Mean-square consensus of discrete-time multi-agent systems with

Research Article

Mean-square consensus of discrete-time multi-agent systems with Markovian switching topologies and persistent disturbances

International Journal of Distributed Sensor Networks 2017, Vol. 13(8) Ó The Author(s) 2017 DOI: 10.1177/1550147717726313 journals.sagepub.com/home/ijdsn

Lipo Mo1, Yintao Wang2, Tingting Pan1 and Yikang Yang3

Abstract This article deals with the leader-following mean-square consensus problem of discrete-time general linear multi-agent systems with Markovian switching topologies and persistent disturbances. Assume that the communication topology is not connected at any time but the union topology is connected. First, the estimators are designed to calculate the states of agents when external disturbance not exists. Based on the error information between the truth-values and estimatedvalues of states, the compensators are proposed to subject to the effect of persistent disturbances. And then, a new mean-square consensus control protocol is proposed for each agent. Second, by using the property of permutation matrix, the original closed-loop system is transferred into an equivalent system. Third, sufficient conditions for meansquare consensus are obtained in the form of matrix inequalities. Finally, numerical simulations are given to illustrate the effectiveness of the theoretical results. Keywords Markovian switching, mean-square consensus, persistent disturbance, multi-agent systems, estimator

Date received: 23 February 2017; accepted: 19 July 2017 Academic Editor: Juan Cano

Introduction Recently, the distributed coordination control problems of the multi-agent systems have attracted intensive attention from different fields’ scholars. As we know, the consensus problem is the essential problem in the distributed coordination control field. A lot of results were reported. For example, in Viesek,1 the model was proposed from statistical mechanics. And then, Ren and Beard2 proved that the multi-agent systems with directed topologies can achieve consensus if the union of the graphs have a spanning tree. For the discretetime systems, Xiao et al.3 solved the consensus problem of a single-leader multi-followers system. In Lin and Jia4 second-order discrete-time agents with nonuniform time-delays were investigated. Then, the output consensus problem was studied for a class of uncertain linear

multi-agent systems.5 More recently, the constrained consensus problems were proposed and some interesting results were obtained.6,7 In the practical engineering applications, the systems often suffer the external disturbances. Using the theory of stochastic stability, Li and Zhang8 investigated the 1

School of Science, Beijing Technology and Business University, Beijing, P.R. China 2 School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, P.R. China 3 School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, P.R. China Corresponding author: Lipo Mo, School of Science, Beijing Technology and Business University, Beijing 100048, P.R. China. Email: [email protected]

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International Journal of Distributed Sensor Networks

mean-square consensus problem of leader-following systems with measurement disturbance, and then, these results were extended to the general linear timeinvariant systems.9 Meanwhile, using the method of robust control, the H‘ consensus problem for multiagent systems with first-order, second-order, and high-order integrators was investigated in Jia and colleagues.10–12 In applications, whether the information exchanges or not between agents is also stochastic, the Markovian switching model was used to deal with this situation. Zhang and Tian13 proved that the switching systems can realize mean-square consensus when the union of graphs has a globally reachable node. In Matei and John,14 the consensus in almost sure sense can be achieved if and only if the union of graphs is strongly connected for both continuous-time and discrete-time systems. Cheng et al.15 studied the mean square and almost sure consensus problem of discrete-time linear multi-agent systems with communication noises and Markovian switching topologies. Then, the distributed output feedback algorithm and containment algorithm were proposed for the Markovian switching multiagent systems.16,17 In the process of military confrontation, powerful jamming signals beamed from enemy missiles. So, researching on how to design control protocols for multi-agent systems with persistent disturbances has great significance. For the leader-following system with unknown external persistent disturbance and connected topology, the leader-following consensus problems were also solved based on the estimated information of states and disturbance.18,19 By studying recent literatures, we found that both the topological structure and the persistent disturbance might destroy the stability of multi-agent systems. Therefore, it is meaningful to investigate the consensus problem when the topology switches stochastically and the system is upset by the persistent disturbances. However, to the best of our knowledge, there are not many results on combining both difficulties. And this problem cannot be solved directly by the previous methods.18,19 In this article, we focus on discussing this problem. Different with the methods on robust H‘ consensus,10–12 the observer technique was employed to estimate the external disturbances. A new control protocol was proposed to drive the multi-agent system with persistent disturbances to achieve mean-square consensus when the topology switches stochastically among some unconnected graphs, and the switch is driven by an ergodic Markovian chain.

