Consensusabilization for Continuous-Time High-Order Multiagent Systems with Time-Varying Delays

Abstract

For the consensus problems of high-order linear multiagent systems with time-varying delays in directed topologies, the LMI based-consensus criterion and NLMI-based consensusabilization (protocol parameters design that makes the multiagent systems achieve consensus) are investigated. Improved Lyapunov-Krasovskii functional is used for establishing the consensus convergence criteria and deriving the corresponding consensus protocol. In order to reduce the conservativeness, some proper free-weighting matrices are added into the derivative of Lyapunov-Krasovskii functional and that only keeps one necessary zoom. The numerical and simulation examples are given to demonstrate the effectiveness of the theoretical results. Compared with existing literatures, the convergence criterion and protocol design proposed have lower conservativeness.

1. Introduction

Recently, much attention is drawn to the consensus problem in multiagent systems [1–3], due to its widely applications ranging from rendezvous [4], flocking [5], formation control [6] to fusion estimation [7]. Furthermore, communication delays in consensus problems are also taken into account, for example, due to the communication congestions or retransmissions.

The study for consensus in multiagent systems with time delays started at first-order integrator systems (e.g., [8–14]), some of which were generalized to second-order (e.g., [15–23]) or high-order (e.g., [24]) integrator systems then. Nevertheless, the system matrix in those systems is a special companion form matrix. Therefore, it is valuable to study the consensus problems of high-order (some paper called it high dimensional) linear systems with time-varying delays.

Compared with the consensus studies on first-order, second-order, and high-order integrator multiagent systems with time delays, relatively few results were conducted in those governed by high-order linear dynamics. For linear SISO multiagent systems, the robust consensus schemes with three kinds of feedback delays were investigated in [25]; and the necessary and sufficient condition for the existence of consensus solution for heterogeneous high-order multiagent systems was given in [26]. It should be noted that both of the above-mentioned studies were conducted in the frequency domain. In terms of linear MIMO multiagent systems, the time-domain methods were employed to analyze the consensus problems. In [27], the high-order linear MIMO consensus problems with single constant time delay under undirected communication topologies were converted into low-dimensional robust stability problems with time delay. The results showed that if the delay was less than a certain bound, the consensus protocols always existed; while [27] only gave the existence criteria on designing protocol parameters instead of algorithms for finding them. By projecting the state onto two subspaces (consensus subspace and consensus complement subspace), high-order linear MIMO consensus problems with single time-varying delay were equivalent simultaneous stabilization problems with low dimensions in [28]. Therefore, a Lyapunov-Krasovskii functional was given to establish both the consensus convergence and the algorithm for protocol parameters design [28]. However, there are still three main aspects that could be enhanced. (1) Conservativeness. The criterion had relatively more conservativeness caused by the structure of the Lyapunov-Krasovskii functional and the zoom in the derivative of this functional. (2) Constraint. All the derivatives 𝜇 of time delays were restricted by 𝜇<1 in [28]. (3) Parameters. The method of protocol design (the protocol has two parameter matrices and sometimes both of them need to be designed at once) could only calculate the second parameter matrix rather than both parameter matrices.

In this paper, we will provide free-weighting matrices [29] based consensus convergence criterion in high-order linear multiagent systems with single time-varying delay under directed topologies. Correspondingly, the protocol parameter matrices will be designed by solving NLMIs (nonlinear matrix inequalities). It should be noted that in undirected communicated topologies, the parameter matrices can be designed without constraining the structures for the solutions on NLMIs; while in directed topologies, some parts of those solutions should be determined in a specific form; otherwise, some nonlinear equality constraints will be added into these NLMIs, which is going to make them hard to solve. Even though we determine the structures of some solutions matrices for directed topologies, the results are better than those in [28], which used stronger constraints in solutions as well as a more conservative Lyapunov-Krasovskii functional.

Compared with [28], our results have three improvements. (1) Conservativeness. By optimizing the structure of Lyapunov-Krasovskii functional and adding proper free-weighting matrices, our criterion has less conservativeness. (2) Constraint. Our criterion could permit the case 𝜇≥1, even when 𝜇 is unknown (3) Parameters. Both two parameter matrices can be calculated in the design of protocol could calculate directly, and for the second parameter matrix design, our method can give a feasible solution in larger time-delay bounds.

Throughout this paper, the notation represents the symmetric part in a symmetric matrix; 𝒟>(≥,<,≤)0 denotes that the matrix 𝒟 is positive definite (positive semidefinite, negative definite, negative semidefinite); ⊗ denotes the Kronecker product; 𝟎 can be an appropriate dimensions zero matrix or vector; for any complex vector 𝐱, any real matrix 𝒟 and 𝜆∈ℂ, we denote 𝐱=[Re(𝐱)^𝑇,Im(𝐱)^𝑇]^𝑇, Λ_𝒟=diag{𝒟,𝒟}, and Υ_𝜆=^{Re(𝜆)𝐈𝑛−Im(𝜆)𝐈𝑛}_{Im(𝜆)𝐈𝑛Re(𝜆)𝐈𝑛}, where 𝐈_𝑛 is an identity matrix with 𝑛×𝑛 dimensions.

