R Postoyan
email: romain.postoyan@univ-lorraine.fr.

W P M H Heemels
email: m.heemels@tue.nl.

Periodic event-triggered control for nonlinear systems R. Postoyan, A. Anta, W.P.M.H. Heemels, P. Tabuada, D. Nešić Abstract-Resources limitations of embedded systems and networked control systems have motivated the development of novel controller implementation paradigms in order to reduce the potential conservatism of traditional clock-driven setups. In this context periodic event-triggered control has recently been proposed. Based on a periodically evaluated and statedependent criterion, it is decided, at every sampling instant, whether the control input needs to be updated or not. In this paper we propose a method for the design of periodic event-triggered controllers for nonlinear systems. We follow an emulation-based approach as we start from a known (continuously evaluated) event-triggered controller and we provide a systematic technique to select the sampling period and to redesign the triggering condition to approximately maintain the guarantees ensured by the original controller. The method is illustrated on a physical system and we compare the obtained results with other available event-based implementations.

I. INTRODUCTION

Embedded systems and networked control systems require the development of novel controller implementation paradigms in order to cope with the ever tighten resources limitations which are typical for these systems. Indeed, the traditional setup, where data is periodically exchanged between the controller and the plant, may be no longer appropriate as it may lead to an excessive usage of the communication channel and of the CPU. In this context, eventtriggered control (ETC) has been proposed as an alternative to the periodic setup [START_REF] Arzén | A simple event-based PID controller[END_REF], [START_REF] Aström | Comparison of periodic and event based sampling for first-order stochastic systems[END_REF]. The underlying idea is to adapt the transmissions and the controller execution to the current state of the plant. In ETC, the plant's states are continuously monitored and the loop is closed whenever a state-dependent criterion is satisfied. In that way, a transmission is triggered only when an event occurs, otherwise the communication channel is not used for control. A number of studies have shown that ETC may be able to significantly reduce the amount of transmissions needed to stabilize a (nonlinear) system, see e.g. [START_REF] Arzén | A simple event-based PID controller[END_REF], [START_REF] Aström | Comparison of periodic and event based sampling for first-order stochastic systems[END_REF], [START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF], [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], [START_REF] Postoyan | A unifying Lyapunovbased 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 | On event design in event-triggered feedback systems[END_REF]. However, a downside of ETC is the requirement to continuously evaluate the triggering condition which is not conceivable when the implementation platform is digital. In practice, the triggering condition can only be verified periodically. In this case, it is decided at each sampling instant whether the control input needs to be updated, which turns the implementation to a periodic event-triggered control (PETC) system, a terminology introduced in [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF], see also [START_REF] Arzén | A simple event-based PID controller[END_REF], [START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF]. The sampling of the triggering mechanism may deteriorate the performance of the ETC system compared to continuously evaluated ETC, in particular when the period cannot be made small enough due to platform limitations. In fact, one may aim for larger sampling periods in order to save (even more) computation and communication resources. There is therefore a strong need for systematic methods to design appropriate PETC strategies. Results regarding the robust stabilization of linear systems under PETC in terms of L 2 -gains are given in [START_REF] Heemels | Model-based periodic eventtriggered control for linear systems[END_REF], [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF].

The purpose of the paper is to propose PETC strategies for general nonlinear systems. We start from a continuously evaluated event-triggered controller which has been designed to ensure a desired specification for a nonlinear continuoustime plant. We then explain how to derive PETC policies which approximately maintain the properties ensured by the original controller. The approach uses the results on isochronous manifolds in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] to analyze the evolution of the original event-triggering condition between two sampling instants. We show that the performance degradations induced by the sampling can be rendered as small as desired at the price of a more complex triggering condition. Compared to [START_REF] Heemels | Model-based periodic eventtriggered control for linear systems[END_REF], [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF], we address nonlinear systems and we emulate a (continuously evaluated) event-triggered controller while the results in [START_REF] Heemels | Model-based periodic eventtriggered control for linear systems[END_REF], [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF] start from a periodically sampled controller. The work in [START_REF] Sanfelice | Lyapunov analysis of sampled-and-hold hybrid feedbacks[END_REF] investigates the effect of sample-and-hold devices on hybrid feedbacks and can thus be used to derive a PETC system based on a continuously evaluated ETC. Our work differs from [START_REF] Sanfelice | Lyapunov analysis of sampled-and-hold hybrid feedbacks[END_REF] in the sense that we do not directly implement the triggering law from the (continuously evaluated) ETC system but we redesign it in order to ensure stronger properties. Furthermore, the presented approach is not only applicable for the purpose of stabilizing a compact attractor as in [START_REF] Heemels | Model-based periodic eventtriggered control for linear systems[END_REF], [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF], [START_REF] Sanfelice | Lyapunov analysis of sampled-and-hold hybrid feedbacks[END_REF]: any ETC system can be turned to a PETC system as long as it satisfies the required conditions.

The results in this paper can be seen as an alternative to the self-triggered control (STC) method of [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. In STC, the measurements are only collected at the triggering instants (and not periodically) and the next time at which the loop must be closed is then pre-computed, see e.g. [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] Velasco | The self triggered task model for real-time control systems[END_REF], [START_REF] Wang | Self-triggered feedback control systems with finite-gain L 2 stability[END_REF]. The techniques in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] may need significant computation resources at each transmission instant to generate long inter-transmission intervals, while we will see that the derived PETC laws are typically simpler from the computational point of view compared to [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. However, PETC may need to more often evaluate the triggering condition. On the other hand, PETC is expected to ensure stronger robustness properties than STC, as shown on an illustrative example, since the state of the plant is more frequently monitored in PETC than in STC.

Notation. Let R = (-∞, ∞), R ≥0 = [0, ∞), Z ≥0 = {0, 1, 2, . . .}, Z >0 = {1, 2, . . .}. For (x, y) ∈ R n+m , (x, y) stands for [x T , y T ] T . A function γ : R ≥0 → R ≥0 is of class K ∞ if it

II. PROBLEM STATEMENT

A. Preliminaries on the event-triggered controller

We consider a dynamical system of the form

ẋ = f (x, u), (1)
where x ∈ R nx is the state which is assumed to be measured, u ∈ R nu is the control input. A control law of the form u = k(x) has been designed in continuous time to satisfy a desired specification. We assume it to be static for the clarity of exposition only; our results also apply to dynamic statefeedback laws as it will be clear in the sequel (see Remark 1 below). We consider a scenario where the controller is implemented on a digital platform. As a consequence, it is only at some time instants t k , k ∈ Z ≥0 , that the measurements are sent to the controller, and that a new control input is computed and transmitted to the plant. When the input is held constant between updates, (1) becomes, for all t

∈ [t k , t k+1 [ ẋ(t) = f (x(t), k(x(t k ))) = f (x(t), k(x(t) + e(t))), (2)
where e represents the difference between the last transmitted value of the state to the controller and the current state

e(t) = x(t k ) -x(t), ∀t ∈ [t k , t k+1 [. (3)
We consider the sampling-induced error on the state x but we could have equivalently chosen to introduce the error induced on the input, i.e. k(x(t k ))k(x(t)) (see Section VII.B in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] for a discussion on this point). In conventional setups, the controller is typically updated in a periodic fashion, i.e. t k+1t k = h where h > 0 is known as the sampling period. Event-triggered control, on the other hand, updates the control input whenever a state-dependent criterion is satisfied. The triggering condition can be defined using a function Γ : R 2nx+nχ → R which depends on the state x, the sampling-induced error e and potentially on some additional variables χ ∈ R nχ which may be introduced by the user (see [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF], [START_REF] Postoyan | Event-triggered and self-triggered stabilization of networked control systems[END_REF] for instance). Without loss of generality, we assume that Γ is negative after a jump and that the next transmission instant occurs as soon as Γ becomes non-negative. Using an impulsive formulation as in [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved and decentralized event-triggering[END_REF], [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF], the closed-loop system under an event-triggering strategy can be defined as

