R Émi Carles

NONLINEAR SCHR ÖDINGER EQUATION AND FREQUENCY SATURATION

Keywords: Nonlinear Schrödinger equation, well-posedness, approximation. 2010 Mathematics Subject Classification. Primary 35Q55, Secondary 35A01, 35B30, 35B45, 35B65

We propose an approach that permits to avoid instability phenomena for the nonlinear Schrödinger equations. We show that by approximating the solution in a suitable way, relying on a frequency cut-off, global well-posedness is obtained in any Sobolev space with nonnegative regularity. The error between the exact solution and its approximation can be measured according to the regularity of the exact solution, with different accuracy according to the cases considered.

INTRODUCTION

We consider the nonlinear Schrödinger equation (1.1) i∂ t u + ∆u = ǫ|u| 2σ u, (t, x) ∈ I × R d ; u |t=0 = u 0 , for some time interval I ∋ 0, with ǫ = 1 (defocusing case) or ǫ = -1 (focusing case). The aim of this paper is to propose an approach to overcome the lack of local well-posedness in Sobolev spaces with nonnegative regularity.

Recall two important invariances associated to (1.1):

• Scaling: if u solves (1.1), then for λ > 0, so does u λ (t, x) := λ 1/σ u λ 2 t, λx . This scaling leaves the Ḣsc x -norm invariant, with s c = d/2 -1/σ. • Galilean: if u solves (1.1), then for v ∈ R d , so does e iv•x-i|v| 2 t/2 u(t, x -vt).

This transform leaves the L 2 x -norm invariant. These two arguments suggest that the critical Sobolev regularity to solve (1.1) is max(s c , 0). Indeed, if s c 0, local well-posedness from H s (R d ) to H s (R d ) for s s c has been established in [START_REF] Cazenave | The Cauchy problem for the critical nonlinear equation in H s , Nonlinear Anal[END_REF], and if s c < 0, local well-posedness from H s (R d ) to H s (R d ) for s 0 has been established in [START_REF] Tsutsumi | L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups[END_REF].

If s c > 0, pathological phenomena have been exhibited for initial data in H s (R d ) with 0 < s < s c : Gilles Lebeau has proved a "norm inflation" phenomenon for the wave equation ∂ 2 t u -∆u + u p = 0, x ∈ R 3 , p ∈ 2N + 1, p 7 ( [START_REF] Lebeau | Non linear optic and supercritical wave equation[END_REF]; see also [START_REF] Étivier | Exemples d'instabilités pour des équations d'ondes non linéaires (d'après G. Lebeau)[END_REF]). The analogous result for (1.1) has been established in [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] and [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF].

Theorem 1.1 (From [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF][START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF]). Let σ 1. Assume that s c = d/2 -1/σ > 0, and let 0 < s < s c . There exists a family (u h 0 ) 0<h 1 in S(R d ) with u h 0 H s (R d ) → 0 as h → 0, a solution u h to (1.1) and 0 < t h → 0, such that:

u h (t h ) H s (R d ) → +∞ as h → 0.
The argument of the proof consists in considering concentrated initial data, u 0 (x) = h s-d/2 (log 1/h) -α a 0 x h with h → 0, and showing that for very short time, the Laplacian can be neglected in (1.1). The above result then stems from its (easy) counterpart in the ODE case, by choosing a suitable α > 0.

In the spirit of [START_REF]Perte de régularité pour les équations d'ondes sur-critiques[END_REF], the above result has been strengthened to a "loss of regularity" in [START_REF] Carles | Loss of regularity for super-critical nonlinear Schrödinger equations[END_REF][START_REF] Carles | Geometric optics and instability for semi-classical Schrödinger equations[END_REF][START_REF] Thomann | Instabilities for supercritical Schrödinger equations in analytic manifolds[END_REF]; the assumptions and conclusion are similar to that in Theorem 1.1, the only difference is that u h (t h , •) is measured in H k (R d ) for any k > s/(1 + σ(s c -s)), thus allowed to be smaller than s. In all the cases mentioned here, the lack of uniform continuity of the nonlinear flow map near the origin is due to the appearance of higher and higher frequencies on a very short time scale. If s c < 0, similar pathological phenomena have been established in H s (R d ) with s < 0, where on the contrary, low frequencies are ignited; see e.g. [START_REF] Bejenaru | Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schrödinger equation[END_REF][START_REF] Carles | Geometric optics and instability for NLS and Davey-Stewartson models[END_REF][START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF][START_REF] Kenig | On the ill-posedness of some canonical dispersive equations[END_REF]. In the rest of this paper, we focus on nonnegative regularity, s 0.

The goal of this paper is twofold. First, we want to investigate a method to remove the pathology mentioned above, causing a lack of well-posedness for (1.1), in a deterministic way, as opposed to the probabilistic approach initiated in [START_REF] Burq | Random data Cauchy theory for supercritical wave equations. I. Local theory[END_REF][START_REF]Random data Cauchy theory for supercritical wave equations. II. A global existence result[END_REF] for the wave equation. The other motivation is related to numerical simulations for (1.1), where high frequencies may be a source of important errors; see for instance [START_REF] Ignat | Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations[END_REF], a reference which will be discussed further into details in Sections 3 and 4.

We show that with a suitable cut-off on the high frequencies of the nonlinearity, the obstructions to local well-posedness vanish, and the problem becomes globally well-posed: the nonlinear evolution of any initial datum in L 2 (R d ) can be controlled a priori, an information which may be useful for numerics, since we do not have to decide if the initial datum belongs to a full measure set or not. This strategy is validated inasmuch as this procedure yields a good approximation of the solution to (1.1) as the cut-off tends to the identity. Note that this approach can be viewed as a deterministic counterpart of the one presented in [START_REF] Burq | Long time dynamics for the one dimensional non linear Schrödinger equation[END_REF] (see also [START_REF] Burq | Large-time dynamics for the one-dimensional Schrödinger equation[END_REF]). There, for the one-dimensional L 2 -supercritical defocusing nonlinear Schrödinger equation, the authors construct a Gibbs measure such that, among other features, the pathological phenomenon described in Theorem 1.1 occurs for a set of initial data whose measure is zero: on the support of the Gibbs measure, the Cauchy problem is globally well-posed, and a scattering theory is available. Both points of view aim at showing that norm inflation in the sense of Theorem 1.1 is a rare phenomenon: in [START_REF] Burq | Long time dynamics for the one dimensional non linear Schrödinger equation[END_REF], the authors give a rigorous meaning to this statement in an abstract way, while we are rather interested in a recipe to avoid instabilities for sure, by a suitable approximation of the equation, which can be used typically for numerical simulations.

Our choice of cutting off the high frequencies instead of, for instance, the values of the function itself is indeed motivated by numerics, where it is standard to filter out high frequencies (sometimes without even saying so). In an appendix, we discuss another approach, consisting in saturating the values of the nonlinearity. One could of course combine both approaches, frequency and physical saturations.

Notations. We define the Fourier transform by the formula

f (ξ) = F (f )(ξ) = 1 (2π) d/2 R d e -ix•ξ f (x)dx, f ∈ S(R d ).
We write a b if there exists C such that a Cb. In the presence of a small parameter h, the notation indicates that C is independent of h ∈ (0, 1].

FROM INSTABILITY TO GLOBAL WELL-POSEDNESS

Let χ : R d → [0, 1] be a smooth function, equal to one on the unit ball, and even: χ(-x) = χ(x) for all x ∈ R d . It may be compactly supported, in the Schwartz class S(R d ), or with a slower decay at infinity. For simplicity, we will not discuss sharp assumptions on χ. We define the frequency "cut-off" Π as the Fourier multiplier

Π(f )(ξ) = χ(ξ) f (ξ).
As pointed out in the introduction, in the examples constructed to prove the lack of local well-posedness, the mechanism of high frequencies amplification occurs at the level of the ordinary differential equation. We discuss some strategies to saturate high frequencies at the ODE level first, with ǫ = 1 for simplicity.

2.1.

Candidates at the ODE level. The first possibility to prevent the appearance of high frequencies by nonlinear self-interaction consists in saturating the whole nonlinearity:

(2.1)

i∂ t v = Π |v| 2σ v .
This can be viewed as an extremely simplified version of the I-method (see e.g. [START_REF] Colliander | Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation[END_REF]). Another choice consists in saturating the high frequencies in the "nonlinear multiplicative potential" only, that is |v| 2σ : for σ ∈ N, we propose two possibilities,

i∂ t v = Π |v| 2σ v, (2.2)
i∂ t v = Π |v| 2 σ v. (2.3)
In the cubic case σ = 1, the last two approaches obviously coincide. These two approaches have two advantages over (2.1):

• They preserve the gauge invariance. If v solves the equation, then so does ve iθ for any constant θ ∈ R. • They preserve the conservation of mass.

To see the second point, rewrite Π(f ) = K * f , with K(x) = (2π) -d/2 χ(-x). Since χ is even and real-valued, so is K, and therefore ∂ t |v| 2 = 0 in (2.2) and (2.3). This identity leads to the conservation of the L 2 -norm at the PDE level.

