Second-Order Consensus in Multiagent Systems via Nonlinear Protocol

This paper focuses on theoretical analysis of second-order consensus in multiagent system. As an extension of the general linear protocol, a nonlinear protocol is designed for multiagent system with undirected communication topology.The nonlinear protocol is also applied to achieve reference velocity consensus. Through choosing the appropriate Lyapunov functions and using LaSalle’s invariance principle, some consensus conditions are derived. Simulation examples are provided to demonstrate the effectiveness of the proposed results.


Introduction
Recently, consensus problems have been investigated extensively in the content of distributed coordinated control of dynamic agents, partly owning to the potential broad application in various areas.A basic and fundamental question in consensus is how to design the effective protocols (or algorithms) such that each agent can achieve a common target.Most of the consensus literatures mainly considered general linear protocols for multiagent systems.However, the relation between agents is usually not linear.In some situations, the states of multiagent systems cannot be observed directly, and some nonlinear functions about the states can be observed.Therefore, it is necessary to design nonlinear protocols for multiagent systems.
Nonlinear functions are always used to describe the coupling relations of nodes in complex networks [1][2][3][4][5][6].On the other hand, some scholars have paid attentions to nonlinear protocols for multiagent systems recently.Bauso et al. [7] designed the distributed nonlinear protocols and proposed a game theoretical approach to solve consensus problems.Xiao et al. [8] presented a new nonlinear protocol for state consensus of multiagent system which provides faster convergence rate than the typical linear protocol [9].Liu et al. [10] discussed the consensus problem under two nonlinear protocols with directed topology.Hui and Haddad [11] developed a thermodynamic framework for addressing consensus for nonlinear multiagent dynamical systems.Shang [12] proposed some sufficient criteria guaranteeing multiagent systems to reach a consensus in finite time under nonlinear protocols.Li and Guan [13] investigated nonlinear consensus protocols for dynamic directed networks of multiagent systems based on the central manifold reduction technique.Shi and Hong [14] considered a group of continuoustime agents with nonlinear agent dynamics and concluded that the agents can flock to a convex target set.Sepulchre [15] introduced consensus problems whose underlying state space is not a linear space but a nonlinear space.Yu et al. [16] studied the consensus problem for cooperative agents with nonlinear dynamics in a directed network.Zhou and Wang [17] derived some sufficient conditions for (global/exponential) semistability for general discrete-time nonlinear protocols.
However, it is worthwhile to note that the abovementioned nonlinear protocols [7][8][9][10][11][12][13]17] are all proposed for the first-order multiagent systems.As we all know, firstorder consensus problems are mainly relative to communication topologies, but, for second-order consensus, both the interaction graph and the coupling strength affect the convergence result.Therefore, the consensus of second-order multiagent systems is more challenging and interesting and attracts researchers' broad attention.For example, Yu et al. [18] considered a second-order consensus problem for multiagent systems with nonlinear dynamics.Furthermore, Song et al. [19] studied the second-order leader-following consensus problem of nonlinear multiagent systems.Ren [20] proposed and analyzed a consensus algorithm for doubleintegrator dynamics with a bounded control input under an undirected interaction graph.Though [20] extended some existing results in consensus algorithms to account for actuator saturation, saturation function which is assumed to be the hyperbolic tangent function is concrete and simple.So far as we know, there is no result about designing some nonlinear protocols for second-order multiagent systems.For this purpose, this paper mainly designs a nonlinear protocol for second-order dynamics with an undirected communication topology as an extension of the linear protocol.Then, the consensus protocol for second-order multiagent system with a group reference velocity available to each individuality is proposed.
The rest of this paper is organized as follows.Section 2 provides some preliminary graph theory, the proposed nonlinear protocol, and two useful lemmas.In Section 3, consensus problem and reference velocity consensus of a secondorder multiagent system with an undirected communication topology are analyzed, respectively.Some numerical examples are given to show the effectiveness and advantage of the theoretical results in Section 4. Finally, a conclusion is provided in Section 5.

Notation and Preliminaries
2.1.Graph Theory.In this paper, the communication topology among  multiagents is assumed to be an undirected graph G = (V, E, A), where V = {V 1 , . . ., V  } is the set of nodes, E ⊆ V × V is the set of edges, and A = [  ] ∈ R × represents the weighted adjacency matrix with nonnegative adjacency elements   .The node or multiagent indexes belong to a finite index set I = {1, 2, . . ., }.An edge   = (V  , V  ) ∈ E in weighted undirected graph G denotes that node  can receive information from  each other.The adjacency elements associated with the edges of an undirected graph are positive   > 0 if   ∈ E,   = 0 otherwise, and have the property that   =   , for all  ̸ = , since   implies   .A path on G from node  1 to   is a sequence of undirected edges in the form of (  ,  +1 ),  = 1, . . .,  − 1.An undirected graph is called connected if there exists a path from any node  to any other node .

