Francesco Pantisano
email: fpantisa@ee.oulu.fi

Mehdi Bennis
email: bennis@ee.oulu.fi

Walid Saad
email: saad@princeton.edu

Mérouane Debbah
email: merouane.debbah@supelec.fr

Cooperative Interference Alignment in Femtocell Networks

Keywords: coalitional games, femtocell networks, game theory, partition form, interference alignment, recursive core

Underlay femtocells have recently emerged as a key technology that can significantly improve the coverage and performance of next-generation wireless networks. In this paper, we propose a novel approach for interference management that enables a number of femtocells to cooperate and improve their downlink rate, by sharing spectral resources and suppressing intra-tier interference using interference alignment. We formulate a coalitional game in partition form among the femtocells and propose a distributed algorithm for coalition formation. Using our approach, the femtocell access points can make individual decisions on whether to cooperate or not, while maximizing a utility function that captures the cooperative gains and the costs in terms of transmit power for information exchange. We show that, using the proposed coalition formation algorithm, the femtocells can self-organize into a network partition composed of disjoint femtocell coalitions, which constitutes the recursive core of the game. Simulation results show significant gains in terms of average payoff per femtocell, reaching up to 30% relative to the non-cooperative scheme.

I. INTRODUCTION

The deployment of femtocell access points (FAPs) is envisioned to be a key solution for providing high wireless data-rates, offloading the macrocell traffic and enhancing the coverage of existing networks (e.g., LTE or WiMAX) [START_REF] Sesia | LTE -the UMTS long term evolution[END_REF] [2]. Nevertheless, the deployment of FAPs underlaid with existing macrocell wireless networks faces several challenges, notably at the level of interference management and resource allocation. In co-channel deployments, FAPs reuse the spectral resources of the macrocell network which can lead to severe interference at both the femtocell and the macrocell tiers [3] [4]. Hence, it is of interest to propose adaptive interference management techniques, namely at the femtocell level [START_REF] Gpp | Service requirements for Home Node B (HNB) and Home eNode B (HeNB)[END_REF].

Recently, interference alignment (IA) has been introduced as a coding technique that can achieve high multiplexing gains within interference limited environments [START_REF] Jafar | Degrees of freedom region of the MIMO X channel[END_REF], [START_REF] Suh | Downlink interference alignment[END_REF]. Essentially, IA is based only on linear precoding at the transmitter side and zero-forcing equalization at the receiver [START_REF] Suh | Downlink interference alignment[END_REF]. In [START_REF] Jafar | Degrees of freedom region of the MIMO X channel[END_REF], [START_REF] Cadambe | Interference alignment and the degrees of freedom of wireless X networks[END_REF], the authors analyze the degrees of freedom resulting from using IA over different channels and for an arbitrary number of antennas per user. The authors in [START_REF] Jafar | Degrees of freedom region of the MIMO X channel[END_REF], [START_REF] Cadambe | Interference alignment and the degrees of freedom of wireless X networks[END_REF] have shown that IA enables wireless users to have an interference-free communications at the cost of each user exploiting only half of the available degrees of freedom. Additional results on the achievability of the total number of degrees of freedom are presented in [START_REF] Gou | Degrees of freedom of the K user MIMO interference channel[END_REF],

The authors would like to thank the Finnish funding agency for technology and innovation, Elektrobit, Nokia and Nokia Siemens Networks for supporting this work. This work has been performed in the framework of the ICT project ICT-4-248523 BeFEMTO, which is partly funded by the EU. [START_REF] Peters | Cooperative algorithms for MIMO interference channels[END_REF]. In essence, [START_REF] Gou | Degrees of freedom of the K user MIMO interference channel[END_REF], [START_REF] Peters | Cooperative algorithms for MIMO interference channels[END_REF] show that, by aligning all interfering signals in the same subspace from the point of view of each receiver, interference can be suppressed simply through zeroforcing equalization. In [START_REF] Tresch | Performance of interference alignment in clustered wireless ad hoc networks[END_REF], an intra-cluster IA technique is applied to a clustered wireless ad hoc network to increase the probability of successful transmission. In summary, existing work in the area of interference alignment has shown that significant performance gains can be achieved by combining IA techniques with cooperative schemes in which FAPs can coordinate their transmission so as to suppress interference among each other [START_REF] Jafar | Degrees of freedom region of the MIMO X channel[END_REF], [START_REF] Cadambe | Interference alignment and the degrees of freedom of wireless X networks[END_REF][START_REF] Gou | Degrees of freedom of the K user MIMO interference channel[END_REF][START_REF] Peters | Cooperative algorithms for MIMO interference channels[END_REF].

The main contribution of this paper is to propose a novel approach for interference management in femtocell networks in which the femtocells can cooperatively perform interference alignment so as to reduce their mutual interference, and, consequently, improve their overall performance. In the proposed approach, we study the cooperative strategies of the femtocells that enables them to form clusters inside which co-tier interference is suppressed using IA. We formulate the problem as a coalitional game in partition form in which the FAPs are the players taking autonomous decisions to create coalitions so as to maximize a utility function, which captures both the benefits from cooperation, in terms of downlink transmission rates, and the costs, in terms of transmit power for information exchange. We solve the game using the concept of a recursive core which is a key solution for coalitional games in partition form. Using simulations, we show that the proposed approach enables the FAPs to cooperate and self-organize into a stable partition while yielding significant performance gains relative to the non-cooperative scheme.

The rest of this paper is organized as follows: Section II presents the proposed system model for cooperative interference alignment. In Section III, we formulate the cooperative femtocells problem as a coalitional game and discuss its properties. In Section IV, we present the simulation results and, finally, conclusions are drawn in Section V. Notations: In the rest of the paper, The log refers to log 2 . Bold uppercase letters (e.g., A) denote matrices, bold lowercase letters (e.g., a) denote column vectors and normal letters (e.g., a) denote scalars. The identity matrix is denoted by 1. The letter E A denotes the expectation operator over A. Symbol C represent the set of complex numbers and (•) † denotes the Hermitian transpose operator.

II. SYSTEM MODEL Consider the downlink transmission in a single macrocell network (e.g., LTE-Advanced or WiMAX) in which N femtocell access points (FAPs) are deployed. These FAPs use an Orthogonal Frequency Division Multiple Access (OFDMA) technique over a portion of the macrocell spectrum that is reused in the femtocell tier (the remaining part of spectrum is assigned to macrocell users). Let N = {1, ..., N } denote the set of all FAPs. Every FAP i ∈ N serves L i femtocell user equipments (FUEs). Let L i = {1, ..., L i } denote the set of FUEs served by an FAP i ∈ N . The FUEs belonging to the same FAP are scheduled on orthogonal subcarriers. Each transmitter (resp. receiver) is equipped with N t (resp. N r ) transmit (resp. receive) antennas. For any transmission between an FAP i and one of its FUEs k ∈ L i , the discretetime received signal at FUE k, at a given time instant, is:

y k = √ γ ik H ik V i s i + j∈N , j =i √ γ jk H jk V j s j + n k , (1)
where γ ik and γ jk denote the signal-to-noise and the interference-to-noise ratio at FAP k ∈ L i , respectively. [H ik ] are N r × N t complex matrices representing the MIMO channels between FAP i and FUE k. s i ∈ C di×1 represents the d i -dimensional signal from FAP i and V i ∈ C Nt×di denotes the associated precoding matrix. d i represents the degrees of freedom of the transmitter-receiver pair (i.e., the multiplexing gain), for the transmitted message. Similarly, s j ∈ C dj ×1 and V j ∈ C Nt×dj are the d j -dimensional signal and the precoding matrix pertaining to interfering FAP j. Further, n k represents the noise vector at receiver k which is considered as a zero mean unit variance circularly symmetric additive white Gaussian noise vector (AWGN).

In existing networks, FUEs belonging to different femtocells are typically scheduled non-cooperatively, i.e., without coordination among the FAPs. Consequently, neighboring FAPs can schedule their transmissions in the downlink over the same subcarrier hence causing interference to one another and limiting the downlink performance of their FUEs. To overcome this problem and reduce the femtocell-to-femtocell interference, FAPs have an incentive to cooperate and coordinate their transmissions using advanced communication techniques such as interference alignment. In this respect, given the signal to noise and interference ratio (SINR) feedbacks from their FUEs, a group of FAPs can decide to form a coalition in which a spatial interference alignment scheme is used so as to mitigate the downlink intra-femtocell interference. In essence, the FAPs that are member of the same coalition S ⊆ N cooperate in order to adjust the spatial structure of their transmitted signals so as to avoid interference among themselves. Note that, even though the FAPs that are members of a given coalition S ⊆ N are able to mitigate interference among each other, they are still affected by non-aligned interference from FAPs (from other coalitions) in N \ S. Thus, given a coalition S of FAPs that are performing cooperative interference alignment, we can separate the contributions pertaining to the different types (aligned or non-aligned) of interference as follows:

y k = √ γ ik H ik V i s i + j∈S, j =i √ γ jk H jk V j s j + l∈ N \S √ γ lk H lk V l s l + n k (2)
Hence, interference alignment is achieved at an FUE k ∈ L i , where FAP i ∈ S, if and only if there exists a zero-forcing interference suppression matrix U k ∈ C Nr×di , such that:

U † k H jk V j = 0, j ∈ S, j = i rank U † k H ik V i = d i , d i > 0 (3)
One must note that, by considering that the channel coefficients in H jk are identically and independently distributed, the existence of a solution for the above IA problem solely depends on the dimensions of the problem (|S|,N t ,N r ) as discussed in [START_REF] Tresch | On the achievability of interference alignment in the k user constant mimo interference channel[END_REF]. For instance, when the target multiplexing gain d i at the FUE is equal to one, a solution to the system (3) exists for the interference channel composed by |S| -1 interfering transmissions plus the useful signal if and only if

N t + N r ≥ |S| .
Moreover, to find the precoding and interference suppression matrices one can use existing iterative algorithms such as in [13, Algorithm 1], [START_REF] Tresch | On the achievability of interference alignment in the k user constant mimo interference channel[END_REF]. Therefore, the interference from members of the same coalition can be suppressed, yielding, after projection, the following signal at receiver k:

ȳk = √ γ ik U † k H ik V i s i + l∈ N \S √ γ lk U † k H lk V l s l +U † k n k , (4)
where the remaining interference term is due to non-aligned transmissions in the coalitions formed outside S, i.e., in N \S.

