Mattia Giaccagli

Vincent Andrieu

Daniele Astolfi

Giacomo Casadei

Sufficient metric conditions for synchronization of leader-connected homogeneous nonlinear multi-agent systems

Keywords: Synchronization, contraction, multi-agent systems, incremental stability

come

INTRODUCTION

Due to the many applications in which agents interact in order to accomplish task and to achieve goals, the control community has devoted a huge attention to networks analysis and to the design of control architecture. For a network to achieve a certain task, we often require the agents to achieve an agreement and to this end, synchronization and consensus are the mathematical features to express such an agreement. For instance, this is the case for power networks [START_REF] Dörfler | Synchronization in complex oscillator networks and smart grids[END_REF], heat networks [START_REF] Scholten | Modeling and control of heat networks with storage: the single-producer multiple-consumer case[END_REF] and robot swarms Olfati-Saber [2006].

First, researchers focused on the problem of networks of linear systems and fundamental results for linear systems can be found in [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF] for homogeneous systems and in [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF] for heterogeneous networks. Nowadays, the focus of the community is placed on networks of nonlinear systems. Many approaches have been considered in the literature of homogeneous nonlinear networks: among the may results, it is worth recalling passivity conditions in [START_REF] Arcak | Passivity as a design tool for group coordination[END_REF], dissipativity in [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF] and ISS in Casadei et al. [2019a,b]. One of the most popular approaches is based on high-gain type arguments (see [START_REF] Isidori | Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory[END_REF], [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF]) inherited from high-gain observers theory. This technique has however well known limits which brought researchers to investigate alternative tools such as nonlinear integral control [START_REF] Pavlov | Controlled synchronization via nonlinear integral coupling[END_REF][START_REF] Pavlov | Synchronization of networked oscillators under nonlinear integral coupling[END_REF] which incorporates both low and high gain arguments.

Naturally, a property associated to synchronization is incremental stability, namely the property that any two solutions of a system converge to each other (see, e.g., [START_REF] Lohmiller | On contraction analysis for non-linear systems[END_REF], [START_REF] Forni | A differential lyapunov framework for contraction analysis[END_REF], [START_REF] Angeli | A lyapunov approach to incremental stability properties[END_REF], [START_REF] Simpson-Porco | Contraction theory on riemannian manifolds[END_REF], [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF] and the references therein). By enforcing this for every system in a network, we will have the systems converge to each other and thus achieve synchronization. More recently, authors have focused on incremental stability as a framework to solve the problem of synchronization and contraction theory as tool for the design of control laws, see, e.g., [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF], [START_REF] Andrieu | LMI conditions for contraction and synchronization[END_REF], [START_REF] Yin | Pinning synchronization of heterogeneous multi-agent nonlinear systems via contraction analysis[END_REF].

In this work we investigate metric-based conditions for exponential synchronization of identical multi-agent systems described by nonlinear dynamics in presence of a leader. With respect to notable articles in the literature such as [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF] (where the state synchronization problem via output exchange for linear systems is addressed with a dynamical regulator) and [START_REF] Pavlov | Controlled synchronization via nonlinear integral coupling[END_REF] (where incremental passivity conditions are proposed for nonlinear dynamics linear in the inputs and with linear outputs), our contribution is twofold. First, we consider input affine dynamics and the agents are allowed to exchange the full-state with their neighborhoods: we generalize the results in [START_REF] Andrieu | Some results on exponential synchronization of nonlinear systems[END_REF] in which the control was linear in the dynamics. Second, we consider the case in which agents are allowed to exchange only a nonlinear output with their neighborhoods while the control is linear in the state-dynamics: inspired by the observer construction given in [START_REF] Andrieu | Characterizations of global transversal exponential stability[END_REF], we generalize the results in [START_REF] Pavlov | Controlled synchronization via nonlinear integral coupling[END_REF] where the output was linear. This paper is organized as follows. In Section 2, we recall some preliminaries of graph theory and contraction theory. Then, we provide the main results in Section 3 concerning the sufficient conditions to achieve synchro-nization via static state-feedback and static output feedback distributed control laws. An illustration is provided in Section 4. Conclusions and future perspectives are drawn in Section 5.

Notation. R denotes the set of real numbers and | • | the standard Euclidean norm. Given a C 1 function P : R n → R n×n and a C 1 vector field ζ : R n → R n , we define the Lie derivative of P along ζ as

L ζ P (x) := d ζ P (x)+P (x) ∂ζ ∂x (x) + ∂ζ ∂x (x) P (x)
where

d ζ P (x) := lim h→0 P (x + hζ(x)) -P (x) h for all x ∈ R n .

PRELIMINARIES

Graph Theory

In a general framework, a communication graph is described by a triplet G = {V, E, A} in which V is a set of N nodes V = {v 1 , v 2 , . . . , v N }, E ⊂ V × V is a set of edges e jk that models the interconnection between nodes with the flow of information from node j to node k weighted by the (k, j)-th entry a kj ≥ 0 of the adjacency matrix A ∈ R N ×N . We denote by L ∈ R N ×N the Laplacian matrix of the graph, defined as

kj = -a kj for k = j, kj = N i=1 a ki for k = j,
where j,k is the (j, k)-th entry of L.

In this work we will consider a network of N homogeneous multi-agent systems, i.e. described by identical dynamics, which are connected according to a directed graph G = {V, E, A}, fulfilling the following assumption. Assumption 1. The graph G is leader connected, namely it contains at least one spanning tree with the leader as a root. Furthermore, its Laplacian L is diagonalizable.

As a consequence, according to Assumption 1, the Laplacian matrix L can be partitioned as

L = 0 0 L 21 L 22 (1)
where the matrix L 22 is a positive definite matrix satisfying (see [START_REF] Godsil | Algebraic graph theory[END_REF])

L 22 ≥ µI ,
(2) for some µ > 0 satisfying µ ≤ λ(L 22 ), where λ(L 22 ) denotes the smallest eigenvalue of L 22 .

Metric conditions for contraction

Consider a system ẋ = f (x)

(3) where f : R n → R n is a C 1 vector field. We indicate with X (x • , t) the trajectory of the system with initial condition x • ∈ R n evaluated at time t. Definition 1. We say that system (3) is Incrementally Globally Exponentially Stable (δGES in short) if there exist positive real numbers k, c such that

|X (x 1 , t) -X (x 2 , t)| ≤ k|x 1 -x 2 | exp(-c t)
for all t ≥ 0 in the time domain of existence of the solution and for all initial conditions x 1 , x 2 ∈ R n .

A sufficient condition for (3) to be δGES, is the existence of a metric for with the associated distance is decreasing along trajectories, as stated in the following proposition. For a proof, see Theorem 1 in [START_REF] Giaccagli | Sufficient conditions for output reference tracking for nonlinear systems: a contractive approach[END_REF] and references therein. Proposition 1. Consider system (3) and assume there exists a symmetric and positive definite matrix function P : R n → R n×n , and positive real numbers p, p, q > 0 satisfying the following

pI ≤ P (x) ≤ pI L f P (x) ≤ -qP (x) for all x ∈ R n . Then, system (3) is δGES.
To conclude the section, we recall the definition of Killing vector field. Definition 2. Consider a C 1 functions P : R n → R n×n , and g : R n → R n . We say that g is a Killing vector field for P if L g P (x) = 0 for all x ∈ R n .

Note that, if both P and g are constant matrices, the Killing vector field conditions is trivially satisfied.

MAIN RESULTS

Synchronization via State-feedback

In this section we consider a network of N identical agents described by the following dynamics ẋi = f

(x i ) + g(x i )u i i = 1, . . . , N, (4)
in which x i ∈ R n is the state and u i ∈ R is the control input. We denote

x := col(x 1 , . . . , x N ) ∈ R N n . Our objective is to design a distributed control law stabilizing the dynamics (4) on the synchronization manifold D defined by

D = {x ∈ R N n | x 1 = x 2 = • • • = x N }, (5)
where the state variables of different systems agree with each other. For every x in R N n , we denote the Euclidean distance to the set D by |x| D . We define our synchronization problem as follows. Problem 1. The control laws u i = φ i (x), i = 1 . . . , N solve the uniform exponential synchronization problem for (4) if the following conditions hold:

(1) For all non-communicating pair (i, j) (i.e., (i, j) / ∈ E),

∂φ i ∂x j (x) = ∂φ j ∂x i (x) = 0 , ∀x ∈ R N n .
(2) For all x ∈ D, φ i (x) = 0 (i.e., φ i is zero on D).

(3) The manifold D of the closed-loop system ẋi = f (x i ) + g(x i )φ i (x), i = 1, . . . , N is uniformly exponentially stable, i.e., there exist positive constants k and c > 0 such that for all x • in R N n and for all t in the time domain of existence of solution

|X (x • , t)| D ≤ k exp(-c t) |x • | D , (6
) where X (x • , t) denotes the solution of (4) initiated from x • .

We can now state the first main result of this work. Theorem 1. Suppose Assumption 1 holds and assume that there exists a C 1 function P : R n → R n×n taking symmetric positive definite values such that the following conditions hold.

• The Control Matrix Function (CMF) condition holds:

L f P (x) -ρP (x)g(x)g(x) P (x) ≤ -qP (x) , p I ≤ P (x) ≤ p I , (7)
for all x ∈ R n and for some positive real numbers p, p, q, ρ > 0. • g is a Killing vector field for P , i.e.,

L g P (x) = 0 , ∀x ∈ R n . (8) • There exists α : R n → R such that ∂α ∂x (x) = P (x)g(x) , ∀x ∈ R n . ( 9
)
Then, the distributed control law u i = φ i (x) with

φ i (x) = -κ N j=1 ij α(x j ) , (10)
solves the global uniform exponential synchronization problem for (4), for any κ ≥ ρ µ , with ρ satisfying (7) and µ satisfying (2).

