Event-triggered cooperative containment control for a class of uncertain non-identical networks

Applied Mathematics and Computation 375 (2020) 125065

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Event-triggered cooperative containment control for a class of uncertain non-identical networks Yu Zhou, Yingnan Pan∗, Shubo Li, Hongjing Liang College of Engineering, Bohai University, Jinzhou 121013, China

a r t i c l e

i n f o

Article history: Received 20 June 2019 Revised 4 January 2020 Accepted 12 January 2020

Keywords: Containment control Event-triggered Multi-agent systems

a b s t r a c t A practical event-triggered design method is developed for cooperative robust containment control problem of multi-agent systems with multiple leaders. The multi-agent systems discussed in this paper are essentially heterogeneous, and each follower is modeled by a integrator with the external disturbance. A containment error also has been expressed to ensure all the outputs of followers converge to the convex hull spanned by the outputs of the leaders. For the case that the undirected communication graph among followers is connected, a distributed containment compensator is designed based on the local state information of neighboring agents. Furthermore, a distributed output feedback controller is designed to solve the event-triggered containment control problem of the continuoustime multi-agent systems. Finally, a numerical example is given to verify the effectiveness of the main results. © 2020 Elsevier Inc. All rights reserved.

1. Introduction Recently, the cooperative control problem of multi-agent systems has attracted many scholars’ attention, which has numerous results about vehicle formations [1,2], mobile autonomous agents [3,4], switching networks [5,6], and optimal control [7–9]. The crucial part of solving cooperative control problem for multi-agent systems is to design the distributed control law, such that the states or outputs information of each agent can reach an agreement. Many existing results had been conducted on consensus problem [10–15] with the rapid development of the distributed control. The limited and unreliable information interaction was considered in [16] among multiple agents for consensus problem based on dynamically changing topologies, and the information can achieve consensus asymptotically if the directed graph contains a spanning tree. Both time-invariant and changing topologies were presented in [17] for the multi-agent systems with consensus seeking conclusion. In [18], the research of input and communication delays had been expanded to the consensus problem for multi-agent systems. The aim of the leader-following problem is to design distributed control law, such that the states or outputs of followers can track the information of leader. In [19], the distributed observers were designed based on second-order followeragents with switching interconnection topology. The case where the leader is signal and active was further considered for leader-following problem in [20], in which the state of the leader was supposed to be unmeasured. Thus, a control law was designed based on the state estimation of each agent with their neighbors. However, some other methods were presented ∗

Corresponding author. E-mail addresses: [email protected] (Y. Zhou), [email protected] (Y. Pan), [email protected] (S. Li), [email protected] (H. Liang). https://doi.org/10.1016/j.amc.2020.125065 0 096-30 03/© 2020 Elsevier Inc. All rights reserved.

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Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

for multi-agent systems, such as event-triggering sampling based consensus [21] and the semi-global leader-following consensus problem [22]. Based on the study of the H∞ control [23,24] and time-varying delays [25,26], these problems are expected to be extended to the multi-agent systems in the future. However, a lot of results lead to the containment control of multi-agent systems. The containment control problem for multi-systems with a group of agents which contain at least two leaders, and there exists a convex hull spanned by the leaders. By designing a distributed control law, the information of the followers can converge to the convex hull. Based on a directed communication topology, a distributed output feedback controller had been designed to solve the containment control problem for the multi-agent systems in [27]. In practical applications, there may exist uncertain parts in systems. To handle this problem for the discrete-time multi-agent systems, a distributed internal model compensator was proposed in [28] based on the containment control. Different from the previous results, Ref. [29] solved the state containment problem by the approach of output regulation framework. However, containment control is also widely used in practice. For example, a group of unmanned vehicles will perform a formation flight mission. Some vehicles play the role of the leaders, and the others can be seen as the followers. The leaders will form a safe flight region, and the followers will perform the main task of formation flight within the safe region. In most control schemes, information communication needs to be continuous, which will lead to excessive energy consumption. Thus, event-triggered strategy [30–34] has been proved to be an effective way to save energy. Asynchronous adaptive event-triggered tracking control had been studied for multi-agent systems in [35]. Under the time-trigger and event-trigger switching, Ref. [36] proposed a distributed hybrid-triggered transmission scheme. Based on the packet losses problem, the event-triggered strategy had been considered in [37] for discrete-time systems. In [38], the real-time scheduling of stabilizing control tasks had been solved based on the event-triggered method. The event-triggered was used to solve an agreement problem of first-order dynamics for multi-agent systems in [39]. Consider the output regulation problem in [40] and the containment control problem in [41], both of these problems were solved based on the method of event-triggered control. Motivated by the above discussions, the event-triggered containment control problem is addressed of the multi-agent systems. We design the event-triggered condition for each agent. Then, the stability of closed-loop system need to be proved such that the problem of event-triggered containment control can be solved. Compared with the method presented for containment control in Ref. [29], this paper further considers the difficulties in practical applications. For example, the uncertain parts and unmeasurable states of the system, the excessive energy consumption. Thus, the contributions of this paper are shown as follows: (i) in this paper, a distributed observer is designed to estimate the information of convex hull, and a novel event-triggered condition is designed for the containment control. In addition, based on the designed event-triggered condition, the effectiveness of the distributed observer is verified by a Lyapunov function. (ii) In many practical applications, such as the system models in [42–44], there may exist the uncertain parts or systems in the dynamics. Thus, a distributed internal model compensator is used to handle the uncertain parts of the systems in this paper. (iii) Different from the observers designed in [28,29], the distributed observer is designed based on the event-triggered strategy with a measurement error in this paper. In other words, the distributed observer presented in this paper is mainly used to estimate the information of the convex hull spanned by the leaders at the trigger time. Then, the problem of event-triggered containment control can be solved by a dynamic output feedback approach. The rest sections are organized as follows: the notations, graph theory and definitions are presented in Section 2. In Section 3, a dynamic model is formulated for multi-agent systems for the containment control problem. The main results are presented in Section 4. In Section 5, we present a numerical example. At last, conclusions are given in Section 6. 2. Preliminaries 2.1. Notation The set of real numbers can be denoted as R and C represents the complex number set. The distance between a and b can be expressed as dist(a, b), in which a, b ∈ Rn . 1N denotes an unit column vector with N rows, and  represents the Kronecker product. I is an unit matrix, and the zero matrix can be seen as 0. λ(A) represents the eigenvalues of square matrix A. 2.2. Graph theory An undirected graph G = (V , E, A ) consists of a nodes set V = {v1 , v2 , . . . , vN } and a edges set E ⊆ V × V and a weighted N×N adjacency matrix A = [ai j ] ∈ R . If there is an edge between vi and vj , i.e., (v j , vi ) ∈ E, then the weighted adjacency ele ment aij > 0, otherwise, ai j = 0. The neighbors set of vi is denoted as Ni = {v j (v j , vi ) ∈ E, i = j }. An undirected path means a sequence of undirected edges, which is a sequence of edges (vi1 , vi2 ), (vi2 , vi3 ), . . . , (vim−1 , vim ) in an undirected graph. In this paper, the paths from leaders to followers are directed, and there exist at least one root node which can be seen as the leader node that has a directed path to each follower. In this paper, a group of agents has been considered with N followers and M − N leaders, and all the leaders have their own path to each follower. R = {vN+1 , vN+2 , . . . , vM } is a leaders set, and F = {v1 , v2 , . . . , vN } is a followers set. An undirected graph G = (F , E, A ) represents the communication topology among the N followers, in which