For ease of analysis, we assume that the signals s l are not known at the receiver and that Gaussian code books are used, which is a common assumption in the literature [START_REF] Mathur | Coalitions in cooperative wireless networks[END_REF]. As a result, the interference Gaussian. Note that, even when the aggregate noise is not Gaussian, this assumption remains reasonable since it constitutes a lower bound on the actual capacity. Further, we focus on the contribution of the interference-plus-noise as in [START_REF] Chandrasekhar | Spectrum allocation in tiered cellular networks[END_REF], and analyze an expression of the achievable rate assuming complete knowledge of the channel conditions. In this respect, we utilize the mutual information between the transmitted and received signal given, for an FAP i ∈ N as follows:

I(s i , ȳk , π N ) = h(ȳ k , π N ) -h(ȳ k , s i , π N ) = log det π e Q ȳk -log det π e Qt k , (5)
where we let tk denote the interference-plus-noise in (4) (i.e., the last two summands), while Q ȳk and Qt k represent the covariance matrices of the received signal and the received interference-plus-noise, respectively. In particular, if we denote

Hlk = √ γ lk U † k H lk V l , ∀(l ∈ N \ S)
, the covariance matrices can be expressed by:

Qt k = E s l ,n k [t k , t † k ] = l∈N \S Hlk Q l H † lk + σ 2 n 1 Q ȳk = E si,s l ,n k [ȳ k , ȳ † k ] = Hik Q i H † ik + Qt k , (6)
where Q i represents the covariance matrix of the signal s i . Moreover, we use the fact that, for the received noise n k ,

E n k [n k , n † k ] = σ 2 n 1. Note that, Qt k is present in the expression of Q ȳk .
In addition, in order to compute the achievable rate, we use that log det(π e(A + B))log det(π e(B)) = log det(1 + AB -1 ) if B is invertible. Since we consider an interference-limited scenario, we neglected the noise term in [START_REF] Jafar | Degrees of freedom region of the MIMO X channel[END_REF]. The average achievable rate at receiver k in coalition S ⊆ N :

R ik s i , ȳk ) = E H log det 1 + Hik Q i H † ik • • l∈ N \S Hlk Q l H † lk -1 (7)
For performing cooperative IA, the FAPs belonging to the same coalition need to exchange information which, in turn, incurs a cost for cooperation. In this context, we consider a cost in terms of the transmit power that each FAP spends to send its information to the other coalition member. This information exchange phase occurs via a wireless broadcast transmission over a control channel such as the X2 interface [START_REF] Gpp | Service requirements for Home Node B (HNB) and Home eNode B (HeNB)[END_REF]. Given the broadcast nature of the wireless channel, we consider that each FAP i ∈ S exchanges its information to the other coalition member by transmitting its data to the farthest cooperating FAP in the coalition. Hence, given a coalition S, the power needed to exchange information between an FAP i ∈ S and the farthest cooperating FAP î ∈ S is:

Pi ( î) = ν 0 • σ 2 n |H i î| 2 , ( 8
)
where ν 0 is the minimum SNR required at the potential coalition partner î, and |H i, î|2 represents the channel gain between FAPs î and i, over the common control channel. Further, FAPs need to satisfy the power constraint

P i = Q si = E[s † i s i ] ≤ P max 1
, while still accounting for the pilot transmit penalty. The constraint on the total transmit power thus becomes:

P i + Pi ( î) ≤ P max . (9)
Clearly, the cost defined in (8) depends on the spatial distribution of the FAPs in the network and on the minimum SINR required. We consider that the FAPs constraint in (9) also stands for the transmission during information exchange (e.g., over the X2 interface). Without loss of generality, the coalition formation framework presented in this article can also be applied to different cost functions 2 .

III. FEMTOCELL COOPERATION AS A COALITIONAL GAME A. Coalitional Game Concepts

In order to mathematically model the femtocell cooperation problem, we formulate a coalitional game between the FAPs. Due to co-channel interference, the rate achieved by the FAPs members of any coalition S that forms in the network is affected by the cooperative behavior of the FAPs outside S, i.e., the FAPs in N \ S. In other words, the performance of a coalition depends on the partition of the network π N (π N is a partition of N ) to which it belongs. Given a partition π N of N , the total rate that any coalition of FAPs S ∈ π N achieves, depends on the external interference that is generated by the coalitions of FAPs in N \S. Hence, given this property, one suitable framework for modeling the femtocell cooperation problem is that of a coalitional game in partition form with transferable utility (TU) defined as follows [START_REF] Saad | A distributed coalition formation framework for fair user cooperation in wireless networks[END_REF]: Definition 1: A coalitional game in partition form with transferable utility (TU) is defined by a pair (N , v) where N is the set of players, and v is a value function that assigns, for every partition π N and every coalition S ⊆ N , S ∈ π N , a real number that represents the total utility (benefit) that players in S can achieve.

Given the set of FAPs N , our next step is to define a suitable value function v(S, π N ) that reflects the total benefit that coalition S achieves when acting within partition π N . For any coalition S, each FAP i ∈ S obtains a certain payment from its served FUEs that is proportional to the amount of rate offered to these FUEs. Hence, any FAP i ∈ S charges a certain price α ik per every unit rate offered to a given FUE k ∈ L i . As a result, given a partition π N , for any coalition S, the value function v(S, π N ) can be defined as the total achieved revenue as follows:

v(S, π N ) = |S| i=1 Li k=1 α ik R ik s i , ȳi , π N ), (10)
where R ik s i , ȳi , π N ) is the rate achieved by FUE k served by FAP i ∈ S and α ik is the price per unit rate that FAP i ∈ S charges to FUE k. Note that, the rate R ik s i , ȳi , π N ) is function of the interference generated, not only by the FAPs inside S but also by the FAPs in N \ S, and, thus, depends on the partition π N that takes place in the network. The dependence of the rate on the actual partition implies that, for a coalition S, (10) is a function of the coalitions that the FAPs in N \ S form and, thus, the game is in partition form.

The value in [START_REF] Peters | Cooperative algorithms for MIMO interference channels[END_REF] represents the total revenue that a coalition S obtains by acting cooperatively controlled by a pricing factor α ik . For example, this revenue can represent payments that the FUEs make to their serving FAPs. Clearly, the revenue is a transferable quantity that can be apportioned in any way between the coalitional members. Hence, using [START_REF] Peters | Cooperative algorithms for MIMO interference channels[END_REF], we clearly have a game with TU and our next step is to propose a fair scheme for dividing the revenue of a coalition S between its members. In fact, for TU games, beyond the total revenue that a coalition S achieves as per [START_REF] Peters | Cooperative algorithms for MIMO interference channels[END_REF], one has to also characterize the payoff x i that each player (i.e., FAP) in the coalition S receives, i.e., the benefit of every member in the coalition. For this purpose, without loss of generality, we adopt an egalitarian fair payoff division method using which the extra value given by the coalition formation is equally divided among the players in the coalition. Thus, the payoff for FAP i in any coalition S can be found from [START_REF] Peters | Cooperative algorithms for MIMO interference channels[END_REF] by an egalitarian fair division, as follows:

x i (S, π N ) = 1 |S|   v(S, π N ) - j∈S v({j} , π N )   + v({i} , π N ).
(11) Note that the payoff is zero if the power used for information exchange in the coalition exceeds the limit given by [START_REF] Gou | Degrees of freedom of the K user MIMO interference channel[END_REF]. We also note that the egalitarian fair distribution does not imply dividing the whole utility equally but rather the extra benefits (relative to the non-cooperative case) equally, while conserving individual rationality, i.e., ensuring that an FAP that is member of S does not get less benefit than when acting non-cooperatively.

Furthermore, given two payoff vectors x, y ∈ R N , we write x > S y if x i ≥ y i for all i ∈ S ⊂ N and for at least one j ∈ S x j > y j . We also define an outcome as couple (x, π N ), where x is a payoff vector resulting from a partition π N . Finally, let Ω(N , v) denote the set of all the possible outcomes of N .

Given these definitions, we have a coalitional game in partition form (N , v) between the FAPs and our objective is to propose a solution that allows the FAPs to self-organize and form coalitions while increasing their revenue (i.e., their total rate) given the costs for cooperation as in [START_REF] Peters | Cooperative algorithms for MIMO interference channels[END_REF].

B. Recursive core

In order to solve the proposed femtocell coalition formation game in partition form, we use the concept of a recursive core introduced and discussed in [START_REF] Kóczy | A recursive core for partition function form games[END_REF]. The recursive core is a key solution concept for coalitional games that have dependence on externalities, i.e., in partition form. In essence, the recursive core draws a parallel with the well-known core concept of games in characteristic form [START_REF] Kóczy | A recursive core for partition function form games[END_REF]. The recursive core is a suitable outcome of a coalition formation process that accounts for externalities across coalitions, which, in the considered game, are represented by the mutual interference between coalitions of FAPs. To define the recursive core we need to first introduce the concept of a residual game: Definition 2: A residual game (R, v) is a coalitional game in partition form defined on a set of players R, after the players in N \ R have already organized themselves in a certain partition. Players outside R are called deviators, while the players in R are called residuals.

