Leader-following consensus for multi-agent systems with nonlinear dynamics subject to additive bounded disturbances and asynchronously sampled outputs

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Leader-following consensus for multi-agent systems with nonlinear dynamics subject to additive bounded

disturbances and asynchronously sampled outputs

Tomas Menard, Syed Ali Ajwad, Emmanuel Moulay, Patrick Coirault, Michael Defoort

To cite this version:

Tomas Menard, Syed Ali Ajwad, Emmanuel Moulay, Patrick Coirault, Michael Defoort. Leader- following consensus for multi-agent systems with nonlinear dynamics subject to additive bounded disturbances and asynchronously sampled outputs. Automatica, Elsevier, 2020, 121, pp.109176.

�10.1016/j.automatica.2020.109176�. �hal-02974230�

Leader-following consensus for multi-agent systems with nonlinear dynamics subject to additive bounded disturbances

and asynchronously sampled outputs

Tomas Ménard ^a , Syed Ali Ajwad ^b , Emmanuel Moulay ^c , Patrick Coirault ^b and Michael Defoort ^d

a

LAC (EA 7478), Normandie Université, UNICAEN, 6 bd du Maréchal Juin, 14032 Caen Cedex, France.

b

LIAS (EA 6315), Université de Poitiers, 2 rue Pierre Brousse, 86073 Poitiers Cedex 9, France.

c

XLIM (UMR CNRS 7252), Université de Poitiers, 11 bd Marie et Pierre Curie, 86073 Poitiers Cedex 9, France.

d

LAMIH, UMR CNRS 8201, Polytechnic University Hauts-de-France, Valenciennes, 59313 France.

Abstract

This paper is concerned with the leader-following consensus problem for a class of Lipschitz nonlinear multi-agent systems with uncertain dynamics, where each agent only transmits its noisy output, at discrete instants and independently from its neighbors. The proposed consensus protocol is based on a continuous-discrete time observer, which provides a continuous time estimation of the state of the neighbors from their discrete-time output measurements, together with a continuous control law. The stability of the multi-agent system is analyzed with a Lyapunov approach and the exponential practical convergence is ensured provided that the tuning parameters and the maximum allowable sampling period satisfy some inequalities. The proposed protocol is simulated on a multi-agent system whose dynamics are ruled by a Chua's oscillator.

Key words: Leader-following consensus, directed graph, sampled data, continuous-discrete time observer

1 Introduction

The study of Multi-Agent Systems (MAS) has been considered by many researchers during the last decades, due to its important practical applications, such as for- mation of UAV, attitude synchronization of spacecraft or distributed sensor networks [20]. MAS are usually characterized by a topology network which reects the possible ways of communication among agents. A fun- damental problem for MAS is to design protocols such that all the agents in the network reach a common value.

This problem can be subdivided into two categories, the

? This paper was not presented at any IFAC meeting. Cor- responding author T. Ménard.

Email addresses: e-mail: tomas.menard@unicaen.fr (Tomas Ménard), e-mail:

syed.ali.ajwad@univ-poitiers.fr (Syed Ali Ajwad), emmanuel.moulay@univ-poitiers.fr (Emmanuel Moulay), patrick.coirault@univ-poitiers.fr (Patrick Coirault), michael.defoort@uphf.fr (Michael Defoort).

leaderless consensus and the leader-following consensus.

In the leaderless consensus problem, the nal common position of the agents cannot be selected. Then, it might be useful to consider a real or virtual leader whose pre- scribed trajectory has to be followed by all the agents.

Many results have been obtained for MAS whose dy- namics are linear, see for instance [13]. However, in many practical cases, MAS are governed by more com- plex dynamics, namely nonlinear dynamics. These non- linear dynamics usually cannot be neglected in order to obtain more accurate control procedures and objectives.

Consensus protocols using the full state information have been considered in [5] for a xed topology or in [29] for time-varying topology. When the state of the agents is only partially available, that is when only the measured output can be used, it is then necessary to use observers in order to reconstruct the state. Such a strategy has been used for example in [14] for a general class of nonlinear MAS.

In the aforementioned nonlinear protocols, the consid- ered signals are assumed to be available continuously in

Preprint submitted to Automatica 3 July 2020

real time. But in most applications, it is more desirable or sometimes only possible to transmit the measure- ments in a discrete way. This may be due to technical constraints or for energy saving [23]. It is then impor- tant to adapt the consensus protocol in order to deal with the sampled signals. A rst idea has been to con- sider time-triggered sampling. Several protocols have been proposed when the state of the agents can be fully measured. Impulsive control for some specic kinds of systems has been considered in [11]. The delay input approach has been used in [30,31] for deterministic sampling period and in [24,28] for stochastic sampling.

When only the output is available at discrete instants, then an observer must be designed. A distributed ob- server protocol with a zero order hold input control has been proposed in [27]. In this work, the sampling peri- ods have to be synchronized between the agents.

Another approach, which allows aperiodic and asyn- chronous sampling periods, is event-triggering based consensus protocol. General linear systems have been considered in [33]. Some specic nonlinear classes of sys- tems have also been considered in the literature. MAS with Lure's nonlinear dynamics have been investigated in [15] and rst order nonlinear systems in [32,16]. Other classes of nonlinear systems have been treated in [17].

Though providing interesting results, event-triggered schemes involve a more complicated set-up and addi- tional parameters to tune. Indeed, a threshold function has to be considered which dictates the sampling in- stants for the data transmission. Furthermore, one has to be careful about the Zeno phenomenon which can occur for some schemes [6].

Most works with discrete signal transmission hold the control input constant between sampling instants. One takes advantage here, of the fact that time-varying control input can be considered. Indeed, only the trans- mitted signals have to be sampled, not the input.

Continuous-discrete time observers, which reconstruct the state in continuous time from discrete-time mea- surements, have been greatly developed these last years, as in [4,12] for dierent classes of nonlinear systems, and are then used here to tackle the problem of leader- following consensus where only discrete-time outputs are transmitted through the network. It should be noted that these works only consider the observer design, the convergence is obtained by assuming that the input belongs to a bounded set and then cannot be applied directly to the problem considered here. This idea has already been exploited in [21,22] for linear systems, following a hybrid approach, where sucient condi- tions for convergence are obtained, based on LMIs. A high-gain approach has been followed in [19,1] for the leaderless and leader-following consensus of MAS with double integrator dynamics. The control part of this high-gain approach is mainly based on [2] where only state feedback is considered. These works are then ex- tended here to the leader-following consensus for a class of systems with nonlinear and uncertain dynamics. The main novelty of the paper is the design of a leader-

following consensus protocol for a multi agent system, whose agent dynamics belong to a class of uniformly observable multi-output nonlinear systems, where only a part of the state is measured, and each agent's output is transmitted at some discrete instants to its neigh- bors independently of the other agents. Several fea- tures of the proposed approach have to be emphasized.

