Open access peer-reviewed chapter

Guaranteed Performance Consensus for Multi-Agent Systems

By Zhong Wang

Submitted: April 22nd 2018Reviewed: July 12th 2018Published: March 6th 2019

DOI: 10.5772/intechopen.80285

Downloaded: 185

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 G=VE, where V=12Nand EV×Vrepresent the agent set and the directed edge set, respectively. A directed edge in a directed graph denoted as ijwhich means that agent ican obtain information from agent j. Definite W=wijRN×Nwith wij0and wii=0is the adjacency matrix, where wij>0if there is a directed edge between agent iand agent j, wij=0otherwise. If wij>0, agent jis called the neighbor of agent i, and all the neighbors of agent iconsist of the neighboring set of agent i. A directed path is a sequence of ordered edges of the form i1i2,i2i3,. The Laplacian matrix of the interaction topology Gis defined as L=lijRN×N, where lii=jiwijand lij=wij. In the current paper, it is assumed that the directed graph Ghas a directed spanning tree. The following lemma shows basic properties of the Laplacian matrix of a directed interaction graph.

Lemma 1 [21]: Let L=lijRN×Nbe the Laplacian matrix of a directed interaction graph G, then (i) Lat least has a zero eigenvalue, and 1Nis an associated eigenvector; that is, L1N=0. (ii) 0 is a simple eigenvalue of L, and all the other nonzero eigenvalues have positive real parts if and only if Ghas a directed spanning tree, i.e., 0=λ1<Reλ2ReλN.

In the current chapter, consider the guaranteed performance consensus problem of a group of Nnonlinear agents with the dynamics of ith agent given by

ẋit=Axit+Mfxit+Buit,E1

where i=1,2,,N,ARd×d,BRd×p,MRd×mare constant real matrices, xitand uitare the state and the control input of the ith agent, respectively, and the function f:Rn×0+Rmis a continuously differentiable vector-valued function representing the nonlinear dynamics of the ith agent which satisfies the following condition:

fξ1tfξ2tαξ1tξ2t,E2

where ξ1t,ξ2tRd,t0and α>0is a constant scalar. In the current paper, a directed graph Gis used to descript the interaction. Based on local relative state information of the neighbor agents, consider the following distributed consensus protocol:

uit=Kj=1Nwijxjtxit,E3

where wijis the weight of the edge between agents jand i. Let xt=x1Ttx2TtxNTtTand gxt=fx1tTfx2tTfxNtTT, and then multi-agent system Eq. (1) with consensus protocol Eq. (3) can be rewritten in a vector form as

ẋt=INALBKxt+INMgxt.E4

Definite εt=ε1Ttε2TtεN1TtTwith εit=xitxi+1t, then existing matrix

H=1100001100001001000011RN1×N,
such that εt=HIdxt. Thus, a performance function can be defined as follows:
JC=i=1N10+εiTtQεitdt,E5

where Qis a given symmetric positive matrix with appropriate dimension. The following lemmas are introduced:

Lemma 2 [22]: For a Laplacian matrix LRN×Nof the graph Gand the matrix H, there exists a matrix URN×N1such that L=UH.

Lemma 3 [22]: If Ghas a directed spanning tree and Uis full column rank, the real part of the eigenvalue of the matrix HUsatisfies ReλHU>0, and there exist a symmetric and positive definite matrix WRN1×N1and a positive scalar γsuch that

HUTW+WHU>γW,

where 0<γ<2minReλHU.

Lemma 4 [23]: For any given ξ1t,ξ2tRdand matrices Sof appropriate dimensions, one has

2ξ1TtSξ2tξ1Ttξ1t+ξ2TtSTSξ2t.

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 limt+xjtxit=0, and there exists a scalar JC>0such that JCJC, where JCis said to be an upper bound of the performance function (Eq. (5)).

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 Ksuch that Eq. (4) achieves guaranteed performance consensus.

Due to the definition εit=xitxi+1t, one has limt+εt=0if and only if limt+xitxi+1t=0with i=1,2,,N1; that is, x1t=x2t==xNtwhen t+. Then, it is obtained that εtis the disagreement vector of multi-agent system (Eq. (4)). Thus, JCin Eq. (5) represents the guaranteed performance of the consensus control for multi-agent system (Eq. (4)).

3. Analysis of guaranteed performance consensus

By εt=HIdxt, multi-agent system (Eq. (4)) can be transformed into

ε̇t=IN1AHUBKεt+IN1Mζxt,E6

where

ζxt=HIdgxt=fx1tfx2tfx2tfx3tfxN1tfxNtE7

and the matrix Usatisfies Lemmas 2 and 3. Moreover, from Eq. (3), one has ut=LKxtwith ut=u1Ttu2TtuNTtT. By εt=HIdxtand Lemma 2, one has

ut=UKεtE8

For JC, the performance function (Eq. (5)) can be rewritten in a vector form as

JC=0+εTtIN1Qεtdt.E9

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 Ghas a directed spanning tree. Nonlinear multi-agent system (Eq. (4)) achieves guaranteed performance consensus if there exists a symmetric and positive definite matrix PRd×dsuch that Ξ<0, where

Ξ=Ξ11PMαIdQωmax1Id00ωminId0ωminQ
with
Ξ11=ATP+PAγPBBTP,
ωmin=minλiWi=12N1,
ωmax=maxλiWi=12N1.

