F Ferrante

F Gouaisbaut

S Tarbouriech

Stabilization of Continuous-Time Linear Systems subject to Input Quantization ‹

Keywords: Linear systems, Quantizers, Convex optimization, Discontinuous control, Numerical simulations, Feedback stabilization

This paper deals with the stabilization of continuous-time linear time-invariant systems subject to uniform input quantization. Specifically, the right-hand side of the closed-loop system is rewritten as a linear system subject to a discontinuous perturbation due to the quantization error. Then, the controller design is performed to achieve finite-time convergence of the closed-loop trajectories towards a compact invariant set surrounding the origin. Furthermore, a computationally tractable design procedure for the proposed controller based on linear matrix inequalities, and some insights on the simulation of the closed-loop system are presented. In addition, the effectiveness of the proposed control design procedure is shown in a numerical example.

Introduction

Recent technology enhancement has enabled the control of dynamical systems via digital controllers. When a continuous-time plant is controlled through a digital controller, side effect as time-delays, asynchronism, quantization, (see [START_REF] Elia | Stabilization of linear systems with limited information[END_REF] and references therein), can turn into an excessive performance degradation like the appearance of limit cycles, chaotic phenomena or even instability of the closed-loop system, [START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF][START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF]. Concerning the effect of quantization in control systems, such a topic has been extensively addressed by researchers over the last years; see, e.g., [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF][START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF][START_REF] Coutinho | Input and output quantized feedback linear systems[END_REF][START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF][START_REF] Fu | The sector bound approach to quantized feedback control[END_REF][START_REF] Liberzon | Switching in Systems and Control[END_REF][START_REF] Sur | Observers for linear systems with quantized outputs[END_REF][START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF] just to cite a few.

This paper pertains to the stabilization of continuoustime linear time-invariant plants with uniformly-quantized input via static state feedback control. Specifically, pursuing the general approach introduced in [START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF], we model the uniform quantizer as a discontinuous static isolated nonlinearity entering into the dynamics of the closed-loop system. At this stage, since the resulting closed-loop system is described by a discontinuous right-hand side differential equa-‹ This work has been supported by ANR project LimICoS contract number 12 BS03 00501.This file contains fixes to some typos in the published paper. The corrections are in blue font and there is a footnote explaining those. Last update: 2 September 2021. If you found this reading useful for your research please cite the published version doi: https://doi.org/10.1016/j.automatica.2015.05.015.

Email addresses: ferrante@laas.fr (F. Ferrante), fgouaisb@laas.fr (F. Gouaisbaut), tarbour@laas.fr (S. Tarbouriech).

tion, the existence of solutions to the closed-loop system is not guaranteed; see [START_REF] Filippov | Differential equations with discontinuous right-hand side[END_REF]. Therefore, to tackle the problem under consideration, we adopt, for the closed-loop system, the notion of solution due to Krasovskii; see [START_REF] Cortés | Discontinuous dynamical systems[END_REF]. Then, by the use of the sector conditions for the uniform quantizer presented in [START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF], coupled through S-procedure (see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]) to a quadratic Lyapunov-like function, we propose a condition to guarantee the finite-time convergence of the closed-loop trajectories towards a compact invariant set surrounding the origin, (asymptotic stability is usually impossible to prove due to the deadzone effect induced by uniform quantization; see, e.g., [START_REF] Ceragioli | Discontinuities and hysteresis in quantized average consensus[END_REF][START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF][START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF]). Afterwards, via the use of the projection lemma (see [START_REF] Pipeleers | Extended LMI characterizations for stability and performance of linear systems[END_REF]), such a condition is turned into a design procedure based on the solutions of a convex optimization problem that in one shot provides: the controller gain, the invariant set wherein the closed-loop trajectories ultimately converge, while minimizing the size of such a set. Moreover, some insights on the simulation of the considered closedloop system are discussed.

It is worthwhile to notice that, although the approach introduced in [START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF], (consisting in modeling quantizers as isolated nonlinearities), has enabled to build constructive design tools for quantized control systems, as for the case of other types of isolated nonlinearities such as, saturation, backlash etc., surprisingly no much work has been done in that direction, except for [START_REF] Coutinho | Input and output quantized feedback linear systems[END_REF][START_REF] Fu | The sector bound approach to quantized feedback control[END_REF][START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF], though the results in [START_REF] Coutinho | Input and output quantized feedback linear systems[END_REF][START_REF] Fu | The sector bound approach to quantized feedback control[END_REF] relate to discrete-time systems subjected to a logarithmic quantizer (deadzone-free). Specifically, the majority of the results available in the literature (see, e.g., [START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF][START_REF] Liberzon | Switching in Systems and Control[END_REF] and the references therein), has dealt with the stability analysis of quantized closed-loop systems involving a controller de-Preprint submitted to Automatica 2 September 2021 signed while ignoring the presence of the quantizer. In this sense, this paper wants to fill this gap by proposing a design technique tailored to quantized systems. The contribution of this paper with respect to [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF] is twofold. On one hand, the novel sector conditions for the uniform quantizer presented in [START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF] are for the first time exploited to design a static state feedback controller in the presence of uniform input quantization. On the other hand, the use of the projection lemma is considered to potentially improve the proposed design technique. Notice also that even if the spirit of the pursued approach is similar to [START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF][START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF], the results presented in this paper allows to deal with multi-inputs systems, derive constructive conditions for the design of a state feedback controller and explicitly characterize the set wherein the closed-loop trajectories ultimately converge, directly from the knowledge of a Lyapunov-like function. Therefore, our results can be considered as complementary with respect to those in [START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF][START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF].

The paper is organized as follows. Section 2 presents the system under consideration and the problem we solve. Section 3 is dedicated to the main results. Section 4 is devoted to numerical issues about the controller design. Moreover, some aspects on the simulation of the closed-loop system are discussed. Finally, Section 5 shows the effectiveness of the presented results in a numerical multi-inputs example.

Notation: The set Bpx, δ qq denotes the δ radius closed Euclidean ball centered at x. I n denotes the identity matrix and 0 denotes the null matrix (equivalently the null vector) of appropriate dimensions. For a matrix A P R nˆm , A1 , A piq denotes its i ´th row, and tracepAq denote its transpose and its trace, and HepAq " A `A1 . The matrix diagtA 1 , A 2 , . . . , A n u is the blockdiagonal matrix having A 1 , A 2 , . . . , A n as diagonal blocks and in symmetric matrices ' stands for symmetric blocks. For a vector x P R n , x piq denotes its i ´th component, x 1 denotes its transpose, |x| stands for the componentwise absolute value operator, signpxq is the componentwise sign function, with signp0q " 0, and txu the componentwise floor operator. The set ∆Z p is the set of the p-tuples of integers multiple of ∆. The symbol x¨, ¨y denotes the standard Euclidean inner product and ˆstands for the standard Cartesian product. For a set U, intpUq denotes the interior of U. The double arrows notation F : R n Ñ R m indicates that F is a set-valued mapping with Fpxq Ă R m . Throughout the paper, a.a. stands for almost all in a Lebesgue-measure sense. For a function f : A Ñ B, rge f :" ty P B : Dx P A such that y " f pxqu.

Preliminary definitions: In this paper we deal with differential inclusions in the form

9 x P Fpxq (1)
where Fpxq : R n Ñ R n . Consider the following definitions given mainly in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF][START_REF] Lin | A smooth converse lyapunov theorem for robust stability[END_REF]. Definition 1 Let I Ă R ě0 be an interval. Given x 0 P R n an absolutely continuous function φ ptq : I Ñ R n is said to be a solution to (1) from x 0 , if φ p0q " x 0 , and 9 φ ptq P Fpφ ptqq for a.a.t P I. Definition 2 A solution φ : I Ñ R n to (1) from x 0 is said to be maximal, if there does not exist any other solution φ : I Ñ R n , with I Ă I and such that φ ptq " φ ptq for every t P I. Moreover, φ is said to be complete, if I " R ě0 . Definition 3 A set A Ă R n is strongly forward invariant for (1), if every maximal solution φ to (1) is complete, and φ p0q P A implies rge φ Ă A .

