## Abstract

The guaranteed performance of the consensus control for multi-agent systems with Lipschitz nonlinear dynamics and directed interaction topologies is investigated, where the directed interaction topology contains a spanning tree. By a special matrix transformation, guaranteed performance consensus problems are transferred into guaranteed performance stabilization problems. Then, the criterions of guaranteed performance consensus for nonlinear multi-agent systems with directed interaction topologies are obtained, and an upper bound of the introduced performance function is given. A numerical simulation is given to demonstrate the effectiveness of the proposed results. Finally, some possible topics about the guaranteed performance consensus problem for nonlinear multi-agent systems are proposed.

### Keywords

- guaranteed performance consensus
- multi-agent system
- nonlinear dynamic

## 1. Introduction

In the past decades, many researchers focused on consensus problems for multi-agent systems due to their wide applications, including formation control of mobile agents [1], synchronization in wireless sensor networks [2], distributed automatic generation control for cyber-physical micro-grid system [3], and rendezvous [4] or flocking [5] of multiple vehicles. After Olfati-Saber and Murray [6] proposed a theoretical framework for the consensus problem, a lot of remarkable conclusions for linear multi-agent systems were presented in the literature, respectively (see the survey papers [7, 8, 9, 10, 11, 12] and the references therein). In fact, many control systems in practical applications are nonlinear. Consensus problems for multi-agent systems with nonlinear dynamic have been investigated in existing works. It should be pointed out that exiting works about nonlinear consensus problem focused on the consensus condition under a control protocol, but the consensus regulation performance was not considered by a performance index.

With the development of the consensus control theory, the guaranteed performance consensus for the multi-agent by the guaranteed performance control approach has received more and more attentions. In the guaranteed performance consensus problems, the consensus regulation performance was explicitly considered by the guaranteed performance function. By the constraint of the performance index, the consensus control can be seen as an optimal or suboptimal problem, and the control process is more affected by choosing appropriate control parameters. In existing literatures about the guaranteed performance consensus problem, such as [11, 12, 13, 14, 15], the dynamic characteristic of each agent in the multi-agent systems was linear. For the linear multi-agent systems, the state-space decomposition approach was widely used to decompose the consensus and disagreement dynamics of multi-agent system, and the disagreement dynamics is the key of guaranteed performance consensus control. Moreover, the guaranteed performance consensus with other control methods has been studied, such as sampled-data control [16], fault-tolerant control [17], event-triggered control [18], tracking control [19], and impulsive control [20].

For the consensus problems of nonlinear multi-agent systems, the intercoupling relationship between the consensus and disagreement dynamics because of the nonlinear dynamic. Then, the state-space decomposition approach is not able to deal with the nonlinear consensus problems. To the best of our knowledge, there are very few research works about the guaranteed performance consensus for nonlinear multi-agent systems. Moreover, the interaction topologies in most of the existing works were undirected, and there were few works about guaranteed performance consensus problem with directed interaction topologies.

In the current chapter, the guaranteed performance consensus for multi-agent systems with nonlinear dynamics is studied by introducing a performance function. By a special matrix transformation, the guaranteed performance consensus problems are transferred into guaranteed performance stabilization problems, and some conclusions about guaranteed performance consensus for nonlinear multi-agent systems are obtained.

## 2. Preliminaries and problem descriptions

In the current paper, the interaction topology among all agents of multi-agent systems can be modeled by a directed graph

*Lemma 1* [21]: Let

In the current chapter, consider the guaranteed performance consensus problem of a group of

where

where

where

Definite

where

*Lemma 2* [22]: For a Laplacian matrix

*Lemma 3* [22]: If

where

*Lemma 4* [23]: For any given

In the squeal, the definitions of the guaranteed performance consensus and consensualization are given, respectively.

*Definition 1*: Multi-agent system (Eq. (4)) is said to achieve guaranteed performance consensus with the performance function (Eq. (5)) if

*Definition 2*: Multi-agent system (Eq. (1)) is said to be guaranteed performance consensualizable by the consensus protocol (Eq. (2)) with the performance function (Eq. (5)), if there exists a gain matrix

Due to the definition

## 3. Analysis of guaranteed performance consensus

By

where

and the matrix

For

The following result presents a sufficient condition for nonlinear multi-agent system (Eq. (4)) to achieve guaranteed performance consensus and designs the consensus control gain matrix for the distributed consensus protocol (Eq. (3)).

