Low Conservative Criteria for Robust Consensus of Multiagent Systems with Delays , Disturbances , and Topologies Uncertainties

Considering the limited communications conditions such as delays, disturbances, and topologies uncertainties, the stability criteria for robust consensus of multiagent systems are proposed in this paper. Firstly, by using the idea of state decomposition and space transformation, the condition for guaranteeing consensus is converted into verifying the robust stability of the disagreement system. In order to deal with multiple time-varying delays and switching topologies, jointly quadratic common LyapunovKrasovskii (JQCLK) functional is built to analyze the robust stability. Then, the numerical criterion can be obtained through solving the corresponding feasible nonlinear matrix inequality (NLMI); at last, nonlinear minimization is used like solving cone complementarity problem.Therefore, the linearmatrix inequality (LMI) criterion is obtained, which can be solved bymathematical toolbox conveniently. In order to relax the conservativeness, free-weighting matrices (FWM) method is employed. Further, the conclusion is extended to the case of strongly connected topologies. Numerical examples and simulation results are given to demonstrate the effectiveness and the benefit on reducing conservativeness of the proposed criteria.


Introduction
Recently, the consensus problem has become an interesting and important topic in the field of formation control, flocking, rendezvous in multiagent systems, fusion estimation, collaborative decision making, and coupled oscillator synchronization.
In some practical applications, communication delays and switching topologies should be considered because of agents moving, communication congestion, or finite transmission distance.The earliest work focused on the conditions for guaranteeing that the agents achieve average consensus with time-delays and switching topologies, for example, frequency domain [1][2][3], time domain [4][5][6][7][8], delay graph [9,10], and max-min value [11,12].These existing methodologies contributed to the average consensus problem with limited networks enormously.
Furthermore, there is also some work focusing on the networks with disturbances, and topologies uncertainties, namely, robust consensus.The stability with noises in the multiagent systems has been discussed in [13][14][15].The consensus problem of high-order agent systems is discussed in [16].The robustness of consensus in single integrator multiagent systems to coupling delays and switching topologies is investigated [17,18].Actually, more researchers are interested in solving the robust consensus problem with both delays and disturbances; see, for example, [3] in frequency domain and [19][20][21][22][23] in time domain.Though the analysis methodologies in frequency domain can provide sufficient and necessary conditions for consensus seeking, it is difficult to find the common or multiple Lyapunov functions in the case of switching interconnections [3].It is only valid for fixed topology in most cases.As a result, the analysis methodologies in time domain through building Lyapunov functional become the preferred one.However, it is still sufficient conditions to achieve consensus.
This paper focuses on the improvement of conservativeness through building Lyapunov-Krasovskii functional.In order to obtain the lower conservativeness, free-weighting matrices (FWM) method [24][25][26] is adopted.This idea has been proved when solving the average consensus problem [8].Meanwhile, this method can also relax the restrictions to the communication conditions and find the numerical criterion.Therefore, it can be employed to deal with the robust consensus with complicated network restrictions, including multiple time-varying communication delays, disturbances with limit energy, and time-varying topologies uncertainties.
The remainder of this paper is organized as follows.Section 2 contains the background and some preliminaries.Section 3 describes the problem.Main result is given in Section 4. Numerical example and simulation results are illustrated in Section 5.At last, Section 6 gives the conclusion.
Throughout this paper, the notation ⋆ represents the symmetric part in a symmetric matrix; consider D > (≥, <, ≤)0 denotes that the matrix D is positive definite (positive semidefinite, negative, negative semidefinite); 0 or 0 denotes zero value or zero matrix with appropriate dimensions.⊗ denotes the Kronecker product.Let   ∈    represent the state value of node   with   dimensions (such as position or temperature, etc.), and the multiagent systems with the state vector  = ( 1 , . . .,   )  and the topology  can be described by   = (, ) (usually denote  by compactness).We say nodes   and   agree in multiagent systems if and only if   =   .The nodes achieve consensus if and only if   =   for all ,  ∈ I,  ̸ = .Whenever the nodes are all in agreement, the common value of all nodes is called the group decision value [1].

Several Definitions and Lemmas.
Before giving the main results, we introduce some definitions and lemmas which play an important role in the proof of our main theoretical results.
Definition 1 (balanced graphs [1]).One says the node   of a digraph  = (V, E, A) is balanced if and only if its in-degree and out-degree equal, that is, deg out (  ) = deg in (  ).A graph  = (V, E, A) is called balanced if and only if all of nodes are balanced or ∑    = ∑    , ∀ ∈ I.
Definition 2 (strongly connected [1]).If there is a directed path from every node to every other node, the graph is said to be strongly connected (connected for undirected graph).
Definition 3 (jointly connected [27]).By the union of a collection of simple graphs { 1 ,  2 , . . .,   ss }, each, with nodes set V, is meant the simple graph  with nodes set V and edges equaling the union of the edge sets of all of the graphs in the collection.One says that such a collection is jointly connected if the union of its members is a strongly connected graph (connected for undirected graph).

