Ilario Alessandro Bosso
email: alessandro.bosso@unibo.it

Antonio Azzollini

Simone Baldi
email: s.baldi@tudelft.nl

Senior Member, IEEE Luca Zaccarian
email: zaccarian@laas.fr

Member, IEEE Ilario A Azzollini
email: ilario.azzollini@unibo.it

Adaptive Hybrid Control for Robust Global Phase Synchronization of Kuramoto Oscillators

Keywords: Kuramoto oscillators, distributed control, hybrid control, adaptive control, robustness

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I. INTRODUCTION

S YNCHRONIZATION and coordination phenomena are ubiquitous in several application domains, including physics, engineering, biology, and social sciences. Particularly studied, in this context, are the dynamical behaviors arising from networks of interacting oscillators. To describe these behaviors, the Kuramoto model [START_REF] Kuramoto | Chemical Oscillations, Waves, and Turbulence[END_REF] is certainly the most popular model due to its ability to capture complex nonlinear phenomena with appealing mathematical simplicity. The study of power networks [START_REF] Guo | Overviews on the applications of the Kuramoto model in modern power system analysis[END_REF], [START_REF] Dorfler | Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators[END_REF] or of connectivity patterns in the human brain [START_REF] Menara | A framework to control functional connectivity in the human brain[END_REF], [START_REF] Qin | Partial phase cohesiveness in networks of networks of Kuramoto oscillators[END_REF] are just some examples where Kuramoto oscillators have been adopted.

In general, synchronization of Kuramoto oscillators may occur with or without a control input affecting the network. Concerning the uncontrolled scenario, significant efforts have been dedicated to studying the impact of couplings (either the network topology or the intensity of connections) on the synchronization properties of the trajectories [START_REF] Ha | On the complete synchronization of the Kuramoto phase model[END_REF]- [START_REF] Zhu | Synchronization of Kuramoto oscillators: A regional stability framework[END_REF]. In the controlled scenario, the emphasis is on finding an appropriate input to achieve synchronization [START_REF] Moreira | Global synchronization of partially forced Kuramoto oscillators on networks[END_REF]. In this context, the objective of leader-follower synchronization, also known as pacemaker-based synchronization [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF], becomes particularly relevant. The typical challenge in achieving leader-follower synchronization is that the controller of each node should employ only locally available quantities and variables shared according to a communication topology.

This work considers a second-order version of the original first-order Kuramoto model, where each oscillator has its own inertia [START_REF] Dorfler | Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators[END_REF], [START_REF] Choi | Complete synchronization of Kuramoto oscillators with finite inertia[END_REF] and is characterized by a phase angle and an angular frequency. We remark that further extensions have been recently proposed, including the third-order Kuramoto model [START_REF] Wu | Synchronization conditions for a third-order Kuramoto network[END_REF], inspired by the transient behavior of power networks, or the generalization of the phase state space given by the Kuramoto model on Stiefel manifolds [START_REF] Ha | Emergent behaviors of highdimensional Kuramoto models on Stiefel manifolds[END_REF], capable of including in a unified framework both the classical model and more complex structures such as the Lohe model [START_REF] Markdahl | Robust synchronization of heterogeneous robot swarms on the sphere[END_REF].

The focus of this work is to achieve global leader-follower phase synchronization in a network of second-order Kuramoto oscillators, without precise information of the model parameters. In the following, we review some representative results in the field, which clarify the motivations for our study.

A. Related works

It has been well recognized in the literature that the non-Euclidean nature of the state space of a Kuramoto model is the main obstruction for achieving global asymptotic convergence to the leader's phase reference. Several strategies have been proposed to deal with this obstruction. For example, a natural approach is to represent the phase of each oscillator as an element of the unit circle. It follows that the ensemble of the phase angles is an element of the N -torus [START_REF] Scardovi | Synchronization and balanc-ing on the N-torus[END_REF].

One of the main advantages of employing the unit circle formalism is that phase synchronization can be reformulated as the attractivity of a compact set. Although this property is beneficial for control design, the N -torus is a non-Euclidean set, meaning that synchronization cannot be handled with the same tools used in linear consensus. In particular, the topological properties of a non-contractible space (i.e., not diffeomorphic to any Euclidean space) pose significant obstacles to global stabilization through continuous feedback. For instance, the continuous-time algorithms in [START_REF] Scardovi | Synchronization and balanc-ing on the N-torus[END_REF] (and their corresponding discrete-time versions) lead to multiple equilibria in the state space, where only one of them corresponds to the desired configuration. The same issue is shared by several applications involving rotations. In the context of rigid body dynamics, only almost global results can be achieved with continuous laws for control [START_REF] Marco | Output regulation on the special Euclidean group SE(3)[END_REF] and observation [START_REF] Mahony | Nonlinear complementary filters on the special orthogonal group[END_REF].

In recent years, it has been shown that robust global stabilization can be achieved on non-contractible spaces through dynamic hybrid (instead of continuous) feedback [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF]. Meaningful results have been proposed, e.g., for unit quaternions [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF] through hysteresis-based techniques and for the N -sphere [START_REF] Casau | Robust global exponential stabilization on the n-dimensional sphere with applications to trajectory tracking for quadrotors[END_REF] via synergistic potential functions. Some efforts have also been dedicated to the unit circle [START_REF] Mayhew | Hybrid control of planar rotations[END_REF]. However, all of the above solutions have been developed in a single-agent scenario and in the absence of uncertain dynamics. One of the first attempts to present hybrid feedback in a multi-agent setting can be found in [START_REF] Mayhew | Quaternion-based hybrid feedback for robust global attitude synchronization[END_REF], for the special case of acyclic communication graphs.

Despite the progress in the field, some additional vital elements are needed for achieving global leader-follower synchronization of uncertain Kuramoto oscillators. Since no specific communication topology is imposed for the network, while the reference is not assumed to be globally available, it is necessary to ensure that each node reconstructs the leader signals. Additionally, the above-cited works have been developed under the assumption of complete knowledge of the parameters. The presence of model uncertainties complicates the asymptotic synchronization goal. Therefore, specific control solutions are needed to ensure robust asymptotic stability of the synchronization set.

B. Main contribution of this work

Motivated by the previous overview, we propose here a distributed scheme that solves the leader-follower problem by combining three components: (i) a distributed observer, used to reconstruct the reference in the nodes not directly connected to the leader; (ii) a hybrid stabilizer used to track the locally estimated reference and ensure, under parametric uncertainties, phase synchronization in a global sense; (iii) an adaptive mechanism to suitably handle the parametric uncertainties. Besides the technological interest of the synchronization problem at stake, for each one of the above components, we provide a solution of independent interest, whose novelty is highlighted next.

(i) About the distributed observer, we follow the idea that the unit circle, used to represent the phase angles, can be naturally embedded in R 2 . With this embedding, since the estimates are designed without being constrained on the unit circle, consensus techniques for Euclidean spaces can be employed to achieve global estimation of the leader signals.

As compared with other solutions in the literature that follow the embedding approach [START_REF] Cai | Leader-following attitude consensus of multiple rigid body systems by attitude feedback control[END_REF], [START_REF] Gui | Global finite-time attitude consensus of leader-following spacecraft systems based on distributed observers[END_REF], here we allow for more general structures of the exosystem: in particular, we consider exosystems admitting a feedback interconnection between the phase and frequency subsystems, whereas the literature in this field only handles cascaded interconnections. Exploiting input-to-state stability (ISS) and small-gain arguments, we prove global asymptotic reconstruction of the reference for a fairly general class of exosystems, which is a contribution of independent interest.

(ii) About the hybrid stabilizers at each node, to ensure compatibility with the adaptive mechanism, we revisit and extend the hysteresis-based hybrid solution originally proposed in [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF] to deal with the topological obstructions associated with the unit circle. In particular, we augment the hybrid feedback with a first-order filter, so that the stabilizing input does not change across jumps, a key property for interlacing the hybrid stabilizer with the continuous-time adaptation commented below. Due to the simplicity of the condition on the filter time constant under which we prove stability, this dynamic extension is of independent interest and can be exploited in future works.