2. Preliminaries and Problem Description

Let 𝒢=(𝒱,ℰ,𝒜) be a directed simple graph of order 𝑁(𝑁>1), where 𝒱={𝜐₁,𝜐₂,…,𝜐_𝑁} denotes the nodes set, ℰ⊂𝒱×𝒱 is the edges set, and 𝒜=[𝑎_𝑖𝑗]∈ℝ^𝑁×𝑁 is the weighted adjacency matrix with 𝑎_𝑖𝑗≥0, where 𝑎_𝑖𝑗>0 if and only if edge (𝜐_𝑗,𝜐_𝑖)∈ℰ. As 𝒢 is simple, 𝑎_𝑖𝑖=0 holds for all 𝑖∈{1,2,…,𝑁}. The neighbors of 𝜐_𝑖 is defined as 𝒩_𝑖={𝜐_𝑗∈𝒱∣(𝜐_𝑗,𝜐_𝑖)∈ℰ}. The in-degree of the node 𝜐_𝑖 are defined as 𝑑_𝑖𝑛(𝜐_𝑖)=∑_{𝜐𝑗∈𝒩𝑖}𝑎_𝑖𝑗. The degree matrix of 𝒢 is a diagonal matrix 𝒟=[𝑑_𝑖𝑗], where 𝑑_𝑖𝑗=0(𝑖≠𝑗), 𝑑_𝑖𝑖=𝑑_𝑖𝑛(𝜐_𝑖). Then Laplacian matrix of 𝒢 is defined as ℒ=[𝑙_𝑖𝑗]=𝒟−𝒜.

Lemma 1 (see [28]). The Laplacian matrix ℒ of 𝒢 satisfies(1)ℒ at least has one zero eigenvalue, and 𝟏_𝑁 is the associated right eigenvector;(2)if 𝒢 has a spanning tree, then 0 is a simple eigenvalue of ℒ, and all the other 𝑁−1 eigenvalues have positive real parts;(3)if 𝒢 does not have a spanning tree, then ℒ at least has two zero eigenvalues with the geometric multiplicity being not less than 2.

Lemma 2 (Schur Complement [24]). For a given symmetric matrix 𝐙 with the form 𝐙=𝐙^𝑇=^𝐙11𝐙12_⋆𝐙22, 𝐙₁₁∈ℝ^𝑟×𝑟, 𝐙₁₂∈ℝ^{𝑟×(𝑛−𝑟)}, 𝐙₂₂∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)}, then 𝐙<0 if and only if 𝐙₁₁<0, 𝐙₂₂−𝐙^𝑇₁₂𝐙⁻¹₁₁𝐙₁₂<0 or 𝐙₂₂<0, 𝐙₁₁−𝐙₁₂𝐙⁻¹₁₁𝐙^𝑇₁₂<0.

In this paper, we consider a group of 𝑁 agents whose dynamics is described by high-order linear systems in continuous-time domain: 𝐱̇_𝑖(𝑡)=𝐀𝐱_𝑖(𝑡)+𝐁𝐮_𝑖(𝑡),𝑖=1,2,…,𝑁, (1) where 𝐱_𝑖(𝑡)=𝑥_𝑖⁽¹⁾(𝑡)𝑥_𝑖⁽²⁾(𝑡)⋯𝑥_𝑖^(𝑛)(𝑡)^𝑇 is the state of agent 𝑖 and 𝐮_𝑖(𝑡)∈ℝ^𝑝 is the consensus protocol, 𝐀∈ℝ^𝑛×𝑛, 𝐁∈ℝ^𝑛×𝑝.

We considered a class of protocols as follows: 𝐮_𝑖(𝑡)=𝐊₁𝐱_𝑖(𝑡)+𝐊_{2𝜐𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝐱_𝑗(𝑡−𝜏(𝑡))−𝐱_𝑖(𝑡−𝜏(𝑡)), (2) where 𝐊₁,𝐊₂∈ℝ^𝑝×𝑛, 𝜏(𝑡)∈[0,ℎ], 𝜏̇(𝑡)∈[0,𝜇].

Let 𝐱(𝑡)=𝐱^𝑇₁(𝑡)𝐱^𝑇₂(𝑡)⋯𝐱^𝑇_𝑁(𝑡)^𝑇. With protocol (2), the dynamics of multiagent systems is 𝐱̇(𝑡)=𝐈_𝑁⊗𝐀+𝐁𝐊₁𝐱(𝑡)−ℒ⊗𝐁𝐊₂𝐱((𝑡[),𝐱(𝑡)(𝑡)[),𝑡−𝜏)),𝑡∈0,+∞=𝜙,𝑡∈−ℎ,0 (3) where 𝜙(𝑡) is a continuously differentiable initial function.

Definition 3 (consensus). For given 𝐊₁,𝐊₂, the multiagent systems (3) achieve consensus if and only if for all 𝑖≠𝑗, lim_𝑡→+∞‖𝐱_𝑖(𝑡)−𝐱_𝑗(𝑡)‖=𝟎.

Definition 4 (consensusabilization). Multiagent systems are said to be consensusabilized by 𝐊₁ and/or 𝐊₂ if and only if the 𝐊₁ and/or 𝐊₂ make the multiagent systems achieve consensus.

3. Main Results

In this section, the consensus convergence criterion and protocol parameters design method for high-order linear multiagent systems with time-varying delays in directed topologies are provided.

Lemma 5 (see [28]). Dynamics (3) achieves consensus if and only if Δ̇_𝑖(𝑡)=𝐀+𝐁𝐊₁Δ_𝑖(𝑡)−𝜆_ℒ(𝑖+1)𝐁𝐊₂Δ_𝑖(𝑡−𝜏(𝑡)),𝑖=1,…,𝑁−1 (4) are asymptotically stable, where 𝜆_ℒ𝑗 is the 𝑗th eigenvalue of the Laplacian Matrix ℒ corresponding to 𝒢, and 0=Re(𝜆_ℒ1)≤Re(𝜆_ℒ2)≤⋯≤Re(𝜆_ℒ𝒩).