ż = g(z) ∀t ∈ [t k , t k+1 [ z(t + k+1 ) = b(z(t k+1 )), (4)
for appropriate g : R nz → R nz and b : R nz → R nz where z = (x, e, χ) ∈ R nz with n z = 2n x + n χ . We typically have b(z) = (x, 0, c(x, e, χ)) where the continuous mapping c defines the behaviour of the designed χ-system at jumps. However, it has to be noted that the assumption that e is reset to 0 after each jump is not needed; such a scenario occurs in decentralized setups, see e.g. [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved and decentralized event-triggering[END_REF], [START_REF] Mazo | Decentralized event-triggered control over wireless sensor/actuator networks[END_REF], [START_REF] Postoyan | Event-triggered and self-triggered stabilization of networked control systems[END_REF], [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF]. The jump instants are defined by the following rule

t k+1 = t k + min{t > 0 : Γ(z(t, b(z(t k )))) ≥ 0}, t 0 = 0, (5)
where z(t, z 0 ) is the solution to ż = g(z) at time t ≥ 0 starting at z 0 . The function Γ : R nz → R in ( 5) is designed in such a way that updating the control input at time instants t k guarantees that for any solution z to (4)-( 5) the following holds

Γ(z(t)) ≤ 0 ∀t > 0, (6)
which ensures stability or required performance for the closed-loop. For instance, stabilizing event-triggered controllers are designed in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] with a triggering law of the form of γ(|e|) ≥ σα(|x|) with γ, α ∈ K ∞ and σ ∈ (0, 1). Hence, we obtain Γ(x, e) = γ(|e|)σα(|x|). Similarly, the conditions of the form |e| ≥ ρ where ρ > 0 used in [START_REF] Arzén | A simple event-based PID controller[END_REF], [START_REF] Aström | Comparison of periodic and event based sampling for first-order stochastic systems[END_REF], [START_REF] Miskowicz | Send-on-delta concept: An event-based data reporting strategy[END_REF], [START_REF] Otanez | Using deadbands to reduce communication in networked control systems[END_REF] to practically stabilize the origin of nonlinear systems give Γ(e) = |e|ρ. It is explained in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] and in Section V of [START_REF] Postoyan | Event-triggered and self-triggered stabilization of networked control systems[END_REF] how Γ can be constructed for other types of event-triggering conditions. We assume that the hybrid system (4)-( 5) has leftcontinuous solutions which exist for all time and for every considered initial condition. In that way, to obtain a solution in the sense of Carathéodory, we flow the continuous dynamics until a jump occurs and then we flow again and so on.

We make the following assumption on system (4)-( 5).

Assumption 1:

The jumps induced by Γ on the eventtriggered system (4)-( 5) are spaced at least by T > 0 units of time, i.e. [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF] where z is solution to ż = g(z) and Ω ⊆ R nz is a compact forward invariant1 set for system (4)-( 5). Assumption 1 is reasonable as the event-triggering condition clearly needs to ensure the existence of a uniform minimum amount of time between two transmissions over a given operating set Ω in order to satisfy the inherent hardware constraints. This condition is verified by most available event-triggering schemes in the literature. The set Ω can be determined using the level set of some Lyapunov function when investigating stabilization problems for example.

T := inf z0∈Ω {t > 0 : Γ(z(t, b(z 0 ))) ≥ 0} > 0,
Remark 1: The forthcoming analysis applies to system (4)- [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF]. We therefore see that the control input u in (1) is not necessarily given by a static law but that it can be the output of a dynamic controller. The states of the controller would then be incorporated into the vector x and we would obtain a model of the form of ( 4)- [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF]. Similarly, the controller may not be implemented using zero-order-holds as any holding function is suitable as long as the problem can be modeled by ( 4)- [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF]. For instance, the model-based technique in [START_REF] Lunze | A state-feedback approach to event-based control[END_REF] would require to introduce an additional variable x s , the estimate of x with the notation of [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], which would correspond to χ in (4).

B. Emulation-based periodic event-triggered control

In PETC, the triggering condition can only be checked periodically every h units. Intuitively, to select h sufficiently small should allow to still maintain the performances ensured by the original event-triggering condition in [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF]. Nevertheless, it may not be possible to select h as small as desired because of the hardware constraints or because we want to use the hardware as little as possible to save resources. We therefore need to develop appropriate triggering conditions for this setup. Using again the impulsive formulation following [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF], the closed-loop system under a PETC strategy can be defined as

ż η = g(z) 1 ∀t ∈ [ tk , tk+1 [ z( t+ k+1 ) η( t+ k+1 ) = b(z( tk+1 )) 0 , (8)
where η ∈ R ≥0 represents the time elapsed since the last jump and its initial condition is taken to be η( t0 ) = 0. The jump instants of (8) are defined as

tk+1 = tk + min{t > 0 : Γ(z(t, b(z( tk )))) ≥ 0 ∧ ∃n ∈ Z ≥0 η(t, 0) = nh }, t0 = 0, (9)
where Γ : R nz → R is a function to be designed and, as before, z(t, z 0 ) denotes the solution to ż = g(z) at time t ≥ 0 starting at z 0 . Since the triggering condition is not checked continuously in [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF], there is no guarantee that Γ remains negative along the solutions to ( 8)-( 9) as in the case of standard event-triggered control. Therefore, the control objectives may not be ensured.

Our aim is to provide tools for the design of h and Γ to guarantee that Γ remains negative along the solutions to ( 8)- [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF] up to an error which can be rendered as small as desired. In that way, the original specifications of the ETC system (4)-( 5) will be approximately maintained.

III. MAIN RESULTS

We first explain how to select the period h and the triggering condition Γ. We then present the analytical results.

A. Design

Under the PETC strategy, the input can only be updated whenever the triggering condition is evaluated, that is, every h units of time. Hence, it is reasonable to select the sampling period to be less than the minimum inter-transmission time of the event-triggered controlled system (4)-( 5) (which does exist in view of Assumption 1). In that way, after a jump, we know that Γ will remain strictly negative at least until the next sampling instant. Therefore, we select h such that

0 < h < T ( 10
)
where T is defined in [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF]. Estimates of T are generally given in the analysis of the event-triggered controlled system (4) to prove the existence of a dwell-time between two transmissions. They can also be numerically evaluated. Since our objective is to guarantee that Γ remains negative along the solutions to ( 8)-( 9) up to an adjustable error, we would like to verify at t = nh, n ∈ Z >0 , whether the condition Γ(z(t)) ≥ 0 may be satisfied for t ∈]nh, (n+ 1)h].

To address this question, we decompose the interval ]nh, (n+ 1)h] into N intervals of length h N , i.e. ]nh, (n + 1)h] = ]nh, nh

+ h N ] ∪ [nh + h N , nh + 2 h N ] ∪ . . . ∪ [nh + (N - 1) h N , (n + 1)h].
We then investigate whether Γ(z(t)) may become non-negative at the time instants t = nh + i h N for i ∈ {1, 2, . . . , N }. Should the condition be violated at some t = nh + i h N (without updating the control action), a jump must occur when t = nh in order to guarantee Γ(z(nh + i h N )) ≤ 0 for all i ∈ {1, 2, . . . , N }. Obviously, Γ(z(t)) may still become positive between these time instants but we will see that the induced error will be of the order of h N which can be reduced by increasing N . Furthermore, the event-triggered control system in (4)-( 5) is often tolerant to small delays (see [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], [START_REF] Wang | On event design in event-triggered feedback systems[END_REF] to mention a few). Hence, it is reasonable to allow the condition (6) to be violated for a small amount of time as it may still ensure satisfactory properties.

To determine at time t = nh, n ∈ Z >0 , whether the condition Γ(z(nh + i h N )) < 0, i ∈ {1, . . . , N }, may be violated, the evolution of the triggering function Γ along the solutions to ż = g(z) needs to be analyzed. This point is addressed by resorting to the techniques of [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] to estimate isochronous manifolds. Like in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF], we make the following assumption.

Assumption 2: The functions g and Γ are p-times continuously differentiable where p ∈ Z >0 and there exist c, ς j ∈ R for j ∈ {0, 1, . . . , p -1} such that the following holds for any z in Ω

L p g Γ(z) ≤ p-1 j=0 ς j L j g Γ(z) + c, (11)
where we have denoted the jth Lie derivative of Γ along the closed-loop dynamics g as L j g , with L 0 g Γ = g, (L g Γ)(z) = ∂Γ ∂z g(z) and L j g Γ = L g (L j-1 g Γ). Note that the condition [START_REF] Lunze | A state-feedback approach to event-based control[END_REF] differs from (V.11) in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] due to the constant c. This allows us to work with a relaxed condition which applies to a larger class of systems2 . Indeed, Assumption 2 always holds (when g and Γ are p-times continuously differentiable) as it suffices to take c = max z∈Ω L p g Γ(z) and ς j = 0 for j ∈ {0, 1, . . . , p -1} to ensure [START_REF] Lunze | A state-feedback approach to event-based control[END_REF] (recall that Ω is compact in view of Assumption 1). Clearly, this particular choice may lead to conservative results as we will explain. We use the property [START_REF] Lunze | A state-feedback approach to event-based control[END_REF] in the following to estimate the set of states for which Γ becomes positive in i h N units of time for system (4)-( 5), which we denote S(i h N ) := {z 0 ∈ Ω : Γ(z 0 ) < 0 and Γ(z(i h N , z 0 )) ≥ 0 }, i ∈ {1, . . . , N }. The parameter p may be increased to derive tighter upper-bounds of L p g Γ(z) in ( 11) and thus tighter estimates of S(i h N ) at the price of a higher computational complexity.

