Robust Consensus for Nonlinear Multiagent Systems with Uncertainty and Disturbance

This paper investigates robust consensus for nonlinear multiagent systems with uncertainty and disturbance. The consensus evolution behavior is studied under general consensus protocol when each node is disturbed by the relative states between the node and its neighbors. At first, the robust consensus condition is obtained and the convergency analysis is given by using Lyapunov stability theory andmatrix theory.Then, the practical consensus is investigated and the boundof the error states is presented. Finally, two numerical simulation examples are given to illustrate the proposed theoretical results.


Introduction
In the past few years, increasing attention has been devoted to the coordination of multiagent systems due to its wide applications for sensor network and multirobot systems [1].As one of the most important problems in the issue of distributed coordination, consensus has even attracted much more interest from researchers.And many profound results are reported [2][3][4][5][6].
Consensus means that the states of the agents reach an agreement on a common physical quantity by implementing an appropriate consensus protocol based on the information from local neighbors [7], which is mainly influenced by the topology of the network and the dynamics of each node.Thereby, on one side, many works about consensus of multiagent systems under switching topology are reported [8][9][10].On the other side, consensus of multiagent systems with nonlinear dynamics or disturbances is intensively investigated [11][12][13][14][15]. External disturbance widely exists in real processes and is a main source of instability and poor performance.When any of the nodes is disturbed, coordinated behavior will be destroyed.Thereby, it is of great significance to investigate distributed coordination for nonlinear multiagent systems with bounded disturbances or stochastic disturbances.Mean square average consensus is investigated for multiagent systems with noisy measurement under directed topology.Necessary and sufficient consensus conditions are established [16].Mean square leader-following consensus for multiagent systems with noisy channels under directed switching topology is investigated [17].Necessary and sufficient average consensus conditions are established for multiple doubleintegrator systems with noisy measurement.It is proven that mean square average consensus can be realized if and only if the topology is balanced and strongly connected [18].Using disturbance observer method, robust leader-following consensus was investigated for nonlinear coupled multiagent systems with external disturbance [19].Containment control problem is studied for general linear multiagent systems with exogenous disturbances.Both the state feedback and output feedback containment protocol are proposed by using the disturbance observer approach [20].Using sliding-mode control method, finite consensus and containment were investigated for second-order nonlinear multiagent systems with disturbances under directed topology [21].Bounded consensus tracking is investigated for linear multiagent systems with actuator saturation and input additive uncertainties and disturbances based on low-and-high gain feedback approach [22].
However, few works investigate the consensus behavior when the relative information of the subsystem is disturbed, which widely exists in the real world.Therefore, in this paper, the consensus evolution behavior is investigated for nonlinear multiagent systems with uncertainty and disturbance.The main contribution in this paper is that the disturbance considered is dependent on the addition of the relative states between each node and their neighbors, and the disturbance satisfies a very mild condition, which is more relaxed than the condition in [21,22].The uncertainty and disturbance function is described as a piecewise continuous function (, ), satisfying ‖(, )‖ 2 ≤  0 (‖‖ 2 ) +  0 .Both robust consensus and practical consensus condition are obtained for  0 = 0 and  0 ̸ = 0, respectively.This leads to the fact that the problems investigated in this paper are theoretically challenging and practically important.
The rest of the paper is organized as follows.Section 2 states the model considered in the paper and gives some basic definitions, lemmas, and assumptions.In Section 3, complete consensus protocol is proposed and the convergency analysis is given and practical consensus protocol is obtained in Section 4. In Section 5, two numerical simulation examples are given for illustrating the theoretical results.Finally, Section 6 concludes the paper.

Robust Leader-Following Consensus for Nonlinear Multiagent Systems with Uncertainty and Disturbance
In this section, the robust leader-following consensus of nonlinear multiagent systems with bounded channel disturbance is investigated.Suppose that there is a virtual leader in the network, whose dynamics are described as Consider the following consensus protocol with the channel disturbance: where  > 0 denotes the control gain to be determined; the nonlinear function  :   ×  →   denotes the actuator input additive uncertainty and disturbance, satisfying (  −   , ) = −(  −   , ).
Assumption 5.The channel disturbance function (, ) is piecewise continuous in  and locally Lipschitz in  and its norm is bounded by a known function.
Theorem 9. Consider a networked multiagent system with  followers and a virtual leader, in which the dynamics are described as ( 1)-( 4).Suppose that Assumptions 1-5 hold and the constant  0 = 0 in Assumption 5.Under the consensus protocol (5), robust consensus of system ( 1) can be achieved if the following two conditions hold: (i) There exists at least one pinned node.
(ii) For a given positive constant , where  1 is the minimal eigenvalue of ( + ).Furthermore, all the following nodes will track the virtual leader asymptotically.

