# Distributed RobustH∞ Consensus Control for Uncertain Multiagent Systems with State and Input Delays

Authors: Ping Li; Kaiyu Qin
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

RobustH∞ consensus control problem is investigated for multiagent systems. Each agent is tackled in a more generalized form, which includes parameter uncertainties, external disturbances, nonidentical time-varying state, and input delays. Firstly, a distributed control protocol based on state feedback of neighbors is designed. By a decoupling method, H∞ consensus control problem for multiagent systems is transformed into H∞ control problem for the decoupling subsystems. Then employing Lyapunov-Krasovskii functional and free-weighting matrices, a lower conservative bounded real lemma (BRL) is derived in terms of linear matrix inequalities (LMIs) such that a class of time-delay system is guaranteed to be globally asymptotically stable with the desired H∞ performance index. Extending BRL, a sufficient delay-dependent condition of lower complexity in terms of the matrix inequalities is obtained to make all agents asymptotically reach consensus with the desired H∞ performance index. Furthermore, an algorithm is elaborately designed to get feasible solution to this condition. Extending this algorithm, an optimization algorithm for control protocol parameter is proposed to improve the disturbance attenuation capacity or allowable delay bounds. Finally, simulation results are provided to illustrate the correctness of the theoretical results and the effectiveness of the algorithms.


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## Body

## 1. Introduction

Consensus problem has attracted a great deal of attention due to its enormous potential applications in many areas such as flocking and swarming modeling [1], cooperative control of unmanned air vehicles [2], and formation control of multirobot systems [3]. In the past decade, a large number of results have been obtained for consensus problem of various multiagent systems, for example, consensus problems of multiagent systems with different dynamics such as double-integrator dynamics in [4] and Lipschitz nonlinear dynamics in [5], consensus problems of multiagent systems with the different network topologies such as random networks in [6] and switching topology in [7], and consensus problems of multiagent systems with the different time-delay such as nonuniform time-varying delays in [8] and input and communication delays in [9].In practical applications, multiagent systems often have disturbances, uncertainties and time-delay such as communication noise, and uncertainties in network parameters and time-delay caused by communication or measuring. Moreover, the existence of these facts might destroy the convergence properties of multiagent systems. Therefore, it is significant to investigate robust consensus problems of multiagent systems, which reflects the effects of these facts on the behaviour of multiagent systems. In the past decade, some interesting results have been obtained for robust consensus problems. Lin et al. investigated robustH∞ consensus analysis of directed networks of first-order agents with time-delay and a class of second-order multiagent systems with uncertainty in [10] and [11], respectively. Li et al. investigated the distributed H2 and H∞ control problems for multiagent systems with linear or linearized dynamics in [12]. Liu and Jia investigated H∞ consensus control problem for multiagent systems with linear coupling dynamics and communication delays in [13]. Wang et al. investigated distributed robust H∞ control problem synthesized with transient performance for a group of autonomous agents governed by uncertain general linear node dynamics in [14]. Hu et al. investigated the H∞ consensus problem for multiagent systems by means of a simultaneous stabilization approach in [15]. Shi and Qin studied rotational motion of multiagent systems with nonuniform time-delays in [16]. Li et al. investigated distributed robust H∞ rotating consensus control problem for directed networks of second-order agents with mixed uncertainties and time-delay in [17]. However, in [10, 11, 17], we could obtain the sufficient conditions where the parameters of the control protocols should be satisfied but their calculation method was not directly given. In [12, 13, 15], the parameter uncertainties and time-varying delay were not considered. In [16], uncertainty and external disturbance were not considered. In [14], time-delay was not considered. In addition, in practical applications, on one hand, each agent itself may be a time-delay system, which includes the state delay itself. On the other hand, due to the introduction of the control protocol, the system often contains the input delay. Furthermore, the state and input delays often are nonidentical and time variant, which makes the control problem more challenging. So far, there exists rare work dealing with robust H∞ consensus control problem for uncertain multiagent systems with nonidentical time-varying state and input delays.Motivated by the previous observations, this paper investigates robustH∞ consensus control problem for uncertain multiagent systems, which is subject to parameter uncertainties, external disturbances, and nonidentical time-varying state and input delays. Firstly, a distributed control protocol based on state feedback of neighbors is designed and the closed-loop dynamics is built. By applying matrix theory tools, multiagent system is decoupled and H∞ consensus control problem for multiagent systems is transformed into H∞ control problem for the decoupling subsystems. Then employing Lyapunov-Krasovskii functional, Jensen integral inequality, and Newton-Leibnitz formula with free-weighting matrices, a lower conservative bounded real lemma (BRL) is derived in terms of LMIs such that a class of time-delay system is guaranteed to be globally asymptotically stable with the desired H∞ performance index. Extending BRL, a sufficient delay-dependent condition in terms of the matrix inequalities, whose complexity is lower because the system is decoupled, is obtained to make all agents asymptotically reach consensus with the desired H∞ performance index. Furthermore, an algorithm is elaborately designed to get feasible solution to this condition by the cone-complementary method. Extending this algorithm, an optimization algorithm for control protocol parameter is proposed to improve the disturbance attenuation capacity or allowable delay bounds. At last, simulation results are provided to illustrate the correctness of the theoretical results and the effectiveness of the algorithms.The following notations will be used throughout this paper.0n denotes the corresponding n×1 column vectors whose elements are all zeros. I and 0 denote identity matrix and zero value or zero matrix with appropriate dimensions, respectively. In and 0n(0n×m) denote the n×n identity matrix and the n×n(n×m) zero matrix, respectively. Rn and Rn×m denote the n-dimensional and the (n×m)-dimensional Euclidean spaces, respectively. · refers to the standard Euclidean norm for vectors. The notations × and ⊗ denote the vector product and the Kronecker product, respectively. The superscript T and -1 stand for matrix transposition and matrix inverse, respectively. In symmetric block matrices, ∗ is used as an ellipsis for terms induced by symmetry. The notation diag{M1,…,Mn} denotes a block diagonal matrix whose diagonal blocks are given by M1,…,Mn.