Proof. Consider, without loss of generality, that x 1 is the leader node in the network, i.e., its dynamics is given by ẋ1 = f (x 1 ). Define n -1 error coordinates e := (e 2 , . . . , e N ) with e i := x i -x 1 and z = x 1 . These errors display the following dynamics

ėi = f (z + e i ) -f (z) -κ   N j=1 ij g(z + e i )α(z + e j )   .
Using the fact that

N j=1 ij g(z + e i )α(z) = g(z + e i )α(z) N j=1 ij = 0 ,
such a term can be added to the e i dynamics to obtain ėi

= f (z + e i ) -f (z) -κ N j=1 ij g(z + e i ) α(z + e j ) -α(z)
all i = 2, . . . , N . Note that in this new error coordinates, the manifold D defined in (5) corresponds simply to the origin of the e-dynamics, i.e. D = {(z, e) ∈ R N n : e = 0}. Consider the function Γ(s, t) defined as Γ(1, 0) = e , Γ(0, 0) = 0 , Γ(s, 0) = γ(s) , where γ is any C 1 with its i-th component Γ i defined as solution of the following ordinary differential equation

∂Γ i ∂t (s, t) = f (Z(t) + Γ i (s, t)) -f (Z(t)) -κ N j=1 ij g(Z(t)+Γ i (s, t))(α(Z(t)+Γ j (s, t))-α(Z(t))).
where Z(t) is the solution at time t of ż = f (z) initiated from z. For each path γ, we consider the function V i , i = 2, . . . , N , defined by This implies for all vector

V i (t) = 1 0 ∂Γ i ∂s (s,
ν in R n d dt ν P (Z(t) + Γ i (s, t))v = ν d f P (Z(t) + Γ i (t))ν -κ(α(Z(t) + Γ j (s, t)) -α(Z(t))) × N j=1 ij ν d g P (Z(t) + Γ i (t))ν
By using the Killing vector field (8) and the integrability assumption ( 9), the time-derivative of V i can be computed as follows

Vi (t) = 1 0 ∂Γ i ∂s (s, t) L f P (Z(t) + Γ i (s, t)) ∂Γ i ∂s (s, t) -κ ∂Γ i ∂s (s, t) N j=1 ij P (Z(t) + Γ i (s, t))g(Z(t) + Γ i (s, t)) × g(Z(t) + Γ j (s, t)) P (Z(t) + Γ j (s, t)) ∂Γ j ∂s (s, t)ds. Note that N i=2 Vi (t) = -κv(t) L 22 v(t) 1 0 N i=2 ∂Γ i ∂s (s, t) L f P (Z(t) + Γ i (s, t)) ∂Γ i ∂s (s, t)ds,
where v is defined as

v(t) =    g(Z(t) + Γ 2 (s, t)) P (Z(t) + Γ 2 (s, t)) ∂Γ2 ∂s (s, t)) . . . g(Z(t) + Γ N (s, t)) P (Z(t) + Γ N (s, t)) ∂Γ N ∂s (s, t))   .
Hence, by using (2), we get,

N i=2 Vi (t) ≤ -µκv(t) v(t) 1 0 N i=2 ∂Γ i ∂s (s, t) L f P (z + Γ i (s, t) ∂Γ i ∂s (s, t)ds, which implies N i=2 Vi (t) ≤ N i=2 1 0 ∂Γ i ∂s (s, t) [L f P (z + Γ i (s, t)) -µκP (Z(t) + Γ i (s, t))g(Z(t) + Γ i (s, t)) g(Z(t) + Γ i (s, t))P (Z(t) + Γ i (s, t))] ∂Γ i ∂s (s, t)ds .
Selecting any κ ≥ ρ µ and employing the CMF condition (7), we obtain

N i=2 Vi (t) ≤ -q N i=2 V i (t) and therefore N i=2 V i (t) ≤ exp(-qt) N i=2 V i (0) . Note that N i=2 V i (t) ≥ p n i=2 e i (t) 2
which yields exponential convergence to D. Furthermore, to obtain inequality (6), we follow [Andrieu et al., 2020, Proposition 4] and consider a sequence of minimizing paths (γ k ) k∈N such that γ k (1) = e, γ k (0) = 0. This concludes the proof.

Remark 1. In the case of a network of linear systems of the form ẋi = Ax i + Bu i , the conditions of Theorem 1 reduces to the following Riccati-like inequality P A + A P -ρP BB P ≤ -qP , with the function α(x) := B P x, and the Killing vector condition (8) automatically satisfied. In other words, we find the conditions in [Li et al., 2009, Section II.C]. Unfortunately, unlike the linear case, the extension of the proposed conditions to the non-leader case are non-trivial due to the presence of the state-dependent metric P . When P is constant, state-synchronization without leader can be achieved by following for instance [START_REF] Zhang | Fully distributed robust synchronization of networked lur'e systems with incremental nonlinearities[END_REF], [START_REF] Andrieu | Some results on exponential synchronization of nonlinear systems[END_REF] or [START_REF] Andrieu | LMI conditions for contraction and synchronization[END_REF].

Synchronization via Output-Feedback

We suppose now that the dynamics of the agents are described by ẋi = f (x i ) + u i i = 1, . . . , N, y i = h(x i ), (11) in which x i ∈ R n is the state and u i ∈ R n is the control input, and y i ∈ R is the exchange output. We denote x := col(x 1 , . . . , x N ) ∈ R N n , y := col(y 1 , . . . , y N ) ∈ R n .

Similarly to the previous section, our objective is to design a distributed control law stabilizing the synchronization manifold D defined in (5). However, we restrict in this section to the case of output synchronization laws, i.e. the agents cannot exchange with their neighbors the full state information x i but only the output y i . In particular, we define such a problem as follows.

Problem 2. The control laws u i = φ i (x i , y), i = 1 . . . , N solve the uniform exponential synchronization problem for (11) if the following conditions hold:

(1) For all non-communicating pair (i, j) (i.e., (i, j) / ∈ E),

∂φ i ∂y j (x i , y) = ∂φ j ∂y i (x j , y) = 0 , ∀x i ∈ R n , y ∈ R N .
(2) For all X ∈ D, φ(x i , y) = 0 (i.e., φ is zero on D).

(3) The manifold D of the closed-loop system ẋi = f (x i ) + φ i (x i , y), i = 1, . . . , N, is uniformly exponentially stable, i.e., there exist positive constants k and λ > 0 such that for all

X • ∈ R N n |X (x • , t)| D ≤ k exp(-ct) |x • | D
, where X (x • , t) denotes the solution initiated from x • , holds for all t in the time domain of existence of solution.

For a leader-connected network of agents described by (11), we have the following set of sufficient conditions to solve the previous problem. Theorem 2. Suppose Assumption 1 holds and suppose that there exists a C 1 function P : R n → R n×n taking symmetric positive definite values such that the following conditions hold.

• The Control Matrix Function (CMF) condition holds:

L f P (x) -ρ ∂h ∂x (x) ∂h ∂x (x) ≤ -qP (x) , p I ≤ P (x) ≤ p I , (12)
for all x ∈ R n and for some positive constants p, p, q, ρ. • The vector field α : R n → R n defined as

α(x) = P -1 (x) ∂h ∂x (x) (13)
is a Killing vector for P , i.e.

L α P (x) = 0 ∀x ∈ R n . ( 14
)
Then the distributed control law u i = φ i (x i , y) with

φ i (x i , y) = -κ N j=1 ij α(x i )y j (15)
solves the global uniform exponential synchronization problem for system (11) for any κ ≥ ρ µ , with ρ satisfying (12) and µ satisfying (2).

Proof. Consider, without loss of generality, that x 1 is the leader node in the network. We define n -1 error coordinates e := (e 2 , . . . , e N ) with e i := x i -x 1 and z := x 1 . The dynamics of these errors reads ėi = f (e i + z) -f (z)

-κ N j=1 ij α(e i + z)(h(e j + z) -h(z))
for all i = 2, . . . , N . Consider now the function Γ(s, t) defined as Γ(1, 0) = e , Γ(0, 0) = 0 , Γ(s, 0) = γ(s) where γ is any C 1 path with its i-th component Γ i defined as solution of the following ordinary differential equation

∂Γ i ∂t (s, t) = f (Z(t) + Γ i (s, t)) -f (Z(t)) -κ N j=1 ij α(Γ i (s, t)+Z(t))(h(Γ j (s, t)+Z(t))-h(Z(t))).
Consider the function V i , i = 2, . . . , N , defined by Consequently, we obtain

V i (t) = 1 0 ∂Γ i ∂s (s,
N i=2 Vi (t) ≤ -q N i=2 V i (t)
for any κ ≥ µ . Finally, since

N i=2 V i (t) ≥ p n i=2 e i (t) 2 ,
it yields the result following the same arguments of the proof of Theorem 1.

Remark 2. In the case of a network of linear systems of the form ẋi = Ax i + u i , y i = Cx i the conditions of Theorem 1 reduces to the following Riccati-like inequality P A + A P -ρC C ≤ -qP with the function α(x) := P -1 C , and the Killing vector condition (8) automatically satisfied. In other words, we find the conditions in [Li et al., 2009, Section II.B].

Remark 3. The conditions of Theorem 2 are inspired by the conditions for observer design based on Riemannian distances, see, e.g., [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a riemannian metric (part i)[END_REF], [START_REF] Andrieu | Some results on exponential synchronization of nonlinear systems[END_REF]. Furthermore, for a constant metric P and systems dynamics (11) in the so-called prime form (or observability canonical form), we recover the conditions of Proposition 1 in [START_REF] Isidori | Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory[END_REF], based on high-gain observer theory.

