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

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

Zhongkui Li and Zhisheng Duan This work was supported by the National Natural Science Foundation of China under grants 61104153, 11332001, 61225013, and a Foundation for the Author of National Excellent Doctoral Dissertation of PR China.Z. Li and Z. Duan are with the State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China (E-mail: zhongkli@pku.edu.cn; duanzs@pku.edu.cn).
###### Abstract

This paper addresses the distributed consensus design problem for linear multi-agent systems with directed communication graphs and external disturbances. Both the cases with strongly connected communication graphs and leader-follower graphs containing a directed spanning tree with the leader as the root are discussed. Distributed adaptive consensus protocols based on the relative states of neighboring agents are designed, which can ensure the ultimate boundedness of the consensus error and adaptive gains in the presence of external disturbances. The upper bounds of the consensus error are further explicitly given. Compared to the existing consensus protocols, the merit of the adaptive protocols proposed in this paper is that they can be computed and implemented in a fully distributed fashion and meanwhile are robust with respect to external disturbances.

{keywords}

Multi-agent system, cooperative control, consensus, distributed control, adaptive control, robustness.

## I Introduction

In the past two decades, rapid advances in miniaturizing of computing, communication, sensing, and actuation have made it feasible to deploy a large number of autonomous agents to work cooperatively to accomplish civilian and military missions. This, compared to a single complex agent, has the capability to significantly improve the operational effectiveness, reduce the costs, and provide additional degrees of redundancy [1, 2]. Having multiple autonomous agents to work together to achieve collective behaviors is usually referred to as cooperative control of multi-agent systems. Due to its potential applications in various areas such as surveillance and reconnaissance systems, satellite formation flying, electric power systems, and intelligent transportation systems, cooperative control of multi-agent systems has received compelling attention from the systems and control community. In the area of cooperative control, consensus is an important and fundamental problem, which means that a group of agents reaches an agreement on certain quantity of interest by interacting with their local neighbors. Advances of various consensus algorithms have been reported in a quite large body of research papers; see [1, 3, 4, 5, 6, 7, 8, 9, 10] and the references therein.

For a consensus control problem, the main task is to design appropriate protocols to achieve consensus. Due to the large number of agents, limited sensing capability of sensors, and short wireless communication ranges, distributed control, depending only on local information of the agents and their neighbors, appears to be a promising tool for handling multi-agent systems. Note that designing appropriate distributed protocols is generally a challenging task, especially for multi-agent systems with complex dynamics, due to the interplay of the agent dynamics, the communication graph among agents, and the cooperative control laws.

Take the consensus problem for multi-agent systems with general continuous-time linear dynamics for instance. In previous works [11, 12, 13, 14, 15, 16], several static and dynamic consensus protocols based on the local state or output information of neighboring agents have been proposed. A common feature in the aforementioned existing papers is that the design of the consensus protocols needs to use some eigenvalue information of the Laplacian matrix associated with the communication graph. Actually, for the simply case with second-order integrator agent dynamics, the design of the consensus protocols does rely on the smallest real part of the nonzero eigenvalues of the Laplacian matrix [13]. However, the smallest real part of the nonzero eigenvalues of the Laplacian matrix is global information of the communication graph, because every agent has to know the whole communication graph to compute it. Therefore, the consensus protocols in the aforementioned works cannot be designed by the agents in a fully distributed way, i.e., relying on only the local information of neighboring agents. To avoid this limitation, two types of distributed adaptive consensus protocols are proposed in [17, 18], which implement adaptive laws to dynamically update the coupling weights of neighboring agents. Similar adaptive schemes are presented in [19, 20] to achieve consensus for multi-agent systems with second-order nonlinear dynamics. Note that the adaptive protocols in [17, 18, 19, 20] are applicable to only undirected communication graphs or leader-follower graphs where the subgraph among followers is undirected. Because of the asymmetry of the Laplacian matrices of directed graphs, designing fully distributed adaptive consensus protocols for general directed graphs is more much challenging. By introducing monotonically increasing functions to provide additional freedom for design, a distributed adaptive consensus protocol is constructed in [21] to achieve leader-follower consensus for the communication graphs containing a directed spanning tree with the leader as the root node. Similar adaptive protocols for directed graphs are developed in [22] for the special case where the agents are described by double integrators. Even though the aforementioned advances have been reported on designing distributed adaptive protocols for general linear multi-agent systems with directed graphs, there are still many important open problems awaiting further investigation. For instance, to design distributed adaptive protocols for the case of general directed graphs without a leader, to examine the robustness issue associated with the adaptive protocols, and to propose distributed dynamic adaptive protocols for the case where only local output information is available, to name just a few.

