Distributed Consensus Tracking for Second-Order Nonlinear Multiagent Systems with a Specified Reference State

This paper mainly addresses the distributed consensus tracking problem for second-order nonlinear multiagent systems with a specified reference trajectory. The dynamics of each follower consists of two terms: nonlinear inherent dynamics and a simple communication protocol relying only on the position and velocity information of its neighbors.The consensus reference is taken as a virtual leader, whose output is only its position and velocity information that is available to only a subset of a group of followers. To achieve consensus tracking, a class of nonsmooth control protocols is proposed which reply on the relative information among the neighboring agents. Then some corresponding sufficient conditions are derived. It is shown that if the communication graph associated with the virtual leader and followers is connected at each time instant, the consensus can be achieved at least globally exponentially with the proposed protocol. Rigorous proofs are given by using graph theory, matrix theory, and Lyapunov theory. Finally, numerical examples are presented to illustrate the theoretical analysis.


Introduction
Consensus problem, one of the most important and fundamental issues in the cooperative control, has attracted great attention from researchers in recent years because of broad applications in various real-world multiagent systems, such as cooperative control of unmanned (air) vehicles, formation control of robots and aircrafts, and design of sensor networks, to name a few (see the survey papers [1][2][3][4][5][6][7] and extensive references therein).The consensus of multiagent systems means to design a network distributed control policy based on the local information obtained by each agent, such that all agents reach an agreement on certain quantities of interest by negotiating with their neighbors.The quantities of interest might represent attitude, position, velocity, temperature, voltage, and so on, depending on different applications.
Numerous interesting results for consensus problem have been obtained in the past decade.In 1986, Reynolds [8] fist proposed a computer animation model to simulate collective behaviors of multiple agents.In [9], Vicsek et al. proposed a discrete-time distributed model to simulate a group of autonomous agents moving in the plane with the same speed but different heading.In [10], Jadbabaie et al. gave a theoretical explanation for the consensus behavior of Vicsek's model based on graph and matrix theories.Ren and Beard [11] extend the work of Jadbabaie et al. [10] to the case of directed graphs and gave some more relaxed conditions.Olfati-Saber and Murray [1] investigated a systematical framework of consensus problems with directed interconnection graphs or time-delay by Lyapunov-based approach.
Note from the literatures which were concerned with consensus problem that the multiagent systems can usually be classified into leaderless and leader-follower systems.In a leaderless consensus problem [12][13][14], there does not exist a (virtual) leader, while in a leader-following consensus problem [15][16][17][18][19], there exists a (virtual) leader.The (virtual) leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones.Therefore, in recent literatures, a (virtual) leader-follower approach has been widely used to the consensus problem [20][21][22][23][24][25].Such a consensus problem with a dynamic (virtual) leader is commonly called consensus tracking problem.Most 2 Mathematical Problems in Engineering of the above works are based on the first-order control protocols that control the velocities of agents rather than their accelerations (which are commonly easier to control and are especially meaningful for analysis of dynamical system in the real world).Hence, it is necessary to study the consensus tracking problem of second-order multiagent systems.In [26], the consensus tracking problem of multiagent systems with double-integrator dynamics was studied.However, [26] requires the availability of the virtual leader's acceleration input to all followers.Compared with previous works of [26].Reference [20] studied a distributed leader-following consensus problem with single-integrator dynamics and double-integrator dynamics under fixed and switching communication topologies.It was shown that the acceleration measurements of the virtual leader and followers are not required, and the consensus tracking can be achieved in finite time.In [27], the authors studied the consensus in multiagent systems with second-order dynamics and sampled data.Note also from the above literatures that most of common models used to study the consensus tracking of the second-order multiagent systems are double integrator models.However, in reality, mobile agents may be governed by more commonly inherent nonlinear dynamics [15,[28][29][30][31]. References [15,28] studied the leaderless consensus problems for secondorder multiagent systems with nonlinear dynamics under fixed network topology.Reference [32] investigated secondorder leader-follower consensus of nonlinear multiagent systems via pinning control.Reference [33] investigated leaderfollowing consensus for second-order multiagent systems with nonlinear inherent dynamics.However, in the above literatures each agent's state only depends on their common inherent dynamics.It is difficult to track a time-varying desired reference state for all agents.
Motivated by the above discussions, the main contributions of this paper lie in that we deal with the consensus tracking for second-order nonlinear multiagent systems with a specified reference state.The dynamics of each follower consists of two terms: nonlinear inherent dynamics and a simple communication protocol relying only on the position and velocity information of its neighbors.The consensus reference is taken as a virtual leader, whose output is only its position and velocity information that is available to only a subset of a group of followers.To achieve consensus tracking, a class of nonsmooth control protocols is proposed which reply on the relative information among the neighboring agents.Then some corresponding sufficient conditions are derived.It is shown that if the communication graph associated with the virtual leader and followers is connected at each time instant, the consensus can be achieved at least globally exponentially with the proposed protocol.Rigorous proofs are given by using graph theory, matrix theory, and Lyapunov theory.Compared with the existing results, this paper has the following advantages.Firstly, with contrast of consensus for multiagent systems with double-integrator dynamics [19,20,[25][26][27], we investigate the consensus tracking problems for multiagent systems with nonlinear inherent dynamics.Secondly, in contrast to the existing results in [15,28,32,33], where each agent's state only depends on their common inherent dynamics, in this paper all agents can well track a time varying desired reference state.Thirdly, in this paper the consensus can be achieved globally exponentially with the proposed protocol, while in [15,[27][28][29][30][31] the consensus can be archived asymptotically.Finally, it is well known that most of real-world systems, for example, biological systems, autonomous vehicles systems, complex systems, and so on, are time-varying systems [34,35].Therefore, in this paper, we consider the consensus tracking problems for time-varying multiagent systems, where the states of all agents are time varying.
The rest of this paper is organized as follows.In Section 2, the relevant notations and preliminaries are presented.In Section 3, the consensus tracking problem to be solved in this paper is described.The main results are presented in Section 4. Numerical examples are shown in Section 5. Finally, some conclusion remakes are given in Section 6.