ILLUSTRATION

Consider the multiagent network system with 5 nodes described in Figure 1 and represented by the Laplacian matrix

L =      0 0 0 0 0 -1 3 -1 0 -1 0 -1 2 -1 0 0 0 -1 1 0 0 -1 0 0 1     
, with µ = 0.18. Each agent dynamics is described by (11) with x i ∈ R 2 , and the functions f, h defined as

f (z) = z 2 + sin(z 1 ) -z 1 + z 2 cos(z i ) + 2 cos(z 1 ) sin(z 1 ) , h(z) = (z 2 + sin(z 1 )) (16)
for any z = (z 1 , z 2 ) ∈ R 2 and for some 0 < < 1. It can be

P (z) = 2[1 -cos(z 1 ) + 2 cos 2 (z 1 )] 1 -2 cos(z 1 ) 1 -2 cos(z 1 ) 2 ,
and constants p = 1, p = 3, q = 3, any ≥ 2. Moreover, the Killing vector property ( 14) is verified with the function α defined as

α(z) = 1 3 -1 + 4 cos(z 1 ) 4 2 cos 2 (z 1 ) -3 cos(z 1 ) + 2 .
Therefore, all conditions of Theorem 2 are satisfied and the control law ( 15) is able to achieve state synchronization of the network with κ ≥ µ = 11.12. The rate of convergence c in item (3) of Problem 2 is approximately computed as c = qµ p/p, corresponding to c = 0.18 √ 3 in our example.

Figures 2, 3 show respectively the trajectories of the first and second state of each node i = 1, . . . , 5 of the system (11), ( 16) where = 1 2 and with the control law (15) with κ = 12 and initial condition x(0) = [[(2, -1), (15, 5), (-7, 16), (8, 14), (-10, -16)]]. After the transient behaviour of the first seconds of the simulation, state-synchronization is achieved to a unique oscillating mode, corresponding to the first node of the graph.

CONCLUSION

We proposed a set of sufficient conditions to solve the problem of state synchronization for a leader-connected network of homogeneous agents described by a nonlinear dynamics into two scenarios: the case of input-affine nonlinear systems in which the agents exchange the full state to their neighbours, and the case of nonlinear systems dynamics linear in the inputs in which the agents are allowed to exchange only a certain nonlinear output. The conditions are based on a metric analysis and guarantees exponential synchronization of the network. This paper is a preliminary work in order to extend of the results in [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF] and in [START_REF] Pavlov | Controlled synchronization via nonlinear integral coupling[END_REF] to the case of input-affine nonlinear systems with nonlinear outputs. Future works will consider also the case of non-leader connected networks and potentially heterogeneous dynamics as in [START_REF] Isidori | Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory[END_REF].

Fig. 1 .

 1 Fig. 1. Network structure

Fig. 3 .

 3 Fig. 3. Dynamics of the second state of node i

  L f P (Z(t) + Γ i (s, t))

	Hence
	N i=2	Vi (t) =	1 (s, t) -κ 0 N i=2 ∂Γ i ∂s N j=1 ij ∂h ∂z	(Z(t))	∂h ∂z (Z(t))	∂Γ i ∂s	(s, t)ds
		=	0	1	N i=2	∂Γ i ∂s	(s, t)L f P (Z(t) + Γ i (s, t))	∂Γ i ∂s	(s, t)
					-κµv(t) v(t) ds
	where we defined
				v(t) =	  	∂h ∂z (Z(t)) ∂Γ2 ∂s (s, t) . . .
									t) P (Z(t) + Γ i (s, t))	∂Γ i ∂s	(s, t)ds .
									Employing the Killing vector field assumption (14) and the
									definition of α in (13), we compute the time-derivative of
									V i as
									Vi (t) =	0	1	∂Γ i ∂s	(s, t) L f P (Z(t) + Γ i (s, t))
									-κ	N j=1	ij	∂h ∂z	(Z(t))	∂h ∂z (Z(t))	∂Γ i ∂s	(s, t)ds

ACKNOWLEDGMENTS

The authors would like to thank Fulvio Forni for many fruitful discussions.


Daniele Astolfi
email: daniele.astolfi@univ-lyon1.fr

Output-feedback repetitive control for minimum-phase nonlinear systems with arbitrarily relative degree

Keywords: Repetitive Control, nonlinear Systems, PDEs, high-gain observers, forwarding

A new design of output-feedback repetitive control scheme for nonlinear minimumphase systems with arbitrary relative degree and globally Lipschitz nonlinearities is proposed. This work extends the recent results in Astolfi et al. [2021]. The delay of the repetitive control scheme is represented by a transport equation. A high-gain observer and a forwarding-based feedback law are employed to steer the desired output to zero in presence of periodic signals (references and/or perturbations) and model uncertainties.

INTRODUCTION

Repetitive control (RC) schemes are based on the main idea that a delay can be used to generate any periodic signal. RC schemes are therefore very popular in regulation problems of periodic signals, i.e. reference tracking and/or disturbance rejection, and used in many control applications in both continuous-time, [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF], and discrete-time domain, [START_REF] Tomizuka | Analysis and synthesis of discrete-time repetitive controllers[END_REF]. See, more recently, [START_REF] Mattavelli | Repetitive-based control for selective harmonic compensation in active power filters[END_REF], [START_REF] Kurniawan | A survey on robust repetitive control and applications[END_REF], [START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF] and references therein. Although this vast success, from the theoretical point of view, few works addressed the problem of using RCschemes for continuous-time nonlinear systems. In view of the difficulties of dealing with infinite-dimensional systems (i.e., the delay), a common approach is to focus on some finite-dimensional RC approximation based on the use of low-pass filters (see, e.g., [START_REF] Weiss | Repetitive control of mimo systems using h ∞ design[END_REF]) or on its equivalent harmonic representation (see, e.g., [START_REF] Astolfi | Francis-wonham nonlinear viewpoint in output regulation of minimum phase systems[END_REF], [START_REF] Ghosh | Nonlinear repetitive control[END_REF]). Recently, a new stability analysis approach for an exact RC-scheme (i.e. infinite-dimensional) has been proposed in the context of nonlinear systems which are strictly dissipative, under the assumption that the relative degree between the control input and the regulated output is zero, see [START_REF] Califano | A stability analysis based on dissipativity of linear and nonlinear repetitive control[END_REF].

This work addresses the problem of designing an outputfeedback RC-scheme for minimum-phase nonlinear systems having a relative degree larger than one and with globally Lipschitz nonlinearities. It relies on the recent developments proposed in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF] where the same class of systems is considered but with a partialstate feedback design. A new design is proposed, allowing to extend the class of systems to which RC-approaches apply. With respect to [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF], it is employed a different transport-equation design which is able to achieve an input-to-state stability property with respect to the estimation error. As a consequence, the state-feedback law, based on forwarding forwarding technique, can be turned into output-feedback by means of classical highgain observers, see, e.g., [START_REF] Atassi | A separation principle for the stabilization of a class of nonlinear systems[END_REF].

Notation. R denotes the space of real numbers and C the space of complex numbers, R + := [0, ∞) and N denotes the space of positive integers, i.e. N := {1, 2, . . . , ∞}. For a function w : (t, x) ∈ R + ×[0, 1] → w(t, x) ∈ R, the notation w t (respectively, w x ) denotes the partial derivative of w w.r.t the variable t (respectively, w.r.t. the variable x). When a function w depends only on the variable of the time t (resp. space x), its derivative is simply denoted by ẇ (resp. w ); L 2 (0, 1) denotes the Hilbert space of realvalued square-integrable functions over the interval (0, 1) and H1 (0, 1) ⊂ L 2 (0, 1) is the Hilbert space of real-valued absolutely continuous functions over [0, 1] with squareintegrable derivative; • L 2 and • H 1 denotes their repsectively induced norms (see, also, [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF]). A function α : R

+ → R + is said to be of class-K ∞ if α is continuous, increasing, α(0) = 0 and lim s→∞ α(s) = ∞. 2. MAIN MOTIVATION Consider a system of the form ẋ = f(x, d(t)) + g(x, d(t))u y = h(x, d(t)) (1)
with system state x ∈ R nx , control input u ∈ R and output y ∈ R. The problem of periodic output regulation consists in designing a regulator for system (1) so that the output y satisfies lim (2) inside the control-feedback loop. The main purpose of (2), based on the so called internal-model principle [START_REF] Paunonen | On infinite-dimensional sylvester equation and the internal model principle[END_REF]), is to generate any desired periodic steadystate input, see for instance [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF]. However, the analysis and the design of a RC-based stabilizerfeedback for systems (1) is not straightforward since the Laplace transformation cannot be used in such a nonlinear framework, and different tools need to be used.

An alternatively approach, pursued in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF], [START_REF] Califano | Stability analysis of nonlinear repetitive control schemes[END_REF], [START_REF] Califano | A stability analysis based on dissipativity of linear and nonlinear repetitive control[END_REF], consists in rewriting the scheme (2) by using an equivalent transport equation representation. In particular, the main result in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF] is that of developing a new RC-feedback control law for systems (1) having a welldefined relative degree and that can be rewritten, after a change of coordinates, into the form

           ż = f (t, z, ξ 1 ), ξi = ξ i+1 , i = 1, . . . , r -1 ξr = q(t, z, ξ 1 , . . . , ξ r ) + u, e = ξ 1 , (3)
where (z, ξ) ∈ R n × R r is the system state, with the z ∈ R n -dynamics being the so-called zero-dynamics and ξ := (ξ 1 , . . . , ξ r ) ∈ R r representing the derivatives of the output e ∈ R that we aim at regulating to zero, and u ∈ R the control input. Note that with respect to the representation (1), we have ξ 1 = h(x, d(t)) -r(t), and the explicit dependence on d, r in (3) is compactly substituted by the dependence on time t. Finally, the zerodynamics in (3) is supposed to possess some good inputto-state properties as specified later. The RC-design for system (3) proposed in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF] has then the form

     η t (t, x) = -1 T η x (t, x) ∀ (t, x) ∈ R + × [0, 1], η(t, 0) = η(t, 1) + θ(t) ∀ t ∈ R + , u(t) = γ(θ(t), η(t, x)) ∀ (t, x) ∈ R + × [0, 1], (4)
where (t, x) ∈ R + × [0, 1] and η ∈ L 2 (0, 1), the signal θ is a linear combination of the state variables ξ of the form

θ := ξ r + r-1 i=1 g r-i a i ξ i ,
and the feedback law γ : R × L 2 (0, 1) → R is designed according to the so-called forwarding approach. The drawback of such an approach is that the knowledge of θ, and therefore of the state components (ξ 1 , . . . , ξ r ), is required.