Notations n

Throughout this article, R represents the set of n dimensions vector and Rm 3 n represents the set of real

m 3 n dimensions matrix. Let In indicates a n 3 n dimensions identity matrix and 0 denotes zero matrix with appropriate dimension.  represents Kronecker product, l(P) denotes the eigenvalue of matrix P, Re(l) represents the real part of eigenvalue, k P k denotes matrix norm and r(P) represents the spectral radius.

Preliminaries A graph G = (V , E, A) consists of node set V = fv1 , v2 , . . . , vN g, edge set E  V 3 V and an adjacency matrix A = ½aij . In graph G, (vj , vi ) 2 E means that agent j is the neighbor of agent i and Ni = fj : (vj , vi ) 2 Eg represents the set of all neighbors of agent i. Adjacency matrix A = ½aij  denotes the connected relationship between nodes, where the elements aij in matrix A is 0 or 1. If (vj , vi ) 2 E, then aij = 1, otherwise aij = P0. The Laplacian P matrix L = D  A, where D = diagf j2N1 a1j , . . . , j2Nn aNj g. For undirected graph, Laplacian matrix is a symmetric matrix. If G is where connected, then 0 = l1  l2      lN , l1 , l2 , . . . , lN are the eigenvalues of L. For the leaderfollowing systems, we define a diagonal matrix D = fd1 , d2 , . . . , dN g, where di = 1, if the leader is a neighbor of agent i, otherwise, di = 0. We use graph G to represent the leader-following system, which consists of N following agents and one leader agent. And sym^ is defined as metric matrix H associated with G H = L + D. In this article, we consider the multi-agent systems with stochastically switching topologies provided that the union of topologies has a globally reachable node. The switch is governed by a Markovian chain fs(k), k 2 Ng, whose state space M = f1, 2, . . . , mg. The transition probability matrix is denoted by G 2 Rm 3 m , and its element g ij = Prfs(k + 1) = jjs(k) = ig, 8i, j 2 M. If the Markovian chain fs(k)g is ergodic, then it has the stationary distribution, denoted by p, which means GT p = p. Accordingly, Markovian switching topology Gs(k) evolves in the set fG1 , G S2 , . . . , Gm g, union graph G is defined as G = s2M Gs , symmetric matrix Hs(k) evolves in the set fH1 , H2 , . . . , Hm g, and the symmetric matrix H associatedP with union graph G is defined as H = s(k)2M Hs(k) . Lemma 1. Let A, B, C, D be the matrices with appropriate dimensions and Kronecker product has the following properties20 1. 2. 3.

(A  B)(C  D) = (AC)  (BD) (A  B)1 = A1  B1 (A  B)T = AT  BT

Mo et al.

3

Lemma 2 (Schur For block symmetric  complement).  C11 C12 , the following three condimatrices C = T C12 C22 10 tions are equivalent: 1. 2. 3.

^ i (k + 1) = F2 w " # X aij (k)(xi (k)  ^xi (k)  (xj (k)  ^xj (k))) + di (k)(xi (k)  ^xi (k)) j2Ni (k)

^ i (k) +w

ð4Þ

C\0 T 1 C11 C12 \0 C11 \0, C22  C12 1 T C22 \0, C11  C12 C22 C12 \0

Lemma 3. If graph G is connected and undirected, the symmetric matrix H is positive definite. And all the eigenvalues of H are positive.21 Lemma 4. Let X , Y , F are the matrices with appropriate dimensions, and F T F  I, then for any e.0, we have XFY + Y T F T X T  eXX T + (1=e)Y T Y .22 Lemma 5. If all eigenvalues of A 2 Rn 3 n have negative real parts, then the eigenvalues of A  In + In  A have negative real parts as well.15

where F2 2 Rq 3 n is also a coefficient matrix to be determined in the following. We adopt the following control protocol in this article, which consists of two parts, one part is based on the traditional control input and the other part is the estimation of disturbances X ui (k) = K aij (k)(xj (k)  xi (k)) j2Ni (k) ð5Þ ^ i (k) + Kdi (k)(x0 (k)  xi (k))  w where K 2 Rq 3 n is the control gain. Using control protocol (5), we can get the following closed-loop system xi (k + 1) = Axi (k) + BK