(iii) Finally, about our hybrid adaptation mechanism, we show that, with an appropriate robust modification of the adaptive law, it is possible to ensure the existence of a robustly globally asymptotically stable attractor for the tracking error system, without requiring any persistency of excitation. This result, which may sound atypical as compared to standard results in the adaptive control literature, represents a notable by-product of this work. Specifically, the powerful characterization of ω-limit sets of well-posed hybrid systems given in the hybrid systems formalism [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF], together with a simple deadzone-based projection mechanism for keeping the parameter estimates in a compact set, enables proving the existence of such a compact globally asymptotically stable attractor.

To conclude, we emphasize that the closed-loop asymptotic stability of the overall control scheme is analyzed through reduction theorems for hybrid systems. In particular, we prove that global phase synchronization is well represented as robust global asymptotic stabilization of a suitable compact set. We remark that the robustness of asymptotic stability is guaranteed in this context by the regularity properties of the feedback law and compactness of the characterized attractor. With respect to this point, let us comment on the interesting distributed quaternion synchronization in [START_REF] Gui | Global finite-time attitude consensus of leader-following spacecraft systems based on distributed observers[END_REF], achieved by combining a sliding-mode distributed observer and a hybrid stabilizer. In that work, unfortunately, the presence of static discontinuities makes it impossible to ensure the robustness properties established in this work.

A preliminary version of this study has been published in [START_REF] Bosso | A hybrid distributed strategy for robust global phase synchronization of secondorder Kuramoto oscillators[END_REF]. In this paper we improve the work [START_REF] Bosso | A hybrid distributed strategy for robust global phase synchronization of secondorder Kuramoto oscillators[END_REF] in several directions. First of all, [START_REF] Bosso | A hybrid distributed strategy for robust global phase synchronization of secondorder Kuramoto oscillators[END_REF] considers the simplified case of known parameters of the Kuramoto oscillators. Therefore, several challenges related to including adaptation laws in a hybrid setting are addressed and solved here for the first time. Moreover, [START_REF] Bosso | A hybrid distributed strategy for robust global phase synchronization of secondorder Kuramoto oscillators[END_REF] only considers a simplified cascaded exosystem structure as in [START_REF] Cai | Leader-following attitude consensus of multiple rigid body systems by attitude feedback control[END_REF], [START_REF] Gui | Global finite-time attitude consensus of leader-following spacecraft systems based on distributed observers[END_REF], whereas in this paper we address nontrivial challenges emerging from feedback interconnections, involving suitable small-gain approaches not required in [START_REF] Bosso | A hybrid distributed strategy for robust global phase synchronization of secondorder Kuramoto oscillators[END_REF]. For example, the results of [START_REF] Bosso | A hybrid distributed strategy for robust global phase synchronization of secondorder Kuramoto oscillators[END_REF] cannot be applied to our simulation example of Section VII.

The paper is organized as follows. Sections II and III are dedicated, respectively, to formulating the model and presenting a formal statement of the control problem. Then, Section IV defines the distributed observer and provides a detailed stability analysis of the estimation error dynamics. On the other hand, Section V introduces the hybrid controller for reference tracking, along with a preliminary stability analysis in conditions of global knowledge of the reference signal. The results of the previous sections are collected and exploited in Section VI, which presents the overall stability analysis based on reduction theorems. In Section VII, we validate the theoretical results through some numerical simulations that confirm the effectiveness of the proposed solution. Finally, Section VIII concludes the article.

Notation

R and Z are the sets of real and integer numbers, while R ≥0 := [0, ∞). The transpose of real-valued vectors and matrices is denoted by (•) , while ⊗ is the Kronecker matrix product. For any positive integer n, I n is the identity matrix of dimension n and 1 n ∈ R n is the vector of all ones. Given column vectors v and w, the notation (v, w) denotes the concatenated vector [v w ] . Finally, diag(a 1 , . . . , a n ) denotes the block-diagonal matrix with diagonal elements a i , i ∈ {1, . . . , n}.

1) Graph Theory: An undirected graph of order N is defined as G := {V, E}, where V := {1, . . . , N } is a finite non-empty set of nodes and E ⊆ V × V is a set of non-ordered pairs of nodes, called edges. For each i ∈ V, N i := {j ∈ V : (i, j) ∈ E} is the set of neighbors of i. An undirected graph G is connected if, taken any arbitrary pair of nodes (i, j), i, j ∈ V, there is a path from i to j. Given a leader node not included in V, we denote with T ⊆ V the set of target nodes, i.e., the set of nodes that receive information from the leader. For an undirected graph G with target nodes T , the adjacency matrix A = [a ij ] ∈ R N ×N is defined as a ij = a ji = 1 if (i, j) ∈ E, i = j, and a ij = 0 otherwise; the Laplacian matrix L = [l ij ] ∈ R N ×N is defined as l ii = j a ij and l ij = -a ij if i = j, while the target matrix T = [τ ij ] ∈ R N ×N is a diagonal matrix such that τ ii = 1 if i ∈ T and τ ii = 0 otherwise. Finally, the matrix B := L + T is denoted leaderfollower matrix. For an undirected and connected graph G with T = 0 (equivalently, such that T = 0), B is positive definite [START_REF] Zhang | Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics[END_REF].

2) Hybrid Dynamical Systems: A hybrid dynamical system can be compactly described as [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]:

H : ẋ ∈ F (x), x ∈ C x + ∈ G(x), x ∈ D (1)
where x ∈ R n is the state, C ⊂ R n is the flow set, F : R n ⇒ R n is the flow map, D ⊂ R n is the jump set, and G : R n ⇒ R n is the jump map. A solution of (1) can either flow according to the differential inclusion ẋ ∈ F (x) when x ∈ C, or jump according to the difference inclusion x + ∈ G(x) when x ∈ D. We refer to [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF], [START_REF] Goebel | Hybrid dynamical systems[END_REF] for the main definitions and tools for the analysis of hybrid systems.

II. MODEL DESCRIPTION

A. Second-Order Kuramoto Network

In this article, we consider a generalization of the celebrated Kuramoto model [START_REF] Kuramoto | Chemical Oscillations, Waves, and Turbulence[END_REF], based on the swing equations described in [START_REF] Dorfler | Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators[END_REF]. More specifically, the second-order Kuramoto network is a system of N nonlinear oscillators, coupled through an undirected and connected graph G = {V, E}:

θi = ω i , i ∈ V m i ωi =-d i ω i + ω ni + u i - j∈Ni k ij sin(θ i -θ j -ϕ ij ), (2)
where, for each i ∈ V, θ i ∈ R and ω i ∈ R are the phase and the frequency, respectively, u i is the control input, m i > 0 is the oscillator's inertia, d i > 0 is a damping constant, and ω ni is the oscillator's natural frequency. In addition, k ij = k ji > 0 and ϕ ij = ϕ ji ∈ [0, 2π) are, respectively, the coupling weight and the phase shift between oscillators i and j. Suppose that the graph G, associated with the physical couplings in (2), also defines the communication topology among the nodes.

Define 2), for some input signals u i (•), i ∈ V, and with initial conditions (θ(0), ω(0)). We say that

θ := θ 1 . . . θ N ∈ R N and ω := ω 1 . . . ω N ∈ R N , then denote by (θ(•), ω(•)) : R ≥0 → R 2N a solution of system (
(θ(•), ω(•)) achieves phase synchronization if lim t→+∞ θ i (t)-θ j (t) ∈ θ : θ = 2kπ, k ∈ Z , ∀i, j ∈ V. (3) Similarly, the solution (θ(•), ω(•)) is said to achieve frequency synchronization if lim t→+∞ ω i (t) -ω j (t) = 0, ∀i, j ∈ V. (4)
For the network (2), our objective is to design a distributed strategy that ensures robust global phase synchronization to a reference trajectory. Namely, our aim is to define feedback laws for the inputs u i based only on local information and network communication such that, for any initialization of system (2), the corresponding solution (θ(•), ω(•)) robustly achieves phase synchronization and convergence to the reference. When we refer to robust synchronization, we mean that (3) is obtained through asymptotic stability of a compact set, with appropriate robustness to perturbations of the closed-loop dynamics. A precise definition of this concept is presented in Section III. Because we do not assume exact knowledge of the local parameters m i , d i , ω ni , k ij , and ϕ ij , we design adaptive controllers that ensure asymptotic convergence in the presence of parametric uncertainties. At the same time, it is well known that the sensitivity of adaptive techniques to nonparametric (unmodeled) perturbations of the dynamics calls for a robust design of the adaptive law and some known bounds of the parametric uncertainty (see, e.g., [START_REF] Ioannou | Robust Adaptive Control[END_REF]Chs. 8 and 9]). Accordingly, we impose the following assumption.

