Christophe Fiter
email: christophe.fiter@univ-lille1.fr

Laurentiu Hetel
email: laurentiu.hetel@ec-lille.fr

Wilfrid Perruquetti
email: wilfrid.perruquetti@inria.fr

Jean-Pierre Richard
email: jean-pierre.richard@ec-lille.fr

State-dependent sampling for online control

maintenance. However, because they are often required to share a limited amount of resources, they generate fluctuations in the sampling intervals, which may result in unstable behaviours. This brings up new theoretical challenges.

In order to take into consideration these new difficulties, several works have studied the stability of sampled-data systems with time-varying sampling ( [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF], [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF], [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], [START_REF] Liu | Wirtinger's inequality and Lyapunov-based sampled-data stabilization[END_REF], [START_REF] Fiter | A state dependent sampling for linear state feedback[END_REF]).

More recently, another research direction has emerged, consisting in controlling dynamically the sampling instants so as to reduce the processor and/or network loads while ensuring the desired control performances. Two main approaches can be found in the literature: -The first approach is the event-triggered control ( [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], [9], [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF]), in which intelligent sensors send information to the controller when special events occur (for example when the system's state leaves some boundary around the equilibrium point). In most works from the literature, this approach requires a dedicated hardware.

-The second approach is the self-triggered control ( [15], [START_REF] Anta | To sample or not to sample: self-triggered control for nonlinear systems[END_REF], [START_REF] Wang | Self-triggering under state-independent disturbances[END_REF]), which emulates event-triggered control without dedicated hardware, by computing at each sampling instant a lower-bound of the next admissible sampling interval. In general in these works, no method is given to compute the Lyapunov function so as to optimize the lower-bound of the sampling intervals (i.e. in the worst case, independently of the system's state).

Still more recently, in [START_REF] Fiter | A state dependent sampling for linear state feedback[END_REF], [START_REF] Fiter | State-dependent sampling for perturbed time-delay systems[END_REF], and [START_REF] Fiter | A robust polytopic approach for statedependent sampling[END_REF], the authors proposed a third dynamic sampling control approach, state-dependent sampling, which consists in designing the state-dependent sampling function (i.e. the self-triggered controller) offline, thanks to a mapping of the state space and linear matrix inequalities (LMIs). An advantage was that these LMIs could be used for optimizing the Lyapunov function, thus solving the issue mentioned previously regarding regular self-triggered control approaches

In the present chapter, we present a novel self-triggered control scheme in the case of LTI systems with unknown, exogenous, state-bounded perturbations. The communication links are assumed to react instantaneously (i.e. there is no communication delay). We guarantee the system's exponential stability for a desired decay-rate using Lyapunov stability conditions and convexification arguments. One of the main contributions and advantages is the optimization of the Lyapunov function, taking into account both the effects of the sampling and the perturbation. It optimizes the lower-bound of the sampling intervals for the proposed self-triggered control scheme.

The chapter is organized as follows. First, we state the problem in Section 2 and propose the main stability result in Section 3. Then, we design the self-triggered controller in Section 4. Finally, simulation results are shown in Section 5 before concluding in Section 6. All the proofs are given in the Appendix.

Notations: R + = {λ ∈ R, λ ≥ 0}, R * = {λ ∈ R, λ = 0}, N * = {n ∈ N, n = 0}, and R * + = R + \ {0}. λ max (M ) denotes the largest eigenvalue of a symmetric matrix M ∈ R n×n . S + n (resp. S + * n )
is the set of positive (resp. positive definite) symmetric matrices P 0 (resp. P ≻ 0) in R n×n . The symmetric elements of a symmetric matrix are denoted by * . . is the Euclidean norm on R n .

2 Problem statement

System description

Consider the perturbed LTI system

ẋ(t) = Ax(t) + Bu(t) + Ew(t), ∀t ∈ R + , (1)
where x : R + → R n , u : R + → R nu , and w : R + → R nw represent respectively the system state, the control function, and the exogenous disturbance. The matrices A, B, and E are constant with appropriate dimensions.

The control is a piecewise-constant state feedback

u(t) = -Kx(t k ), ∀t ∈ [t k , t k+1 ), ∀k ∈ N, (2)
where the gain K is fixed and such that A -BK is Hurwitz. The sampling instants t k are defined by the self-triggered control law

t k+1 = t k + τ (x(t k )), ∀k ∈ N, (3)
where τ : R n → R + is a state-dependent sampling function to be designed. The disturbance is assumed to be state-bounded:

∃W ≥ 0, w(t) 2 ≤ W x(t k ) 2 , ∀t ∈ [t k , t k+1 ), ∀k ∈ N. (4)
Such a perturbation can represent model uncertainties, local nonlinearities, or some kind of measurement noises. We denote by S the closed-loop system {(1),( 2),(3),(4)}.

Objectives

Our goal is to design a sampling function τ as large as possible and with a maximal lower-bound, while ensuring the exponential stability of S for a desired decay-rate α, also called α-stability (i.e. such that there exists a scalar γ for which all trajectories satisfy x(t) ≤ γe -αt x 0 for any initial condition x 0 ). To this aim, we use the well known Lyapunov exponential stability property:

Proposition 1. (See [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] for example) Consider a scalar α > 0 and a quadratic Lyapunov candidate function

V (x) = x T P x, ∀x ∈ R n , with P ∈ S + * n . If the condition V (x) + 2αV (x) ≤ 0 (5)
is satisfied for all trajectories of S, then the system is globally α-stable.

Our objectives can then be formulated as:

Objective 1: Sampling map design Given a quadratic Lyapunov function V , design a lower-bound approximation of the optimal sampling function τ V opt (x) = max τ (x) such that (5) holds.

Objective 2: Lyapunov function design Find a quadratic Lyapunov function V such that there exists a sampling function τ satisfying (5) with a minimum value τ * = inf x∈R n τ (x) as large as possible.

Main stability result

In this section, we provide sufficient stability conditions that depend on the time variable σ tt k and on the sampled-state x x(t k ).

The following result is obtained by bounding the effects of the perturbations on the system's behaviour thanks to (4), by using the well known inequality ( [2])

x T y + y T x ≤ ε -1 x T x + εy T y, (6)
which is satisfied for any vectors x and y of same dimension, and any scalar ε > 0.

Theorem 1. Consider a tuning parameter ε > 0, scalars α > 0 and W ≥ 0, and a sampling function τ : R n → R + . Then, the system S is globally α-stable if there exist a matrix P ∈ S + * n and scalars η ≥ 0 and µ ≥ 0 such that

(1 + ε -1 )E T P E ηI, Q 5 µI, (7
)

and x T Π(σ)x ≤ 0, ∀x ∈ R n , ∀σ ∈ [0, τ (x)], (8)
with

Π(σ) = Λ(σ) T Q 1 Λ(σ) + Λ(σ) T Q 2 + Q T 2 Λ(σ) + Q 3 + Q 4 (σ), (9)
Λ(σ) = I + σ 0 e sA ds(A -BK), (10)
Q 1 = (1 + 2αε)(A T P + P A) + εA T P A + 2(α + ε + 2εα 2 )P, Q 2 = -(I + ε(A + 2αI) T )P BK, Q 3 = εK T B T P BK + W ηI, Q 4 (σ) = σW µλ max (E T E)f A (σ)I, Q 5 = A T P + P A + ε -1 A T P A + (1 + 2α + ε -1 )P, (11)
and

f A (σ) = 1 λmax(A+A T ) e λmax(A+A T )σ -1 if λ max (A + A T ) = 0, σ otherwise. ( 12
)
Remark 1: Since A-BK is assumed to be Hurwitz (i.e. the LTI system (1) without perturbation is stable with the continuous state feedback u(t) = -Kx(t)), one can show that there always exist parameters P , η and µ, such that the conditions ( 7) are satisfied and such that Π(0) ≺ 0, provided that the tuning parameter ε, the decayrate α, and the perturbations upper-bound W are small enough (if ε, α, and W are close to 0, then Π(0) ≃ (A -BK) T P + P (A -BK)). With such parameters, we can thus find a sampling function τ satisfying the stability conditions of Theorem 1, which is lower-bounded by a strictly positive scalar. An algorithm enlarging that lower-bound will be provided in the next section.

