CC BY
0Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 979087,12 pages http://dx.doi.org/10.1155/2013/979087
Research Article
Distributed Robust Hœ Consensus Control of Multiagent Systems with Communication Errors Using Dynamic Output Feedback Protocol
Xi Yang and Jinzhi Wang
State Key Laboratory for Turbulence and Complex System, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China
Correspondence should be addressed to Jinzhi Wang; jinzhiw@pku.edu.cn
Received 17 April 2013; Accepted 8 May 2013
Academic Editor: Guanghui Wen
Copyright © 2013 X. Yang and J. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 prescribed HM performance are presented. Numerical examples are given to illustrate the theoretical results.
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]designedrobuststaticoutputfeedbackcon-trollers 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 with Hœ 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 Hœ performance is transmitted into an Hœ control problem of another reduced-order system. It is shown that consensus with Hœ performance can be achieved if there exists a common dynamic output feedback controller which can be realized by solving Hœ problem for N-1 linear dynamic systems simultaneously, where N is thenumberofagents. Secondly, interms 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. Section 2 introduces basic notations and reviews some useful results on graph theory and robust Hœ control theory. Section 3 formulates the problem and conditions for reaching consensus with Hœ 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 R"x" and C"x" be the set of«x« real matrices and n x n complex matrices, respectively. Matrices, if not explicitly stated, are assumed to have compatible dimensions. In and 0nxm are the n x n identity matrix and the n x m zero matrix, respectively. For a matrix A, ||A||2 is the induced 2-
norm of the vector norm, and ||A||2 = a(A). The notation <r(A) is the maximal singular value of matrix A. Notations A , 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 R" spanned by {v1, v2,... vm|, where v; e R", i = 1,2, ...,m.
The notation ® represents the Kronecker product. For a vector
x e C", |x| = is the Euclidean norm. 1N denotes
the N x 1 column vector whose elements are all ones. The space of piecewise continuous functions in Rm that are square integrable over [0,+to) is denoted by L™[0, to), for any v(i) e L™[0, to), and its normalized energy is defined by
IIvWIIl = (Jo™ lv(^)|2^^) / . Let be a state space
realization of G(s) = C(s7 - A)-1£ + D.
2. Preliminaries
2.1. Graph Theory. Directed graphs are used to model the information interaction among agents. Let GN = (V№ En, An) be a directed weighted graph, where V N = {1,2,..., N} is the node set, EN c VN x VN is the edge set, and A N e RNxN is a weighted adjacency matrix with nonnegative elements a^. An edge of GN is denoted by («, j) which means that agent j can directly get information from agent «. (j,«) e EN if and only if a^ > 0, otherwise a^ = 0. If ( (,j) e &N ^ (j, () e then &N is said to be an undirected graph. In this paper, we assume that there are no self-cycles in GN; that is, ait = 0, i = 1,2, ...,N. The in-degree and out-degree of the f'th agent are, respectively, defined as din(i) = ^ a;j and dout(') = a;i. Let dmax = maXj{din(«)}. Correspondingly, the Laplacian matrix of graph Gn is denoted by Ln = AN - AN e RNxN,whereAN = [A ] is a diagonal matrix with A = din(i).
A sequence ( <'2), ( <'2, (3),..., ((fc-1, (fc) of edges is called a directed path from node i1 to node <'fc. GN is called a strongly connected digraph if for any«, j e VN,there is adirectedpath from i to j. Gn has a directed spanning tree if there exists a node r e VN (a root) such that all other nodes can be linked to r via a directed path. A directed graph is called balanced if
Zj=iaij = L=jaji for all< e V.
Below are well-known results for the Laplacian matrix.
Lemma 1 (see [3]). The Laplacian matrix LN of a directed graph has at least one zero eigenvalue with an associated eigenvector 1N.
Lemma 2 (see [3]). The Laplacian matrix LN 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]). Let LN be the Laplacian matrix of a directed graph GN, then there exists an orthogonal matrix U =
[1n/Vn U] e RNxN such that
0 -^LU
U LnU =
Furthermore, if G is a balanced graph, then
utlnu =
0 UJLNU
2.2. Robust Hm Control Theory. Consider that the nth-order linear time-invariant (LTI) system is described as follows:
x (t) = Ax (t) + B1w (t) + B2u (t), z(t) = C1 x (t) + D11w (t) + D12u (t), (3)
y(t)=C2X(t) + D21a(t)+D22U(t),
where x(t) e R" is the state, w(t) e Lq2[0, >x) is the external disturbance, u(t) e Rm is the control input, z(t) e R^ is the regulated output, y(t) e r' is the measured output, and A, Bt, C;, and Dtj, 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.