Assumption 1. There exists a scalar ρ > 0, known to each node i ∈ V, such that:

m i ≤ ρ, d i ≤ ρ, |ω ni | ≤ ρ, ∀i ∈ V, k ij ≤ ρ, ∀i ∈ V, ∀j ∈ N i , (5)
where the bound ρ is taken to be the same for all parameters for simplicity of notation.

B. Quaternion-Inspired Representation

For control design, we propose to rewrite system (2) in a more convenient form. Motivated by the equivalence modulo 2π of the phases θ i , also reflected in the phase synchronization condition (3), we choose to represent θ i on the unit circle S 1 := α β ∈ R 2 : α 2 + β 2 = 1 . Recall that the compact set S 1 has Lie group structure that is isomorphic to the group of planar rotations SO( 2)

:= {R ∈ R 2×2 : R R = I 2 , det(R) = 1}.
In view of such an isomorphism, we define the function R : S 1 → SO(2), which maps any α β ∈ S 1 into the corresponding rotation matrix:

R α β := α -β β α . (6)
Function R(•) is useful to define the group multiplication between any ξ, ξ ∈ S 1 as R(ξ) ξ = R( ξ)ξ (note that S 1 is Abelian, i.e., the group operation is commutative), where the identity element is given by e := 1 0 .

From the above definitions, we introduce the following representation for θ i :

ζ i := η i i := cos(θ i /2) sin(θ i /2) ∈ S 1 , (8)
corresponding to a unit quaternion for planar rotations (cf. [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF] for the parameterization adopted for 3D rotations). We refer to [START_REF] Bosso | A hybrid distributed strategy for robust global phase synchronization of secondorder Kuramoto oscillators[END_REF] for the relation between ( 8) and the representation employed in [START_REF] Bosso | Global frequency synchronization over networks of uncertain second-order Kuramoto oscillators via distributed adaptive tracking[END_REF] and [START_REF] Goebel | Hybrid dynamical systems[END_REF]Ex. 34]. Using ( 6) and ( 8), the phase dynamics on SO(2) and S 1 is obtained as

d dt R(ζ i ) = 1 2 ω i JR(ζ i ), ζi = 1 2 ω i Jζ i , i ∈ V, (9)
where

J := 0 -1 1 0
∈ SO(2). Let T N := N i=1 S 1 denote the N -torus. The network dynamics (2) can be conveniently rewritten on T N × R N as follows:

ζi = 1 2 ω i Jζ i m i ωi = -d i ω i + ω ni + u i - j∈Ni k ij φ(ζ i ) Jφ(ζ j )cos(ϕ ij ) + j∈Ni k ij φ(ζ i ) φ(ζ j ) sin(ϕ ij ), i ∈ V, (10)
where φ : S 1 → S 1 is defined as

φ(ζ i ) := R(ζ i )ζ i = η 2 i -2 i 2η i i , ζ i := η i i ( 11
)
and corresponds to the double angle formula from

ζ i := [cos(θ i /2) sin(θ i /2)] to [cos(θ i ) sin(θ i )]
. Note that, with the proposed representation ( 8), the condition (3) corresponding to phase synchronization coincides with

lim t→+∞ R(ζ i (t)) ζ j (t) ∈ {-e, e}, ∀i, j ∈ V. ( 12
)
Remark 1. In some applications, such as those involving rotary encoders, θ i is provided by sensors that "wrap" the angles in [0, 2π) (equivalently, in [-π, π)). In this scenario, if (8) is used to compute ζ i from the available sensor measurement, call it θ s i , special care must be taken to ensure that a continuous trajectory of the vector [cos(θ s i ) sin(θ s i )] (uniquely corresponding to any θ s i ∈ [0, 2π)) is mapped into a continuous trajectory of ζ i . More specifically, for any θ s i ∈ [0, 2π), there are two possible values of ζ i , expressed through the half-angle formula:

ζ i ∈ {-ζ * i , ζ * i }, ζ * i :=    1+cos(θ s i ) 2 1-cos(θ s i ) 2    . ( 13
)
The same issue arises for unit quaternions. In that context, a path-lifting mechanism has been proposed in [START_REF] Mayhew | On path-lifting mechanisms and unwinding in quaternion-based attitude control[END_REF] to ensure that a continuous selection of the two quaternions is obtained for a "measured" rotation matrix. For simplicity, we avoid embedding a similar mechanism as [START_REF] Mayhew | On path-lifting mechanisms and unwinding in quaternion-based attitude control[END_REF] by considering ζ i available for measurement. Including the path-lifting mechanism does not affect the results of this paper.

III. PROBLEM STATEMENT

A. Leader Exosystem

Since our objective involves the synchronization of the network to a reference signal, we consider a formulation of the tracking problem based on a pacemaker (see, e.g., [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF], [START_REF] Bosso | Global frequency synchronization over networks of uncertain second-order Kuramoto oscillators via distributed adaptive tracking[END_REF]). Specifically, the graph G is augmented with an additional node, named leader system, which delivers to the network some reference signals. The references are generated through an autonomous exosystem of the form

ζ = 1 2 c w Jζ ẇ = s(ζ , w )    (ζ , w ) ∈ K ⊂ S 1 × R n , ( 14
)
where ζ ∈ S 1 is the phase reference, w ∈ R n is a state such that the frequency reference is given by c w ∈ R, while c ∈ R n is a constant vector and s(•) :

S 1 × R n → R n is a nonlinear function. Furthermore, K is a compact set of admissible initial conditions (ζ (0), w (0)).
The feedback structure in [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF] suggests that, different from the solutions using unit quaternions (such as [START_REF] Cai | Leader-following attitude consensus of multiple rigid body systems by attitude feedback control[END_REF], [START_REF] Gui | Global finite-time attitude consensus of leader-following spacecraft systems based on distributed observers[END_REF]), we do not restrict the structure of exosystem ( 14) to a cascade between the w -subsystem and the ζ -subsystem. The following assumption describes the properties related to [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF]. Assumption 2. For system [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF], it holds that: 1) the compact set K is forward invariant; 2) the map s(•) is globally Lipschitz, with Lipschitz constant s ≥ 0; 3) c and s(•) are known to each node i ∈ V.

The global Lipschitz condition in Assumption 2 is instrumental in achieving global asymptotic stability, cf. [START_REF] Isidori | Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory[END_REF]. As we shall see in Section IV, this Lipschitz continuity property allows designing the controllers for each node i ∈ V without the explicit knowledge of the compact set K , even though the knowledge of s is required for tuning the controller gains.

(k34, ϕ34) (k13, ϕ13) (k12, ϕ12) (k45, ϕ45) (k56, ϕ56) (k16, ϕ16) ( ζ2, ŵ2)
Fig. 1. Interaction and communication scheme. The same graph will be employed for the numerical example in Section VII.

As a final requirement for our design, we impose a standard assumption describing the communication topology among the leader ( 14) and the network [START_REF] Chopra | On exponential synchronization of Kuramoto oscillators[END_REF]. Assumption 3. System (14) interacts, by communicating the reference (ζ , w ), with at least one node of graph G, which defines both the physical couplings and the communication topology. More specifically, it holds that T = ∅ (equivalently, T = 0). Remark 2. Since G is undirected and connected, Assumption 3 implies that the leader-follower matrix B := L+T is positive definite.

Fig. 1 shows a scheme of the interaction and communication pattern underlying our distributed architecture.

The control problem of this work, stated in the following, aims at ensuring global asymptotic stability of a compact set corresponding to phase synchronization as in [START_REF] Zhu | Synchronization of Kuramoto oscillators: A regional stability framework[END_REF] and convergence to ζ , i.e.:

lim t→+∞ R(ζ i (t)) ζ ∈ {-e, e}, ∀i ∈ V. (15)
In particular, we seek for a hybrid adaptive controller whose data satisfy the so-called hybrid basic conditions of [

A s := (ζ , w ), (ζ i , ω i ), i ∈ V : R(ζ i ) ζ ∈ {-e, e}, ω i = c w , ( 16
)
where e is defined in [START_REF] Zhang | Synchronization of Kuramoto oscillators in small-world networks[END_REF], and note that A s is compact because (14) evolves in the compact set K . Then, design a distributed adaptive strategy, only based on the local measurements (ζ i , ω i ) and the information exchange according to graph G, such that the second-order Kuramoto network (10) achieves robust global phase synchronization to the reference ζ . Namely, the closed-loop dynamics is such that there exists a robustly globally KL asymptotically stable compact set (in the sense of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Def. 7.18]), whose projection in the plantexosystem direction coincides with the compact set A s in (16).