Self-triggered controller design

The self-triggered control scheme we propose in this chapter is based on the stability conditions from Theorem 1, which involve a few LMIs (7) (considering that ε is a tuning parameter) as well as the more complex set of conditions:

x T Π(σ)x ≤ 0, ∀x ∈ R n , ∀σ ∈ [0, τ (x)].
These stability conditions can not be checked "as is" in real time, since they involve an infinite number of inequalities regarding the time variable σ. Therefore, in order to obtain checkable conditions, we propose to use: -A discretization of the time interval, with a step ∆ ∈ R * + (which can be either chosen by the user or imposed by the system); -Convex embeddings around the matrix function Π over every time interval [j∆, (j+ 1)∆] (with j ∈ N), so as to guarantee the desired performances between any two successive discrete instants. Indeed, since the matrix function Π is continuous, then for any j ∈ N it is possible to design a convex polytope with a finite number of vertices Π

(j) i (∆) (i ∈ {0, • • • , N }) such that for all x ∈ R n , x T Π (j) i (∆)x ≤ 0, ∀i ∈ {0, • • • , N } ⇒ x T Π(σ)x ≤ 0, ∀σ ∈ [j∆, (j + 1)∆] . (13)

Convex embedding design based on Taylor polynomials

Here, we adapt the result from [START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF] to design a convex embedding satisfying [START_REF] Lunze | A state-feedback approach to event-based control[END_REF]. The approach is based on an N -order Taylor series approximation of Π.

The vertices are defined for i ∈ {0, • • • , N } and j ∈ N as

Π (j) i (∆) = Π(j) i (∆) + ν (j) (∆)I, (14)
with

Π(j) i (∆) = i k=0 Ψ (j) k (∆)∆ k , (15)
Ψ (j) 0 (∆) = Γ (j) 1 (∆) T Q 1 Γ (j) 1 (∆) + Γ (j) 1 (∆) T Q 2 + Q T 2 Γ (j) 1 (∆) + Q 3 + L (j) 0 (∆), Ψ (j) 1 (∆) = Γ (j) 2 (∆) T (Q 1 Γ (j) 1 (∆) + Q 2 ) + (Γ (j) 1 (∆) T Q T 1 + Q T 2 )Γ (j) 2 (∆) + L (j) 1 (∆), Ψ (j) k≥2 (∆) = Γ (j) A k-1 k! Γ (j) 2 (∆) +Γ (j) 2 (∆) T k-1 i=1 (A i-1 ) T i! Q 1 A k-i-1 (k -i)! Γ (j) 2 (∆) + L (j) k (∆), (16)
and

Γ (j) 1 (∆) = I + N j (∆)(A -BK), Γ (j) 2
(∆) = N ′ j (∆)(A -BK), N j (∆) = j∆ 0 e As ds, N ′ j (∆) = AN j (∆) + I. (17)
If λ max (A + A T ) = 0, the matrices L k (j) are defined as

L (j) 0 (∆) = W µλ max (E T E)(j∆) 2 I, L (j) 1 (∆) = 2W µλ max (E T E)j∆I, L (j) 2 (∆) = W µλ max (E T E)I, L (j) k≥3 (∆) = 0. (18)
Otherwise, if λ max (A + A T ) = 0, they are defined as

L (j) 0 (∆) = W µ λ max (E T E) λ max (A + A T ) j∆ e λmax(A+A T )j∆ -1 I, L (j) 1 (∆) = W µ λ max (E T E) λ max (A + A T ) e λmax(A+A T )j∆ 1 + j∆λ max (A + A T ) -1 I, L (j) k≥2 (∆) = W µ λ max (E T E) λ max (A + A T ) e λmax(A+A T )j∆ j∆ (λ max (A + A T )) k k! + (λ max (A + A T )) k-1 (k -1)! I. (19)
Finally,

ν (j) (∆) ≥ max σ∈[j∆,(j+1)∆] λ max Π(σ) - N k=0 Ψ (j) k (∆)(σ -j∆) k . ( 20
)
Remark 2: The matrices Ψ i (∆) defined in ( 14) satisfy [START_REF] Lunze | A state-feedback approach to event-based control[END_REF]

: if x T Π (j) i (∆)x ≤ 0 for all i ∈ {0, • • • , N }, then x T Π(σ)x ≤ 0 for all σ ∈ [j∆, (j + 1)∆].

Design of the sampling function τ for given parameters

In this subsection, we consider that the parameters P ∈ S + * n , η ≥ 0, µ ≥ 0, and ε > 0 satisfying (7) are given.

Then, using Theorem 2, it is possible to design a lower-bound estimation τ of the maximal state-dependent sampling function satisfying the stability condition (8) from Theorem 1 as:

τ (x) = j(x)∆, (21)
with j(x) = min j ∈ N | ∃i ∈ {0, • • • , N }, x T Π (j) i (∆)x > 0 . ( 22
)
Remark 3: It is important to guarantee that j(x) > 0 (and thus τ (x) > 0) for any state x ∈ R n . To this aim, one may simply check if the inequality Π (0) i (∆) 0 is satisfied for all i ∈ {0, • • • , N }. If this is not the case, it may be that the discretization step time ∆ is too large, or that the tuning parameter ε, the decayrate α, or the perturbations upper-bound W are also too large (see Remark 1). In the following subsection, we will show how to design all these parameters so as to optimize the lower-bound of the sampling function.

Remark 4: Note that the matrices Π (j) i (∆) can be computed offline. The number of online computations required to perform the self-triggered control scheme (i.e. to compute j(x)) is upper-bounded by n(n + 1)(N + 1) τ (x) ∆ multiplications and

(n + 1)(n -1)(N + 1) τ (x) ∆ additions. The online complexity is thus O( τ (x) ∆ N n 2 ), which is comparable to the one obtained in the self-triggered control scheme from [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF] for example. Note that choosing a larger step ∆ will reduce the number of computations, but will end in a loss of precision for the sampling function (21). Therefore, a trade-off has to be made between precision and online complexity. In this subsection, we propose an LMI-based algorithm that computes the different parameters involved in the self-triggered control scheme (the Lyapunov matrix P ∈ S + * n and the additional parameters ε > 0, η ≥ 0, and µ ≥ 0) so as to optimize the lower-bound of the sampling function (21). It is based on the following property: Theorem 3. Consider a tuning parameter ε > 0, a time-step ∆ > 0, and scalars α > 0 and W ≥ 0. Consider a matrix P ∈ S + * n , scalars η ≥ 0 and µ ≥ 0, and a sampling function τ : R n → R + such that the conditions from Theorem 1 hold. If there exists an integer j * ∈ N * such that the matrix inequalities Π (j) i (∆) 0 are satisfied for all i ∈ {0, • • • , N } and all j ∈ {0, • • • , j * -1}, then the system S is globally α-stable with the sampling function τ : R n → R + defined as τ (x) = max(τ (x), j * ∆).

The conditions

Π (j) i (∆)
0 in Theorem 3 depend from the parameters P , η, µ, and ε. Therefore, it becomes clear that maximizing the sampling function lowerbound reduces to searching the parameters that maximize τ * j * ∆. This can be done as follows.

Algorithm:

Step 1: In this step, we consider the polytopic description [START_REF] Mazo | On self-triggered control for linear systems: guarantees and complexity[END_REF] with upper-bounds on the estimation errors ν (j) (∆) = 0. Then, for a given ε > 0, the search for P , η and µ maximizing τ * in Theorem 3 reduces to an LMI problem (from ( 11) and ( 14) to [START_REF] Wang | Self-triggering under state-independent disturbances[END_REF], we can see that the matrices Π (j) i (∆) are linear in P , η and µ), which can be solved using LMI solvers. It is possible to find the appropriate parameter ε with a linear search algorithm.

Step 2: Next, we compute the value of the upper-bounds ν (j) (∆) and of the matrices

Π (j)
i (∆) that correspond to the obtained parameters P , η, µ, and ε.

Step 3: Finally, the lower-bound of the sampling function can be designed as

τ * = j * ∆, with j * = min j ∈ N | ∃i ∈ {0, • • • , N }, Π (j) i (∆) ≻ 0 .
This parameter optimization presents several advantages, among which the guarantee of a large lower-bound τ * = j * ∆ for the sampling intervals, and the reduction of the online complexity, since it not necessary anymore to check the inequalities x T Π (j) i (∆)x > 0 for j ∈ {0, • • • , j * -1} when computing j(x) in (22).

Numerical example

Consider the Batch Reactor system from [START_REF] Mazo | On self-triggered control for linear systems: guarantees and complexity[END_REF] with some additional perturbation: We will use the polytopic description presented in Section 4.1, with a polynomial approximation degree N = 5 in all the results that are presented.

ẋ(t) =    

Simulation results

Here, we consider a time-step ∆ = 0.005s, a desired decay-rate α = 0.1, and an upper-bound W = 0.01 on the perturbations (i.e. such that w(t) ≤ 0.1 x(t k ) ).

First, we apply the algorithm proposed in Section 4.3 to enlarge the lower-bound τ * of the sampling function. We obtain τ * = 3∆ = 0.015s, with the parameters Then, we design the sampling function (which defines the self-triggered controller) as proposed in Section 4.2.

P = 10
Figure 1 presents the simulation results obtained with the designed self-triggered controller and a perturbation satisfying w(t) ≤ 0.1 x(t k ) .

