# Distributed RobustH∞ Consensus Control of Multiagent Systems with Communication Errors Using Dynamic Output Feedback Protocol

Authors: Xi Yang; Jinzhi Wang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper studies robust consensus problem for multiagent systems modeled by an identical linear time-invariant system under a fixed communication topology. Communication errors in the transferred data are considered, and only the relative output information between each agent and its neighbors is available. A distributed dynamic output feedback protocol is proposed, and sufficient conditions for reaching consensus with a prescribedH∞ performance are presented. Numerical examples are given to illustrate the theoretical results.


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## Body

## 1. Introduction

Consensus problem of multiagent systems has been a popular subject in system and control theory due to its widespread applications such as satellite formation flying, cooperative unmanned air vehicles, and mobile robots [1–3]. The study of consensus problem focuses on designing a distributed protocol using information which can only be obtained and shared locally to ensure that the resulting closed-loop system has the desired characteristics. A number of solutions that are based on relative states between each agent and its neighbors to the consensus problem have been proposed up to now. The theoretical framework of solving consensus for multiagent system was suggested by [4], providing the convergence analysis of a consensus protocol for a network of single integrators with directed fixed/switching topologies. Later, under different cases of communication topologies such as fixed, switching, and with communication delays, many different types of protocols have been proposed for different types of agent dynamics to reach global asymptotical consensus [2, 3, 5–16].Recently, solving consensus problem for the multiagent systems by using output information has attracted particular attention due to its theoretical significance and wide applications. Reference [17] constructed a dynamic output feedback protocol based on a observer for the synchronization of a network of identical linear state space models under a possibly time-varying and directed interconnection, where each agent needs to obtain all the observer’s state information of its neighbors. Based on the low gain approach, [18] proposed a consensus protocol which only used the relative outputs for N identical linear dynamics with fixed directed communication topologies. Consensus problem with ℒ2 external disturbance under switching undirected communication topologies was studied by [19], where a dynamic output feedback protocol was proposed for subjecting the external disturbances. Reference [20] studied the output consensus problem for a class of heterogeneous uncertain linear SISO multiagent systems, where each agent’s output information and the relative outputs with its neighbors were used to design the controller. Reference [21] designed robust static output feedback controllers to achieve consensus for undirected networks of heterogeneous agents modeled as nonlinear systems of relative degree two.It can be seen that there is a common assumption in the literatures mentioned above that each agent can receive accurate measurements of relative states or outputs between its neighbors and itself all the time. However, in some practical situations, agents cannot perfectly sense their neighbors due to the existence of sensor failures or some other communication constraints. In view of this, we consider the consensus problems for the multiagent systems with communication errors. It is required to point out that the measurement for communication errors we considered is limited to some errors in the transferred data not including loss of communication. The robustness analysis of first-/second-order leader-follower consensus with communication errors is studied by [22, 23]. Some robustness issues for systems with external disturbances or model uncertainties are investigated by some other researchers [19, 24–28], which are different from the robust consensus problem stated in this paper.Motivated by the above-mentioned works, we study the consensus problem for linear multiagent systems to attenuate the communication errors by using dynamic output feedback controller. The agent dynamics considered here are general stabilizable and detectable linear systems, and a dynamic consensus protocol is proposed which uses only the relative output information between each agent and its neighbors. The main contributions of this paper can be summarized as two aspects. Firstly, in order to describe the effects of communication errors on consensus, a concept called consensus withH∞ performance is introduced which can characterize the effects of communication errors on the difference between the state of each agent and the average of states of all agents. The problem of consensus with H∞ performance is transmitted into an H∞ control problem of another reduced-order system. It is shown that consensus with H∞ performance can be achieved if there exists a common dynamic output feedback controller which can be realized by solving H∞ problem for N-1 linear dynamic systems simultaneously, where N is the number of agents. Secondly, in terms of the N-1 linear systems, a sufficient condition based on linear matrix inequalities for the existence of the controller is provided, and the approach to construct the corresponding controller is given.The rest of this paper is organized as follows. Section2 introduces basic notations and reviews some useful results on graph theory and robust H∞ control theory. Section 3 formulates the problem and conditions for reaching consensus with H∞ performances that are derived. The existence for a dynamic output feedback protocol and a method to construct such controller are proposed in Section 4. Numerical simulations are provided in Section 5. Section 6 concludes the paper.Notations. Let ℝn×n and ℂn×n be the set of n×n real matrices and n×n complex matrices, respectively. Matrices, if not explicitly stated, are assumed to have compatible dimensions. In and 0n×m are the n×n identity matrix and the n×m zero matrix, respectively. For a matrix A, ∥A∥2 is the induced 2-norm of the vector norm, and ∥A∥2=▵σ(A). The notation σ(A) is the maximal singular value of matrix A. Notations AT, A-1, and A* represent the transpose, the inverse, and the complex conjugate transpose of matrix A, respectively. Let Im(A) and ker(A) be the image space and kernel of A. A<0 (A>0) means that the matrix A is negative (positive) definite. Span{v1,v2,…vm} is a subspace of ℝn spanned by {v1,v2,…vm}, where vi∈ℝn,i=1,2,…,m. The notation ⊗ represents the Kronecker product. For a vector x∈ℂn,|x|=▵x*x is the Euclidean norm. 1N denotes the N×1 column vector whose elements are all ones. The space of piecewise continuous functions in ℝm that are square integrable over [0,+∞) is denoted by ℒ2m[0,∞), for any v(t)∈ℒ2m[0,∞), and its normalized energy is defined by ∥v(t)∥ℒ2=(∫0∞‍|v(t)|2dt)1/2. Let [ABCD] be a state space realization of G(s)=C(sI-A)-1B+D.