B. Control Architecture

Through the parametrization ( 8), ( 10), [START_REF] Dörfler | On the critical coupling for Kuramoto oscillators[END_REF], Problem 1 addresses the synchronization goal in (3), ( 4), in a convenient scenario wherein the set to be stabilized is compact. Topological obstructions associated to the non-Euclidean nature of the phase dynamics make Problem 1 challenging. In fact, the N -torus is non-contractible, i.e., it is not diffeomorphic to any Euclidean space, and convergence to [START_REF] Wu | Synchronization conditions for a third-order Kuramoto network[END_REF] requires convergence to a disconnected set of points. Two main issues arise in this context.

• If the control laws u i are designed to stabilize only one of the points of ( 16), the trajectories in the coordinates θ i display the so-called unwinding phenomenon [START_REF] Mayhew | On path-lifting mechanisms and unwinding in quaternion-based attitude control[END_REF], which causes unnecessary motion in cases where the system is initialized close to synchronization. • If a static discontinuous feedback is employed, it is not possible to ensure robust KL asymptotic stability because the closed-loop system does not satisfy the hybrid basic assumptions. This fact, in practice, translates into chattering and high disturbance sensitivity [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF]. In view of these considerations, we employ a hybrid dynamic feedback to robustly globally asymptotically stabilize a compact set comprising A s in [START_REF] Wu | Synchronization conditions for a third-order Kuramoto network[END_REF]. As discussed in the introduction, the proposed control strategy is built upon the interconnection of a distributed observer for exosystem [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF], a hybrid stabilizer for globally tracking the observer estimates, and an adaptive law to handle parametric uncertainties under Assumption 1. More specifically, our design is based on the following steps.

• A distributed observer is designed so that certain local estimates ( ζi , ŵi ) of (ζ , w ) ∈ K are defined as elements of R 2+n . The ensuing estimation error dynamics is described by two feedback-interconnected subsystems, associated with the phase and the frequency estimation errors, respectively. These subsystems are proven to be ISS and then combined through small-gain arguments.

The design and the stability analysis of the observer are discussed in Section IV. • For each agent i, we implement an adaptive hybrid mechanism to ensure R(ζ i ) ζi → {-e, e}, ω i → c ŵi , for all i ∈ V. The design is first performed assuming global knowledge of the leader signals (Section V). Supposing that ω i can be assigned as a virtual input ω vi , a hysteresis-based controller is used to show global phase synchronization while ensuring that ω vi is constant across jumps. Then, a backstepping-based adaptive controller is designed to guarantee ω i → ω vi . Finally, exploiting the cascade structure between the estimation error subsystem and the tracking subsystem, the effectiveness of the overall control solution is proven through reduction theorems (Section VI).

IV. DISTRIBUTED OBSERVER

In order to solve Problem 1, we propose the following distributed observer:

ζi = 1 2 c ŵi J ζi -k ζ e ζ i ẇi = s( ζi , ŵi ) -k w e wi i ∈ V, (17)
where ζi ∈ R 2 , ŵi ∈ R n are the local estimates of (ζ , w ) (14) at node i, k ζ and k w ∈ R are gains to be designed, while

e ζ i := j∈Ni a ij ( ζi -ζj ) + τ ii ( ζi -ζ ) e wi := j∈Ni a ij ( ŵi -ŵj ) + τ ii ( ŵi -w ) i ∈ V, (18)
are the local estimation errors, in which a ij and τ ii are the entries of the adjacency matrix A and the target matrix T , respectively. Observer ( 17) is distributed as it is only driven by locally available quantities [START_REF] Markdahl | Robust synchronization of heterogeneous robot swarms on the sphere[END_REF]. To represent the variables for the overall network in a compact form, it is convenient to use the Kronecker product. In particular, define the overall states 18) can be written as [START_REF] Azzollini | Adaptive leader-follower synchronization over heterogeneous and uncertain networks of linear systems without distributed observer[END_REF]:

ζ := [ ζ 1 . . . ζ N ] ∈ R 2N and ŵ := [ ŵ 1 . . . ŵ N ] ∈ R N n ,
e ζ = (B ⊗ I 2 ) ζ, e w = (B ⊗ I n ) w, (19)
where B := L+T is the leader-follower matrix, which satisfies B = B > 0 as discussed in Remark 2. In the following, denote with σ(B) > 0 the smallest singular value of B.

A. Phase Subnetwork

We start by analyzing the (ζ , ζ)-subsystem (referred to as phase subnetwork) and the phase estimation error ζ. From ( 14), [START_REF] Ha | Emergent behaviors of highdimensional Kuramoto models on Stiefel manifolds[END_REF], the phase subnetwork obeys dynamics

ζ = 1 2 c w Jζ ζi = 1 2 c w J ζi + 1 2 c wi J ζi -k ζ e ζ i , i ∈ V. (20)
For notational convenience, define Ω := diag(c w1 , . . . , c wN ) = diag((I N ⊗ c ) w), [START_REF] Mahony | Nonlinear complementary filters on the special orthogonal group[END_REF] which allows writing the dynamics of ζ in compact form as follows

ζ = 1 2 c w (I N ⊗ J) + ( Ω ⊗ J) ζ -k ζ e ζ . ( 22
)
As a consequence, the dynamics of the phase estimation error ζ := ζ -1 N ⊗ ζ can be computed from ( 19), [START_REF] Marco | Output regulation on the special Euclidean group SE(3)[END_REF], and [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF] as:

ζ = 1 2 c w (I N ⊗ J) -k ζ (B ⊗ I 2 ) + 1 2 ( Ω ⊗ J) ζ + 1 2 ( Ω ⊗ J)(1 N ⊗ ζ ), (23)
with inputs given by ζ , w , and w (through Ω in ( 21)). The next proposition provides an ISS characterization for system [START_REF] Casau | Robust global exponential stabilization on the n-dimensional sphere with applications to trajectory tracking for quadrotors[END_REF].

Proposition 1. For any scalar gain k ζ > 0, system (23) is finite-gain exponentially input-to-state stable with respect to the input w, uniformly in the inputs (ζ , w ). Namely, for any solution (ζ (•), w (•)) of the exosystem (14) and any w(•) ∈ L ∞ , the solutions of (23) satisfy, for all t ≥ 0:

| ζ(t)| ≤ max e -1 2 σ(B)k ζ t | ζ(0)|, |c| w(•) ∞ σ(B)k ζ . (24)
Proof: For any solution (ζ (•), w (•)) of the exosystem (14), system (23) can be regarded as a time-varying system with input w. It is convenient to rewrite the last term of ( 23) as

( Ω ⊗ J)(1 N ⊗ ζ ) = Ω1 N ⊗ Jζ = diag((I N ⊗ c ) w)1 N ⊗ Jζ = diag(1 N )(I N ⊗ c ) w ⊗ Jζ = (I N ⊗ Jζ c ) :=Z w = Z w, (25)
where we used the identity ((I N ⊗ c )w) ⊗ v = (I N ⊗ vc )w, which holds for any vectors c , w, v, of compatible dimensions. Consider the Lyapunov function candidate

V ζ := 1 2 | ζ| 2 (26)
whose derivative along the system trajectories results in

Vζ = -k ζ ζ (B ⊗ I 2 ) ζ + 1 2 ζ Z w + 1 2 ζ c w (I N ⊗ J) + ( Ω ⊗ J) ζ, = -k ζ ζ (B ⊗ I 2 ) ζ + 1 2 ζ Z w, (27)
where we employed the fact that I N ⊗ J and Ω ⊗ J are skew symmetric. Since k ζ > 0, we obtain

Vζ ≤ -σ(B)k ζ | ζ| 2 + 1 2 |Z || ζ|| w|. ( 28
)
The following computations yield |Z | = |c|:

|Z | = |I N ||Jζ c | = |Jζ c | = |Jζ c | F = Tr(cζ J Jζ c ) = Tr(cc ) = |c|, (29)
where |Jζ c | = |Jζ c | F since the rank of Jζ c is 1 by construction. Applying ( 28) and ( 29) yields

| ζ| ≥ |c| σ(B)k ζ | w| =⇒ Vζ ≤ - σ(B)k ζ 2 | ζ| 2 , (30)
which leads to (24) via standard ISS calculations [37, Thm. 10.4.1].