## 2. Preliminaries

In this section, some preliminary knowledge of graph theory is introduced for the following analysis (referring to [18]). Let G(V,E,A) be an undirected graph of order n, where V={s1,…,sn} is the set of nodes, E⊆V×V is the set of edges, and A=[aij]∈Rn×n is a weighted adjacency matrix. The node indexes belong to a finite index set I={1,2,…,n}. An edge of G is denoted by eij=(si,sj), where the first element si of eij is said to be the tail of the edge and the other sj to be the head. The adjacency elements are defined as aii=0 and aij=aji⩾0. aij>0 if and only if there is an edge between node vi and node vj. The Laplacian of the undirected graph is defined as L=Δ-A∈Rn×n, where Δ=[Δij] is a diagonal matrix with Δii=∑j=1naij. The set of neighbors of node si is denoted by Ni={sj∈V:(si,sj)∈E}. If there is a path from every node to every other node, the graph is said to be connected.


## 3. Problem Statement

Consider a multiagent system consisting ofn agents. Each agent is regarded as a node in a graph G. Each edge (si,sk)∈E corresponds to an available information channel between agents si and sk. Suppose that the ith agent si(i∈I,I≜{1,2,…,n}) has the dynamics as follows: (1)x˙it=A+ΔAtxit+Ad+ΔAdtxit-d1t+B1uit-d2t+B2wit,(2)0<d1t⩽τ1,d˙1t⩽μ1,(3)0<d2t⩽τ2,d˙2t⩽μ2,where xi(t)=[xi1,…,xim]T∈Rm denotes the state of the ith agent si, ui(t)=[ui1,…,uim1]T∈Rm1 denotes the control input (or control protocol) of the ith agent si, and A, Ad, B1, and B2 are constant matrices with compatible dimensions. d1(t) denotes the state delay of the system itself, which can be considered as the summation of computation time and execution time. d2(t) denotes the input delay, which is caused by communication and measuring. wi(t)∈Rm2 denotes the external disturbances belonging to L2[0,∞). ΔA(t) and ΔAd(t) are matrix valued function representing time-varying parameter uncertainties. The parameter uncertainties are assumed to be norm bounded and ΔA(t)=GF(t)E and ΔAd(t)=GdF(t)Ed, where G, Gd, E, and Ed are known constant matrices with appropriate dimensions, which represent the structure of uncertainties, and F(t) is an unknown matrix function with Lebesgue measurable elements satisfying FT(t)F(t)⩽I for all t⩾0.Remark 1.
The system model (1) is a more generalized form. On one hand, it can cover the first-order dynamics in [10], second-order dynamics in [4, 11], and high-order dynamics in [19]. On the other hand, compared with [12, 13, 15], it includes the parameter uncertainties and nonidentical time-varying delay, and compared with [8, 9, 16], it includes parameter uncertainties and external disturbance.Definition 2.
The multiagent system (1) reaches consensus, if and only if the states of agents satisfy(4)limt→+∞xit-xjt=0∀i,j∈I.Define a vector functionz(t)=z1Tt,…,znTtT∈Rmn, where ziT(t)=zi1,…,zimT∈Rm measures the disagreement of the state of the ith agent to the average state of all agents for i∈I and its value is as follows: (5)zit=xit-1n∑j=1nxjt.Note that ifz(t)=0 can be satisfied, then xi(t)=xj(t) holds ∀i,j∈I. That is to say, the multiagent system (1) reaches consensus. This implies that z(t) can quantitatively reflect the disagreement degree of all agents on their states and the norm of z(t) indicates consensus performance. Therefore, z(t) is defined as the controlled output of the multiagent system (1) for analysing its consensus behaviour.Let(6)xt=x1Tt,…,xnTtT∈Rmn,ut=u1Tt,…,unTtT∈Rm1n,wt=w1Tt,…,wnTtT∈Rm2n.Then combining the dynamic equation (1) with controlled output defined in (5) yields the following system in matrix form:(7)x˙t=In⊗A+ΔAtxt+In⊗Ad+ΔAdtxt-d1t+In⊗B1ut-d2t+In⊗B2wt,zt=C⊗Imxt,where (8)C=Ciji,j=1n,Cij=n-1n,i=j,-1n,i≠j.According to robust control theory, the attenuating ability of consensus performance for the multiagent system (1) against external disturbances can be quantitatively measured by the H∞ norm of the closed-loop transfer function matrix Twz(s) from the external disturbance w(t) to the controlled output z(t). In order to obtain prescribed performance against external disturbances, we need to design a distributed state feedback protocol ui(t) such that(9)Twzs∞<γholds for a prescribed H∞ disturbance attenuation index γ>0, or equivalently, the closed-loop system satisfies the following dissipation inequality:(10)∫0∞zTtzt-γ2wTtwtdt<0,∀wt∈L20,∞.