Before passing to the PDE case, we conclude this section by showing that even at the ODE level, cutting off high frequencies in the initial data does not suffice to prevent the appearance of higher frequencies in the solution for positive time. For a ∈ S(R d ) and s > 0, consider v h the solution to

i∂ t v h = |v h | 2σ v h ; v h (0, x) = h s-d/2 a x h . Then v h |t=0 is bounded in H s (R d ), uniformly in h ∈ (0, 1], and if a is compactly supported (in B(0, R)), then v h |t=0 is compactly supported (in B(0, R/h)). Since ∂ t |v h | 2 = 0, we have the explicit formula v h (t, x) = h s-d/2 a x h exp -ith 2σ(s-d/2) a x h 2σ .
We check that for t > 0, as h → 0, the homogeneous Sobolev norms behave like

v h (t) Ḣk ≈ h s-2kσ(s-d/2)-k t k , at least for k ∈ N. The above quantity is unbounded as h → 0 if k > s 1 + 2σ(s -d/2) . Therefore, if s < d/2, v h (t, •) is unbounded in H s (R d )
for t > 0, as h → 0: cutting off the high frequencies in the initial data does not suffice to control the frequency support of the solution. On the other hand, the models (2.2) and (2.3) prevent the appearance of high frequencies by nonlinear self-interaction. The above mechanism is essentially the one that leads to the norm inflation phenomenon in [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF][START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF][START_REF] Lebeau | Non linear optic and supercritical wave equation[END_REF], except that in those papers, the approximation by an ODE is used only on a time interval where the H s -norm becomes unbounded, but not the H k -norm for any k < s. The above mechanism at the PDE level leads to the loss of regularity [START_REF] Carles | Loss of regularity for super-critical nonlinear Schrödinger equations[END_REF][START_REF] Carles | Geometric optics and instability for semi-classical Schrödinger equations[END_REF][START_REF]Perte de régularité pour les équations d'ondes sur-critiques[END_REF][START_REF] Thomann | Instabilities for supercritical Schrödinger equations in analytic manifolds[END_REF], where indeed k is allowed to be smaller than s, as recalled in the introduction. Roughly speaking, the appearance of oscillations is quite similar to the above ODE example; in the PDE case, the numerology is different, and the proof is more intricate.

2.2.

Choice at the PDE level. We consider now the equations (2.4)

i∂ t u + P (D)u = ǫΠ |u| 2σ u,

and

(2.5)

i∂ t u + P (D)u = ǫ Π |u| 2 σ u,
where P (D) is a Fourier multiplier with a real-valued symbol P : R d → R,

P (D)f = P (ξ) f (ξ).
The L 2 -norm of u is formally independent of time:

(2.6) d dt R d |u(t, x)| 2 dx = 0.
In view of this conservation and of Young inequality

(2.7) Π(f ) L ∞ K L ∞ f L 1 ,
the option (2.5) seems more interesting than (2.4), and we have the following result.

Theorem 2.1. Let σ ∈ N, ǫ ∈ {±1}, P : R d → R and χ ∈ S(R d ) even and real-valued.

• For any

u 0 ∈ L 2 (R d ), (2.5) has a unique solution u ∈ C(R; L 2 (R d )) such that u |t=0 = u 0 . Its L 2 -norm is independent of time: (2.6) holds. • If in addition u 0 ∈ H s (R d ), s ∈ N, then u ∈ C(R; H s (R d )). • The flow map u 0 → u is uniformly continuous from the balls in L 2 (R d ) to C(R; L 2 (R d )). More precisely, for all u 0 , v 0 ∈ L 2 (R d ), there exists C depending on σ, K L ∞ , u 0 L 2 and v 0 L 2 such that for all T > 0, (2.8) u -v L ∞ ([-T,T ];L 2 (R d )) u 0 -v 0 L 2 (R d ) e CT ,
where u and v denote the solutions to (2.5) with initial data u 0 and v 0 , respectively. • More generally, let s ∈ N. For all u 0 , v 0 ∈ H s (R d ), there exists C depending on σ, K W s,∞ , u 0 H s and v 0 H s such that for all T > 0,

(2.9) u -v L ∞ ([-T,T ];H s (R d )) u 0 -v 0 H s (R d ) e CT .
Remark 2.2. As pointed out in [START_REF] Cazenave | Continuous dependence for NLS in fractional order spaces[END_REF], even if the solution is constructed by a fixed point argument, the continuity of the flow map is not trivial in general. In the case of Schrödinger equations (1.1), continuity of the flow map in H s (R d ) is known only in a limited number of cases: see [START_REF] Tsutsumi | L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups[END_REF] for s = 0, [START_REF] Kato | On nonlinear Schrödinger equations[END_REF] for s = 1 and s = 2, and [START_REF] Cazenave | Continuous dependence for NLS in fractional order spaces[END_REF] for 0 < s < 1.

Proof. First, recall that S(t) = e -itP (D) is a unitary group on Ḣs (R d ), s ∈ R. Duhamel's formula associated to (2.5) reads

(2.10)

u(t) = S(t)u 0 -iǫ t 0 S(t -τ ) K * |u| 2 σ u (τ )dτ.
The local existence in L 2 stems from a standard fixed point argument in

X(T ) = {u ∈ C([-T, T ]; L 2 (R d )); u L ∞ ([-T,T ];L 2 ) 2 u 0 L 2 }.
Denote by Φ(u)(t) the right hand side of (2.10). In view of (2.7), for t ∈ [-T, T ],

Φ(u)(t) L 2 u 0 L 2 + T -T K * |u| 2 σ u (τ ) L 2 dτ u 0 L 2 + T -T K * |u(τ )| 2 σ L ∞ u(τ ) L 2 dτ u 0 L 2 + K σ L ∞ T -T u(τ ) 2σ+1 L 2 dτ.
By choosing T > 0 sufficiently small, we see that X(T ) is stable under the action of Φ.

Note that in the case of the model (2.4), the above estimate would have to be adapted, forcing us to work in a space smaller than X(T ) (L 2 regularity in space would no longer be sufficient in general). Contraction is established in the same way:

Φ(u)(t) -Φ(v)(t) L 2 T -T K * |u| 2 σ u (τ ) -K * |v| 2 σ v (τ ) L 2 dτ T -T K * |u| 2 σ -K * |v| 2 σ u L 2 dτ + T -T K * |v| 2 σ (u -v) L 2 dτ.
Using the estimate |a σ -b σ | (|a| σ-1 + |b| σ-1 )|a -b|, and (2.7) again, we infer

Φ(u)(t) -Φ(v)(t) L 2 K σ L ∞ T -T u 2σ-1 L 2 + v 2σ-1 L 2 u -v L 2 u L 2 dτ + K σ L ∞ T -T v 2σ L 2 u -v L 2 dτ,
where all the functions inside the integrals are implicitly evaluated at time τ . Choosing T > 0 possibly smaller, Φ is a contraction on X(T ). Note that this small time T depends only on σ, K L ∞ and u 0 L 2 . Since the L 2 -norm of u is preserved (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] for the rigorous justification), the construction of a local solution can be repeated indefinitely, hence global existence and uniqueness at the L 2 level.

Global existence in H s (R d ) for s ∈ N then follows easily, thanks to the estimate

K * |u| 2 σ u H s |α|+|β|=s ∂ α K * |u| 2 σ ∂ β u L 2 K σ W s,∞ u σ L 2 u H s .
The continuity of the flow map in L 2 is obtained by resuming the estimate written to establish the contraction of Φ: for t > 0,

u(t) -v(t) L 2 u 0 -v 0 L 2 + K σ L ∞ t 0 u 2σ L 2 + v 2σ L 2 u -v L 2 dτ u 0 -v 0 L 2 + K σ L ∞ u 0 2σ L 2 + v 0 2σ L 2 t 0 u -v L 2 dτ,
where we have used the conservation of the L 2 -norm. Proceeding similarly for t < 0, Gronwall lemma then yields (2.8) for C depending only of σ, K L ∞ , u 0 L 2 and v 0 L 2 . Finally, (2.9) is obtained in a similar fashion.

Remark 2.3. The proof of continuity of the flow map is extremely easy. This is in sharp constrast with the case of the equation without frequency cut-off. In the case of Schrödinger equations (P (ξ) = -|ξ| 2 ), continuity is more intricate to establish (see [START_REF] Tsutsumi | L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups[END_REF]), and is true only for L 2 -subcritical nonlinearities, σ 2/d, from [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF].

We note that even for large σ, global well-posedness in L 2 is available, in sharp contrast with the nonlinear Schrödinger equation (1.1). Even in the focusing case ǫ = -1, the high frequency cut-off prevents finite time blow-up. In (2.9), consider v 0 = v = 0 and s = 1 for instance: by comparison with the case of (1.1), we see that the constant C necessarily depends on K (or equivalently on χ), and is unbounded as χ converges to the Dirac mass. The frequency cut-off Π removes the instabilities, and prevents finite time blow-up.

Remark 2.4 (Hamiltonian structure in the cubic case). If σ = 1, (2.4) and (2.5) coincide. We have the equivalence χ even and real-valued ⇐⇒ K even and real-valued. This implies that under the assumption of Theorem 2.1, (2.5) has an Hamiltonian structure, and the conserved energy is

H(u) = R d u(x)P (D)u(x)dx + ǫ 2 K(x -y)|u(y)| 2 |u(x)| 2 dxdy.

CONVERGENCE IN THE SMOOTH CASE

Suppose that P (D) converges to ∆ and that Π converges to Id: does the solution to (2.5) converge to the solution of NLS? We show that this is the case under suitable assumptions on these convergences, at least in the case where the solution to the limiting equation (1.1) is very smooth. In the sequel, the convergence is indexed by h ∈ (0, 1]. Proposition 3.1. Let σ ∈ N. We assume that P and Π verify the following properties:

• There exist α, β 0 such that P h (ξ) = -|ξ| 2 + O h α ξ β . • χ h (ξ) = χ (hξ), with χ ∈ S(R d ; [0, 1]
) even, real-valued, χ = 1 on the unit ball.