Consider a coalitional game (N , v) and let S be a certain coalition of deviators. Then, let R = N \ S denote the set of residual players. The residual game (R, v) is defined as a game in partition form over the set R. Clearly, a residual game is still in partition form and it can be solved as an independent game, regardless of how it was generated as discussed in [START_REF] Kóczy | A recursive core for partition function form games[END_REF]. To better present this concept, we will provide an intuitive introduction. For instance, when some deviators reject an existing partition and decide to reorganize themselves into a different partition, their decisions will, in general, affect the payoff of the residual players. As a result, the residual players form a new game that is part of the original game (e.g., the game over the whole set N ), but with a certain part of the partition (composed by deviators) already fixed. In consequence, one of the main attractive properties of a residual game is its consistency as well as the possibility of dividing any coalitional game in partition form into a number of residual games which, in essence, are easier to solve. In fact, any game in partition form can be seen as a collection of residual games, each one of which can be solved as if it was the original one. The solution of a residual game is known as the residual core which is defined as follows:

Definition 3: The residual core of a residual game (R, v) is a set of possible game outcomes, i.e., partitions of R that can be formed.

One can see that given any coalitional game (N , v), residual games are smaller than the original one and therefore computationally easier to analyze. Given any coalitional game (N , v), the recursive core solution can be found by recursively playing residual games, which, in fact, yields the following definition as per [START_REF] Kóczy | A recursive core for partition function form games[END_REF]Definition 2]:

Definition 4: The recursive core of a coalitional game (N , v) is inductively defined in four main steps:

1) Trivial Partition. Let (N , v) be a coalitional game. The recursive core of a coalitional game where N = {i} is composed by the only outcome with the trivial partition composed by the single player i: C({i} , v) = (v(i), i). 2) Inductive Assumption. Proceeding recursively, suppose the recursive core C(R, v) for each game with at most N -1 players has been defined. Now, we define the assumption A(R, v) about the game (R, v) as follows:

A(R, v) = C(R, v), if C(R, v) = 0 ; A(R, v) = Ω(R, v), otherwise. 3) Dominance. An outcome (x, π N ) is dominated via a coalition S if for at least one (y N \S , π N \S ) ∈ A(N \S, v)
there exists an outcome ((y S , y N \S ), π S ∪ π N \S ) ∈ Ω(N , v) such that (y S , y N \S ) > S x. 4) Core Generation. The recursive core of a game of |N | players is the set of undominated outcomes and we denote it by C(N , v). Note that, in Definition 4, the concept of dominance in step 3) inherently captures the fact that the value of a coalition depends on the belonging partition. Hence, it can be expressed in the following way: given a current partition π N and the respective payoff vector x, an undominated coalition S represents a deviation from π N in such a way that the resulting outcome ((y S , y N \S ), π S ∪ π N \S ) is more rewarding for the players of S, compared to x.

Since a partition uniquely determines the payoffs of all the players in the game, the recursive core can be seen a set of partitions that allow the players to organize in a way that provides them with the highest payoff. It is worth stressing that the recursive core verifies the properties of rationality, welldefinition and efficiency [START_REF] Kóczy | A recursive core for partition function form games[END_REF]. In detail, with rationality it is intended that players never choose an inferior (i.e., dominated) strategy, therefore, they always pursue a profitable strategy. The recursive core is also well-defined because when it exists, its solution is unique. Furthermore, efficiency is a consequence of the fact that there is no preference in the set composed by the recursive core, and thus, all the included partitions are equivalent in terms of individual payoff. Moreover, In [START_REF] Kóczy | A recursive core for partition function form games[END_REF]Lemma 10], the author proves that the term core is justified, since the recursive core is, as previously mentioned, a natural generalization of the core in characteristic form, to games with externalities. Consequently, when applied to a game in characteristic function form, the recursive core coincides with the core, as classically defined [START_REF] Ray | A Game-Theoretic Perspective on Coalition Formation[END_REF].

Given these properties, once a partition in the recursive core takes place, the players have no incentive to abandon it, because any deviation would be detrimental. As a result,

Algorithm 1 Distributed coalition formation algorithm for interference alignment in femtocell networks

Initial State: The network is partitioned by πN = N = {1, . . . , N } with non-cooperative FAPs Proposed Coalition Formation Algorithm repeat Phase I -Interferer Discovery Each FAP i computes the current payoff xi(S, πN ) as in [START_REF] Tresch | Performance of interference alignment in clustered wireless ad hoc networks[END_REF] Each FAP collects RSSI of the neighboring FAPs from each of its own FUEs For each of the occupied subcarriers, each FUE sorts the interfering FAPs from the stronger to the weaker Phase II -Femtocell Coalition Formation for all Interferers in the list do Each FAP sequentially engages in pairwise negotiations with the strongest discovered interfering FAPs, to identify potential coalition partners, which satisfy the power constraint in [START_REF] Gou | Degrees of freedom of the K user MIMO interference channel[END_REF] The payoff in ( 11) is updated, accounting for the new cost Pi( î) Each FAP joins the interferer which ensures the maximum payoff until any further growth of the coalition does not result in a payoff enhancement of at least one FAP, without decreasing the other FAPs payoffs. end for Outcome of this phase: Convergence to a stable partition in the recursive core; Phase III -Coalition-level Interference Alignment Within each coalition, femtocell-to-femtocell interference alignment operations described in Section II are initiated a partition in the recursive core is also stable since it is a partition which ensures the highest possible payoff for each one of the players who have no incentive to leave this partition.

In [START_REF] Kóczy | A recursive core for partition function form games[END_REF], the author shows that the existence of the recursive core requires at least one residual core (and not all of them) to be nonempty. This means that at least a subset of the players in the network must have defined a preference on how to organize themselves, i.e., how to partition the network. In detail, an empty residual core reflects a case in which the players of the corresponding residual game do not identify any preferred network partition, or in our proposed FAP cooperation scenario, can equivalently choose between cooperating or not.

Therefore, for the proposed FAPs coalitional game, the emptiness of a residual core does not happen and this can be justified as follows. As per Definition 4, the recursive core is evaluated through a sequence of residual games over subsets of players (i.e., FAPs in our case) in the network. When a given residual core is empty, it is still possible to solve a larger game which contains it, as a residual game, in a nested fashion. Hence, the existence of the recursive core is in fact guaranteed as long as one can find at least one residual core that is nonempty. Thus, the recursive core is a solution concept that exists for any game in partition form, unless all the residual cores are empty. This latter case is very unlikely since it would represent a network in which FAPs are indifferent (i.e., have the same payoff) between states in which they are actually suppressing interference (e.g., cooperatively using interference alignment).

In a nutshell, for the proposed FAPs coalitional game, one can use the concept of residual cores in order to find a partition in the recursive core, i.e., a stable and efficient partition, as will be further described in the next section. To reach a partition in the recursive core, the FAPs can use Algorithm 1.

IV. SIMULATION RESULTS

For system-level simulations, we consider a single hexagonal macrocell with a radius of 500m within which N FAPs are randomly deployed. Each FAP i ∈ N serves L i = 2 FUEs scheduled over L i orthogonal subcarriers which are typical values as in [START_REF] Gpp | Service requirements for Home Node B (HNB) and Home eNode B (HeNB)[END_REF]. FAPs and FUEs are equipped with N t = N r = 2 antennas. An open access policy is adopted at each FAP. We set each FAP's maximum transmit power to P max = 10 dBm, which includes both the power for data transmission and the cost for cooperation in [START_REF] Cadambe | Interference alignment and the degrees of freedom of wireless X networks[END_REF]. Transmissions are affected by distance dependent path loss shadowing according to the 3GPP specifications [START_REF] Gpp | Physical layer aspects for evolved UTRA (Release 7)[END_REF]. Moreover, a wall loss attenuation of 12 dB affects femtocell-to-femtocell transmissions. The considered macrocell has 110 available subcarriers, each one having a bandwidth of 180 KHz, and dedicates 16 OFDMA subcarriers to femtocell transmissions. For both femto users and MUEs, we assume that power control fully compensates for the path loss. The signal-to-noise ratio required for information exchange is ν 0 = 5 dB. The parameters α ik are set to 1 by simulation choice. To leverage channel variations, statistical results are averaged on 10000 simulation rounds.

Figure 1 depicts the average individual FAP payoff as a function of the number of subcarriers dedicated to the femtocell tier, for N = {100, 200, 300}. This figure shows that the ratio between number of FAPs and number of dedicated subcarriers plays a key role, as interference becomes critical over a high congested spectrum. For all of the considered network sizes, the cooperative and non-cooperative strategies lead to similar payoffs when a large portion of spectrum is available, due to the low density of interferers for a given subcarrier. Conversely, the benefit from coalition formation becomes relevant in a high congested spectrum and, in the case of N = 300 allows for a maximum spectrum saving of 38% with respect to the non-cooperative case. In addition, Figure 1 demonstrates that the proposed coalitional game model has a significant advantage over the non-cooperative case, increasing with N and resulting in an improvement of up to 30% for N = 300 and 16 subcarriers.

