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

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.


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][6][7][8][9][10][11][12][13][14].Different from [5][6][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, secondorder 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][17][18][19][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][22][23][24][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 (  ( − ) −   ( − )) . ( 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 quasiconsensus; 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 multiagent systems with a protocol of communication delay hold similarly.Furthermore, some simulations are given to verify the theoretical results. Notations.Let R  denote the -dimensional Euclidian space.Let 0  and   be the  ×  zero matrix and identity matrix, respectively.If  is a matrix, then rank  and () denote the rank and the spectrum of , respectively.Let 1 and 0 denote the  × 1 column vectors of all ones and all zeros, respectively.The symbol ⊗ denotes the Kronecker product.
Remark 2. For convenience, we assume  = 1 in the following discussion; however, all the results hereafter remain valid for  > 1 by the Kronecker product.

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.
Proof.See the appendix.
Proof.See the appendix.
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  0 zero eigenvalues and all of other eigenvalues are negative.
(Necessity).By reduction to absurdity, suppose that the sufficient condition dose not hold; then there are two cases may be hold.
Case 2. Φ has exactly  0 zero eigenvalues, but it has at least one eigenvalue with nonnegative real part.
If Case 1 holds, then rank(lim If Case 2 holds, then lim  → ∞   2  ̸ = 0 ℓ− 0 , and rank(lim  → ∞  Φ ) ≥ rank(lim  → ∞   1  )+rank(lim  → ∞   2  ) >  0 .However, it follows from Φ  (the dominant terms of  Φ ) that if the consensus is achieved, then rank(lim  → ∞  Φ ) ≤  0 .This is a contradiction.Remark 7. From Theorem 6, if (4) achieves the consensus, then Φ has exactly  0 zero eigenvalues.Therefore,  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][17][18][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.
When  ∈ (0, +∞),   () crosses the negative real axis for the first time at   1 , one has where Therefore, one has the following result equivalently.
Remark 14.If one considers consensus protocol with communication delays as follows: 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  is also

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.