Javeria Ahmed
email: javeria.ahmed@etu.univ-orleans.fr

Matthieu Fruchard
email: matthieu.fruchard@univ-orleans.fr

Estelle Courtial
email: estelle.courtial@univ-orleans.fr

Youssoufi Touré
email: youssoufi.toure@univ-orleans.fr

Low-power High Gain Observers for Wake Flow Rebuild

Low-power High Gain Observers for Wake Flow Rebuild

Javeria Ahmed, Matthieu Fruchard, Estelle Courtial and Youssoufi Touré Abstract-This paper deals with the observation of the wake flow behind a bluff body based on reduced order models. Two Galerkin models with different truncation orders are considered to capture the intermittent character of the vortex shedding. The system output is the oscillation amplitude obtained thanks to pression measurements on the sides and on the back of the bluff body. For flow control purposes, the state estimation has to be efficient and robust to noises and dynamic changes. A high gain observer is modified i) by extending the state with the introduction of fictitious outputs in order to avoid an observability singularity and to ensure the existence of the inverse jacobian matrix and ii) by integrating a high gain parameter whose power is limited to 2 instead of n to address the sensitivity issue of the standard high gain observer. Simulation results illustrate the efficiency of the proposed observers and highlight promising applications for flow control at high Reynolds numbers.

I. INTRODUCTION

A. Context and motivation

The flow at the rear of a bluff body is a common occurrence such as the flows at the back of an airplane, a submarine or a vehicle. The turbulent wakes downstream the bluff body can be very complex, exhibiting coherent structures with different scales. The strongly intermittent behavior of the flow depends on the Reynolds number. At high Reynolds numbers, vortex shedding occurs in the wake, leading to a significant pressure drop on the rear surface of the body. This phenomenon gives rise to structural vibrations, acoustic noise and increases the drag force. The control of vortex shedding is then of major interest for engineering applications. Active control in fluid mechanics is a promising and challenging research area. Promising because lower energy consumption and better performances are expected by decreasing the drag force. Challenging because observation and control require both real-time tractable and robust models with respect to dynamics changes in the wake flow [START_REF] Ma | Dynamics and lowdimensionality of a turbulent near wake[END_REF], [START_REF] Östh | On the need for a nonlinear subscale turbulence term in pod models as exemplified for a high-reynolds-number flow over an ahmed body[END_REF]. Fluid flows are governed by the Navier-Stokes equations known for their complexity (strong nonlinearities, high dimensionality and time-delays). This kind of modeling can not be used when real-time applications are targeted [START_REF] Brunton | Closed-loop turbulence control: Progress and challenges[END_REF]. Linear models are unable to recover unstabilities characterizing the wake flow [START_REF] Sipp | Dynamics and control of global instabilities in open-flows: a linearized approach[END_REF], [START_REF] Rowley | Model reduction for flow analysis and control[END_REF]. Within this context, reduced order models approaches have been widely used to address the extraction of the coherent structures dynamics in fluid flows: Proper Orthogonal Decomposition [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF] and more recently Dynamic Mode Decomposition [START_REF] Rowley | Spectral analysis of nonlinear flows[END_REF], [START_REF] Schmid | Dynamic mode decomposition of numerical and experimental data[END_REF] using projection of the Navier-Stokes equations onto a modal basis. Galerkin models have proven to be a good compromise between simplicity and robustness. They are based on a Karhunen-Loève expansion around the steady or time-averaged Navier-Stokes solution u s . The unsteady flow u(x,t) is then approximated by an orthonormal Galerkin series:

u(x,t) ≈ u s (x) + 2N ∑ i=1 a i (t)u i (x), (1)
where u i (x) denote the orthonormal spatial modes and a i (t) are the temporal modes amplitudes. For time periodic flows, structures are described by pairs of modes, acting as modal oscillators. Since the first two modes (N = 1) convey 95% of the turbulent kinetic energy, is it necessary to truncate the equation (1) at higher order? However, one mode plays a crucial role to robustify the low-order Galerkin model with respect to Reynolds range and transient dynamics [START_REF] Noack | A hierarchy of low-dimensional models for the transient and posttransient cylinder wake[END_REF],

[10]: it's the shift mode, noted a ∆ , modeling the energy flow between the mean flow and the vortex shedding. In the sequel, two Galerkin models are considered: a minimal Galerkin representation [START_REF] Noack | A hierarchy of low-dimensional models for the transient and posttransient cylinder wake[END_REF] using the dominant von Karman modes a 1 , a 2 and the shift mode a ∆ (the decay rate µ > 0 is supposed to be constant):

    a 1 a 2 a ∆ µ     =     (µ -a ∆ )a 1 -a 2 (µ -a ∆ )a 2 + a 1 a 2 1 + a 2 2 -a ∆ 0     (2)
and a more refined Galerkin representation including higher damped (µ ′ > 0) modes (a 3 , a 4 ) interacting with the first two ones through the shift mode: 

       a 1 a 2 a 3 a 4 a ∆ µ         =         (µ -a ∆ )a 1 -a 2 (µ -a ∆ )a 2 + a 1 (-µ ′ -a ∆ )a 3 -a 4 (-µ ′ -a ∆ )a 4 + a 3 a 2 1 + a 2 2 + a 2 3 + a 2 4 -a ∆ 0         (3)
The latter model, referred as a hybrid model [START_REF] Noack | A hierarchy of low-dimensional models for the transient and posttransient cylinder wake[END_REF], [START_REF] Luchtenburg | A generalized mean-field model of the natural and high-frequency actuated flow around a high-lift configuration[END_REF] between the empirical Galerkin model (2) and higher dimensional invariant-manifold model, is expected to be more precise. Considering the oscillation amplitudes A i = a 2 2i-1 + a 2 2i , the above models become two systems of order 3 and 4 respectively. With the notation x min = A 1 a ∆ µ ∈ X ⊂ R 3 , system (2) becomes:

S min : ẋmin =   2x 1 (x 3 -x 2 ) x 1 -x 2 0   = f (x min ) (4)
and using the state vector

x hyb = A 1 A 2 a ∆ µ ∈ X ⊂ R 4 , system (3)
becomes

S hyb : ẋhyb =     2x 1 (x 4 -x 3 ) 2x 2 (-µ ′ -x 3 ) x 1 + x 2 -x 3 0     = f (x hyb ) (5)
For the sake of simplicity, the state vector x will be used for both x min and x hyb in the sequel. The only state component accessible to measurement is the oscillation amplitude A 1 . So the output is y = h(x) = x 1 in both cases.

B. Problem statement

Rebuilding the unmeasured states is mandatory to i) understand the macro-physics at work in the wake, i.e. the dominant structure dynamics, and ii) synthesize control laws aiming at stabilizing the wake around some desired trajectory. A first prerequisite is to study the conditions guaranteeing the observability of the sytem.

Proposition 1 Systems S min and S hyb respectively given by (4) and [START_REF] Rowley | Model reduction for flow analysis and control[END_REF]

are observable on Ω min = {x ∈ X ⊂ R 3 : |x 1 | > ε} and Ω hyb = {x ∈ X ⊂ R 4 : |x 1 | > ε, |x 2 + x 3 + µ ′ | > ε ′ } for any ε, ε ′ > 0, respectively.
Proof: Let O denote the smallest vector space containing h and closed under the Lie derivative L f . A system is said weakly observable if dim dO(x) = dim X where dO = Span{dτ, τ ∈ O} is the observability co-distribution.

Computing the differentials of the successive Lie derivatives of the output y = h(x) along the system vector field f leads to the observability co-distributions dO min and dO hyb for systems S min and S hyb respectively, whose determinants are:

|dO min | = -4x 2 1 |dO hyb | = -16x 3 1 (x 2 + x 3 + µ ′ ). (6)
An observability singularity appears in particular when x 1 = 0, whereas the control aims at forcing the oscillation amplitude to zero. The designed observer should be able to handle this singularity in order to synthesize control laws based on the estimated states. Among the observer approaches, the High Gain Observer (HGO) is well-adapted to estimate nonlinear system states and exploits a fast error convergence depending on the high gain parameter. However, it has some flaws which need to be addressed properly. The power of its high gain parameter increases up to the dimension of the system causing some sensitivity issues. Besides, a non-singular diffeomorphism between the system expressed in natural coordinates and the system in its normal form is required.

C. Contribution

The present paper proposes a solution for the aforementioned issues. The first issue is addressed by limiting the power of the high gain parameter. As suggested by [START_REF] Astolfi | A high-gain nonlinear observer with limited gain power[END_REF] for high dimensional physical systems, the proposed observer limits the high gain parameter power to two instead of n at the expense of the increased observer dimension from n to 2n-2. In order to deal with the singularity issue that happens during the inversion of the jacobian matrix to get back the observer in original coordinates, the HGO is modified by extending the state with the introduction of fictitious outputs [START_REF] Bernard | Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism[END_REF]. In this way, we avoid the singularity by making the system observable and ensuring the existence of the inverse jacobian matrix everywhere in space R n . This paper is organized as follows: we briefly recall the design of high gain observers in section 2. Then the design of the proposed observer is detailed in section 3. Section 4 is devoted to the simulation results of the observation of the wake flow behind a bluff body. The observer efficiency for the two reduced order models are compared. Finally concluding remarks and on-going work are discussed in section 5.

