Romain Postoyan
email: romain.postoyan@cran.uhp-nancy.fr

Adolfo Anta
email: anta@control.tu-berlin.de

Dragan Nešić

Paulo Tabuada
email: tabuada@ee.ucla.edu

A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems

We present a prescriptive framework for the event-triggered control of nonlinear systems. Rather than closing the loop periodically, as traditionally done in digital control, in event-triggered implementations the loop is closed according to a state-dependent criterion. Event-triggered control is especially well suited for embedded systems and networked control systems since it reduces the amount of resources needed for control such as communication bandwidth. By modeling the event-triggered implementations as hybrid systems, we provide Lyapunov-based conditions to guarantee the stability of the resulting closed-loop system and explain how they can be utilized to synthesize event-triggering rules. We illustrate the generality of the approach by showing how it encompasses several existing event-triggering policies and by developing new strategies which further reduce the resources needed for control.

I. INTRODUCTION

, a simple strategy is proposed for nonlinear systems. The idea is the following. Assuming the continuous-time closed-loop system is input-to-state stable (ISS) with respect to measurement errors, a triggering condition is derived to guarantee that the Lyapunov function V for the continuous-

time system always decreases at a given rate when control tasks are executed at discrete time instants. It is shown that there does exist a constant minimal time interval between executions that reinforces the idea that event-triggered control is expected to generate larger inter-event intervals than periodic rules. This translates into a lower usage of the communication bandwidth and the computational resources. Noting that the monotonic decrease of V is not necessary to guarantee asymptotic stability for the obtained hybrid systems, a triggering rule is developed in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] to ensure that V appropriately decreases at each transmission instant. This method was shown to potentially exhibit larger inter-event intervals compared to [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF].

In this paper, we present a prescriptive framework for the event-triggered control of nonlinear systems. We model the problem as a hybrid system using the formalism of [START_REF] Goebel | Solution to hybrid inclusions via set and graphical convergence with stability theory applications[END_REF], as in [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved event-triggering[END_REF]. We start by identifying the key features of the strategy in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] in terms of a hybrid Lyapunov function and use them to introduce the main idea of our approach. The proposed framework relies on Lyapunov-based conditions that can be used to synthesize event-triggering rules to guarantee asymptotic stability properties. Our approach encompasses the strategies in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] for which we propose new stability analyses and relax some of the required conditions. We also develop a family of new triggering rules inspired by [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] which allows us to trade performance for longer interevent times. In the companion paper [START_REF] Postoyan | Event-triggered and self-triggered stabilization of distributed networked control systems[END_REF], we show how this framework can be applied to distributed networked control systems subject to scheduling.

The proofs are omitted for space reasons and can be found in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF].

II. NOTATION AND DEFINITIONS

Let R = (-∞, ∞), R ≥0 = [0, ∞), R >0 = (0, ∞), Z ≥0 = {0, 1, 2, . . .}, Z >0 = {1, 2, . . .}. A function γ : R ≥0 → R ≥0 is of class K if it is continuous, zero at zero and strictly increasing, and it is of class K ∞ if in addition it is unbounded. A continuous function γ : R 2 ≥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. Additionally, a function β : R 3 ≥0 → R ≥0 is of class KLL, if β(•, •, t) ∈ KL and β(•, t, •) ∈ KL for any t ∈ R ≥0 . For (x, y) ∈ R n+m , the notation (x, y) stands for [x T , y T ] T . The distance of a vector x to a set A ⊂ R n is denoted by |x| A = inf{|x -y| : y ∈ A}.

We will consider locally Lipschitz Lyapunov functions (that are not necessarily differentiable everywhere), therefore we will use the Clarke derivative which is defined as follows

for V : R n → R ≥0 : V • (x; v) := lim h→0 + sup y→x V (y+hv)-V (y) h
, that corresponds to the usual derivative when V is continuously differentiable. We define the generalized gradient of f : R n → R at x as:

∂f (x) := ζ ∈ R n : f • (x; v) ≥ ζ, v ∀v ∈ R n
, that matches the classical notion of gradient when f is differentiable. We will write hybrid systems using the models proposed in [START_REF] Goebel | Solution to hybrid inclusions via set and graphical convergence with stability theory applications[END_REF], that are of the form:

ẋ = f (x) x ∈ C, x + = g(x) x ∈ D, (1)
where x ∈ R n is the state and C, D ⊂ R n are respectively the flow and the jump sets. Hence, any hybrid system is defined by a tuple (C, D, f, g). The solutions of ( 1) are defined on so-called hybrid time-domains. 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 (1) For all j ∈ Z ≥0 and almost all t ∈ R ≥0 such that (t, j) ∈ dom φ we have:

times 0 = t 0 ≤ t 1 ≤ . . . ≤ t J . E is a hybrid time domain if for all (T, J) ∈ E, E ∩ ([0, T ] × {0, 1, . . . ,
φ(t, j) ∈ C, φ(t, j) = f (φ(t, j)). (2)
For (t, j) ∈ dom φ such that (t, j + 1) ∈ dom φ, we have φ(t, j) ∈ D, φ(t, j + 1) = g(φ(t, j)). Assuming f and g are continuous and C, D are closed, system (1) possesses solutions that may be non-unique, see [START_REF] Goebel | Solution to hybrid inclusions via set and graphical convergence with stability theory applications[END_REF]. We are interested in the following stability definition.

Definition 1. The closed set A ⊂ R n is semiglobally asymptotically stable (S-GAS) for system [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF] if for any ∆ ∈ R >0 there exists β ∆ ∈ KLL such that for any solution φ to [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF]

with |φ(0, 0)| A ≤ ∆: |φ(t, j)| A ≤ β ∆ (|φ(0, 0)| A , t, j) for all (t, j) ∈ dom φ.
Remark 1. It is shown for continuous-time systems in Proposition 3.4 in [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF] that a closed set A is S-GAS if and only if it is globally asymptotically stable. We note that the stability bound in Definition 1 does not imply forward completeness and this is referred to as pre-asymptotic stability according to [START_REF] Cai | Smooth Lyapunov functions for hybrid systems part II: (pre)asymptotically stable compact sets[END_REF].

We will show that two successive jumps (that will correspond to data transmissions in our study) are always separated by a certain uniform amount of time as long as the solution is not in the stable set A.

Definition 2. For any forward invariant set 1 A ⊂ R n for system [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF], we say that solutions to (1) have a semiglobal dwell time on R n \A if, for any ∆ ∈ R >0 , there exists τ (∆) ∈ R >0 such that for any solution φ to [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF]

with |φ(0, 0)| A ≤ ∆, j ∈ Z ≥0 : φ(t, j) / ∈ A ∀t ∈ [t j , t j+1 ] ⇒ t j+1 -t j ≥ τ (∆) , (2)
1 A set A ⊂ R n is forward invariant for system (1), if for any solution φ to (1) with φ(0, 0) ∈ A we have φ(t, j) ∈ A for all (t, j) ∈ dom φ.

where dom φ = ([t j , t j+1 ], j).

III. PROBLEM STATEMENT

A. System models

Consider the following plant:

ẋP = f P (x P , u), (3)
where x P ∈ R nP is the plant state, u ∈ R nu the control input for which a stabilizing dynamic state-feedback controller is designed:

ẋC = f C (x C , x P ), u = g C (x C , x P ), (4)
where x C ∈ R nC is the controller state. On digital platforms, transmission between the sensors, the controller and the actuators only occur at some transmission instants t j , j ∈ Z >0 . The problem can then be modeled as follows:

ẋP = f P (x P , û) ∀t ∈ [t j-1 , t j ] ẋC = f C (x C , xP ) ∀t ∈ [t j-1 , t j ] u = g C (x C , xP ) ẋP = fP (x P , x C , xP , û) ∀t ∈ [t j-1 , t j ] u = fC (x P , x C , xP , û) ∀t ∈ [t j-1 , t j ] xP (t + j ) = x P (t j ) û(t + j ) = u(t j )                    (5)
where xP and û denote the variables respectively generated from the most recently transmitted plant state and control input. They are usually kept constant between two transmission instants i.e. xP (t) = x P (t j-1 ) and û(t) = u(t j-1 ) for t ∈ [t j-1 , t j ] that corresponds to fP = 0 and fC = 0. However, other implementations are possible. At each transmission instant, the controller receives x P (t j ), updates xP (t + j ) = x P (t j ), sends the control input u(t j ) and the actuators update û(t + j ) = u(t j ). We suppose that this process occurs in a synchronized manner and leave the study of the effects of the eventual induced delays for future work.

Traditionally, the sequence of t j , j ∈ Z >0 , is periodic, i.e. t j -t j-1 = T where T ∈ R >0 . The stability of system ( 5) is then guaranteed by selecting T sufficiently small, see [START_REF] Aström | Computer-controlled systems, theory and design[END_REF], [START_REF] Laila | Open and closed loop dissipation inequalities under sampling and controller emulation[END_REF], [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] to mention a few. In this study, we abandon this paradigm and implicitly define the transmission instants by a rule based on the states of system [START_REF] Cai | Smooth Lyapunov functions for hybrid systems part II: (pre)asymptotically stable compact sets[END_REF]. Rewriting the problem using the hybrid formalism in [START_REF] Goebel | Solution to hybrid inclusions via set and graphical convergence with stability theory applications[END_REF], similar to Section II.C in [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved event-triggering[END_REF], we obtain:

ẋ = f (x, e) ė = g(x, e) (x, e) ∈ C, x + = x e + = 0 (x, e) ∈ D, (6)
f (x, e) = f P (x P , g C (x C , xP ) + e u ) f C (x C , xP ) g(x, e) =          fP (x P , x C , xP , g C (x C , xP ) + e u ) -f P (x P , g C (x C , xP ) + e u ) fC (x P , x C , xP , g C (x C , xP ) + e u ) -∂gC ∂xC (x C , xP )f C (x C , xP ) -∂gC ∂ xP (x C , xP ) × fP (x P , x C , xP , g C (x C , xP ) + e u )          (7)
and are assumed to be continuous.

