Mohamed Maghenem
email: mmaghene@ucsc.edu.

Antonio Loría

Emmanuel Nuño
email: emmanuel.nuno@cucei.udg.mx.

Elena Panteley
email: panteley@lss.supelec.fr.

Consensus-based formation control of networked nonholonomic vehicles with delayed communications

Keywords: Formation control, persistency of excitation, Lyapunov design, nonholonomic systems

For a network of nonholonomic vehicles communicating according to an undirected connected graph, a consensus-based formation control problem is solved via a smooth time-varying, proportionalderivative, δ-persistently-exciting, controller. It is assumed that the communication between agents is affected by time-varying, non-differentiable, communication delays and uniform global asymptotic stability is demonstrated. This goes beyond the more-often encountered property of nonuniform convergence and, what is more, for the first time in the literature, a strict Lyapunov-Krasovskiȋ functional is provided.

I. INTRODUCTION

For a group of (first or second order) integrators the problem of consensus, which consists in the state variables of all agents converging to a common value, is well studied and solved under many different scenarios [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF]. For instance, for simple integrators interconnected over an undirected, static and connected graph, the consensus equilibrium corresponds to the (weighted) average to the initial conditions. However, the problem is more complex if one considers the agents' dynamics [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF], [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF], network constraints, such as communication delays [START_REF] Zhao | Distributed consensus of multiple Euler-Lagrange systems networked by sampled-data information with transmission delays and data packet dropouts[END_REF], [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF], or nonholonomic constraints that restrict the systems' motion [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF].

For autonomous nonholonomic vehicles, which can obviously not converge to a common position, the objective may be formulated as the requirement that the vehicles converge to specific positions to form a geometric pattern anywhere on the plane. While the formation is predetermined by defining a position vector originating at the center of the formation, the said center is not imposed a priori, but it results from the initial positions, the network's topology and the dynamic behavior of the agents. We refer to this goal as consensus-based formation control.

As a set-point stabilization problem, consensus-based formation control for nonholonomic systems is not solvable via smooth timeinvariant feedback. Akin to [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF], [START_REF] Lizárraga | Obstructions to the existence of universal stabilizers for smooth control systems[END_REF] for the case of one robot, in [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF] necessary and sufficient conditions for consensus of nonholonomic vehicles are established.

From a control-theory viewpoint, it is relevant to make a distinction according to the model that is used as well as the state variables that are regulated. In terms of modelling, in many works a kinematicsonly (hence simplified) model is used -see e.g., [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF], [START_REF] Dong | Cooperative control of multiple nonholonomic mobile agents[END_REF], [START_REF] Peng | Distributed consensusbased formation control for multiple nonholonomic mobile robots with a specified reference trajectory[END_REF], and [START_REF] González | Stability analysis of nonholonomic multiagent coordinate-free formation control subject to communication delays[END_REF]. In this note we employ a force-controlled (hence full) model -cf. [START_REF] Sarkar | Control of mechanical systems with rolling constraints: application to dynamic control of mobile robots[END_REF], [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF]. Concerning the regulated variables, in some works the orientation of the vehicles is completely disregarded [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF], so formation control applies only to the Cartesian positions of a point on the plane. In this note, as in [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF], we address the problem of partial consensus which pertains to the case in which all the agents are required to converge to a formation around an unknown center and the orientation of each vehicle is required to be stabilized to a desired given reference. In [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF], [START_REF] Peng | Distributed consensusbased formation control for multiple nonholonomic mobile robots with a specified reference trajectory[END_REF], [START_REF] Maghenem | Distributed fullconsensus control of nonholonomic vehicles under non-differentiable measurement delays[END_REF], [START_REF] Nuño | Consensus-based formation control of nonholonomic robots with communication delays[END_REF] the full consensus problem is addressed; that is, it is also required for the robots to acquire a common, non-a priori given, orientation. Other problems' classifications may be made rather in terms of the communications topology that is employed, as well as on the nature of available measurements, regardless of the model used [START_REF] Oh | A survey of multi-agent formation control[END_REF].

Measurements may come from global-positioning systems [START_REF] Peng | Distributed consensusbased formation control for multiple nonholonomic mobile robots with a specified reference trajectory[END_REF], from proximity sensors [START_REF] González | Stability analysis of nonholonomic multiagent coordinate-free formation control subject to communication delays[END_REF], or both [START_REF] Dong | Cooperative control of multiple nonholonomic mobile agents[END_REF]. The first yield usually reliable measurements with respect to a fixed frame and may be used to achieve formation consensus from initially far-away positions; the latter deliver relative distances and orientations and are useful to maintain the formation while tracking a trajectory. In this note it is assumed that global-positioning measurements are available and that each robot communicates its position and orientation to a set of neighbors through a wireless network. Consequently, the information that each agent receives is affected by a delay.

Consensus of nonholonomic vehicles under delayed measurements is addressed, for instance, in [START_REF] Dong | Cooperative control of multiple nonholonomic mobile agents[END_REF], [START_REF] González | Stability analysis of nonholonomic multiagent coordinate-free formation control subject to communication delays[END_REF] and [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF]. In [START_REF] Dong | Cooperative control of multiple nonholonomic mobile agents[END_REF] the delays are assumed to be constant, in [START_REF] González | Stability analysis of nonholonomic multiagent coordinate-free formation control subject to communication delays[END_REF] the delays are assumed to be constant and equal for all agents, and in [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF], they are assumed to be differentiable with bounded derivatives (up to the fourth). In this note, we relax these hypotheses by assuming that the delays are timevarying and non-differentiable, and bounded. Such relaxations are clearly motivated by realistic scenarios such as in the case of package dropouts, intermittent communications, blackouts, etc. [START_REF] Zhao | Distributed consensus of multiple Euler-Lagrange systems networked by sampled-data information with transmission delays and data packet dropouts[END_REF], [START_REF] Abdessameud | Synchronization of Lagrangian systems with irregular communication delays[END_REF].

The main contribution, however, is of a more theoretical nature. We prove uniform global asymptotic stability for the closed-loop system, which is a much stronger property than mere (non-uniform) convergence, as most-often encountered in the literature (including most of the previous references). Establishing uniform global asymptotic stability is crucial for time-varying nonlinear systems because, unlike convergence, it guarantees robustness of the system with respect to bounded disturbances. Furthermore, for the first time in the literature, we provide a strict Lyapunov-Krasovskiȋ functional. A significant use of the latter is in incorporating further restrictions in the control problem, such as obstacle avoidance, but this is not addressed here.

This note is a byproduct of [START_REF] Maghenem | Stability and stabilization of networked time-varying systems[END_REF] and it builds upon the previous articles [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF], [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF], and [START_REF] Maghenem | Consensus-based formation control of nonholonomic robots using a strict Lyapunov function[END_REF]. In [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF] only convergence is demonstrated under more restrictive conditions on the time delays. In [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF] and [START_REF] Maghenem | Consensus-based formation control of nonholonomic robots using a strict Lyapunov function[END_REF] uniform global asymptotic stability is established, but in the absence of delays. We also emphasize that our controller, which is inspired from that in [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF], is smooth time-varying and of the δ-persistentlyexciting class [START_REF] Loría | UGAS of skew-symmetric timevarying systems: application to stabilization of chained form systems[END_REF]. The crafting of the strict Lyapunov-Krasovskiȋ functional relies on the powerful Mazenc construction [START_REF] Malisoff | Constructions of Strict Lyapunov functions[END_REF], [START_REF] Mazenc | Strict Lyapunov functions for time-varying systems[END_REF].

In the next section we describe the networked systems' model and we formulate the problem at hand. The main result is presented in Section III and some simulation results are provided in Section IV. Concluding remarks are presented in Section V.

II. MODEL AND PROBLEM FORMULATION

Let us consider a swarm of N autonomous vehicles modeled as unicycles, hence, with kinematics

ẋi = vi cos(θi) (1a) ẏi = vi sin(θi) (1b) θi = ωi, i ≤ N (1c)
and dynamics

vi = uvi (2a) ωi = uωi. (2b)
The variables xi and yi denote Cartesian coordinates of a fixed point on the vehicle relative to a fixed frame, θi denotes its orientation with respect to the horizontal axis, and vi and ωi denote the forward and angular velocities respectively. The pair (uvi, uωi) corresponds to the control inputs, which may depend on the current vehicle's local coordinates and time, but also on the coordinates of its neighbors. Remark 1: We emphasize that, in contrast with a greater body of literature (e.g., [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF], [START_REF] Dong | Cooperative control of multiple nonholonomic mobile agents[END_REF], [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF], and [START_REF] González | Stability analysis of nonholonomic multiagent coordinate-free formation control subject to communication delays[END_REF]) in which the dynamics (2) is neglected, we assume here that the nonholonomic vehicle is torque controlled. From a physical viewpoint, there exists a direct relation between the wheels' input torque and the controls (uvi, uωi). For simplicity, however, we model the dynamics using an integrator. There is little loss of generality in such simplification since, typically, Equations (2) correspond to those of a fully-actuated (hence feedback linearizable) Lagrangian system [START_REF] Sarkar | Control of mechanical systems with rolling constraints: application to dynamic control of mobile robots[END_REF], [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF].

It is assumed that the ith vehicle communicates its positions (xi, yi) and its orientation θi to a group of neighbors over a network; the topology of which is assumed to be defined by a connected, undirected, and static graph. The connectivity of the latter is defined by the Laplacian L

:= [ ij ] ∈ R N ×N , where ij = j∈N i aij if i = j -aij if otherwise, (3)
aij > 0 if the ith and jth vehicles communicate with each other, and aij = 0 otherwise. Remark 2: By construction, L has a zero row sum, i.e., L1N = 0, where 1N ∈ R N corresponds to the vector [1 • • • 1] . Moreover, since the interconnection graph is connected, undirected and static, L is symmetric, it has a single zero-eigenvalue, and the rest of the spectrum of L is positive. Thus, rank(L) = N -1.

