 𝑖 and agent 𝑗, 𝜃_𝑖𝑗(𝑘)=1; when packet dropout takes place between agent 𝑖 and agent 𝑗, 𝜃_𝑖𝑗(𝑘)=0, which can be described as 𝜃_𝑖𝑗(𝑘)=0,withprobability𝑝_𝑖𝑗,1,withprobability1−𝑝_𝑖𝑗, (3) where 𝑝_𝑖𝑗 is the occurrence probability of packet dropout between 𝑖 and 𝑗, and for any 𝑗∉𝑁_𝑖, 𝜃_𝑖𝑗(𝑘)≡0, 𝑝_𝑖𝑗≡1.

Besides, due to the occurrence of packet dropout, the topology of MAS (1) is no longer fixed. Denote |𝑁_𝑖| as the number of agents in set 𝑁_𝑖, 𝑆 as the total number of the possible topologies. It is easy to see that 𝑆=2^{∑𝑁𝑖=1|𝑁𝑖|/2}; then the topology of the MAS is changing among the set {𝐺₁,…,𝐺_𝑆}. Suppose that at certain time instant 𝑘, 𝜃_𝑖𝑗(𝑘)=𝜃^𝑚_𝑖𝑗 and the network topology is 𝐺_𝑚, then the occurrence probability of graph 𝐺_𝑚, 𝑚=1,…,𝑆, is 𝜋_𝑚=_{𝑖=1,𝑖<𝑗}𝜃^𝑚_𝑖𝑗1−𝑝_𝑖𝑗+1−𝜃^𝑚_𝑖𝑗𝑝_𝑖𝑗. (4)

Based upon the above analysis, the following distributed consensus protocol is employed in this paper: 𝑢_𝑖(𝑘)=𝐾_{𝑗∈𝑁𝑖}𝜃_𝑖𝑗(𝑘)𝑎_𝑖𝑗𝑥_𝑗(𝑘)−𝑥_𝑖(𝑘),𝑖=1,…,𝑁, (5) where 𝐾 is the state feedback gain matrix to be designed.

Substituting (5) into (1), the dynamic equation of agent 𝑖 can be rewritten as 𝑥_𝑖(𝑘+1)=𝐴𝑥_𝑖(𝑘)+𝐵𝐾_{𝑗∈𝑁𝑖}𝜃_𝑖𝑗(𝑘)𝑎_𝑖𝑗𝑥_𝑗(𝑘)−𝑥_𝑖(𝑘). (6) Then, the states of MAS (1) evolve according to the following linear system: 𝑥(𝑘+1)=𝐼_𝑁⊗𝐴−𝐿_𝜎(𝑘)⊗(𝐵𝐾)𝑥(𝑘), (7) where 𝑥=[𝑥^𝑇₁,…,𝑥^𝑇_𝑁]^𝑇, 𝜎(⋅)∶𝑍⁺→{1,2,…,𝑆} is a stochastic process driven by an i.i.d. process and 𝐿_𝜎(𝑘)∈{𝐿_𝑚}, where 𝐿_𝑚=[𝑙^𝑚_𝑖𝑗]_𝑁×𝑁, 𝑚=1,…,𝑆, is the Laplacian matrix associated with 𝐺_𝑚, with 𝑙^𝑚_𝑖𝑗=^𝑁_𝑗=1𝜃^𝑚_𝑖𝑗𝑎_𝑖𝑗,𝑖=𝑗,−𝜃^𝑚_𝑖𝑗𝑎_𝑖𝑗,𝑖≠𝑗. (8) Obviously, the Laplacian matrix 𝐿_𝑚 is a symmetric matrix with zero row sum that is, ∀𝑖≠𝑗, 𝑙_𝑖𝑗=𝑙_𝑗𝑖, and ∑^𝑁_𝑗=1𝑙^𝑚_𝑖𝑗=0.

Before presenting our main results, the concepts of mean-square stability and mean-square consensus are introduced first, and along with them are some useful lemmas which play significant roles in the derivation of our results.