B. Frequency Subnetwork

Starting again from ( 14), ( 17), the frequency subnetwork obeys dynamics

ẇ = s(ζ , w ) ẇi = s( ζi , ŵi ) -k w e wi , i ∈ V. (31)
We can then write the dynamics of ŵ as

ẇ = S( ζ, ŵ) -k w e w , (32)
where

S( ζ, ŵ) :=     s( ζ1 , ŵ1 ) . . . s( ζN , ŵN )     . (33)
Therefore, using [START_REF] Scardovi | Synchronization and balanc-ing on the N-torus[END_REF], the dynamics of the frequency estimation error w :

= ŵ -(1 N ⊗ w ) is given by ẇ = S( ζ, ŵ) -1 N ⊗ s(ζ , w ) -k w (B ⊗ I n ) w, (34)
which, in view of ζ = ζ +1 N ⊗ζ , ŵ = w+1 N ⊗w , is a nonautonomous system with inputs given by ζ , w , and ζ. In the following, we present a result that follows the same structure as Proposition 1, now applied to the frequency subnetwork.

Proposition 2. For any scalar gain k w > s /σ(B), system (34) is finite-gain exponentially input-to-state stable with respect to the input ζ, uniformly in the inputs (ζ , w ). Namely, for any solution (ζ (•), w (•)) of the exosystem (14) and any ζ(•) ∈ L ∞ , the solutions of system (34) satisfy, for all t ≥ 0:

| w(t)| ≤ max e -1 2 (σ(B)kw-s )t | w(0)|, 2 s ζ(•) ∞ σ(B)k w -s . (35)
Proof: For any solution (ζ (•), w (•)) of the exosystem (14), system (34) can be regarded as a time-varying system with input ζ. Consider the Lyapunov function candidate

V w := 1 2 | w| 2 , (36)
whose derivative along [START_REF] Mayhew | On path-lifting mechanisms and unwinding in quaternion-based attitude control[END_REF] is

Vw = -k w w (B ⊗ I n ) w + w S( ζ, ŵ) -1 N ⊗ s(ζ , w ) = -k w w (B ⊗ I n ) w + N i=1 w i s( ζi , ŵi ) -s(ζ , w ) . ( 37
)
By Assumption 2, it holds that

|s( ζi , ŵi ) -s(ζ , w )| ≤ s (| ζi | + | wi |), (38)
therefore we conclude that

Vw ≤ -σ(B)k w | w| 2 + s N i=1 | wi | 2 + | wi || ζi | ≤ -σ(B)k w -s | w| 2 + s | w|| ζ|. (39)
Finally, from [START_REF] Khalil | Nonlinear Systems[END_REF] we obtain the following ISS characterization:

| w| ≥ 2 s σ(B)k w -s | ζ| =⇒ Vw ≤ - σ(B)k w -s 2 | w| 2 , ( 40
)
which proves the finite-gain exponential ISS bound [START_REF] Isidori | Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory[END_REF] through [START_REF] Isidori | Nonlinear control systems II[END_REF]Thm. 10.4.1].

C. Overall Observer Analysis

We conclude the section with a stability result for the feedback interconnection between the phase estimation error dynamics [START_REF] Casau | Robust global exponential stabilization on the n-dimensional sphere with applications to trajectory tracking for quadrotors[END_REF] and the frequency estimation error dynamics [START_REF] Mayhew | On path-lifting mechanisms and unwinding in quaternion-based attitude control[END_REF].

Theorem 1. For any choice of the scalar gains k ζ and k w such that

k ζ > 0, k w > s /σ(B), k ζ σ(B)(k w σ(B) -s ) -2 s |c| > 0, (41)
the zero-equilibrium ( ζ, w) = 0 of the overall estimation error system (23), ( 34) is globally exponentially stable.

Proof: From Proposition 1, which holds for k ζ > 0, and Proposition 2, valid for k w σ(B) > s , we obtain that both [START_REF] Casau | Robust global exponential stabilization on the n-dimensional sphere with applications to trajectory tracking for quadrotors[END_REF] and [START_REF] Mayhew | On path-lifting mechanisms and unwinding in quaternion-based attitude control[END_REF] are finite-gain exponentially ISS. Therefore, global exponential stability is ensured from ( 24), [START_REF] Isidori | Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory[END_REF], and [37, Thm. 10.6.1], through the following small-gain condition:

2 s |c| k ζ σ(B)(k w σ(B) -s ) < 1, (42)
which is ensured by [START_REF] Arcak | Input-to-state stability for a class of Lurie systems[END_REF].

Remark 3. In the special case where exosystem (14) is a cascade, i.e., s = s(w ), conditions (41) collapse to k ζ > 0, k w > s /σ(B). Additionally, if s(w ) = S w w , where S w is a Poisson stable matrix as in [START_REF] Cai | Leader-following attitude consensus of multiple rigid body systems by attitude feedback control[END_REF], [START_REF] Gui | Global finite-time attitude consensus of leader-following spacecraft systems based on distributed observers[END_REF], [START_REF] Bosso | A hybrid distributed strategy for robust global phase synchronization of secondorder Kuramoto oscillators[END_REF], (37

) becomes Vw = -k w w (B ⊗ I n ) w, therefore conditions (41) become k ζ > 0, k w > 0.
V. SYNCHRONIZATION WITH GLOBAL KNOWLEDGE OF THE LEADER SIGNALS In this section, we design a tracking controller for the simplified setup where the observer estimation errors are zero. This approach will be motivated in Section VI by the reduction arguments of the stability analysis.

Firstly, we compute the local tracking error dynamics. Define the phase and frequency tracking errors as

ζi := ηi ¯ i := R(ζ i ) ζi ∈ R 2 ωi := c ŵi -ω i ∈ R i ∈ V. ( 43
)
In these coordinates, the control objective in Problem 1 corresponds to imposing (¯ i , ωi ) → 0, for all i ∈ V. From ( 10), ( 17), and R(ζ i ) J = JR(ζ i ) , we can compute the phase error dynamics as

ζi = d dt R(ζ i ) ζi + R(ζ i ) ζi = -R(ζ i ) dR(ζ i ) dt R(ζ i ) ζi + R(ζ i ) ζi = 1 2 ωi J ζi -k ζ R(ζ i ) e ζ i . i ∈ V (44)
Similarly, the dynamics of the frequency error ωi is computed from ( 10), [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF], and ( 17) as

m i ωi = ψ i -u i -m i k w c e wi , i ∈ V, (45)
where we defined

ψ i := m i c s( ζi , ŵi ) + d i ω i -ω ni + j∈Ni k ij φ(ζ i ) Jφ(ζ j ) cos(ϕ ij ) - j∈Ni k ij φ(ζ i ) φ(ζ j ) sin(ϕ ij ). ( 46
)
Observe that, with ζi = ζ and ŵi = w (i.e., ζ = 0, w = 0, equivalently, e ζ = 0, e w = 0), the quantities in (43

) become ζi = R(ζ i ) ζ ∈ S 1 and ωi = c w -ω i ∈ R.
In view of this reduction argument, we begin the design by assuming that the exosystem signals (ζ , w ) are globally known for feedback. This scenario corresponds to the requirement T = V, which will be removed in Section VI.

