Mahmoud Abdelrahim

Romain Postoyan

Jamal Daafouz

Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics ⋆

Keywords: Event-triggered, Singular Perturbation, Hybrid Systems

Controllers are often designed based on a reduced or simplified model of the plant dynamics. In this context, we investigate whether it is possible to synthesize a stabilizing event-triggered feedback law for networked control systems (NCS) which have two time-scales, based only on an approximate model of the slow dynamics. We follow an emulation-like approach as we assume that we know how to solve the problem in the absence of sampling and then we study how to design the eventtriggering rule under communication constraints. The NCS is modeled as a hybrid singularly perturbed system which exhibits the feature to generate jumps for both the fast variable and the error variable induced by the sampling. The first conclusion is that a triggering law which guarantees the stability and the existence of a uniform minimum amount of time between two transmissions for the slow model may not ensure the existence of such a time for the overall system, which makes the controller not implementable in practice. The objective of this contribution is twofold. We first show that existing event-triggering conditions can be adapted to singularly perturbed systems and semiglobal practical stability can be ensured in this case. Second, we propose another technique that combines event-triggered and time-triggered results in the sense that transmissions are only allowed after a predefined amount of time has elapsed since the last transmission. This technique has the advantage, under an additional assumption, to ensure a global asymptotic stability property and to allow the user to directly tune the minimum inter-transmission interval. We believe that this technique is of its own interest independently of the two-time scale nature of the addressed problem. The results are shown to be applicable to a class of globally Lipschitz systems.

Introduction

The increasing popularity of embedded systems and networked control systems has motivated the development of new implementation paradigms in order to handle the resources limitations of these systems. Indeed, although periodic sampling is appealing from the analysis and implementation point of view, it may yield a conservative solution as it may unnecessarily use the network. Eventtriggered control has been proposed as an alternative where it is the occurrence of an event, typically a varia-⋆ The material of this paper was partially presented at the 9th IFAC Symposium on Nonlinear Control Systems (NOL-COS 2013), Toulouse, France. This work has been supported by the European 7th Framework Network of Excellence 'Highly-complex and networked control systems' (HY-CON2) (grant agreement no. 257462) and also by the ANR under the grant COMPACS (ANR-13-BS03-0004-02).

Email addresses: othmanab1@univ-lorraine.fr (Mahmoud Abdelrahim), romain.postoyan@univ-lorraine.fr (Romain Postoyan), jamal.daafouz@univ-lorraine.fr (Jamal Daafouz). tion of the plant state and not a clock, which closes the feedback loop [START_REF] Årzén | A simple event-based PID controller[END_REF], [START_REF] Åström | Comparison of periodic and event based sampling for first-order stochastic systems[END_REF]. This may allow to significantly reduce the utilization of the resources compared to the periodic implementation, see e.g. [START_REF] Donkers | Outputbased event-triggered control with guaranteed L∞-gain and improved and decentralised event-triggering[END_REF], [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF], [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF], [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF]. Available techniques rely on the knowledge of an accurate model of the plant (which may be affected by uncertainties or external disturbances). However, the controller is often designed based on a reduced or simplified model of the plant dynamics. For two time-scale systems for instance, singular perturbation theory can be used to approximate the slow and the fast dynamics, see [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Kokotović | Singular Perturbation Methods in Control: Analysis and Design[END_REF]. In this context, it is possible to design the controller based only on the slow model, when the origin of the fast model is asymptotically stable, for stabilizable linear time-invariant (LTI) systems [START_REF] Kokotović | Singular Perturbation Methods in Control: Analysis and Design[END_REF], classes of nonlinear systems (see Section 5.4 in [START_REF] Khalil | Nonlinear Systems[END_REF]) and linear time-varying sampled data systems with periodic sampling [START_REF] Pan | H ∞ -optimal control for singularly perturbed systems with sampled-state measurements[END_REF]. In this paper, we investigate whether this approach is applicable for event-triggered control.

We consider the scenario where the controller communicates with a two-time scale nonlinear system via a dig-ital communication channel. Our objective is to design a stabilizing event-triggered feedback law based only on an approximate model of the slow dynamics. This problem is motivated by the fact that engineers often neglect the fast stable dynamics in practice when these are stable and design the feedback law based only on the slow model. To the best of our knowledge, this is the first paper in that direction. It is important to stress that the stability of the approximate slow and fast models is not sufficient to conclude the overall stability of the original system even for continuous-time systems, see [START_REF] Lobry | Singular perturbations methods in control theory[END_REF] for a counter example. Hence, although the approximate fast subsystem is assumed to be stable and the approximate slow dynamics is stabilized by the event-triggered controller, a thorough analysis has to be made to ensure the stability of the original system.

We cast the overall problem as a hybrid singularly perturbed system with the formalism of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]. The stability of this type of systems is analysed in [START_REF] Sanfelice | On singular perturbations due to fast actuators in hybrid control systems[END_REF], [START_REF] Wang | Analysis for a class of singularly perturbed hybrid systems via averaging[END_REF], [START_REF] Wang | Averaging in singularly perturbed hybrid systems with hybrid boundary layer systems[END_REF]. In this study, we address a design problem as we construct the flow and jump sets (i.e. the triggering condition) and we propose different stability analyses under a different set of assumptions. We highlight a specific challenge which arises with the event-triggered implementation: the state of the fast model experiences a jump at each transmission due to the change of variables we introduce to separate the slow and the fast dynamics using singular perturbation theory. These jumps induce non-trivial difficulties in the stability analysis. That is a feature of the problem which is not present in available results on event-triggered control where only the sampling-induced error is reset to zero at each transmission, see e.g. [START_REF] Donkers | Outputbased event-triggered control with guaranteed L∞-gain and improved and decentralised event-triggering[END_REF], [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF], [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF], [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF].

We follow an emulation-like approach to design the event-triggered controllers (see [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF], [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]). We first synthesize a stabilizing controller for the approximate slow model obtained by singular perturbation theory, in the absence of communication constraints. Afterwards, we take into account the effect of the network and we design the event-triggering condition. The first observation we make is that, even if the triggering law guarantees the asymptotic stability of the origin of the slow model and the existence of a strictly positive lower bound on the inter-transmission times, such a time is no longer guaranteed to exist for the overall system. As a consequence, the controller is not implementable in practice. We then propose two classes of event-triggered controllers which overcome this issue. The first policy relies on the event-triggering conditions [START_REF] Donkers | Outputbased event-triggered control with guaranteed L∞-gain and improved and decentralised event-triggering[END_REF], [START_REF] Mazo | Decentralized event-triggered control with one bit communications[END_REF] but it requires to fully modify the stability analysis to handle the features of the problem due to the two-time scale nature of the system. We show that a semiglobal practical stability property holds where the adjustable parameter appears in the event-triggering condition. The second technique combines the event-triggered implementation of [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] with the time-triggered results in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], like in [START_REF] Mazo | Decentralized event-triggered control over wireless sensor/actuator networks[END_REF], [START_REF] Tallapragada | Decentralized event-triggering for control of nonlinear systems[END_REF], [START_REF] Tallapragada | Event-triggered decentralized dynamic output feedback control for LTI systems[END_REF], [START_REF] Wang | Asynchronous task execution in networked control systems using decentralized event-triggering[END_REF], in the sense that transmissions are only allowed after a predefined amount of time has elapsed since the last transmission. This allows us to directly tune the minimum transmission interval. We show that a global asymptotic stability property is satisfied in this case, under an additional assumption. The results are shown to be applicable to a class of globally Lipschitz systems, which include stabilizable LTI systems as a particular case.

The remainder of the paper is organised as follows. The problem is stated in Section 3. The main assumptions are presented in Section 4. In Section 5, we state the main results. In Section 6, we show that the proposed eventtriggered control strategies are applicable to a class of globally Lipschitz systems. The proofs are given in the Appendix.

Preliminaries

We

denote R = (-∞, ∞), R ≥0 = [0, ∞), Z ≥0 = {0, 1, 2, ..}. The Euclidean norm is denoted as |.|.
We use the notation (x, y) to represent the vector

[x T , y T ] T for x ∈ R n and y ∈ R m . A continuous func- tion γ : [0, ∞) → R ≥0 is of class K if it is zero at zero, strictly increasing, and it is of class K ∞ if in addition γ(s) → ∞ as s → ∞. A continuous func- tion γ : R ≥0 × R ≥0 → R ≥0 is of class KL if for each t ∈ R ≥0 , γ(., t) is of class K,
and, for each s ∈ R ≥0 , γ(s, .) is decreasing to zero. We denote the minimum and maximum eigenvalues of the symmetric positive definite matrix A as λ min (A) and λ max (A) respectively. We will consider locally Lipschitz Lyapunov functions (that are not necessarily differentiable everywhere), therefore we will use the generalized directional derivative of Clarke which is defined as follows. For a locally Lipschitz function V : R n → R ≥0 and a vector υ ∈ R n , V • (x; υ) := lim sup h→0 + , y→x (V (y + hυ) -V (y))/h. For a C 1 function V , V • (x; υ) reduces to the standard directional derivative ∇V (x), υ , where ∇V (x) is the (classical) gradient. We will use the following result which corresponds to Proposition 1.1 in [START_REF] Nešić | A Lyapunov-based small-gain theorem for hybrid ISS systems[END_REF].

