Novel Distributed Robust Adaptive Consensus Protocols for Linear Multi-agent Systems with Directed Graphs and External Disturbances \thanksreffootnoteinfo

# Novel Distributed Robust Adaptive Consensus Protocols for Linear Multi-agent Systems with Directed Graphs and External Disturbances \thanksreffootnoteinfo

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###### Abstract

This paper addresses the distributed consensus protocol design problem for linear multi-agent systems with directed graphs and external unmatched disturbances. A novel distributed adaptive consensus protocol is proposed to achieve leader-follower consensus for any directed graph containing a directed spanning tree with the leader as the root node. It is noted that the adaptive protocol might suffer from a problem of undesirable parameter drift phenomenon when bounded external disturbances exist. To deal with this issue, a distributed robust adaptive consensus protocol is designed to guarantee the ultimate boundedness of both the consensus error and the adaptive coupling weights in the presence of external disturbances. Both adaptive protocols are fully distributed, relying on only the agent dynamics and the relative states of neighboring agents.

China]Yuezu Lv, China]Zhongkui Li, China]Zhisheng Duan, HK]Gang Feng

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China

Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China

Key words:  Multi-agent systems, cooperative control, consensus, distributed control, adaptive control, robustness.

11footnotetext: This work was supported by the National Natural Science Foundation of China under grants 61104153, 11332001, 61225013, a Foundation for the Author of National Excellent Doctoral Dissertation of PR China, and the State Key Laboratory of Complex System Intelligent Control and Decision. Corresponding author Zhongkui Li.

## 1 Introduction

In recent years, the consensus problem of multi-agent systems has been an emerging research topic in the field of control, due to its wide applications in many areas such as satellite formation flying, cooperative unmanned systems, and distributed reconfigurable sensor networks [1]. There has been remarkable progress in achieving consensus for different scenarios; see [1, 2, 3, 4, 5, 6] and the references therein. For the consensus problem, the critical task is to design distributed consensus protocols based on local information, i.e., local state or output information of each agent and its neighbors.

In this paper, we consider the consensus problem of multi-agent systems with general linear time invariant dynamics. Previous works [7, 8, 9, 10, 11, 12] have presented various static and dynamic consensus protocols, which are proposed in a distributed fashion, using only the local information of each agent and its neighbors. However, those consensus protocols involves some design issues. To be specific, the design of those consensus protocols generally requires the knowledge of some eigenvalue information of the Laplacian matrix associated with the communication graph, that is, the smallest nonzero eigvenvalue of the Laplacian matrix for undirected graphs and the smallest real part of the nonzero eigenvalues of the Laplacian matrix for directed graphs. Note that the nonzero eigenvalue information of the Laplacian matrix is global information in the sense that each agent has to know the entire communication graph to compute it. Therefore, although these consensus protocols are proposed and can be implemented in a distributed fashion, they cannot be designed by each agent in a distributed fashion. In other words, those consensus protocols in [7, 8, 9, 10, 11, 12] are not fully distributed.

To remove the limitation of requiring global information of the communication graph, distributed adaptive consensus protocols are reported in [13, 14, 15, 16], which, depending on only local information of each agent and its neighbors, are fully distributed. It is worth noting that the adaptive protocols in [13, 14, 15, 16] are applicable to only undirected communication graphs or leader-follower graphs where the subgraphs among followers are undirected. Due to the asymmetry of the Laplacian matrices, it is much more difficult to design distributed adaptive consensus protocols for general directed communication graphs. By including monotonically increasing functions to provide additional design flexibility, a distributed adaptive consensus protocol is derived in [17] to achieve consensus for general leader-follower directed graphs containing a directed spanning tree. The robustness of the distributed adaptive protocols with respect to uncertainties or external disturbances is an important issue. The adaptive protocol in [17] can only be modified to be applicable to external disturbances satisfying the matching condition; see [19]. To the best of our knowledge, how to design distributed robust adaptive consensus protocols for the case with directed graphs and general unmatched disturbances is still open.

The rest of this paper is organized as follows. The mathematical preliminaries used in this paper is summarized in Section 2. The distributed adaptive consensus protocol is designed in Section 3 for general directed leader-follower graphs. A novel robust adaptive consensus protocol is presented in Section 4 to deal with external disturbances. Simulation results are presented in Section 5. Section 6 concludes this paper.