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

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A = [ai j ] represents the weighted adjacency matrix of G , the Laplacian matrix of G is denoted as L = D − A, and D =    diag{ Nj=1 a1 j , Nj=1 a2 j , . . . , Nj=1 aN j }. 2.3. Definition Convex hull [45]: If there exist any x, y ∈ C and λ ∈ [0, 1], such that a convex C ⊆ RN satisfies (1 − λ )x + λy ∈ C. Then, we can obtain the following convex hull equation.



Co(X ) =

N 

αi xi | xi ∈ R , αi ≥ 0, p

N 

i=1



αi = 1 .

i=1

in which, X = {x1 , x2 , . . . , xN } can be seen as a set in V ⊆ R p . 3. Problem formulation The continuous-time multi-agent systems consist of N followers’ dynamics and M − N exosystems will be considered in this section. The dynamics of followers are given as follows:

⎧ ⎨

M 

⎩

l=N+1

x˙ i (t ) = A¯ i xi (t ) + B¯ i ui + yi = C¯i xi (t ), i ∈ F,

E¯il ωl ,

(1)

where xi ∈ Rn is the state of ith agent, and ui ∈ Ru is the input of ith agent, ωl ∈ Rq are the exosystems’ states which can express the reference inputs or the disturbances, and the measured output yi ∈ R p . The following matrices A¯ i , B¯ i , E¯il , C¯i are uncertain parts with appropriate dimensions.

A¯ i = Ai + Ai , B¯ i = Bi + Bi , E¯il = Eil + Eil , C¯i = Ci + Ci , in which Ai , Bi , Eil and Ci are considered as the disturbance matrices. The uncertain parts of the systems can be written as the following form

 = (vecA, vecB, vecC, vecEN+1 , . . . , vecEM−N )T , where  ∈ RNn(n+u+ p+(M−N )q ) and

A = (A1 , . . . , AN ), B = (B1 , . . . , BN ), C = (C1 , . . . , CN ), EN+1 = (E1,N+1 , . . . , EN,N+1 ), EM−N = (E1,M−N , . . . , EN,M−N ). The M − N leaders’ dynamics and reference outputs are generated with the following form:



ω˙ l = Sωl , yrl = Fr ωl .

(2)

in which, S ∈ Rq×q is expressed as the matrix of leaders, l ∈ R, and yrl ∈ R p is the reference output. Definition 1. The containment control problem of linear multi-agent systems based on event-triggered strategy can be solved if the properties are satisfied:

(1 ) By designing an appropriate distributed control law based on event-triggered strategy, the closed-loop system is asymptotically stable without considering the external disturbances. (2 ) There exist an open neighborhood W of  = 0. For all  ∈ W, there exist a convex hull spanned by the outputs of leaders, and each follower’s output will converge to the convex hull with the initial conditions xi (0) and ωl (0), i.e., lim dist(yi , co(yrl , l ∈ R )) = 0, i ∈ F.

t→∞

(3)

The following containment error is shown as:

e˜i =

 j∈Ni

ai j ( yi − y j ) +

M 

ail (yi − yrl ), i ∈ F.

l=N+1

(4)

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Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

Let e˜ = col(e˜1 , e˜2 , . . . , e˜N ), x = col(x1 , x2 , . . . , xN ), then M 

e˜ = (H I p )C¯x(t ) −

(Hl  Fr )ω¯ l ,

(5)

l=N+1

where, C¯ is a diagonal block matrix with elements C¯1 , C¯2 , . . . , C¯N , Hl = 1/(M − N )L +A0l , A0l = diag{a1l , a2l , . . . , aNl }, and  H= M ¯ l = 1N  ωl . l=N+1 Hl , ω According to the Lemma 7 presented in [29], the matrix H can be proved to be an M-matrix. Lemma 1 [29]. Consider the systems presented in (1) and (2), the convergence of the followers to the convex hull spanned by the leaders can be expressed as limt→∞ e˜ = 0. According to Lemma 1, if limt→∞ e˜i = 0, we can obtain the following equation:

lim C¯i xi (t ) = lim

t→∞

t→∞

M 

ζil Fr ωl , i ∈ F,

(6)

l=N+1

in which, ζil ∈ R is the element in i-th row of ζl = H−1 Hl 1N . To solve the problem of the event-triggered containment control for multi-agent systems, the following assumptions is needed in this paper. Assumption 1. The topology among followers is undirected, and there exist at least one leader that has a path to each follower. Assumption 2. (Ai , Bi ) is stabilizable and (Ci , Ai ) is detectable. Assumption 3. There are no negative real parts in all the eigenvalues of S. Lemma 2 [46]. For any vectors c1 , c2 , . . . , the following inequality holds:

 2   n n    2  c i  ≤ n c i .  i=1  i=1

(7)

4. Main result The internal model method will be used to handle the uncertain parts, and some important lemmas will be given to solve the final problem in this section. The pairs (G1 , G2 ) incorporates a p-copy internal model of the matrix S and

G1 = block diag(γ1 , γ2 , . . . , γ p ), G2 = block diag(δ1 , δ2 , . . . , δ p ),

(8)

the γ i and δ i are the matrix and vector respectively to be designed later, in which i = 1, . . . , p, and (γ i , δ i ) is controllable. What’s more, the characteristic polynomial of γ i divided by the minimal polynomial of S. Lemma 3 [28]. Under Assumption 3, if the matrices pair (G1 , G2 ) incorporates a p-copy internal model of S, and let



Ai =

Ai

0

G2Ci

G1

Bi

, Bi =

0

,

then the pair (Ai , Bi ) is stabilizable. Based on the event-triggered strategy, the following designed dynamic measurement output feedback controller can be written as:



η˙ i = Sηi + Kη



 i



M 

ηi t k − η ( ) + i j tk

j∈Ni

   i  i  ail ηi tk − ωl tk ,

l=N+1

ξ˙i = G1i ξi + G2i (yi − Fr ηi ), ui = Ki ξi , i ∈ F, in which, ξi ∈ follows:

Rn+ psm ,



G 1i =

(9)

Ki = (K1i , K2i ) is the gain matrix, and

Ai + Bi K1i + LiCi

Bi K2i

0

G1

, G 2i =



−Li G2

,

t0i , t1i , . . . ,

are the triggering times. G1i and G2i are designed as

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

5

where G1 ∈ R psm ×psm , G2 ∈ R psm ×p . Lemma 4. The pairs (G1 , G2 ) incorporates a p-copy internal model of S. Let

min det (λI − S ) = λsm + a1 λsm−1 + · · · + asm−1 λ + asm be the minimal polynomial of S. γ i and δ i in (8) can be chosen as the following form:

⎛

γi

0 ⎜ 0 ⎜ . =⎜ ⎜ .. ⎝ 0 −asm

··· ··· .. . ··· ···

1 0 .. . 0 −asm−1

⎛ ⎞

0 0 .. . 0 −a2

⎞

0 0 ⎟ .. ⎟ ⎟ . ⎟, 1 ⎠ −a1

0

⎜0 ⎟ ⎜.⎟ ⎟ δi = ⎜ ⎜ .. ⎟, ⎝0 ⎠

(10)

1 where i = 1, 2, . . . , p, γi ∈ Rsm ×sm , and δi ∈ Rsm ×1 .  i  Lemma 5. Let εi (t ) = ηi (t ) − M l=N+1 ζil ωl , define a measurement error ei (t ) = εi tk − εi (t ), and the following algebraic Riccati equation has a unique solution P > 0:

P S + ST P + 2P 2 + μ2 In = 0,

(11)

in which μ2 is chosen as a positive constant and (S, Iq ) is stabilizable. Suppose Assumption 1 is satisfied, let the matrix Kη = μ1 P and choose μ1 > 0 such that 0 < μ1 λi ≤ 1, where λi is the eigenvalue of H. Then, the the error ei tends to 0 as the time tends to infinity based on the following triggering condition:

e¯i (t ) − βi ε¯i (t ) = 0,

       where e¯i (t ) =  Nj=1 hi j ei (t ), ε¯i (t ) =  Nj=1 hi j εi (t ), and βi2 = σ μ2 /8μ21 ρ 2 N Nj=1 h2i j < 1 with σ ∈ (0, 1 ), ρ = P 2 .

(12)

Proof. The derivative of ε i can be rewritten in the following compact form:



ε˙ = (IN  S ) η −

M 



  H−1 Hl  In ω ¯ l (t ) + (IN  Kη )(H  In )(e(t ) + ε (t ) )

l=N+1



= IN  S + H  Kη

   ε (t ) + H  Kη e(t ),

(13)

    where ε = ε1T , ε2T , . . . , εNT T , e = eT1 , eT2 , . . . , eTN T and ω ¯ l (t ) = (ωlT (t ), ωlT (t ), . . . , ωlT (t ) )T . Given the following Lyapunov function:

V (ε ) = ε T (IN  P )ε .

(14)

Since the matrix Kη = μ1 P, the derivative of V(ε ) can be obtained as follows:

V˙ = ε˙ T (IN  P )ε + ε T (IN  P )ε˙ =

      ε T IN  ST P + PS + H  2μ1 P2 ε + ε T H  2μ1 P2 e.

(15)

Based on the properties of M-matrix which were presented in Lemma 7 of [29] and Assumption 1, the matrix H has no zero and negative eigenvalues. However, matrix H is symmetric, and an orthogonal matrix U ∈ RN×N is presented such that

U HU T = J = diag{λ1 , λ2 , . . . , λN }.



 T T

U U T = IN and H = UT JU. Let εˆ = (U  In )ε = εˆ1T , εˆ2T , . . . , εˆN

V˙ = εˆ =

 T

N  i=1

V˙ ≤

IN 





(16) , then (15) can be rewritten as

   ST P + P S + J  2μ1 P 2 εˆ + ε T H  2μ1 P 2 e

n  N    εˆiT (t ) ST P + PS + 2μ1 λi P2 εˆi (t ) + εiT (t )hi j 2μ1 P2 ei (t ), i=1 j=1

n N     εˆiT (t ) ST P + PS + 2P2 εˆi (t ) + μ1 2εiT (t ) hi j P2 ei (t ).