A. Phase Synchronization

Assume initially that ω i can be arbitrarily assigned by the feedback controller as a virtual input ω vi . With e ζ = 0, the dynamics (44) thus reduces to

ζi = 1 2 (c ŵi -ω vi )J ζi , i ∈ V, (47)
where ω vi is the virtual input that should ensure ¯ i → 0. We refer to this objective as phase synchronization with the reference ζ . Define Q := {-1, 1} and choose any gain k > 0 and a hysteresis margin δ ∈ (0, 1). For each i ∈ V, a hysteresisbased hybrid dynamic controller that achieves global phase synchronization is given by qi = 0, ( ζi

, q i ) ∈ C κ q + i = -q i , ( ζi , q i ) ∈ D κ ω vi = c ŵi + kq i ¯ i , i ∈ V ( 48
)
where q i ∈ Q is the controller state and

C κ := {( ζi , q i ) ∈ S 1 × Q : ηi q i ≥ -δ} D κ := {( ζi , q i ) ∈ S 1 × Q : ηi q i ≤ -δ}. ( 49
)
The closed-loop error dynamics then corresponds to

   ζi = - 1 2 kq i ¯ i J ζi qi = 0, ζi q i ∈ C κ , ζ+ i = ζi q + i = -q i , ζi q i ∈ D κ , (50)
which provides an autonomous hybrid dynamics having state ( ζi , q i ) = ((η i , ¯ i ), q i ) ∈ S 1 × Q and such that q i ηi = -1 is not included in the flow set C κ (because δ < 1). The next lemma is a straightforward generalization of [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF].

Lemma 1. The attractor A κ := {(ζ, q) ∈ S 1 × Q : ζ = qe} is uniformly globally asymptotically stable for (50).

Proof: Choose the Lyapunov function

V κ ( ζi , q i ) := 2(1 -q i ηi ), (51)
which is positive definite and radially unbounded with respect to A κ . Denoting Vκ = ∇V κ , [ ζi qi ] and

∆V κ = V κ ( ζ+ i , q + i ) -V κ ( ζi , q i ), straightforward calculations yield Vκ = -k¯ 2 i < 0, ∀( ζi , q i ) ∈ C κ \A κ ∆V κ = 4q i ηi ≤ -4δ < 0, ∀( ζi , q i ) ∈ D κ , (52)
implying UGAS from standard hybrid Lyapunov theory. For a convenient design of the backstepping-based adaptive controller defined in the next subsection, we propose now a dynamically extended version of (48) to ensure that ω vi remains constant across jumps. Specifically, we augment the controller with a first-order filter of the feedback kq i ¯ i :

qi = 0 λi = -h(λ i -kq i ¯ i ) ( ζi , q i , λ i ) ∈ C λ q + = -q i λ + i = λ i ( ζi , q i , λ i ) ∈ D λ i ∈ V, (53)
where h is a positive gain and the sets C λ , D λ are defined as the next generalization of (49):

C λ := ( ζi , q i , λ i ) ∈ S 1 × Q × R : ηi + λ i ¯ i k q i ≥ -δ D λ := ( ζi , q i , λ i ) ∈ S 1 × Q × R : ηi + λ i ¯ i k q i ≤ -δ . (54)
We can then replace by λ i the term kq i ¯ i in the selection of ω vi of (48), namely we choose:

ω vi = c ŵi + λ i , i ∈ V, (55)
which remains constant across jumps. The closed-loop error dynamics for each node i, obtained from the interconnection of (47), (53), and (55), is described by:

         ζi = - 1 2 λ i J ζi qi = 0 λi = -h(λ i -kq i ¯ i )   ζi q i λ i   ∈ C λ ,        ζ+ i = ζi q + i = -q i λ + i = λ i   ζi q i λ i   ∈ D λ . (56)
The next result generalizes the argument of Lemma 1.

Proposition 3. For any h > k, the attractor

A λ := {(ζ, q, λ) ∈ S 1 ×Q×R : ζ = qe, λ = 0}
is uniformly globally asymptotically stable for the hybrid system (56).

Proof: Define λi := λ i -kq i ¯ i , then consider the Lyapunov function

V λ ( ζi , q i , λ i ) := 2k 2 (1 -q i ηi ) + λ2 i . (57)
Note that V λ is positive definite with respect to A λ and radially unbounded relative to

S 1 × Q × R. Denote Vλ = ∇V λ , [ ζi qi λi ] . For all ( ζi , q i , λ i ) ∈ C λ , it holds that Vλ = -k 2 q i λ i ¯ i + λi (-2h λi + kq i λ i ηi ) = -k 3 ¯ 2 i + k 2 λi ¯ i (η i -q i ) -(2h -kq i ηi ) λ2 i ≤ - |¯ i | | λi | k 3 -k 2 -k 2 2h -k |¯ i | | λi | . (58)
From δ < 1, for any point in C λ we have that λi = 0 and ¯ i = 0 implies that ( ζi , q i , λ i ) ∈ A (in particular, the point with λ i = 0 and q i = -η i does not belong to C λ ), then Vλ < 0, for all ( ζi , q i , λ

i ) ∈ C λ \A λ , if k 3 (2h -k) -k 4 = 2k 3 (h-k) > 0, i.e., h > k. On the other hand, denote ∆V λ = V λ ( ζ+ i , q + i , λ + i ) -V κ ( ζi , q i , λ i )
, then for all ( ζi , q i , λ i ) ∈ D λ we have:

∆V λ = 4k 2 q i ηi + ( λi + 2kq i ¯ i ) 2 -λ2 i = 4k 2 q i ηi + λ i ¯ i k ≤ -4k 2 δ < 0, (59)
thus concluding UGAS for the attractor A λ .

B. Global Adaptive Synchronization

Taking advantage of the hybrid system defined in (53), we propose to achieve global synchronization to the reference ζ using an adaptive backstepping controller where, in place of the feedback ω vi = c ŵi + λ i in (55), we ensure ω i → ω vi by design of the control input u i .

In place of the frequency tracking error ωi in (43), consider the error variable

z i := c ŵi + λ i -ω i = ωi + λ i ∈ R, i ∈ V. (60)
We can rewrite the error dynamics (44) and (45) using variables z i as follows:

ζi = 1 2 (z i -λ i )J ζi -k ζ R(ζ i ) e ζ i m i żi = ψ i -u i -m i k w c e wi + h(λ i -kq i ¯ i ) i ∈ V. (61)
Using ( 46), the second equation can also be rewritten as follows

m i żi = Ψ i p i -u i -m i k w c e wi , i ∈ V, (62)
with regressor Ψ i and parameter vector p i ∈ R 3+2|Ni| given by:

Ψ i :=                  c s( ζi , ŵi )-h(λ i -kq i ¯ i ) ω i 1 φ(ζ i ) Jφ(ζ j1 ) . . . φ(ζ i ) Jφ(ζ j |N i | ) φ(ζ i ) φ(ζ j1 ) . . . φ(ζ i ) φ(ζ j |N i | )                  , p i :=                  m i d i -ω ni k ij1 cos(ϕ ij1 )
. . .

k ij N i cos(ϕ ij |N i | ) -k ij1 sin(ϕ ij1 ) . . . -k ij N i sin(ϕ ij |N i | )                 
(63) where we denoted N i = {j 1 , . . . , j |Ni| }. By Assumption 1, it follows that

|p i | ∞ = max{|p i1 |, . . . , |p i(3+2|Ni|) |} ≤ ρ. (64)
The control of system (61) is based on the augmentation of control law (53) with the following adaptive state-input selections:

u i = Ψ i pi + k z z i ṗi = γΨ i z i -γν dz(p i ) p+ i = pi , i ∈ V, (65)
where k z , γ, and ν are positive gains, while dz : R 3+2|Ni| → R 3+2|Ni| is a decentralized dead-zone function defined as [38, §3.4]:

dz(ξ) :=       ξ 1 -ρsat ξ1 ρ . . . ξ 3+2|Ni| -ρsat ξ 3+2|N i | ρ       , (66)
where sat(y) := max{-1, min{1, y}}. Exploiting (64), it can be verified that, for all p i and all ξ ∈ R 3+2|Ni| :

(ξ -p i ) dz(ξ) ≥ 0. (67)
Moreover, there exist positive scalars r and µ such that, for all p i and all ξ ∈ R 3+2|Ni| :

|ξ| ≥ r =⇒ (ξ -p i ) dz(ξ) ≥ µ|ξ| 2 . ( 68
)
In view of our reduction arguments (e ζi = 0 and e wi = 0 in (61)), the closed-loop dynamics obtained from the interconnection of (61), ( 62), (53), and (65), having state x i := ( ζi , q i , λ i , z i , pi ), is expressed, for each i ∈ V, as follows:

                   ζi = 1 2 (z i -λ i )J ζi qi = 0 λi = -h(λ i -kq i ¯ i ) m i żi = -k z z i -Ψ i (p i -p i ) ṗi = γΨ i z i -γν dz(p i ) x i ∈ C i ,                  ζ+ i = ζi q + i = -q i λ + i = λ i z + i = z i p+ i = pi x i ∈ D i , (69)
where

C i := C λ × R 4+2|Ni| and D i := C z × R 4+2|Ni| .
In the following, we focus on the stability properties of the closed-loop system obtained through the interconnection of the exosystem [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF] and the local error dynamics (69). For this interconnection, we are going to show that there exists a globally asymptotically stable compact attractor wherein all oscillators are synchronized with the reference ζ . More specifically, this compact attractor, named A 0 , is a set wherein ζi = q i e, z i = 0, for all i ∈ V, and satisfying A 0 ⊂ K 0 , with K 0 compact, defined as:

K 0 :={(ζ , w , x 1 , . . . , x N )∈K × i∈V (S 1 ×Q×R 5+2|Ni| ): ζi = q i e, λ i = 0, z i = 0, |p i | ≤ r, ∀i ∈ V}, (70)
where r > 0 as per (68). In the sequel, we call K 0 synchronization set.