Problem Statement

Consider the following continuous-time linear system with quantized input

" 9 x " Ax `B qpuq xp0q " x 0 (2)
where x P R n , u P R p , x 0 P R n are respectively the state, the input of the system and the initial state. A, B are real matrices of suitable dimensions, and qp¨q is the uniform quantizer, which is described by the static nonlinear functions defined as qpuq :"

∆ signpuq Z |u| ∆ ^(3)
where ∆ is a positive given real scalar characterizing the quantization error bound. Assuming that the state x is fully accessible, we want to stabilize system (2) via the following control law u " Kx. Therefore, by defining the function Ψpuqqpuq ´u, the closed-loop system reads as

" 9 x " pA `BKqx `BΨpKxq xp0q " x 0 . (4)
Notice that, due to the presence of the uniform quantizer, the right-hand side of ( 4) is a discontinuous function of the state, then there is no guarantees about the existence of solutions when intended in a classical sense; see [START_REF] Cortés | Discontinuous dynamical systems[END_REF]. To this end, as in [START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF], in this paper we focus on Krasovskii solutions to system (4), that is the solutions to the following differential inclusion 9

x P K ppA `BKqx `BΨpKxqq (5) where the Krasovskii operator K is defined by 1

K p f pxqq - Ş δ ą0 co f pBpx, δ qq.
The existence of such solutions is guaranteed under the very mild requirement of local boundedness of the right-hand side of (4), obviously verified in our case. Moreover, as pointed out in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF][START_REF] Hájek | Discontinuous differential equations, I[END_REF], Krasovskii solutions are arbitrarily close to the solutions to (4) obtained by perturbing the state x with arbitrarily small perturbations; see [START_REF] Hájek | Discontinuous differential equations, I[END_REF]. This fact provides a further reason to consider Krasovskii solutions, beyond the merely issue concerning the existence of solutions. Remark 1 As pointed out in [START_REF] Liberzon | Switching in Systems and Control[END_REF][START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF], the presence of the uniform quantizer defined in (3), due to its deadzone effect, can represent a real obstacle to the asymptotic stabilization of the closed-loop system. Namely, one should be aware that if the matrix A is not Hurwitz, then the asymptotic stability of the origin for the closed-loop system (4) cannot be achieved via any choice of the gain K.

Concerning closed-loop system (4), notice that according to [START_REF] Paden | A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators[END_REF] every solution to ( 4) is a Krasovskii solution to2 3

9 x P pA `BKqx `B p ą i"1 K `ΨpK piq xq ˘. (6)
In the sequel, for notational simplicity, we denote the righthand side of (6) as W pxq, so that (6) reads as 9

x P W pxq. Now we are in position to state the problem we solve.

Problem 1 Given the matrices A, B of adequate dimensions and a positive real quantization error bound ∆, determine a gain K and a compact set S u Ă R n containing the origin, such that ' S u is strongly forward invariant for (6); ' For every initial condition x 0 P R n zS u , the resulting maximal solutions to (6) are bounded, complete and converge in a finite time into S u .

Main results

To solve Problem 1, we want to exploit the sector conditions for the uniform quantizer presented in [START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF], which are recalled below. Lemma 1 For every u P R p and every v P Ś p i"1 K pΨpu piq qq, the following relations are verified

v 1 S 1 v ´tracepS 1 q∆ 2 ď 0 (7) v 1 S 2 pv `uq ď 0 ( 8
)
for any diagonal positive definite matrices S 1 , S 2 P R pˆp .

The next result gives a first condition to solve Problem 1. Proposition 1 If there exist a symmetric positive definite matrix P P R nˆn , two diagonal positive definite matrices S 1 , S 2 P R pˆp , a matrix K P R pˆn and a positive scalar τ such that

N " « HepPpA `BKqq `τP PB ´K1 S 2 ' ´S1 ´2S 2 ff ă 0 (9)
tracepS 1 q∆ 2 ´τ ď 0 [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] then K, and S u " tx P R n : x 1 Px ď 1u [START_REF] Cortés | Discontinuous dynamical systems[END_REF] are solution to Problem 1.

PROOF. Consider the following quadratic function V pxq "

x 1 Px, we want to prove that there exists a positive real scalar β such that x∇V pxq, wy ď ´β x 1 x @x P R n zintpS u q, w P W pxq.

(12) By S-procedure arguments, this can be obtained by showing that for every x P R n , there exists a positive real scalar τ such that x∇V pxq, wy ´τp1 ´x1 Pxq ď ´β x 1 x @w P W pxq. [START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF] On the other hand, due to (6), for every w P W pxq there exists a v P Ś p i"1 K pΨpK piq xqq such that w " pA `BKqx `Bv, thus still by S-procedure arguments, by continuity of the function Kx (see [START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF]), thanks to Lemma 1, ( 13) is ensured by proving that for each x P R n and v P R p , x∇V pxq, pA `BKqx `Bvy ´τp1 ´x1 Pxq ´v1 S 1 vt

racepS 1 q∆ 2 ´2v 1 S 2 pv `Kxq ď ´β x 1 x. ( 14
)
By straightforward calculations the left-hand side of the above relation can be rewritten as follows

« x v ff 1 N « x v ff `tracepS 1 q∆ 2 ´τ. (15)
Thus in view of ( 9) and [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF], it follows that there exists a small enough positive scalar β such that for every x P R n , w P W pxq, one has x∇V pxq, wy ´τp1 ´x1 Pxq ď ´β px 1 x `v1 vq, which in turn gives [START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF].

To conclude, by following standard arguments (see, e.g., [START_REF] Benedetti | Positive invariance and differential inclusions with periodic right-hand side[END_REF][START_REF] Khalil | Nonlinear Systems[END_REF]), from [START_REF] Coutinho | Input and output quantized feedback linear systems[END_REF] it follows that, S u is strongly forward invariant for [START_REF] Castelan | L 2 -stabilization of continuous-time linear systems with saturating actuators[END_REF], every solution to (6) is bounded (then complete; see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]Proposition 6.10.]) and converges to S u at most in4 λ max pPq β lnpV px 0 qq units of time. l As it appears in Proposition 1, condition ( 9) is nonlinear in the decision variables, thus from a numerical standpoint the solution to Problem 1 directly via Proposition 1 appears unlikely; see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. For this, the next result allows to make a first step toward a convex design procedure based on the result given in Proposition 1. Corollary 1 If there exist a symmetric positive definite matrix J P R nˆn , a matrix Y P R pˆn , a matrix F P R nˆn , two diagonal positive definite matrices S 1 , S 2 P R pˆp and a positive scalar τ such that (10) is verified and

» - - - ´HepFq J `AF `BY ´F1 B ' τJ `HepAF `BY q ´Y 1 S 2 `B ' ' ´S1 ´2S 2 fi ffi ffi fl ă 0 (16)
then K " Y F ´1 and the set S u " tx P R n : x 1 Px ď 1u, with P " F 1´1 JF ´1, are solution to Problem 1.

PROOF. The proof is inspired by [START_REF] Pipeleers | Extended LMI characterizations for stability and performance of linear systems[END_REF]. From Proposition 1, notice that N " W 1 QW , where

W " » - - - A `BK B I 0 0 I fi ffi ffi fl , Q " » - - - 0 P 0 ' τP ´K1 S 2 ' ' ´S1 ´2S 2 fi ffi ffi fl .
Thus, (9) can be rewritten equivalently as W 1 QW ă 0. Moreover, being S 1 and S 2 positive definite, U 1 QU ă 0, with U 1 " " 0 0 I ı , is obviously satisfied. Thus, by the projection lemma; see [START_REF] Pipeleers | Extended LMI characterizations for stability and performance of linear systems[END_REF], the satisfaction of (9), whenever S 1 and S 2 are required to be positive definite, is equivalent to find a matrix X such that5

Q `pW K r q 1 X U K r `pU K r q 1 X 1 W K r ă 0 (17)
where, U K r and W K r are some matrices having as rows a basis of the row-null space, respectively of U and W . Now, by selecting U K r "

" I 2n 0 2nˆp ı and W K r " " ´I A `BK B ı ,
and by partitioning X "

" X 1 X 2 ı , where X 1 , X 2 P R nˆn , from (17) one gets » - - - ´HepX 1 q P ´X2 `X1 1 pA `BKq X 1 1 B ' HepX 1 2 pA `BKqq `τP X 1 2 B ´K1 S 2 ' ' ´S1 ´2S 2 fi ffi ffi fl ă 0.
(18) At this stage, by setting in the above expression X 1 " X 2 " X, then by pre-and-post multiplying the lefthand side of the resulting matrix by diagtX 1´1 , X 1´1 , Iu and diagtX ´1, X ´1, Iu and finally by setting X ´1 " F, J " F 1 PF and Y " KF yields the left-hand side of ( 16). Then, since the satisfaction of ( 16) implies the satisfaction of ( 9), thanks to Proposition 1 the assertion is proven. l 4 Numerical Issues