The objective of this work is to extend such an approach to the context of output feedback, in which the sole use of e = ξ 1 is employed for the RC-feedback law (4). To this end, recall that a common routine in output-feedback control is to replace the state information ξ by an estimate ξ provided by an observer. In the context of minimumphase systems, such an observer is typically chosen as a high-gain observer, see, e.g., [START_REF] Atassi | A separation principle for the stabilization of a class of nonlinear systems[END_REF]. However, for this to work, an input-to-state stability (ISS) with respect to estimation error is needed, see [START_REF] Andrieu | A unifying point of view on output feedback designs for global asymptotic stabilization[END_REF]. As a consequence, when considering the feedback (4), such an approach cannot be followed since simple computations can show that the ISS property with respect to estimation error acting at the boundary conditions η(t, 0) may not be verified, see, for instance [START_REF] Tanwani | Input-tostate stabilization in h 1-norm for boundary controlled linear hyperbolic pdes with application to quantized control[END_REF]. The main result of this work is therefore that of proposing a variation of the scheme (4) by developing a pure output feedback RC-scheme for systems of the form (3), with the signal θ acting in a distributed way and not at the boundary of the η-dynamics.

MAIN RESULT

Following [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF], consider a system of the form (3) and suppose that the following assumptions hold. Assumption 1. The function q in (3) is globally Lipschitz, C 2 in its arguments and periodic with respect to the first argument.

Assumption 2. The zero-dynamics ż = f (t, z, 0) of system (3) admits a unique C 2 T -periodic bounded solution z(t) which is globally uniformly stable and input-to-state stable.

In particular, there exists a positive definite function

V : R × R n → R + and class K ∞ functions α, ᾱ and real numbers α, ρ > 0 satisfying α(|z -z(t)|) ≤ V (t, z -z(t)) ≤ ᾱ(|z -z(t)|) (5) for all (t, z) ∈ R + × R n and ∇V (t, z -z(t)), f (t, z, ξ 1 ) -f (t, z(t), 0) ≤ -α|z -z(t)| 2 + ρ|ξ 1 | 2 (6) for all (t, z, ξ 1 ) ∈ R + × R n × R.
Under previous assumptions, an output-feedback repetitivecontrol based regulator can be designing by combining the following elements.

• A high-gain observer of the form:

   ξi = ξi + i b i (e -ξ1 ), i = 1 . . . , r -1 ξr = u s + r b r (e -ξ1 ), ( 7a
)
where ≥ 1 is the so-called high-gain parameter, the parameters b i > 0, i = 1, . . . , r have to be properly chosen and the term u s is an extra input to be defined. Note that following [START_REF] Wang | Output stabilization for a class of nonlinear systems via high-gain observer with limited gain power[END_REF], the high-gain observer (7a) could be also substituted by a low-power high-gain observer [START_REF] Astolfi | A high-gain nonlinear observer with limited gain power[END_REF]. • The core of the RC scheme, also denoted as internalmodel unit in output regulation theory, see, e.g., [START_REF] Astolfi | Francis-wonham nonlinear viewpoint in output regulation of minimum phase systems[END_REF], embedding a delay represented as a transport equation:

     η t (t, x) = -1 T η x (t, x) + exp(βx) θ η(t, 0) = η(t, 1) η(0, x) = η 0 (x) (7b) defined on (t, x) ∈ R + × [0, 1],
with β being a parameter to be chosen and θ a signal to be defined. • The forwarding-based (see, e.g., [START_REF] Marx | Forwarding design for stabilization of acoupled transport equation/ode with a cone-bounded input nonlinearity[END_REF][START_REF] Marx | Forwarding techniques for the global stabilization of dissipative infinitedimensional systems coupled with an ode[END_REF], [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF]) stabilizing feedback control law:

     u = u s + u im u s = -κ θ u im = µ 1 0 (η(t, x) -M (x) θ)M (x)dx, (7c)
where µ > 0 is a positive parameters, θ is defined as

θ := ξr + r-1 i=1 g r-i a i ξi , (7d)
with g > 0, a i > 0, i = 1, . . . , r -1, parameters to be defined, and the function M : [0, 1] → R is defined as the solution of the following two-boundary value problem

1 T M (x) = κM (x) + exp(βx), M (0) = M (1) (8)
which can be explicitly computed as

M (x) = exp(kT x)M + T exp(βx)-exp(κT x) β-κT , β = κT, exp(kT x)M + xT exp(κT x), β = κT, M =    exp(β)+exp(κT ) (κT -β)(1-exp(κT )) , β = κT, T exp(κT ) 1-exp(κT ) , β = κT. (9)
The following main result can be stated. Theorem 1. Suppose Assumptions 1 and 2 hold. Let the parameters a i , b i of the regulator (7) be such that the corresponding polynomial

p a (λ) := λ r-1 + a r-1 λ r-2 + • • • + a 2 λ + a 1 p b (λ) := λ r + b r λ r-1 + • • • + b 2 λ + b 1
are Hurwitz and let µ > 0 and β = 0 be fixed. Then, there exists g ≥ 1 and, for any g > g , there exists κ ≥ 1 such that, for any κ ≥ κ there exists ≥ 1 (the values of g , κ , depends only on the Lipschitz constants of f, q, the parameters α, ρ defined in Assumptions 1, 2, and the choice of a 1 , b i ) such that the following statements hold true for any ≥ .

(1) For any initial condition (z 0 , ξ 0 , ξ0 , η 0 ) ∈ R n × R r × L2 (0, 1), the closed-loop system (3), ( 7), ( 8), admits a unique solution (z, ξ, ξ, η) ∈ C 0 ([0, ∞); R n+2r × L 2 (0, 1)), which is bounded forward in time, namely

|z(t)| + |ξ(t)| + | ξ(t)| + η(t, •) L 2 ≤ δ, ∀ t ≥ 0
for some δ > 0.

(2) Any solution of the closed-loop system the closed-loop system (3), ( 7), ( 8), starting from R n × R r × L 2 (0, 1) satisfies lim t→∞ e(t) = 0.

The result of Theorem 1 mainly states that the proposed design (7) guarantees asymptotic regulation to zero of the error e, namely periodic reference tracking and disturbance rejection is achieved for systems (1) that can be rewritten, via a change of coordinates, in the form (3), with a pure output-feedback design which uses only the regulated output e. Furthermore, the proposed design is robust with respect to model uncertainties. Indeed, the control law ( 7) is parametrized by some parameters. and their values depend only on the Lipschitz constants of the functions f, q and the ISS-properties α, ρ of the zero-dynamics (see Assumption 2). In other words, the exact knowledge of the function f, q characterizing system (3), and so of f, g, h of system (1), is not needed for the design of the proposed RC-scheme.

As one can see comparing equations (7b) with (4), the main difference with respect to the design proposed in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF] is that the driving term of the transport equation is acting in a distributed fashion and not at the boundary. This, in turns, is able to guarantee some ISS properties with respect to the estimated θ. In practice, once the high-gain observer (7a) is converged, one recovers a state-feedback equation η t (t, x) = -1 T η x (t, x) + exp(βx)θ which in turns guarantee the signal θ (and therefore ξ 1 ) to converge asymptotically to zero. Such a modification is however not trivial to analyze as one has to prove that the internal-model property of (7b) is still verified, i.e. the system

η t (t, x) = -1 T η x (t, x), η(t, 0) = η(t, 1), u ss im = µ 1 0 η(t, x)M (x)dx, (10)
is such that the output u ss im can generate any arbitrarily periodic signal.

Finally, note that for unitary systems of the form ż = f (t, z, e) ė = q(t, z, e) + u the proposed control law (7) reads

     η t (t, x) = -1 T η x (t, x) + exp(βx)e, η(t, 0) = η(t, 1), η(0, x) = η 0 (x), u = -κe + µ 1 0 M (x)(η(t, x) -M (x)e)dx,
with M chosen as in (8).

PROOF OF THEOREM 1

The proof is organized as follows. First, it is shown that system (7b) possesses the so-called "internal-model property", i.e. system (10) can generate any periodic signal. Then, a suitable change of coordinates for the closed-loop dynamics is proposed. Existence of solution is briefly discussed and the stability and convergence analysis is finally addressed. Fundamental properties developed in the works [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF], [START_REF] Marx | Forwarding design for stabilization of acoupled transport equation/ode with a cone-bounded input nonlinearity[END_REF] and [START_REF] Marx | Forwarding techniques for the global stabilization of dissipative infinitedimensional systems coupled with an ode[END_REF] will be used throughout the proof.

Internal Model Property

Consider system (7b), (7c). When needed, it will be used the following equivalent operator-form (see, e.g. [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]) given by

η = Sη + G θ u = -κ θ + µM * (η -M θ) ( 11
)
where S = -1 T ∂ x , G = exp(βx) and with M * denoting the adjoint of M so that M * η = 1 0 η(t, x)M (x)dx. The representation (11) can be be obtained following for instance [Marx et al., 2021, Example 2,4].