## 4. Main Results

In this section, we will solveH∞ consensus control problem for multiagent systems (7). Firstly, we will design a distributed control protocol with an undetermined feedback matrix K and build the closed-loop dynamics of the system. By applying matrix theory tools, multiagent system is decoupled and H∞ consensus control problem for multiagent systems is transformed into H∞ control problem for the decoupling subsystems. Then employing Lyapunov-Krasovskii functional and free-weighting matrices, a BRL will be derived in terms of LMIs such that a class of time-delay system is guaranteed to be globally asymptotically stable with the desired H∞ performance index. Extending BRL, a sufficient delay-dependent condition in terms of the matrix inequalities is obtained to make all agents asymptotically reach consensus with the desired H∞ performance index. Furthermore, an algorithm is elaborately designed to get feasible solution to this condition. Extending this algorithm, an optimization algorithm for control protocol parameter is proposed to improve the disturbance attenuation capacity or allowable delay bounds.


### 4.1. Protocol Design

To solveH∞ consensus control problem for the multiagent systems (1), the distributed control protocol is given as(11)uit=K∑sj∈Niaijxjt-xitfor i∈I, where K∈Rm1×m is an undetermined feedback matrix and aij are the adjacency elements of interaction graph G.By substituting the protocol (11) into system (7), the closed-loop dynamics of system (1) can be written as(12)x˙t=In⊗A+ΔAtxt+In⊗Ad+ΔAdtxt-d1t-L⊗B1Kxt-d2t+In⊗B2wtzt=C⊗Imxt, where L is the Laplacian matrix of the graph G.On the basis of the above analysis, it is clear that the multiagent system (1) reaches consensus while satisfying the desired H∞ disturbance attenuation index γ by control protocol (11), if and only if the closed-loop system (12) reaches consensus with the desired H∞ disturbance attenuation index γ. In the next section, we will find design rules for K such that system (1) reaches consensus with the H∞ disturbance attenuation index γ.


### 4.2. Some Necessary Lemma

Lemma 3 (see [20]).
LetL be the Laplacian of an undirected graph G. Then L has at least one zero eigenvalue and all of the nonzero eigenvalues are positive. Furthermore, matrix L has exactly one zero eigenvalue if and only if the graph G is connected, and the eigenvector associated with zero is 1.Lemma 4 (see [10]).
Consider the matrixC. The following statements hold.(a)
The eigenvalues ofC are 1 with multiplicity n-1 and 0 with multiplicity 1. The vectors 1nT and 1n are the left and right eigenvectors of C associated with the zero eigenvalue, respectively.(b)
There exists an orthogonal matrixU∈Rn×n such that UTCU=In-1000 and the column is 1n/n. Let Ξ1∈Rn×n be the Laplacian of any directed graph; then UTΞ1U=υ10,υ1∈Rn×(n-1).
For convenience, denoteU=U1U¯1, where U¯1=1n/n is the last column of U and U1∈Rn×(n-1) is the remaining part.Lemma 5 (Schur complement formula [21]).
For a given symmetric matrixS with the form S=S11S12∗S22, S11∈Rr×r, S12∈Rr×(n-r), and S22∈R(n-r)×(n-r); then S<0 if and only if S11<0 and S22-S12TS11-1S12<0 or S22<0 and S11-S12S22-1S12T<0.


### 4.3. System Decoupling

Theorem 6 (system decoupling).
Assume that the interaction graphG is connected. For a given γ>0, the closed-loop system (12) can reach consensus with the desired H∞ disturbance attenuation index γ, if the following n-1 subsystems are simultaneously asymptotically stable with Tw^iz^i(s)∞<γ for i=1,2,…,n-1.(13)δ^˙it=A+ΔAtδ^it+Ad+ΔAdtδ^it-d1t-λiB1Kδ^it-d2t+B2w^it,z^it=δ^it,where λi(i=1,2,…,n-1) are the positive eigenvalues of the Laplacian matrix L and δ^i(t), w^i(t), z^i(t)∈Rm.Proof.
See Appendix.