Controller design

The implicit objective in solving Problem 1 is to shrink the size of the set S u as much as possible, so as to keep the closed-loop trajectories sufficiently close to the origin. Thus, as shown in [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF], one can select among all the feasible solutions to condition [START_REF] Filippov | Differential equations with discontinuous right-hand side[END_REF], that one providing the smallest size of the set S u . To achieve this objective, one need firstly to choose a suitable measure for the set S u . Now, being the ultimate set an ellipsoid, several size criteria can be adopted. A typical choice adopted in the literature consists in minimizing the volume of the set S u which is proportional to a detpP ´1q; see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. However, since the variable P does not directly appear in [START_REF] Filippov | Differential equations with discontinuous right-hand side[END_REF], the adoption of the latter criterion is not handy and it can even lead to nonconvex problems. To overcome this issue, one can implicitly minimize the volume of S u by minimizing tracepP ´1q. Then, to link the matrix P to the other decision variables the following further constraint is added

« N F ' J ff ě 0 (19)
which, for every symmetric positive definite matrix N, is equivalent to P ´1 ď N. Essentially, the latter constraint guarantees that tracepP ´1q ď tracepNq, so as the minimization tracepP ´1q can be performed via the minimization of tracepNq, and N is a new decision variable. At this stage, it is important to note that ( 16) is still nonlinear in the decision variables. On the other hand, relation ( 16) becomes linear with respect to the decision variables once τ and S 2 are fixed. The selection of τ and S 2 is not a difficult task and it can be performed through a grid search. Therefore, Problem 1 can be solved through the solution of the following convex optimization problem, along with a grid search stage aimed at selecting the values of τ and S 2 providing the best value for the objective function.

minimize

F,Y,S 1 ,N,J tracepNq subject to N ą 0, S 1 ą 0, J ą 0, (16), (10), (19). (20)
Nevertheless, due to the dead-zone effect induced by the quantizer, by solving the above optimization problem, one may run into solutions characterized by a high norm control gain, situation that need to be ruled out to envision the physical construction of the proposed controller. To this end, we implicitly limit the control gain by constraining the eigenvalues of the matrix A `BK to lay in suitable subsets of the open-left half complex plane. This kind of additional constraints can be easily expressed in a linear matrix inequality form; see [START_REF] Chilali | H 8 design with pole placement constraints: an LMI approach[END_REF]. A typical choice is to consider as region the closed circle centered in p´ω, 0q with radius r ą 0, where ω is a positive real scalar, i.e., tz P C : |z `ω| ď ru; such a condition is guaranteed by adding the following constraint; see [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF],

« ´rQ ωF `AF `BY ' ´rpF 1 `F ´Qq ff ă 0 ( 21
)
where Q is a symmetric matrix with adequate dimensions. Remark 2 The introduction of the slack variable F allows to decouple the matrix P defining the set S u from the gain K. This in turn allows to introduce an extra variable, i.e., Q, in [START_REF] Khalil | Nonlinear Systems[END_REF]. On the one hand, this in general leads to a less conservative design technique when further constraints like [START_REF] Khalil | Nonlinear Systems[END_REF] are considered; see [START_REF] Castelan | L 2 -stabilization of continuous-time linear systems with saturating actuators[END_REF]. On the other hand, this approach potentially enables to obtain better result in term of size of the set S u , being the variable J, (and then indirectly P), not subject to any additional constraint.

Notice that, as long as the gain K is known, condition ( 9) is linear in the decision variables, modulo the variable τ. Therefore, as a second step, by combining Proposition 1 along with a griding procedure for the scalar τ, by solving a finite number of linear matrix inequalities, one may attempt to tighten the set S u . This is a typical approach pursued in the literature; see, e.g., [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF]. In particular, throughout this further stage one may also account for different measures for the set S u than the trace criterion considered during the design stage.

Numerical integration

To overcome the issues on the existence of solutions to the closed-loop system, we addressed the study of such a system by mean of the notion of Krasovskii solution, and resting on this notion we shown some stability and stabilization results for the closed-loop system. Thus, at this stage, the question that naturally arises is how to simulate the closed-loop system while taking into account the notion of solution adopted throughout the paper. For this purpose, we need to introduce the notion of approximated polygonal arc and Euler solution, which are both given in [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]. Definition 4 (Polygonal approximation) Given x 0 P R n and T ą 0, consider the following construction ' Fix an arbitrary partition of the interval r0, T s, 0 ă t 1 ă t 2 ă ¨¨¨ă t N , with t N " T and max kPt0,...,N´1u

tt k`1 ´tk u ď ε.

' Compute x k`1 " x k `pt k`1 ´tk qpAx k `B qpKx k qq, for k " 0, . . . , N and xp0q " x 0 . ' Build the affine function such that x ε pt k q " x k for k " 0, . . . , N. The function x ε ptq is said to be an ε-polygonal approximation for closed-loop system (4).

Definition 5 A function ϕ E ptq is said to be an Euler solu- tion to (4) if it is the uniform limit for ε Ñ 0 of a polygonal approximation x ε ptq obtained by some partition of the interval r0, T s. The interest in considering Euler solutions stems from the fact that, as proven in [START_REF] Bressan | Singularities of stabilizing feedbacks[END_REF], Euler solutions are Krasovskii solutions. In particular, notice that, among all the possible polygonal approximations one can consider, the simplest and straightforwardly attainable through a numerical procedure arises from selecting a uniform partitioning of the time interval. Namely, let N be an arbitrarily positive integer, then fix δ " T N , and for k " 1, 2, . . . , N ´1, t k`1 " t k `T N . Thus, the sequence of polygonal approximations tx T N u 8 N"1 , if converges uniformly, it converges uniformly to a Krasovskii solution to [START_REF] Bressan | Singularities of stabilizing feedbacks[END_REF]. Therefore, for N sufficiently large, the function x T N can represent a good approximation of a Krasovskii solution to the considered system. However, often given x 0 P R n , there may exist multiple Krasovskii solutions from x 0 , and some of them may not be Euler solutions. Moreover, establishing whether the considered sequence uniformly converges whenever N approaches infinity could be nontrivial. Therefore, this aspect is still worth of further investigations.

Numerical example

Consider the following example derived from [START_REF] Amato | Stabilization of bilinear systems via linear state feedback control[END_REF] for which A " " ´0.5

‰

. As a second step, we perform an analysis stage via Proposition 1, while considering as measure for the set S u its volume (the volume of the set S u can be minimized by considering as convex objective ´logdetpPq; see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]). An iterative research for the scalar τ allows to select the more convenient value for such a parameter, that is τ " 7.



. In Fig. 2, the evolution of the closed-loop system from different initial conditions is reported. Simulations show that trajectories converge towards two equilibria, (obviously also the origin is an equilibrium point for the closed-loop system, though unstable).

It is interesting to notice that these two equilibrium points belong respectively to the two surfaces Kx " r∆ ´∆s 1 and Kx " r´∆ ∆s 1 wherein the function qpKxq is discontinuous.

It can be shown that the two mentioned equilibria are actually Krasovskii equilibria, and they can be computed as illustrated in [START_REF] Ceragioli | Discontinuities and hysteresis in quantized average consensus[END_REF]. Fig 1 reports a closed-loop trajectory converging towards an equilibrium. Specifically, in approaching the equilibrium point, the state slides onto different quantization boundaries making the control inputs switch repeatedly among the two adjacent quantization levels (chattering). Simulations also show the accuracy provided by Euler integration in capturing the peculiar behaviors due to the discontinuity introduced by the quantizer, as the two Krasovskii equilibria. The evolution of the closed-loop system from x 0 " p0.5, 0.5, 0.5q: Above the control inputs (qpu 1 q (solid-black), qpu 2 q (solid-gray), and the two quantization-free inputs (K p1q xptq (dashed-black), K p2q xptq (dashed-gray). Below the closed-loop states: x 1 (solid), x 2 (dashed), x 3 (dotted).

Conclusion

In this paper we tackled the stabilization problem for linear systems subject to input quantization, with respect to Krasovskii solutions. Specifically, via the use of the novel sector conditions for the uniform quantizer presented in [START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF], and the adoption of the projection lemma, a design procedure based on the solution of a convex optimization problem was provided. In addition some hints on the numerical simulations of the closed-loop system are discussed. Finally, the effectiveness of the proposed methodology is displayed in a multi-inputs example.