## 2 Mathematical Preliminaries

Let be the set of real matrices and the superscript donates transpose for real matrices. represents the identity matrix of dimension and denotes the identity matrix of an appropriate dimension. 1 donates a column vector with all entries equal to 1. represents a diagonal matrix with elements , on its diagonal while donates the minimal eigenvalue of . The matrix inequality means and are symmetric matrices and is positive definite. represents the Kronecker product of matrices and . A nonsingular -matrix means that , and all eigenvalues of have positive real parts.

A directed graph consists of a node set and an edge set , in which an edge is represented by an ordered pair of distinct nodes. For an edge , node is called the parent node, node the child node, and is a neighbor of . A path from node to node is a sequence of ordered edges of the form (, ), . A directed graph contains a directed spanning tree if there exists a node called the root such that the node has directed paths to all other nodes in the graph.

Suppose there are nodes in the directed graph . The adjacency matrix of is defined by if and 0 otherwise. The Laplacian matrix is defined as and , .

###### Lemma 1 ([20])

Zero is an eigenvalue of with 1 as a right eigenvector and all nonzero eigenvalues have positive real parts. Furthermore, zero is a simple eigenvalue of if and only if has a directed spanning tree.

###### Lemma 2 ([21, 22])

Consider a nonsingular -matrix . There exists a diagonal matrix so that and .

###### Lemma 3 ([23])

If and are nonnegative real numbers and p and q are positive real numbers such that , then , and the equality holds if and only if .

###### Lemma 4 ([24])

For a system , where is locally Lipschitz in and piecewise continuous in , assume that there exists a continuously differentiable function such that along any trajectory of the system,

 α1(∥x∥)≤V(x,t)≤α2(∥x∥), ˙V(x,t)≤−α3(∥x∥)+ϵ,

where is a constant, and are class functions, and is a class function. Then, the solution of is uniformly ultimately bounded.

## 3 Distributed Adaptive Consensus Protocol Design

Consider a group of identical agents with general linear dynamics, consisting of followers and a leader. The dynamics of the -th agent are described by

 ˙xi=Axi+Bui,i=0,⋯,N, (1)

where is the state, is the control input, and are constant matrices with compatible dimensions.

Without loss of generality, let the agent in (1) indexed by 0 be the leader whose control input is assumed to be zero, i.e., , and the other agents be the followers. The communication graph among the agents is assumed to satisfy the following assumption.

###### Assumption 1

The graph contains a directed spanning tree with the leader as the root node.

Under Assumption 1, the Laplacian matrix associated with can be partitioned as . In light of Lemma 1 and the definition of -matrix, it is easy to verify that is a nonsingular -matrix.

The objective of this paper is to design distributed consensus protocols such that the agents in (1) achieve leader-follower consensus in the sense of ,

Based on the relative states of neighboring agents, we propose a distributed adaptive consensus protocol to each follower as

 ui=(ci+ρi)Kξi, (2) ˙ci=ξTiΓξi,i=1,⋯,N,

where , denotes the time-varying coupling weight associated with the -th follower with , and are the feedback gain matrices, and are smooth functions to be determined.

Let and . Then, we get

 ξ=(L1⊗IN)(x−1⊗x0). (3)

Since the graph satisfies Assumption 1, it follows from Lemma 1 that is a nonsingular -matrix and that the leader-follower consensus problem is solved if and only if asymptotically converges to zero. Hereafter, we refer to as the consensus error. Substituting (2) into (1) yields

 ˙ξ=[IN⊗A+L1(C+ρ)⊗BK]ξ, (4) ˙ci=ξTiΓξi,i=1,⋯,N,

where and .

The following theorem provides a result on the design of the adaptive consensus protocol (2).

###### Theorem 1

Suppose that the communication graph satisfies Assumption 1. Then, the leader-follower consensus problem of the agents in (1) can be solved under the adaptive protocol (2) with , , and , where is a solution to the following linear matrix inequality (LMI):

 PAT+AP−2BBT<0. (5)

Moreover, each coupling weight converges to some finite steady-state value.

Proof Consider the Lyapunov function candidate:

 V1=N∑i=112gi(2ci+ρi)ρi+λ02N∑i=1~c2i, (6)

where is a positive definite matrix such that , denotes the smallest eigenvalue of , and , where is a positive constant to be determined later. It follows from Assumption 1 and Lemma 1 that is a nonsingular -matrix. Thus we know from Lemma 2 that such a positive definite matrix does exist. Since , it follows from that for any . Then, it is not difficult to see that is positive definite.