### 4.4. Delay-DependentH∞ Control

In order to investigate theH∞ stability, let us consider a class of time-delay system as follows:(14)x˙t=Axt+Adxt-dt+Ahxt-ht+Bwwt,zt=Cxt,0<dt⩽τ1,d˙t⩽μ1,0<ht⩽τ2,h˙t⩽μ2.A bounded real lemma (BRL) will be derived in the following part.Lemma 7 (BRL).
For a givenγ>0, the time-delay system (14) is asymptotically stable with Twz(s)∞<γ, if there exist positive definite matrices P, Q1, Q2, R1, R2, S1, and S2 and matrices M1, M2, N1, N2, U1, U2, V1, and V2 with appropriate dimensions such that(15)Γ=Γ0Γ1Γ2Γ3Γ4Γ5∗-S0000∗∗-τ1S1000∗∗∗-τ1S100∗∗∗∗-τ2S20∗∗∗∗∗-τ2S2<0,where S=τ1S1+τ2S2, (16)Γ0=Γ11Γ12Γ13-M1-U1PBw∗Γ220-M200∗∗Γ330-U20∗∗∗-R100∗∗∗∗-R20∗∗∗∗∗-γ2I,Γ1=SA,SAd,SAh,0,0,SBwT,Γ2=τ1M1T,τ1M2T,0,0,0,0T,Γ3=τ1N1T,τ1N2T,0,0,0,0T,Γ4=τ2U1T,0,τ2U2T,0,0,0T,Γ5=τ2V1T,0,τ2V2T,0,0,0T,Γ11=PA+ATP+Q1+Q2+R1+R2+N1+N1T+V1+V1T+CTC,Γ12=PAd+M1-N1+N2T,Γ13=PAh+U1-V1+V2T,Γ22=μ1-1Q1+M2+M2T-N2-N2T,Γ33=μ2-1Q2+U2+U2T-V2-V2T.Proof.
See  Appendix.Theorem 8.
Assume that the interaction graphG is connected. For given positive scalar constants γ, by distributed protocol (11), the multiagent system (1) can reach consensus with the desired H∞ disturbance attenuation index γ, if there exist positive definite matrices Q1,Q2,R1,R2,S1,S2,T,X∈Rm×m, matrices M1,M2,N1,N2,U1,U2,V1,V2∈Rm×m,  Y∈Rm1×m, and positive scalars ε1, ε2 such that (17)Φi=Φ0iΠ1iTΠ2THTHdT∗-T+ε1GGT+ε2GdGdT000∗∗-XT-1X00∗∗∗-ε1I0∗∗∗∗-ε2I<0,where (18)Φ0i=Φ0xi0Φ1Φ2Φ3Φ4Φ5∗-S00000∗∗-τ1S10000∗∗∗-τ1S1000∗∗∗∗-τ2S200∗∗∗∗∗-τ2S20∗∗∗∗∗∗-I,Φ0xi=Φ11Φ12Φ13i-M1-U1B2∗Φ220-M200∗∗Φ330-U20∗∗∗-R100∗∗∗∗-R20∗∗∗∗∗-γ2I,S=τ1S1+τ2S2,Φ11=AX+XAT+Q1+Q2+R1+R2+N1+N1T+V1+V1T+ε1GGT+ε2GdGdT,Φ12=AdX+M1-N1+N2T,Φ13i=-λiB1Y+U1-V1+V2T,Φ22=μ1-1Q1+M2+M2T-N2-N2T,Φ33=μ2-1Q2+U2+U2T-V2-V2T,Φ1=τ1M1T,τ1M2T,0,0,0,0T,Φ2=τ1N1T,τ1N2T,0,0,0,0T,Φ3=τ2U1T,0,τ2U2T,0,0,0T,Φ4=τ2V1T,0,τ2V2T,0,0,0T,Φ5=X,0,0,0,0,0T,Π1i=AX+ε1GGT+ε2GdGdT,AdX,-λiB1Y,0,0,B2,0,0,0,0,0,0,Π2=0,0,0,0,0,0,τ1S1+τ2S2,0,0,0,0,0,H=EX,0,0,0,0,0,0,0,0,0,0,0,Hd=0,EdX,0,0,0,0,0,0,0,0,0,0.Furthermore, the feedback matrix K in the proposed control protocol (11) can be designed by K=YX-1.Proof.
See Appendix.