Remark 6. This necessary and sufficient condition converts the consensus problem in dynamics (3) into a stability problem of dynamics (4). According to Lemma 1, if 𝒢 has a spanning tree, there is no 𝜆_ℒ𝑖=0(𝑖=2,…,𝑁). Therefore, 𝐊₂ can make it possible to stabilize systems (4); while if 𝒢 does not have a spanning tree, then ℒ at least has two zero eigenvalues. Without loss of generality, we assume that 𝜆_ℒ2=0. Thus, (4) can be rewritten as Δ̇₂(𝑡)=𝐀+𝐁𝐊₁Δ₂(𝑡),Δ̇₃(𝑡)=𝐀+𝐁𝐊₁Δ₃(𝑡)−𝜆_ℒ3𝐁𝐊₂Δ₃((𝑡,⋮Δ̇𝑡−𝜏))_𝑁(𝑡)=𝐀+𝐁𝐊₁Δ_𝑁(𝑡)−𝜆_ℒ𝑁𝐁𝐊₂Δ_𝑁(𝑡−𝜏(𝑡)). (5) If (4) is asymptotically stable, 𝐀+𝐁𝐊₁ should be Hurwitz. In this case, multiagent systems (3) can achieve consensus even if there is no cooperation at all. Therefore, Lemma 5 is adequate to the case with or without spanning trees in communication topology. Therefore, the connectivity assumptions can be neglected.

3.1. Consensus Convergence Criteria

Theorem 7. Given the upper bound in communication delay ℎ and its derivative 𝜇, multiagent systems (3) achieve consensus if there exist proper-dimensional 𝐏_𝑖=𝐏^𝑇_𝑖>0, 𝐐_𝑖=𝐐^𝑇_𝑖≥0, 𝐑_𝑖=𝐑^𝑇_𝑖≥0, 𝐙_𝑖=𝐙^𝑇_𝑖>0, 𝐗_𝑖=𝐗^𝑇_𝑖≥0 and free-weighting matrices 𝐍_𝑖=𝐍_𝑖1𝐍_𝑖2𝐍_𝑖3,𝐒_𝑖=𝐒_𝑖1𝐒_𝑖2𝐒_𝑖3,𝑖=1,…,𝑁−1 (6) such that the following LMIs hold: Ψ_𝑖1=𝐗_𝑖𝐍_𝑖⋆𝐙_𝑖≥0,Ψ_𝑖2=𝐗_𝑖𝐒_𝑖⋆𝐙_𝑖Ψ≥0,_𝑖3=Φ_𝑖ℎ𝐀^𝑇_𝑖𝐙_𝑖⋆−ℎ𝐙_𝑖<0, (7) where Φ_𝑖=Φ_𝑖1+Φ_𝑖2+Φ^𝑇_𝑖2+ℎ𝐗_𝑖Φ_𝑖1=^{𝐏𝑖Λ𝐀+𝐁𝐊1+Λ𝑇𝐀+𝐁𝐊1𝐏𝑖+𝐐𝑖+𝐑𝑖−Υ𝜆ℒ(𝑖+1)𝐏𝑖Λ𝐁𝐊2𝟎}_{𝐐⋆−1−𝜇𝑖𝟎⋆⋆−𝐑𝑖},Φ_𝑖2=𝐍_𝑖−𝐍_𝑖+𝐒_𝑖−𝐒_𝑖,𝐀_𝑖=Λ_{𝐁𝐊𝐀+1}−Υ_{𝜆ℒ(𝑖+1)}Λ_𝐁𝐊2𝟎. (8)