Denote by u h the solution to (2.5) with P h and χ h , such that u h |t=0 = u |t=0 . Suppose that the solution to (1.1)

satisfies u ∈ L ∞ ([0, T ]; H s+β ), for some s > d/2. Then u -u h L ∞ ([0,T ];H s ) h min(α,β) .
Example 3.2. The above assumption on P h is satisfied with α = 1 and β = 2 in the following cases:

• P h (ξ) = -|ξ| 2 1 + h|ξ| 2 . • P h (ξ) = - 1 h arctan h|ξ| 2 .
The second example is borrowed from [START_REF] Debussche | Modified energy for split-step methods applied to the linear Schrödinger equation[END_REF], where this truncated operator appears naturally when discretizing the Laplacian for numerical schemes.

Remark 3.3. In this result, no assumption is needed on the possible decay of χ at infinity.

Proof. Let w h = u -u h : it satisfies w h |t=0 = 0 and i∂ t w h + P h (D)w h = ǫ Π h |u| 2 σ u -ǫ Π h |u h | 2 σ u h + (P h (D) -∆) u + ǫ |u| 2σ -Π h |u| 2 σ u,
where we have denoted by Π h the Fourier multiplier of symbol χ h . Denote by R h (u) the second line, which corresponds to a source term. In view of the assumption on P h , there exists C independent of h ∈ (0, 1] such that

P h (D)f -∆f H s Ch α f H s+β ∀f ∈ H s+β (R d ).
We also have, by Plancherel formula,

(1 -Π h ) f 2 H s = R d (1 -χ(hξ)) 2 ξ 2s | f (ξ)| 2 dξ |ξ|>1/h ξ 2s | f (ξ)| 2 dξ h 2β |ξ|>1/h ξ 2s+2β | f (ξ)| 2 dξ h 2β f 2 H s+β . Therefore, R h (u) L ∞ ([0,T ];H s ) h min(α,β) u L ∞ ([0,T ];H s+β ) . Now since s > d/2, H s (R d
) is an algebra, and there exists C independent of h such that

Π h |u| 2 σ u -Π h |u h | 2 σ u h H s C χ σ L 1 u 2σ H s + u h 2σ H s u -u h H s ,
where the Young inequality that we have used is not the same as in Section 2:

K * f L 2 K L 1 f L 2 .
This is essentially the only way to obtain an estimate independent of h ∈ (0, 1]. Indeed,

Π h (f ) = K h * f , with K h (x) = 1 (2π) d/2 h d χ -x h .
The result then stems from a bootstrap argument: so long as

u h L ∞ ([0,t];H s ) 1 + u L ∞ ([0,T ];H s ) , Gronwall lemma yields u -u h L ∞ ([0,t];H s ) h min(α,β) u L ∞ ([0,T ];H s+β ) .
Therefore, up to choosing h sufficiently small, this estimate is valid up to t = T .

Such a convergence result can be compared to the one proved in [START_REF] Ignat | Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations[END_REF] to prove the convergence of numerical approximations. The approach there is a bit different though, inasmuch as the frequency cut-off does not affect the nonlinearity (as in (2.5)), but the initial data: consider v h solution to

i∂ t v h + P h (D)v h = ǫ|v h | 2σ v h ; v h |t=0 = Π h u 0 .
Then in [START_REF] Ignat | Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations[END_REF], the discrete analogue to Π h u -v h is proven to be small. Proposition 3.1 differs from the results in [START_REF] Ignat | Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations[END_REF] on several aspects:

• The context in [START_REF] Ignat | Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations[END_REF] is discrete.

• Only the low frequency part of u, Π h u, is shown to be well approximated.

• The regularity assumption on u may be much weaker. As mentioned above, the second point is due to the choice of the model. However, as discussed in Section 2.1, controlling the high frequencies of the initial data must not be expected to ensure a control of high frequencies of the solution v h for positive time.

The third point is due to the use of Strichartz estimates in [START_REF] Ignat | Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations[END_REF]. In the next section, we show that in the presence of dispersion (with P h (ξ) = -|ξ| 2 ), Proposition 3.1 can be adapted to rougher data.

CONVERGENCE USING DISPERSIVE ESTIMATES

We first recall the standard definition.

Definition 4.1. A pair (p, q) = (2, ∞) is admissible if p 2, q 2, and 2 p = d 1 2 - 1 q .
We shall consider (2.5) when P (D) is exactly the Laplacian, and not an approximation as in Proposition 3.1. The reason is that when P is bounded, then no Strichartz estimate is available, as we now recall. Let S(•) be bounded on H s for all s 0. By Sobolev embedding, for all (p, q) (not necessarily admissible) with 2 q < ∞, there exists C > 0 such that for all u 0 ∈ H d/2-d/q (R d ), and all finite time interval I,

S(•)u 0 L p (I;L q (R d )) C S(•)u 0 L p (I;H d/2-d/q (R d )) C u 0 L p (I;H d/2-d/q (R d )) = C|I| 1/p u 0 H d/2-d/q (R d ) .
If the Fourier multiplier P is bounded, the above estimate cannot be improved, in sharp contrast with the result provided by Strichartz estimates. Proposition 4.2 (From [START_REF]On Schrödinger equations with modified dispersion[END_REF]). Let d 1, and P ∈ L ∞ (R d ; R). Denote S(t) = e -itP (D) . Suppose that there exist an admissible pair (p, q), an index k ∈ R, a time interval I ∋ 0, |I| > 0, and a constant C > 0 such that

S(•)u 0 L p (I;L q (R d )) C u 0 H k (R d ) , ∀u 0 ∈ H k (R d ).
Then necessarily, k 2/p = d/2 -d/q.

We now state the main result of this section. Theorem 4.3. Let σ ∈ N and T > 0. We assume that χ h (ξ) = χ (hξ), with χ ∈ S(R d ) even, real-valued, χ = 1 on B(0, 1). Let u solve (1.1), and consider the solution u h to

i∂ t u h + ∆u h = ǫ Π h |u h | 2 σ u h ; u h |t=0 = u 0 . 1. Suppose that σ = 1 and d 2. If u ∈ L ∞ ([0, T ]; L 2 ) ∩ L 8/d ([0, T ]; L 4 ), then u -u h L ∞ ([0,T ];L 2 ) -→ h→0 0.
2. Suppose that σ = 1 and d = 3.

• If u, ∇u ∈ L ∞ ([0, T ]; L 2 ) ∩ L 8/d ([0, T ]; L 4 ), then u -u h L ∞ ([0,T ];H 1 ) -→ h→0 0. • If u ∈ L ∞ ([0, T ]; H s ), with s > 3/2, then u -u h L ∞ ([0,T ];L 2 ) h s and u -u h L ∞ ([0,T ];H 1 ) h s-1 . 3. Suppose that σ 1 and d 2. If u ∈ L ∞ ([0, T ]; H s ), with s 1 and s > d/2, then u -u h L ∞ ([0,T ];L 2 ) h s and u -u h L ∞ ([0,T ];H 1 ) -→ h→0 0. If in addition s > 1, then u -u h L ∞ ([0,T ];H 1 ) h s-1 .
Remark 4.4. Suppose u 0 sufficiently smooth. If ǫ = +1 (defocusing case), the bounds for u are known in several cases, with T > 0 arbitrarily large. On the contrary, if ǫ = -1 (focusing case), T may have to be finite, bounded by a blow-up time. See e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Ginibre | On the global Cauchy problem for some nonlinear Schrödinger equations[END_REF]. Typically, if σ = d = 1, then the assumption of the first point is fulfilled for all T > 0 as soon as u 0 ∈ L 2 (R), for ǫ ∈ {±1}, from [START_REF] Tsutsumi | L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups[END_REF], and if σ 1, d 2, the assumption of the third point is fulfilled for all T > 0 as soon as u 0 ∈ H s (R d ), for ǫ = +1, from [START_REF] Ginibre | On the global Cauchy problem for some nonlinear Schrödinger equations[END_REF].

Proof. For fixed h > 0, Theorem 2.1 shows that u h ∈ C(R; H k ), with k = 0, 1 or s according to the cases considered in the assumptions of the theorem. Of course, the bounds provided by Theorem 2.1 blow up as h → 0 if k > 0.