During this 10s simulation, the average sampling interval is τ average = 0.051s, which is more than three times the value of the optimized lower-bound τ * . Fig. 1 Inter-execution times τ (x(t k )) and Lyapunov function V (x) = x T P x evolution

Advantages of the sampling function's lower-bound optimization

In order to illustrate the interest of the sampling function's lower-bound optimization, we consider the simpler case of the batch reactor system without perturbation nor desired decay-rate, with the same time-step ∆ = 0.005s.

Using the optimization algorithm from Section 4.3, we get a lower-bound of the sampling function τ * = 36∆ = 0.18s, whereas with a regular approach to compute the Lyapunov function without taking into account the sampling (for example by solving the LMI (A -BK) T P + P (A -BK) ≺ 0, as in [START_REF] Mazo | On self-triggered control for linear systems: guarantees and complexity[END_REF]), we get τ * = 3∆ = 0.015s.

Optimizing the lower-bound of the sampling function provides three main advantages.

1) It guarantees a large lower-bound for the sampling intervals (0.18s instead of 0.015s in this case).

2) It allows for using larger time-steps if one needs to reduce the number of online computations (in the present case, we can fix ∆ up to 0.18s with the optimization, and up to 0.015s without).

3) It allows for reducing the number of online computations, since it is not necessary to check the condition x T Π (j) i (∆)x ≤ 0 for j ∈ {0, • • • , j * -1} (here, it allows for saving 33n(n + 1)(N + 1) additions and 33(n + 1)(n -1)(N + 1) multiplications during each sampling interval... provided that the sampling interval in the non-optimized case is not smaller than 36∆ = 0.18s).

Conclusion

We have introduced a novel self-triggered control approach based on convex embeddings that ensures the exponential stability with a desired decay-rate for perturbed LTI systems with linear state feedback. An interesting feature of the approach is that it allows for enlarging the lower-bound of the sampling function, by computing an appropriate Lyapunov function, thanks to LMIs. Extensions to nonlinear systems are currently under study. (25) Then, the idea of the proof is to find an upper-bound of the left part (25) that is independent of the perturbation w. Using the notations

M 1 = A T P + P A + 2αP, M 2 (σ) = -P BK + M 1 Λ(σ), M 3 = P E, M 4 (σ) = E T P T Λ(σ),
we can rewrite the left part of equation ( 25) as

G w (σ, x) = x T (Λ(σ) T M 1 Λ(σ) -Λ(σ) T P BK -K T B T P Λ(σ))x +J w (σ) T M 1 J w (σ) + J w (σ) T M 2 (σ)x + x T M 2 (σ) T J w (σ) +J w (σ) T M 3 w(σ) + w(σ) T M T 3 J w (σ) +w(σ) T M 4 (σ)x + x T M 4 (σ) T w(σ).
Then, using inequality (6), we get

Jw(σ) T M 2 (σ)x + x T M 2 (σ) T Jw(σ) ≤ Jw(σ) T ε -1 [P + A T P A]Jw(σ) +x T ε[Λ(σ) T P Λ(σ) + [(A + 2αI)Λ(σ) -BK] T P [(A + 2αI)Λ(σ) -BK]]x, Jw(σ) T M 3 w(σ) + w(σ) T M T 3 Jw(σ) ≤ Jw(σ) T P Jw(σ) + w(σ) T E T P Ew(σ), w(σ) T M 4 (σ)x + x T M 4 (σ) T w(σ) ≤ w(σ) T ε -1 E T P Ew(σ) + x T Λ(σ) T εP Λ(σ)x,
and thus obtain (with matrices Q i defined in [START_REF] Liu | Wirtinger's inequality and Lyapunov-based sampled-data stabilization[END_REF]):

G w (σ, x) ≤ x T Λ(σ) T Q 1 Λ(σ) + Λ(σ) T Q 2 + Q T 2 Λ(σ) + εK T B T P BK x +w(σ) T (1 + ε -1 )E T P Ew(σ) + J w (σ) T Q 5 J w (σ).
(26) Using ( 7) and ( 4), we get

w(σ) T (1 + ε -1 )E T P Ew(σ) ≤ ηw(σ) T w(σ) ≤ W ηx T x. (27)
From ( 7), we also have

J w (σ) T Q 5 J w (σ)≤µ σ 0 e A(σ-s) Ew(s)ds T σ 0 e
A(σ-s) Ew(s)ds .

Then, using Jensen's inequality ( [START_REF] Gu | Stability of time-delay systems[END_REF], Proposition B.8), the inequality (2.2) in [START_REF] Loan | The sensitivity of the matrix exponential[END_REF], along with classic inequalities and assumption (4), one gets

J w (σ) T Q 5 J w (σ) ≤ σµ σ 0 w(s) T E T e A(σ-s) T e A(σ-s) Ew(s)ds ≤ σµ σ 0 e (σ-s)λmax(A+A T ) w(s) T E T Ew(s)ds ≤ σµλ max (E T E) σ 0 e (σ-s)λmax (A+A T ) w(s) 2 ds ≤ σW µλ max (E T E) σ 0 e λmax(A+A T )s ds x 2 = σW µλ max (E T E)f A (σ)x T x, (28)
with f A (σ) defined in [START_REF] Loan | The sensitivity of the matrix exponential[END_REF]. Implementing inequalities ( 27) and ( 28) in (26) shows that G w (σ, x) ≤ x T Π(σ)x, with Π(σ) defined in (9), and thus that Proposition 1 stability conditions are satisfied if x T Π(σ)x ≤ 0 for all x ∈ R n and σ ∈ [0, τ (x)].

Proof (Theorem 2). Let ∆ ∈ R * + , j ∈ N, and x ∈ R n .

Step (1): Compute the Taylor expansion of the matrix function Π over the time interval [j∆, (j + 1)∆]. Remember that Π(σ) involves the term Λ(σ) defined in [START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF]. Using the property with the notations from [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. Using this last equation, one can show that (the computational details are omitted)

Π(σ) = ∞ k=0 Ψ (j) k (∆)σ ′k ,
with the matrices Ψ (j) k (∆) defined in [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF]. Here, the matrices L (j) k (∆) that appear in [START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF] come from the Taylor expansion of the term Q 4 (σ) = σW µλ max (E T E)f A (σ)I, and are defined in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] and [START_REF] Wang | Self-triggering under state-independent disturbances[END_REF].

A polynomial approximation of order N of Π on the interval [j∆, (j + 1)∆] can therefore be expressed as Π(∆,j)

N (σ) = N k=0 Ψ (j) k (∆)σ ′k .
Step ( 2 With this, it is clear that if x T Π(∆,j) N (σ) + ν (j) (∆)I x ≤ 0, then x T ∆(σ)x ≤ 0.

Step (3): Since the function

x T Π(∆,j) N (.) + ν (j) (∆)I x : [j∆, (j + 1)∆] → R is polynomial, we can use the convex embedding design from [START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF], Section 3, to prove that if we have x T Π (j) i (∆)x ≤ 0 for all i ∈ {0, • • • , N }, with Π Proof (Theorem 3). The matrix inequalities and Theorem 2 ensure that the stability conditions of Theorem 1 are satisfied for the sampling function τ (x) = max(τ (x), j * ∆).

  defined in[START_REF] Postoyan | A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems[END_REF] are the coefficients of the Taylor polynomial of Π over [j∆, (j + 1)∆]. The constants ν (j) (∆) defined in (20) represent upper-bounds of the Taylor series approximation error over each time intervals [j∆, (j + 1)∆]. Theorem 2. Consider ∆ ∈ R * + , j ∈ N, and x ∈ R n . The polytope vertices Π (j)

4. 3

 3 Optimization of the parameters (maximization of the lower-bound τ * of the sampling function)

  0.0580 0.2611 -0.1701 0.0580 0.2443 0.1029 0.1101 0.2611 0.1029 0.2854 -0.1314 -0.1701 0.1101 -0.1314 0.2861     , ε = 0.0285, η = 0.0254, and µ = 0.3614.

e

  A(σ-s) Ew(s)ds,

a+b 0 e 0 e 0 e 0 e

 0000 As ds = a As ds + b As ds A a As ds + I , which is valid for any scalars a and b, we can rewrite Λ(σ) as a function of σ ′ = σ -j∆ ∈ [0, ∆]: Λ(σ) = I + N j (∆) + σ ′ 0 e As dsN ′ j (∆) (A -BK)

  ): The approximation error term R ν (j) (∆)I, with ν (j) (∆) defined in (20).

  k (∆)∆ k +ν (j) (∆)I, then we have x T Π(∆,j)N (σ) + ν (j) (∆)I x ≤ 0for all σ ∈ [j∆, (j + 1)∆], and therefore Π(σ) ≤ 0.