Preliminaries
2.1.Notations.The following notations will be used throughout this paper: Let   denote the  ×  identity matrix, let 0 × denote the  ×  zero matrix, and let 1  = [1, 1, . . ., 1]  ∈   (1 for short, when there is no confusion). min () and  max () are the smallest and the largest eigenvalues of the matrix , respectively.

Graph Theory Notions.
Using graph theory, we can model the interaction topology in multiagent systems consisting of  agents.Let G = (V, E, A) be a weighted graph of order  with the finite nonempty set of nodes V = {V 1 , . . ., V  }, the set of edges E ⊆ V × V, and a weighted adjacency matrix A = (  ) × .Here, each node V  in V corresponds to an agent , and each edge (V  , V  ) ∈ E in a weighted directed graph corresponds to an information link from agent  to agent , which means that agent  can receive information from agent .In contrast, the pairs of nodes in weighted undirected graph are unordered, where an edge (V  , V  ) ∈ E denotes that agent  and  can receive information from each other.The weighted adjacency matrix A of a weighted directed graph is defined such that   = 0 for any and   = 0 otherwise.The weighted adjacency matrix A of a weighted undirected graph is defined analogously except that The Laplacian matrix  = (  ) × of graph G is defined by   = −  for  ̸ = , and   = ∑  =1, ̸ =    , ,  ∈ {1, . . ., }.For an undirected graph,  is symmetric positive semidefinite.However,  is not necessarily symmetric for a directed graph.
Lemma 1 (see [36]).Assume G is a weighted undirected graph with Laplacian matrix ; then the following two statements are equivalent: (i) the matrix  has an eigenvalue zero with multiplicity 1 and corresponding eigenvector 1, and all other eigenvalues are positive; (ii) G is connected.
Mathematical Problems in Engineering 3 2.3.Nonsmooth Analysis Background.Consider the following vector differential equation: where () = [ 1 (),  2 (), . . .,   ()]  and (()) is not necessarily continuous.A Filippov solution of ( 1) on where  1 could be ∞, is absolutely continuous function () satisfying the following differential inclusion: where In the following, a Lyapunov stability theorem in terms of the set-valued map L K  is stated.

Problem Description
Consider a multiagent system consisting of  agents.In what follows, all agents are assumed in one-dimensional space for the simplicity of presentation.However, all results hereafter are still valid for the -dimensional ( > 1) by the Kronecker product [39].Here, the dynamics of each agent in the group is given by where  ( = 1, . . ., ),   ∈   and V  ∈   denote the position and velocity vectors, (,   , V  ) ∈   denotes the inherent nonlinear dynamics, and   () denotes the control input.When (,   , V  ) ≡ 0, the multiagent system has double-integrator dynamics.
The consensus problem of the multiagent system ( 5) is to design control inputs   ,  = {1, . . ., } such that lim for any  = 1, 2, . . ., , where  0 ∈   and V 0 ∈   are, respectively, the position vector and velocity vector of the virtual leader, which does not have to be an actual agent but is specified by where (,  0 , V 0 ) ∈   describes the nonlinear dynamics of the virtual leader, and (,  0 , V 0 ) ∈   is responsible for controlling trajectory of the virtual leader.We assume that |(,  0 , V 0 )| ≤  0 , where  0 is a positive constant.
Hereafter, the  agents in systems ( 5) are called followers, and their communication topology is represented by the graph G. Suppose that the virtual leader and all followers share the same nonlinear inherent dynamics, and these nonlinear inherent dynamics satisfy a Lipchitz-type condition given by Assumption 4 as follows, which is satisfied in many well-known systems.Remark 6.In recent reference [40], the robust cooperative tracking for multiple nonidentical second-order nonlinear systems is investigated.The nonlinear inherent dynamics (,   , V  ) in this paper satisfies the Lipchitz condition (8), while the nonlinear function (,   , V  ) in [40] is required to be continuously differentiable (the derivative ḟ (,   , V  ) exists in its domain and is itself a continuous function).