For a group of N vehicles, we formulate the formation control problem, consisting in making the vehicles take positions according to a geometric pattern centered at a position on the plane not known a priori, as a consensus problem. To that end, let zc := [xc, yc] denote the unknown coordinates of the said center and, for each i ≤ N , let δi := [δxi, δyi] denote a vector compatible with physical constraints. Then, defining

zi := xi -δx i yi -δy i , z := z 1 • • • z N , v := [v1 • • • vN ] ∈ R N , (4)
one may set as control objective that

lim t→∞ z(t) = 1N ⊗ zc, lim t→∞ v(t) = 0 (5)
A more complete formulation, however, is the so-called called partial consensus [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF]. In this reference it is required that, in addition to [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF], the orientation angles converge to respective given constant references θ di , that is,

lim t→∞ θ(t) = θ d lim t→∞ ω(t) = 0 (6)
where

θ := [θ1 • • • θN ] ∈ R N , idem θ d and ω.
For nonlinear non-autonomous systems, however, the properties ( 5) and ( 6) are fairly weak since they do not include neither stability nor uniformity and, hence, offer no guarantee of robustness with respect to external disturbances. In contrast to this, uniform (global) asymptotic stability guarantees that the system is totally stable [START_REF] Krasovskii | Problems of the theory of stability of motion[END_REF]. For this reason in this note we solve the more challenging problem of rendering the consensus set

Spc := {z = 1N ⊗ zc, θ = θ d , v = ω = 0}
uniformly globally asymptotically stable, for the system

ż = Φ(θ)v (7a) v = uv (7b) θ = ω (7c) ω = uω, (7d)
where

Φ(θ) = diag[φ(θi)], φ(θi) := [cos(θi) sin(θi)] , uv := [uv1 • • • uvN ] , and uω := [uω1 • • • uωN ] .
Remark 3: Uniform global asymptotic stability of Spc implies ( 5) and ( 6), but not the opposite. Now, for the purpose of analysis via Lyapunov's direct method, we redefine the consensus problem for [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF] as one of stabilization of an equilibrium, as opposed to that of a set. To that end, let ei = φ(θi)

j∈N i aij(zi -zj), (8)
si = φ(θi) ⊥ j∈N i aij(zi -zj), (9)
where φ(θi

) ⊥ = [sin(θi) -cos(θi)] or, in compact form, e = Φ(θ) Lz, s = Φ(θ) ⊥ Lz, (10)
where

L := L ⊗ I2, Φ(θ) ⊥ = diag[φ(θi) ⊥ ], e := [e 1 • • • e N ] ,
and

s := [s 1 • • • s N ]
. The advantage of this change of variable is that, since the matrix Φ(θ) Φ(θ) ⊥ is invertible for all θ ∈ R N and the communication graph is connected,

Lz = 0 ⇔ (e, s) = (0, 0). (11)
Thus, defining θ := θ -θ d , uniform global asymptotic stability of the set Spc is equivalent to that of the equilibrium point (e, s, θ, v, ω) = (0, 0, 0, 0, 0). This is established in the next section.

III. MAIN RESULTS

A. Control design and main statement

We use a certainty-equivalent version of the δ-persistently-exciting controller proposed in [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF], in the absence of delays; this yields

uv = -K dt v -Kpte d , (12)
uω = -K dθ ω -K pθ θ -p(t)κ(s d , e d ), (13)
where K dt , Kpt, K dθ , and K pθ are diagonal positive definite matrices, θ := θ -θ d , κ : R 2N → R N is defined as

κ(s d , e d ) := 1 2 [s 2 d1 + e 2 d1 , • • • , s 2 dN + e 2 dN ]
,

and
e di = φ(θi) j∈N i aij(zi(t) -zj(t -Tij(t))), (14)
s di = φ(θi) ⊥ j∈N i aij(zi(t) -zj(t -Tij(t))). ( 15
)
The term p(t)κ(s d , e d ) is an essential component of the control law [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF]. It is assumed that this term is δ-persistently-exciting [START_REF] Loría | A nested Matrosov theorem and persistency of excitation for uniform convergence in stable non-autonomous systems[END_REF] that is, for any δ > 0 there exist T δ and µ δ > 0 such that

e d s d ≥ δ =⇒ t+T δ t κ(e d , s d )|p(τ )|dτ ≥ µ δ ∀ t ≥ 0, (16)
where | • | denotes the Euclidean norm. In turn, for the condition [START_REF] Nuño | Consensus-based formation control of nonholonomic robots with communication delays[END_REF] to hold, it is necessary to impose that the function p : R ≥0 → R be persistently exciting in the usual sense, that is, that there exist T and µ > 0 such that

t+T t |p(τ )|dτ ≥ µ ∀ t ≥ 0. (17)
Furthermore, we impose the following conditions on the function t → p(t) and its derivatives: Assumption 1: There exists bp > 0 such that

max |p|∞, | ṗ|∞, |p|∞, |p (3) |∞ ≤ bp, (18)
where |p|∞ := sup t≥0 |p(t)| and, moreover, ṗ(t) is persistently exciting with excitation parameters (Tp, µp). Theorem 1: Consider the system (7) in closed loop with the controller ( 12)-( 13) with K dt , Kpt, K dθ , and K pθ diagonal positive definite and let Assumption 1 hold. Suppose, in addition, that there exists T * > 0 such that Tij(t) ∈ [0, T * ] for all i, j ≤ N , t ≥ 0 and the matrices K dt and Kpt satisfy

1 -1 + N 2 ā2 T * λM (KptK -1 dt ) ≥ 0, (19)
where λM (•) denotes the largest eigenvalue, ā := max{aij}. Then, the origin in the closed-loop state space, i.e, {(e, s, θ, v, ω) = (0, 0, 0, 0, 0)}, is uniformly globally asymptotically stable. The proof of Theorem 1 is based on Lyapunov's direct method. For clarity of exposition, it is divided in two main parts: first, we derive the closed-loop equations and, then, we present the main guidelines to construct a strict Lyapunov-Krasovskiȋ functional.

B. The closed-loop equations

The first two error equations are obtained by differentiating on both sides of the the two equations in [START_REF] Peng | Distributed consensusbased formation control for multiple nonholonomic mobile robots with a specified reference trajectory[END_REF] and using (7a) and

Φ(θ) = -Φ(θ) ⊥ ω, Φ(θ) ⊥ = Φ(θ)ω, (20)
where

ω := diag[ωi] ∈ R N ×N , to obtain ė = -ωs + Φ(θ) LΦ(θ)v (21)
ṡ = ωe + Φ(θ) ⊥ LΦ(θ)v. (22)
Then, by direct substitution of ( 12) and ( 13) in (7b) and (7d), respectively, we obtain

v = -K dt v -Kpte d (23a) θ = ω (23b) ω = -K dθ ω -K pθ θ -p(t)κ(s d , e d ). (23c)
Note the dependence on the delayed errors e d and s d in (23a) and (23c) -for the purpose of analysis we stress that, after ( 14) and ( 15), e d and s d may be expressed in function of e and s as

e d = e + Φ(θ) A( żt), (24a)
s d = s + Φ(θ) ⊥ A( żt), (24b)
where żt is short notation for żt(∆) := ż(t+∆), with ∆ ∈ [-T * , 0] and the vector A( żt) ∈ R 2N is defined as

A( żt) :=    A1( żt) . . . AN ( żt)    , Ai( żt) = j∈N i aij t t-T ji (t) żj(τ )dτ.
where

κ d = 1 2     A 1 A1 + 2e1φ 1 A1 + 2s1φ ⊥ 1 A1 . . . A N AN + 2eN φ N AN + 2sN φ ⊥ N AN     . (26)
Remark 4: If measurements are not affected by delays, A i ( żt) = 0 and we recover the error-dynamics equations from [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF], [START_REF] Maghenem | Consensus-based formation control of nonholonomic robots using a strict Lyapunov function[END_REF].

Remark 5: The smooth time-varying stabilization mechanism may be roughly explained as follows. On one hand, the term -p(t)κ(s, e) acts as a perturbation to the angular dynamics, (23c), as long as either of the errors, e d and s d are away from a δ-neighbourhood of the origin. This is necessary because without the term -p(t)κ(s d , e d ), even in the absence of delays, θ would converge exponentially fast to the reference θ d . Now, on {v = 0, ω = 0}, we have ė(t) = ṡ(t) = 0 -see ( 21), [START_REF] Maghenem | Consensus-based formation control of nonholonomic robots using a strict Lyapunov function[END_REF] without guarantee that consensus has been reached, i.e., that both, e(t) = 0 and s(t) = 0 -see Eq. [START_REF] González | Stability analysis of nonholonomic multiagent coordinate-free formation control subject to communication delays[END_REF]. On the other hand, the control input uv steers v, e d , and s d to zero, rendering the "perturbation" -p(t)κ(s d , e d ) smaller in norm and thereby allowing for θ to converge to zero as well. For further insight on δ-persistentlyexciting controllers see [START_REF] Loría | UGAS of skew-symmetric timevarying systems: application to stabilization of chained form systems[END_REF], [START_REF] Loría | A new persistency-of-excitation condition for UGAS of NLTV systems: Application to stabilization of nonholonomic systems[END_REF], [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF].

Next, akin to latter reference, for the purpose of analysis let us introduce the new errors e θ = θ + q(t)κ(s, e), eω = ω + q(t)κ(s, e)

where q : R ≥0 → R N ×N is a differentiable function defined dynamically as

q + K dθ q + K pθ q = p(t)IN , (28)
where K dθ and K pθ are diagonal positive definite matrices and IN denotes the N × N identity matrix. Therefore, in view of (18) there exists bq > 0 such that max |q|∞, | q|∞, |q|∞, |q (3) |∞ ≤ bq.

In the new equivalent coordinates [START_REF] Loría | A nested Matrosov theorem and persistency of excitation for uniform convergence in stable non-autonomous systems[END_REF], the closed-loop dynamics equations become

Ẋt =   -K dt -Kpt 0 0 0 qκ -ēω 0 -qκ + ēω 0   Xt +   0 Φ L Φ ⊥ L   Φv -   KptΦ 0 0   A( żt) (30a) Ẋr = 0 IN -K pθ -K dθ Xr + q q ēΦ + sΦ ⊥ LΦv - 0 p κ d , (30b)
where κ :=diag[s 2 i + e 2 i ], ēω :=diag[eω i ], Xt := [v e s ] , and Xr := [e θ e ω ] . The rest of the proof consists in establishing uniform global asymptotic stability of the origin for [START_REF] Ioannou | Robust adaptive control[END_REF].