*Theorem 1*: Assume that

In this case, for the distributed consensus protocol (Eq. (3)), the consensus control gain matrix

*Proof*: Consider the following Lyapunov functional candidate:

where

Let

By Lemma 4 and Eq. (2), one can see that

By Eqs. (2) and (7), one can see that

From Lemma 3 and Eq. (13),

where the fact that

Define

where

It should be pointed out that if

By the Schur complement, if

When the nonlinear multi-agent system (Eq. (4)) achieves guaranteed performance consensus, the performance of consensus control is described by the performance function (Eq. (5)). Then, an upper bound of the performance function (Eq. (5)) is able to determine.

*Theorem 2*: Assume that

*Proof*: From the proof of Theorem 1, it is obtained that

when nonlinear multi-agent system (Eq. (4)) achieves guaranteed performance consensus. For Eq. (18), integrating both sides along with

Since

In the existing works [11, 12, 13, 14, 15, 16, 17, 18, 19, 20], the guaranteed performance consensus problems for linear multi-agent systems have been studied. Theorem 1 gives a sufficient condition for nonlinear multi-agent system (Eq. (4)) to achieve guaranteed performance consensus. Moreover, the directed topology is considered in the current chapter, but the topologies in [11, 12, 13, 14, 15, 16, 17, 18, 19] were undirected, and the directed case problem was dealt with by the sampled-data control in [20].

## 4. Design of guaranteed performance consensus

*Theorem 3*: Multi-agent system (Eq. (1)) is said to be a guaranteed performance consensualizable by consensus protocol (4) if there exists

In this case, the control gain matrix satisfies

*Proof*: The method of changing variables is used to determine

Theorem 3 presents the LMI conditions for controller design of guaranteed performance consensus. The feasibility of these LMI conditions can be checked by using the MATLAB’s LMI Toolbox.

## 5. Simulations

A nonlinear multi-agent system composed of four agents is analyzed to demonstrate the effectiveness of the proposed approach, where all agents are labeled from 1 to 4. The dynamics of each agent is described in Eq. (1) with

It can be seen that

In the performance function in Eq. (5),

By the definition of the matrix

Then, the matrix

Thus, according to Theorem 3, one has

In Figures 2 and 3, the state trajectories of the nonlinear multi-agent system are shown, and one can see that the state of all agents is convergent. By Theorem 3, an upper bound of the guaranteed performance function is

## 6. Conclusions

In this chapter, the guaranteed performance consensus problems for nonlinear multi-agent systems with directed interaction topologies were studied. A special matrix transformation was introduced, and guaranteed performance consensus problems were transferred into guaranteed performance stabilization problems. Sufficient conditions for guaranteed performance consensus control were obtained, and an upper bound was given.

The directed topology was assumed to be fixed and connected in the guaranteed performance consensus problem, and then the application of conclusions of the current paper is limited. Therefore, the influence of the general switching topologies for the guaranteed performance consensus problem is the possible topic. The existing work [24, 25] assumed that the switching topologies were strongly connected and balanced, but the joint-connected switching topology cases have not been studied.

The analysis method was used to analyze the guaranteed performance consensus problem with Lipschitz-type nonlinearities in the chapter, but this method cannot be directly applied to the problem with the other kind of nonlinearities. Therefore, the analysis approach for the guaranteed performance consensus problem with general nonlinear dynamic should be given in future works.

Sliding mode control (SMC) technique has a strong robustness for external noises, such as [26, 27], and then the SMC technique is a possible method for guaranteed performance consensus problems to improve the robustness of the obtained consensus controller in future works.

Moreover, it is usually desirable to design a controller which not only achieves formation but also ensures an adequate level of performance in many practical formation control problems. Then, the idea of guaranteed performance control can be introduced into formation control problems for multi-agent systems. To this end, the guaranteed performance consensus algorithm is an effective approach, such as the guaranteed performance formation control in [28].

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Numbers 61703411 and 61374054 and by the Youth Foundation of High-Tech Institute of Xi’an under Grant Number 2016QNJJ004. The authors would like to thank Jianxiang Xi and Tang Zheng for providing numerical analysis and simulation.

## Conflict of interest

No potential conflict of interest was reported by the authors.