Proof. Since (4) contains complex vectors and matrices, it is required to be transformed into the following form by decompositing the real and imaginary parts: Δ̇_𝑖(𝑡)=Λ_{𝐁𝐊𝐀+1}Δ_𝑖(𝑡)−Υ_{𝜆ℒ(𝑖+1)}Λ_𝐁𝐊2Δ_𝑖(𝑡−𝜏(𝑡)),𝑖=1,…,𝑁−1. (9)
Define the following 𝑁−1 quadratic Lyapunov-Krasovs-kii functional candidates: 𝑉_𝑖(𝑡)=𝑉_𝑖1(𝑡)+𝑉_𝑖2(𝑡)+𝑉_𝑖3(𝑡)+𝑉_𝑖4(𝑡),𝑖=1,…,𝑁−1, (10) where 𝑉_𝑖1(𝑡)=Δ^𝑇_𝑖(𝑡)𝐏_𝑖Δ_𝑖(𝑡), 𝑉_𝑖2(𝑡)=∫^𝑡_{𝑡−𝜏(𝑡)}Δ^𝑇_𝑖(𝑠)𝐐_𝑖Δ_𝑖(𝑠)𝑑𝑠, 𝑉_𝑖3(𝑡)=∫^𝑡_𝑡−ℎΔ^𝑇_𝑖(𝑠)𝐑_𝑖Δ_𝑖(𝑠)𝑑𝑠, 𝑉_𝑖4(𝑡)=∫⁰_−ℎ∫^𝑡_𝑡+𝜃Δ̇^𝑇_𝑖(𝑠)𝐙_𝑖Δ̇_𝑖(𝑠)𝑑𝑠𝑑𝜃, and 𝐏_𝑖=𝐏^𝑇_𝑖>0, 𝐐_𝑖=𝐐^𝑇_𝑖≥0, 𝐑_𝑖=𝐑^𝑇_𝑖≥0, 𝐙_𝑖=𝐙^𝑇_𝑖>0, 𝐗_𝑖=𝐗^𝑇_𝑖≥0, 𝑖=1,…,𝑁−1.
By Newton-Leibniz formula, for any given free-weighting matrices with proper dimensions 𝐍_𝑖=𝐍_𝑖1𝐍_𝑖2𝐍_𝑖3,𝐒_𝑖=𝐒_𝑖1𝐒_𝑖2𝐒_𝑖3,𝑖=1,…,𝑁−1, (11) we have 𝑉_𝑖1(𝑡)=2𝜁^𝑇_𝑖1(𝑡)𝐍_𝑖Δ_𝑖(𝑡)−Δ_𝑖((𝑡−𝑡−𝜏))_{𝑡(𝑡)𝑡−𝜏}Δ̇_𝑖(𝑠)𝑉𝑑𝑠=𝟎,_𝑖2(𝑡)=2𝜁^𝑇_𝑖1(𝑡)𝐒_𝑖Δ_𝑖(𝑡−𝜏(𝑡))−Δ_𝑖(𝑡−ℎ)−^𝑡_{(𝑡)𝑡−𝜏}Δ̇_𝑖(𝑠)𝑑𝑠=𝟎, (12) where 𝜁_𝑖1(𝑡)=[Δ^𝑇_𝑖(𝑡),Δ^𝑇_𝑖(𝑡−𝜏(𝑡)),Δ^𝑇_𝑖(𝑡−ℎ)]^𝑇.
For any appropriate 𝐗_𝑖=𝐗^𝑇_𝑖≥0, 𝑖=2,…,𝑁, (13) holds 𝑉_𝑖3(𝑡)=ℎ𝜁^𝑇_𝑖1(𝑡)𝐗_𝑖𝜁_𝑖1(𝑡)−^𝑡_{𝑡−𝜏(𝑡)}𝜁^𝑇_𝑖1(𝑡)𝐗_𝑖𝜁_𝑖1(𝑡)−𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝜁^𝑇_𝑖1(𝑡)𝐗_𝑖𝜁_𝑖1(𝑡)𝑑𝑠=𝟎. (13)
Taking the time derivative of 𝑉_𝑖(𝑡) along the trajectory of (9), it is obtained that 𝑉̇_𝑖1(𝑡)=Δ^𝑇_𝑖(𝑡)𝐏_𝑖Λ_{𝐁𝐊𝐀+1}+Λ^𝑇_{𝐀+𝐁𝐊1}𝐏_𝑖Δ_𝑖(𝑡)−2Δ^𝑇_𝑖(𝑡)𝐏_𝑖Υ_{𝜆ℒ(𝑖+1)}Λ_𝐁𝐊2Δ_𝑖((𝑡,𝑉̇𝑡−𝜏))_𝑖2(𝑡)=Δ^𝑇_𝑖(𝑡)𝐐_𝑖Δ_𝑖(𝑡)−(1−𝜏̇(𝑡))Δ^𝑇_𝑖(𝑡−𝜏(𝑡))𝐐_𝑖Δ_𝑖((𝑡,𝑉̇𝑡−𝜏))_𝑖3(𝑡)=Δ^𝑇_𝑖(𝑡)𝐑_𝑖Δ_𝑖(𝑡)−Δ^𝑇_𝑖(𝑡−ℎ)𝐑_𝑖Δ_𝑖(),𝑉̇𝑡−ℎ_𝑖4(𝑡)=ℎΛ_{𝐁𝐊𝐀+1}Δ_𝑖(𝑡)−Υ_{𝜆ℒ(𝑖+1)}Λ_𝐁𝐊2Δ_𝑖(𝑡−𝜏(𝑡))^𝑇𝐙_𝑖×Λ_{𝐁𝐊𝐀+1}Δ_𝑖(𝑡)−Υ_{𝜆ℒ(𝑖+1)}Λ_𝐁𝐊2Δ_𝑖(𝑡−𝜏(𝑡))−^𝑡_𝑡−ℎΔ̇^𝑇_𝑖(𝑠)𝐙_𝑖Δ̇_𝑖(𝑠)𝑑𝑠. (14)(15)(16)(17)
Adding 3 zero items into the derivative, we have 𝑉̇_𝑖(𝑡)=𝑉̇_𝑖1(𝑡)+𝑉̇_𝑖2(𝑡)+𝑉̇_𝑖3(𝑡)+𝑉̇_𝑖4(𝑡)+𝑉_𝑖1(𝑡)+𝑉_𝑖2(𝑡)+𝑉_𝑖3(𝑡). (18)
With 𝜏̇(𝑡)≤𝜇, (18) can be zoomed; we can utilize the decomposition of Δ̇^𝑡_𝑡−ℎ^𝑇_𝑖(𝑠)𝐙_𝑖Δ̇_𝑖(𝑠)𝑑𝑠=^𝑡_{𝑡−𝜏(𝑡)}Δ̇^𝑇_𝑖(𝑠)𝐙_𝑖Δ̇_𝑖(𝑠)+𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}Δ̇^𝑇_𝑖(𝑠)𝐙_𝑖Δ̇_𝑖(𝑠)𝑑𝑠 (19) in (17) as a contributor to make the zoomed (18) be quadratic forms: 𝑉̇_𝑖(𝑡)≤𝜁^𝑇_𝑖1(𝑡)Φ_𝑖1+ℎ𝐀^𝑇_𝑖𝐙_𝑖𝐀_𝑖𝜁_𝑖1(𝑡)−^𝑡_{𝑡−𝜏(𝑡)}𝜁^𝑇_𝑖2()𝐗𝑡,𝑠^𝑖𝐍_𝑖⋆𝐙_𝑖𝜁_𝑖2()−𝑡,𝑠𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝜁^𝑇_𝑖2()𝐗𝑡,𝑠^𝑖𝐒_𝑖⋆𝐙_𝑖𝜁_𝑖2(𝑡,𝑠)𝑑𝑠, (20) where 𝜁_𝑖2(𝑡,𝑠)=[𝜁^𝑇_𝑖1(𝑡),Δ̇^𝑇_𝑖(𝑠)]^𝑇. For formula (20), if Φ_𝑖1+ℎ𝐀^𝑇_𝑖𝐙_𝑖𝐀_𝑖<0 and Ψ_𝑖1=𝐗_𝑖𝐍_𝑖⋆𝐙_𝑖≥0,Ψ_𝑖2=𝐗_𝑖𝐒_𝑖⋆𝐙_𝑖≥0, (21) then 𝑉̇_𝑖(𝑡)<−𝜀‖Δ_𝑖(𝑡)‖² holds for any 𝜀>0, 𝑖=1,…,𝑁−1.
By Lemma 2, Φ_𝑖1+ℎ𝐀^𝑇_𝑖𝐙_𝑖𝐀_𝑖<0 equals to Ψ_𝑖3<0. Thus, if (7), holds then (9) is asymptotically stable and multiagent systems (3) achieve consensus.

