Jean Auriol
email: jean.auriol@mines-paristech.fr

Florent Di Meglio
email: florent.dimeglio@mines-paristech.fr

Trajectory tracking for a system of two linear hyperbolic PDEs with uncertainties

Keywords: Hyperbolic Partial Differential Equations; Uncertain linear system, Robustness, Tracking, Backstepping

) it is then possible to derive sufficient conditions that guarantee robustness. Using the same target system we derive a simple algorithm that ensures robust output tracking.

INTRODUCTION

This article details the robustness properties of the controller proposed in [START_REF] Vazquez | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF]). This control law (applied at only one boundary) ensures, in the ideal case, the stabilization of a system of two heterodirectional linear first-order hyperbolic Partial Differential Equations (PDEs) in minimum time. This article provides sufficient conditions that guarantee the exponential stability of the controlled system in presence of uncertainties. Moreover, we solve the problem of output tracking for this uncertain system.

Most physical systems involving a transport phenomenon can be modeled using hyperbolic partial differential equations (PDEs): traffic flow [START_REF] Amin | On stability of switched linear hyperbolic conservation laws with reflecting boundaries[END_REF]), heat exchangers [START_REF] Xu | Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems[END_REF]), open channel flow [START_REF] Coron | A lyapunov approach to control irrigation canals modeled by saint-venant equations[END_REF], [START_REF] De Halleux | Boundary feedback control in networks of open channels[END_REF]) or multiphase flow [START_REF] Di Meglio | Dynamics and control of slugging in oil production[END_REF], [START_REF] Djordjevic | Boundary actuation structure of linearized two-phase flow[END_REF], [START_REF] Dudret | Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models[END_REF]).

The backstepping approach [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF]; [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]; [START_REF] Vazquez | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF]) has enabled the design of stabilizing full-state feedback laws for these systems. These controllers are explicit, in the sense that they are expressed as a linear functional of the distributed state at each instant. The (distributed) gains can be computed offline. If the gap between backstepping design and existence results for stabilizing controllers is now partially bridged [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic pdes[END_REF]), many properties (including robustness properties) of the backstepping controllers remain unknown.

A natural question towards an engineering use of backstepping is whether these control designs are robust to the presence of uncertainties in the system parameters?

To investigate this question, one can rely on the numerous contributions focusing on stability conditions [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2× 2 hyperbolic systems over a bounded interval[END_REF]; [START_REF] Diagne | Lyapunov exponential stability of 1-d linear hyperbolic systems of balance laws[END_REF]).

The main contribution of this paper is to derive sufficient robustness conditions guaranteeing the exponential stability of an uncertain system subject to a backstepping controller. Moreover, we derive a control law that guarantees the (robust) output tracking of the uncertain system to a reference signal χ(t).

Our approach is the following. Using two successive Volterra transformations, the uncertain system combined with a nominal control law is mapped to a target system with a well-known structure. The main idea is to use backstepping-like transformation not only for control design, but also for analysis. Then, using existing results, we derive sufficient conditions that guarantee exponential stability. For the tracking problem, we derive from the obtained target system an integral equation satisfied by the output. Analyzing the properties of this integral equation we derive, using a Recursive Least Squares algorithm, a simple control law that ensures robust exponential tracking.

The paper is organized as follow. In Section 2 we introduce the model equations and the notations; in particular we distinguish the nominal system (that was used to design the controller) and the real system with uncertainties. In Section 3, using the backstepping approach and successive Volterra transformations, we present a robustness result as a sufficient condition that guarantees the exponential stability of the uncertain system. Some complementary remark are given in subsection 3.5. In section 4 we focus on the tracking problem, expressing the output as the solution of an integral equation. An algorithm ensuring robust output tracking is then presented in Section 5. Finally some simulation results are given in Section 6.

PROBLEM DESCRIPTION

We first consider the nominal-ideal problem consisting of the following 2-states linear hyperbolic system

2.1 Nominal problem u t (t, x) + λu x (t, x) = σ +-v(t, x) (1) v t (t, x) -µv x (t, x) = σ -+ u(t, x) (2) evolving in {(t, x)| t ≥ 0, x ∈ [0, 1]}, with the following linear boundary conditions u(t, 0) = qv(t, 0), v(t, 1) = ru(t, 1) + U (t)
(3) The inside-domain coupling terms σ -+ and σ +-, the boundary coupling terms q and ρ and the velocities λ and µ are assumed to be constant. Moreover, we assume that -µ < 0 < λ and q = 0 (4) The initial conditions denoted u 0 and v 0 are assumed to belong to L 2 ([0, 1]). For this system, a feedback control law has been designed in [START_REF] Coron | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF]). This control law ensures the stabilization of the system in finite time. More precisely, the following theorem holds. Theorem 1. [START_REF] Coron | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF]). Consider system (1)-( 2) with boundary conditions (3) and the following feedback law

U (t) = -ru(t, 1) + 1 0 [K(1, ξ)u(t, ξ) + L(1, ξ)v(t, ξ)]dξ
(5) where the kernels K and L are defined by the following well-posed system of equations µK x (x, ξ) -λK ξ (x, ξ) = σ -+ L(x, ξ) (6) µL x (x, ξ) + µL ξ (x, ξ) = σ +-K(x, ξ) (7) and the following set of boundary conditions

K(x, x) = - σ -+ λ + µ L(x, 0) = qλ µ K(x, 0) (8) Then v(t, 0) ≡ 0 if t ≥ t F = 1 λ + 1 µ .
In what follows, we are interested in the robustness of the above controller to uncertainties in the system parameters.

System under consideration

We now consider the following uncertain system

u t (t, x) + λu x (t, x) = σ+-v(t, x) (9) v t (t, x) -μv x (t, x) = σ-+ u(t, x) (10) evolving in {(t, x)| t > 0, x ∈ [0, 1]}, with the following linear boundary conditions u(t, 0) = qv(t, 0), v(t, 1) = ru(t, 1) + δ U U (t) (11) where λ = λ + δ λ , μ = µ + δ µ (12) σ+-= σ +-+ δ σ +-, σ-+ = σ -+ + δ σ -+ (13) q = q + δ q , r = r + δ r (14)
with -μ < 0 < λ and q = 0. The terms δ λ and δ µ represent uncertainties on the velocities whereas the terms δ σ -+ , δ σ +-, δ q and δ r represent uncertainties on the coupling parameters. The term δ U = 0 represents an uncertainty on the actuation. All of these uncertainties are assumed constant.

Formulation of the problem

The first goal of this article is to derive sufficient conditions on the uncertainties for exponential stability of system (1)-( 3) with the feedback law (5). This is the topic of Section 3. Then we derive a control law that ensures the exponential convergence of the output v(t, 0) to a reference continuous function χ(t). This is the topic of Section 4.

ROBUSTNESS: GENERAL SUFFICIENT CONDITIONS FOR A SYSTEM OF TWO EQUATIONS

In this section, we present the main new idea of the paper. We first apply the classical backstepping transformation to the uncertain system ( 9)-( 11). Because of the uncertainties, there remain integral coupling terms at the boundary of the target system. We analyze these by means of a second Volterra transformation. This new approach highlights the potential of backstepping as an analysis tool, rather than only a control design tool. With these two successive transformations, system (9)-( 11) is mapped to a system with a simpler structure. Then, it becomes possible, using existing results, to derive sufficient conditions for robustness.

First Volterra transformation: Removing inside-domain couplings

We first consider the following classical Volterra transformation from [START_REF] Coron | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF])

α(t, x) = u(t, x) - x 0 (M uu (x, ξ)u(ξ) + M uv (x, ξ)v(ξ))dξ (15) β(t, x) = v(t, x) - x 0 (M vu (x, ξ)u(ξ) + M vv (x, ξ)v(ξ))dξ (16)
where the kernels M uu , M uv , M vu , M vv are defined on

T = {(x, ξ) ∈ [0, 1] 2 | ξ ≤ x} by the following set of hyperbolic PDEs: 0 = λM uu x (x, ξ) + M uu ξ (x, ξ) λ + M uv (x, ξ)σ -+ (17) 0 = λM uv x (x, ξ) -M uv ξ (x, ξ)μ -M uv (x, ξ)σ +- (18) 0 =μM vu x (x, ξ) -M vu ξ (x, ξ) λ -M vv (x, ξ)σ -+ (19) 0 =μM vv x (x, ξ) + M vv ξ (x, ξ)μ -M vu (x, ξ)σ +- (20
) with the following set of boundary conditions: [START_REF] Coron | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF]), this transformation is well-posed, i.e there exists M uu , M uv , M vu and M vv in L ∞ (T ). Moreover these kernels are invertible and we denote N αα , N αβ , N βα and N ββ the inverse kernels.

M vu (x, x) = - σ-+ λ + μ , μM vv (x, 0) = λqM vu (x, 0) (21) M uv (x, x) = - σ+- λ + μ , μM uv (x, 0) = λqM uu (x, 0) (22) As proved in
Remark 1. The inverse transformation of ( 15)-( 16) can be written u(t, x) = α(t, x)

+ x 0 (N αα (x, ξ)α(ξ) + N αβ (x, ξ)β(ξ))dξ (23) v(t, x) = β(t, x) + x 0 (N βα (x, ξ)α(ξ) + N ββ (x, ξ)β(ξ))dξ (24)
The transformation ( 15),( 16) maps the initial system ( 9)-( 11) to the following target system

α t (t, x) + λα x (t, x) = 0 (25) β t (t, x) -μβ x (t, x) = 0 (26) with the following linear boundary conditions α(t, 0) = qβ(t, 0) (27) β(t, 1) = Aα(t, 1) - 1 0 L α (ξ)α(t, ξ) + L β (ξ)β(t, ξ)dξ (28) with A = r + δ r -δ U r (29) L α (ξ) = N βα (1, ξ) -AN αα (1, ξ) -δ U K vu (1, ξ) -δ U 1 ξ N αα (ν, ξ)K(1, ν) + N βα (ν, ξ)L(1, ν)dν (30) L β (ξ) = N ββ (1, ξ) -AN αβ (1, ξ) -δ U K vv (1, ξ) -δ U 1 ξ N αβ (ν, ξ)K(1, ν) + N ββ (ν, ξ)L(1, ν)dν (31)
Proof 1. The result directly follows from inserting ( 23),( 24) into ( 9),(10)

Second Volterra transformation: removing the integral terms

In this section, we analyze the stability of ( 25)-( 28). Denoting

Ū (t) = - 1 0 L α (ξ)α(t, ξ) + L β (ξ)β(t, ξ)dξ (32)
boundary condition (28) rewrites β(t, 1) = Aα(t, 1) + Ū (t) (33) This highlights the fact that ( 25)-( 28) has the structure of a closed loop system with (32) as the control law. We now derive the backstepping transformation that would have resulted in this closed-loop system. It reads

Φ(t, x) = α(t, x) (34) Ψ(t, x) = β(t, x) - x 0 (P α (x, ξ)α(ξ) + P β (x, ξ)β(ξ))dξ (35)
where the kernels P α and P β are defined on

T = {(x, ξ) ∈ [0, 1] 2 | ξ ≤ x} by the following set of hyperbolic PDEs μP α x -P α ξ λ = 0 (36) μP β x + P β ξ μ = 0 (37)
with the boundary conditions

P α (1, ξ) = -L α (ξ), P β (1, ξ) = -L β (ξ) (38) P α (x, 0) λq = μP β (x, 0) (39)
Lemma 1. Consider system (36)-( 39). There exists a unique solution P α and P β in L ∞ (T ).