Consensus Tracking Protocols
To satisfy (6), the following control input ( 9) is proposed for each follower: where  is a nonnegative constant,  is a positive constant,   , ,  = 1, . . ., , is the (, )th entry of the adjacency matrix A associated with , and sgn(⋅) is the signum function.In addition,  0 > 0 ( = 1, . . ., ) if the virtual leader's position is available to follower , and  0 = 0 otherwise.Substituting (9) to (5) gives Let  =  + diag( 10 , . . .,  0 ), where  is the Laplacian matrix of G. Proof.Since the communication graph G is connected and the virtual leader is a neighbor of at least one follower, it follows that the matrix  is symmetric positive definite.Therefore, the matrix  can be diagonalized as  =  −1 Δ, where Δ = { 1 ,  2 , . . .,   },   > 0 with   being the th eigenvalue of .Then we can obtain a matrix Π such that

Lemma 7. Suppose that the communication graph G is connected and at least one follower can receive information from the virtual leader. Let 𝐻 = [
where 0 × is the  ×  zero matrix.It is easy to see that Π is symmetric and has the same eigenvalues as .
Let  be an eigenvalue of Π.Because Δ is a diagonal matrix, it follows from (11) that As the matrix Π is symmetric, it follows that the roots of ( 12) are all real.Therefore, they are positive if and only if the following conditions are satisfied: And,  is positive definite if Next, we consider the matrix .Similarly with the analysis of the matrix , there exists a matrix  such that is symmetric and has the same eigenvalues as .
Remark 9.In this paper, we assume that there exists a virtual leader, which does not have to be an actual agent and can be a specified reference state.In fact, the conclusion obtained in this paper can also be extended to the leader-less case where there exists no virtual leader.Because the proof is similar to that of Theorem 8 in this paper, it is therefore omitted here.

Numerical Results
In order to demonstrate the effectiveness of the theoretic results, some simulations are given in this section.six followers indexed by 1 to 6, respectively.The communication topology is given in Figure 1(a).The nonlinear inherent dynamics of each follower is given as follows: It is easy to verify that (,   , V  ) satisfies Assumption 4. Here, the Lipschitz constants are chosen as  1 = 3 and  2 = 3.The trajectory of virtual leader is chosen as  0 () = [, sin()]  ∈  2 .It follows that the dynamics of virtual leader is given as Furthermore, From (33), let us take  0 = 3.
Therefore, the matrix  can be derived from the topology given in Figure 1(a): It is easy to obtain that  min () = 0.1284.By Theorem 8, when  ≤ ((8 Tracking errors of x positions , respectively, and the initial position and velocity states of the virtual leader are  0 (0) = [0, 0]  and V 0 (0) = [1, 1]  .It can be seen that all followers ultimately track the virtual leader.Simulation results verify the theoretical analysis very well.

Leaderless Case.
In this part, let us consider a group of six agents without virtual leader.The communication topology is given in Figure 1(b).In order to compare with leader-follower case, the nonlinear inherent dynamics of each agent is chosen as (31), which is the same as that in leader-follower case.It can be seen from Figures 6 and 7 that the tracking errors ultimately converge to zero.We see from Figure 5 that consensus tracking is achieved.
Remark 10.For the leaderless consensus case, the final states of each agent are determined by the communication topology, the control gains, and the initial value of each agent.The final states of each agent cannot be specified.However, for the leader-follower case, there exists a virtual leader that determines the final state, and the control objective is to guarantee that the final states of all agents reach the state of the virtual leader.From the above simulations, the conclusion obtained in this paper can be extended to the leaderless case.

Conclusion
The consensus tracking problem for second-order multiagent systems with nonlinear inherent dynamics and a time varying reference state has been studied in this paper.A class of nonsmooth control protocols has been proposed, and the  corresponding sufficient conditions have been obtained.It is found that if the communication graph associated with the virtual leader and followers is connected, the consensus can be achieved globally exponentially with the proposed protocol.
There are still a number of related interesting problems that deserve further infestation, for example, the consensus tracking problem for high-order multiagent systems with nonlinear inherent dynamics, the consensus tracking problem under directed communication topology and with time delays, and so on, some of which will be studied in the near future.

5. 1 .Figure 1 :
Figure 1: The communication topologies for (a) a group of six followers and one virtual, and (b) a group of six agents without virtual leader.

Figure 2 :
Figure 2: Position states of the followers and virtual leader under the communication topology of Figure 1(a).
denotes the intersection over all sets of Lebesgue measure zeroes, B(, ) denotes the open ball of radius  centered at , and co(Ω) denotes the convex closure of the convex hull of the set Ω.