Syed Tomas Menard
email: tomas.menard@unicaen.fr

Ali Ajwad

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Michael Defoort
email: michael.defoort@uphf.fr

Tomas Ménard

Syed Ali Ajwad
email: syed.ali.ajwad@univ-poitiers.fr

Leader-following

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

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The study of Multi-Agent Systems (MAS) has been considered by many researchers during the last decades, due to its important practical applications, such as formation of UAV, attitude synchronization of spacecraft or distributed sensor networks [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF]. MAS are usually characterized by a topology network which reects the possible ways of communication among agents. A fundamental problem for MAS is to design protocols such that all the agents in the network reach a common value. This problem can be subdivided into two categories, the leaderless consensus and the leader-following consensus. In the leaderless consensus problem, the nal common position of the agents cannot be selected. Then, it might be useful to consider a real or virtual leader whose prescribed trajectory has to be followed by all the agents. Many results have been obtained for MAS whose dynamics are linear, see for instance [START_REF] Hong | Tracking control for multiagent consensus with an active leader and variable topology[END_REF]. However, in many practical cases, MAS are governed by more complex dynamics, namely nonlinear dynamics. These nonlinear dynamics usually cannot be neglected in order to obtain more accurate control procedures and objectives. Consensus protocols using the full state information have been considered in [5] for a xed topology or in [START_REF] Wen | Consensus tracking of multi-agent systems with lipschitz-type node dynamics and switching topologies[END_REF] for time-varying topology. When the state of the agents is only partially available, that is when only the measured output can be used, it is then necessary to use observers in order to reconstruct the state. Such a strategy has been used for example in [START_REF] Hu | Consensus of nonlinear multi-agent systems with observer-based protocols[END_REF] for a general class of nonlinear MAS. In the aforementioned nonlinear protocols, the considered signals are assumed to be available continuously in real time. But in most applications, it is more desirable or sometimes only possible to transmit the measurements in a discrete way. This may be due to technical constraints or for energy saving [START_REF] Ploennigs | Comparative study of energy-ecient sampling approaches for wireless control networks[END_REF]. It is then important to adapt the consensus protocol in order to deal with the sampled signals. A rst idea has been to consider time-triggered sampling. Several protocols have been proposed when the state of the agents can be fully measured. Impulsive control for some specic kinds of systems has been considered in [START_REF] He | Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control[END_REF]. The delay input approach has been used in [START_REF] Wen | Consensus of multiagent systems with nonlinear dynamics and sampled-data information: a delayed-input approach[END_REF][START_REF] Wen | H∞ pinning synchronization of directed networks with aperiodic sampleddata communications[END_REF] for deterministic sampling period and in [START_REF] Shen | Sampled-data synchronization control of dynamical networks with stochastic sampling[END_REF][START_REF] Wan | Distributed nodeto-node consensus of multi-agent systems with stochastic sampling[END_REF] for stochastic sampling. When only the output is available at discrete instants, then an observer must be designed. A distributed observer protocol with a zero order hold input control has been proposed in [START_REF] Wan | Distributed observer-based stabilization of nonlinear multi-agent systems with sampled-data control[END_REF]. In this work, the sampling periods have to be synchronized between the agents. Another approach, which allows aperiodic and asynchronous sampling periods, is event-triggering based consensus protocol. General linear systems have been considered in [START_REF] Zhao | Observer-based adaptive sampled-data event-triggered distributed control for multi-agent systems[END_REF]. Some specic nonlinear classes of systems have also been considered in the literature. MAS with Lure's nonlinear dynamics have been investigated in [START_REF] Huang | Eventtriggered consensus tracking of multi-agent systems with lur'e nonlinear dynamics[END_REF] and rst order nonlinear systems in [START_REF] Xie | Event-triggered average consensus for multi-agent systems with nonlinear dynamics and switching topology[END_REF][START_REF] Liu | Distributed eventtriggered xed-time consensus for leader-follower multiagent systems with nonlinear dynamics and uncertain disturbances[END_REF]. Other classes of nonlinear systems have been treated in [START_REF] Liu | Cooperative global robust output regulation for a class of nonlinear multi-agent systems by distributed event-triggered control[END_REF]. Though providing interesting results, event-triggered schemes involve a more complicated set-up and additional parameters to tune. Indeed, a threshold function has to be considered which dictates the sampling instants for the data transmission. Furthermore, one has to be careful about the Zeno phenomenon which can occur for some schemes [START_REF] Ding | An overview of recent advances in event-triggered consensus of multiagent systems[END_REF]. Most works with discrete signal transmission hold the control input constant between sampling instants. One takes advantage here, of the fact that time-varying control input can be considered. Indeed, only the transmitted signals have to be sampled, not the input. Continuous-discrete time observers, which reconstruct the state in continuous time from discrete-time measurements, have been greatly developed these last years, as in [4,[START_REF] Hernández-González | A cascade observer for a class of MIMO non uniformly observable systems with delayed sampled outputs[END_REF] for dierent classes of nonlinear systems, and are then used here to tackle the problem of leaderfollowing consensus where only discrete-time outputs are transmitted through the network. It should be noted that these works only consider the observer design, the convergence is obtained by assuming that the input belongs to a bounded set and then cannot be applied directly to the problem considered here. This idea has already been exploited in [START_REF] Phillips | A hybrid consensus protocol for pointwise exponential stability with intermittent information[END_REF][START_REF] Phillips | Robust distributed synchronization of networked linear systems with intermittent information[END_REF] for linear systems, following a hybrid approach, where sucient conditions for convergence are obtained, based on LMIs. A high-gain approach has been followed in [START_REF] Ménard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF][START_REF] Ajwad | Observer based leader-following consensus of second-order multi-agent systems with nonuniform[END_REF] for the leaderless and leader-following consensus of MAS with double integrator dynamics. The control part of this high-gain approach is mainly based on [2] where only state feedback is considered. These works are then extended here to the leader-following consensus for a class of systems with nonlinear and uncertain dynamics. The main novelty of the paper is the design of a leader-following consensus protocol for a multi agent system, whose agent dynamics belong to a class of uniformly observable multi-output nonlinear systems, where only a part of the state is measured, and each agent's output is transmitted at some discrete instants to its neighbors independently of the other agents. Several features of the proposed approach have to be emphasized. Firstly, only the sampled outputs have to be transmitted through the network, it is not necessary to transmit the inputs. Secondly, the data sent by the agents are not needed to be synchronized, each agent can send its measurements independently from its neighbors, provided that the maximum allowable sampling period is bounded. This allows to reduce the overall bandwidth of the network. Thirdly, the proposed protocol has only three tuning parameters, namely c, λ, θ, where c is the coupling force, λ > 0 represents the speed of convergence of the control part and θ > 0 represents the speed of convergence of the observer part. Then, the tuning of the proposed scheme is relatively simple and can be adapted easily by the practitioner since the eect of the modication of each parameter has a direct physical meaning. Fourth, the class of nonlinear systems considered here is challenging since it is quite large and it has not yet been considered in the literature for aperiodic and asynchronous sampling periods. A new protocol is thus proposed here to tackle this problem.

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

A long version of the note, which contains complete proofs can be found in [START_REF] Ménard | Leader-following consensus for multiagent systems with nonlinear uncertain dynamics and asynchronously sampled outputs[END_REF].

Preliminaries

In this paper, the following notations will be used. The symbol = means equal by denition. The set of n × n real matrices is denoted by R n×n . The transpose for real matrices is represented by the superscript T . I n is the identity matrix of dimension n, 0 m×n is the zero matrix of dimension m × n and 0 n = 0 n×n . The Kronecker product of matrices A and B is A ⊗ B. For a symmetric matrix M , ρ max (M ) and ρ min (M ) respectively denotes the maximum and minimum eigenvalue of M . The notation diag(w 1 , . . . , w q ), with w i ∈ R m×m , i = 1, . . . , q, q, m ∈ N, is used for the diagonal by block matrix with w 1 , . . . , w q on its diagonal. The positive deniteness of a matrix M is denoted M > 0. The vector of dimension N ∈ N with all entries equal to 1 is denoted 1 N .