C. Stability analysis

We proceed now to construct a Lyapunov-Krasovskiȋ functional for [START_REF] Ioannou | Robust adaptive control[END_REF], by relying on the Lyapunov function proposed in [START_REF] Maghenem | Consensus-based formation control of nonholonomic robots using a strict Lyapunov function[END_REF] for the system ( 21)- [START_REF] Loría | UGAS of skew-symmetric timevarying systems: application to stabilization of chained form systems[END_REF] subject to (e d , s d ) = (e, s), that is, without delays.

First, we observe that the translational part of the system, i.e., Eq. (30a), admits the following non-strict Lyapunov-Krasovskiȋ functional

V (v, z, żt) = v K -1 pt v + z Lz + 0 -T * t t+θ | ż(τ )| 2 dτ dθ. ( 31
)
After [14, Lemma 1], we have

λ2(L)z Lz ≤ |e| 2 + |s| 2 ≤ λN (L)z Lz, (32)
where λi(L) denotes the ith eigenvalue of L so, in view of the inequality

0 -T * t t+θ | ż(τ )| 2 dτ dθ ≤ T * t t-T * | ż(τ )| 2 dτ,
it follows, according to [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF], that the functional V is positive definite and radially unbounded with respect to Xt. Furthermore, the timederivative of V along the trajectories of (30a) is

V = -2v K -1 pt K dt v + 2v Φ(θ) A( żt) + T * |v| 2 - t t-T * | ż(τ )| 2 dτ ≤ -[2 -T * λM (KptK -1 dt )]v K -1 pt K dt v + 2v Φ(θ) A( żt) - 1 ā2 N N j=1 N i=1 a 2 ij t t-T ij | żi(τ )| 2 dτ. ( 33
)
Now, on one hand, after Jensen's inequality, we have

- t t-T ij | żi(τ )| 2 dτ ≤ - 1 T * ij t t-T ij żi(τ )dτ 2 (34)
while, on the other, the second term on the right-hand side of (33

) satisfies 2v Φ(θ) A( żt) ≤ (N/2 )|A( żt)| 2 + λM (KptK -1 dt )v K -1 pt K dt v for any > 0. Let = N 2 āT * . Then, using |A( żt)| 2 ≤ N N j=1 N i=1 a 2 ij t t-T ij żi(τ )dτ 2 (35)
and [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF], we obtain

V ≤ -v K -1 pt K dt v -Y ( żt) (36) Y ( żt) := 1 2ā 2 N T * N j=1 N i=1 a 2 ij t t-T ij żi(τ )dτ 2
which is negative semidefinite. That is, V is a non-strict Lyapunov-Krasovskiȋ functional for the system (30a).

Next, following the Mazenc construction method of strict Lyapunov functions [START_REF] Mazenc | Strict Lyapunov functions for time-varying systems[END_REF], [START_REF] Malisoff | Constructions of Strict Lyapunov functions[END_REF], we introduce a function (t, Xt, V ) → W (t, Xt, V ). We recall the function proposed in [START_REF] Maghenem | Consensus-based formation control of nonholonomic robots using a strict Lyapunov function[END_REF]:

W = γ(V )V + V κ(s, e) Υ(t)κ(s, e) + α(V )e v -c1V e qs + c1bqλN (L)V 2 + λN (L) + |Kpt| α(V )V (37)
where bq > 0 is defined in [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF], γ and α are non decreasing functions mapping R ≥0 to R>0 chosen as polynomials of V -see the appendix for details-, and

Υ(t) := 1 + 2b 2 q T IN - 2 T t+T t m t q(τ ) 2 dτ dm. ( 38
)
Remark 6: We stress for further development that e v and e s are quadratic terms that, in view of the triangle inequality, [START_REF] Salvo-Rossi | Joint endto-end loss-delay hidden markov model for periodic UDP traffic over the internet[END_REF], and (32), satisfy e v + e s ≤ cV for some appropriate value of c > 0. Therefore, in view of [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF],

α(V )e v -c1V e qs ≤ c α (V ) (39)
for some continuous non-decreasing function α : R ≥0 → R>0. The specific definitions of c and α are irrelevant, but in the sequel we use extensively this type of bound. The computation of Ẇ involves lengthy but straightforward steps that are relegated to the appendix -cf. [START_REF] Maghenem | Stability and stabilization of networked time-varying systems[END_REF]. We rather underline the purpose of the terms that compose W and the main steps in the differentiation of this function.

The purpose of the term γ(V )V is to generate in Ẇ the term -γ(V )v K -1 pt K dt v, whose negativity is "enforced" by the factor γ(V ) -cf. (36). The second term on the right-hand side of (37), generates the negative term -µ T V |κ(e, s)| 2 . This is in virtue of the fact that

Υ(t) ≤ - 2 T t+T t q(τ ) 2 dτ + 2b 2 q IN ∀ t ≥ 0 ( 40
)
and q is persistently exciting, as per Assumption 1. Indeed, this follows because q is solution to 28), [START_REF] Ioannou | Robust adaptive control[END_REF].

q (3) + K dθ q + K pθ q = ṗ(t)IN -cf. (
Next, the respective derivatives of the third and fourth terms on the right hand side of (37) satisfy, respectively,

d dt α(V )e v ≤ - ∂α ∂V e vv K dt K -1 pt v + α(V ) v Φ LΦv +s qκv -e K dt v -e Kpte -v ēωs (41) d dt -c1V e qs ≤ -e qsV c1 + c1[e qs]v K dt K -1 pt v -c1V v Φ LΦ qs + s κ q2 s -e q2 κe +e qΦ ⊥ LΦv -s ēω qs + e qēωe . ( 42
)
Note the non-positive terms -α(V )e Kpte and -c1V s κ q2 s in (41) and ( 42) respectively; all the other terms are of undefined sign and may be bounded as those in (39). On the other hand, in view of (36), the total derivative of the last two terms that define W is negative semidefinite and serves in bounding from above the terms of undefined sign in (41) and (42). Thus, for an appropriate choice of γ(V ) and α(V ) -see the appendix for details-, we obtain

Ẇ ≤ - 1 2 γ(V )v K dt K -1 pt v - 1 4 α(V )e Kpte - µ 2T V |κ(s, e)| 2 -α(V )v ēωs + c1V s qēωs -qc1V e qēωe -α(V )e KptΦ(θ) A( żt) -γ(V ) -c1e qs + ∂α ∂V e v Y ( żt). (43)
Now, for any σ > 0 we have

α(V )v ēωs ≤ λ N (L)|K -1 dt Kpt| σ α(V )V v K dt K -1 pt v + σ 4 α(V )|eω| 2 , ( 44
)
c1V e ω q (ss -ēe) ≤ c1V bq σ 4 |eω| 2 + 4 σ |κ(s, e)| 2 . ( 45
) Furthermore, -α(V )e KptΦ(θ) A( żt) ≤ α(V ) 16 e Kpte+4α(V )λM (Kpt) |A| 2 ,
and

|A| 2 ≤ N N j=1 N i=1 a 2 ij t t-T ij żi(τ )dτ 2 hence, |A( żt)| 2 ≤ 2N 2 T * ā2 Y ( żt)
and for an appropriate choice of γ(V ) and α(V ), the sum of the last two terms on the right-hand side of (43) is non-positive. Furthermore, after (44)-(45), if we set

σ(V ) = 4λN (L) K -1 dt Kpt α(V )V γ(V ) + 16T c1bq µ (46)
we obtain

Ẇ ≤ - 1 4 γ(V )v K dt K -1 pt v - 1 8 α(V )e Kpte - µ 4T V |κ(s, e)| 2 + σ 4 α(V ) + bqc1V |eω| 2 . ( 47
)
Note that, apart from the last term on the right-hand side of (47), Ẇ is negative definite in Xt. The next step in the construction of a Lyapunov-Krasovskiȋ functional for the closed-loop system is to design a function of Xr whose derivative contains terms that are sufficiently negative to dominate over the last term on the right-hand side of (47). Let Z(Xr) = c2 e ω eω + e θ K pθ e θ + e θ eω (48)

which is positive definite in Xr for an appropriate choice of c2 > 0.

For clarity of exposition in the computation of Ż along the trajectories of (30b) let, for the time being, κ d ≡ 0. Then, the derivative of Z along the trajectories of (30b) satisfies

Ż(Xr) ≤ - c2 4 e ω K dθ eω + 1 4 e θ K pθ e θ + c2 2 ēΦ LΦv 2 + c2 2 sΦ ⊥ LΦv 2 (49)
Furthermore, using the inequalities

ēΦ LΦv 2 ≤ λN (L)V Φ LΦ 2 ∞ K -1 dt Kpt v K dt K -1 pt v, sΦ ⊥ LΦv 2 ≤ λN (L)V Φ ⊥ LΦ 2 ∞ K -1 dt Kpt v K dt K -1 pt v and defining ρ3(V ) = c2λN (L) K -1 dt Kpt 2 Φ LΦ 2 ∞ + Φ ⊥ LΦ 2 ∞ , we obtain Ż(Xr) ≤ - c2 4 e ω K dθ eω + 1 4 e θ K pθ e θ + 2ρ3(V )V v K dt K -1 pt v.
By a simple inspection one may see that Ẇ +[ • ] Ż can be rendered negative semidefinite if Z(Xr) is scaled appropriately by a factor depending on V . Let V → ρ1(V ) be a strictly increasing function of V such that ρ1(0) > 0. Then, the total derivative of ρ1(V )Z(Xr) along the trajectories of (30b) yields

d dt ρ1(V )Z(Xr) ≤ -ρ1(V ) c2 4 e ω K dθ eω + 1 4 e θ K pθ e θ + ρ1(V )ρ3(V )V v K dt K -1 pt v + 2ρ1(V )Kptc2e ω κ d + ρ1(V )Kpte θ κ d + Z(Xr) V (50)
and we recall that V satisfies (36). Then, we use the identity

κ d (e d , s d ) = 1 2    |A1| 2 . . . |AN | 2    + ēΦ A( żt) + sΦ ⊥ A( żt)
as well as the inequalities