In Fig. 2, we show the average and the average maximum size of the FAP coalitions in the recursive core for a target SNR for information exchange of ν 0 = 5 dB. Fig. 2 shows that, for small networks, the FAPs have low incentive to cooperate, and, thus, the recursive core is mainly populated by singleton coalitions. In contrast, as N grows, the density of FAPs and the interference level increases, thus, a cooperative strategy becomes more rewarding, as seen through the increase in the average size of the coalitions. As a matter of fact, by forming coalitions, nearby interfering FAPs can still utilize the same subcarriers and suppress interference via alignment. However, the maximum size of a coalition is limited by the cost for cooperation as well as the inherent efficiency of IA.

Figure 3 shows the growth of the number of coalitions, i.e., the size of a partition in the recursive core, while the number of FAPs increases. Additionally, the average number of iteration in the proposed algorithm is observed. The network is initially organized in a non-cooperative structure where each FAP represents a singleton coalition, therefore the number of coalitions equals the number of FAPs (grey dotted line in Figure 3) and, since interferers are out of range of cooperation, the number of iterations is minimum. Initially, for N < 50 cooperation seldom occurs, due to the large distance between mutual interferers. As N increases (50 < N < 300), the network topology changes with the emergence of new coalitions. The number of iterations depend on the number of potential coalitional partners which satisfy the constraint in [START_REF] Gou | Degrees of freedom of the K user MIMO interference channel[END_REF]. Therefore, Figure 2 and Figure 3 show that the incentive towards cooperation becomes significant when the femtocells' spectrum becomes more congested and femtocells are densely deployed in the network.

V. CONCLUSIONS In this paper, we have presented a cooperative framework aimed at interference mitigation in a femtocell network. We have formulated the problem as a coalitional game in partition form and proposed a distributed coalition formation algorithm that enables FAPs to decide on whether to cooperate or not, while accounting for the cost in terms of power for information exchange. We have shown that the proposed algorithm reaches a stable partition, lying in the recursive core of the studied game. Within every formed coalition, we have adopted an interference mitigation technique based on the alignment of interfering signals from members of the same coalition. Results have shown that the femtocell performance is critically limited by the interference, therefore, the proposed cooperative strategy among interfering femtocells can provide significant gains, in terms of average payoff per femtocell, reaching up to 30% relative to the non-cooperative case.

Fig. 1 .

 1 Fig. 1. Average individual user payoff as a function of the number of subcarriers dedicated to the femtocell tier for different number of FAPs.

Fig. 2 .

 2 Fig. 2. Average and average maximum coalition size as function of the number of FAPs.

Fig. 3 .

 3 Fig.3. Average number of coalitions and average number of iterations per FAP until convergence to a stable partition in the recursive core.

Clearly, the same constraint applies to the covariances of the interferers j, and it is still denoted with subscript j instead of i.

For example, since femtocells are connected to each other through the X2 interface, the proposed cost can be replaced by a cost for synchronization or for trading additional information such as bandwidth usage or priority policies.


Alexander Kuzmin
email: alexander.kuzmin@pobox.spbu.ru

Anatoly Ryabinin
email: anatoly.ryabinin@pobox.spbu.ru

Sensitivity of Transonic Flow past a Symmetric Airfoil to Free-Stream Perturbations

Keywords: local supersonic regions, shock waves, instability, bifurcations, self-exciting oscillations

Turbulent transonic flow past a flat-sided symmetric airfoil with an elliptic nose is studied numerically. Solutions of the RANS equations are obtained with ANSYS CFX-13 solver using SSG Reynolds-stress and k-ω SST turbulence models. The solutions reveal flow bifurcations in a narrow band of the free-stream Mach number at zero angle of attack. Flow sensitivity to both stationary and time-dependent variations of the angle of attack is analyzed.

Introduction

In 1990s and 2000s, numerical simulations based on the Euler and RANS equations unveiled instability of shock waves on airfoils whose curvature is small enough in the midchord region [START_REF] Jameson | Airfoils admitting non-unique solutions of the Euler equations[END_REF][START_REF] Hafez | Some anomalies of numerical simulation of shock waves. Part II: effect of artificial and real viscosity[END_REF][START_REF] Kuzmin | Aerodynamic surfaces admitting jumps of the lift coefficient in transonic flight[END_REF]. Such airfoils admit flow bifurcations and jumps of the lift coefficient under slight variations of the free-stream conditions. Recently, Kuzmin [START_REF] Kuzmin | Self-sustained oscillations and bifurcations of transonic flow past simple airfoils[END_REF][START_REF] Kuzmin | Bifurcations of transonic flow past flattened airfoils[END_REF][START_REF] Kuzmin | Bifurcations of transonic flow past simple airfoils with elliptic and wedge-shaped noses[END_REF] demonstrated bifurcations of turbulent flow past simple symmetric airfoils constituted by elementary curves.

In this paper, we employ ANSYS CFX-13 finite-volume solver and the SSG Reynolds stress turbulence model in order to verify results obtained in [START_REF] Kuzmin | Bifurcations of transonic flow past simple airfoils with elliptic and wedge-shaped noses[END_REF] for a flat-sided airfoil with an elliptic nose.

Formulation of the problem and a numerical method

We consider the airfoil constituted by an elliptic arc, two parallel segments, and two circular arcs of radius R [START_REF] Kuzmin | Bifurcations of transonic flow past simple airfoils with elliptic and wedge-shaped noses[END_REF]:

y(x) = ±0.045 √ (2 -x/0.3)x/0.3 , 0 ≤ x ≤ 0.3 ; (1)
y(x) = ±0.045 , 0.3 ≤ x ≤ 0.7 ;

(2)

y(x) = ∓b ± √ b 2 + 0.3 2 -(x -0.7) 2 , 0.7 ≤ x ≤ 1 ; (3)
where b = R -0.045 and R = 1.0225 (see Fig. 1). Hereinafter, (x, y) are Cartesian coordinates that are non-dimensionalized by the airfoil length L = 0.5 m. A lens-type computational domain is bounded by two circular arcs Γ i :

x(y) = (-1) i (-105+ √ 145 2 -y 2 ), -100 ≤ y ≤ 100, i = 1, 2.
The width and height of the domain are 80 and 200 lengths of the airfoil chord, respectively.

On the inflow part Γ 1 of the boundary, we prescribe the static temperature T ∞ = 250 K and flow velocity components U ∞ , V ∞ , where V ∞ = U ∞ tanα and α is the angle of attack. On the outflow part Γ 2 , the static pressure p ∞ = 108, 040 Pa is imposed. The turbulence level is We use the standard value of 28.96 kg/kmol for the molar mass, the value of 1004.4 J/(kg K) for specific heat at constant pressure, and 1.831 × 10 -5 kg/(m s) for the molecular dynamic viscosity. It follows from the above that the specific heat at constant volume is 717.3 J/(kg K), and the sound speed a ∞ is 317.02 m/s.

Unstructured surface meshes were generated with ANSYS Workbench platform [START_REF]ANSYS Products[END_REF]. The meshes were composed of quadrangles in a vicinity of the airfoil and triangles outside it. Mesh points were clustered at the shock waves locations, in the wake, and in the boundary layer. A distance from the first grid point to the airfoil was set to 4 × 10 -6 L. This provides the nondimensional distance y + ≈ 1 in the region of the boundary layer separation from the airfoil. As CFX-13 solver needs 3D meshes with at least one cell in the z-direction, they were generated by an extrusion of the 2D meshes.

Solutions of the Reynolds-averaged Navier-Stokes equations were obtained with CFX-13 solver [START_REF]ANSYS Products[END_REF] using a second-order accurate high-resolution discretization scheme for convective terms [START_REF] Barth | The design and application of upwind schemes on unstructured meshes[END_REF]. An implicit second-order accurate backward Euler scheme was employed for the time-stepping.

Verification of the numerical method

First, the method was applied to a benchmark problem on oscillatory transonic flow past a 18% thick circular-arc airfoil at zero angle of attack and Re = 1.1 × 10 7 . Flow oscillations arise due to instability of the shock-induced boundary-layer separation in the aft region. A comparison of our computational results with available experimental and numerical data [START_REF] Geissler | Transition and turbulence modelling for dynamic stall and buffet[END_REF][START_REF] Mcdevitt | Transonic flow about a thick circular-arc airfoil[END_REF] showed a good agreement. For instance, at M ∞ = 0.75 the calculated amplitude of lift coefficient oscillations was 0.30 when the classical k -ω SST turbulence model was employed, and 0.36 when the SSG Reynolds stress model was used. The latter agrees well with the amplitude of 0.37 obtained numerically in [START_REF] Geissler | Transition and turbulence modelling for dynamic stall and buffet[END_REF] using the Spalart-Allmares and Baldwin-Lomax turbulence models.

For gradually decreasing M ∞ , computations demonstrated an onset of flow oscillations over the circular-arc airfoil in the range as follows:

0.723 ≤ M ∞ ≤ 0.780, α = 0.
A slight discrepancy with the test observations [START_REF] Mcdevitt | Transonic flow about a thick circular-arc airfoil[END_REF], which showed the flow oscillations at 0.732 ≤ M ∞ ≤ 0.777, can be explained by a special adaptation of test section used in the experiments. In fact, the experiments were performed in a wind-tunnel with a small size of the test section whose walls were adapted to simulate streamlines over the airfoil in infinite atmosphere at M ∞ = 0.775. Therefore, at smaller values of M ∞ , the adaptation inadequately simulated the streamlines and admitted an adverse influence of the walls on the flow. For the flat-sided airfoil ( 1)-( 3), numerical solutions of the problem formulated in Section 2 also showed self-exciting oscillations of the separated boundary layers in the aft region. Figure 2 presents maxima and minima of the lift coefficient in the course of periodic oscillations of an asymmetric flow past airfoil (1)-( 3) at α = 0. These results were obtained using the classical k -ω SST turbulence model and four different meshes. A technique for achieving asymmetric regimes at zero angle of attack is described below in Section 4.

