1 Introduction

Distributed Robust Control of Linear Multi-Agent Systems with Parameter Uncertainties 111Zhongkui Li and Xiangdong Liu are with the School of Automation, Beijing Institute of Technology,  Beijing 100081, China (e-mail: zhongkli@gmail.com). Zhisheng Duan is with State Key Lab for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China (e-mail: duanzs@pku.edu.cn). Lihua Xie is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798 (e-mail: elhxieg@ntu.edu.sg).

Zhongkui Li, Zhisheng Duan, Lihua Xie, Xiangdong Liu

July 5, 2019

Abstract: This paper considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighboring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.

Keywords: Multi-agent system, distributed control, robustness, control, parameter uncertainty.

## 1 Introduction

The coordination control problems of multi-agent systems have received increasing attention from various scientific communities, for its broad applications in such fields as satellite formation flying, sensor networks, and air traffic control, to name just a few [1]. Due to the spatial distribution of actuators, communication constraint, and limited sensing capability of sensors, centralized controllers are generally too expensive or even infeasible to implement in practice. Therefore, distributed control strategies based on only local information have been proposed and extensively studied in the last decade.

Formation control of multiple autonomous vehicles is considered in [2], where a Nyquist-like criterion is derived. Distributed linear quadratic regulator (LQR) control of a set of identical decoupled dynamical systems is discussed in [3]. A decomposition approach is proposed in [4] to solve the distributed control of identical coupled linear systems. A general framework of the consensus problem for networks of integrator agents with fixed and switching topologies is addressed in [5]. The conditions derived in [5] are further relaxed in [6]. The consensus problems of networks of double- and high-order integrators are investigated in [7, 8, 9]. Consensus algorithms are designed in [10, 11] for a group of agents with quantized communication links and limited data rate. Distributed consensus control of multi-agent systems with general linear dynamics is studied in [12, 13, 14, 15]. Distributed consensus and control problems are investigated in [16, 17] for networks of agents subject to external disturbances and model uncertainties. Flocking algorithms are studied in [18, 19] for a group of autonomous agents.

The aforementioned works have a common assumption that the dynamics of the agents are all identical, i.e., the multi-agent systems are homogeneous. Such an assumption may be restrictive in many circumstances. It is necessary and important to study the cooperative control problems of heterogenous multi-agent systems consisting of nonidentical agents. Previous works along this line include [20, 21, 22, 23, 24]. Specifically, scalar robust stability conditions are derived in [20, 21] for a network of heterogeneous agents by using the notion of S-hull. Similar results are given in [22] using tools from the integral quadratic constraint (IQC). In [23], a distributed controller based on the internal model is designed for the output regulation of heterogeneous linear multi-agent systems. Neural adaptive tracking control of first-order nonlinear systems with unknown dynamics and disturbances is investigated in [24].

This paper is concerned with the distributed robust control problems of multi-agent systems with general linear dynamics subject to norm-bounded parameter uncertainties. It is assumed that all the agents have the same nominal dynamics but subject to different parameter uncertainties. Thus, the resulting multi-agent systems are weakly heterogeneous, which fits well into the gap between the commonly-studied homogeneous multi-agent systems and the heterogeneous multi-agent systems as investigated in [20, 21, 22]. Typical examples belonging to this scenario are the mass-spring systems [25] with different or uncertain spring constants, the Lorenz-type chaotic systems that cover the Lorenz, Chen, Lü systems as special cases with the change of a key parameter [26], and the discrete-time double integrators with unknown model parameters which have applications in synchronization of a network of clocks [27].