The following results, concerning some observability and stability properties of the operators in (11), are stated.

Lemma 1. For any β = 0, the pair (S * , G * ) defined in (11) is approximately observable in infinite time 2 . Proof. In order to prove the desired result it suffices to check that the adjoint system, namely

     T η t (t, x) + η x (t, x) = 0 η(t, 0) = η(t, 1) y(t) = 1 0 exp(βx)η(t, x)
dx is approximately observable from the output y. To this end, suppose that y is constantly equal to zero. Then, by computing an integration by parts, this implies

y(t) = 1 0 exp(βx)η(t, x)dx = 1 β exp(βx)η(t, x) 1 0 -1 β 1 0 exp(βx)η x (t, x)dx = 1 β (exp(β) -1)η(t, 0) + T β y(t). If β = T , it yields 0 = y(t) = 1 β (exp(β) -1)η(t, 0) + y(t) for all t ≥ 0. Otherwise, for β = T , 0 = y(t) = 1 0 exp(βx)η(t, x)dx = exp(β)-1 β-T η(t, 0)
for all t ≥ 0. In both cases, one obtain η(t, 0) = 0 for all t ≥ 0. Therefore, following the proof in [Marx et al., 2020, Theorem 2], the latter implies that η(t, x) = 0 for all (t, x) ∈ R + × [0, 1] concluding the desired result. More details are given also in [Marx et al., 2021, Example 5]. An alternative proof consists in following [Russell, 1978, Section 4] and using an eigenfunction decomposition of the operators S, G.

Lemma 2. For any β = 0, the pair (S, M * ) defined in ( 11) is approximately observable in infinite time.

Proof. The definition of M given in ( 8) is solution of the Sylvester equation -κM = SM + G.

(12) Indeed, the spectrum of -κ and S is disjoint and the solution is unique. In particular, it is given by ( 9). Recall that by Lemma 1, the pair (S * , G * ) is approximately observable, the operator S is skew-adjoint, and the matrix

-κ + λ 1 1 0
is full rank for any λ in the spectrum of S. Hence one can apply Proposition 1 of [START_REF] Marx | Forwarding techniques for the global stabilization of dissipative infinitedimensional systems coupled with an ode[END_REF] to conclude the proof. Lemma 3. For any β = 0, µ > 0, the operator

A cl = -κ -µM * M M * G S
generates a strongly stable C 0 semigroup.

Proof. The proof is a direct application of Theorems 1, 2 and Proposition 1 of [START_REF] Marx | Forwarding techniques for the global stabilization of dissipative infinitedimensional systems coupled with an ode[END_REF] and it is therefore just sketched. As noted in the previous lemma, M given in ( 8) is solution of the Sylvester equation ( 12). Indeed, the spectrum of -κ and S is disjoint and the solution is unique, as given by ( 9). Hence, the system θ = -κ θ + µM * (η -M θ) η = Sη + G θ is transformed, by using the change of coordinates

η → η := η -M θ into a system in triangular form χ = Ãcl χ, Ãcl := -κ µM * 0 (S -µMM * )
,

where χ := ( θ, η). Now, by Lemma 1, the pair (S * , G * ) is approximately observable in infinite time and therefore, in view of Proposition 1 of [START_REF] Marx | Forwarding techniques for the global stabilization of dissipative infinitedimensional systems coupled with an ode[END_REF], also the pair (S, M * ) is approximately observable. As a consequence, since the operator S is skew-adjoint, the operator (S -µMM * ) is dissipative. Furthermore using LaSalles Invariance Principle for infinite-dimensional systems (see, e.g., Theorem 3.1 in [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF]) and the lower triangular structure of Ãcl , one can conclude that the origin of χ is globally asymptotically stable, i.e., the operator A cl generates a strongly stable semigroup, concluding the proof. Technical details follow from [START_REF] Marx | Forwarding techniques for the global stabilization of dissipative infinitedimensional systems coupled with an ode[END_REF].

Finally, similarly to [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF], Lemma 1], the following result concerning the internal-model property of the feedback regulator ( 7) can be stated. Lemma 4. For any C 2 T -periodic function ψ : R + → R and any µ, β = 0, there exists η0 ∈ L 2 (0, 1), such that the solution to the system

ηt (t, x) = -1 T ηx (t, x), η(t, 0) = η(t, 1), η(0, x) = η0 (x), y(t) = µ 1 0 η(t, x)M (x)dx (13) defined on (t, x) ∈ R + × [0, 1], with M defined by (8), satisfies y(t) ≡ ψ(t) for all t ∈ R + .
Proof. Since the function ψ is periodic and C 2 , the signal ψ can thought as generated by an autonomous infinitedimensional system of the form w t (t, x) -1 T w x (t, x) = 0, w(t, 1) = w(t, 0), w(0, x) = w 0 (x), ψ(t) = w(t, 1), with (t, x) ∈ R + × [0, 1] and suited initial conditions w 0 (x) ∈ L 2 (0, 1). Equivalently, it is represented with the operator form ẇ = Sw, ψ = Ew. Now, by using the operator-form introduced above, consider the following system

   ẇ = Sw θ = -κ θ -Ew + M * (η -M θ) η = S η + G θ
Let study now the existence of a solution Π θ , Π η to the following infinite-dimensional Sylvester equation

Π θ S = -(κ + M * M)Π θ + M * Π η -E Π η S = SΠ η + GΠ θ . (14)
Since the operator A cl defined in Lemma 3 generates a strongly stable semigroup, the Sylvester equation ( 14) is well defined, see Lemma 1 in [START_REF] Paunonen | On infinite-dimensional sylvester equation and the internal model principle[END_REF] and the existence of a solution is ensured. Furthermore, it is possible to verify that the conditions of Theorem 6 in [START_REF] Paunonen | On infinite-dimensional sylvester equation and the internal model principle[END_REF] are verified. Therefore, by Theorem 5 in [START_REF] Paunonen | On infinite-dimensional sylvester equation and the internal model principle[END_REF], one can conclude from the second equation of ( 14) that Π θ = 0, and thus from the first equation of ( 14) that M * Π η -E = 0. Selecting η(0) = Π η w(0) and coming back in the explicit form (13) yields the desired result since y = M * Π η w = Ew = ψ.

Change of coordinates

Given z from Assumption 2, let ψ(t) := -q(t, z(t), 0, . . . , 0) and let µ > 0 be fixed. In view of Assumptions 1 and 2, ψ is C 2 and T -periodic. Then, let η0 be the initial conditions given by Lemma 4 so that

µ 1 0 η(t, x)M (x)dx = ψ(t) = -q(t, z(t), 0). ( 15
)
Now, consider the following changes of coordinates for the closed-loop system dynamics (3)-( 7)

z → ζ := z -z(t) ξ i → X i := g r-i ξ i , i = 1, . . . , r -1, ξ r → θ := ξ r + r-1 i=1 r-i a i ξ i , ξi → ε i := (r-i) ( ξi -ξ i ), i = 1, . . . , r, η → φ := η(•, x) -η(•, x) -M (x) θ and denote X := (X 1 , . . . , X r-1 ) ∈ R r-1 , ε := (ε 1 , . . . , ε r ) ∈ R r . First, note that θ reads θ = θ + ε r + r-1 i=1 a i g r-i ε i = θ + Ψε with Ψ := a 1 g r-1 , . . . , a r-1 g , 1 . (16)
The input u is then transformed into 15). As a consequence, the closed-loop system dynamics reads

u = -κθ + µ 1 0 φ(t, x)M (x)dx -(κ + µm)Ψε + ψ(t) with m := 1 0 M (x) 2 dx. in light of (
                   ζ = F (t, ζ, g 1-r C r-1 X ) Ẋ = g(A r-1 -B r-1 K a )X + gB r-1 θ θ = ∆(t, ζ, X , θ) -(κ + µm)Ψε -κθ + y ε = (A r -K b C r )ε -B r [∆(t, ζ, X , θ) + y] y = µ 1 0 φ(t, x)M (x)dx (17a)
where the triplet (A i , B i , C i ) is in prime form, i.e.,

A i := 0 i-1,1 I i-1 0 0 • • • 0 , B i := (0 i-1,1 1) , C i := (1 0 1×i-1
) , K a = (a 1 , . . . , a r-1 ), K b = (b 1 , . . . , b r ), the function F, ∆ are defined as

F (t, ζ, g 1-r C r-1 X ) := f (t, ζ + z(t), g 1-r C r-1 X ) -f (t, z(t), 0), ∆(t, ζ, X , θ) := q(t, ζ + z(t), g 1-r X 1 , . . . , g -1 X r-1 , θ -K a X ) -q(t, z(t), 0) + g r-2 i=1 a i X i+1 + ga r-1 (θ - r-1 i=1 a i X i ),
and the φ-dynamics is given by

           φ t (t, x) = -1 T φ x (t, x) -M (x)y -M (x) ∆(t, ζ, X , θ) -(κ + µm)Ψε φ(t, 0) = φ(t, 1) φ(0, x) = φ 0 (x) (17b) defined on (t, x) ∈ R + × [0, 1], with φ 0 (x) = η 0 (x) -η0 (x) -M (x) θ(0) ∈ L 2 (0, 1).
Recall that the function q is globally Lipschitz by Assumption 1. Furthermore, ∆(t, 0, 0, 0) = 0. As a consequence, by using the definition of the matrix Ψ in ( 16), there exists a L > 0, independent of g, , κ such that the following inequalities hold

|Ψ a | ≤ L, |(κ + µm)Ψε| ≤ Lκ|ε|, |∆(t, ζ, X , θ)| ≤ L |ζ| + g|X | + g|θ| , (18)
for all (ζ, X , θ, ε) ∈ R n+2r and for any g ≥ 1 and ≥ max{g, 1}. Finally, system (17) can be also compactly written as ṗ = F(t, p) (19) with p = (ζ, X , θ, ε, φ) ∈ R n+2r × L 2 (0, 1), and associated norm |p|

:= |ζ| + |X | + |θ| + |ε| + |φ| L 2 (0,1) .