As seen from Fig. 2, the bifurcation band of U ∞ , in which there exists asymmetric flow at α = 0, depends insignificantly on the mesh refinement. At the same time, the amplitude of lift coefficient oscillations is sensitive to the mesh. Indeed, at the right end of the bifurcation band, about U ∞ = 274 m/s, computations on meshes composed of 127,348 or 47,704 elements yield considerable amplitudes, whereas computations on meshes of 169,394 or 93,814 elements demonstrate no oscillations.

Employment of the k -ϵ turbulence model instead of k -ω SST caused a decay of the oscillations for all meshes. In this case, the dependence of C L on U ∞ looks like the one for the inviscid flow (see the dashed curve in Fig. 3).

On the other hand, employment of the SSG Reynolds stress turbulence model, which has six additional transport equations [START_REF] Speziale | Modeling the pressure-strain correlation of turbulence: an Invariant Dynamical Systems approach[END_REF], enhances the amplitude of the lift coefficient oscillations. Moreover, the results obtained using this model on meshes of 169,394 and 127,348 elements are close enough (see Fig. 3).

One can conclude from the above that the SSG Reynolds stress turbulence model along with the mesh of 169,394 elements describe most adequately the oscillations of C L for airfoil (1)-( 3). Therefore we used them for a detailed analysis of the transonic flow in the next section.

Numerical results for flow past airfoil (1)-(3) at zero angle of attack α = 0

In order to attain the asymmetric flow regime respective to Figs. Maximum and minimum of the lift coefficient oscillations in this asymmetric regime are indicated by squares in Fig. 5a. A typical dependence of C L on time is displayed in Fig. 6. In the band (4), there exists another asymmetric flow, with a single supersonic region on the lower surface and a double supersonic region on the upper surface of the airfoil (see the sketch on the right in Fig. 5a). This flow regime was obtained in a similar way by using the uniform initial and boundary conditions determined by U ∞ = 273.5 m/s and the negative α = -0.1 deg.

Dependence of the lift coefficient on the angle of attack α

In case of time-dependent angle of attack, the computational mesh must be refined in the region between the inflow boundary Γ 1 and the airfoil in order to provide a good accuracy of the unsteady flow simulation. To avoid a dramatic increase of the total number of mesh nodes, we displaced the inflow boundary downstream and set it at a distance of 4 chord lengths from the airfoil,

First, we examined a dependence of C L on stationary changes of α. Figure 7a depicts the calculated lift coefficient as a function of the angle α at the free-stream velocity U ∞ = 270 m/s. As seen, slight changes of α in a vicinity of zero can result in considerable jumps of C L , from -0.2 to 0.2 and vice versa. This is caused by a joint effect of the oscillating boundary layers in the aft region and alternate coalescence/rupture of local supersonic regions on both sides of the airfoil.

Figure 7b presents C L as a function of α at a larger velocity, U ∞ = 276 m/s. It can be seen that variations of C L under the same changes of α are crucially reduced. This is due to the fact that, on each side of the airfoil, the single supersonic regions persist and don't rupture unless |α| exceeds 0.97 deg.

Finally, we imposed a time-dependent angle of attack α(t) that switches between 0.3 and -0.3 deg every 0.16 s, with a period T = 0.32 s (see Fig. 8). This means that the vertical component V ∞ of the free-stream velocity switches between about 1.4 and -1.4 m/s if U ∞ = 270÷276 m/s. Figure 8a shows considerable variations of the lift coefficient C L (t) calculated at U ∞ = 270 m/s. An increase of U ∞ to 276 m/s entails a dramatic reduction of the variations (Fig. 8b). In the latter case, the frequency of lift coefficient oscillations concerned with the unsteady boundarylayer separation turns out to be high, 1750 Hz. That is why, to make visible crowded maxima and minima of the plot C L (t), the time interval in Fig. 8b has been limited to just one period T .

Conclusion

A comparison of three turbulence models, k -ω SST, k -ϵ, and SSG Reynolds stress, showed that the latter is most appropriate for the numerical simulation of the oscillatory transonic flow past airfoil (1)-( 3). The study confirmed that there exist bifurcation bands of the free-stream velocity U ∞ and angle of attack α, in which airfoil (1)-(3) admits up to three different flow regimes, and slight changes of U ∞ or α can result in considerable jumps of the lift coefficient.

Figure 1 :

 1 Figure 1: Sketch of the airfoil (1)-(3).

Figure 2 :

 2 Figure 2: Maxima C L,max and minima C L,min of the lift coefficient oscillations in time as functions of U ∞ in asymmetric flow over airfoil (1)-(3) at zero angle of attack. Computations are performed on four different meshes using the classical k -ω SST turbulence model.

Figure 3 :

 3 Figure 3: Maxima C L,max and minima C L,min of the lift coefficient oscillations in time as functions of U ∞ in asymmetric flow over airfoil (1)-(3) at zero angle of attack. Computations are performed using three different models of turbulence and for inviscid flow.

Figure 4 :

 4 Figure 4: Mach number contours in an asymmetric flow past the airfoil (1)-(3) at an instant t for U ∞ = 273.5 m/s, α = 0, Re = 1.1 × 10 7 .

2 and 3 , 4 )Figure 5 :

 345 Figure 5: Margins of the lift coefficient oscillations versus U ∞ for the airfoil (1)-(3) at α = 0, Re = 1.1 × 10 7 : (a) asymmetric flow regimes, (b) symmetric flow regimes.

Figure 6 :

 6 Figure 6: Dependence of the lift coefficient on time in an asymmetric flow past the airfoil (1)-(3) at U ∞ = 273.5 m/s, α = 0, Re = 1.1 × 10 7 .

Figure

  Figure 5b presents margins of the lift coefficient oscillations for flows that are symmetric about the axis y = 0 on average in time. Computations were performed using the uniform free-stream at α = 0 as an initial state. For 273 ≤ U ∞ , m/s ≤ 274, a symmetric flow past airfoil (1)-(3) does not exist because of the intrinsic instability of closely spaced local supersonic regions. The Reynolds number based on L = 0.5 m and U ∞ = 273.2 m/s is 1.1 × 10 7 . The frequency of lift coefficient oscillations is about 160 Hz for the asymmetric flow regimes and the symmetric regime at 270 ≤ U ∞ , m/s ≤ 272.9. Meanwhile, for the symmetric flow at larger velocities (see Fig. 5b), the frequency increases from 147 Hz to 169 Hz as U ∞ increases from 274.1 to 277 m/s.

Figure 7 :

 7 Figure 7: Margins of the lift coefficient oscillations versus stationary variations of the angle of attack α in case of the reduced distance from the inflow boundary Γ 1 to airfoil (1)-(3): (a) U ∞ = 270 m/s, (b) U ∞ = 276 m/s.

  Γ 1 : x(y) = 1248 -√ 1252 2 -y 2 , -100 ≤ y ≤ 100. A hybrid mesh of 94, 183 elements was generated in the modified domain. As in Section 4, we employed the SSG Reynolds stress turbulence model and boundary data T ∞ = 250 K, p ∞ = 108, 040 Pa.

Figure 8 :

 8 Figure 8: Lift coefficient as a function of time for the airfoil (1)-(3) at the unsteady angle of attack; (a) U ∞ = 270 m/s, (b) U ∞ = 276 m/s. The modification of the computational domain shifted the bifurcation band (4) to smaller values of U ∞ : 269.7 ≤ U ∞ , m/s ≤ 272.7.(5)

Acknowledgement.

This work was supported by the Russian Foundation for Basic Research under a grant no. 11-08-00643a.


Nicolas Marchand
email: nicolas.marchand@gipsa-lab.inpg.fr

Sylvain Durand
email: sylvain.durand@inrialpes.fr

Jose Fermi

Guerrero Castellanos
email: fguerrero@ece.buap.mx

† S Durand

‡ J F Guerrero Castellanos

A General Formula for the Stabilization of Event-Based Controlled Systems

In this paper, a universal formula is proposed for event-based stabilization of general nonlinear systems affine in the control. The feedback is derived from the original one proposed by Sontag. Under the assumption of the existence of a smooth Control Lyapunov Function, it enables smooth (except at the origin) global asymptotic stabilization of the origin while ensuring that the sampling interval do not contract to zero. Indeed, for any initial condition within any given closed set the minimal sampling interval is proved to be strictly positive. Under homogeneity assumptions the control can be proved to be smooth anywhere and the sampling intervals bounded below for any initial condition.

INTRODUCTION

The classical so-called discrete time framework of controlled systems consists in sampling the system uniformly in time with a constant sampling period T and in computing and updating the control law every time instants t k = k • T . This field, denoted as the time-triggered case (or the synchronous case in sense that all the signal measurements are synchronous), has been widely investigated for linear control systems (see [START_REF] Åström | Computer Controlled Systems[END_REF] and the references therein), even in the case of delays, sampling jitter and measurement loss that can be seen as a kind of asynchronicity [START_REF] Cervin | How does control timing affect performance? analysis and simulation of timing using jitterbug and TrueTime[END_REF]. In the case of nonlinear control systems, one way to address a discrete-time feedback is to implement a continuous time control algorithm with a sufficiently small sampling period [START_REF] Hsu | The effect of discretized feedback in a closed loop system[END_REF]. However, the hardware used to sample and hold the plant measurements or compute the feedback control action may make it impossible to reduce the sampling period to a level that guarantees acceptable closed-loop performance. Other way to tackle this problem is the application of sampled-data control algorithms based on an approximate discrete-time models of the process [START_REF] Nešić | A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models[END_REF] which is not a trivial task. Another proposed approach consist to modify a continuous time stabilizing control using for instance Sontag's general formula to obtain a redesigned control suitable for sampled-data implementation [START_REF] Nešić | Lyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation[END_REF].