II. RECALL ON HIGH GAIN OBSERVERS

High gain observers [START_REF] Gauthier | A simple observer for nonlinear systems. application to bioreactors[END_REF], [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF] are widely used since their arbitrarily fast exponential decay and practical stability arbitrary bound, in case of unmodeled dynamics, make possible to synthesize real-time output feedback stabilizing control laws for systems with few state measurements. We briefly recall the standard high gain observer, the low-power high gain observer and the observer in natural coordinates.

Proposition 2 (High gain observer) Consider the nonlinear system ẋ = f (x), y = h(x) fulfilling the observability condition with x ∈ X ⊂ R n , y ∈ R. There exists an injective immersion φ from X ⊂ R n to R m with m ≥ n such that the nonlinear system can be rewritten in the observability normal form with z = φ (x) as:

ż = A m z + B m F(z), y = C m z = z 1 ( 7
)
where the matrices A m , B m and C m are in prime form: A m is a (m × m) square matrix whose non null entry is the unitary superdiagonal, B m (C T m ) is a m-dimensional column vector whose non null entry is its unitary last (first) component. Then a high gain observer is given by

ż = A m ẑ + B m F s (ẑ) + ∆ m K(y -C m ẑ) ( 8
)
with ∆ m = diag(θ , ..., θ m ) where the high gain parameter θ > θ 0 ≥ 1 is related to the Lipschitz constant of F s , F s is a saturated function of F and K is chosen such that the matrix

(A m -KC m ) is Hurwitz.
Remark 1 Two well-known limitations of the high gain observer are related to the increasing powers of the high gain θ and to the inversion of φ . First, as dimension increases, so does θ m , and it may result in an innovation term that dominates the observer dynamics and an increased sensitivity to noise on the last states, especially at high frequencies. Second, since the observer is built in the z-space, recovering an estimation useful for control purposes often requires to go back to the natural coordinates x, meaning that φ has to be inverted. This inversion is in general not easy, especially if m > n.

In a recent study [START_REF] Astolfi | A high-gain nonlinear observer with limited gain power[END_REF], the authors proposed a new observer of dimension (2m -2) using a high gain limited to power 2 instead of designing an observer of dimension m with a gain that grows up to power m. The limited gain to power 2 reduces the numerical implementation problems of the classic high gain observer and also reduces the sensitivity to noise measurements.

Proposition 3 (Low-power high gain observer) Under assumptions and notations of Proposition 2, let

ζi = A 2 ζ i + Nζ i+1 + ∆ 2 K i e i , i ≥ 1 . . . ζm-1 = A 2 ζ m-1 + B 2 F s (ζ ) + ∆ 2 K m-1 e m-1 (9
)

with ζ i ∈ R 2 , ζ 0,2 = y given by convention, e i = ζ i-1,2 -ζ i,1 and K i = k i1 k i2 T . Let N = 0 2,1 B 2 , E i = -K i C T 2 and Q i = 0 2,1 K i .
Choosing the high gain θ > θ 0 as in Proposition 2 and gains K i such that the matrix M

M =                E 1 N 0 . . . . . . 0 Q 2 E 2 N . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . Q i E i N . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . Q m-2 E m-2 N 0 . . . . . . 0 Q m-1 E m-1                (10)
is Hurwitz then

ζ = ϒ(ζ , y) ( 11
)
with ϒ given by (9) provides a low-power high gain observer for system [START_REF] Rowley | Spectral analysis of nonlinear flows[END_REF]. We can use either ẑ = Pζ or ẑ′ = P ′ ζ with the block diagonal projectors P given by I 2 and (m -1) matrices B T 2 , and P ′ given by (m -1) matrices C 2 and I 2 .

Remark 2 Despite this dimension increase, the low-power (∆ 2 depends only on θ , θ 2 ) high gain observer advantage is twofold: innovation term predominance is limited and sensitivity to output noise is consequently reduced, especially at high frequencies which is of particular interest considering wake flow dynamics.

Both aforementioned observers make use of a generically nonlinear mapping φ : x → z whose inversion is required to infer the observed state x in the original coordinates of the system dynamics. Such an inversion can be obtained through optimization but often at the price of costly computations. Furthermore, the non unicity of the inverse when m > n may raise numerical issues. An alternative approach is to build the observer in the natural coordinates x as suggested initially by [START_REF] Deza | High gain estimation for nonlinear systems[END_REF].

(Observer in natural coordinates)

If the transformation φ : X ⊂ R n → Π ⊂ R q mapping the original system ẋ = f (x) to a target system admitting an observer in the form π = ϕ(π, x, y) is a diffeomorphism, then an observer in natural coordinates is given by:

ẋ = ∂ φ ∂ x ( x) -1 ϕ(φ ( x), x, y). ( 12
)
Remark 3 Obviously, building an observer in natural coordinates is more difficult when q = m > n using a high gain observer [START_REF] Bernard | Tools for observers based on coordinate augmentation[END_REF], or when q = 2(m -1) for low-power high gain observer, even if m = n.

III. MAIN RESULTS

To address the observability singularity avoidance, we follow the core idea initiated in [START_REF] Andrieu | Dynamic extension without inversion for observers[END_REF] and then enhanced and structured in [START_REF] Bernard | Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism[END_REF]. We follow a three steps procedure. The first step consists in augmenting the initial n dimensional domain of the mapping φ to equal the dimension q = 2(m-1) of the low-power observer. The resulting augmented mapping φ a is now a diffeomorphism on a subset of R q . We then proceed to an extension of φ a into an injective immersion φ e on R q using fictitious outputs to artificially remove the observability singularity. The third step lies in the augmentation of φ e into a global diffeomorphism φ φ φ where we augment the natural coordinates by the coordinate τ ⊂ R N-n so that φ φ φ (x, 0) = φ e (x).

A. A meaningful example

Let us illustrate the previous section results as well as the observability singularity issue on the simple empirical Galerkin system (4). First we define φ ∈ C 1 (Ω min ⊂ X , R 3 ) using

z = φ (x) =   x 1 2x 1 (x 3 -x 2 ) 2x 1 (2(x 3 -x 2 ) 2 -x 1 + x 2 )   (13)
so φ maps (4) on the observability canonical form [START_REF] Rowley | Spectral analysis of nonlinear flows[END_REF] with m = 3 and

F 3 (x) = 2x 1 [4(x 3 -x 2 ) 3 -2(x 3 -x 2 )(4x 1 -3x 2 ) + x 1 -x 2 ]
. One can build an observer using Proposition 2 and get back to natural coordinates through the explicit expression of φ -1 :

x = φ -1 (ẑ) =     ẑ1 ẑ1 ẑ3 -ẑ 2 2 +2ẑ 3 1 2ẑ 2 1 ẑ1 ẑ2 +ẑ 1 ẑ3 -ẑ 2 2 +2ẑ 3 1 2ẑ 2 1     (14)
which is undefined at the singularity ẑ1 = x1 = 0. One can also get an observer in the natural coordinates using Propositions 2 and 4:

ẋ = ∂ φ ∂ x ( x) -1 (A 3 φ ( x) + B 3 F s ( x) + ∆ 3 K(y -C 3 x)) (15)
as long as φ is a diffeomorphism. Yet we know from Proposition 1 that φ is singular at x1 = 0 and there is no guarantee that the observer will not leave Ω min . Now using Propositions 2 and 3, we get a low-power high gain observer

ζ = ϕ(ζ , x, y) =     ζ 12 + θ k 11 (y -ζ 11 ) ζ 22 + θ 2 k 21 (y -ζ 11 ) ζ 22 + θ k 12 (ζ 12 -ζ 21 ) F 3s (ζ ) + θ 2 k 22 (ζ 12 -ζ 21 )     (16)
and we can get back to the natural coordinates using either ẑ = Pζ or ẑ′ = P ′ ζ combined with [START_REF] Gauthier | A simple observer for nonlinear systems. application to bioreactors[END_REF]. Note that such an observer exhibits the same redundancy as in using the lowpower high gain observer since we can use either x = P xa or x′ = P ′ xa . This observer also faces the problem linked with the observability singularity at x1 = 0.