In this paper, distributed controllers are proposed for both the continuous- and discrete-time uncertain multi-agent systems, which rely on the relative states between neighboring agents and the absolute states of a subset of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability (which are referred to as distributed quadratic stabilization problems) are equivalent to the control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which decouples the feedback gain design from the communication topology and does not involve any conservatism. The maximal allowable uncertainty bound is given as a corollary. A sufficient existence condition in terms of linear matrix inequalities is further derived for the distributed discrete-time controller. Note that such a condition is conservative to some extent, because the eigenvalues of the stochastic matrix associated with the communication graph are treated as an uncertainty. Moreover, distributed quadratic stabilization with disturbance attenuation is considered for uncertain linear multi-agent systems subject to external disturbances, which can be reduced to the scaled control problems of a set of independent systems whose dimensions are equal to that of a single agent. Design procedures for distributed controllers are further given for both the continuous- and discrete-time cases.

The rest of this paper is organized as follows. Some basic notation and useful results of the graph theory are reviewed in Section 2. The distributed robust control problems of continuous- and discrete-time multi-agent systems are investigated in, respectively, Sections 3 and 4. Simulation examples are given for illustration in Section 5. Conclusions are drawn in Section 6.

## 2 Graph Theory and Notation

Let be the set of real matrices. The superscript means the transpose for real matrices. represents the identity matrix of dimension . Matrices, if not explicitly stated, are assumed to have compatible dimensions. Denote by the column vector with all entries equal to one. represents a block-diagonal matrix with matrices on its diagonal. The matrix inequality (respectively, ) means that is positive definite (respectively, positive semi-definite). denotes the Kronecker product of matrices and . A matrix is Hurwitz (in the continuous-time case) if all of its eigenvalues have negative real parts, while is Schur stable (in the discrete-time case) if all of its eigenvalues have magnitude less than 1.

An undirected graph is a pair , where is the set of nodes and is the set of unordered pairs of nodes, called edges. Two nodes , are adjacent, or neighboring, if is an edge of graph . A path on from node to node is a sequence of ordered edges of the form , . A graph is called connected if there exists a path between every pair of distinct nodes, otherwise is disconnected.

The adjacency matrix associated with the undirected graph is defined by , if , and otherwise. The Laplacian matrix is defined as and , . Let be a double-stochastic matrix associated with with the additional assumption that , if and otherwise. It is straightforward to verify that zero is an eigenvalue of with as the corresponding eigenvector and all nonzero eigenvalues are positive. One is an eigenvalue of with as the corresponding eigenvector and all other eigenvalues of are in the open unit disk [6].

## 3 Distributed Robust Control of Uncertain Continuous-Time Multi-Agent Systems

Consider a network consisting of continuous-time linear agents subject to parameter uncertainties, described by

 ˙xi =(A+ΔAi)xi+Bui,i=1,⋯,N, (1)

where and are, respectively, the state and the control input of the -th agent, and are constant matrices with compatible dimensions, and is an unknown matrix which represents the time-varying uncertainty associated with the -th agent and is assumed to be in the form of , where is the uncertainty satisfying

 FTiFi≤δ2I,i=1,⋯,N, (2)

with elements of being Lebesgue measurable and a given constant, and and are known constant matrices which characterize the structure of the uncertainty.

The communication topology among the agents is represented by an undirected graph . It is assumed here that only a subset of agents know their own states but each agent can measure the relative states with respect to its neighbors. Without loss of generality, assume that the first agents have access to their state information.

The following assumption will be used in the sequel.

Assumption 1. The undirected communication graph is connected and at least one agent knows its own state.

Based on the relative state information between neighboring agents and the absolute states of a portion of agents, a distributed controller is proposed as

 (3)

where is the coupling strength, is the feedback gain matrix, is the -th entry of the adjacency matrix associated with , and are constant scalars, satisfying , and , .

Let and . Then, the closed-loop network dynamics resulting from (1) and (3) can be rewritten as

 ˙x =(IN⊗A+cˆL⊗BK+(IN⊗D)Δ(IN⊗E))x, (4)

where , is the Laplacian matrix of , and .

First, the notion of quadratic stability is introduced.

Definition 1 [28, 29]. The system (1) with is quadratically stable if there exists a positive-definite matrix such that for all admissible uncertainty ,

 P(A+ΔAi)T+(A+ΔAi)P<0.