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

A = (a ij ) ∈ R N ×N is dened by a ii = 0 and a ij = 1 if (j, i) ∈ E and a ij = 0 otherwise. The Laplacian matrix L ∈ R N ×N is dened as L ii = j =i a ij , L ij = -a ij for i = j.
Denition 1 [3, Def. 1.2 and Th. 2.3-G20 p.133-134] A non singular real matrix Q ∈ Z n×n is called an Mmatrix if all its diagonal elements are positive, all of its o-diagonal elements are non positive, and each of its eigenvalues has positive real part.

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

tor ω = ω 1 . . . ω n T , such that ΩQ + Q T Ω > 0,
where

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

d dt n i=1 γiv 2 i (t) ≤ n i=1 -aiv 2 i (t) + bi t t-δ v 2 i (s)ds + k, (1)
for all t ≥ 0, where

γ i > 0, a i > 0, b i ≥ 0, i = 1, . . . , n, k ≥ 0 and δ < min ( √ 2 -1) 2 min i=1,n ai bi , 1 √ 2 min i=1,n γi ai . (2)
Then, the following inequality holds true

n i=1 γ i v 2 i (t) ≤ ςe -ϑt + k ϑ , (3)
with ϑ = 1 2 min i=1,...,n ai γi

and ς = n i=1 γ i v 2 i (0).
The proof of Lemma 2 can be found in [START_REF] Ménard | Leader-following consensus for multiagent systems with nonlinear uncertain dynamics and asynchronously sampled outputs[END_REF].

Lemma 3 [1, Lemma 4] (i) Let M ∈ R n×n be a symmetric positive denite matrix. Then, one has

x T M y ≤ √ x T M x y T M y for all x, y ∈ R n . (ii) Let M ∈ R n×n be a symmetric matrix. Then, one has ρ min (M )x T x ≤ x T M x ≤ ρ max (M )x T x for all x ∈ R n . (iii) One has n i=1 √ α i ≤ √ n n i=1 α i for all α i ≥ 0. (iv) Let A ∈ R n×n
be a symmetric positive denite matrix and B ∈ R m×m be a symmetric semi-denite matrix, then the following inequality holds

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

Dynamical model of the agents

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

ẋ(1) i (t) = x (2) i (t) + ϕ 1 t, x (1)
i (t) + ε (1) i (t),
(5) . . .

ẋ(q-1) i (t) = x (q) i (t) + ϕ q-1 t, x (1)
i (t), . . . , x (q-1) i (t) + ε (q-1) i (t), ẋ(q) i (t) = u i (t) + ϕ q t, x (1)
i (t), . . . , x (q) i (t) + ε (q)
i (t),

y i = x (1) i + w i , where x (k) i ∈ R m , k = 1, . . . , q, is the state, u i ∈ R m is the input, y i ∈ R m the output, ε (k) i : R → R m , k = 1, . . . , q, the dynamics uncertainties, w i : R → R m the noise, ϕ k : R × R km → R m the nonlinearities and q, m ∈ N.
System (5) is then made up of q blocks, each one of size m.

Let us denote

x i = x (1) i T • • • x (q) i T T ∈ R n
the state of the i-th agent, with n = qm, and

ε i = ε (1) i T . . . ε (q) i T T
. System (5) can be written in the following compact form

ẋi (t) = Ax i (t) + ϕ(t, x i (t)) + Bu i (t) + ε i (t), y i = Cx i + w i , (6)
with A ∈ R n×n , B ∈ R n×m and C ∈ R m×n the corre- sponding matrices and ϕ = ϕ T 1 . . . ϕ T q T .
Remark 1 The class of systems considered here is closely related to the class of uniformly observable systems, which has been introduced in [START_REF] Gauthier | Observability for any u(t) of a class of nonlinear systems[END_REF] for Single Input Single Output (SISO) systems. Indeed, the system is composed of an integrator chain and the nonlinear part has a triangular structure. While there exists canonical form for SISO systems, no such canonical form exists for Multi Input Multi Output systems. Nevertheless, several classes of uniformly observable systems have been considered in the literature such as in [START_REF] Farza | Observer design for a class of mimo nonlinear systems[END_REF]9]. The class of systems considered here is the same as in [START_REF] Farza | Observer design for a class of mimo nonlinear systems[END_REF] (where more details are given about possible required change of coordinates) except that the input is supposed to act linearly and without zero dynamics. Assumption 1 The function ϕ is globally Lipschitz in x uniformly with respect to t, that is there exists

L ϕ > 0 such that ϕ(t, x 1 ) -ϕ(t, x 2 ) ≤ L ϕ x 1 -x 2 , (7)
for all t ∈ R and

x 1 , x 2 ∈ R n .
The aim is to design a protocol such that all the agents converge toward a leader, denoted agent 0, whose dynamics are given by

ẋ0 (t) = Ax 0 (t) + ϕ(t, x 0 (t)) + ε 0 (t), (8)
y 0 = Cx 0 + w 0 .
Remark 2 It should be noted that the uncertainties on the dynamics of the leader can also represent an unknown input. Indeed, given the structure of the dynamics (see equation ( 5)), the uncertainty on the dynamics of x (q) 0 can be decomposed as an unknown leader input u 0 on one part and some perturbations on the dynamics on the other part.

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

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

ε (k) i (t) ≤ δ k ε , ∀t ≥ 0, i = 0, . . . , N, k = 1, . . . , q, ( 9
)
w i (t) ≤ δ w , ∀t ≥ 0, i = 0, . . . , q. (10)
Due to the uncertainties, the agents cannot converge exactly toward the leader but, instead, in a ball around the leader. The problem to be solved here is dened more precisely below.

Denition 2 The leader-following consensus problem is said to be exponentially practically solved if there exist α, β > 0 and γ ≥ 0 such that x i (t)-x 0 (t) ≤ αe -βt +γ, for all i = 1, . . . , N .

Communication constraints

The communication topology between followers i = 1, . . . , N is denoted G and its adjacency and Laplacian matrices A = (a ij ) and L respectively. The output of the leader is supposed to be transmitted only at some agents. More precisely, one denes

D = diag(d 1 , . . . , d N )
, where d i = 1 if agent i has access to the output of the leader and d i = 0 otherwise. One denes the general index ν ij for i = 1, . . . , N, j = 0, . . . , N as ν ij = 1 if agent i receives the output of agent j and 0 otherwise. This means that if ν ij = 1, then the output of agent j is transmitted to agent i at time instants t i,j k k∈N

. These sampling instants are supposed to verify

t i,j 0 < t i,j 1 < • • • < t i,j k < . . . and τm < t i,j k+1 -t i,j k < τM , (11)
for all k ∈ N and τ m , τ M > 0. Note that the lower bound τ m is just considered in order to explicitly avoid the Zeno phenomena. It is not a restrictive bound since it can be taken as small as desired.

Example 1 Consider a MAS composed of a leader (denoted 0) and two agents (denoted 1 and 2) such as depicted in Figure 1. Agent 1 receives the transmitted output y 0 of the leader at time instants t 1,0 k for k ∈ N. This is used to reconstruct the state of the leader in continuous time. Agent 1 also uses its own output at time instants t 1,1 k for k ∈ N, to reconstruct its own state in continuous time. Similarly, Agent 2 receives the output of agent 1 at time instants

t 2,1
k and uses its own output at time instants t 2,2 k . Note that the time sequences t 1,0 k k∈N

, t 1,1 k k∈N , t 2,1 k k∈N , t 2,2
k k∈N can be chosen freely, and in particular independently from each other, as long as they verify [START_REF] He | Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control[END_REF]. One denotes G the digraph corresponding to all the agents i = 1, . . . , N together with the leader. The Laplacian matrix L of G is given by Lemma 4 [26, Lemma 9] The matrix H is a nonsingular M-matrix if and only if the communication topology G has a directed spanning tree.