Firstly, only the sampled outputs have to be transmit- ted through the network, it is not necessary to transmit the inputs. Secondly, the data sent by the agents are not needed to be synchronized, each agent can send its measurements independently from its neighbors, pro- vided that the maximum allowable sampling period is bounded. This allows to reduce the overall bandwidth of the network. Thirdly, the proposed protocol has only three tuning parameters, namely ¯ c, λ, θ , where c ¯ is the coupling force, λ > 0 represents the speed of conver- gence of the control part and θ > 0 represents the speed of convergence of the observer part. Then, the tuning of the proposed scheme is relatively simple and can be adapted easily by the practitioner since the eect of the modication of each parameter has a direct physical meaning. Fourth, the class of nonlinear systems consid- ered here is challenging since it is quite large and it has not yet been considered in the literature for aperiodic and asynchronous sampling periods. A new protocol is thus proposed here to tackle this problem.

The paper is organized as follows. Some notations and existing results are recalled in Section 2. The class of considered MAS is depicted in Section 3. The proposed protocol together with a convergence result is reported in Section 4. Section 5 contains an example illustrating the performances of the proposed protocol. Finally, Sec- tion 6 concludes the paper.

A long version of the note, which contains complete proofs can be found in [18].

2 Preliminaries

In this paper, the following notations will be used. The

symbol

⁴

= means equal by denition. The set of n × n

real matrices is denoted by R ^n×n . The transpose for real

matrices is represented by the superscript T . I n is the

identity matrix of dimension n , 0 m×n is the zero matrix

of dimension m × n and 0 _n

⁴

= 0 _n×n . The Kronecker

product of matrices A and B is A ⊗ B . For a symmetric

matrix M , ρ _max (M ) and ρ _min (M ) respectively denotes

the maximum and minimum eigenvalue of M . The no-

tation diag (w ₁ , . . . , w _q ) , with w _i ∈ R ^m×m , i = 1, . . . , q ,

q, m ∈ N, is used for the diagonal by block matrix with

w 1 , . . . , w q on its diagonal. The positive deniteness of

a matrix M is denoted M > 0 . The vector of dimension

N ∈ N with all entries equal to 1 is denoted 1 N .

A directed graph G is a pair (V, E) , where V is a nonempty nite set of nodes and E ⊆ V × V is a set of edges, in which an edge is represented by an ordered pair of distinct nodes. For an edge (i, j) , node i is called the parent node, node j the child node, and i is a neigh- bor of j . A graph with the property that (i, j) ∈ E implies (j, i) ∈ E is said to be undirected. A path on G from node i 1 to node i l is a sequence of ordered edges of the form (i k , i k+1 ) , k = 1, . . . , l − 1 . A directed graph has or contains a directed spanning tree if there exists a node called the root, which has no parent node, such that there exists a directed path from this node to every other node in the graph. Suppose that there are N nodes in a graph. The adjacency matrix A = (a _ij ) ∈ R ^N

^×N

is dened by a ii = 0 and a ij = 1 if (j, i) ∈ E and a ij = 0 otherwise. The Laplacian matrix L ∈ R ^N

^×N

is dened as L ii = P

j6=i a _ij , L ij = −a ij for i 6= j .

Denition 1 [3, Def. 1.2 and Th. 2.3-G20 p.133-134]

A non singular real matrix Q ∈ Z ^n×n is called an M - matrix if all its diagonal elements are positive, all of its o-diagonal elements are non positive, and each of its eigenvalues has positive real part.

Lemma 1 [3, Th. 2.3-H24 p.134] Suppose that Q ∈ Z ^n×n is an M -matrix. Then, there exists a positive vec- tor ω = h

ω 1 . . . ω n

i T

, such that ΩQ + Q ^T Ω > 0 , where Ω = diag (ω ₁ , . . . , ω _n ) .

Lemma 2 Let n ∈ N and v i : R → R, i = 1, . . . , n be some functions C ¹ on (0, +∞) and such that v _i (t) = 0 for t < 0 , verifying

d dt

n

X

i=1

γ

i

v

²i

(t)

!

≤

n

X

i=1

−a

i

v

²i

(t) + b

i

Z

t t−δ

v

²i

(s)ds

+ k, for all t ≥ 0 , where γ i > 0 , a i > 0 , b i ≥ 0 , i = 1, . . . , n (1) , k ≥ 0 and

δ < min ( √

2 − 1)

2 min

i=1,n

a

i

b

i

, 1

√ 2 min

i=1,n

γ

i

a

i

. (2)

Then, the following inequality holds true

n

X

i=1

γ i v ² _i (t) ≤ ςe

^−ϑt

+ k

ϑ , (3)

with ϑ = ¹ ₂ min i=1,...,n

a

i

γ

_i

and ς = P n

i=1 γ i v ² _i (0) . The proof of Lemma 2 can be found in [18].

Lemma 3 [1, Lemma 4]

(i) Let M ∈ R ^n×n be a symmetric positive denite matrix. Then, one has x ^T M y ≤ √

x ^T M x p y ^T M y for all x, y ∈ R ⁿ .

(ii) Let M ∈ R ^n×n be a symmetric matrix. Then, one has ρ min (M )x ^T x ≤ x ^T M x ≤ ρ max (M )x ^T x for all x ∈ R ⁿ .

(iii) One has P n i=1

√ α i ≤ √ n pP n

i=1 α i for all α i ≥ 0 . (iv) Let A ∈ R ^n×n be a symmetric positive denite ma- trix and B ∈ R ^m×m be a symmetric semi-denite matrix, then the following inequality holds

ρ min (A)I n ⊗ B ≤ A ⊗ B ≤ ρ max (A)I n ⊗ B. (4) 3 Problem statement

3.1 Dynamical model of the agents

One considers a group of N agents whose dynamics are nonlinear. More precisely, the i -th agent dynamics, i = 1, . . . , N , are given by

˙

x ⁽¹⁾ _i (t) = x ⁽²⁾ _i (t) + ϕ ₁

t, x ⁽¹⁾ _i (t)

+ ε ⁽¹⁾ _i (t), (5) ...