B. Augmentation of the mapping φ

Since the low-power high gain observer underlies a mapping φ from x ∈ R n to ζ ∈ R q with q = 2(m -1) > m ≥ n, the mapping is not surjective for m > 2. A first step is to augment the mapping into a diffeomorphism by augmenting the mapping domain. Since the state of the observer of Proposition 3 is redundant in the form (using ẑ = Pζ and ẑ′

= P ′ ζ ) ζ =                ζ 1 ζ 2 . . . ζ n-2 ζ n-1                =                ζ 1 1 ζ 1 2 ζ 2 1 ζ 2 2 . . . ζ n-2 1 ζ n-2 2 ζ n-1 1 ζ n-1 2                =                ẑ1 ẑ2 ẑ3 . . . ẑn-1 ẑn                =                ẑ′ 1 ẑ′ 2 . . . ẑ′ n-2 ẑ′ n-1 ẑ′ n               , (17)
a = x 1 x 2 x ′ 2 x 3 . . . x ′ n-1 x n T ∈ R q .
Proposition 5 For systems S min and S hyb , defining respectively the natural coordinate augmented state x a and φ a as

φ a : x a =     x 1 x 2 x ′ 2 x 3     → z a =     x 1 2x 1 (x 3 -x 2 ) 2x 1 (x 3 -x ′ 2 ) 2x 1 (2(x 3 -x 2 ) 2 -x 1 + x 2 )     (18)
and φ a :

x a = x 1 x 2 x ′ 2 x 3 x ′ 3 x 4 T → z a given by z a =           x 1 2x 1 (x 4 -x 3 ) 2x 1 (x 4 -x ′ 3 ) 4x 1 (x 4 -x 3 ) 2 -2x 1 (x 1 + x 2 -x 3 ) 4x 1 (x 4 -x 3 ) 2 -2x 1 (x 1 + x ′ 2 -x 3 ) 2x 1 4(x 4 -x 3 ) 3 + (x 1 + x 2 -x 3 ) -2x 2 (-µ ′ -x 3 ) -2(x 4 -x 3 )(3(x 1 + x 2 -x 3 ) -x 1 (x 4 -x 3 ))           (19) φ a is a diffeomorphism on Ω a,min = {x a ∈ R 4 : |x 1 | > ε}, respectively on Ω a,hyb = {x a ∈ R 6 : |x 1 | > ε, |x 2 + x 3 + µ ′ | > ε ′ }.
Proof: To lighten notations, one can write

∂ φ a ∂ x a (x a ) = 1 0 1,q-1 * J . ( 20
)
For system S min , differentiating φ a leads to

J min = 2   -x 1 0 x 1 0 -x 1 x 1 x 1 -4x 1 q 0 0 4x 1 q 0   ( 21
)
with q 0 = x 3x 2 , so we have

∂ φ a ∂ x a (x a ) = x 3 1
and φ a is a diffeomorphism on Ω a,min . Similarly for the system S hyb , differentiating φ a leads to

J hyb = 2       0 0 -x 1 0 x 1 0 0 0 -x 1 x 1 -x 1 0 -4x 1 q 1 + x 1 0 4x 1 q 1 0 -x 1 -4x 1 q 1 + x 1 0 4x 1 q 1 x 1 q 2 0 x 1 q 3 0 x 1 q 4       (22) with q 1 = x 4 -x 3 , q 2 = 1 -(-µ ′ -x 3 ) -6q 1 , q 3 = -12q 2 1 + 6q 1 + 8x 2 + 8x 1 -6x 3 -1 and q 4 = 12q 2 1 -2(4x 1 + 3x 2 -3x 3 ). The determinant is given by ∂ φ a ∂ x a (x a ) = x 5 1 (x 2 + x 3 + µ ′ ),
so φ a is a diffeomorphism on Ω a,hyb . Invertibility of the jacobian matrix of φ a is now possible as long as the estimated augmented state remains away from the observability singularity. Yet there is no guarantee that the estimated state stays in this set. The next step addresses this concern.

C. Extension of φ a into an injective immersion on R q

To avoid loosing any access to some unmeasured states because of the observability rank loss pinpointed in Proposition 1, for instance the loss of information about x 2 and x 3 when x 1 = 0 for system S min , a way to get around the observability singularity is to provide fictitious outputs aiming at getting information about these states around the singularity.

Proposition 6 For systems S min and S hyb , adding respectively fictitious outputs y

f ∈ R p y f (x) =   ρ(x)x 2 ρ(x)x ′ 2 ρ(x)x 3   , y f (x) =         ρ(x)x 2 ρ(x)x ′ 2 ρ(x)x 3 ρ(x)x ′ 3 ρ(x)x 4 ρ ′ (x)         (23)
, where ρ and ρ ′ are differentiable functions that satisfy for all positive δ :

ρ(x 1 ) : ρ(x 1 ) + δ x 2 1 > 0, ∀x 1 ∈ R ρ ′ (x 2 , x 3 ) : ∂ ρ ′ ∂ x 2 + ∂ ρ ′ ∂ x 3 + x 2 + x 3 + µ ′ > 0, ( 24
)
to the state z a results in an extended state z e = z a y f .

Extending the transformation φ e mapping x a ∈ R q to z e ∈ R q+p results in an injective immersion on the domain R q .

Proof: The Jacobian matrix of φ e is given by

∂ φ e ∂ x a (x a ) =   1 0 1,q-1 * J * J f   ( 25
)
where J is J min given by (21) (J hyb given by ( 22)) for system S min (S hyb ) respectively, and J f is given by

J f ,min = ρI 3 , J f ,hyb = ρI 5 ρ ′ 2 0 ρ ′ 3 0 0 ( 26
)
where

ρ ′ i = ∂ ρ ′ ∂ x i .
Since by construction 2x 2 1 + ρ(x) > 0 and

x 2 + x 3 + µ ′ + ρ ′ 2 (x) + ρ ′ 3 (x) > 0, it is straightforward that ∂ φ e
∂ x e (x e ) has a constant rank q for any x a ∈ R q .

D. Augmentation of φ e into a global diffeomorphism

The observability singularity has been solved, but using Proposition 4 requires a diffeomorphism to get an observer in natural coordinates. This last step consists in augmenting the immersion φ e defined on the full domain R q to R q+p into a global diffeomorphism φ φ φ on R q+p , which requires an augmentation of the natural coordinates x a ∈ R q to x x x =

x a τ ∈ R q+p . We follow here the Jacobian completion initially proposed in [START_REF] Andrieu | Dynamic extension without inversion for observers[END_REF].

Proposition 7 Let τ ∈ R p and x x x = x a τ ∈ R q+p . Defining φ φ φ : x x x → z z z = z e + Ψ(x a )τ = φ e (x a ) + Ψ(x a )τ (27)
with matrix Ψ(x a ), the completion of the jacobian matrix

∂ φ e ∂ x a : ∂ φ φ φ ∂ x x x = ∂ φ e ∂ x a + ∂ Ψ ∂ x a Ψ(x a ) (28)
is fulfilled for systems S min and S hyb using

Ψ min (x a ) =     0 1,3 H 1 -B T 3 x 1 I 3     with H 1 = I 2 -1 2,1 (29)
and

Ψ hyb (x a ) =       0 1,4 0 H 2 0 4,1 -B T 5 -2x 1 x 1 I 5 0 5,1 0 1,4 1       with H 2 = 0 2 I 2 -1 2,1 I 2 0 2 -1 2,1 .
(30) Then φ φ φ defines a global diffeomorphism.

Proof: Since Ψ depends only on x 1 , differentiating φ φ φ using (25) leads to

∂ φ φ φ ∂ x x x (x x x) =   1 0 1,q-1 * J * J f Ψ(x 1 )   (31)
Due to expressions (29) and (30), it follows that

∂ φ φ φ ∂ x x x (x x x) = (ρ + 2x 2 1 ) 3 ∂ φ φ φ ∂ x x x (x x x) = (ρ + 2x 2 1 ) 4 (2x 2 1 (ρ ′ 2 + ρ ′ 3 + 2(x 2 + x 3 + µ ′ )) + ρ) (32
) for S min and S hyb , respectively. By construction of functions ρ and ρ ′ , these Jacobian determinants are strictly positive for all x x x ∈ R q+p .

E. Low-power high gain observer in natural coordinates

Proposition 8 The dynamic system

ẋ x x = ∂ φ φ φ ∂ x x x (x x x) -1 ϒ(φ φ φ (x x x), y) -K p φ φ φ p (x x x) ( 33
)
is a converging observer for system S min (S hyb ) in their natural coordinates on any bounded subset of R q+p with (q, p) = (4, 3) (with (q, p) = (6, 6)). ϒ is defined by Proposition 3, φ φ φ is defined by Propositions 5, 6 and 7, K p is a positive definite diagonal matrix of dimension p and the notation φ φ φ p stands for the last p components of φ φ φ .

Proof: It can be checked that φ φ φ -1 ∈ C 1 , hence it is L-Lipschitz on any compact of R q+p . It follows that, using z z z * = φ φ φ (x a , 0) = φ e (x a ): Simulations are performed using the low-power high gain observer in natural coordinates given by Proposition 8. The first simulation is led on system S min and illustrates the robustness with respect to noise. The x 1 dynamics and the output y are corrupted by the addition of zero mean Gaussian white noises of standard deviations 0.2 and 0.1 respectively. The former noise is intended to model turbulence and impact of other unmodeled dynamics on the main mode x 1 . System S min given by (4) has been simulated with the initial conditions and observer parameters given in Table I. Fictitious outputs required to get a full-ranked Jacobian matrix in Proposition 6 are defined using