P

aij (k)(xj (k)  xi (k)) +

j2Ni (k)

^ i (k)) BKdi (k)(x0 (k)  xi (k)) + B(wi (k)  w

System model Consider a multi-agent system consisting of N + 1 agents. Assume that the dynamic equation of following agent i is as follows xi (k + 1) = Axi (k) + B½ui (k) + wi (k), i 2 1, . . . , N ð1Þ

where xi (k) 2 Rn and ui (k) 2 Rq represent state and control input of agent i, respectively; wi (k) 2 Rq represents persistent disturbances; and A 2 Rn 3 n , B 2 Rn 3 q are constant matrices. Assume the dynamic equation of leader is as follows x0 (k + 1) = Ax0 (k)

ð2Þ

ð6Þ

Definition 1. We say the control protocol (5) can solve the leader-following mean-square consensus problem of system (1) with (2) if the following equations hold lim Ejjxi (k)  x0 (k)jj2 = 0,

k!‘

i = 1, . . . , N

ð7Þ

The purpose of this article is to study the meansquare leader-following consensus condition for multiagent systems with persistent disturbances and the Markovian switching topologies.

Main results

~ i (k) = w ^ i (k)  wi (k), and Let ~xi (k) = xi (k)  ^xi (k), w where x0 (k) 2 Rn represents the state of the leader. ei (k) = xi (k)  x0 (k). Denote ~x(k) = (~xT1 (k), . . . , ~xTN (k))T , The state estimation of agent i is denoted by ^xi (k) X ^xi (k + 1) = A^xi (k)+BK aij (k)(^xj (k)  ^xi (k))+BKdi (k) j2Ni (k)

3 (x0 (k)  ^xi (k)) + F1

"

X j2Ni (k)

#

ð3Þ

aij (k)(xj (k)  bx j (k)  (xi (k)  bx i (k))) + di (k)(b x i (k)  xi (k))

where F1 2 Rn 3 n is a coefficient matrix. ^ i (k), and its The estimation of wi (k) is denoted as w dynamic equation is as follows

~ TN (k))T , and e(k) = (eT1 (k), . . . , ~ (k) = (~ w wT1 (k), . . . , w T T en (k)) . According to equations (3)–(6), we can obtain that

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International Journal of Distributed Sensor Networks e(k + 1) = (IN  A)e(k)  (Ls  BK)e(k)  (Ds  BK)e(k)  (IN  B)~ w(k) ~x(k + 1) = (IN  A)~x(k)  (Ls  BK)~x(k)  (Ds  BK)~x(k) + (Ls  F1 )~x(k) + (Ds  F1 )~x(k)  (IN  B)~ w(k) ~ (k + 1) = (Ls  F2 )~x(k) + (Ds  F2 )~x(k) + (IN  Iq )~ w w(k)

Then, the above three equations can be written in the following matrix form 2

3

2

  BK f e(k + 1) A 6 7 6 4 ~x(k + 1) 5 = 4 0 ~ (k + 1) w 0 2 3 e(k) 6 7 ~ 4 x(k) 5 ~ (k) w

s

3

0 s

  BK f +F f1 A s f F2

s

 B 7 5 B Iq

 = IN  A, BK f s = Hs  BK, B  = IN  B, where A s s f2 = Hs  F2 , and Iq = IN  Iq , and f1 = Hs  F1 , F F the switching signal fs, s 2 Mg is a Markovian chain. Then, the leader-following mean-square consensus problem of systems (1) and (2) is converted to the mean-square stability problem of system (8). For undir s , Hs , s 2 M are symected and connected graph G metric matrices, and there exists a set of orthogonal ^ s = UsT Hs Us matrices fUs 2 Rn 3 n , s 2 Mg, such as H s s s s = diag(l1 , l2 , . . . , lN ), where li (i = 1, . . . , N ) are the eigenvalues of matrix Hs . According to Lemma 3, it is easy to see that lsi .0 (i = 1, . . . , N ). 3 2 T e(k) Us  I n Let 4 x(k) 5=4  (k) w

32

3 e(k) 5 4 ~x(k) 5, T ~ (k) w Us  I q

UsT  In

then system (8) can be changed into the following form 3 2 s   BK c e(k + 1) A 6 7 6 4 x(k + 1) 5 = 4 0 2