0 1 2 y0 t 1,0 k y1 t 2,1 k y1 t 1,1 k y2 t 2,2 k
L = 0 0 1×N -d L + D = 0 0 1×N -d H , (12)
Assumption 3 The communication topology G between the agents and the leader contains a directed spanning tree.

Remark 3 If Assumption 3 holds true, then according to Lemma 4 and Lemma 1, there exists a positive vector ω = ω 1 . . . ω N such that ΩH + H T Ω > 0 where Ω = diag(ω 1 , . . . , ω N ). Then, in the rest of the paper, one denotes

= ρ min ΩH + H T Ω , ( 13
)
ω min = min(ω 1 , . . . , ω N ), (14)
ω max = max(ω 1 , . . . , ω N ). [START_REF] Huang | Eventtriggered consensus tracking of multi-agent systems with lur'e nonlinear dynamics[END_REF] 4 Main result

Consensus protocol

The proposed protocol is given, for i = 1, . . . , N , by

u i (t) = d i cK c Γ λ (x i,0 (t) -xi,i (t)) (16)
+ cK c Γ λ N j=1 a ij (x i,j (t) -xi,i (t)), ∀t ≥ 0,
where xi,j is the estimate of x j by agent i and is given, [START_REF] Liu | Cooperative global robust output regulation for a class of nonlinear multi-agent systems by distributed event-triggered control[END_REF] with

for t ∈ t i,j k , t i,j k+1 , k ∈ N, by ẋi,j (t) = Ax i,j (t) + ϕ(t, xi,j (t)) -θ∆ -1 θ K o z i,j (t),
z i,j (t) = e -θK o 1 (t-t i,j k ) C xi,j t i,j k -y j t i,j k , (18)
where c, λ, θ > 0 are the tuning parameters, and

Γ λ = diag λ q I m , λ q-1 I m , . . . , λI m , (19)
∆ θ = diag I m , 1 θ I m , . . . , 1 θ q-1 I m , (20)
K o = P -1 C T , K c = B T Q, ( 21
)
where q is the number of blocks of System (5), P, Q ∈ R n×n are the symmetric positive denite solutions of the following matrix equalities

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

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

K o = K o 1 I m . . . K o q I m T , (24)
K c = K c 1 I m . . . K c q I m , (25)
where

K o 1 , . . . , K o q , K c 1 , . . . , K c
q ∈ R are equal to

K o i = q i , K c i = q q -i + 1 , i = 1, . . . , q, (26)
and n k are the binomial coecients.

Remark 5 The term z i,j (t), dened in ( 18) corresponds to a prediction of the output error (C xi,j (t) -y j (t)) on each interval t i,j k , t i,j k+1 (see [START_REF] Farza | Continuous-discrete time observers for a class of mimo nonlinear systems[END_REF] section III.B for more details).

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

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

Then, System (6) is characterized by

K o = 2 1 , K c = 1 2 , Γ λ = λ 2 0 0 λ , ∆ θ = 1 0 0 1 θ .
Given the topology of the considered MAS:

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

ẋ1,0 (t) = Ax 1,0 (t) + 0 ϕ 2 (x 1,0 (t)) -θ∆ -1 θ K o e -2θ(t-t 1,0 k ) x(1) 1,0 t 1,0 k -y 0 t 1,0 k , for t ∈ t 1,0 k , t 1,0 k+1 , ẋ1,1 (t) = Ax 1,1 (t) + 0 ϕ 2 (x 1,1 (t)) -θ∆ -1 θ K o e -2θ(t-t 1,1 k ) x(1) 1,1 t 1,1 k -y 1 t 1,1 k , for t ∈ t 1,1 k , t 1,1 k+1 ,
where x1,0 and x1,1 are the estimates of x 0 and x 1 respectively. The input of agent 1 is then given by:

u 1 (t) = 2cλ 2 x(1) 1,0 (t) - x(1) 1,1 (t) +cλ x(2) 1,0 (t) - x(2) 1,1 (t) .
• Agent 2 has to reconstruct the state of agent 1 and its own state. Similarly as for agent 1, agent 2 has to run two observers: one whose state x2,1 is an estimate of x 1 and one whose state x2,2 is an estimate of x 2 .The input of agent 2 is then given by:

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

Convergence result

The convergence of the proposed consensus protocol is now analyzed.

Theorem 1 Consider the MAS (5)-( 8) subject to Assumptions 1, 2 and 3 and the consensus protocol ( 16)-( 17)- [START_REF] Ménard | Leader-following consensus for multiagent systems with nonlinear uncertain dynamics and asynchronously sampled outputs[END_REF]. If the tuning parameters λ, θ, c ≥ 1 are chosen such that

c ≥ c * = max ωmax , 1 , λ ≥ λ * = 24Lϕ √ n max   ρ Qω M ρ Qω m , ρ P M ρ P m   , θ ≥ λc 2 ξ * , with ξ * = 36 K c 2 (N +1) 3 h 2 max max √ ρ P M √ ρ P m , ρ P M ρ Qω m , ρ Qω M ρ P m , ρ P m = ρ min (P ), ρ P M = ρ max (P ), ρ Qω m = ω min ρ min (Q), ρ Qω M = ω max ρ max (Q), h max = max ij |H ij |, , ω min , ω max
respectively dened by ( 13)-( 14)- [START_REF] Huang | Eventtriggered consensus tracking of multi-agent systems with lur'e nonlinear dynamics[END_REF], and the upper bound on the sampling periods τ M veries

τ M < σ * c(θ + L ϕ ) , (27)
with

σ * = √ 2-1 8 min( √ ωmin, √ ρ P m ) K o (N +1) 3 2 hmax √ ρ P M
, then the consensus error x i -x 0 veries

x i -x 0 ≤χ 1 θ q-1 e -λ 8 t + χ 2 λ -1 θ q δ w + τ M δ 1 ε + χ 3 λ -1 q k=1 θ q-k δ k ε , (28)
for i = 1, . . . , N , where χ 1 , χ 2 , χ 3 ≥ 0 are independent of the tuning parameters λ, θ, c and given by

χ1 = ρ Qω M ρ Qω m N i=1 xi(0) -x0(0) + ρ P M ρ Qω m N i=1 N j=0 xi,j(0) -xj(0) , χ2 = 8 ρ P M (N + 1) K o ρ Qω m , χ3 = 8 2N ρ Qω M + (N + 1) ρ P M ρ Qω m .
Remark 6 Equation [START_REF] Wan | Distributed nodeto-node consensus of multi-agent systems with stochastic sampling[END_REF] states that even if the exponential practical consensus is reached, not all the uncertainties eect on the tracking error can be lowered through the tuning of λ and θ. Indeed, it is only possible for the uncertainties appearing on the last block of system (5) (that is only ε (q) i are non zero) and if q ≥ 2. This is done by increasing λ and θ, since the corresponding term χ 3 λ -1 δ q ε goes to zero as λ increases. In particular, if the leader has an unknown but bounded non zero input, then this input can be seen as a dynamic uncertainty on the last block dynamics x (q) 0 . Therefore, the tracking error can be set as low as desired by increasing λ and θ.