Lemma 1 Consider two continuously differentiable functions U 1 : R n → R and U 2 : R n → R. Let A := {x : U 1 (x) > U 2 (x)}, B := {x : U 1 (x) < U 2 (x)} and Γ := {x : U 1 (x) = U 2 (x)}. For any υ ∈ R n , the func- tion U : x → max{U 1 (x), U 2 (x)} satisfies U • (x; υ) = ∇U 1 (x), υ for all x ∈ A, U • (x; υ) = ∇U 2 (x), υ for all x ∈ B and U • (x; υ) = max{ ∇U 1 (x), υ , ∇U 2 (x), υ } for all x ∈ Γ.

Problem statement

Consider the following nonlinear time-invariant singularly perturbed system ẋ = f (x, z, u), ǫ ż = g(x, z, u),

where x ∈ R nx and z ∈ R nz are the states, u ∈ R nu is the control input and ǫ > 0 is a small parameter. We use singular perturbation theory to approximate the slow and the fast dynamics. We rely on the following standard assumption (see (11.3)- (11.4) in [START_REF] Khalil | Nonlinear Systems[END_REF]).

Assumption 1 The equation g(x, z, u) = 0 has n ≥ 1 isolated real roots

z = h i (x, u), i = 1, 2, ..., n, (2)
where h i is continuously differentiable.

In that way, the substitution of the ith root z = h(x, u) into the x-system yields the corresponding approximate slow model ẋ = f (x, h(x, u), u).

To investigate stability, it is more convenient to write system (1) with the coordinates (x, y) where

y := z -h(x, u) (4)
is introduced to shift the quasi-steady-state of z to the origin. Then we derive the approximate fast dynamics

dy dτ = g(x, y + h(x, u), u), (5)
where τ := (tt 0 )/ǫ and x ∈ R nx is treated as a fixed parameter.

In this study, we want to stabilize system (1) using a controller which is implemented over a network. We opt for an event-triggered implementation in the sense that transmissions are not triggered by a clock but according to a state-dependent criterion, which may reduce the utilization of the network compared to the periodic approach. Moreover, we concentrate on the case where the approximate fast dynamics ( 5) is stable and we aim at designing the feedback law based only on the slow model (3) as explained in Section 1.

We follow an emulation-like approach as we first assume that the slow model ( 3) can be stabilized by a controller of the form u = k(x). Afterwards, we take into account the effects of the network and we synthesize appropriate triggering conditions. The controller receives the state measurements only at the transmission instants t i , i ∈ Z ≥0 and we consider zero-order-hold devices. In that way

u(t) = k(x(t i )) for all t ∈ [t i , t i+1
). The sequence of transmission instants t i , i ∈ Z ≥0 is defined by the event-triggering condition we will design. We introduce the sampling-induced error e ∈ R nx as in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], which is defined by e(t) := x(t i )x(t) for all t ∈ [t i , t i+1 ) and which is reset to zero at each transmission instant. The state feedback controller is therefore given by

u = k(x + e). (6)
Hence, in view of (4), the variable y becomes

y = z -h(x, k(x + e)). (7)
As a consequence, the system in the (x, y) coordinates is, for all

t ∈ [t i , t i+1 ), i ∈ Z ≥0 ẋ = f x, y + h(x, k(x + e)), k(x + e) =: f x (x, y, e) (8)
ǫ ẏ = g x, y +h(x,k(x+e)), k(x+e) -ǫ ∂h ∂x f x (x, y, e) =: f y (x, y, e), (9)
and at each transmission

x(t + i+1 ) = x(t i+1 ) (10)
y(t + i+1 ) = z(t + i+1 ) -h x(t + i+1 ), k(x(t + i+1 ) + e(t + i+1 )) = z(t i+1 ) -h x(t i+1 ), k(x(t i+1 )) = y(t i+1 ) + h x(t i+1 ), k(x(t i+1 ) + e(t i+1 )) -h x(t i+1 ), k(x(t i+1 )) =: h y (x(t i+1 ), y(t i+1 ), e(t i+1 )). ( 11
)
It has been shown in [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF] that additional variables may be useful when designing the event-triggering condition. This will be the case for one of the strategies we propose in Section 5. We denote these extra variables by a single vector τ ∈ R nτ . The problem is modeled using the hybrid formalism of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] (like in [START_REF] Donkers | Outputbased event-triggered control with guaranteed L∞-gain and improved and decentralised event-triggering[END_REF], [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF])

q = F (q) for q ∈ C, q + = G(q) for q ∈ D, (12)
where q = (x, y, e, τ ) ∈ R nq , n q = 2n x + n y + n τ ,

F (q) :=        f x (x, y, e) 1 ǫ f y (x, y, e) -f x (x, y, e) f τ (x, y, e, τ )        , G(q) :=        x h y (x, y, e) 0 h τ (x, y, e, τ )        , (13)
and f τ and h τ are designed vector fields which respectively define the dynamics of τ on flows and at jumps. The flow set C and the jump set D in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] are defined according to the event-triggering condition which we will synthesize in the following. The system flows on C where the triggering condition is not satisfied and experiences a jump on D where the triggering condition is verified. When q ∈ C ∩ D, the system can either jump or flow, the latter only if flowing keeps q in C. The flow map F and the jump map G are assumed to be continuous and the sets C and D will be closed (which ensures that system ( 12) is well-posed, see Chapter 6 in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]). We note that the state variable y experiences a jump on the set D according to [START_REF] Tallapragada | Decentralized event-triggering for control of nonlinear systems[END_REF].

We briefly recall some basics about the hybrid formalism of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]

. A set E ⊂ R ≥0 × Z ≥0 is called a compact hybrid time domain if E = ∪ j∈{0,...,J}
([t j , t j+1 ], j) for some finite sequence of times 0 = t 0 ≤ t 1 ≤ ... ≤ t J and it is a hybrid time domain if for all (T, J) ∈ E, E∩([0, T ]×{0, 1, ..., J}) is a compact hybrid time domain. A function φ : E → R nq is a hybrid arc if E is a hybrid time domain and if for each j ∈ Z ≥0 , t → φ(t, j) is locally absolutely continuous on I j := {t : (t, j) ∈ E}. A hybrid arc φ is a solution to [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] if: (i) φ(0, 0) ∈ C ∪ D; (ii) for any j ∈ Z ≥0 , φ(t, j) ∈ C and φ(t, j) = F (φ(t, j)) for almost all t ∈ I j ; (iii) for every (t, j) ∈ dom φ such that (t, j + 1) ∈ dom φ, it holds that φ(t, j) ∈ D and φ(t, j + 1) = G(φ(t, j)). A solution φ to ( 12) is maximal if there does not exist another solution ψ to [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] such that φ is a truncation of ψ to some proper subset of dom ψ.

Our objective is to design event-triggering conditions for system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], which is equivalent to defining the sets C and D, to guarantee stability properties for system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] and the existence of a uniform amount of time between two jumps, which is essential in practice as the hardware cannot generate transmissions that are arbitrarily close in time. Moreover, we want to ignore the fast dynamics, which will be assumed to be stable, and design triggering conditions based only on the slow variables, i.e. the state x, and the sampling-induced error e and potentially an additional designed variable τ .

Assumptions

We present the assumptions made on system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. We will show in Section 6 that all the conditions are satisfied by a class of globally Lipschitz systems. The approximate slow and fast models (3) and ( 5) are now in view of ( 8) and ( 9)

ẋ = f x, h(x, k(x + e)), k(x + e) =: f s (x, e) (14)
dy dτ = g x, y +h(x,k(x+e)), k(x+e) =: g f (x, y, e). ( 15
)
First, we assume that the slow system ( 14) is input-tostate stable (ISS) with respect to e.

Assumption 2 There exist a continuously differentiable function V x : R nx → R ≥0 and class K ∞ functions α x , α x , γ 1 with γ 1 continuously differentiable and α 1 > 0 such that for all (x, e) ∈ R 2nx the following is satisfied

α x (|x|) ≤ V x (x) ≤ α x (|x|) ∂Vx ∂x f s (x, e) ≤ -α 1 V x (x) + γ 1 (|e|). ( 16
)
Since we design the triggering condition based only on the slow dynamics, to guarantee the overall stability of the closed-loop system, we need to make some assumptions on the stability of the approximate fast model [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF] on flows. In particular, we assume that the following stability property holds for the approximate fast dynamics like in [START_REF] Khalil | Nonlinear Systems[END_REF].

Assumption 3 There exist a continuously differentiable function V y : R ny → R ≥0 and class K ∞ functions α y , α y and α 2 > 0 such that for all (x, y, e) ∈ R 2nx+ny

α y (|y|) ≤ V y (x, y) ≤ α y (|y|) ∂Vy ∂y g f (x, y, e) ≤ -α 2 V y (x, y). (17)
Assumption 3 implies that the origin of the approximate fast dynamics ( 15) is globally asymptotically stable. Note that Assumption 3 does not imply that the origin of the fast dynamics ( 15) is globally exponentially stable as the functions α y , α y can be nonlinear. We impose the following conditions on the interconnections between the slow and fast dynamics ( 14), [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF].