## 2. Preliminaries

### 2.1. Graph Theory

Directed graphs are used to model the information interaction among agents. Let𝒢N=(𝒱N,ℰN,𝒜N) be a directed weighted graph, where 𝒱N={1,2,…,N} is the node set, ℰN⊆𝒱N×𝒱N is the edge set, and 𝒜N∈ℝN×N is a weighted adjacency matrix with nonnegative elements aij. An edge of 𝒢N is denoted by (i,j) which means that agent j can directly get information from agent i. (j,i)∈ℰN if and only if aij>0, otherwise aij=0. If (i,j)∈ℰN⇔(j,i)∈ℰN, then 𝒢N is said to be an undirected graph. In this paper, we assume that there are no self-cycles in 𝒢N; that is, aii=0, i=1,2,…,N. The in-degree and out-degree of the ith agent are, respectively, defined as din(i)=∑j=1N‍aij and dout(i)=∑j=1N‍aji. Let dmax=maxi{din(i)}. Correspondingly, the Laplacian matrix of graph 𝒢N is denoted by LN=ΔN-𝒜N∈ℝN×N, where ΔN=[Δij] is a diagonal matrix with Δii=din(i).A sequence(i1,i2),(i2,i3),…,(ik-1,ik) of edges is called a directed path from node i1 to node ik. 𝒢N is called a strongly connected digraph if for any i,j∈𝒱N, there is a directed path from i to j. 𝒢N has a directed spanning tree if there exists a node r∈𝒱N (a root) such that all other nodes can be linked to r via a directed path. A directed graph is called balanced if ∑j≠i‍aij=∑i≠j‍aji for all i∈𝒱.Below are well-known results for the Laplacian matrix.Lemma 1 (see [3]).
The Laplacian matrixLN of a directed graph has at least one zero eigenvalue with an associated eigenvector 1N.Lemma 2 (see [3]).
The Laplacian matrixLN of a directed graph has a simple zero eigenvalue with an associated eigenvector 1N, and all of the other eigenvalues have positive real parts if and only if the directed graph has a directed spanning tree.Lemma 3 (see [24]).
LetLN be the Laplacian matrix of a directed graph 𝒢N, then there exists an orthogonal matrix U-=[1N/NU]∈ℝN×N such that
(1)U-TLNU-=[01NTNLU0UTLU].
Furthermore, if 𝒢 is a balanced graph, then
(2)U-TLNU-=[000UTLNU].


### 2.2. RobustH∞ Control Theory

Consider that thenth-order linear time-invariant (LTI) system is described as follows:
(3)x˙(t)=Ax(t)+B1ω(t)+B2u(t),z(t)=C1x(t)+D11ω(t)+D12u(t),y(t)=C2x(t)+D21ω(t)+D22u(t),
where x(t)∈ℝn is the state, ω(t)∈ℒ2q[0,∞) is the external disturbance, u(t)∈ℝm is the control input, z(t)∈ℝp is the regulated output, y(t)∈ℝl is the measured output, and A, Bi, Ci, and Dij, for i,j=1,2, are known real constant matrices of appropriate dimensions. Without loss of generality, we assume that D22=0, (A,B2) is stabilizable and (C2,A) is detectable.Thencth-order dynamic output feedback (DOF) controller is described as follows:
(4)x˙c(t)=Akxc(t)+Bky(t),u(t)=Ckxc(t)+Dky(t),
where xc(t)∈ℝnc is the controller state and Ak, Bk, Ck, and Dk are constant matrices with appropriate dimensions.LetSpc(s)=[AcBcCcDc] be the transfer function from ω(t) to z(t) of the closed-loop system obtained from (3) and (4), where
(5)Ac=A0+B-KC-,Bc=B0+B-KD-21,Cc=C0+D-12KC-,Dc=D11+D-12KD-21,A0=[A000],B0=[B10],C0=[C10],B-=[0B2I0],C-=[0IC20],D-12=[0D12],D-21=[0D21],K=[AkBkCkDk].
The H∞ problem for the given LTI system (3) is to find a DOF controller (4) such that the closed-loop system is internally stable and ∥Spc(s)∥∞<γ for some constant γ>0.To facilitate the consensus protocol design and stability analysis, several results of theH∞ problem are recalled as follows.Lemma 4 (see [29]).
Givenγ>0, there exists a DOF controller (4) which can solve the H∞ problem for the LTI system (3) if and only if there exist symmetric matrices X>0 and Y>0 such that
(6)(a)[NX00I]*[XA+A*XXB1C1*B1*X-γID11*C1D11-γI][NX00I]<0,(b)[NY00I]*[YA*+AYYC1*B1C1Y-γID11B1*D11*-γI][NY00I]<0,(c)[XIIY]≥0,
where NX and NY are full-rank matrices whose images satisfy
(7)Im(NX)=ker([C2D21]),Im(NY)=ker([B2*D12*]).As the results shown by [29], the DOF controller (4) solving the H∞ problem for the LTI system (3) can be constructed as follows.(i)
FindX>0 and Y>0 which satisfy Lemma 4.(ii)
LetXc=[XX2X2*I], where X2∈ℝn×nc satisfying X-Y-1=X2X2*.(iii)
Solve the the following inequality:(8)HXc+Q*K*PXc+PXc*KQ<0.
For a feasible solution K, where
(9)PXc=[B-*Xc0D-12*],Q=[C-D-210],HXc=[XcA0+A0*XcXcB0C0*B0*Xc-γ0ID11*C0D11-γ0I],
and A0,B0,C0,B-,C-,D-12,D-21 are defined by (5). The solution K provides the state space realization for a feasible controller (4) which can solve H∞ problem for system (3).Lemma 5 (see [30]).
Letγ>0, G(s)=[ABCD] with A is Hurwitz stable, and
(10)H≜[A+BR-1D*CBR-1B*-C*(I+DR-1D*)C-(A+BR-1D*C*)*],
where R=γ2I-D*D. Then, the following conditions are equivalent:(1)
∥G(s)∥∞<γ,(2)
σ(D)<γ and H have no eigenvalues on the imaginary axis,(3)
there exists a matrixX=X*>0 such that
(11)[XA+A*XXBC*B*X-γID*CD-γI]<0.