Remark 8. Theorem 7 has only one zoom in the derivative of Lyapunov-Krasovskii functional which is necessary, but Theorem 2 in [28] has three zooms, which make the criterion have more conservativeness. Numerical examples and Simulations, in Section 4.1, will validate that our Theorem 7 has less conservativeness.

Remark 9. Setting 𝐐_𝑖=𝟎 for all 𝑖∈{1,2,…,𝑁−1}, we can get the delay-dependent and rate-independent consensus convergence criterion; while Theorem 2 in [28] requires that the upper bounds of 𝜇 are limited by 𝜇<1.

3.2. Protocol Parameters Design Algorithm in Undirected Communication Topology

In this section, the algorithms in designing protocol parameter matrices in (2) under undirected communication topologies are investigated (including single parameter matrix design and both parameter matrices design).

Theorem 10 (double protocol parameter matrices design for undirected communication topology). Assume that the topology 𝒢 is an undirected graph. Given the upper bound in communication delay ℎ and its derivative 𝜇, if there exist proper-dimensional 𝐋=𝐋^𝑇>0, 𝐖_𝑖=𝐖^𝑇_𝑖≥0, 𝐔_𝑖=𝐔^𝑇_𝑖≥0, 𝐆_𝑖=𝐆^𝑇_𝑖≥0𝐘_𝑖=𝐘^𝑇_𝑖=𝐘^(𝑖)₁₁𝐘^(𝑖)₁₂𝐘^(𝑖)₁₃⋆𝐘^(𝑖)₂₂𝐘^(𝑖)₂₃⋆⋆𝐘^(𝑖)₃₃≥0 (22) and free-weighting matrices 𝐕₁,𝐕₂, 𝐌_𝑖=𝐌_𝑖1𝐌_𝑖2𝐌_𝑖3,𝐓_𝑖=𝐓_𝑖1𝐓_𝑖2𝐓_𝑖3,𝑖=1,…,𝑁−1 (23) such that the following NLMIs hold: Ψ_𝑖1=𝐘_𝑖𝐌_𝑖⋆𝐋𝐆_𝑖⁻¹𝐋≥0,Ψ_𝑖2=𝐘_𝑖𝐓_𝑖⋆𝐋𝐆_𝑖⁻¹𝐋≥0,Ψ_𝑖3=Π^(𝑖)₁₁Π^(𝑖)₁₂Π^(𝑖)₁₃ℎ𝐋𝐀^𝑇+ℎ𝐕^𝑇₁𝐁^𝑇⋆Π^(𝑖)₂₂Π^(𝑖)₂₃−ℎ𝜆_ℒ(𝑖+1)𝐕^𝑇₂𝐁^𝑇⋆⋆Π^(𝑖)₃₃𝟎⋆⋆⋆−ℎ𝐆_𝑖<0, (24)(25)𝑖=1,…,𝑁−1, where Π^(𝑖)₁₁=𝐀𝐋+𝐋𝐀^𝑇+𝐁𝐕₁+𝐕^𝑇₁𝐁^𝑇+𝐖_𝑖+𝐔_𝑖+𝐌_𝑖1+𝐌^𝑇_𝑖1+ℎ𝐘^(𝑖)₁₁,Π^(𝑖)₁₂=−𝜆_ℒ(𝑖+1)𝐁𝐕₂−𝐌_𝑖1+𝐓_𝑖1+𝐌^𝑇_𝑖2+ℎ𝐘^(𝑖)₁₂,Π^(𝑖)₁₃=−𝐓_𝑖1+𝐌^𝑇_𝑖3+ℎ𝐘^(𝑖)₁₃,Π^(𝑖)₂₂=−1−𝜇𝐖_𝑖−𝐌_𝑖2−𝐌^𝑇_𝑖2+𝐓_𝑖2+𝐓^𝑇_𝑖2+ℎ𝐘^(𝑖)₂₂,Π^(𝑖)₂₃=−𝐓_𝑖2−𝐌^𝑇_𝑖3+𝐓^𝑇_𝑖3+ℎ𝐘^(𝑖)₂₃,Π^(𝑖)₃₃=−𝐔_𝑖−𝐓_𝑖3−𝐓^𝑇_𝑖3+ℎ𝐘^(𝑖)₃₃, (26)
then multiagent systems (1) can be consensusabilized by protocol (2), and the parameters are from 𝐊₁=𝐕₁𝐋⁻¹, 𝐊₂=𝐕₂𝐋⁻¹.