Remark 2. Our assumptions allow for triggering rules that depend both on x and e. However, the specific choice of triggering rule needs to be done according to the implementation scenario. In the case of dynamic controllers, a triggering rule depending on x C requires continuous communication between the sensors and the controller. This is difficult to achieve in practice since sensors do not have, in general, access to the state of the controller. We have chosen to present the problem in a general setting because it allows to recover as particular cases the stabilization using a static controller (as in Sections III-B, V-A for example) and the cases where only the plant states or the inputs are sampled.

The main problem addressed in this paper is to define the triggering condition, i.e. the flow and jump sets C and D in [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved event-triggering[END_REF], in order to minimize the resource usage while ensuring asymptotic stability properties. We now introduce the main idea of the framework presented hereafter in Section IV by interpreting the work in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] using a hybrid Lyapunov function.

B. Main idea

We first revisit the work in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] where a static controller u = g C (x P ) is assumed to render the closed-loop system (3) input-to-state stable (ISS) with respect to the samplinginduced errors (that can be considered as measurement errors at this stage since, when the controller is static, the samplinginduced error can be seen to be only due to the sampling of the measurements, i.e. e = e xP ). This is equivalent to the following assumption (see Theorem 1 in [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF]) where x = x P (as the controller is static).

Assumption 1. There exists a smooth Lyapunov function

V : R nx → R and α V , α V , α, γ ∈ K ∞ such that for all x ∈ R nx : α V (|x|) ≤ V (x) ≤ α V (|x|), (8)
and for all (x, e) ∈ R nx+ne :

∂V ∂x f (x, e) ≤ -α(V (x)) + γ(|e|). (9)
Since zero-order-hold devices are used in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], we have g(x, e) = -f (x, e) in [START_REF] Goebel | Solution to hybrid inclusions via set and graphical convergence with stability theory applications[END_REF] and the model ( 6) is here:

ẋ = f (x, e) ė = -f (x, e) (x, e) ∈ C, x + = x e + = 0 (x, e) ∈ D. (10)
From ( 9), we deduce that σα(V (x)) ≥ γ(|e|) with σ ∈ (0, 1) implies:

∂V ∂x f (x, e) ≤ -(1 -σ)α(V (x)). (11)
In that way, the triggering rule in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] can be written as σα(V (x)) ≤ γ(|e|), that we rewrite as:

V (x) ≤ α -1 (σ -1 γ(|e|)) =: γ(|e|). (12)
At each transmission instant, e is reset to 0, so we have γ(|e + |) = 0 ≤ V (x) and V decreases monotonically according to [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. The next transmission occurs as soon as ( 12) is satisfied. The flow and the jump sets in [START_REF] Löfberg | Pre-and post-processing sum-of-squares programs in practice[END_REF] can be defined as follows:

C = (x, e) : γ(|e|) ≤ V (x) D = (x, e) : γ(|e|) ≥ V (x) . ( 13
)
To guarantee the existence of a minimum interval of time between two transmissions when (x, e) = 0, the following conditions are used in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF].

Assumption 2. For any compact set S ⊂ R nx+ne , there exist L 1 , L 2 ∈ R ≥0 such that for all (x, e) ∈ S:

|f (x, e)| ≤ L 1 (|x| + |e|) (14) α -1 V • γ(|e|) ≤ L 2 |e|. (15)
The stability analysis of system (10) can be done using the following Lyapunov function (assuming γ is locally Lipschitz): R(x, e) = max{V (x), γ(|e|)}. Indeed,

• Property (a): R is positive definite and radially unbounded in view of (8) and since γ ∈ K ∞ . • Property (b): R decreases on C according to [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF].

• Property (c): R does not increase at jumps since x + = x and e + = 0. • Property (d): it was shown in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] using Assumption 2 that there does exist a uniform minimal time interval between two successive transmission instants (as long as (x, e) = 0) for solutions that start in a compact set that contains the origin. In other words, solutions to [START_REF] Löfberg | Pre-and post-processing sum-of-squares programs in practice[END_REF] have a semiglobal dwell time on R nx+ne \{0} according to Definition 2. We show in Section IV that these four properties guarantee asymptotic stability properties for system [START_REF] Löfberg | Pre-and post-processing sum-of-squares programs in practice[END_REF] and that they can be used to build up other event-triggering conditions. Note that similar ingredients are used to prove the stability of other types of hybrid systems in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], [START_REF] Nešić | Stability properties of reset systems[END_REF] for example.

IV. A LYAPUNOV-BASED FRAMEWORK

Before stating the main result of this section, it is important to note that auxiliary variables may be introduced to define the triggering condition. Indeed, it is common in the hybrid literature to introduce additional variables like clocks to ensure or analyse the stability e.g. [START_REF] Cai | Smooth Lyapunov functions for hybrid systems part II: (pre)asymptotically stable compact sets[END_REF], [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. We will see in Section V-A that the strategy in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] can be interpreted using our framework by making use of an additional variable which is employed to build up a decreasing threshold on the known Lyapunov function for the system in the absence of sampling. We also show in Section V-B that the eventtriggered policy in Section III-B can be redesigned to exhibit larger inter-event intervals thanks to the use of an auxiliary variable. Therefore, we denote by a single vector variable η ∈ R nη the additional variables which may be needed for describing the system that are neither x nor e.

In that way, to define a triggering condition ensuring desired stability properties for the overall system is tantamount to defining appropriate flow and jump sets C and D for the following hybrid system: ẋ = f (x, e) ė = g(x, e) η = h(x, e, η) q ∈ C, x + = x e + = 0 η + = l(x, e, η) q ∈ D, [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] where q = (x, e, η) ∈ R nq , h, l are continuous and C, D are closed subsets of R nq . We use q = F (q) and q + = G(q) to denote [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF].

The stability of system ( 16) can be guaranteed by means of the following theorem. It can be seen as a variation of the results in [START_REF] Cai | Smooth Lyapunov functions for hybrid systems part II: (pre)asymptotically stable compact sets[END_REF].

Theorem 1. Consider system [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] and suppose G(D) ⊂ (C ∪ D) and that there exist a locally Lipschitz function R : R nq → R and a continuous function υ : R nη → R nυ with n υ ≤ n η such that the following conditions hold.

(i)

There exist α R , α R ∈ K ∞ such that for any q ∈ C ∪D, α R (|(x, e, υ(η))|) ≤ R(q) ≤ α R (|(x, e,

υ(η))|). (ii)

There exists α R ∈ K ∞ such that for all q ∈ C: R • (q; F (q)) ≤ -α R (R(q)). (iii) For all q ∈ D, R(G(q)) ≤ R(q). (iv) Solutions to [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] have a semiglobal dwell time on R nq \A, where A = q : (x, e, υ(η)) = 0 . Then the set A is S-GAS.

Theorem 1 provides a Lyapunov-based prescriptive framework for developing event-triggered control strategies for nonlinear systems as we show in Section V. Other triggering rules may be derived by following the guidelines below for instance. We illustrate each item with the example of Section III-B for the sake of clarity.

1) Select a locally Lipschitz function R : R nq → R that satisfies item (i) of Theorem 1. Usually, R is built using a known Lyapunov function V for the continuoustime system (3)-( 4) in the absence of sampling and a positive definite radially unbounded function W (e) that has to be designed. Typically, W is chosen by investigating the robustness property of the closed-loop system ẋ = f (x, e) with respect to e that is assumed to hold. The sets C and D have not been defined so far but item (i) of Theorem 1 needs to hold on C ∪ D. This apparent contradiction is overcome as follows.

When there is no variable η, as it is the case so far, we typically have C ∪ D = R nx+ne and we do not need to know C and D to verify item (i) of Theorem 1.

Section III-B: we took W (e) = γ(|e|) where γ(|e|) is defined in [START_REF] Nešić | Stability properties of reset systems[END_REF] that is deduced from the ISS property stated in Assumption 1. We considered R(q) = max{V (x), γ(|e|)} with q = (x, e), that does satisfy item (i) of Theorem 1 on R nx+ne . This corresponds to Property (a). 2) Choose α R ∈ K ∞ for item (ii) of Theorem 1. Obviously, if R(x, 0) = V (x), the decreasing rate α R will have to be less than the decreasing rate of V in order to allow some flow before entering the set D.