The objective in this section is to design a distributed controller (3) such that the closed-loop network (4) is quadratically stable for all admissible uncertainties , . This problem is referred to as distributed quadratic stabilization problem.

Lemma 1 [29]. The system (1) with is quadratically stable for all admissible uncertainties satisfying (2) if and only if is Hurwitz and .

Lemma 2 [30, 14]. For a graph satisfying Assumption 1, the matrix in (4) is positive definite.

The following presents a necessary and sufficient condition for the distributed quadratic stabilization problem.

Theorem 1. Under Assumption 1, the closed-loop network (4) is quadratically stable for all admissible uncertainties , , satisfying (2), if and only if the matrices are Hurwitz and , , where , , and , , are the eigenvalues of .

Proof. (Necessity) Consider a special case where the certainties in (1) are the same, i.e., . Thus, the system (4) can be rewritten as

 ˙x =(IN⊗A+cˆL⊗BK+IN⊗DFE)x. (5)

Because Assumption 1 holds, it follows from Lemma 2 that is positive definite. Let be such a unitary matrix that . Let . Then, it follows from (5) that

 ˙ξ =(IN⊗A+cΛ⊗BK+IN⊗DFE)ξ. (6)

Note that the state matrix of (6) is block diagonal. Therefore, the network (5) is quadratically stable if and only if the following systems:

 ˙ξi=(A+cλiBK+DFE)ξi,i=1,⋯,N, (7)

are simultaneously quadratically stable, which by Lemma 1 implies that are Hurwitz and , .

(Sufficiency) From (2), it follows that the uncertainty in (4) satisfies . In light of Lemma 1, the system (4) is quadratically stable for all admissible uncertainties , , satisfying (2), if is Hurwitz and , where

 T(s)≜(IN⊗E)(sI−IN⊗A−cˆL⊗BK)−1(IN⊗D). (8)

Note that

 (U⊗In)T(s)(UT⊗In) =(IN⊗E)(sI−IN⊗A−cUTˆLU⊗BK)−1(IN⊗D) (9) =diag(T1(s),⋯,TN(s)).

By the definition of the norm [31], it follows readily from (9) that

 ∥T(s)∥∞=maxi=1,⋯,N∥Ti(s)∥∞.

Therefore, if the matrices are Hurwitz and , , then the system (4) is quadratically stable for all admissible uncertainties , , satisfying (2).

Remark 1. Although the uncertainty in (4) is structural, it is shown in Theorem 1 that the distributed quadratic stabilization problem of (4) is equivalent to the control problems of a set of decoupled linear systems having the same dimensions as a single agents, which is essentially due to the fact that the nominal dynamics of the agents are identical.

Remark 2. Different from most of the existing references focusing on homogeneous multi-agent systems, the current paper considers the uncertain multi-agent systems where the agents have the same nominal dynamics but subject to different parameter uncertainties. The resulting agent network (4) is thus weakly heterogeneous, which fits well into the gap between the commonly-studied homogeneous multi-agent systems and the heterogeneous multi-agent systems as investigated in [20, 21, 22, 23]. Typical examples belonging to this scenario are the mass-spring systems [25] with different or uncertain spring constants, the Lorenz-type chaotic systems [26], and the discrete-time double integrators with unknown model parameters [27].

Next, an algorithm is presented to determine the distributed controller (3).

Algorithm 1. Under Assumption 1, a distributed controller (3) can be constructed as follows:

• Solve the following linear matrix inequality (LMI):

 ⎡⎢⎣AP+PAT−τBBTδDPETδDT−I0EP0−I⎤⎥⎦<0 (10)

to get a matrix and a scalar . Then, choose .

• Select the coupling strength , where

 cth=τmini=1,⋯,N(λi). (11)

Theorem 2. Under Assumption 1, the closed-loop network (4) with the distributed controller constructed by Algorithm 1 is quadratically stable for all admissible uncertainties , , satisfying (2).