Assumption 4 There exist a class K ∞ function γ 2 and β 1 , β 2 , β 3 > 0 such that for all (x, y, e) ∈ R 2nx+ny the following hold

∂Vx ∂x [f x (x, y, e) -f s (x, e)] ≤ β 1 V x (x)V y (x, y) ∂Vy ∂x - ∂Vy ∂y ∂h ∂x f x (x,y,e) ≤ β 2 V x (x)V y (x,y)+β 3 V y (x,y) +γ 2 (|e|), (18)
where V x and V y come from Assumptions 2 and 3 respectively. In addition, there exists L > 0 such that, for all

s ≥ 0 γ 2 • γ -1 1 (s) ≤ Ls, (19)
where γ 1 comes from Assumption 2.

Conditions [START_REF] Tallapragada | Event-triggered decentralized dynamic output feedback control for LTI systems[END_REF] represent the effect of the deviation of the original system (12) from the slow and fast models [START_REF] Pan | H ∞ -optimal control for singularly perturbed systems with sampled-state measurements[END_REF], [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF] respectively and are related to (11.43) and (11.44) in [START_REF] Khalil | Nonlinear Systems[END_REF].

Finally, we assume that the dynamics of V y along jumps of system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] satisfies the following condition.

Assumption 5 There exist λ 1 , λ 2 > 0 such that for all (x, y, e) ∈ R 2nx+ny

V y (x, h y (x, y, e)) ≤ V y (x, y) + λ 1 γ 1 (|e|) + λ 2 γ 1 (|e|)V y (x, y), (20)
where V x , γ 1 and V y come from Assumptions 2 and 3 respectively.

Assumption 5 is an algebraic condition which only requires the knowledge of h y (which is defined in [START_REF] Nešić | A Lyapunov-based small-gain theorem for hybrid ISS systems[END_REF]) and γ 1 and V y from Assumptions 2 and 3 respectively: we do not need to know the triggering condition to check it.

Remark 1 Assumptions 3, 4 may require [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Tallapragada | Event-triggered decentralized dynamic output feedback control for LTI systems[END_REF] to hold regardless the magnitude of the sampling-induced error e. We show in Section 6 that all these conditions are satisfied by a class of globally Lipschitz systems which encompasses LTI systems as a particular case and for which these results are new.

Main results

First, we show that the design of triggering conditions of the same form as in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] for the slow model may not ensure the existence of a strictly positive minimum amount of time between two jumps for the overall system. We then present our main results.

A first observation

In view of Assumption 2, a first attempt would be to define a triggering condition of the form γ 1 (|e|) ≥ σα 1 V x (x) where σ ∈ (0, 1) like in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. The flow and jump sets are in this case

C = {q : γ 1 (|e|) ≤ σα 1 V x (x)} D = {q : γ 1 (|e|) = σα 1 V x (x)}. ( 21
)
The results in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] guarantee the global asymptotic stability of the origin of the slow model ( 14) and the existence of a uniform (semiglobal) amount of time between two jumps (under some conditions). However, this triggering rule no longer ensures a minimum time of flow between two jumps for system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Indeed G(D) ∩ D = {q : x = e = 0} = ∅. Thus, any solution in G(D) ∩ D may jump an infinite number of times, which makes the controller not realizable in practice. In the sequel, we first apply existing strategies in order to overcome this issue and we investigate how to modify the stability analysis and what kind of stability property one may expect. We also propose another strategy that allows to guarantee global asymptotic stability.

Semiglobal practical stabilization

The most straightforward approach to enforce a lower bound on the inter-jumps for system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] is to add a dead-zone to the triggering condition in Section 5.1, i.e.

γ 1 (|e|) ≥ max{σα 1 V x (x), ρ}, (22)
where ρ > 0 is a design parameter. The flow and jump sets in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] are then

C = {q : γ 1 (|e|) ≤ max{σα 1 V x (x), ρ}} D = {q : γ 1 (|e|) = max{σα 1 V x (x), ρ}} (23)
and we do not need to introduce an extra variable τ , i.e. q = (x, y, e). Although this type of triggering conditions has already been used in [START_REF] Donkers | Outputbased event-triggered control with guaranteed L∞-gain and improved and decentralised event-triggering[END_REF], [START_REF] Mazo | Decentralized event-triggered control with one bit communications[END_REF] for example, the fact that the state y experiences jumps and that we rely on different assumptions require to fully modify the stability analysis and leads to the following result.

Theorem 1 Consider system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] with the flow and jump sets defined in [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF]. Suppose that Assumptions 1-5 hold. Then, for any ∆, ρ > 0, there exist β ∈ KL, κ ∈ K ∞ and ǫ * > 0 such that for any ǫ ∈ (0, ǫ * ) and any solution φ = (φ x , φ y , φ e ) with |φ(0, 0)| ≤ ∆,

|φ(t, j)| ≤ β(|φ(0, 0)|, t + j) + κ(ρ) ∀(t, j) ∈ dom φ, (24)
and all inter-transmission times are lower-bounded by a strictly positive constant ρ ξ(∆) , where ξ : R ≥0 → R >0 is a continuous increasing function, i.e. for all j ∈ Z ≥0 sup I jinf I j ≥ ρ ξ(∆) , where I j = {t : (t, j) ∈ dom φ}. Furthermore, all maximal solutions to (12) are complete.

Theorem 1 ensures a semiglobal practical stability property for system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Indeed, given an arbitrary (large) ball of initial conditions centered at the origin and of radius ∆ and any constant ρ, there exists ǫ sufficiently small such that solutions to ( 12), ( 23) converge towards a neighbourhood of the origin whose 'size' can be rendered arbitrarily small by reducing ρ (at the price of shorter inter-transmission intervals).

Global asymptotic stabilization

We propose another strategy to design the eventtriggering condition to ensure a global asymptotic stability property under an extra assumption. The idea is to combine the event-triggered technique of [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] with the time-triggered results of [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] such that we allow transmissions only after a fixed amount of time T * has elapsed since the last one.

We suppose that Assumptions 1-5 are satisfied with γ 1 (s) = γ1 s 2 and γ 2 (s) = γ2 s 2 for some γ1 , γ2 ≥ 0 and for s ≥ 0. We introduce an extra variable τ ∈ R ≥0 to model the elapsed time between two successive jumps. Hence, τ = 1 on flows and τ + = 0 at jumps. Consequently, q = (x, y, e, τ ) and f τ (x, y, e, τ ) = 1, h τ (x, y, e, τ ) = 0 in [START_REF] Tallapragada | Decentralized event-triggering for control of nonlinear systems[END_REF]. We define the flow and jump sets as follows

C := {q : γ1 |e| 2 ≤ σα 1 V x (x) or τ ∈ [0, T * ]} D := q : γ1 |e| 2 = σα 1 V x (x) and τ ≥ T * or γ1 |e| 2 ≥ σα 1 V x (x) and τ = T * . (25)
While the idea of merging event-triggered and timetriggered techniques is intuitive, the stability analysis is non-trivial as we need to build a common hybrid Lyapunov function for the two approaches. It has to be emphasized that the constant T * allows us to directly tune the minimum inter-transmission interval provided it is smaller than the bound given below.

Inspired by [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], we make the following additional assumption on system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF].

Assumption 6 There exist M, N ≥ 0 such that, for all (x, y) ∈ R nx+ny and for almost all e ∈ R nx ∇|e|, -f x (x, y, e) ≤ M |e| + N ( V x (x) + V y (x, y)), where V x and V y come from Assumptions 2 and 3 respectively.

The constant T * in ( 25) is selected such that T * < T , like in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], where

T :=        1 Mr arctan(r) M 2 < γ1N 2 α1 1 M M 2 = γ1N 2 α1 1 Mr arctanh(r) M 2 > γ1N 2 α1 (26) with r := γ1N 2 α1M 2 -1 ,
where M, N come from Assumption 6 and α 1 , γ1 come from Assumption 2. We obtain the following result. Theorem 2 Consider system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] with the flow and jump sets defined in (25) and suppose the following hold.

(1) Assumptions 1-6 hold with γ 1 (s) = γ1 s 2 and γ 2 (s) = γ2 s 2 with γ1 , γ2 ≥ 0, for s ≥ 0.

(2) The constant T * in ( 25) is such that T * ∈ (0, T ).

Then there exist β ∈ KL and ǭ > 0 such that for any ǫ ∈ (0, ǭ) and any solution φ = (φ x , φ y , φ e , φ τ )

|(φ x (t, j), φ y (t, j))| ≤ β(|φ(0, 0)|, t+j) ∀(t, j) ∈ dom φ. (27)
Moreover, all maximal solutions to (12) are complete.

We see that Theorem 2 ensures a global asymptotic stability property and that it requires an additional condition to hold, namely Assumption 6, compared to Section 5.2.