Section III-B: we have taken α R (s) = (1 -σ)α(s) ≤ α(s) for s ∈ R ≥0 , since σ ∈ (0, 1). 3) Define the flow and the jump sets to be closed and such that items (i)-(iii) of Theorem 1 hold and G(D) ⊂ (C ∪ D). For instance, when items (i) and (iii) of Theorem 1 are satisfied for all q ∈ R nq , we can directly take the following sets: C = q : R • (q; F (q)) ≤ -α R (R(q)) and D = q : R • (q; F (q)) ≥ -α R (R(q)) which ensure item (ii) of Theorem 1 and

G(D) ⊂ (C ∪ D) = R nq .
Section III-B: the flow and the jump sets in [START_REF] Otanez | Using deadbands to reduce communication in networked control systems[END_REF] guarantee that items (ii)-(iii) of Theorem 1 holds in view of [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] and since x + = x and e + = 0, that is equivalent to Properties (b)-(c). We note that C ∪ D = R nx+ne . 4) Study the existence of dwell times. Among other techniques, Lemma 1 below can be used for this purpose. The existence of dwell times notably depends on the triggering condition and the vector field F that is usually assumed to satisfy some Lipschitz properties.

If the existence of a dwell time is guaranteed, the desired result is obtained. Otherwise, variable η may be introduced, then go back to 1) and modify the function2 R. The way the variable η may be chosen will become clearer in the light of Section V. The non-existence of dwell time may also be due to the fact that the decreasing rate of R along flows, α R in 2), is too strong, thus choose a different function αR ∈ K ∞ such that αR (s) < α R (s) for any s ∈ R >0 .

Section III-B: the existence of semiglobal dwell-time solutions is guaranteed in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] using Assumption 2, as stated in Property (d). The following lemma provides a tool for verifying the existence of dwell times which is used in the proofs of the theorems of Section V (that are not provided in this paper). Lemma 1. Consider system [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] and suppose the following holds.

(i) G(D) ⊂ (C ∪ D) and items (i)-(iii) of Theorem 1 are satisfied. (ii) For any q ∈ D, G(0, e, η) ∈ A ∪ M where A = q : (x, e, υ(η)) = 0 and M ⊂ (C\D) is a forward invariant set. (iii) There exists a locally Lipschitz function ψ : Θ(µ) → R ≥0 , where Θ(µ) = {q ∈ C ∪ D : R(q) ≤ µ and x = 0} for µ > 0, such that: (iii-a) There exists a ∈ R ≥0 such that for any q ∈ D with G(q) ∈ Θ(µ): ψ(G(q)) ≤ a.

(iiii-b) There exists b > a such that for any solution φ to [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF], t j ≤ t with (t, j) ∈ dom φ: ψ(φ(t, j)) < b ⇒ φ(t, j) ∈ C\D . (iii-c) There exists a continuous non-decreasing function λ : R >0 → R ≥0 such that for all q ∈ Θ(µ): ψ • (q; F (q)) ≤ λ(ψ(q)). Then solutions to [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] have a semiglobal dwell time on R nq \A. Remark 3. The triggering condition that satisfies the conditions of Theorem 1 respects the practical requirement that there does exist a uniform minimum time interval between two transmissions according to item (iv) of Theorem 1. The only region of the state space where this may not be guaranteed is when (x, e, υ(η)) = 0, but this will only occur if the system is initialized in the stable set.

V. APPLICATIONS

We already know that the framework allows us to capture the work in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], we show in this section that it is also the case for the strategy in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF]. Afterwards, new triggering rules are proposed.

A. Event-triggered strategy in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] As in Section III-B, the controller is static (x = x P ) and implemented using zero-order-hold devices. It is considered that Assumption 1 is satisfied with α linear that is α(s) = ᾱs with ᾱ ∈ R >0 . The triggering rule is defined to guarantee that V (x(t j )) always decreases at a certain rate compared to V (x(t j-1 )). In that way, the control loop is closed in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] as soon as the condition below is violated, for t ≥ t j-1 :

V (x(t)) ≤ -σ ᾱV (x(t j-1 ))(t -t j-1 ) + V (x(t j-1 )) = (-σ ᾱ(t -t j-1 ) + 1)V (x(t j-1 )), (17)
where σ ∈ (0, 1). Since zero-order-hold devices are considered, we have x(t) = x(t j-1 ) and e(t) = e xP (t) = x(t j-1 ) -x(t). Consequently, ( 17) is equivalent to, for t ≥ t j-1 :

V (x(t)) ≤ (-σ ᾱ(t -t j-1 ) + 1)V (x(t) + x(t) -x(t)) = (-σ ᾱ(t -t j-1 ) + 1)V (x(t) + e(t)). (18)
To model ( 17) using the hybrid formulation ( 16), we introduce the variable η ∈ R as the solution of η = -σ ᾱ on flows and η + = 1 at jumps. We see that η(t) = -σ ᾱ(t -t j-1 ) + 1 for t ∈ [t j-1 , t j ] (j ∈ Z >0 ). In that way, we can reformulate (18) using the following algebraic inequality:

V (x) ≤ ηV (x + e). (19)
The problem can then be modeled as follows:

ẋ = f (x, e) ė = -f (x, e) η = -σ ᾱ    q ∈ C, x + = x e + = 0 η + = 1    q ∈ D, (20)
where q = (x, e, η),

C = q : V (x) ≤ ηV (x + e) and η ∈ [ε, 1] D = D 1 ∪ D 2 , (21)
with D 1 = q : V (x) ≥ ηV (x + e) and ∂V ∂x f (x, e) ≥ -σ ᾱV (x) and D 2 = q : η = ε where ε ∈ (0, 1) is arbitrary small. The condition ∂V ∂x f (x, e) ≥ -σ ᾱV (x) has been added in the definition of D 1 to avoid Zeno behaviour since after a jump V (x) = ηV (x + e) holds. Indeed, it is not necessary to jump again since V (x) will decrease faster than ηV (x + e) for some time according to [START_REF] Laila | Open and closed loop dissipation inequalities under sampling and controller emulation[END_REF]. The lower bound ε on η is used to guarantee that the threshold on V (x) defined by ηV (x+e) (see ( 19)) never reaches the origin when V (x + e) = 0. This condition adds no conservatism as by setting ε sufficiently small, the triggering condition η = ε will not be satisfied in practice before q reaches D 1 .

We recover Theorem 3.2 in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] and relax some of the required conditions. Its proof is based on Theorem 1 with R(q) = max{V (x), ηV (x + e)} and υ(η) = 0. Theorem 2. Consider system (20) and suppose Assumption 1 holds with α(s) = ᾱs ( ᾱ ∈ R >0 ) and Assumption 2 is satisfied. Then the set A = {q : (x, e) = 0} is S-GAS and solutions to (20) have a semiglobal dwell time on R nq \A.

We note that the conditions of Theorem 2 are more general than those of Theorem 3.2 in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] as γ in ( 9) is allowed to be a nonlinear function. In addition, condition [START_REF] Postoyan | Event-triggered and self-triggered stabilization of distributed networked control systems[END_REF] in this paper extends [START_REF] Cai | Smooth Lyapunov functions for hybrid systems part II: (pre)asymptotically stable compact sets[END_REF] in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] and allows us to consider more general types of Lyapunov functions, such as quadratic, which is not the case in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF].

B. New triggering rules

In Section V-A, the triggering condition is obtained by defining a decreasing threshold on V (see [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]). In this subsection, we propose an alternative that consists in defining a similar threshold for an appropriate function W for the e-system. We suppose that the dynamic controller (4) has been designed so that Assumption 1 applies. Thus, by using the ISS property of the x-system, we will be able to show that when W remains below a given decreasing threshold, system (16) satisfies asymptotic stability properties.

We define our threshold variable η ∈ R as the solution of the following differential equation on flows:

η = -δ(η), (22)
where δ is any class-K ∞ function, and at jumps,

η + = γ(|e|) =: W (e), (23)
where γ(s) = α -1 (σ -1 γ(s)) for s ∈ R ≥0 , with σ ∈ (0, 1) as in [START_REF] Nešić | Stability properties of reset systems[END_REF]. We note that W is positive definite and radially unbounded. An obvious choice of triggering rule is: W (e) ≥ η. Nevertheless, in the case where W (e) ≤ V (x), V decreases according to [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] and therefore we do not need to close the loop. This suggests considering the following triggering condition instead:

W (e) ≥ max{η, V (x)}. (24)
The problem can be modeled as follows:

ẋ = f (x, e) ė = g(x, e) η = -δ(η)    q ∈ C, x + = x e + = 0 η + = W (e)    q ∈ D, (25)
where q = (x, e, η), C = q : max{V (x), η} ≥ W (e) and η ≥ 0 D = q : max{V (x), η} ≤ W (e) and η ≥ 0 .

The following theorem ensures the stability of system (25). Its proof is an application of Theorem 1 with R(q) = max{V (x), γ(W (e)), η} and υ(η) = η.

Theorem 3. Consider system (25), suppose the following conditions hold.

(i) Assumptions 1-2 apply.

(ii) Function γ is locally Lipschitz.

(iii) For any compact set S ⊂ R nx+ne , there exist L 3 ≥ 0 such that for all (x, e) ∈ S: |g(x, e)| ≤ L 3 (|x| + |e|). Then q = (x, e, η) = 0 is S-GAS and solutions to (25) have a semiglobal dwell time on R nq \{0}.

Contrary to Section V-A, we note that α in ( 9) is allowed to be nonlinear. In addition, we do not focus on zero-orderhold devices that is why condition (iii) of Theorem 3 is introduced in order to guarantee the existence of dwell times. We show on an example in Section VI that the inter-event intervals can be enlarged to some extent compared to Section V-A by playing with the initial value of η.