As in the proof of Proposition 3.1, let w h = u -u h . The equation satisfied by w h is simpler than in the proof of Proposition 3.1, since P h (D) = ∆:

i∂ t w h + ∆w h = ǫ Π h |u| 2 σ u -ǫ Π h |u h | 2 σ u h + ǫ |u| 2σ -Π h |u| 2 σ u. Resume the notations R h (u) = ǫ |u| 2σ -Π h |u| 2 σ u and Π h (f ) = K h * f , with K h (x) = (2π) -d/2 h -d χ(-x/h).
From young inequality, we have, for all q ∈ [1, ∞],

(4.1) Π h (f ) L q K h L 1 f L q χ L 1 f L q ,
an estimate which is uniform in h > 0. Introduce the Lebesgue exponents

q = 2σ + 2 ; p = 4σ + 4 dσ ; θ = 2σ(2σ + 2) 2 -(d -2)σ .
The pair (p, q) is admissible, and

(4.2) 1 q ′ = 2σ q + 1 q ; 1 p ′ = 2σ θ + 1 p . For t > 0, denote L j t L k = L j ([0, t]; L k (R d ))
. From Strichartz estimates (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]),

w h L p t L q ∩L ∞ t L 2 Π h |u| 2 σ u -Π h |u h | 2 σ u h L p ′ t L q ′ + R h (u) L p ′ 1 t L q ′ 1 u 2σ L θ t L q + u h 2σ L θ t L q w h L p t L q + R h (u) L p ′ 1 t L q ′ 1 ,
where we have used Hölder inequality and (4.1), and where (p 1 , q 1 ) is an admissible pair whose value will be given later. If σ = 1 and d 2, then θ p, and we infer

w h L p t L q ∩L ∞ t L 2 t 1/θ-1/p u 2σ L p t L q + u h 2σ L p t L q w h L p t L q + R h (u) L p ′ 1 t L q ′ 1 .
In the first case of the theorem, we assume u ∈ L p ([0, T ]; L q ), since p = 8/d and q = 4 for σ = 1. We use again a bootstrap argument: so long as u h L p t L q 2 u L p t L q , we divide the interval [0, T ] into finitely many small intervals so the first term of the right hand side is absorbed by the left hand side (recall that p is finite), and we have

w h L p t L q ∩L ∞ t L 2 R h (u) L p ′ 1 t L q ′ 1 .
The bootstrap argument is validated provided that R h (u)

L p ′ 1 T L q ′ 1 → 0 as h → 0.
If we have only σ < 2 d-2 , then by Sobolev embedding,

u L θ t L q t 1/θ u L ∞ t H 1 .
In the same way as above,

∇w h L p t L q ∩L ∞ t L 2 ∇ Π h |u| 2 σ u -Π h |u h | 2 σ u h L p ′ t L q ′ + ∇R h (u) L p ′ 1 t L q ′ 1
The first term of the right hand side is controlled by

(4.3) Π h |u| 2 σ ∇u -Π h |u h | 2 σ ∇u h L p ′ t L q ′ + u∇ Π h |u| 2 σ -u h ∇ Π h |u h | 2 σ L p ′ t L q ′
Introducing the factor Π h |u| 2 σ ∇u h , the first term is estimated by

Π h |u| 2 σ ∇w h L p ′ t L q ′ + Π h |u| 2 σ -Π h |u h | 2 σ ∇u h L p ′ t L q ′ Π h |u| 2 σ L θ/2 t L q/2 ∇w h L p t L q + u 2σ-2 L θ t L q + u h 2σ-2 L θ t L q |u| 2 -|u h | 2 L θ/2 t L q/2 ∇u h L p t L q u 2σ L θ t L q ∇w h L p t L q + u 2σ-1 L θ t L q + u h 2σ-1 L θ t L q w h L θ t L q ∇u h L p t L q t 2σ/θ u 2σ L ∞ t H 1 ∇w h L p t L q + t 2σ/θ u 2σ-1 L ∞ t H 1 + u h 2σ-1 L ∞ t H 1 w h L ∞ t H 1 ∇u h L p t L q
Proceeding similarly for the other term in (4.3), splitting [0, T ] into finitely many time intervals where the terms containing w h on the right hand side can be absorbed by the left hand side, and using a bootstrap argument, we end up with

w h L p t W 1,q ∩L ∞ t H 1 R h (u) L p ′ 1 t W 1,q ′ 1 .
Therefore, it suffices to show that for some admissible pair (p 1 , q 1 ), the source term converges to

0 in L p ′ 1 ([0, T ]; L q ′ 1 ) (if σ = 1 and d 2) or in L p ′ 1 ([0, T ]; W 1,q ′ 1
) (in the other cases), so the bootstrap argument is completed. In addition, the rate of converge of the source term, if any, yields a rate of convergence for w h . The theorem then stems from the following lemma, in which (p, q) is given by (4.2). Lemma 4.5. Let T > 0. The source term R h (u) can be controlled as follows. 1. Suppose that σ = 1 and d

2. If u ∈ L ∞ ([0, T ]; L 2 ) ∩ L 8/d ([0, T ]; L 4 ), then R h (u) L p ′ ([0,T ];L q ′ ) -→ h→0 0.
2. Suppose that σ = 1 and d = 3.

• If u, ∇u ∈ L ∞ ([0, T ]; L 2 ) ∩ L 8/d ([0, T ]; L 4 ), then R h (u) L p ′ ([0,T ];W 1,q ′ ) -→ h→0 0. • If u ∈ L ∞ ([0, T ]; H s ), with s > 3/2, then R h (u) L 1 ([0,T ];L 2 ) h s and R h (u) L 1 ([0,T ];H 1 ) h s-1 . 3. Suppose that σ 1 and d 2. If u ∈ L ∞ ([0, T ]; H s ), with s 1 and s > d/2, then R h (u) L 1 ([0,T ];L 2 ) h s and R h (u) L 1 ([0,T ];H 1 ) -→ h→0 0. If in addition s > 1, then R h (u) L 1 ([0,T ];H 1 ) h s-1 .
Proof of Lemma 4.5. For the first case, we use Hölder inequality, in view of (4.2):

R h (u) L p ′ T L q ′ = (1 -Π h ) |u| 2 u L p ′ T L q ′ (1 -Π h ) |u| 2 L θ/2
T L q/2 u L p T L q . We note that for σ = 1, q = 4, so by Plancherel Theorem,

(1 -Π h ) (|u| 2 ) 2 L 2 = R d (1 -χ(hξ)) 2 |F (|u| 2 )(ξ)| 2 dξ |ξ|>1/h |F (|u| 2 )(ξ)| 2 dξ. By assumption, u ∈ L p ([0, T ]; L 4 ) ⊂ L θ ([0, T ]; L 4 ), thus |u| 2 ∈ L θ/2 ([0, T ]; L 2 ), and by Plancherel Theorem, F (|u| 2 ) ∈ L θ/2 ([0, T ]; L 2 ).
The first point of the lemma then stems from the Dominated Convergence Theorem.

For the first case of the second point, we note that now θ > p, so the above argument must be adapted, and we have to estimate the gradient of R h (u) in the same space as above. Since L ∞ ([0, T ];

H 1 (R 3 )) ⊂ L θ ([0, T ]; L 4 (R 3 )), the Dominated Convergence Theorem yields R h (u) L p ′ T L q ′ -→ h→0 0.
We now estimate ∇R h (u). Write

∇R h (u) L p ′ T L q ′ (1 -Π h ) |u| 2 L θ/2 T L 2 ∇u L p T L 2 + (1 -Π h )∇ |u| 2 L (1/θ+1/p) -1 T L 2 u L θ T L 2 (1 -Π h ) |u| 2 L ∞ T L 2 ∇u L p T L 2 + (1 -Π h )∇ |u| 2 L (1/θ+1/p) -1 T L 2 u L ∞ T L 2
By the same argument as above,

(1 -Π h ) |u| 2 L ∞ T L 2 ∇u L p T L 2 -→ h→0 0.
We note that u bounded in L ∞ ([0, T ]; H 1 (R 3 )) ⊂ L θ ([0, T ]; L 4 (R 3 )), and ∇u bounded in L p T L 4 , so ∇|u| 2 is bounded in L

(1/θ+1/p) -1 T L 2 . Invoking Plancherel Theorem and the Dominated Convergence Theorem like above, we infer

(1 -Π h )∇ |u| 2 L (1/θ+1/p) -1 T L 2 u L ∞ T L 2 -→ h→0 0.
This completes the proof for the first case of the second point.

For the remaining cases, we use that

H s (R d ) is embedded into L ∞ (R d ): for fixed t, R h (u)(t) L 2 u(t) 2σ-2 L ∞ + Π h (|u(t)| 2 ) σ-1 L ∞ (1 -Π h )(|u(t)| 2 ) L 2 u(t) L ∞ u(t) 2σ-1 L ∞ (1 -Π h )(|u(t)| 2 ) L 2 u(t) 2σ-1 H s (1 -Π h )(|u(t)| 2 ) L 2 .
Like in the proof of Proposition 3.1, we use the estimate

(4.4) (1 -Π h ) f L 2 h s f H s ,
and since H s (R d ) is an algebra, R h (u) L ∞ ([0,T ];L 2 ) h s u 2σ+1 L ∞ ([0,T ];H s ) .

To conclude the proof, we estimate ∇R h (u) in L 2 (R d ). We compute

∇R h (u) = σ|u| 2σ-2 (1 -Π h ) ∇ |u| 2 u + σ |u| 2σ-2 -Π h (|u| 2 ) σ-1 Π h ∇ |u| 2 u + |u| 2σ -Π h |u| 2 σ ∇u,
where the second line is zero is σ = 1. We estimate successively, thaks to (4.1),

|u| 2σ-2 (1 -Π h ) ∇ |u| 2 u L 2 u 2σ-1 L ∞ (1 -Π h ) |u| 2 H 1 , |u| 2σ -Π h |u| 2 σ ∇u L 2 u 2σ-2 L ∞ (1 -Π h ) |u| 2 L ∞ ∇u L 2 ,
and, if σ 2,

|u| 2σ-2 -Π h (|u| 2 ) σ-1 Π h ∇ |u| 2 u L 2 u 2σ-4 L ∞ (1 -Π h ) |u| 2 L 2 ∇ |u| 2 L 2 u L ∞ u 2σ-2 L ∞ (1 -Π h ) |u| 2 L 2 ∇u L 2 .
Since we have H s (R d ) ֒→ L ∞ (R d ), we end up with

∇R h (u) L 2 u 2σ-2 H s (1 -Π h ) |u| 2 H 1 .
If s > 1, (4.4) yields, since in addition s > d/2,

(1 -Π h ) |u| 2 H 1 h s-1 |u| 2 H s h s-1 u 2 H s .
If s = 1 (a case which may occur only if d = 1, since s > d/2), we write This completes the proof of Theorem 4.3, by choosing (p 1 , q 1 ) = (p, q) or (∞, 2).

