Distributed Consensus Control Protocols for Heterogeneous Multi-Agent Systems With Time-Varying Topologies

Distributed group consensus control problems of heterogeneous multi-agent systems with time-varying topologies and communication delay are investigated in this paper. Two different types of control protocols without relative velocity information are designed. By reformulating the original problem and utilizing the property of Metzler matrix, sufficient group consensus criteria under some assumptions for the two proposed protocols are developed, respectively. Simulation examples are exploited to illustrate the results.

time delays. In [22], Feng investigated two group consensus protocols for delay-free discrete-time heterogeneous MASs. In [23], Ji proposed a novel couple-group consensus protocol and showed that the consensus of the system is independent of the communication delay. In [24]- [26], group consensus control of heterogeneous MASs with communication delays and input delays were investigated. As far as we know, there are rare works on group consensus control of heterogeneous MASs with time-varying topologies, especially with timevarying communication delay.
Motivated by this fact, this paper investigates the group consensus problems for heterogeneous MASs with timevarying topologies and communication delay. Comparing with the works in [21] and [22], this paper investigates the heterogeneous systems with time delay, which makes the systems more realistic and more challenging. In addition, different from [23]- [26] in which the delays are time-invariant, time-varying communication delay, directed topologies and control parameters are considered.
The main contributions of this paper are summarized as follows. Firstly, two novel group consensus protocols for heterogeneous MASs with time-varying delays are proposed. Particularly, the second protocol is more general than the first one. Secondly, we not only consider the fixed topology with uniform control parameters, but also consider the timevarying topologies and multiple control parameters, which is more realistic in engineering practice. Last but not least, under some mild assumptions and by using the property of Metzler matrix, we show that the group consensus can be achieved with any arbitrary bounded communication delays even if the digraph has no spanning tree.

II. GRAPH THEORY
Let G = (V, ε, A) be a directed graph, where V = {s 1 , · · · , s n } represents the vertex set, ε ⊆ V × V represents the edge set, and A = [a ij ] n×n represents the weighted adjacency matrix( [27]). L = {1, 2, · · · n} and e ij = (s i , s j ) denote the node indexes set and the edge e ij in the digraph, respectively. N i = {s j ∈ V : s i , s j ∈ ε} represents the neighbours of agent i. Moreover, the elements of adjacency matrix A are denoted as a ii = 0 and a ij > 0 if e ij ∈ ε. In digraph G, if for every pair of s i , s j , i = j, there always exists a path from s i to s j and a path from s j to s i , then digraph G is said to be strongly connected. A digraph contains a spanning tree if there exists a vertex called root vertex such that there exists a directed path from it to every other vertex. L n = [l ij ] n×n denotes the Laplacian matrix of digraph G, where According to the definition of L n we have L n 1 n = 0, where 1 n = (1, · · · , 1) T ∈ R n . Lemma 1: [27] A matrix (t) is called to be a Metzler matrix if its off-diagonal elements are nonnegative.

III. PROBLEM FORMULATION
Consider a heterogeneous MAS consisting of m first-order agents and n − m second-order agents. The dynamic equation of the first-order agent is given bẏ where 1 = {1, 2, · · · , m}, x i (t) ∈ R is the position state of the ith agent and u i (t) ∈ R is its control input, i ∈ 1 . The dynamic equation of the second-order agent is given bẏ where 2 = {m + 1, m + 2, · · · , n}, x i (t) ∈ R and v i (t) ∈ R represent the position state and velocity state of ith agent, u i (t) ∈ R is its control input, i ∈ 2 . Definition 1: Given the heterogeneous systems with m first-order agents (2) and n − m second-order agents (3), the system can achieve group consensus with protocol u i (t), if and only if, for any initial condition, the following conditions hold:

IV. MAIN Results
In this section, two control protocols are developed. Particularly, the second protocol is more general than the first one in expression.

A. GROUP CONSENSUS PROTOCOL I
In this subsection, the group consensus problem for heterogeneous MAS with time-varying topologies and communication delay will be discussed. In particular, the following control protocol is proposed: where k 1 , k 2 , k 3 , α 1 , α 2 are all positive constants and represent scaling parameters, τ ij (t) ≥ 0 is the communication delay from jth agent to ith agent. By substituting (4) and (5) into system (2) and system (3), respectively, it follows thaṫ To proceed further, we denote Then, by taking the derivative of (8) and (9) with respect to t, we obtaiṅ (11) where N i is the neighbors of agent i.
In what follows, the group consensus problem of the heterogeneous MAS will be discussed by examining the properties of Metzler matrix. For this, we denote η i (t) = 2 α 2 k 3 ϕ i (t)+ ξ i (t), and system (11) becomeṡ Then, we define where If there is no communication delay, i.e. τ ij (t) = 0, then, by considering (13), (10) and (12) becomeṡ where I n is an n-dimensional identity matrix and 0 is a zero matrix with appropriate dimension, and ; L 1 (t) and L 2 (t) are the Lapacian matrixes of first-order agents and second-order agents, respectively; D 12 (t) and D 21 (t) are described by and the adjacency matrix of the heterogeneous MAS is given by In fact, (t) is a Metzler matrix, which will be shown later. Similarly, if the communication delay exists, suppose there are altogether M different communication delays, which denoted by τ ε (t) ∈ {τ ij (t), i, j ∈ 1 ∪ 2 }(ε = 1, 2, · · · , M ). Then system (14) becomeṡ where 0 (t) = diag (t) and the ijth element of ε (t), ε = 1, 2, · · · , M , is either zero or equal to the weight of edge e ij .
Proof: Since k 1 > 0, k 2 > 0, k 3 > 0, α 1 > 0, α 2 > 0, then we can verify that the off-diagonal elements of (t) are nonnegative under Assumption 1. Note that the Laplacian matrix of the heterogeneous MAS L(t) = L 1 (t) −A 12 (t) −A 21 (t)L 2 (t) then we can prove that the summation of each row of (t) is equal to zero. This completes the proof.