 (k + 1) w 2 3 e(k) 6 7 4 x(k) 5  (k) w

0

0 s

c +F c1 A  BK s c2 F

s

0

A  li BK B 0 fsi = @

0 A  li BK + li F1

1 B C B A,

li F2

0

Iq

ð11Þ

i = 1, . . . , N

Without loss of generality, we only need to consider the stability of the ith block matrix. Denote the ith  i (k)T )T and the system block as ei (k) = (ei (k)T , xi (k)T , w equation of the ith block is ð8Þ

2

simultaneously. Using permutation transformation, Fs can be changed into the block diagonal form as diag(fs1 , . . . , fsN ), where

3  B 7 5 B

ei (k + 1) = fsi ei (k),

i = 1, . . . , N

ð12Þ

To continue, we need the following lemma.  s is connected for some Lemma 6. Suppose that G s 2 M. If there exists a positive-definite symmetric   P1 P2 and ei .0 (i = 1, 2, 3), such that matrix P = PT2 P3 the following matrix inequalities hold AT P1 A  P1  AT P1 B(I + ln tBT P1 B)1 BT P1 A\0 ð13Þ 3 2 X1 I B K T BP2 K T BT P1 X2 1 7 6 P 6 I  2 2 0 0 0 0 7 7 6 ln m 7 6 7 6 e 3 7 6 BT 0  I 0 0 0 7 6 2 2 ln m 7\0 6 7 6 T e 2 7 6 P2 BK 0 0  I 0 0 2 7 6 l 7 6 n 7 6 e1 7 6 P1 BK 0 0 0  I 0 2 5 4 ln T 0 0 0 0 X3 X2 ð14Þ

I q 1 T



F1 F2



ð10Þ

take K = t(I + ln tB P1 B) B P1 A, = mP1   In , where X1 = AT P1 A  P1  l1 AT P1 BK  l1 K T 0 BT P1 A  l1 mA  l1 mAT + l2n mBK + l2n mK T BT + l2n K T BT P1 BK, X2 =  AT P1 B + AT P2  P2 , P = P1 1 + 1 T 1 T T T P1 P (P  P P P ) , X = B P B  P B  B P 2 3 2 3 1 2+ 1 2 1 2 e1 BT P1 B + e2 I + e3 I, t.1=l1 is large enough, and m is small enough. Then, r(Fs )\1.

coefficient matrix 1 s   c 0 B A  BK   BK cs+F c1 s B  A. Fs = @ 0 A c2 s Iq 0 F Note that the eigenvalues of the matrix are not changed by exchanging the row and corresponding column

Proof. First, let us discuss the spectral radius of the matrix (11). Note the form of equation (11), we only need to discussthe spectral radius of theblock matrices A  li BK + li F1 B . A  li BK and Iq li F2

T

ð9Þ

cs=H b s  BK, F c1 s = H b s  F1 , F c2 s = H b s where BK F2 , and s 2 M.  (k)T )T , equation (9) will Denote e(k) = (e(k)T , x(k)T , w be reduced to the following form e(k + 1) = Fs e(k)

where

0

Mo et al.

5

By taking K = t(I + ln tBT P1 B)1 BT P1 A with t.1=l1 . For the block matrix A  li BK, we have (A  li BK)T P1 (A  li BK)  P1

the block matrix  Similarly,  for   0 0 0 li K T BT P2 + , there exists li PT2 BK 0 0 0 e2 .0 such that

= AT P1 A  P1  li tAT P1 B(I + ln tBT P1 B)1 BT P1 A

   0 0 0 li K T BT P2 + li PT2 BK 0 0 0 !   2 T T 0 0 1 ln K B P2 PT2 BK 0  e2 + e2 0 I 0 0



 li tAT P1 B(I + ln tBT P1 B)1  BT P1 A + l2i t 2 AT P1 B(I + ln tBT P1 B)1 T

1 T

T

B P1 B(I + ln tB P1 B) B P1 A \AT P1 A  P1  AT P1 B(I + ln tBT P1 B)1 BT P1 A  AT P1 B(I + ln tBT P1 B)1 (I + ln tBT P1 B)1 BT P1 A \AT P1 A  P1  AT P1 B(I + ln tBT P1 B)1 BT P1 A

From condition (13), it is easy to see that

 And, 0 li mBT that 

T

(A  li BK) P1 (A  li BK)  P1 T

T

1 T

T

\A P1 A  P1  A P1 B(I + ln tB P1 B) B P1 A\0

According to the Lyapunov stability theorem, the above inequality is equivalent to r(A  li BK)\1. For the second block matrix, we have 

A  li BK + li F1

B



 =

B

A

0



 +

0

li K T BT P1 B

0 0

!