The ncth-order dynamic output feedback (DOF) controller is described as follows:
xc (t) = Akxc (t)+Bky(t), u(t) = Ckxc (t) + Dky(t),
where xc(t) e R" is the controller state and Ak, Bk, Ck, and Dk are constant matrices with appropriate dimensions.
Let Spc(s) =
be the transfer function from
a(t) to z(t) of the closed-loop system obtained from (3) and (4), where
Ac = A Q + BKC,
Cc = C0 + DnKC,
Bc = BQ +BKD21,
Dc = Dn +D12KD21,
A 0 0 0
0 B2' , c = 0 I
I 0 C2 0
Cq = [ci 0], di2 = [0 d12] ,
0 * II \Ak Bk'
D21. Ck Dk.
The Hm problem for the given LTI system (3) is to find a DOF controller (4) such that the closed-loop system is internally stable and (s)llm < f for some constant y > 0.
To facilitate the consensus protocol design and stability analysis, several results of the Hm problem are recalled as follows.
Lemma 4 (see [29]). Given y > 0, there exists a DOF controller (4) which can solve the Hm problem for the LTI system (3) if and only if there exist symmetric matrices X > 0 and Y > 0 such that
Nx 0 0 I
XA + A*X XB1 C\ b;X -yi D*1 C1 D11 -yi
Nx 0 0 I
X I I Y
YA* + AY YC: B
-yi Dn
Dh ~yl\
where Nx and NY are full-rank matrices whose images satisfy Im (Nx)= ker ([C2 D21]), Im (Ny) = ker ([B*2 D'n]).
As the results shown by [29], the DOF controller (4) solving the Hm problem for the LTI system (3) can be constructed as follows.
(i) Find X > 0 and Y > 0 which satisfy Lemma 4.
(ii) Let Xc = [X X2 ], where X2 e R"x"' satisfying X -Y-1 = X2X22.
(iii) Solve the the following inequality:
HXc + Q2K2PXc + PXcKQ < 0. For a feasible solution K, where
Px_ = \B X 0 D
Q=\C D21 0
XcAq + AQXc XcBq
B*Xc Cn
-Yq1 D*1
D11 -yolj
and A0, B0, C0, B, C, D12, D21 are defined by (5). The solution K provides the state space realization for a feasible controller (4) which can solve Hm problem for system (3).
Lemma 5 (see [30]). Let y > 0, G(s) = Hurwitz stable, and
with A is
A + BR-1 D*C
BR-1B*
-C* (I + DR-1D*)C -(A + BR-1D*C*)*
where R = y21 - D2D. Then, the following conditions are equivalent:
(1) ||G(s)||m < y,
(2) a(D) < y and H have no eigenvalues on the imaginary axis,
(3) there exists a matrix X = X2 > 0 such that
~XA + A2X XB C2
B2X -yi D2 <0. (11)
. C D -yl_
3. Consensus Problem
Consider a multiagent system consisting of N identical agents with linear dynamics described by
X; (i) = AX; (i) + BM; (i) (i) = CX; (i)
i = 1,
where x;(i) e R" is the state, m; (i) e Rm is the control input, y;(i) e R^ 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 M;(i) solves the consensus problem for the multiagent system (12) if the states of the agents satisfy
lim [x; (i) - X,. (i)l = 0, Vi, j e VN
i —» L J J
for any initial states.
Assume that the communication topology among the N 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]:
V; (i)=AfcV; (i)+fifc ^fly [y, (i)-7i (i)j
Mi (i)=CfcV; (i) + Dk Xfly [y, (i)-y; (i)j
where v;(i) e R"c, nc is a preassigned dimension of the coordinating law, and fly is the element of the corresponding adjacency matrix AN. The system matrix
Ak Bk Ck Dk
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 the ith 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
& (i) = "i (i), y (*) = & (i).
i = 1,2,...,N,
where £(f), y;(i) e R. Let £(f) = (i). Then, the above system can be rewritten as the form of (12) with
X;(i) =
(Î) C; (Î)
0 1 0 0
C= [1 0].
Figure 1: Communication topology.
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
-50.1039 -0.485
Ck = [0.1039 0.485],
D = 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).