Remark 7 Theorem 1 only provides sucient conditions for the proposed consensus scheme ( 16)- [START_REF] Liu | Cooperative global robust output regulation for a class of nonlinear multi-agent systems by distributed event-triggered control[END_REF]. Indeed, the consensus may be obtained even if the bounds given by Theorem 1 are not respected. Conservativeness is a general drawback when considering general classes of nonlinear systems with a Lyapunov approach. Nevertheless, the convergence analysis gives some useful hints for the tuning of the control parameters. In fact, the bounds σ * , c * , λ * , ξ * only depend on the structure of the system (number of blocks q and size of each block m) and the topology of the network G. Given the inequalities in Theorem 1, the coupling force should be tuned rst as for a classical consensus protocol. Then, λ (representing the speed of convergence of the control part) should be chosen high enough to dominate the nonlinear Lipschitz term. The parameter θ of the observer part should be higher than the parameter of the control part λ. This is due to the fact that the input is not transmitted through the network, only the output is transmitted. Finally, the maximum bound on the sampling periods will have to be chosen small if the Lipschitz constant or if θ takes high values. This corresponds to a kind of Shannon condition: if the system is fast, the sampling periods have to be small. Furthermore, given the bounds on the consensus error given by equation [START_REF] Wan | Distributed nodeto-node consensus of multi-agent systems with stochastic sampling[END_REF], it can be seen that increasing λ and θ will decrease the eect of some uncertainties (those on the dynamics of x (q) i ) but it will increase the eect of the noise, then a trade-o has to be considered between lowering the eect of some uncertainties and amplifying the noise.

A full version of the proof with detailed computations can be found in [START_REF] Ménard | Leader-following consensus for multiagent systems with nonlinear uncertain dynamics and asynchronously sampled outputs[END_REF].

Proof. The proof of Theorem 1 is split into three steps. First, new coordinates are considered in Step 1. Then,in Step 2, some candidate Lyapunov functions are dened and over-valuations of their derivatives are obtained. Finally, it is shown in Step 3, that, if the dierent inequalities of Theorem 1 are veried, then Lemma 2 can be applied and the leader-following consensus problem is solved.

Step 1. Let us dene e i = Γ λ (x i -x 0 ) and ēi,j = ∆ θ (x i,jx j ). Then, using the equalities

Γ λ AΓ -1 λ = λA, Γ λ B = λB, ∆ θ A∆ -1 θ = θA, ∆ θ B = 1 θ q-1 B, C∆ -1 θ = C
and the notation φ(t, x, y) = ϕ(t, x) -ϕ(t, y), one gets

ėi = λAe i + Γ λ ( φ(t, x i , x 0 ) + ε i -ε 0 ) + λBu i , (29) ėi,j = θ(A -K o C)ē i,j + ∆ θ φ(t, xi,j , x j ) (30) - 1 θ q-1 Bu j -θK o (z i,j -C ēi,j ) -∆ θ ε j ,
for i = 1, . . . , N and j = 0, . . . , N , where z i,j is dened by [START_REF] Ménard | Leader-following consensus for multiagent systems with nonlinear uncertain dynamics and asynchronously sampled outputs[END_REF] and the inputs are given by u 0 = 0 m×1 and

u j = cK c d j Γ λ ∆ -1 θ ēj,0 - N k=1 H jk (e k + Γ λ ∆ -1 θ ēj,k ) , for j = 1, . . . , N . Further denoting η c = e T 1 . . . e T N T
, the dynamics of the e i can be written in compact form as follows

ηc = λ(I N ⊗ A)η c -cλ[H ⊗ (BK c )]η c + Φ λ + Ψ λ,ε + Θ, with Φ λ =      Γ λ φ(t, x 1 , x 0 ) . . . Γ λ φ(t, x N , x 0 )      , Ψ λ,ε =      Γ λ (ε 1 -ε 0 ) . . . Γ λ (ε N -ε 0 )      , and Θ =      cλBK c Γ λ ∆ -1 θ d 1 ē1,0 - N k=1 H 1k ē1,k . . . cλBK c Γ λ ∆ -1 θ d N ēN,0 - N k=1 H N k ēN,k      .
Step 2. One now introduces the candidate Lyapunov functions, one that depends on the control error η c and one that depends on the observer errors ēi,j .

One thus denes rst

Vc (η c ) = (η c ) T [Ω ⊗ Q]η c
, where Ω is dened in Remark 3. Then, for the observer error part, one considers

Vo (η o ) = N i=1 N j=0 ν ij V o (ē i,j ), with V o (ē i,j ) = ēT
i,j P ēi,j , where the vector η o contains all the ēi,j such that

ν ij = 1. It can be shown that, if c ≥ c * = max{ω max / , 1}, then Vc ≤ -λ Vc + 2k 1 Vc + 2ck 2 λ Γ λ ∆ -1 θ Vc Vo + 2k 3 Vc q k=1 λ q-k+1 δ k ε , with k 1 = L ϕ √ n ρ Qω M ρ Qω m , k 2 = (N +1) K c h max ρ Qω M ρmin(P ) , k 3 = 2N ρ Qω M ,

and

Vo ≤ -θ Vo + 2k4 Vo + 2k5 Vo t t-τ M Vo(s)ds + 2k6 Vo t t-τ M Vc(s)ds + 2ck7 θ q-1 Vo Vc + 2ck8 Γ λ ∆ -1 θ θ q-1 Vo + 2k9 Vo q k=1 δ k ε θ k-1 + 2θk10 Vo δw + τM δ 1 ε , with k 4 = L ϕ √ n ρmax(P ) ρmin(P ) , k 5 = θ(N + 1) K o (L ϕ + θ) ρmax(P ) ρmin(P ) , if q ≥ 2, k 5 = θ(N + 1) K o ρmax(P ) ρmin(P ) c√ N + 1h max λ + L ϕ , if q = 1, k 6 = 0, if q ≥ 2, k 6 = cθ(N +1) 3 2 hmax √ ωmin , if q = 1, k 7 = K c (N + 1) 3 2 h max ρmax(P ) ρ Qω m , k 8 = K c (N + 1) 3 2 h max ρmax(P )
ρmin(P ) , k 9 = ρ max (P )(N + 1) and k 10 = ρ max (P ) K o (N + 1).

Step 3. Since c ≥ c * ≥ 1, if the inequalities λ ≥ λ * and

ξ ≥ c2 ξ * hold true, with λ * = 24L ϕ √ n max
we obtain

d dt ξ 3 2 Vc + θ q Vo ≤ - ξ 3 2 λ 4 Vc - θ q+1 4 Vo + k 5 θ q t t-τ M Vo ds + k 6 θ q t t-τ M Vc ds + k 3 q k=1 ξ 3 2 λ q-k+1 δ k ε + k 9 q k=1 θ q-k+1 δ k ε + θ q+1 k 10 δ w + τ M δ 1 ε . (31)
since ξ * ≥ max (6k 2 ) 2 , (5k 7 ) 2 , 24k 8 and λ * ≥ max {20k 1 , 24k 4 }. If τ M veries inequality [START_REF] Wan | Distributed observer-based stabilization of nonlinear multi-agent systems with sampled-data control[END_REF], then one can apply Lemma 2 and inequality ( 28) is obtained by using the following inequalities

λ ρ Qω m x i -x 0 ≤ Vc (η c ) ≤ λ q ρ Qω M N i=1 x i -x 0 , Vo (η o ) ≤ ρ max (P ) N i=1 N j=0
xi,j .