0 0

the block matrix li mB , there exists e3 .0 such 0 

 +

l2n m2 BBT 0

0 li mB 0 0 ! 0



  e3

0 0

0 I

 ð17Þ

0

From the above inequalities (14)–(16), we have X11 XT12

X12 X22



 

X11 XT2

X2 X3

 ð18Þ

where X11 =AT P1 A  l1 AT P1 BK  l1 K T BT P1 A  l1 mA l1 mAT +l2n mBK +l2n mK T BT +l2n K T BT P1 BK +(1=e1 ) l2n K T BT P1 P1 BK +l2n m2 P +(1=e2 )l2n K T BT P2 PT2 BK + (1=e3 )l2n m2 BBT  P1 . According to Lemma 2, inequality (14) is equivalent to 2 6 6 6 4



li BT P1 BK 0 0 0   0 0 1 l2n K T BT P1 P1 BK  e1 + T e1 0 B B 0

1 + e3

 for  0 + 0



where X11 = AT P1 A  li AT P1 BK  li K T BT P1 A  li mA  li mAT + l2i mBK + l2i mK T BT + l2i K T BT P1 BK + l2i m2 P T T T T T T P1 , X12 =  A P1 B +A P2 +li K B P1 B  li K B P2 +li mB  P2 , and X22 =BT P1 B  PT2 B   BT P2 .  0 0 For the block matrix li BT P1 BK 0   0 li K T BT P1 B , by Lemma 4, there exists e1 .0 + 0 0 such that 0

0 li mBT



li F2 Iq 0 Iq     li BK 0 l i F1 0 + + 0 0 l i F2 0     F1 1 In =  mP , then By taking F2 0     A  li BK + li F1 B T P1 P2 li F2 Iq PT2 P3     A  li BK + li F1 B P1 P2  li F2 Iq PT2 P3   X11 X12 = XT12 X22



0 0

ð16Þ

ð15Þ

X11  e11 l2n K T BT P1 P1 BK P1 BK XT2

K T BT P 1 e1  2I ln 0

X2

3

7 0 7 7\0 5 X3

ð19Þ

It is  easy to seethat inequality (19) is equivalent to X11 X2 \0, which implies that matrix XT2 X3 

A  li BK + li F1



li F2 A  li BK + li F1 li F2

 P1 P 2 Iq PT2 P3    B P1 P2  \0 Iq PT2 P3 B

T 

the of matrix  Therefore, eigenvalues A  li BK + li F1 B are within the unit circle. In Iq li F2 summary, r(fsi )\1. For the block diagonal matrix Fs = diag(fs1 , . . . , fsN ) with r(Fsi )\1, we know r(Fs )\1.

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International Journal of Distributed Sensor Networks

Remark 1. When (A, B) is controllable, we can find a matrix K such that Re(A  li BK)\1, and then, we can find a positive-definite 1 such   matrix P  that inequality I F1 (13) holds. Let =  mP1 n , for any P2 , it is F2 0 clear that there exist three positive scalars e1 , e2 , and e3 such that inequalities (15), (16), and (17) hold from Lemma 4. We can also find that matrix P3 guarantees matrix P is positive definite. Then, by selecting a large enough scalar t and a small enough scalar m, inequalities (13) and (14) are relatively easy to realize. Next, let us state our main result of this article; here, we suppose that each topology does not connected, but the union of topologies has a globally reachable node. Theorem 1. Consider system (1) with (2). Suppose that the Markovian chain fs(k)g is ergodic and union graph  has a globally reachable node. If there exists a G   P1 P2 and positive-definite symmetric matrix P = PT2 P3 positive scalars e1 , e2 , and e3 , such that AT P1 A  P1  AT P1 B(I + ln t BT P1 B)1 BT P1 A\0 ð20Þ

1 1 1 T 1  AT P1 B+AT P2 P2 , P=P 1 +P1 P2 (P3 P2 P1 P2 ) , 1 X3 =BT P1 BPT2 BBT P2 +e1 BT P1 B+e2 I +e3 I, t.1=l is large enough, and m is small enough. Then, the leader-following mean-square consensus problem can be solved by protocol (5).