Definition 1 (see [23]). A closed-loop system is said to be mean-square stable, if for any 𝜀>0, there exists 𝛿(𝜀)>0 such that for any 𝑘>0, 𝐸(‖𝑥(𝑘)‖²)<𝜀 holds when 𝐸(‖𝑥(0)‖²)<𝛿(𝜀). In addition, if lim_𝑘→∞𝐸(‖𝑥(𝑘)‖²)=0 for any initial condition, then the closed-loop system is said to be globally mean-square asymptotically stable.

Definition 2. Multiagent system (1) with protocol (5) is said to reach consensus in mean-square sense, if for any initial distribution and initial states, lim_𝑘→∞𝐸(‖𝑥_𝑖(𝑘)−𝑥_𝑗(𝑘)‖²)=0 holds for all 𝑖,𝑗∈{1,…,𝑁}. Furthermore, if there exists a proper state feedback gain matrix 𝐾 such that the Multiagent system can achieve mean-square consensus, then we say that system (1) is mean-square consensusable under protocol (5).

Lemma 3 (see [21]). For any vector 𝑧(𝑘)∈ℝ^𝑛, denote 𝑧(𝑘)∶=𝐸(𝑧(𝑘)⊗𝑧(𝑘)), then there is 𝐸‖𝑧(𝑘)‖²≤‖𝑧(𝑘)‖₁≤𝑛𝐸‖𝑧(𝑘)‖². (9)

Lemma 4 (see [6]). For an undirected graph with 𝑛 vertices, the Laplacian is a positive-semidefinite symmetric matrix, and the spectrum of a graph Laplacian is on the form 0=𝜆₁(𝐿)≤𝜆₂(𝐿)≤⋯≤𝜆_𝑛−1(𝐿)≤𝜆_𝑛(𝐿), (10) with 𝜆₂(𝐿)>0 if and only if the graph that corresponds to 𝐿 is connected.

Lemma 5. ∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 has only one zero eigenvalue, if and only if the union graph of the topology set {𝐺₁,…,𝐺_𝑆} is connected.

Proof. Since ∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 can be treated as a Laplacian matrix of a graph which has the same edges with the union graph of the topology set {𝐺₁,…,𝐺_𝑆}, then together with Lemma 4, it can be obtained that Lemma 5 holds.

Lemma 6 (see [24]). For any 𝐴∈ℝ^𝑛×𝑛, there exist orthogonal matrix 𝑈 and an upper triangular matrix 𝑉, such that 𝑈^𝑇𝜇𝐴𝑈=𝑉=¹𝑣₁₂⋯𝑣_1𝑛𝜇₂⋯𝑣_2𝑛𝜇⋱⋮_𝑛, (11) where 𝜇_𝑖, 𝑖=1,…,𝑛, are the eigenvalues of matrix A.

3. Necessary and Sufficient Conditions for Mean-Square Consensus

Introduce the following variable: 𝛿_𝑖(𝑘)=𝑥_𝑖(𝑘)−1𝑁^𝑁_𝑗=1𝑥_𝑗(𝑘). (12)

Rewrite (12) in a compact form, which is 𝛿(𝑘)=𝐶⊗𝐼_𝑛𝑥(𝑘), (13) where 𝐶=𝐼_𝑁−(1/𝑁)11^𝑇 and 𝛿=[𝛿^𝑇₁,…,𝛿^𝑇_𝑁]^𝑇∈𝑅^𝑁𝑛 is the so-called disagreement vector.

It is easy to verify that 𝛿 evolves according to the following disagreement dynamics: 𝛿(𝑘+1)=𝐼_𝑁⊗𝐴−𝐿_𝜎(𝑘)⊗(𝐵𝐾)𝛿(𝑘), (14) where 𝛿(𝑘) is defined in (7).

Since 𝐸(‖𝛿(𝑘)‖²)=𝐸(𝑥^𝑇(𝑘)(𝑄⊗𝐼_𝑛)𝑥(𝑘)), where 𝑄=𝐶^𝑇𝐶=𝐼_𝑁−(1/𝑁)11^𝑇, it is easy to verify that 𝐸‖𝛿(𝑘)‖²=12𝑁_𝑖≠𝑗𝐸𝑥_𝑗(𝑘)−𝑥_𝑖(𝑘)². (15)

Equation (15) implies that the mean-square consensus problem can be solved if and only if lim_𝑘→∞𝐸(‖𝛿(𝑘)‖²)=0. From Definitions 1 and 2, it is easy to see that the consensus problem of MAS (1) is converted to the mean square stability analysis of system (14).