x a (t) -xa (t) + τ(t) ≤ √ 2 x a -xa τ ≤ √ 2 x a 0 -xa τ ≤ √ 2 φ φ φ -1 (z z z * ) -φ φ φ -1 (ẑ z z) ≤ L √ 2 z z z * -ẑ z z ≤ L √ 2 φ e (x a ) -φ φ φ ( xa , τ) . ( 34
ρ : x → max(0, ε 2 -x 2 1 ) 2 . ( 36
)
Despite the initial condition x 1 (0) = 0, the proposed observer is well defined and exhibits a good convergence property irrespective of the singularity. However, some peaking phenomenon can be noticed at the beginning of the simulation on Figure 1(e) because of the inappropriate choice for initial conditions that "activates" the fictitious outputs. It implies changes in the dynamics of the extension τ, and in turn on the estimates since the observer aims at making φ φ φ (x x x) = φ e ( xa ) + Ψ( xa )τ converge, whilst φ e ( xa ) is a better representation of the estimates. Then the error asymptotically converges to zero and the role of fictitious outputs vanishes outside the x1 > |ε|, hence displaying the effectiveness of the proposed observer, particularly with respect to noise filtering. Simulation parameters for system S hyb described by (5) are given in Table II. As highlighted in Proposition 1, the system S hyb can face an additional observability singularity for x 2 + x 3 + µ ′ = 0, so we use functions ρ defined by (36) and ρ ′ (x) = (1/(x 2 + x 3 + µ ′ ) -1/ε ′ ) 2 in order to maintain x2 + x3 + µ ′ > 0. As it can be noticed on Figure 1(b), the initial dynamics of S are driven by the growth rate parameter x 4 = µ, here assumed to be a constant. S hyb behaves as a double oscillator whose main mode amplitude x 1 asymptotically follows a transcritical bifurcation, whilst the enslaved mode amplitude x 2 is damped thanks to the term (-µ ′ ). During the time interval [10s; 16s], a control input is applied on the dynamics of the second mode as suggested in [START_REF] Luchtenburg | A generalized mean-field model of the natural and high-frequency actuated flow around a high-lift configuration[END_REF] in order to excite this oscillator. As expected the coupling -through the shift mode x 3 in the third equation of ( 5)-results in an energy flow from the main mode x 1 to the mode x 2 , forcing x 1 to go to zero. During the transient phase, Figure 1(d) illustrates the convergence of the state estimates to the real states despite initial values close to zero. As in the first simulation, avoiding the singularity induces some peaking: it is expected since the observer is then built on a system that does not represent exactly the physical model. The estimation tracking is ensured as x 1 goes to zero due to the control input.

V. CONCLUSIONS AND FUTURE WORKS

We have proposed an observer capable of being sufficiently robust with respect to noises and efficient to rebuild the system states while avoiding singularities as shown by the simulation results. The modified low-power high gain observer has provided a solution to many challenges : i) a rebuild of turbulent wake flow from an unique measured output based on reduced order models, ii) an observability singularity avoidance, iii) a robustness in regard to high frequency noises and sudden changes of the dynamics. It would be interesting to study if the redundancy induced by the low-power high gain observer may be exploited to reduce the proposed observer dimension. The on-going works are on one hand, the experimental validation of the observer in real conditions (the experimental setup is being fully equipped by sensors and actuators) to get a better knowledge of physical phenomena and, on the other hand, the design of feedback laws based on the estimated state to control the flow in order to reduce the drag force. Both perspectives are of great interest for applications in fluid mechanics.

  error is drawn using the previous color convention. Due to the low-power high gain observer redundancy, there are two estimates xi and x′

			1.2
			1.1
																																			1.5
			1
			0.9
			0.8
	Smin States	0.4 0.6 0.5 0.7																															Shyb States	0.5 1
			0.3
			0.2
			0.1
			0																																0
				0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15 t(s)	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30		0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30
																																											t (s)
															(a) S min states																		(b) S hyb states
			1
		0.9																																2
		0.8
																																			1.5
		0.7
	Smin Estimated states	0.4 0.6 0.3 0.5																															Shyb Estimated States	0 1 0.5
		0.2
		0.1																																-0.5
			0	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30		0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30
																			t (s)																								t (s)
										(c) Observer estimation for S min											(d) Observer estimation for S hyb
		0.8
		0.7																																1.5
		0.6																																1
		0.5
	Estimation error	0.2 0.4 0.3																															Estimation Error	0 -0.5 0.5
		0.1
																																			-1
			0
	0 -0.1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	-1.5	0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30
																			t (s)																								t (s)
	(e) Estimation error for S min x1 in black solid line, ( x2 , x′												(f) Estimation error for S = ( x1 x2	x′ 2 . . . x4 ) is plotted with
																																																	)
																																		Since φ e is injective and (9) defines a converging observer
																																		from Proposition 3, it follows that
																																							lim t→∞		x a (t) -xa (t) + τ(t) = 0.	(35)
																																							IV. SIMULATION RESULTS
																																									x 0		0 0.51 0.8	T
																																									x0		0.1 0.1 0.2	T
																																								K 1 , K 2		0.18 0.1	,	0.18 0.02
																																									K p					I 3
																																									ε				0.5
																																									θ					2
																																											TABLE I
																																			INITIAL CONDITIONS AND PARAMETERS FOR SYSTEM S min

hyb Fig. 1. Estimation using the low-power high gain observer in natural coordinates. From top to bottom, on the subfigures: (a)-(b) the states of the physical systems S min (S hyb ): x 1 , x 2 , x 3 (and x 4 ) are depicted in black, red, green and blue solid lines; (c)-(d) state estimation xT a 2 ) in red solid and dash dotted lines, ( x3 , x′ 3 ) in green solid and dash dotted lines, and x4 in blue; (e)-(f) the estimation i for 2 ≤ i ≤ n -1.

  [START_REF] Ma | Dynamics and lowdimensionality of a turbulent near wake[END_REF]

						T
	K 1 , K 2 , K 3	1.8 1	,	2.4 0.4	,	2 0.2
	K p		0.1I 6
	θ ε, ε ′		2 0.1, 0.5
		TABLE II
	INITIAL CONDITIONS AND PARAMETERS FOR SYSTEM S hyb

VI. ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support from the Region Centre Val de Loire (France) under the project APR-IA COMODO.


Guofeng Lu

Christelle Despas

Liang Liu

Grégoire Herzog
email: gregoire.herzog@univ-lorraine.fr

Ametryn

Keywords: Liquid-liquid interface, preconcentration, pesticide, electroanalysis, proton transfer

Introduction

The improvement of sensitivity and the lowering of detection limits at the interface between two immiscible electrolyte solutions (ITIES) are a constant drive for the development of new accumulation and preconcentration procedures. Stripping voltammetry at the ITIES has taken advantage through the preconcentration of ions close to the interface during the accumulation period [START_REF] Mareček | Electrolysis at the interface between two immiscible electrolyte solutions by means of a hanging electrolyte drop electrode[END_REF][START_REF] Mareček | Determination of calcium, barium and strontium ions by differential pulse stripping voltammetry at a hanging electrolyte drop electrode[END_REF]. It is achieved by the application of a potential difference across the interface and with the help of an ionophore [START_REF] Scanlon | Electrochemical Detection of Oligopeptides at Silicon-Fabricated Micro-Liquid|Liquid Interfaces[END_REF][START_REF] Herzog | Detection of Food Additives by Voltammetry at the Liquid-Liquid Interface[END_REF] or of the organic electrolyte anion [START_REF] Alvarez De Eulate | Adsorptive Stripping Voltammetry of Hen-Egg-White-Lysozyme via Adsorption-Desorption at an Array of Liquid-Liquid Microinterfaces[END_REF][START_REF] O'sullivan | Stripping voltammetric detection of insulin at liquid-liquid microinterfaces in the presence of bovine albumin[END_REF][START_REF] Alvarez De Eulate | Ion-transfer electrochemistry of rat amylin at the water-organogel microinterface array and its selective detection in a protein mixture[END_REF][START_REF] Alvarez De Eulate | Detection of haemoglobin using an adsorption approach at a liquid-liquid microinterface array[END_REF][START_REF] Herzog | Ion-Transfer Voltammetric Behavior of Protein Digests at Liquid|Liquid Interfaces[END_REF], which lower the Gibbs free energy of transfer to allow ion transfer within the potential window available. Stripping voltammetric detection at the ITIES was reported for choline [START_REF] Lee | Stripping Voltammetric Determination of Choline Based on Micro-Fabricated Composite Membrane[END_REF], oligopeptides [START_REF] Scanlon | Electrochemical Detection of Oligopeptides at Silicon-Fabricated Micro-Liquid|Liquid Interfaces[END_REF], food additives [START_REF] Herzog | Detection of Food Additives by Voltammetry at the Liquid-Liquid Interface[END_REF], protein digests [START_REF] Herzog | Ion-Transfer Voltammetric Behavior of Protein Digests at Liquid|Liquid Interfaces[END_REF], and proteins such as hen-egg-white-lysozyme [START_REF] Alvarez De Eulate | Adsorptive Stripping Voltammetry of Hen-Egg-White-Lysozyme via Adsorption-Desorption at an Array of Liquid-Liquid Microinterfaces[END_REF], insulin [START_REF] O'sullivan | Stripping voltammetric detection of insulin at liquid-liquid microinterfaces in the presence of bovine albumin[END_REF], amylin [START_REF] Alvarez De Eulate | Ion-transfer electrochemistry of rat amylin at the water-organogel microinterface array and its selective detection in a protein mixture[END_REF], and haemoglobin [START_REF] Alvarez De Eulate | Detection of haemoglobin using an adsorption approach at a liquid-liquid microinterface array[END_REF]. Stripping voltammetric methods at the ITIES can be accompanied with the miniaturisation of the ITIES down to the micro-and to the nanoscale to benefit from the improved mass transfer due to radial diffusion and to the reduced iR drop and capacitive current [START_REF] Liu | Electrochemistry at micro-and nanoscopic liquid/liquid interfaces[END_REF][START_REF] Herzog | Stripping voltammetry at micro-interface arrays: A review[END_REF][START_REF] Herzog | Recent developments in electrochemistry at the interface between two immiscible electrolyte solutions for ion sensing[END_REF][START_REF] Arrigan | Electroanalytical Ventures at Nanoscale Interfaces Between Immiscible Liquids[END_REF].