Future research directions include the extension of this methodology to the case of the logarithmic quantizer, and to the case of static state feedback control via quantized state.

  Fig.1. The evolution of the closed-loop system from x 0 " p0.5, 0.5, 0.5q: Above the control inputs (qpu 1 q (solid-black), qpu 2 q (solid-gray), and the two quantization-free inputs (K p1q xptq (dashed-black), K p2q xptq (dashed-gray). Below the closed-loop states: x 1 (solid), x 2 (dashed), x 3 (dotted).

  up showing the two equilibrium points (x).

Fig. 2 .

 2 Fig.2. The evolution of the closed-loop system from different initial conditions. The trajectory is obtained by integrating the closed-loop model with an Euler first order method with step size 10 ´3.

  As a first step, by considering τ " 1.8, S 2 " diagt1.4 10 ´6, 4.3 ¨10 ´5u, selected via a grid search, and by considering the further constraint in[START_REF] Khalil | Nonlinear Systems[END_REF] with ω " 10, r " 8.5, the solution to the optimization problem (20) yields K "

	1.5 4 4.3 6 5 3.2 6.8 7.2	ı	, B "	"	´0.7 ´1.3 0 ´4.3 0.8 ´1.5		and assume ∆ "
	0.5. " ´0.71 1.9 ´27
	4.3 4.1 4.3

Missing co in the published version.

In the published version there is a missing B in[START_REF] Castelan | L 2 -stabilization of continuous-time linear systems with saturating actuators[END_REF].

Systems (4) and[START_REF] Castelan | L 2 -stabilization of continuous-time linear systems with saturating actuators[END_REF] are swapped in the published version. The error does not propagate and the (correct) argument used in the paper is that solutions to[START_REF] Castelan | L 2 -stabilization of continuous-time linear systems with saturating actuators[END_REF] are Krasovskii solutions to (4) and not the other way around.

Typo in the published version, P should be P.

Typo in[START_REF] Fu | The sector bound approach to quantized feedback control[END_REF], W 1 K r should be pW K r q 1 , same for U 1 K r .


Yasmine Hamdouche
email: yasminehamdouche@gmail.com

Jean Christophe Meile

Marc Lebrun

Tagro Guehi

Renaud Boulanger

Corinne Teyssier

Didier Montet

Impact of turning, pod storage and fermentation time on microbial ecology and volatile composition of cocoa beans

Keywords: cocoa beans fermentation, microbial communities, volatile compounds, SPME-GC-MS, PCR-DGGE

Cocoa quality depends on several parameters, such as cocoa variety, environmental growth conditions, cultivation technique, and post-harvest treatments applied to coca beans. In this work, we studied the impact of cocoa post-harvest processing on both microbial communities structure and volatile composition. Cocoa beans samples were fermented in wooden boxes in Ivory Coast at different time intervals with turning and without turning, and derived from pods stored for two different duration times. Cocoa beans were analyzed using a molecular fingerprinting method (PCR-DGGE) in order to detect variations in microbial communities' structure; this global analysis was coupled to SPME-GC-MS for assessing cocoa volatile profiles. The results showed that the main parameter that influenced microbial communities structure was fermentation time, followed by turning, whereas, pods storage duration had a minor impact. Similar results were obtained for aromatic profile, except for pods storage duration that significantly affected volatile compound production. Global statistical analysis using Canonical Correspondence Analysis (CCA), showed the relationship between microbial communities and volatile composition. Furthermore, this study allowed the identification of discriminating microbial and chemical markers of cocoa post-harvest processing.

Introduction

Cocoa is ranked among the most traded agricultural products in the world. Cocoa beans production is estimated to 4,552,000 tons in 2016/2017; 73% of which is produced by Africa, mainly by Ivory Coast, which is the cocoa first producer, followed by Ghana, Cameroon and Nigeria (ICCO, 2017).

Cocoa has an intrinsic quality potential that depends on several parameters, such as cocoa variety, age of the cocoa tree, soil chemical composition, post-harvest treatments, and industrial process [START_REF] Kongor | Factors influencing quality variation in cocoa (Theobroma cacao) bean flavour profile-A review[END_REF]. In particular, postharvest processing applied on beans can enhance or degrade the sanitary and/or organoleptic qualities of cocoa [START_REF] Saltini | Optimizing chocolate production through traceability: A review of the influence of farming practices on cocoa bean quality[END_REF]. The transformation of cocoa goes through three main stages (pods opening, fermentation, and drying). The crucial step is fermentation, which is a spontaneous process involving microbial flora succession responsible for aroma precursor's production inside the beans through biochemical reactions. Fermentation lasts usually between 5 to 7 days. Yeasts are the first type of microorganism involved followed by Lactic Acid Bacteria (LAB), and then Acetic Acid Bacteria (AAB) [START_REF] Schwan | The microbiology of cocoa fermentation and its role in chocolate quality[END_REF]. Several recent studies tried to understand the role of these microorganism groups in cocoa fermentation and to control this phenomenon by inoculating selected strains previously isolated from cocoa (Lefeber et al., 2012[START_REF] Crafack | Influencing cocoa flavour using Pichia kluyveri and Kluyveromyces marxianus in a defined mixed starter culture for cocoa fermentation[END_REF]. The major results of these works showed that yeast strains have a greater impact on the improvement of cocoa aromatic quality when compared to bacterial strains [START_REF] Crafack | Influencing cocoa flavour using Pichia kluyveri and Kluyveromyces marxianus in a defined mixed starter culture for cocoa fermentation[END_REF]2014;[START_REF] Ho | Yeasts are essential for cocoa bean fermentation[END_REF][START_REF] Koné | Contribution of predominant yeasts to the occurrence of aroma compounds during cocoa bean fermentation[END_REF][START_REF] Hamdouche | Impact of Acetic Bacteria on Cocoa Fermentation[END_REF].

Several studies on the impact of postharvest treatments on cocoa beans composition showed that fermentation types [START_REF] Papalexandratou | Spontaneous organic cocoa bean box fermentations in Brazil are characterized by a restricted species diversity of lactic acid bacteria and acetic acid bacteria[END_REF]Visintin et al., 2016) and times [START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF] significantly affected the microbiota associated with cocoa beans, while pods storage (pulp preconditioning) and turning affected chemical composition and physical appearance of cocoa (Nazaruddin et al., 2006;[START_REF] Guehi | Effect of turning beans and fermentation method on the acidity and physical quality of raw cocoa beans[END_REF].

Aroma profile significantly contributes to cocoa quality and thus is crucial for the acceptability of chocolate. This profile varied mainly with fermentation and drying time [START_REF] Rodriguez-Campos | Dynamics of volatile and non-volatile compounds in cocoa (Theobroma cacao L.) during fermentation and drying processes using principal components analysis[END_REF][START_REF] Rodriguez-Campos | RI: Retention Index calculated RT: Retention time[END_REF]. Aroma precursors previously formed during fermentation and drying steps were highly developed during roasting. Roasting reduces acidity by volatile acids evaporation and removes bitter, acidic and astringent flavors [START_REF] Beckett | The Science of Chocolate[END_REF][START_REF] Ramli | Influence of roasting conditions on volatile flavor of roasted Malaysian cocoa beans[END_REF]. Maillard reactions generate a valuable source of aromatic compounds as pyrazines, pyrroles, furans and thiazoles [START_REF] Belitz | Coffee, tea, cocoa[END_REF]. The 2methylpyrazine and 2, 3, 5, 6 tetramethylpyrazine are the main volatile compounds present in roasted cocoa. Both are responsible for the sweet and strong cocoa flavor respectively (Lefeber et al., 2012).

The most used techniques for volatile compounds extraction are distillation, direct solvent extraction, solid phase extraction (SPE), headspace methods and solid phase micro-extraction (SPME). After concentration, gas chromatography is the most suitable analytical method. This technique allows the separation of a very large number of compounds on a fused silica capillary column. Different detection methods coupled with gas chromatography allow volatile compounds detection and identification: flame ionization, flame photometry, atomic emission, and mass spectrometry. SPME-GC-MS for the headspace analysis of volatile compounds allowed sensitive and representative analysis of cocoa products with high reproducibility [START_REF] Ducki | Evaluation of solid-phase micro-extraction coupled to gas chromatography-mass spectrometry for the headspace analysis of volatile compounds in cocoa products[END_REF].