Example

In this section, we present a numerical example, consisting of systems whose dynamics are the same as a Chua's oscillator, in order to illustrate the performances of the proposed protocol. Indeed, as in [START_REF] Song | Second-order leader-following consensus of nonlinear multi-agent systems via pinning control[END_REF], let us consider the MAS where each agent's dynamics are given by

       ẋ(1) i (t) = x (2) i (t), ẋ(2) i (t) = f t, x (1)
i (t), x (2)
i (t) + u i (t) + ε (2) i (t), y i = x (1) i + w i , i = 1, . . . , 10, (32)
where x (1)

i = x (1,1) i x (1,2) i x (1,3) i T ∈ R 3 , x (2) i = x (2,1) i x (2,2) i x (2,3) i T ∈ R 3
are the position and velocity of agent i, respectively, u i , y i , ε (2) i , w i ∈ R 3 are the input, output, uncertainty and noise of each agent, and the nonlinear function f is given by

f t, x (1) i , x (2) i =     α x (2,2) i -x (2,1) i -h x (2,1) i x (2,1) i -x (2,2) i + x (2,3) i -βx (2,2) i -γx (2,3) i -β sin ωx (1,1) i     ,
where α = 10, β = 19.53, γ = 0.1636, = 0.2, ω = 0.5 and h is a piece-wise linear function given by The communication graph of the MAS System (32) is described in Figure 2 and one assumes that only agents 3 and 5 receive the measured position of the leader which is referenced as agent 0. The leader-following consensus protocol ( 16)-( 17) has been implemented in Matlab for minimum and maximum bound on the sampling periods equal to τ m = 0.02s and τ M = 0.04s respectively. The sampling periods have been set following a uniform distribution on [τ m , τ M ] independently for each edge.

The rst sampling periods corresponding to the transmission of the output of agent 2 to agent 1 are reported in Figure 3. The tuning parameters c, θ, λ have been chosen by trial and error and taken equal as c = 1, θ = 20 and λ = 2.

A rst simulation has been conducted with no uncertainty on the dynamics and no noise on the outputs, that is ε (2) i = 0 and w i = 0. The position of the dierent agents are reported in Figure 5a)c)e). Furthermore, the estimation error of agent 2 by agent 1 is depicted in Figure 6a)c). The position tracking mean error

1 N N i=1 x (1) i (t) -x (1
) 0 (t) is reported in Figure 7a). As expected by the theory, both the tracking errors and the observation errors go to zero exponentially. It is worth to be noted that only the outputs, namely the position of the agents, are transmitted through the network and according to the topology described in Figure 2. The velocities and inputs are not transmitted and then unknown by neighbors agents. Furthermore, each agent transmits its output at time instants independently from its neighbors. Another simulation has been conducted, in the same conditions but with a non zero uncertainty only on the leader's dynamics, corresponding to an unknown leader input (that is ε (2) 0 ≡ 0 and ε (2) i ≡ 0 for i = 1, . . . , 10) and noise on the transmitted outputs. More precisely, the uncertainty on the leader is given T and the noise on the outputs of the agents and the leader w i are centered white noise with variance equal to 0.1. The second component of the noisy and non noisy outputs of the leader are reported in Figure 4. The positions of the agents are reported in Figure 5b)d)f). The estimation error of agent 2 by agent 1 is reported in Figure 6b)d). The position tracking mean error 1

N N i=1 x (1) i (t) -x (1) 0 (t)
is reported in Figure 7b). the position tracking error does not converge exactly to zero, the eect of the uncertainties has been lowered by taking θ and λ suciently high. It should be noted that in the case of noisy measurements, very high values of θ will lead to an amplication of the noise in the reconstructed state. Thus, a trade-o on the value of θ and λ has to be done. Indeed, suciently high values of these parameters have to be considered to attenuate the eect of the uncertainties but not too high such that the noise is not amplied too much. Nevertheless, despite uncertainties and noise, the proposed leader following consensus protocol still performs well as illustrated in the simulation.

Conclusion

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

(1) i -x

(1) 0 .

Fig. 1 .

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

where d = d 1

 1 . . . d N T . One needs the following result.

- 1

 1 with parameters a = -1.4325 and b = -0.7831.

Fig. 2 .

 2 Fig. 2. Communication graph of the MAS.

Fig. 3 .

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

Fig. 4 .

 4 Fig. 4. Second component of y0 with noise (red) and without noise (blue) by ε (2) 0 (t) = cos(t) cos(2t) cos(3t)

10 Fig. 5 .

 105 Fig. 5. Positions of the agents for τm = 20ms, τM = 40ms, θ = 20 and λ = 2.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Estimation error of the state of agent 2 by agent 1. sampled position data. Journal of the Franklin Institute, 356(16):1003110057, 2019. [2] A. Bédoui, M. Farza, M. MSaad, and M. Ksouri. Robust nonlinear controllers for bioprocesses. IFAC Proceedings Volumes, 41(2):1554115546, 2008. [3] A. Berman and R.J. Plemmons. Nonnegative matrices in the mathematical sciences, volume 9. SIAM, 1994.[4] I. Bouraoui, M. Farza, T. Ménard, R.B. Abdennour, M. MSaad, and H. Mosrati. Observer design for a class of uncertain nonlinear systems with sampled outputs application to the estimation of kinetic rates in bioreactors.Automatica, 55:7887, 2015.[5] M. Defoort, A. Polyakov, G. Demesure, M. Djemai, and

Preprint submitted to Automatica3 July 2020

The nonlinear function ϕ is supposed to verify the following assumption.


Alexey N Bilyachenko

Mikhail M Levitsky

Alexey I Yalymov

Alexander A Korlyukov

Viktor N Khrustalev

Anna V Vologzhanina

Lidia S Shul'pina

Nikolays I Konnikov

Alexander E Trigub

V Dorovatovskii

Xavier Bantreil

Frødøric Lamaty

Jørôme Long

Joulia Larionova

Igor E Golub

Elena S Shubina

Georgiy B Shul'pin

Cage-likeF e,Na-Germsesquioxanes:Structure,M agnetism,and Catalytic Activity

Abstract: As eries of four unprecedented heterometallic metallagermsesquioxanes were synthesized. Their cage-like architectures have au nique type of molecular topology consisting of the hexairon oxo{ Fe 6 O 19 }c ore surrounded in at riangular manner by three cyclic germoxanolates [PhGe-(O)O] 5 .T his structural organization induces antiferromagnetic interactions between the Fe III ions through the oxygen atoms.E valuated for this first time in catalysis,t hese compounds showed ah igh catalytic activity in the oxidation of alkanes and the oxidative formation of benzamides from alcohols.

The design of metal-based polycyclicc age-like compounds has attracted ag reat deal of attention from researchers worldwide over several decades owing to their structural diversity and their potential technological applications especially in catalysis. [1] Numerous metal-oxygen clusters, [1b,c,j] metal-organic frameworks [1d,l,m] or metal-organic polyhedrons [1f,n] have been designed by using av ariety of selfassembling or templated reactions.A mong those,awide family of polynuclear cage-like metallasilsesquioxanes (CLMSs) architectures with interesting structural topologies [2] and important catalytic activity (e.g.i ne poxidation, oxidation and polymerization) [3] has been synthesized through ar eaction involving transition-metal ions and silsesquioxane [RSiO 1.5 ] n ligands.S urprisingly,e xamples of cage-like architectures involving another metalloid close to silicon, namely germanium, are scarce despite the great potential of this element in organometallic chemistry.T oour knowledge there is only aseries of compounds obtained from germasesquioxane [RGeO 1.5 ] 2 (R = HOOCCH 2 CH 2 )and various cations. [4] Their hydrothermal synthesis requires relatively hard conditions consisting of several days of reactions at 110 8 8C. It is worth mentioning that to date focus has only been on the synthesis,t hermogravimetric analysis,l uminescence, and sorption properties of these architectures and no investigation of the catalytic activity of metallagermanates has been reported.

Herein, we report an alternative and simplified synthetic approach to design aseries of unprecedented individual cagelike metallagermsesquioxanes under mild conditions.I n addition, the catalytic activity and magnetic properties of the metallagermsesquioxanes will be discussed.