∇ (1 -Π h ) |u| 2 2

  Now since ∇ |u| 2 = 2 Re ū∇u and u ∈ H 1 (R) ֒→ L ∞ (R), ∇u ∈ L 2 (R), we conclude thanks to the Dominated Convergence Theorem.

CNRS & UNIV. MONTPELLIER 2, UMR5149, MATH ÉMATIQUES, CC051, 34095 MONTPELLIER, FRANCE E-mail address: Remi.Carles@math.cnrs.fr

This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01).

APPENDIX A. PHYSICAL SATURATION OF THE NONLINEARITY

Instead of cutting off the high frequencies, one may be tempted by saturating the nonlinear potential, by replacing |u| 2 not by Π(|u| 2 ) but by f (|u| 2 ) where f is smooth, equal to the identity near the origin, and constant at infinity. Note also that a saturated nonlinearity may be in better agreement with physical models (recall however that (1.1) appears in rather different physical contexts, such as quantum mechanics, optics, and even fluid mechanics), since typically the power-like nonlinearity in (1.1) may stem from a Taylor expansion; see e.g. [START_REF] Lannes | High-frequency nonlinear optics: from the nonlinear Schrödinger approximation to ultrashortpulses equations[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation, Self-focusing and wave collapse[END_REF]. More precisely, let

The analogue of the Fourier multiplier Π h is defined as

and we replace (2.5) with

so the formal conservation of the L 2 -norm still holds. We could also consider

In both cases, the main aspect to notice is that f h is bounded and z → f h (|z| 2 ) σ z is globally Lipschitzean. We infer the analogue of Theorem 2.1, at least in the L 2 case. • For any

. More precisely, for all u 0 , v 0 ∈ L 2 (R d ), there exists C depending on σ, h, u 0 L 2 and v 0 L 2 such that for all T > 0,

where u h and v h denote the solutions to (A.2) with data u 0 and v 0 , respectively.

We check that the following conservation of energy holds:

Proving the analogue of Proposition 3.1 is easy in the case (A.1), since the last source term for the error w h is now

and under the assumptions of Proposition 3.1, u ∈ L ∞ ([0, T ]×R d ), so there exists h 0 > 0 such that for 0 < h h 0 ,

Therefore, this source term simply vanishes for h sufficiently small. In the case (A.3), we can use the relation

and Schauder lemma to get a source term which is O(h) in H s (R d ), for s > d/2. Proposition A.2. Let σ ∈ N. We assume that P is such that there exist α, β 0 with

Denote by u h the solution to (A.2) with P h and f h , such

In the case (A.1), proving an analogue to Theorem 4.3 seems to be more delicate though, and we choose not to investigate this aspect here. On the other hand, in the case (A.3), using the estimate (A.4), Strichartz estimates and Hölder inequalities with the "standard" Lebesgue exponents (in the same fashion as in the proof of Theorem 4.3, see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]), we have, with steps similar to those presented in the proof of Theorem 4.3:

Theorem A.3. Let σ ∈ N and T > 0. Let u solve (1.1), and consider u h solution to h.


Xuan Hung Nguyen
email: xhnguyen@laas.fr

Guy Juanole
email: juanole@laas.fr

Design of Networked Control Systems (NCSs) on the basis of interplays between Quality of Control and Quality of Service

The goal of this paper is to consider a co-design approach between the controller of a process control application and the frame scheduling of a Local Area Network (LAN). More precisely we present a bi-directional relation between the Quality of Control (QoC) provided by the controller and the Quality of Service (QoS) provided by the LAN (relation noted QoS⇋QoC). We present, first, the implementation of the relation QoS→QoC on the basis of a compensation method for time delays called dominant pole method, and second, the implementation of the relation QoC→QoS on the basis of the hybrid priorities for the frame scheduling. The final objective is the implementation of the bidirectional relation QoS⇋QoC which is the combination of the both relations QoS→QoC and QoC→QoS in order to have a more efficient NCSs design.

I. INTRODUCTION

The study and design of Networked Control Systems (NCSs) is a very important research area today because of its multidisciplinary aspect (Automatic Control, Computer Science, Communication Network). The current objective of NCS design today is to consider a co-design in order to have an efficient control system [START_REF] Branicky | Networked Control Systems Co-Simulation for Co-Design[END_REF], [START_REF] Song | Networked Control Systems: From Independent Designs of the Network QoS and the Control to the Co-design[END_REF].

Several works [START_REF] Marti | Analysis and design of real-time control systems with flexible timing constraints[END_REF], [START_REF] Branicky | Scheduling and Feedback Co-Design for Networked Control Systems[END_REF], [START_REF] Cervin | Integrated control and real-time scheduling[END_REF] have considered the aspects Automatic Control and Computer Science (Task scheduling) and have shown the interest of a co-design approach.

Here we consider the aspects Automatic Control and Local Area Network (LAN) by focusing on the frame scheduling implemented in the MAC layer (the messages of the controlcommand applications are encapsulated in frames). The aim is to show the interest of a co-design approach, of the frame scheduling in the LAN and of the controller of a process control application, on the basis of a bidirectional relation between the Quality of Control (QoC) provided by the controller and the Quality of Service (QoS) provided by the LAN (QoC⇋QoS), i.e. we have both QoC→QoS (QoS is QoC driven i.e. Application performance aware dynamic QoS adaptation) and QoS→QoC (QoC is QoS driven i.e. Network performance aware dynamic adaptation). Some works [START_REF] Velasco | A CAN Application Profile for Control Optimization in Networked Embedded Systems[END_REF], [START_REF] Juanole | Networked Control Systems: Definition and Analysis of a Hybrid Priority Scheme for the Message Scheduling[END_REF], [START_REF] Nguyen Xuan | Networked Control System (NCS) on a network CAN: on the Quality of Service (QoS) and Quality of Control (QoC) provided by different message scheduling schemes based on hybrid priorities[END_REF] have shown the interest of the relation QoC→QoS . In particular, in our works [START_REF] Juanole | Networked Control Systems: Definition and Analysis of a Hybrid Priority Scheme for the Message Scheduling[END_REF], [START_REF] Nguyen Xuan | Networked Control System (NCS) on a network CAN: on the Quality of Service (QoS) and Quality of Control (QoC) provided by different message scheduling schemes based on hybrid priorities[END_REF], we have shown the interest of the implementation of this relation on the basis of the concept of hybrid priority for the frame scheduling in the MAC layer.

In [START_REF] Nguyen Xuan | Networked Control Systems Quality of Service aware Quality of Control adaptation[END_REF], we have shown the interest of the relation QoS→QoC where the controller is aware on-line of the time delay in the control loop and adapts, in consequence, its parameters.

In this paper, we want to show the interest to associate the relation QoC→QoS with a relation QoS→QoC. We consider several simple continuous linear process control applications (DC motor) implemented on a CAN network [START_REF] Gmbh | Can specification 2[END_REF] with a Proportional-Derivative (PD) controller.

This paper includes the following sections: the section 2 presents the general context of the study; the section 3 presents, by summarizing results obtained in [START_REF] Nguyen Xuan | Networked Control Systems Quality of Service aware Quality of Control adaptation[END_REF], the implementation of the relation QoS→QoC; the section 4 presents the main ideas [START_REF] Juanole | Networked Control Systems: Definition and Analysis of a Hybrid Priority Scheme for the Message Scheduling[END_REF] and the main results of the implementation of the relation QoC→QoS; the section 5 presents the implementation of the bidirectional relation QoS⇋QoC and the recapitulation of all the obtained results; the conclusion is represented in the section 6. The considered process control application is a continuous linear application, the model of which is given on Fig. 1. The process to control has the transfer function G(s) = 1000 s(s+1) . We have a Proportional Derivative (PD) controller in order to have a phase margin of 45 • which imposes the following values: K = 0.7291; T d = 0.0297 s. The input reference is a unity position step R(s) = 1/s. The output is Y (s).

II. CONTEXT OF THE STUDY

A. Process control application which is considered

Y(s) G(s) K(1+sT d ) R(s)
The transfer function F(s) of the closed loop system is

F(s) = K(1 + T d s)G(s) 1 + K(1 + T d s)G(s) (1)
F(s) = 1000K(1 + T d s) s 2 + (1 + 1000KT d )s + 1000K (2) = ω 2 n (1 + T d s) s 2 + 2ζω n s + ω 2 n (3)
where ω n is the natural pulsation and ζ is the damping coefficient (ω n = 27 rad/s; ζ = 0.4); the two poles p 1,2 are -ζω n ± jω n 1 -ζ 2 , i.e. p 1,2 = -11 ± j24.5.

The performances are: settling time (at 2%) t s = 284 ms; overshoot O = 33.8%. The time response is represented in Fig. 2.

B. On the implementation of a process control application through a network

The implementation through a network which requires the sampling of the output y is represented on Fig. 3. We call h the sampling period which is defined by considering the following formula ω n h ∈ [0.1; 0.6] [START_REF] Åström | Computer-controlled systems: Theory and design[END_REF].

N E T W O R K Process to control h y(t) u(t)
We have three computers: the computer C1 where we have the sensor task which receives the numerical information provided by the Analog Digital (AD) Converter after the sampler; the computer C2 where we have the controller task which works out the control signal u from the difference between the reference r and the output y; the computer C3 where we have the actuator task which provides the control signal u in a numerical way to the Digital Analog (AD) converter; the DA converter then transmits it in an analog way to the Zero Order Hold (ZOH).