A recent work revealed that cocoa aroma compounds composition varied according to cocoa variety (Menezes et al., 2016), but no study showed the impact of pod storage and turning on microbial ecology and the associated impact on aroma profiles. The main aims of this original study were: (i) to measure the impact of the variation of three parameters (pod storage, fermentation time and turning) during cocoa post-harvest processing on microbial and volatile profiles; (ii) to identify discriminating microbial and/or aroma markers of each process; (iii) to establish relationship between microbial communities associated with cocoa beans and volatile compounds produced.

Material and Methods

1. Cocoa beans sampling

Four hundred cocoa pods were harvested in November 2014 in Akoupé (Ivory Coast). The number of harvested pods was divided in two batches (200 pods each), the first batch was stored during 2 days and the other one during 8 days. After storage, all pods were opened using a machete. For each pods opening delay, approximately 30 kg of pulp-bean mass were extracted manually, recovered and distributed in two wooden boxes for fermentation during 7 days. Two types of processing were applied on these cocoa beans: fermentation with and without turning. The turning was done at 2 and 4 fermentation days, this choice was made following the artisanal practices applied. One kilogram of cocoa beans was collected at 2, 4, 5, 6 and 7 days of fermentation and sun dried during 7 days on wooden drying racks. During drying step, no turning was done. The sampling was performed in sterile conditions by using sterile gloves and plastic bags. Samples were sent by plane to Cirad laboratory, Montpellier, France and stored at room temperature. The microbial and chemical analyses were carried out immediately.

Microbial ecology analysis

1. Microbial DNA extraction

Microbial DNA was extracted from 10 g of fermented and dried cocoa beans. Two technical repetitions were assessed for each sample. The extraction was performed according to the method described previously by [START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF]. The quantity and purity of extracted DNA were measured by a UV spectrophotometer (Biospec-Nano), and by electrophoresis through a 0.8% agarose gel in 1X TAE buffer (containing 40 mM Tris-HCl, pH 7.4, 20 mM sodium acetate, 1.0 mM Na 2 -EDTA; Eppendorf) with a 1 Kb molecular weight ladder (Promega). The agarose gels were run at 100 V for 30 min and stained for 15 min in Gel Red solution (50 mg.mL ¯1; Biotium), then observed and photographed on a UV transilluminator, using a black and white CCD camera (Scion Company) and Gel Smart 7.3 system software (Clara Vision).

PCR amplification of extracted DNA

For bacterial communities analyses: A 160 bp fragment of the V3 variable region of 16S rDNA was amplified using gc-338f (5'-GCG CCG CCG CGC GCG GCG GGC GGG GCG GGG GCA CGG GGG GCA TCC TAC GGG AGG CAG CAG-3', Sigma) and 518r (5'-ATT ACC GCG GCT GCT GG-3', Sigma) DNA primers [START_REF] Leesing | Identification and validation of specific markers for traceability of aquaculture fish for import/export[END_REF][START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF]2016). A 40-bp GC-clamp (Sigma) was added to the forward primer in order to ensure that the DNA fragment will remain partially double stranded and that the region screened is in the lowest melting domain [START_REF] Sheffield | Attachment of a 40 bp G+C rich sequence (GC-clamp) to genomic DNA fragments by polymerase chain reaction results in improved detection of single-base changes[END_REF]. Each mixture (final volume 50 µL) contained about 100 ng of DNA template, DNA primers at 0.2 mM, deoxyribonucleotide triphosphate (dNTPs) at 200 µM, 1.5 mM MgCl 2 , 5 µL of 10X of Taq reaction buffer MgCl 2 free and 1.25 unit of a-Taq polymerase (Promega). In order to increase the specificity of amplification and to reduce the formation of spurious by-products, DNA was amplified by a "touchdown" PCR previously described by [START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF].

For fungal communities analyses: a 250 bp fragment of D1/D2 region of the 26S rDNA was amplified using universal DNA primers gc-NL1 (5'-CGC CCG CCG CGC GCG GCG GGC GGG GCG GGG GCC ATA TCA ATA AGC GGA GGA AAA G-3') and LS2 (5'-ATT CCC AAA CAA CTC GAC TC-3', Sigma) [START_REF] Cocolin | Direct profiling of the yeast dynamics in wine fermentations[END_REF]. A 30-bp GC-clamp was added to the forward primer. PCR mixture performed as previously described for bacterial DNA analyses. The amplification program was described previously by [START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF].

Electrophoresis in 2% (w/v) agarose gel with TAE 1X buffer stained with Gel Red (as described above) was used to verify and quantify PCR products using a standard (DNA mass ladder 100 bp, Promega). The migration lasted about 30 min at 100 V.

3. Denaturing gradient gel electrophoresis (DGGE) analysis

PCR products were separated by DGGE, using a Bio-Rad Dcode universal mutation detection system (Bio-Rad Laboratories, USA) according to the procedure described by Muyzer et al. (1993) and improved by [START_REF] Leesing | Identification and validation of specific markers for traceability of aquaculture fish for import/export[END_REF]. Forty microliters (µL) of each PCR product were loaded onto 8% (w/v) polyacrylamide gels (acrylamide/N,N΄-methylene bisacrylamide, 37.5/1, Promega) in 1X TAE.

Electrophoresis experiments were performed at 60°C using a denaturing gradient ranging from 30 to 60% (100% corresponded to 7 M urea and 40% (v/v) formamide, Promega). The migration was carried out at 20 V for 10 min and then at 80 V for 12 h. Then the gel was stained for 30 min with Gel Red solution and then photographed as described above in 2.2.1 section.

4. Purification and identification of DGGE bands

Detected DNA bands were cut from the DGGE gel with a sterile scalpel. The DNA of each band was then eluted in 100 µL TE buffer (10 mM Tris-HCl; pH 8.0, Promega) at 4°C overnight. The DNA purification was carried out according to the protocol described by [START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF]2016). Bacterial and fungal purified DNA bands were amplified using 338f and NL-1 (without GC clamp) and 518r and LS-2 primers respectively. PCR amplicons were sent for sequencing to GATC Biotech (Germany).

The raw sequences data were edited using Bioedit Software and compared to the GenBank database using the BLAST program [START_REF] Altschul | Gapped BLAST and PSI-BLAST: a new generation of protein databases search programs[END_REF] (http://www.ncbi.nlm.nih. gov/BLAST/). Sequences having a percentage of identity of 97% or greater were considered to belong to the same species (Palys et al., 1997).

5. Image and statistical analysis

In each lane, banding patterns were standardized with two bacterial reference DNA patterns (Lactobacillus plantarum and Escherichia coli DNAs) for bacteria and two yeast DNA reference patterns of yeasts (Lodderomyces elongisporus and Candida apicola DNAs). Each band relative positions to reference DNA were measured and recorded.

The images of DGGE gels were analyzed using ImageQuant TL software V. 2003 (Amersham Biosciences, USA). In DGGE analysis, the revealed banding pattern is considered as an image of all the major microbial populations. An individual band refers to a unique 'sequence type' or phytotype [START_REF] Kowalchuk | Analysis of ammonia-oxydizing bacteria of the beta subdivision of the class Proteobacteria in coastal sand dunes by denaturing gradient gel electrophoresis and sequencing of PCR amplified 16S ribosomal DNA fragments[END_REF]. DGGE fingerprinting was marked by the presence and absence of co-migrating bands and their intensity using ImagQuant TL software V. 2003.

Pair wise community similarities were quantified using Dice similarity coefficient [START_REF] Heyndrickx | Applicability of combined amplified ribosomal DNA restriction analysis (ARDRA) patterns in bacterial phylogeny and taxonomy[END_REF], then generated data were exploited by Principal Component Analysis (PCA) and Canonical analysis of correspondence (CAC) using XLSTAT (V. 2014).

The bands intensities (or volume as expressed in "grey level x pixel" unit) measured by ImageQuant were statistically compared using ANOVA.

CAC is a method that encompasses several statistical analyses; Ter Braak (1986) developed it for the first time. In our case, it was used to describe the linear relationships that exist between two groups of variables measured on the same individuals. The first group represents the variables to be explained (pod storage duration, fermentation time and turning) and the second group shows the explanatory variables that are the microbial communities (DNA bands identified by PCR-DGGE).

3. Volatile compound analysis

3. 1. Sample preparation and SPME-HS conditions

The shell and the bean were both analysed, because the objective of the work was to evaluate the volatile composition of exported merchant cocoa (fermented and dried beans). In food industry, before the shelling, fermented and dried beans are roasted in order to develop the chocolate aromas, these aromas depend essentially on the precursors born during fermentation.