### 4.5. Algorithm of Solving Delay-Dependent Condition

In the following section, an algorithm will be designed to get the feasible solution to the matrix inequality condition (17).Due to the existence of nonlinear entryXT-1X, the matrix inequality condition (17) is not yet in the form of an LMI. Therefore, we cannot directly use LMI method to solve the matrix inequality (17). But we can turn this problem into the LMI optimization problem by the cone-complementary linearization algorithm [22].Define a new positive definite matrix variableW such that W⩽XT-1X. It is easy to derive that W-1-X-1TX-1⩾0. Furthermore, by defining X¯≜X-1,  T¯≜T-1, and  W¯≜W-1 and using Lemma 5 (Schur complement formula), we can turn condition (17) into (19)Φ0iΠ1iTΠ2THTHdT∗-T+ε1GGT+ε2GdGdT000∗∗-W00∗∗∗-ε1I0∗∗∗∗-ε2I<0,(20)W¯X¯X¯T¯⩾0,X¯IIX⩾0,T¯IIT⩾0,W¯IIW⩾0.Then, we may solve the following optimization problem and find a feasible solution satisfying W⩽XT-1X:(21)minTraceX¯X+T¯T+W¯Ws.t.19,20.In a word, in order to solve the matrix inequality condition (17), an algorithm is designed as follows.Algorithm 9.
(1)  Solve the LMI (19) and (20) for given positive scalar constants τ1, τ2, μ1, μ2, and γ. There exists a feasible solution set {X¯0,X0,T¯0,T0,W¯0,W0} and set k=0.
(2)  Solve the following optimization problem for the variables(22)X¯,X,T¯,T,W¯,WminTraceX¯kX+T¯kT+W¯kW+X¯Xk+T¯Tk+W¯Wks.t.19,20and set X¯k+1=X¯,  Xk+1=X,  T¯k+1=T¯,  Tk+1=T,  W¯k+1=W¯, and  Wk+1=W.
(3)  IfW⩽XT-1X for the above solution set, then save the current X, Y and exit. Otherwise, set k=k+1, go to Step (2), and repeat the optimization for a prescribed maximum iterative number kmax until finding a feasible solution satisfying W⩽XT-1X. If such a solution does not exist, then exit.If a feasible solution set is found by Algorithm9, given the disturbance attenuation index γ and delay parameters τ1, τ2, μ1, and μ2, by distributed protocol (11), the multiagent system (1) can reach consensus with the desired H∞ disturbance attenuation index γ and the feedback matrix can be constructed by K=YX-1.


### 4.6. Optimization forH∞ Control Parameters

According to Theorem8, assume that delay parameters τ1, τ2, μ1, and μ2 have been given; in order to achieve a minimum H∞ disturbance attenuation index γ, we can solve the following optimization problem for the positive definite decision variables Q1,Q2,R1,R2,S1,S2,T,X∈Rm×m, matrices M1,M2,N1,N2,U1,U2,V1,V2∈Rm×m,  Y∈Rm1×m, and positive scalars ε1,ε2:(23)minγs.t.17.Because the matrix inequality condition (17) is not yet in the form of an LMI, we cannot find global minima for the optimization problem (23) using convex optimization algorithms. However, by the similar methods to Algorithm 9, we may still obtain a suboptimal controller for the optimization problem (23) using an iterative algorithm presented in the following sequel.Algorithm 10.
(1) Assume that the delay parametersτ1, τ2, μ1, and μ2 have been given. Choose a sufficiently large initial γ such that there exists a feasible solution set {X¯0,X0,T¯0,T0,W¯0,W0} to the LMIs (19) and (20). Set k=0.
(2) Solve the optimization problem (22) and set X¯k+1=X¯,  Xk+1=X,  T¯k+1=T¯,  Tk+1=T,  W¯k+1=W¯, and Wk+1=W.
(3)  IfW⩽XT-1X for the above solution set, then set γmin=γ and return to Step (1) after decreasing γ=γ-Δγ, where Δγ is predefined step-sizes. Otherwise, set k=k+1, go to Step (2), and repeat the optimization for a prescribed maximum iterative number kmax until finding a feasible solution satisfying W⩽XT-1X. If such a solution does not exist, then exit.
If a feasible solution set is found by Algorithm10, for the given delay parameters τ1, τ2, μ1, and μ2, by distributed protocol (11), the multiagent system (1) can reach consensus with the suboptimal H∞ disturbance attenuation index γmin and the suboptimal feedback matrix can be constructed by K=YX-1.Remark 11.
Assuming that the disturbance attenuation indexγ and delay parameters μ1 (or the proportion coefficient of μ1 and τ1), τ2, and μ2 have been given, we can raise the maximum allowable delay bound to τ1max and the corresponding feedback matrix K=YX-1 by the similar methods to Algorithm 10. Similarly, we also can raise the maximum allowable delay bound to τ2max and the corresponding feedback matrix K=YX-1.


## 4.1. Protocol Design

To solveH∞ consensus control problem for the multiagent systems (1), the distributed control protocol is given as(11)uit=K∑sj∈Niaijxjt-xitfor i∈I, where K∈Rm1×m is an undetermined feedback matrix and aij are the adjacency elements of interaction graph G.By substituting the protocol (11) into system (7), the closed-loop dynamics of system (1) can be written as(12)x˙t=In⊗A+ΔAtxt+In⊗Ad+ΔAdtxt-d1t-L⊗B1Kxt-d2t+In⊗B2wtzt=C⊗Imxt, where L is the Laplacian matrix of the graph G.On the basis of the above analysis, it is clear that the multiagent system (1) reaches consensus while satisfying the desired H∞ disturbance attenuation index γ by control protocol (11), if and only if the closed-loop system (12) reaches consensus with the desired H∞ disturbance attenuation index γ. In the next section, we will find design rules for K such that system (1) reaches consensus with the H∞ disturbance attenuation index γ.