λM (Kpt)c2ρ1(V ) |eω| |A| 2 ≤ λM (Kpt)c2 2 1 |eω| 2 |A| 2 + λM (Kpt)c2ρ1(V ) 2 1 2 |A| 2 , 1 2 λM (Kpt)ρ1(V ) |e θ | |A| 2 ≤ λM (Kpt) 4 1 |e θ | 2 |A| 2 + λM (Kpt)ρ1(V ) 2 1 4 |A| 2 , 2ρ1(V )Kptc2e ω ēΦ + sΦ ⊥ A( żt) ≤ 2 λM (Kpt)ρ1(V )c2 |eω| 2 + λM (Kpt)c2ρ1(V ) e 2 + s 2 |A| 2 , ρ1(V )Kpte θ ēΦ + sΦ ⊥ A( żt) ≤ λM (Kpt)ρ1(V ) |e θ | 2 + ρ1(V )λM (Kpt) e 2 + s 2 2 |A| 2 ,
which hold for any , 1 > 0, to obtain

d dt ρ1(V )Z(Xr) ≤ -ρ1(V ) c2 4 e ω K dθ eω + 1 4 e θ K pθ e θ + ρ1(V )ρ3(V )V v K dt K -1 pt v + Z(Xr) V + λM (Kpt)ρ1(V ) [2c2 + 1] |eω| 2 + |e θ | 2 + λM (Kpt) 2 1 c2 + 1 2 |eω| 2 + |e θ | 2 |A| 2 + λM (Kpt)ρ1(V ) c2 + 1 2 ρ1(V ) 1 2 + λN (L)V |A| 2 . (51)
We see that for sufficiently large values of , the sum of the first and fourth terms on the right-hand side of ( 51) is non-positive and so is the sum of the third and fifth terms, in view of (36) and

Z(Xr) ≥ min {1, λm(K pθ )} |e θ | 2 + |eω| 2 , |A| 2 ≤ N N j=1 N i=1 a 2 ij t t-T ij żi(τ )dτ 2 .
Therefore, we have

d dt ρ1(V )Z(Xr) ≤ -ρ1(V ) 3c2 16 e ω K dθ eω + 3 16 e θ K pθ e θ + λM (Kpt)ρ1(V ) c2 + 1 2 ρ1(V ) 1 2 + λN (L)V |A| 2 + ρ1(V )ρ3(V )V v K dt K -1 pt v. (52)
Thus, in view of (47) and ( 52), the total derivative of V(t, Xt, Xr) = W (t, V, v, e, s) + ρ1(V )Z(Xr) + ρ2(V )V along the trajectories of [START_REF] Ioannou | Robust adaptive control[END_REF], where

ρ2(V ) = 2λM (Kpt)ρ1(V )[1 + c2]N 2 ā2 T * [λN (L) V + 1ρ1(V )] + 2ρ1(V )ρ3(V )V, satisfies V ≤ - 1 8 γ(V )v K dt K -1 pt v - 1 16 α(V )e Kpte - µ 8T V |κ(s, e)| 2 -ρ1(V ) c2 16 e ω K dθ eω + 1 16 e θ K pθ e θ , (53)
which is negative definite since, by design, ρ1(0) > 0. Also, V(t, Xt, Xr) is positive definite. The result follows.

IV. SIMULATION RESULTS

We present now some illustrative numerical simulations using four differential wheeled mobile robots. The desired formation pattern is fixed to be a rhomboid with an open-chain interconnection graph; its Laplacian and schematic representation are given in Figure 1. The initial positions and the relative distances with regards to the center of the formation are given in Table I. The control gains have been set to: K dt = 60I4; Kpt = 30I4; K dθ = 30I4; and K pθ = 10I4. The persistently exciting function p(t) is given by p(t) = -3 + 2π sin(0.5t). For simplicity, the time-varying delay is the same for all the interconnections and is depicted in Fig. 2, its bound has been taken as T * = 0.37. These delays follow a normal Gaussian distribution with a mean of 0.3 and a variance of 0.0003 -cf. [START_REF] Salvo-Rossi | Joint endto-end loss-delay hidden markov model for periodic UDP traffic over the internet[END_REF]. Under this scenario, condition [START_REF] Nuño | Leaderless consensus-based formation control of multiple nonholonomic mobile robots with interconnecting delays[END_REF] is satisfied. Fig. 3 shows the xi, yi trajectories of the nonholonomic mobile robots, while Figs. 4 and 5 depict the relative positions zi and the orientation of the robots. From these plots it can be observed that the robots reach the target formation and their respective orientations converge to the desired values. The "zigzaggy" motion is common in stabilization of nonholonomic robots -cf. [START_REF] Dong | Cooperative control of multiple nonholonomic mobile agents[END_REF], [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF], [START_REF] González | Stability analysis of nonholonomic multiagent coordinate-free formation control subject to communication delays[END_REF], [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF].

1 2 3 4 L =     1 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 1    

V. CONCLUSIONS

A decentralized smooth time-varying controller that guarantees partial-consensus among groups of nonholonomic mobile robots interacting over a network that is affected by time-varying communication delays was presented. Uniform global asymptotic stability of the consensus set is formally established via constructive proof (a strict Lyapunov-Krasovskiȋ functional is provided). From a practical viewpoint, however, the solution presented has the limitation of relying on global-positioning systems which, in a variety of concrete robotics scenarii, are unavailable. Current research focuses on consensus control using relative-positioning measurements, including under the influence of delays.

Furthermore, κ(

  e d , s d ) in (23c) may be expressed as κ(e d , s d ) = κ(e, s) + κ d (e, s, θ, żt)

Fig. 1 .

 1 Fig. 1. Interconnection undirected graph and Laplacian matrix

Fig. 3 .Fig. 4 .

 34 Fig. 3. Trajectories and formation of the network of mobile robots.

Fig. 5 .

 5 Fig. 5. Orientation of each mobile robot in the network.

  Fig. 2. Time delay in the network interconnection.

				TABLE I
	INITIAL CONDITIONS AND DESIRED ORIENTATIONS AND RELATIVE
		x i (0) y i (0)	θ i (0)	δx i	δy i	θ di
	1	5	-7	-(1/3)π	-2	0	-1
	2	-4	7	(2/3)π	0	2	1
	3	-2	-5	(4/3)π	2	0	0
	4	7	3	0	0	-2	0.5
	0.36
	0.32
	0.28
	0.24
		2		4	6	8	10

This article was supported by: Dept. STITS of University of Paris Saclay, by the Government of Russian Federation (grant 08-08), by the French National Research Agency via the project HANDY (grant number ANR-18-CE40-0010), by ECOS-Nord contract no. M14M02, by the Mexican CONACyT Basic Scientific Research grant CB-282807, and by CEFIPRA under the grant number 6001-A.

APPENDIX

Detailed computation of Ẇ , after (37)

We evaluate the total derivative of each term on the right-hand side of (37). First, since γ is non-decreasing, we have

Now, for the second term, we have

V κ q(t) 2 κ + 2V κ Υ ēΦ LΦv + sΦ ⊥ LΦv and since q is persistently exciting with parameters(µ, T ) and the following identities hold: κ ΥēΦ LΦv = e κΥΦ LΦv and κ ΥsΦ ⊥ LΦv = s κΥΦ ⊥ LΦv, we obtain

Then, after the triangle inequality, the following hold for any > 0:

for which we also used the inequalities s κs + e κe ≤ 2κ κ,

Next, using the inequality V κ q(t) 2 κ ≤ V s q(t) 2 κs + b 2 q λN (L)V 2 K -1 pt e Kpte, we obtain:

For the third term on the right-hand side of (37), we have

in which we used the inequalities

Rearranging terms, we obtain

Next, we evaluate the derivative of the fourth term on the righthand side of (37). We obtain

for which we used the inequalities:

and

Therefore, using the inequality

Putting the bounds (57), ( 58), ( 56), (55), and (54) together, we obtain , we obtain


Session n.3: Re-assembled and driven microbial communities of complex ecosystems DECIPHERING YEAST-YEAST INTERACTIONS IN OENOLOGICAL FERMENTATION

HARLE Olivier (1), MOURET Jean-Roch (1), NIDELET Thibault (1) *

(1) SPO, INRA, Montpellier SupAgro, Univ Montpellier, Montpellier, France * Corresponding Author: thibault.nidelet@inra.fr

Text of the abstract:

In natural or anthropized environments, microbial species are part of an ecosystem and interact in a complex network in advantageous or disadvantageous way. Until recently, process optimization in agriculture or food processing was mostly based on the selection of single strains. However, this paradigm is now being challenged and the scientific community is increasingly seeking to exploit and optimize consortia consisting of several strains. Indeed, many studies have shown that more diverse anthropized environments have many advantages in terms of resilience, disease resistance or yield of production (Barot et al. 2017). Efforts are now being made to design optimal consortia of various species and strains whose interactions will be exploited to maximize a given criterion. In this context, the so-called 'transgressive' interactions, are the most interesting to exploit. Transgressive interactions are observed when a mixture of entities has a better (or lower) performance than the best (or worst) of the entities cultivated separately. These transgressive interactions, which are the cause of 'over yielding', are already exploited and studied in agriculture (Barot et al. 2017), but are still very little exploited in microbial ecosystems. Microbial ecosystems are widely exploited in anthropogenic environments. They are used in pollution control, environmental protection, agriculture, pharmacology and food production (Ciani et al. 2015). Among the food ecosystems, oenology that has great importance, both economic and societal use regularly the addition at the beginning of fermentation 'starters' composed of selected yeasts. It is estimated that 80% of the oenological fermentations in the world are conducted with their use (Marsit et al. 2015). Most often these fermentation starters are composed of a single yeast strain of the species Saccharomyces cerevisiae selected for its ability to complete the fermentation. However, in recent years and under the pressure of consumers for more aromatic wines, multispecies starters have emerged which most often combine, a strain of S. cerevisiae allowing to complete the fermentation and a strain of a different species, often from a different genus than Saccharomyces, bringing a greater variety of flavours (Ivey et al. 2013). Nevertheless, the composition and the protocol of inoculation of these multi-strains starters are still very empirical and only based on input/output balance without taking into account the dynamics of the microbial populations and their interactions. This lack of knowledge about the yeast-yeast interactions prevents improving a reasoned design of multi-strain starters (Song et al. 2014).