The time derivative of along the trajectory of (4) is given by

 ˙V1 =N∑i=1[2gi(ci+ρi)ξTiP−1˙ξi+giρi˙ci] (7) +λ0N∑i=1(ci−α)˙ci =ξT[(C+ρ)G⊗(P−1A+ATP−1) −(C+ρ)(GL1+LT1G)(C+ρ)⊗Γ]ξ +ξT(ρG⊗Γ)ξ+ξT[λ0(C−αI)⊗Γ]ξ ≤ξT[(C+ρ)G⊗(P−1A+ATP−1) −λ0(C+ρ)2⊗Γ]ξ+ξT(ρG⊗Γ)ξ +ξT[λ0(C−αI)⊗Γ]ξ.

By using Lemma 3, we can get that

 ξT(ρG⊗Γ)ξ≤ξT(λ02ρ2⊗Γ)ξ+ξT(12λ0G2⊗Γ)ξ, (8)

and

 ξT(λ0C⊗Γ)ξ≤ξT(λ02C2⊗Γ)ξ+ξT(λ02I⊗Γ)ξ. (9)

Substituting (8) and (9) into (7) yields

 ˙V1≤ ξT[(C+ρ)G⊗(P−1A+ATP−1) (10) −(λ02(C+ρ)2+λ0αI−λ02−12λ0G2)⊗Γ]ξ.

Choose , where will be determined later. Then, it follows from (10) that

 ˙V1 ≤ξT[(C+ρ)G⊗(P−1A+ATP−1) (11) −(λ02(C+ρ)2+λ0^αI)⊗Γ]ξ ≤ξT[(C+ρ)G⊗(P−1A+ATP−1) −√2^αλ0(C+ρ)⊗Γ]ξ.

Let and choose to be sufficiently large such that . Then we can get from (11) that

 ˙V1 ≤~ξT(IN⊗(AP+PAT−2BBT))~ξ (12) ≤0,

where the last inequality comes directly from the LMI (5). Therefore, we can get that is bounded and so is each . Noting that , we can know that each coupling weight converges to some finite value. Noting that is equivalent to and thereby . By LaSalle’s Invariance principle [25], it follows that the consensus error asymptotically converges to zero. That is, the consensus problem is solved.

###### Remark 1

Contrary to the consensus protocols in [7, 8, 10, 11] which use the nonzero eigenvalues of the Laplacian matrix, the design of the proposed adaptive protocol (2) relies on only the agent dynamics and the relative states of neighbors, which can be conducted by each agent in a fully distributed way. As shown in [7], a necessary and sufficient condition for the existence of the solution to the LMI (5) is that is stabilizable. Therefore, the existence condition of an adaptive protocol (2) satisfying Theorem 1 is that is stablizable.

## 4 Distributed Robust Adaptive Consensus Protocols

Theorem 1 in the previous section shows that the adaptive protocol (2) is applicable to any directed graph satisfying Assumption 1 for the case without external disturbances. In many circumstances, the agents might be subject to various external disturbances, for which case it is necessary and interesting to investigate whether the adaptive protocol (2) is robust.

The dynamics of the -th agent are described by

 ˙xi=Axi+Bui+ωi,i=0,⋯,N, (13)

where denotes external disturbances associated with the -th agent, which satisfy the following assumption.

###### Assumption 2

There exist positive constants such that , , and .

Note that due to the existence of disturbances in (13), the relative states will not converge to zero any more but rather can only be expected to converge into some small neighborhood of the origin. Since the derivatives of the adaptive gains in (2) are of nonnegative quadratic forms in terms of the relative states, in this case it is easy to see from (2) that will keep growing to infinity, which is called the parameter drift phenomenon in the classic adaptive control literature [18]. Therefore, the adaptive protocol (2) is not robust in the presence of external disturbances.

In the following, we aim to make modification on (2) to propose a distributed robust adaptive protocol which can guarantee the ultimate boundedness of the consensus error and adaptive weights for the agents in (13). We propose a new robust distributed adaptive consensus protocol as follows:

 ui=(di+ρi)Kξi, (14) ˙di=−φi(di−1)2+ξTiΓξi,i=1,⋯,N,

where denotes the time-varying coupling weight associated with the -th follower with , , , are small positive constants, and the rest of the variables are defined as in (2).