## 2.1. Graph Theory

Directed graphs are used to model the information interaction among agents. Let𝒢N=(𝒱N,ℰN,𝒜N) be a directed weighted graph, where 𝒱N={1,2,…,N} is the node set, ℰN⊆𝒱N×𝒱N is the edge set, and 𝒜N∈ℝN×N is a weighted adjacency matrix with nonnegative elements aij. An edge of 𝒢N is denoted by (i,j) which means that agent j can directly get information from agent i. (j,i)∈ℰN if and only if aij>0, otherwise aij=0. If (i,j)∈ℰN⇔(j,i)∈ℰN, then 𝒢N is said to be an undirected graph. In this paper, we assume that there are no self-cycles in 𝒢N; that is, aii=0, i=1,2,…,N. The in-degree and out-degree of the ith agent are, respectively, defined as din(i)=∑j=1N‍aij and dout(i)=∑j=1N‍aji. Let dmax=maxi{din(i)}. Correspondingly, the Laplacian matrix of graph 𝒢N is denoted by LN=ΔN-𝒜N∈ℝN×N, where ΔN=[Δij] is a diagonal matrix with Δii=din(i).A sequence(i1,i2),(i2,i3),…,(ik-1,ik) of edges is called a directed path from node i1 to node ik. 𝒢N is called a strongly connected digraph if for any i,j∈𝒱N, there is a directed path from i to j. 𝒢N has a directed spanning tree if there exists a node r∈𝒱N (a root) such that all other nodes can be linked to r via a directed path. A directed graph is called balanced if ∑j≠i‍aij=∑i≠j‍aji for all i∈𝒱.Below are well-known results for the Laplacian matrix.Lemma 1 (see [3]).
The Laplacian matrixLN of a directed graph has at least one zero eigenvalue with an associated eigenvector 1N.Lemma 2 (see [3]).
The Laplacian matrixLN of a directed graph has a simple zero eigenvalue with an associated eigenvector 1N, and all of the other eigenvalues have positive real parts if and only if the directed graph has a directed spanning tree.Lemma 3 (see [24]).
LetLN be the Laplacian matrix of a directed graph 𝒢N, then there exists an orthogonal matrix U-=[1N/NU]∈ℝN×N such that
(1)U-TLNU-=[01NTNLU0UTLU].
Furthermore, if 𝒢 is a balanced graph, then
(2)U-TLNU-=[000UTLNU].


## 2.2. RobustH∞ Control Theory

Consider that thenth-order linear time-invariant (LTI) system is described as follows:
(3)x˙(t)=Ax(t)+B1ω(t)+B2u(t),z(t)=C1x(t)+D11ω(t)+D12u(t),y(t)=C2x(t)+D21ω(t)+D22u(t),
where x(t)∈ℝn is the state, ω(t)∈ℒ2q[0,∞) is the external disturbance, u(t)∈ℝm is the control input, z(t)∈ℝp is the regulated output, y(t)∈ℝl is the measured output, and A, Bi, Ci, and Dij, for i,j=1,2, are known real constant matrices of appropriate dimensions. Without loss of generality, we assume that D22=0, (A,B2) is stabilizable and (C2,A) is detectable.Thencth-order dynamic output feedback (DOF) controller is described as follows:
(4)x˙c(t)=Akxc(t)+Bky(t),u(t)=Ckxc(t)+Dky(t),
where xc(t)∈ℝnc is the controller state and Ak, Bk, Ck, and Dk are constant matrices with appropriate dimensions.LetSpc(s)=[AcBcCcDc] be the transfer function from ω(t) to z(t) of the closed-loop system obtained from (3) and (4), where
(5)Ac=A0+B-KC-,Bc=B0+B-KD-21,Cc=C0+D-12KC-,Dc=D11+D-12KD-21,A0=[A000],B0=[B10],C0=[C10],B-=[0B2I0],C-=[0IC20],D-12=[0D12],D-21=[0D21],K=[AkBkCkDk].
The H∞ problem for the given LTI system (3) is to find a DOF controller (4) such that the closed-loop system is internally stable and ∥Spc(s)∥∞<γ for some constant γ>0.To facilitate the consensus protocol design and stability analysis, several results of theH∞ problem are recalled as follows.Lemma 4 (see [29]).
Givenγ>0, there exists a DOF controller (4) which can solve the H∞ problem for the LTI system (3) if and only if there exist symmetric matrices X>0 and Y>0 such that
(6)(a)[NX00I]*[XA+A*XXB1C1*B1*X-γID11*C1D11-γI][NX00I]<0,(b)[NY00I]*[YA*+AYYC1*B1C1Y-γID11B1*D11*-γI][NY00I]<0,(c)[XIIY]≥0,
where NX and NY are full-rank matrices whose images satisfy
(7)Im(NX)=ker([C2D21]),Im(NY)=ker([B2*D12*]).As the results shown by [29], the DOF controller (4) solving the H∞ problem for the LTI system (3) can be constructed as follows.(i)
FindX>0 and Y>0 which satisfy Lemma 4.(ii)
LetXc=[XX2X2*I], where X2∈ℝn×nc satisfying X-Y-1=X2X2*.(iii)
Solve the the following inequality:(8)HXc+Q*K*PXc+PXc*KQ<0.
For a feasible solution K, where
(9)PXc=[B-*Xc0D-12*],Q=[C-D-210],HXc=[XcA0+A0*XcXcB0C0*B0*Xc-γ0ID11*C0D11-γ0I],
and A0,B0,C0,B-,C-,D-12,D-21 are defined by (5). The solution K provides the state space realization for a feasible controller (4) which can solve H∞ problem for system (3).Lemma 5 (see [30]).
Letγ>0, G(s)=[ABCD] with A is Hurwitz stable, and
(10)H≜[A+BR-1D*CBR-1B*-C*(I+DR-1D*)C-(A+BR-1D*C*)*],
where R=γ2I-D*D. Then, the following conditions are equivalent:(1)
∥G(s)∥∞<γ,(2)
σ(D)<γ and H have no eigenvalues on the imaginary axis,(3)
there exists a matrixX=X*>0 such that
(11)[XA+A*XXBC*B*X-γID*CD-γI]<0.