Deriving ( 34) and ( 35) with respect to space and time leads to the following target system Φ t (t, x) + λΦ x (t, x) = 0 (40) Ψ t (t, x) -μΨ x (t, x) = (μ + λ)P α (x, x)Φ(t, x) (41) with the boundary conditions Φ(t, 0) = qΨ(t, 0) (42) Ψ(t, 1) = AΦ(t, 1) (43) The stability of this system is much easier to analyze than ( 25)-( 28), as illustrated in the next section.

General sufficient robustness conditions

Let us introduce the following notations

χ = (Φ Ψ) T , Λ = diag{ λ, -μ} (44)
S 0 = 1 -q 0 0 , S 1 = 0 0 -A 1 (45)
and

M (x) = 0 0 (μ + λ)P α (x, x) 0 (46)
Applying the result given in ( [START_REF] Diagne | Lyapunov exponential stability of 1-d linear hyperbolic systems of balance laws[END_REF], Theorem 1)) yields the following theorem Theorem 2. The system ( 9)-( 11) with control law (5) remains exponentially stable in presence of uncertainties if there exist ζ > 0, π 1 > 0 and π 2 > 0, such that

(1) The boundary quadratic form

χ T (t, 1) χ T (t, 0) ΛΠ(1) 0 0 -ΛΠ(0) χ(t, 1) χ(t, 0) where Π(x) = π 1 e -λζx 0 0 π 2 e μζx (47)
is positive definite under the constraint of the linear boundary condition S 0 w(t, 0) + S 1 w(t, 1) = 0, ∀t ≥ 0 along the solutions of system (1)-( 3)

(2) The matrix -ζ λ 0 0 μ Π(x)+M T (x)Π(x)+Π(x)M (x) is negative definite ∀x ∈ [0, 1] Proof 2. (Sketch)
The complete proof is given in [START_REF] Diagne | Lyapunov exponential stability of 1-d linear hyperbolic systems of balance laws[END_REF]). The LMIs given above are obtained deriving the following Lyapunov candidate function

V (t) = 1 0 χ T (t, x)Π(x)χ(t, x)dx (48)
For general systems of the form ( 9)-( 11) it is rather clear that more explicit conditions can be derived only on a caseby-case basis, when the specific structure or the numerical values of the different matrices are specified. In the next section we analyze further this sufficient condition in a specific case.

Sufficient condition when A = 0

We now assume that A = 0, which is the case when the right boundary condition is perfectly known. Then, the following theorem derived from [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2× 2 hyperbolic systems over a bounded interval[END_REF]), holds.

Theorem 3. The system (9)-( 11) is exponentially stable if

q0 1 0 (| ( λ + μ) μ P α (ξ, ξ)|)dξ < 1 (49)
This condition is equivalent to

1 0 (|q 0 L α (ξ)| + |L β (ξ)|)dξ < 1 (50)
where L α and L β are defined by ( 30)-( 31).

Proof 3. This proof is derived from ((Bastin and Coron, 2011, Theorem 1, Theorem2)). A sufficient condition so that the system is ( 9)-( 11) exponentially stable is q

< 1 η(1)
where η is the solution (defined on [0, 1]) of

(C) : η (x) = | ( λ + μ) μ P α (x, x)|, η(0) = 0 (51)
Integrating ( 36) and (37) along their characteristic lines and using the corresponding boundary conditions yields: 52) Using this expression and integrating equation (51) yields condition (50) Remark 2. Comparing with Theorem 2, condition (50) only requires an inequality to check as opposed to LMIs to be solved. In Section 6, we illustrate the simplicity of use of this equation by means of a toy problem.

P α (x, x) =      L α ((1 + λ μ )x - λ μ ) if x ≥ λ μ + λ μ λq 0 L β (1 -( μ λ + 1)x) else (

Complementary remarks

In this section, we give some fundamental complementary remarks that are not detailed here due to the space constraints.

Bounds for the uncertainties As is, Theorem 2 gives a sufficient condition for stability for a fixed value of each uncertain parameter. The following theorem gives a result for uncertainty within a certain interval for one uncertain parameter. Theorem 4. Consider system (9)-( 11) where we have one uncertainty δ ∈ [-δ max , δ max ] corresponding to only a single parameter. If ( 9)-( 11) remains exponentially stable for δ = -δ max and δ = δ max then it remains exponentially stable for any δ ∈ [-δ max , δ max ] Proof 4. We only give here a sketch of the proof due to lack of space: first, using Lumer-Phillips' Theorem (see [START_REF] Lumer | Dissipative operators in a banach space[END_REF]) for instance), one can prove that the operators corresponding to extreme values of the uncertainty and their adjoints are dissipative and closed. Then, one can show by computation that dissipativity and closure are preserved for uncertainties within the extreme values. Finally, Lumer-Phillips' theorem yields the result.

Let us now consider system (9)-( 11) with uncertainties on each parameters. We have no more than 7 uncertain coefficients. We can rewrite this set of coefficients (x 1 , . . . , x 7 ) (for instance x 1 = σ +-, x 2 = λ). The order in which these coefficients are taken does not matter. Let us now denote for each coefficient x i the known bound for the corresponding uncertainty as δ i . Finally, we denote I = {-1, 1}. We then have the following result: Corollary 1. If for all set r = (r 1 , r 2 , . . . r 7 ) ∈ I 7 , system (9)-( 11) where the coefficients (x 1 . . . x 7 ) are replaced by (x 1 + r 1 δ 1 , . . . x 7 + r 7 δ 7 ) satisfies Theorem 2, then for all s i ∈ [-1, 1] system ( 9)-( 11) where the coefficients (x 1 . . . x 7 ) are replaced by (x 1 + s 1 δ 1 , . . . , x k + s 7 δ 7 ) is exponentially stable. Remark 3. This theorem means that knowing that the system remains exponentially stable for set of uncertainties located on the 2 7 vertices of an 7-orthotope (i.e a hyperrectangle) implies that the system remains exponentially stable for uncertainties inside the 7-orthotope.

Robustness of the output feedback controller

We have made as an implicit assumption that the distributed states u and v are available as measurements. Actually, in [START_REF] Vazquez | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF]), the control law ( 5) is combined with an observer to synthesize an output feedback controller that requires only measurements from the uncollocated boundary. The robustness of the controller does not imply that of the output feedback controller. We also need to prove that the observer remains robust to the model uncertainties, but this topic is out of the scope of this paper.

TRACKING PROBLEM: ANALYSIS OF THE

OUTPUT V (T, 0)

In this section we focus on the following trajectory tracking problem. Given χ(t) a reference continuous function, we want to find a control law U (t) so that the output v(t, 0) converges exponentially to χ(t). In the case of the nominal problem it is proved in [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF]) that for (1)-( 2) with boundary conditions (3) and the following feedback law

U (t) = -ru(t, 1)+χ(t + 1 µ ) + 1 0 [K(1, ξ)u(t, ξ) + L(1, ξ)v(t, ξ)]dξ (53) Then v(t, 0) ≡ χ(t) if t ≥ t F = 1 λ + 1 µ .
In this section we analyze the properties of the output v(t, 0) of the uncertain system with the control law (53) for a particular reference trajectory χ(t). More precisely, we prove that if χ is a polynomial function of time, then x(t) = v(t, 0) is a polynomial function of the same order.

An integral equation for v(t, 0)

First, notice that following the same approach as above with control law (53) the uncertain system ( 9)-( 11) with control law ( 53) is equivalent to ( 40)-( 43) where ( 43) is replaced by

Ψ(t, 1) = AΦ(t, 1) + δ U χ(t + 1 µ ) (54)
Using the method of characteristics (i.e integrating ( 40)-( 41) along their characteristic lines and using the boundary conditions ( 42)-( 54)), yields

Φ(t, x) = Φ(t - x λ , 0) = qΨ(t - x λ , 0) (55) Ψ(t + 1 μ , 0) = Ψ(t, 1) + 1 μ 0 ( λ + μ)P α (1 -μs, 1 -μs)Φ(t + s, 1 -μs)ds (56)
Combining these two equations along with the boundary condition (39) and denoting y(t) = Ψ(t, 0) and R(s) = -q( λ + μ)P α (1 -μs, 1 -μs) yields

y(t + 1 μ ) =Aqy(t - 1 λ ) + δ U χ(t + 1 µ ) - 1 μ 0 R(s)y(t + s - 1 λ + μ λ s)ds (57)
In the following we will denote C 1 = 1 -Aq + 1 μ 0 R(s)ds and we assume that C 1 = 0. We make the following natural assumption. Assumption 1. The uncertain system is assumed to remain exponentially stable, i.e. we assume that the conditions of Theorem 2 are satisfied. It yields that the solutions of (57) with χ(t) ≡ 0 converge exponentially to zero.

Constant reference trajectory

We start by the simplest case assuming that χ(t) = a 0 for all t > 0, with a 0 ∈ . Denoting z(t) = y(t) -δ U C1 a 0 yields

z(t + 1 μ ) =Aqz(t - 1 λ ) - 1 μ 0 R(s)z(t + s - 1 λ + μ λ s)ds
Using Assumption 1, we have that z converges exponentially to 0 and that consequently y converges exponentially to δ U a0 C1 .

χ(t) is polynomial function

We now assume that χ(t) = n k=0 a k t k (a k ∈ ) is a polynomial function. We prove that it implies that y(t) converges exponentially to a polynomial function of the same order. To do so, let us first notice that Equation ( 57) is linear. We recall the following lemma. Lemma 2. If a C 1 function y(t) is such that y (t) converges exponentially to 0 then y converges exponentially to a constant. More precisely, if a C k function y(t) is such that y (k) (t) converges exponentially to a constant then y converges to a polynomial function of degree k. Proof 5. We have that

∀t > 0, y(t) -y(0) = t 0 y (τ )dτ (58)
Since y converges exponentially to 0, there exist γ > 0 such that y (t)e γt converges to 0. Consequently the integral t 0 y (τ )dτ converges exponentially to a constant. So does y(t).