We have two flows of frames: the Sensor-Controller flow which concerns the frames going from the sensor to the controller (noted f sc flow and call " f sc frame" a frame of this flow); the Controller-Actuator flow which concerns the frames going from the controller to the actuator (noted f ca flow and call " f ca frame" a frame of this flow).

The sensor task is Time-Triggered (i.e. controlled by the sampler clock). The controller task is Event-Triggered (the events are the receptions of the f sc frames). The actuator task is also Event-Triggered (the events are the receptions of the f ca frames).

The running of the process control application is characterized by several time delays:

• The computational and functional time delays in the execution of the tasks in the computers C1, C2, C3.

• The communication network delays in the communications of the f sc frame (denoted τ sc ) and the f ca frame (denoted τ ca ). During each sampling period, the time delay τ sc is the time difference between the sampling instant and the reception instant of the f sc frame by the controller; the time delay τ ca is the time elapsed from the sending instant of the f ca frame by the controller till the reception instant of this frame by the actuator.

• The time delay induced by the block ZOH can be seen as a communication time delay τ ZOH ≈ h 2 [START_REF] Franklin | Feedback Control of Dynamic Systems[END_REF]. In this work, we only consider the time delays τ sc , τ ca and τ ZOH . The computational and functional delays are neglected. The time delay of the closed loop in each sampling period is

τ = τ sc + τ ca + τ ZOH .
The model of the implementation of a process control application on a network can be represented by the continuous model given on Fig. 4 where a time delay tau is represented as e -tau.s . The transfer function F(s) is now:

Y(s) G(s) R(s) ( ) ca ZOH s e τ τ - + K(1+Tds)
F(s) = K(1 + T d s)e -(τ ca +τ ZOH )s G(s) 1 + K(1 + T d s)e -τs G(s) (4)
The ) . By calling a = 2/τ and b = 2/(τ ca + τ ZOH ), we get finally the transfer function as follows:

F(s) = 1000Ka(1 + T d s)(1 + s/a)(1 -s/b) f 3 (s)(1 + s/b) ( 5
)
where

f 3 (s) = s 3 + (1 + a -1000KT d )s 2 + (1000KT d a + a - 1000K)s + 1000Ka.
From Equation 5, we have 4 poles (3 poles p 1 , p 2 , p 3 of the polynomial f 3 (s), p 4 = -2/(τ ca + τ ZOH ) and 3 zeros

(z 1 = -1/T d , z 2 = -2/τ, z 3 = 2/(τ ca + τ ZOH )).

C. Global control system which is considered

We now present the system which will be analyzed in this work by means of the simulator TrueTime [START_REF] Ohlin | TrueTime 1.5 -Reference Manual[END_REF], a toolbox based on Matlab/Simulink which allows to simulate real-time distributed control systems.

1) General considerations:

We consider the implementation of 4 process control applications (noted P 1 , P 2 , P 3 , P 4 ) like the one specified in the subsection II-A through a CAN network. The controller tasks, the sensor tasks and the actuator tasks of the 4 process control applications are all in different computers. Then we have 12 computers connected to the network and we have 4 f sc flows and 4 f ca flows which share the network.

The sampling period is h = 10 ms for the 4 applications and we suppose also that the sampling instants are the same for the 4 applications (synchronous samplings).

The CAN network has a throughput in the physical layer of 125 Kbits/s and the lengths of the data field of the f sc frames (sensor data) and of the f ca frames (command data) are two bytes (16 bits) which give frames of 75 bits [START_REF] Navet | Evaluation de performances temporelles et optimisation de l'ordonnancement de tâches et messages[END_REF]. But as we used the old version of TrueTime (which considers frames with lengths multiples of a byte), we have taken frames of 80 bits (i.e. 10 bytes). Then the durations of the f sc frames (noted D sc ) and the f ca frames (noted D ca ) are identical and equal to 0.64 ms.

The scheduling of the frames is done in the MAC layer by means of priorities which are represented by the Identifier (ID) field of the frame. The ID field is at the beginning of the frames.

Generally, the priorities are static priorities, i.e. each flow has an unique priority (specified a priori out of line) and all the frames of this flow have the same priority.

Concerning the priorities, we will consider here either static priorities (i.e. each flow has an unique priority specified out of line) or hybrid priorities (as we said in the introduction) i.e. with two priority levels. One level represents the flow priority which is a static priority. The other level represents the frame transmission urgency. The urgency can be the same for all the frames of the flow and, in this case, the transmission urgency is also a static priority. The urgency can vary (for example, if the conditions of the application, which uses the flow, change) and, in this case, the transmission urgency is a dynamic priority. This concept has a great interest during the transient behavior of systems [START_REF] Zuberi | Scheduling messages on controller area network for real-time CIM applications[END_REF], [START_REF] Zuberi | Design and Implementation of Efficient Message Scheduling for Controller Area Network[END_REF], [START_REF] Walsh | Scheduling of Networked Control Systems[END_REF], [START_REF] Yepez | The large error first (lef) scheduling policy for real-time control systems[END_REF].

The consideration of hybrid priorities requires to structure the field ID in two levels (Fig. 5) where the level 1 represents the flow priority and the level 2 represents the urgency priority. In the context of the competition based on these hybrid priorities, the frame scheduling is executed by comparing first the bits of the level 2 (urgency predominance). If the urgencies are identical, the level 1 (static priorities which have the uniqueness properties) resolves the competition.

2) Static priorities associated to the f sc and f ca flows: Call P sc and P ca the priorities of respectively the f sc flow and the f ca flow of the process control application noted P. It has been shown in a previous work [START_REF] Juanole | Fundamental Considerations for Implementing Control Systems on a CAN Network[END_REF] that we must have P ca > P sc . In the context of a network shared by several applications, this condition allows the controller to act faster after the reception of a f sc frame. In this study, we consider this result.

Concerning the 4 process control applications P 1 , P 2 , P 3 , P 4 , for each process P i , the priorities of the f cai and f sci flows are P cai > P sci and the priorities of the flows of the 4 processes are arranged in the following order: [START_REF] Velasco | A CAN Application Profile for Control Optimization in Networked Embedded Systems[END_REF] i.e. the process P i is considered more important than the process P j with i < j.

P ca1 > P ca2 > P ca3 > P ca4 > P sc1 > P sc2 > P sc3 > P sc4
3) Criteria of the performance evaluation: The Quality of Control (QoC) is evaluated, in particular, with a cost function ITSE (Integral of Time-weighted Square Error) noted 2 dt with T > t s in order to cover the transient regime duration. We consider T = 500 ms and we get the value of J for the system in Section II-A (noted J 0 ) is J 0 = 9.4562.10 -4 . J 0 will be considered as the reference value for the study done along the paper.

J with J = T 0 t(r(t) -y(t))
When we consider the process control applications implemented on the network, the performance criteria is represented by the term J-J 0 J 0 % = ∆J J 0 %. The higher the value ∆J J 0 % is, the more degraded the QoC is.

We will also consider the overshoot (O%) and the time response (y(t)) in order to characterize QoC.

4) Control law:

The controller computes the control signal u k (which includes the Proportional component P k and the Derivate component D k ) by using the formula [START_REF] Juanole | Networked Control Systems: Definition and Analysis of a Hybrid Priority Scheme for the Message Scheduling[END_REF]. [START_REF] Juanole | Networked Control Systems: Definition and Analysis of a Hybrid Priority Scheme for the Message Scheduling[END_REF] where e k = r ky k , N is a constant ( [START_REF] Åström | Computer-controlled systems: Theory and design[END_REF], page 307).

   P k = Ke k D k = T d Nh+T d D k-1 + NKT d Nh+T d (e k -e k-1 ) u k = P k + D k

III. RELATION QOS→QOC

The implementation of the relation QoS→QoC is on the basis of a compensation method for time delays. Here we consider the dominant pole method. The strict study of this relation must be based on the use of static priorities (in this way QoS is not linked to QoC).

A. Main ideas

The QoS, which is considered here, is the time delay in the loop (τ = τ sc + τ ca + τ ZOH ) during each sampling period. With the knowledge of τ, the controller can compensate this time delay by modifying the parameters K and T d in such a way to maintain the same type of transient behavior for the process control application as before the implementation on the network (i.e. characterized by the two poles of the transfer function of Equation 3). As the transfer function of the system implemented on the network (Equation 5) has 4 poles (p 1 , p 2 , p 3 , p 4 ), the modification of K and T d , according to the dominant pole method, must keep the main role for the 2 poles of Equation 3 (i.e. poles p 1,2 = R± jI (with R = -11 and I = 24.5) which are called the dominant poles) and integrate the conditions which give an insignificant role to the poles p 3 , p 4 (called insignificant poles). In order to be insignificant, the poles p 3 and p 4 must have their real part very smaller than that of the dominant poles.

Note that, if the dominant pole method maintains the transient behavior type, however it cannot maintain the overshoot value.

Actually, the numerator of Equation 5 has now three zeros:

z 1 = -1/T d , z 2 = -2/τ, z 3 = 2/(τ ca + τ ZOH ).
Thus we have to evaluate the influence of these zeros on the overshoot.

The dominant pole method is characterized by two important points: The obtaining of the loop delay by the controller; The computations in the controller for the maintenance of the dominant poles.

1) The obtaining of the loop delay by the controller: We consider the following hypotheses:

• The computer, where the sensor task is, knows the static priority of the f ca flow.