Proof. Since fs(k)g is an ergodic Markovian chain and  is a union graph which has a globally reachable G  are all posinode, the eigenvalues of the symmetric H tive. P According to Lemma 6, we can obtain r( m i = 1 Fi )\1. By the property of ergodic Markovian chain, we have that lim s(k) = p, where p = k!‘

(p1 , . . . , pmP )T and pi .0, i = 1, . . . , m. P So, we also m pi Fi )\1. Denote have r( m i = 1 i = 1 p i Fi = Pm I + e i = 1 pi Ci , where e is a small enough positive number, and then, it is easy to see that P l( m p i = 1 i Ci )\0. According to Theorem 3.9 of Costa et al.,23 if r½(GT  I(2n + q)2 )diag(Fi  Fi )\1, system (12) is a mean-square stability. Q = (GT  I(2n + q)2 ) For Fi = I + eCi , denote diag(Fi  Fi ), and then, we have

Y

= GT  I(2n + q)  I(2n + q) 0 1 g11 (I(2n + q)  C1 + C1  I(2n + q) )    gm1 (I(2n + q)  Cm + Cm  I(2n + q) ) B C .. .. .. C + eB . . . @ A g (I(2n + q)  C1 + C1  I(2n + q) )    gmm (I(2n + q)  Cm + Cm  I(2n + q) ) 0 1m 1 g11 C1  C1    g m1 Cm  Cm B C .. .. .. C + e2 B . . . @ A g 1m C1  C1    g mm Cm  Cm 2

X1 6 6 I 6 6 BT 6 6 T  6 P2 BK 6 6 P BK 4 1  XT2

I

1

 lP2 m2 n 0

T

T

T

B 0

K BP2 0

K B P1 0

 l2em3 2 I

0

0

n

0

0

 le22 n

0

0

0

I

0  le12 I n

0

0

0

0

3

X2 7 0 7 7 0 7 7 7\0 0 7 7 0 7 5 X3 ð21Þ

 = t(I + ln tBT P1 B)1 BT P1 A and K and     take F1 I n , where li (i = 1, 2, . . . , N ) are =  mP1 F2 0  the eigenvalue of the union matrix H, T T T T      X1 = A P1 A  P1  l1 A P1 BK  l1 K B P1 A l1 mA 2 mBK  2 mK 1 mAT + l  +l  T BT + l2 K  X2 =  T BT P1 BK, l n n n

For small enough e, the last term can be treated as a perturbation term, which can be neglected compared with the first two terms; thus, we only need to discuss the matrix P1 :¼ (GT  I(2n + q)  I(2n + q) ) diagfI(2n + q)2 + e(I(2n + q)  Ci + Ci  I(2n + q) )g

It is easy to see that l(GT  I(2n + q)  I(2n + q) ) = 1 with algebraic multiplicity (2n + q)2 , and its eigenvector is p  a, where a is a (2n + q)2 -component vector. Denote l is an eigenvalue of matrix P1 , and m is defined by m = (1  l)=e; we can see that m is a eigenvalue of matrix GT  I(2n + q)  I(2n + q) diag(I(2n + q)  Ci + Ci  I(2n + q) ) and its eigenvector is b, b = ½bT1 , . . . , bTm T . Multiplying P1 by p  a + eb, the ith block matrix is

Mo et al. m X

7

h i gji I(2n + q)2 + e(I(2n + q)  Cj + Cj  I(2n + q) )

PT  I2n + q diag(Fi ) has at least q unit eigenvalues, and thus, limt!‘ E½e(k) 6¼ 0, which is contrary to mean-square consensus.

i=1

(pj  a + ebj ) = (1 + em)(pi a + ebi )