Example 6 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:
V; (i) = A^; (i) + Bk X Oij [yj (Î) - y (Î) + ^y (i)j ,
(i) = CkV; (i) + Dk X % [y, (i) - y; (i) + Wy (i)j
i = 1,...,N,
Without communication errors
With communication errors
30 Time
100 Time
100 Time
(a) (b)
Figure 2: The disagreement states between xt and x1 without/with communication errors, i = 2,3,... ,6.
where w^ e L^lO, +ot) represents the communication error when the ith agent gets information from the jth agent. For convenience, denote
x (t)=[xT (t),..., xN (t)]T e RN(n+n\ x, (t) = [xT (t), vT (t)]T e Rn+n, i=l,2,...,N,
y(t) = [yT(t),yT(t),---,yTN(Of e RNp, d (t)=[dT (t), dT (t),..., dN (t)]T e RN(N-1^ , d; M = K (t),..., 0%-!) (t), û^+d (t),..., uJn]T (t)
e R(n-1)p, i=\,2,...,N.
Then, the overall dynamics result in the system (12) with the DOF control protocol (19) can be written as
x(t) = (IN ®A-Ln ®BC) x (t) + (Dn ®b) d (t), (21)
A BCk 0 Ak . , B = BDk .Bk C=[C 0],
" Si 01xN-1 °1xN-1 ' ' ' ''' 01xN-1 01xN-1 e rnxn(n-i)
-01xN-1 01xN-1 ' ' ' SN -
[1,1 ' (¡-1) li(i+1) ' In] i=l,...,N,
and I¡j 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 graph GN 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)1 N.
By Lemma 3, there exists an orthogonal matrix U] e RNxN such that
¿N K =
0 êJLNU 0
where LN = UTLNU. It is obvious that the eigenvalues of LN are equal to the nonzero eigenvalues of LN, which means that all of the eigenvalues of LN are in the right half-plane. Here, the matrix U satisfies UT1N = 0, UTU = 7N-1, and UUT = JN - (1/N)1NlN according to [e1 U] being an orthogonal matrix.
Let C0 = [7" 0]. Define an output vector z(t) as
z (t) = (UT « Co) x (t) = (UT « 7") x (t), (24)
where z(i) = [z;(i),
■ . , zN-
!(i)]T e R(N-1)", z;(i) e R", and
x(i) = [x; (i),X2(i),...,xN(i)]J. Then, |z (i)!2 = |(UT ®Co) x (i)|2 = |x(i)- 1n ®Xo (i)l2
where x0(t) = (1/N) £j=1 x;(t), which means that z(t) can measure the difference between the state of each agent and the average state of all agents.
Let x(t) = (UT ® 7"+"c )x(t). Using the DOF control protocol (19), the system dynamics with the output z(t) can be represented as
x(i) = Ax(i) + Bd (i), z (i) = CX (i),
Definition 7. Given a scalar y > 0. The system (12) with the DOF control protocol (19) is called to achieve consensus with Hœ performance if the following conditions hold.
(1) It can reach consensus when d(i) = 0;
(2) ||Gdz(s)||œ < y, where Gdz(s) is the transfer function matrix of system (26) from d(i) to z(i) and the output z(i) 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 Hœ performance.
Theorem 8. Given a scalar y > 0. Assume that the fixed communication topology GN has a spanning tree. The system (12) with the DOF control protocol (19) achieves consensus with Hœ performance if there exists a matrix Xc = X* > 0 such that
xc (a-a;bc) + (a-a;bc)0xc xcb C0* -0
B xc -%7 0
Co 0 -yo7
where y0 = dm1axy and A; is the nonzero eigenvalue ofLaplacian matrix Ln, i = 2, 3,..., N.
Proof. It is known that the system (12) with the DOF control protocol (19) achieves consensus with HTO performance if and only if A is Hurwitz and
llGd, ML HIC^-A^BL <y> (29)
where A = 7N-1 ® A - LN ® BC, B = UTDN ® B, and C = 7n-1 ® co.
From the fact that the null-space of matrix UT ® 7" is Span {1N ® 7"}, we know that
lim z (i) = lim (UT ® 7") x (i) = 0
' —» f —» ^ '
where A, B, and C are defined by (26).
According to Lemma 5 and (28), we have A - A;BC is Hurwitz stable, and
A - A;BC
-C0Co -(A-A;BC)0
if and only if there exist x0(t) e R" such that limt^+TOx;(i) = 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 HTO norm of the transfer function matrix Gdz(s) from d(t) to z(t), which is defined
by ||Gdz(s)|L = sup^R{HGdZ(»ll2} = ^p^R^dzO'^K
that results in the following definition.
has no eigenvalues on the imaginary axis; that is, for any w e R,
(jw7 -( A - A;B C)) ^ - (y-2B B0 ) ft = 0,
C0OO>ft + (jw7 + (A - A;BC)0) ft = 0,
if and only if ft =0 and ft = 0, where ft, ft e C"+".