If y (k) converges exponentially to a constant d 0 , then (y (k-1) (t) -d 0 t) converges exponentially to a constant d 1 . So, (y (k-1) (t) -d 0 t -d 1 ) converges exponentially to zero. By recursion, one can easily prove that y converges exponentially to a polynomial function of degree k Using the linearity, we only need to prove that ∀k > 0, if χ(t) = t k then y converges to a polynomial function of degree k. Let us formally derive k times equation ( 57) with respect to time (we look for solutions that have the same regularity as χ). It yields

y (k) (t + 1 μ ) =Aqy (k) (t - 1 λ ) + δ U k! - 1 μ 0 R(s)y (k) (t + s - 1 λ + μ λ s)ds (59)
Consequently (using the previous subsection), y (k) converges exponentially to k!δ U C1 . Using Lemma 2 yields that y converges to a polynomial function of degree k. Consequently, we have the following theorem. Theorem 5. If χ(t) = n k=0 a k t k , then y(t) converges exponentially (i.e is equivalent) to the function n k=0 b k t k . Moreover, injecting this expression in (57) yields to an explicit expression of each b k in function of the parameters of the problems (µ, μ, A, λ, R...).

Recursive formulation of polynomial outputs

In this subsection we consider the two polynomial functions χ 1 (t) = t n and χ 2 (t) = t n-1 (n > 0). As seen above the corresponding outputs y 1 (t) and y 2 (t) converge to polynomial functions that can respectively be expressed as

P 1 (t) = n k=0 a k t k , P 2 (t) = n-1 k=0 b k t k (60)
We then have the following theorem Theorem 6. The coefficients of P 2 can be expressed as functions of the coefficients of P 1 :

∀0 ≤ k ≤ n -1, b k = k + 1 n a k+1 (61)
Proof 6. The (recursive) proof is omitted due to space constraints.

Theorem 6 is extremely important since, combining it with linearity, it means that if one knows the polynomial function to which x converges when χ(t) = t n then,it is possible to know the output for any polynomial function of degree n or lower as input.

General expression of the output for

χ(t) = n k=0 a k t k
We assume here that χ(t) = n k=0 a k t k is a polynomial function of degree n. We denote A = (a 0 . . . a n )

T . Using Theorem 5, we have that the output x(t) converges to a polynomial function of degree n :

n k=0 b k t k . Denoting B = (b 0 . . . b n )
T one has, due to Theorem 5

B = F A (62)
where F is an upper triangular matrix depending only on the uncertainties. Moreover, Theorem 6 implies that F is entirely defined by its last column and that ∀ 1 ≤ i ≤ j ≤ n + 1,

F i,j = i j F i+1,j+1 (63)
Remark 4. F is always invertible since, as proved above, it is upper triangular and all its diagonal coefficients are non-null

If one is able to determine F , then, inverting it, it becomes possible to track any polynomial function. This is the purpose of the next section.

ROBUST OUTPUT TRACKING

The purpose of this subsection is for a given polynomial signal χ 0 (t) = n k=0 a k t k to derive the control law U (t) such that x(t) converges exponentially to χ 0 (t).

Identification of the matrix F

The first step is to identify the coefficients of the constant upper-triangular matrix F defined in section 4.4. To do so let us send χ(t) = χ 0 (t) as an input. As seen above the output v(t, 0) = x(t)converges exponentially to the polynomial function n j=0 b j t j . Using a Recursive Least Squares algorithm (RLS) on the measured output it is possible to identify the coefficients b j . We then denote B = (b 0 . . . b n )

T and A = (a 0 . . . a n )

T Using the results of the previous section yields B = F A (64) F is entirely defined by its last column:

F n : (F 1,n+1 . . . F n+1,n+1 ) T (65)
Consequently, knowing A and B, it becomes possible to explicitly determine F n and then F .

Tracking of χ 0 (t)

We now want x(t) to converge to χ 0 . Let us define the coefficients Q j as

∀0 ≤ j ≤ k Q j = ((F -1 ) (a 0 a 1 . . . a k )
T ) j (66) These coefficients are well defined since F is invertible. We define Q = (Q 1 , . . . Q n ) T . We now consider the input function χ 1 (t) defined by

χ 1 (t) = n j=0 Q j χ j (t) (67)
Using the property of linearity and the results of the previous sections, we have that x(t) converges to the polynomial function R(t) whose coefficients are defined by ∀0 ≤ j ≤ n R j = (F Q) j (68) It yields ∀0 ≤ j ≤ n, R j = a j and that consequently R = χ 0 . We then have the following theorem: Theorem 7. Considering an arbitrary polynomial function P (t) = n j=0 a j t j , and the input

χ(t) = n j=0 ((F -1 ) (a 0 a 1 . . . a n ) T ) j t j (69)
then the output v(t, 0) of system ( 9)-( 11) with the control law (5) converges exponentially to χ 0 .

Practical implementation

The proposed algorithm is divided in two parts

• First we send as an input χ 0 (t) and we use the RLS algorithm on the output. • Once we have enough points so that the estimation is accurate, we can compute F and χ 1 so that the output converges to χ 0 .

This approach is summarized on Figure 1

Extension to any continuous function

Let us consider an arbitrary continuous function f (t) defined on the compact [0, T ]. Using Stone-Weierstrass' theorem it is then possible to approximate it by a polynomial function P f . It is then sufficient to track this polynomial function P f . Another solution would be to consider sliding window so that F can be approximated by a polynomial function of small degree on each window. The size of the first window must be large enough to be sure that the RLS algorithm converges, in order to identify the correct F .

SIMULATION RESULTS

In this section we illustrate our tracking result with simulations on a toy problem. The numerical values of the parameters are as follow

λ = 1, µ = 1 σ +-= 1, σ -+ = 1 (70) q = -1, r = δ r = 0, δ q = -0.25, δ U = 1.1 (71) δ λ = 0.25, δ µ = 0.25, δ σ +-= -δ σ -+ = 0.25 (72)
For this set of uncertainties we get

1 0 (|q 0 L α (ξ)| + |L β (ξ)|)dξ = 0.121 < 1 (73)
Consequently, the sufficient condition provided by Theorem 2 is satisfied and the uncertain system is exponentially stable. We choose as reference χ 0 (t) = 0.0225t 2 -10 -3 t 3 so that χ 0 (0) = χ 0 (0) = χ 0 (20) = 0 and χ 0 (20) = 3. Figure 2 pictures the reference signal, and the output v(t, 0) in three different cases. The first one is using the backstepping controller without adaption (i.e the one defined by ( 5)).

The second one is the adaptive backstepping controller that is derived in the previous sections. Finally we consider a simple PI controller that is tuned to ensure the tracking in the nominal case. In the case of the adaptive controller, the data from the output are obtained using the RLS algorithm during the 10 first seconds of the simulation. In the same time the input is χ 0 (t). Since the system is slightly different from the nominal one, the output progressively diverges from the reference. At time t = 7, the matrix F is computed and the input is corrected to take into account uncertainties. Due to the dynamic of the system the effect on the output is visible 1 second later. As expected the output quickly converges to the reference signal.

CONCLUSION

Using the backstepping approach on an uncertain system of first-order hyperbolic linear PDEs controlled at one boundary, we have presented a change of coordinates that enables to derive sufficient conditions guaranteeing the The presented result raises several important questions about the properties of the backstepping controller. In particular, would it be possible to determine some of the uncertainties comparing the real output (obtained with the RLS algorithm) with the theoretical one?

The generalization of this result to derive sufficient conditions guaranteeing robustness of the output feedback controller for a general class of systems of hyperbolic PDEs will be the purpose of future contribution.

Fig. 2 .

 2 Fig.2. Time evolution of the output for different controllers exponential stability of the system. Then, expressing the output v(t, 0) as the solution of a simple integral equation and analyzing the properties of this integral equation, we have derived an algorithm based on a RLS method to track any continuous function.The presented result raises several important questions about the properties of the backstepping controller. In particular, would it be possible to determine some of the uncertainties comparing the real output (obtained with the RLS algorithm) with the theoretical one?The generalization of this result to derive sufficient conditions guaranteeing robustness of the output feedback controller for a general class of systems of hyperbolic PDEs will be the purpose of future contribution.

  . Remark 5. The time required on phase 1 strongly depends on the sampling rate of the measurement.

		|	(t)	+	U(t)	System	y(t)
				-			u(x,t)
	(t) 0 |	Analysis		Backstepping controller	v(x,t)
					RLS
	Fig. 1. Description of the algorithm


R Zouari

T Domenech

B Vergnes

E Peuvrel-Disdier

TIME-EVOLUTION OF THE STRUCTURE OF ORGANOCLAY/ POLYPROPYLENE NANOCOMPOSITES AND APPLICATION OF THE TIME/TEMPERATURE SUPERPOSITION PRINCIPLE

We investigated the rheological properties of nanocomposites composed of polypropylene, organoclay and maleic anhydride grafted polypropylene in small amplitude oscillatory shear.

Samples were prepared in two steps: a masterbatch was first obtained by melt extrusion and then diluted into polypropylene using an internal mixer. Three formulations were investigated.

The measurement of the storage and loss moduli evolution with time showed that these materials were not stable: the nanostructure obtained after steady shear continuously changed with time, due to the disorientation of the clay platelets and the build-up of a 3D network. The kinetics of the structure built-up (followed via the melt yield stress) showed a two-step process. This feature was found to be valid whatever the nanocomposite formulation. Such evolution of the structure is generally assumed to violate the time-temperature superposition principle. We demonstrate in this paper that the time-temperature equivalence always exists if the same nanostructure is probed. This was achieved by using different annealing times for different temperatures or annealing the samples at the highest temperature before measuring at lower values. The time-temperature equivalence evidences that the temperature does not induce any chemical change within the material whose properties remain governed by the same physical phenomena.

I. INTRODUCTION

Since the pioneering work of Toyota on polyamide nanocomposites, nanocomposites based on organically modified layered silicates (organoclays) have received a considerable attention for some years [START_REF] Alexandre | Polymer-layered silicate nanocomposites: preparation, properties and use of a new class of materials[END_REF]; [START_REF] Palvidou | A review on polymer-layered silicate nanocomposites[END_REF]; [START_REF] Paul | Polymer nanotechnology: nanocomposites[END_REF]]. Nanocomposites based on organoclays exhibit for very low clay loadings enhanced properties like mechanical properties [START_REF] Krishnamoorti | Structure and dynamics of polymer-layered silicate nanocomposites[END_REF]; [START_REF] Ren | Linear viscoelasticity of disordered polystyrenepolyisoprene block copolymer based layered-silicate nanocomposites[END_REF]], flame retardancy and increased thermal stability [START_REF] Gilman | Flammability and thermal stability studies of polymer-layered silicate (clay) nanocomposites[END_REF]], as well as lower permeability to gases [START_REF] Messersmith | Synthesis and barrier properties of poly(caprolactone)-layered silicate nanocomposites[END_REF]; [START_REF] Xu | New biomedical poly(urethane uera)-layered silicate nanocomposites[END_REF]] when compared to pure polymers. One of the key scientific and technical challenges in these materials is the control of the particulate structure to be reached in order to achieve the desired properties.