• The CAN network is not overloaded i.e. each process control application can send its two frames ( f sc and f ca frames) during each sampling period. Then the value (τ ca + τ ZOH ) is the same for each process control application in each sampling period. Considering the priorities which have been specified in the formula 6, we have (τ sc + τ ca ) of P i which is smaller than (τ sc + τ ca ) of P j with i < j. Each sensor task can evaluate the time delay (τ sc + τ ca ) of the concerned process control application (by noting the time difference between the sampling instant t k = kh (with k = 0, 1, 2...) where a f sc frame must be sent and the instant of the reception of the corresponding f ca frame). Then, at the next sampling instant t k+1 , the sensor task will put in the data field of the f sc frame this value (τ sc + τ ca ) which will be used by the controller.

From the knowledge of (τ sc + τ ca ), the controller gets, by adding the value τ ZOH = h/2, the value of the loop time delay τ characterizing the period starting at t k (call τ k this loop delay). Remark: the loop time delay, used by the controller during each sampling period, is the loop time delay evaluated at the previous sampling period (but it is not important as the time delay (τ sc + τ ca ) is the same whatever the sampling period may be).

In brief, the controller, in the sampling period starting at t k , receives the f sc frame including the value (τ sc + τ ca ) and the sampled output value y k (sampled at t k ):

• With the knowledge of (τ sc +τ ca ), the controller computes the loop time delay τ k and then computes on-line the new parameters K and T d using the dominant pole method.

• With the knowledge of these new parameters K and T d and the value of y k , the controller computes the control signal u k and send it in a f ca frame.

Remark about the first sampling period: at the first sampling instant t 0 = 0, as the sensor has no information about τ sc and τ ca , the controller will not get such information and then will use only τ ZOH i.e. the loop time delay τ 0 = τ ZOH .

2) The computations in the controller for the maintenance of the dominant poles: Consider the polynomial f 3 (s) in the denominator of Equation 5, this polynomial concerns the poles p 1 , p 2 , p 3 and can be written as (sp 1 )(sp 2 )(sp 3 ), which can be re-written by considering the values of p 1,2 = R ± jI:

(s -p 1 )(s -p 2 )(s -p 3 ) = s 3 -(2R + p 3 )s 2 + (2Rp 3 + R 2 + I 2 )s -(R 2 + I 2 )p 3 (8)
By identifying f 3 (s) with Equation 8, we get the relations which allow to determine the values of p 3 , K and T d :

             p 3 = - a 3 + (2 + 2R)a 2 -(R 2 + I 2 )a a 2 -2Ra + R 2 + I 2 K = - (R 2 + I 2 )p 3 1000a T d = 1 + a + p 3 + 2R 1000K (9)
We replace the value of K in Equation 5 by this one found in Equation 9and taking into account for the relation

R 2 + I 2 = ω 2
n , we have now the transfer function F(s):

F(s) = ω 2 n (1 + T d s)(1 + s/a)(1 -s/b) (s 2 + 2ζω n s + ω 2 n )(1 -s/p 3 )(1 + s/b) (10)
Note that p 3 and p 4 = -2 τ ca +τ ZOH are real poles.

3) Conditions for the insignificance of the poles p 3 and p 4 : We take the conditions expressed by [START_REF] Kuo | Automatic Control System[END_REF] (page 281) i.e. the real part is five times smaller than the real part of the dominant poles, then:

• p 3 ≤ 5R which gives τ < 40 ms and as τ ZOH = 5 ms we need τ sc + τ ca < 35 ms.

• p 4 ≤ 5R which gives τ ca + τ ZOH < 36 ms and as τ ZOH = 5 ms we need τ ca < 31 ms. These conditions are always satisfied as the sampling period h is 10 ms and τ sc + τ ca is always smaller than h. 4) Effect of the zeros: "When a zero gets closer to the origin, the overshoot increases" [START_REF] Dorf | Modern Control Systems[END_REF], [START_REF] Kwoon | Transient Response Analysis and Compensation of the Second Order System with One RHP Real Zero[END_REF].

For a negative zero (call z 0 the absolute value of this zero), in [START_REF] Dorf | Modern Control Systems[END_REF] (page 226), the zero has a little effect and can be neglected if z 0 /ζω n ≥ 5.

For a positive zero (call z 0 the value of this zero), in [START_REF] Kwoon | Transient Response Analysis and Compensation of the Second Order System with One RHP Real Zero[END_REF], the overshoot is evaluated as follows: the second order system without zero. We can thus consider that if z 0 2ζω n > 5 and ( z 0 ω n ) 2 > 5 (these conditions induce that z 0 > 110 (with ω n = 27, ζ = 0.4)), the positive zero has a little effect and can be neglected.

O = e - ζ(Φ+π) √ 1-ζ 2 1 + 2ζω n z 0 + ( ω n z 0 ) 2 where Φ = tan -1 √ 1-ζ 2
We consider now the three zeros:

• The negative zero z 2 = -2/τ: The effect of z 1 depends on the value of T d ; the higher the value of T d is, the more closed the zero is to the origin, thus the zero has the stronger effect.

as τ = τ sc + τ ca + τ ZOH is smaller than h + h/2 =

Remark about the transfer function after the compensation of the time delay:

Considering the insignificance of the poles p 3 , p 4 and the negligible effect of the zeros z 2 and z 3 , the transfer function in Equation 10 can be written as Equation 11.

F(s) = ω 2 n (1 + T d s) s 2 + 2ζω n s + ω 2 n (11)
We see that we have the same form as the transfer function of the system without delays (Equation 3) in Section II-A but the value of T d is now changed.

B. Validation of the dominant pole method

Table I shows the results obtained by considering different loop delays τ. We see the maintain of the dominant poles resulting from the action on K and T d (decrease of K and increase of T d ). We also see the increase of the overshoot (mainly due to the increase of the zero z 1 = -1/T d ; the role of the zeros z 2 , z 3 and of the poles p 3 , p 4 is negligible).

C. Considering the implementation of the four process control applications on CAN network

We want to show here the interest of the dominant pole method (which is based on an adaptive controller i.e. the parameters of the controller are modified according to the time delay) in comparison with the case where we do not use this method (i.e. we have a fixed controller).

The comparison between the case "adaptive controller" (noted Adapt-Controller) and the case "fixed controller" (noted Fix-Controller) is presented on Table II

ITSE in the subsection II-A) and on Figures 6 and 7 (presentation of the output time responses).

From the results of the case "fixed controller": the performances follow the order of the priorities which is normal (lower is the priority, then higher is the time delay and less good is the performance).

From the results of the case "adaptive controller": it improves the results (what was waited for !) which are less dispersed (the influence of the priorities is reduced); however we cannot have identical performances for all the process control applications because of the effect of the zero -1/T d (as the applications which have less priority have higher delays, the compensation requires higher values of T d which induces that the zero -1/T d moves closer to the origin). We note also the effect of the zero -1/T d by the increase of O (Fig. 7) compared with the O (Fig. 2) of the control system without network in Section II-A.

IV. RELATION QOC→QOS

The implementation of this relation is based on the use of hybrid priorities for the scheduling of the frames. The controller is a fixed controller (the parameters of the controller are presented in Section II-A)

A. On the dynamic priority part (level 2 of the identifier field) used in hybrid priorities

The specification of a dynamic priority requires, at first, to choose a variable of the process control application, which is relevant for characterizing the frame transmission urgency, and then to express a way (how and when) for translating this variable into a priority.

Once this priority has been obtained, we have to finally specify when and how this priority is implemented in the frames of the f sc and f ca flows.

1) Dynamic priority specification:

The variable that we consider here is the control signal u [START_REF] Juanole | Networked Control Systems: Definition and Analysis of a Hybrid Priority Scheme for the Message Scheduling[END_REF], the value of which is representative of the strength of the action which is requested to the actuator (higher is this value that means that higher is the error (e = ry) and then more quickly the action of the actuator must be done). Then the control signal u is appropriate to obtain from it the dynamic priority.

The control signal u is translated into a priority on the basic of an increasing function of |u| (Fig. 8, Equation 12) characterized by a saturation for a value |u| which is 2/3|u| max i.e. less than |u| max . The idea is to be very reactive before the need becomes very important. The choice 2/3|u| max is an arbitrary choice.

The dynamic priority is computed by the controller each time a frame of the f sc flow is received.

Dynamic priority 0 max 3 2 u ) (u f max P u max u Fig. 8. Function of u. f (u) =    P max |u| 2 3 |u| max , 0 ≤ |u| ≤ 2 3 |u| max P max , |u| > 2 3 |u| max (12)
2) Implementation of the dynamic priority in the frames of the f sc and f ca flows: Recall us that, as CAN is a bus, any computer can read the frames which are sent. We furthermore consider here that the computer, where the sensor task of a process control application is, knows the static priority of the f ca flow (level 1 of the ID field of each frame of the f ca flow) of this process control application.

The presented implementation scheme has two objectives:

1) The sensor task sends, at the beginning of each sampling period (i.e. at t k = kh), a f sc frame with a dynamic priority which has been computed by the controller during the previous sampling period.

2) The sequence "sending of a f sc frame by the sensor task; reception of this frame by the controller; sending of the f ca frame by the controller; reception of this frame by the actuator" must be an atomic action.