However, such an accumulation method is not suited for non-charged species present in environmental samples, such as many pesticides. We propose here a method for the accumulation and the detection of neutral lipophilic molecules at the ITIES, using ametryn as a model molecule. Ametryn (AME) is a selective herbicide of the triazine family used in fruits and vegetables farming, which does not degrade easily and can accumulate in soil and in natural waters [START_REF] Jacomini | Assessment of ametryn contamination in river water, river sediment, and mollusk bivalves in São Paulo State, Brazil[END_REF][START_REF] Liu | Comprehensive analysis of degradation and accumulation of ametryn in soils and in wheat, maize, ryegrass and alfalfa plants[END_REF]. Because of its relatively high solubility in water (209 mg L -1 at 25 °C), residues of ametryn are found in rivers causing concern for drinking water. Triazine detection and quantification can be achieved by high performance liquid chromatography [START_REF] Koohpaei | Optimization of solid-phase extraction using developed modern sorbent for trace determination of ametryn in environmental matrices[END_REF][START_REF] Li | Quantitative analysis of triazine herbicides in environmental samples by using high performance liquid chromatography and diode array detection combined with second-order calibration based on an alternating penalty trilinear decomposition algorithm[END_REF][START_REF] Zambonin | Determination of triazines in soil leachates by solid-phase microextraction coupled to gas chromatography-mass spectrometry[END_REF], surface-enhanced Raman spectroscopy [START_REF] Chen | Quantitative determination of ametryn in river water using surface-enhanced Raman spectroscopy coupled with an advanced chemometric model[END_REF] and by electrochemical methods [START_REF] Guse | Electrochemical reduction of atrazine: NMR evidence for reduction of the triazine ring[END_REF][START_REF] Ignjatović | Polarographic behavior and determination of some S-triazine herbicides[END_REF][START_REF] Shoji | Atrazine sensor based on molecularly imprinted polymermodified gold electrode[END_REF][START_REF] Islam | Microchip Capillary Electrophoresis Based Electroanalysis of Triazine Herbicides[END_REF][START_REF] Pospíšil | Electrochemistry of s-triazine herbicides: Reduction of atrazine and terbutylazine in aqueous solutions[END_REF][START_REF] Skopalová | Polarographic behaviour of some s-triazine herbicides and their determination by adsorptive stripping voltammetry at the hanging mercury drop electrode[END_REF][START_REF] Tavares | Determination of ametryn in soils via microwave-assisted solvent extraction coupled to anodic stripping voltammetry with a gold ultramicroelectrode[END_REF][START_REF] Islam | Fast detection of triazine herbicides on a microfluidic chip using capillary electrophoresis pulse amperometric detection[END_REF][START_REF] Švorc | Green electrochemical sensor for environmental monitoring of pesticides: Determination of atrazine in river waters using a boron-doped diamond electrode[END_REF][START_REF] Bonin | Electrochemical Reduction of Hexahydro-1,3,5-trinitro-1,3,5-triazine in Aqueous Solutions[END_REF][START_REF] Luo | A voltammetric sensor based on GO-MWNTs hybrid nanomaterialmodified electrode for determination of carbendazim in soil and water samples[END_REF][START_REF] Ya | Electrochemical Determination of Carbendazim in Food Samples Using an Electrochemically Reduced Nitrogen-Doped Graphene Oxide-Modified Glassy Carbon Electrode[END_REF]. Triazine molecules are reduced at mercury [START_REF] Ignjatović | Polarographic behavior and determination of some S-triazine herbicides[END_REF][START_REF] Pospíšil | Electrochemistry of s-triazine herbicides: Reduction of atrazine and terbutylazine in aqueous solutions[END_REF][START_REF] Skopalová | Polarographic behaviour of some s-triazine herbicides and their determination by adsorptive stripping voltammetry at the hanging mercury drop electrode[END_REF], carbon [START_REF] Švorc | Green electrochemical sensor for environmental monitoring of pesticides: Determination of atrazine in river waters using a boron-doped diamond electrode[END_REF][START_REF] Bonin | Electrochemical Reduction of Hexahydro-1,3,5-trinitro-1,3,5-triazine in Aqueous Solutions[END_REF][START_REF] Luo | A voltammetric sensor based on GO-MWNTs hybrid nanomaterialmodified electrode for determination of carbendazim in soil and water samples[END_REF][START_REF] Ya | Electrochemical Determination of Carbendazim in Food Samples Using an Electrochemically Reduced Nitrogen-Doped Graphene Oxide-Modified Glassy Carbon Electrode[END_REF] and gold [START_REF] Shoji | Atrazine sensor based on molecularly imprinted polymermodified gold electrode[END_REF] electrodes, or oxidised at gold electrodes [START_REF] Islam | Microchip Capillary Electrophoresis Based Electroanalysis of Triazine Herbicides[END_REF][START_REF] Tavares | Determination of ametryn in soils via microwave-assisted solvent extraction coupled to anodic stripping voltammetry with a gold ultramicroelectrode[END_REF][START_REF] Islam | Fast detection of triazine herbicides on a microfluidic chip using capillary electrophoresis pulse amperometric detection[END_REF]. Reduction of ametryn is occurring at a potential of -1.0 V [START_REF] Ignjatović | Polarographic behavior and determination of some S-triazine herbicides[END_REF], overlapping with the reduction of dissolved oxygen, and thus requiring the purging with nitrogen to avoid interferences from oxygen. Electrochemical detection of triazine is also possible at the ITIES by ion transfer voltammetry [START_REF] Juárez | Electrochemical Study of s-Triazine Herbicides Transfer Across the Water/1,2-Dichloroethane Interface[END_REF][START_REF] Velázquez-Manzanares | Triazine Herbicides Transfer at the Water/1,2-Dichloroethane Interface[END_REF][START_REF] Velázquez-Manzanares | Electrochemical Behavior of Simetryn Herbicide at Water|1,2-Dichloroethane Interface[END_REF]. Such an electrochemical mechanism does not involve electron transfer and is not hindered by the presence of dissolved oxygen. However, species need to be in an ionic form to be detectable at the ITIES. Triazines are lipophilic weak bases, which require very acidic conditions (pH < 2) to be fully protonated.

We report here a protocol for the accumulation and detection of lipophilic molecules at a single µITIES supported by a glass capillary, using AME as a model target analyte (pKa = 4.1, log P = 2.63 [START_REF] Yang | pH-dependence of pesticide adsorption by wheatresidue-derived black carbon[END_REF]). This protocol could be generalised to molecules with the following properties: they should (i) be lipophilic (log P >> 0), and (ii) and they should be neutral around pH 7 (pKa < 7) and susceptible to be protonated. The protocol is based on the accumulation of AME from a solution close to neutral pH into an organic phase through spontaneous transfer at open-circuit potential. For the second step, the capillary containing the organic phase is transferred to a mildly acidic (pH 3) AME-free solution for a voltammetric detection through proton transfer assisted by AME. The presence of neutral molecules dissolved in the organic phase can indeed facilitate the transfer of proton across the interface by lowering the Gibbs free energy of transfer [START_REF] Velázquez-Manzanares | Kinetics of facilitated proton transfer by hydrophobic aromatic amines across the water/1,2-dichloroethane interface[END_REF][START_REF] Rimboud | Proton transfer across liquid{divides}liquid interfaces facilitated by the disproportionation reaction of a lutetium bisphthalocyanine: A voltammetric study at the microinterfaces between water and nitrobenzene or 1,6-dichlorohexane[END_REF][START_REF] Benvidi | Facilitated proton transfer by 2-acetylpyridine-4-phenyl-3thiosemicarbazone across water/1,2-dichloroethane interface[END_REF][START_REF] Holub | Proton transfer across a liquid/liquid interface facilitated by phospholipid interfacial films[END_REF][START_REF] Bingol | Facilitated proton transfer by a novel 2-aminothiazole derivative across the water/1,2-dichloroethane interface[END_REF][START_REF] Dassie | Facilitated proton transfer or protonated species transfer reactions across oil|water interfaces[END_REF][START_REF] Vega Mercado | Facilitated proton transfer across liquid | liquid interfaces under forced hydrodynamic conditions. Determination of partition coefficients of neutral weak bases[END_REF][START_REF] Burgoyne | Electrochemical Detection of Pseudomonas aeruginosa Quorum Sensing Molecules at a Liquid|Liquid Interface[END_REF][START_REF] Burgoyne | Detection of Pseudomonas aeruginosa quorum sensing molecules at an electrified liquid|liquid micro-interface through facilitated proton transfer[END_REF].

We propose here to use the proton transfer, assisted by the AME preconcentrated in the organic phase, as the analytical signal, allowing the accumulation and detection neutral species by electrochemistry at the ITIES.