## 4.2. Some Necessary Lemma

Lemma 3 (see [20]).
LetL be the Laplacian of an undirected graph G. Then L has at least one zero eigenvalue and all of the nonzero eigenvalues are positive. Furthermore, matrix L has exactly one zero eigenvalue if and only if the graph G is connected, and the eigenvector associated with zero is 1.Lemma 4 (see [10]).
Consider the matrixC. The following statements hold.(a)
The eigenvalues ofC are 1 with multiplicity n-1 and 0 with multiplicity 1. The vectors 1nT and 1n are the left and right eigenvectors of C associated with the zero eigenvalue, respectively.(b)
There exists an orthogonal matrixU∈Rn×n such that UTCU=In-1000 and the column is 1n/n. Let Ξ1∈Rn×n be the Laplacian of any directed graph; then UTΞ1U=υ10,υ1∈Rn×(n-1).
For convenience, denoteU=U1U¯1, where U¯1=1n/n is the last column of U and U1∈Rn×(n-1) is the remaining part.Lemma 5 (Schur complement formula [21]).
For a given symmetric matrixS with the form S=S11S12∗S22, S11∈Rr×r, S12∈Rr×(n-r), and S22∈R(n-r)×(n-r); then S<0 if and only if S11<0 and S22-S12TS11-1S12<0 or S22<0 and S11-S12S22-1S12T<0.


## 4.3. System Decoupling

Theorem 6 (system decoupling).
Assume that the interaction graphG is connected. For a given γ>0, the closed-loop system (12) can reach consensus with the desired H∞ disturbance attenuation index γ, if the following n-1 subsystems are simultaneously asymptotically stable with Tw^iz^i(s)∞<γ for i=1,2,…,n-1.(13)δ^˙it=A+ΔAtδ^it+Ad+ΔAdtδ^it-d1t-λiB1Kδ^it-d2t+B2w^it,z^it=δ^it,where λi(i=1,2,…,n-1) are the positive eigenvalues of the Laplacian matrix L and δ^i(t), w^i(t), z^i(t)∈Rm.Proof.
See Appendix.


## 4.4. Delay-DependentH∞ Control

In order to investigate theH∞ stability, let us consider a class of time-delay system as follows:(14)x˙t=Axt+Adxt-dt+Ahxt-ht+Bwwt,zt=Cxt,0<dt⩽τ1,d˙t⩽μ1,0<ht⩽τ2,h˙t⩽μ2.A bounded real lemma (BRL) will be derived in the following part.Lemma 7 (BRL).
For a givenγ>0, the time-delay system (14) is asymptotically stable with Twz(s)∞<γ, if there exist positive definite matrices P, Q1, Q2, R1, R2, S1, and S2 and matrices M1, M2, N1, N2, U1, U2, V1, and V2 with appropriate dimensions such that(15)Γ=Γ0Γ1Γ2Γ3Γ4Γ5∗-S0000∗∗-τ1S1000∗∗∗-τ1S100∗∗∗∗-τ2S20∗∗∗∗∗-τ2S2<0,where S=τ1S1+τ2S2, (16)Γ0=Γ11Γ12Γ13-M1-U1PBw∗Γ220-M200∗∗Γ330-U20∗∗∗-R100∗∗∗∗-R20∗∗∗∗∗-γ2I,Γ1=SA,SAd,SAh,0,0,SBwT,Γ2=τ1M1T,τ1M2T,0,0,0,0T,Γ3=τ1N1T,τ1N2T,0,0,0,0T,Γ4=τ2U1T,0,τ2U2T,0,0,0T,Γ5=τ2V1T,0,τ2V2T,0,0,0T,Γ11=PA+ATP+Q1+Q2+R1+R2+N1+N1T+V1+V1T+CTC,Γ12=PAd+M1-N1+N2T,Γ13=PAh+U1-V1+V2T,Γ22=μ1-1Q1+M2+M2T-N2-N2T,Γ33=μ2-1Q2+U2+U2T-V2-V2T.Proof.
See  Appendix.Theorem 8.
Assume that the interaction graphG is connected. For given positive scalar constants γ, by distributed protocol (11), the multiagent system (1) can reach consensus with the desired H∞ disturbance attenuation index γ, if there exist positive definite matrices Q1,Q2,R1,R2,S1,S2,T,X∈Rm×m, matrices M1,M2,N1,N2,U1,U2,V1,V2∈Rm×m,  Y∈Rm1×m, and positive scalars ε1, ε2 such that (17)Φi=Φ0iΠ1iTΠ2THTHdT∗-T+ε1GGT+ε2GdGdT000∗∗-XT-1X00∗∗∗-ε1I0∗∗∗∗-ε2I<0,where (18)Φ0i=Φ0xi0Φ1Φ2Φ3Φ4Φ5∗-S00000∗∗-τ1S10000∗∗∗-τ1S1000∗∗∗∗-τ2S200∗∗∗∗∗-τ2S20∗∗∗∗∗∗-I,Φ0xi=Φ11Φ12Φ13i-M1-U1B2∗Φ220-M200∗∗Φ330-U20∗∗∗-R100∗∗∗∗-R20∗∗∗∗∗-γ2I,S=τ1S1+τ2S2,Φ11=AX+XAT+Q1+Q2+R1+R2+N1+N1T+V1+V1T+ε1GGT+ε2GdGdT,Φ12=AdX+M1-N1+N2T,Φ13i=-λiB1Y+U1-V1+V2T,Φ22=μ1-1Q1+M2+M2T-N2-N2T,Φ33=μ2-1Q2+U2+U2T-V2-V2T,Φ1=τ1M1T,τ1M2T,0,0,0,0T,Φ2=τ1N1T,τ1N2T,0,0,0,0T,Φ3=τ2U1T,0,τ2U2T,0,0,0T,Φ4=τ2V1T,0,τ2V2T,0,0,0T,Φ5=X,0,0,0,0,0T,Π1i=AX+ε1GGT+ε2GdGdT,AdX,-λiB1Y,0,0,B2,0,0,0,0,0,0,Π2=0,0,0,0,0,0,τ1S1+τ2S2,0,0,0,0,0,H=EX,0,0,0,0,0,0,0,0,0,0,0,Hd=0,EdX,0,0,0,0,0,0,0,0,0,0.Furthermore, the feedback matrix K in the proposed control protocol (11) can be designed by K=YX-1.Proof.
See Appendix.