The common idea is that a stable fully dispersed state is necessary to obtain enhanced properties.

Due to its characteristics, there is a large interest in polypropylene based nanocomposites.

However, even though organomodified nanoclays are used, the dispersion in a non-polar matrix is difficult and necessitates to use a compatibilizer (polypropylene grafted with maleic anhydride (PP-g-MA) in most studies) [START_REF] Solomon | Rheology of Polypropylene/clay hybrid materials[END_REF]; [START_REF] Lertwimolnun | Influence of compatibilizer and processing conditions on the dispersion of nanoclay in a polypropylene matrix[END_REF]; [START_REF] Nazockdast | Linear and non-linear melt-state viscoelastic properties of polypropylene/organoclay nanocomposites[END_REF]].

The dispersion state and the nanocomposite structure can vary from simple intercalation of the silicate layers by the polymer chains to total exfoliation (individual layers). Generally, a mixed structure with intercalated tactoids and partially exfoliated platelets is observed, possibly forming a 3D network. The nanoclay dispersion state and associated nanocomposite structure can be characterized via complementary techniques, like X-ray scattering to evaluate the interlayer spacing. It can also be visualized by transmission electronic microscopy (TEM) even if this technique remains very local and not really adapted to observe a 3D structure. On another hand, rheometry constitutes a powerful technique to detect and characterize a percolated system as proven in the literature [START_REF] Krishnamoorti | Rheology of end-tethered layered silicate nanocomposites[END_REF], Loiseau and Tassin (2005), [START_REF] Vermant | Quantifying dispersion of layered nanocomposites via melt rheology[END_REF]; [START_REF] Cassagnau | Melt rheology of organoclay and fumed silica nanocomposites[END_REF]; [START_REF] Mobuchon | Structural analysis of non-aqueous layered silicate suspensions subjected to shear flow[END_REF]; [START_REF] Elasmi | Linear and nonlinear rheology of polymer/layered silicate nanocomposites[END_REF]]. In particular, linear viscoelastic measurements at low frequencies are sensitive to the interactions between particles inside the polymer and their evolution within the system.

In fact, a percolated network shows a solid-like behavior with a low frequency domain characterized by the presence of an elastic plateau and an increase of the complex viscosity. [START_REF] Lertwimolnun | Influence of compatibilizer and processing conditions on the dispersion of nanoclay in a polypropylene matrix[END_REF] were the first to propose to use the Carreau-Yasuda law with a yield stress to represent the viscosity variation at low frequency. They have shown that the yield stress value is correlated to the degree of exfoliation of the nanocomposite. [START_REF] Vergnes | The use of apparent yield stress to characterize exfoliation in polymer nanocomposites[END_REF] has recently shown the interest in using this parameter to describe the nanocomposite structure.

Although nanocomposite materials received a lot of interest in the last years, the exact nature of the nanocomposite structure is still an open question. Depending on the nanoclay state (from intercalated to fully exfoliated), the rheological behavior lies between a liquid-like and a solid-like one. The solid-like behavior is associated to the presence of a 3D structure where the nanoclays form a more or less soft network. This network can be seen as a nanoclay network formed by platelets and tactoids, as attractive forces between the clay platelets exist and play a role in the network properties. Moreover, it has been shown that this nanostructure can change with time or temperature, which renders the comprehension of the system more and more complicated. For sake of simplicity, the nanocomposite structure will be referred to nanoclay network or 3D network in the following.

Most of the studies dedicated to the investigation of the linear viscoelastic properties of nanocomposites assume that the sample structure does not change during small amplitude oscillatory shear experiments. This stability generally comes from chemical or thermal treatments of the samples [START_REF] Krishnamoorti | Rheology of end-tethered layered silicate nanocomposites[END_REF]; [START_REF] Nazockdast | Linear and non-linear melt-state viscoelastic properties of polypropylene/organoclay nanocomposites[END_REF]].

Under these conditions, no remarkable structural changes occur within the material, at least in the experiment time scale, for different temperatures. Thereby, some investigations showed that the phenomenological time-temperature superposition principle (TTS) applies and a mastercurve could be obtained to describe the rheological behavior of the nanostructured material on a broader frequency range than that allowed by the rheometer [START_REF] Solomon | Rheology of Polypropylene/clay hybrid materials[END_REF]]. To the best of our knowledge, studies dealing with time evolutive nanostructured materials did not mention any time-temperature equivalence and the authors investigated rheological features associated to the sample at a fixed temperature with well defined protocols [START_REF] Reichert | Morphological stability of Poly(propylene) nanocomposites[END_REF]; [START_REF] Ren | Disorientation kinetics of aligned polymer layered silicate nanocomposites[END_REF]; [START_REF] Mobuchon | Effect of flow history on the structure of a non-polar polymer/clay nanocomposite model system[END_REF][START_REF] Mobuchon | Structural analysis of non-aqueous layered silicate suspensions subjected to shear flow[END_REF]]. However, some authors found some failures by applying TTS, principally in the low frequency range [START_REF] Zhao | Rheological characterization of polystyrene-clay nanocomposites to compare the degree of exfoliation and dispersion[END_REF]; Treece and Oberhauser (2007a,b)]. [START_REF] Reichert | Morphological stability of Poly(propylene) nanocomposites[END_REF], [START_REF] Wagener | A rheological method to compare the degree of exfoliation of nano-composites[END_REF][START_REF] Wagener | A rheological method to compare the degree of exfoliation of nano-composites[END_REF]Drozdov et al. (2010) mentioned that the structure change within the nanocomposite leads to a violation of the TTS principle. [START_REF] Reichert | Morphological stability of Poly(propylene) nanocomposites[END_REF] have checked the validity of the TTS principle applied to extruded and injection molded PP nanocomposite samples with an unstable morphology and concluded that the TTS principle does not hold unless a thermal treatment (ageing) is applied to the samples at the highest tested temperature during 200 min, to ensure that they are in a thermodynamically stable state.

The aim of the present work is to report about the development of the nanostructure during an annealing process and the corresponding applicability of TTS principle. The conditions related to a successful application of the TTS principle will be presented as well as information about the mechanisms leading to the formation of a 3D network within the polymer matrix.

II. EXPERIMENTAL PART

A. Materials

The organoclay was provided by Laviosa Chimica Mineraria (Italy). The Dellite ® 67G organomodified montmorillonite (OMMT) was chosen for its compatibility with polyolefins.

Its surfactant is a dimethyl dihydrogenated tallow alkyl ammonium and its cation exchange capacity (CEC) is 115 meq/100 g. Scanning Electron Microscopy (SEM) observations of the Dellite ® 67G at different magnifications showed agglomerates with a diameter lying between 20 and 40 µm. It gives an idea of how important is the role of the dispersion process to reach a nanoscale structure, starting with 40 µm agglomerates to end up with single platelets of 1 nm in thickness and 500 nm in lateral dimensions.

The selected matrix was an injection grade isotactic polypropylene (PP) produced by LyondellBasell under reference Moplen HP400R (M w = 233 300 g/mol).

A polypropylene grafted with maleic anhydride (PP-g-MA) was used as a compatibilizer. The selected PP-g-MA is produced by Eastman as Epolene G-3015. Its maleic anhydride content is 3% in weight.

B. Samples preparation

The nanocomposites were obtained by dilution into the polypropylene matrix of masterbatches containing the organoclay. Two masterbatches were prepared by twin screw extrusion on a laboratory extruder Brabender PL2000, in the same processing conditions: processing temperature 200°C, flow rate 5 kg/h, screw speed 200 rpm. The first one contained only PP-g-MA (80 wt%) and OMMT (20 wt%), the second contained 20 wt% of OMMT, 40 wt% of PP-g-MA and 40 wt% of PP. The ratio PP-g-MA/OMMT was thus 4 and 2, respectively. These masterbatches were then diluted into the PP matrix using an internal mixer (Haake Rheomix 600) at 180°C, with a rotor speed of 100 rpm during 10 min. The final compositions of the three nanocomposites investigated in this study are indicated in Table 1.

Samples A and B have the same OMMT content (5 wt%) and differ by the PP-g-MA/OMMT ratio and thus the level of exfoliation. Samples B and C have got the same PP-g-MA/OMMT ratio, but different amounts of OMMT (5 and 7 wt%, respectively). In all cases, OMMT content is much higher than the percolation threshold, measured at 3 wt% for samples B and C. This threshold was not measured for sample A. It is probably slightly higher, as the exfoliation is expected to be lower, but smaller than 5%. All the samples extracted from the internal mixer were then compression moulded (at 180°C and 25 MPa for 10 min, followed by cooling down to room temperature) in order to obtain disks with controlled thickness (e = 1 mm) and diameter (D = 25 mm)

C. X-ray diffraction measurements

Wide Angle X-ray Diffraction (WAXD) experiments were conducted on the compressed samples with a Philips X'pert Pro diffractometer with Cu Kα radiation of wavelength 1.54 Ǻ.

The accelerating voltage was 40 kV. Diffraction spectra were obtained over 2θ range lying between 2 and 10° and the interlayer spacing d 001 was calculated using the Bragg's equation:

θ λ sin 2d =
, where λ is the wavelength.

D. Morphological characterization

Microscopic scale observations of the nanocomposites structure were achieved using a Scanning Electron Microscope (SEM) Philips XL30 ESEM, equipped with a LaB 6 cathode, under an acceleration voltage of 15 kV. Pictures have been acquired using a back scattered electron detector to allow high phase contrast between matrix and OMMT. Prior to observations, samples were successively polished to access core-structure on a flat surface (surface roughness less than 1 µm) and metallized (Au/Pd coating of 100 Å) to enhance samples electronic conductivity.

Observations at the nanoscopic scale were carried out with a Transmission Electron Microscope (TEM) Philips CM12, equipped with a LaB 6 cathode, under an acceleration voltage of 120 kV. Ultrathin sections of the samples were prepared at -100°C from the cross section of an injection moulded tensile bar (in the central zone), using an Leica Ultracut ultracryomicrotome mounted with Diatome Cryotrim and Cryo diamond knives. The sections, with nominal thickness of 50 nm, were transferred to 200 mesh copper grids and dried at room temperature using filter paper.

E. Rheological characterization

The rheological measurements were conducted on an ARES (TA Instruments) straincontrolled rotational rheometer. Measurements were performed at 180°C, 200°C and 220°C, using either a parallel plate or a cone and plate geometry, with a cone angle of 0.0401 rad.