Remark 2: The self-triggering formulas in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] tend to provide accurate estimates of the time instants when Γ becomes non-negative provided the bound ( 11) is tight, which may be difficult to achieve in practice. In the proposed PETC approach, the evolution of Γ is investigated over shorter horizons than in STC, namely time-intervals of length h. Hence, the bound in [START_REF] Lunze | A state-feedback approach to event-based control[END_REF] does not necessarily need to be accurate to provide satisfactory results as we expect these estimates to be tighter whenever times are shorter based on previous experience [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF].

Assumption 2 allows to bound the evolution of Γ by a linear differential equation for which the analytical solution can be computed as stated in the lemma below which directly follows from Lemma V.2 in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF].

Lemma 1: Under Assumption 2, for all solutions z to ż = g(z) with initial condition

z 0 ∈ Ω such that z(t, z 0 ) ∈ Ω for any t ∈ [0, h], Γ(z(t, z 0 )) ≤ y 1 (t, y 0 ) for any t ∈ [0, h],
where y 1 is the first component of the solution of the linear differential equation below

   ẏj = y j+1 , j ∈ {1, 2, . . . , p -1} ẏp = p-1 j=0 ς j y j+1 + y p+1 ẏp+1 = 0 (12) with y 0 = (Γ(z 0 ), L g Γ(z 0 ), . . . , L p-1 g Γ(z 0 ), c).
In that way, for a given state z 0 ∈ Ω, if y 1 (i h N , y 0 ) is positive for some i ∈ {1, 2, . . . , N }, then Lemma 1 implies that Γ(z(i h N , z 0 )) may be positive. On the other hand, if

y 1 (i h N , y 0 ) is negative, Lemma 1 ensures that Γ(z(i h N , z 0 ))
is negative. The term y 1 (i h N , y 0 ) can be computed off-line by noting that y(i h N , y 0 ) = R(i h N , z 0 ) where

R(i h N , z 0 ) := (1 0 . . . 0 ) exp(A p i h N )      Γ(z0) LgΓ(z0) . . . L p-1 g Γ(z0) c      with A p :=         0 1 0 . . . 0 0 0 0 0 1 . . . 0 0 0 . . . . . . . . . . . . 0 0 0 . . . 1 0 0 0 0 0 . . . 0 1 0 ς 0 ς 1 ς 2 . . . ς p-2 ς p-1 1 0 0 0 . . . 0 0 0         . (13)
Hence, we define Γ(z) for any z ∈ Ω as

Γ(z) := max R(i h N , z) | i ∈ {1, . . . , N } . (14)
Every h units of time the current state z is measured and we verify whether Γ(z) is positive in which case the control input is updated. Conversely, if Γ(z) is negative, then the control input is not updated. Although inspired by the selftriggering techniques, we do not propose a self-triggered emulation of the event-triggered strategy. Indeed, we do not compute the next time at which a jump should occur, but we simply verify at each sampling instant whether the state of system (8) lies in the set {z : Γ(z) ≥ 0}. Note that this set can be computed off-line in view of ( 13) and [START_REF] Otanez | Using deadbands to reduce communication in networked control systems[END_REF]. It also has to be noticed that when T h ≥ 2, we do not need to verify the triggering condition for the next ⌊ T h ⌋ -1 sample times following a control input update according to Assumption 1, which allows to further reduce computations.

Before presenting the analytical guarantees, we need to assume that the solutions to ż = g(z) which start in Ω remain in this set for the next h units of times. The condition is linked to the robustness of system (4)-( 5) to delays and is reasonable as we have already mentioned in this section that ETC systems are usually tolerant to delays.

Assumption 3: Consider the system ż = g(z). For any

z 0 ∈ Ω, z(t, z 0 ) ∈ Ω for all t ∈ [0, h].

B. Analytical result

The theorem below states that to choose h such that [START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF] holds and Γ as in [START_REF] Otanez | Using deadbands to reduce communication in networked control systems[END_REF] ensures that Γ will be negative along the solutions to ( 8)-( 9) up to an error which can be rendered arbitrarily small by increasing N . The proof is given in the appendix.

Theorem 1: Consider system (8)-( 9) with h which satisfies [START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF] and Γ defined in [START_REF] Otanez | Using deadbands to reduce communication in networked control systems[END_REF] and suppose Assumptions 1-3 hold. Then for any solution z to (8)-( 9) which is initialized in Ω, the following hold.

(i) Γ(z(i h N )) < 0 for any i ∈ Z >0 . (ii) There exists M ≥ 0 which is independent of (h, N ) such that Γ(z(t)) ≤ h N M for any t > 0. Theorem 1 shows that, for a given h, the parameter N can be increased to reduce the error induced by the sampling, which corresponds to the term h N M in item (ii) of Theorem 1. On the other hand, to increase N leads to more computations since N conditions need to be evaluated at each sampling instant in view of [START_REF] Otanez | Using deadbands to reduce communication in networked control systems[END_REF]. It is interesting to note that constant ratios h N lead to the same Γ and thus the same performance guarantees in view of item (ii) of Theorem 1. For a given performance specification, we are therefore free to select both h and N small or large. The former would typically lead to less control input updates and frequent but simple checks of the triggering condition, while the latter would generate more complex but less frequent checks and we may expect that the control input will be more often transmitted to the controller. This degree of freedom allows the designer to adapt the PETC strategy according to the setup constraints.

Remark 3: When the requirement that Γ remains negative along the solutions to ( 8)-( 9) for any positive time is essential, it is possible to modify the strategy proposed above by adding a term to [START_REF] Miskowicz | Send-on-delta concept: An event-based data reporting strategy[END_REF] which upper-bounds the intersample behaviour of y 1 between i h N and (i + 1) h N using the techniques of [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF] for instance.

IV. ILLUSTRATIVE EXAMPLE

We apply the results of Section III to a rigid body previously studied in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. The model is (see [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF]) ẋ1 = u 1 , ẋ2 = u 2 , ẋ3 = x 1 x 2 , and we consider the controller synthesized in [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF] in order to stabilize the origin:

u 1 = -x 1 x 2 -2x 2 x 3 -x 1 -x 3 and u 2 = 2x 1 x 2 x 3 + 3x 2 3 -x 2 .
The implementation of the controller on a digital platform leads to the following closed-loop system ẋ1 = -(x 1 + e 1 )(x 2 + e 2 ) -2(x 2 + e 2 )(x 3 + e 3 ) -(x 1 + e 1 ) -(x 3 + e 3 ) ẋ2 = 2(x 1 + e 1 )(x 2 + e 2 )(x 3 + e 3 ) + 3(

x 3 + e 3 ) 2 -(x 2 + e 2 ) ẋ3 = x 1 x 2 , (15)
where e i (t) = x i (t k )x i (t) for t ∈ [t k , t k+1 [, with t k , k ∈ Z ≥0 , the sequence of transmissions and i ∈ {1, 2, 3}, as in Section II-A. In order to stabilize the origin of (15), we take the triggering condition as in [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] which is obtained using the Lyapunov function V (x) := 1 2 (x

1 +x 3 ) 2 + 1 2 (x 2 -x 2 3 ) 2 + x 2 3 : Γ(x, e) := |e| 2 -0.79 2 σ 2 |x| 2 , where x = (x 1 , x 2 , x 3
), e = (e 1 , e 2 , e 3 ) and σ = 0.8.