To resolve this problem, we decide to precisely follow population dynamics and metabolite productions during oenological fermentations realized by unique or mixed populations of yeasts. This detailed analysis of the fermentation dynamics will help to better understand the mechanisms of interactions and the influence of yeast-yeast interactions on the metabolic composition of the final product.

Material and Methods

We performed fermentations in oenological conditions (200 g/L of sugars, 200 mg/L of assimilable nitrogen) mediated by one or two species of yeasts. S. cerevisiae was tested against 5 different species of yeasts: Metschnikowia pulcherrima, Metschnikowia fructicola, Hanseniaspora opuntiae, Hanseniaspora uvarum and Torulaspora delbrueckii. We inoculate 9.10 5 cells/mL of the nonsaccharomyces species and 1.10 5 cells/mL of S. cerevisiae. We also performed isolated cultures of each specie inoculated at 10 6 cells/mL. During the fermentation process, CO 2 production was online monitored (every 20 minutes). Population density, viability and proportion of S. cerevisiae in mixed cultures were offline measured every 12 hours during the first week and every 24h during the second week with a flow cytometer. At the same time scale, consumption of glucose and fructose, production of ethanol, glycerol, succinate, acetate and alpha keto-glutarate were measured by HPLC. Finally, the residual concentrations of nitrogen compounds (ammonium and amino-acids) were evaluated at the end of the fermentation. All these data were analysed to identify the intensity of interactions and the mechanisms underlying them.

Results

In this short paper, we analysed competitions separately in function of the non-saccharomyces species that is involved and only compare them to the S. cerevisiae species that is our central reference we will also leave aside the detail of the consumption of resources. In the case of Torulaspora delbrueckii (figure 1 bottom line), the dynamic in both isolated and mixed cultures are a very similar. We have almost the same growth rate, carrying capacity and viability. Furthermore, the viability is almost equal to 100% throughout the fermentation process indicating that there is no mortality in our culture conditions. Yet the proportion of S. cerevisiae increase during the fermentation passing from 10% to around 25% of the culture, indication a slightly better growth rate of S. cerevisiae. These measures indicate a simple case of resource competition of two species with very similar behaviours.

The competitions with the two Hanseniaspora species have a similar dynamic (figure 1 first and second line). The two Hanseniaspora isolated cultures have an exponential growth phase followed by a long phase of mortality where the viability decreases up to 25% of living cells. The mixed cultures between Hanseniaspora species and S. cerevisiae have and intermediate dynamic between both isolated culture with a similar mortality that explains the increase in frequency until an almost fixation of S. cerevisiae after 200h of fermentation. These dynamics indicate a simple competition for resources as with T. delbrueckii but this time with differential mortalities between the Hanseniaspora species and S. cerevisiae.

The competition with Metschnikowia species is different (figure 1, lines 3 and 4). In the isolated cultures, there is a phase of exponential growth followed by a stationary phase where the viability remains constant during the entire fermentation. On the opposite, in the mixed cultures, we observed a decrease in viability during the stationary phase. It is not clear if this decrease in viability is linked with the death of both species or only one. The fact that during the stationary phase the frequency of S. cerevisiae remains constant seems to indicate that both species are equally affected. Further investigation will be necessary to clarify the mechanism undergoing the mortality that only appears in the mixed fermentations.

Conclusion

This work makes it possible a better understanding of yeast-yeast interactions in oenological conditions. It seems that most interactions are indirect and mediated by the competition for resources. This observation leads to the possibility to predict with mathematical models and with data from isolated cultures only, the fermentation dynamics of mixed cultures. In the future, using such models will allow optimization of yeast consortia: frequencies of inoculation, possibilities to perform simultaneous or sequential inoculations.

Keywords

Oenology, Fermentation, yeast-yeast interactions, population dynamics, non-saccharomyces species

Barot, S., Allard, V., Cantarel, A., Enjalbert, J., Gauffreteau, A., Goldringer, I., Lata, J.-C., Roux,

Figure1:

  Figure1: Global monitoring of the kinetics of the living population (left), the proportion of the S. cerevisiae in mixed cultures (center) and the viability (right) across the fermentation. All points represent a sample and the curve lines a smoothing of the data. The full lines are for isolated cultures and the dashed lines for mixed cultures. The light gray colours represent pure culture of 'non-Saccharomyces' strains and the darker, to the corresponding culture in competition with S. cerevisiae. The isolated cultures of S. cerevisiae are represented in black. Sc, Mp, Mf, Ho, Hu, Td, Sc vs Mp, Sc vs Mf, Sc vs Ho, Sc vs Hu, Sc vs Td correspond respectively to cultures with: isolated S. cerevisiae, isolated M. pulcherrima, isolated M. fructicola, isolated H. opuntiae, isolated H.uvarum, isolated T. delbrueckii, M. pulcherrima mixed with S. cerevisiae, M. fructicola mixed with S. cerevisiae, H. opuntiae mixed with S. cerevisiae, H. uvarum mixed with S. cerevisiae and T. delbrueckii mixed with S. cerevisiae.


Mohamed Maghenem

Romain Postoyan

Antonio Loria

Antonio Loría

Elena Panteley

Lyapunov-based synchronization of networked systems: From continuous-time to hybrid dynamics

Keywords: Control engineering, Synchronization, Networks of dynamical systems, Safety, Hybrid dynamical systems, Lyapunov methods

come

Introduction

Generally speaking, for a network of dynamical systems, inducing synchronization consists in designing a controller (often one for each node in the network) so that an output or a state variable of each system acquires, asymptotically, a common behavior. For example, the spatial positions of network-interconnected autonomous vehicles (such as drones, mobile robots, platoons of cars, etc.) may be controlled to form a specified geometric pattern, a formation of sorts; the currents and voltages in a network of DC-AC inverters may be controlled to reach the same frequency of oscillation.

Depending on the nature of interaction among the nodes in the network, we distinguish between centralized and decentralized control schemes. In the first case, each node receives global information dictating its desired behavior. This may consist in a desired set-point (a specific position or oscillating frequency), a reference Email address: mmaghene@ucsc.edu (Mohamed Maghenem) trajectory, or a desired dynamical behavior in more complex cases. In centralized control approaches, each node has an independent task to ensure, which is defined by a central entity. These approaches are often used when the network is composed of a relatively small number of nodes, but abundant computation and communication capacities may be required for the control entity. Decentralized approaches, as we consider in this article, overcome these limitations. There, the control input for each node depends only on local information provided by a given set of nodes called neighbors. The network topology, which determines how the information among the neighbors flows, is typically defined using a communication graph [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF]. The communication graph plays a key role in the analysis and design of networked systems.

Decentralized synchronization in networked systems is an active and challenging research topic that has continuously attracted undivided attention from many researchers over the previous two decades -see, for instance, [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF][START_REF] Tsitsiklis | Distributed asynchronous deterministic and stochastic gradient optimization algorithms[END_REF][START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF][START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF] in which both continuous and discrete-time networks are considered. Such is the case in engineering disciplines where technological solutions involve groups of systems required to cooperate to complete a common task. Some common examples of these are related, but are not limited, to robotic manipulation and the use of autonomous vehicles [START_REF] Rodriguez-Angeles | Mutual synchronization of robots via estimated state feedback: a cooperative approach[END_REF][START_REF] Dong | Cooperative control of multiple nonholonomic mobile agents[END_REF][START_REF] Eun | Cooperative control of multiple unmanned aerial vehicles using the potential field theory[END_REF][START_REF] Belleter | Leader-follower synchronisation for a class of underactuated systems[END_REF][START_REF] Liu | Robust attitude coordination control for satellite formation with matched perturbations and measurement noises[END_REF], but they are also common in energy networks [START_REF] De Persis | A power consensus algorithm for dc microgrids[END_REF], for instance. Other disciplines include neuroscience, in which networks and sub-networks (clusters) formed due to neuronal population activities and interactions are studied [START_REF] Wendling | Relevance of nonlinear lumped-parameter models in the analysis of depth-eeg epileptic signals[END_REF], social science, in which a subject of study is that of opinion networks formed due to the interactions among individuals [START_REF] Parsegov | A new model of opinion dynamics for social actors with multiple interdependent attitudes and prejudices[END_REF], and geography, in which groups of populations that interact when reacting to catastrophes are studied [START_REF] Cantin | Nonidentical coupled networks with a geographical model for human behaviors during catastrophic events[END_REF]. While the systems models involved in each of these instances are different, a common denominator is the presence of a network composed of nodes having a dynamic behaviors that interact locally with the aim at reaching a common global goal. From a systems-theory viewpoint, at least two primary challenges can be identified when contributing to synchronization problems. The first one arises from recognizing that communication among the nodes is subject to constraints; this includes scenarios where the transfer of information is unreliable, time-dependent, or subjected to delays. The second challenge stems from considering the complexity of the nodes' dynamics; these may be linear, nonlinear or hybrid in nature and, moreover, may vary throughout the network -such is the case of heterogeneous networks. Some of these aspects have been addressed in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF], in scenarios pertaining to electromechanical engineering and robotics. The main originality of [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF] is to propose advanced Lyapunov-based approaches inspired by techniques for stand-alone nonlinear time-varying systems to analyze the coordination problems after transforming it into global stabilization of a closed, but unbounded, set. Then, Lyapunov-based techniques, via the construction of strict and differentiable Lyapunov functions, are proposed for the network's model after interconnection. This allows to derive several systematic methods to analyze the coordination task, the network's performance, and its robustness with respect to perturbations and delays affecting the transfer of information among the nodes. Note that, in related literature, the coordination task is usually analyzed via trajectory-based approaches or using weak Lyapunov functions, in the sense that the Lyapunov function does not systematically allow the verification the coordination task. In that way, for autonomous mobile robots, different coordination tasks are studied in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF], such as consensus and trajectory-tracking formation control or, for networks of oscillators, the frequency synchronization problem has been addressed. In regards to the nature of interaction and transfer of information among the nodes, several scenarios are considered; these include time-varying interconnections as well as communication delays. As far as the intrinsic dynamics of the nodes is considered, only some continuous-time models, such as single and double integrators, time-varying planar oscillators, and nonholonomic unicycles, have been considered.