Thes (4; Scheme S1 in the Supporting Information) in higher yields (up to 26 %). Single-crystal X-ray diffraction analyses of 1-4 revealed similar structural organization with the cage-like molecular architecture.T he complexes are formed by ac entral hexairon(III) oxo {Fe 6 O 19 }c ore,r eminiscent of the "Lindqvist polyanion", coordinated by three pentameric germoxanolate [PhGe(O)O À ] 5 ligands,f orming an unprecedented three-pointed star geometry.

In the representative crystal structure of 1,the octahedral iron ions are located in vertexes of an octahedron (Figure 1). They are connected through the central and peripheral oxygen atoms.F e À O(Fe) distances are in the range of 1.99-2.05 ,t hose of Fe À O(Ge) and Fe À Fe are in the range of 1.87-1.90 and 3.16-3.27 ,respectively.Inturn, the values of the FeÀOÀFe angle depend on the position of the oxygen atom. It is close to 908 8 when the oxygen occupies the central position and significantly exceeds 908 8 in the other cases (]Fe À O1 À Fe :8 8.1-91.78 8; ]Fe À O À Fe :1 01.9-108.28 8). The Pentameric germoxanolate fragment of 1 adopts ac rown configuration (Figure S1), with the main structural parameters:G e À O(Ge) 1.740-1.791 , ]Ge À O À Ge 117.6-129.08 8. Note that the only example of cyclic germanoxane fragments in metallagermsesquioxane structures describe hexameric units. [4a] Thestructural organization of the related compounds 2-4 is similar to 1.T he difference consists in additional coordination of the respective bidentate ligands (1,10-phenanthroline,2 ,2'-bipyridine,n eocuproine) to the Na ions (Figure S2). Noteworthy is that sodium ions are located in external positions to the cage and, thus,a re responsible for the supramolecular packing (Figure S3). According to the higher yields of 2-4 in comparison to 1,b identate ligands might give additional stabilization of the metallagermsesquioxane cage and facilitate its crystallization. These results show that germsesquioxanes indeed provide great potential for creation of cage metalladerivatives,e ven when using simple alkoxygermane as an initial reagent. Drawing an analogy between Fe-based sesquioxanes of germanium and silicon, we would like to emphasize that known types of cage ironsilsesquioxanes (Refs. [2a,c,j]) have quite different structural features in comparison to germanium-based compounds 1-4.This suggests aunique role of germanium centers in cagecompound formation. Results of EXAFS study on iron and germanium K-edges for 1-4 confirmed that the oxidation states of iron and germanium are + 3a nd + 4, respectively (Figures S4-S5, Tables S3-S4).

Themagnetic properties of these compounds,investigated by using aSQUID magnetometer working in the temperature range 1.8-300 Kupto7T, indicate that they are paramagnetic behavior in this temperature range.The temperature dependence of the magnetic susceptibility for 1 is given as an example (Figure S7, ESI). Theroom temperature value of cT,equal to 14.29 cm 3 Kmol À1 ,i sc onsiderably lower than the theoretical one of 26.25 cm 3 Kmol À1 ,e xpected for six non-interacting Fe III ions (S = 5/2 and g = 2.00). Such ad iscrepancy most likely originates from the occurrence of strong antiferromagnetic interactions between the spin carriers that are still operative at room temperature.T his is indeed confirmed by the temperature dependence of cT,w hich reveals ap rogressive decrease upon cooling to reach 3.80 cm 3 Kmol À1 at 1.8 K. Antiferromagnetic interactions between the Fe III ions can be mediated through both the central and bridging oxygen atoms,a sf ound in similar hexanuclear architectures. [5] For temperatures lower than 10 K, as teep decrease of cT occurs and can be assigned to the zero-field splitting and/or intermolecular interactions between the polynuclear complexes.I na ddition, the non-null value of cT at low temperature indicates the presence of an on-diamagnetic ground state that should be expected for ap ure S = 0s pecies.T his could reflect either the presence of thermally populated excited states or antagonist magnetic interactions mediated by the different bridging oxygen atoms.The field dependence of the magnetization at 1.8 Kfor 1 further confirms the nondiamagnetic state with am agnetization value of 2.72 m B without any evidence of the saturation indicating the presence of am agnetic anisotropy (Figure S8). Alternating current (AC) measurements performed as the temperature dependence of the in-phase (c')and out-of-phase (c'')susceptibilities measured in azero DC magnetic field show an increase upon cooling but without the presence of am aximum, precluding an in-depth characterization of the relaxation dynamics (Figure S9).

We also present herein the first applications of cage metallagermsesquioxanes in catalysis.Ascomplexes 1-4 have similar molecular architectures,weconcentrated on the study of one of them as an example,namely,compound 2.Itiswell established that iron-based compounds [START_REF]Iron Catalysis in Organic Chemistry[END_REF] are capable of catalyzing the oxidation of hydrocarbons with peroxides.W e have found that cyclohexane and other alkanes as well as benzene can be oxidized to the alkyl hydroperoxides and phenol, respectively,byhydrogen peroxide in the presence of catalytic amounts of 2 and CF 3 COOH (some examples are shown in Figure 2, the experimental details are described in the Supporting Information). Theo xidation with H 2 O 2 catalyzed by complex 2 was very efficient as it gave oxygenates from alkanes in avery high yield of 44 %. This yield is one of the highest values obtained so far in metal-catalyzed oxidations of cyclohexane with H 2 O 2 since usually yields are not higher than 5-30 %. Fore xample,c yclohexaneo xidation [6c] with H 2 O 2 catalyzed by ah exanuclear iron(III) carboxylate gave maximum yield of oxygenates of 29 %.

TheS -shape mode of the initial rate W 0 dependence on the concentration of 2 attests that the reaction order with respect to 2 is not constant and gradually changes from an apparent second order to first order,when [2] 0 changes in the interval from 1.5 10 À4 to 3 10 À4 m (Figure 3).

On the basis of these data, amechanism can be proposed which includes as tage of the starting complex monomerization. [2c] Further dimerization of these particles affords species which are responsible for the catalytic activity of the system.

Dependence of the initial oxidation rate on the concentration of the CF 3 COOH co-catalyst is given in Figure S10. Thed ependence of W 0 on the initial concentration of cyclohexane approaches ap lateau at [CyH] 0 > 0.4 m (Fig- ure S11) indicating acompetitive interaction of the oxidizing species with the alkane and acetonitrile. [8] Ther ate dependence on the initial concentration of hydrogen peroxide is linear at [H 2 O 2 ] 0 < 1.5 m (Figure S12).

Complex 2 was also evaluated in the formation of amides directly from alcohols and amines,inthe presence of tert-butyl hydroperoxide as the terminal oxidant. Fort his purpose,w e chose the newly developed metal-catalyzed microwave-activated amidation [9,2c] between benzylic alcohol and (AE)-amethylbenzylammonium chloride (Scheme 1). Theu se of complex 2 allowed amide 5 to be obtained in ayield of 71 %. This result was compared to that obtained with amore simple iron salt, FeCl 2 •4 H 2 O, which gave the product in ay ield of only 51 %. This result clearly shows that the germsesquioxane structure exerts astabilizing effect around the metallic center, resulting in an improved catalytic activity.