We compare the amount of transmissions respectively generated by the event-triggered controller, the corresponding self-triggered controller in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] and a periodic event-triggered controller designed by following the procedure in Section III. Assumption 1 is satisfied with T = 0.08 (which has been numerically computed) for Ω = {(x, e) : V (x) ≤ 5 and Γ(x, e) ≤ 0} and Assumption 3 also holds. Regarding Assumption 2, we note that the system vector fields and the triggering condition are smooth. In addition, (11) is verified with 3 p = 3, ς 0 = -748.4986, ς 1 = -1.0008, ς 2 = 4.3166 and c = 0. We can thus apply the method presented in Section III as all the conditions of Theorem 1 are ensured. We have selected N = 1 and h < T . In that way, the expression of Γ in ( 14) reduces to Γ(z) = R(h, z) where R defined in [START_REF] Miskowicz | Send-on-delta concept: An event-based data reporting strategy[END_REF] with the coefficients given above. The selftriggered controller is designed as in Section VI.B in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] where system (15) was augmented to be homogeneous. In that way, the transmission times are defined by the formula in (V.30) in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] with 3 p = 3, χ 0 = -73.2528, χ 1 = 1.7157, χ 2 = 1.8299 and t * = 0.2. Table I provides the average intertransmission times for 100 points in Ω whose x-components are equally spaced along the sphere centered at 0 and of radius 1 and whose e-components are set to 0. PETC generates inter-transmission times which are smaller than in ETC but larger than in STC. Moreover, we expect the average inter-transmission time to increase when the sampling period h decreases as suggested by Table I.

One might wonder the interest of designing a specific function Γ to derive a periodic event-triggered controller compared to the case where we simply take Γ = Γ, in particular when N = 1. Indeed, it can be noted that to take Γ = Γ also ensures the item (ii) of Theorem 1 by following similar lines as in the proof of Theorem 1, but it does not guarantee the item (i) of Theorem 1. We plotted for that purpose the evolution of Γ for these two options in Figure 1 with h = 0.079 and the same initial conditions. We see that Γ remains negative all the time with Γ given by ( 14), which implies that the periodic event-triggered controller ensures the same specification as the event-triggered controller, while Γ often reaches positive values when Γ = Γ.

On the other hand, we would expect PETC to be more robust to measurement noises, external disturbances and model uncertainties than STC. Indeed, the state is periodically monitored in PETC and the evolution of Γ is only predicted in some sense on a horizon of length h, while the STC method of [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] considers the evolution of Γ on longer time intervals. To illustrate these points, we have added a constant external additive disturbance to each equation in [START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] from t = 0.5 to t = 1. The disturbance value was set to 12 and we have run 100 simulations with the same initial conditions as for Table I. The simulations have shown that PETC with h = 0.01 asymptotically stabilizes the origin for 91% of the simulations while STC only does it for 3%.

These observations confirm the expected robustness features of PETC compared to STC.

V. CONCLUSIONS A method for the design of periodic event-triggered controllers for nonlinear systems has been presented. The approach consists in emulating a known event-triggered controller. It has been explained how the sampling period should be selected and how the triggering condition needs to be redesigned to approximately ensure the same performances as in ETC. In that way, this study offers guidelines for the implementation of event-triggered controllers in practice. It would be interesting in the future to analytically investigate the robustness properties of the proposed PETC strategies. On the other hand, we have focused on the case where an event-triggered controller has been a priori designed. In [START_REF] Heemels | Model-based periodic eventtriggered control for linear systems[END_REF], [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF], the problem is tackled from a different point of view: the periodic event-triggered controller is derived from a periodically sampled controller. Analogous results for nonlinear systems would be very appealing though challenging.

Fig. 1 .

 1 Fig.1. Evolution of Γ for the periodic event-triggered controlled system[START_REF] Postoyan | A unifying Lyapunovbased framework for the event-triggered control of nonlinear systems[END_REF] with different choices for Γ.

The set Ω is said to be forward invariant for the impulsive system (4)-(5) if z 0 ∈ Ω implies that the corresponding solution z with z(t 0 ) = z 0 to (4)-(5) lies in Ω for all time larger than t 0 .

It is possible to derive self-triggering rules when[START_REF] Lunze | A state-feedback approach to event-based control[END_REF] holds with c = 0 by following the approach of[START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. Nevertheless, the next transmission time will generally be given by an implicit formula in this case which may be difficult to solve on-line.

He is partially supported by the European 7th Framework Network of Excellence "Highly-complex and networked control systems" (HYCON2) (grant agreement no. 257462). He is supported by the Innovational Research Incentives Scheme under the VICI grant "Wireless control systems: A new frontier in automation" (No. 11382) awarded by NWO (The Netherlands Organization for Scientific Research) and STW (Dutch Science Foundation), and the European 7th Framework Network of Excellence "Highly-complex and networked control systems" (HYCON2) (grant agreement no. 257462). P. Tabuada is with the Cyber-Physical Systems Laboratory of the Department of Electrical Engineering at the His work was partially supported by the NSF award 1035916. D. His work is supported by the Australian Research Council under the Discovery Grant and Future Fellowship schemes.

APPENDIX

Proof of Theorem 1. Let z 0 ∈ Ω, we consider the solution (z, η) to ( 8)- [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF] with initial condition z(t 0 ) = z 0 , η(t 0 ) = 0 and initial time t 0 = 0. Denote t max ∈ R ≥0 ∪ {∞} the maximum existence time of (z, η). Let z be the solution to (4)-( 5) with initial condition z(t 0 ) = z 0 . Note that z is defined for all positive time in view of Section II-A. According to ( 5) and ( 9), both z and z jump at t = 0. Hence z and z remain equal until one of the two solutions jumps again since they have the same dynamics during flow. We know that z will not jump before t = h, neither will z on the interval ]0, h] since h < T where T is the minimum inter-transmission time (see Assumption 1). Hence, since z and z are left-continuous, z(t) = z(t) for any t ∈]0, h] from which we deduce that t max ≥ h, Γ(z(t)) = Γ(z(t)) < 0, and z(t) ∈ Ω for any t ∈]0, h] as Ω is forward invariant for system (4)-( 5) by virtue of Assumption 1. Consider t = h. If Γ(z(h)) ≥ 0, then a jump occurs and we follow similar arguments as above to conclude that t max ≥ 2h and that Γ(z(t)) < 0 for all t ∈]h, 2h]. Suppose now that Γ(z(h)) < 0. No jump occurs for system (8)- [START_REF] Heemels | Periodic eventtriggered control for linear systems[END_REF]. Consequently, z(t) remains in Ω for t ∈ [h, 2h] in view of Assumption 3. We deduce from Lemma 1 and the definition of Γ in (14) that

as Γ(z(h)) < 0. Suppose there exists t ∈]h

Using the fact that z(t * ) ∈ Ω, we define M = max y∈Ω {|L g Γ(y)|} which is well-defined since Ω is compact and L g Γ is continuous in view of Assumptions 1 and 2, respectively. We note that M is independent of h and N . We derive that Γ(z(t)) ≤ Γ(z(h+i h N ))+ h N M . Using that Γ(z(h+i h N )) < 0 in view of ( 16), we obtain Γ(z(t)) ≤ h N M . Hence for any t ∈ [h, 2h], it holds that Γ(z(t)) ≤ h N M . The desired properties are ensured by induction.


Arnaud Cacucci

Stéphane Loffredo

Valérie Potin
email: valerie.potin@u-bourgogne.fr.highlights

Luc Imhoff

Nicolas Martin

Imhoff Luc

Interdependence of structural and electrical properties in tantalum/tantalum oxide multilayers

Keywords: Tantalum oxide, thin film, reactive sputtering, gas pulsing, metal, semi-conductor, electricals properties

Dc reactive sputtering was used to deposit tantalum metal/oxide periodic nanometric multilayers using the innovative technique namely, the reactive gas pulsing process (RGPP). Different pulsing periods were used for each deposition to produce metal-oxide periodic alternations included between 5 and 80 nm.

Structure, crystallinity and chemical composition of these films were systematically investigated by Transmission Electron Microscopy (TEM) and Energy-dispersive X-ray (EDX) spectroscopy techniques. Moreover, electrical properties were also studied by the Van der Pauw technique.

Introduction

Tantalum oxide compound has great technological interest, particularly because of its optical and electrical performances [1][2][3]. This kind of material has attracted huge attention in recent years due to its potential applications as high refractive index materials in multilayered interference filters [4], as anti-reflecting coatings for solar cells [5], as optical waveguides [6] and as capacitor dielectric in high density dynamic random access memories (DRAMs) [7][8][9]. Compared to silicon dioxide, which has a dielectric constant of 3.9, stoichiometric tantalum oxide Ta 2 O 5 compounds can provide about 5 to 6 times increase in the capacitance density (dielectric constant of Ta 2 O 5 is [20][21][22][23][24][25][26][27]. This could lead to a high packing density in the next generations of DRAMs without reducing the thickness of the dielectric materials to unachievable values. Because of such technological attraction, tantalum oxide thin films have been grown by a large variety of deposition methods including anodization, thermal oxidation, ion-assisted deposition, chemical vapour deposition (CVD) and physical vapour deposition (PVD) [10][11][12][13][14][15]. The TaO x films prepared by these different techniques showed widely disperse properties depending on the process parameters, temperature grown, post-annealing conditions, etc. However, the influence of the oxygen concentration on the characteristics of the films has insufficiently been investigated, especially for films prepared by reactive sputtering [16][17]. It mainly comes from the difficulties to synthesize tantalum oxides with tuneable element concentrations due to instabilities of the reactive process [18][19][20]. Such a drawback restrains the range of reachable chemical compositions and consequently the resulting properties.