The case of globally Lipschitz systems

In this section, we show that all the conditions of Section 4 are verified by a class of globally Lipschitz systems, which includes LTI systems as a particular case. We assume that the vector fields f, g and k are globally Lipschitz and that the stability of the slow and the fast model can be verified using quadratic functions V x and V y . Under these conditions, the proposition below states that Assumptions 1-6 hold. Hence, the triggering rules presented in Sections 5.2 and 5.3 can be applied. Proposition 1 Consider system (1), [START_REF] Kokotović | Singular Perturbation Methods in Control: Analysis and Design[END_REF]. Suppose the following hold [START_REF] Abdelrahim | Eventtriggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] The functions f, g and k are globally Lipschitz.

(2) Assumption 1 is verified with h globally Lipschitz.

(3) There exist positive definite and symmetric real matrices P 1 , P 2 such that the functions V x : x → x T P 1 x and V y : y → y T P 2 y satisfy for all (x, y, e) ∈ R 2nx+ny

∂V x ∂x f s (x, 0) ≤ -ᾱ1 V x (x) (28) ∂V y ∂y g f (x, y, e) ≤ -α 2 V y (x, y), (29)
where ᾱ1 , α 2 > 0. Then, Assumptions 2-6 are satisfied.

The choice of the triggering condition between Section 5.2 and 5.3 has to be done on a case-by-case basis according to the desired specifications. The technique in Section 5.3 offers stronger a priori guarantees on the minimum inter-event times, however the strategy in Section 5.2 may lead to less transmissions in average, see for simulation results on the autopilot event-triggered control of an F-8 aircraft [START_REF] Abdelrahim | Eventtriggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF].

Conclusion

We have investigated the event-triggered stabilization of nonlinear singularly perturbed systems based only on the slow dynamics. Two classes of controllers have been developed which ensure different asymptotic stability properties. We believe that this work can be extended along two important directions. First, the design of event-triggered controllers for singularly perturbed systems with potentially unstable fast dynamics can be pursued based on the model [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Second, it would be interesting to use similar ideas as in this paper for the event-triggered control of systems based on a simplified model of the plant dynamics obtained by other means like model reduction.

with ǫ ∈ (0, ǫ * ) where ǫ * > 0 will be defined in the following. Let q ∈ C, it holds that, in view of ( 8) and ( 9) (we omit the arguments of f x , g, V x , V y in the following for space reasons)

∇V (q), F (q) = ∂Vx ∂x f x + √ ǫ ∂Vy ∂x f x + √ ǫ ǫ ∂Vy ∂y f y = ∂Vx ∂x f s + ∂Vx ∂x [f x -f xs ] + √ ǫ -1 ∂Vy ∂y g f + √ ǫ ∂Vy ∂x - ∂Vy ∂y ∂h ∂x f x . (31) In view of the definition of the set C, γ 1 (|e|) ≤ max{σα 1 V x (x), ρ} and γ 2 (|e|) ≤ γ 2 •γ -1 1 max{σα 1 V x (x), ρ} . The condition (19) ensures that γ 2 (|e|) ≤ L max{σα 1 V x (x), ρ}. Using Assumptions 2-5, we derive that ∇V (q), F (q) ≤ -χ T Aχ + (1 + √ ǫL)ρ (32)
where χ := ( V x (x), V y (x, y)) and

A := α 1 (1 -σ(1 + √ ǫL)) -(β 1 + √ ǫβ 2 )/2 -(β 1 + √ ǫβ 2 )/2 α 2 √ ǫ -1 - √ ǫβ 3 . (33)
Let µ ∈ (0, α 1 (1σ)). The following conditions ensure that A ≥ µdiag(1, √ ǫ), where diag(1, √ ǫ) is the diagonal matrix with elements (1, √ ǫ) on the diagonal

       α 1 (1 -σ(1 + √ ǫL)) ≥ µ α 1 (1 -σ(1 + √ ǫL)) -µ α 2 √ ǫ -1 - √ ǫβ 3 - √ ǫµ ≥ (β 1 + √ ǫβ 2 ) 2 /4 (34)
The inequalities in (34) are always satisfied by selecting ǫ * > 0 (and so ǫ) sufficiently small. Consequently

∇V (q), F (q) ≤ -µχ T diag(1, √ ǫ)χ + (1 + √ ǫL)ρ = -µV (q)+(1+ √ ǫL)ρ, from which we deduce that V (q) ≥ 2 (1+ √ ǫL) µ ρ implies ∇V (q), F (q) ≤ -µ 2 V (q). (35)
Let q ∈ D, V (G(q)) = V x (x) + √ ǫV y (x, h y (x, y, e)). view of Assumption 5 and the definition of the that

∇V y (x, y), (f x , f y ) = 1 ǫ ∂Vy ∂y g + ∂Vy ∂x - ∂Vy ∂y ∂h ∂x f x ≤ -α2 ǫ V y (x, y) + β 2 V x (x)V y (x, y) +β 3 V y (x, y) + γ 2 (|e|) ≤ -( α2 ǫ -β 2 -β 3 )V y (x, y) + β 2 V x (x) +γ 2 (|e|)
(45) and, as shown before,

V y (x,h y (x,y,e)) ≤ V y (x,y)+λ 1 γ 1 (|e|)+λ 2 γ 1 (|e|)V y (x,y) ≤ (1 + ǫ 1 4 λ 2 )V y (x, y) + (λ 1 + ǫ -1 4 λ 2 )γ 1 (|e|) (46)
By following similar lines as above, we deduce that, by taking ǫ * sufficiently small, the y-system is ISS with respect to x and e. As a consequence, in view of (43), ( 45) and ( 46) we derive that

|φ y (t, j)| ≤ max β y (|φ(0, 0)|, t + j), ϑ y (ρ)} ( 47
)
for some β y ∈ KL and θ y ∈ K ∞ . The property (24) then follows from (43), ( 44) and (47). Equations ( 43), ( 44) and (47) ensure that φ cannot explode in finite time, neither it can flow out of C ∪ D since G(D) ⊂ C. Noting that system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF] does not admit trivial solution 2 , we conclude that maximal solutions to ( 12), ( 23) are complete according to Proposition 6.10 in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF].

2
Proof of Theorem 2. Like in the proof of Theorem 1 in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], we introduce the solution

ζ to ζ = -1 -2M ζ -µ -µζ(τ ) + γ1 α1-µ (N ζ) 2 =: f ζ (τ ) (48)
with ζ(0) = ϑ -1 , ϑ ∈ (0, 1) and µ ∈ (0, α 1 ). We denote T (µ, ϑ) the time it takes for ζ to decrease from ϑ -1 to ϑ. This time T (µ, ϑ) is a continuous function of µ, ϑ which is decreasing in µ, ϑ (by invoking the comparison principle). On the other hand, we note that T (µ, ϑ) → T as (µ, ϑ) tends to (0, 0) by following similar lines as in the proof of Claim 1 in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], where T is defined in (26). As a consequence, since T * < T , there exist µ, ϑ such that T * ≤ T (µ, ϑ) which we fix.

We define

R(q) := V x (x)+dV y (x, y)+max{0, γ1 ζ(τ )|e| 2 } ∀q ∈ R nq (49)
where d ∈ 0, min{ γ1 γ2 µ,

R • (q; F (q)) ≤ -(χ, |e|) T A 1 (χ, |e|), (50)
where χ := ( V x (x), V y (x, y)),

A 1 := α1 -1 2 (β1+dβ2) -γ1N ζ(τ ) * d ǫ α2-dβ3 -γ1N ζ(τ ) * * υ(τ ) , υ(τ ) := -γ 1 -dγ 2 - γ1 f ζ (τ ) -2γ 1 M ζ(τ )
                                   0 ≤ α1 -µ 0 ≤ (α1 -µ)d( 1 ǫ α2 -β3 -µ) ≥ 1 4 (β1 + dβ2) 2 0 ≤ (α1 -µ) × d( 1 ǫ α2 -β3 -µ)(υ(τ ) -µγ1ζ(τ )) -(γ1ζ(τ )N ) 2 + 1 2 (β1 + dβ2) × -1 2 (β1 + dβ2)(υ(τ ) -µγ1ζ(τ )) -(γ1ζ(τ )N ) 2 -γ1N ζ(τ ) × 1 2 (β1 + dβ2)γ1ζ(τ )N + γ1N ζ(τ )d( 1 ǫ α2 -β3 -µ) . (51)
The first two inequalities above are respectively verified by definition of µ and by taking ǫ sufficiently small. For the last inequality to hold, it suffices to select ǫ sufficiently small provided that

R • (q; F (q)) ≤ -µR(q). (52)
Suppose now that ζ(τ ) < 0, hence γ1 |e| 2 ≤ σα 1 V x (x) in view of the definition of the set C. Using Assumptions 2-6 and Lemma 1,

R • (q; F (q)) ≤ -χ T A 2 χ, (53)
where

A 2 := α1(1-σ(1+dγ2 γ-1 1 )) -1 2 (β1+dβ2) -1 2 (β1+dβ2) d ǫ α2-dβ3
. By following similar arguments as above and since d < 1-σ σ γ1 γ2

and R(q) = V x (x) + dV y (x, y) in this case, we derive that (52) holds by selecting ǫ sufficiently small. When ζ(τ ) = 0, (52) is verified in view of Lemma 1 and the results obtained for the cases where ζ(τ ) > 0 and ζ(τ ) < 0.