To overcome these drawbacks, event-based control -also called event-triggered control -has been recently proposed. In this control strategy, the control task is executed after the occurrence of an external event, generated by an event mechanism. Thus in this scheme, the term sampling time denotes a time interval between two consecutive events (e.g. level crossings of the measure). Hence, two successive sampling instants may not be equidistant in time. Let us first consider general nonlinear systems of the form:

ẋ = f (x, u) (1)
where x ∈ X ⊂ R n , u ∈ U ⊂ R p , and f a Lipschitz function vanishing at the origin. For sake of simplicity, we only consider in this paper null stabilization with initial time instant t 0 = 0. By event-based feedback, it is usually meant a set of the two following functions:

• an event function e : X × X → R that indicates if one needs (when e ≤ 0) or not (when e > 0) to recompute the control. In its more general form, the event function e takes the current state x as input and a memory m of its value last time e became negative. A memoryless version is also possible, that is an event function e : X → R that only requires the current value of the state. There is a priori no constraint on the regularity of e.

• a feedback function k. We talk about static event-based feedback when k : X → U . The time (then k : X × R + → U ) or simply the sampling index (then k : X × N → U ) can be added to build a dynamic event-based feedback.

Additionally, e can depend upon the time (then e : X ×R + → {0, 1}) or simply upon the sampling index (then e : X ×N → {0, 1}) and one then talks about dynamic event-based feedback. Classical sampled feedback of period T can be seen as a dynamic event-based feedback with e(x, t) ≤ 0 if and only if t/T ∈ N. Typical event-detection mechanisms are functions on the variation of the state (or at least the output) of the system, like in [START_REF] Årzén | A simple event-based PID controller[END_REF][START_REF] Sandee | Event-driven control as an opportunity in the multidisciplinary development of embedded controllers[END_REF][START_REF] Durand | Further results on event-based PID controller[END_REF][START_REF] Sánchez | Comparative study of event-based control strategies: An experimental approach on a simple tank[END_REF][START_REF] Sánchez | On the application of different event-based sampling strategies to the control of a simple industrial process[END_REF]. Although the event-triggered control is wellmotivated and allows to relax the periodicity for computations of the control law, only few works report theoretical results about the stability, convergence and performance of event-triggered control systems. In [START_REF] Åström | Comparison of Riemann and Lebesque sampling for first order stochastic systems[END_REF] for instance, it is proved that such an approach reduces the number of sampling instants for the same final performance. Recent works deal with the problem of scheduling the control task for continuous-time linear systems [START_REF] Åström | Comparison of Riemann and Lebesque sampling for first order stochastic systems[END_REF][START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF][START_REF] Lunze | A state-feedback approach to event-based control[END_REF][START_REF] Donkers | Output-based event-triggered control with guaranteed L ∞gain and improved event-triggering[END_REF] and discretetime linear system [START_REF] Eqtami | Event-triggered control for discretetime systems[END_REF] where stability and some robustness proprieties such as ISS and L ∞performance are exploited. Furthermore, in [START_REF] Eqtami | Event-triggered control for discretetime systems[END_REF] a Model Predictive Control scheme is used where the event-triggered policies are used for relaxing the computationally demanding algorithms. Some of the above contributions do not need the memory of the last sample, the event function e is memoryless, that is, it can simply be formulated as a function of X .

An alternative approach consists in taking e related to the variation of a Lyapunov function -and consequently to the state too -between the current state and the previous sample, like in [START_REF] Mazo | On self-triggered control for linear systems: Guarantees and complexity[END_REF][START_REF] Velasco | On Lyapunov sampling for event-driven controllers[END_REF], or in taking e related to the time derivative of the Lyapunov function. Convergence and stability in the nonlinear case is studied in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF][START_REF] Anta | Self-triggered stabilization of homogeneous control systems[END_REF][START_REF]To sample or not to sample: Self-triggered control for nonlinear systems[END_REF]. The main contribution from the above mentioned works is the existence of a minimal time between consecutive executions of the control task guaranteeing desired levels of performances in the absence of accumulation points. However, in these works is assumed the Input-to-State Stability (ISS) property of the system which is a very strong assumption. Moreover, these techniques are developed for two classes of nonlinear control systems, namely, state-dependent homogeneous systems and polynomial systems.

The solution of (1) with event-based feedback (e, k) starting in x 0 ∈ X at t = 0 is then defined as the solution of the differential system (when it exists, a discussion follows on that subject):

ẋ = f (x, k(m)) (2)
m(x) = x if e(x, m) ≤ 0, x = 0 m elsewhere (3)
with:

x(0) = x 0 and m(0

) = x(0) (4)
If f is assumed to be Lipschitz, and events are punctual, a unique solution in the Caratheodory sense always exists without any smoothness assumption on k similarly to [START_REF] Clarke | Asymptotic controllability implies feedback stabilization[END_REF] when . However, this solution may not exists for all t ≥ 0 as shown in item 3 of section 2. Let t → x(t, x 0 ) denote this solution. Given an event function e, and a feedback k defined as above, for any initial condition x(t = 0) = x 0 it fully defines a sampling set T e,k,x0 := {t 0 , t Notations: In the following, B(d, x) will stand for the ball of radius d centered at x and B(d) for the ball of radius d centered at the origin. x(t; x 0 , t 0 , u) will denote the solution of a differential system starting in x 0 at t 0 with control u. For sake of simplicity, u will be omitted when trivial and x(t; x 0 ) will stand for x(t; x 0 , 0).

WHAT CAN HAPPEN WITH EVENT-BASED CONTROL ?

To illustrate different phenomena that can arise with event-based feedback systems, we consider the simple linear integrator ẋ = u. All event functions considered in these examples are assumed to be memoryless that is to be just functions of X . Between two sampling instants t i and t i+1 , u remains constant so that: and the trajectory x(t) = 0 is defined for all t ∈ [0, +∞[.

x i+1 = x i + (t i+1 -t i ) • u, x i

k(x)

= -x, e(x) = 0 when |x| = exp(-κ), κ ∈ Z and initial condition x 0 = 1. Assuming that at time t i of the sampling set T e,k,x0 , the state of the system is x i = ± exp(-κ i ), then at the next sampling instant t i+1 , the state of the system becomes

x i+1 = ± exp(-κ i+1 ) = ± exp(-κ i -1
). The sampling is therefore periodic of period 1 -exp(-1):

T e,k,x0 := {j • (1 -exp(-1)), j ∈ N}
The trajectory is well-defined for all t ∈ [0, +∞[.

3. k(x) = -x, e(x) = 0 when |x| = exp(-κ • |κ|), κ ∈ Z and initial condition x 0 = 1.
In that case, one can calculate that t i = i -i j=1 exp(-2j + 1). And when i tends to infinity, t i tends to t := exp(-1)

1-exp(-2) . The trajectory is then well-defined only for all t ∈ [0, t[.

4. k(x) = -x 3 , e(x) = 0 when |x| = exp(-κ), κ ∈ Z and initial condition x 0 = 1. In that case, t i+1 -t i = x i+1 -x i -x 3 i = exp(2i) • [1 -exp(-1)]
and when i tends to infinity, t i+1 -t i also tends to infinity. Although the inter-sampling duration tends to infinity, the trajectory is well-defined for all t ∈ [0, +∞[ as in case 2.

k(

x) = -x • (1 -exp(-1)) • (1 -log |x|) if |x| ≤ 1, k(x) = -x • (1 -exp(-1)
) elsewhere, and k(0) = 0 by continuity, e(x) = 0 when |x| = exp(-κ), κ ∈ Z and initial condition x 0 = 1.

Here, one can prove that t i+1 -t i = 1 i+1 and therefore t i+1 -t i tends to zero when i tends to infinity as in case 3 but the t i 's do not converge to a finite limit.

Consider now the unstable system ẋ = (x + u) 3 . The control u being constant between each sampling instant, the solution is the one of a Bernoulli differential system whose solution is:

x i+1 = x i + u 1 -2(t i+1 -t i ) • (x i + u) 2 -u
Then, taking 6. k(x) = -2x, e(x) = 0 when |x| = exp(-κ), κ ∈ Z and initial condition x 0 = 1. Then the inter-sampling duration is:

t i+1 -t i = exp(2i) 2 • 1 - 1 (2 -exp(-1)) 2
and when i tends to infinity, t i+1 -t i also tends to infinity. However, the origin of the closed loop system can be proved to be asymptotically stable and the trajectories well-defined on [0, +∞[ for all initial condition.

In cases 1 to 6, the system can trivially be proved to be globally null-asymptotically stable taking V (x) := 1 2 x 2 . Cases 1 and 2 show that the sampling set is clearly initial condition dependent. Case 2 to 5 show that for the same system and initial condition, the sampling can be periodic, contractile or expansile (with a finite or infinite limit) depending upon the event function or the feedback. Case 6 shows the inconsistency of the Shannon criteria in the event based paradigm and in particular that the inter-sampling duration can infinitely increase even when insuring the stability of the closed loop of an open-loop unstable system.