## 3.H∞ Consensus Problem

Consider a multiagent system consisting ofN identical agents with linear dynamics described by
(12)x˙i(t)=Axi(t)+Bui(t)yi(t)=Cxi(t)i=1,…,N,
where xi(t)∈ℝn is the state, ui(t)∈ℝm is the control input, yi(t)∈ℝp is the measured output, and A, B, and C are constant matrices with compatible dimensions. It is assumed that (A,B) is stabilizable and (C,A) is detectable, and without loss of generality, B is of full column rank. We say that the control input ui(t) solves the consensus problem for the multiagent system (12) if the states of the agents satisfy
(13)limt→+∞[xi(t)-xj(t)]=0,∀i,j∈𝒱N
for any initial states.Assume that the communication topology among theN agents is represented by a fixed directed graph 𝒢N=(𝒱N,ℰN,𝒜N). Based on the relative output information between the agents, the following dynamic output feedback (DOF) control protocol is used by [18]:
(14)v˙i(t)=Akvi(t)+Bk∑j=0N‍aij[yj(t)-yi(t)]ui(t)=Ckvi(t)+Dk∑j=0N‍aij[yj(t)-yi(t)]i=1,…,N,
where vi(t)∈ℝnc, nc is a preassigned dimension of the coordinating law, and aij is the element of the corresponding adjacency matrix 𝒜N. The system matrix
(15)K=[AkBkCkDk]
of the DOF control protocol (14) need to be designed to make the multiagent system (12) achieve consensus. A general method for constructing the system matrix K was presented by [18].However, if there exist communication errors between theith agent and the jth agent, i,j=1,2,…,N, then the performance of consensus will be affect by these errors, as illustrated by the example given below.Example 6.
We consider double-integrator systems given by(16)ξ¨i(t)=ui(t),yi(t)=ξi(t),i=1,2,…,N,
where ξi(t),yi(t)∈ℝ. Let ξ˙i(t)=ζi(t). Then, the above system can be rewritten as the form of (12) with
(17)xi(t)=[ξi(t)ζi(t)],A=[0100],B=[01],C=[10].
The weighted communication topology with 6 agents is shown in Figure 1. Using the results presented in [18], the DOF control protocol (14) can be constructed with
(18)Ak=[-10-1-50.1039-0.485],Bk=[1050],Ck=[0.10390.485],Dk=0.
It is known that the consensus is asymptotically achieved when there are no communication errors with the designed protocol (see Figure 2(a)). However, communication errors are inevitable. Assume that a 1% error appears in all of the communication channels. Simulation results show that, under the same protocol, the system diverges in the sense that the position state of each agent is far away from the position state of leader (node 1) as can be seen in Figure 2(b).Figure 1
Communication topology.The disagreement states betweenxi and x1 without/with communication errors, i=2,3,…,6.
(a)
(b)Example6 implies that, under the influence of communication errors, consensus cannot be achieved for each agent with the given control protocol. This provides motivation to design an appropriate DOF control protocol to attenuate the effects of communication errors on the consensus performance. In this paper, we assume that there exist communication errors in the transferred data; that is, the DOF control protocol takes the following form:
(19)v˙i(t)=Akvi(t)+Bk∑j=0N‍aij[yj(t)-yi(t)+ωij(t)],ui(t)=Ckvi(t)+Dk∑j=0N‍aij[yj(t)-yi(t)+ωij(t)]i=1,…,N,
where ωij∈ℒ2p[0,+∞) represents the communication error when the ith agent gets information from the jth agent. For convenience, denote
(20)x(t)=[x1T(t),…,xNT(t)]T∈ℝN(n+nc),xi(t)=[xiT(t),viT(t)]T∈ℝn+nc,i=1,2,…,N,y(t)=[y1T(t),y2T(t),…,yNT(t)]T∈ℝNp,d(t)=[d1T(t),d2T(t),…,dNT(t)]T∈ℝN(N-1)p,di(t)=[ωi1T(t),…,ωi(i-1)T(t),ωi(i+1)T(t),…,ωiNT]T(t)∈ℝ(N-1)p,i=1,2,…,N.
Then, the overall dynamics result in the system (12) with the DOF control protocol (19) can be written as
(21)x˙(t)=(IN⊗A--LN⊗B-C-)x(t)+(DN⊗B-)d(t),
where
(22)A-=[ABCk0Ak],B-=[BDkBk],C-=[C0],DN=[δ101×N-1⋯01×N-101×N-1δ2⋯01×N-1⋮⋮⋱⋮01×N-101×N-1⋯δN]∈ℝN×N(N-1),δi=[li1⋯li(i-1)li(i+1)⋯liN],i=1,…,N,
and lij is the element of the Laplacian matrix LN.In order to characterize the effects of the communication errors on consensus performance, we need to define a controlled output for the multiagent system (12) as follows.Assume that the fixed directed communication graph𝒢N has a spanning tree, and according to Lemma 2, the Laplacian matrix LN of graph 𝒢N has a simple zero, and all of the other eigenvalues are in the right half-plane. Let e1=(1/N)1N. By Lemma 3, there exists an orthogonal matrix [e1U]∈ℝN×N such that
(23)[e1TUT]LN[e1U]=[0e1TLNU0L-N],
where L-N=UTLNU. It is obvious that the eigenvalues of L-N are equal to the nonzero eigenvalues of LN, which means that all of the eigenvalues of L-N are in the right half-plane. Here, the matrix U satisfies UT1N=0,UTU=IN-1, and UUT=IN-(1/N)1N1NT according to [e1U] being an orthogonal matrix.LetC0=[In0]. Define an output vector z(t) as
(24)z(t)=(UT⊗C0)x(t)=(UT⊗In)x(t),
where z(t)=[z1T(t),…,zN-1T(t)]T∈ℝ(N-1)n, zi(t)∈ℝn, and x(t)=[x1T(t),x2T(t),…,xNT(t)]T. Then,
(25)|z(t)|2=|(UT⊗C0)x(t)|2=|x(t)-1N⊗x0(t)|2,
where x0(t)=(1/N)∑i=1Nxi(t), which means that z(t) can measure the difference between the state of each agent and the average state of all agents.Letx^(t)=(UT⊗In+nc)x(t). Using the DOF control protocol (19), the system dynamics with the output z(t) can be represented as