Substituting (14) into (13), it follows that

 ˙ξ=[IN⊗A+L1(D+ρ)⊗BK]ξ+(L1⊗In)ω (15) ˙di=−φi(di−1)2+ξTiΓξi,i=1,⋯,N,

where , , and the rest of the variables are defined as in (4).

In light of Assumption 2, we have that

 ∥ω∥≤ ⎷N∑i=1(vi+v0)2. (16)

Note that and when in (15). Then, it is not difficult to see that for any .

The following theorem presents a result on design of the robust adaptive consensus protocol (14).

###### Theorem 2

Suppose that Assumptions 1 and 2 hold. Then, both the consensus error and the coupling weights , , in (15) are uniformly ultimately bounded under the adaptive protocol (14) with , , and , where is a solution to the LMI:

 AQ+QAT+εQ−2BBT<0, (17)

where . The upper bound of the consensus error will be given in the proof.

Proof Consider the Lyapunov function candidate:

 V2=N∑i=112gi(2di+ρi)ρi+λ02N∑i=1~d2i, (18)

where , where is a positive constant to be determined later, and the rest of the variables are defined as in (6). Since , for any , and , it can be similarly shown as in the proof of Theorem 1 that is positive definite.

By following similar steps in deriving Theorem 1, we can obtain the time derivative of along (15) as

 ˙V2 ≤ξT[(D+ρ)G⊗(Q−1A+ATQ−1−Γ)]ξ (19) +2ξT[(D+ρ)GL1⊗Q−1]ω −ξT[φ(D−I)2G⊗Q−1]ξ −N∑i=1λ0φi(di−α)(di−1)2,

where is chosen to be sufficiently large as in the proof of Theorem 1 and .

By choosing and using Lemma 3, we can get that

 −(di−α)(di−1)2 (20) =−(di−1)3+(α−1)(di−1)2 =−(di−1)3+[(34)23(di−1)2][(43)23(α−1)] ≤−(di−1)3+12(di−1)3+1627(α−1)3 =−12(di−1)3+1627(α−1)3.

Note that

 2ξT[(D+ρ)GL1⊗Q−1]ω (21) =2ξT[(D−I)√φG⊗√Q−1](√φ−1GL1⊗√Q−1)ω +2ξT(1√2√G⊗√Q−1)(√2GL1⊗√Q−1)ω +2ξT(1√2√ρG⊗√Q−1)(√2ρGL1⊗√Q−1)ω ≤ξT[(D−I)2φG⊗Q−1]ξ+∥(√φ−1GL1⊗√Q−1)ω∥2 +12ξT(G⊗Q−1)ξ+2∥(√GL1⊗√Q−1)ω∥2 +12ξT(ρG⊗Q−1)ξ+2∥(√ρGL1⊗√Q−1)ω∥2,

where we have used Lemma 3 several times to get the last inequality, and

 2∥(√ρGL1⊗√Q−1)ω∥2 (22) ≤2∥√ρ√G⊗In∥2∥(G14L1⊗√Q−1)ω∥2 ≤12∥√ρ√G⊗In∥4+2∥(G14L1⊗√Q−1)ω∥4 ≤12ξT(ρG⊗Q−1)ξ+2∥(G14L1⊗√Q−1)ω∥4,

where we have used matrix norm properties to get the first inequality, and Lemma 3 to get the second inequality, and to get the last inequality we have used the fact that

 ∥√ρ√G⊗In∥4 =maxi=1,⋯,N(√ρi√gi)4 ≤N∑i=1(√ρi√gi)4=ξT(ρG⊗Q−1)ξ.

Substituting (20), (21), and (22) into (19) yields

 ˙V2 ≤ξT[(D+ρ)G⊗(Q−1A+ATQ−1−Γ)]ξ (23) +ξT[(ρ+12)G⊗Q−1]ξ−N∑i=1λ02φi(di−1)3+Π1 ≤W(ξ)−12ξT(G⊗Q−1)ξ−N∑i=1λ02φi(di−1)3+Π1,

where we have used the fact that ,

 Π1 =N∑i=116λ027φi(α−1)3+∥(√φ−1GL1⊗√Q−1)ω∥2 +2∥(√GL1⊗√Q−1)ω∥2+2∥(G14L1⊗√Q−1)ω∥4.

and

 W(ξ) ≜ξT[(D+ρ)G⊗(Q−1A+ATQ−1+Q−1−Γ)]ξ ≤0.