Experimental Section

Chemicals & Reagents: The organic phase electrolyte was 1,2-dichloroethane (1,2-DCE, 99% Sigma-Aldrich) containing 10 mM Bis (triphenylphosphoranylidene) ammonium tetrakis (4chlorophenylborate) (BA + TPBCl -). The organic salt was prepared by metathesis of Bis (triphenylphosphoranylidene) ammonium chloride (BA + Cl -, 97% Aldrich) and potassium tetrakis (4chlorophenylborate) (K + TPBCl -, 98% Fluka). All aqueous solutions were prepared in purified water (resistivity of 18.2 MΩ cm). For pH adjustment, solutions of HCl or NaOH were used. Sodium chloride (NaCl, 99% Aldrich) was used as the aqueous phase electrolyte. Ametryn (AME, 99%) was obtained from Sigma-Aldrich and used without further purification. Stock solutions of 2 mM AME in ethanol were prepared and stored in the dark at 4°C. 50 mM tetramethylammonium chloride (TMA + Cl -, 98% Aldrich) was prepared in water to characterise the size of single pore capillary. 50 mM tetrabutylammonium chloride (TBA + Cl -, 97% Aldrich) was also prepared in water and was used to calibrate the half-wave Galvani potential of AME. Real samples were spring waters collected in the watershed of the Strengbach River (Vosges, France). They were stored at 4°C and filtered using 0.22 µm filter from Millipore before electrochemical and ion chromatography experiments.

Electrochemical analyses were performed in these samples spiked with ametryn at selected concentrations.

Preparation of capillaries: µITIES is established by pulling borosilicate glass capillaries, (inner diameter = 0.86 mm and outer diameter was 1.5 mm) according to the fabrication protocol described by Stockmann et al [START_REF] Stockmann | Hydrophobic alkylphosphonium ionic liquid for electrochemistry at ultramicroelectrodes and micro liquidliquid interfaces[END_REF]. Briefly, a home-made vertical capillary puller was used to seal through heating the capillary, which contained a gold wire (Ø = 50 µm). The capillary was split into two pieces, each contained a piece of the sealed gold wire. The capillary was then placed in an aqua regia solution of 3:1 HCl (36.5-38.0 %, Aldrich): HNO3 (70 %, Aldrich), which dissolved the gold wire leaving a microchannel of uniform diameter. After rinsing thoroughly with water, the µITIES tip was polished with sandpaper (granulation: 4000). The inside walls of the capillary were silanized with chlorotrimethylsilane (98%, Aldrich) prior cyclic voltammetry experiments to prevent the leakage of the organic phase.

Electrochemical Measurements: Cyclic voltammetry (CV) was performed with a potentiostat (302N) controlled by NOVA software (Version 1.11) from Autolab (Netherlands). All the measurements were done in customary set-up for ion transfer voltammetry with a silver wire serving as both a pseudo-reference electrode and a counter electrode for the organic phase. For the aqueous phase, a platinum mesh was used as a counter electrode, a silver-silver chloride wire (Ag | AgCl) as a reference electrode. All measurements were carried out at room temperature (25°C). The potential scale was calibrated with the ion transfer of TBA + , whose formal transfer potential at the aqueous -DCE interface was established at

∆ 𝑜 𝑤 𝜙 𝑇𝐵𝐴 + 0 = -0.270 V (∆𝐺 𝑇𝐵𝐴 + 0 = -26 𝑘𝐽 𝑚𝑜𝑙 -1 ) [47].
AC voltammetry experiments were measured with an Autolab PGSTAT302N (Metrohm), equipped with a FRA32 frequency response analyzer, connected to the electrochemical cell. The potentiostat was controlled by NOVA 2.1.4 software. The frequency selected was 1 Hz, the amplitude was 10

mV, and the step potential was 10 mV. We assumed that the electrochemical cell behaves as a Randles circuit. The analytical signal, Y, used to plot the AC voltammograms was determined by equation ( 1), as it was described previously [START_REF] Suárez-Herrera | Quantitative Analysis of Redox-Inactive Ions by AC Voltammetry at a Polarized Interface between Two Immiscible Electrolyte Solutions[END_REF]:

𝑌 = - 10 6 𝜔𝑍 𝐼𝑚 𝐴 -𝐶 𝑑 ( 1
)
where  is the angular frequency (rad s -1 ), ZIm is the imaginary part of the impedance (), A is the surface area of the interface (A = 1.96 × 10 -5 cm 2 ) and Cd is the double layer capacitance (µF cm -2 ) determined in the absence of analyte. Professional chromatography software controlled the 882 Compact IC plus instrument and its peripherals.

Ion chromatography analysis:

Results and Discussion

Ametryn transfer voltammetry at µITIES

The electrochemical behavior of AME at a single µITIES was investigated by cyclic voltammetry. In pH conditions where AME was protonated, AMEH + , (pH << pKa), the direct transfer of AMEH + from the aqueous phase to the organic phase was observed (Equation 1). The direct ion transfer was confirmed by the asymmetric diffusion profile of the cyclic voltammogram (Figure 1A). When the pH was raised to 2.1, the shape of the voltammogram changed to a peak-shaped response on both forward and reverse scans (Figure 1B), indicating that a linear diffusion is limiting ion transfer.

This change of diffusion profile was linked to a different transfer mechanism. With the increase of pH, the neutral form of AME transferred spontaneously to the organic phase and assisted the transfer of protons to the organic phase according to the following reaction:

𝐴𝑀𝐸 𝑜𝑟𝑔 + 𝐻 𝑎𝑞 + ↔ 𝐴𝑀𝐸𝐻 𝑜𝑟𝑔 + (II)
In such conditions, the proton assisted transfer was limited by the linear diffusion of AME in the organic phase, conferring the peak-shaped voltammograms in both forward and reverse scans. These observations were confirmed by the scan rate studies conducted for 320 µM AME at pH 1.1 and 70 µM at pH 2.1 (Figure 2). At pH 1.1, the asymmetric behaviour was preserved when the scan rate increased.

The limiting current for the direct transfer of AMEH + was constant although the reverse peak current was linear with the square root of the scan rate. At pH 2.1, both forward and reverse currents varied linearly with the square root of the scan rate, confirming limitation of the diffusion of AME in the organic side. When pH was further increased to pH 4, the shape of the voltammogram showed again an asymmetric diffusion profile (Figure 1C), although it was still associated to the proton transfer assisted by AMEorg.

However, the low concentration of H + became the limiting factor and was determining the CV shape.

On the forward scan, the transfer current reached a steady state, resulting from the radial diffusion of H + in the aqueous phase.

The influence of pH on the electrochemical behaviour of AME was then investigated by cyclic voltammetry for gradually increasing pH values from 1.1 to 2.9 (Figure 3). The transfer current increased with the pH and the transfer potential is shifted towards more positive potential difference values. When pH > pKa -log P, the variation of transfer potential should vary linearly with the pH with a slope of

2.3𝑅𝑇 𝑧𝐹

where R was the universal gas constant (in J K -1 mol -1 ), T the temperature (in K), z the charge transferred (+1) and F the Faraday constant (96485.15 C). The slope of the variation of transfer potential was measured at 76 mV per pH unit, which is greater than the theoretical slope of 59 mV per pH unit (Figure 3B). However, this variation of transfer potential with pH is supporting the assisted proton transfer in these pH conditions. For an initial concentration of 70 µM of AME in the aqueous phase, a transfer current of 0.69 nA was recorded at pH 1.1, while it rose to 6.2 nA at pH 2.9 for the same concentration. This important increase of transfer currents found their origin in the asymmetric diffusion profiles observed at µITIES, as well as in the volume differences between the organic and the aqueous phases. Variation of peak current as a function of pH was also observed at macroITIES for prometryn, another molecule of the triazine family [START_REF] Juárez | Electrochemical Study of s-Triazine Herbicides Transfer Across the Water/1,2-Dichloroethane Interface[END_REF], yet in that report, the peak current decreased as the pH increased. The difference of this trend with the one observed for AME at µITIES, was due to the difference of volume ratios between the organic and aqueous phases. In the case of macroITIES, the volume ratio is close to 1, so the spontaneous transfer of prometryn to the organic phase led to a prometryn concentration in the organic phase lower than in the aqueous phase, while, here, the concentration of [AME]org rose significantly as the volume of the organic phase, Vorg, was smaller than the volume of the aqueous phase, Vaq.

AME distribution across the liquid-liquid interface

When dissolved in an aqueous solution, AME was dissociated according to reaction (III), for which the dissociation constant, Ka, was written as described by equation ( 2).