## 4.5. Algorithm of Solving Delay-Dependent Condition

In the following section, an algorithm will be designed to get the feasible solution to the matrix inequality condition (17).Due to the existence of nonlinear entryXT-1X, the matrix inequality condition (17) is not yet in the form of an LMI. Therefore, we cannot directly use LMI method to solve the matrix inequality (17). But we can turn this problem into the LMI optimization problem by the cone-complementary linearization algorithm [22].Define a new positive definite matrix variableW such that W⩽XT-1X. It is easy to derive that W-1-X-1TX-1⩾0. Furthermore, by defining X¯≜X-1,  T¯≜T-1, and  W¯≜W-1 and using Lemma 5 (Schur complement formula), we can turn condition (17) into (19)Φ0iΠ1iTΠ2THTHdT∗-T+ε1GGT+ε2GdGdT000∗∗-W00∗∗∗-ε1I0∗∗∗∗-ε2I<0,(20)W¯X¯X¯T¯⩾0,X¯IIX⩾0,T¯IIT⩾0,W¯IIW⩾0.Then, we may solve the following optimization problem and find a feasible solution satisfying W⩽XT-1X:(21)minTraceX¯X+T¯T+W¯Ws.t.19,20.In a word, in order to solve the matrix inequality condition (17), an algorithm is designed as follows.Algorithm 9.
(1)  Solve the LMI (19) and (20) for given positive scalar constants τ1, τ2, μ1, μ2, and γ. There exists a feasible solution set {X¯0,X0,T¯0,T0,W¯0,W0} and set k=0.
(2)  Solve the following optimization problem for the variables(22)X¯,X,T¯,T,W¯,WminTraceX¯kX+T¯kT+W¯kW+X¯Xk+T¯Tk+W¯Wks.t.19,20and set X¯k+1=X¯,  Xk+1=X,  T¯k+1=T¯,  Tk+1=T,  W¯k+1=W¯, and  Wk+1=W.
(3)  IfW⩽XT-1X for the above solution set, then save the current X, Y and exit. Otherwise, set k=k+1, go to Step (2), and repeat the optimization for a prescribed maximum iterative number kmax until finding a feasible solution satisfying W⩽XT-1X. If such a solution does not exist, then exit.If a feasible solution set is found by Algorithm9, given the disturbance attenuation index γ and delay parameters τ1, τ2, μ1, and μ2, by distributed protocol (11), the multiagent system (1) can reach consensus with the desired H∞ disturbance attenuation index γ and the feedback matrix can be constructed by K=YX-1.


## 4.6. Optimization forH∞ Control Parameters

According to Theorem8, assume that delay parameters τ1, τ2, μ1, and μ2 have been given; in order to achieve a minimum H∞ disturbance attenuation index γ, we can solve the following optimization problem for the positive definite decision variables Q1,Q2,R1,R2,S1,S2,T,X∈Rm×m, matrices M1,M2,N1,N2,U1,U2,V1,V2∈Rm×m,  Y∈Rm1×m, and positive scalars ε1,ε2:(23)minγs.t.17.Because the matrix inequality condition (17) is not yet in the form of an LMI, we cannot find global minima for the optimization problem (23) using convex optimization algorithms. However, by the similar methods to Algorithm 9, we may still obtain a suboptimal controller for the optimization problem (23) using an iterative algorithm presented in the following sequel.Algorithm 10.
(1) Assume that the delay parametersτ1, τ2, μ1, and μ2 have been given. Choose a sufficiently large initial γ such that there exists a feasible solution set {X¯0,X0,T¯0,T0,W¯0,W0} to the LMIs (19) and (20). Set k=0.
(2) Solve the optimization problem (22) and set X¯k+1=X¯,  Xk+1=X,  T¯k+1=T¯,  Tk+1=T,  W¯k+1=W¯, and Wk+1=W.
(3)  IfW⩽XT-1X for the above solution set, then set γmin=γ and return to Step (1) after decreasing γ=γ-Δγ, where Δγ is predefined step-sizes. Otherwise, set k=k+1, go to Step (2), and repeat the optimization for a prescribed maximum iterative number kmax until finding a feasible solution satisfying W⩽XT-1X. If such a solution does not exist, then exit.
If a feasible solution set is found by Algorithm10, for the given delay parameters τ1, τ2, μ1, and μ2, by distributed protocol (11), the multiagent system (1) can reach consensus with the suboptimal H∞ disturbance attenuation index γmin and the suboptimal feedback matrix can be constructed by K=YX-1.Remark 11.
Assuming that the disturbance attenuation indexγ and delay parameters μ1 (or the proportion coefficient of μ1 and τ1), τ2, and μ2 have been given, we can raise the maximum allowable delay bound to τ1max and the corresponding feedback matrix K=YX-1 by the similar methods to Algorithm 10. Similarly, we also can raise the maximum allowable delay bound to τ2max and the corresponding feedback matrix K=YX-1.