VI. ILLUSTRATIVE EXAMPLE

To illustrate the benefits of the strategy presented in Section V-B, we revisit the example considered in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. The simplified version of the considered nonlinear system is:

ẋ = dx 2 -x 3 + u, ( 27
)
where d is an unknown possibly time-varying parameter satisfying |d| < 1. The stabilizing control law considered in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] was u = -2x. We select V (x) = 1 2 x 2 as our Lyapunov function that satisfies Assumption 1 with 3 α V (s) = α V (s) = 1 2 s 2 , α(s) = 0.84s and γ(s) = 2.66s 2 for s ∈ R ≥0 . We consider 200 random initial conditions distributed in the interval [-1, 1]. The parameter d takes for each initial condition a random value in the interval [0, 1]. We compare the average number of executions required under the technique in [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] as extended in Section V-A, the event-triggered strategy proposed in Section V-B and the periodic strategy in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] in Table I, for different values for the design parameter η(0, 0). We select σ = 10 -3 in (20) and δ(s) = 0.5s with s ∈ R ≥0 in (22). It can be observed that the average number of executions is considerably lower under the event-triggered strategies. Moreover we note that the policy in Section V-B generates less executions than [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] and that it can be adjusted by means of the design parameter η(0, 0). It is however not easy to compare the performance guarantees under the two different policies, since Theorem 2 and Theorem 3 ensure different stability properties.

We can explore deeper the role played by the initial condition of the auxiliary variable η(0, 0) and its effect on performance and number of executions. Simulations have shown that smaller values of η(0, 0) imply a faster decay, at 3 The Yalmip software ( [START_REF] Löfberg | Pre-and post-processing sum-of-squares programs in practice[END_REF]) was used to compute α and γ. [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] [18] Section V-B η(0, 0) = 0.1 η(0, 0) = 1 η(0, 0) = 2 [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF].

the expense of more executions. Hence the design parameter η(0, 0) represents the tradeoff between performance and resource usage. Similar conclusions can be drawn for the decay function δ in (22). The technique in Section V-B exhibits great potential for real-time scheduling, since both the initial value for the auxiliary variable and the differential equation in ( 22) can be designed according to the available resources. For instance, functions δ with slow increasing slopes could be chosen in case of overload in the network or in the processor executing the controller.

  where x = (x P , x C ) ∈ R nx , e = (e xP , e u ) ∈ R ne denotes the sampling-induced error with e xP = xP -x P , e u = û-u. The sets C and D are closed and respectively denote the flow and the jump sets, they are defined according to the triggering condition. Typically, the system flows on C and experiences a jump on D where the triggering condition is satisfied. When (x, e) ∈ C ∪D, the system can either jump or flow, the latter only if flowing keeps (x, e) in C. Functions f and g are defined as (where we can replace xP by x P +e xP ):

TABLE I AVERAGE

 I NUMBER OF EXECUTIONS OVER 200 INITIAL CONDITIONS FOR A SIMULATION TIME OF 20S FOR THE EXAMPLE IN

	54.34	18.47	16.88	15.27	13.31

It may be the case that the new R no longer satisfies item (i) of Theorem 1 on R nq , so identify S ⊆ R nη such that item (i) of Theorem 1 is satisfied on R nx+ne × S, afterwards make sure (C ∪ D) ⊂ R nx +ne × S.

This work was supported by the Australian Research Council under the Future Fellowship and Discovery Grants Schemes, the NSF awards 0820061 and 0834771 and the Alexander von Humboldt Foundation R. Postoyan is with the Centre de Recherche en Automatique de Nancy, UMR 7039,


Romain Postoyan
email: romain.postoyan@cran.uhp-nancy.fr

Paulo Tabuada
email: tabuada@ee.ucla.edu

Dragan Nešić

Adolfo Anta
email: anta@control.tu-berlin.de

Event-triggered and self-triggered stabilization of distributed networked control systems

Event-triggered and self-triggered control have recently been proposed as implementation strategies that considerably reduce the resources required for control. Although most of the work so far has focused on closing a single control loop, some researchers have started to investigate how these new implementation strategies can be applied when closing multiplefeedback loops in the presence of physically distributed sensors and actuators. In this paper, we consider a scenario where the distributed sensors, actuators, and controllers communicate via a shared wired channel. We use our recent prescriptive framework for the event-triggered control of nonlinear systems to develop novel policies suitable for the considered distributed scenario. Afterwards, we explain how self-triggering rules can be deduced from the developed event-triggered strategies.

I. INTRODUCTION

Today's control systems are frequently implemented over networks as these types of structures present many advantages in terms of flexibility and cost. In this setup, controllers communicate with sensors and actuators through the network, not in a continuous fashion but rather at discrete time instants when the channel is available for the control system. Traditionally, the time interval between two successive transmissions is constrained to be less than a fixed constant T , which is called the maximum allowable transmission interval (MATI) (see e.g. [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], [START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF]). In order to achieve a desired performance, T is generally chosen as small as technology and network load permit. This strategy, although easy to implement and analyze, represents a conservative solution that may unnecessarily overload the communication channel. Indeed, one would expect that the transmission instants should not satisfy a prefixed bound but rather be based on the current state of the system, the channel occupancy and the desired performance. Drawing intuition from this idea, event-triggered control has been developed to reduce the need for feedback while guaranteeing satisfactory levels of performance. It involves closing the loop whenever a predefined state-dependent triggering condition is satisfied, e.g. [START_REF] Arzén | A simple event-based PID controller[END_REF], [START_REF] Aström | Comparison of Riemann and Lebesgue sampling for first order stochastic systems[END_REF], [START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF], [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. This technique reduces resource usage such as communication bandwidth or computational time and provides a high degree of robustness since the state is continuously monitored. Most work in this direction has focused so far on the so-called one packet transmission problem, which corresponds to the case when all the states are sent together in a single packet. This generally implies the collocation of all sensors and, for multiple-input control systems, collocation of all actuators as well. Such an assumption does not hold in many cases. Thus, significant work on distributed event-triggered control has recently appeared in [START_REF] Mazo | Decentralized event-triggered control over wireless sensor/actuator networks[END_REF], [START_REF] Wang | Asymptotic stability in distributed eventtriggered networked control systems with delays[END_REF]. These papers focus on a setup where sensors decide locally when they need to transmit their measurements. Proposed solutions may however be conservative as they allow several sensors to transmit simultaneously.

In this paper, we consider a network setup where a central coordinator is available and only one group of sensors or actuators (that forms a node) can transmit at each transmission instant. This central coordinator grants access to the node selected according to a given scheduling protocol whenever a predefined triggering rule is satisfied. We follow the same approach as in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF], which is a particular case of this study since we considered networks that have one node only. Modeling the problem using the hybrid formalism of [START_REF] Goebel | Solution to hybrid inclusions via set and graphical convergence with stability theory applications[END_REF], we apply the prescriptive framework in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] to synthesize event-triggering rules for networked control systems (NCS). It has to be noted that existing strategies are not directly applicable since only a subset of sensors and actuators get access to the network at a transmission instant. Hence, we adapt a policy developed in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] and develop new eventtriggering rules for classes of NCS governed by uniformly globally asymptotically stable (UGAS) protocols (see [START_REF] Nešić | Input-to-state stability of networked control systems[END_REF]). Examples of UGAS protocols are the round-robin protocol or the try-once-discard protocol [START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF] as shown in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF]. To the best of the authors' knowledge, we believe that this is the first time that event-triggered control is addressed for such NCS.

The scheme we introduce in this paper can be seen as a centralized event-triggering policy for physically distributed control systems, which in practice would require constant communication between the sensors/actuators and the coordinator (since the decision depends on the current value of the system states). To overcome this problem, we show how self-triggered implementations (see [START_REF] Anta | To sample or not to sample: self-triggered control for nonlinear systems[END_REF], [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF], [START_REF] Wang | Self-triggered feedback control systems with finite-gain L 2 stability[END_REF]) can be derived from a known event-triggering strategy by applying the techniques in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. Under this paradigm, the coordinator decides the next time instant at which communication should occur based on the last received information.

The proofs are omitted for space reasons and can be found in [START_REF] Postoyan | Event-triggered and self-triggered stabilization of distributed networked control systems[END_REF].

II. PROBLEM STATEMENT

Due to space constraints, we omit the description of the notation and definitions used throughout this paper and refer the reader to Section II in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF]. Consider the plant:

ẋP = f P (x P , u), (1)
where x P ∈ R nP is the plant state and u ∈ R nu the control input. The following stabilizing dynamic state-feedback controller is designed:

ẋC = f C (x C , x P ), u = g C (x C , x P ), (2)
where x C ∈ R nC is the controller state. We consider a scenario where the controller (2) communicates with the plant (1) via a shared centralized network. Since state measurements and control inputs are no longer continuously available but only at given transmission instants t j , j ∈ Z >0 , systems (1) and (2) become:

ẋP = f P (x P , û) ∀t ∈ [t j-1 , t j ] ẋC = f C (x C , xP ) ∀t ∈ [t j-1 , t j ] u = g C (x C , xP ), (3)
where xP and û denote the variables respectively generated from the most recent transmitted plant state and control input through the network. Between two transmission instants, they are generated by the in-network processing algorithm modeled by functions fP and fC :

ẋP = fP (x P , x C , xP , û) ∀t ∈ [t j-1 , t j ] u = fC (x P , x C , xP , û) ∀t ∈ [t j-1 , t j ]. (4)
Zero-order-hold devices are often used so that xP and û are kept constant on [t j-1 , t j ] i.e. fP = 0 and fC = 0. Nevertheless, we allow for other types of implementations.