Plate diameter was 25 mm and the gap for the parallel plate geometry was 0.5 mm. In order to minimize the deformation history and all phenomena associated to the preparation and the loading of the sample in the gap, a permanent shear was applied to the sample, prior to any measurement. It consisted in shearing the material at 1 s -1 during 155 s to destroy the initial structure and to reach a steady state. The applied shear rate was chosen to be large enough to break the filler network, but not too large in order to avoid any expulsion of the nanocomposite which could happen at shear rates larger than a few s -1 . The strain applied to the sample is much larger than what would be necessary to reach the steady state (30 strain units, for example, in Lertwilmolnum et al. (2007)). However the application of a larger strain has no effect since the steady state is reached. Then, the same shear rate was applied again, during the same time but in the reverse direction. This subsequent shear was essentially performed for experimental reasons. Indeed, the rotating plateau position must be accurately reinitialized in order to switch from the permanent shear mode to the oscillatory one. These conditions were selected as leading to similar structures whatever the nanocomposite initial state was. This type of protocol, but with different values of shear rate and time, is often used to obtain a reproducible initial state for all the samples [Lertwimolnun et al. (2007), [START_REF] Kim | Processing-property relationships of polycarbonate/graphene composites[END_REF]]. The rheological measurements were carried out in the linear domain, determined by monitoring storage modulus in dynamic strain sweep experiments at low frequency (0.1 rad/s). Linear melt state viscoelastic behaviour was investigated using a decreasing frequency sweep, from 100 to 0.03 rad/s. The decreasing frequency sweep was preferred to the increasing one in order to limit some artefacts when starting from the lower frequency. Indeed the structure evolves with time and this evolution is faster at short times. This means that the structure evolves during the time necessary to measure the lower frequency. For this reason, the frequency lower value was fixed at 0.03 rad/s in order to limit this effect. In some cases, annealing phases were included into the experimental protocol: the sample was then let at rest in the rheometer during a certain time at a fixed temperature.

III. RESULTS AND DISCUSSION

A. X-ray diffraction

Figure 1 shows the WAXD pattern of the OMMT (Dellite © 67G) and that of nanocomposites.

As it can be seen, the organoclay exhibits three characteristic peaks at 2θ between 2 and 8°.

The interlayer spacings associated to those peaks and given by Bragg's equation are indicated in the figure. It should be noted that the WAXD peaks of the matrix do not appear in this angular range. When the organoclay is mixed with PP containing a certain amount of compatibilizer, polymer chains become able to penetrate the layer galleries, giving rise to an increase of the interlayer distance, which is reflected on the WAXD patterns by a shift of the peaks to lower angles. In the present case, interlayer spacing increases from 3.37 nm for the OMMT to 3.70 nm for nanocomposites A and C and 3.87 nm for nanocomposite B. The interlayer distance appears to be function of the PP-g-MA/OMMT ratio: the two nanocomposites with the same PP-g-MA/OMMT ratio present the same intercalation distance, when the nanocomposite B, with the higher PP-g-MA/OMMT ratio, results in a more intercalated structure. As the d 001 peak is clearly visible on the XRD pattern, we may assume that the nanocomposites are only partially exfoliated.

B. Morphological characterization

The large amount of OMMT in the masterbatches (20 wt%) leads to the presence of large agglomerates, of average diameter in the range 22-25 µm (Figures 2a and2b). Besides these agglomerates, TEM observations show a certain level of exfoliation and a high concentration of platelets and tactoids (Figures 2c and2d). As expected, exfoliation seems to be more important for the sample with the higher PP-g-MA content.

After dilution, the distance between tactoids and/or exfoliated platelets tends to increase, as seen in Figure 3. Observations at a higher magnification, not presented here, show that the average number of platelets per tactoid decreases with the PP-g-MA/OMMT ratio, from 7 to 2, respectively, for nanocomposites A and B. This is also depicted via the variation of the specific density of particles for the two nanocomposites: from 6 to 12 particles.µm -2 for nanocomposites A and B, respectively. At the same OMMT content, nanocomposite B with a higher PP-g-MA/OMMT ratio presents a better exfoliation level than nanocomposite A. The organoclay concentration at a fixed PP-g-MA/OMMT ratio does not play a significant role in the conditions we tested as nanocomposites B and C are characterized by tactoids of a few platelets. Apparent particle lengths comprised between 90 and 400 nm are observed for the three nanocomposites samples (A, B and C), which is shorter than the one observed in the masterbatches (see Figures 2c and2d). This indicates a possible breakage of the platelets during the mixing process.

C. Time-evolution of the rheological behaviour

Figure 4 presents the variations of the storage modulus G' associated to the neat matrix (PP/PP-g-MA) and to the nanocomposite A (85/10/5), after different annealing times (20 to 300 min). As mentioned above, before each annealing, all samples were sheared at a constant shear rate of 1 s -1 according to the protocol previously described to obtain a reproducible initial state. It can clearly be seen in Figure 4 that the nanocomposite samples exhibit a pronounced non-terminal behaviour, characterized by an elastic low frequency plateau, indicating interactions between the nanofillers (exfoliated platelets and tactoids). However, this plateau increases with annealing time. It means that the 3D network existing in the system is not stable and evolves with time. When moving to the high frequency domain, the contribution of the matrix polymer chains becomes dominant and, therefore, the curves for different annealing times superimpose perfectly. According to these observations, it is now interesting to study in more details the time evolution of the rheological behaviour of the nanostructured material for different frequencies.

D. Kinetics of the time-evolution of the nanostructure

In a first step, we focus on nanocomposite A. During the pre-shear at 1 s -1 , the existing nanostructure resulting from the sample preparation and loading is destroyed, and the platelets and tactoids are oriented in the flow direction (Figure 5). The increase of moduli during annealing time is thus attributed to the build-up of a new nanostructure, reforming after the loss of the previous orientation. In order to evaluate the build-up kinetics of this nanostructure, experiments were performed on the nanocomposite samples at different frequencies and different temperatures, using the scenario drawn in Figure 6. It consists in loading the sample at a given temperature T (180, 200 or 220°C), then applying flow and flow reversal shear at 1 s -1 , each during 155 s, and then measuring for a long period of time (up to 20 h) the loss and storage moduli in frequency sweeps (in the linear domain), between 100 and 0.03 rad/s (with 4 points per decade). Each frequency sweep taking around 18 min, we could then have the time evolution associated to these frequencies every 18 min. Total measuring time was around 70 000 s (around 20 h), leading to 65 frequency sweeps. The thermal stability of the matrix was obviously checked before the experiments. This experimental protocol enables to obtain, for each temperature, the variations of storage and loss moduli versus frequency for different ageing times, as presented in Figure 7. We observe the progressive development of the low frequency plateau on G', whereas a weaker effect can be seen on G". The perfect superposition of the data at high frequency proves the absence of degradation, even at very long time. From these data, we can extract the time evolution of G' for several frequencies, as presented in Figure 8 in double logarithmic scale.

At high frequency, G' is stable as it reflects the behaviour of the polymer matrix. But, as soon as the frequency decreases, the time-evolution of G' is more and more visible. Moreover, we can clearly see that this evolution follows a two-step process: above a critical time t crit , around 35 000 s, the kinetics is largely accelerated.

Such changes of G' with time have been shown by [START_REF] Ren | Disorientation kinetics of aligned polymer layered silicate nanocomposites[END_REF] who found for PS/OMMT nanocomposites a power law evolution (G' α t β ), with an experimental exponent β between 0.1 and 0.25. Similar observations were made by [START_REF] Kim | Processing-property relationships of polycarbonate/graphene composites[END_REF] on graphene/polycarbonate nanocomposites, with β values between 0.3 and 1.4. To our knowledge, only Treece and Oberhausen (2007a,b) and [START_REF] Kim | Processing-property relationships of polycarbonate/graphene composites[END_REF] showed a two-step process. Treece and Oberhausen found a time-evolution with two slopes (β 1 and β 2 ), similar to the ones shown in Fig. 8. Depending on the amount of nanoclay and the preparation procedure, β 1 values were between 0.02 and 0.17, and β 2 between 0.26 and 1.17. In our case, it can be seen in Figure 9 that β values strongly depend on the frequency used for the experiments. However, β 2 is always higher than β 1 , and both values are all the more high as the frequency is low. According to Treece and Oberhausen (2007a,b), we assume that the first kinetics is related to the disorientation of tactoids and platelets, and the second one to the aggregation of these entities in a 3D network, through van der Waals attractive forces (Fig. 5).

As shown in Fig. 9, the description of the kinetics using the slope of G' is inaccurate, as it depends on the frequency. It is the same problem when the slope of the complex viscosity η* is used to quantify the level of exfoliation [START_REF] Wagener | A rheological method to compare the degree of exfoliation of nano-composites[END_REF]]. Moreover, this parameter has no physical meaning. To overcome these problems, we suggested [START_REF] Lertwimolnun | Influence of compatibilizer and processing conditions on the dispersion of nanoclay in a polypropylene matrix[END_REF], [START_REF] Vergnes | The use of apparent yield stress to characterize exfoliation in polymer nanocomposites[END_REF]] to fit the complex viscosity curve by a Carreau-Yasuda law with a yield stress:

( ) ( 1)/ 0 0 * ( ) [1 ] a m a σ η ω η λω ω - = + + (1)
where 0 σ is the melt yield stress, 0 η the zero-shear viscosity, λ the time constant, a the Yasuda parameter and m the dimensionless power law index. This model contains five parameters that are adjusted for obtaining the best fit with the experimental data (Microsoft Solver in Excel is used for this purpose). It has been shown that the value of the melt yield stress 0 σ , which is unique and unambiguously defined for a fixed viscosity curve, is a good indicator of the exfoliation level (or more generally of the interactions between the nanofillers) [START_REF] Lertwimolnun | Influence of compatibilizer and processing conditions on the dispersion of nanoclay in a polypropylene matrix[END_REF], 2006, 2007)]. By treating the data of Fig. 7 with the formalism of Eq. ( 1) and extracting the values of the melt yield stress, we can now plot its evolution with time, as shown in Figure 10. As for the storage modulus, we observe a kinetics in two steps. The first one (platelets disorientation) can be described by a square root:

0 1 C t σ = when t < t crit (2)
and the second one (network aggregation) by a linear expression:

0 1 2 ( ) crit crit C t C t t σ = + - when t > t crit (3)
where C 1 and C 2 are constants.

In order to confirm these results, experiments were repeated using sample B. As said before, this 5 wt% OMMT nanocomposite presents a higher level of exfoliation compared to sample A (see Fig. 3) thanks to a higher PP-g-MA content. Figure 11 shows the time evolution of melt yield stress for these two nanocomposites. The same behaviour is observed, with the same kind of evolution, but with a faster progression for sample B: the critical time is shorter and the melt yield stress is always higher than for sample A. We may assume that it is a consequence of a higher exfoliation level, inducing more interactions between individualized platelets.