In conclusion, afamily of unprecedented cage-like Fe,Nagermsesquioxanes having acentral hexairon(III) oxo {Fe 6 O 19 } core coordinated by three pentameric germoxanolate ligands has been obtained. Their structural arrangement induces antiferromagnetic interactions between Fe III ions through the oxygen bridges resulting in ap aramagnetic behavior. This class of compounds was evaluated for the first time for its catalytic activity.C omplex 2,d isplayed ah igh efficiencyi n catalysis for alkane oxidation and amide formation from benzylic alcohol and amine.Afurther investigation on the design of new metallagermsesquioxanes with different structural organization and catalytic properties is currently in progress.T hese results,c oncerning an influence of composition and structure of such cage complexes on catalytic behavior (including possible synergetic effect of metal and germanium ions as catalytic centers) will be presented in the near future.

  ingle crystals of Fe,Na-phenylgermsesquioxane [(Ph 5 Ge 5 O 10 ) 3 Fe 6 (OH) 3 (O)Na 2 ]•(EtOH) 2 •H 2 O 1 (10 %y ield) were obtained by ar eaction between phenyltrimethoxygermane PhGe(OMe) 3 ,N aOH and anhydrous FeCl 3 in ethanol with further crystallization. Thea ddition of N-containing ligands,s uch as phenanthroline,b ipyridine,o rn eocuproine to this reaction allowed us to isolate complexes [(Ph 5 Ge 5 O 10 ) 3 Fe 6 (OH) 3 (O)Na 2 (phen)]• (EtOH) 4 •( H 2 O) 0.5 (2), [(Ph 5 Ge 5 O 10 ) 6 Fe 6 (OH) 3 OFe 6 (OH) 2 (O) 2 Na 3 (bipy)]• (EtOH) 14.5 •( C 5 H 5 N) 0.5 •( H 2 O) (3), and [(Ph 5 Ge 5 O 10 ) 3 Fe 6 -(OH) 3 (O)Na 2 (neocuproine)]• (EtOH) 2 •( H 2 O) 1.5

Figure 1 .

 1 Figure 1. Top: Molecular structure of 1.Bottom:The structure of Fe 6 O 19 core of 1.F ull color version is given in ESI (Figures S1,S2).

Figure 2 .

 2 Figure 2. Accumulation of oxygenates(cyclohexanol, curves 1, and cyclohexanone, curves 2) in the oxidation of cyclohexane (0.46 m)w ith H 2 O 2 (50 %, aqueous, 1.5 m)c atalyzed by 2 (5 10 À4 m)inthe presence of CF 3 COOH (0.05 m). Solvent was acetonitrile (total volume of the reaction solution was 5mL) at 40 8 8C. Concentrations of cyclohexanone and cyclohexanol were determined before (curves a) and after (curves b) reduction of the aliquots with solid PPh 3 (for this method, see Ref. [7]).

Figure 3 .

 3 Figure 3. Dependence of the initial rate of oxygenate (the sum cyclohexanol + cyclohexanone) formation W 0 on the initial concentration of 2 in the oxidation of cyclohexane (0.46 m,c urve 1) and benzene (0.45 M, curve 2) with hydrogen peroxide (50 %aqueous) in the presence of CF 3 COOH (0.05 m)inMeCN at 40 8 8C. Concentrations of cyclohexanoneand cyclohexanol were determined after reduction of the aliquots with solid PPh 3 .

2016W iley-VCH Verlag GmbH &C o. KGaA, Weinheim Angew.C hem. Int. Ed. 2016, 55,1 5360 -15363

Angew.C hem. Int.E d. 2016, 55,1 5360 -15363 2016 Wiley-VCH Verlag GmbH &C o. KGaA, Weinheim www.angewandte.org

www.angewandte.org 2016 Wiley-VCH Verlag GmbH &C o. KGaA, Weinheim Angew.C hem. Int. Ed. 2016, 55,1 5360 -15363

Acknowledgements

This work was supported by RFBR (projects 16-03-00609, 16-03-00206, 16-03-00254, 16-29-05180, 16-53-150008). French Embassy in Moscow,B alard Foundation, University of Montpellier,C NRS (Programme de Recherche Conjoint), and the Plateforme DAnalyse et de CaractØrisation (PAC Balard), and "Science without Borders Program, Brazil-Russia", CAPES(grant A017-2013). Keywords: cage compounds •homogeneous catalysis • metallagermsesquioxanes •oxidation •structure elucidation


Jessica Labarre-Lainé

Redouane Beniazza

Valérie Desvergnes

Yannick Landais
email: y.landais@ism.u-bordeaux1.fr

A Convergent Access to Bis-spiroacetals Through a Sila-Stetter-ketalization Cascade

An NHC-catalyzed sila-Stetter reaction between aliphatic acylsilanes and vinylketones bearing silyl ether substituents affords functionalized 1,4-diketones, which upon treatment under acidic conditions leads to the corresponding bis-spiroacetals. The two-step sequence may be also carried out in a one-pot operation leading to high yields of the desired bis-spiroacetals. § These authors contributed equally to the work.

(1) For a review on the isolation, structure and biogenesis of these marine toxins, see:

The bis-spiroacetal skeleton is found in various bioactive marine phycotoxins including pinnatoxin A 1, spirolide C 2 (Figure 1) or pteriatoxins (not shown). 1 These toxins have led recently to intensive research due to their potent acute neurotoxicity. 2 The nature of the tricyclic system, embedded in a complex macrocycle, varies with the size of the oxygenated rings (5 and 6membered rings) forming the bis-spiroacetal. Several strategies directed toward the elaboration of these bisspiroacetals have been devised en route to the total synthesis of 1 and 2. 3,4 A straightforward disconnection leading to this skeleton implies the generation of a 1,4diketone, which upon ketalization under acidic conditions provides the desired tricyclic framework. 4

Figure 1. Bis-spiroacetal skeleton in marine phycotoxins

This spiroacetalization has been elegantly pioneered by Kishi 5 then by Inoue-Hirama, 6 Nakamura-Hashimoto 7 and Zakarian 8 in their respective total syntheses of pinnatoxin (5) Mc Cauley, J. A.; Nagasawa, K.; Lander, P. A.; Mischke, S. G.; Semones, M. A.; Kishi, Y. J. Am. Chem. Soc. 1998, 120, 7647-7648. (6) Sakamoto, S.; Sakazaki, H.; Hagiwara, K.; Kamada, K.; Ishii, K.; Noda, T.; Inoue, M.; Hirama, M. Angew. Chem., Int. Ed. 2004, 43, 6505-6510. (7) Nakamura, S.; Inagaki, J.; Sugimoto, T.; Kudo, M. ; Nakajima, M.; Hashimoto, S. Org. Lett. 2001, 3, 4075-4078. ( 1, as well as by Ishihara et al. in their preparation of the bis-spiroacetal core of spirolide B. 9 The 1,4-diketone motif may be constructed by a number of ways, often requiring several steps. 10 In the search for a convergent strategy, allowing the presence of functionality and protecting groups on the carbon backbone, the Stetter reaction appeared as an attractive method to elaborate efficiently the bis-spiroacetal I present in toxins 1 and 2 (Scheme 1). It was thus envisioned that the coupling of an aldehyde (or an acylsilane) III and a vinyl ketone such as IV in the presence of a suitable organocatalyst (usually a N-heterocyclic carbene (NHC)) would deliver the 1,4diketone framework II. A careful choice of the protecting groups on the alcohol functions within the chain would then allow the construction of the desired bis-spiroacetal I after acid-catalyzed deprotection and ketalization. We report here that NHC-mediated Stetter and sila-Stetter processes followed by the acid-catalyzed ketalization effectively offer a straightforward access to the bisspiroacetal skeleton. A cascade process also allows both events to be carried out in a one-pot operation.