(∆) T (A k-1 ) T k! (Q 1 Γ (j) 1 (∆) + Q 2 ) + (Γ (j) 1 (∆) T Q T 1 + Q T 2 )

Acknowledgements This work was supported by Ministry of Higher Education and Research, Nord-Pas de Calais, and FEDER through the 'CPER 2007-2013', by the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n o 257462: HYCON2 Network of Excellence "Highly-Complex and Networked Control Systems", and by the INTER-REG IV A 2 Mers Seas Zeen Cross-border Cooperation Program 2007-2013 under SYSIASS 6-20 project.

Appendix

Proof (Theorem 1). Denoting x(t) = Λ(σ)x + J w (σ) (the solution of S with initial value x and perturbation w), with x the sampled-state x(t k ), σ the time tt k , and the matrices

and J w (σ) = σ 0


Huan Zhao

Frédéric De Geuser

Alisson Kwiatkowski Da Silva

Agnieszka Szczepaniak

Baptiste Gault

Dirk Ponge

Dierk Raabe

Segregation assisted grain boundary precipitation in a model Al-Zn-Mg-Cu alloy

Keywords: Al-Zn-Mg-(Cu) alloy, Atom probe tomography, Grain-boundary segregation, Grain-boundary precipitation, Precipitate-free zones

Understanding the composition evolution of grain boundaries and grain boundary precipitation at near-atomic scale in aluminum alloys is crucial to tailor mechanical properties and to increase resistance to corrosion and stress corrosion cracking. Here, we elucidate the sequence of precipitation on grain boundaries in comparison to the bulk in a model Al-Zn-Mg-Cu alloy. We investigate the material from the solution heat treated state (475°C), through the very early stages of aging to the peak aged state at 120°C and further into the overaged regime at 180°C. The process starts with solute enrichment on grain boundaries due to equilibrium segregation accompanied by solute depletion in their vicinity, the formation of Guinier-Preston (GP) zones in the solute-enriched grain boundary regions, and GP zones growth and transformation. The equilibrium segregation of solutes to grain boundaries during aging accelerates this sequence compared to the bulk. Analysis of the ~10 nm wide precipitate-free zones (PFZs) adjacent to the solute-enriched grain boundaries shows that the depletion zones are determined by (i) interface equilibrium segregation; (ii) formation and coarsening of the grain boundary precipitates and (iii) the diffusion range of solutes in the matrix. In addition, we quantify the difference in kinetics between grain boundary and bulk precipitation. The precipitation kinetics, as observed in terms of volume fraction, average radius, and number density, is almost identical next to the depletion zone in the bulk and far inside the bulk grain remote from any grain boundary influence. This observation shows that the region influenced by the grain boundaries does not extend beyond the PFZs.

Introduction

It is well understood that grain boundaries differ in structure and properties from the grain interiors. These differences cause chemical partitioning phenomena between the two regions.

More specific solutes segregate to grain boundaries, driven by the Gibbs adsorption isotherm, an effect which modifies the grain boundary composition compared to the bulk. Such equilibrium segregation phenomena play an important role for transport and phase transformations [START_REF] Raabe | Grain boundary segregation engineering in metallic alloys: A pathway to the design of interfaces[END_REF][START_REF] Frolov | Effect of interface phase transformations on diffusion and segregation in high-angle grain boundaries[END_REF][START_REF] Kuzmina | Grain boundary segregation engineering and austenite reversion turn embrittlement into toughness: Example of a 9wt.% medium Mn steel[END_REF][START_REF] Harmer | The Phase Behavior of Interfaces[END_REF][START_REF] Lejcek | Grain Boundary segregation in Metals[END_REF], as well as for the material's strength and strain hardening [START_REF] Nie | Periodic segregation of solute atoms in fully coherent twin boundaries[END_REF], mechanical creep [START_REF] Buban | Grain boundary strengthening in alumina by rare earth impurities[END_REF], liquid metal embrittlement [START_REF] Duscher | Bismuth-induced embrittlement of copper grain boundaries[END_REF][START_REF] Luo | The role of a bilayer interfacial phase on liquid metal embrittlement[END_REF] and corrosion behavior [START_REF] Song | Stress corrosion cracking and hydrogen embrittlement of an Al-Zn-Mg-Cu alloy[END_REF]. Thorough investigations of solutes behavior at grain boundaries at the near-atomic scale have been reported e.g. for Fe alloys [START_REF] Krakauer | Subnanometer scale study of segregation at grain boundaries in an Fe(Si) alloy[END_REF][START_REF] Isheim | An atom-probe tomographic study of the temporal evolution of the nanostructure of Fe-Cu based high-strength low-carbon steels[END_REF][START_REF] Raabe | Segregation engineering enables nanoscale martensite to austenite phase transformation at grain boundaries: A pathway to ductile martensite[END_REF][START_REF] Kwiatkowski Da Silva | Competition between formation of carbides and reversed austenite during tempering of a mediummanganese steel studied by thermodynamic-kinetic simulations and atom probe tomography[END_REF][START_REF] Herbig | Atomic-scale quantification of grain boundary segregation in nanocrystalline material[END_REF]. Detailed segregation studies were published on ultrafine grained Al alloys [START_REF] Sha | Segregation of solute elements at grain boundaries in an ultrafine grained Al-Zn-Mg-Cu alloy[END_REF][START_REF] Liddicoat | Nanostructural hierarchy increases the strength of aluminium alloys[END_REF][START_REF] Sha | Strength, grain refinement and solute nanostructures of an Al-Mg-Si alloy (AA6060) processed by highpressure torsion[END_REF][START_REF] Chen | Microstructural evolution, strengthening and thermal stability of an ultrafine-grained Al-Cu-Mg alloy[END_REF][START_REF] Sauvage | Atomic-scale analysis of the segregation and precipitation mechanisms in a severely deformed Al-Mg alloy[END_REF], better general understanding of grain boundary segregation in conventional coarse-grained Al alloys is required, particularly in high-strength Al-alloys with multiple substitutional elements. Here we address this topic for a model Al-Zn-Mg-Cu alloy, representing the essential engineering class of 7xxx ultrahigh strength Al alloys. In these materials grain boundary segregation plays an essential role for several reasons: First, they are well known to become highly susceptible to stress corrosion cracking and subject to brittle intergranular fracture under certain conditions of heat treatment [START_REF] Knight | Correlations between intergranular stress corrosion cracking, grain-boundary microchemistry, and grain-boundary electrochemistry for Al-Zn-Mg-Cu alloys[END_REF][START_REF] Su | Influence of RRA Treatment on the Microstructure and Stress Corrosion Cracking Behavior of the Spray-Formed 7075 Alloy[END_REF][START_REF] Rao | Stress corrosion cracking behaviour of 7xxx aluminum alloys: A literature review[END_REF][START_REF] Pao | Environmentally assisted fatigue-crack growth in 7075 and 7050 aluminum alloys[END_REF]. The embrittlement has been attributed to the formation of the coarse, incoherent precipitates along grain boundaries, which leads to softening and substantially reduces the fracture resistance of these materials [START_REF] Joshi | Effect of heat treatment on solute concentration at grain boundaries in 7075 aluminum alloy[END_REF][START_REF] Viswanadham | Grain boundary segregation in Al-Zn-Mg alloys-Implications to stress corrosion cracking[END_REF][START_REF] Vasudevan | Grain boundary ductile fracture in precipitation hardened aluminum alloys[END_REF]. Second, interfacial segregation in these alloys is generally associated with the formation of ~100 nm wide precipitation free zones (PFZs) as the grain boundaries trap the solutes located adjacent to them [START_REF] Ogura | Atom probe tomography of nanoscale microstructures within precipitate free zones in Al-Zn-Mg(-Ag) alloys[END_REF][START_REF] Starke | The causes and effects of "denuded" or "precipitate-free" zones at grain boundaries in aluminum-base alloys[END_REF]. This well-known phenomenon is characterized by strong elemental partitioning among the solute-enriched grain boundaries, the depleted regions adjacent to them and the unaffected grain interiors [START_REF] Doherty | Diffusive phase transformation in the solid state[END_REF]. Similar aspects apply to the quenched-in vacancies to which the interfaces and precipitates can act as sinks [START_REF] Embury | The nucleation of precipitates: the system Al-Zn-Mg[END_REF][START_REF] Lorimer | Further results on the nucleation of precipitates in the Al-Zn-Mg system[END_REF]. This spatially highly confined compositional variation at and around grain boundaries, which is essentially driven by the Gibbs adsorption isotherm, creates very high local electrochemical and mechanical contrast, often reducing the alloy's corrosion resistance and mechanical properties [START_REF] Ogura | Effect of PFZ and Grain Boundary Precipitate on Mechanical Properties and Fracture Morphologies in Al-Zn-Mg[END_REF][START_REF] Kawabata | Ductile fracture in the interior of precipitate free zone in an Al-6.0% Zn-2.6% Mg alloy[END_REF][START_REF] Park | Microchemical analysis of precipitate free zones in 7075-A1 in the T6, T7 and RRA tempers[END_REF]. Both phenomena are essential as further tuning of high strength Al alloys as key materials for light weight transportation solutions is limited more by the weakness of the internal interfaces rather than by insufficient strength of the grain interiors.