Proof. By the Schur Complement Lemma [32], the LMI (10) is feasible for some matrix and scalar if and only if the following Riccati inequality holds:

 AP+PAT−τBBT+δ2DDT+PETEP<0. (12)

From step 2) in Algorithm 1, it follows that , . Noting , it can be obtained that

 (A+cλiBK)P +P(A+cλiBK)T+δ2DDT+PETEP (13) =AP+PAT−cλiBBT+δ2DDT+PETEP <0.

By the Bounded Real Lemma [31], it follows from (13) that are Hurwitz and , . Therefore, Theorem 1 implies that the network (4) in this case is quadratically stable for all admissible uncertainties , , satisfying (2).

Remark 3. As shown in the proof of Proposition 1 in [14], by using Finsler’s Lemma [33], it is not difficult to see that there exist and such that (10) holds if and only if there exists a such that . That is, is Hurwitz and . Therefore, it follows from Theorems 1 and 2 that the feasibility of the LMI (10) with respect to and is not only sufficient but also necessary for the existence of a controller (3) satisfying Theorem 1. The largest allowable uncertainty bound can be obtained by the following optimization problem:

 maximizeδ>0 (14) subject to\quad LMI (???), with P>0, τ>0.

### 3.2 Distributed Quadratic H∞ Control

This subsection extends to consider a network of uncertain linear agents subject to external disturbance, given by

 ˙xi =(A+ΔAi)xi+Bui+B2ωi, (15) zi =Cxi,i=1,⋯,N,

where and are, respectively, the exogenous disturbance and the performance variable of the -th agent, and the rest of the variables are defined as in (1).

The distributed controller is still given as in (3). Let and . Then, the closed-loop system resulting from (15) and (3) can be written as

 ˙x =(IN⊗A+cˆL⊗BK+(IN⊗D)Δ(IN⊗E))x+(IN⊗B2)ω, (16) z =(IN⊗C)x.

This section is to design a distributed controller (3) such hat the closed-loop system (16) is quadratically stable and meanwhile achieves a prescribed level of disturbance attenuation in the -norm sense for all admissible uncertainties , , satisfying (2).

Definition 2 [29]. The system (15) with is quadratically stable with a disturbance attenuation level if there exists a positive-definite matrix such that for all admissible uncertainty ,

 P(A+ΔAi)T+(A+ΔAi)P+1γ2B2BT2+PCTCP<0.

Lemma 3 [29]. There exists a positive-definite matrix such that

 P(A+H1F1E1)T+(A+H1F1E1)P<0

for all admissible uncertainty satisfying if and only if there exists a scalar such that

 PAT+AP+1ϵPET1E1P+ϵϱ2H1HT1<0.

Theorem 3. Suppose that Assumption 1 holds. Then, the closed-loop network (16) is quadratically stable with disturbance attenuation for all admissible uncertainties , , satisfying (2), if the following systems:

 ˙xi =(A+cλiBK)xi+[ϵ12δDγ−1B2]¯ωi, (17) ¯zi =[ϵ−12EC]xi,i=1,⋯,N,

are simultaneously asymptotically stable with unitary disturbance attenuation, where , , , and is a scalar to be chosen.

Proof. As stated in the proof of Theorem 1, the uncertainty in (16) is structural and satisfies . In view of Definition 2, the network (16) is quadratically stable with disturbance attenuation for all admissible uncertainties , , satisfying (2), if there exists a positive-definite matrix such that

 PST+SP+1γ2B2BT2+PCTCP<0, (18)

where

 S =IN⊗A+cˆL⊗BK+(IN⊗D)Δ(IN⊗E), B2 =IN⊗B2,C=IN⊗C.

Let be such a unitary matrix that . Multiplying the left and right sides of (18) by and , respectively, gives

 ˜P˜ST+˜S˜P+1γ2B2BT2+˜PCTC˜P<0, (19)

where

 ˜S =IN⊗A+cΛ⊗BK+(IN⊗D)˜Δ(IN⊗E), ˜Δ =(UT⊗In)Δ(U⊗In),˜P=(UT⊗In)P(U⊗In).