Let q ∈ D. Suppose that τ = T * (note that γ1 |e| 2 ≥ σα 1 V x (x) in this case). In view of Assumption 5

R(G(q)) = V x (x) + dV y (x, h y (x, y, e)) ≤ V x (x) + d V y (x, y) + λ 1 γ1 |e| 2 + λ 2 γ1 |e| 2 V y (x, y) . Us- ing that γ1 |e| 2 V y (x, y) ≤ 1 √ d γ1 |e| 2 + √ dV y (x, y) and since d ≤ √ d ≤ 1, it holds that R(G(q)) ≤ V x (x)+dV y (x, y)+ √ d(λ 1 +λ 2 )γ 1 |e| 2 +dλ 2 √ dV y (x, y) ≤ V x (x) + dV y (x, y) + √ d(λ 1 + λ 2 )γ 1 |e| 2 + λ 2 √ d(V x (x) + dV y (x, y)). We take d sufficiently small such that √ d(λ 1 + λ 2 )γ 1 |e| 2 ≤ γ1 ζ( T (µ, ϑ))|e| 2 = γ1 ϑ|e| 2 (since ζ( T (µ,ϑ)) = ϑ). As a consequence R(G(q)) ≤ (1 + λ 2 √ d)(V x (x) + dV y (x, y) + γ1 ζ( T (µ, ϑ))|e| 2 ). Since in this case we transmit at τ = T * ≤ T (µ, ϑ), then ζ(τ ) ≥ ζ( T (µ, ϑ)), as ζ(τ ) is a decreasing function, and we obtain R(G(q)) ≤ (1 + λ 2 √ d)R(q).
When τ > T * , it holds that γ1 |e| 2 = σα 1 V x (x) in view of (25). Hence, by following similar lines as above, we deduce that

R(G(q)) ≤ (1 + λ √ d)R(q), (54)
where λ = max{λ 2 , (λ 1 + λ 2 )σα 1 }. Thus, (54) holds for all q ∈ D (since λ 2 ≤ λ).

Finally, we use similar arguments as in Proposition 3.29 in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] to conclude. In view of ( 52) and ( 54 ) > 0). As a result, like in the proof of Proposition 3.29 in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], we obtain, for all (t, j) ∈ dom φ R(φ(t, j)) ≤ e -ψ(t+j) R(φ(0, 0)).

(55)

By using Assumptions 2-3 and the fact that ζ(τ ) ∈ [ϑ, ϑ -1 ], we deduce from (55) that (27) holds. Let φ = (φ x , φ y , φ e , φ τ ) be a maximal solution to ( 12)-(25). We note that φ is non-trivial by using similar arguments as in the proof of Theorem 1. In view of (55), φ x and φ y cannot explode in finite time. Since φ e (t, j) = φ x (t j , j)-φ x (t, j) for any (t j , j), (t, j) ∈ dom φ and j ≥ 1, φ e cannot explode in finite time. The same conclusion holds for φ τ in view of its dynamics, see [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Hence, φ cannot explode in finite-time. In addition, G(D) ⊂ C. As a consequence, φ is complete according to Proposition 6.10 in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]. 2

Proof of Proposition 1. Let L be a common Lipschitz constant for the functions f, g, k and h, which exists in view of items (1), (2) of Proposition 1.

• Assumption 2: In view of ( 14) and item (3) of Proposition 1, we have, for all (x, e) ∈ R 2nx , ∂Vx ∂x f s (x, e) ≤ -ᾱ1 V x (x) + 2|x||P 1 ||f s (x, e)f s (x, 0)|. Since f is globally Lipschitz, |f s (x, e)f s (x, 0)| ≤ L|e|. As a consequence, ∂Vx ∂x f s (x, e) ≤ -ᾱ1 V x (x) + 2 L|P 1 ||x||e|. Using the fact that 2 L|P 1 ||x||e| ≤ 1 2 ᾱ1 λ min (P 1 )|x| 2 + 2(ᾱ 1 λ min (P 1 )) -1 L2 |P 1 | 2 |e| 2 and the fact that λ min (P 1 )|x| 2 ≤ V x (x) ≤ λ max (P 1 )|x| 2 , since P 1 is positive definite and symmetric, it holds that ∂Vx ∂x f s (x, e) ≤ -ᾱ1 2 V x (x) + 2 L2 |P1| 2 ᾱ1λmin(P1) |e| 2 . Hence Assumption 2 holds with α 1 = ᾱ1 2 and γ 1 (s) = 2 L2 |P 1 | 2 (ᾱ 1 λ min (P 1 )) -1 s 2 for s ≥ 0.

• Assumption 3 follows directly from item (3) of Proposition 1.

• Assumption 4: In view of items ( 1), (3) of Proposition 1 and since λ min (P 1 )|x| 2 ≤ V x (x) and λ min (P 2 )|y| 2 ≤ V y (x, y), it holds that, for all (x, y, e) ∈ R 2nx+ny

Thus, the first condition of Assumption 4 is verified with β 1 = 2 L|P 1 |(λ min (P 1 )λ min (P 2 )) -1/2 . On the other hand, in view of items ( 1), (3) of Proposition 1 and using the fact that |e||y| ≤ |y| 2 + |e| 2 and using that f x (0, 0, 0) = 0 since the origin of system ( 14) is asymptotically stable in view of (28), it holds that, for all (x, y, e) ∈ R 2nx+ny • Assumption 6: In view of ( 12)-( 13) and item [START_REF] Abdelrahim | Eventtriggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] of Proposition 1 and using that f x (0, 0, 0) = 0, it holds that, for all (x, y) ∈ R nx+ny and for almost all e ∈ R nx , ∇|e|, -f x (x, y, e) ≤ L(|x| + |y| + |e|).

Hence, Assumption 6 is verified with M = L and N = L max{(λ min (P 1 )) -1/2 , (λ min (P 2 )) -1/2 }. 2

d ǫ α 2

 2 (α 1µ)(υµγ 1 ζ(τ )) -(γ 1 N ζ(τ )) 2 > 0 which is equivalent to, by definition of υ and definition of f ζ in (48), (α 1µ)(γ 1 µdγ 2 ) > 0 which holds by definition of d and µ. Consequently, by selecting ǫ sufficiently small

  ∂V x ∂x [f x (x,y,e)-f s (x,e)] ≤ 2 L|P 1 ||x||y|.

  x, y, e) ≤ 2|P 2 ||y|| ∂h ∂x ||f x (x, y, e)| ≤ β 2 V x (x)V y (x, y)+β 3 V y (x, y) + 2 L2 |P 2 ||e| 2 , where β 2 = 2 L2 |P 2 |(λ min (P 1 )λ min (P 2 )) -1/2 and β 3 = 4 L2 |P 2 |(λ min (P 2 )) -1 . Hence, the second condition of Assumption 4 holds with γ 2 (s) = 2 L2 |P 2 |s 2for s ≥ 0. The third condition is satisfied withL = ᾱ1 λ min (P 1 )|P 2 ||P 1 | -2 .• Assumption 5: We denote h xe (x, e) := h(x, k(x + e)) and h x (x) := h(x, k(x)). In view of[START_REF] Nešić | A Lyapunov-based small-gain theorem for hybrid ISS systems[END_REF] and the definition of V y , V y (x, h y (x, y, e)) = h T y (x, y, e)P 2 h y (x, y, e) ≤V y (x, y)+|P 2 ||h xe (x, e)-h x (x)| 2 +2|P 2 ||y||h xe (x, e)h x (x)|. Since h is globally Lipschitz, it holds that |h xe (x, e)h x (x)| ≤ L|e|. As a consequence, V y (x, h y (x, y, e)) ≤ V y (x, y) + L2 |P 2 ||e| 2 + 2 L|P 2 ||y||e| and Assumption 5 is satisfied with λ 1 = 1 2 ᾱ1 λ min (P 1 )× |P 2 ||P 1 | -2 and λ 2 = (ᾱ 1 λ min (P 1 )|P 2 | 2 ) 1/2 × (λ min (P 2 )|P 1 | 2 ) -1/2 .