Proof. If multiagent systems (3) achieve consensus, then according to Theorem 7, inequalities (7) hold. Note that ℒ is symmetrical due to a undirected 𝒢; we can halve the dimension of Ψ_𝑖1 and Ψ_𝑖2 in (7). Further, Ψ_𝑖3 can be rewritten as follows: Ψ_𝑖3=Φ_𝑖ℎ𝐀^𝑇_𝑖𝐙_𝑖⋆−ℎ𝐙_𝑖<0, (27) where Φ_𝑖=Φ_𝑖1+Φ_𝑖2+Φ^𝑇_𝑖2+ℎ𝐗_𝑖Φ_𝑖1=^{𝐏𝑖𝐀+𝐁𝐊1+𝐀+𝐁𝐊1𝑇𝐏𝑖+𝐐𝑖+𝐑𝑖−𝜆ℒ(𝑖+1)𝐏𝑖𝐁𝐊2𝟎}_{𝐐⋆−1−𝜇𝑖𝟎⋆⋆−𝐑𝑖},Φ_𝑖2=𝐍_𝑖−𝐍_𝑖+𝐒_𝑖−𝐒_𝑖,𝐀_𝑖=𝐀+𝐁𝐊₁−𝜆_ℒ(𝑖+1)𝐁𝐊₂𝟎. (28)
Then employing the congruent transformation with diag{𝐏_𝑖⁻¹,𝐏_𝑖⁻¹,𝐏_𝑖⁻¹,𝐙_𝑖⁻¹} for Ψ_𝑖3 in (27), we can derive diag𝐏_𝑖⁻¹,𝐏_𝑖⁻¹,𝐏_𝑖⁻¹,𝐙_𝑖^−1𝑇Ψ_𝑖3diag𝐏_𝑖⁻¹,𝐏_𝑖⁻¹,𝐏_𝑖⁻¹,𝐙_𝑖⁻¹<0, (29) where 𝑖=1,…,𝑁−1. Set 𝐏₁⁻¹=𝐏₂⁻¹=⋯=𝐏⁻¹_𝑁−1=𝐋, and let 𝐖_𝑖=𝐋𝐐_𝑖𝐋, 𝐔_𝑖=𝐋𝐑_𝑖𝐋, 𝐆_𝑖=𝐙_𝑖⁻¹ and 𝐘_𝑖=𝐋𝐗_𝑖𝐋=𝐘^(𝑖)₁₁𝐘^(𝑖)₁₂𝐘^(𝑖)₁₃⋆𝐘^(𝑖)₂₂𝐘^(𝑖)₂₃⋆⋆𝐘^(𝑖)₃₃≥0,𝐌_𝑖=diag𝐋,𝐋,𝐋𝐍_𝑖𝐋=𝐌_𝑖1𝐌_𝑖2𝐌_𝑖3,𝐓_𝑖=diag𝐋,𝐋,𝐋𝐒_𝑖𝐋=𝐓_𝑖1𝐓_𝑖2𝐓_𝑖3, (30) where 𝑖=1,…,𝑁−1 (note that the 𝐏_𝑖,𝐐_𝑖,𝐑_𝑖,𝐙_𝑖,𝐗_𝑖,𝐍_𝑖,𝐒_𝑖 are half dimensional compared to those in Theorem 7). Subsequently, with 𝐊₁𝐋=𝐕₁ and 𝐊₂𝐋=𝐕₂, formula (29) can be rewritten as (25). Similarly, by using congruent transformation with diag{𝐋,𝐋,𝐋} in dimension-halved Ψ_𝑖1 and Ψ_𝑖2, respectively, inequalities (24) can be derived. Note that (24) and (25) are equivalent to (7); the protocol parameters are from 𝐊₁=𝐕₁𝐋⁻¹, 𝐊₂=𝐕₂𝐋⁻¹.

Remark 11. To avoid divergence of the multiagent systems, sometimes we need to design 𝐊₁,𝐊₂ at the same time, even though it is proven in [28] that the 𝐊₁ is responsible for the state trajectories of those multiagent systems which achieve consensus. NLMIs (24) and (25) can be solved by using Cone Complementarity Linearization (CCL) method (see [30, 31]).

Remark 12. In order to design the 𝐊₂ only, we can set 𝐊₁ equal to a certain constant matrix in Theorem 10. Accordingly, our Theorem 10 can be compared with Theorem 3 in [28].

3.3. Protocol Parameters Design Algorithm in Directed Communication Topology

In this section, the algorithms in designing protocol parameter matrices in (2) under directed communication topologies are proposed.

Note that if the topology is directed, the structure of free-weighting matrices 𝐕₁,𝐕₂ is limited by Λ_𝐊1=𝐕₁𝐋⁻¹ and Λ_𝐊2=𝐕₂𝐋⁻¹ (actually, the limitation caused by diagonal structures of Λ_𝐊1,Λ_𝐊2). That means even though we calculate the free-weighting matrices 𝐕₁,𝐕₂ by solving NLMIs, we could not always acquire a diagonal structured Λ_𝐊1,Λ_𝐊2. More importantly, both the diagonal blocks in 𝐊₁ (as well as in 𝐊₂) are required to be equal. This will add nonlinear equality constraints in NLMIs, which are hard to solve.

Therefore, we determine the structures of 𝐕₁,𝐕₂ as well as 𝐋 in advance. Theorem 13 gives the corresponding NLMIs.