For matrix LN, there are two unitary matrices V1 and V2 such that
Vi°LnVi = r= li,l e R
(N-1)X(N-1)
V20LnV2 = S = [5y] e R1
(N-1)X(N-1)
where T and S are upper triangular, with diagonal entries tH = Xi+1 and Sjj = X*+1, respectively, i = 1,2,... ,N - 1. Now, suppose that jw0 is an eigenvalue of
IN-1 ®A-T®BC
Y-2In-i ®BB*
In-I ® C0 C0
-(In-1
®A-S* ®
>^1,(N-1) 2,1,
Then, there exists a vector ^ = [q11,q12,
Ilv->1UN-1)]T = 0, where nKl e Cn+n, k = 1,2, I = 1, 2,...,N- 1, such that
(jœ0I-(Â-\i+1BC))qu
+ I ttjBcmj -(Y-^B')^ = 0,
cocom,i + {j^o1 + {A- ^iBcy)^
- I ^(BcYm,! = 0, j=i+i
where i = 1,2,... ,N - 1. When i = N -1, from (31), it is easy to get that q1tN-1 = q2,n-1 = 0. Then, when i = N - 2, (34) can be rewritten as
[jw0I -(A- XN-1 BC)) KN-2 - (Y0)2BB*)^2N-2 = 0
C*0C0%N-2 + {Po1 +{A- XN-1 BCy^l]2,N-2 = 0
which implies that q1N-2 = q2,N-2 = 0. Similarly, it can be known that = q2 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
IN-1 ®a-ln ®bc
y^V1V2 ®BB
-v2v' ®c'c0 -(IN-1 ®a-ln®bc)'
V:1 ®In+nr
V2 ®In+nr
V ®In+nc
V ®In+nc
which means that matrix H has no pure imaginary eigenvalues. Moreover, matrices A - XtBC are Hurwitz stable, i = 2,3,..., N, which implies that matrix A = IN-1 ®A-LN ®BC is Hurwitz stable. Noting that V1 and V2 are unitary matrices, then there must exist two matrices Y1 and Y2 such that Y]Y* = V^*, Y*2Y2 = V2V*',and\\Y1\\2 = \\Y2\\2 = 1. Thus, according to Lemma 5, it can be obtained that
\(Y2 ®C0)(sI-(lN-1 ®A-Ln ®BC))-1 ®B)\\ = ||c(s/-A)-1 (In-1 ®I)L <Yo.
In addition, it is easy to know that |KDn||2 <\\Dn\\2 = max {fitf Then, we have
= \\c(s/-A) 1b\
Il ^ ' llœ < dmjc(sl - A) 1 (IN-1 ® 5)\\ < y.
This completes the proof.
Remark 9. If there are no communication errors, that is, d(i) = 0, then from Theorem 8, it is only required that A - XiBC is Hurwitz for all nonzero eigenvalues Xi 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 Lemma 5, there exists Xc > 0 such that the inequality (28) holds, i = 2, 3,..., N, if and only if there exists a common nrth-order DOF controller
K (s) =
solving Hm problem with performance y0 for N-1 nth-order LTI systems
Z (s) = In 0 0
. -W D21 0
where A, B, and C are the state space matrices for system (12), D21 = Ip, and Xi is the nonzero eigenvalue of Laplacian matrix Ln, i = 2,3,... ,N.
4. Dynamic Output Feedback Design for Hm Consensus
In this section, we determine the system matrix K of the DOF control protocol (19) for the multiagent system (12) to achieve consensus with Hm performance. According to Remark 10, it is required to design a common DOF controller (40) to solve Hm problem for N-1 LTI systems. Notice that the common DOF controller is difficult to obtain; thus, we firstly consider the Hm problem for the systems ^¡(s), i = 2,3,... ,N.
Let Nx and NY be full-rank matrices whose images satisfy
Im (Nx) = ker ([C Ip]),
Im(Ny) = ker([B' 0]).
< ^max.
Denote Nx = [N* N* ] *, where Nc and Nj have n and p rows, respectively.