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

Proof: Consider the following Lyapunov functional candidate:

Vt=εTtWPεt,E10

where PRd×dis a symmetric and positive definite matrix and Wsatisfies Lemma 3. Then, the time derivation of the Lyapunov functional candidate Vtalong the trajectory of Eq. (6) is

V̇t=2εTtIN1AHUBKTWPεt+2ζTxtWMTPεt.E11

Let K=BTP, and then Eq. (11) can be transformed into

V̇t=εTtWATP+PAHUTW+WHUPBBTPεt+2ζTxtWMTPεt.E12

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

2ζTxtWMTPεtζTxtζxt+εTtW2PMMTPεt,E13

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

ζTxtζxtα2εTtεt,E14

From Lemma 3 and Eq. (13), V̇tsatisfies

V̇tεTtWATP+PAγPBBTP+α2ωmin1Id+ωmaxPMMTPεt,E15

where the fact that

ωminIN1WωmaxIN1
is used. ωminand ωmaxare the minimum and the maximum eigenvalues of W, respectively.

Define

̇t=V̇t+J˜C,E16

where J˜C0and

J˜C=εTtIN1Qεt.

It should be pointed out that if ̇t0, then V̇t0. Then, one can see that

̇tεTtWATP+PAγPBBTP+α2ωmin1Id+ωmaxPMMTP+ωmin1Qεt.E17

By the Schur complement, if Ξ<0, one has

̇t0
and ̇t=0if and only if εt0. Then, by J˜C0, if Ξ<0, V̇t0and V̇t=0if and only if εt0. Thus, limt+εt=0holds; that is,
x1t=x2t==xNt
when t+. Therefore, if there exists Pthat satisfies Ξ<0, then guaranteed performance consensus for multi-agent system (Eq. (4))with K=BTPis achieved.

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 Ghas a directed spanning tree. If nonlinear multi-agent system (Eq. (4)) with a symmetric and positive definite matrix PRd×dachieves guaranteed performance consensus, then the performance function (Eq. (5)) has an upper bound:

JC=xT0HTWHPx0.

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

J˜CV̇tE18

when nonlinear multi-agent system (Eq. (4)) achieves guaranteed performance consensus. For Eq. (18), integrating both sides along with t0+gives

JC=0+J˜Cdt0+V̇tdt.E19

Since limt+Vt=0, one has JCV0. Thus, JC=V0is an upper bound of the quadratic performance function (Eq. (5)). From Eq. (10) and εt=HIdxt, the result of Theorem 2 is obtained.

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 d-dimensional matrix P˜=P˜T>0such that Ξ˜<0, where

Ξ˜=Ξ˜11MαP˜P˜Qωmax1Id00ωminId0ωminQ
with
Ξ˜11=P˜AT+AP˜γBBT,
ωmin=minλiWi=12N1,
ωmax=maxλiWi=12N1.

In this case, the control gain matrix satisfies K=BTP˜1, and the guaranteed performance function has an upper bound:

JC=xT0HTWHP˜1x0.

Proof: The method of changing variables is used to determine K. Pre- and post-multiplying Ξ<0by Π=diagP1IdIdIdand ΠT=diagPTIdIdId, respectively, one has

Ξ¯11MαP1IdP1Qωmax1Id00ωminId0ωminQ<0
with Ξ¯11=P1AT+APTγBBT. Setting P˜=P1, one has Ξ˜<0. From Theorem 1, if Ξ˜<0are feasible, then multi-agent system (Eq. (4)) can achieve consensus. Therefore, by Definition 2 and Theorem 2, the conclusion of Theorem 3 can be obtained.

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

A=011.50.6,B=01,
M=0.7000.9,fxit=0.1sinxit0.04sinxit.

It can be seen that α=0.1in Eq. (2). The initial states of all agents are

x10=2.21.5,x20=1.20.7,x30=00.2,x40=1.60.3.

In the performance function in Eq. (5),

Q=0.45000.3
are given. A directed interaction topology Gis given in Figure 1, where the weights of edges of the interaction topology are 1 and the Laplacian matrix of Gis
L=1001120101101001

Figure 1.

The interaction topology G .

By the definition of the matrix H, it is obtained that

H=110001100011.

Then, the matrix

U=111111010111
and minReλHU=1satisfy the conditions of Lemma 2. In this simulation, γ=1.6is chosen. By Lemma 3 and LMI Toolbox of MATLAB, the matrix
W=1.04330.08120.47870.08120.43200.13250.47870.13250.9919.

Thus, according to Theorem 3, one has

P˜=0.55340.04810.04810.7672
and
K=0.11391.3105.

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 JC=7.8284. Figure 4 presents the guaranteed performance function and the upper bound. From Figure 3, one can see that there exists conservatism induced by the approach taken to compute the upper bound on the performance, where the actual performance JTand then the conservatism can be depicted by ΔJ=JCJT. It is clear that the multi-agent system achieves guaranteed performance consensus although there exists conservatism.

Figure 2.

State trajectories of x i 1 t i = 1 2 3 4 .

Figure 3.

State trajectories of x i 2 t i = 1 2 3 4 .

Figure 4.

Trajectories of the performance function.

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.

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Zhong Wang (March 6th 2019). Guaranteed Performance Consensus for Multi-Agent Systems, Multi-Agent Systems - Control Spectrum, Vladimir Shikhin, IntechOpen, DOI: 10.5772/intechopen.80285. Available from:

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