The results contained in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF] concern exclusively interconnected systems described by continuous-time models. However, the problems solved therein lead to more general open questions that transcend these systems and are, in our opinion, of sufficient significance to awake the interest of a wide readership. The problems that we describe are related to the study of synchronization problems described previously, but in the context of hybrid networked systems. In particular, it is of major importance to handle the presence of discrete phenomena that are coupled with the continuous-time dynamics of the nodes that compose the network. A prominent example that makes this problem relevant is that of digital or intermittent control strategies that interact with continuous-time models, which captures constrained control actions as in power networks [START_REF] Torquati | A hybrid predictive control algorithm for tracking in a single-phase DC/AC inverter[END_REF][START_REF] Beneux | Robust stabilization of switched affine systems with unknown parameters and its application to DC/DC flyback converters[END_REF], in valve-based control systems [START_REF] Panzani | Hybrid control of a hydro-pneumatic tractor suspension[END_REF], or, in networked control systems [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF][START_REF] Postoyan | Tracking control for nonlinear networked control systems[END_REF]. It also captures the creation (resp. loss) of links and the addition (resp. removal) of nodes as in social networks [START_REF] Morȃrescú | Opinion dynamics with decaying confidence: Application to community detection in graphs[END_REF][START_REF] Frasca | Hybrid models of opinion dynamics with opinion-dependent connectivity[END_REF][START_REF] Mariano | A hybrid model of opinion dynamics with memory-based connectivity[END_REF]. Furthermore, in the intrinsic behavior of the nodes, one must also consider systems affected by discontinuities such as impacts and instantaneous jumps like in mechanical systems with impacts [START_REF] Westervelt | Hybrid zero dynamics of planar biped walkers[END_REF][START_REF] Jiménez-Leudo | Juggler system: Hybrid model and implementation[END_REF] or in cyber-physical systems [START_REF] De Persis | A Lyapunov redesign of coordination algorithms for cyber-physical systems[END_REF][START_REF] Borgers | Tradeoffs between quality-of-control and quality-of-service in large-scale nonlinear networked control systems[END_REF][START_REF] Nowzari | Event-triggered communication and control of networked systems for multi-agent consensus[END_REF]. Hence, such research aims at covering unexplored aspects which, nonetheless, are naturally motivated by intrinsic characteristics of modern systems networks.

The rest of this paper is organized as follows. In the next section, we describe the main contributions of [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF]. In section 3, we discuss the relevance and challenges related to synchronization in hybrid networks. Finally, in Section 4, we provide some research directions.

Synchronization in Continuous-Time Networked Systems

Main Idea

In [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF], original Lyapunov-based approaches are developed to analyze networks where the nodes are interconnected via decentralized control laws. More precisely, the proposed approaches consist in transforming the coordination task into global stabilization of a closed set. The analysis of the coordination task has been conducted using Lyapunov's direct method. That is, original constructions of strict and differentiable Lyapunov functions, for classes of nonlinear time-varying systems modeling the network after interconnection, are provided. Compared to related literature, where the coordination task is usually analyzed using trajectory-based approaches or using weak Lyapunov functions, in the sense that the Lyapunov function does not systematically allow the verification the coordination task, the solutions proposed in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF] allow a systematic analysis of the coordination task as well as the network's performance, e.g., in terms of convergence rates. Moreover, a systematic robustness analysis with respect to perturbations and delays affecting the transfer of information among the nodes is also carried out. Thanks to these approaches, various important coordination problems have been solved in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF]. These problems have been solved for different graph topologies, depending on how the information flows among the nodes, and for diverse target trajectories.

Leader-Follower Formation Control for Nonholonomic Vehicles

In leader-follower formation problems for mobile vehicles, the coordination task consists in making the vehicles maintain a given geometric posture in terms of both, positions and orientations, relative to that of a leader and track its trajectory [START_REF] Wang | Distributed adaptive control for consensus tracking with application to formation control of nonholonomic mobile robots[END_REF]. For a network of nonholonomic vehicles and due to the nonholonomic restriction, solving this problem under a general leader's trajectory, using a unique continuous feeedback control law, is a challenging problem, especially to establish uniform asymptotic stability. Indeed, it is shown in [START_REF] Lizárraga | Obstructions to the existence of universal stabilizers for smooth control systems[END_REF] that nonholonomic systems are not asymptotically stabilizable to arbitrary leaders trajectories having piecewise continuous velocities via continuous controllers, not even if the controllers are time-varying (the possibility of using discontinuous controllers is not analyzed in [START_REF] Lizárraga | Obstructions to the existence of universal stabilizers for smooth control systems[END_REF] and we are not aware of any other work for that matter). In the literature, for the simpler one-leader-and-one-follower case, the tracking problem has been studied in e.g., [START_REF] Wang | Simultaneous stabilization and tracking of nonholonomic mobile robots: A Lyapunov-based approach[END_REF][START_REF] Lee | Track-ing control of unicycle-modeled mobile robots using a saturation feedback controller[END_REF][START_REF] Morin | Control of nonholonomic mobile robots based on the transverse function approach[END_REF][START_REF] Do | Simultaneous tracking and stabilization of mobile robots: an adaptive approach[END_REF][START_REF] Maghenem | A unique robust controller for tracking and stabilisation of non-holonomic vehicles[END_REF] under particular scenarios of the leader's trajectories. To the best of our knowledge, the only article that allows for general leader's trajectories and for a network of nonholonomic vehicles is [START_REF] Wang | Distributed adaptive control for consensus tracking with application to formation control of nonholonomic mobile robots[END_REF]. The control design method in the latter reference follows the framework proposed in [START_REF] Morin | Practical stabilization of driftless systems on lie groups: the transverse function approach[END_REF]. However, it is only established that the formation errors converge to an arbitrarily small compact ball centered at the origin.

In [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF], see also the associated publications [START_REF] Maghenem | A unique robust controller for tracking and stabilisation of non-holonomic vehicles[END_REF][START_REF] Maghenem | A cascades approach to formation-tracking stabilization of force-controlled autonomous vehicles[END_REF][START_REF] Maghenem | A robust δpersistently exciting controller for leader-follower trackingagreement of multiple vehicles[END_REF][START_REF] Maghenem | Formation-tracking control of autonomous vehicles under relaxed persistency of excitation conditions[END_REF][START_REF] Maghenem | Cascades-based leader-follower formation-tracking and stabilization of multiple nonholonomic vehicles[END_REF], decentralized feedback control laws are proposed to solve the leader-follower formation control problem for a network of nonholonomic mobile robots. The interconnections among the nodes is assumed to be a directed spanning-tree topology; namely, each node has only one leader and can have multiple followers. The formation pattern is specified in terms of both the positions and the orientations. The network of vehicles is shown to asymptotically reach the desired geometric pattern around the leader and to track the leader's trajectory. Although the communication graph is simple, the originality of this work consists in providing strong stability guarantees; namely, global uniform asymptotic stability, for general leader's trajectories and using continuous-time feedback laws. That is, the leader's trajectory is allowed to con-verge to a static location (parking problem), and can also have a general time-varying velocity profile.

Leaderless Formation Control for Nonholonomic Vehicles

In the context of the previously described problems, the network is assumed to contain a leader. However, in some applications, the formation does not have a leader. Hence, the coordination task consists in making the vehicles maintain a given geometric posture, in terms of positions and orientations, around a center point. This problem has been studied for a network of nonholonomic vehicles in [START_REF] Peng | Distributed consensus-based formation control for multiple nonholonomic mobile robots with a specified reference trajectory[END_REF][START_REF] Yang | Smooth time-varying formation control of multiple nonholonomic agents[END_REF][START_REF] Dong | Consensus of multiple nonholonomic systems[END_REF][START_REF] Ajorlou | Distributed consensus control of unicycle agents in the presence of external disturbances[END_REF][START_REF] Nuño | Strict Lyapunov-Krasovski ȋ functionals for undirected networks of Euler-Lagrange systems with timevarying delays[END_REF] using decentralized continuous control laws. In the aforementioned references, it is shown that the nodes converge asymptotically to the required formation. However, the stability properties of the formation errors have not been analyzed. While ensuring convergence to the desired formation is important, it is also essential to have guarantees about the convergence times, in particular that two nearby initial conditions have approximately to the same convergence rate, as well as to make sure that when the robots are "close" to form the formation, they preserve this property for all times. One way to capture these features is to ensure that uniform global asymptotic stability properties are satisfied.

The leaderless formation control problem for a network of nonholonomic mobile vehicles is studied in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF], see also [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunovs direct method[END_REF][START_REF] Maghenem | Partialconsensus control of nonholonomic vehicles with delayed local measurements[END_REF][START_REF] Maghenem | Distributed full-consensus control of nonholonomic vehicles under non-differentiable measurement delays[END_REF]. The communication graph, in this case, is assumed to be a general bidirectional graph. A continuous and time-varying decentralized interconnection algorithm is proposed and the network after interconnection is showed to converge to the desired formation in terms of both positions and orientations. A significant contribution consists in establishing uniform global asymptotic stability of the formation error for the closed-loop network. Furthermore, in contrast to most of the previous-cited works, where only the kinematics model is considered, we use a model augmented by two integrators, which represent the feedback-linearized velocity dynamics. Furthermore, original tools, via the construction of strict Lyapunov functions, are proposed to analyze stability of the network in closed loop. Using these tools, in [START_REF] Maghenem | Partialconsensus control of nonholonomic vehicles with delayed local measurements[END_REF][START_REF] Maghenem | Distributed full-consensus control of nonholonomic vehicles under non-differentiable measurement delays[END_REF], we allowed the transfer of information among the nodes to be affected by general time-varying delays. The same approach was used in [START_REF] Nuño | Strict Lyapunov-Krasovski ȋ functionals for undirected networks of Euler-Lagrange systems with timevarying delays[END_REF] for a network of Euler-Lagrange systems.