  := max{λ 2 , (λ 1 +λ 2 )σα 1 }. Let q ∈ C and consider the case where ζ(τ ) > 0. In view of Assumptions 2-6 and Lemma 1

	1-σ σ	γ1 γ2 ,	ϑ 2 (λ1+λ2) 2 , (e µT * -1) 2 λ 2	, 1}
	and λ

  and * stands for the symmetric components of A 1 . The following conditions ensure that A 1 ≥ µdiag(1, d, γ1 ζ(τ )) according to Sylvester's criterion, where diag(1, d, γ1 ζ(τ )) is the diagonal matrix with elements (1, d, γ1 ζ(τ )) on the diagonal

This comes from the fact that C\D is the interior of C. Hence, the tangent cone (see definition 5.12 in[START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]) is R n and (VC) in Proposition 6.10 in[START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] holds for any point in C\D

Appendix

Proof of Theorem 1. We define the function (like in the proof of Theorem 1 in [START_REF] Sanfelice | On singular perturbations due to fast actuators in hybrid control systems[END_REF])

set D, V y (x, h y (x, y, e)) ≤ V y (x, y) + λ 1 γ 1 (|e|) + λ 2 γ 1 (|e|)V y (x, y) = V y (x, y) + λ 1 max{σα 1 V x (x), ρ} + λ 2 max{σα 1 V x (x), ρ}V y (x, y) ≤ V y (x, y)+λ 1 σα 1 V x (x)+ λ 1 ρ + λ 2 (σα 1 V x (x) + ρ)V y (x, y). Using that max{σα

V y (x, y). As a consequence, by taking ǫ * sufficiently small: ǫ

√ ǫV y (x, y). After some direct computations, we derive that V (G(q)) ≤ (1 + ǫ

Let ∆ > 0 and φ = (φ x , φ y , φ e ) be a solution to [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF] such that |φ(0, 0)| ≤ ∆. Assume without loss of generality 1 that φ e (0, 0) = 0. By invoking standard comparison principles, we obtain from ( 35), for all (t, 0) ∈ dom φ,

with θ := (1 + 2λ) max{2 (1+L) µ , 1 (where we have used the fact ǫ * is sufficiently small such that ǫ * ≤ 1). The length of the inter-jump interval is lower bounded by the time it takes for γ 1 (|φ e |) to grow from 0 to ρ in view of [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF]. We aim at establishing a lower bound of the latter time, which we denote τ (∆). In view of (37), (φ x (t, 0), φ y (t, 0)) lie in the compact set {(x, y) : V (x, y, 0) ≤ max α(∆), θρ}} for all (t, 0) ∈ dom φ, where α(|(φ x (t, j), φ y (t, j))|) = α x (|φ x (t, j)|) + α y (|φ y (t, j)|) (we use again the fact that ǫ * ≤ 1 as well as ( 16) and ( 17)). As φ e (t, 0) = φ x (0, 0)-φ x (t, 0), we deduce that φ(t, 0) lie in a compact set S(∆) for all (t, 0) ∈ dom φ. Since γ 1 is continuously differentiable by assumption, φ is continuous between two jump instants, f x is continuous and S(∆) is compact

1 If that is not the case, the inequality obtained later in (41) will hold for any (t, j) ∈ dom φ with j ≥ 1. A bound on V (φ) on the interval [0, t1] can then be derived using ( 35) and (36) to upper-bound on V (φ) on the whole domain dom φ. Note that if φ never jumps, the bound on the inter-jump times used in (39) trivially holds and (41) will be verified.

for some ξ(∆) > 0, which ensures the property on the inter-jump intervals stated below (24). Hence τ (∆) ≥ ρ ξ(∆) . To compensate the potential increase of V at jumps in view of (36), we will use the following condition

which is always satisfied by selecting ǫ * sufficiently small such that 4 µ ln 1 + 2(ǫ * )

The inequality (39) ensures e -µ 2 t1 (1 + 2ǫ 1 4 λ) < 1 where t 1 is such that (t 1 , 0), (t 1 , 1) ∈ dom φ (and verifies t 1 ≥ τ (∆)). As a consequence, from (36) and (37), we deduce that

which is the same right hand-side as in (37). Hence, we apply the same arguments as above to derive that t 2t 1 ≥ τ (∆), where (t 2 , 1), (t 2 , 2) ∈ dom φ. Consequently, by induction t ≥ τ (∆)j for any (t, j) ∈ dom φ. By using the comparison principle and the fact that (1 + 2ǫ 1 4 λ)e -µ 2 τ (∆) ≤ 1, we obtain in view of ( 35) and (36), for any (t, j) ∈ dom φ V (φ(t, j)) ≤ max e -µ 2 t (1 + 2λǫ

) j V (φ(0, 0)), θρ}.

(41) The condition (39) ensures, since t ≥ τ (∆)j for any (t, j) ∈ dom φ, that e -µ 2 t (1 + 2ǫ

. As a consequence, for any (t, j) ∈ dom φ, V (φ(t, j)) ≤ max e -γ(t+j) V (φ(0, 0)), θρ}.

In view of ( 16) and ( 17), for any (t, j) ∈ dom φ, α x (|φ x (t, j)|) ≤ max e -γ(t+j) α(φ(0, 0)), θρ}

x max e -γ(t+j) α(|φ(0, 0)|), θρ} , (43) Using that γ 1 (|e|) ≤ max{σα 1 V (x), ρ} for any q ∈ C ∪ D ∪ G(D), we deduce that for any (t, j) ∈ dom φ |φ e (t, j)| ≤ max β e (|φ(0, 0)|, t + j), ϑ e (ρ)} (44)

for some β e ∈ KL and θ e ∈ K ∞ . We are left with the y-component of φ. In view of Assumptions 3-4, it holds


MD Jacques Bouget
email: jacques.bouget@chu-rennes.fr

MD Emmanuel Oger

PhD Nathalie Nicolas

Emergency admissions for major haemorrhage associated with antithrombotics: a cohort study

Keywords: antithrombotic, major bleeding, anticoagulant, antiplatelet, emergency department, intracranial haemorrhage

published or not. The documents may come

Emergency admissions for major haemorrhage associated with antithrombotics: a cohort study. Word count 5,375 /6,000 Introduction Antithrombotics, i.e. oral and parenteral anticoagulants as well as antiplatelet agents, have demonstrated significant benefits in preventing venous or arterial thrombotic events, especially in coronary disease, stroke, atrial fibrillation, venous thromboembolism and mechanical heart valves [1]. These drugs are commonly prescribed and their long term use is increasing, particularly in elderly. Bleeding represent the most well-known and feared complications of antithrombotics. Numerous studies on adverse drug events reported anticoagulants as the first medication class implicated in haemorrhage and specifically intracranial haemorrhage (ICH) which often results in substantial morbidity and mortality [2]. Using adverse drug events from a National Surveillance System, Budnitz et al demonstrated that warfarin and oral antiplatelet agents are respectively the first and the third medications leading to emergency department and hospitalization in adults 65 years of age or older [3]. For antiplatelet agents, gastrointestinal bleeding and intracranial haemorrhage are well known [4][5][6]. Emergency departments are unique to describe antithrombotic-related major haemorrhage. We report the results of a prospective cohort that aimed to describe antithrombotic-related major haemorrhage, diagnostic process, clinical and therapeutic management as well as 1month outcome.

Material and methods

Patient selection and definitions

Patients older than 16 years admitted in the emergency department of our teaching hospital with antithrombotic-related major haemorrhage were consecutively included between January 1, 2011 and December 31, 2012. Patients were identified through haemorrhagic symptoms at emergency admission. Computerised requests based on several related-haemorrhagic diagnostic codes and specific emergency therapies were made every month on electronic health records. Then, criteria of major bleeding were required and antithrombotic treatment searched. According to the French National Authority for Health ( Haute Autorité de Santé -HAS) [7], major bleeding was defined by at least one of the following criteria : hemodynamic instability (systolic arterial pressure < 90 mmHg or mean arterial pressure < 65 mm Hg), signs of shock, uncontrollable bleeding, need for transfusions of red cell packs, need for haemostatic procedure (embolization, endoscopic procedure, surgery), or a life-threatened bleeding such as intracranial, intraspinal, intraocular, retroperitoneal, pericardial, thoracic bleeding, compressive muscular hematoma, acute gastrointestinal bleeding. In addition to these criteria, we considered major bleeding in case of epistaxis if at least two procedures of nasal packing were needed and in case of hematuria if the bleeding continued during more than 12 hours despite bladder washing. All antithrombotic drugs, alone and in combination were eligible whatever their indication and their dosage. Patients with major bleeding events associated with antithrombotic during hospitalization, patients with intentional overdose with antithrombotic drugs and patients with multi-trauma were excluded. For all included patients, the following clinical and biological data were collected from electronic health record: demographics, medical history, treatments with their indication and duration, type of bleeding manifestation, vital signs at admission, contributory procedures that led to diagnostic of major bleeding (CT scan, endoscopy) , biological data on admission, treatments done in the emergency ward, blood transfusions, specific reversal treatment, haemostatic procedure, time between admission and diagnosis, time between admission and reversal therapy, outcomes, length of stay in hospital, and decision about antithrombotic treatment after the haemorrhagic event. At 1 month, clinical data about hemorrhagic or ischemic event and vital status were asked. The study protocol was approved by the ethical committee of our hospital.

Data processing and analysis

Three subgroups were defined depending on antithrombotic medications: group1: Oral anticoagulant (OA) group including patients with oral vitamin K antagonist and new oral anticoagulant, alone and in combination with other antithrombotic drugs; group 2: Antiplatelet (AP) group including patients with antiplatelet therapy only, either alone or dual therapy; group 3: Parenteral anticoagulant (PA) group including patients with unfractionated heparin (UFH), low molecular weight heparin (LMWH), fondaparinux, danaparoid, alone and in combination with antiplatelet therapy.