Clearly, it follows from (2) that satisfies

 ˜ΔT˜Δ=(UT⊗In)ΔTΔ(U⊗In)≤δ2I.

By Lemma 3, it follows that (19) holds if there exists a scalar such that

 ˜P(IN⊗A+cΛ⊗BK)T+(IN⊗A+cΛ⊗BK)˜P+1ϵ˜P(IN⊗ETE)˜P (20) +ϵδ2IN⊗DDT+1γ2B2BT2+˜PCTC˜P<0.

Note that all the matrices in (20) except are block diagonal. Therefore, if there exist positive-definite matrices such that

 Pi(A+cλiBK)T+(A+cλiBK)Pi+1ϵPiETEPi+ϵδ2DDT (21) +1γ2B2BT2+PiCTCPi<0,i=1,⋯,N,

then the matrix satisfies (18). By the Bounded Real Lemma [31], it is easy to see that (21) is equivalent to that the systems in (17) are simultaneously asymptotically stable with unitary disturbance attenuation. This completes the proof.

Remark 4. Theorem 3 casts the robust control problem of high-dimensional network (16) to the scaled control problems of a set of independent low-dimensional linear systems in (17), thereby reducing the computational complexity significantly. When the external disturbances do not exist, Theorem 3 is reduced to Theorem 1.

Algorithm 2. Under Assumption 1, a distributed controller (3) can be constructed as follows:

• Solve the following LMI:

 ⎡⎢ ⎢⎣AQ+QAT−~τBBT+1γ2B2BT2+ϵδ2DDTQCTQETCQ−I0EQ0−ϵI⎤⎥ ⎥⎦<0 (22)

to get a matrix and scalars , . Then, choose .

• Select the coupling strength , where is defined in (11).

Remark 5. It is worth noting that in Algorithms 1 and 2, the feedback gain design of (3) is decoupled from the communication topology and only the smallest eigenvalue of is used to select the coupling strength . One consequence of this decoupling property is that the controller designed for one given connected communication graph can be used directly to any other connected graphs, with the only task of appropriately selecting the coupling strength . By selecting the coupling strength to be relatively large, the distributed controller (3) constructed by these algorithms maintains certain degree of robustness with respect to variations of the communication graph , such as adding or removing edges or agents in , in which case although the eigenvalues , , are changed, , , can still be larger than (or ).

Theorem 4. Suppose that Assumption 1 holds. Then, the closed-loop network (16) with the distributed controller (3) constructed by Algorithm 2 is quadratically stable with disturbance attenuation for all admissible uncertainties , , satisfying (2)

Proof. The proof is similar to that of Theorem 2 and thus is omitted.

Corollary 1. As , the LMI condition (22) in Algorithm 2 is reduced to (10) in Algorithm 1.

Proof. By the Schur Complement lemma [32], the LMI (22) is equivalent to

 QAT+AQ−~τBBT+1γ2B2BT2+ϵδ2DDT+1ϵQETEQ+QCTCQ<0.

Letting , , and gives

 PAT+AP−τBBT+1γB2BT2+δ2DDT+PETEP+1γPCTCP<0. (23)

When , (23) implies that

 PAT+AP−τBBT+δ2DDT+PETEP<0,

which is equivalent to (10).

## 4 Distributed Robust Control of Uncertain Discrete-Time Multi-Agent Systems

This section considers a network consisting of discrete-time agents with parameter uncertainties, described by

 xi(k+1) =(A+ΔAi)xi(k)+Bui(k),i=1,⋯,N, (24)

where and are, respectively, the state and the control input at the time instant of the -th agent, and denotes the time-varying uncertainty associated with the -th agent, which is assumed to be in the form of , with satisfying (2).