Synchronization in Heterogeneous Networks Using Strong Interconnections

In general, the intrinsic dynamics of the nodes are not necessarily the same. Such is the case, for example, of networks of oscillators, in which the intrinsic frequency of the nodes is not identical [START_REF] Wilson | Synchronization of heterogeneous oscillator populations in response to weak and strong coupling[END_REF]. We refer to such networks as heterogenous. For a network of nonlinear systems, in which all the nodes are identical and the interconnection gain is sufficiently high, it is expected that the nodes synchronize; namely, that the state variables reach a common value. Furthermore, it is intuitively clear that a synchronized behavior of such a network corresponds, roughly speaking, to all the nodes adopting the motion of a single isolated unit [START_REF] Aminzare | Cluster synchronization of diffusively coupled nonlinear systems: A contraction-based approach[END_REF]. However, in the case of heterogeneous networks, the paradigm of asymptotic synchronization is much more complex: the existence of a synchronization manifold is not guaranteed [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF]. Hence, in lieu of adopting the behavior of one of the nodes, some type of internal model is necessary and sufficient for synchronization [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF] and, in general, only "practical" synchronization is usually ensured in the literature considering heterogeneous networks [START_REF] Montenbruck | Practical synchronization with diffusive couplings[END_REF][START_REF] Liu | Output synchronization of dynamical networks with incrementally-dissipative nodes and switching topology[END_REF][START_REF] Panteley | A stability-theory perspective to synchronisation of heterogeneous networks[END_REF][START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF][START_REF] Lee | A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling[END_REF][START_REF] Panteley | Practical dynamic consensus of stuartlandau oscillators over heterogeneous networks[END_REF]. In [START_REF] Panteley | A stability-theory perspective to synchronisation of heterogeneous networks[END_REF][START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF] a novel framework of analysis of heterogeneous networks was developed upon the premise that the collective behavior of network-interconnected systems is dichotomic: it consists, on the one hand, in a "weighted averaged" motion determined by the so-called emergent dynamics and, on the other, in the dynamics of the synchronization errors of each individual unit in the network, relatively to the collective behavior. Then, we say that the network synchronizes or, more precisely, achieves dynamic consensus if all the nodes adopt the emergent behavior in steady state.

This approach was pursued and extended in [START_REF] Maghenem | Singularperturbations-based analysis of synchronization in heterogeneous networks: A case-study[END_REF][START_REF] Maghenem | Singularperturbations-based analysis of dynamic consensus in heterogeneous networks[END_REF] 1 , where it is assumed that the emergent dynamics admits a limit cycle with certain stability properties. It is shown that each individual interconnected node also admits a limit cycle with the same frequency and the same stability properties as the one in the emergent dynamics. The different nodes are interconnected via a general directed graph. For the first time in the literature of synchronization, the analysis was based on singular-perturbation theory. Indeed, due to the strong interconnection, it is showed that the emergent dynamics corresponds to a slow system while the dynamics of the synchronization errors form a fast one.

Consensus Under Time-Varying Graphs

A last contribution of [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF] is dedicated to the consensus problem where the graph is time-varying. Considering time-varying graphs captures different scenarios in which the transfer of information among the nodes is unreliable, the information flow through a restricted amount of channels, or, when the nodes have a limited sensing range. Consensus problem under time-varying and switching graphs has been widely studied in the literature, see e.g., [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF][START_REF] Teel | Smooth time-varying stabilization of driftless systems over communication channels[END_REF][START_REF] Moreau | Stability of continuous-time distributed consensus algorithms[END_REF][START_REF] Ren | Synchronization of coupled harmonic oscillators with local interaction[END_REF][START_REF] Martin | Continuous-time consensus under persistent connectivity and slow divergence of reciprocal interaction weights[END_REF][START_REF] Anderson | Convergence and state reconstruction of time-varying multi-agent systems from complete observability theory[END_REF][START_REF] Alvarez-Jarquin | Consensus under switching spanning-tree topologies and persistently exciting interconnections[END_REF][START_REF] Goebel | A unifying convex analysis and switching system approach to consensus with undirected communication graphs[END_REF], mostly for nodes with single or double integrator dynamics, under different graph-connectivity conditions. The approach in the aforementioned references uses non-systematic trajectory-based approaches via non-smooth Lyapunov functions, which makes the analysis of robustness challenging.

In [START_REF] Maghenem | Lyapunov functions for persistently-excited cascaded time-varying systems: Application in consensus analysis[END_REF][START_REF] Chowdhury | On the estimation of algebraic connectivity in graphs with persistently exciting interconnections[END_REF][START_REF] Maghenem | Decentralized synchronization of time-varying oscillators under time-varying bidirectional graphs[END_REF] 2 , a novel analysis approach for time-varying consensus protocols employing the notions of persistency of excitation and strict Lyapunov functions are provided. Those approaches rely on stability theory of so-called gradient-descent and model-reference-adaptivecontrol systems, known in adaptive control theory [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF]. That is, the classical consensus paradigm is recast into a problem of stability analysis for systems with persistency of excitation. The analysis relies on the edgeagreement transformation based on the relationship between node and edge-agreement protocol [START_REF] Mesbahi | Graph theoretic methods in multiagent networks[END_REF]. The interconnection graph is assumed time-varying while maintaining a mild persistent connectivity among the nodes. The time-varying interconnections, individually, are assumed to be persistently exciting. Furthermore, networks composed of single-integrator nodes, double-integrator nodes, and time-varying planar oscillators are studied. Finally, different convergence rate estimates, for the consensus error coordinates, are provided.

Towards Hybrid Networked Systems

All the results mentioned in Section 2 concentrate on networked systems with continuous-time dynamics. It appears that in many situations, the overall system also exhibits discontinuous/jump dynamics either at the nodes' level or at the network's level. Before describing hybrid networks, in the next section, hybrid dynamical systems are introduced according to the modeling framework of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, stability, and robustness[END_REF].

Hybrid Dynamical Systems

In [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, stability, and robustness[END_REF], a hybrid dynamical system is seen as the following combination of a constrained differential equation and a constrained difference equation

(x, u) ∈ C ẋ = f (x, u) (x, u) ∈ D x + = g(x, u). ( 1
)
The state of the hybrid system, represented by x, can change according to the differential equation ẋ = f (x, u) while in the flow set C, and it can change according to a difference equation x + = g(x, u) while in the jump set D. When x ∈ C ∩ D, the solution can either jump or flow if flowing keeps the solution in C. The notation ẋ represents the time derivative of the state x, while x + represents the value of the state after an instantaneous jump. The strength of this formalism resides on the possibility of capturing, in a single representation, heterogeneous dynamics involving both continuous and discrete evolutions. It allows to cover many real-world applications that cannot be modeled precisely as purely continuous-time systems nor as purely discrete-time systems. Finally, it allows the use and the extension of state-of-the-art tools to analyze continuous and discrete-time systems (stability, robustness, safety, etc). Indeed, a range tools for analysis and control design of systems described by [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF] are by now available in the literature [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, stability, and robustness[END_REF].

Motivation

Hybrid phenomena at the node's level are common in the case of mechanical systems with impacts, such as walking robots [START_REF] Westervelt | Hybrid zero dynamics of planar biped walkers[END_REF][START_REF] Lopes | Synchronization of a class of cyclic discrete-event systems describing legged locomotion[END_REF], juggling mechanisms [START_REF] Jiménez-Leudo | Juggler system: Hybrid model and implementation[END_REF][START_REF] Jiménez-Leudo | Juggling robots network-hybrid model and multi-agent synchronization[END_REF], and classes of cyber-physical systems [START_REF] De Persis | A Lyapunov redesign of coordination algorithms for cyber-physical systems[END_REF][START_REF] Deng | Communication synchronization in cluster-based sensor networks for cyber-physical systems[END_REF]. Furthermore, hybrid phenomena at the network's level are common in many applications. For example, when interconnecting nodes under constrained control actions, which is the case in power converters within energy grids [START_REF] Torquati | A hybrid predictive control algorithm for tracking in a single-phase DC/AC inverter[END_REF][START_REF] Beneux | Robust stabilization of switched affine systems with unknown parameters and its application to DC/DC flyback converters[END_REF] and in valve-based control systems [START_REF] Panzani | Hybrid control of a hydro-pneumatic tractor suspension[END_REF], where the input switches among a finite set of operation modes. In networks of event-triggered and self-triggered control systems [START_REF] Postoyan | Tracking control for nonlinear networked control systems[END_REF][START_REF] De Persis | A Lyapunov redesign of coordination algorithms for cyber-physical systems[END_REF][START_REF] Dimarogonas | Distributed event-triggered control for multi-agent systems[END_REF][START_REF] Fan | Distributed eventtriggered control of multi-agent systems with combinational measurements[END_REF][START_REF] Dimarogonas | Distributed self-triggered control for multi-agent systems[END_REF], where the input is updated only at particular events and remains constant between events. Finally, in network control systems [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF][START_REF] Borgers | Tradeoffs between quality-of-control and quality-of-service in large-scale nonlinear networked control systems[END_REF][START_REF] Nowzari | Event-triggered communication and control of networked systems for multi-agent consensus[END_REF], where all the data (inputs and measurements) flow through the same channel; hence, the measurements and the control actions are updated only at particular intervals of time.

While there are now numerous approaches to analyze and design continuous-time and discrete-time interconnected systems, there is a need for methodological tools, which are adapted to hybrid phenomena in the literature.