PRELIMINARY DEFINITIONS FOR EVENT-BASED SYSTEMS

Usually, the event is of null measure, in the sense that the control is recomputed only at distinct x (in the countable sense). However, taking e(x) = 0 for all x ∈ X would mean that one recomputes the control at each x and therefore that one applies a classical continuous-time feedback. On the sets of non null measure where e(x) = 0, the solution is understood in the classical sense (with all possible solution existence problems if the field is discontinuous). Elsewhere, the solution can be intended in the Caratheodory sense. To go further on that, we define: Definition 3.1 (Well-defined event-based control). An event-based control (k, e) will be said well-defined if and only if for any initial condition x 0 at t = 0, the solution t → x(t; x 0 ) exists for all t ≥ 0. Property 3.1 (Minimal Sampling Interval -MSI). An event-based control (k, e) will be said to fulfill the Minimal Sampling Interval property (MSI) if and only if for any initial condition x 0 at t = 0, there exists a non zero minimal sampling interval τ (x 0 ) defined by:

τ (x 0 ) := inf i∈N,ti∈T (x0)
t i+1 -t i > 0

In that case, the control is piecewise constant between each time sample and we define:

• x i , i ∈ I ⊂ N with x 0 := x(t = 0) as the series of successive values of the state at which e is negative for a given initial condition x 0

• t i , i ∈ I ⊂ N with t 0 := 0 as the corresponding series of time instants

The aim of definition 3.1 is to exclude solutions with sampling intervals converging to zero at some time (case 3 of section 2) or to infinity (case 5 of section 2). It quite trivially follows: Theorem 3.2. An MSI event-based control is well-defined.

Proof. of Theorem 3.2 The proof is trivial since if the event-based control is MSI, then T (x 0 ) is either finite or countably infinite and the t i are isolated in R + . The solution t → x(t; x 0 ) hence exists for all t ≥ 0 in the Caratheodory sense.

This minimal sampling period is useful for implementation purpose but also when the feedback k is discontinuous for robustness purpose [START_REF] Sontag | Clocks and insensitivity to small measurement errors[END_REF]. However, it would be more suitable to have such a bound less depending upon initial condition:

τ := inf i∈N,ti∈T (x0),x0∈X
t i+1 -t i > 0 Properties 3.1 to 3.3 can be specified using the qualifying term "global" when X = R n in opposition to the term "local" that was omitted above for sake of simplicity. Now that the above notions for event-based controlled systems are appropriately defined, notions like stability, asymptotic stability and stabilizability naturally follow since they rely on the resulting trajectory. The question that arises then is: does a universal formula for uniformly discrete event-based feedback stabilization exist similarly to the continuous time case ? This is the purpose of the next section.

A UNIVERSAL FORMULA FOR EVENT-BASED STABILIZATION

In the sequel, the analysis is restricted to systems affine in the control:

ẋ = f (x) + g(x)u = f (x) + i g i (x)u i (5)
where f and g are smooth functions with f vanishing at the origin. We assume that a Control Lyapunov Function (CLF) exists for system [START_REF] Nešić | Lyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation[END_REF], that is a smooth and positive definite function V : X → R so that for each x = 0 there is some u ∈ U such that:

∂V ∂x f (x) + ∂V ∂x g(x)u < 0 (6)
In addition, one may require that the CLF V fulfills the small control property [START_REF]Mathematical control theory, deterministic finite dimensional systems[END_REF], that is that for each ε > 0 there is some δ > 0 such that for any x in the ball B(δ)\ {0}, there is some u with u ≤ ε such that (6) holds. Then, it is known that it is possible to design a smooth feedback control that asymptotically stabilizes the system. This is known as the Sontag's universal formula:

Theorem 4.1 (Sontag's universal formula). If there exists a CLF for system (5), then the feedback k : X → U , smooth on X \ {0} is such that:

∂V ∂x f (x) + ∂V ∂x g(x)k(x) < 0, x ∈ X \ {0} (7)
for k defined by: if y 2 = 0 0 if y 2 = 0 [START_REF] Durand | Further results on event-based PID controller[END_REF] with q : R → R is any real analytic function such that q(0) = 0 and y 2 q(y 2 ) > 0 whenever y 2 = 0. Moreover, if the CLF satisfies the so called small control property, then taking q(y 2 ) := y 2 , the control is continuous at the origin [START_REF]Mathematical control theory, deterministic finite dimensional systems[END_REF].

k i (x) := -b i (x)φ(a(x), β(x)), i ∈ {1, . . .
The main purpose of this paper is to establish that a universal formula also exists in the event-based context up to a slight modification of the original formula proposed by Sontag: Theorem 4.2. If there exists a CLF for system (5), then the event-based feedback (e, k) defined below is semi-uniformly MSI, smooth on X \ {0}, and such that:

∂V ∂x f (x) + ∂V ∂x g(x)k(m) < 0, x ∈ X \ {0} (9)
where m is defined by (3) and:

k i (x) := -b i (x)γ(x) (10) e(x, m) := -a(x)-b(x)k(m)-σ a(x) 2 +β(x)α(x) (11)
where a(x), b(x) and β(x) are as in Theorem 4.1, α : X → R + is any smooth function strictly positive on S := {x ∈ X |β(x) = 0}, σ ∈]0, 1[, and

γ(x) := a(x)+ √ a(x) 2 +β(x)α(x) β(x) if x ∈ S 0 if x / ∈ S (12)
As in Theorem 4.1, if the CLF satisfies the so called small control property, then the control is continuous at the origin. Moreover, if there exists some smooth function w : X → R + strictly positive on S and such that w(x)β(x) -a(x) ≥ 0 on S, then the control is smooth on X with the choice:

α(x) := w(x) 2 β(x) -2w(x)a(x) (13)
Before giving the proof of Theorem 4.2, let us explain the ideas behind the construction of feedback [START_REF] Sánchez | On the application of different event-based sampling strategies to the control of a simple industrial process[END_REF]. In the event function, a(x) + b(x)k(m) is the time derivative of the Lyapunov function V and therefore the event function detects when the Lyapunov function V stops to be enough decreasing. In the control, the term a(x) outside the square root is here to compensate the autonomous evolution of the CLF. The term a(x) inside the square-root is linked to the one outside by smoothness considerations. In the term β(x)α(x), β(x) is added for smoothness reasons in connection with the β(x) at the denominator and α(x) plays a fundamental role by tuning how fast the CLF must decrease when an event occurs. This is the term that enables to avoid too close successive events.

We next focus on homogeneous systems that gave rise to an important literature for general nonlinear systems (see for instance [START_REF] Faubourg | Control Lyapunov functions for homogeneous Jurdjevic-Quinn systems[END_REF][START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF] and the references therein) and more recently for event-based approaches (mainly in [START_REF]To sample or not to sample: Self-triggered control for nonlinear systems[END_REF][START_REF] Mazo | On self-triggered control for linear systems: Guarantees and complexity[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]). However, in these event-based contributions, ISS is assumed contrary to the proposed result. We shortly recall some definitions given for general nonlinear system (see the cited references for more detailed definitions and properties): Definition 4.1. For n positive real numbers r i , i ∈ {1, . . . , n}, d > -min i r i , all λ > 0, and Λ r = diag(λ r1 , λ r2 , . . . , λ rn ),

1. a function V : X → R is homogeneous of degree d if for all x ∈ X , V (Λ r x) = λ d V (x) 2. a function e : X × X → R is homogeneous of degree d if for all x, m ∈ X , e(Λ r x, Λ r m) = λ d e(x, m) 3. a vector field h : X → X is homogeneous of degree d if for all x ∈ X , h(Λ r x) = λ d Λ r h(x)
4. a controlled system of the form [START_REF] Åström | Computer Controlled Systems[END_REF]

with feedback u = k(x) is homogeneous of degree d if k is such that x → f (x, k(x)) is homogeneous of degree d 5. an event-based controlled system of the form (2-4) is homogeneous of degree (d 1 , d 2 ) if k is such that (x, m) → f (x, k(m)
) and (x, m) → e(x, m) are respectively homogeneous of degree d 1 and d 2 .

6. the notation • Λ r will denote the homogeneous p-norm, that is:

x Λ r = n i=1 |x i | p r i 1 p
with p sufficiently large so that the norm is smooth except at the origin.

Property 4.3. Consider now an event-based controlled dynamical system homogeneous of degree (d 1 , d 2 ) then as long as the trajectory exists and is unique, the solution is such that:

x(t; Λ r x 0 ) = Λ r x(λ d1 t; x 0 ) (14)
Proof. of Property 4.3 Between events, ( 14) holds by homogeneity of f (x, k(m)). e being homogeneous of degree d 2 , e(Λ r x, Λ r m) = λ d2 e(x, m). λ being strictly positive, e(Λ r x, Λ r m) and e(x, m) have the same sign, therefore events along the trajectories of t → x(t; Λ r x 0 ) appear for the same t as for t → Λ r x(λ d1 t; x 0 ).

For homogeneous systems, Theorem 4.2 becomes:

Theorem 4.4. Assume that f and each g i are homogeneous respectively of degree d f and d g , identical for all g i , i ∈ {1, . . . , p} with d g < d f . Assume in addition that the CLF V is homogeneous of degree d V , then the following feedback proposed in [START_REF] Tsinias | Remarks on feedback stabilizability of homogeneous systems[END_REF], that corresponds to [START_REF] Sánchez | On the application of different event-based sampling strategies to the control of a simple industrial process[END_REF] with α as in [START_REF] Lunze | A state-feedback approach to event-based control[END_REF] and

w(x) = x d f -2dg-d V Λ r , k i (x) := -νb i (x) x d f -2dg-d V Λ r (15)
with ν > 0 sufficiently large and e as in [START_REF] Åström | Comparison of Riemann and Lebesque sampling for first order stochastic systems[END_REF] is such that:

1. the event-based controlled system is homogeneous of degree (

d f , d V + d f )
2. the event-based control is smooth and uniformly MSI 3. the CLF is strictly decreasing for all x ∈ X \ {0}

The end of the section is dedicated to the proofs of Theorems 4.2 and 4.4.