𝐴𝑀𝐸𝐻 𝑎𝑞 + ↔ 𝐴𝑀𝐸 𝑎𝑞 + 𝐻 𝑎𝑞 + (III)

𝐾 𝑎 = [𝐴𝑀𝐸] 𝑎𝑞 [𝐻 + ] 𝑎𝑞 [𝐴𝑀𝐸𝐻 + ] 𝑎𝑞 (2)
Since AME is a lipophilic molecule, its neutral form was distributed spontaneously between the organic and aqueous phase as follows:

𝐴𝑀𝐸 𝑎𝑞 ↔ 𝐴𝑀𝐸 𝑜𝑟𝑔 (IV) 𝑃 = [𝐴𝑀𝐸] 𝑜𝑟𝑔 [𝐴𝑀𝐸] 𝑎𝑞 (3)
Equation ( 4) describes the mass conservation:

𝑉 𝑜𝑟𝑔 * [𝐴𝑀𝐸] 𝑜𝑟𝑔 + 𝑉 𝑎𝑞 * [𝐴𝑀𝐸] 𝑎𝑞 + 𝑉 𝑎𝑞 * [𝐴𝑀𝐸𝐻 + ] 𝑎𝑞 = 𝑛 (4)
Where n was the total number of moles initially present in the system, Vorg and Vaq were the volumes of the organic and aqueous phases, respectively. Previous studies have investigated the effect of the ratio of the volumes of the two phases on the transfer potential and current at either macroscopic ITIES [START_REF] Vega Mercado | Facilitated proton transfer across liquid | liquid interfaces under forced hydrodynamic conditions. Determination of partition coefficients of neutral weak bases[END_REF][START_REF] Kubota | Ion-transfer Voltammetry of Local Anesthetics at an Organic Solvent / Water Interface and Pharmacological Activity vs . Ion Partition Coefficient Relationship[END_REF][START_REF] Gobry | Generalization of ionic partition diagrams to lipophilic compounds and to biphasic systems with variable phase volume ratios[END_REF][START_REF] Zhang | Studies of Effect of Phase Volume Ratio on Transfer of Ionizable Species across the Water/1,2-Dichloroethane Interface by a Three-Electrode Setup[END_REF][START_REF] Garcia | Novel electrochemical approach to the determination of the partition coefficient of neutral weak bases[END_REF][START_REF] Fernández | Electroanalytical procedure to resolve a sample solution containing tetracycline and its toxic degraded product: Anhydrotetracycline[END_REF][START_REF] Garcia | Effect of ligand protonation on the facilitated ion transfer reactions across oil/water interfaces. IV. Buffer solution effect[END_REF] and thick film modified electrodes [START_REF] Zanotto | Facilitated proton transfer-electron transfer coupled reactions at thick-film modified electrodes[END_REF][START_REF] Zanotto | An integrated theoreticalexperimental approach to understand facilitated proton transfer-electron transfer coupled reactions at thick-film modified electrodes[END_REF]. By combining equations ( 2), ( 3) and ( 4), we can express the concentrations [AME]aq, [AME]org and [AMEH + ]aq at equilibrium. We assumed that in the absence of the application of interfacial potential, the protonated species AMEH + and H + were only present in the aqueous phase due to their negligible solubility in the organic phase.

[𝐴𝑀𝐸] 𝑎𝑞 =

The distribution of AME in the biphasic system as a function of pH was shown in Figure 4

[𝐻 + ] 𝑎𝑞 = [𝐴𝑀𝐸] 𝑜𝑟𝑔 (9)
The equation ( 11) was observed when: 𝑝𝐻 = 𝑝𝐾𝑎 -𝑙𝑜𝑔 𝑃 [START_REF] Lee | Stripping Voltammetric Determination of Choline Based on Micro-Fabricated Composite Membrane[END_REF] indicating that this limit was independent of the volumes Vorg and Vaq. When pH < pKa-log P, the electrochemical signal was the result of the direct transfer of 𝐴𝑀𝐸𝐻 𝑎𝑞 + , resulting in the steady state voltammogram (as shown in Figure 1). The limiting current, Ilimiting, was defined by equation

𝐼 𝑙𝑖𝑚𝑖𝑡𝑖𝑛𝑔 = 4𝑧𝐹𝐷 𝑎𝑞 𝐴𝑀𝐸𝐻 + [𝐴𝑀𝐸𝐻 + ] 𝑎𝑞 𝑟 (11)
Although Ilimiting is directly proportional to the [𝐴𝑀𝐸𝐻 + ] 𝑎𝑞 , the very acidic conditions for the detection of AMEH + via direct transfer are not preferred. However, the upper limit of the second domain (shown in Figure 4) is the solution of the quadratic equation:

𝑉 𝑎𝑞 ([𝐻 + ] 𝑎𝑞 𝑙𝑖𝑚 ) 2 + 𝐾 𝑎 (𝑃𝑉 𝑜𝑟𝑔 + 𝑉 𝑎𝑞 )[𝐻 + ] 𝑎𝑞 𝑙𝑖𝑚 -𝑛𝑃𝐾 𝑎 = 0 (12) [𝐻 + ] 𝑎𝑞 𝑙𝑖𝑚 = -𝐾 𝑎 (𝑃𝑉 𝑜𝑟𝑔 +𝑉 𝑎𝑞 )+√𝐾 𝑎 2 (𝑃𝑉 𝑜𝑟𝑔 +𝑉 𝑎𝑞 ) 2 +4𝑛𝑃𝐾 𝑎 𝑉 𝑎𝑞 2𝑉 𝑎𝑞 (13)
When pKa-log P < pH < pHlim, the electrochemical signal was the result of the proton facilitated by the neutral form of AME dissolved in the organic phase, AMEorg. As in this pH range, [H + ]aq > [AME]org, the transfer current is proportional to the [AME]org:

𝐼 𝑝 = 2.69 × 10 5 𝑧 3/2 𝐴(𝐷 𝑜𝑟𝑔 𝐴𝑀𝐸 ) 1/2 [𝐴𝑀𝐸] 𝑜𝑟𝑔 𝜐 1/2 (14)
When the Vorg tends to 0, the upper limit depends on the initial concentration, [𝐴𝑀𝐸] 𝑎𝑞 𝑖𝑛𝑖 = 𝑛/𝑉 𝑎𝑞 , which is given by equation ( 15):

𝑙𝑖𝑚 𝑉 𝑜𝑟𝑔 →0 [𝐻 + ] 𝑎𝑞 𝑙𝑖𝑚 = -𝐾 𝑎 +√𝐾 𝑎 2 +4[𝐴𝑀𝐸] 𝑎𝑞 𝑖𝑛𝑖 𝑃𝐾 𝑎 2 (15)
The evolution of pHlim as a function of [𝐴𝑀𝐸] 𝑎𝑞 𝑖𝑛𝑖 was plotted in Figure 5 and the three domains were defined by equations ( 10) and ( 15). When pH > pHlim, [H + ]aq < [AME]org so the proton assisted transfer is limited by the proton concentration. As a result, a steady state voltammogram is observed and the limiting current is given by the following equation:

𝐼 𝑙𝑖𝑚𝑖𝑡𝑖𝑛𝑔 = 4𝑧𝐹𝐷 𝑎𝑞 𝐻 + [𝐻 + ] 𝑎𝑞 𝑟 (16)
The analysis above could qualitatively explain the shape of voltammogram of AME at different pH. It may also provide an estimation on the limiting current of peak current. Nevertheless, it should also be noted that the actual condition is a mixed transfer of [AMEH + ]aq, [AME]aq and [H + ], so the current expressions are valid only under strict approximations when only one ion is considered, and they should not be overinterpreted.

Proton transfer assisted by ametryn in buffered solutions

Results shown in section 3.2 have demonstrated that both accumulation and peak current for the proton assisted transfer are pH-dependent. Accumulation will be optimal at pH >> pKa, and ametryn detection will be optimal at pH < pHlim as defined by equation [START_REF] Jacomini | Assessment of ametryn contamination in river water, river sediment, and mollusk bivalves in São Paulo State, Brazil[END_REF]. Such contradictory conditions would require a time-consuming two-step protocol. Furthermore, detection conditions would vary greatly with the nature of the target analytes pKa and P. To circumvent these limitations, we have investigated the effect of the addition of buffer to the aqueous solution. The presence of buffer in the aqueous solution ensures a constant concentration of protons available for the transfer across the interface.