The precipitation sequence in the bulk of Al-Zn-Mg-Cu alloys has been studied by several groups and is well understood [START_REF] Löffler | Decomposition processes in Al-Zn-Mg alloys[END_REF][START_REF] Sha | Early-stage precipitation in Al-Zn-Mg-Cu alloy (7050)[END_REF][START_REF] Marlaud | Influence of alloy composition and heat treatment on precipitate composition in Al-Zn-Mg-Cu alloys[END_REF][START_REF] Berg | GP-zones in Al-Zn-Mg alloys and their role in artificial aging[END_REF][START_REF] Ortner | On the structure and composition of GP zones in high purity AlZnMg alloys[END_REF][START_REF] Chinh | The effect of Cu on mechanical and precipitation properties of Al-Zn-Mg alloys[END_REF][START_REF] Ma | Mechanical behavior and strengthening mechanisms in ultrafine grain precipitationstrengthened aluminum alloy[END_REF][START_REF] Sha | Influence of equal-channel angular pressing on precipitation in an Al-Zn-Mg-Cu alloy[END_REF], in terms of the individual processes of solid solution → GP zones→ metastable η'→ stable η (MgZn2). In addition, the final structure of grain boundary precipitates has been studied [START_REF] Li | Precipitation sequence of η phase along low-angle grain boundaries in Al-Zn-Mg-Cu alloy during artificial aging[END_REF][START_REF] Unwin | The nucleation and initial stages of growth of grain boundary precipitates in Al-Zn-Mg and Al-Mg alloys[END_REF][START_REF] Butler | In situ observations of the nucleation and initial growth of grain boundary precipitates in an Al-Zn-Mg alloy[END_REF]. Most of the grain boundary dependent phenomena discussed in these works refer to changes in kinetics, i.e. the precipitation and coarsening phenomena observed are generally faster on the interfaces than in the bulk grains.

Yet, it must also be considered that equilibrium grain boundary segregation can create locally a different composition, characterized by much higher and/or different solute content than in the grain interior. Segregation changes the local thermodynamic state at grain boundaries in a sense that it can in principle influence the sequence, composition and type of the precipitates that are formed. However, details regarding segregation, the evolution of grain boundary chemistry, the early stages of grain boundary precipitation and the formation of the PFZs in Al-Zn-Mg-Cu alloys are lacking. These knowledge gaps hinder precise understanding of the interplay between decoration, formation, growth, and kinetics of grain boundary precipitation, and hence prevent tailoring grain boundaries and avoiding the adjacent precipitation free regions for the design of materials with better mechanical and corrosion properties.

Atom probe tomography (APT) provides three-dimensional (3D), near-atomic scale analytical mapping of materials, and is uniquely positioned to measure the local chemical composition and 3D morphology of individual grain boundary and precipitates [START_REF] Blavette | An atom probe for threedimensional tomography[END_REF][START_REF] Seidman | Three-Dimensional Atom-Probe Tomography: Advances and Applications[END_REF][START_REF] Gault | Atom probe microscopy investigation of Mg site occupancy within δ′ precipitates in an Al-Mg-Li alloy[END_REF][START_REF] Miller | Atom probe tomography[END_REF][START_REF] Reich | Evolution of Ω phase in an Al-Cu-Mg-Ag alloy-a threedimensional atom probe study[END_REF]. More intense deployment of its capabilities to investigate grain boundaries in Al-based alloys has been hindered by the lack of reliable methods to prepare site-specific specimens, i.e. APT tips containing specific features such as grain boundaries. Most APT specimens in this field have so far been prepared with the aid of a focused-ion-beam (FIB) that makes use of a Ga source. Ga implantation has however been known to cause sample embrittlement issues in Al-alloys [START_REF] Marya | Temporary embrittlement followed by increase in ductility after gallium penetration in cold rolled aluminium[END_REF][START_REF] Unocic | Effect of gallium focused ion beam milling on preparation of aluminium thin foils[END_REF], as it gets readily trapped at interfaces, dislocations and grain boundaries. Also it can cause embrittlement, which is attributed to the formation of a low-temperature Al-Ga eutectic [START_REF] Benson | Crack growth behavior of a high strength aluminum alloy during LME by gallium[END_REF]. The recent development of an alternative ion source for FIBs, based on a Xe plasma, has thus been used in the current study to alleviate this stringent limitation [START_REF] Xiao | Investigation of the deformation behavior of aluminum micropillars produced by focused ion beam machining using Ga and Xe ions[END_REF].

The objective of the present work is to investigate grain boundary precipitation in comparison to bulk precipitation, from the solution heat treated state (475°C), through the very early stage of aging (120°C/0.5 hours), peak aged state (120°C/24 hours) and further into the overaged regime (120°C/24 hours +180°C/6 hours). In that context we also followed the kinetics of partitioning of the solutes from the matrix to the interfaces, the formation and growth of the precipitates in the bulk and at the grain boundaries, solute segregation to grain boundaries and the evolution of PFZs, so as to unveil the effects of segregation on grain boundary precipitation in a model Al-Zn-Mg-Cu alloy in a more holistic fashion. We made use of a parameter-free methodology to efficiently extract information from APT data such as matrix and precipitate composition, volume fraction, and number density introduced in ref. [START_REF] Engdahl | Investigation of fine scale precipitates in Al-Zn-Mg alloys after various heat treatments[END_REF]. APT is combined with transmission electron microscopy (TEM), and electron backscattered diffraction (EBSD) characterization and hardness measurements to unveil the effects of segregation on grain boundary precipitation in a model Al-Zn-Mg-Cu alloy.

Materials and methods

The alloy used in this investigation is a model Al-Zn-Mg-Cu alloy. It was laboratory-cast in a vacuum induction furnace to a rectangular ingot with dimensions of 200mm×190mm×40mm.

The ingot was homogenized in an Ar-atmosphere furnace for 1.5 hours at 475 °C and subsequently water quenched. Table 1 summarizes the chemical composition of the homogenized ingot measured by wet chemical analysis. Zr was added to the alloy for forming dispersoids which help refining the grain size. for 24 hours produces the peak aged state. The peak aged sample was further aged for 6 hours at a higher temperature of 180°C to obtain overaged material. The different aging times were selected in order to follow the kinetics of precipitation according to the hardness curve [START_REF] Engdahl | Investigation of fine scale precipitates in Al-Zn-Mg alloys after various heat treatments[END_REF].

APT specimens were prepared to investigate the precipitation at the grain boundaries. EBSD was first used to locate high-angle grain boundaries. APT specimens were then prepared from such regions by site-specific preparation as outlined in ref. [START_REF] Thompson | In situ site-specific specimen preparation for atom probe tomography[END_REF] on a FEI Helios Plasma focused ion beam (PFIB) using of a Xe source on commercial silicon micro-tip coupons (Cameca Inc.) as support. Specimens were sharpened by using a 30 kV Xe + ion beam, followed by a 5 kV final cleaning step to remove regions with higher concentrations of implanted highenergy Xe + ions. APT analyses were performed on a Cameca Instrument Inc. Local Electrode Atom Probe (LEAP) 5000XS (straight flight path). The detection efficiency of this instrument is approx. 80% owing to the improved multi-channel plate detector equipped with a fine holearray. The measurements were performed at a base temperature of 50K in voltage-pulsed mode.

The pulse fraction, pulse rate and detection rate of the measurement were 20%, 250 kHz and 1% repetition rate, respectively. Datasets containing 70-150 million ions were acquired for each aging state. APT reconstruction and analysis were carried out using the Integrated Visualization and Analysis software (IVAS). The reconstruction parameters were calibrated according to the partial structural information within the dataset following the protocol described by Gault et al. [START_REF] Gault | Advances in the calibration of atom probe tomographic reconstruction[END_REF]. TEM observations were performed on a JEOL JEM-2200 FS, operated at 200kV conducted on a PFIB-prepared sample.