Remark 7. If the system matrix 𝐴 is Schur stable, which means all eigenvalues of 𝐴 are located in the unit circle, MAS (1) would definitely reach consensus without the design of consensus protocol, since under this condition, for any 𝑖∈{1,…,𝑁}, 𝑥_𝑖 will converge to zero, and lim_𝑘→∞𝐸(‖𝑥_𝑖(𝑘)−𝑥_𝑗(𝑘)‖²)=0 holds for sure. In view of this, we assume that not all eigenvalues of 𝐴 are located in the unit circle.

Theorem 8. For any initial distribution and initial states, Multiagent system (1) with the Bernoulli packet dropout between agents can achieve mean-square consensus under protocol (1), if and only if 𝜌𝐴−𝜆_𝑖𝐵𝐾<1, (16) where 𝜆_𝑖, 𝑖=2,…,𝑁, are the eigenvalues of matrix ∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 except one zero eigenvalue.

Proof. Since for each 𝜎(𝑘)=𝑚, 𝑚=1,…,𝑆, 𝐿_𝜎(𝑘) is a symmetric matrix with zero row sum, and 𝑤^𝑇=(1/𝑁)1^𝑇 and 1 are, respectively, the left and right eigenvectors of 𝐿_𝜎(𝑘) associated with zero eigenvalue, we can construct an invertible matrix 𝑇=1𝑇₁ with 𝑇⁻¹=𝑤𝑇₂^𝑇𝑇, such that 𝑇⁻¹𝐿_𝜎(𝑘)𝑇=𝐿_𝜎(𝑘)=0𝐿00_𝜎(𝑘), (17) where 𝑇∈ℝ^𝑁×𝑁, 𝑇₁∈ℝ^{𝑁×(𝑁−1)}, 𝑇₂∈ℝ^{(𝑁−1)×𝑁}, and 𝐿_𝜎(𝑘)=𝑇₂𝐿_𝜎(𝑘)𝑇₁∈𝑅^{(𝑁−1)×(𝑁−1)}.
Introduce the following variable: 𝜑(𝑘)=(𝑇⁻¹⊗𝐼_𝑛)𝛿(𝑘), where 𝜑=[𝜑^𝑇₁,…,𝜑^𝑇_𝑁]^𝑇∈𝑅^𝑁𝑛; then (14) can be denoted in terms of 𝜑 as 𝜑(𝑘+1)=𝐼_𝑁⊗𝐴−𝐿_𝜎(𝑘)⊗(𝐵𝐾)𝜑(𝑘). (18)
Note that 𝜑₁=(𝑤^𝑇⊗𝐼_𝑛)𝛿≡0, which implies that state 𝜑₁ is always stable; thus, divide 𝜑(𝑘) into two parts; that is, 𝜑(𝑘)=[𝜑^𝑇₁(𝑘),𝜑^𝑇(𝑘)]^𝑇, with 𝜑=[𝜑^𝑇₂,…,𝜑^𝑇_𝑁]^𝑇, it can be derived from (17) and (18) that 𝜑(𝑘+1)=𝐴_𝜎(𝑘)𝜑(𝑘), (19) where 𝐴_𝜎(𝑘)=𝐼_𝑁−1⊗𝐴−𝐿_𝜎(𝑘)⊗(𝐵𝐾).
Similar to [21], let 𝜉(𝑘)=𝐸(𝜑(𝑘)⊗𝜑(𝑘)); then 𝜉(𝑘) evolves according to 𝜉(𝑘+1)=^𝑆_𝑚=1𝜋_𝑚𝐴_𝑚⊗𝐴_𝑚𝜉(𝑘). (20)
From Lemma 3, system (14) is mean-square stable if and only if system (20) is stable. Then, from the discrete-time system stability theory, it can be concluded that the necessary and sufficient condition for mean-square stability of (14) is 𝜌(∑^𝑆_𝑚=1𝜋_𝑚(𝐴_𝑚⊗𝐴_𝑚))<1.