Remark 4. The set K 0 is compact. Indeed, (ζ , w ) ∈ K is in a compact by assumption, while the only components of x i := ( ζi , q i , λ i , z i , pi ) that are possibly unbounded are λ i , z i , and pi . Therefore, from the conditions in (70), compactness follows immediately.

Remark 5. Since no persistency of excitation is necessarily satisfied by the regressor Ψ i in (63), it might be surprising that a globally asymptotically stable attractor can be found with the considered adaptive controller. This result is possible because we make use of the analysis tools in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Chs. 6 and 7] instead of the standard tools for adaptive control (see, e.g., [39, §8.3]). In particular, we leverage the result [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Cor. 7.7], which states that, under some regularity properties includingwell posedness, the ω-limit set from a compact set of initial conditions is locally asymptotically stable.

Theorem 2. For any selection of the tuning parameters k > 0, δ ∈ (0, 1), h > k, k z > 0, γ > 0, and ν > 0, there exists a compact set A 0 , contained in the synchronization set K 0 of (70), that is robustly globally KL asymptotically stable in the sense of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Def. 7.18] for the interconnection among [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF] and (69).

Robustness of our stability result follows from compactness of A 0 and well posedness of the considered hybrid dynamics.

Proof: First, we prove that the closed-loop solutions are bounded and forward complete. The state (ζ , w ) of the exosystem [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF] evolves in the bounded forward invariant set K , thus it is bounded. Moreover, q i is bounded by construction. In the scenario with known leader signals considered in this section, it holds that ζi :

= R(ζ i ) ζi = R(ζ i ) ζ i ∈ S 1 , thus ζi is bounded. Since |kq i ¯ i | ≤ k we obtain, for |λ i | ≥ k: d dt |λ i | = -h λ i |λ i | (λ i -kq i ¯ i ) ≤ -h(|λ i | -k) ≤ 0, ∀i ∈ V (71)
therefore λ i is bounded. Boundedness of (z i , pi ) is established by using the following Lyapunov function

V i (z i , pi ) := 1 2 m i z 2 i + 1 2γ |p i -p i | 2 , ∀i ∈ V. (72)
Along the closed-loop solutions, we obtain from (69),

Vi = -k z z 2 i -z i Ψ i (p i -p i ) + (p i -p i ) Ψ i z i -ν dz(p i ) = -k z z 2 i -ν(p i -p i ) dz(p i ), ∀i ∈ V. (73)
Therefore, using (68) and the properties of the dead-zone function, we obtain, for all i ∈ V,

Vi ≤ -k z z 2 i ≤ 0, if |p i | ≤ r, Vi ≤ -k z z 2 i -νµ|p i | 2 < 0, if |p i | ≥ r. (74)
Property (74) shows forward invariance of the sublevel sets of V i , i ∈ V, thus (z i , pi ) is contained in a compact set, for all i ∈ V. Exploiting boundedness of the components of x i , i ∈ V, established above, we also conclude by [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Prop. 6.10] that solutions are forward complete, thus they are precompact. From (72)-(74) and using the fact that z i and pi do not change across jumps, we can apply [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Cor. 8.4] to obtain that all solutions approach the largest weakly invariant subset of the set

U := {(ζ , w , x 1 , . . . , x N ) ∈ K × i∈V (S 1 × Q × R 5+2|Ni| ): z i = 0, |p i | ≤ r, ∀i ∈ V}. (75)
The closed-loop dynamics restricted to the set U in (75) is given, for all i ∈ V, by z i = 0, ṗi = 0, and (56). In view of Proposition 3, the solutions globally approach K 0 ⊂ U as per (70) and (75). For a set of initial conditions of the form K ε := K 0 + εB, where ε > 0 is arbitrary, it holds that A 0 := Ω(K ε ) ⊂ K 0 ⊂ Int(K ε ), where Ω(K ε ) denotes the ω-limit set of K ε . By [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Cor. 7.7], A 0 is asymptotically stable (therefore Lyapunov stable) with basin of attraction containing K ε . From the previous arguments, A 0 is globally attractive, which, together with its Lyapunov stability, gives GAS. Since the hybrid dynamics satisfies the hybrid basic conditions of [28, Assumption 6.5] and A 0 is compact, then GAS of A 0 implies robust global KL asymptotic stability from [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Thm. 7.21].

As customary in adaptive control, convergence of the estimated parameters pi to the true parameters p i cannot be guaranteed. This in turn makes it difficult, if at all possible, to give an explicit representation of the attractor A 0 . Even without its explicit representation, the mere existence of A 0 is sufficient to complete the design through reduction theorems in the next section.

VI. MAIN RESULT

We finally present the complete hybrid observer-based controller for each node i, obtained by combining the distributed observer [START_REF] Ha | Emergent behaviors of highdimensional Kuramoto models on Stiefel manifolds[END_REF] and the local hysteresis-based controller (53), and the local adaptive controller (65). Note that, in this context, we no longer assume ζ = 0, w = 0 (equivalently, e ζ = 0, e w = 0), thus the dynamics of the tracking errors ( ζi , z i ) in (61) is not simplified as in the scenario with known leader signals. The robustness property established in Theorem 2 is naturally inherited here due to well posedness of the hybrid dynamics.

Define the overall state at node i as

χ i := ( ζi , ŵi , ζi , q i , λ i , z i , pi xi ) ∈ R n+4 × Q × R 5+2|Ni| , (76) R(ζ i ) ζi , where ζ i ∈ S 1 , therefore ζi is bounded because | ζi | ≤ | ζi |.
Since ζi is bounded, so is kq i ¯ i . Indicate with ki the upper bound of kq i ¯ i , for a given set of initial conditions, and boundedness of λ i is proven by parallel derivations to (71).

To analyze (z i , pi ), as in the proof of Theorem 2, consider the Lyapunov function

V i (z i , pi ) := 1 2 m i z 2 i + 1 2γ |p i -p i | 2 , i ∈ V. (79)
From ( 67) and (68), respectively for each i ∈ V, similar steps to those in (73), (74) yield:

Vi = -k z z 2 i -ν(p i -p i ) dz(p i ) -m i k w z i c e wi ≤ - k z 2 z 2 i + 1 2k z |m i k w c e wi | 2 , |p i | ≥ r =⇒ Vi ≤ - k z 2 z 2 i -νµ|p i | 2 + 1 2k z |m i k w c e wi | 2 . ( 80
)
These two bounds provide, respectively,

|z i | > m i k w |c| k z |e wi | =⇒ Vi (z i , pi ) < 0, |p i | > max r, m i k w |c| √ 2k z νµ |e wi | =⇒ Vi (z i , pi ) < 0. (81)
The two implications above prove that neither z i nor pi can grow unbounded because e w = (B ⊗ I n ) w is bounded. Therefore, we conclude global boundedness of solutions.