(26)x^˙(t)=A~x^(t)+B~d(t),z(t)=C~x^(t),
where A~=IN-1⊗A--L-N⊗B-C-, B~=UTDN⊗B-, and C~=IN-1⊗C0.From the fact that the null-space of matrixUT⊗In is Span{1N⊗In}, we know that
(27)limt→+∞z(t)=limt→+∞(UT⊗In)x(t)=0
if and only if there exist x0(t)∈ℝn such that limt→+∞xi(t)=x0(t), which implies that consensus of the multiagent system (12) can be achieved asymptotically. However, it is obvious that z(t) cannot approach zero as t tending to infinity due to the existence of communication error d(t), which indicates that consensus cannot be achieved for the system (12) with the DOF control protocol (19). Inspired by the analysis above, it is reasonable to evaluate the effects of communication error on consensus of the system (12) with the DOF control protocol (19) by using the effects of communication error d(t) on the output z(t) of system (26). Notice that the latter can be quantitatively measured by the H∞ norm of the transfer function matrix Gdz(s) from d(t) to z(t), which is defined by ∥Gdz(s)∥∞=supω∈ℝ{∥Gdz(jω)∥2}=supω∈ℝ{σ(Gdz(jω))}, that results in the following definition.Definition 7.
Given a scalarγ>0. The system (12) with the DOF control protocol (19) is called to achieve consensus with H∞ performance if the following conditions hold.(1)
It can reach consensus whend(t)=0;(2)
∥Gdz(s)∥∞<γ, where Gdz(s) is the transfer function matrix of system (26) from d(t) to z(t) and the output z(t) is defined by (24).A sufficient condition is given in the following theorem to ensure that the multiagent system (12) with the DOF control protocol (19) can achieve consensus with H∞ performance.Theorem 8.
Given a scalarγ>0. Assume that the fixed communication topology 𝒢N has a spanning tree. The system (12) with the DOF control protocol (19) achieves consensus with H∞ performance if there exists a matrix Xc=Xc*>0 such that
(28)[Xc(A--λiB-C-)+(A--λiB-C-)*XcXcB-C0*B-*Xc-γ0I0C00-γ0I]<0,
where γ0=dmax-1γ and λi is the nonzero eigenvalue of Laplacian matrix LN, i=2,3,…,N.Proof.
It is known that the system (12) with the DOF control protocol (19) achieves consensus with H∞ performance if and only if A~ is Hurwitz and
(29)∥Gdz(s)∥∞=∥C~(sI-A~)-1B~∥∞<γ,
where A~, B~, and C~ are defined by (26).
According to Lemma5 and (28), we have A--λiB-C- is Hurwitz stable, and
(30)H-i≜[A--λiB-C-γ0-2B-B-*-C0*C0-(A--λiB-C-)*]
has no eigenvalues on the imaginary axis; that is, for any ω∈ℝ,
(31)(jωI-(A-λiB-C-))η1-(γ0-2B-B-*)η2=0,C0*C0η1+(jωI+(A--λiB-C-)*)η2=0,
if and only if η1=0 and η2=0, where η1,η2∈ℂn+nc.
For matrixL-N, there are two unitary matrices V1 and V2 such that
(32)V1*L-NV1=T=[tij]∈ℝ(N-1)×(N-1),V2*L-N*V2=S=[sij]∈ℝ(N-1)×(N-1),
where T and S are upper triangular, with diagonal entries tii=λi+1 and sii=λi+1*, respectively, i=1,2,…,N-1. Now, suppose that jω0 is an eigenvalue of
(33)H≜[IN-1⊗A--T⊗B-C-γ0-2IN-1⊗B-B-*-IN-1⊗C0*C0-(IN-1⊗A--S*⊗B-C-)*].
Then, there exists a vector η=[η1,1T,η1,2T,…,η1,(N-1)T,η2,1T,η2,2T,…,η2,(N-1)T]T≠0, where ηk,l∈ℂn+nc,k=1,2,l=1,2,…,N-1, such that
(34)(jω0I-(A--λi+1B-C-))η1,i+∑j=i+1N-1‍tijB-C-η1,j-(γ0-2B-B-*)η2,i=0,C0*C0η1,i+(jω0I+(A--λi+1B-C-)*)η2,i-∑j=i+1N-1‍sij(B-C-)*η2,j=0,
where i=1,2,…,N-1. When i=N-1, from (31), it is easy to get that η1,N-1=η2,N-1=0. Then, when i=N-2, (34) can be rewritten as
(35)(jω0I-(A-λN-1B-C-))η1,N-2-(γ0-2B-B-*)η2,N-2=0,C0*C0η1,N-2+(jω0I+(A--λN-1B-C-)*)η2,N-2=0,
which implies that η1,N-2=η2,N-2=0. Similarly, it can be known that η1,i=η2,i=0 for all i=1,2,…,N-1. This contradicts our assumption, and hence matrix H has no eigenvalues on the imaginary axis.
Notice that(36)H~≜[IN-1⊗A--L-N⊗B-C-γ0-2V1V2*⊗B-B-*-V2V1*⊗C0*C0-(IN-1⊗A--L-N⊗B-C-)*]=[V1⊗In+nc00V2⊗In+nc]H[V1*⊗In+nc00V2*⊗In+nc],
which means that matrix H~ has no pure imaginary eigenvalues. Moreover, matrices A--λiB-C- are Hurwitz stable, i=2,3,…,N, which implies that matrix A~=IN-1⊗A--L-N⊗B-C- is Hurwitz stable. Noting that V1 and V2 are unitary matrices, then there must exist two matrices Y1 and Y2 such that Y1Y1*=V1V2*,Y2*Y2=V2V1*, and ∥Y1∥2=∥Y2∥2=1. Thus, according to Lemma 5, it can be obtained that
(37)∥(Y2⊗C0)(sI-(IN-1⊗A--L-N⊗B-C-))-1(Y1⊗B-)∥∞=∥C~(sI-A~)-1(IN-1⊗B-)∥∞<γ0.
In addition, it is easy to know that(38)∥UTDN∥2≤∥DN∥2=maxi{δiδiT}≤dmax.
Then, we have
(39)∥Gdz(s)∥∞=∥C~(sI-A~)-1B~∥∞<dmax∥C~(sI-A~)-1(IN-1⊗B-)∥∞<γ.
This completes the proof.Remark 9.
If there are no communication errors, that is,d(t)=0, then from Theorem 8, it is only required that A--λiB-C- is Hurwitz for all nonzero eigenvalues λi of the Laplacian matrix LN. In this case, consensus can be achieved asymptotically for the multiagent system (12) with the DOF control protocol (19), which is the result shown by [18].Remark 10.
In fact, from Lemma5, there exists Xc>0 such that the inequality (28) holds, i=2,3,…,N, if and only if there exists a common ncth-order DOF controller
(40)𝒦(s)=[AkBkCkDk]
solving H∞ problem with performance γ0 for N-1nth-order LTI systems
(41)Σi(s)=[A0BIn00-λiCD210],
where A, B, and C are the state space matrices for system (12), D21=Ip, and λi is the nonzero eigenvalue of Laplacian matrix LN,i=2,3,…,N.