Theorem 13 (double protocol parameter matrices design for directed communication topology). Assume the topology 𝒢 is a directed graph. Given the upper bound in communication delay ℎ and its derivative 𝜇, if there exist proper-dimensional 𝐋=𝐋^𝑇>0, 𝐖_𝑖=𝐖^𝑇_𝑖≥0, 𝐔_𝑖=𝐔^𝑇_𝑖≥0, 𝐆_𝑖=𝐆^𝑇_𝑖≥0𝐘_𝑖=𝐘^𝑇_𝑖=𝐘^(𝑖)₁₁𝐘^(𝑖)₁₂𝐘^(𝑖)₁₃⋆𝐘^(𝑖)₂₂𝐘^(𝑖)₂₃⋆⋆𝐘^(𝑖)₃₃≥0 (31) and free-weighting matrices 𝐕₁,𝐕₂, 𝐌_𝑖=𝐌_𝑖1𝐌_𝑖2𝐌_𝑖3,𝐓_𝑖=𝐓_𝑖1𝐓_𝑖2𝐓_𝑖3,𝑖=1,…,𝑁−1 (32) such that the following NLMIs hold: Ψ_𝑖1=𝐘_𝑖𝐌_𝑖⋆𝐋𝐆_𝑖⁻¹𝐋≥0,Ψ_𝑖2=𝐘_𝑖𝐓_𝑖⋆𝐋𝐆_𝑖⁻¹𝐋≥0,Ψ_𝑖3=Π^(𝑖)₁₁Π^(𝑖)₁₂Π^(𝑖)₁₃ℎΛ_𝐀𝐋^𝑇+ℎΛ_𝐁𝐕₁^𝑇⋆Π^(𝑖)₂₂Π^(𝑖)₂₃−ℎΥ_{𝜆ℒ(𝑖+1)}Λ_𝐁𝐕₂^𝑇⋆⋆Π^(𝑖)₃₃𝟎⋆⋆⋆−ℎ𝐆_𝑖<0, (33)𝑖=1,…,𝑁−1, where Π^(𝑖)₁₁=Λ_𝐀𝐋+Λ_𝐀𝐋^𝑇+Λ_𝐁𝐕₁+Λ_𝐁𝐕₁^𝑇+𝐖_𝑖+𝐔_𝑖+𝐌_𝑖1+𝐌^𝑇_𝑖1+ℎ𝐘^(𝑖)₁₁,Π^(𝑖)₁₂=−Υ_{𝜆ℒ(𝑖+1)}Λ_𝐁𝐕₂−𝐌_𝑖1+𝐓_𝑖1+𝐌^𝑇_𝑖2+ℎ𝐘^(𝑖)₁₂,Π^(𝑖)₁₃=−𝐓_𝑖1+𝐌^𝑇_𝑖3+ℎ𝐘^(𝑖)₁₃,Π^(𝑖)₂₂=−1−𝜇𝐖_𝑖−𝐌_𝑖2−𝐌^𝑇_𝑖2+𝐓_𝑖2+𝐓^𝑇_𝑖2+ℎ𝐘^(𝑖)₂₂,Π^(𝑖)₂₃=−𝐓_𝑖2−𝐌^𝑇_𝑖3+𝐓^𝑇_𝑖3+ℎ𝐘^(𝑖)₂₃,Π^(𝑖)₃₃=−𝐔_𝑖−𝐓_𝑖3−𝐓^𝑇_𝑖3+ℎ𝐘^(𝑖)₃₃ (34) and the structures of 𝐋,𝐕₁,𝐕₂ are 𝐋=𝐋₁₁𝐋₁₂⋆𝐋₂₂>0,𝐕₁=𝐕⁽¹⁾₁₁𝐕⁽¹⁾₁₂𝐕⁽¹⁾₂₁𝐕⁽¹⁾₂₂,𝐕₂=𝐕⁽²⁾₁₁𝐕⁽²⁾₁₂𝐕⁽²⁾₂₁𝐕⁽²⁾₂₂, (35) where 𝐋₁₁=𝐋₂₂, 𝐋₁₂=𝟎, 𝐕^(𝑖)₁₁=𝐕^(𝑖)₂₂=𝟎, 𝐕^(𝑖)₁₂=𝐕^(𝑖)₂₁=𝟎, 𝑖=1,2, then multiagent systems (1) can be consensusablized by protocol (2), and the parameters are from 𝐊₁=𝐕⁽¹⁾₁₁𝐋⁻¹₁₁, 𝐊₂=𝐕⁽²⁾₁₁𝐋⁻¹₁₁.

Proof. The process of proof is similar to that in Theorem 10. The differences are as follows: (1) the dimension of matrices is doubled compared to those in Theorem 10; (2) the structures of 𝐋,𝐕₁,𝐕₂ are restricted by (35) for the sake of eliminating the nonlinear equality constraints.

Remark 14. Similar to Remark 11, we can set 𝐊₁ equal to a certain constant matrix in Theorem 13 so that the 𝐊₂ can be designed under the condition that 𝐊₁ has already been chosen.

4. Numerical Examples and Simulations

In order to validate the theoretical results proposed in this paper, numerical examples and simulations are involved to illustrate the effectiveness. Figure 1 reveals some topologies in multiagent systems. For simplicity, assume that their adjacency matrices are limited to 0, 1 matrices.

(a)

(b)

Figure 1 

Two directed graphs.

4.1. Comparisons on Consensus Convergence Criteria

Firstly, we will compare the conservativeness of the criteria with [28]. Additionally, the applicability for unknown 𝜇 in our criterion will be illustrated.

Consider multiagent systems (3) with the following parameters: 𝐊𝐀=010001−3−1−2,𝐁=000110₁,=312−1−1−1,𝐊₂=001541. (36)(37)

The initial function is 𝜙(𝑡)≡𝜙^𝑇₁(𝑡),𝜙^𝑇₂(𝑡),𝜙^𝑇₃(𝑡),𝜙^𝑇₄(𝑡)^𝑇,𝑡∈[−ℎ,0), (38) where ℎ is the upper bound on communication delay 𝜏(𝑡) (see formula (3)) and 𝜙₁(𝑡)≡3−9−9,𝜙₂(𝑡)≡,𝜙−3−3−3₃(𝑡)≡−2−2−2,𝜙₄(𝑡)≡333. (39)

Results of comparisons are shown in Table 1.

Table 1 

Allowable upper bound on communication delay for third-order linear multiagent systems with topology 𝒢_𝑎, 𝒢_𝑏.

Remark 15. The comparison results illustrate that allowable upper bound on communication delays given by Theorem 7 is larger than those in [28] (to be more precise, ranging from 13.51% to 55.67%); the simulation results in Figure 2 validate the effectiveness. More importantly, our criterion has more advantages when 𝜇 becomes larger. Especially, unknown 𝜇 is able to judge by our criterion.

(a)

(b)

(c)

Figure 2 

State trajectories of 3rd-order multiagent systems in 𝒢𝑎 with dynamics (36), protocol parameters (37), and communication delay 𝜏(𝑡)=0.149/2+(0.149/2)sin((2×0.9/0.149)𝑡).

For Theorem 2 in [28], the number of Lyapunov-Krasovskii functional candidates is reduced to 2 with the aim of saving computing time, even though this would lead to some conservativeness. Similarly, in the current paper, we could also reduce the candidates’ number to 2 to get a higher computation efficiency. But still, for a more persuaded comparison, we select the all (𝑁−1) Lyapunov-Krasovskii functional candidates.