Proof. Choose the Lyapunov candidate function as
Differentiating  with respect to  along (9), one can obtain According to Lemma 4 and Assumption 5, one has According to Lemma 7, we get According to (12), one has V < 0, which means that the error systems ( 9) are locally asymptotically stable.Then   () →  0 (), for  → ∞.This means that Theorem 9 holds.

Robust
Leader-Following Practical Consensus.In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case where  0 ̸ = 0 in Assumption 5.
Theorem 10.Consider a networked multiagent system with  followers and a virtual leader, in which the dynamics are described as ( 1)-( 4).Suppose that Assumptions 1-5 hold and the constant  0 ̸ = 0 in Assumption 5.Under the consensus protocol (5), robust practical consensus of system (1)

can be achieved if the following two conditions hold:
(i) There exists at least one pinned node.
(ii) For a given positive constant , where  1 is the minimal eigenvalue of ( + ).

Robust Leaderless Consensus for Nonlinear Multiagent Systems with Uncertainty and Disturbance
In this section, the robust leaderless consensus of nonlinear multiagent systems with bounded channel disturbance is investigated.Consider the following consensus protocol: where  > 0 and  :  ×   →   are defined the same way as Section 3. Denoting  = ∑  =1   /,  = (∑  =1 (,   ) + ∑  =1   )/.Since the Laplacian matrix  is symmetric and zero-row-sum, one has It follows that ẋ = 0. Let x =   − ; according to (9), the error system can be described as 4.1.Robust Leaderless Consensus.In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case where  0 = 0 in Assumption 5.
Theorem 11.Consider a networked multiagent system with  follower agents, in which the dynamics are described as (1).Suppose that Assumptions 1-5 hold and the constant  0 = 0 in Assumption 5.Under the consensus protocol (22), robust consensus of system ( 1) can be achieved if where  2 () is the minimal eigenvalue of  and  is a given positive constant.

Robust Leaderless Practical Consensus.
In this subsection, robust consensus for nonlinear multiagent systems is investigated in the case where  0 ̸ = 0 in Assumption 5.
Theorem 12. Consider a networked multiagent system with  follower agents, in which the dynamics are described as (1).Suppose that Assumptions 1-5 hold and the constant  0 ̸ = 0 in Assumption 5.Under the consensus protocol (22), robust practical consensus of system (1) can be achieved if where  2 () is the minimal eigenvalue of  and  is a given positive constant.

Simulations
Consider a network with 8 nodes and a virtual leader.The topology is described as Figure 1.We consider both complete consensus and practical consensus for nonlinear first-order multiagent systems for the cases where  0 = 0 and  0 ̸ = 0, respectively.The dynamics of each agent are described as ẋ  = 2  sin 3 +   .
For the complete consensus case, choosing the nonlinear disturbance function is chosen as (, ) = 0.02 sin ; according to Figures 2 and 4, we can know that the consensus for the proposed network can be achieved.For the practical consensus case, choosing the nonlinear disturbance function is chosen as (, ) = 0.02 sin  + 0.2 cos ; according to Figures 3 and 5, it can be found that there is a tiny error in the trajectories, which means that the practical consensus is achieved.

Conclusions
In this paper, both the robust consensus and the practical consensus problem are investigated for nonlinear multiagent systems with uncertainty and disturbance.The and disturbance function is piecewise continuous and is described as (, ), satisfying ‖(, )‖ 2 ≤  0 (‖‖ 2 ) +  0 .Both complete consensus and practical consensus condition are obtained and the analysis is presented by using hybrid tools from matrix theory and Lyapunov stability theory.

Figure 5 :
Figure 5: Leader-following practical consensus of state trajectories.