˙

x ^(q−1) _i (t) = x ^(q) _i (t) + ϕ _q−1

t, x ⁽¹⁾ _i (t), . . . , x ^(q−1) _i (t) + ε ^(q−1) _i (t),

˙

x ^(q) _i (t) = u i (t) + ϕ q

t, x ⁽¹⁾ _i (t), . . . , x ^(q) _i (t)

+ ε ^(q) _i (t), y i = x ⁽¹⁾ _i + w i ,

where x ^(k) _i ∈ R ^m , k = 1, . . . , q , is the state, u i ∈ R ^m is the input, y _i ∈ R ^m the output, ε ^(k) _i : R → R ^m , k = 1, . . . , q , the dynamics uncertainties, w i : R → R ^m the noise, ϕ k : R × R ^km → R ^m the nonlinearities and q, m ∈ N.

System (5) is then made up of q blocks, each one of size m .

Let us denote x i =

x ⁽¹⁾ _i ^T

· · ·

x ^(q) _i ^{T T}

∈ R ⁿ the state of the i -th agent, with n = qm , and ε i =

ε ⁽¹⁾ _i T

. . .

ε ^(q) _i T T

. System (5) can be writ- ten in the following compact form

x ˙ i (t) = Ax i (t) + ϕ(t, x i (t)) + Bu i (t) + ε i (t),

y i = Cx i + w i , (6)

with A ∈ R ^n×n , B ∈ R ^n×m and C ∈ R ^m×n the corre- sponding matrices and ϕ =

ϕ ^T ₁ . . . ϕ ^T _q ^T

.

Remark 1 The class of systems considered here is closely related to the class of uniformly observable sys- tems, which has been introduced in [10] for Single Input Single Output (SISO) systems. Indeed, the system is composed of an integrator chain and the nonlinear part

3

has a triangular structure. While there exists canonical form for SISO systems, no such canonical form exists for Multi Input Multi Output systems. Nevertheless, sev- eral classes of uniformly observable systems have been considered in the literature such as in [8,9]. The class of systems considered here is the same as in [8] (where more details are given about possible required change of coordinates) except that the input is supposed to act linearly and without zero dynamics.

The nonlinear function ϕ is supposed to verify the fol- lowing assumption.

Assumption 1 The function ϕ is globally Lipschitz in x uniformly with respect to t , that is there exists L _ϕ > 0 such that

kϕ(t, x 1 ) − ϕ(t, x 2 )k ≤ L ϕ kx 1 − x 2 k, (7) for all t ∈ R and x ₁ , x ₂ ∈ R ⁿ .

The aim is to design a protocol such that all the agents converge toward a leader, denoted agent 0 , whose dy- namics are given by

˙

x 0 (t) = Ax 0 (t) + ϕ(t, x 0 (t)) + ε 0 (t), (8) y ₀ = Cx ₀ + w ₀ .

Remark 2 It should be noted that the uncertainties on the dynamics of the leader can also represent an unknown input. Indeed, given the structure of the dynamics (see equation (5)), the uncertainty on the dynamics of x ^(q) ₀ can be decomposed as an unknown leader input u ₀ on one part and some perturbations on the dynamics on the other part.

The dynamics uncertainties and noises of the agents and the leader are supposed to be uniformly bounded.

Assumption 2 There exist constants δ _ε ¹ , . . . , δ _ε ^q ≥ 0 and δ w ≥ 0 such that

kε ^(k) _i (t)k ≤ δ ^k _ε , ∀t ≥ 0, i = 0, . . . , N, k = 1, . . . , q, (9) kw _i (t)k ≤ δ _w , ∀t ≥ 0, i = 0, . . . , q. (10)

Due to the uncertainties, the agents cannot converge exactly toward the leader but, instead, in a ball around the leader. The problem to be solved here is dened more precisely below.

Denition 2 The leader-following consensus problem is said to be exponentially practically solved if there exist α, β > 0 and γ ≥ 0 such that kx i (t)−x 0 (t)k ≤ αe

^−βt

+γ , for all i = 1, . . . , N .

3.2 Communication constraints

The communication topology between followers i = 1, . . . , N is denoted G and its adjacency and Laplacian matrices A = (a _ij ) and L respectively.

The output of the leader is supposed to be transmit- ted only at some agents. More precisely, one denes D = diag (d ₁ , . . . , d _N ) , where d _i = 1 if agent i has access to the output of the leader and d i = 0 otherwise.

One denes the general index ν _ij for i = 1, . . . , N, j = 0, . . . , N as ν ij = 1 if agent i receives the output of agent j and 0 otherwise. This means that if ν _ij = 1 , then the output of agent j is transmitted to agent i at time instants

t ^i,j _k

k∈

N

. These sampling instants are supposed to verify

t

^i,j₀

< t

^i,j₁

< · · · < t

^i,j_k

< . . . and τ

m

<

t

^i,j_k+1

− t

^i,j_k

< τ

M

, for all k ∈ N and τ m , τ M > 0 . Note that the lower (11) bound τ m is just considered in order to explicitly avoid the Zeno phenomena. It is not a restrictive bound since it can be taken as small as desired.

Example 1 Consider a MAS composed of a leader (denoted 0 ) and two agents (denoted 1 and 2 ) such as depicted in Figure 1.

Agent 1 receives the transmitted output y 0 of the leader at time instants t ^1,0 _k for k ∈ N. This is used to reconstruct the state of the leader in continuous time. Agent 1 also uses its own output at time instants t ^1,1 _k for k ∈ N, to reconstruct its own state in continuous time. Similarly, Agent 2 receives the output of agent 1 at time instants t ^2,1 _k and uses its own output at time instants t ^2,2 _k . Note that the time sequences

t ^1,0 _k

k∈

N

, t ^1,1 _k

k∈

N

,

t ^2,1 _k

k∈

N

, t ^2,2 _k

k∈

N

can be chosen freely, and in par- ticular independently from each other, as long as they verify (11).

0 1

2

y

0

t

^1,0_k

y

1

t

^2,1_k

y

1

t

^1,1_k

y

2

t

^2,2_k

Fig. 1. Data transmission along the directed graph G.

One denotes G ˜ the digraph corresponding to all the

agents i = 1, . . . , N together with the leader. The

Laplacian matrix L ˜ of G ˜ is given by L ˜ = 0 0 1×N

− d ˜ L + D

!

4

= 0 0 1×N

− d ˜ H

!

, (12)

where d ˜ =

d ₁ . . . d _N T

. One needs the following result.

Lemma 4 [26, Lemma 9] The matrix H is a non- singular M-matrix if and only if the communication topology G ˜ has a directed spanning tree.

Assumption 3 The communication topology G ˜ between the agents and the leader contains a directed spanning tree.