Existence of solutions

The following lemma concerning the existence of solutions to ( 17) is stated. Its proof is similar to Lemma 2 in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimumphase systems[END_REF]. Details are omitted for space reasons. Lemma 5. (Well-posedness of ( 19)). For any t 0 > 0 and any initial conditions p 0 ∈ R n+2r × L 2 (0, 1), respectively p 0 ∈ R n+2r × H 1 (0, 1), with the compatibility condition φ 0 (0) = φ 0 (1)), there exists a sufficiently small constant τ > 0 such that system (19) admits a unique solution satisfying p ∈ C 0 ([t 0 , t 0 +τ ]; R n+2r ×L 2 (0, 1)), respectively p ∈ C 1 ([t 0 , t 0 +τ ]; R n+2r ×L 2 (0, 1))∩C 0 ([t 0 , t 0 +τ ]; R n+2r × H 1 (0, 1)).

Stability of the origin of the closed-loop system

Once the existence of solutions has been established thanks to Lemma 5, it is possible to prove that the origin of ( 19) is globally asymptotically stable. To this end, consider the Lyapunov functional

W (t, p) = g 2 U 1 (t, ζ, X ) + U 2 (θ, ε) + U 3 (φ) U 1 (t, ζ, X ) = V (t, ζ) + X P a X , U 2 (θ, ε) = θ 2 + ε P b ε, U 3 (φ) = 2µ
1 0 φ(t, x) 2 dx, where the matrices P a , P b are defined as solution of the following Lyapunov matrix equations

P a (A r-1 -B r-1 K a ) + (A r-1 -B r-1 K a ) P a = -2I, P b (A r -K b C r ) + (A r -K b C r ) P b = -2I, since the matrices (A r-1 -B r-1 K a ) and (A r -K b C r ) are
Hurwitz due to the choice of the parameters a i , b i in the statement of the theorem. In view of Assumption 2, the function W satisfies, for all (t, p)

∈ R + × R n+2r × L 2 (0, 1), α W (|p|) ≤ W (t, p) ≤ ᾱW (|p|), ( 20
) for some class K ∞ functions α W , ᾱW . Then, compute the time derivative of U 1 . Using Assumption 2, it yields U1 ≤ -α|ζ| 2 + ρg 1-r |C r-1 X | 2 -2g|X | 2 + 2gX P a B r-1 θ ≤ -α|ζ| 2 -(g -ρ)|X | 2 + g|P a | 2 θ 2 for all g ≥ 1,
where in the second step it is used the Young's inequality and the fact that g 1-r ≤ 1 for all g ≥ 1. Then, consider the time derivatives of U 2 . By using the Lipschitz inequalities in (18), it gives U2 = -2κθ 2 + 2θy + 2θ∆(t, ζ, X , θ) -2θ(κ + µm)Ψε with c 1 (g) = 2 + 2L 2 + L 2 g 2 + 2Lg, and c 2 (g, κ) = 1 + κ 2 + L 2 + 2L 2 g 2 . Finally, following similar computations in [Marx et al., 2021, Appendix], and using the property 1 0 φ x (t, x)φ(t, x)dx = 0, due to the boundary conditions φ(t, 1) = φ(t, 0), and recalling the definition of y in (17), the time-derivative of U 3 gives U3 = -4y 2 -2y ∆(t, ζ, X , θ) -(κ + µm)Ψε ≤ -3y 2 + 2L 2 |ζ| 2 + g 2 |X | 2 + g 2 |θ| 2 + κ 2 |ε| 2 ). Finally, by combining all previous inequalities together, the time derivative of W gives Ẇ ≤ -

(g 2 α -2L 2 -2)|ζ| 2 -(g 3 -ρg 2 -2L 2 g 2 -2)|X | 2 -(2κ -c 1 (g) -g 3 |P a | 2 -2L 2 g 2 )θ 2 -(2 -c 2 (g, κ) -2L 2 κ 2 )|ε| 2 -y 2 .
Hence, since c 1 (g) depends only g and c 2 (κ) on κ, it is always possible to select g, κ and large enough so that there exists a positive number > 0 such that Ẇ ≤ -|ζ|

2 + |X | 2 + |θ| 2 + |ε| 2 -( 2 0 φ(t, x)M (x)dx) 2 ≤0.
Asymptotic stability of p = 0 in the R n+2r × L 2 (0, 1)topology can be therefore proved following [Marx et al., 2021, Theorem 1] and in particular by using the following arguments: completness of solutions, applying Barbalat's lemma, precompactness of solutions and LaSalle's invariance principle arguments for infinite-dimensional systems (see [Slemrod, 1989, Theorem 3.1]). The proof concludes by noting that the origin of ( 17) implies X = 0 and therefore e = X 1 = 0.

CONCLUSIONS

A new output-feedback RC-scheme for minimum-phase nonlinear systems with arbitrarily relative degree is proposed. This work is cast into a global context (i.e. with globally Lipschitz nonlinearities) but similar results could be proved into a semi-global framework by relaxing these assumptions. Future works envision also to enlarge the class of nonlinear systems by considering systems in the original coordinates with sector-bounded or monotonic nonlinearities.

  r(t) = 0 for some T -periodic reference r ∈ R and in presence of possible T -periodic disturbances d ∈ R n d . In Repetitive-Control design, the solution to such a problem is typically pursued by including a regulator of the form 1 R(s) := exp(-T s) 1 -exp(-T s)

-

  2 |ε| -2ε B r ∆(t, ζ, X , θ) + y ≤ -(2κ -c 1 (g))θ 2 -(2 -c 2 (g, κ))|ε| + 2y 2 + 2|ζ| 2 + 2|X | 2

Here s ∈ C represents the Laplace variable.

See, e.g.,[Tucsnak and Weiss, 2009, Definition 6.1.1] for a precise definition of approximate observability.

ACKNOWLEDGEMENT

The author thanks Swann Marx and Vincent Andrieu for their precious help in preparing this work.

This research was partially supported by the French Grant ANR ODISSE (ANR-19-CE48-0004-01).


Corentin Delacour
email: corentin.delacour@lirmm.fr

Stefania Carapezzi

Madeleine Abernot
email: madeleine.abernot@lirmm.fr

Gabriele Boschetto

Nadine Azemard
email: nadine.azemard@lirmm.fr

Jeremie Salles
email: jeremie.salles@lirmm.fr

Thierry Gil
email: thierry.gil@lirmm.fr

Aida Todri-Sanial

Oscillatory Neural Networks for Edge AI Computing

Keywords: Oscillatory neural networks (ONN), Internet-of-Things (IoT), Edge Artificial Intelligence (edge AI), Neuromorphic Computing, Beyond-CMOS devices, Technology Computer-Aided Design (TCAD), FPGA

In this paper, we showcase the innovative concept of implementing Oscillatory Neural Networks (ONNs) for neuromorphic computing with beyond-CMOS devices based on vanadium dioxide to mimic neurons and resistors to emulate synapses. We explore ONN technology potentials from device to analog circuit-level simulations. We report that ONN behaves like an associative memory and can implement energy-based models such as Hopfield Neural Networks on edge devices. Finally, as a proof of concept, a reconfigurable digital ONN is implemented on FPGA for pattern recognition tasks.

I. INTRODUCTION

Classical computing based on the von Neumann paradigm suffers from the communication bottleneck between memory and processor, which causes a significant limitation for data processing. To keep up with the ever-increasing data growth from connected IoT devices and intelligent agents [START_REF] Merenda | Edge machine learning for AIenabled IoT devices: a review[END_REF], referred to as "data deluge," there are tremendous efforts to develop a novel computing paradigm inspired by brain-like computing to process the data where data is available meaning at the edge devices. This has resulted in industry and academic communities investigating in-memory computing by exploring novel material, devices, circuits, and architecture design styles This work was supported by the European Union's Horizon 2020 research and innovation programme, EU H2020 NEURONN (www.neuronn.eu) project under Grant 871501.

to enable massive parallelism and energy-efficient computing [START_REF] Kendall | The building blocks of a brain-inspired computer[END_REF].

Over the last decades, computing with artificial neural networks (ANNs) and convolutional neural networks (CNNs) have revolutionized Artificial Intelligence (AI) in every aspect of society from healthcare, autonomous driving, entertainment, among others. Yet, these powerful neural networks run on classical systems such as CPUs, GPUs, or TPUs, which suffer from energy inefficiency and deployment at the edge [START_REF] Yang | A method to estimate the energy consumption of deep neural networks[END_REF]. Alternatively, a new class of artificial neural networks based on coupled oscillators or Oscillatory Neural Networks (ONN) draws a lot of interest as a low power computing paradigm that can enable online learning and inference at the edge [START_REF] Csaba | Coupled oscillators for computing: a review and perspective[END_REF]. ONN is based on analog computing, where information is encoded on the phase relations between oscillators with the key concept of "let physics compute" [START_REF] Hoppensteadt | Pattern recognition via synchronization in phase-locked loop neural networks[END_REF], [START_REF] Raychowdhury | Computing with networks of oscillatory dynamical systems[END_REF].