In this paper, we intend to address the first two aforementioned problems. Specifically, we address the distributed adaptive consensus protocol design problem for general linear multi-agent systems with directed graphs and external disturbances. We consider both the cases where the communication graph among the agents is strongly connected and contains a directed spanning tree with the leader as the root. For the case with leader-follower graphs containing a directed spanning tree with the leader as the root, we revisit the distributed adaptive protocol in [21]. It is pointed out that in the presence of external disturbances, the adaptive gains of the adaptive protocol in [21] will slowly grow to infinity, which is the well-known parameter drift phenomenon in the adaptive control literature [23]. To deal with this instability issue associated with the adaptive protocol in [21], we propose a novel distributed adaptive consensus protocol, by using modification technique [23]. This novel adaptive protocol is designed in fully distributed fashion to ensure the ultimate boundedness of both the consensus error and the adaptive coupling gains. That is, the proposed adaptive protocol is robust in the presence of external disturbances. The upper bound of the consensus error is also explicitly given. The case with strongly connected communication graphs is further studied. A distributed robust adaptive protocol is also presented, which can guarantee the ultimate boundedness of the consensus error and the adaptive coupling gains in the presence of external disturbances. A sufficient condition for the existence of the adaptive protocols proposed in this paper is that each agent is stabilizable.

The rest of this paper is organized as follows. Mathematical preliminaries required in this paper is summarized in Section 2. Distributed robust adaptive consensus protocols are presented in Sections 3 and 4 for multi-agent systems with strongly connected graphs and directed leader-follower graphs, respectively. Simulation examples are presented for illustration in Section 5. Conclusions are drawn in Section 6.

## Ii Mathematical Preliminaries

In this paper, we use the following notations and definitions: represents the set of real matrices. denotes the identity matrix of dimension . denotes a column vector of appropriate dimension with its entries equal to one. For real symmetric matrices and , means that is positive (semi-)definite. denotes the Kronecker product of the matrices and . Denote by the largest singular value of a matrix . A matrix is termed as a nonsingular -matrix, if , , and all the eigenvalues of have positive real parts.

A directed communication graph is a pair , where is a nonempty finite set of vertices (i.e., nodes) and is a set of edges, where an edge is represented by an ordered pair of distinct vertices. For an edge , is called the parent vertex, is called the child vertex, and is a neighbor of . If a directed graph having the property that implies for any , then it is an undirected graph. A directed path from vertex to vertex is a sequence of ordered edges in the form , . A directed graph contains a directed spanning tree if there exists a vertex called the root, which has no parent vertex, such that there exist directed paths from the vertex to all other vertices in the graph. A directed graph is strongly connected if there is a directed path between every distinct vertices. A directed graph has a directed spanning tree if it is strongly connected, but not vice versa.

For the directed graph , its adjacency matrix is defined such that , if and otherwise. The Laplacian matrix associated with is defined as and , .

###### Lemma 1 ([24])

Zero is an eigenvalue of with as a right eigenvector and all the nonzero eigenvalues have positive real parts. Zero is a simple eigenvalue of if and only if contains a directed spanning tree.

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

Suppose that is strongly connected. Let be the positive left eigenvector of associated with the zero eigenvalue and . Then, is the symmetric Laplacian matrix associated with an undirected connected graph. Let by any vector with positive entries. Then, where denotes the smallest nonzero eigenvalue of .