Problem Description
3.1.Agent Dynamics and Consensus Protocol.We assume that the agents are depicted by a first-order integrator dynamics, as considered in [1,19].This results in a tractable mathematical problem, although some ideas can be extended to high-order or linear models.Suppose that agent  can be described by where x  () is the state of agent , u  () is the consensus protocol and   () is the external disturbance.Moreover, For the purpose of reaching robust consensus, the consensus protocol is adopted as follows: where Δ  denotes the uncertainties of the communication topologies.In practical terms, it may be influenced by communication congestion or the transmission distance.Parameter  0 is the positive constant coefficient and   ,  = 1, 2, . . .,  are the time-varying communication delays.), where   =   0 and  +1 =    ss satisfy   +1 −    ≥  dell (0 ≤  <  ss ) for some integer  ss ≥ 0 and given constant  dell > 0 such that, during each of such subintervals, the interconnection topology does not change.

Switching
Let () : [0, ∞] → {1, 2, . . .,  ss } denote the switching signal with successive times to describe the topology switches between subintervals; then the collective dynamics of the multiagent network with  communication delays and disturbances (including uncertainties and disturbances) can be depicted in matrix form where x() ∈ R ×1 is state vector, () ∈ R ×1 is disturbance vector, and z() is output vector.The efficient matrix of output vector in ( 5) is defined as follows:

Definition of Robust Consensus. Multiagent systems (5) achieve robust consensus if and only if lim
here,  > 0.

Main Results
In this section, we will give the delay-dependent NLMI criteria for robust consensus seeking of multiagent system including jointly connected switching topologies and strongly connected topologies.Then, nonlinear minimization is employed to deal with the NLMI criterion like solving cone complementarity problem.Lastly, the iterative LMI algorithm is given, which the max tolerant upper bounds can be obtained by Matlab LMI toolbox directly.Before giving the main results of convergence property, we have the following assumptions: (A1) the switching topologies of multiagent systems are jointly connected; (A2) the switching topologies of multiagent systems are strongly connected; (A3) its corresponding graph is balanced; (A4) for all jointly connected topologies determined by switching signal , the sum of the in-degree or outdegree of all nodes is equals to each other.

. , 𝑚 with proper dimensions and common free-weighting matrices
Where then the protocol (4) globally asymptotically solves the robust consensus of multiagent systems (5) satisfying the specified  ∞ index .
Proof.See the Appendix.
The above conclusion derived from Theorem 7 is also valid for the strongly connected topologies case.We omit the detailed proof procedure because it is similar to that in Theorem 7.

, 𝑚 with proper dimensions and common free-weighting matrices
. .,   } holds; then the protocol (4) globally asymptotically solves the robust consensus of multiagent systems (5) satisfying the specified  ∞ index .
Remark 9.The definition of Ξ  in Corollary 8 is similar to that in Theorem 7.
Remark 10.If the feasibility of the NLMI ∑  =1 Ξ  < 0 exists, then the derivative of the Lyapunov-Krasovskii functional is negative definite.Therefore, the systems have robust stability.In addition, the bound condition of communication can be obtained using mathematical package.Since the existence of the term −  R −1  ,  = 1, 2, . . ., , the allowable solution of NLMI ∑  =1 Ξ  < 0 cannot be obtained by LMI toolbox directly.
Next, the iterative LMI algorithm is proposed to solve the problem (13).Therefore, Theorem 7 and Corollary 8 can be solved directly by the existing LMI-based methods or mathematical software package.
Algorithm 11 (maximum of the upper bound on network delay   with specified  ∞ index ).
Step 3. Solve the following minimum problem about matrices variables Υ   , Υ  , R   , R  satisfying the restriction in (13): Let Step 4. Justify ∑  =1 Ξ  < 0 or not.If yes, increase   and return to Step 2; otherwise, check up the iterative time.If it exceeds the specified number, terminate the algorithm; otherwise, let  =  + 1 and return to Step 3.

Remark 12.
As the literature [31] pointed, it is difficult to get the global solution of the nonlinear minimum problem (13).In order to get suboptimal maximum upper bound on delay, the inequality ∑  =1 Ξ  < 0 is adopted in the iterative algorithm.
Remark 13.Similarly, the above iterative algorithm also can be used to get the maximum upper bound on delay derived from Corollary 8.Moreover, the minimum of the index  with specified the upper bound the minimum of the index , with specified the delay, can be obtained by the iterative algorithm.