Nonlinear Protocols for
Second-Order Dynamics.Consider a second-order multiagent system with the following dynamics: where   () ∈ R is the position state, V  () ∈ R is the velocity state, and   () ∈ R is the control input (or protocol).Given the dynamical system (1), it is said that protocol   () asymptotically solves the consensus problem, if the states of agents satisfy lim for any   (0) and V  (0).The linear protocol [21,22] is proposed as follows: where   is the (, ) entry of the weighted adjacency matrix A ∈ R × , and parameters ,  are the coupling strengths.
It is clear that protocol ( 3) is based on a linear function of difference between each agent and other ones.Linear protocol ( 3) is simple and easy to use.However, sometimes in order to improve convergence rate or model a bounded control input, the linear protocol is often needed to be extended.Furthermore, in some cases, the state feedback   () −   () may be unobservable; instead, we can observe a nonlinear function (  () −   ()) of   () −   ().Naturally, a nonlinear protocol for second-order system (1) is designed as follows: where   is the (, ) entry of the weighted adjacency matrix A and (⋅) and (⋅) are two continuous functions.
In order to ensure existence, uniqueness of solution, and the operative property of nonlinear functions (⋅) and (⋅), the following sets are defined.At the end of this section, two useful lemmas about (⋅) and (⋅) are introduced.
where the facts that   =   and (⋅) is an odd function that is, (  −   ) = −(  −   ), are used to obtain the equalities.

Consensus Analysis
In this section, the consensus and reference velocity consensus of second-order multiagent system (1) with the designed nonlinear protocol will be analyzed.
When nonlinear protocol ( 4) is simplified as follows: and the Lyapunov function ( 9) is chosen, the following conclusion is easily obtained.
Remark 7. The second-order multiagent system with linear protocol can be written as compact matrix form, and the consensus conditions can be derived by analysing the eigenvalues of the compact matrix [22].When the secondorder multiagent system achieves consensus, the positions and velocities will, respectively, converge to a constant value which is relative to the initial position and velocity and the left eigenvector of the Laplacian matrix (see [21,22]).
However, the nonlinear protocol is different from the linear protocol.The second-order multiagent system with nonlinear protocol cannot be written as a compact matrix form.So, the proof of consensus is different from the method used in the linear protocol and is more difficult.When the consensus of the second-order multiagent system with nonlinear protocol is reached, the value which the positions and velocities converge to cannot be decided.The value is relative to the nonlinear function, the communication graph, and the initial position and velocity.

Consensus with Group Reference Velocity.
In practice, given a reference velocity V * (), the positions and velocities of the agents in multiagent system (1) are often required to achieve consensus and converge to V * (); that is,   () →   (), V  () → V  () → V * (), for all  ̸ = .Hence, the following nonlinear protocol is designed with a group reference velocity V * () as Given a group reference velocity V * (), second-order multiagent system (1) with protocol ( 19) is expressed as Theorem 8.For nonlinear protocol (19), consider functions (⋅) ∈ Φ and (⋅) ∈ Ψ.When information topology graph G is undirected and connected, multiagent system (20)   ( From Lemma 2, it is obtained that Since function (⋅) ∈ Ψ and the information topology graph G is undirected and connected, there exist the following facts that Combing (25), it follows that V() ≤ 0. When V() ≡ 0, it is clear that V  () ≡ V  (), V  () ≡ V * ().Then, it is derived that As a result, it is concluded that x () ≡ x () which is equivalent to   () ≡   ().Making use of LaSalle's invariance principle, it is concluded that   () →   (), V  () → V  () → V * (), as  → ∞.The proof is completed.

Numerical Examples
In this section, a number of simulations are given to illustrate the effectiveness of the results proposed in the paper.
Example 9. Consider the communication topology graph G as described in Figure 1.Graph G has eight nodes, and the edges denote the information interchange between agents.For simplicity, it is assumed that graph G has 0-1 weights.To simulate numerical examples, choose and () =  3 in the nonlinear protocol (4), and it is clear that  ∈ Φ and  ∈ Ψ.Therefore, the nonlinear protocol (4) can be achieved asymptotically for multiagent system (1) from Theorem 4. Figure 2 shows that the velocity and position trajectories of second-order multiagent system (8) converge to consensus, respectively.In addition, Figure 3 describes that both the velocity and position trajectories of second-order multiagent system (1) with protocol (18) achieve consensus.Choose a reference velocity V * () = sin(3 * ) + cos(), and using protocol (19), multiagent system (1) can achieve consensus asymptotically.Figure 4 describes the trajectories of velocity and position of multiagent system (1) with protocol (19), respectively.It is seen that the trajectories of velocity follow the reference velocity V * () successfully.

Conclusions
This paper provided consensus analysis for the second-order multiagent system with nonlinear protocol.As an extension of linear protocols, a nonlinear protocol was designed to achieve consensus.In addition, the protocol was also used to achieve reference velocity consensus for second-order multiagent system.The analysis mainly relied on some tools from algebraic graph and control theory.Through using the Lyapunov theory and LaSalle's invariance principle, sufficient conditions for consensus are derived.Simulation examples show that when the nonlinear function satisfies the assumption, the second-order multiagent can achieve consensus and reference velocity consensus.