N  i=1

i=1

j=1

(17)

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Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

The inequality 2 a

b ≤ κ a 2 + 1/κ b can be used for any κ > 0, and by using εˆiT εˆi = εiT εi , a, b ∈ Rn . 2

⎛

V˙ ≤ −μ2

N 

εi (t ) 2 + ⎝μ1

i=1 N   μ2 + =− i=1

κ



2 ⎞

 j=1



 N  N   

εi (t ) 2 + κμ1 ρ 2  hi j ei (t )

n 1 i=1

i=1

⎠

N  μ1 

εi (t ) 2 + κμ1 ρ 2 e¯2i (t ). κ i=1

(18)

Remark 1. There exist many results in solving the containment control problem based on the directed graphs, such as [47,48]. However, this paper studied the containment control problem based on the undirected graphs. Because the equation εˆiT εˆi = εiT εi is used for the derivation of Lyapunov function (14), and εˆiT εˆi = εiT εi works because the matrix U is orthogonal. In fact, the matrix H is asymmetric if the graph is directed and we cannot guarantee whether U is orthogonal in the diagonalization process of matrix H, which will affect the subsequent derivation and stability proof. Therefore, we study the containment control problem based on the undirected graph in this paper. Furthermore, the following inequation will enforced by the triggering condition (12):

e¯i (t ) − βi ε¯i (t ) ≤ 0.

(19)

According to Lemma 2, it follows from (19) that

βi2 ε¯i2 (t )  2   N  2 = βi  hi j εi (t )  j=1 

e¯2i (t ) ≤

≤

βi2 N

N  h2i j εi (t ) 2 .

(20)

j=1

Then (18) can be rewritten as follows: N  μ1 

εi (t ) 2 + κμ1 ρ 2 e¯2i (t ) κ i=1 i=1  N  N N    μ μ2 − 1 εi (t ) 2 + κμ1 ρ 2 βi2 N h2i j εi (t ) 2 ≤− κ i=1 i=1 j=1   N N   μ =− μ2 − 1 − κμ1 ρ 2 βi2 N h2i j × εi (t ) 2 . κ i=1 j=1

N   μ2 − V˙ ≤ −

(21)

Since μ2 , μ1 , κ > 0, it yields N  μ2 V˙ ≤ − (1 − σ ) εi (t ) 2 < 0. 2 i=1



Thus, system (13) is asymptotically stable. We can obtain lim εi (t ) = 0, lim ηi (t ) − t→∞

gering condition (12), limt→∞ ei = 0. The proof is completed.



t→∞

M

  = 0, and from the trig-

l=N+1 ζil ωl (t )

mentioned in Definition 1 will be proved by the controller (9). Next, the properties  Let ξi = ξ1Ti , ξ2Ti T , the closed-loop network dynamics resulting from (1), (2) and (9) can be written as

x˙ i (t ) = A¯ i xi (t ) + B¯ i K1i ξ1i + B¯ i K2i ξ2i +

M 

E¯il ωl ,

l=N+1

ξ˙1i = −LiCi xi (t ) + (Ai + Bi K1i + LiCi )ξ1i + Bi K2i ξ2i + Li Fr ηi (t ), ξ˙2i = G2C¯i xi (t ) + G1 ξ2i − G2 Fr ηi (t ),   M N  N     η˙ i (t ) = Sηi (t ) + Kη ail (ηi (t ) − ωl ) + Kη hi j e i . ηi (t ) − η j (t ) + j∈Ni

l=N+1

i=1 j=1

(22)

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

Let



  η = η1T , η2T , . . . , ηNT T ,  T T T  T T  ξ1 = ξ11 , ξ12 , . . . , ξ1TN , ξ2 = ξ21 , ξ22 , . . . , ξ2TN T ,  T T  T T x = xT1 , xT2 , . . . , xTN

T

7

e = e1 , e2 , . . . , eN

,

,





A¯ = block diag A¯ 1 , A¯ 2 , . . . , A¯ N ,





B¯ = block diag B¯ 1 , B¯ 2 , . . . , B¯ N ,





C¯ = block diag C¯1,C¯2 , . . . , C¯N ,





E¯l = block diag E¯1l , E¯2l , . . . , E¯2l , K1 = block diag{K11 , K12 , . . . , K1N }, K2 = block diag{K21 , K22 , . . . , K2N }, L = block diag{L1 , L2 , . . . , LN }. The compact forms of closed-loop (22) can be rewritten with the following forms: M 

x˙ = A¯ x + B¯ K1 ξ1 + B¯ K2 ξ2 +

¯ l, E¯l ω

l=N+1

ξ˙1 = −LCx + (A + BK1 + LC )ξ1 + BK2 ξ2 + L(IN  Fr )η, ξ˙2 = (IN  G2 )C¯x + (IN  G1 )ξ2 − (IN  G2 Fr )η, M        η˙ = (IN  S )η + H  Kη η − Hl  Kη ω ¯ l + H  Kη e. 