Remark 3 If Assumption 3 holds true, then according to Lemma 4 and Lemma 1, there exists a positive vec- tor ω =

ω 1 . . . ω N

such that ΩH + H ^T Ω > 0 where Ω = diag(ω 1 , . . . , ω N ) . Then, in the rest of the paper, one denotes

% = ρ _min ΩH + H ^T Ω

, (13)

ω _min = min(ω ₁ , . . . , ω _N ), (14) ω max = max(ω 1 , . . . , ω N ). (15) 4 Main result

4.1 Consensus protocol

The proposed protocol is given, for i = 1, . . . , N , by u _i (t) = d _i ¯ cK ^c Γ _λ (ˆ x _i,0 (t) − x ˆ _i,i (t)) (16)

+ ¯ cK ^c Γ λ N

X

j=1

a ij (ˆ x i,j (t) − x ˆ i,i (t)), ∀t ≥ 0,

where x ˆ _i,j is the estimate of x _j by agent i and is given, for t ∈

t ^i,j _k , t ^i,j _k+1 , k ∈ N, by

˙ˆ

x i,j (t) = Aˆ x i,j (t) + ϕ(t, x ˆ i,j (t)) − θ∆

⁻¹

_θ K ^o z i,j (t), (17) with

z i,j (t) = e

^−θK1o

^(t−t

^i,jk

⁾ Cˆ x i,j

t ^i,j _k

− y j

t ^i,j _k

, (18) where c, λ, θ > ¯ 0 are the tuning parameters, and

Γ λ = diag λ ^q I m , λ ^q−1 I m , . . . , λI m

, (19)

∆ θ = diag I m , 1

θ I m , . . . , 1 θ ^q−1 I m

, (20)

K ^o = P

⁻¹

C ^T , K ^c = B ^T Q, (21)

where q is the number of blocks of System (5), P, Q ∈ R ^n×n are the symmetric positive denite solutions of the following matrix equalities

P + P A + A ^T P = C ^T C, (22) Q + QA + A ^T Q = QBB ^T Q, (23) (see [2] for more details).

Remark 4 Given the structure by block of A, B, C , one can show directly (see [9] section 2.2 and [2] section 3 for more details) that K ^o and K ^c can be written as follows

K ^o =

K ₁ ^o I _m . . . K _q ^o I _m T

, (24)

K ^c =

K ₁ ^c I m . . . K _q ^c I m

, (25) where K ₁ ^o , . . . , K _q ^o , K ₁ ^c , . . . , K _q ^c ∈ R are equal to

K _i ^o = q i

!

, K _i ^c = q q − i + 1

!

, i = 1, . . . , q, (26)

and n k

!

are the binomial coecients.

Remark 5 The term z i,j (t) , dened in (18) corresponds to a prediction of the output error (C x ˆ _i,j (t) − y _j (t)) on each interval

t ^i,j _k , t ^i,j _k+1 (see [7] section III.B for more details).

Example 2 Consider the same topology as in Example 1 and assume that the dynamics of the agents are ruled by a second order system, more precisely x ˙ ⁽¹⁾ _i = x ⁽²⁾ _i ,

˙

x ⁽²⁾ _i = u i + ϕ 2 (x i ) , y i = x ⁽¹⁾ _i ∈ R. It means here that there are two blocks ( q = 2 ) and the dimension of each agent is equal to n = 2 . One further has d ₁ = 1 , d ₂ = 0 , a 11 = a 12 = a 22 = 0 and a 21 = 1 .

Then, System (6) is characterized by A = 0 1 0 0

! , B = 0

1

!

and C = 1 0

. Furthermore, the solution of Equa-

tions (22) and (23) are equal to P = 1 −1

−1 2

! and

Q = 1 1

1 2

!

and the gains are given by K ^o = 2 1

! ,

K ^c = 1 2

, Γ _λ = λ ² 0 0 λ

!

, ∆ _θ = 1 0 0 ¹ _θ

! . Given the topology of the considered MAS:

5

• Agent 1 has to reconstruct the state of the leader and its own state. Then agent 1 has to run two observers:

˙ˆ

x _1,0 (t) = A x ˆ _1,0 (t) + 0 ϕ 2 (ˆ x 1,0 (t))

!

− θ∆

⁻¹

_θ K ^o e

^−2θ

( ^t−t

^1,0_k

) ˆ x ⁽¹⁾ _1,0

t ^1,0 _k

− y 0

t ^1,0 _k

, for t ∈ h

t ^1,0 _k , t ^1,0 _k+1 ,

˙ˆ

x 1,1 (t) = A x ˆ 1,1 (t) + 0 ϕ ₂ (ˆ x _1,1 (t))

!

− θ∆

⁻¹

_θ K ^o e

^−2θ

( t−t

^1,1_k

) ˆ x ⁽¹⁾ _1,1

t ^1,1 _k

− y ₁ t ^1,1 _k

, for t ∈ h

t ^1,1 _k , t ^1,1 _k+1 ,

where x ˆ 1,0 and x ˆ 1,1 are the estimates of x 0 and x 1

respectively. The input of agent 1 is then given by:

u ₁ (t) = 2¯ cλ ² ˆ

x ⁽¹⁾ _1,0 (t) − x ˆ ⁽¹⁾ _1,1 (t) +¯ cλ

ˆ

x ⁽²⁾ _1,0 (t) − x ˆ ⁽²⁾ _1,1 (t) .

• Agent 2 has to reconstruct the state of agent 1 and its own state. Similarly as for agent 1, agent 2 has to run two observers: one whose state x ˆ _2,1 is an estimate of x 1 and one whose state x ˆ 2,2 is an estimate of x 2 .The input of agent 2 is then given by:

u ₂ (t) = 2¯ cλ ² ˆ

x ⁽¹⁾ _2,1 (t) − x ˆ ⁽¹⁾ _2,2 (t) +¯ cλ

ˆ

x ⁽²⁾ _2,1 (t) − x ˆ ⁽²⁾ _2,2 (t) . It should be noted that the only parameters which must be tuned are ¯ c, λ and θ since it is clear from (24)-(26) that K ^o and K ^c only depend on the structure of the system (i.e. the number of blocks q and the size of each block m ).

4.2 Convergence result

The convergence of the proposed consensus protocol is now analyzed.