###### Lemma 3 ([21])

For a nonsingular -matrix , there exists a positive diagonal matrix such that . Moreover, can be given by , where .

###### Lemma 4 (Young’s Inequality, [25])

If and are nonnegative real numbers and and are positive real numbers such that , then .

###### Lemma 5 ([26])

For a dynamical system , where is locally Lipschitz in and piecewise continuous in , suppose that there exists a continuous and 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 positive constant, and are class functions, and is a class function. Then, the solution of is uniformly ultimately bounded.

## Iii Distributed Robust Adaptive Protocols for Leader-follower Graphs in The Presence of External Disturbances

### Iii-a Problem Formulation and Motivation

In this paper, we consider a group of identical agents with general linear dynamics. The dynamics of the -th agent are described by

 ˙xi =Axi+B[ui+ωi],i=1,⋯,N, (1)

where is the state vector, is the control input vector, and are known constant matrices with compatible dimensions, and denotes external disturbances associated with the -th agent, which satisfies the matching condition and the following assumption.

Assumption 1: There exist positive constants such that , .

In this section, we consider the case where there are followers and one leader. Without loss of generality, let the agent in (1) indexed by 1 be the leader whose control input is assumed to be zero, i.e., , and the agents indexed by , be the followers. The communication graph among the agents is assumed to satisfy the following assumption.

Assumption 2: The graph contains a directed spanning tree with the leader as the root node.

Since the vertex indexed by 1 is the leader and it does not have any neighbor, the Laplacian matrix associated with can be written into

 L=[001×(N−1)L2L1], (2)

where and . In light of Lemma 1, it is easy to see that is a nonsingular -matrix if satisfies Assumption 2.

This intention of this section is to solve the consensus problem for the agent in (1), i.e., to design distributed consensus protocols under which the states of the followers converge to the state of the leader in the sense of ,

For the case without external disturbances, i.e., , , a distributed adaptive consensus protocol based on the relative states of neighboring agents was proposed for each follower as [21]:

 ui =¯ciρi(ξTiQξi)Kξi, (3) ˙¯ci =ξTiΓξi,i=2,⋯,N,

where , , denotes the time-varying coupling gain (weight) associated with the -th follower with , is the -th entry of the adjacency matrix associated with , and are the feedback gain matrices, are smooth and monotonically increasing functions which satisfies that for , and is a solution to the following algebraic Riccati equation (ARE):

 ATQ+QA+I−QBBTQ=0. (4)
###### Lemma 6 ([21])

Suppose that the communication graph satisfies Assumption 2. Then, the consensus problem of the agents in (1) is solved by the adaptive protocol (3) with , , and . Moreover, each coupling gain converges to some finite steady-state value.

Lemma 6 shows that the adaptive protocol (3) can achieve consensus for the case where the agents in (1) are not perturbed by external disturbances. The adaptive protocol (3) constructed by Lemma 6, contrary to the consensus protocols in [11, 12, 14, 27, 15], depends on only the agent dynamics and the relative states of neighboring agents, and thereby can be computed and implemented by each agent in a fully distributed way.

Note that in many circumstances, the agents may be subject to certain external disturbances, for which case it is necessary and interesting to investigate whether the adaptive protocol (3) designed by Lemma 6 is still applicable, i.e., whether (3) is robust with respect to external disturbances. Due to the existence of nonzero in (1), 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 are of nonnegative quadratic forms in terms of the relative states, in this case it is easy to see from (3) that will keep growing to infinity, which is known as the parameter drift phenomenon in the classic adaptive control literature [23]. Therefore, the adaptive protocol (3) is not robust in the presence of external disturbances.