Let xc = xT , ξ1T , ξ2T

T



, and vl = ω ¯ lT , ηT

M 

x˙ c = A¯ c xc +

(23)

l=N+1

T

, with ζ il defined in (6), then it yields:

B¯ cl vl ,

(24)

l=N+1

in which,

A¯ c =

⎛ ⎝

A¯

B¯ K1

B¯ K2

−LC

A + BK1 + LC

BK2

0

IN  G1

(IN  G2 )C¯

⎛¯ B¯ cl =

⎞

0

El

⎝0

1 L M−N

0

−IN 

⎞ ⎠,

(IN  Fr ) ⎠. 1 G F M−N 2 r

A transformational matrix which is used to prove  the stability  of matrix  A¯ c is given with the form T1 = ( (T 1 )T (T 2 )T . . . (T 3N )T )T , in which T 3k−2 = INk  In 0 0 , T 3k−1 = 0 INk  In 0 , T 3k = 0 0



INk  I psm with k = 1, 2, . . . , N and INk is the k-row of IN . Thus, a transformation matrix of Ac can be obtained as follows:

Aˆ c = T1 Ac T1−1 , and the matrix for each agent can be expressed as

⎛ ¯ Ai Aˆ ci = ⎝−LiCi

B¯ i K1i

B¯ i K2i

Ai + Bi K1i + LiCi

G2C¯i

0

⎞

Bi K2i ⎠. G1

Then a new matrix which can be seen as the normal form of Aˆ ci is given with the following form:

⎛

Ai

Bi K1i

Aˆ nci = ⎝−LiCi G2Ci

Bi K2i

Ai + Bi K1i + LiCi



0



and Aˆ c = block diag Aˆ nc1 , Aˆ nc2 , . . . , Aˆ ncN .

⎞

Bi K2i ⎠, G1

8

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

Next, the matrix Aˆ ci will be diagonalized to prove the stability of matrix Ac , then a diagonal matrix A¯ ci transformed from Aˆ nci is expressed as

⎛

Ai + Bi K1i

Bi K2i

Bi K1i

G2Ci

G1

0

0

0

Ai + LiCi

A¯ ci = T2 Aˆ nci T2−1 = ⎝

 in which, T2 =



In

0

0

0

0

I psm

−In

In

Ai + Bi Ki , with Ai =

0

Ai

0

G2 Ci

G1

⎞

⎠,

, and the matrix block of the diagonal matrix can be rewritten as

Ai +Bi K1i

Bi K2i

G2 Ci

G1

Bi

, Bi =

0

= A¯ 11 = ci

. According to Lemma 3, there exist a gain matrix Ki = (K1i , K2i ) such that A¯ 11 = ci

Ai + Bi Ki is Hurwitz. Under Assumption 2, (Ci , Ai ) is detectable, therefore the matrix A¯ nci is Hurwitz, which means A¯ ci is Hurwitz. It follows that, the closed-loop matrix A¯ c is Hurwitz. Theorem 1. Under Assumptions 1, 2 and 3. For linear multi-agent systems (1) and (2), the designed distributed output feedback controller (9) can solve the containment control problem based on event-triggered strategy. Proof. To solve the containment control problem, the following M Sylvester equations are considered:

Xl (I2N  S ) = A¯ c Xl + B¯ cl , l ∈ R, where, Xl ∈

RN (2n+ psm )×2Nq .

⎛

Xl11

Xl = ⎝Xl21 Xl31

Xl12

⎞

(25)

Since A¯ c is Hurwitz with Assumption 3, the unique solution of (25) can be expressed as follows:

Xl22 ⎠, Xl32

with Xl11 , Xl12 , Xl21 , Xl22 ∈ RNn×Nq , Xl31 , Xl32 ∈ RN psm ×Nq . Thus, (25) can be rewritten as the following forms:





Xl31 (IN  S ) = (IN  G1 )Xl31 + (IN  G2 ) C¯ Xl11 ,



Xl32 (IN  S ) = (IN  G1 )Xl32 + (IN  G2 ) CXl12 − IN  Let

⎛ Xl31 = ⎝

···

X l11 .. .

X l (N p)1

···

⎛ C¯Xl11 = l = ⎝

l11





1 Fr . M−N

(27)

⎞

X l1N .. ⎠, . X l (N p)N ···

.. .

l (N p)1

(26)

···

l1N 

.. .

⎞ ⎠,

l (N p)N

in which X li j ∈ Rsm ×q , li j ∈ R1×q . Since the matrices G1 , G2 and γi = γ , σi = σ are defined in (8) and (10), respectively, Eq. (26) can be rewritten as

X li j S = γ X li j + σ li j ,

(28)

li j

in which Xk ∈ R1×q , k = 1, . . . , sm is the kth row of Xlij . It follows from (10) and (28) that: j Xkli j S = Xkli+1 , k = 1, 2, . . . , sm − 1,

(29)

li j = Xslimj S + asm X1li j + asm −1 X2li j + · · · + a1 Xslimj ,

(30)

which yields:

  li j = X1li j Ssm + a1 Ssm −1 + · · · + asm Iq .

By Lemma 4, Ssm + a1 Ssm −1 + · · · + asm Iq = 0 and li j = 0, i.e.

C¯Xl11 = 0.

(31)

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

9

From the Eq. (27), the following equation can be obtained by a similar proof method to (26):

C¯Xl12 − IN 

1 Fr = 0. M−N

(32)

The compact form of eˆi = C¯i xi − Fr ηi is eˆ = C¯x − (IN  Fr )η. The error e˜ can be rewritten as M 

e˜ = (H I p )C¯x −

(Hl  Fr )ω¯ l

l=N+1 M 

= (H I p )C¯x − (H I p )(IN  Fr )η + (H I p )(IN  Fr )η −

 = (H I p )eˆ + (H  Fr )

η−



M 

 H Hl  Iq ω ¯l



(Hl  Fr )ω¯ l

l=N+1

−1

l=N+1

= (H I p )eˆ + (H  Fr )ε .

(33)

In order to prove limt → ∞ e˜ = 0, we will further consider the equation eˆi = C¯i xi − Fr ηi , the error eˆi can be rewritten as:

eˆ = C¯x − (IN  Fr )η M 

= C¯c xc −

Dcl vl ,

l=N+1





where C¯c = C¯

(34)



0 , Dcl = 0



C¯c Xl − Dcl = C¯c Xl11

IN 



1 M−N Fr



C¯c Xl12 − 0



= C¯c Xl11

C¯c Xl12 − IN 



. Thus, combining (31) and (32), one gets

IN  1 F M−N r

1 F M−N r





= 0.