Results

A typical APT dataset of the Al-Zn-Mg-Cu alloy containing a grain boundary in the asquenched state is shown in Fig. 1. The EBSD map shows the high angle grain boundary where several APT samples were prepared from the as-quenched state. Fig. 4 shows the result of the APT analysis of the specimen in peak aged state, i.e. after 24 hours at 120°C. A high number density (1×10 24 m -3 ) of small precipitates is homogeneously distributed in the bulk. In Fig. 4(b), a close-up on one precipitate in the bulk near the {111} planes of the Al matrix shows that its morphology is near-spherical with a diameter of around 7 nm. The corresponding composition presented in Fig. 4(c) indicates that these bulk precipitates are mainly GP zones, which profoundly contribute to strengthening. and η.) We assume that within the grain boundary, there is a transition of GP zones to η' plates, while in the adjacent bulk grain the GP zones prevail. The large η' precipitates nucleate and grow on the grain boundary through the accumulation of GP zones, as they have a much lower number density (10 15 m -2 ) compared to the initial stage of aging (10 16 m -2 ). Based on the evidence that no elongated precipitate exists within the grain boundary at this state, we suggest that the elongated precipitates observed after 2 hours of aging form as a transition phase between the small spherical GP zones and the larger η' plates.

The precipitate free zones shown in Fig. 4(a) are still noticeable in the peak aged state. In addition to the APT analysis, TEM was used to map the nanostructure in the vicinity of the grain boundary. The electron micrograph in Fig. 4(f) agrees well with the APT result in Fig. 4(a) and shows a comparable size of the coarse precipitates and the PFZs.

In the overaged state, one single large precipitate (~40 nm in length) is observed along the grain boundary as shown in Fig. 5(a). Some of the bulk precipitates show morphology changes which are characterized by a transition from the small spherical shape in the earlier stage towards coarse denser plates upon aging. The chemical composition of the coarse precipitates from the bulk show less Al (~30 at %) compared to 60 at% Al in the precipitates in the peak aged state, suggesting that they are η'. These observations indicate that in the over-aged state, some of the bulk precipitates transform from GP zones to η' precipitates. This finding is in contrast to the same phase transition that was observed for the grain boundary precipitates already in the peak aged state. We recently described an analysis protocol for APT datasets that combines the determination of the matrix composition using nearest neighbor distributions called DIAM [START_REF] Geuser | Determination of matrix composition based on solute-solute nearest-neighbor distances in atom probe tomography[END_REF] with a radial distribution function (RDF)-based protocol to characterize the evolution of the precipitates over the course of aging. The key idea of the joint application of these two methods is that it opens up a parameter-free and versatile analysis pathway and can be applied to comparable nanostructured systems [START_REF] Engdahl | Investigation of fine scale precipitates in Al-Zn-Mg alloys after various heat treatments[END_REF].

Here, this approach allows us to quantify the evolution of the precipitation sequence from three regions, namely, region I (far and unaffected from the grain boundary), region II (region adjacent to the grain boundary, essentially containing the PFZs), and region III (directly on the GB). The different regions are depicted in Fig. 6, which take the peak aged state as an example.

For evaluating the precipitation state in region I, we use the information extracted form APT datasets obtained from the grain interiors discussed in more detail in ref. [START_REF] Engdahl | Investigation of fine scale precipitates in Al-Zn-Mg alloys after various heat treatments[END_REF]. For regions II and III, we use cuboidal shaped regions-of-interest to split the tomographic reconstruction into two subsets, i.e., one containing the adjacent bulk after extraction of the grain boundary zone (region II), and another one containing the grain boundary only (region III). The PFZs existing in the vicinity of the grain boundary are essentially included in region II. However, it must be considered that the PFZs and the grain boundaries in the APT reconstructions may vary in size by about 10% along the GB plane so that some of the analysis features can overlap. of each element within the mass spectrum. The matrix composition is derived from the DIAM method [START_REF] Geuser | Determination of matrix composition based on solute-solute nearest-neighbor distances in atom probe tomography[END_REF]. In region II, the global composition of Zn eventually decreases from 2.6 to 2.3 at% in the peak aged state and more significantly to 1.7 at% in the overaged state; The global composition of Mg also show loss from 2.2 to 1.7 at% in the overaged state. This can be attributed to depletion of solutes in the PFZs enabling the segregation to be followed in region II, i.e. in diffusional reach. The observed increase in the solute composition after 0.5 hours in region II is likely related to artifacts in the investigated dataset that exhibits a large crystallographic pole. This effect is known to sometimes affect the distribution of solutes [START_REF] Gault | Impact of directional walk on atom probe microanalysis[END_REF].

However, when excluding this specific region, we clearly see the formation of the precipitates through their consumption of solutes (global minus matrix), and the evolution of the matrix composition during aging is essentially similar in regions I and II . 60 at % Al, 20 at % Zn, 18 at% Mg and 2 at% Cu, which does not reach that of the η' phase and can thus be considered as the GP zones prevail. Only when the specimen is overaged at 180 °C, the composition of the precipitates reaches a content which matches that of the η' phase. This structural transition is associated with significant capillary driven competitive particle coarsening, which also leads to a degradation of the mechanical properties [START_REF] Engdahl | Investigation of fine scale precipitates in Al-Zn-Mg alloys after various heat treatments[END_REF]. It should also be noted that 2 at% Cu is comparatively richer in the very early clusters/ GP zones, indicating that Cu aids the formation of GP zones formation in the initial stages of aging as previous study also reported [START_REF] Chinh | The effect of Cu on mechanical and precipitation properties of Al-Zn-Mg alloys[END_REF]. The Cu content (2 at%) remains unchanged after long time aging at 120°C, while it exhibits a strong increase to 5 at% in the overaged state both in regions I and II. This is explained by the low diffusivity of Cu, together with its solubility in the GP zones according to a previous study [START_REF] Marlaud | Evolution of precipitate microstructures during the retrogression and re-ageing heat treatment of an Al-Zn-Mg-Cu alloy[END_REF].

aged state as Fig. 4 shows, and the precipitates' composition presented in Fig. 7(b) also reveals less Al (45 at %), more Zn (25 at%) and Mg (25 at%) compared to the earlier stages of aging. We can also observe that in the initial stage of aging (0.5 hours at 120°C aging), the mean radius (~5 nm in size) of the clusters/GP zones in region III is already larger compared to those in regions I and II (~3 nm in size). During long time aging, the precipitates in region III shows a larger radius and interparticle spacing, and a lower number density.

Chemical composition on grain boundaries and within PFZs during aging

Direct analysis of the chemical composition of the grain boundaries provides additional information enabling better understanding of grain boundary segregation and precipitation.

The global composition of the grain boundary regions and the available solute composition as a function of aging time were quantified separately. We analyzed the global grain boundary composition by taking the whole grain boundary volumes into account including the grain boundary precipitates. We quantified the solute composition as averaged on the grain boundaries at positions between the grain boundary precipitates, along 25nm-diameter cylindrical regions-of-interest taken across the grain boundaries.

Grain boundary precipitation sequence and PFZs formation

Based on the APT results, our view of grain boundary precipitation and PFZs formation is schematically shown in Fig. 10, where solute atoms, nanoclusters, and precipitates are shown. The GP zones distributed along the grain boundary are on average around 5 nm in size after 0.5 hours at 120°C, i.e. they are substantially larger than the GP zones with 3 nm in size formed in the bulk. The fast formation of GP zones on the grain boundary rapidly trap the solutes both from the interface itself but also from the region adjacent to the grain boundary. These processes are kinetically supported by the highly abundant quenched-in vacancies which heal out very rapidly, thereby providing additional solutes transport. This local partitioning between the Gibbs-decorated grain boundary and the region adjacent to them reduces the remaining solute composition in the near-interfacial zones. This effect kinetically reduces the availability of solutes in this region to a level insufficient to form precipitates, hence causing formation of PFZs. This means that the first step in this process is determined by a local equilibrium mechanism, i.e. driven by the Gibbs isotherm decoration of the grain boundary by Zn, Mg and Cu. The second step is also due to a local equilibrium effect, namely, the spinodal decomposition of the segregated solutes into enriched and depleted grain boundary regions.

The third step, namely, the solute depletion of the regions adjacent to the interface, is related to kinetics and governed by the mass transport of those solutes that are in reach for a given time and temperature so as to follow the Gibbs adsorption isotherm and the associated confined spinodal decomposition at interfaces [START_REF] Kwiatkowski Da Silva | Phase nucleation through confined spinodal fluctuations at crystal defects evidenced in Fe-Mn alloys[END_REF]. As mentioned above the formation of these PFZs is further influenced by the fact that the grain boundaries act as vacancy sinks, an effect which further reduces kinetics in the PFZs as suggested by Nicholson et al. [START_REF] Ogura | Atom probe tomography of nanoscale microstructures within precipitate free zones in Al-Zn-Mg(-Ag) alloys[END_REF][START_REF] Embury | The nucleation of precipitates: the system Al-Zn-Mg[END_REF][START_REF] Unwin | The origin of the grain boundary precipitate free zone[END_REF][START_REF] Rometsch | Heat treatment of 7xxx series aluminium alloys-Some recent developments[END_REF][START_REF] Ogura | Quantitative characterization of precipitate free zones in Al-Zn-Mg(-Ag) alloys by microchemical analysis and nanoindentation measurement[END_REF].