Sensors and actuators are grouped into l nodes depending on their spatial location. At each transmission instant, a single node gets access to the channel according to the scheduling protocol and transmit its data. We model this process as follows:

xP (t + j ) = x P (t j ) + h P (j, e(t j )) û(t + j ) = u(t j ) + h u (j, e(t j )),

where e = (e xP , e u ), with e xP = xPx P and e u = ûu, denoting the networked-induced error. Thus, vector e is partitioned as e = (e 1 , . . . , e l ). At each transmission instant t j , functions h P , h u are typically such that if the node i gets access to the network, the corresponding error e i experiences a jump while the other components of e remain unchanged; usually e i (t + j ) = 0 but this assumption is not needed in general. In that way, h P and h C can be used to model common scheduling protocols such as round-robin (RR) or try-once-discard (TOD) [START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF], see [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF] for more details.

The sequence of transmission instants t j , j ∈ Z >0 , is traditionally defined such that ε ≤ t jt j-1 ≤ T (as in [START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF], [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]), where T ∈ R >0 denotes the MATI and ε ∈ R >0 is an arbitrary small constant which models the fact that there exists a minimum amount of time between two transmissions. In this study, we resort to a different paradigm: transmissions are triggered according to a criterion that depends on the variables of the overall system. The underlying idea behind this paradigm is to reduce the usage of the communication bandwidth by transmitting data only when needed to ensure the desired stability properties.

We consider the following implementation architecture. Sensors, actuators and controllers exchange information through a network, where the schedule is dynamically decided by a central coordinator. The order at which the network is assigned to each node is defined by means of the protocol. The time instants at which communication needs to be established are decided by the central coordinator according to the current state of the dynamical system, in the spirit of event-triggered control. This central node receives information from the sensors and the controller and evaluates a so-called triggering rule in order to decide whether communication is needed to guarantee stability for the control system. Since the triggering condition usually depends on the state of the plant and the controller, such setup requires in general continuous communication between the sensors, the controller and the central coordinator. To overcome this handicap, we propose in this paper a selftriggered implementation that emulates the designed eventtriggered policy, where the next transmission is decided based on the last data received by the coordinator (see Section V). Since transmission times are known in advance under this policy, the self-triggered policy also facilitates the schedulability analysis for the network.

We model the problem using the hybrid formalism of [START_REF] Goebel | Solution to hybrid inclusions via set and graphical convergence with stability theory applications[END_REF], similar to [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved event-triggering[END_REF]. We group together the states of the plant and the controller in the variable x = (x P , x C ) ∈ R nx and we denote by κ ∈ Z ≥0 the counter variable that may be required to model protocols such as round-robin for instance (see [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF]). It has to be noted that additional variables may also be introduced for designing the triggering rule. For instance, we will see in Section IV-A that the event-triggering strategy in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] is not applicable to the considered distributed NCS unless we introduce an appropriate auxiliary variable. We will also show in Section IV-B that the time-triggered policy in [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] can be modified to obtain event-triggering rules thanks to the use of a clock-like variable. Thus, we denote by η ∈ R nη all auxiliary variables. The model can be written as:

ẋ = f x (x, e) ė = f e (x, e) κ = 0 η = f η (x, e, κ, η)      q ∈ C x + = x e + = h e (κ, e) κ + = κ + 1 η + = h η (x, e, κ, η)      q ∈ D, (6)
where q = (x, e, κ, η). We use q = f q (q) and q + = h q (q) to denote [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]. The sets C and D are closed, included in R nq (n q = n x + n e + 1 + n η ) and respectively denote the flow and the jump set. Typically, the system flows on C and experiences a jump on D, where the triggering condition is satisfied. When q ∈ C ∪ D, the system can either jump or flow, the latter only if flowing keeps q in C. We call e + = h e (κ, e) the protocol where h e = (h P , h u ) as in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]. Functions f η , h η ,

f x :(x, e) → f P (x P , g C (x C , xP ) + e u ) f C (x C , xP ) f e :(x, e) →       fP (x P , x C , xP , g C (x C , xP ) + e u ) -f P (x P , g C (x C , xP ) + e u ) fC (x P , x C , xP , g C (x C , xP ) + e u ) -∂gC ∂xC (x C , xP )f C (x C , xP ) -∂gC ∂ xP (x C , xP ) ×f P (x P , x C , xP , g C (x C , xP ) + e u )       (7)
where xP = x P + e xP , are assumed to be continuous.

The main problem addressed in this paper is to define appropriate event-triggering rules, that is, to define appropriate flow and jump sets C and D for system [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] in order to ensure asymptotic stability properties of (6) while reducing the number of transmissions as much as possible. Afterwards, we explain how self-triggering conditions may be derived from a known event-triggering criterion.

III. A PRESCRIPTIVE FRAMEWORK FOR THE

EVENT-TRIGGERED CONTROL OF NCS We recall in this section the framework of [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF], originally developed for sampled-data systems. It is based on the following theorem that provides sufficient conditions that ensure asymptotic stability properties for system [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]. It can be regarded as a variation of the general results in [START_REF] Cai | Smooth Lyapunov functions for hybrid systems part II: (pre)asymptotically stable compact sets[END_REF].

Theorem 1. Consider system [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] and suppose h q (D) ⊂ (C ∪ D) and that there exist a locally Lipschitz function R : R nq → R and a continuous function υ : R nη → R nυ with n υ ≤ n η such that the following conditions hold:

(i) There exist α R , α R ∈ K ∞ such that for any q ∈ C ∪D: α R (|(x, e, υ(η))|) ≤ R(q) ≤ α R (|(x, e, υ(η))|). (ii) There exists α R ∈ K ∞ such that for all 1 q ∈ C: R • (q; f q (q)) ≤ -α R (R(q)). (iii) For all q ∈ D, R(h q (q)) ≤ R(q).
(iv) Solutions to [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] have a semiglobal dwell time2 on R nq \A, where A = q : (x, e, υ(η)) = 0 . Then the set A is S-GAS.

Theorem 1 can be used as a framework for the synthesis of event-triggering rules for [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]. The main idea is to design the triggering criterion so that there exists a Lyapunov function for the overall system (6) that decreases on flows, does not increase at jumps and guarantees the existence of a minimal interval of times between two jumps outside the stable set. This approach has been used to investigate the stability of other types of hybrid systems (e.g., see [START_REF] Nešić | Stability properties of reset systems[END_REF], [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]). General guidelines on how to apply Theorem 1 to synthesize triggering conditions for system (6) can be found in Section IV in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF]. The main difference with [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] is that the edependency of the Lyapunov function R will depend on the considered scheduling protocol as we show it in Section IV.

IV. EVENT-TRIGGERED STRATEGIES

We apply the framework of Section III to synthesize eventtriggering rules for NCS. Two strategies are proposed but others can be developed by using Theorem 1.

A. Using a threshold-like variable

First, we show that the event-triggering strategy proposed in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] for sampled-data systems is not directly applicable to distributed NCS. Hence, we redesign this technique as in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] by introducing an auxiliary variable. It has to be noted that the method in Section V.B in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] cannot be applied 'off-the-shelf' here as we need to adapt the strategy to the protocol. We suppose that the controller (2) has been designed to make the closed-loop system (1) input-to-stable w.r.t. to networked-induced error, which is equivalent to the following assumption (see Theorem 1 in [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF]).

Assumption 1. There exists a smooth Lyapunov function

V : R nx → R, α V , α V , α, γ ∈ K ∞ such that for all x ∈ R nx : α V (|x|) ≤ V (x) ≤ α V (|x|)
, and for all (x, e) ∈ R nx+ne : ∇V (x), f x (x, e) ≤ -α(V (x)) + γ(|e|).

(8) We suppose that the protocol is uniformly globally asymptotically stable (UGAS) [START_REF] Nešić | Input-to-state stability of networked control systems[END_REF], i.e., that the following holds.

Assumption 2. There exist W : R ≥0 × R ne → R ≥0 , α W , α W ∈ K ∞ and ρ ∈ [0, 1) such that for all (κ, e) ∈ Z ≥0 × R ne the following is satisfied:

α W (|e|) ≤ W (κ, e) ≤ α W (|e|) (9) W (κ + 1, h e (κ, e)) ≤ ρW (κ, e). ( 10
)
The round-robin and try-once-discard protocols have been shown to satisfy this property in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], as well as other protocols (see [START_REF] Nešić | Input-to-state stability of networked control systems[END_REF] for instance). We note that when ρ = 0 we recover the situation in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] where all nodes transmit at each transmission instant (i.e. h e (κ, e) = 0 in ( 6)) (notice that ( 9) implies [START_REF] Krstić | Lean backstepping design for a jet engine compressor model[END_REF] in that case).