Let x˜c = xc −

x˜˙ c = x˙ c −

(35)

M

N+1 Xl vl ,

M 

then

Xl v˙ l

l=N+1

(IN  S )ω¯ l Xl ,  l=N+1 l=N+1     in which  = (IN  S )η + H  Kη ε + H  Kη e, and



0 (IN  S )ω¯ l (IN  S )ω¯ l    = + .  H  Kη ε + H  Kη e (IN  S )η = A¯ c xc +

M 

B¯ cl vl −

M 



Thus, the closed-loop system can be rewritten as:



M  0 (IN  S )ω¯ l    Xl − Xl H  Kη ε + H  Kη e (IN  S )η l=N+1 l=N+1 l=N+1



M M M M     0 0 ¯ ¯ = Ac xc + Xl (I2N  S )vl − Xl Xl e. ε− Bcl vl −

x˜˙ c = A¯ c xc +

M 

B¯ cl vl −

l=N+1

M 



l=N+1

l=N+1

H  Kη

l=N+1

H  Kη

According to (25), the derivative of x˜c is expressed by the following equation:

x˜˙ c = A¯ c x˜c + ε + e, where,  = −

M



Let ϕ = x˜Tc



ϕ˙ =



l=N+1 Xl

(36)

0 . H  Kη

 ε T T , it follows (13) and (36) that

A¯ c



0

IN  S + H  Kη



 e, ϕ+

and A¯ c has been proved to be Hurwitz.

H  Kη

(37)

10

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

Fig. 1. The topology graph.

Let S1 = block diag{S + μ1 λ1 P, . . . , S + μ1 λN P } and S2 = block diag{S + P, . . . , S + P }, one has









IN  S + H  Kη = U T  Iq



IN  S + J  Kη (U  Iq )

= U T  Iq S1 (U  Iq ). Then, an inequation can be obtained by (16) with the following form:





IN  S + H  Kη ≤ U T  Iq S2 (U  Iq ), in which U is an orthogonal matrix, U U T = IN and H = UT JU. λ1 , . . . , λN are the eigenvalues of matrix H and 0 < μ1 λi ≤ 1, μ1 > 0. Kη = μ1 P The equation which is obtained from (11) is given as follows:

P (S + P ) + (S + P )T P = −μ2 Iq < 0, where P is chosen as a positive matrix, therefore S + P = S2 < 0 and IN  S + H  Kη < 0. From Lemma 1 of [49], limt→∞ ϕ = 0. Thus, limt→∞ x˜c = 0. Then, the following error eˆ is given from (34) with the following form:

eˆ = C¯c xc −

M 

Dcl vl

l=N+1



= C¯c x˜c +



M 

Xl vl

l=N+1

= C¯c x˜c +

M 

M 

−

Dcl vl

l=N+1



C¯c Xl − Dcl

 vl

l=N+1

= C¯c x˜c .

(38)

Finally, we have limt → ∞ e˜ = 0. By a presented Lyapunov function and a designed distributed control law, the containment control problem of multi-agent systems based on event-triggered strategy is solved.  Remark 2. Compared with the existing results [13,44] which considered the conservatism issue, our main concern is the containment control problem in this paper. Thus, the uncertain part only requires the minimum region which guarantees the stability of the closed-loop system. Theorem 2. Suppose all the assumptions hold and consider the systems (1) and (2). No agent will exhibit the Zeno behavior based on the controller (9) and triggering condition (12). Proof. We first give the following condition to guarantee that e¯i (t ) − βi ε¯i (t ) ≤ 0:

e¯i (t ) ≤



βi 2 + 2β

2 i

  ε¯i tki ,

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

11

2.5 follower1 follower2 follower3 follower4 leader5 leader6

2

1.5

Outputs

1

0.5

0

−0.5

−1

−1.5

−2

0

10

20

30

Time(Sec) Fig. 2. The output trajectories of six agents which consist of two leaders (the red line and the green line) and four followers (the blue lines). (For interpretation of the reference to color in this figure legend, the reader is referred to the web version of this article.)

which can can be obtained from:

 2   N N    e¯2i (t ) ≤  hi j ei (t ) + hi j εi (t ) 2 2 + 2βi  j=1  j=1 ⎛ 2  2 ⎞    N N  βi2 ⎝    ⎠ ≤ h e t + h ε t ( ) ( )     i j i i j i 1 + βi2  j=1   j=1  βi2

=

 βi2  2 e¯ (t ) + ε¯i2 (t ) , 1 + βi2 i

it yields

e¯2i (t ) ≤ βi2 ε¯i2 (t ). Then the time derivative of e¯i (t ) is

e¯T e¯˙ i de¯i (t ) = i ≤ ε¯˙ i (t ) dt

e¯i

    N   =  hi j ε˙ i (t )  j=1       N M    =  hi j Sηi + Kη ψi − ζil Sωl   j=1  l=N+1

12

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

0.03

0.025

0.02

0.015

Error

0.01

0.005

0

−0.005

−0.01

−0.015

0

6

12

18

24

30

Time(Sec) Fig. 3. Error of ω51 (tki ) − ω51 (t ).

    N N    =  hi j Sεi (t ) + hi j Kη ψi   j=1  j=1     N N   i     =  hi j S εi tk − ei (t ) + hi j Kη ψi   j=1  j=1     N N  i    ≤ S e¯i (t ) +  hi j Sεi tk + hi j Kη ψi ,  j=1  j=1 in which ψi =

 j∈Ni

               j j i −ω tj ηi tki − η j t j η + M a t and k t = arg max t t ≤ t, j ∈ N . Then, we ( )  i i  il l l=N+1 k k k

have

k (t )

 ik  S (t −t i ) k e −1 ,

S

      i where k = max  Nj=1 hi j Sεi tki + Nj=1 hi j Kη ψi , t ∈ tki , tki +1 .   β Let ik = ! i ε¯i tki , it yields e¯i (t ) ≤

2+2βi2

 ik  S (t i −t i ) k+1 k e −1

S



S ik 1 tki +1 − tki ≥ ln + 1 ,

S

ik 



e¯i tki +1 = ik ≤

k (t )

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

13

0.015

0.01

0.005

Error

0

−0.005

−0.01

−0.015

−0.02

0

6

12

18

24

30

Time(Sec) Fig. 4. Error of ω52 (tki ) − ω52 (t ).