The purpose of this paper is to show how structuration of thin multilayers can influence the electrical properties of TaO x films. The role of the total period and the parameter defined as the ratio met / ox (where met is the thickness of metal sub-layer and ox that of the oxide) on the properties of the films is principally investigated. A systematic increase of the oxygen injection time into the pulsing process leads to clear alternation of α-Ta/aTa 2 O 5 .

Materials and methods

Thin films were simultaneously deposited onto (100) silicon wafer and glass substrates by dc reactive magnetron sputtering from a tantalum metallic target (purity 99.6 at. % and 51 mm diameter) in an Ar + O 2 gas mixture. The target was sputtered with a constant current density J = 50 A m -2 . The distance between the target and the substrate was fixed at 65 mm. The gas flow rates were controlled by a homemade system [21]. All depositions were carried out with an argon flow rate of 2 sccm and a constant pumping speed of 10 L s -1 , which produced an argon partial pressure of 0.28 Pa. Oxygen mass flow rate was periodically controlled versus time according to a rectangular signal by the reactive gas pulsing process, namely RGPP [22]. The pulsing period T of the oxygen flow rate was changed from 22 to 433 s to produce ten different periodic metal/oxide alternations (Ta pure , Ta1 to Ta9, samples). The maximum O 2 flow rate was fixed at 1.6 sccm during the t on injection time. Otherwise it was completely stopped (no oxygen injection) during the t off time. The expected tantalum/tantalum oxide period is in-between 5 to 100 nm, with = met + ox , where met and ox are the thicknesses of the metal and oxide sub-layers, respectively. As a result, the duty cycle α defined as the ratio t on /T takes values from 26 to 66% of T in order to tune the ox / met ratio. Moreover, the deposition procedure always started with the O-rich layer and finished with the Ta-rich in order to reach a total thickness close to 260 nm.The local structure was examined by high resolution transmission electron microscopy to check the thickness, interface and local order in each sub-layer. The chemical composition was determined by energy-dispersive X-ray spectroscopy implemented in a TEM JEOL 2100 LaB 6 . Moreover, dc electrical resistivity ρ of the films deposited on glass substrates was performed versus temperature from 300 to 470 K with a homemade system based on the Van der Pauw method.

Results and discussion

HRTEM results

The study of cross-sectional view by TEM allows the observation of distinct alternations of tantalum and tantalum oxide sub-layers for all samples. Indeed, clear interfaces of different chemical phases can be observed and measured by TEM images irrespective of the Λ period for each sample (Fig. 1a and1b). TEM allows a nanometric measurement (± 1 nm) for total thickness (e tot ) and subnanometric (± 0.2 nm) for period and sub-layer thicknesses (Λ and λ, respectively).

Measurements of total period are included between 5.4 and 78.6 nm, with e tot ranging from 203 to 330 nm (Fig. 1a and1b). Dc reactive sputtering with the RGPP technique has performed sub-layer thickness lower than 2 nm as it can be observed in figure 1a by TEM analyses. Indeed, this technique allows obtaining met included between 1.9 nm and 50.8 nm, and ox between 3 nm and 27.8 nm, respectively.

The energy-dispersive X-ray spectroscopy analysis was used to determine the chemical composition of sub-layers of metal (dark bands) and oxide (bright bands).

In the O-rich sub-layers, the amount of Ta is 27 at. % and 73 at. % for O with a variation of less than 4 % between all samples. This reveals an oxide phase of composition close to Ta 2 O 5 . For the Ta-rich sub-layers, the EDX measurements show that this layer is mainly composed by tantalum with 90 at. % of Ta and 10 at. % of O. The composition of the metal sub-layer does not fall below 81 at. %. So the occurrence of two phases can be possible with some inclusions of tantalum oxide in the pure metallic sub-layer.

Association of HRTEM images and diffraction patterns allows the observation of some eventual crystalline phases. By comparing experimental and simulated (JEMS software) patterns, the determination and identification of crystals are accurately done. HRTEM pictures highlight that Ta-rich sub-layer is partially crystallized and that the O-rich layer is completely amorphous. Moreover, the systematic HRTEM study reveals that the crystallization starts for met > 10 nm with small crystallite size about 3 -5 nm and finishes with a quasi total crystallization for the biggest periods. Zone axes (ZA) [111] of g-Ta have been indexed for all crystallites (Fig. 1.c) with interplanar distances typical of pure Ta (d 111 = 2.33 ). Moreover, the diffraction pattern presented in figure 1.d points out the g-Ta cubic phase. TEM study allows concluding that the Ta/TaO x multilayers deposited by dc magnetron sputtering are all composed by g-Ta/a-Ta 2 O 5 for Tarich and O-rich sub-layers, respectively.

Conductivity measurements and discussion

Characterization of gas sensor is most often carried out by conductivity measurements varying temperature. For the gas sensor characterization, it would be ideal to measure only the electrical contribution of gas adsorbed quantity by freeing itself of the sensitive material contribution while varying the temperature. Obtain such hybrid and invariant electrical behavior versus temperature can be interesting for gas sensor application.

Every sample has been characterized by Van der Pauw analyses with two cycles of measurements. They have been carried out for thin films grown on glass substrate to avoid the contribution of silicon conductivity. The cycle starts at room temperature, then, the substrate is heated up to 200 °C using 2 °C min -1 increment.

A plateau of 10 min is set at 200 °C before coming back to room temperature with the same ramp of temperature. These analyses (Table 1) all exhibit a linear evolution of the electrical resistivity versus temperature with 0 included between 1.77×10 -06 and 4.10×10 -06 m.

The Ta1 sample rather exhibits a semiconducting resistivity value and behaviour with 0 = 1.82×10 -04 m. All samples have a negative slope when ρ is plotted versus temperature, but two samples (Ta8 and Ta9) get a quasi constant resistivity versus temperature. In fact, they all have a metallic-like behaviour but with positive or negative temperature coefficients of resistivity according to the Λ period size (TCR = g 0 in Table 1). For metals, the variation of resistivity with temperature is a function of TCR. This latter is expressed as a percentage of resistivity variation in temperature relative to the initial value 0 obtained at T 0 = 25 °C.

( )

000 1- ρρ α  =+  TT ( 1 )
In order to better understand the electrical properties of tantalum/tantalum oxide multilayers, we first assumed that the charge carrier concentration and mobility are high. To have an equivalent parameter in terms of structure which will be noted , mobility and charge carrier concentration can be brought back to the amount of metal present in the met sub-layers under the amount of insulator in the ox sublayers. The parameter defined as = met / ox , can be calculated without taking into account the metal concentration as it is a constant in all sub-layers.

Van der Pauw analyses pointed out that any samples have a positive TCR like the Ta pure reference sample. Moreover, resistivity measurements highlight that the ratio χ must be higher than 3 to obtain a metallic-like behaviour with a quasi zero TCR value. By studying the data provided in table 1, no clear dependence between the values of , met , ox , and can be pointed out. Only the structural parameter χ may explain the change of electrical response. Indeed, when χ increases, ρ decreases to tend towards the behaviour of the Ta pure reference sample. The insertion of the χ parameter is therefore crucial to understand the relationship of interdependence between electrical properties and structural of multilayered thin films.

Conclusion

Dc reactive sputtering deposition of tantalum and tantalum oxide periodic nanometric multilayers was successful with the reactive gas pulsing process. The transmission electron microscope study showed sharp interfaces between the sublayers. EDX and TEM analyses proved that the Ta-rich sub-layers was composed by crystalline g-Ta for met > 10 nm, whereas the O-rich sub-layers are composed by amorphous a-Ta 2 O 5 . The study of the electrical properties was carried out by Van der Pauw method, pointing out a metallic behaviour for met > 1.9 nm. Sublayers thickness and composition were accurately determined by TEM introducing the χ parameter, which allows explaining the electrical behaviour. The interdependence relationship was demonstrated between the electrical properties and the structure of thin g-Ta/aTa 2 O 5 periodic multilayers.