Two hundred grams of unshelled dried cocoa beans from Ivory Coast were put in liquid nitrogen and ground using a waring blender (SEB, France), to obtain cocoa flour, which was stored at -80°C until analysis. Three grams of cocoa powder were introduced into vial (10 mL), 30 µL of internal standard of 1-butanol (202.5 mg/L) (Sigma) were added, and the vial was sealed with a septum cap. The extraction of three technical repetitions was performed using a 50/30 µm divinylbenzene/carboxene/polydimethylsiloxane fiber (DVB/CAR/PDMS, Supelco), and using the technique of solid phase micro-extraction in the headspace (SPME-HS) previously described by [START_REF] Rodriguez-Campos | Effect of fermentation time and drying temperature on volatile compounds in cocoa[END_REF] and modified in our lab by [START_REF] Koné | Contribution of predominant yeasts to the occurrence of aroma compounds during cocoa bean fermentation[END_REF]. These authors had conditioned the fiber at 250°C for 3 min then exposed it to the sample headspace at 50°C for 45 min.

3. Gas Chromatography Analytical Conditions

The analysis of the volatile compounds extracted from cocoa powder was performed using an Agilent 6890N gas chromatography-mass spectrometer (GC-MS) in automatic injection mode, on a polar capillary column DBWAX, 30 m length × 0.25 mm internal diameter × 0.25 µm film thickness (Agilent, Palo Alto, CA, USA). The GC-MS conditions have been established in Cirad laboratory, the injection temperature was 250°C. The GC oven temperature was initially set at 40°C for 5 min, increased to 140°C at a rate of 2°C/min and then increased at a rate of 10°C/min to 250°C. The carrier gas was high purity Helium at 1 mL.min -1 . Splitless injection mode was used at 250°C for 2 min. The selective mass detector was a quadrupole (Hewlett Packard, Model 5973), with an electronic impact ionization system at 70 eV and at the temperature source at 230°C.

3. 3. Volatile compounds identification

For the identification of compounds, pure standards were not used, this is why the identification was just tentative. The identification was done: (i) by comparing the mass spectra with commercial database Wiley275.L, HP product no. G1035A); (ii) and by comparing of the linear retention index calculated (RI) with those found in the literature data.

Retention index (RI) was calculated in the same chromatographic conditions using series of nalkane C8-C20 (Supelco, Bellefonte, USA). Peak areas were used for relative quantification of compounds using the MSD Chemstation software (version E.02.02.1431, Agilent Technologies).

3 .4. Statistical analysis

The chromatographic data obtained by SPME-GC-MS were statistically analyzed using ANOVA and Principal Component Analysis (PCA). As described previously, Canonical analysis of correspondence (CAC) was applied to GC-MS and DGGE data, in order to describe relationship between microbial communities (explanatory variables) and volatile compounds produced (variables to be explained).

Non-volatile organic acids analysis

Organic acids extraction

The extraction was carried out according to the protocol of [START_REF] Holm | The effects of the organic acids in cocoa on the flavour of chocolate[END_REF]. Five grams of cocoa powder were mixed with 50 mL of warm distilled water (75°C) in a 100 mL flask. The resulting suspension was well mixed and then cooled to room temperature. The volume was supplemented with distilled water up to 100 mL, then filtered through a pleated filter paper (Whatman) and a 0.45 µm filter (Sigma-Aldrich), then transferred to an HPLC vial (1 mL) (Thomson ).

Organic acids quantification by HPLC

Concentrations of acetic acid, citric acid, lactic acid, malic acid, oxalic acid and tartaric acid were determined by a High Performance Liquid Chromatography (HPLC) with conductivity using AS11-HC column (Dionex). The mobile phase, at a flow rate of 0.6 mL.min -1 consisted of ultrapure water (eluent A), NaOH 0.1 N (eluent B) and NaOH 0.002 N (eluent C). A volume of 0.02 mL for each sample was injected and run together with external standards. All samples were analyzed in triplicate and the data were controlled and processed with ClarityChrom software and analyzed using ANOVA.

Results and discussion

1. Microbial community structures

1. 1. Variation in bacterial community structures

Microbial DNA profiles were generated by DGGE from all cocoa fermentations carried out, with and without turning, with a delay of pod breaking of 2 and 8 days, respectively. Bacterial two fermentation days, a few DNA bands (Y, B V, W) were observed and identified by sequencing as belonging to Enterobacteriaceae, Lactobacillus fermentum, and Acetobacter pasteurianus taxa respectively (Fig. 1.a,b ; Table 1). At days 4 and 5, additional DNA bands (A, E, F, J, L, M, N) were detected and identified as belonging to Lactobacillus curvatus, Bacillus megaterium, Bacillus sp, Acetobacter sp and Providencia sp species, respectively.

All eight bacterial taxa remained detected until the last day of fermentation. This is concordant with previously reported data [START_REF] Lefeber | Dynamics and species diversity of communities of lactic acid bacteria and acetic acid bacteria during spontaneous cocoa bean fermentation in vessels[END_REF][START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF], which showed that bacterial communities of cocoa undergo drastic changes after 4 days of fermentation and remain pretty similar throughout the process length.

Moreover, at the shortest fermentation time, only L. fermentum, A. pasteurianus and species belonging to Enterobacteriaceae taxa were detected and confirmed the presence of LAB and AAB in the initial phase of fermentation [START_REF] Schwan | The microbiology of cocoa fermentation and its role in chocolate quality[END_REF][START_REF] Papalexandratou | Hanseniaspora opuntiae, Saccharomyces cerevisiae, Lactobacillus fermentum, and Acetobacter pasteurianus predominate during well-performed Malaysian cocoa bean box fermentations, underlining the importance of these microbial species for a successful cocoa bean fermentation process[END_REF][START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF].

Comparison of bacterial DNA profiles of fermented cocoa beans with and without turning showed that the majority of revealed bands were common in the two types of fermentation processing whatever the pod storage time. However, some bands (M, E, F) identified as Bacillus genus were mainly revealed in DGGE profiles obtained from cocoa beans fermented with turning (Fig. 1.a,b ; Table 1). Furthermore, some bands (G, H) were detected only in fermentation with turning and were identified as belonging to Acetobacter sp. These observations were explained by the better development of aerobic microorganisms due to aeration obtained by turning. The abundance of Bacillus genera populations seems to be the consequence of temperature elevation after 2 days and at the turning time, which promoted the development of thermophilic and aerobic species [START_REF] Lima | Microbiota of cocoa powder with particular reference to aerobic thermoresistant spore-formers[END_REF]. Previous works, carried out on fermentation of different cocoa beans masses, showed that the fermentation of the large cocoa bean-pulp mass decreased the aeration, which slows down microbial activities [START_REF] Camu | Influence of turning and environmental contamination on the dynamics of populations of lactic acid and acetic acid bacteria involved in spontaneous cocoa bean heap fermentation in Ghana[END_REF][START_REF] Guehi | Effect of turning beans and fermentation method on the acidity and physical quality of raw cocoa beans[END_REF]. It is important to note that A. pasteurianus species (V, W) was not present during fermentation with turning, at 5, 6 and 7 days (Fig. 1.b-Table 1).

This result was unexpected because A. pasteurianus was the most prevalent AAB species involved in spontaneous cocoa from Ivory coast, Brazil and Malaysia [START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF][START_REF] Papalexandratou | Spontaneous organic cocoa bean box fermentations in Brazil are characterized by a restricted species diversity of lactic acid bacteria and acetic acid bacteria[END_REF]2013) and was tested as starter culture in cocoa fermentation (Moens et al., 2014;[START_REF] Lefeber | Kinetic analysis of strains of lactic acid bacteria and acetic acid bacteria in cocoa pulp simulation media toward development of a starter culture for cocoa bean fermentation[END_REF][START_REF] Rodriguez-Campos | RI: Retention Index calculated RT: Retention time[END_REF]. First of all, we thought that this unexpected result came from technical error, but the most surprising thing about this observation, was the correlation of this result with the one obtained from volatile compounds analysis. The disappearance of A. pasteurianus species was correlated with acetic acid content decreases during fermentations with turning. If the technical error is sparing, the result could be explained by the fact that the temperature increased during fermentation with turning (by increasing activity AAB). In our fermentation assays, the temperatures could have risen such as it became lethal for this thermosensible species. However, since, temperatures were not measured in our studies, we could not conclude about this hypothesis.