Note that for any positive , we have the following assertion:

 δλ02(di−α)2≤δλ02(di−1)2+δλ02(α−1)2 (24) =[(3λ0φi4)23(di−1)2][(2λ09φ2i)13δ]+δλ02(di−α)2 ≤λ02φi(di−1)3+2λ0δ327φ2i+δλ02(α−1)2,

where we have used the fact that and to get the first inequality, and Lemma 3 to get the last inequality. From (23) and (24), we can obtain that

 ˙V2 ≤−δV2+δV2+W(ξ)−12ξT(G⊗Q−1)ξ (25) −N∑i=1λ02φi(di−1)3+Π1 ≤−δV2+¯W(ξ)−12ξT(G⊗Q−1)ξ+Π,

where

 Π=N∑i=1[2λ0δ327φ2i+δλ02(α−1)2]+Π1,

and

 ¯W =ξT[(D+ρ)G⊗(Q−1A+ATQ−1 +(1+δ)Q−1−Q−1BBTQ−1)]ξ,

By choosing such that , we can obtain that . Then, it follows from (25) that

 ˙V2≤−δV2−12ξT(G⊗Q−1)ξ+Π. (26)

In light of Lemma 4, we can conclude from (26) that both the consensus error and the adaptive gains are uniformly ultimately bounded. Further, from (26), we can get that if . Therefore, converges to the set

 D1=⎧⎪⎨⎪⎩ξ:∥ξ∥2≤2Πλmin(Q−1)mini=1,⋯,Ngi⎫⎪⎬⎪⎭ (27)

with a convergence rate faster than .

###### Remark 3

As shown in Proposition 1 in [8], there exists a satisfying (17) if and only if is controllable. Thus, a sufficient condition for the existence of (14) satisfying Theorem 2 is that is controllable, which, compared to the existence condition of (2) satisfying Theorem 1, is more stringent. Theorem 2 shows that the modified adaptive protocol (14) does ensure the ultimate boundedness of both the consensus error and the adaptive gains . That is, the adaptive protocol (14) is robust in the presence of external disturbances. The upper bound of the consensus error as given in (27) depends on the dynamics of each agent, the communication graph, the upper bounds of the disturbances, and the parameters . We should choose appropriately small to get an acceptable upper bound of .

###### Remark 4

Compared to the robust adaptive protocol in [19] which are only applicable to the case with matching disturbances, the adaptive protocol (14) works for general external disturbances. This is a favorable consequence of introducing novel additive functions , rather than multiplicative ones as in [19], into (14). It is worth noting that the procedures in showing Theorem 2 is quite different from those in [19].

## 5 Simulation

Consider a network of second-order integrators, described by (1), with

 xi=[xi1xi2], A=[0100], B=[01].

The communication graph is given as in Fig. 1, which clearly satisfies Assumption 1.

Solving the LMI (5) by using the LMI toolbox of Matlab gives a feasible solution matrix . Then, the feedback gain matrices of (2) are given by

 K=[−0.8543−2.5628],Γ=[0.72982.18932.18936.5678].

Let , . The consensus errors , of the second-order integrators under the adaptive protocol (2) are depicted in Fig. 2 and the adaptive coupling weights are shown in Fig. 3.

Further, consider the case where the second-order integrators are perturbed by external disturbances. For illustration, the disturbances associated with the agents are assumed to be , , , , , , , and the control input of the leader is assumed to be . Solving the LMI (17) with gives and then , . In (14), let and , . The consensus errors , under the robust adaptive protocol (14) are depicted in Fig.4 and the coupling weights are shown in Fig. 5, both of which are obviously bounded.

## 6 Conclusion

In this paper, we have presented novel distributed adaptive consensus protocols for linear multi-agent systems with external disturbances and directed graphs containing a directed spanning tree with the leader as the root. The adaptive consensus protocols, depending on only the agent dynamics an the relative state information of neighboring agents, can be designed and implemented in a fully distributed way. One contribution of this paper is that the new distributed adaptive protocol is robust in the presence of general bounded external disturbances. An interesting topic for future investigation is to design fully distributed adaptive protocols for nonlinear multi-agent systems or the case with local output information of each agent and its neighbors.

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