Haemorrhagic events were firstly divided in 5 groups: intracranial haemorrhage, gastrointestinal bleeding, muscular hematoma, internal bleeding including pericardial, thoracic, peritoneal bleeding, and external bleeding including hematuria, epistaxis, scalp injury, vascular injury. Then, two subgroups were defined: patients with intracranial haemorrhage and patients with other major bleeding The primary outcome was 1-month mortality. Between subgroups comparisons were performed using student's t test for parametric data, Mann-Whitney U test for non parametric data, and the chi-square test for qualitative data. Crude relative risks were estimated along with 95% confidence interval. Prognostic factors for death at 1-month were defined in the overall population and in the subgroups of patients with intracranial haemorrhage and patients with other bleeding than intracranial haemorrhage. Multivariate logistic models were run in subgroups of patients with intracranial haemorrhage and of patients with other major bleeding to assess the predictors of death. Statistical analysis was conducted using SAS software version 9.3 (SAS Institute Inc, Cary, NC, USA). A p value of .05 was considered statistically significant.

Results

Population characteristics

In 2011 and 2012, 98,377 adult patients were admitted in our emergency department. Nine hundred and thirteen patients (0.9%) with major bleeding while receiving at least one antithrombotic drug were analysed, which represented 1.2 patients per day. During the study period, there were no monthly variations. The mean age was 80 ± 10 years ( range : 21-103); median, 82 years) with a sex ratio of 1.1. Hospitalization was required for 728 patients (79%) with a mean length of stay of 8.5 ± 14.2 days, including 1.3 ± 3.9 days in intensive care units. Demographic characteristics and indications of antithrombotics, according to the therapeutic subgroups, are reported in table 1. Among 913 patients, 152 (17.2%) had a combination of antithrombotic agents. The duration of prescription was unknown in 46.3% of cases. Gastrointestinal bleeding and intracranial haemorrhage (ICH) represented more than 70% of major haemorrhagic events (table 2). There was a statistically significant association between types of antithrombotics and types of haemorrhage (p < .0001): AP-related haemorrhages were mostly ICH (n = 164, 39%) and gastrointestinal bleeding (n = 179, 42%) whereas OArelated haemorrhages were mostly external haemorrhages or muscular hematomas (n = 147, 34%), gastrointestinal bleeding (n = 140, 33%) and ICH (n = 134, 31%). The mechanism of the ICH, known in 307 patients, was traumatic in 44.6% of cases, mostly a fall from standing height, spontaneous in 45.6%, impossible to determine in 9.4%. In traumatic ICH, there were more patients with AP than OA (60% versus 40%). Frequencies of AP and OA were similar in those patients with spontaneous ICH (51.5% versus 48.5%).

Predictors of 1-month mortality

At 1 month, data from 903 documented medical records were analysed. In this population, 179 patients died at 1 month (19.8 %). Fifty-five patients died in the emergency ward. Death occurred within the first 3 days of hospitalization in 81 patients. The distribution of death according to the type of haemorrhagic events showed that intracranial haemorrhage was responsible of the most number of death (115/179 patients, 64.2%). Among 267 patients with a known ICH mechanism and a known vital status at 1 month, 1-month mortality was higher in spontaneous ICH than in traumatic ICH: 45.9% (62/135) vs. 26.5% (35/132), p = 0.001. Among 314 patients with ICH, 27 (8.6%) had limitation of life support; 22 patients (7.7%) out of the 287 remaining patients underwent surgery: 18 out of those 22 patients were alive at one month (81.8%) compared to 178 out of 265 patients who did not have surgery (67.2%), p = 0.15. Among 342 patients with GI bleeding, 61 had an invasive therapy (endoscopy, n = 42; surgery, n = 20; embolisation, n = 3; of note 4 patients had two procedures); 59 out of those 61 patients were alive at one month (96.7%) compared to 249 out of 281 patients who did not have invasive therapy (88.6%), p = 0.055.One-month mortality was not associated with the type of antithrombotic agents (table 3). In univariate analysis, prognostic factors for 1-month mortality in case of intracranial haemorrhage were age, Glasgow coma scale (GCS) score, the type of antithrombotic agents (OA), time between admission and diagnosis and time between admission and biological results (table 4). OA was associated with a 1.4-fold increased risk for 1-month mortality compared to AP: RR = 1.5 (95%CI, 1.1 to 2.0); relative risk for PA compared to AP was 1.2 (95%CI, 0.6 to 2.5). Prognostic factors for 1-month mortality after other haemorrhagic events were age, mean arterial pressure at admission and type of antithrombotic agents (table 4). PA was associated with a 1.7-fold increased risk for 1-month mortality compared to AP: RR = 1.7 (95%CI, 0.8 to 3.3); relative risk for OA compared to AP was 0.8 (95%CI, 0.5 to 1.4). Adjusting for age and GCS through multivariate logistic regression, OA remained statistically associated with ICH 1-month mortality compared to AP (p = 0.0451) and PA did not reached statistical significance for other haemorrhagic event 1-month mortality compared to AP (p = 0.57). Among surviving patients at 1 month (724), antithrombotic treatment was continued without modifications in 382 patients (52.8%), with modifications in 125 patients (17.3%), discontinued in 171 patients (23.6%), not determined in 46 patients (6.3%). Antithrombotic treatment was stopped in all patients with ICH. Oral anticoagulant group: characteristics, clinical and therapeutic management Demographic and clinical characteristics of the 3 treatment groups are reported in table 5. Most of oral anticoagulants were vitamin K antagonists (427/429). Vitamin K antagonists were fluindione in 68% of cases, warfarine in 38% of cases and acenocoumarol in 3% of cases. In 63 patients, OA was in combination with AP. Only two patients were taking new oral anticoagulant (dabigatran etexilate) for prevention of stroke in atrial fibrillation and they were included at the end of 2012: a 82 years old man with major fatal hematuria, and a 80 years old man with gastrointestinal bleeding, alive at 1 month. At admission, the international normalized ration (INR) was in the therapeutic range (between 2 and 3) in 41.9% of cases, under the range in 18.6% and above in 39.5%. Median [p25-p75] INR value was slightly higher among patients with ICH who died within one month compared to those who were alive: 2.6 [2.1-3.2] vs. 2.4 [1.9-2.9] (p = 0.066). The same pattern was observed in other type of haemorrhage: 3.7 [2.4-7.4] vs. 2.8 [2.2-4.2] (p = 0.031). Reversal therapy was initiated in only 215 patients (50.5%); the complete recommended treatment, that is prothrombin complex concentrate (PCC) at a dose equal or superior to 20 IU/kg in association with 5 or 10 mg of vitamin K, was prescribed only in 80 patients (19.1%). This reversal treatment was not significantly associated with 1-month mortality considering only patients with no limitation of life support: 5/21 (23.8%) vs. 11/40 (27.5%) in patients with ICH and 6/70 (8.6%) vs. 7/68 (10.3%) for other type of haemorrhage. Two-hundred and two patients needed blood transfusions (mean 2.3 units of red cells), for gastrointestinal bleeding in 101 patients, muscular hematomas in 39 patients, epistaxis in 17 patients. Fifty patients required surgery. At 1 month, OA therapy was continued in 53% of cases and stopped in 24% of cases. In 23% of cases, treatment was changed, switch with antiplatelet agents or stop of combination.

Antiplatelet group: characteristics, clinical and therapeutic management

In the antiplatelet group (420 patients), aspirin represented 88% of cases, clopidogrel 20% of cases, dual therapy 9% of cases. Demographic and clinical characteristics, number of red cell transfusions, hospitalization rate and mortality rate were not different compared to patients from the oral anticoagulant group (table 5). No platelet transfusion was prescribed. At 1 month, antiplatelet agents were continued in 61% of cases, discontinued in 24% of cases. In 15% of cases, the treatment was modified, that is stop of the dual therapy or switch with other antiplatelet drugs. Parenteral anticoagulant group: characteristics, clinical and therapeutic management In the parenteral anticoagulant group (64 patients), drugs were: LMWH in 43 patients, UFH in 5 patients, fondaparinux in 15 patients and danaparoid in one patient; the dose was curative in 37 patients (57.8%), mainly with LMWH (n = 23) and fondaparinux (n = 10) whereas only four patients had curative UFH dose. The main indication was the venous thromboembolic disease. Glasgow coma scale score, mean arterial pressure, mean haemoglobin level at admission were different with patients of other groups (not significant). No reversal treatment (protamin sulfate) was initiated. At 1 month, parenteral anticoagulants were continued in 22 patients, stopped in 20 patients.