Tables

Figure 1 Cross

 1 Figure 1 Cross section TEM image of a) Ta1 and b) Ta6 samples, c) HRTEM picture of g-

Figure 1

 1 Figure 1

Table 1

 1 Structural parameters and resistivity results from Van der Pauw analyses for Ta1 to Ta9 with Ta pure as reference sample.

	name	e tot	Λ	λ met	λ ox	χ	ρ 0	g 0	ρ (470K)
	of sample	(± 2 nm)	(± 0,1 nm) (± 0,1 nm)	(± 0,1	(± 0.05)	( m)	(K -1 )	( m)
					nm)
	Ta1	330	5,4	1,9	3,5	0,54	1,82×10 -04 -2,62×10 -03 9,84×10 -05
	Ta2	278	5,6	2,6	3	0,87	4,12×10 -06 -2,36×10 -04 3,95×10 -06
	Ta3	264	8	4,3	3,7	1,16	3,25×10 -06 -2,11×10 -04 3,13×10 -06
	Ta4	244	9,8	5,8	4	1,45	2,90×10 -06 -4,93×10 -04 2,65×10 -06
	Ta5	239	14,7	9,3	5,4	1,72	2,80×10 -06 -9,39×10 -04 2,34×10 -06
	Ta6	237	78,6	50,8	27,8	1,83	2,60×10 -06 -3,30×10 -04 2,45×10 -06
	Ta7	220	20,2	13,7	6,5	2,11	2,23×10 -06 -5,12×10 -04 2,03×10 -06
	Ta8	272	16	12	4	3,00	1,87×10 -06 -1,53×10 -04 1,82×10 -06
	Ta9	272	45.3	35,3	10	3,53	1,76×10 -06 -1,94×10 -04 1,71×10 -06
	T pure	300	-	-	-	-	1,90×10 -07	1,20×10 -03	2,30×10 -07


Abstract-Traffic Sign Recognition (TSR) is now relatively well-handled by several approaches. However, traffic signs are often completed by one (or several) supplementary placed below. They are essential for correct interpretation of main sign, as they specify its applicability scope. difficulty of supplementary sub-sign recognition potentially infinite number of classes, as nearly any can be written on them. In this paper, we propose and evaluate a hierarchical approach for recognition of supplementary signs, in which the "meta-class" of the sub-sign (Arrow, P Text or Mixed) is first determined. The classification is based on the pyramid-HOG feature, completed by dark area proportion measured on the same pyramid. large database of images with and without supp shows that the classification accuracy of our approach 95% precision and recall. When used on output of our sub specific detection algorithm, the global correct detection and recognition rate is 91%.

I. INTRODUCTION

ars are now often equipped with A Assistance Systems (ADAS) for helping the driver and avoiding accidents. Many of those ADAS can benefit from, or even require, vision-based Traffic Sign detection and Recognition (TSR), and a correct interpretation signs. For instance, in order to develop a smart Cruise Control (ACC) that would automatically tune target cruising speed depending on the current speed needs reliable detection and recognition of speed This is a mandatory complement to speedextracted from GPS cartographic data, which always complete nor systematically up furthermore cannot handle temporary speed limits for road works, nor variable speed limits, due to the pre-defined digital cartographic data. Until now, most TSR published research and existing systems rely on the same process. At first, a detection stage aims at extracting from the image all the possible locations Anne-Sophie Puthon and Fabien Moutarde are with the Robotics Laboratory (CAOR), Mines ParisTech, 60 Bd Saint-Michel, F cedex 06, FRANCE, Phone: (+33) 1.40.51.92.92, Fax: (e-mail: Anne-Sophie.Puthon@ensmp.fr, Fabien.Moutarde@ensmp.fr) Fawzi Nashashibi is with IMARA project at INRIA, Domaine de Voluceau -Rocquencourt, B.P. 105 -F-78153 Le Chesnay, FRANCE, 1.39.63.52.56 (e-mail: Fawzi.Nashashibi@inria.fr).

Recognition of correct interpretation of traffic signs

Anne-Sophie Puthon based Traffic Sign detection and correct interpretation of these For instance, in order to develop a smart Adaptive would automatically tune target current speed limit, one ecognition of speed-limit signs.

-limit information which is neither always complete nor systematically up-to-date, and temporary speed limits for road the static nature of search and existing systems rely on the same process. At first, a detection stage aims at extracting from the image all the possible locations are with the Robotics Michel, F-75272 Paris (+33) 1.43.26.10.51 @ensmp.fr, Fabien.Moutarde@ensmp.fr) Domaine de Voluceau , FRANCE, Phone: (+33) of speed signs. Two different types of information can be used. Colour information (see e.g. [START_REF] De La Escalera | recognition and analysis for intelligent vehicles[END_REF], [START_REF] Bahlmann | A System for Traffic Sign Detection, Tracking, and Recognition Using Color, Shape[END_REF], the detection easier but less robust, and detection working on greyscale images [START_REF] Moutarde | Robust on-vehicle real-time visual detection of American and European speed limit signs, with a modular Traffic Signs Recognition system[END_REF]) increases robustness to variation of conditions and to fading away of colo then tested through a classification step in order to eliminate false alarms and recognize the sun tracking helps in validating the results from frame to frame. Figure 1. Examples of traffic signs that cannot be interpreted correctly without taking into account the supplementary sign located below.

However, the exact meaning of traffic signs, in particular speed-limit signs, is often altered by one (or several) supplementary sign(s) placed below (see supplementary signs are essential for correct interpretation of main signs, as they specify their applicability scope, such

Recognition of supplementary signs for correct interpretation of traffic signs

Sophie Puthon, Fabien Moutarde and Fawzi Nashashibi different types of information can be formation (see e.g. [START_REF] De La Escalera | recognition and analysis for intelligent vehicles[END_REF], [START_REF] Bahlmann | A System for Traffic Sign Detection, Tracking, and Recognition Using Color, Shape[END_REF], and [START_REF] Broggi | Time Road Signs Recognition[END_REF]) makes but less robust, and shape-based yscale images (e.g. [START_REF] Gavrila | Traffic sign recognition revisited 21st DAGM symposium fur Musterekennung[END_REF], [START_REF] Barnes | Real speed sign detection[END_REF], [START_REF] García-Garrido | Fast traffic sign detection and recognition under changing lighting conditions[END_REF], and to variation of illumination colours. Each candidate is then tested through a classification step in order to eliminate false alarms and recognize the sun-sign type. Finally a tracking helps in validating the results from frame to frame.

Examples of traffic signs that cannot be interpreted correctly without taking into account the supplementary sign located below.

However, the exact meaning of traffic signs, in particular limit signs, is often altered by one (or several) supplementary sign(s) placed below (see Figure 1). These supplementary signs are essential for correct interpretation of main signs, as they specify their applicability scope, such as type of vehicle (car, truck, etc.), distance extension, specific lane, particular dates or timeslots, or weather condition (rain, snow, etc.).

II. RELATED AND PREVIOUS WORK

The detection and recognition of supplementary signs is a task very similar to usual Traffic Sign Recognition. It might even, wrongly, be thought simpler due to the fact that detection can b position and size of the main sign it is related to.

However there are several difficulties s supplementary signs: 1/ the aspect ratio (height/width) of the rectangle wildly varies between sub-sign types (see Figure 2/ supplementary signs are much less normalized than main signs, and significantly vary from country to country; 3/ the contrast between sub-signs and their is often low. In greyscale images, the colour of supplementary signs can easily be confused to sky, and contrary to most main signs coloured border around them; 4/ supplementary sign recognition is a classification task with a potentially infinite number of classes sub-signs contain rather "free" text.

Figure 2. Sub-sign width/height ratio distribution for our database (see Table 1).

Some or all of the above difficulties could explain why, despite its practical importance, supplementary sign detection is not a trendy field and only very few published. To our best knowledge, the first paper including some work on supplementary sub-sign detection and recognition is that of Hamdoun et al. [8 proposed to use a rectangle edge detection for localizing potential sub-signs in a rather large region signs. However their recognition, using a neural network applied to rescaled images and tested only for "Exit Arrow" supplementary sign type as type of vehicle (car, truck, etc.), distance extension, specific lane, particular dates or timeslots, or weather WORK detection and recognition of supplementary signs is a task very similar to usual Traffic Sign detection and be thought to be simpler due to the fact that detection can be guided by the main sign it is related to.