When comparing DGGE bacterial profiles obtained from stored pods during 2 and 8 days, DNA bands (R, K, X) specific to cocoa pods stored 8 days were detected and identified as belonging to Klebsiella pneumoniae, Bacillus sp, and Acinetobacter sp (Fig. 1.b compared to Fig. 1.a). These bacterial taxa could be present on pods surface and developed during storage time, which rendered them detectable by PCR-DGGE. To our knowledge, no work has investigated the pod storage effect on cocoa microbial ecology except the study of our colleagues [START_REF] Kedjebo | Effect of post-harvest treatments on the occurrence of ochratoxin A in raw cocoa beans[END_REF] who studied the effect of post-harvest treatment on the occurrence of ochratoxin A (OTA) in raw cocoa beans from Ivory Coast. They showed that OTA content in raw cocoa beans increased with pods storage duration (0, 4 and 7 days) which reflected the increase of ochratoxinogenic fungi activity (Aspergillus strains) during cocoa pods storage time. Therefore, their conclusion confirmed our results that showed a highest microbial contamination with long pod storage after fermentation.

Principal Component Analysis (PCA) was applied on bacterial DGGE profiles to represent the dataset variance. We observed that fermented cocoa samples tended to group according to fermentation time along the first component axis (F1 representing 53.44% of the total variance). For 2 day storage samples, two main groups containing cocoa beans samples turned fermented during 2 and 4 days on the left side of the axis and those fermented during 5, 6 and 7 days on the right side of the axis could be observed (Fig. 2). The picture is similar for 8 days storage of non-turned samples. Eight-day storage of turned samples appeared as two main groups: 2 day fermented on the left side of the axis and the rest of the samples on the right side. 2 days storage unturned samples tend to group according to the fermentation duration.

The secondary axis F2 (12.58% variance) separated groups of cocoa beans samples fermented by turning (bottom part) from those fermented without turning (upper part, Fig. 2). This result showed that bacterial communities varied mainly according to fermentation time followed by processing (turning). PCA did not show an obvious effect of storage time on bacterial ecology of cocoa comparing with the impact of the two other parameters (fermentation time and turning). The strong dynamics and activity of the microbial populations observed during the fermentation does not seem to be linked to any other environmental parameter.

1. 2. Variation in fungal community structures

During cocoa fermentations, dominant species (1, 10, and 11) were detected on DGGE profiles and identified as belonging to Hanseniaspora opuntiae, Pichia kudriavzevii and Pichia manshurica species (Fig. 3 -Table 2). These species had previously been detected in fermented cocoa beans from the same location in Ivory Coast [START_REF] Hamdouche | Dynamics of microbial ecology during cocoa fermentation and drying: Towards the identification of molecular markers[END_REF][START_REF] Koné | Contribution of predominant yeasts to the occurrence of aroma compounds during cocoa bean fermentation[END_REF] and in other countries and continents [START_REF] Crafack | Influencing cocoa flavour using Pichia kluyveri and Kluyveromyces marxianus in a defined mixed starter culture for cocoa fermentation[END_REF]de Melo Periera et al., 2013;[START_REF] Papalexandratou | Hanseniaspora opuntiae, Saccharomyces cerevisiae, Lactobacillus fermentum, and Acetobacter pasteurianus predominate during well-performed Malaysian cocoa bean box fermentations, underlining the importance of these microbial species for a successful cocoa bean fermentation process[END_REF]. Minor variations were observed comparing DGGE profiles obtained from cocoa fermented with the two processes due to intrinsic variations in fungal diversity during fermentation (Fig. 3.a.b). Generally, we did not observe specific fungal communities associated to a process probably due to the fact that turning or aeration do not drastically favor fungal populations growth.

In DGGE profiles obtained from fermented cocoa beans previously stored longer (8 days), four specific bands were detected (4, 5, 7, 9) and identified as Hanseniaspora uvarum and Pichia sp (Fig. 3.b). The results of PCR amplifications coupled to DGGE profiles of both bacterial and yeast communities showed more DNA bands in cocoa pods stored longer (8 days), which probably means that microbial species abundance increased with storage time (as discussed above with bacterial data).

PCA on fungal communities associated with cocoa, represented by F1 and F2 axis (58. 26%) did not allow to discriminate cocoa beans samples, this result was expected, as mentioned before (Fig. 1 in supplementary material).

Canonical analysis of correspondence (CAC) was applied in order to describe the relation between variations in microbial community structures and changes in cocoa post-harvest parameters. CAC results showed that microbial communities associated with turning parameter consisted mainly of bacterial communities: Bacillus, LAB and Acetobacter species (Fig. 4). This result was expected for AAB because they are aerobic microorganisms, which are usually affected by the aeration. For Bacillus, it could be explained by the increasing of the temperature after turning due to the thermophilic aptitude of this genus. No microbial species was found linked to the parameter of pods storage time (Fig. 4) as pods storage duration was not related to the composition of microbiota.

2. Volatile compounds analysis

The results obtained on the volatile compounds analysis might not be accurate because of the limit of the identification and quantification method used.

Twenty-six volatile compounds were detected in fermented and dried cocoa beans by SPME-GC-MS (Table 3). These compounds were separated in 7 families: aldehydes, ketones, esters, alcohols, acids, pyrazines and phenols.

2. 1. Variation of aldehyde compounds

Four aldehydes compounds were identified on all fermented cocoa beans analyzed, which were isobutanal, 2 and 3-methylbutanal, benzaldehyde. Benzaldehyde was the dominant aldehyde, followed by 2 and 3-methylbutanal in all cocoa fermentations (Fig. 5.a.g.m and Fig. 6.a.g). These aldehydes were previously identified on fermented cocoa beans (Serra-Bonvehi, 2005;[START_REF] Rodriguez-Campos | Effect of fermentation time and drying temperature on volatile compounds in cocoa[END_REF][START_REF] Ho | Yeasts are essential for cocoa bean fermentation[END_REF]. In all fermentations, aldehydes content increased with fermentation time. These results were correlated with those obtained by [START_REF] Rodriguez-Campos | Effect of fermentation time and drying temperature on volatile compounds in cocoa[END_REF]. Statistical test (ANOVA) showed significant differences in aldehydes level (p<0.05) between the first stage of fermentation (2 and 4 days) and the last stage (5, 6 and 7 days) (Fig. 5.a.g.m and Fig. 6.a.g).

ANOVA test showed significant differences for aldehydes content between cocoa beans fermented with and without turning (Fig. 7.a.b), with the highest level obtained in the case of fermentation with turning at 7 days. These observations were probably related to the presence of LAB that were detected throughout the fermentation stages mostly in the case of turning (Fig. 1.a and b). Aldehydes compounds could be produced by enzymatic reactions from amino acids mainly by LAB [START_REF] Jinap | Effect of drying on acidity and volatile fatty acids content of cocoa beans[END_REF]. The 2 and 3-methylbutanal are usually produced from isoleucine and leucine, while benzaldehyde is produced from phenylalanine by LAB [START_REF] Jinap | Effect of drying on acidity and volatile fatty acids content of cocoa beans[END_REF]Smit et al., 2005;[START_REF] Bonnarme | Methylthioacetaldehyde, a possible intermediate metabolite for the production of volatile sulphur compounds from L-methionine by Lactococcus lactis[END_REF]. During beans mixing, temperature increased and reached its maximum [START_REF] Camu | Influence of turning and environmental contamination on the dynamics of populations of lactic acid and acetic acid bacteria involved in spontaneous cocoa bean heap fermentation in Ghana[END_REF], which increased the enzymatic activity. As regard to pod storage time effect, the aldehydes level had not significant difference (p>0.05) between the two pod storage delays (Fig. 7.a.b).