Illustrative Example.
We now offer an illustrative example to show the effectiveness of the proposed method.Consider the jointly connected and balanced topologies with two communication delays; see Figure 1.For simplicity, suppose that their adjacency matrices are limited to 0 and 1 matrices.In the following, two cases with different connectivities are discussed, respectively.
Assume that the time-varying communication delays satisfy  1 <  2 and let For given index  = 1, the allowable upper bound on communication delay can be obtained by Theorem 7 and Algorithm 11.In Table 1, four-factor three-level orthogonal experiment results are given by using Matlab LMI toolbox.
Remark 14.The major factor affecting  2 is the positive constant coefficient  0 .Then the second one is  1 .However, their derivatives  1 ,  2 seem not to affect the results unduly; so we can increase  0 or reduce  1 to get bigger  2 .This also describing the relationship between each factor and the upper bound on communication delay.
Remark 15.Since the delay derivative has little effect on the communication delay, we can obtain the delaydependent/derivative-independent criterion through excluding the second item in (A.4).Plenty of experiments show that their numerical results are basically the same.
With the above conditions, the sixth group in Table 1 is selected.Further, we select the delay  2 () = 0.917| sin((0.5/0.917))|s for Case I and delay  2 () = 5.197| sin((0.5/5.197))|s for Case II.Corresponding simulation experiments are given in Figures 2(a In Figures 3 and 4, the state evolvement of multiagent systems with different  1 () and  0 is given, respectively.Comparing Figure 3 with Figure 2(a), we can find out that the vibration of multiagent systems achieving consensus is acute and getting worse when increasing  1 () gradually.Particularly, multiagent systems even cannot achieve robust consensus from Figure 3(c).In addition, the similar conclusion can be obtained when decreasing  0 ; see Figures 2(b) and 4.These simulation results show the correctness of Remark 14.

Conservativeness Comparisons.
In order to show the benefit of our results, we will compare the conservativeness of criteria with the existing results for strongly connected case [19].Assume that there is a uniform communication delay in multiagent systems ( = 15,  = 1, and  0 = 1), and the topology can be seen in Figure 5. Table 2 lists the allowable upper bounds on communication delay for different  when  = 2.The comparison result shows that the criterion derived from this paper is better than [19].Moreover, the upper bound on the time-varying delay can be achieved with the proposed method, while the result in [19] is only valid for invariant delay.The major reason is the FWM approach which does not magnify the derivative of Lyapunov-krasovskii functional excessively.

Conclusions
The numerical criteria for robust consensus of multiagent systems were provided in this paper.It can be used to justify the convergence property of multiagent systems with multiple time-varying communication delays, disturbances with limit energy, and time-varying topologies uncertainties.Mainly, the following contributions were concluded in this paper.(1) The stability criteria can be obtained to determine the robust consensus through justifying the existence of common Lyapunov-Krasovskii functional.(2) The existence of common Lyapunov-Krasovskii functional is equivalent to solving feasible solution of NLMI.In order to assure lower conservativeness than the existing one [19], FWM method was employed to justify whether the derivative of the functional is negative definite.(3) Nonlinear minimization methods were used to convert the NLMI criteria into the problem which can be solved easily by the existing LMI-based approaches.(4) Moreover, the proposed conclusions also characterized the relationship among communication delays, their derivative, and the coefficient of the consensus protocol for any given  index.It can be regarded as an important basis to depress the influence of limited communication conditions effectively.Proof.Considering all jointly connected topologies determined by  : [0, ∞] → {1, 2, . . .,  ss }, define the following common jointly quadratic Lyapunov-Krasovskii (CJQLK) functional: where P = P  > 0, Q  = Q   ≥ 0, and R  = R   ≥ 0,  = 1, 2, . . .,  are common matrices with appropriate dimensions.

Consider
where we have the following equivalent relationship according to Lemmas 4 and 5: where Ξ  (A.17) According to Lemma 6, we have If ∑  =1 Ξ  < 0, then the condition (A.13) holds.This means that the multiagent systems are stable with nonzero  † 2 () ∈ L 2 [0, ∞) and a given  > 0. The proof procedure of (C2), that is, the stability of system (A.It is clear that (A.20) can be derived from Ξ  < 0. That means, if ∑  =1 Ξ  < 0 holds, multiagent systems are stable under the condition  † 2 () = 0. Therefore, we can conclude that systems (A.1) are robust stable satisfying specified  ∞ index .As a result, the multiagent systems asymptotically achieve robust consensus.The proof of Theorem 7 is complete.

2. 1 .
Graph Theory.Let  = (V, E, A) denote the relationship between multiple agents with the set of nodes V = { 1 , . . .,   }, the set of edges E ⊂ V × V, and adjacent matrix A = [  ].The node indices belong to a finite index set I = {1, 2, . . ., }.The edge can be depicted by (  ,   ), and the value of   corresponds to the edge of the graph; that is, (  ,   ) ∈ E ⇔   > 0. The neighbors of node   are defined by N  = {  ∈ V : (  ,   ) ∈ E}.Let L = [  ] denote the Laplacian matrix corresponding to the topology of the graph, where   = ∑  =1   and   = −  ,  ̸ = .

Figure 1 :
Figure 1: Jointly connected topologies G  , G  and G  , G  .
) and 2(b) for different cases.It is clear from Figure2that multiagent systems achieving robust consensus with multiple delays, impulse disturbances, and time-varying topologies uncertainties.The simulation results further confirm the effectiveness of the proposed theory.
Remark 17.The proposed criteria for robust consensus seeking can be employed to maximize  2 for different  2 ,  1 , and  1 and  0 with specified ; on the other hand, they also can be used to minimize  for given  1 ,  1 ,  2 ,  2 , and  0 .

Table 2 :
Allowable upper bound on communication delay for firstorder multi-agent systems with fixed topology   when  = 2.