Similar to (3), a distributed controller based on the relative states between neighboring agents and the absolute states of a subset of agents is proposed as

 ui(k)=K(N∑j=1eij(xi(k)−xj(k))+^dixi(k)),i=1,⋯,N, (25)

where is the feedback gain matrix, is the -th entry of the double-stochastic matrix associated with , and are constant scalars, satisfying , and , .

Let and . Then, the closed-loop network dynamics resulting from (24) and (25) can be rewritten as

 x(k+1) =(IN⊗A+(IN−˜D)⊗BK+(IN⊗D)Δ(IN⊗E))x(k), (26)

where and .

The objective in this section is to solve the discrete-time distributed quadratic stabilization problem for (26), i.e., to design a distributed controller (25) such that the closed-loop network (26) is quadratically stable for all admissible uncertainties , .

The notion of quadratic stability for (24) is introduced below.

Definition 3 [34]. The system (24) with is quadratically stable if there exists a positive-definite matrix such that for all admissible uncertainty ,

 (A+ΔAi)TP(A+ΔAi)−P<0.

Lemma 4 [34]. The system (24) with is quadratically stable for all admissible uncertainties satisfying (2) if and only if is Schur stable and .

Lemma 5. Suppose that the graph satisfies Assumption 1. Then, the matrix in (26) have all of its eigenvalues located inside the unit circle.

Proof. Consider the following new row-stochastic matrix:

 ¯¯¯¯D=[˜DˆD1N01×N1].

According to the definition of the directed spanning tree [6], the graph associated with has a directed spanning tree if satisfies Assumption 1. Therefore, by Lemma 3.4 in [6], if Assumption 1 holds, then 1 is a simple eigenvalue of and all the other eigenvalues of are in the open unit disk, which further implies that all the eigenvalues of lie in the open unit disk.

The following gives a sufficient condition for solving the distributed quadratic control problem of (26).

Theorem 5. Under Assumption 1, the closed-loop network (26) is quadratically stable for all admissible uncertainties , , satisfying (2), if and only if the matrices are Schur stable and , , where , , and , , are the eigenvalues of .

Furthermore, if there exist matrices , , and a scalar such that

 ⎡⎢ ⎢ ⎢ ⎢⎣−Q(AQ+BW)TQETWTAQ+BW−Q+δ2DDT+τκ2BBT00EQ0−I0W00−τI⎤⎥ ⎥ ⎥ ⎥⎦<0, (27)

where , then there exists a distributed controller (25) such that (26) is quadratically stable. Specifically, the feedback gain matrix of (25) is given by .

Proof. The equivalence between the quadratic stability of (26) and can be checked by following similar steps in the proof of Theorem 1. Next, it will be shown that the feasibility of the LMI (27) guarantees the existence of a controller (25) satisfying the quadratic stability of (26). In virtue of the discrete-time Bounded Real Lemma [31], the matrices are Schur stable and , , if and only if there exist matrices such that

 ^ATPi^A−Pi+ETE+δ2^ATPiD(I−δ2DPiDT)−1DPi^A<0,i=1,⋯,N, (28)

where . Noting that , , (28) clearly hold for , if there exists a such that

 ~ATP~A−P+ETE+δ2~ATPD(I−δ2DPDT)−1DP~A<0, (29)

where , for all . It is not difficult to see that (29) can be rewritten as

 ~AT(P−1−δ2DDT)−1~A−P+ETE<0.

By letting and applying the Schur Complement Lemma [32], the above inequality is equivalent to

 ⎡⎢ ⎢⎣−Q−1~ATET~AQ−δ2DDT0E0−I⎤⎥ ⎥⎦<0,

which can be rewritten as

 ⎡⎢⎣−Q−1(A+BK)TETA+BKQ−δ2DDT0E0−I⎤⎥⎦+⎡⎢⎣0B0⎤⎥⎦~Δ[K00]+⎡⎢⎣KT00⎤⎥⎦~Δ[0BT0]<0. (30)

Using Lemma 3, (30) holds if and only if

 ⎡⎢ ⎢ ⎢ ⎢⎣−Q−1(A+BK)TETKTA+BKQ−δ2DDT+τκ2BBT00E0−I0K00−τI⎤⎥ ⎥ ⎥ ⎥⎦<0, (31)

for some scalar . Multiplying the both sides of (31) by and noting leads directly to (27).