 

consider the case that ε¯i tki = 0, which means ik > 0. Then,

tki +1

− tki

1 ≥ ln

S



S ik + 1 > 0. ik

 

On the other hand, if ε¯i tki = 0 as k tends to infinity, we have

    N N    ε¯˙ i (t ) =  hi j Sεi (t ) + hi j Kη ψi  = 0  j=1  j=1

and

  ε¯i tki lim ≥ 1 − βi k→∞ ε¯i (t )

based on the

        ε¯i tki ε¯i tki ε¯i tki ε¯i tki ≤ ε ¯ t ≤ , where ≤ ε ¯ t ≤ can be obtained from e¯i (t ) − βi ε¯i (t ) ≤ 0 and ( ) ( ) i i 1+βi 1−βi 1+βi 1−βi

e¯i (t ). It follows from (39) that:

    N   ik ≤ S ε¯i tk + max  hi j Kη ψi   j=1     N  i      = S ε¯i tk + max  hi j Sεi t    j=1  i   = S ε¯i tk + S ε¯i t  i

(39)

(40)

    ε¯i t i − ε¯i (t ) ≤ k

14

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

0.02 0.015 0.01 0.005

Error

0 −0.005 −0.01 −0.015 −0.02 −0.025 −0.03

0

6

12

18

24

30

Time(Sec) Fig. 5. Error of ω53 (tki ) − ω53 (t ).





in which t ∈ tki , tki +1 . Based on the (40),

  √ βi 2 ε¯i tki ik 2+2β  i i   lim ≥ k→∞ i S ε ¯ tk + S ε¯i t 



i k βi (1 − βi ) ≥ 

S (2 − βi ) 2 + 2βi2 

As a result, the inter-event times tki +1 − tki



are lower bounded by a time τ which is obtained as follows

τ = lim tki +1 − tki k→∞

S ik 1 ≥ ln + 1

S

ik   1 βi (1 − βi ) ≥ ln +1 >0 

S

(2 − βi ) 2 + 2βi2 Finally, the Zeno behaviour will not appear, and the proof is completed.



Remark 3. In this paper, the event-triggered containment control problem has been solved. According to the advantages of event-triggered strategy in practical application, and the study of the fuzzy systems [50,51], H∞ control [52–54], transition probability problem [55,56] and quantized measurements [57], we will further study the event-triggering strategy based on the above results in the future.

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

15

0.015

0.01

0.005

0

Error

−0.005

−0.01

−0.015

−0.02

−0.025

−0.03

0

6

12

18

24

30

Time(Sec) Fig. 6. Error of ω61 (tki ) − ω61 (t ).

5. Simulation A presented numerical example will show the effectiveness of the main results in Section 4. Fig. 1 describes a communication topology graph among a group of agents, in which agents 1–4 represent the followers and the agents 5, 6 can be seen as the leaders. According to the Fig. 1, a following Laplacian matrix is shown as:

⎛

2 ⎜−1 L =⎝ 0 −1

−1 2 0 −1

0 0 1 −1

⎞

−1 −1⎟ . −1⎠ 3

H5 and H6 are expressed as

⎛

2 ⎜−0.5 H5 = ⎝ 0 −0.5

⎛

1 ⎜−0.5 H6 = ⎝ 0 −0.5

⎞

−0.5 1 0 −0.5

0 0 1.5 −0.5

−0.5 −0.5⎟ , −0.5⎠ 2

−0.5 2 0 −0.5

0 0 1.5 −0.5

−0.5 −0.5⎟ . −0.5⎠ 2

⎞

16

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

0.02

0.015

0.01

Error

0.005

0

−0.005

−0.01

−0.015

0

6

12

18 Time(Sec)

Fig. 7. Error of ω62 (tki ) − ω62 (t ).

Consider the uncertain parts of the dynamics, systems (1) can be rewritten as:

⎧ M ⎪ ⎨x˙ i (t ) = (Ai + Ai )xi (t ) + (Bi + Bi )ui +  (Eil + Eil )ωl , l=N+1 ⎪ ⎩ yi = (Ci + Ci )xi (t ), i ∈ F, with

Ai =

Bi =



Ci = i

0 1

0.25 ∗ i , A i = 0

1 0

0 , B i = 1







0 0



Ei5 = Ei6 =

1 0

0 0

0 , 0

0 , 0

0 , Ci = 0.001



0 0.001



0 ,

0 , Eil = 0.25i



0 0

0 0

0 , 0

in which Ai , Bi , Ci and Eil represent for the uncertain parts. The matrices for systems (2) is shown as follows:

 S=

0.92 0.92 1

0 0 0



−1  −0.92 , Fr = 0 −0.92

1



0 .

24

30

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

17

0.03 0.025 0.02 0.015

Error

0.01 0.005 0 −0.005 −0.01 −0.015 −0.02

0

6

12

18

24

30

Timen(Sec) Fig. 8. Error of ω63 (tki ) − ω63 (t ).