## 5. Simulation Results

To illustrate the correctness of the theoretical results and the effectiveness of the algorithms, numerical simulations will be given in this section. Consider a multiagent system with four agents and the dynamics of each agent is described by (1), with (24)Ad=00.200000.500000.5000.50,B1=00101101,B2=000.500.20.500.2,A=01000020000200-20,G=Gd=00.0100000.0200000.03000.010,Ft=diagsin⁡10t,sin⁡20t,cos⁡10t,cos⁡20t,E=Ed=I4. The connected interaction graph G is shown in Figure 1, whose nonzero weighting factors aij are all 1. So the corresponding Laplacian matrix is (25)L=2-10-1-12-100-12-1-10-12. So its smallest and largest nonzero eigenvalues are λ1=2 and λ3=4, respectively. We assume that the external disturbance (26)wt=1,1.2,0.8,1.5,1,1.4,2,1.2Tεt,εt=1,0⩽t⩽10,otherwise and the initial state of the multiagent system is taken as x(t=0)=016.Figure 1
The communication topology.Example 12.
Suppose that theH∞ performance index γ=1, the delay d1(t)=0.1sin⁡t, and d2(t)=0.15cos⁡t (so τ1=μ1=0.1 and τ2=μ2=0.15). By Theorem 8 and Algorithm 9, we can figure out the feedback matrix K in the control protocol (11) as follows: (27)K=YX-1=0.51031.07010.8234-0.1057-0.3108-0.6331-0.31410.8557.On the one hand, Figure2 shows the trajectories of zi(t)=[zi1,…,zi4]T when γ=1, d1(t)=0.1sin⁡t, and d2(t)=0.15cos⁡t for all i=1,2,3,4, where zik=xik(t)-1/n∑j=1nxjk(t),  k=1,2,3,4. It is clear that consensus is asymptotically achieved for the multiagent systems. On the other hand, Figure 3 shows the energy relation of the controlled output and the external disturbance. Obviously, the closed-loop system satisfies ∫0∞[zT(t)z(t)-γ2wT(t)w(t)]dt<0; that is, the H∞ disturbance attenuation index is achieved. Therefore, applying the distributed protocol (11) and calculating the feedback matrix K by Theorem controller synthesis theorem and Algorithm 9, the multiagent system can reach consensus while satisfying the desired H∞ disturbance attenuation index γ=1. So we validate the correctness of Theorem 8 and the effectiveness of Algorithm 9.Figure 2
The trajectories ofzi(t)=[zi1,…,zi4]T in Example 12.Figure 3
The energy trajectories of the controlled outputz(t) and the external disturbance w(t) in four examples.Example 13.
Suppose that the delayd1(t)=0.1sin⁡t and d2t=0.15cos⁡t (so τ1=μ1=0.1 and τ2=μ2=0.15). Using the distributed protocol (11) and optimizing the feedback matrix K by Algorithm 10, we can figure out the suboptimal H∞ disturbance attenuation index γmin=0.7 and the corresponding feedback matrix as follows: (28)K=YX-1=0.66771.24830.7142-0.1105-0.3815-0.6790-0.18720.8701.Figure3 shows the energy relation of the controlled output and the external disturbance. Obviously, the H∞ disturbance attenuation index is smaller than its value in Example 12. That is to say, by Algorithm 10, the disturbance attenuation capacity of the multiagent system (1) is obviously improved. So the effectiveness of Algorithm 10 is validated.


## 6. Conclusion

This paper studies robustH∞ consensus control problem for multiagent systems with parameter uncertainties, external disturbances, nonidentical time-varying state, and input delays. The main contributions of this paper are as follows: first, considering a more generalized form, which includes parameter uncertainties, external disturbances, nonidentical time-varying state, and input delays; second, a lower conservative BRL; third, a lower complexity delay-dependent control condition; fourth, an algorithm for calculating parameters of control protocol; fifth, an optimization algorithm for improving the disturbance attenuation capacity or allowable delay bounds. In further research, we will research robust H∞ consensus control problem for multiagent systems with heterogeneous uncertainties and nonuniform time-varying delays.



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