Theorem 1 Consider the MAS (5)-(8) subject to As- sumptions 1, 2 and 3 and the consensus protocol (16)- (17)-(18). If the tuning parameters λ, θ, ¯ c ≥ 1 are chosen such that

¯

c ≥ c

^∗

= max ω

max

% , 1

,

λ ≥ λ

^∗

= 24L

ϕ

√ n max



 q

ρ

^Qω_M

q

ρ

^Qωm

, p ρ

^P_M

p ρ

^P_m



 , θ ≥ λ¯ c

²

ξ

^∗

,

with ξ

^{∗ 4}

= 36kK ^c k ² (N +1) ³ h ² _max max

√

ρ

^P_M

√

ρ

^P_m

, ^ρ

^PM

ρ

^Qω_m

, ^ρ

Qω M

ρ

^P_m

, ρ ^P _m = ρ min (P ) , ρ ^P _M = ρ max (P ) , ρ ^Qω _m = ω min ρ min (Q) ,

ρ ^Qω _M = ω max ρ max (Q) , h max = max ij |H ij | , %, ω min , ω max

respectively dened by (13)-(14)-(15), and the upper bound on the sampling periods τ _M veries

τ M < σ

^∗

¯

c(θ + L ϕ ) , (27)

with σ

^∗

=

^√

2−1 8

_min(

^√

_ω

min

, √

ρ

^P_m

)

kK^ok(N

+1)

³2

h

_max

√

ρ

^P_M

, then the con- sensus error kx _i − x ₀ k veries

kx _i − x ₀ k ≤χ ₁ θ ^q−1 e

^−λ8

^t + χ ₂ λ

⁻¹

θ ^q δ _w + τ _M δ _ε ¹

+ χ 3 λ

⁻¹

q

X

k=1

θ ^q−k δ _ε ^k

!

, (28)

for i = 1, . . . , N , where χ 1 , χ 2 , χ 3 ≥ 0 are independent of the tuning parameters λ, θ, c ¯ and given by

χ

1

= q

ρ

^Qω_M

q

ρ

^Qωm N

X

i=1

kx

i

(0) − x

0

(0)k

+ p ρ

^P_M

q

ρ

^Qωm N

X

i=1 N

X

j=0

kˆ x

i,j

(0) − x

j

(0)k,

χ

2

= 8 p

ρ

^P_M

(N + 1)kK

^o

k q

ρ

^Qωm

,

χ

3

= 8

2N

q

ρ

^Qω_M

+ (N + 1) p ρ

^P_M

q

ρ

^Qωm

.

Remark 6 Equation (28) states that even if the expo- nential practical consensus is reached, not all the uncer- tainties eect on the tracking error can be lowered through the tuning of λ and θ . Indeed, it is only possible for the uncertainties appearing on the last block of system (5) (that is only ε ^(q) _i are non zero) and if q ≥ 2 . This is done by increasing λ and θ , since the corresponding term χ 3 λ

⁻¹

δ _ε ^q goes to zero as λ increases.

In particular, if the leader has an unknown but bounded non zero input, then this input can be seen as a dynamic uncertainty on the last block dynamics x ^(q) ₀ . Therefore, the tracking error can be set as low as desired by increas- ing λ and θ .

Remark 7 Theorem 1 only provides sucient condi-

tions for the proposed consensus scheme (16)-(17). In-

deed, the consensus may be obtained even if the bounds

given by Theorem 1 are not respected. Conservativeness

is a general drawback when considering general classes of

nonlinear systems with a Lyapunov approach. Neverthe-

less, the convergence analysis gives some useful hints for

the tuning of the control parameters. In fact, the bounds

σ

^∗

, c

^∗

, λ

^∗

, ξ

^∗

only depend on the structure of the system

(number of blocks q and size of each block m ) and the

topology of the network G ˜ . Given the inequalities in The- orem 1, the coupling force should be tuned rst as for a classical consensus protocol. Then, λ (representing the speed of convergence of the control part) should be chosen high enough to dominate the nonlinear Lipschitz term.

The parameter θ of the observer part should be higher than the parameter of the control part λ . This is due to the fact that the input is not transmitted through the net- work, only the output is transmitted. Finally, the maxi- mum bound on the sampling periods will have to be cho- sen small if the Lipschitz constant or if θ takes high val- ues. This corresponds to a kind of Shannon condition: if the system is fast, the sampling periods have to be small.

Furthermore, given the bounds on the consensus error given by equation (28), it can be seen that increasing λ and θ will decrease the eect of some uncertainties (those on the dynamics of x ^(q) _i ) but it will increase the eect of the noise, then a trade-o has to be considered between lowering the eect of some uncertainties and amplifying the noise.

A full version of the proof with detailed computations can be found in [18].

Proof. The proof of Theorem 1 is split into three steps.

First, new coordinates are considered in Step 1. Then, in Step 2, some candidate Lyapunov functions are dened and over-valuations of their derivatives are obtained. Fi- nally, it is shown in Step 3, that, if the dierent inequal- ities of Theorem 1 are veried, then Lemma 2 can be applied and the leader-following consensus problem is solved.

Step 1. Let us dene e i = Γ λ (x i −x 0 ) and ¯ e i,j = ∆ θ (ˆ x i,j − x j ) . Then, using the equalities Γ λ AΓ

⁻¹

_λ = λA , Γ λ B = λB , ∆ θ A∆

⁻¹

_θ = θA , ∆ θ B = _θ

q−1

¹ B , C∆

⁻¹

_θ = C and the notation ϕ(t, x, y) ˜ =

⁴

ϕ(t, x) − ϕ(t, y) , one gets

˙

e _i = λAe _i + Γ _λ ( ˜ ϕ(t, x _i , x ₀ ) + ε _i − ε ₀ ) + λBu _i , (29)

˙¯

e i,j = θ(A − K ^o C)¯ e i,j + ∆ θ ϕ(t, ˜ x ˆ i,j , x j ) (30)

− 1

θ ^q−1 Bu j − θK ^o (z i,j − C¯ e i,j ) − ∆ θ ε j ,

for i = 1, . . . , N and j = 0, . . . , N , where z i,j is dened by (18) and the inputs are given by u ₀ = 0 _m×1 and u _j =

¯ cK ^c

d j Γ λ ∆

⁻¹

_θ e ¯ j,0 − P N

k=1 H jk (e k + Γ λ ∆

⁻¹

_θ ¯ e j,k ) , for j = 1, . . . , N . Further denoting η ^c = h

e ^T ₁ . . . e ^T _N i T

, the dynamics of the e _i can be written in compact form as follows

˙

η ^c = λ(I N ⊗ A)η ^c − ¯ cλ[H ⊗ (BK ^c )]η ^c + Φ λ + Ψ λ,ε + Θ,

with Φ λ =











Γ λ ϕ(t, x ˜ 1 , x 0 ) ...

Γ _λ ϕ(t, x ˜ _N , x ₀ )











, Ψ λ,ε =











Γ λ (ε 1 − ε 0 ) ...

Γ _λ (ε _N − ε ₀ )









 ,

and Θ =











¯

cλBK ^c Γ λ ∆

⁻¹

_θ

d 1 ¯ e 1,0 − P N

k=1 H 1k ¯ e 1,k

...

¯

cλBK ^c Γ λ ∆

⁻¹

_θ

d N ¯ e N,0 − P N

k=1 H N k e ¯ N,k









 .

Step 2. One now introduces the candidate Lyapunov functions, one that depends on the control error η ^c and one that depends on the observer errors e ¯ _i,j .