4.2. Validations for Protocol Design Algorithm

Firstly, our Theorem 10 is proposed to design the parameter matrices in protocol (2) under undirected communication topologies, including (1) calculating 𝐊₂ with a known 𝐊₁, (the comparisons with Theorem 3 in [28] are made in Table 2); (2) determining both of the parameter matrices 𝐊₁ and 𝐊₂ (while in [28], 𝐊₁ could not be calculated automatically by its theorems) and results are shown in Table 3.

Table 2 

Protocol parameter 𝐊₂ designs for third-order linear multiagent systems with topologies 𝒢_𝑎, 𝒢_𝑏 in different ℎ and 𝜇.

Table 3 

Protocol parameters 𝐊₁, 𝐊₂ designs for third-order linear multiagent systems with topologies 𝒢_𝑎, 𝒢_𝑏 in different ℎ and 𝜇.

Secondly, for directed communication topologies, the parameter matrices of protocol (2) are determined by our Theorem 13. Similarly, both cases (designing 𝐊₂ only or calculating 𝐊₁,𝐊₂ at the same time, see Tables 2 and 3) are employed to validate the effectiveness of Theorem 13.

In this section, all the numerical examples and simulation results share the same dynamics and initial function as those in Section 4.1 (that is the multiagent systems (3) with parameters (36) and initial function (38)).

In Table 2, the parameter matrix 𝐊₁ is determined as 𝐊₁=311−5−1−2. (40)

As we can see in this table, for undirected topology 𝒢_𝑏, comparisons are made between [28, Theorem 3] and our Theorem 10. Taking the situation 𝜇=0.9 as an example, the allowable upper bound on communication delay when designing a feasible 𝐊₂ in [28, Theorem 3] is ℎ=0.026, and the protocol parameter matrix is 𝐊₂=000.0866−0.00200.22300. (41)

However, in our Theorem 10, the allowable upper bound is ℎ=0.295, which increases one order of magnitude than that in [28, Theorem 3]. The protocol parameter matrix is 𝐊₂=000.254−0.6121.0310, (42) and the effectiveness is validated by simulations in Figure 3.

(a)

(b)

(c)

Figure 3 

State trajectories of 3rd-order multiagent systems in 𝒢𝑏 with protocol parameters (40) and (42) and communication delay 𝜏(𝑡)=0.295/2+(0.295/2)sin((2×0.9/0.295)𝑡).

Similarly, for directed topology 𝒢_𝑎, the comparisons between [28, Theorem 3] and our Theorem 13 are made in Table 2; while it should be noted that in the situation 𝜇=0.5, the allowable upper bound on communication delay in our Theorem 13 is ℎ=0.050, which is less than that in [28, Theorem 3] (ℎ=0.064). However, this just holds in the case that 𝜇 is small. When 𝜇 becomes 0.9, [28, Theorem 3] cannot find a feasible 𝐊₂ any more (actually, the allowable upper bound on 𝜇 for [28, Theorem 3] is 0.766). Correspondingly, we can see that the allowable upper bound for [28, Theorem 3] under communication topology 𝒢_𝑏 experiences a dramatic decrease when 𝜇 becomes larger. Therefore, [28, Theorem 3] is not robust enough for 𝜇. By contrast, ℎ in our Theorem 13 is relatively stable when 𝜇 varies. For 𝒢_𝑏, our Theorem 13 calculates the upper bound for communication delay ℎ=0.294 in the situation 𝜇=0.5, while ℎ=0.295 in 𝜇=0.9. It seems unreasonable, but it does happen when solving the NLMIs (24) and (25). This is because the CCL algorithm is not globally optimal.

Table 3 gives the information about the results for 𝐊₁,𝐊₂ designs under undirected communication topology 𝒢_𝑏 (using Theorem 10) and directed communication topology 𝒢_𝑎 (using Theorem 13), respectively. It should be noted that 𝐊₁ and 𝐊₂ are calculated at the same time rather than 𝐊₁ is determined in advance. To validate the effectiveness of our algorithms, we select the item with ℎ=0.1,𝜇=0.9 to carry on the simulation, and 𝐊₁=,𝐊311.9991−1.0017−3.2781−2₂=.000.000070.00007−0.000080 (43)

Figure 4 shows the effectiveness in our results with time-varying delay 𝜏(𝑡)=0.1/2+(0.1/2)sin((2×0.9/0.1)𝑡).

(a)

(b)

(c)

Figure 4 

State trajectories of 3rd-order multiagent systems in 𝒢𝑎 with protocol parameters design (ℎ=0.1, 𝜇=0.9).

5. Conclusions

We investigated the consensus problems for high-order linear multiagent systems with time varying delays, which include a consensus convergence criterion and two protocol parameters design methods (for undirected and directed communication topologies, resp.). Mainly, the following contributions were concluded in this paper:(1)a high-order linear multiagent systems consensus convergence criterion is proposed. By optimizing the structure on Lyapunov-Krasovskii functional and adding proper free-weighting matrices, the criterion has less conservativeness than the existing results;(2)the consensus convergence criterion is validated when 𝜇≥1, and even can apply with a unknown 𝜇; while [28] could only fit for 𝜇<1;(3)the protocol parameters design algorithm can give the two parameters, directly while the algorithm in [28] could only give 𝐊₂ but the undetermined 𝐊₁ would be set in advance;(4)under the circumstance that the 𝐊₁ is determined in advance, our theorems have less conservativeness and more robustness when the derivative of communication delay varies.

However, for the parameters designing part, the structures of matrices 𝐕₁,𝐕₂ are constrained diagonal forms for the sake of eliminating some nonlinear inequality constraints. This has already brought some conservativeness because some feasible solutions for NLMIs are excluded manually. Therefore, the structure of 𝐕₁,𝐕₂ will be investigated further so that our protocol parameters design could be better used in directed topologies with less conservativeness.