Remark 6. By similar steps in proving Corollary 1, it is not difficult to show that the LMI condition (27) is equivalent to , when , i.e., the uncertainty bound is sufficiently small. As pointed out in [34, 35], a sufficient condition for the existence of satisfying is that is stabilizable and , where are the unstable eigenvalues of . Therefore, under such a condition, the LMI (27) is feasible for the case where the uncertainty bound in (2) is sufficiently small. The largest allowable uncertainty bound can be obtained by maximizing in (27).

Remark 7. Theorem 5 involves certain conservatism, which is introduced by treating the eigenvalues , , as uncertainties with the bound . By selecting to be relatively large, the distributed controller (25) constructed by Theorem 5 maintains certain degree of robustness with respect to variations of the communication graph , such as adding or removing edges in .

Next, consider a network of uncertain discrete-time agents subject to external disturbance, given by

 xi(k+1) =(A+ΔAi)xi(k)+Bui(k)+B2ωi(k), (32) zi(k+1) =Cxi(k),i=1,⋯,N,

where , , and the rest of the variables are defined as in (24).

Let and . Then, it follows from (32) and (25) that the closed-loop system can be written as

 x(k+1) =(IN⊗A+(IN−˜D)⊗BK+(IN⊗D)Δ(IN⊗E))x(k)+(IN⊗B2)ω(k), (33) z(k+1) =(IN⊗C)x(k).

Theorem 6. Suppose that Assumption 1 holds. Then, the closed-loop network (16) is quadratically stable with disturbance attenuation for all admissible uncertainties , , satisfying (2), if the following systems:

 xi(k+1) =(A+(1−~λi)BK)xi(k)+[ϵ12δDγ−1B2]~ωi(k), (34) ~zi(k+1) =[ϵ−12EC]xi(k),i=1,⋯,N,

are simultaneously asymptotically stable with unitary disturbance attenuation, where , , , and is a scalar to be chosen.

Further, if there exist matrices , , and a scalar such that

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−Q(AQ+BW)TQETQCTWTAQ+BW−Q+δ2DDT+τκ2BBT000EQ0−ϵI00CQ00−I0W000−τI⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦<0, (35)

where is defined as in (27), then the feedback gain matrix of the controller (25) is given by .

Proof. The proof of the first part is similar to that of Theorem 3 and the proof of the second part is similar to that of Theorem 5. Both are omitted here for brevity.

## 5 Simulation Examples

In this section, simulation examples are provided to validate the effectiveness of the theoretical results.

Example 1. Consider a network of mass-spring systems with a common mass but different spring constants, described by

 m¨yi+kiyi=ui,i=1,⋯,N, (36)

where are the displacements from certain reference positions and , are the spring constants, which are assumed to be in the form of

 ki=k0+Δki,i=1,⋯,N, (37)

where is the identical nominal spring constant and are the uncertainties satisfying . Denote by the state vector of the -th agent. Then, (36) and (37) can be rewritten as

 ˙xi=(A+DΔkiE)xi+Bui,i=1,⋯,N, (38)

with

 A=[01−k0m0],B=[01],D=[0−1m],E=[10],

The objective is to design a distributed controller (3) such that the closed-loop network is quadratically stable for all . Let and . Solving the LMI (10) with by using the Sedumi toolbox [36] gives a feasible solution:

 P=[1.6448−2.3499−2.34999.7007],τ=64.0444.

Thus, the feedback gain matrix is chosen as . Assume that the communication topology is given in Fig. 1, with only the first node knowing its own state. In (3), let , and , . Then, the minimal eigenvalue of the matrix