With aging time increasing (between 2 hours and 24 hours at 120°C), solutes from the bulk keep segregating, forming a flux of solutes towards grain boundaries. The growth and transformation of GP zones within grain boundaries are accelerated by the solute segregation from the matrix and enhanced solute diffusion along the grain boundaries. Solute segregation creates a gradient of composition from the bulk to the grain boundaries, also promoting the growth and coarsening of grain boundary precipitates. These effects result in precipitate growth and transformation into thick η' plates with 15 nm in average length within the grain boundaries in the peak aged state, while the bulk contains mainly spherical GP zones with an average size of only 4 nm. Upon long time aging, GP zones can still form within the PFZs close to the interior of the grains with some solutes supply. This effect reduces the width of the PFZs to 8 nm compared to 11 nm in the initial state of aging.

During over aging (6 hours at 180°C after peak aged), some of the bulk precipitates transform to η' precipitates, resulting in a mixed distribution of GP zones and η' precipitates in the bulk.

On the grain boundaries where the solute content exceeds that in the bulk, the growing of η' phase keeps on trapping solutes from the grain boundaries and from the matrix. This makes the η' phase grows to a size of around 40 nm in length on the grain boundaries compared to an average size of 6 nm in the bulk.

Cu has been known to be an essential addition for improving fracture toughness and stress corrosion cracking resistance in Al-Zn-Mg alloys [START_REF] Maloney | Effects of Cu on precipitation in Al-Zn-Mg alloys, Materials science forum[END_REF][START_REF] Starke | Application of modern aluminum alloys to aircraft[END_REF][START_REF] Polmear | Aluminium Alloys--A Century of Age Hardening[END_REF]. In the present work, a much narrower PFZs width of ~10 nm in the Al-Zn-Mg-Cu alloy is observed compared to ~100 nm in the Al-Zn-Mg alloy in the peak aged state as previous study shown [START_REF] Ogura | Atom probe tomography of nanoscale microstructures within precipitate free zones in Al-Zn-Mg(-Ag) alloys[END_REF][START_REF] Ogura | Quantitative characterization of precipitate free zones in Al-Zn-Mg(-Ag) alloys by microchemical analysis and nanoindentation measurement[END_REF]. We consider the reason to this is the presence of Cu, which possess attractive interactions with Zn and Mg atoms and vacancies, thus to likely enhance the formation of GP zones in the vicinity of grain boundaries and hence makes the PFZs width narrower. These soft PFZs, that can localize strain, has an adverse effect on fracture toughness and stress corrosion cracking resistance. After quenching, cold working prior to aging can also support an increase of resistance against grain boundary embrittlement since the presence of dislocation before aging might introduce a trapping of solutes in the bulk due to segregation to the dislocations [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF], thus balancing the effect of segregation to the grain boundaries.

Another opportunity to improve the stress corrosion resistance and avoid the weakness of PFZs is alloying with beneficial solutes that can be rejected by the grain boundaries due to the segregation of Zn and/or Mg and/or Cu (anti-segregation). Solute segregation to the grain boundaries can influence the cohesion at grain boundaries and raise the grain boundary embrittlement susceptibility as the previous results considered [START_REF] Kuzmina | Grain boundary segregation engineering and austenite reversion turn embrittlement into toughness: Example of a 9wt.% medium Mn steel[END_REF][START_REF] Buban | Grain boundary strengthening in alumina by rare earth impurities[END_REF][START_REF] Song | Stress corrosion cracking and hydrogen embrittlement of an Al-Zn-Mg-Cu alloy[END_REF][START_REF] Lejček | Solute segregation and classification of [100] tilt grain boundaries in α-iron: consequences for grain boundary engineering[END_REF]. Mg segregation can also enhance hydrogen absorption as a result of the Mg-H interaction, resulting in H embrittlement and leading to the grain boundary embrittlement [START_REF] Song | Stress corrosion cracking and hydrogen embrittlement of an Al-Zn-Mg-Cu alloy[END_REF]. Anti-segregation of beneficial solutes could be an interesting strategy to occupy atomic positions within PFZs to prevent their weakening. However, more evidence is required for confirmation these suggestions.

Conclusions

Atom probe tomography has been successfully used to study the atomic-scale mechanism associated with the effects of segregation on grain boundary precipitation and PFZs formation in a model Al-Zn-Mg-Cu alloy during aging. The present observations suggest that segregation have an important influence on grain boundary precipitation. Our experimental evidence leads to the following conclusions:

1. Solute segregate to the grain boundaries with a grain boundary excess of about 2 for Zn 2 for Mg, and 3 for Cu after solution heat treated at 475°C. The local increase of solute composition brings the local composition of the grain boundaries closer to the spinodal decomposition point so that compositional fluctuation on the grain boundaries lead to

Fig. 1 (

 1 b) displays a 2D desorption map directly extracted from the APT dataset, (111) poles of the top and bottom grains are indexed as the dashed circle shows.

Fig. 1 .

 1 Fig. 1. Atom probe tomography analysis for a sample in the as-quenched state: (a) EBSD IPF map showing the grain boundary (GB) for site-specific APT tip preparation; (b) Desorption map showing indexed crystallographic poles of two grains; (c) Atom maps of all elements in the as-quenched state; (d) Composition profile across the grain boundary in a 20nmdiameter cylinder. (Al, Zn, Mg, and Cu are depicted in grey, dark cyan, olive, and dark red, respectively.)

Fig. 2 .

 2 Fig. 2. Atom probe tomography analysis after aging for 0.5 hours at 120°C. (a) Precipitates visualized by adding isocomposition surfaces of 6 at% Zn; (b) Distribution of solutes within the grain boundary (of the region shown in a as indicated by the red arrow); (c) 1D composition profile showing the chemistry of the selected grain boundary precipitate (GBP) (of the region shown in b as indicated by dashed red circle); (d) 1D composition profile across the grain boundary was computed for a region between grain boundary precipitates, along a 25nm-diameter cylindrical region-of interest. (Al, Zn, Mg, and Cuare depicted in grey, dark cyan, olive, and dark red, respectively.)

Fig. 3 .

 3 Fig. 3. Atom probe tomography analysis showing the nanostructure after aging for 2 hours at 120°C. (a) Precipitates with 8 at% Zn iso-composition surfaces; (b) Close up map of the grain boundary precipitates (of the region shown in a as indicated by the solid red rectangle); (c) 1D composition profile showing the chemistry of the selected grain boundary precipitate (of the region shown in b as indicated by the dashed red rectangle). (Al and Zn are depicted in grey and dark cyan, respectively.)

Fig. 4 .

 4 Fig. 4. Atom probe tomography analysis taken on a specimen after aging for 24 hours at 120°C. (a) Precipitates highlighted in terms of 10 at% Zn iso-composition surfaces. (b) Close up image of one precipitate in the bulk, taken with a view along the {111} planes of the Al matrix (of the region in a indicated by the solid yellow rectangle); (c) 1D composition profile showing the chemistry of the selected bulk precipitate; (d) Image of the precipitates within the grain boundary (of the region shown in a as indicated by the solid red rectangle); (e) 1D composition profile showing the chemistry of the selected grain boundary precipitate (of the region shown in d as indicated by dashed red rectangle); (f) Electron micrograph shows the comparable size of the precipitates and of the PFZs in the peak aged state. (Al and Zn are depicted in grey and dark cyan, respectively.)

Fig. 5 .

 5 Fig. 5. Atom probe tomography analysis in the overaged state: (a) Precipitates highlighted in terms of 10 at% Zn isocomposition surfaces; (b) 1D composition profile showing the chemistry of the selected grain boundary precipitate (of the region shown in a as indicated by solid red rectangle); (c) 1D composition profile showing the chemical composition of the selected bulk precipitate (of the region shown in a as indicated by solid yellow rectangle). (Al and Zn are depicted in grey and dark cyan, respectively.)

Fig. 6 .Fig. 7 (

 67 Fig. 6. The three regions in the peak aged samples (24 hours at 120°C). The images show region I (far and unaffected from the grain boundary), region II (adjacent to the grain boundary, essentially containing the PFZs), and region III (directly on the grain boundary). (Regions I, II, and III are depicted in blue, orange, and yellow, respectively.)

Fig. 7 (

 7 Fig. 7(b) presents the evolution of the precipitates' composition during aging within regions I,

Fig. 7 .

 7 Fig. 7. (a) Evolution of the global composition measured in the entire datasets (discs, the solid line is a guide to the eye) and the matrix composition (diamonds, the dashed line is a guide to the eye) for the different solutes as function of aging time (b) Evolution of the precipitates' composition (discs, the solid line is a guide to the eye) for the different solutes as a function of aging time. (Regions I, II, and III are depicted in blue, orange, and yellow, respectively.)

Fig. 8 .