However there are several difficulties specific to the aspect ratio (height/width) of the rectangle wildly Figure 2); supplementary signs are much less normalized than main signs, and significantly vary from country to their surroundings low. In greyscale images, the background can easily be confused contrary to most main signs, there is no supplementary sign recognition is a classification task ite number of classes, as many ratio distribution for our database Some or all of the above difficulties could explain why, supplementary sign very few works were knowledge, the first paper including sign detection and 8] in 2008, who a rectangle edge detection for localizing region below the main , using a feed forward s, was developed rrow" supplementary sign type (one of the most frequent in France have presented in 2010 a work handling signs: "Exit Arrow", "Truck", "Bei Nässe" ( and "7.5t". They however had to train a Support Vector Machine (SVM) classifier which makes difficult to extend their all types of supplementary signs. Liu supplementary signs detection (based on Hough voting) and recognition (using applied to Fourier and wavelet descriptors presented in 2011. Finally, Puthon proposed and tested a new original and performant technique for the detection step, based on region contrasted seeds.

In the present paper, we propose and evaluate a general framework for recognition of supplementary signs.

III. RECOGNITION OF TRAFFIC SU

Recognition of supplementary signs is not very different from TSR in general, except for the existence of a potentially very large number of classes, as several sub types consist of "free" text. The undefined and large number of classes pleads for the use of a hierarchical classification approach as adopted by Liu et al. in [START_REF] Liu | An Efficient Real-Time Speed Limit Signs Recognition Based on Rotation Invariant Feature[END_REF] approach the addition of a new type of sub require retraining their whole cascade of classifiers. For this reason, and also for taking into accoun "text" supplementary signs, we have decided to implement a hand-crafted hierarchical recognition in which we first determine the "meta-class" of the sub "Arrow", "Pictogram", "Text", and " The idea is that for "text" meta-class, interpretation could be done in a further stage by applying character recognition. The first step is the choice of the features (or "descriptors") that should be computed on the candidate region and fed into classifiers. We focus on global descriptors which describe the entire image contrary to local ones, such as interest points, localizing information around discriminative regions. A solution is to use the pixel values to a fixed size is first applied, as was done in [START_REF] Hamdoun | Detection and Recognition of End-of-Speed-Limit and Supplementary Signs for Improved European Speed Limit Support[END_REF] and [START_REF] Nienhüser | Fast and Reliable Recognition of Supplementary Traffic Signs[END_REF] However the drawback is that "bounding" error in the detection stage can translate into a total change of the resulting descriptor.

For this "offset robustness" issue, it seems better to use descriptors that integrate information on large enough (one of the most frequent in France). Nienhüser et al. [START_REF] Nienhüser | Fast and Reliable Recognition of Supplementary Traffic Signs[END_REF] a work handling more types of subei Nässe" ("when raining") d to train a one-against-all Vector Machine (SVM) classifier for each type, which makes difficult to extend their approach for handling Liu et al. [START_REF] Liu | An Efficient Real-Time Speed Limit Signs Recognition Based on Rotation Invariant Feature[END_REF] also include detection (based on gradients and Hough voting) and recognition (using a tree of binary SVMs applied to Fourier and wavelet descriptors) in their system Puthon et al. [START_REF] Moutarde | Subsign detection with region-growing from contrasted seeds[END_REF] in 2012 have proposed and tested a new original and performant technique , based on region-growing from

In the present paper, we propose and evaluate a general of supplementary signs.

C SUPPLEMENTARY SIGNS

Recognition of supplementary signs is not very different from TSR in general, except for the existence of a potentially very large number of classes, as several sub-sign The undefined and large number use of a hierarchical classification in [START_REF] Liu | An Efficient Real-Time Speed Limit Signs Recognition Based on Rotation Invariant Feature[END_REF]. However, in their addition of a new type of sub-sign would require retraining their whole cascade of classifiers. For this reason, and also for taking into account the specificity of we have decided to implement a hierarchical recognition in which we first class" of the sub-sign, between ext", and "Mixed" (see Figure 3). class, interpretation could be done in a further stage by applying character recognition. the 4 "meta-classes" used for supplementary signs.

the features (or "descriptors") that should be computed on the candidate region and fed into We focus on global descriptors which describe contrary to local ones, such as interest points, localizing information around discriminative regions. A solution is to use the pixel values, provided that rescaling applied, as was done in [START_REF] Hamdoun | Detection and Recognition of End-of-Speed-Limit and Supplementary Signs for Improved European Speed Limit Support[END_REF] and [START_REF] Nienhüser | Fast and Reliable Recognition of Supplementary Traffic Signs[END_REF]. any slight offset or "bounding" error in the detection stage can translate into a total change of the resulting descriptor.

For this "offset robustness" issue, it seems better to use descriptors that integrate information on large enough chunks of the sub-sign. Resizing to very few pixels (e.g. 8x8) might sound tempting, but it would not retain enough information to discriminate categories. We therefor to the "pyramid-HoG" (pHoG) descriptor, proposed by Bosch et al. [START_REF] Bosch | Representing shape with a spatial pyramid kernel[END_REF], which has the advantage o sensitive to image offset, while retaining pertinent and discriminating information (see Figure 4). The main parameter for pHoG is the number of levels the pyramid. For determining its optimal value, we compared performances for L=0, 1, 2 and 3. on Figure 5, increasing L from 0 to 2 provides significant improvement to classification accuracy attained when training Gaussian kernel SVMs but going above not bring more recognition performance. L=2 the best performance and a relatively small number of values. Another critical parameter for pHoG of bins for computing the histogram of gradient orientations; similar analysis of the impact of the number K of bins for computing the histogram has led us to fix it to the smallest value K=8.

Figure 5 Influence of the number L of levels of the pyramid shows much better recall and precision than L=0 and 1, and L=3 brings no further improvement. Evaluation is done on kernel SVMs trained on variants of the feature.

By looking closer at frequent confusions that were produced when training SVM classifier on only pHoG descriptor, we realized that most of them seemed due to the fact that contains only gradient information, while correct sign. Resizing to very few pixels (e.g. 8x8) might sound tempting, but it would not retain enough information to discriminate categories. We therefore turned HoG) descriptor, proposed by [START_REF] Bosch | Representing shape with a spatial pyramid kernel[END_REF], which has the advantage of being less sensitive to image offset, while retaining pertinent and HoG) descriptor used. On the left, with no subdivision, in middle with L=1 level of subdivision, on the right with L=2 levels.

HoG is the number of levels L of the pyramid. For determining its optimal value, we 2 and 3. As can be seen provides significant to classification accuracy attained when oing above L=2 does L=2 thus provides the best performance and a relatively small number of HoG is the number K of bins for computing the histogram of gradient orientations; similar analysis of the impact of the number K of bins for computing the histogram has led us to fix it to the smallest

of the number L of levels of the pyramid: L=2 shows much better recall and precision than L=0 and 1, and L=3

Evaluation is done on Gaussian kernel SVMs trained on variants of the feature. ions that were produced HoG descriptor, we realized that most of them seemed due to the fact that pHoG contains only gradient information, while correct discrimination between our meta-classes would clearly be made easier if using also some colorimetric info have therefore enhanced pHoG by adding, for each sub region of the pyramid, the proportion of "dark" pixels (as illustrated on Figure 6). This "dark proportion" is computed using the same binarization technique as [START_REF] Moutarde | Subsign detection with region-growing from contrasted seeds[END_REF] for extracting contrasted seed from which is grown detected region. A morphological reconstruction used to find highly contrasted pixels in the image, then "dark" pixels are selected as pixels with intensity included in the [t 1 ;t 2 ] interval, with t 1 =µ+0.5σ and the mean of pixel values, and σ Examples of pixels considered as "dark" by this method are shown on second column of Figure 6 We nickname pHoG-Dp this new proposed descriptor concatenating the standard pHoG with the "darkness proportion" information computed on the same pyramid. new pHoG-Dp descriptor allows significant classification improvement, as can be seen on precision figure 7.

Figure 7. Comparison of classification performance of 3 descriptors: pixels of rescaled image ( pHoG (middle in dashed red), and pHoG dash-point).

classes would clearly be easier if using also some colorimetric information. We HoG by adding, for each subregion of the pyramid, the proportion of "dark" pixels (as This "dark proportion" is computed technique as that used by us in extracting contrasted seed from which is grown detected region. A morphological reconstruction algorithm is ly contrasted pixels in the image, then "dark" pixels are selected as pixels with intensity included in and t 2 =µ+1.5σ (where µ is the standard deviation). Examples of pixels considered as "dark" by this method are 6.

darkness proportion » computed on : left, the original image, and its inverse version; middle, visualization of the darkness proportions

regions for the 3 levels of the pyramid; the resulting vector.