Thus, for all i = 1, 2, . . . , m, we can obtain that " e

m X

gji bj +

i=1

+ e2

m X

# gji pj (I(2n + q)  Cj + Cj  I(2n + q) )a

i=1

m X

g ji (I(2n + q)  Cj + Cj  I(2n + q) )bj

i=1

= ebi + empi a + e2 mbi

Summing up the above formula from i = 1 to m, we have e I(2n + q) 

m X

pj Cj +

j=1

m X

! pj Cj  I(2n + q) a

j=1

  + e2 I(2n + q)  Cj + Cj  I(2n + q) bj m X = ema + e2 m bi i=1

e, m depends on I(2n + q)  PmFor small Penough m p C + p C  I(2n + q) ; according to j j j j j=1 j=1  Lemma 5, the eigenvalues of matrix I (2n + q) Pm Pm p C + p C  I have negative real j j j j (2n + q) j=1 j=1 P parts as the same as matrix m i = 1 p i Ci , thus Re(m)\0. Then, l(P1 ) = 1 + em\1, which implies that r(P)\1. System (1) with (2) under Markovian switching topologies with a globally reachable node can achieve the mean-square consensus. It means that lim Ejje(k)jj2 = 0, which implies that the estimation k!‘ error of states and the tracking error converge to zero in the sense of mean square, that is lim Ejjx(k)  ^x(k)jj2 = 0

k!‘

lim Ejjxi (k)  x0 (k)jj2 = 0

,

i = 1, . . . , N

k!‘

Remark 2. In fact, the condition that the union topology has a globally reachable node is also a necessary condition for achieving mean-square consensus. Denote di (k) = E½ei (k)1s(k) = i  (i = 1, 2, . . . , m), we can get d(k + 1) = Snj= 1 pji Fj dj (k). Then, for any topology  i , the corresponding matrix Hi has at least one zero G eigenvalue, and by exchanging the row and corresponding column of Fi simultaneously, we can get that 0 Fi is similar1 to diagffi , f2 , . . . , fn g, where A 0 B F1 = @ 0 A B A, and it is obvious that f1 has 0 0 Iq q unit eigenvalues. For all i = 1, 2, . . . , n,

Remark 3. In Li and Zhang,8 by the sampling technique, the continuous-time system is transformed into an equivalent discrete-time system, and the mean-square average consensus of the system with measurement noises is obtained. While Li and Zhang8 do not consider the Markovian switching topologies. In practical systems, communication topologies are always switching. Thus, the problem considered herein has a wider scope of application. Remark 4. Li et al.24 consider the containment control of leader-following multi-agent systems with Markovian switching network, and there are multiple leaders in the system model. This article considers the mean-square consensus problem of discrete-time multiagent systems with a leader and persistent disturbances under Markovian switching topologies; the order of each agent’s dynamics in Li et al.24 is one, and this article considers the general linear multi-agent systems. Liu and Jia25 used the H‘ approach to study the consensus problem of high-order multi-agent systems with external disturbances, and the protocol in Liu and Jia25 does not have the estimation of disturbance. Li and colleagues24,26 consider the case with multiple leaders, and we will extend our results to the case with multiple leaders in our following work. Remark 5. In Cheng et al.,15 noises are inevitable in real communication environments, while in applications, persistent noise might occur along with the input signal. In this article, the effect of persistent disturbance on closed-loop systems is considered, and the analysis method here is totally different with the one in Cheng et al.15 In the future, we will study the consensus problem when communication noise and persistent disturbance exist simultaneously.

Numerical simulations In this section, we give the numerical simulation to check the effectiveness   of the above results. Suppose 0:8 0:1 0:1 A= ,B= . Assume that there are 0 0:9 0:1 four followers and one leader. Assume that topological structures switch between the following three topologies, as shown in Figure 1, and the persistent disturbances wi (k) = 1 (i = 1, 2, 3, 4) for all k. It is obvious that these graphs are not connected, but the union topology has a globally reachable node. The errors of tracking and state estimations are shown

8

Figure 1. Jointly connected topological structures between agents.

International Journal of Distributed Sensor Networks systems with Markovian switching topologies and persistent disturbances. A mean-square consensus control protocol consisting of estimators and compensators is proposed to subject to the effect of the persistent disturbances and the stochastic switches. Using the permutation matrix, sufficient conditions for mean-square consensus are obtained in the form of matrix inequalities. We will extend our results to the case with multiple leaders in our following work. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation (NNSF) of China (Grant No. 61304155) and the Beijing Municipal Government Foundation for Talents (Grant No. 2012D005003000005).

References Figure 2. The state errors.

Figure 3. The state estimations.

in Figures 2 and 3, respectively, which show that the multi-agent systems can achieve consensus.

Conclusion In this article, we investigate the leader-following meansquare consensus problem of discrete-time multi-agent

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