VII. NUMERICAL EXAMPLE

For the numerical analysis, we consider a Kuramoto model composed of six oscillators, whose parameters and initial conditions are reported in Tab. I. In particular, the graph of the network is depicted in Fig. 1, where the coupling parameters have been assigned as k 12 = 0.5, k 13 = 3, k 14 = 1, k 16 = 1.5, k 34 = 2, k 45 = 2.5, k 56 = 2, ϕ 12 = π/2, ϕ 13 = π/3, ϕ 14 = π/4, ϕ 16 = π/3, ϕ 34 = π/5, ϕ 45 = π/4, ϕ 56 = π/2. We suppose to have a rough knowledge of the parameter bounds by letting ρ = 25 in (5). It follows that Assumptions 1 and 3 hold. The leader exosystem [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF] has been chosen as Response of exosystem (82) (in blue) and corresponding asymptotic behavior (in violet).

d dt ζ 1 ζ 2 = 1 2 (w 1 + w 3 )J ζ 1 ζ 2 d dt   w 1 w 2 w 3   =    0 w 3 -w 2 + 1 -1 2 |w 3 | tanh(w 3 ) + 3 2 ζ 2    , (82)
The Kuramoto model has been implemented according to (2), with the angles θ i wrapped between -2π and 2π in order to ensure boundedness of the simulation variables. Then, for the computation of the feedback laws, the variables ζ i have been computed according to [START_REF] Schmidt | Frequency synchronization and phase agreement in Kuramoto oscillator networks with delays[END_REF]. The tuning parameters have been selected as k ζ = 50, k w = 50, δ = 0.5, k = 1, k z = 5, h = 2, γ = 1, ν = 1. Note that (41) is verified since σ(B) = 0.1136. The initial conditions for controller (77) have been randomly chosen, where in particular the logic variables q i have been initialized in the set Q := {-1, 1}.

In Figs. 5, 6 we report the results of a simulation run. Fig. 5 shows the behavior of the distributed observer, which rapidly converges to the exosystem signals. On the other hand, Fig. 6 depicts the tracking performance. In Fig. 6-(e), we also report the evolution of p1 , showing that the parameters of the adaptive controllers converge to constant values. Finally, we employ wrapped angles to depict the phase tracking performance in Figs. 6-(

ϑ i := mod (θ i + π, 2π) -π, i ∈ V, ( 83
)
where ϑ is the angular reference corresponding to ζ , while ϑ i is θ i wrapped in the interval [-π, π).

VIII. CONCLUSIONS

We introduced an adaptive hybrid control strategy for the robust global phase synchronization of second-order Kuramoto oscillators. The objective of phase synchronization was cast into a leader-follower tracking problem, where the leader system is modeled as an autonomous nonlinear exosystem. Under fairly mild assumptions on the network topology and the exosystem dynamics, we proved that our design, which comprises a distributed observer and an adaptive hybrid stabilizer, ensures robust global stability of a compact synchronization set. In particular, robust adaptive stabilization was ensured without requiring persistency of excitation conditions. Future efforts will be dedicated to relaxing the information requirements (e.g., by removing the frequency measurements) and the connectivity properties of the network. Furthermore, it will be worth generalizing the approach to a broader class of nonlinear oscillators.

  so that the overall estimation errors are ζ := ζ -1 N ⊗ ζ and w := ŵ-1 N ⊗w . Furthermore, define e ζ := [e ζ1 . . . e ζ N ] ∈ R 2N and e w := [e w1 . . . e w N ] ∈ R N n , which from (

Fig. 3 .

 3 Fig. 3. Response of exosystem (82) initialized in ζ (0) = [1 0] , w (0) = [2 0 0] .

  f), 6-(g). In particular, we defineϑ := 2atan2 (ζ 2 , ζ 1 ) ,

Fig. 5 .

 5 Fig. 5. Closed-loop simulation results. (a): distributed observer phase estimation (reference in blue); (b): distributed observer frequency estimation (reference in blue).

2 Fig. 6 .

 26 Fig. 6. Closed-loop simulation results. (a): phase tracking errors; (b): phase tracking errors, zoomed in [0, 0.8]s to highlight the jumps during the initial transient; (c): filtered inputs λ i ; (d): frequency tracking errors z i ; (e): evolution of p1 ; (f): arc distance between θ i and ϑ ; (g): phase angles, wrapped in the interval [-π, π) (reference in blue).

  Note that (ζ 1 , ζ 2 , w 1 ) are bounded by construction. On the other hand, boundedness of (w 2 , w 3 ) is proven by direct application of[START_REF] Arcak | Input-to-state stability for a class of Lurie systems[END_REF] Thm. 2]. We remark that, from the chosen initial conditions, the solution converges to a periodic orbit as depicted in Figs.3, 4. It can be easily shown that s( ζ , w ) is globally Lipschitz, since the derivative of the nonlinear term is bounded for all w

	with initial conditions ζ (0) = [1 0] and w (0) = [2 0 0] .
	For completeness, we briefly prove that Assumption 2 is
	satisfied. The existence of K is guaranteed by proving bound-
	edness of solutions of (82).

[START_REF] Dorfler | Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators[END_REF]

. From the numerical evaluation of the differential of s over the values of (ζ , w ), we established a Lipschitz constant s = 2.129.

then the local controllers that solve Problem 1 are given as follows, for each i ∈ V:

and:

where e ζ i , e wi are given in [START_REF] Scardovi | Synchronization and balanc-ing on the N-torus[END_REF], ηi , ¯ i are defined in (43), z i is defined in (60), Ψ i is given in (63), dz is given in (66), while the tuning parameters are the stabilizer gains k, h, k z , γ, ν, the observer gains k ζ , k w , and the hysteresis margin δ.

The closed-loop system is given by the interconnection of the second-order Kuramoto network [START_REF] Chopra | On exponential synchronization of Kuramoto oscillators[END_REF], the exosytem [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF], and the local controllers (77). For such system, we exploit reduction theorems to show that there exists a compact attractor A that is robustly globally KL asymptotically stable. As for Theorem 2, we show that A is a subset of a compact set K that we may call again synchronization set, with a slight abuse of notation, because its elements enjoy phase synchronization to the reference ζ :

Note that the projection of K in the direction of

The main result of this work is given by the following statement, which provides formal guarantees for the effectiveness of the controllers (77). Theorem 3. For any selection of the tuning parameters k > 0, δ ∈ (0, 1), h > k, k z > 0, γ > 0, ν > 0 and k ζ , k w satisfying (41), there exists a compact set A, contained in K of (78), that is robustly globally KL asymptotically stable for the interconnection among (10), [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF], and (77).

Proof: We begin by highlighting the cascade-structure of the closed-loop error dynamics. As shown in Section IV, the distributed observer dynamics is collected in the estimation error subsystems [START_REF] Casau | Robust global exponential stabilization on the n-dimensional sphere with applications to trajectory tracking for quadrotors[END_REF] and [START_REF] Mayhew | On path-lifting mechanisms and unwinding in quaternion-based attitude control[END_REF]. We can establish a cascade interconnection between the system (23), [START_REF] Mayhew | On path-lifting mechanisms and unwinding in quaternion-based attitude control[END_REF], [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF], with output (ζ , w , ζ, w), and the tracking error dynamics (69). We highlight that whenever ( ζ, w) = 0 the closed-loop system is described by the dynamics with known leader signals ( 14), (69). The overall interconnection of these subsystems is shown in Fig. 2.

Asymptotic stability of the attractor A is proven through reduction theorems. By Theorem 1, we showed that the closed (but not compact) attractor

corresponding to the scenario with known leader signals, is UGAS. On the set Â, we recover the dynamics (69), thus by Theorem 2 there exists an attractor A ⊂ K that is UGAS relative to Â. By [40, Cor 4.8], A is uniformly asymptotically stable for the overall closed-loop system, with basin of attraction given by all the initial conditions generating bounded solutions.

We conclude the proof by showing that all solutions of the closed-loop system are bounded, which then implies UGAS and then robust global KL asymptotic stability from [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Thm. 7.21]. First note that the state (ζ , w ) of the exosystem [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF] evolves in the bounded forward invariant set K , thus it is bounded. Similarly, q i is bounded by construction. Due to Theorem 1, ( ζ, w) converge to zero, therefore ( ζi , ŵi ) are bounded for all i ∈ V. It remains to show that ζi , λ i , z i , and pi are bounded, for all i ∈ V. Concerning ζi , recall that ζi := Alessandro Bosso (Member, IEEE) received the master's degree in automation engineering and the Ph.D. degree in automatic control from the University of Bologna, Italy, in 2016 and 2020, respectively.

Currently, he is a Postdoctoral Researcher at the Department of Electrical, Electronic and Information Engineering (DEI), University of Bologna. His research interests include nonlinear control with input and state constraints, nonlinear and distributed adaptive control, hybrid systems, and sensorless control of electric machines.

Ilario