 8 Fig. 8. Evolution of the precipitates during aging in terms of (a) The volume fraction; (b) The radius; (c) The number density per volume in the regions I and II as shown on left axis, and precipitate number density per area in region III as shown on right axis; (d) The average distance between precipitates. (Region I, II, and III are depicted in blue, orange, and yellow, respectively.)

Fig. 9 .

 9 Fig. 9. Evolution of the grain boundary composition: (a) Measured in the entire grain boundaries including the grain boundary precipitates; (b) Solute composition between grain boundary precipitates, along 25nm-diameter cylindrical regions of interest taken across the grain boundaries; (c) The available solute composition within thePFZs; (d) PFZ widths variation during aging. (Zn, Mg, and Cu are depicted in dark cyan, olive, and dark red, respectively.)

Fig. 10 .

 10 Fig. 10. Schematic diagrams of the grain boundary precipitation and PFZ formation during aging: (a) As-quenched state; (b) Initial state of aging (0.5 hours at 120°C); (c) Peak aged; (d) Overaged. (Zn, Mg, and Cu are depicted in dark cyan, olive, and dark red, respectively.)

Fig. 11 .

 11 Fig. 11. Schematic diagrams showing grain boundary composition C allowing the phase transformation during aging: (a) Segregation in the as-quenched state; (b) Phase decomposition in the initial state of aging (0.5 hours at 120°C); (c) η' formation in the peak aged state; (d) Precipitates coarsening in the overaged state.

Table 1 .

 1 Bulk chemical composition of a model Al-Zn-Mg-Cu alloy obtained from wet chemical analysis.

	Alloy	Zn	Mg	Cu	Zr	Fe	Si	Al
	(wt %)	6.22	2.46	2.13	0.155	0.021	<0.01	Rest
	The homogenized ingot was then hot-rolled from 40 to a 3 mm sheet at 450 °C and
	subsequently solution heat treated for 1 hour at 475 °C and then water quenched. The quenched
	material was immediately aged at 120 o C for different times and subsequently water quenched
	to room temperature. The aging times were 0.5 hours, 2 hours and 24 hours. Aging at 120°C

Acknowledgements

H. Zhao would like to acknowledge the Chinese Scholarship Council for the PhD scholarship granted to support this work. U. Tezins and A. Sturm are acknowledged for their support in the use of the atom probe and PFIB facility at MPIE. The authors are grateful for the Max-Planck Society and the BMBF for the funding of the Laplace and the UGSLIT projects respectively.

earlier formation of GP zones compared to the grain interior in the initial stage of aging at 120°C .

2.

With aging time increasing (between 2 hours and 24 hours at 120°C), solutes from the bulk keep segregating, forming a flux of solutes towards grain boundaries. The growth and transformation of GP zones within the grain boundaries are accelerated by the solute segregation from the matrix and enhanced solute diffusion along the grain boundaries.

3. Analysis of the ~10 nm wide precipitate-free zones (PFZs) adjacent to the enriched grain boundaries shows that the depletion zone is determined by (i) interface equilibrium segregation; (ii) formation and coarsening of the grain boundary precipitates and (iii) the diffusion range of the solutes in the matrix.

4. The precipitation kinetics, as observed by the volume fraction, size, and number density, are almost identical next to the depletion zone in the bulk and far inside the bulk grain remote from any grain boundary influence. This confirms that the region of influence of grain boundaries does not extend beyond the PFZs.


Guillaume Laurent

Christian Bonhomme

Improving solid-state NMR

ouverte pluridisciplinaire HAL, est destinée au dépôt

8 Non 2 PSNR

 82 Improving solid-state NMR sensitivity using instrumentation, fast acquisition and post-processing G. LAURENT, C. BONHOMME Laboratoire de Chimie de la Matière Condensée de Paris (LCMCP) 4 place Jussieu, 75005, Paris, France guillaume.laurent@sorbonne-universite.fr EUROMAR 2018, July 1 st to 5 th 2018, Nantes, c q u i s i t i o n 4 -P r o c e s s i n g Sensitivity 2 -Instrumentation Magic Angle Coil Spinning (MACS) [1-3] Micro-quantities (100 µg) PSNR / sqrt(t) / g → x Complementary techniques:• MACS: sensitivity increased by 8• NUS: sensitivity increased by 2• SVD: sensitivity increased by 2Next step:• Combining MACS + NUS + SVD: sensitivity increased by 32• Stanislas Von Euw and Virgile Barret-Vivin are thanked for providing the samples • Laboratory head is acknowledged for its confidence • Colleagues are recognized for their encouragements and fruitful discussions leopards.com.au PSNR max = H signal h noise _ peak _ peak /


Foreword: CERME10 in lovely Dublin

Susanne Prediger 1 , Viviane Durand-Guerrier 2 1 Dortmund, Germany, prediger@math.uni-dortmund.de 2 Montpellier, France, viviane.durand-guerrier@univ-montp2.fr Since its beginnings in 1998, ERME, the European Society for Research in Mathematics Education was dedicated to supporting the so-called "three C's": communication, cooperation, and collaboration among researchers in Europe and beyond. The major occasion for the ERME spirit to come to life is the biannual congress, CERME. In this year, the 10 th congress of ERME, CERME10, took place in Dublin from February 1 st to February 5 th , 2017.

The congress took place in Dublin in Croke Park, the stadium that is home of the Irish national sports of Gaelic football and hurling. Although the conference was bigger than ever, the 774 participants felt few compared to the 80 000 people who fit into the stadium on large sporting and other occasions. However, our local chair, Thérèse Dooley, her co-chair Maurice OReilly and all their colleagues did a fantastic job in making us feel at home and not lost in the huge venue. Their hospitality and engagement were praised by all participants.

The program of the congress was organized by the International Program Committee, chaired by Ghislaine Gueudet and the vice-chair Andreas Eichler in a very well structured, transparent and highly efficient way. Under their guidance, the IPC developed a substantial program with two very interesting plenaries, one presented by Elena Nardi (entitled "From Advanced mathematical thinking to university mathematics education: A story of emancipation and enrichment") and the other by Lieven Verschaffel (entitled "Towards a more comprehensive model of children's number sense"). In a panel on "Solid findings in mathematics education: What are they and what are they good for?" this 'hot' topic was discussed from different perspectives. Marianna Bosch, Tommy Dreyfus, Caterina Primi, and Gerry Shiel made up the panel. All of the plenary activities contributed substantially to the success of the conference.

However, the core and the heart of each CERME are the seven sessions in the Thematic Working Groups, which offer the main place for the spirit of inclusion realized in communication and cooperation. The 23 Thematic Working Groups were organized by 84 group leaders, an impressive number of people who invest their energy and time in the success of the congress. Several external conference organizers expressed their surprise that during the sessions, nobody was wandering around in the corridors. Of course not, we said, they are communicating and cooperating! And we become aware again that this intensity of work is specific, and perhaps even unique, to CERME.

Most of the CERME group leaders have taken this responsibility for several years and have established a long-term collaboration with substantial academic outcomes. This group of people engaged in the enormous effort of managing the process of quality development for 474 submitted papers and 94 posters, numbers much larger than ever before. countries in terms of numbers of participants were Germany (127), United Kingdom (60), Norway (55), France (47), Italy (47), Ireland (41), Spain (39), Sweden (38), Israel (32), and the US (30). Austria, Belgium, Croatia, Cyprus, Czech Republic, Denmark, Faroe Islands, Finland, Greece, Hungary, Iceland, Kosovo, Malta, Netherlands, Poland, Portugal, Russia, Slovakia, Switzerland, Turkey, and Ukraine were included in the European countries. Among the non-European countries were Algeria, Argentina, Australia, Bangladesh, Brazil, Cameroon, Canada, Chile, Hong Kong, Kenya, Iran, Japan, Lebanon, Malawi, Mexico, New Zealand, Nigeria, Singapore, South Africa, Thailand, and Tunisia. It must be the specific style of the congress and the ERME spirit which attracts so many people from all over the world! With the increasing numbers and diversity, the challenge of compiling proceedings is getting more and more complex. We thank the chairs who served as editors for this complex process and for finalizing it so quickly. Such a huge and complex congress as CERME could not be conducted without the engagement of more than 15% of all participants (including TWG leaders, IPC members, LOC members and ERME board members). We thank everybody who has contributed to the ongoing work behind the scenes which allowed the congress to be a real success. Specific thanks go to Ghislaine Gueudet, Andreas Eichler, Thérèse Dooley and Maurice OReilly for their hard work with a wonderful outcome.

We encourage interested researchers to meet us at the next CERME that will take place from February 5 th to February 10 th 2019, in Utrecht (the Netherlands).

Susanne Prediger, ERME President since February 2017

Viviane Durand-Guerrier, ERME President until February 2017

Proceedings of CERME10