Denote 𝐴_𝑚=Φ−Ψ_𝑚𝐾 with Φ=𝐼_𝑁−1⊗𝐴, Ψ_𝑚=𝐿_𝑚⊗𝐵, 𝐾=𝐼_𝑁−1⊗𝐾, and 𝐾=𝜀𝐾^′; then there is ∑^𝑆_𝑚=1𝜋_𝑚(𝐴_𝑚⊗𝐴_𝑚)=Φ⊗Φ−𝜀(∑^𝑆_𝑚=1𝜋_𝑚Φ⊗(Ψ_𝑚𝐾^′)+∑^𝑆_𝑚=1𝜋_𝑚(Ψ_𝑚𝐾^′)⊗Φ) + 𝜀²∑^𝑆_𝑚=1𝜋_𝑚(Ψ_𝑚𝐾^′)⊗(Ψ_𝑚𝐾^′). The above equation can be treated as a perturbation of Φ⊗Φ by two terms depending on 𝜀. For small enough 𝜀, the above formula is equal to ∑^𝑆_𝑚=1𝜋_𝑚(𝐴_𝑚⊗𝐴_𝑚)=Φ⊗Φ−𝜀(∑^𝑆_𝑚=1𝜋_𝑚Φ⊗(Ψ_𝑚𝐾^′)+∑^𝑆_𝑚=1𝜋_𝑚(Ψ_𝑚𝐾^′)⊗Φ). Meanwhile, (∑^𝑆_𝑚=1𝜋_𝑚𝐴_𝑚)⊗(∑^𝑆_𝑚=1𝜋_𝑚𝐴_𝑚)=Φ⊗Φ−𝜀(∑^𝑆_𝑚=1𝜋_𝑚Φ⊗(Ψ_𝑚𝐾^′)+∑^𝑆_𝑚=1𝜋_𝑚(Ψ_𝑚𝐾^′)⊗Φ)+𝜀²∑^𝑆_𝑚=1(𝜋_𝑚Ψ_𝑚𝐾^′)⊗(𝜋_𝑚Ψ_𝑚𝐾^′). Likewise, by applying perturbation argument, for a small enough 𝜀, 𝜌(∑^𝑆_𝑚=1𝜋_𝑚(𝐴_𝑚⊗𝐴_𝑚))<1 is equivalent to 𝜌^𝑆_𝑚=1𝜋_𝑚𝐴_𝑚⊗^𝑆_𝑚=1𝜋_𝑚𝐴_𝑚<1. (21)
Next, it will be proved that (21) holds if and only if (16) is satisfied.
It can be seen from (18) that ∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 has the same eigenvalue with ∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 and ∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 except one zero eigenvalue. Besides, based on Lemma 6, there exists an orthogonal matrix 𝑈∈𝑅^{(𝑁−1)×(𝑁−1)} such that 𝑈^𝑇𝑆_𝑚=1𝜋_𝑚𝐿_𝑚𝜆𝑈=²𝜆∗⋯∗₃⋮𝜆⋯∗_𝑁, (22) where 𝜆_𝑖, 𝑖=2,…,𝑁, are the eigenvalues of matrix ∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 except one zero eigenvalue. It is easy to verify that ∑^𝑆_𝑚=1𝜋_𝑚𝐴_𝑚=𝐼_𝑁−1⊗𝐴−∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚⊗(𝐵𝐾) is similar to 𝐴−𝜆₂𝜆𝐵𝐾∗⋯∗𝐴−₃𝜆𝐵𝐾⋯∗⋱⋮𝐴−_𝑁𝐵𝐾. (23)
Based on the matrix theory, matrix ∑^𝑆_𝑚=1𝜋_𝑚𝐴_𝑚 has the same eigenvalue with matrix (23), obviously 𝜌(∑^𝑆_𝑚=1𝜋_𝑚𝐴_𝑚)<1, if and only if (16) is satisfied. Then, together with the properties of the Kronecker product, it is safe to say that (21) holds if and only if (16) is satisfied, which completes the proof.