The novelty in ONNs is to harness the rich dynamics of coupled oscillators to compute in the phase domain rather than amplitude domain as in other artificial or spiking neural networks. This means that the signal amplitude can be extremely low, resulting in a low power computing solution [START_REF] Shukla | Ultra low power coupled oscillator arrays for computer vision applications[END_REF]. Oscillators physically emulate neuronal oscillatory behavior (e.g., brain waves), and resistors emulate synaptic coupling between neurons. Such architecture enables in-memory computing where both processing (oscillators) and memory (synapses) constitute a truly parallel analog computing architecture.

ONN computational capacity has been studied and proved by mathematicians for decades [START_REF] Hoppensteadt | Pattern recognition via synchronization in phase-locked loop neural networks[END_REF], but hardware implementations were limited. The emergence of energy-efficient and

ΔΦ 12 = 180°I
NPUT OUTPUT ONN Fig. 1. ONN as associative memory, like a Hopfield Neural Network (60 oscillators in this example). Every oscillator is coupled to all others via resistors. Gray-scale images represent the ONN phase state. ONN stores pattern 'D,' which can be retrieved from noisy input patterns. ONN phase state evolves such that the conceptual Hopfield energy [START_REF] Hoppensteadt | Pattern recognition via synchronization in phase-locked loop neural networks[END_REF] reaches a minimum corresponding to a stored pattern. compact oscillators based on vanadium dioxide (VO 2 ) incites a new interest in ONN implementation. Up to 6 VO 2 -oscillators have been coupled experimentally [START_REF] Shukla | Ultra low power coupled oscillator arrays for computer vision applications[END_REF], and it is expected that this number will increase with the progress on device fabrication.

In this work, we perform device to circuit-level ONN simulations based on VO 2 -oscillators. We have developed a design tool flow to allow material properties, device to circuit-level modeling for exploring the potential of ONN computing. We report on performances and energy-efficiency up to 60 coupled VO 2 -oscillators for pattern recognition, like a Hopfield Neural Network (HNN) [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF]. Finally, as a proofof-concept, we developed a digital ONN architecture and implemented it in an FPGA for testing AI workloads.

II. ONN COMPUTING PARADIGM A. Phase-based computing

A large number of physical phenomena can be modeled in terms of interacting oscillator dynamics. In biology, cellular processes such as the glycolytic system of muscle cells have oscillatory behavior [START_REF] Kim | A design principle underlying the synchronization of oscillations in cellular systems[END_REF]. From neuroscience, there is evidence that biological oscillators are synchronized through coupling via signaling messengers like hormones [START_REF] Kim | A design principle underlying the synchronization of oscillations in cellular systems[END_REF]. Also, neurons can sometimes be represented as oscillators [START_REF] Stiefel | Neurons as oscillators[END_REF] and recent brain studies [START_REF] Bauer | Synchronisation of neural oscillations and cross-modal influences[END_REF] have demonstrated that signals from a sensory cortex can affect activity in another one by synchronizing neural oscillations. This suggests that phase-based computing possesses benefits that can be biomimetically exploited in hardware, such as ONNs.

R ext

In ONNs, information is encoded in the phase between oscillators, and the first oscillator is used as reference [START_REF] Todri-Sanial | How Frequency Injection Locking Can Train Oscillatory Neural Networks to Compute in Phase[END_REF]. ONN input is the initial ONN phase state, which evolves until it settles to a stable fixed point. ONN output is given by the final oscillator phases that take binary values ∆Φ out {0 • ; 180 • }. ONN phase state evolution can be interpreted as the minimization of an energy function in the phase space [START_REF] Hoppensteadt | Pattern recognition via synchronization in phase-locked loop neural networks[END_REF], as in HNNs. Conceptually, ONN inference consists of exploring the energy landscape to reach the minima that correspond to stable phase states. ONN's initial state corresponds to an initial point in energy space that reaches a local minimum after a settling time.

B. ONN as an associative memory

ONN behaves like an associative memory when oscillators are fully connected, i.e., every oscillator is connected to all others via resistors. In this case, ONN training consists of storing patterns that can be retrieved with partial input information (like a noisy input) [START_REF] Hoppensteadt | Pattern recognition via synchronization in phase-locked loop neural networks[END_REF]. Conveniently, binary phase states ∆Φ = 180 • and ∆Φ = 0 • can be represented as black and white pixels of an image, respectively (a pixel corresponds to a single oscillator). In this work, we explore pattern recognition application by simulating coupled VO 2oscillators as an associative memory (Fig. 1).

III. ONN DEVICES TO CIRCUITS A. VO 2 device

Vanadium dioxide (VO 2 ) is a transition metal oxide that undergoes a phase change at 340 K, close to room temperature: it switches from an insulating monoclinic M 1 phase to a metallic rutile R phase, with a sharp change in resistivity up to 5 orders of magnitude. Such insulator-to-metal transition (IMT) is reversible, making VO 2 suitable for switches and memory devices. The mechanism of the IMT is not yet fully understood. However, it is now believed that a strong electronelectron correlation and an electron-phonon interaction both play an important role: this suggests a Mott-Peierls transition mechanism [START_REF] Cocker | Phase diagram of the ultrafast photoinduced insulator-metal transition in vanadium dioxide[END_REF]. Moreover, the properties of the phase transition of VO 2 can be tuned, for instance, by doping with metal atoms, by introducing defects, and by induced stress at the interface of grain boundaries [START_REF] Chen | Unraveling mechanism on reducing thermal hysteresis width of VO2 by Ti doping: a joint experimental and theoretical study[END_REF].

Experimental results indicate that self-heating initiates the IMT of VO 2 in biased devices [START_REF] Zimmers | Role of thermal heating on the voltage induced insulator-metal transition in VO2[END_REF]. Electrothermal simulations at the device level assist in 1) understanding the underlying device physics and 2) switching mechanism. We have developed a dedicated technology computer-aided design (TCAD) procedure to simulate VO 2 devices. We use doping density to mimic the temperature-dependent change of resistivity inside the VO 2 channel. The match between experimental and simulated data provides a good fit (Fig. 2), confirming that self-heating prompts IMT transition.

In ONN circuits, IMT and metal-to-insulator transitions (MIT) of VO 2 devices are exploited to design relaxation oscillators [START_REF] Corti | Scaled resistively-coupled VO 2 oscillators for neuromorphic computing[END_REF]. When VO 2 device's voltage V is above a threshold V H , its current flow induces IMT switching, and the device stays in the metallic state as long as V remains larger than a lower threshold V L . When V < V L , VO 2 transitions to an insulating state (MIT). Thus, VO 2 has a hysteresis in its I -V curve (Fig. 3b) that is used in oscillatory circuits.

To embed the VO 2 device description into circuit-level simulations, we developed a tool flow TCAD-SPICE, where the role of TCAD is to assist compact modeling for circuit simulations. Specifically, VO 2 metallic and insulating resistances, associated with both voltage thresholds, serve as input to a SPICE VO 2 model [START_REF] Maffezzoni | Modeling and simulation of vanadium dioxide relaxation oscillators[END_REF].

(b) V DD1 V DD2 SW Δt Time ΔΦ in = 360°Δ t T osc VO 2 R S C P V DD1 V out1 VO 2 R S C P V DD2 V out2 R C SW (a)
Input: Delay on V DD

B. VO 2 -oscillator

Similar to Schmitt-trigger based oscillators, we use VO 2 hysteresis behavior to design compact relaxation oscillators. We bias the VO 2 device with a resistor R S in series (Fig. 3a). To obtain oscillations, the load line I L intercepts VO 2 I -V characteristic in its Negative Differential Region (NDR). In this case, the output capacitor C P charges through the VO 2 device in the metallic state and discharges through R S when the VO 2 device is in an insulating state. V L and V H VO 2 threshold voltages define the oscillation amplitude. For a given VO 2 device, R S and C P set the oscillation frequency (in this work R S =20 kΩ and C P =500 pF gives f=46.3 kHz). The oscillating output signal is the voltage across C P , and the input is the supply voltage starting time. To set an initial input phase between two oscillators, we delay the starting time of the second oscillator's supply voltage with respect to the first oscillator.

C. Two coupled-oscillators circuit

Fig. 4a shows the smallest ONN consisting of two oscillators coupled by a resistor R C . We insert switches in series with R C to decouple oscillators during initialization [START_REF] Delacour | Mapping Hebbian Learning Rules to Coupling Resistances for Oscillatory Neural Networks[END_REF]. To set an input phase state ∆Φ in between two oscillators, we delay the second oscillator's supply voltage starting time, as shown in Fig. 4b. Once all oscillators are turned on, we close the switches to couple the oscillators. After a few cycles, both oscillators settle to a stable phase state, and we can measure their phase difference. Both ∆Φ in and R C influence the final phase state outcome ∆Φ out . Fig. 5a shows ∆Φ out obtained for various R C and inputs ∆Φ in . We observe that a small R C << R 0 C induces ∆Φ out = 0 • , whereas a large R C >> R 0 C leads to ∆Φ out = 180 • , as already noticed in [START_REF] Corti | Scaled resistively-coupled VO 2 oscillators for neuromorphic computing[END_REF]. These phase states can be represented by two distinct patterns stored in ONN via R C values. Particularly, for R C = R 0 C , we equally retrieve both patterns by adjusting ∆Φ in (Fig. 5b). In other words, two coupled oscillators behave like an associative memory, where R C contains the memory, and ∆Φ in is the input to recall a stored pattern. In the next section, we demonstrate this concept to 60 coupled oscillators using the same initialization procedure and scaling coupling resistances.