One thus denes rst V ¯ c (η ^c ) = (η ^c ) ^T [Ω ⊗ Q]η ^c , where Ω is dened in Remark 3.

Then, for the observer error part, one considers V ¯ o (η ^o ) = P N

i=1

P N

j=0 ν _ij V _o (¯ e _i,j ) , with V _o (¯ e _i,j ) = ¯ e ^T _i,j P e ¯ _i,j , where the vector η ^o contains all the e ¯ i,j such that ν ij = 1 . It can be shown that, if c ¯ ≥ c

^{∗ 4}

= max{ω max /%, 1} , then

V ˙¯ _c ≤ −λ V ¯ _c + 2k ₁ V ¯ _c + 2¯ ck ₂ λkΓ λ ∆

⁻¹

_θ k p V ¯ _c p

V ¯ _o

+ 2k 3

p V ¯ c q

X

k=1

λ ^q−k+1 δ _ε ^k

! ,

with k 1 = L ϕ

√ n r

ρ

^Qω_M

ρ

^Qω_m

, k 2 = (N +1)kK ^c kh max

r

ρ

^Qω_M

ρ

min

(P) , k ₃ = 2N

q

ρ ^Qω _M , and

V ˙¯

o

≤ −θ V ¯

o

+ 2k

4

V ¯

o

+ 2k

5

p V ¯

o

Z

t t−τ_M

q V ¯

o

(s)ds

+ 2k

6

p V ¯

o

Z

t t−τ_M

q V ¯

c

(s)ds + 2¯ ck

7

θ

^q−1

p V ¯

o

p V ¯

c

+ 2¯ ck

8

kΓ

λ

∆

⁻¹_θ

k θ

^q−1

V ¯

o

+ 2k

9

p V ¯

o q

X

k=1

δ

^kε

θ

^k−1

!

+ 2θk

10

p V ¯

o

δ

w

+ τ

M

δ

_ε¹

,

with k 4 = L ϕ

√ n

q ρ

max

(P)

ρ

_min

(P) , k 5 = θ(N + 1)kK ^o k(L ϕ + θ)

q ρ

max

(P)

ρ

_min

(P ) , if q ≥ 2 , k 5 = θ(N + 1)kK ^o k q

ρ

max

(P) ρ

_min

(P)

¯ c √

N + 1h max λ + L ϕ

, if q = 1 , k 6 = 0 , if q ≥ 2 , k ₆ = ^{¯ cθ(N +1)}

3 2

h

_max

√

ω

min

, if q = 1 , k ₇ = kK ^c k(N + 1)

³²

h max

q _ρ

max

(P)

ρ

^Qω_m

, k 8 = kK ^c k(N + 1)

³²

h max

q _ρ

max

(P) ρ

min

(P) , k 9 = p

ρ max (P)(N + 1) and k 10 = p

ρ max (P )kK ^o k(N + 1) .

Step 3. Since ¯ c ≥ c

^∗

≥ 1 , if the inequalities λ ≥ λ

^∗

and ξ ≥ ¯ c ² ξ

^∗

hold true, with λ

^{∗ 4}

= 24L _ϕ √

n max

p ρ

^Qω_M

√

ρ

^Qω_m

,

√

ρ

^P_M

√

ρ

^P_m

and ξ

^{∗ 4}

= 36kK ^c k ² (N +1) ³ h ² _max max √

ρ

^P_M

√

ρ

^P_m

, ^ρ

^PM

ρ

^Qω_m

, ^ρ

Qω M

ρ

^P_m

,

7

we obtain d dt

ξ

³²

p V ¯ c + θ ^q p V ¯ o

≤ − ξ

³²

λ 4

p V ¯ c − θ ^q+1 4

p V ¯ o

+ k ₅ θ ^q Z t

t−τ

M

p V ¯ _o ds + k ₆ θ ^q Z t

t−τ

M

p V ¯ _c ds

+ k ₃

q

X

k=1

ξ

³²

λ ^q−k+1 δ _ε ^k

! + k ₉

q

X

k=1

θ ^q−k+1 δ ^k _ε

!

+ θ ^q+1 k 10 δ w + τ M δ _ε ¹

. (31)

since ξ

^∗

≥ max

(6k 2 ) ² , (5k 7 ) ² , 24k 8 and λ

^∗

≥ max {20k 1 , 24k 4 } . If τ M veries inequality (27), then one can apply Lemma 2 and inequality (28) is obtained by using the following inequalities

λ q

ρ ^Qω m kx i − x 0 k ≤

q V ¯ c (η ^c ) ≤ λ ^q q

ρ ^Qω _M

N

X

i=1

kx i − x 0 k,

q V ¯ _o (η ^o ) ≤ p

ρ _max (P)

N

X

i=1 N

X

j=0

k˜ x _i,j k.

5 Example

In this section, we present a numerical example, consist- ing of systems whose dynamics are the same as a Chua's oscillator, in order to illustrate the performances of the proposed protocol. Indeed, as in [25], let us consider the MAS where each agent's dynamics are given by



 



 



˙

x ⁽¹⁾ _i (t) = x ⁽²⁾ _i (t),

˙

x ⁽²⁾ _i (t) = f

t, x ⁽¹⁾ _i (t), x ⁽²⁾ _i (t)

+ u _i (t) + ε ⁽²⁾ _i (t), y i = x ⁽¹⁾ _i + w i , i = 1, . . . , 10,

(32) where x ⁽¹⁾ _i =

⁴

x ^(1,1) _i x ^(1,2) _i x ^(1,3) _i T

∈ R ³ , x ⁽²⁾ _i

⁴

=

x ^(2,1) _i x ^(2,2) _i x ^(2,3) _i T

∈ R ³ are the position and veloc- ity of agent i , respectively, u i , y i , ε ⁽²⁾ _i , w i ∈ R ³ are the input, output, uncertainty and noise of each agent, and the nonlinear function f is given by

f

t, x

⁽¹⁾_i

, x

⁽²⁾_i

=







 α

x

^(2,2)_i

− x

^(2,1)_i

− h x

^(2,1)_i

x

^(2,1)_i

− x

^(2,2)_i

+ x

^(2,3)_i

−βx

^(2,2)_i

− γx

^(2,3)_i

− β sin ωx

^(1,1)_i







 ,

where α = 10 , β = 19.53 , γ = 0.1636 , = 0.2 , ω = 0.5 and h is a piece-wise linear function given by h

x ^(2,1) _i

= ^a−b ₂

x ^(2,1) _i + 1 −

x ^(2,1) _i − 1

with pa- rameters a = −1.4325 and b = −0.7831 .