In view of ( 8), for any σ ∈ K ∞ with σ(s) < s for s > 0, we have that γ(|e|) ≤ σ • α(V (x)) implies:

∇V (x), f x (x, e) ≤ -(I -σ) • α(V (x)). (11)
Instead of comparing |e| and |x| to derive the triggering rule as in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF], we use the Lyapunov function W (κ, e) which is characteristic of the protocol. According to (9), we have that |e| ≤ α -1 W (W (κ, e)). Therefore we can conclude that γ(W (κ, e)) ≤ V (x) (where γ(s

) = α -1 • σ -1 • γ • α -1 W (s), for s ≥ 0) implies γ(|e|) ≤ σ • α(V (x))
, that in return ensures [START_REF] Löfberg | Pre-and post-processing sum-of-squares programs in practice[END_REF]. Following the main idea of [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], a first attempt to define the triggering rule is:

γ(W (κ, e)) ≥ V (x). (12)
In this way, the flow and the jump sets of the corresponding system (6) are:

C = (x, e, κ) : γ(W (κ, e)) ≤ V (x) D = (x, e, κ) : γ(W (κ, e)) ≥ V (x) . (13)
The problem with this policy is that we have no guarantee that (x, e, κ) enters into C after a jump. Indeed, while in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] after each jump e is reset to 0, here typically only a subvector e i is reset to zero after each transmission (see Section II).This may not be enough for γ(W (κ, e)) to become less than V (x).

As a consequence, the triggering rule (12) may generate several transmissions in a row before entering into C that is unrealistic and contradicts item (iv) of Theorem 1. To overcome this drawback, we introduce a variable η ∈ R ≥0 with the following dynamics:

η = -δ(η), η + = γ(W (κ, e)), (14)
where δ is any locally Lipschitz class-K ∞ function. The system is now modeled as:

ẋ = f x (x, e) ė = f e (x, e) κ = 0 η = -δ(η)      q ∈ C, x + = x e + = h e (κ, e) κ + = κ + 1 η + = γ(W (κ, e))      q ∈ D, (15)
where q = (x, e, κ, η) and the sets C and D are defined as C = q : max{V (x), η} ≥ γ(W (κ, e)) and η ≥ 0 D = q : max{V (x), η} ≤ γ(W (κ, e)) and η ≥ 0 .

(16) The variable η can be regarded as a decreasing threshold on γ(W ) in view of ( 14), and thus we enter into C after a jump. Indeed, we have that η + = γ(W (κ, e)) ≥ γ(ρW (κ, e)) ≥ γ(W (κ + , e + )) according to [START_REF] Krstić | Lean backstepping design for a jet engine compressor model[END_REF], ( 14) and since ρ < 1 and γ is strictly increasing, therefore q + ∈ C. We are now able to apply Theorem 1 to guarantee stability properties for system [START_REF] Nešić | Stability properties of reset systems[END_REF] and the existence of dwell times. Its proof is based on Theorem 1 with the Lyapunov function R(q) = max{V (x), γ(W (κ, e)), η} and υ(η) = η. Theorem 2. Consider system [START_REF] Nešić | Stability properties of reset systems[END_REF] and suppose the following conditions hold.

(i) Assumptions 1-2 are satisfied.

(ii) Function γ(W ) is locally Lipschitz in e.

(iii) For any compact set S ⊂ R nx+ne , there exist L 1 , L 2 ∈ R ≥0 such that for any (x, e) ∈ S, κ ∈ Z ≥0 ,

| ∇V (x), f x (x, e) | ≤ L 1 (V (x) + γ(W (κ, e))) |γ(W ) • (e; f e (x, e))| ≤ L 2 (V (x) + γ(W (κ, e)

)).

(iv) There exists ǫ ∈ [0, 1) such that lim s→0 γ(ρs) γ(s) = ǫ. Then A = {q : (x, e, η) = 0} is S-GAS and solutions to [START_REF] Nešić | Stability properties of reset systems[END_REF] have a semiglobal dwell time on R nq \A.

Items (iii) and (iv) need to be added to guarantee the existence of dwell times compared to Theorem 3 in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF]. It is easy to check that condition (iv) of Theorem 2 holds when γ is polynomial or homogeneous of degree k ∈ Z >0 .

B. Using a clock-like variable

In [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], NCS with time-triggered execution are modeled as a hybrid system similar to (6) by introducing a clock variable τ that would correspond to η = τ in [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]. The flow and the jump sets are defined as τ being bigger or not than a given fixed bound T known as the MATI. This constant T corresponds to the time it takes for the solution of the ordinary differential equation ζ = -2Lζγ(ζ 2 + 1) to decrease from ρ -1 to ρ, where L and γ are some constants (see [START_REF] Cai | Smooth Lyapunov functions for hybrid systems part II: (pre)asymptotically stable compact sets[END_REF] in [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]) and ρ ∈ (0, 1) is given by Assumption 2, which is assumed to hold. In this subsection, we reduce the conservativeness of the strategy in [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] by making the ordinary differential equation that defines ζ state-dependent. This allows us to consider a larger class of systems and to potentially enlarge the inter-execution intervals compared to [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]. We suppose that Assumption 2 is satisfied with W locally Lipschitz in e and that the following holds.

Assumption 3. There exist a locally Lipschitz function V : R nx → R and continuous functions H : R nx → R ≥0 , L, G : R nx+ne → R ≥0 , α V , α V ∈ K ∞ , ̺ : R → R continuous, positive definite such that the following conditions holds.

(i) For all (x, e, κ) ∈ R nx+ne × Z ≥0 , W • (e; f e (x, e)) ≤ L(x, e)W (κ, e) + H(x), [START_REF] Postoyan | Event-triggered and self-triggered stabilization of distributed networked control systems[END_REF] where W comes from Assumption 2.

(ii) For all x ∈ R nx : α V (|x|) ≤ V (x) ≤ α V (|x|).
(iii) For all (x, e, κ) ∈ R nx+ne × Z ≥0 :

V • (x; f x (x, e)) ≤ -̺(|x|) -̺(|e|) -H 2 (x) +G(x, e)W 2 (κ, e). (18)
In [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], L and G are supposed to be constant that implies that system ẋ = f x (x, e) is L 2 -gain stable from W to H. Making L and G state-dependent allows us to enlarge the studied class of systems and to eventually obtain less conservative upper bounds in ( 17) and ( 18) that will help to enlarge the inter-event intervals. Model ( 6) becomes here:

ẋ = f x (x, e) ė = f e (x, e) κ = 0 η = -2ηL(x, e) -η 2 -G(x, e)      q ∈ C x + = x e + = h e (κ, e) κ + = κ + 1 η + = a      q ∈ D, (19)
where q = (x, e, κ, η), a is any constant in (ρ, ∞) and η ∈ R plays the role of ζ mentioned above and is called a clock-like variable (see [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]). The sets C and D are:

C = q : η ∈ [aρ 2 , a] , D = q : η = aρ 2 , ( 20
)
Note that, instead of setting η + to ρ -1 at jumps as in [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], we consider any a ∈ (ρ, ∞) that may help generating larger inter-execution intervals. Remark 1. We focus in this paper on NCS for which only one node among the l nodes communicates at each transmission instant so that ρ > 0. When ρ = 0, as in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF], we can redefine the sets in [START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF] as follows: C = q : η ∈ [b, c] , D = q : η = b where 0 < b < c some constants.

The following theorem ensures the stability of system [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] and the existence of dwell times. It can be seen as an application of Theorem 1 with R(q) = V (x) + ηW 2 (κ, e) and υ(η) = 0. Theorem 3. Consider system [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] and suppose Assumption 2-3 hold with W locally Lipschitz in e. Then the set A = {q : (x, e) = 0} is S-GAS and solutions to [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] have a semiglobal dwell time on R nq \A.

V. SELF-TRIGGERED CONTROL

As mentioned before, the proposed event-triggering schemes in Section IV require the coordinator to continuously evaluate the triggering condition. This may induce a significant cost in terms of communication, computation time and power. To overcome these drawbacks, a possible solution lies in the self-triggered strategy. Self-triggered control considers the mathematical model of the control system and the last measurement of the plant states and/or the last control input in order to derive the next transmission instant. It represents a model-based emulation of event-triggered control in the sense that it identifies the time instants at which the jump condition is satisfied. In this section, we suppose that an event-triggering strategy has been designed such that the following conditions hold.