Remark 9. The contribution of Theorem 8 lies in the fact that it converts the consensus problem of MAS (1) into a set of matrices with the same dimensions as a single agent, which reduces the computational complexity greatly. Besides, while in most of the existing results each agent is guaranteed to achieve consensus with agent 1, see [7, 11, 21] for example, or to achieve average consensus, see [5], for example, in this paper a new disagreement vector is introduced to solve the consensus problem so that each agent will reach consensus on the average states of all agents, that is, dynamic average consensus.

4. Consensusability Analysis

Theorem 8 presents the numerical solution to consensus problem, which also implies the design method of consensus protocol. However, in practice, we are equally interested in that under what conditions MAS (1) has the ability to reach consensus and inequality (16) has a solution at the same time, that is, what are the consensusable conditions? In this section, we focus on addressing this problem.

Theorem 10. Multiagent system (1) with the Bernoulli packet dropout between agents can achieve mean-square consensus under protocol (5), if and only if the union graph of the topology set {𝐺₁,…,𝐺_𝑆} is connected, and (𝐴,𝐵) is stabilizable.

Proof. Necessity: define 𝐸(𝐿_𝜎(𝑘))=∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 as the expected Laplacian matrix and its corresponding graph as the expected graph. Since 𝐸(𝐿_𝜎(𝑘)) is a symmetric matrix, there exists an invertible matrix 𝑇=1𝐹 with 𝑇⁻¹=𝑤𝐸^𝑇𝑇, such that 𝑇⁻¹𝐸𝐿_𝜎(𝑘)0Δ𝑇=Λ=00, (24) where 𝑇∈ℝ^𝑁×𝑁, 𝐹∈ℝ^{𝑁×(𝑁−1)}, 𝐸∈ℝ^{(𝑁−1)×𝑁}, 𝑤^𝑇, which is defined in (18), is also the left eigenvalue of 𝐸(𝐿_𝜎(𝑘)) associated with zero eigenvalue, and Δ∈𝑅^{(𝑁−1)×(𝑁−1)} is a diagonal matrix with its diagonal elements being 𝜆_𝑖, 𝑖=2,…,𝑁.
Denote 𝜙(𝑘)=𝐸(𝛿(𝑘)); then 𝜙(𝑘) evolves according to 𝜙(𝑘+1)=𝐼_𝑁⊗𝐴−𝐸𝐿_𝜎(𝑘)⊗(𝐵𝐾)𝜙(𝑘), (25) where 𝜙=[𝜙^𝑇₁,…,𝜙^𝑇_𝑁]^𝑇∈𝑅^𝑁𝑛.
Similar to the proof in Theorem 8, introduce the following variable transformation: 𝜁(𝑘)=(𝑇⁻¹⊗𝐼_𝑛)𝜙(𝑘); then the dynamic equation of 𝜁(𝑘) can be written as 𝜁(𝑘+1)=𝐼_𝑁⊗𝐴−Λ⊗(𝐵𝐾)𝜁(𝑘), (26) where 𝜁=[𝜁^𝑇₁,…,𝜁^𝑇_𝑁]^𝑇∈𝑅^𝑁𝑛.
Notice that 𝜁₁(𝑘)=(𝑤^𝑇⊗𝐼_𝑁)𝐸(𝛿(𝑘))≡0, divide 𝜁(𝑘)=(𝑤^𝑇⊗𝐼_𝑁)𝐸(𝛿(𝑘))≡0 into two parts, that is, 𝜁(𝑘)=[𝜁^𝑇₁(𝑘),𝜁̃^𝑇(𝑘)]^𝑇, from (19) and (21), it can be derived that 𝜁̃()=𝜆𝑘+1𝐴−₂𝜆𝐵𝐾0⋯0𝐴−₃𝜆𝐵𝐾⋯0⋱⋮𝐴−_𝑁𝜁̃(𝑘)𝐵𝐾. (27)
If the union graph of the topology set {𝐺₁,…,𝐺_𝑆} is not connected, the expected graph is definitely not connected. It can be obtained from Lemma 3 that 0=𝜆₂≤𝜆₃≤⋯≤𝜆_𝑁. Due to the fact that not all eigenvalues of 𝐴 are located in the unit circle, there exists an initial state such that lim_𝑘→∞𝜁(𝑘)≠0 and lim_𝑘→∞𝐸(𝛿(𝑘))≠0 thus; system (14) is not mean-square stable since 𝐸(‖𝛿(𝑘)‖²)≥‖𝐸(𝛿(𝑘))‖², and consequently, Multiagent system (1) cannot achieve mean-square consensus.
Besides, if (𝐴,𝐵) is unstabilizable, it is impossible to guarantee that 𝜌(𝐴−𝜆_𝑖𝐵𝐾)<1, 𝑖=2,…,𝑁, even if the topology condition is satisfied, and system (27) is still not guaranteed to be stable.
Sufficiency: It can be derived from Lemma 5 that when the union of the graphs in the topology set is connected, all eigenvalues of ∑^𝑆_𝑚=1𝜋_𝑚𝐿_𝑚 are positive; that is, 𝜆_𝑖>0, 𝑖=2,…,𝑁. Therefore, when the union graph of the topology set {𝐺₁,…,𝐺_𝑆} is connected and (𝐴,𝐵) is stablizable, there exists a proper gain matrix 𝐾 such that 𝜌(𝐴−𝜆_𝑖𝐵𝐾)<1, 𝑖=2,…,𝑁. The proof is completed.