The communication graph of the MAS System (32) is described in Figure 2 and one assumes that only agents 3 and 5 receive the measured position of the leader which is referenced as agent 0 . The leader-following consensus

1

2 3 4 5 6

10 9 8 7

0

Fig. 2. Communication graph of the MAS.

protocol (16)-(17) has been implemented in Matlab for minimum and maximum bound on the sampling peri- ods equal to τ _m = 0.02s and τ _M = 0.04s respectively.

The sampling periods have been set following a uniform distribution on [τ m , τ M ] independently for each edge.

The rst sampling periods corresponding to the trans- mission of the output of agent 2 to agent 1 are reported in Figure 3.

The tuning parameters ¯ c, θ, λ have been chosen by trial

0 0.01 0.02 0.03 0.04

5 10 15 20 25 30

Fig. 3. Sampling periods for the transmission from agent 2 to agent 1 .

and error and taken equal as c ¯ = 1 , θ = 20 and λ = 2 . A rst simulation has been conducted with no uncer- tainty on the dynamics and no noise on the outputs, that is ε ⁽²⁾ _i = 0 and w i = 0 . The position of the dier- ent agents are reported in Figure 5a)c)e). Furthermore, the estimation error of agent 2 by agent 1 is depicted in Figure 6a)c). The position tracking mean error

1 N

P N i=1

x ⁽¹⁾ _i (t) − x ⁽¹⁾ ₀ (t)

is reported in Figure 7a). As expected by the theory, both the tracking errors and the observation errors go to zero exponentially.

It is worth to be noted that only the outputs, namely the position of the agents, are transmitted through the network and according to the topology described in Figure 2. The velocities and inputs are not transmitted and then unknown by neighbors agents. Furthermore, each agent transmits its output at time instants inde- pendently from its neighbors.

Another simulation has been conducted, in the same conditions but with a non zero uncertainty only on the leader's dynamics, corresponding to an unknown leader input (that is ε ⁽²⁾ ₀ 6≡ 0 and ε ⁽²⁾ _i ≡ 0 for i = 1, . . . , 10 ) and noise on the transmitted outputs.

More precisely, the uncertainty on the leader is given

5 10 15 20 25 30 35 40 45 -1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Fig. 4. Second component of y

0

with noise (red) and without noise (blue)

by ε ⁽²⁾ ₀ (t) =

cos(t) cos(2t) cos(3t) T

and the noise on the outputs of the agents and the leader w _i are centered white noise with variance equal to 0.1 . The second com- ponent of the noisy and non noisy outputs of the leader are reported in Figure 4. The positions of the agents are reported in Figure 5b)d)f). The estimation error of agent 2 by agent 1 is reported in Figure 6b)d). The po- sition tracking mean error _N ¹ P N

i=1

x ⁽¹⁾ _i (t) − x ⁽¹⁾ ₀ (t) is reported in Figure 7b). While the position tracking error does not converge exactly to zero, the eect of the uncertainties has been lowered by taking θ and λ suciently high. It should be noted that in the case of noisy measurements, very high values of θ will lead to an amplication of the noise in the reconstructed state.

Thus, a trade-o on the value of θ and λ has to be done.

Indeed, suciently high values of these parameters have to be considered to attenuate the eect of the uncertain- ties but not too high such that the noise is not amplied too much. Nevertheless, despite uncertainties and noise, the proposed leader following consensus protocol still performs well as illustrated in the simulation.

6 Conclusion

The problem of leader-following consensus for a class of nonlinear systems which can only transmit their outputs at discrete aperiodic and asynchronous instants has been considered in this paper. A consensus protocol has been proposed based on a continuous-discrete time observer that reconstruct the states of the neighbors in continu- ous time, from sampled measurements only, and a con- tinuous control law. It has been shown theoretically that if the tuning parameters fulll some sucient conditions then the convergence of the MAS with the proposed pro- tocol is ensured under directed topology. Furthermore, in case of bounded uncertainties on the dynamics and bounded noise on the output, an exponential practical consensus is guaranteed. The performances of the ap- proach have been illustrated with simulations on a MAS whose agents dynamics are given by a Chua's oscillator.

References

[1] S.A. Ajwad, T. Menard, E. Moulay, M. Defoort, and P. Coirault. Observer based leader-following consensus of second-order multi-agent systems with nonuniform

5 10 15 20 25 30 35 40 45

-140 -120 -100 -80 -60 -40 -20 0 20

(a) Positions x

^(1,1)_i

, i = 0, . . . , 10

5 10 15 20 25 30 35 40 45

-80 -60 -40 -20 0 20

(b) Positions x

^(1,1)_i

, i = 0, . . . , 10

5 10 15 20 25 30 35 40 45

-8 -6 -4 -2 0 2 4 6

(c) Positions x

^(1,2)_i

, i = 0, . . . , 10

5 10 15 20 25 30 35 40 45

-8 -6 -4 -2 0 2 4 6

(d) Positions x

^(1,2)_i

, i = 0, . . . , 10

5 10 15 20 25 30 35 40 45

-40 -20 0 20 40 60 80 100 120 140

(e) Positions x

^(1,3)_i

, i = 0, . . . , 10

5 10 15 20 25 30 35 40 45

-40 -20 0 20 40 60 80

(f) Positions x

^(1,3)_i

, i = 0, . . . , 10

Fig. 5. Positions of the agents for τ

m

= 20ms , τ

M

= 40ms , θ = 20 and λ = 2 .

5 10 15 20 25 30 35 40 45

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

(a) Position error

xˆ⁽¹⁾_1,2−x⁽¹⁾₂

5 10 15 20 25 30 35 40 45

0 0.1 0.2 0.3 0.4 0.5 0.6

(b) Position error

xˆ⁽¹⁾_1,2−x⁽¹⁾₂

5 10 15 20 25 30 35 40 45

0 1 2 3 4 5 6 7 8

(c) Velocity error

xˆ⁽²⁾_1,2−x⁽²⁾₂

5 10 15 20 25 30 35 40 45

0 2 4 6 8 10 12

(d) Velocity error

xˆ⁽²⁾_1,2−x⁽²⁾₂

Fig. 6. Estimation error of the state of agent 2 by agent 1 . sampled position data. Journal of the Franklin Institute, 356(16):1003110057, 2019.

[2] A. Bédoui, M. Farza, M. MSaad, and M. Ksouri. Robust nonlinear controllers for bioprocesses. IFAC Proceedings Volumes, 41(2):1554115546, 2008.

[3] A. Berman and R.J. Plemmons. Nonnegative matrices in the mathematical sciences, volume 9. SIAM, 1994.