## 4. Dynamic Output Feedback Design forH∞ Consensus

In this section, we determine the system matrixK of the DOF control protocol (19) for the multiagent system (12) to achieve consensus with H∞ performance. According to Remark 10, it is required to design a common DOF controller (40) to solve H∞ problem for N-1 LTI systems. Notice that the common DOF controller is difficult to obtain; thus, we firstly consider the H∞ problem for the systems Σi(s), i=2,3,…,N.LetNX and NY be full-rank matrices whose images satisfy
(42)Im(NX)=ker([CIp]),Im(NY)=ker([B*0]).
Denote NX=[Nc*NI*]*, where Nc and NI have n and p rows, respectively.Lemma 11.
If there exist matricesX=X*>0 and Y=Y*>0 such that
(43)(i)[Nc00I]*[XA+A*XInIn-γ0I][Nc00I]-[γmNI*NI000]<0,(44)(ii)NY*[YA*+AYYY-γ0I]NY<0,(45)(iii)[XIIY]≥0,
where γm=γ0{mini|λi|2}, then there exists a DOF controller 𝒦i(s)=[AkiBkiCkiDki] which can solve the H∞ problem with a given performance γ0>0 for the LTI system Σi(s) given by (41), i=2,3,…,N.Proof.
According to Lemma4, we have that there exists a DOF controller 𝒦i(s) to solve the H∞ problem for the LTI system Σi(s) with a given index γ0>0 if and only if there exist matrices Xi=Xi*>0 and Y=Y*>0 such that
(46)(a)[NXi00I]*[XiA+A*Xi0In0-γ0I0In0-γ0I][NXi00I]<0,(b)[NY00I]*[YA*+AYY0Y-γ0I000-γ0I][NY00I]<0,(c)[XiIIY]≥0,
where NXi spans the kernel of [-λiCIp]. Notice the following facts:
(47)[-λiCIp][Nc-λiNI]=-λi(CNc+IpNI)=0,
and, for all nonzero λi,
(48)rank[Nc-λiNI]=rank[NcNI].
Thus, we can choose NXi=[Nc-λiNI]. Let Xi=X for all i=2,3,…,N. Then, it is easy to obtain that
(49)[NXi00I]*[XiA+A*Xi0In0-γ0I0In0-γ0I][NXi00I]=[Nc00I]*[XA+A*XInIn-γ0I][Nc00I]-[γ0|λi|2NI*NI000]≤[Nc00I]*[XA+A*XInIn-γ0I][Nc00I]-[γmNI*NI000]<0
and (b), (c) naturally hold from (44), (45), respectively. This completes the proof.Remark 12.
Lemma11 gives a sufficient condition for the existence of the controller 𝒦i(s) which can solve the H∞ problem for the system Σi(s), i=2,3,…,N. As the results stated in Section 2.2, if there exist X=X*>0 and Y=Y*>0 satisfying Lemma 11, then the DOF controller 𝒦i(s) can be obtained by solving the following inequality:
(50)HXc+Q(λi)*Ki*PXc+PXc*KiQ(λi)<0,
for Ki,i=2,3,…,N, where
(51)HXc=[XcA0+A0*Xc0C0*0-γ0I0C00-γ0I],PXc=[B^*Xc00],Q(λi)=[C^(λi)D^0],(52)A0=[A000],C0=[I0],B^=[0BI0],C^(λi)=[0I-λiC0],D^=[0I],(53)Ki=[AkiBkiCkiDki],Xc=[XX2X2*I],X2X2*=X-Y-1.
Obviously, if there exists a common K that makes the inequality (50) hold for all i=2,3,…,N, then there exists a common ncth-order DOF controller 𝒦(s) which can solve the H∞ problem for the LTI systems Σi,i=2,3,…,N. Thus, we have the following result.Theorem 13.
Given a scalarγ>0, and let γ0=dmax-1γ. Assume that the fixed communication topology 𝒢N has a spanning tree. Then, there exists a DOF control protocol (19) for the system (12) achieving consensus with H∞ performance if(i)
there exist matricesX>0 and Y>0 satisfying (43), (44), and (45),(ii)
there exists a matrixK satisfying the following LMIs:
(54)HXc+Q(λi)*K*PXc+PXc*KQ(λi)<0,
for all nonzero eigenvalues λi of Laplacian matrix LN, where HXc, PXc, and Q(λi) are given in (51) and Xc is defined by (53).Proof.
From the analysis above, the conditions (i) and (ii) hold which implies that there exist matricesK and Xc>0 such that (28) holds due to the fact that (54) is exactly (28). The proof is completed by using Theorem 8 directly.Remark 14.
If we want controllers of ordernc less than n, it is only required to add the additional constraint
(55)rank[XIIY]≤n+nc,
to (i) of Theorem 13, which can be obtained by using Corollary 7.8 given by [29] and Theorem 13 directly.Remark 15.