IV. ONN PATTERN RECOGNITION

To showcase the associative properties of a larger VO 2 -ONN architecture, we perform circuit-level simulations of 60 fully-coupled VO 2 -oscillators. There are 60x59/2=1770 different synapses, all modeled by ideal resistors that values are determined during the ONN training [START_REF] Delacour | Mapping Hebbian Learning Rules to Coupling Resistances for Oscillatory Neural Networks[END_REF]. In hardware, synapses could be implemented by a resistive cross-bar array with programmable resistors or memristors [START_REF] Kendall | The building blocks of a brain-inspired computer[END_REF]. ξ 1 ξ 2 ξ 3 ξ 4 ξ 5 ξ 6

Fig. 6. Six training patterns ξ k with 60 black and white pixels used in this example. ξ k i = 1 if pixel i is white and ξ k i = -1 otherwise. Synaptic weights are computed using the Hebbian rule [START_REF] Hoppensteadt | Pattern recognition via synchronization in phase-locked loop neural networks[END_REF] and are then mapped to coupling resistances to perform circuit simulations.

A. Training and Inference

We study a simple pattern recognition application where six patterns ξ k are stored in ONN representing letters 'A' to 'F' (Fig. 6). To train the ONN, we use the Hebbian rule [START_REF] Hoppensteadt | Pattern recognition via synchronization in phase-locked loop neural networks[END_REF], which gives synaptic coefficients based on ξ k . We map Hebbian coefficients to coupling resistances, as described in section 3 (Fig. 5a).

The recognition operation consists of associating an input pattern to one of the training patterns. A black pixel i corresponds to ∆Φ in i = 180 • that is set by delaying V DDi by T osc /2. A white pixel corresponds to an oscillator in phase with the reference oscillator, thus without any delay. Gray pixels are initialized by delaying oscillators between 0 and T osc /2. When the input is sufficiently close to one of the training patterns, ONN retrieves it after a few oscillation cycles. Sometimes, ONN converges toward spurious patterns when the input is too noisy (Fig. 7a). In that case, ONN cannot be considered as accurate.

B. ONN performances

We add random noise to training patterns to generate a test set, where noisy pixels take values between -1 and +1. We vary the amount of noise from 0 to 50% of the number of pixels, and we report on ONN recognition accuracy as in Fig. 7b. Pattern recognition is unacceptable if a single output pixel differs from the original training pattern. If we tolerate two wrong output pixels, the recognition accuracy increases, meaning that ONN sometimes locks to spurious patterns that are close to the training patterns. Evaluating ONN accuracy with the test set, we observe that ONN has more than 60% of accuracy for test images with up to 30% of noise.

During the test set evaluation, we report an average settling time of 3.9 oscillation cycles (i.e., the required time for the ONN to reach a stable phase state), in accordance with the 4-5 cycles observed in experiments [START_REF] Corti | Coupled VO 2 Oscillators Circuit as Analog First Layer Filter in Convolutional Neural Networks[END_REF]. ONN operates fast thanks to coupled oscillators that work in parallel. We compute the average energy associated with ONN inference and report it on Table I. We notice that the total energy consumption for 60 VO 2 -oscillators is E 60 ≈ 3.9 x 60 x E /cycle , with E /cycle the energy consumed by a single oscillator during one cycle. Hence, we expect a linear energy scaling for larger ONNs. pattern 'E' with 20 randomly altered pixels is applied to the ONN and successfully identified after a few oscillations. Right: pattern 'D' with 28 randomly altered pixels is not identified as the ONN converges to an unknown phase state. b) ONN recognition accuracy with respect to the number of noisy input pixels. For each configuration, 100 simulations are performed using random test patterns. Performances decrease significantly for more than 30 % of input fuzzy pixels.

Based on this observation, we compute the ONN energy for 28x28=784 oscillators required to perform pattern recognition on the MNIST dataset.

In this work, we simulate "slow" oscillators (f=46.3 kHz) to reproduce experiments that were conducted with a large load capacitance C P =500 pF and high supply voltage V DD =2.5V. However, if VO 2 devices are scaled down with smaller load capacitances and lower supply voltage, we expect a reduced energy footprint [START_REF] Corti | Coupled VO 2 Oscillators Circuit as Analog First Layer Filter in Convolutional Neural Networks[END_REF]. Thus, we apply our ONN energy estimation for scaled VO 2 devices reported in literature [START_REF] Shukla | Ultra low power coupled oscillator arrays for computer vision applications[END_REF], [START_REF] Corti | Coupled VO 2 Oscillators Circuit as Analog First Layer Filter in Convolutional Neural Networks[END_REF].

Moreover, we compare our ONN energy prediction with an example of a state-of-the-art edge processor [START_REF]Edge tpu performance benchmarks[END_REF], which implements an HNN for pattern recognition. In HNN [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF], at least N(N-1) MAC (multiple-accumulate) operations are required to compute the HNN state. We estimate the processor energy consumption as N(N-1)E M AC and assuming 1 MAC ≈ 2 OP. Our ONN energy estimation is optimistic as we do not consider all the peripheral circuits to initialize and read-out ONN. Nevertheless, the trend shows that ONN can be a very competitive architecture for less than 800 neurons, as ONN with scaled VO 2 devices [START_REF] Corti | Coupled VO 2 Oscillators Circuit as Analog First Layer Filter in Convolutional Neural Networks[END_REF] would allow a 4000x energy reduction compared to state-of-the-art edge processor [START_REF]Edge tpu performance benchmarks[END_REF].

C. ONN on FPGA

We implement a digital ONN-on-FPGA to quantitatively assess the performance metrics of ONN architectures in realworld applications. Our ONN-on-FPGA is inspired by a hybrid analog-digital solution [START_REF] Jackson | An oscillatory neural network with programmable resistive synapses in 28 nm CMOS[END_REF], but we have implemented a fully digital architecture [START_REF] Abernot | Digital Implementation of Oscillatory Neural Network for Image Recognition Application[END_REF]. Digital neurons oscillate at 488 kHz. We use pattern recognition as a testbed as it is a typical edge AI task. For instance, one could embed low-power ONNs into edge camera systems for image post-processing, noise filtering, or object detection at the interface with sensors. Here, we perform image recognition with a 60-neurons ONN-on-FPGA using the same training set previously defined (Fig. 6).

ONN synaptic weights are computed offline via Matlab using the Hebbian learning rule [START_REF] Hoppensteadt | Pattern recognition via synchronization in phase-locked loop neural networks[END_REF]. Weights are subsequently hardcoded inside the FPGA. Fig. 8 describes the experimental procedure that has been set up. Images are streamed by a cell phone and captured by a camera. Then images are re-scaled to 60 pixels to fit the ONN size and used as input to the ONN-on-FPGA. Finally, the output image is re-scaled to be displayed on an HDMI screen once synchronization is achieved.

We evaluate ONN-on-FPGA performances on a test set composed of 30 corrupted training patterns. Only 6 test patterns could not be retrieved correctly (80% recognition accuracy). We measured the time required from ONN initialization to the end of the computation. It has resulted in an average of 13µs (7.8µs initialization and 5.2 µs settling time), demonstrating the real-time computation capability of our ONN-on-FPGA.

V. CONCLUSIONS

In this work, we presented an innovative concept for neuromorphic computing based on oscillatory neural networks implemented with beyond-CMOS devices such as vanadium dioxide (VO 2 ) to emulate neurons and resistors to implement synapses. We explored the ONN technology from devices, circuits to architecture-level implementation. To do so, we developed a dedicated TCAD procedure for VO 2 device simulation to capture its resistive switching mechanism in circuit simulations. Then, we investigated the dynamics of two VO 2 -oscillators coupled by a resistor, showing that they naturally behave like an associative memory. We also assessed the performances of larger ONNs and showed that ONN is a competitive paradigm for edge applications thanks to their energy efficiency. Finally, we reported on an ONN-on-FPGA as a proof-of-concept for real-world pattern recognition applications.

As future directions for our work, we aim to complete the entire simulation toolchain from Density Functional Theory (DFT) to TCAD to circuit simulation by designing customized compact models. This will provide precise assessments of

Fig. 2 .

 2 Fig. 2. Experimental and simulated I-V curves (up to IMT switching) of VO 2 device (thickness 200 nm, width 100 µm and length 5 µm) [22]. Rext is a 5 kΩ external resistor.

Fig. 3 .

 3 Fig. 3. a) VO 2 -oscillator circuit. b) Example of VO 2 I -V curve and evolution of the device voltage. VO 2 hysteresis behavior is used to produce oscillations by biasing the VO 2 device in its Negative Differential Region (NDR).

Fig. 4 .

 4 Fig. 4. a) Two identical VO 2 -oscillators coupled by a resistor R C that can be disconnected via switches. b) The starting-time of the second oscillator is delayed via V DD2 to set an initial phase, and the switches are closed to compute.

Fig. 5 .

 5 Fig. 5. a) ∆Φ out extracted after convergence for every set of parameters (R C , ∆Φ in ). Positive weights are mapped to small coupling resistances, whereas negative weights are implemented with large R C . At R C = R 0 C , two distinct states are memorized, corresponding to the right-hand side patterns. b) Output voltages of the two oscillators for R C = R 0 C . Left: ∆Φ in = 80 • and the oscillators converge to a stable in-phase state. Right: ∆Φ in = 100 • and the oscillators converge to a stable out-of-phase state.

Fig. 7 .

 7 Fig.7. a) Left: pattern 'E' with 20 randomly altered pixels is applied to the ONN and successfully identified after a few oscillations. Right: pattern 'D' with 28 randomly altered pixels is not identified as the ONN converges to an unknown phase state. b) ONN recognition accuracy with respect to the number of noisy input pixels. For each configuration, 100 simulations are performed using random test patterns. Performances decrease significantly for more than 30 % of input fuzzy pixels.

Fig. 8. Workflow of image recognition procedure by digital ONN-on-FPGA; close to each image are indicated the pixel sizes.

ONN power, performance, area, and reliability. Finally, we are exploring the real-time capability of ONN-on-FPGA for other edge AI applications such as autonomous motion-control systems for robotics.