Assumption 4. The conditions of Theorem 1 hold and item (iii) is satisfied on C ∪ D for system [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] with flow and jump sets of the form C = {q : Γ(q) ≤ 0} and D = {q : Γ(q) ≥ 0} where Γ : R nq → R ≥0 . Formally speaking, the event-triggered control strategy in Assumption 4 requires data transmission at the following time instant, for j ∈ Z >0 :

τ (j) = inf{t > t j-1 , (t, j-1) ∈ dom φ : Γ(φ(t, j-1)) = 0},
where φ is a solution to [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]. In order to guarantee stability, data transmission needs to occur no later than τ (j). When only the previous measurement of the state is available, the computation of τ (j) in an exact way is in most cases not possible. The self-triggered strategy computes a lower bound for τ (j) at which the next jump will occur. In [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF], this lower bound is taken to be τ (j) = λ(j)t * , where λ(j) is strictly positive and satisfies for each (t j , j) ∈ dom φ:

n-1 i=0 ς i (L i fq Γ)(φ(t j , j))λ i (j) = 0, (21)
where we have denoted the ith Lie derivative of Γ along f as

L i f Γ = L f (L i-1 f Γ), (L f Γ)(x) = ∂Γ ∂x f (x)
and L 0 f Γ = f . The parameters t * ∈ R >0 and ς i ∈ R are coefficients computed from f q and Γ. We assume that Γ and f q are smooth functions in (x, e, η). By abuse of notation we consider the ith Lie derivative L i fq Γ even though f q and Γ may not be differentiable in κ (this is justified since κ = 0 on flows). Equation ( 21) corresponds to the bound provided in Theorem V.4 in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. Guidelines for the design parameter t * are provided in Section VIII-A in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF], and the set of parameters ς i have to be chosen to satisfy inequality (V.12) in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. We refer to [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] for a more detailed description of the roles played by these coefficients and how to compute them from the expressions of q and Γ. The parameter n > 1 in (21) represents a design choice that trades the accuracy of the bound for the computational complexity. In other words, high values of n imply times τ (j) closer to τ (j) (and therefore less transmissions), but at the cost of solving a more complex algebraic equation. Since we may now transmit before Γ(q) = 0, we suppose item (iii) of Theorem 1 holds on C ∪ D (and not only D) in order to guarantee that the considered Lyapunov function still does not increase at jumps. This additional condition typically comes for free as it is the case in Section IV for instance. The problem is modeled as follows:

q = f q (q) τ1 = 1 τ2 = 0    q ∈ C q + = h q (q) τ + 1 = 0 τ + 2 = max{λ(κ)t * , ε}    q ∈ D, (22)
where τ 1 ∈ R ≥0 is a clock variable and τ 2 ∈ R ≥0 is used to define the next transmission instant, q = (q, τ 1 , τ 2 ), and the sets C and D are:

C = q : τ 1 ∈ [0, τ 2 ] , D = q : τ 1 ≥ τ 2 . ( 23
)
It is shown in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] that the formula in [START_REF] Wang | Self-triggered feedback control systems with finite-gain L 2 stability[END_REF] can be used to design τ (j) that is very close to τ (j). Nevertheless, to prevent from the situation where the self-triggering technique of [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] generates conservative times because of an inadequate parameters choice in [START_REF] Wang | Self-triggered feedback control systems with finite-gain L 2 stability[END_REF], we introduce an arbitrary small constant ε ∈ R >0 in [START_REF] Wang | Asymptotic stability in distributed eventtriggered networked control systems with delays[END_REF] to guarantee the existence of a minimal interval of time between two transmissions. This is justified since Assumption 4 implies the existence of such a constant time (semiglobally), in view of item (iv) of Theorem 1. The following theorem shows that the properties ensured by the considered event-triggering strategy are maintained under the proposed self-triggering rules. Theorem 4. Let f q and Γ be smooth functions in (x, e, η). Under Assumption 4, the set A = { q : (x, e, υ(η)) = 0} is S-GAS for system [START_REF] Wang | Asymptotic stability in distributed eventtriggered networked control systems with delays[END_REF] and solutions to [START_REF] Wang | Asymptotic stability in distributed eventtriggered networked control systems with delays[END_REF] have a semiglobal dwell time on R n q \ A.

Self-triggering rules can be derived for the event-triggered strategies developed in Section IV as follows. For Section IV-B, we take Γ(q) = ηaρ 2 . The function Γ is smooth and we assume that f x , f e , L, G are smooth functions in (x, e, η). Embedding system [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] into [START_REF] Wang | Asymptotic stability in distributed eventtriggered networked control systems with delays[END_REF] allows us to obtain a self-triggered control technique. The conclusions of Theorem 4 apply as all the required conditions hold. For the event-triggered control of Section IV-A, we cannot define Γ(q) = γ(W (κ, e)) -max{V (x), η} as Γ will not be smooth. Therefore, we define Γ 1 (q) = γ(W (κ, e))η and Γ 2 (q) = γ(W (κ, e)) -V (x). By assuming that γ(W ) and V are smooth in e and x respectively, we see that Γ 1 and Γ 2 are smooth in (x, e, η). We modify the jump equation for τ 2 in [START_REF] Wang | Asymptotic stability in distributed eventtriggered networked control systems with delays[END_REF] as follows: τ + 2 = max{λ 1 (κ)t * , λ 2 (κ)t * } where λ i satisfies [START_REF] Wang | Self-triggered feedback control systems with finite-gain L 2 stability[END_REF] with Γ = Γ i , i ∈ {1, 2}. Under this setup, the conditions of Theorem 4 are satisfied provided that f x , f e , δ are smooth.

VI. ILLUSTRATIVE EXAMPLE

To illustrate the proposed strategy, we consider the control of a jet engine compressor. We borrow the model from [START_REF] Krstić | Lean backstepping design for a jet engine compressor model[END_REF]:

ẋ1 = -x 2 - 3 2 x 2 1 - 1 2 x 3 1 , ẋ2 = u, (24)
where x 1 represents the mass flow, x 2 is the pressure rise and u is the throttle mass flow. In this model the origin has been translated to the desired equilibrium point, hence the objective is to steer (x 1 , x 2 ) to zero. The control law u = 4x 1 -4x 2 -9 2 x 2 1 -3 2 x 3 1 is designed to stabilize the system. This controller is connected to the two sensors measuring x 1 and x 2 through a network under the TOD protocol. For simplicity, we consider that the controller is connected to the actuator. Nonetheless, as pointed out previously in this paper, the developed framework accounts for the more realistic case of the controller and the actuator not being collocated. We implement the event-triggering strategy proposed in Section IV-A. Assumption 1 is satisfied with V

(x) = 1 2 x 2 1 + 1 2 (x 2 -3x 1 ) 2 , α(s) = -0.066s, γ (
s) = 4.37 • 10 4 s 2 + 9.10 • 10 6 s 4 for s ≥ 0. Assumption 2 is satisfied for W (e) = |e| for the TOD protocol, where e 1 and e 2 represent the network-induced errors for x 1 and x 2 , and with ρ = l-1 l = 1 √ 2 as the number of nodes is l = 2 (see Proposition 5 in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF]). We select σ(s) = 0.9s in [START_REF] Löfberg | Pre-and post-processing sum-of-squares programs in practice[END_REF] and thus we obtain γ(W (κ, e)) := 7.34 • 10 5 |e| 2 + 1.52 • 10 8 |e| 4 . The Yalmip software [START_REF] Löfberg | Pre-and post-processing sum-of-squares programs in practice[END_REF] was used to compute α, γ and γ(W (κ, e)). We now construct a linear differential equation for the auxiliary variable η: η = -0.01η, with initial condition η(0, 0) = 5000. Periodic [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] Event-triggered Self-triggered 0.010 0.061 0.046 of η(0, 0) would enlarge the transmission times, at the cost of a degradation in performance. It can be verified that items (ii)-(iv) of Theorem 2 hold (we use the fact that γ is a polynomial function to show item (iv) of Theorem 2). We derive as well a self-triggered emulation of the eventtriggered technique of Section IV-A. As detailed at the end of Section V, we consider Γ 1 (q) := γ(W (κ, e))η and Γ 2 (q) := γ(W (κ, e)) -V (x). The design parameters for the self-triggering condition are reported in Table I. In both cases we considered t * = 10 -3 .

We compare the event-triggered strategy herein proposed with a periodic implementation. In order to compute a period, we apply the technique in [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]. For operating ball of radius 1, the obtained period is T = 0.010. For the comparison, we consider 200 different initial conditions randomly distributed in a ball of radius 1. Table II shows the average intertransmission time under the three different strategies. Both the event-triggered and the self-triggered strategy outperform the periodic approach. The gap between the event-triggered and the self-triggered inter-transmission times is due to the conservativeness of the technique in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. To further illustrate the proposed approach, we depict as well the evolution of the transmission times and the network-induced error under the self-triggered strategy for a particular initial condition (x(0, 0) = (0.95, -0.14), e(0, 0) = (0, 0)). The plot shows how the network grants access to the node with the largest error. Likewise, the transmission times vary according to the current state of the plant. This fact suggests that the rigid periodic paradigm overloads unnecessarily the network, and the flexibility and adaptability of event-triggered control (and consequently self-triggered control) is able to relax this requirement and reduce network usage.

VII. CONCLUSION

We have used the prescriptive framework in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] originally developed for sampled-data systems to synthesize novel event-triggering rules for distributed NCS. We have then shown how self-triggering conditions can then been derived (under some conditions) by applying the techniques in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. This work represents the first step towards a more fundamental question in distributed networked control: given a set of physically distributed sensors and actuators, how should communication between the different nodes be scheduled? Ad-hoc solutions to this problem include the TOD protocol, where the node with the largest network-induced error information is granted access to the communication channel. The presented framework can be further explored to design communication protocols that decide the order at which each node needs to send information.

Fig. 1 .

 1 Fig. 1. Evolution of the transmission times and the networked-induced error under the self-triggered strategy.

  It is expected that larger values

	n	ς 0	ς 1	ς 2	ς 3
	Γ 1 : 4	-8.06 • 10 3	-226.07	481.76	-258.47
	Γ 2 : 3	-1.46 • 10 3	-1.21 • 10 3	4.94 • 10 3	-
			TABLE I
	PARAMETERS FOR THE SELF-TRIGGERED TECHNIQUE.

TABLE II AVERAGE

 II INTER-TRANSMISSION TIME FOR 200 INITIAL CONDITIONS.