Handling Hybrid Phenomena in Networked Systems

In the literature, fundamental links between algebraic graph theory and synchronization have been established for different scenarios of continuous and discrete-time networked systems. However, the tools that are available to date to analyze synchronization are not always applicable when both continuous and discrete phenomena coappear. Indeed, assume that the interaction among the hybrid nodes occurs only during their continuous-time, or only during their discrete-time evolution. Such is the case for mechanical nodes with impacts [START_REF] Westervelt | Hybrid zero dynamics of planar biped walkers[END_REF] or for networks under any event-based or intermittent interconnections [START_REF] Phillips | Robust distributed synchronization of networked linear systems with intermittent information[END_REF]. In this case, one needs to show that the network's attitude to synchronize during one of the two modes (flowing or jumping) is robust with respect to the other mode along which the interconnection is lost. However, most of the existing techniques to show synchronization for purely continuoustime or purely discrete-time networks are trajectory-based approaches [START_REF] Martin | Continuous-time consensus under persistent connectivity and slow divergence of reciprocal interaction weights[END_REF][START_REF] Hendrickx | A new condition for convergence in continuous-time consensus seeking systems[END_REF] and cannot be easily extended in order to analyze robustness. Another example, where classical tools do not apply, is that of hybrid networks operating in clusters. In this case, sub-networks are formed due to their strong interconnections. The nodes within the same sub-network tend to synchronize quickly (fast dynamics) before interacting with the other sub-networks (slow dynamics) [START_REF] Su | Decentralized adaptive pinning control for cluster synchronization of complex dynamical networks[END_REF][START_REF] Lu | Cluster synchronization in networks of coupled nonidentical dynamical systems[END_REF][START_REF] Wu | Cluster synchronization of linearly coupled complex networks under pinning control[END_REF][START_REF] Martin | New cut-balance conditions in networks of clusters[END_REF]. The global hybrid network exhibits a multi-time-scale dynamics [START_REF] Kokotović | Singular perturbation methods in control: analysis and design[END_REF][START_REF] Chow | Time scale modeling of sparse dynamic networks[END_REF]. As discussed in Section 2.4, singular-perturbation theory is useful to analyze synchronization in the presence strong interconnections. However, for stand-alone hybrid singularly perturbed systems and even for a one-system case, only few results are available, see e.g., [START_REF] Sanfelice | On singular perturbations due to fast actuators in hybrid control systems[END_REF][START_REF] Malloci | Stability and stabilization of two time scale switched systems in discrete time[END_REF][START_REF] Ben Rejeb | Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems[END_REF] for more discussion.

The main originality of [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF] consists in providing a range of advanced Lyapunov-based approaches for setstability, inspired by techniques for nonlinear time-varying systems, for interconnected continuous-time systems. Similar Lyapunov-based tools exist for hybrid systems [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, stability, and robustness[END_REF]. In particular, a range of tools are available for the analysis of set stability. Hence, there is hope to follow similar approaches when analyzing hybrid networks. For this purpose, several key questions would need to be addressed.

1. The modeling of the interconnections among the nodes, which is not a trivial task and requires special care to preserve the so-called basic conditions for the network in closed loop [START_REF] Biemond | Incremental stability of hybrid dynamical systems[END_REF][START_REF] Phillips | Robust distributed synchronization of networked linear systems with intermittent information[END_REF]. Hybrid basic conditions are essential for hybrid systems to ensure some robustness properties, see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, stability, and robustness[END_REF].

2. The current approaches to address hybrid interconnected systems consist in modeling the overall network as a hybrid system and to apply related results to conclude about stability like in [START_REF] De Persis | A Lyapunov redesign of coordination algorithms for cyber-physical systems[END_REF]. For instance, it would be much more natural to derive global properties based on the local properties of each node and the interconnection graph, as suggested in [START_REF] Teel | Asymptotic stability for hybrid systems via de-composition, dissipativity, and detectability[END_REF].

3. The persistence of excitation property, which have been extensively used in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF], extends to the notions of persistent flow and persistent jump in the hybrid systems [START_REF] Prieur | Relaxed persistent flow/jump conditions for uniform global asymptotic stability[END_REF], which need to be better understood in terms of their implication on synchronization.

4. The possibility of a mismatch between the times when the hybrid nodes jump would considerably increase the difficulty to analyze synchronization. This fact has been mentioned in the particular context of hybrid observers in [START_REF] Bernard | Robust Observer Design for Hybrid Dynamical Systems with Linear Maps and Approximately Known Jump Times[END_REF].

Challenging Problems

The aforementioned questions are general and fundamental. It therefore seems appropriate to follow a divideand-conquer strategy by considering, separately, situations where the hybrid phenomena are due to the node dynamics, from those where these are due to the control strategy as discussed in the following.

Synchronizing Intrinsically Hybrid Nodes

Networks of hybrid nodes include networks of switched systems (like power converters), mechanical systems with impacts, or cyber-physical systems. Because such networks are challenging, it would be relevant to first concentrate on situations where the interaction among the systems occurs only during their continuous-time evolution, which is the case in walking robots, or only during their discrete-time evolution, which is the case of juggling mechanisms. In each of the previously-cited scenarios, we need to provide robustness-analysis tools in order to conclude when the network's attitude to synchronize during one of the two modes of evolution (flowing or jumping) is robust with respect to the other mode, where the interconnections are not present, and also with respect to the mismatch in the jump times as the nodes do not necessarily jump at the same time. The qualitative Lyapunov-based techniques for time-varying systems proposed in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF], as described in Section 2 and [START_REF] Chowdhury | On the estimation of algebraic connectivity in graphs with persistently exciting interconnections[END_REF][START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF], is relevant here and would need to be adapted to this context.

Synchronization Using Hybrid Controllers

A node may also be hybrid because of the used local controller. There, a hybrid controller is a hybrid system admitting the system or part of the system's state variables as inputs and generates, as outputs, the system's control inputs. First, this class of controllers can overcome fundamental limitations of continuous-time controllers e.g. [START_REF] Hustig-Schultz | A robust hybrid heavy ball algorithm for optimization with high performance[END_REF][START_REF] Brodtkorb | Hybrid controller concept for dynamic positioning of marine vessels with experimental results[END_REF][START_REF] Beerens | Reset integral control for improved settling of pid-based motion systems with friction[END_REF]. They also offer the possibility to solve the combined stability-safety control problem robustly [START_REF] Sanfelice | Robust hybrid controllers for continuous-time systems with applications to obstacle avoidance and regulation to disconnected set of points[END_REF]. More importantly, hybrid controllers offer the flexibility to model different scenarios of constrained interconnection among the nodes [START_REF] Casadei | About disconnected topologies and synchronization of homogeneous nonlinear agents over switching networks[END_REF] and they naturally arise in presence of sampling [START_REF] De Persis | A Lyapunov redesign of coordination algorithms for cyber-physical systems[END_REF][START_REF] Nowzari | Event-triggered communication and control of networked systems for multi-agent consensus[END_REF][START_REF] Nowzari | Event-triggered communication and control of networked systems for multi-agent consensus[END_REF][START_REF] Phillips | Robust distributed synchronization of networked linear systems with intermittent information[END_REF][START_REF] Biemond | Incremental stability of hybrid dynamical systems[END_REF][START_REF] Poveda | Hybrid mechanisms for robust synchronization and coordination of multi-agent networked sampled-data systems[END_REF].

A concrete research problem, related to synchronization in networks of non-hybrid nodes using hybrid interconnections, is to analyze networks operating in clusters under event-based interconnections. In this case, the global network may exhibit a multi-time-scale dynamics. Moreover, since the interconnections are intermittent, the resulting closed-loop network exhibits hybrid phenomena. A promising research direction that contribute to this question, consists in extending the singular-perturbation-based framework proposed in [START_REF] Maghenem | Singular-perturbations-based analysis of synchronization in heterogeneous networks: A case-study[END_REF][START_REF] Bıyık | Area aggregation and time-scale modeling for sparse nonlinear networks[END_REF] as described in Section 2.4. The tools developed in the aforementioned references exploit the continuity of the stability properties of classes of singularly perturbed systems under small perturbations. Showing such a continuity property for classes of hybrid singularly perturbed systems is an open question to the best of our knowledge.

Safety Specifications in Hybrid Networks

Usually, in many real-world applications, safety requirements need to be guaranteed when solving synchronization problems. Safety is also named obstacle avoidance in the particular context of mobile robotics. Safety while synchronizing a network of dynamical systems means that the trajectories of the network in closed loop, when starting from a given set of initial conditions, remain within a safe region of the state space while the variables of interest converge to the agreement. Barrier functions are analogues to Lyapunov functions are have been usually used to guarantee safety in dynamical systems [START_REF] Prajna | A framework for worst-case and stochastic safety verification using barrier certificates[END_REF][START_REF] Ames | Control barrier function based quadratic programs for safety critical systems[END_REF]. They have been also used for networked systems in [START_REF] Coogan | A dissipativity approach to safety verification for interconnected systems[END_REF][START_REF] Verginis | Closed-form barrier functions for multi-agent ellipsoidal systems with uncertain lagrangian dynamics[END_REF][START_REF] Dimarogonas | Sufficient conditions for decentralized potential functions based controllers using canonical vector fields[END_REF][START_REF] Lindemann | Control barrier functions for multi-agent systems under conflicting local signal temporal logic tasks[END_REF][START_REF] Jagtap | Compositional construction of control barrier functions for interconnected control systems[END_REF]. Recently, in [START_REF] Maghenem | Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions[END_REF], barrier functions are used to show safety for hybrid systems in the form of H. However, to the best of our knowledge, establishing distributed safety criteria for hybrid networks has not been done yet.

Conclusion

This article reviews the main contributions presented in [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF] where original Lyapunov-based approaches for set stability have been developed for the synchronization of networked continuous-time systems. The approach of [START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF] together with the recent advances on the Lyapunov stability theory for hybrid systems allow us to envision a range of exciting perspectives for the synchronization of hybrid interconnected systems. The main motivations of this research direction are discussed and some open problems are proposed.

Preprint prepublished online in Annual Reviews in Control, https://doi.org/10.1016/j.arcontrol.2020.06.003

The contributions in[START_REF] Maghenem | Singularperturbations-based analysis of synchronization in heterogeneous networks: A case-study[END_REF][START_REF] Maghenem | Singularperturbations-based analysis of dynamic consensus in heterogeneous networks[END_REF] have not been included in[START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF] and[START_REF] Maghenem | Singularperturbations-based analysis of dynamic consensus in heterogeneous networks[END_REF] will be submitted soon.

The works in[START_REF] Chowdhury | On the estimation of algebraic connectivity in graphs with persistently exciting interconnections[END_REF][START_REF] Maghenem | Decentralized synchronization of time-varying oscillators under time-varying bidirectional graphs[END_REF] have not been included[START_REF] Maghenem | Stability and stabilisation of networked systems[END_REF].

Acknowledgment

This work was partially funded by ANR via grant HANDY, number ANR-18-CE40-0010. The first author feels deeply grateful to Prof. Ricardo G. Sanfelice for introducing him to the area of hybrid systems.