Discussion

With more than one patient per day admitted in our emergency ward on a two-year period, our study highlighted the magnitude of major bleeding associated with antithrombotics, whatever the antithrombotic drug. The patient's median age was high (82 years old), reflecting the increasing use of antithrombotic in elderly in all their indications, as previously reported [4]. Particularly, the number of patients with atrial fibrillation will progressively increase in the next years. Consequently, the number of major bleeding will still increase in this frail population, leading to an important morbidity and mortality. Despite this increased haemorrhagic risk secondary to age, the elderly still have the most important benefits of antithrombotics, even after a major bleeding event [8]. In survivors, we observed that 53 % of the oral anticoagulant group and 61% of the antiplatelet group continued the same antithrombotic treatment 1 month after the haemorrhagic event.

In our study, 152 patients were taking more than one antithrombotic drugs with various combinations. Even if the combination of antithrombotics increase the haemorrhagic risk [9, 10] the various combination would have complicated the analysis of our study population. Thus, to compare the different antithrombotic agents, we classified our patients in only three therapeutic groups, considering that the haemorrhagic risk was most important with oral and parenteral anticoagulants than with antiplatelet agents [1,11]. We identified 64 patients with major bleeding associated with parenteral anticoagulant, especially heparins and fondaparinux. To our knowledge, this was the first report on major bleeding in an ambulatory population with these antithrombotics. Usually, most data came from multicenter clinical trial [13]. Our results showed that the corresponding population was different from patients of others therapeutic groups, i.e. more severity, increased mortality rate in patients with mainly venous thromboembolic indication. In this group, the distribution of the different haemorrhagic complications was equivalent to other group of antithrombotics, as previously reported [START_REF] Garcia | Parenteral anticoagulants. Antithrombotic therapy and prevention of thrombosis, 9th ed: American College of Chest Physicians. Evidence-based clinical practice guidelines[END_REF]. In the therapeutic management of the hemorrhagic events, there were an important number of blood transfusions, with numerous haemostatic procedures (surgery, endoscopic procedure). Indications of blood transfusions were left to physician appreciation. In the oral anticoagulant group, reversal therapy was used in only 50.5 % of cases, despite the national guidelines [7]. In ICH associated with oral anticoagulant, reversal therapy with PCC and vitamin K was specially used because of the poor prognosis of this bleeding event. Nevertheless, the mortality rate was not different with or without reversal therapy in case of ICH. In the antiplatelet group, no platelet transfusion was needed, according to the recent systematic review in traumatic and spontaneous ICH [START_REF] Batchelor | A meta-analysis to determine the effect on survival of platelets transfusions in patients with either spontaneous or traumatic antiplatelet medicationassociated intracranial haemorrhage[END_REF]. Gastrointestinal bleeding was the first major haemorrhagic events in our study, with a similar number associated with OA and AP. Even if this bleeding risk was well known, international trial and meta-analysis have reported various risk levels according to the antithrombotic drugs. Relative risk of gastrointestinal risk was estimated at 1.8 and 1.6 for respectively aspirin and clopidogrel, two-fold higher for oral anticoagulant therapy, especially when the INR value is high [5, [START_REF] Reynolds | Warfarin anticoagulation and outcomes in patients with atrial fibrillation : a systematic review and meta-analysis[END_REF][START_REF] Lanas | Low doses of acetylsalicylic acid increase risk of gastrointestinal bleeding in a meta-analysis[END_REF]. In our OA group, INR value was not a prognostic factor. Dangerous combinations such as OA + AP or dual antiplatelet therapy induced a more 2-fold higher risk of gastrointestinal bleeding [10,12].

In 314 patients (34.4 %), the haemorrhagic event was intracranial, spontaneous or traumatic, leading to a dramatic prognosis (64.2% of dead patients at 1 month had IHC). The number of patients with OA and AP was similar, but, the multivariate OR for death in ICH patients was significantly unfavourable at one month in OA versus AP (OR = 1.89 (95%CI 1.01-3.54). This high mortality has been reported elsewhere [START_REF] Beynon | Clinical review : traumatic brain injury in patients receiving antiplatelet medication[END_REF][START_REF] Lauer | Intracerebral haemorrhage associated with antithrombotic treatment: translational insights from experimental studies[END_REF] and seemed to be dependant to the mechanism of the ICH and to the antithrombotic medication. In a multicenter Italian study in emergency departments, Baldi and al reported a mortality rate of 44% in post traumatic IHC (41% with AP and 51% with OA), and 58% in spontaneous ICH (51% with AP and 69% with OA [4]. Like others, we found that age and Glasgow coma scale will be strong prognostic factors in patients with intracranial haemorrhage. In case of ICH, time between admission and diagnosis and time between admission and first biological results were significantly shorter in patients who died, which could be explained by a more severe condition on admission. In head trauma, oral anticoagulant led to 10-fold higher incidence of intracranial haemorrhagic lesions (3 versus 29%), reaching a mortality rate near to 60% [START_REF] Leiblich | Emergency management of minor head injury in anticoagulated patients[END_REF]. Old age increased furthermore this rate (OR = 1.2 for each increase of 10 years) [START_REF] Moore | Impact of age and anticoagulation: need for neurosurgical intervention in trauma patients with mild traumatic brain injury[END_REF]. As reported by Dowlatshahi et al [START_REF] Dowlatshahi | Poor prognosis in warfarin-associated intracranial haemorrhage despite anticoagulation reversal[END_REF] and as shown in our results, the mortality did not seem to be affected by complete reversal therapy. For antiplatelet drugs, there is a great heterogeneity for establishing a strong link between ICH and this medication, because of various inclusion criteria, different methodology, severity at inclusion, mechanism, outcomes. Nevertheless, antiplatelet drugs appeared to be a risk factor for IHC in patients with minor head trauma. In a recent meta-analysis, Bachelor et al [START_REF] Batchelor | A meta-analysis to determine the effect of preinjury antiplatelet agents on mortality in patients with blunt head trauma[END_REF] reported an important overall relative risk of aspirin in all head trauma (OR = 2.43), higher than clopidogrel (OR = 1.55); other studies have reported opposite results, with a more important risk with clopidogrel than with aspirin [24,[START_REF] Nishijima | Risk of traumatic intracranial haemorrhage in patients with head injury and preinjury warfarine or clopidogreluse[END_REF]. Whatever these various risk assessments, the old age and the Glasgow coma score appeared to be strong prognosis factors, as shown in our results [START_REF] Pomero | Short-term prognosis of intracranial haemorrhage in patients on oral anticoagulant or antiplatelet drugs[END_REF]. In spontaneous ICH, the systematic review of Thompson et al reported an increase of mortality rate induced by antiplatelet agents (40% in univariate analysis, 27% in multivariate analysis) and a higher poor prognosis (respectively 29 and 10%) [START_REF] Thompson | Prior antiplatelet therapy and outcome following intracerebral hemorrhage. A systematic review[END_REF].

Strengths

The selection method used in our study would have contributed to increase the number of haemorrhagic events. We selected patients firstly from the haemorrhagic symptom at emergency admission, in contrast with most studies on this subject which screened patients from the medication. Our method was quite similar with the study reported by Budnitz et al, in which coders reviewed clinical records of each emergency department visit to identify adverse drug events [3]. Heterogeneity of codes compelled to use a list of about forty different diagnostic codes relative to haemorrhagic events and pathologies. This first step was then completed by the search for the criteria of major bleeding and finally by the identification of the antithrombotic medication, information often missed in the emergency medical record. This method could explain the equal number of major bleeding with oral anticoagulant and antiplatetet drugs, with a quite equivalent distribution of types of haemorrhage, particularly intracranial haemorrhage and gastrointestinal bleeding. In the same way, it could explain the relative important group of patients with parenteral anticoagulants.

Limitations

Our study was monocentric and in part retrospective. Computer requests were made every month in a retrospective manner, which could explain a lost of some data concerning medical history and antithrombotic duration. However, this design could provide an unbiased observation of real life behaviour because patients' management was not influenced by the knowledge of an ongoing study. Because of the heterogeneity of diagnostic codes, we listed about 40 different symptoms and diseases related to haemorrhagic events. We used criteria of major bleeding recommended by the French National Authority for Health (Haute Autorité de Santé -HAS) [7] rather than criteria of the International Society on Thrombosis and Haemostasis (ISTH) defined by Schulman et al [START_REF] Schulman | Definition of major bleeding in clinical investigations of antihemostatic medicinal products in non-surgical patients[END_REF]. We thought that all these criteria were close to each other, which would not have modified our population. In addition, we thought that the fall in haemoglobin level of 20g/L or more is a less suitable criterion for an emergency recruitment.

Conclusion

Our study shows the magnitude of major bleeding in a very old population taking antithrombotics, admitted in the emergency department of a teaching hospital. Oral anticoagulants and antiplatelet drugs are equally implicated in this iatrogenic pathology. The severity is attested by a high mortality (about 20%) at one month after the haemorrhagic event. Intracranial haemorrhage, age and Glasgow coma scale at admission are prognostic factors. In case of intracranial haemorrhage, oral anticoagulant is a predictor of death. The high frequency of these events and the important mortality rate suggest a great vigilance in risk benefit imbalance of antithrombotic medication for individual patients, especially for elderly.

Table 2

 2 Distribution of haemorrhagic events

	Type of haemorrhage