### Iii-B Distributed Robust Adaptive Consensus Protocols

The objective of this subsection is to made modifications on (3) in order to present novel distributed robust adaptive protocols which can guarantee the stability or ultimate boundedness of the consensus error in the presence of external disturbances. By utilizing the modification technique [23], we propose a new distributed adaptive consensus protocol as follows:

 ui =ciρi(ξTiQξi)Kξi, (5) ˙ci =−φi(ci−1)+ξTiΓξi,i=2,⋯,N,

where , , are small positive constants and the rest of the variables are defined as in (3).

Let , where . Then, it follows that

 ξ=(L1⊗In)⎡⎢ ⎢⎣x2−x1⋮xN−x1⎤⎥ ⎥⎦, (6)

where is defined as in (2). Because is a nonsingular matrix for satisfying Assumption 2, the consensus problem is solved if and only if asymptotically converges to zero. Hereafter, is referred to as the consensus error. In light of (1) and (5), it is not difficult to verify that and satisfy the following dynamics:

 ˙ξ =[IN−1⊗A+L1ˆC^ρ(ξ)⊗BK]ξ+(L1⊗B)ω, (7) ˙ci =−φi(ci−1)+ξTiΓξi,

where , , and .

In light of Assumption 1, it is easy to see that

 ∥˜ω∥≤ ⎷N∑i=2(υi+υ1)2. (8)

By the second equation in (7), we have

 ci(t) =ci(0)e−φit+∫t0e−φi(t−s)(φi+ξTiΓξi)ds (9) =(ci(0)−1)e−φit+1+∫t0e−φi(t−s)ξTiΓξids ≥1,

where we have used the fact that to get the last inequality.

The following theorem designs the adaptive protocol (5).

###### Theorem 1

Suppose that the communication graph satisfies Assumption 2. Then, both the consensus error and the coupling gains , , in (7), under the adaptive protocol (5) with , , and designed as in Lemma 6, are uniformly ultimately bounded. Furthermore, if is chosen to be small enough such that , then exponentially converges to the residual set

 D1≜⎧⎪⎨⎪⎩ξ:∥ξ∥2≤2Π(τ−δ)λmin(Q)mini=2,⋯,Nqi⎫⎪⎬⎪⎭, (10)

where

 Π≜^λ024N∑i=2φi(α−1)2+12^λ0σ2max(GL1)N∑i=2(υi+υ1)2, (11)

, , , and denotes the smallest eigenvalue of .

{proof}

Consider the following Lyapunov function candidate:

 V1=N∑i=2ciqi2∫ξTiQξi0ρi(s)ds+^λ024N∑i=2~c2i, (12)

where . As mentioned earlier, for a communication graph satisfying Assumption 2, is a nonsingular -matrix, which, by Lemma 3, implies that and . Furthermore, by noting that are monotonically increasing functions satisfying for and that for as in (9), it is not hard to get that is positive definite with respect to and , .

The time derivative of along the trajectory of (7) can be obtained as

 ˙V1 =N∑i=2ciqiρi(ξTiQξi)ξTiQ˙ξi (13) +N∑i=2˙ciqi2∫ξTiQξi0ρi(s)ds +^λ012N∑i=2(ci−α)[−φi(ci−1)+ξTiΓξi].

In the rest of this proof, we will use and instead of and , respectively, whenever without causing any confusion.

Observe that

 N∑i=2ciqiρiξTiQ˙ξi =ξT(ˆC^ρG⊗Q)˙ξ (14) =12ξT[ˆC^ρG⊗(QA+ATQ) −ˆC^ρ(GL1+LT1G)ˆC^ρ⊗QBBTQ]ξ +ξT(ˆC^ρGL1⊗QB)˜ω ≤12ξT[ˆC^ρG⊗(QA+ATQ)−^λ0ˆC2^ρ2⊗QBBTQ]ξ +ξT(ˆC^ρGL1⊗QB)˜ω,

where we have used the fact that to get the first inequality.