Then, the matrix pair (G1 , G2 ) is chosen as follows:

 G1 =

0 −1 0

1 0 0



 

0 −1 , G2 = 0

0 0 . 1

From the proof of Theorem 1, the matrix A¯ ci should be stable. Since the matrix A¯ 11 = Ai + Bi Ki is Hurwitz, and the gain ci matrix Ki = (K1i , K2i ), by the calculation of the Riccati equation, the gain matrices K1i and K2i with following values are shown respectively as

K11 =

K12 =

K13 =

K14 =

K21 =

K22 =

−2.6229

−0.3785

−0.3785

−1.0227

−3.3417

−0.5630

−0.5630

−1.1163

−3.8482

−0.7785

−0.7785

−1.2497

−4.2483

−1

−1

−1.4142

,

,

,

,

−1.2297

0.6868

−2.6329

0.1222

0.0331

0.2985

−0.9276

1.0659

−2.3395

0.0589

−0.0074

0.1386

,

,

18

Y. Zhou, Y. Pan and S. Li et al. / Applied Mathematics and Computation 375 (2020) 125065

K23 =

K24 =

−0.7291

1.2115

−2.1430

0.0236

−0.0087

0.0558

,

−0.5918

1.2844

−2.0060

2.3674e − 16

2.54911e − 15

−2.0408e − 15

.

Under Assumption 2, the pairs (Ci , Ai ) is detectable. Thus, the gain matrix Li can be obtain as follows:

L1 = L3 =









−1.4856 , L2 = −2.4142 −1.3014 , L4 = −1.3874

−1.3450 , −1.6180

−1.2808 . −1.2808

From the definition which has been presented in Section 3, a controller should be designed to make the closed-loop system stable, and all the outputs of the followers will converge to the convex hull spanned by the reference output of leaders. In Fig. 2, the results have been shown with blue lines which represent followers, the green line and the red line represent the the leader 5 and leader 6, respectively. Finally, we can see that the outputs of the four followers gradually converge to the regional range formed by the two leaders, which is consistent with the purpose and definition of this paper. To show the triggered situations and measurement errors, we will present the following figures. Figs. 3–8 represent the errors of ωl j (tki ) − ωl j (t ) where i = 1, 2, 3, 4, l = 4, 5 and j = 1, 2, 3, so the ωlj represents the jth element of ωl . 6. Conclusion In this paper, we has considered the containment control problem of linear multi-agent systems by event-triggered strategy. To solve this problem, an output feedback controller has been presented based on the event-triggered strategy. Then, we have proposed a triggering condition and proved the effectiveness of the observer by Lyapunov stability condition. What’s more, a distributed internal model compensator has been used to handle the uncertain parts of dynamics, then the stability of the closed-loop system has been proved. Finally, the problem of event-triggered containment control has been solved for the multi-agent systems. Acknowledgments This work was partially supported by the National Natural Science Foundation of China (61703051), and the PhD Start-up Fund of Liaoning Province (20170520124), and the Project of Liaoning Province Science and Technology Program under Grant 2019-KF-03-13. References [1] J.A. Fax, R.M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Autom. Control 49 (9) (2004) 1465–1476. [2] H. Zhang, H. Liang, Z. Wang, T. Feng, Optimal output regulation for heterogeneous multiagent systems via adaptive dynamic programming, IEEE Trans. Neural Netw. Learn. Syst. 28 (1) (2015) 18–29. [3] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control 48 (6) (20 03) 988–10 01. [4] Z. Lin, M. Broucke, B. Francis, Local control strategies for groups of mobile autonomous agents, IEEE Trans. Autom. Control 49 (4) (2004) 622–629. [5] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49 (9) (2004) 1520–1533. [6] H.G. Tanner, A. Jadbabaie, G.J. Pappas, Flocking in fixed and switching networks, IEEE Trans. Autom. Control 52 (5) (2007) 863–868. [7] H. Zhang, T. Feng, H. Liang, Y. Luo, LQR-based optimal distributed cooperative design for linear discrete-time multiagent systems, IEEE Trans. Neural Netw. Learn. Syst. 28 (3) (2015) 599–611. [8] H. Zhang, J. Zhang, G. Yang, Y. Luo, Leader-based optimal coordination control for the consensus problem of multiagent differential games via fuzzy adaptive dynamic programming, IEEE Trans. Fuzzy Syst. 23 (1) (2015) 152–163. [9] H. Zhang, F. Teng, Q. Sun, Q. Shan, Distributed optimization based on a multiagent system disturbed by general noise, IEEE Trans. Cybern. 49 (8) (2018) 3209–3213. [10] H. Jiang, H. He, Data-driven distributed output consensus control for partially observable multiagent systems, IEEE Trans. Cybern. 49 (3) (2018) 848–858. [11] Y. Zheng, J. Ma, L. Wang, Consensus of hybrid multi-agent systems, IEEE Trans. Neural Netw. Learn. Syst. 29 (4) (2018) 1359–1365. [12] Z. Li, Y. Tang, T. Huang, J. Kurths, Formation control with mismatched orientation in multi-agent systems, IEEE Trans. Netw. Sci. Eng. 6 (3) (2018) 314–325. [13] D. Mukherjee, D. Zelazo, Robustness of consensus over weighted digraphs, IEEE Trans. Netw. Sci. Eng. 6 (4) (2018) 657–670. [14] H. Liang, L. Zhang, Y. Sun, T. Hunag, Containment control of semi-markovian multi-agent systems with switching topologies, IEEE Trans. Syst. Man Cybern. Syst. (2019), doi:10.1109/TSMC.2019.2946248. [15] X. Chang, G. Yang, Nonfragile H∞ filtering of continuous-time fuzzy systems, IEEE Trans. Signal Process. 59 (4) (2011) 1528–1538. [16] W. Ren, R.W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (5) (2005) 655–661. [17] W. Ren, R.W. Beard, E.M. Atkins, A survey of consensus problems in multi-agent coordination, in: Proceedings of the 2005 American Control Conference, 2005, IEEE, 2005, pp. 1859–1864. [18] Y. Tian, C. Liu, Consensus of multi-agent systems with diverse input and communication delays, IEEE Trans. Autom. Control 53 (9) (2008) 2122–2128. [19] Y. Hong, G. Chen, L. Bushnell, Distributed observers design for leader-following control of multi-agent networks, Automatica 44 (3) (2008) 846–850.

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