Remark 1:
If there is no communication delay, the heterogeneous system can be described as system (14). From Theorem 1, we can verify that (t) is a Metzler matrix with zero row sums. Then, by applying Lemma 1 of [28], we conclude that system (14) can reach asymptotic consensus.
Theorem 2: For any given constants σ 1 > 0 and σ 2 > 0, suppose that every digraph of the system (16) keep fixed at σ 1 time, the union of the digraphs has a spanning tree at every bounded time interval T , T < σ 2 , Assumption 1 holds for system (16), and time delay is bounded. Then, under protocols (4) and (5), heterogeneous systems (2) and (3) can reach group consensus.
Proof: For solving problem (16), we construct the following Lyapunov function (ii) at least one component of f i (σ 2 ), i ∈ F, is in the following interval: where |F| represents the number of agents in F. It is straight forward to verify that G 1 ∈ [χ min σ 1 , χ max σ 1 ] and G 2 ∈ [χ min σ 1 , χ max σ 1 ], which means that all the components of f i (σ 2 ), i ∈ E are strictly contained in [χ min σ 1 , χ max σ 1 ] and at least one component of which is strictly contained in [χ min σ 1 , χ max σ 1 ] and at the same time remove it from f i (σ 2 ), i ∈ F, that means we can denote a new disjoint sets E 1 = E + m 1 and F 1 = F − m 1 , where m 1 ∈ F, f m 1 (σ 2 ) is contained in (19). By repetitive application of this operation, we can finally get Eσ =L and Fσ = ∅ at timeσ . Since min σ > χ min σ 1 , max σ < χ max σ 1 , Lyapunov function has decreased over the time interval [σ 1 ,σ ], which means V (f (σ )) − V (f (σ 1 )) < 0. According to the analysis approach in [28], we can verify that system (16) can achieve consensus asymptotically, i.e., Therefore, we can conclude that the heterogeneous systems (2) and (3) can achieve group consensus under protocols (4) and (5). This completes the proof.
Remark 2: Theorem 2 reveals that if the communication delays are bounded, then the weights of the delays do not affect whether the consensus of the heterogeneous systems can be reached. However, the weights of the delays will affect the consensus rate of the heterogeneous systems.
Remark 3: In this paper, we assume the union of the digraphs has a spanning tree at every bounded time interval T , then the consensus of the heterogeneous systems can be reached regardless of the existence of the spanning trees for the corresponding graphs. Obviously, if the topology is fixed and connected, the heterogeneous systems is also able to achieve group consensus.

B. GROUP CONSENSUS PROTOCOL II
In this subsection, we shall consider a more general case, and the corresponding control protocol is given by where κ i > 0, β i > 0, γ i > 0, α 1 > 0, α 2 > 0 are scaling parameters, τ ij (t) ≥ 0 is the communication delay from jth agent to ith agent. By repeating the same procedure as that in (9), we denotē Then we havė To proceed, we setη i (t) = 2 α 2 γ iφ i (t) +ξ i (t). Then, system (24) becomeṡ Therefore, the heterogeneous system under protocols (20) and (21) becomeṡ where¯ 0 (t) = diag(¯ (t)) and¯ ε (t), ε = 1, 2, · · · , M , is either zero or equal to the weight of edge e ij . Obviously, .¯ (t) is defined as follows:  Theorem 3: For any given constants σ 1 > 0 and σ 2 > 0, suppose that every digraph of the system (26) keep fixed at σ 1 time, the union of the digraphs has a spanning tree at every bounded time interval T whose time length is smaller than σ 2 , Assumption 2 holds for system (26), and time delay is bounded. Then under protocols (20) and (21), heterogeneous systems (2) and (3) can reach group consensus.
Proof: The proof of Theorem 3 is omitted, since it is similar to that of Theorem 2.

V. SIMULATION RESULTS
In this section, the effectiveness of the obtained results will be demonstrated by carrying out some numerical simulations. Consider a heterogeneous MAS with random initial conditions, and the weight of each edge is set as 1.
To compare the difference between protocol I and protocol II, we take the control parameters of protocol I as k 1 = 2, k 2 = 1, k 3 = 2, α 1 = 1, α 2 = 2, and the initial conditions and communication topologies are same as protocol II. Velocity states of second-order agents with protocol II are shown in Fig. 7, it is not hard to see that the consensus speed of protocol II is faster than that of protocol I, and this problem will be discussed in our future work.

VI. CONCLUSION
A group consensus control problem for heterogeneous MAS was investigated in this paper. Different from most works in the existing literature, time-varying topologies and communication delay were considered. Two distributed consensus control protocols were developed. Sufficient group consensus criteria for the heterogeneous MAS were obtained by using the property of Metzler matrix. The main results were verified by carrying out several numerical examples. For future study, the group consensus control of heterogeneous MAS with both of communication and input delays should be addressed.