Since are monotonically increasing functions satisfying that for , it follows that

 N∑i=2˙ciqi∫ξTiQξi0ρi(s)ds ≤N∑i=2˙¯ciqi∫ξTiQξi0ρi(s)ds (15) ≤N∑i=2˙¯ciqiρiξTiQξi ≤N∑i=2˙¯ciq3i3^λ20+N∑i=223^λ0˙¯ciρ32i(ξTiQξi)32 ≤N∑i=2˙¯ciq3i3^λ20+N∑i=123^λ0˙¯ciρ32i(1+ξTiQξi)32 ≤N∑i=1(q3i3^λ20+23^λ0ρ2i)(ξTiQBBTQξi),

where we have used the fact that ( is defined in (3)) to get the first inequality, used the well-known mean value theorem for integrals to obtain the second inequality, and used Lemma 4 to get the third inequality.

Substituting (14) and (15) into (13) yields

 ˙V1 ≤12ξT[ˆC^ρG⊗(QA+ATQ)]ξ (16) −N∑i=1[^λ0(12c2iρ2i−112ci−13ρ2i)+112(^λ0α−2q3i^λ20)]ξTiQBBTQξi +^λ024N∑i=1φi[−~c2i+(α−1)2]+ξT(ˆC^ρGL1⊗QB)˜ω,

where we have used the following fact:

 −(ci−α)(ci−1)=−~ci(~ci+α−1)≤−12~c2i+12(α−1)2. (17)

By noting that and , , and that , where , it follows from (16) that

 ˙V1 ≤12ξT[ˆC^ρG⊗(QA+ATQ)]ξ−^λ012N∑i=1(c2iρ2i+2^α)ξTiQBBTQξi (18) +^λ024N∑i=1φi[−~c2i+(α−1)2]+ξT(ˆC^ρGL1⊗QB)˜ω.

Note that

 2ξT(ˆC^ρGL1⊗QB)ω =2ξT(√^λ012ˆC^ρ⊗QB)(√12^λ0GL1⊗I)˜ω (19) ≤^λ012ξT(ˆC2^ρ2⊗QBBTQ)ξ+12^λ0∥(GL1⊗I)˜ω∥2 ≤^λ012ξT(ˆC2^ρ2⊗QBBTQ)ξ+12^λ0σ2max(GL1)N∑i=2(υi+υ1)2,

where we have used (8) to get the last inequality. Then, substituting (19) into (18) gives

 ˙V1 ≤12ξT[ˆC^ρG⊗(QA+ATQ) (20) −^λ024(ˆC2^ρ2+4^αI)⊗QBBTQ)]ξ−^λ024N∑i=2φi~c2i+Π ≤12W(ξ)−^λ024N∑i=1φi~c2i+Π,

where we have used the assertion that if to get the last inequality, is defined as in (28), and

 W(ξ) ≜ξT[ˆC^ρG⊗(QA+ATQ−QBBTQ)]ξ =−ξT(ˆC^ρG⊗I)ξ ≤0.

Therefore, we can verify that is negative definite. In virtue of the results in [26], we get that both the consensus error and the adaptive gains are uniformly ultimately bounded.

Note that (18) can be rewritten into

 ˙V1 ≤−δV1+δV1+12W(ξ)−^λ024N∑i=2φi~c2i+Π. (21)

Because are monotonically increasing and satisfy for , as shown in (15), we have

 N∑i=2ciqi∫ξTiQξi0ρi(s)ds ≤N∑i=2ciqiρiξTiQξi (22) =ξT(ˆC^ρG⊗Q)ξ.

Substituting (22) into (21) yields

 ˙V1 ≤−δV1+12~W(ξ)−^λ024N∑i=2(φi−δ)~c2i (23) −τ−δ2ξT(ˆC^ρG⊗Q)ξ+Π,

where Because , we can obtain that . Then, it follows from (41) that

 ˙V1 ≤−δV1−τ−δ2λmin(Q)(mini=2,⋯,Nqi)∥ξ∥2+Π, (24)

where we have used the facts that , , , , , and . Obviously, it follows from (41) that if Then, we can get that if then exponentially converges to the residual set in (10) with a convergence rate faster than .