Scheme 1. Retrosynthetic analysis

Although the pioneering work by Stetter 11 and others has shown that this coupling may be carried out starting from a large variety of aldehydes, few examples have been reported to date on aliphatic partners. Moreover, acyloins resulting from the homo-coupling of the aldehyde are often present as by-products. Recent work by Scheidt et al 12 however showed that this could be circumvented, using acylsilanes instead of aldehydes, although mostly aromatic acylsilanes were tested during this work. We thus studied first the influence of the nature of the partner III in the Stetter reaction, using as precursors, aldehyde 3 or acylsilane 4a and vinylketone 5a in the presence of NHC-catalyst precursor A. 13 As summarized in Table 1, using aldehyde 3 and various amounts of catalyst invariably led to a mixture of both the (9) Ishihara, J.; Ishizaka, T.; Suzuki, T.; Hatakeyama, S. Tetrahedron Lett. 2004, 45, 7855- (13) A screening of other thiazolium salts led to similar results (supporting information). Catalyst A was finally selected, being the easiest to handle.

desired product 6a and the corresponding acyloin 7 (Entries 1-3, Table 1). In contrast, we were pleased to observe that acylsilane 4a led to 6a in good yield, without a trace of 7 (entries 4-5), thus notably extending the scope of the sila-Stetter reaction. 12 Increasing the quantity of acylsilane however produced small amount of 7 (entry 6). Optimization of the sila-Stetter reaction showed that decreasing the quantity of acylsilane was detrimental to the conversion, with 1.5 to 2 equivalents leading to optimal results. Dry iso-propanol (4 equiv) and DBU as a base were also shown to provide the highest yields. With these optimized conditions in hand, the methodology was extended to a large variety of aliphatic acylsilanes 14 and enones 15 precursors (supporting information), as illustrated in Scheme 2. Scheme 2. sila-Stetter reaction between aliphatic acylsilanes 4 and vinylketones 5 t-Butyldimethylsilyl and benzyl substituents were selected as orthogonal protecting groups for alcohol functions, so that they can be deprotected selectively if needed (vide infra). 1,4-Diketones were obtained in good to excellent yields. 16 Reaction time ranged between 3 and 24 hours depending on the nature of the substrate. 1.5 equivalent of acylsilane was generally used except when indicated. As mentioned above, the reaction was carried out using 15 mol% of salt A, except for the preparation of 6d, 6j-k and 6o-q, where 30 mol% were employed. It is noteworthy that substitution on the carbon chain is allowed on both partners, but -substitution in acylsilane was shown to slow down the process, leading to lower yields. 17 Removal of the t-butyldimethylsilyl protecting group was then carried out under standard acidic conditions, leading to a spontaneous cyclization, [5][6][7][8] producing the desired bis-spiroacetals in moderate to good yields as a mixture of diastereomers (d.r. estimated using both 1 H NMR and GC-MS) (Scheme 3). 18 Various acidic conditions were tested (TfOH, HCl, MgBr2, BF3•OEt2, TMSOTf, Sc(OTf)3) but camphorsulfonic acid (CSA) in CH3CN at room temperature was found optimal in terms of yields. In some cases, major diastereomers were obtained pure after chromatography, but their structures could not be attributed based solely on 1D and 2D NMR data. Attempts at obtaining crystals for X-ray diffraction studies unfortunately failed.

Scheme 3. Spiroacetalization of 1,4-diketones 6e, 6i, 6n-q When the deprotection was carried out using CSA in MeOH as reported by Ishihara et al, 9 bis-acetal 9 was obtained as a mixture of diastereomers instead of the corresponding bis-spiroacetal (Scheme 4). Interestingly, the structure of the major isomer could be determined through X-ray diffraction studies.

(16) It was also possible to perform reaction between acylsilanes 4 and other Michael acceptors such as acrylonitrile (51%), methylacrylate (36%) and vinylsulfone (26%) (unoptimized yields) (Supporting Information).

(17) Increasing the amount of acylsilane 4h to 2.5 equiv resulted in the isolation of 6l but in only 42% yield.

(18) Thermodynamic ratio under the given conditions. Having demonstrated the efficiency of the sila-Stetter reaction and the acid-mediated spiroacetalization of the resulting 1,4-diketones, we then developed a one-pot process, in order to avoid the purification of the sensitive diketone intermediate. The sila-Stetter reaction was performed using as above precatalyst A, DBU and i-PrOH. After heating the mixture at 75°C until complete disappearance of enone 5, CSA (50 mol%) was added and the reaction mixture stirred at room temperature. This afforded the required bis-spiroacetals 8a,c,e,i in good yields (Scheme 5). It is worthy of note that 8e and 8i were obtained with slightly better yields than those observed using the two-pot protocol (77% vs 53% and 68% vs 60% respectively). The major diastereomer for 8a and 8c was obtained pure in each case, and the relative configuration was attributed based on literature reports. 3e,19 Scheme 5. One-pot sila-Stetter/spiroacetalization cascade Finally, orthogonal protecting groups on 1,4-diketones allow for the mono-deprotection and the selective formation of tetrahydrofurans. For instance, treatment of 1,4-diketone 6d under Lewis acidic conditions, triggered the selective deprotection of the TBS group, followed by the cyclisation and the formation of an oxonium intermediate, which was eventually trapped in situ by a nucleophilic allylsilane, leading to THF 10 in good yield (Scheme 6). Scheme 6. One-pot mono-deprotection/cyclisation/allylation of 1,4-diketone 6d

In summary, we reported here the sila-Stetter coupling between a series of aliphatic acylsilanes and vinylketones, which afforded in generally good yields, 1,4-diketones bearing ether substituents on the chain. Subsequent acidmediated deprotection of silyl ether protecting groups triggered an efficient spiroacetalization, leading to bisspiroacetal motifs, which are present in marine natural products such as pinnatoxin 1. Both successive transformations may be carried out in a one-pot operation with high efficiency. Application of this strategy to the synthesis of bis-spiroacetal fragments of spirolides and analogues is now underway and will be reported in due course.

  ) (a) Lu, C.-D.; Zakarian, A. Org. Lett. 2007, 9, 3161-3163. (b) Stivala, C.; Zakarian, A. J. Am. Chem. Soc. 2008, 130, 3774-3776.

(

  14) Clark, C. T.; Milgram, B. C.; Scheidt, K. A. Org. Lett. 2004, 6, 3977-3980. (15) Nahm S.; Weinreb S. M. Tetrahedron Lett., 1981, 22, 3815-3818.

Scheme 4 .

 4 Scheme 4. Spiroacetalization of 6i in MeOH

(

  19) (a) Mudryk, B.; Shook, C. A.; Cohen, T. J. Am. Chem. Soc. 1990, 112, 6389-6391. (b) Weldon, A. J.; Tschumper, G. S. J. Org. Chem. 2006, 71, 9212-9216. (c) McGarvey, G. J.; Stepanian, M. W.; Bressette, A. R.; Ellena, J. F. Tetrahedron Lett. 1996, 37, 5465-5468.

Table 1 .

 1 Stetter versus sila-Stetter reaction

	entry 3 or 4a (equiv) A (mol %) t (h)	6a/7 a	yield b (%)
	1	3 (1)	30	2	81/19	61
	2	3 (1.5)	15	2.5	72/28	68
	3	3 (2)	30	2	65/35	77
	4	4a (1) c	30	1.5	100/0	67
	5	4a (1.5) c	15	3	100/0	89
	6	4a (2) c	30	3	80/20	90

a Measured by 1 H NMR of the crude reaction mixture. b Isolated yield of 6a after column chromatography. c 4 equiv of i-PrOH were used.

Acknowledgment. We thank Dr. V. Liautard, Dr. S. Toumieux and D. Morgant (ISM, Bordeaux) for preliminary experiments, CNRS, MNERT, ANR (Spirosyn N°11-BS07-006-02) for generous support, Dr. C. Guillou, L. Chabaud (ICSN, Gif s/Yvette) and Dr. F. Robert (ISM, Bordeaux) for fruitful discussions. We are grateful to Dr. B. Kauffmann (IECB, Bordeaux) for X-ray diffractions studies, J.M. Lasnier and N. Pinaud (ISM, Bordeaux) for NMR experiments.

Supporting Information Available: Experimental procedures and spectroscopic data for compounds 4-10 and morpholine and Weinreb amide precursors are available in the electronic supporting information. Copies of 1 H and 13 C NMR spectra for all new compounds are also supplied. This material is available free of charge via the Internet at http://pubs.acs.org.