Remark 11. In [21], for two-order continuous-time MASs with 𝐴=⁰¹₀₀, 𝐵=⁰₁, which implies that (𝐴,𝐵) is stablizable, Zhang and Tian prove that the MAS can reach mean-square consensus if and only if the union graph is connected, while in this paper, it is verified theoretically that the stabilizability of (𝐴,𝐵) is essential for achieving consensus.

5. Numerical Simulation

Consider an MAS consisted of four agents, with its prior fixed weighted communication topology being 𝑁₁={2}, 𝑁₂={1,3}, 𝑁₃={2,4}, 𝑁₄={3}, 𝑎₁₂=2, 𝑎₂₃=2, 𝑎₃₄=3; then there are eight graphs in the topology set {𝐺₁,…,𝐺₈}. Assume that the packet dropout probability between each pair of agents is 0.1.

Each agent is described by a three-order difference equation, with 001𝐴=0.600.20.30.30.4001.2,𝐵=. (28) Obviously, each agent is unstable since the eigenvalues of 𝐴 are 0.6, 0.3, and 1.2, and (𝐴,𝐵) is stablizable. By applying Theorem 8, it can be obtained that 𝐾=[0.1111−0.00000.1944], such that 𝜌(𝐴−𝜆_𝑖𝐵𝐾)=0.8910<1. Suppose the initial conditions are 𝑥₁₀=232^𝑇, 𝑥₂₀=21−1^𝑇, 𝑥₃₀=1−2−3^𝑇, and 𝑥₄₀=−12−2^𝑇; the simulation results are shown in Figure 1.

Figure 1 

The disagreement values between agents under consensus protocol (5).

6. Conclusion

We have studied the consensus problem of general linear discrete-time Multiagent systems (MASs) with random packet dropout that happens during information exchange between agents. The packet dropout phenomenon is characterized as being a Bernoulli random process. A distributed consensus protocol with weighted graph is proposed to address the packet dropout phenomenon. By constructing a new disagreement vector, a new framework is established to solve the consensus problem. Then, through introducing a common linear transformation for the switching system, together with the control theory, the perturbation argument, and the matrix theory, the necessary and sufficient condition for MASs to reach mean-square consensus is derived in terms of stability of an array of low-dimensional matrices. Moreover, mean-square consensusable conditions with regard to network topology and agent dynamic structure are also provided.

It is worth pointing out that this paper is only a first step; the consensus problem of linear discrete time MASs, which may be affected by many other factors including time-delay, and noise, is not fully investigated, and this is of our research interest in the future.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grants 61074124 and 61374171, the Project-Sponsored by SRF for ROCS, SEM, the Fundamental Research Funds for the Central Universities, HUST, under Grant 2012TS020, and the Graduate Fund of Innovation and Entrepreneurship, Hust, under Grant HF-11-18-2013.