Proof. of Theorem 4.2 We begin the proof by establishing that γ is smooth on X \ {0}. For this, consider the algebraic equation:

F (x, p) := β(x)p 2 -2a(x)p -α(x) = 0 (16)
Note first that p = γ(x) is a solution of ( 16) for all x ∈ X . It is easy to prove that the partial derivative of F with respect to p is always strictly positive on X \ {0}:

∂F ∂p := 2β(x)p -2a(x)
Indeed, when β(x) = 0, equation ( 6) gives ∂F ∂p = -2a(x) > 0 and when β(x) = 0, equation [START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF] gives ∂F ∂p = 2 a(x) 2 + β(x)α(x) > 0, ∂F ∂p never vanishes at each point of the form {(x, γ(x))|x ∈ X \ {0}}. Furthermore, F is smooth with respect to x and p since so are functions a, β and α. Therefore, using the implicit function theorem, γ is smooth on X \ {0}.

The decrease of the CLF is trivial to prove. Indeed, for each x i , i ∈ N:

dV dt (x i ) = ∂V ∂x (x i )f (x i ) + ∂V ∂x (x i )g(x i )k(x i ) = -a(x i ) 2 + β(x i )α(x i ) < 0 for all x = 0
With the updated control, the event function becomes strictly positive: e(x i , To prove that the event-based control is MSI, we have to prove that for any initial condition in a priori given set, the sampling intervals are bounded below. First of all, notice that events occur only when V vanishes or when x = 0 and therefore from equation ( 9) there is no event on the set {x ∈ X |β(x) = 0} ∪ {0}. We therefore restrict the study to the set S\ {0} where α is strictly positive by assumption. Let us rewrite the time derivative of the CLF along the trajectories:

x i ) = (1 - σ) a(x i ) 2 + β(x i )α(x i ) > 0.
dV dt (x) = a(x) + b(x)k(m) = -a(x) 2 +β(x)α(x)+b(x)(k(m)-k(x)) (17)
where, defining for m ∈ S, ϑ m := V (m) and the set V ϑm := {x ∈ X |V (x) ≤ ϑ m }, x belongs to V ϑx ⊂ V ϑm . Note that although m must belong to S, this is not necessarily the case for x. First see that for t = t i , x = m and therefore, since α is strictly positive on S and a necessarily non zero on the frontier of S (except possibly at the origin):

dV dt (x) = -a(m) 2 + β(m)α(m) ≤ -inf m∈S s.t. V (m)=ϑm a(m) 2 + β(m)α(m) =: -χ(ϑ m ) < 0
Considering now the second time derivative of the CLF:

V (x) = ∂a ∂x (x) + k(m) T ∂b T ∂x (x) (f (x) + g(x)k(m))
By continuity of all the functions involved, both terms can be bounded for all x ∈ V ϑm by the following upper bounds ̺ 1 (ϑ m ) and ̺ 2 (ϑ m ):

̺ 1 (ϑ m ) := sup m∈S s.t. V (m)=ϑm x∈V ϑm ∂a ∂x (x) + k(m) T ∂b T ∂x (x) ̺ 2 (ϑ m ) := sup m∈S s.t. V (m)=ϑm x∈V ϑm f (x) + g(x)k(m)
Therefore, V is strictly negative at any event instant t i and can not vanish until the time τ (ϑ m ) is elapsed and this minimal sampling is only depending on the level ϑ m of the CLF in m:

τ (ϑ m ) ≥ χ(ϑ m ) ̺ 1 (ϑ m )̺ 2 (ϑ m ) > 0 (18)
which ends the proof, the event-based feedback (10-11) is semi-uniformly MSI.

To prove the continuity of k at the origin, we only need to consider the points where β(x) = 0 since we already have k(x) = 0 if β(x) = 0. Considering first the subset where a(x) > 0, we have:

k(x) ≤ |a(x)| + a(x) 2 + β(x)α(x) β(x) ≤ 2 |a(x)| β(x) + α(x)
With the small control property, for any ε > 0, there is δ > 0 such that for any x ∈ B(δ)\ {0}, there exists some u with u ≤ ε such that a(x) + b(x)u < 0. Therefore |a(x)| < β(x)ε. The continuity of α at the origin where it vanishes yields that for the same ε there is a δ ′ > 0 such that for all x ∈ B(δ ′ )\ {0}, α(x) ≤ ε. Therefore, for any x ∈ B(min(δ, δ ′ ))\ {0}:

k(x) ≤ 3ε
Now, if a(x) ≤ 0, using the triangular inequality:

0 ≤ a(x) + a(x) 2 + β(x)α(x) ≤ β(x)α(x)
And therefore using again the continuity of α:

k(x) ≤ α(x) ≤ ε
which ends the proof of the continuity. Finally, with α as in ( 13), the control becomes k i (x) = -b i (x)w(x) which is obviously smooth on X .

Proof. of Theorem 4.4 Take ν such that:

ν > sup x∈{x∈X | x Λ r =1,a(x)≥0} a(x) β(x) x d f -2dg-d V Λ r (19)
As in [START_REF] Tsinias | Remarks on feedback stabilizability of homogeneous systems[END_REF], k is homogeneous of degree d f -d g and the system is therefore homogeneous of degree d f . In addition e is homogeneous of degree d V + d f . Item 1 therefore holds. Thanks to Theorem 4.2, item 3 also holds and k is smooth on X . To finish the proof, remains to establish that the event-based feedback is uniformly MSI. For this, we invoke the homogeneity of the Lyapunov level sets V ϑm together with [START_REF] Donkers | Output-based event-triggered control with guaranteed L ∞gain and improved event-triggering[END_REF], it follows that for all ϑ m > 0:

τ (ϑ m ) = τ (1)

EXAMPLES

Consider the linear time-invariant system:

ẋ = Ax + Bu (20)
Take P , a positive definite matrix solution of the Riccati equation P A+A T P -4εP BB T P = -P (that can be proved to exist as soon as (A, B) is a stabilizable pair). Then V (x) := x T P x is a CLF for system [START_REF]To sample or not to sample: Self-triggered control for nonlinear systems[END_REF] since for all x = 0, u = -2εB T P x renders V strictly negative. Since a(x) = x T (P A + A T P )x, b(x) = 2x T P B and β(x) = 4x T P BB T P x, the Riccati equation gives:

εβ(x) -a(x) = x T P x = V (x) ≥ 0 ∀x = 0 Therefore, taking w(x) = ε, and α(x) according to [START_REF] Lunze | A state-feedback approach to event-based control[END_REF] the control is smooth everywhere and linear: k(x) = -εb(x) T

We next consider the nonlinear system proposed in [START_REF] Jankovic | CLF based designs with robustness to dynamic input uncertainties[END_REF]:

ẋ1 = -x 5 1 + x 1 x 2 ẋ2 = x 2 2 + u (21)
which admits V (x) = 1 2 x 2 as CLF and a(x) = -x 6 1 + x 2 1 x 2 + x and σ = 0.1, it gives the trajectories of the states, control and event function represented in Figure 1. Now, taking w(x)

= 1 2 (1 + x 2 ) 2 , w(x)β(x) -a(x) = x 3 1 x 2 1 -1 2 -1 2 1 2 x 3 1 x 2 + 1 2 x 4 2
is strictly positive for all x = 0. Therefore with α(x) as in ( 13), the control is smooth. The resulting trajectories with σ = 0.1 are represented in Figure 2.

CONCLUSION

In this paper, we proposed an extension of the universal formula for smooth feedback stabilization to event-based controlled systems. A modification of the original formula is necessary to ensure that there is a minimal sampling interval between two consecutive events avoiding phenomena like accumulation points. As in the original work, if the Control Lyapunov Function fulfills the small control property, then the control is continuous at the origin. With additional homogeneity assumptions, the control can be proved to be smooth everywhere and the minimal sampling intervals bounded below for all initial conditions.

  denoting the value of the state when the i th event occurs. With the following different feedback laws and event functions and initial conditions, it gives: 1. k(x) = -x, e(x) = 0 when |x| = exp(-κ), κ ∈ Z and initial condition x 0 = 0. Then T e,k,x0 := {0}

Property 3 . 3 (Property 3 . 4 (

 3334 Semi-uniformly MSI event-based control). An event-based control (k, e) will be said semi-uniformly MSI if and only if for any δ > 0:τ (δ) := inf i∈N,ti∈T (x0),x0∈B(δ)t i+1 -t i > 0 Uniformly MSI event-based control). An event-based control (k, e) will be said uniformly MSI if and only if:

  , p} where a(x) := ∂V ∂x f (x), b(x) := ∂V ∂x g(x), β(x) := b(x)

3 2 ,

 2 b(x) = x 2 . Taking α(x) = β(x) + a(x) 2

Figure 1 :

 1 Figure 1: System (21) under event-based control with α(x) = β(x) + a(x) 2 . The red dots represent the events instants.

Figure 2 :

 2 Figure 2: System (21) under smooth event-based control. The red dots represents the events instants.

  Therefore, by smoothness of the Lyapunov function V , f and g, it clearly follows that dV dt (x(t; x i , t i )) < 0 for all t ∈ [t i , t i+1 [, that is until the next event occurs. t i+1 is necessarily bounded since, if not, the Lyapunov function V should converge to a constant value where dV dt = 0. The event function precisely prevents this phenomena detecting dV dt is close to vanish and updates the control if it happens.