###### Remark 1

It is well known that there exists a unique solution to the ARE (4) if is stabilizable [28]. Therefore, a sufficient condition for the existence of an adaptive protocol (5) satisfying Theorem 1 is that is stabilizable. The consensus protocol (5) can also be equivalently designed by solving the linear matrix inequality: , as in [11, 21]. In this case, the parameters in (5) can be chosen as , , and . Similar to the adaptive protocol (3) in Lemma 6, the adaptive protocol (5) in Theorem 1, depending only on the agent dynamics and the relative states of neighboring agents, can be constructed and implemented in a fully distributed fashion.

###### Remark 2

Theorem 1 shows that the modified adaptive protocol (5) can ensure the ultimate boundedness of the consensus error and the adaptive gains for the agents in (1), implying that (5) is indeed robust in the presence of bounded external disturbances. From (10), it can be observed that the upper bound of the consensus error depends on the communication graph, the upper bounds of the external disturbances, and the parameters of the adaptive protocol (5). Roughly speaking, should be chosen to be relatively small in order to ensure a smaller bound for .

## Iv Distributed Robust Adaptive Protocols for Strongly Connected Graphs in The Presence of External Disturbances

The results in the previous section are applicable to the case where there exists a leader. In this section, we extend to consider the case where the communication graph among the agents is directed and does not contain a leader.

The dynamics of the agents are still described by (1). The communication graph among the agents is assumed to be strongly connected in this section.

Based on the relative states of neighboring agents, we propose the following distributed adaptive consensus protocol:

 ui =diρi(ζTiQζi)Kζi, (25) ˙di =−φi(di−1)+ζTiΓζi,i=1,⋯,N,

where , denotes the time-varying coupling gain associated with the -th agent with , , , are small positive constants, and the rest of the variables are defined as in (5).

Let and . Then, , where denotes the Laplacian matrix associated with . Since is strongly connected, it is well known via Lemma 1 that the consensus problem is solved if and only if asymptotically converges to zero. Hereafter, we refer to as the consensus error. In virtue of (1) and (25), it is not difficult to get that and satisfy the following dynamics:

 ˙ζ =[IN⊗A+LˆD~ρ(ζ)⊗BK]ζ+(L⊗B)ω, (26) ˙di =−φi(di−1)+ζTiΓζi,

where , , and .

###### Theorem 2

Suppose that the communication graph is strongly connected and Assumption 1 holds. Then, both the consensus error and the coupling gains , , in (26), under the adaptive protocol (25) with , , and designed as in Theorem 1, are uniformly ultimately bounded. Furthermore, if is chosen to be small enough such that , then exponentially converges to the residual set

 D2≜⎧⎪⎨⎪⎩ζ:∥ζ∥2≤2Ξ(τ−ε)λmin(Q)mini=1,⋯,Nri⎫⎪⎬⎪⎭, (27)

where is the positive left eigenvector of associated with the zero eigenvalue, denotes the smallest nonzero eigenvalue of , , , and

 Ξ≜λ2(ˆL)24NN∑i=1φi(α−1)2+12Nλ2(ˆL)σ2max(RL)N∑i=1υ2i. (28)
{proof}

Consider the following Lyapunov function candidate:

 V2=N∑i=1diri2∫ζTiPζi0ρi(s)ds+λ2(ˆL)24NN∑i=1~d2i, (29)

where . Similarly as shown in (9), it is easy to see that for . Furthermore, by noting that are monotonically increasing functions satisfying for , it is not difficult to see that is positive definite.

The time derivative of along the trajectory of (26) is given by

 ˙V2 =N∑i=1diriρi(ζTiPζi)ζTiQ˙ζi (30) +N∑i=1˙diri2∫ζTiQζi0ρi(s)ds +λ2(ˆL)12NN∑i=1(di−β)[−φi(di−1)+ζTiΓζi].

By using (26) and making some mathematical manipulations, we can get that

 N∑i=1diriρiζTiQ−1˙ζi =ζT(ˆD~ρR⊗Q)˙ζ (31) =12ζT