Nanocomposite C has the same PP-g-MA/OMMT ratio as sample B, but a higher OMMT content (7 wt.%). Figure 12 shows once again the same time evolution, with higher critical time and melt yield stress values, probably due to the more concentrated structure. It seems thus that the behaviour described by equations ( 2) and ( 3) is common to these nanocomposite systems, with a first kinetics related to the loss of orientation of the platelets or tactoids, and a second one related to the formation of a 3D network. It is important to notice that, even at very long times, the structure of the nanocomposites does not seem to stabilize. Treece and Oberhausen (2007a,b) suggested therefore to consider these materials as belonging to the class of soft colloidal glasses.

In order to go further in the understanding of this time evolution, we investigated the effect of temperature on the kinetics. Experiments were conducted on the nanocomposite A at three different temperatures (180, 200 and 220°C). As it can be seen in Figure 13, the general tendency of evolution given by Eqs. ( 2) and ( 3) remains valid, but the kinetics is modified by the temperature: when the temperature increases, the critical time t crit is shorter and both kinetics are more rapid, C 1 and C 2 being functions of the temperature.

The different values of parameters C 1 , C 2 and t crit are reported in Table 2. If the first kinetics is assumed to be due to the disorientation of the platelets by Brownian motion, t crit should be related to the rotary diffusion coefficient D r0 [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF], [START_REF] Ren | Disorientation kinetics of aligned polymer layered silicate nanocomposites[END_REF]]:

0 3 0 3 4 ( ) B r k T D T d η = (4)
where k B is the Boltzmann constant, T the absolute temperature and d the diameter of the tactoid. Figure 14 compares the values of t crit and the time related to the rotary diffusion (1/D r0 ) for the three temperatures. The rotary diffusion coefficient is calculated using the Newtonian viscosity of the PP/PP-g-MA matrix and for a range of tactoid diameters, between 100 and 500 nm. It can be seen in Fig. 14 that the two parameters vary in the same way with the temperature. However the time related to the rotary diffusion is much shorter than the time corresponding to the end of the first step of the structure evolution. This was also noted by [START_REF] Mobuchon | Effect of flow history on the structure of a non-polar polymer/clay nanocomposite model system[END_REF]. If the tactoid diameter d is evaluated from t crit , we obtain d ≈ 1.35 µm, which is much higher than the values of 90-400 nm estimated from TEM observations. For these dimensions, the rotary diffusion time would be around 1000 s, which is much shorter than the times reported here. Consequently, even though Brownian motion of the platelets and tactoids may play a role in the first part of the kinetics, other mechanisms are also to be considered.

E. Validity of the time-temperature superposition principle

The rheological properties of the PP/PP-g-MA/organoclay nanocomposites shown before prove their ability to form a 3D network. Moreover, the continuous evolution of this nanostructure shows that it never reaches a thermodynamically stable state in the conditions we tested. Therefore, the validity of the time-temperature superposition principle can be questioned. In a first time, TTS principle was checked on nanocomposite A using an isothermal frequency sweep, after shearing each sample in flow and flow reversal during 155 s and allowing it to relax for the same period of 20 min. The results obtained are shown on Figure 15a. At high frequency, we observe a classical decrease of modulus with temperature.

But, at low frequency, the plateau is all the more high as the temperature is high.

Consequently, by shifting the curves along the frequency axis to apply TTS principle, we observe in Figure 15b a complete failure of this principle. At high frequency, the curves superimpose perfectly, as the polymer chains govern the rheological properties, but, at low frequency, the curves diverge. Such behaviour was expectable as it is typical for samples with properties changing during the experiments. We have seen before (Fig. 13) that, for a fixed annealing time of 20 min, the kinetics of structure formation were different according to the temperature, leading to materials with different nanotructures. If we want to correctly apply TTS principle, we have to select, for each temperature, an annealing time providing the same nanostructure, i.e. the same yield stress (Fig. 13). As an example, let us consider that the reference state is the one obtained after 38 520 s at 180°C (horizontal dotted line in Fig. 13).

The same yield stress (252 Pa) is obtained after 3360 s at 200°C and 2160 s at 220°C. The data (G' and G") corresponding to these different times are presented in Fig. 16. If we now apply the TTS principle to these data, using only the frequency shift factor a T , it is possible to obtain a perfect master curve for G'and G" (Figure 17).

This time-temperature equivalence indicates that, for a same nanostructure, all the processes, including the non-terminal behaviour related to the organoclay and also the high frequency relaxations, change with temperature in the same manner. Therefore, increasing temperature only accelerates structure build-up and we can conclude that there are equivalences or similarities between the nanostructures characterized by the same melt yield stress.

Subsequent time-temperature superposition based on another value of yield stress (367 Pa)

was also successfully tested. The shift factors a T used to construct the mastercurves associated to the neat matrix and to the nanocomposites are nearly identical (see Table 3), leading to very close values of activation energy (40 kJ/mol, see Table 4).

This result implies that the temperature dependence being probed is that of the polymer. Since the silicate layers do not have a temperature dependant relaxation, the only relaxation process dependent on temperature is that of the polymer segments. Such a result, implying that the addition of inorganic clays significantly affects the magnitude of G' and G" but not the characteristics of the time-temperature superposition, has been previously observed for nanostructured materials with different kinds of matrices and nanofillers [START_REF] Krishnamoorti | Rheology of end-tethered layered silicate nanocomposites[END_REF]; [START_REF] Ren | Linear viscoelasticity of disordered polystyrenepolyisoprene block copolymer based layered-silicate nanocomposites[END_REF]; [START_REF] Solomon | Rheology of Polypropylene/clay hybrid materials[END_REF]]. Since temperature only accelerates the formation of the 3D network, it should be possible to get mastercurves by annealing the samples at the highest temperature (220°C in our case) in order to induce fast morphological changes of the nanostructure, before performing measurements at the selected temperature. This protocol has been successfully used by [START_REF] Reichert | Morphological stability of Poly(propylene) nanocomposites[END_REF] to generate a mastercurve associated to PP/organoclay nanocomposites with unstable microstructure. The experimental protocol is as follows: after pre-shear in forwards and backwards directions, each for 155 s at 1 s -1 , the sample is let at rest at 220°C for 40 min. Then, the temperature is decreased, down to the selected temperature (180 or 200°C) in approximately 13 min, and the frequency sweep is started, from 100 to 0.01 rad/s. It is to be noted that, for these experiments, we used parallel plate geometry, and the thermal dilatation of the plates has been taken into account in order to avoid artefacts associated with lack of material in the gap. Figure 18 evidences that, using a thermal treatment at the highest temperature, the TTS principle holds on the whole tested frequency range. Although less perfect than when probing different annealing times (Fig. 17), the superposition is really good relatively to the one obtained with the classical protocol as shown in Figure 15. It confirms the conclusion that the rheological behaviour of the nanocomposite remains governed by the same physical phenomena, which are only accelerated with temperature increase.

IV. CONCLUSION

We have investigated the melt linear viscoelastic properties of layered silicate based polymer nanocomposites under oscillatory shear. We have studied the evolution of the storage modulus G' at different frequencies as it reflects the evolution of the system at multiscale. At high frequency, where the polymer chains govern the behaviour of the material, G' is almost stable. On the contrary, at low frequency, G' presents a monotonic increase with time, associated to the continuous build-up of a structure. The melt yield stress (obtained by fitting the viscosity curve with a Carreau-Yasuda model with a yield stress) was used to follow the nanostructure evolution and avoid any frequency dependence of the parameter. We have quantified the kinetics of this structure build-up and shown that it was a two-step process. The first one is attributed to platelets disorientation and the second one, faster, to the aggregation of platelets and tactoids in a 3D network. This structure evolution was observed for different formulations of nanocomposites based on PP, PP-g-MA and organoclay whose structure is partially exfoliated and intercalated and where the nanoclays form a more or less soft network. This feature should be valid for other nanocomposites characterized by a similar structure. It varies drastically with temperature. Consequently, it violates the time-temperature superposition principle. However, we proved that the time-temperature equivalence exists if we anneal samples for different times, depending on temperature, in order to obtain similar nanostructures, quantified by a same melt yield stress. This time-temperature equivalence evidences that, for a fixed nanostructure, the only relaxation process involved is that of the polymer chains, and that the behaviour of the nanocomposite is governed by the same physical phenomena for the whole temperature range investigated here. Finally, as temperature does not induce any change through chemical interactions within the system, a thermal treatment applied to the nanocomposite at a high temperature could be a solution to obtain time-temperature equivalence. The curve obtained at 180°C is taken as reference.

Aknowledgement

  Activation energy for the matrix and the nanocomposite A for two values of the

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Microscopy observations of the masterbatches.

Figure 3 .Figure 4 .

 34 Figure 3. TEM observations of the nanocomposites :

Figure 5 .

 5 Figure 5. Schematic description of structure changes during shear flow (from (a) to (c)) and

Figure 6 .Figure 7 .

 67 Figure 6. Experimental protocol used to probe the evolution of the structure using successive

Figure 8 .

 8 Figure 8. Time evolution of storage modulus G' of nanocomposite A (85/10/5) for different

Figure 9 .

 9 Figure 9. Variation of the power law exponents of the time evolution of G' with frequency.

Figure 10 .

 10 Figure 10. Time evolution of the melt yield stress 0 σ of nanocomposite A (85/10/5) at

Figure 11 .

 11 Figure 11. Comparison of the time evolution of the melt yield stress 0 σ for nanocomposites

Figure 12 .

 12 Figure 12. Comparison of the time evolution of the melt yield stress 0 σ for nanocomposites

Figure 13 .

 13 Figure 13. Time evolution of the melt yield stress 0 σ of nanocomposite A (85/10/5) for

Figure 14 .

 14 Figure 14. Comparison between critical time t crit and rotary diffusion time for different

Figure 15 .

 15 Figure 15. Variations of storage modulus G' with frequency for three temperatures (: 220°C

Figure 16 .

 16 Figure 16. Variations of storage modulus G' (a) and loss modulus G" (b) with frequency, for

Figure 17 .

 17 Figure 17. Mastercurves of storage modulus G' (a) and loss modulus G" (b) obtained by TTS

Figure 18 .

 18 Figure 18. Variations of storage modulus G' for three temperatures (: 220°C : 200°C :

Fig. 1 .

 1 Fig. 1. Zouari et al.

Fig. 6 .

 6 Fig. 6. Zouari et al.

Fig. 7 .

 7 Fig. 7. Zouari et al.

Fig. 9 .

 9 Fig. 9. Zouari et al.

Fig. 12 .

 12 Fig. 12. Zouari et al.

Fig. 13 .

 13 Fig. 13 . Zouari et al.

Fig. 18 .

 18 Fig. 18 . Zouari et al.

Table 1 .

 1 Composition of the nanocomposites

	Sample	PP (wt%)	PP-g-MA (wt%) OMMT (wt%) PP-g-MA/OMMT
	A	85	10	5	2
	B	75	20	5	4
	C	65	28	7	4

Table 2 .