[4] I. Bouraoui, M. Farza, T. Ménard, R.B. Abdennour, M. MSaad, and H. Mosrati. Observer design for a class of uncertain nonlinear systems with sampled outputs application to the estimation of kinetic rates in bioreactors.

Automatica, 55:7887, 2015.

[5] M. Defoort, A. Polyakov, G. Demesure, M. Djemai, and

9

5 10 15 20 25 30 35 40 45 0

5 10 15 20

(a) Simulation without noise and uncertainty

5 10 15 20 25 30 35 40 45

0 2 4 6 8 10 12 14 16 18

(b) Simulation with noise and uncertainty

Fig. 7. Position tracking mean error

_N^{1 PN}i=1

x⁽¹⁾_i −x⁽¹⁾₀ .

K. Veluvolu. Leader-follower xed-time consensus for multi- agent systems with unknown non-linear inherent dynamics.

IET Control Theory & Applications, 9(14):21652170, 2015.

[6] L. Ding, Q.L. Han, X. Ge, and X.M. Zhang. An overview of recent advances in event-triggered consensus of multiagent systems. IEEE transactions on cybernetics, 48(4):11101123, 2018.

[7] M. Farza, M. M'Saad, M.L. Fall, E. Pigeon, O. Gehan, and K. Busawon. Continuous-discrete time observers for a class of mimo nonlinear systems. IEEE Transactions on Automatic Control, 59(4):10601065, 2014.

[8] M. Farza, M. M'Saad, and L. Rossignol. Observer design for a class of mimo nonlinear systems. Automatica, 40(1):135 143, 2004.

[9] M. Farza, M. MSaad, M. Triki, and T. Maatoug. High gain observer for a class of non-triangular systems. Systems &

Control Letters, 60(1):2735, 2011.

[10] J.P. Gauthier and G. Bornard. Observability for any u(t) of a class of nonlinear systems. IEEE Transactions on Automatic Control, 26(4):922926, 1981.

[11] W. He, G. Chen, Q.L. Han, and F. Qian. Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control. Information Sciences, 380:145158, 2017.

[12] O. Hernández-González, M. Farza, T. Ménard, B. Targui, M. MSaad, and C.M. Astorga-Zaragoza. A cascade observer for a class of MIMO non uniformly observable systems with delayed sampled outputs. Systems & Control Letters, 98:86 96, 2016.

[13] Y. Hong, J. Hu, and L. Gao. Tracking control for multi- agent consensus with an active leader and variable topology.

Automatica, 42(7):11771182, 2006.

[14] J. Hu, J. Cao, J. Yu, and T. Hayat. Consensus of nonlinear multi-agent systems with observer-based protocols. Systems

& Control Letters, 72:7179, 2014.

[15] N. Huang, Z. Duan, G. Wen, and Y. Zhao. Event- triggered consensus tracking of multi-agent systems with lur'e nonlinear dynamics. International Journal of Control, 89(5):10251037, 2016.

[16] J. Liu, Y. Yu, J. Sun, and C. Sun. Distributed event- triggered xed-time consensus for leader-follower multiagent systems with nonlinear dynamics and uncertain disturbances.

International Journal of Robust and Nonlinear Control, 28(11):35433559, 2018.

[17] W. Liu and J. Huang. Cooperative global robust output regulation for a class of nonlinear multi-agent systems by distributed event-triggered control. Automatica, 93:138148, 2018.

[18] T. Ménard, S.A. Ajwad, E. Moulay, P. Coirault, and M. Defoort. Leader-following consensus for multi- agent systems with nonlinear uncertain dynamics and

asynchronously sampled outputs (long version). arXiv Preprint arXiv:2007.00921, 2020.

[19] T. Ménard, E. Moulay, P. Coirault, and M. Defoort.

Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods. Automatica, 99:237245, 2018.

[20] R. Olfati-Saber and R.M. Murray. Consensus problems in networks of agents with switching topology and time-delays.

IEEE Transactions on Automatic Control, 49(9):15201533, 2004.

[21] S. Phillips, Y. Li, and R.G. Sanfelice. A hybrid consensus protocol for pointwise exponential stability with intermittent information. IFAC-PapersOnLine, 49(18):146151, 2016.

[22] S. Phillips and R.G. Sanfelice. Robust distributed synchronization of networked linear systems with intermittent information. Automatica, 105:323333, 2019.

[23] J. Ploennigs, V. Vasyutynskyy, and K. Kabitzsch.

Comparative study of energy-ecient sampling approaches for wireless control networks. IEEE Transactions on Industrial Informatics, 6(3):416424, 2010.

[24] B. Shen, Z. Wang, and X. Liu. Sampled-data synchronization control of dynamical networks with stochastic sampling.

IEEE Transactions on Automatic Control, 57(10):26442650, 2012.

[25] Q. Song, J. Cao, and W. Yu. Second-order leader-following consensus of nonlinear multi-agent systems via pinning control. Systems & Control Letters, 59(9):553562, 2010.

[26] Q. Song, F. Liu, J. Cao, and W. Yu. Pinning-controllability analysis of complex networks: an m-matrix approach. IEEE Transactions on Circuits and Systems I: Regular Papers, 59(11):26922701, 2012.

[27] Y. Wan, J. Cao, A. Alsaedi, and T. Hayat. Distributed observer-based stabilization of nonlinear multi-agent systems with sampled-data control. Asian Journal of Control, 19(3):918928, 2017.

[28] Y. Wan, G. Wen, J. Cao, and W. Yu. Distributed node- to-node consensus of multi-agent systems with stochastic sampling. International Journal of Robust and Nonlinear Control, 26(1):110124, 2016.

[29] G. Wen, Z. Duan, G. Chen, and W. Yu. Consensus tracking of multi-agent systems with lipschitz-type node dynamics and switching topologies. IEEE Transactions on Circuits and Systems I: Regular Papers, 61(2):499511, 2014.

[30] G. Wen, Z. Duan, W. Yu, and G. Chen. Consensus of multi- agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach. International Journal of Robust and Nonlinear Control, 23(6):602619, 2013.

[31] G. Wen, W. Yu, M. Chen, X. Yu, and G. Chen. H

∞

pinning synchronization of directed networks with aperiodic sampled- data communications. IEEE Transactions on Circuits and Systems I: Regular Papers, 61(11):32453255, 2014.

[32] D. Xie, S. Xu, Y. Chu, and Y. Zou. Event-triggered average consensus for multi-agent systems with nonlinear dynamics and switching topology. Journal of the Franklin Institute, 352(3):10801098, 2015.

[33] L.N. Zhao, H.J. Ma, L.X. Xu, and X. Wang. Observer-based

adaptive sampled-data event-triggered distributed control for

multi-agent systems. IEEE Transactions on Circuits and

Systems II: Express Briefs, 2019.