Thus, we can choose N^ = [ _A N j. ^^ 2,3,..., N. Then, it is easy to obtain that
. Let X = X for all i =
Lemma 11. If there exist matrices X = X* >0 and Y = Y* > 0 such that
X 0" * XA + A*X I« " X 0
0 I" I« -7oI 0 I
7mW 0 0 0
(ii) N
YA * + AY 7
^ -yo'J
Ny < 0,
X I I Y
XiA + A*Xi 0 I„ 0 -7oI 0
I« 0 -7oI
N 0 0 I
XA + A*X L
ymNi*Ni 0
N 0 0 I
X 0" * XA + A*X I« " X 0
0 I. I« -7oI 0 I
Kf(s) =
= 7o min
Afc Bfc
Cfc Di
A;| }, then there exists a DOF controller which can solve the HTO problem with a
given performance y0 > 0 for the LTIsystem Z;(s) given by (41), i = 2, 3,..., N.
Proof. According to Lemma 4, we have that there exists a DOF controller K;(s) to solve the HTO problem for the LTI system Zj(s) with a given index y0 >0 if and only if there exist matrices X; = X* >0 and Y = Y* >0 such that
Ny 0 0 I
XiA + A*Xi 0 I« 0 -7oI 0 I« 0 -7oI
YA * + AY Y 0 Y -7oI 0
0 0 -7oI
Ny 0 0 I
and (b), (c) naturally hold from (44), (45), respectively. This completes the proof. □
Remark 12. Lemma 11 gives a sufficient condition for the existence of the controller K;(s) which can solve the Hœ problem for the system Z;(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 (s) can be obtained by solving the following inequality:
HXc + Q(A,) K*Px +iXK,Q(A,)<0, for K;, i = 2,3,..., N, where
0 C* "
XCA o + A oXc 0
-7oI 0 0 -7oI]
PXc = [-B*XC 0 0], Q(A,)=[C(A,) D 0],
X, I I Y
A 0 0 0
Co = [I 0],
C(A,) =
0 I" ___ 0"
-A,C 0" , D = I"
where spans the kernel of [-A;C Tpj.Noticethe following facts:
[-A,C ^
= -A, (CNc + IpNr) = 0, (47)
and, for all nonzero A;,
r NC X!
rank C = rank C
"A'fc Bi ■ , XC = " X X2"
k DiJ C X2* I J
X2X* =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 Hœ problem for the LTI systems i = 2, 3,..., N. Thus, we have the following result.
Theorem 13. Given a scalar y > 0, andlet y0 = 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 Hm performance if
(i) there exist matrices X > 0 and Y > 0 satisfying (43), (44), and (45),
(ii) there exists a matrix K satisfying the following LMIs:
HXr +Q(\i)'KtPXr + P* KQ(\,)<0,
for all nonzero eigenvalues Xi of Laplacian matrix LN, where Hx , Px , and Q(Xt) 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 matrices К 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 order nc less than n, it is only required to add the additional constraint
X I I Y
<n + nr
to (i) of Theorem 13, which can be obtained by using Corollary 7.8 given by [29] and Theorem 13 directly.
Remark 15. Theorem 13 gives the sufficient conditions under which there exists a DOF control protocol such that the multiagent system (12) achieve consensus with a given Hm 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: Construct Xc > 0 as (53).
Step 3: Solve the N-1 LMIs (54) for a common feasible solution K.
Remark 16. Assume that there is no communication error in the system. As shown in Remark 9, 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 XiC and C are exactly equal for all nonzero Xi, 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 matrices X = X* > 0 and Y = Y* > 0 such that
N*x (A*X + XA)Nx <0, N** (AY + YA*)Ny < 0,
X I I Y
> 0, rank
X I I Y
<n + nc,
where Nx and NY span the kernels of C and B*, respectively,
(2) there exists a matrix К satisfying the following LMIs:
A0P + PA0 + С(Х^К*Ё*Р + PÊKC (Xt) < 0, (57)
for all nonzero eigenvalues Xi of Laplacian matrix LN, where
and X2 satisfies X2X* = X-Y-1 and A0, B, and C(Xt) 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 Theorem 13 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 that a, and p, are the real part and imaginary part of Xi, respectively, where Xi is the nonzero eigenvalue of Laplacian matrix LN, i = 2,3,..., N. Let a0 = min;{a;}, a0 = max, {a,}, po = min,{$}, and^, = max,{^} =
Theorem 17. Given a scalar y > 0, andlety0 = d-^^y. 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 Hm performance if
(i) there exist matrices X > 0 and Y > 0 satisfying (43), (44), and (45);
(ii) there exists a matrix K such that
HXc + Q(«k + ih)*K*PXc + PXcKQ (ak + jpk) < 0 (59)
for (ak,^k) e {(ao,Po),(ao,~Po),(Ko,Po),(ao,~Po)},
where Hx , Px , and Q((*k + jfa) 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 (ak,fa) e {K,po),(«o»Po),@o,Po), (a0,p0)}, then LMI (54) holds for all nonzero eigenvalues X; = a; + jfa of Laplacian matrix LN, i = 2,3,..., N. Notice that
HXc + Q((X; + j^)*K*PXc + P* KQ (a, + jfr)
Xc (A-(ai + jßi)BC) + (A-(ai + jß,)BC)'Xc XCB C0*
-Vol 0 0 -YoU
which, in virtue of the Schur Complement Lemma, is equivalent to
Xc (A - & + jßt)BC) + (A- & + jßt)BC):Xc + У-1 (XcBB:Xc + C'0C0)<0,
20 15 10 5 0 -5
iv 0 -2 -4
30 40 50 Time
15 10 5 0 -5 -10
30 40 50 Time
(a) Position states of the 6 agents
(b) Velocity states of the 6 agents
Figure 3: The disagreement states between x and x without communication error d(i), i = 2,3,..., 6.