 2 Parameters of Eqs. (2) and (3) for the various nanocomposites

		Temperature	C 1	C 2	t crit
	Sample
		(°C)	(Pa.s -0.5 )	(Pa.s -1 )	(s)
	A	180	1.3	0.0037	50 300
	A	200	3.9	0.025	38 500

Table 3 .

 3 Shift factor a T for the matrix and the nanocomposite A for two values of the melt

	yield stress
	Temperature
	(°C)

Fig. 2. Zouari et al.

Fig. 3. Zouari et al.

Fig. 4. Zouari et al.

Fig. 5. Zouari et al.

Fig. 8. Zouari et al.

Fig. 10. Zouari et al.

Fig. 11. Zouari et al.

Fig. 14 . Zouari et al.

Fig. 15. Zouari et al.

Fig. 16. Zouari et al.

Fig. 17. Zouari et al.

Denmark) for gently providing the masterbatch. Financial support by the European Commission through project Nanotough-213436 is gratefully acknowledged. The authors are also grateful to the referees for interesting suggestions allowing to improve the paper.


Mathieu Hébert
email: mathieu.hebert@univ-st-etienne.fr

Juan Martínez-García

Thomas Houllier

Hayk Yepremian

Nico- Las Crespo-Monteiro

Francis Vocanson

Alain Trémeau

Nathalie Destouches

Characterization by hyperspectral imaging and hypercolor gamut estimation for structural color prints

Keywords: Structural colors, Multiview prints, Specular surface, Hyperspectral imaging, Nano-technologies

A recently developed color printing system on glass plates, based on dot-by-dot laser irradiation generating the growth of metallic nanoparticles in a special coating, produces structural colors depending strongly on the illumination and observation configuration. The difficulty for an exhaustive color characterization of the printing technology comes not only from the goniochromaticity of the samples, but also from their very high specularity, to which classical measurement instruments are not adapted. Moreover, as the light-matter interaction relies on a number of optical phenomena (surface plasmon resonance, interferences, diffraction, effects of polarization of light) for which no predictive model is available today, their characterization requires measurement of many printed samples. In this paper, we present a characterization method based on multispectral imaging and on spectral prediction for halftone colors that permitted a first gamut estimation in three specific illumination/viewing configurations.

Introduction: laser-induced color lusters on glass

Recent progresses in nanotechnologies enable the coloration of glass with interesting visual rendering. This is for example the case of the technology developed by the laboratoire Hubert Curien, called PICSULP [START_REF] Martinez | Multi-color properties of silver glaze images photo-engraved on glass plates[END_REF], where a coating containing silver [START_REF] Tricot | Flexible photochromic TiO2:Ag thin film elaborated by printing techniques[END_REF] is deposited on the glass plate, then irradiated by a laser beam in order to anneal the coating and cluster the metallic ions into metallic nanoparticles (NPs). Goniochromatic coloration of the glass plate surface is thus obtained thanks to various optical phenomena: the presence of silver NPs generates surface plasmon resonance, therefore wavelengthselective absorption as in stained glass [START_REF]Controlling the Quantum World: The Science of Atoms, Molecules, and Photons. Chapter 6[END_REF]; the organization of the NPs along one plane parallel to the coating-air interface generates interferences as in thin films; the NPs can even be aligned along parallel lines, as shown in Figure 1-a, which produces diffraction effects visible at grazing angles, and also gives to the sample a dichroic spectral behavior, i.e. polarization sensitive colors [START_REF] Destouches | Dichroic colored luster of laserinduced silver nanoparticle gratings buried in dense inorganic films[END_REF][START_REF] Destouches | Permanent dichroic coloring of surfaces by laser-induced formation of chain-like self-organized silver nanoparticles within crystalline titania films[END_REF]. These optical effects are influenced by several physical parameters: the nanoparticle shape, size and spatial organization, as well as the coating thickness and refractive index that evolve during the laser treatment. Four command parameters can be varied: exposure (tuned by the laser scanning speed), power, wavelength, and focusing of the laser beam. Polarization of the laser can also be varied. In opposition to classical printing techniques on paper or transparency films where a color patch has "one color" almost independent of the viewing conditions, a glass plate uniformly treated by the PICSLUP technology with constant laser command parameters can display many colors according to the illumination/observation configuration (IOC). Figure 2 shows this dramatic color change through the example of a fourcolor image observed in three IOCs. Therefore, a sample cannot be attributed one color, but should be assigned a set of colors that we propose to call "hypercolor". A second difference with conventional printing is that no model allows predicting the color viewed in a given IOC as a function of the printing command parameters. An exhaustive color characterization therefore requires a huge amount of samples produced each one with different laser command parameters, incrementally varied, to be measured in several IOCs. This represents too many measurements for a reasonable calibration process. Moreover, radiances reflected or transmitted in the regular directions (given by Snell's laws) differ from the ones scattered in the rest of the hemisphere by several orders of magnitude: the sensitivity of the measurement setup must therefore be adapted to different IOCs.

In this paper, we present a color characterization method that is not exhaustive, but already representative of the color generation capacities of the PICSLUP technology. It relies on a set of primary colors printed with different laser parameters on one glass plate, and three measurement geometries: specular reflection, off-specular reflection, and normal transmission. Spectral measurements are carried out by using a self-made hyperspectral imaging system. From this set of primary colors, secondary colors can be predicted by juxtaposing lines of different primary colors, and color gamuts can be computed for each of the selected measurement geometries.

Selection of samples and geometries

In a previous study [START_REF] Martinez | Color calibration of an RGB digital camera for the microscopic observation of highly specular materials[END_REF], an exhaustive color characterization has been performed for the 0°:0° specular reflection mode by measuring the XYZ color coordinates of the reflected radiances under a A illuminant thanks to a calibrated camera mounted on top of a microscope. This geometry was the only one allowed by the microscope. The colorimetric calibration of the camera was developed in order to suit the very high specularity of the samples by using of a specific color chart presenting as high specularity as the samples. Microscopic patches of 10656 hypercolors, similar to those displayed in Figure 1.b, were printed and analyzed in this way by excluding the edges of each patch. For the present study, we selected a subset of 330 hypercolors printed over 1 × 1 mm squares, with the following combinations of laser command parameters [START_REF] Martinez | Multi-color properties of silver glaze images photo-engraved on glass plates[END_REF]: 2 wavelengths (488 and 647 nm), 3 powers (300, 400, and 600 mW), 11 laser scanning speeds (30000, 3200, 2000, 1400, 1200, 1000, 900, 800, 600, 500, and 300 μm/s) and 5 focus distances (0, 0.5, 0.8, 1.8, and 2 mm).

These 330 hypercolors were measured with the hyperspectral imaging setup described in the next section, in three different geometries : 15°:15° specular reflectance (illumination by collimated light at 15° from the normal of the sample, and observation in the specular direction), 45°:0° off-specular reflectance (illumination at 45° from the normal of the sample, observation along the normal), 0°:0° transmittance (illumination by collimated light at the normal of the sample, and direct observation of the transmitted collimated beam).

Hyperspectral imaging system

The optical bench assembled for the hyperspectral measurements is shown in Fig. 3. The illumination module was placed on a rotating arm permitting to vary the incidence angle at both sides of the sample; it comprised an optical fiber connected to a light source (HL-2000-HP 20W halogen lamp by Ocean Optics) placed at the focus of a lens in order to obtain a collimated light beam of 4 cm diameter. The detection module comprised a lens-free 12-bit gray-level camera, a set of 31 interferential filters of 10 nm spectral width centered every 10 nm from 400 to 700 nm, and a telecentric lens made of two lenses whose foci meet each other, the sample being placed in the other focal plane of the first lens, the detector of the camera in the one of the second lens [START_REF] Lequime | A goniometric light scattering instrument with high-resolution imaging[END_REF]. The advantage of the telecentric lens is that it allows the simultaneous measurement of many patches, in strictly similar measurement conditions. This is a crucial point for our very specular samples since a very small angular variation may modify considerably the detected radiance (variations up to 50% can be easily observed, which is still small in comparison to the 10 10 dynamic of the BRDF of the samples). In this context, the telecentric lens is crucial to ensure that all points of the sample are viewed under the same angle. With a conventional lens, the image of a uniform hypercolor patch would be non-uniform because of the goniochromaticity of the sample and the oblicuity of the captured rays at the border of the image. The calibration of the spectral bench was done by using a neutral color reference sample reflecting or transmitting similar radiance as the studied samples. In specular reflection mode, the reference sample was a silicon mirror whose spectral reflectance was measured with a spectrophotometer in the 15°:15° geometry. Once placed in the hyperspectral bench, for each band-pass filter (denoted by its central wavelength λ), the aperture and exposure of the camera was adjusted in order to obtain a maximum signal without saturation of the camera. These camera parameters as well as the captured images I ref (x, y, λ) of the reference sample were stored, then the images I(x, y, λ) of the PICLSUP print were captured by using for each band-pass filter the same camera parameters as those used for the reference sample. Lastly, we considered for each pixel (x, y) and each waveband λ, the ratio I(x, y, λ) / I ref (x, y, λ), which corresponds to a reflectance factor.

In off-specular reflection mode, the reference sample was a green diffusing ceramic tile, whose spectral reflectance was comparable in all wavebands to the one of the samples (these latter exhibiting a greenish hue in this IOC). Since the BRDF of our specular samples is extremely low at 45° from the illumination direction, we illuminated them with a high conical irradiance coming from an integrating sphere with a 500 lm halogen lamp in it. Similar procedure as for the specular reflection mode was then applied.

In transmittance mode, the reference sample was simply void. Once again, similar procedure as for the specular reflection mode was applied. By way of illustration, we show in Figure 4 the images of the sample and the reference sample captured in two wavebands around 460 nm and 600 nm. Note that spatial variations of the illuminance, lens transmittances, vigneting and other aberrations introduced by the system, are systematically corrected as they are similar in both the sample and reference images. From the 31 pairs of images obtained in each mode, we can reconstruct the hyperspectral image with a wavelength sampling of 10 nm, which is a rather good spectral (10 nm) and spatial (1024 × 768 pixels) resolution compared to hyperspectral imaging systems with variable lighting geometry existing today [START_REF] Seroul | Hyperspectral imaging system for in-vivo quantification of skin pigments[END_REF][START_REF] Jolivot | Skin parameter map retrieval from a dedicated multispectral imaging system applied to dermatology/cosmetology[END_REF]. Before obtaining the hyperspectral image, an important step is the registration of the different images because the band-pass filters can introduce a small shift between the different images captured [START_REF] Zhang | A Novel 3D Multispectral Vision System based on Filter Wheel Cameras[END_REF]. An image processing applet was developed to perform registration automatically.