Considering the objective 2, the controller puts in the level 2 of the ID field the maximum priority P max [START_REF] Nguyen Xuan | Réseaux de communication et applications de contrôle-commande[END_REF] (in this way and remember that the f ca frames have static priorities higher than those of the f sc frames, the frame of the f ca flow is sent immediately). Considering the objective 1, the controller puts, in the data field of the f ca frame, the dynamic priority which has been computed (in this way, the sensor task, as it knows the static priority of the f ca flow, learns the dynamic priority, and it will use this dynamic priority in the next sampling period). Remark: consequently to this behavior, at the first sampling instant (t 0 = 0), the sensor has no knowledge about a dynamic priority to use. So we propose to use the maximum value P max .

B. Interest of the hybrid priority compared with the static priority

We want here to compare, at first, the QoS which is provided by the different types of priorities and then to show the consequences on the QoC which is obtained.

1) Quality of Service:

The QoS is here characterized by the mean value of the communication delay (τ sc + τ ca ). We note D this mean value. Call D i the time delay (τ sc + τ ca ) during the period starting at the instant t i (i.e. period [t i ,t i+1 ]) and call n the number of the sampling period during the simulation. We have D =

∑ n i=1 D i n
. We present on Table III the QoS in terms of D of the 4 processes provided by the static and hybrid priorities. For the static priorities, the higher the priority is, the smaller the time delay is, which is logic. For the hybrid priorities, we obtain time delays more balanced than these with static priorities. This is the result of the predominant role of the parts "dynamic priorities" compared with the parts "static priorities" (as all the processes have the same type of variable urgencies because they have the same transfer function, that explains the balanced aspect).

V. RELATION QOS⇋QOC

A. Ideas

The idea is to combine the frame scheduling scheme based on the hybrid priority (i.e. the relation QoC→QoS) and the method of compensation of time delay (i.e. the relation QoS→QoC) in order to have a more efficient control system. However, concerning the loop time delay compensation, in the sampling period k, we cannot consider here that the controller can use the value of the loop time delay of the sampling period (k -1) because now, taking into account for the dynamic priority used by the sensor task, the time delay (τ sc + τ ca ) changes every period.

Then the controller must make the delay compensation in the sampling period k by knowing the loop time delay of this sampling period k. We explain now this implementation.

B. Principle of the implementation of the relation QoS⇋QoC

This principle, relatively to the sampling period starting at t k , is represented on Fig. 10 where we indicate the content of the f sc and f ca frames and the computations done by the controller.

At the instant t k , the sensor task generates the f sc frame which includes in the ID field the dynamic priority P k-1 i.e. dynamic priority computed by the controller in the previous period (this expresses the implementation of the relation QoC→QoS). The data field of this frame includes the value of the instant t k and the sampled value of the output y k (these values represent the contribution of the sensor task to the relation QoS→QoC).

When the controller task receives this frame, it undertakes the following actions (we suppose that these actions are instantaneous):

• With the value of t k , it deduces (by comparison to its local clock; we suppose that the clocks are well synchronized) the value of the time delay τ sc ; furthermore, as the level 2 of the ID field of the f ca frame includes P max , the f ca frame is sent immediately without waiting time (then τ ca = D ca ); and as the value of τ ZOH is known (τ ZOH = h/2), the controller has the knowledge of the loop time delay τ in this sampling period and can then compute the new parameters K and T d . This compensation expresses the implementation of the relation QoS→QoC.

• Then, with the value y k and the new parameters K and T d , the controller computes the value of the control signal u k . With this value u k , the controller computes the dynamic priority P k which will be used in the next sampling period by the sensor task (this characterizes the setting up of the relation QoC→QoS for the next sampling period).

C. Performance evaluation and summary of obtained results

We still consider the implementation of 4 applications (P 1 , P 2 , P 3 , P 4 ) studied in the previous sections.

The results (∆J/J 0 %) of the implementation of the relation QoS⇋QoC and the other relations (static priority + Fix-Controller, QoS→QoC, QoC→QoS) are represented on Table V. These results which are represented graphically on Fig. 11 obviously show the balanced performances provided by the hybrid prirority (QoC→QoS) compared with the static priority and by the bidirectional relation QoS⇋QoC compared with the relation QoS→QoC.

From these results, we can say that, if we have a constraint of performance which cannot be exceeded (not too small), the hybrid priority allows to implement more applications than the static one and, extensionally, the bidirectional relation QoS⇋QoC allows to implement more applications than the relation QoS→QoC. The visualizations of the time responses provided by the different relations are represented on Fig. 12.

VI. CONCLUSION

In this paper, we have presented three following points:

• The first is the implementation of the relation QoS→QoC on the basis of the dominant pole method and of static priorities for the frame scheduling. This relation QoS→QoC shows the interest of the delay compensation to improve the QoC performances provided by static priorities only.

• The second is the implementation of the relation QoC→QoS on the basis of hybrid priorities for the frame scheduling. The hybrid priority is characterized by, on one hand, a static part representing the uniqueness of the flow, and on the other hand, a dynamic part that represents the urgency of transmission (this dynamic part is expressed from a function of the control signal). This relation QoC→QoS (i.e. hybrid priority) gives, in comparison to the static priority case, balanced performances for the different process control applications.

• The final is the implementation of the relation QoS⇋QoC. This implementation can be considered as a combination of the relations QoS→QoC and QoC→QoS.

The results, which are obtained, show the interest of the joint action of the hybrid priorities and the delay compensation (by the hybrid priorities, we introduce the balance aspect compared with the static priority; by the delay compensation, we maintain the balance aspect while improving the QoC and then we can consider the possibility to implement more applications than with the relation QoS→QoC). The relation QoS⇋QoC is a co-design approach for Networked Control Systems. The further work should be the following points: the utilization of other compensation methods for time delays for the relation QoS→QoC (for example, maintenance of the phase margin); the consideration of other types of controller (PID for example) and the consideration of other types of process control applications.

Still furthermore, the study of this relation (QoS⇋QoC) might be also improved by the consideration of theoretical problems (in particulars, stability conditions when the online control law parameters change from sampling period to sampling period).

Fig. 1 .

 1 Fig. 1. Continuous control system.

Fig. 2 .

 2 Fig. 2. Time response.

Fig. 3 .

 3 Fig.3. Implementation of a process control application through a network.

Fig. 4 .

 4 Fig. 4. Control system with time delays.

Fig. 5 .

 5 Fig. 5. ID field structure (hybrid priority).

Fig. 6 .Fig. 7 .

 67 Fig. 6. Fix-Controller: time responses.

P

Fig. 11 .

 11 Fig. 11. Graphic representation of the QoC (∆J/J 0 %) of the 4 applications.

  P

  QoS⇋QoC: hybrid priority + Adapt-Controller.

Fig. 12 .

 12 Fig. 12. Time responses.

TABLE I VALIDATION

 I OF THE DOMINANT POLE METHOD (τ ca = 1 MS, τ ZOH = 5 MS).

	τ sc (ms)	τ (ms)	K	T d (s)	p 1,2	p 3	p 4	O (%)	z 1	z 2	z 3
	1 2 3	7 8 9	0.6119 0.5959 0.5799	0.0365 0.0377 0.0389	-11 ± j24.5 -11 ± j24.5 -11 ± j24.5	-242 -206 -178	-333 -333 -333	40.66 41.74 42.84	-27.3 -26.5 -25.7	-285 -250 -222	333 333 333

z 0 +ζω n . We see that if z 0 2ζω n >> 1 and ( z 0 ω n ) 2 >> 1, the overshoot can be rewritten as O = e -ζπ √ 1-ζ 2 i.e. we have the overshoot of

  15 ms, z 2 is smaller than We see that |z 2 |/ζω n > 5. Thus the zero z 2 can be neglected. • The positive zero z 3 = 2/(τ ca + τ ZOH ): because of τ ca + τ ZOH < 15 ms, z 3 is bigger than 2 0.015 = 133, thus z 3 > 110. Hence the zero z 3 can be neglected.

	-2 0.015 = -133.

• The negative zero z 1 = -1/T d : note that this zero is based on a parameter of the controller (T d ).

  (we use the criteria

		P 1	P 2	P 3	P 4
	Fix-Controller	75.3	107	150	208
	Adapt-Controller (relation QoS→QoC)	47.6	62.8	80.6	101

TABLE III QUALITY

 III OF SERVICE: D MS.2) Quality of Control:The QoC is represented on Table IV ( ∆J J 0 %) and on Fig.9(y(t)). The balance of QoS induces a balance of QoC. This is an interesting result (compared with the static priority use) in the perspective of implementing more applications by considering a performance threshold.

		Controller Computations of • loop time delay • control parameters K and T d τ • control signal u k	ID field P max	Data field u k P k	N E T W O	Actuator	u(t)	Process to control
		• dynamic priority P k				R K	ID field	Data field	h	y(t)
							C A N	P k-1	y k	instant t k	Sensor	yk
								t
				Fig. 10. Principle of the implementation of the relation QoS⇋QoC.
	1.8	y(t)
	1.6	P1, P2, P3, P4
	1.4
	1.2
	1
	0.8
	0.6
	0.4
	0.2
	0 0	100	200		300	400
		time (ms)
		Fig. 9. Time responses (hybrid priority).
			P 1	P 2	P 3	P 4
		Static priority	75.3	107	150	208
		Hybrid priority (relation QoC→QoS)	122	125	131	145

TABLE IV QUALITY

 IV OF CONTROL: ∆J/J 0 %.

TABLE V QOC

 V SUMMARY (∆J/J 0 %).

ACKNOWLEDGEMENTS

This work has been partly funded by the project OSEO AMIC-TCP (Architecture Multiplexage Informatique Communication pour Transport en Commun de Personnes). We want to express here our sincere thanks.