Theorem13 gives the sufficient conditions under which there exists a DOF control protocol such that the multiagent system (12) achieve consensus with a given H∞ performance. When the conditions are satisfied, the procedure to construct the DOF control protocol is presented as follows.Sept 1: Solve LMIs (43), (44), and (45) for getting a solution: X>0 and Y>0.Step 2: ConstructXc>0 as (53).Step 3: Solve theN-1 LMIs (54) for a common feasible solution K.Remark 16.
Assume that there is no communication error in the system. As shown in Remark9, in this case, the given problem is to design a stabilizing controller 𝒦(s) defined by (40) for the LTI systems (41) with D21=0. Using Theorem 5.8 given by [29] and the fact that the kernels of λiC and C are exactly equal for all nonzero λi, reproducing the steps of the proof of Theorem 13, we have the following results. Assume that the fixed communication topology 𝒢N has a spanning tree, then there exists a DOF control protocol (14) with order nc for the system (12) achieving consensus if (1)
there exist matricesX=X*>0 and Y=Y*>0 such that
(56)NX*(A*X+XA)NX<0,NY*(AY+YA*)NY<0,[XIIY]≥0,rank[XIIY]≤n+nc,
where NX and NY span the kernels of C and B*, respectively,(2)
there exists a matrixK satisfying the following LMIs:
(57)A0*P+PA0+C^(λi)*K*B^*P+PB^KC^(λi)<0,
for all nonzero eigenvalues λi of Laplacian matrix LN, where
(58)P=[XX2X2*I]
and X2 satisfies X2X2*=X-Y-1 and A0, B^, and C^(λi) are given in (51). Moreover, using the method similar to that stated in Remark 15, we can construct the DOF controller for the multiagent system (12) reaching consensus.Notice that condition (ii) in Theorem13 implies that we need to solve N-1 LMIs after constructing Xc, which increase the difficulty of the numerical calculation if the size of the multiagent system N is large. We give the following conditions, which can reduce the computational complexity for getting the DOF control protocol by solving four LMIs.Denote thatαi and βi are the real part and imaginary part of λi, respectively, where λi is the nonzero eigenvalue of Laplacian matrix LN,i=2,3,…,N. Let α0=mini{αi},α-0=maxi{αi}, β0=mini{βi}, and β-0=maxi{βi}=-β0.Theorem 17.
Given a scalarγ>0, and let γ0=dmax-1γ. Assume that the fixed communication topology 𝒢N has a spanning tree. Then, there exists a DOF control protocol (19) for the system (12) achieving consensus with H∞ performance if (i)
there exist matricesX>0 and Y>0 satisfying (43), (44), and (45);(ii)
there exists a matrixK such that
(59)HXc+Q(αk+jβk)*K*PXc+PXc*KQ(αk+jβk)<0
for (αk,βk)∈{(α0,β0),(α0,β-0),(α-0,β0),(α-0,β-0)}, where HXc, PXc, and Q(αk+jβk) are given in (51) and Xc is defined by (53).Proof.
To complete the proof, we only need to show that if LMI (59) holds for (αk,βk)∈{(α0,β0),(α0,β-0),(α-0,β0),(α-0,β-0)}, then LMI (54) holds for all nonzero eigenvalues λi=αi+jβi of Laplacian matrix LN,i=2,3,…,N.
Notice that(60)HXc+Q(αi+jβi)*K*PXc+PXc*KQ(αi+jβi)=[Xc(A--(αi+jβi)B-C-)+(A--(αi+jβi)B-C-)*XcXcB-C0*B-*Xc-γ0I0C00-γ0I]<0
which, in virtue of the Schur Complement Lemma, is equivalent to
(61)Xc(A--(αi+jβi)B-C-)+(A--(αi+jβi)B-C-)*Xc+γ0-1(XcB-B-*Xc+C0*C0)<0,
where A-, B-, and C- are defined by (22). For convenience, we denote
(62)ℋ(αi,βi)=Xc(A--(αi+jβi)B-C-)+(A--(αi+jβi)B-C-)*Xc+γ0-1(XcB-B-*Xc+C0C0*)=H0+αiH1+jβiH2,
where H0=XcA-+A-*Xc+γ0-1XcB-B-*Xc+γ0-1C0C0*, H1=-XcB-C--C-*B-*Xc, and H2=-XcB-C-+C-*B-*Xc.
In fact, there must exists∈[0,1],r∈[0,1] such that αi=sα0+(1-s)α-0,βi=rβ0+(1-r)β-0. When r≥s,
(63)ℋ(αi,βi)=H0+[sα0+(1-s)α-0]H1+j[rβ0+(1-r)β-0]H2=sℋ(α0,β0)+(r-s)ℋ(α-0,β0)+(1-r)ℋ(α-0,β-0)<0.
Similarly, when s≥r, we have
(64)ℋ(αi,βi)=rℋ(α0,β0)+(s-r)ℋ(α0,β-0)+(1-s)ℋ(α-0,β-0)<0.
This completes the proof.