At the end of the process, we have the spectral and spatial information of the prints. A good way to visualize their colors in a computer display is to render them by assuming an illuminant to obtain their CIE-XYZ values and then obtain sRGB colors by using standard transformations. In Figure 5, we can see a color version of the hyperspectral images obtained in the three measurement geometries, by assuming a D65 illuminant. We can notice the dramatic change in color of the same printed areas according to the IOC used. Some patches look scratched in the specular reflectance mode: a surface pealing effect due to thermal and mechanical strains produced by the laser treatment in specific parameter ranges can roughen the surface, which thus reflects a much lower radiance in the specular direction. Each hyperspectral image obtained contains one spectrum per pixel. By averaging the spectra of pixels belonging to a same patch, we could obtain the spectra of all the printed patches and calculate the corresponding CIE 1976 LAB color coordinates by assuming a D65 illuminant (Figure 6). In off-specular reflection mode, the volume of colors that can be obtained have a moderate size, with a strong presence in the green hue which are lacking in the other configurations. The specular reflection mode produces a wider range of chromaticities in the samples. However, the variation in lightness in that geometry is rather limited, presenting L* values between 40 and 65 units. The colors in the back side of the samples in this configuration have a much smaller volume, with very reduced chromatic and lightness variations. In transmission mode, the color gamut is very reduced, consisting mainly of very light and pale colors of orange hue.

Gamut extension by halftoning

In contrast with ink-based printers which need halftoning in order to reproduce many colors from their limited set of inks (four inks in quadrichromy), the PICSLUP technology can produce a wide variety of hypercolors in each point, and cover the plate with one uniform hypercolor (continuous tone printing). However, since only 330 hypercolors have been characterized, we propose to use halftoning to obtain additional, intermediate hypercolors. Since the set of printable halftones is considerable, we wanted to know whether it is possible to predict their color in the OIC instead of measuring them. We thus propose to verify the validity of a classical prediction model for halftone colors in case of low light scattering in the printing support: the Spectral Neugebauer model [START_REF] Balasubramanian | A spectral Neugebauer model for dot-on-dot printers[END_REF].

A set of 30 different 2 × 2 mm patches where printed on the same glass support, with different combinations of five primaries and different proportions of area coverage; line halftoning was used by juxtaposing lines of 20 µm width. An image of the squares, captured with the hyperspectral imaging system in the 15º:15º specular reflection configuration is shown in Figure 7, as well as the composition of each of the printed patches. The leftmost column corresponds to patches wholly covered by one primary. These are used as inputs in the prediction models in order to estimate the colors obtained from the combination of several primaries. The remaining 25 patches correspond to areas printed with the combinations of two or three different primaries. The primaries used are specified by the numbers in the brackets, and the ratio of the area covered by each of them is specified below the primaries. For example, the square located in the 2 nd row and 5 th column is made with the primaries 1 and 5, with a ratio of 2 lines of Primary 1 for each line of Primary 5. The spectral properties of the samples were measured in the three IOCs presented in Section 3. The Spectral Neugebauer model predicts, for a given IOC, the spectrum of a halftone hypercolor, ( ) halftone S λ , as a function of the spectra ( ) i S λ of the primaries i and their respective surface coverage a i , as:

( ) ( ) 1/n halftone i i i S a S λ = λ  (1)
Figure 8 shows the measured and predicted spectra of the first halftone patch, located in the first row and second column of the sample (see Figure 7), for each of the different measurement configurations. The prediction accuracy, assessed by the CIE94 color difference between the measured and predicted colors by assuming a D65 illuminant, is rather good, considering that the just-noticeable-difference threshold of the human visual system is approximately 1 unit. The prediction accuracy for the Neugebauer model over the set of samples that we produced is slightly poorer than the one usually achieved in color reproduction by digital printers (e.g. inkjet on photo-quality paper), e.g. the Yule-Nielsen modified Spectral Neugebauer model which is one of the more efficient [START_REF] Hébert | Review of spectral reflectance models for halftone prints: Principles, Calibration, and prediction accuracy[END_REF]. In the present case, as expected since our samples are non-scattering, no improvement is achieved by using the Yule-Nielsen modified Spectral Neugebauer model. The PICSLUP printing system is probably less reproducible than commercial digital printers, especially due to small artifacts which may have small but noticeable impact on the spectral properties of the prints. Moreover, the measurement process is also subject to some imprecision and noise. Being given these constraints, we can be satisfied with the prediction accuracy that is achieved.

By using the Spectral Neugebauer model, we can predict the colors that can be produced in each geometrical configuration by the combination of the numerous primaries produced by the PICSLUP system, shown in Fig. 6. To obtain the complete set of colors that can be produced with the PICSLUP system, we exhaustively predicted the colors produced by the halftoning mixture of every pair of primaries from the set of 330 hypercolor patches. We found that in all the IOCs, the predicted colors obtained by halftoning with each pair of primaries draw an almost straight line between them, as a consequence of the linear equation ( 1). This volume of colors is very close to the convex hull of the set of primaries used, i.e., the minimum convex volume that contains all the primaries in a given configuration. The color gamut in each of the geometries can therefore be approximated in a simple manner as the convex hull defined by the primaries used. The extended color gamuts obtained for each of the six measurement configurations thanks to use of halftoning are presented in Figure 9. Despite the photometric constraints and limitations that exist for characterizing the PICSLUP technology, we demonstrated that line halftoning was possible in our system by obtaining a satisfactory accuracy in the prediction of the colors that can be produced by mixing different primaries. The worst accuracy was obtained in the off-specular reflection mode where the reflected radiance was very low, which made the measurements prone to errors due to stray light or diffraction effects. We can see by comparing Figures 6 and9 that the volume of producible colors notably increased and became more convex and continuous. This is important for color imaging, where a continuous sets of colors are necessary to produce gradients present in real images.

Multiview imaging

Another advantage introduced by halftoning is that the amount of hypercolors is considerably increased, displaying each one a different set of colors in the different IOCs. It is then easier to obtain parameters to print areas that visually match a given color A in one geometry while matching another target color B in a different geometry. This opens the possibility for some interesting imaging design and security applications. For example, we can hide patterns in a background by selecting different sets of command parameters for the patterns and the background that display the same color in one geometry and reveal different colors in another geometry.

By way of illustration, we designed a 2-geometry pattern from samples measured by our hyperspectral imaging system. Figure 10 shows the colors produced in two different geometrical configurations by the four sets of parameters selected. These colors are then used to design the spatial arrangement of the areas that will be printed with each of the selected parameter sets. Figure 11 shows the spatial arrangement we designed to produce our 2-geometry pattern, as well as the resulting colors produced displayed in the two geometries. This approach can be extended to multicolor images, and it is probable that more than two different images can be displayed. The strong color variations that polarizing filters can generate is also a way of increasing the number of possible configurations [START_REF] Destouches | Permanent dichroic coloring of surfaces by laser-induced formation of chain-like self-organized silver nanoparticles within crystalline titania films[END_REF]. However, multi-gamut mapping would be necessary to automatically determine the hypercolors, if any, that would be able to display the different targeted images, and no multi-gamut mapping technique is available today: a challenge for the incoming years.

Conclusions

In the paper, we presented a spectral and color characterization of glass plates coated with a layer containing composite layer nanostructured by laser irradiation, which exhibits different "hypercolors" (i.e., sets of colors displayed in different illumination/observation configurations) in each point. Despite the difficulty of characterizing the color appearance of these strongly goniochromatic, transparent and specular samples, we could find a suitable measurement method thanks to hyperspectral imaging, where the camera is preceded by a telecentric lens. Several patches can thus be simultaneously measured, while being all viewed under the same angle independently of their position in the sample, thereby in the image. Special attention must be paid to the choice of the reference sample, which should have similar reflectance or transmittance as the studied sample in the considered measurement geometry in order to avoid a too low or excessive light signal detected. We then showed that, once knowing the spectral reflectance or transmittance of pure hypercolors (primaries), we can estimate, with rather good accuracy in this context, the spectral properties of halftone hypercolors obtained by juxtaposing thin lines of different primaries. This finally enables estimating to the color gamut of the printing technology in the considered illumination/observation geometry, and opens interesting perspectives for the production of multi-view still images.

Fig. 1 .

 1 Fig. 1. Printed glass plate observed at (a) nanometric scale where we can see the NPs aligned along parallel lines (scale bar: 2 µm), (b) at microscopic scale where we can see many segments each one printed with different laser parameters (scale bar: 200 µm), and (c) at macroscopic scale in specular reflection mode (scale bar : 2 mm).

Fig. 2 .

 2 Fig. 2. Color image printed with PICSLUP (1 x 1 cm), showing the dramatic color change according to the illumination/observation geometry. Left: Specular reflection mode (45°:45° geometry); Center: Off-specular reflection mode (45°:0° geometry); Right: Transmission mode (0°:0° geometry) [1].

Fig. 3 .

 3 Fig. 3. Scheme of the hyperspectral imaging system for the measurement of samples, here in transmission mode.

Fig. 4 .

 4 Fig. 4. (a, c) Images of a sample printed by the PICSLUP system and (b, d) images of the reference sample (void) (b,d) obtained in transmission mode at 460 nm (a, b), and 600 nm (c, d).

Fig. 5 :

 5 Fig. 5: sRGB color images rendered from the hyperspectral images in three different measurement geometries. The areas of the squares are: 1 mm × 1 mm.

Fig. 6 .

 6 Fig. 6. Colors obtained for the 45°:0° off-specular reflection geometry (left), 15°:15° specular reflection geometry (center) and 0°:0° transmission geometry (right) of the 330 printed hypercolor patches, represented in the CIE 1976 L*a*b* color space (bottom line) and projected onto the (a*,b*)-plane (top line).

Fig. 7 .Fig. 8 .

 78 Fig.7. Image of the squares printed for halftoning prediction testing, captured with the hyperspectral imaging bench. Specification of squares printed for halftoning evaluation. The squares on the leftmost column were printed by using only one primary, while the other 25 were printed by combining the primaries in the parenthesis in the proportions shown.

Fig. 9 .

 9 Fig. 9. Estimated color gamut volumes for the PICSLUP technology in the CIE 1976 L*a*b* color space for the 45°:0° off-specular reflection geometry (left), 15°:15° specular reflection geometry (center) and 0°:0° transmission geometry (right).

Fig. 10 .

 10 Fig. 10. Left and middle: Rendered sRGB color of four different hypercolors P1, P2, P3 and P4, in the 45º:0º off-specular reflection (left) and 0º:0º transmission (middle) geometries. Right: Spatial arrangement designed with the four hypercolors in order to produce the 2-geometry pattern displayed in Figure 11.

Fig. 11 .

 11 Fig. 11. Rendered sRGB colors of a 2-geometry pattern observed in the 45º:0º off-specular reflection (left) and 0º:0º transmission (middle) geometries.

Acknowledgement

This work was supported by the French National Research Agency (ANR) within the program "Investissements d'Avenir" (ANR-11-IDEX-0007), in the framework of project PHOTOFLEX n°ANR-12-NANO-0006 and the LABEX MANUTECH-SISE (ANR-10-LABX-0075) of Université de Lyon.