We consider the R • (q; fq(q)), the Clarke derivative of R (see[START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF]), by abuse of notation, although R is not necessarily locally Lipschitz in κ. This is justified since the component of fq(q) corresponding to κ is 0.

See Definition 2 in[START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF]

This work was supported by the Australian Research Council under the Future Fellowship and Discovery Grants Schemes, the NSF awards 0820061 and 0834771 and the Alexander von Humboldt Foundation. R. Postoyan is with the Centre de Recherche en Automatique de Nancy, UMR 7039,


Lina Maria Rojas-Barahona

Matthieu Quignard
email: matthieu.quignard@loria.fr

An Incremental Architecture for the Semantic Annotation of Dialogue Corpora with High-Level Structures. A case of study for the MEDIA corpus. *

The semantic annotation of dialogue corpora permits building efficient language understanding applications for supporting enjoyable and effective human-machine interactions. Nevertheless, the annotation process could be costly, time-consuming and complicated, particularly the more expressive is the semantic formalism. In this work, we propose a bootstrapping architecture for the semantic annotation of dialogue corpora with rich structures, based on Dependency Syntax and Frame Semantics.

Introduction

We propose a cooperative architecture that incrementally generates and improves the annotation of the French MEDIA dialogue corpus with high-level semantics (HLS), as a result of the cooperation of several linguistic modules. MEDIA is a French corpus that has collected about 70 hours of spontaneous speech from the task of hotel room reservation. It contains transcribed utterances 1 that have been manually segmented 2 and annotated with a flat semantics i.e.,concept-value pairs [START_REF] Bonneau-Maynard | MEDIA: A semantically annotated corpus of task oriented dialogs in French[END_REF]. * This work is supported by the Agence Nationale de la Recherche (ANR) in France and is part of the French Project PORT-MEDIA.

1 Utterances with ellipsis, disfluencies, false starts, reformulations, repetitions and ungrammaticalities and special characters such as the symbol '*' that indicates uncertainty due to noise in the communication channel. 2 The term Segment means sequence of words in utterances.

The HLS semantics, namely the MultiModal Interface Language formalism (MMIL) [START_REF] Denis | Extending MMIL Semantic Representation: Experiments in Dialogue Systems and Semantic Annotation of Corpora[END_REF], augments the expressivity of the flat semantics by representing communicative actions, predicates, arguments and fine-grained features. Communicative actions are components built up from two types of entity (i.e. events and participants), which are linked together by relations and described by sets of features (attribute-value pairs). It is possible to identify in entities a set of main features, which can be domain-specific. For the semantic annotation, components are mapped to segments in utterances. Figure 1 shows the canonical representation of an utterance in the corpus in compliance with the specifications for the annotation 3 .

The Architecture

The architecture (Figure 2) for the automatic annotation has been formulated as a post-interpretation process that takes place after the syntactic analysis and semantic role labeling (SRL). Two linguistic resources interact within the architecture, the corpus and the Frames 4 . Four linguistic modules are involved in the annotation: the Part-Of-Speech (POS) tagger, the parsing, the semantic-role labeling (SRL) and the HLS Builder. The common knowledge base comprises two knowledge-bases (one for the domain and the other for the HLS formalism) together with a relational database management system (RDBMS). The knowledge bases assure the coherence of the an-3 http://www.port-media.org/doku.php?id= mmil_for_annotating_media 4 Frames is the process in which the frames and frame elements (FE) are defined. It shows a request to reserve: the communicative action is Request the main event is Reserve. Note that the beneficiary and the patient are two different roles, the beneficiary is the person, not necessarily the same speaker, who will use the object reserved (e.g. rooms). The patient is the speaker. The segmentation of the HLS Component is presented in the Table, the component is mapped to the whole utterance. The finegrained segmentation of features is shown for the Participant 2.

notation while the database assures persistence and data integrity. The database stores the corpus, the frames, the results at each level of analysis, as well as the progress in the annotation. The persistence permits progressively optimizing the algorithms until the desired annotation is obtained and integrated into the corpus files. The corpus manager is in charge of the resources management. Last but not least, two annotation tools were built: one for the SRL gold standard (web-based) and the other for the HLS gold standard (standalone). Syntactic Analysis. We decided to employ statistical approaches that could learn the irregularities of spoken language: the French Tree-Tagger5 and the dependency-based MALT-PARSER [START_REF] Nivre | The CoNLL 2007 Shared Task on Dependency Parsing[END_REF]. The parser has been trained with 1449 utterances annotated according to the annotation guidelines described in [START_REF] Cerisara | Analyse syntaxique du franc ¸ais parlé. Journée thématique ATALA Quels analyseurs syntaxiques pour le franc ¸ais[END_REF]. Definition of Frames. Frame Semantics, [START_REF] Baker | The Berkeley FrameNet Project[END_REF] arranges common background knowledge for situations by grouping verbal, nominal causative and non-causative predicates. Nevertheless, paraphrases are more used in spoken language than explicitly uttered nouns, adjectives or verbs for referring to a situation (e.g.'ask', 'request' or 'demand'). Here we introduce the term: Frame Evoking Phrase (FEP) for evoking frames and we include syntactic templates that mirror these phrases in frames and frame elements (FE). Table 1 summarizes the differences between PORT-MEDIA frames and FrameNet [START_REF] Baker | The Berkeley FrameNet Project[END_REF] Semantic Role Labeling. We built a rule-based semantic role labeling for detecting frames and FE (roles) by using dependency tree-template pattern matchers that exploit the information already compressed in frames. The SRL detects the boundaries of FEP and FE by measuring the syntactic and semantic similarity between the utterance and the frame.

HLS Builder.The HLS Builder is the last phase in the annotation process: it is rule-based and it takes utterances in the corpus with their flat semantics, de-pendency trees and predicates-arguments and builds the HLS representation (See Figure 1), according to the specifications for the annotation and the knowledge bases. The dialogue act and main event in HLS components can be detected from the predicates. Similarly, secondary events and participants with their features can be detected from the roles and the flat semantics.

Evaluation and Discussion

For evaluating the system we separately computed the accuracy of its linguistic components. The parser achieved a label attachment score (LAS) [START_REF] Nivre | The CoNLL 2007 Shared Task on Dependency Parsing[END_REF] of 86.16%, with a training set of 1097 utterances and a test-set of 100 utterances. The SRL was evaluated with metrics adapted from the CONLL 2005 evaluation [START_REF] Carreras | Introduction to the CoNLL-2005 shared task: Semantic role labeling[END_REF] for supporting FEP and allowing overlapped FEP for different frames. The LAS was computed by comparing the semantic dependencies of system's and gold's propositions 6 and their segments. The gold standard comprises 115 utterances annotated with the major frames in the domain: Request, Reserve and Attributes. The F1-measure computed for propositions with exactly the same segments was 56.66%. When verifying whether the segments the same syntactic governor, the SRL achieves a better score: 71.30%. Finally, varying the number of excluded words in both segments 7 yielded a constant increase of the F1-measure until a maximum of 84.27%. The HLS annotation was evaluated by measuring the similarity between gold's and system's components with a gold standard of 330 complex utterances related to the reservation task. When rigorously measuring the equality of components 8 , we obtained a F1-measure of 57.79%. Measuring equality of components without being so rigorous with features' segmentation, yielded a slightly higher score 63.31%. Finally, when measuring equality of components by taking 6 A proposition is a structure containing the predicate, their arguments and the semantic relation between them.

7 From 1 to n words not common in both segments.

8 Two HLS components are equal if their entities and relations are equal. Two entities are equal if they have the same segment and features (feature name and feature value) and if these features are mapped to the same segments in the utterance. Two relations are equal if they have the same source and target entities as well as the same name into account only the main features of entities, we obtained a higher score: 70.65%.

We proposed an architecture for corpus management that allows incremental updates over persistent information until a more accurate semantic annotation is obtained. The preliminary results show a general agreement when defining the main features and the main entities in HLS components and a disagreement when segmenting fine-grained features. We observed that the system tends to create new entities when it detects repetitions or references in long utterances. Defining a more precise segmentation policy in the manual annotation guidelines, augmenting the training data for parsing, as well as integrating reference resolution and disambiguation techniques, will enhance the annotation process. An appealing research direction would be to integrate and evaluate machine learning components in the architecture.

Figure 1 :

 1 Figure 1: HLS representation for the French utterance "je voudrais faire une réservation d' une chambre pour une personne à Niort" (So I would like to make a reservation for a room for one person in Niort).It shows a request to reserve: the communicative action is Request the main event is Reserve. Note that the beneficiary and the patient are two different roles, the beneficiary is the person, not necessarily the same speaker, who will use the object reserved (e.g. rooms). The patient is the speaker. The segmentation of the HLS Component is presented in the Table, the component is mapped to the whole utterance. The finegrained segmentation of features is shown for the Participant 2.

Figure 2 :

 2 Figure 2: General Architecture for the HLS Annotation.

Table 1 :

 1 ). Static Characteristics of Frames in FrameNet and in PORT-MEDIA.

		FrameNet	PORT-MEDIA
		Lexical Units	Lexical Units, POS tags and templates
	Frames
			MEDIA Flat Semantics
		Lexical Units, Phrase Type	Lexical Units, POS tags, templates
	Frame Elements	and Grammatical Function	and dependency relation
		Semantic Type	Semantic Type
			and MEDIA flat semantics

http://www.ims.uni-stuttgart.de/