## 5. Numerical Example

An example is shown to verify the results obtained in the above section. The agent dynamics and the communication topology are given in Example6, and the H∞ performance index is chosen as γ=10. According to the results presented in Section 4, we have
(65)X=[1.8218-1.2978-1.29783.6435],Y=[1.2145-0.9109-0.91093.6435],Xc=[1.8218-1.2978-0.86880.2312-1.29783.64351.81480.1107-0.86881.81481.000000.23120.110701.0000].
Then, by solving the LMIs (54), we can get a feasible controller (40) with
(66)Ak=[-7.0377-5.6811-0.6005-5.2415],Bk=[-9.5464-0.5110],Ck=[3.45202.9461],Dk=4.9523,
and the H∞ norm of system (26) is ∥Gdz(s)∥∞=3.0707. With the designed DOF control protocol, the disagreement states between xi and x1 without communication error d(t) are shown in Figure 3, i=2,3,…,6, which implies that the consensus can be reached when d(t)=0. The communication error d(t)∈ℒ2[0,∞) is supposed to be
(67)d(t)={θsin(t),t∈[0,40],0,t>40,
where θ=[1.110.581.481.592.122.530.421.314.00.14]T. Under zero initial condition, the state trajectories of the six agents are depicted in Figure 4, and the corresponding energy trajectories of d(t) and z(t) are given in Figure 5. It is noted obviously that ∥z(t)∥ℒ2≤∥d(t)∥ℒ2. Thus, the multiagent system with the given DOF controller can achieve the consensus with the given H∞ performance, which validates the effectiveness of the proposed protocol and demonstrates the correctness of the obtained theoretical results.The disagreement states betweenxi and x1 without communication error d(t), i=2,3,…,6.
(a)
Position states of the 6 agents
(b)
Velocity states of the 6 agentsThe state trajectories of the multiagent system with communication errord(t).
(a)
Position states of the 6 agents
(b)
Velocity states of the 6 agentsFigure 5
Energy trajectories of the controlled outputz(t) and the communication error d(t).


## 6. Conclusions

This paper is devoted to the consensus problem for multiagent systems molded by linear time-invariant systems under fixed directed communication topologies and subject to communication errors in the transferred data. A dynamic output feedback control algorithm is proposed. The theoretical analysis shows that if there exists a common dynamic output feedback controller which can solveH∞ problem for N-1 linear time-invariant systems of order n, then the consensus with a desired H∞ level can be reached. By using H∞ theory, a sufficient condition in terms of linear matrix inequalities is given to ensure the existence for such a controller. A procedure for the controller design is presented.



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*Source: 979087-2013-06-10.xml*