Figure 5 shows the cyclic voltammograms of 8 µM ametryn initially dissolved in the aqueous phase at pH 6.8, 0.1, 1, and 10 mM phosphate buffer. At 0.1 mM phosphate buffer, the proton transfer assisted by ametryn was not observed. The buffer concentration was not high enough and limited the transfer reaction. However, when the phosphate buffer concentration was 1 mM, a peak for the transfer of 8 µM ametryn was observed, and thus allowing the one-step detection of ametryn. Increasing the concentration of phosphate buffer to 10 mM resulted in a loss of ametryn signal due to the rise of background current (inset of Figure 5). All subsequent experiments were done in the presence of 1 mM phosphate buffer (pH 6.8). A more sensitive electrochemical method was required to overcome the mediocre limit of detection reached with cyclic voltammetry (Figure 6). Recently, AC voltammetry was described by Suárez-Herrera and Scanlon [START_REF] Suárez-Herrera | Quantitative Analysis of Redox-Inactive Ions by AC Voltammetry at a Polarized Interface between Two Immiscible Electrolyte Solutions[END_REF] as an quantitative analytical method at the ITIES. It was used here to improve the sensitivity of the ametryn detection (Figure 7). The frequency of the AC signal was set at a value of 1 Hz, which favoured the Faradaic response of the proton assisted transfer limited by AME diffusion in the organic phase. An electrochemical cell was set with 10 mM LiCl + 1 mM phosphate buffer as an aqueous phase. A blank cyclic voltammogram and a blank AC voltammogram were recorded prior to additions of ametryn to the aqueous phase. After each ametryn addition, AC voltammograms were recorded after 30 minutes of waiting time at open-circuit potential to allow the partition of neutral lipophilic ametryn to reach equilibrium. AC voltammograms for ametryn concentrations ranging from 0.5 to 2.5 µM are shown in Figure 7. A peak at 0.16 V, corresponding to the proton transfer assisted by AME, was measured and its height increased linearly with the concentration of AME (Figure 7B). The sensitivity of 33.6 µF cm -2 µM -1 allowed to reach a limit of detection of 0.2 µM (calculated as 3 times the standard deviation divided by the sensitivity). Such a limit of detection is 40 times lower than with cyclic voltammetry. Three river water samples (CS3, CS4 and SG) were then analysed by AC voltammetry. The ionic composition of the three river samples was determined by ion chromatography. Phosphate concentration was measured at 27.7 µM for CS3 and below the limit of detection for ion chromatography for the other two samples (CS4 and SG). Since the phosphate concentration in the samples was naturally insufficient, 1 mM phosphate buffer was added, and the pH was set 6.8. AC voltammograms for increasing concentrations are shown after subtraction of the blank (Figure 8). As for the model solution, the analytical signal increased linearly with the concentration of AME in the aqueous phase for all three real samples investigated. Analytical figures of merit for the real samples are gathered in Table 1. Concentrations of Li + , Na + , NH4 + , K + were also determined by ion chromatography since the presence of those ions could interfere with proton assisted transfer. Their concentrations in the samples tested are given in Table 1. Among those four cations found naturally in the river samples, NH4 + has the lowest standard transfer potential with ∆ 𝑜 𝑤 𝜙 𝑁𝐻 4 + 0 = 0.531 V and is hence the most likely to interfere with the proton assisted transfer. The standard transfer potential of K + is slightly higher with ∆ 𝑜 𝑤 𝜙 𝐾 + 0 = 0.542 V, followed by Na + (∆ 𝑜 𝑤 𝜙 𝑁𝑎 + 0 = 0.606 V) and Li + (∆ 𝑜 𝑤 𝜙 𝐿𝑖 + 0 = 0.620 V) [START_REF] Zhou | Improvement in the assessment of direct and facilitated ion transfers by electrochemically induced redox transformations of common molecular probes[END_REF]. For Li + , K + and NH4 + , the concentrations are between 3.6 and 37.2 µM. The natural concentrations of these three ions did not significantly alter the limits of detection achieved as they were comprised in the range 0.2 < LOD < 0.31 µM. The real samples tested were naturally more concentrated in Na + , but the concentration was artificially increased to 1.5 mM due to the addition of the sodium salt of the phosphate buffer. The proximity of the proton transfer assisted by AME with the Na + transfer may hamper the limit of detection.

Table 1: Analytical performance and chemical composition of the solution analysed.

Analytical performances

Conclusion

We investigated the cyclic voltammetry of cationic lipophilic model target analyte, ametryn, at a single µITIES as a function of pH. Both direct ion transfer and proton assisted transfer were observed depending on the pH of the aqueous phase. When pH < pKa -log P, the transfer of direct ion transfer of the protonated ametryn, AMEH +, was predominant. Spontaneous transfer of the neutral form of ametryn, AME, from the aqueous to the organic phase led to a switch of ion transfer mechanism from direct AMEH + transfer to H + transfer assisted by AMEorg. The presence of AME in the organic phase assisted the transfer of H + across the interface, which was either limited by the diffusion of AME in the organic phase (when pKa -log P < pH < pHlim) or by the concentration of H + in the aqueous phase (when pH > pHlim). Furthermore, the microscopic dimension of the interface and the volume ratio Vaq / Vorg >> 1 favoured the accumulation of AME in the organic phase and a preconcentration effect was thus observed. The addition of 1 mM phosphate buffer to the aqueous phase provided an excess source of protons available to transfer across the ITIES, assisted by the presence of AME in the organic phase. In such conditions, the detection of sub-micromolar concentrations of AME in real samples was possible.

This simple protocol may be applicable to lipophilic ionisable molecules and could be implemented for flow-cell analysis [START_REF] Yoshida | A flow electrolysis cell with a thin aqueous phase and a thin organic phase for the absolute determination of trace ionic species[END_REF] or for stripping voltammetry using paper-based electrochemical systems [START_REF] Podrazka | Paper-Based System for Ion Transfer Across the Liquid-Liquid Interface[END_REF].

  Both anionic and cationic compositions of real samples were determined by ion chromatography using a Metrohm 882 Compact IC plus instrument with a conductivity detector. A high-pressure pump, sequential (Metrohm CO2 suppressor MCS) and chemical (Metrohm suppressor MSM II for chemical) suppression modules were installed on the instrument. The cation separation was performed on a Metrosep C6-150/4.0 column packed with silica gel particles functionalized with carboxyl groups (5 µm particles diameter) and preceded by a guard column (Metrosep C6/4.0 guard) and an RP 2Guard column to remove traces of organic compounds. The mobile phase consisted of a solution of HNO3 (1.7 mmol L -1 ) and picolinic acid (1.7 mmol L -1 ) in ultrapure water (18.2 MΩ cm at 293 K). The flow rate was 0.9 mL min -1 and the sample loop volume was 5 µL. Phosphate determination was performed on a MetrosepA Supp 16-250/4.0 column packed with quaternary ammonium-functionalized polystyrene-divinylbenzene copolymer (4.6 µm particles diameter) and preceded by a guard column (Metrosep A supp 16Guard/2,0) and an RP 2Guard column to remove traces of organic compounds. The mobile phase consisted of a solution of Na2CO3 (7.5 mmol L -1 ) and NaOH (0.75 mmol L -1 ) in ultrapure water (18.2 MΩ cm at 293 K). The flow rate was 0.8 mL min -1 , and the sample loop volume was 20 µL. The MagIC Net™ 2.3

  In the forward scan, the current reached a steady-state due to the radial diffusion of AMEH + towards the µITIES. On the reverse scan, the transfer current took the shape of a peak, linked to the linear diffusion of the AMEH + in the organic phase, restricted by the side walls of the glass capillary. At pH = 1.1, the transfer potential for AMEH + , ∆ 𝑜 𝑤 𝜙 𝐴𝑀𝐸𝐻 + 1/2 = -0.152 V, was determined.

Figure 1 .

 1 Figure 1. Cyclic voltammetry of AME (A) at pH 1.1 ([AME] = 320 µM); (B) at pH 2.1 ([AME] = 70 µM); (C) at pH 4 ([AME] = 50 µM).  = 10 mV s -1 .

Figure 2 .

 2 Figure 2. Cyclic voltammetry of AME (A) at pH 1.1 ([AME] = 320 µM); (B) at pH 2.1 ([AME] = 70 µM). Scan rates were 10, 15, 20, and 25 mV s -1 .

Figure 3 .

 3 Figure 3. (A) Cyclic voltammograms obtained for variable pH value (from pH1 to pH3) of aqueous solution within 70µM AME. (B) Evolution of the half-wave potential of transfer of AME as a function of pH.

  using the following experimental conditions: n = 25 nmol, Vaq = 2 mL and Vorg = 50 µL. With pH ranging from 0 to 6, three domains were identified: (i) a low pH domain where the predominating form of AME was AMEH + aq; (ii) a medium pH domain where the neutral form of AME in the organic phase dominated but [AME]org remained lower than [H + ]aq; (iii) and a high pH domain where [AME]org > [H + ]aq. The boundaries of domains were marked by the following equations: [AMEH + ]aq= [AME]org (8)

Figure 4 .

 4 Figure 4. Concentration distribution of [AME]org (blue line based on equation 7), [AMEH + ]aq (red line based on equation 6), [AME]aq (black line based on equation 5), and [H + ]aq (green line). n = 25 nmol, Vaq = 2 mL and Vorg = 50 µL.

Figure 5 .

 5 Figure 5. Distribution of the three domains as a function of the pH and [𝐴𝑀𝐸] 𝑎𝑞 𝑖𝑛𝑖 . The solid lines separating the domains were established with equations (10) and (15). The shape of the data points represents the types of ion transfer observed experimentally: () AMEH + ion transfer, () H + assisted by AMEorg, and () H + limited assisted transfer.

Figure 6 .

 6 Figure 6. Cyclic voltammograms of 8 µM AME in a 10 mM LiCl aqueous solution containing 0.1 (red curve), 1 (black curve) and 10 mM (inset) phosphate buffer. Scan rate was 10 mV s -1 . The CVs were recorded after the electrochemical cell was left 30 min at open-circuit potential to reach equilibrium.

Figure 7 .

 7 Figure 7. (A) Background subtracted AC voltammograms for increasing concentrations of AME (0.5 -2.5 µM) in the aqueous phase (10 mM LiCl + 1 mM phosphate buffer, pH 6.8). Experiments were done in triplicate. Inset: AC voltammograms before background subtraction. (B) Calibration curve for AME obtained from the AC voltammograms. Error bars are smaller than symbols when not visible.

Figure 8 .

 8 Figure 8. AC voltammograms for increasing spiked concentrations of AME (0.5 -2.5 µM) in real sample CS3 from the Strengbach river (Vosges, France). 1 mM phosphate buffer (pH 6.8) was added to the sample. 30 minutes of waiting time after each addition prior to AC voltammetry run. The voltammograms were background subtracted.

  of detection was calculated as 3 times the standard deviation of the intercept divided by the sensitivity. b : These sample names correspond to sampling points on the Strengbach river. c : concentration below the limit of detection of ion chromatography. d : determined by IC before addition of phosphate buffer.

Acknowledgments

GL is grateful to the China Science Council for sponsoring his PhD (N°201708070029). The authors thank the French PIA project "Lorraine Université d'Excellence" (Reference No. ANR-15-IDEX-04-LUE).