Dp this new proposed descriptor concatenating the standard pHoG with the "darkness proportion" information computed on the same pyramid. The Dp descriptor allows significant classification n be seen on precision-recall curves of

Comparison of classification performance of descriptors: pixels of rescaled image (bottom curve, in blue),

HoG-Dp (top curve, in green .

IV. EXPERIMENTS AND RESULTS

In order to evaluate the classification accuracy of our approach, we use a large database of sub-signs as described in Table 1. As mentioned in previous section, supplementary signs were grouped in 4 "meta-classes": Pictogram, Mixed.

Table 1. Content of the supplementary signs dat training and testing.

Category Examples Training

After training Gaussian kernel SVMs, the classification accuracy is satisfactory, as can be seen on shows precision-recall curves computed separately for each meta-class. Note that this first evaluation is for recognition alone, as it is done on test images extracted after manual labelling, which emulates a "perfect" detection step.

Figure 8. Precision-recall curves separately for each meta computed on test database (from manually bounding-boxes).

ESULTS

In order to evaluate the classification accuracy of our signs as described As mentioned in previous section, supplementary , the classification accuracy is satisfactory, as can be seen on Figure 8, which separately for each Note that this first evaluation is for recognition done on test images extracted after manual elling, which emulates a "perfect" detection step.

recall curves separately for each meta-class, computed on test database (from manually-labeled

More details on the classification accuracy reached can be found on the confusion matrix computed on test examples of our database, shown in Table 2. The Text category is the one with the lowest precision, which can be explained by some confusions, in particular with guard rails (see top line of Figure 9). The slightly lower precision for " can be explained by some confusions arising in particular with pictograms contained in main signs lying in the searched sub-region (see bottom line of All the above evaluations were conducted on a database in which the bounding boxes around su stems from manually-made "ground truth". From the application point of view, it is essential to also test performance of our recognition approach when applied on the candidate sub-sign regions found by our sub detection technique. As described in more details in [START_REF] Moutarde | Subsign detection with region-growing from contrasted seeds[END_REF], our detection method is based on region highly contrasted seeds (hopefully lying on border of black pictogram or text of the sub-sign).

Figure 10 shows, for our global chain consisting of detection by region-growing from contrasted seeds followed by recognition with SVM applied on p f(FPPP) curves separately for eac (FPPP = False Positive Per Positive). One can see that maximum recall is rather low (~45%) for Arrow essentially due to lower detection rate for this type of sub signs. Maximum recall for Pictogram and Mixed categories are respectively ~65% and ~72%. The Text meta the best maximum recall ~92%, but also the highest false accuracy reached can be computed on test examples of The Text category is the one on, which can be explained by some confusions, in particular with guard rails (see top line of The slightly lower precision for "Pictogram" type explained by some confusions arising in particular with pictograms contained in main signs lying in the bottom line of Figure 9).

Confusion matrix.

Pict conducted on a database in which the bounding boxes around supplementary signs made "ground truth". From the application point of view, it is essential to also test formance of our recognition approach when applied on found by our sub-sign detection technique. As described in more details in [START_REF] Moutarde | Subsign detection with region-growing from contrasted seeds[END_REF], our detection method is based on region-growing applied from ully lying on border of black for our global chain consisting of detection growing from contrasted seeds followed by cognition with SVM applied on pHoG-Dp, the recall = separately for each meta-class obtained Positive). One can see that maximum recall is rather low (~45%) for Arrow category, rate for this type of subsigns. Maximum recall for Pictogram and Mixed categories respectively ~65% and ~72%. The Text meta-class has the best maximum recall ~92%, but also the highest false positive rate, which was expected given that this class had the lowest precision in table 2. The recall rates may seem low, but these are frame-by-frame evaluations, including first "human-detectable" appearance of sub often too few pixels to be easily classified by recognition step. In practice however, one does not need to detect and recognize on every frame, but only on enough frame confident in output.

Figure 10. Frame-by-frame evaluation of our complete detection recognition chain: Recall=f(False Positive Per separately for each meta-class

In the final application, performance should only count the percentage of physical sub-signs that are "validated" their appearance. Our validation criterion same sub-sign is detected and recognized (with same class!) at least 3 times during the total time-window visibility. The resulting correct detection validation rates are reported in Table 3. As one can see the global recall is from 70% for the "Arrow" meta for the "Text" meta-class, with an average recall of 91% the same setting, the false alarm rate for signs sub-sign placed below is ~12%. positive rate, which was expected given that this class had the lowest precision in table 2. The recall rates may seem frame evaluations, including detectable" appearance of sub-sign which has often too few pixels to be easily classified by recognition

In practice however, one does not need to detect and recognize on every frame, but only on enough frames to be frame evaluation of our complete detection + Per Positive) curves class.

hould only count the signs that are "validated" during on adopted is that sign is detected and recognized (with same class!) window of sub-sign + recognition + . As one can see the rrow" meta-class to 99% recall of 91%. For the false alarm rate for signs without any We have presented in this paper an original framework for hierarchical recognition of supplementary signs. We use feature pHoG-Dp, a new proposed original variant of "pyramid HoG", in which we have added the "proportion of dark pixels" evaluated on the same pyramid of sub as the histograms of gradient. Furthermore, to cope with potentially very large number of supplementary sign types, we proposed to apply a hierarchical classification in which the meta-class of sub-sign (Arrow, is first determined.

An experimental evaluation conducted on a large data sub-signs showed that accuracy of our proposed framework can reach 95% recall and precision on perfectly cropped sub signs. Furthermore, the global performance when chaining our recognition after our detection wa all supplementary signs types (with a false alarm rate of 12% on signs without sub-signs).

Future works include training and evaluation of second stage of the hierarchical classifier (in order to recognize precise type of supplementary sign), and optical character recognition (OCR) on the sub identified as "text" type.

C

  on (TSR) is now relatively handled by several approaches. However, traffic signs are supplementary sign(s) correct interpretation of sign, as they specify its applicability scope. The main recognition is the potentially infinite number of classes, as nearly any information can be written on them. In this paper, we propose and evaluate hierarchical approach for recognition of supplementary signs, Arrow, Pictogram, The classification is based HOG feature, completed by dark area same pyramid. Evaluation on a of images with and without supplementary signs our approach reaches When used on output of our sub-sign specific detection algorithm, the global correct detection and now often equipped with Advanced Driving (ADAS) for helping the driver and avoiding accidents. Many of those ADAS can benefit from,

Figure 3 .

 3 Figure 3. Illustration of the 4 "meta supplementary signs.

Figure 4 .

 4 Figure 4. Illustration of pyramid-HoG (pHoG) descriptor used the left, with no subdivision, in middle with L=1 level of subdivision, on the right with L=2 levels.

Figure 6 .

 6 Figure 6. Illustration of the « darkness proportion same pyramid as pHoG: left, the original image, and its binarized version; middle, visualization of the darkness proportions in the various sub-regions for the 3 levels of the pyramid; right, the resulting vector

  classes": Text, Arrow, Content of the supplementary signs database used for

Figure 9 .

 9 Figure 9. Some examples of negatives confused with "Text" type supplementary sign (top line), and of negatives confused "pictogram" type (bottom line).

Table 2 .

 2 Confusion matrix.

				Classifier
	Ground Truth	Neg.	Text Arrow
	Negative	4766 107	0
	Text	48	632	2
	Arrow	2	4	90
	Pictogram	23	8	0
	Mixed	24	14	0
	Precision 98% 83% 98%

  . Mixed Recall

	49	22	96%
	13	2	91%
	2	1	91%
	440	2	93%
	9	474	91%
	86% 95%	95%

Some examples of negatives confused with "Text" type supplementary sign (top line), and of negatives confused with

"pictogram" type (bottom line).

Table 3 .

 3 Correct detection + recognition + validation rates for each sub-sign meta-class, and false alarm rate on signs withoutsub-signs.

	Meta-class Text Arrow Pictogram Mixed
	Total	102	23	55	49
	detection +	101	16	53	39
	recognition	99%	70%	95%	80%

ACKNOWLEDGEMENTS

This work was conducted in the framework of the SpeedCam project (ANR-09-VTT-11), funded by the ANR (Agence Nationale de la Recherche).

DISCUSSION AND PERSPECTIVES

We have presented in this paper an original framework for supplementary signs. We used as Dp, a new proposed original variant of yramid HoG", in which we have added the "proportion of same pyramid of sub-regions Furthermore, to cope with the