4 2 0 -2 -4
30 40 50 Time
(a) Position states of the 6 agents
30 40 50 Time
(b) Velocity states of the 6 agents
Figure 4: The state trajectories of the multiagent system with communication error d(i).
where A, £, and C are defined by (22). For convenience, we denote
H (a„£,) = Xc (A - (« + ;A)IC)
+ (A-(a, +j^,)BC)*Xc + y-1 (XC55*XC +cocq*) = ho + a,Hi
where H0 = XCA + A Xc + y-1 Xc££ Xc + y-1 C0C°, H1 = -Xc£C - CTXc, and H2 = -XcBC + CTXc.
In fact, there must exist s e [0,1], r e [0,1] such that« = sa0 + (1 - s)a0, A = r^0 + (1 - r)A0. When r > s,
H (a,,^,) = Ho + [s« + (1-s)«0]H1
+ j[r^o + (1-r)^o]H2 = sH («o,A)) + (r-s) H («0,^o)
+ (1-r) H («o,^o)<0. Similarly, when s > r,we have
H (a,, £) = rH («o, Ao) + (s - r) H («o Jo)
+ (1-s) H (a0, A0) < 0. This completes the proof.
5. Numerical Example
(64) □
An example is shown to verify the results obtained in the above section. The agent dynamics and the communication
topology are given in Example 6, and the HTO performance index is chosen as y = 10. According to the results presented in Section 4, we have
1.8218 -1.2978 -1.2978 3.6435
1.2145 -0.9109 -0.9109 3.6435
1.8218 -1.2978 -0.8688 0.2312 -1.2978 3.6435 1.8148 0.1107 -0.8688 1.8148 1.0000 0
0.2312 0.1107
1.0000
Then, by solving the LMIs (54), we can get a feasible controller (40) with
-7.0377 -5.6811 -0.6005 -5.2415
-9.5464 -0.5110
Cfc = [3.4520 2.9461],
Dfc = 4.9523,
and the Hœ norm of system (26) is ||Gdz(s)||œ = 3.0707. With the designed DOF control protocol, the disagreement states between and x1 without communication error d(i) are shown in Figure 3, i = 2,3,..., 6, which implies that the consensus can be reached when d(i) = 0.Thecommunication error d(i) e L2 [0, œ) is supposed to be
d (i) =
0 sin (t), te[0,40]: 0, t > 40,
where 0 = [1.11 0.58 1.48 1.59 2.12 2.53 0.42 1.31 4.0 0.14]T. Under zero initial condition, the state trajectories of the six agents are depicted in Figure 4, and the corresponding energy trajectories of d(i) and z(i) are given in Figure 5.
--- lldWIli, - Mt)IlL2
Figure 5: Energy trajectories of the controlled output z(i) and the communication error d(i).
It is noted obviously that ||z(i)||L2 < ||d(i)||L2. Thus, the multiagent system with the given DOF controller can achieve the consensus with the given HTO performance, which validates the effectiveness of the proposed protocol and demonstrates the correctness of the obtained theoretical results.
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 solve HTO problem for N - 1 linear time-invariant systems of order n, then the consensus with a desired HTO level can be reached. By using HTO 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.
Acknowledgment
This work is supported by National Natural Science Foundation of China under Grants 90916003 and 61074026.
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