2. 2. Variation of ketone compounds

For this family, the detected compounds were 2-heptanone, 2-pentanone, 2-nonanone, acetophenone, and acetoin. This later had been found as the dominant ketone in all cocoa samples (Fig. 5.b.h.n and Fig. 6.b.h) as reported by [START_REF] Rodriguez-Campos | Dynamics of volatile and non-volatile compounds in cocoa (Theobroma cacao L.) during fermentation and drying processes using principal components analysis[END_REF]. In general, acetoin quantity decreased with fermentation time. This compound is mainly produced from pyruvate by alcoholic fermentation and from 2,3-butanediol by yeast [START_REF] Pretorius | Tailoring wine yeast for the new millennium: novel approaches to the ancient art of winemaking[END_REF]. Acetoin abundance in the first stage of fermentation could be correlated to yeasts intervention in this stage. The quantity of others ketones varied slightly during fermentation. These compounds were naturally present in cocoa beans [START_REF] Ho | Yeasts are essential for cocoa bean fermentation[END_REF] and their production was not affected by fermentation, except for 2-pentanone, which was probably used as substrate by microorganisms during fermentation. Ketones were significantly important (p<0.05) in cocoa beans fermented without turning (Fig. 7.a,b). Some ketones that were probably present naturally [START_REF] Afoakwa | Flavor formation and character in cocoa and chocolate: a critical review. Critical reviews in food science and nutrition, 48(9), 840-857. pre-conditioning and fermentation[END_REF], Rodriguez et al., 2012), were degraded and catabolized in the case of the aerated fermentation process, where more diversity and abundance of bacteria were observed by PCR-DGGE (Fig. 1.a). There was no difference in ketones level for the two pod storage durations, except in the case of fermentation with turning, whither the dominance of ketones was observed in pods stored at the shorter time (2 days) (Fig. 7.a.b).

2. Variation of alcohol compounds

The alcohols compounds found in cocoa beans samples were: ethanol, 2-pentanol, isopentanol, 2-heptanol, 2,3-butanediol, and 2-phenethylethanol. Among them, 2phenethylethanol and 2,3 -butanediol were dominant during the four types of fermentations (Fig. 5.c.i.o and Fig. 6.c.i); as already showed by [START_REF] Rodriguez-Campos | Dynamics of volatile and non-volatile compounds in cocoa (Theobroma cacao L.) during fermentation and drying processes using principal components analysis[END_REF]. Ethanol was surely produced by the majority of yeasts [START_REF] Ho | Yeasts are essential for cocoa bean fermentation[END_REF][START_REF] Koné | Contribution of predominant yeasts to the occurrence of aroma compounds during cocoa bean fermentation[END_REF]. It was probably the same in this study where it was generated by all detected yeasts by PCR-DGGE (Fig. 3). Some alcohols could be elaborated by both bacteria and yeasts such as 2,3butanediol. Ethanol content presented significant differences (p<0.05) between fermentation times (2 and 4 days) and (5, 6, and 7 days) (Fig. 5.c.i.o and Fig. 6.c.i). Its concentration was higher during the first four days of fermentation and then decreased rapidly, because this compound is produced at the first (anaerobic) fermentation stage, which lasted 48 h, subsequently it was oxidized in acetic acid [START_REF] Schwan | The microbiology of cocoa fermentation and its role in chocolate quality[END_REF] mainly by AAB during the second (aerobic) stage.

The abundance of these alcohols was significantly different (p<0.05) during the different fermentation process, with the highest level obtained in cocoa fermented beans without turning mainly for ethanol (Fig 7.a,b,Fig. 5.c.i.o and Fig. 6.c.i). In aerated conditions, the contribution of oxygen allows alcohols oxidation. For example, most alcoholic compounds are oxidized to ester compounds, as phenethylethanol that is a precursor of aldehyde and is oxidized to phenylethyl acetate with the presence of oxygen by LAB (Smit et al., 2005). Furthermore, alcohol compounds could be evaporated with the aeration brought by turning process. This result agreed with those obtained by [START_REF] Camu | Influence of turning and environmental contamination on the dynamics of populations of lactic acid and acetic acid bacteria involved in spontaneous cocoa bean heap fermentation in Ghana[END_REF] that showed the evaporation of ethanol during the aeration of cocoa beans mass, as well as favoring the AAB growth that oxidized ethanol into acetic acid.

Total alcohols level was significantly higher in cocoa obtained from pods stored longer (Fig

2. 4. Variation of ester compounds

The six detected esters were ethyl acetate, isobutyl acetate, isoamyl acetate, phenylethyl acetate, methyl isopentanoate, and methyl isovalerate (Fig . 5.d.j.p and Fig. 6.d.j). Ethyl acetate was the dominant ester in all fermentations. According to Rodriguez-Campos et al.

(2012), esters concentration decreased with fermentation time, but in our case, no significant difference was observed (p>0.05) depending on fermentation time except for ethyl acetate, which dominated in the first stage of fermentation (2 days) (Fig . 5.d.j.p and Fig. 6.d.j). This ester seems to be produced only at the beginning of cocoa fermentation as reported previously [START_REF] Rodriguez-Campos | Dynamics of volatile and non-volatile compounds in cocoa (Theobroma cacao L.) during fermentation and drying processes using principal components analysis[END_REF], and also in wine fermentation [START_REF] Rojas | Acetate ester formation in wine by mixed cultures in laboratory fermentations[END_REF] mainly in the first day of fermentation. This may be due to the abundance of glucose and ethanol during this stage, because they are the main substrates for ethyl acetate synthesis (Yong et al., 1981).

Moreover, Ethyl acetate is a product of esterification from acetic acid and ethanol (Pretorious, 2000); this concept explains the correlation between the amount of this ester, acetic acid and ethanol quantities mainly in the case of fermentation with aeration, in which a logical evolution of the quantities was observed for the most of volatile compounds detected in this study. A large quantity of these compounds was observed at the beginning of fermentation and decreased sharply after 2 days (Fig. 5.c.i.o.e.k.q and 6.c.i.e.k). The concentration of this ester was significantly higher in cocoa fermented without turning (Fig 7.a,b). Esters were synthetized by yeasts from alcohols during anaerobic phase [START_REF] Rojas | Measurement of alcohol acetyltransferase and ester hydrolase activities in yeast extracts[END_REF], which explained their dominance in fermentation without aeration. This observation could explain the high level of esters in this process. Concerning pods storage time, esters were higher as mainly methyl isovalerate in cocoa obtained from pods, which were stored less time (Fig. 7.a.b). That could be explained by the decrease of sugar and protein during cocoa pod storage [START_REF] Guehi | Effect of turning beans and fermentation method on the acidity and physical quality of raw cocoa beans[END_REF][START_REF] Afoakwa | Changes in acidification, sugars and mineral composition of cocoa pulp during fermentation of pulp preconditioned cocoa (Theobroma cacao) beans[END_REF] that could be substrates for esters synthesis.

Substrates reduction during long storage (8 days) allowed probably the decrease of the esters content.

2. 5. Variation of acid compounds

Three acids were detected in fermented cocoa beans, of which acetic acid was largely dominant comparing with isobutyric and isovaleric acids. Acetic acid quantity decreased significantly during fermentation realized with turning (Fig. 5.e.k.q and Fig. 6.e.k). In the case of fermentation without turning, the evolution of acetic acid was irregular because its quantity was not stable, decreased and increased during fermentation (Fig. 7.a,b). The illogical evolution of acetic acid amount in this case, could be explained by the fact that there was no beans turning, therefore no homogenization in the metabolites produced and beans composition. Whatever the pods storage delay, a normal evolution of acetic acid amount during fermentation with turning was observed. According to the literature, acetic acid increased and reached its maximum around 72h of cocoa fermentation then decreased after 88 h (Shwan and Wheal, 2004; Lefeber and al., 2012). This data confirmed our result, which showed that turning of cocoa beans allowed to realize fermentation with a normal evolution over time. The concentration of this acid was significantly higher in cocoa beans fermented without turning. To explain this, one possibility is that residual or minimal acetic fermentation (with AAB growth and/or activity) occurs in unmixed cocoa, and the rise in temperature plus the aeration favors the volatilization nutrients depletion [START_REF] Camu | Influence of turning and environmental contamination on the dynamics of populations of lactic acid and acetic acid bacteria involved in spontaneous cocoa bean heap fermentation in Ghana[END_REF] during fermentation with turning.

During fermentation, assimilation of citric acid by LAB generates a slight rise in pH to 4.5 and temperature to 37°C of cocoa pulp (Camu et al., 2007) which create a favorable environment for the growth of AAB involved in the second phase of fermentation (acetic fermentation). During this phase, AAB are responsible of the oxidation of ethanol to acetic acid [START_REF] Schwan | The microbiology of cocoa fermentation and its role in chocolate quality[END_REF]. Our results on acetic acid bacterial communities structure was related to acetic acid production, because low acid acetic level was observed in parallel with the disappearance of the main AAB, which was A. pasteurianus (band V and W) (Fig. 1) in the absence of aeration.

We also observed that volatile acids content decreased among time in cocoa beans and were very low after a pod storage of 8 days (Fig. 7.a.b). This observation is in agreement with the works of Tomlins et al. (1993) and [START_REF] Jinap | Effect of drying on acidity and volatile fatty acids content of cocoa beans[END_REF], who observed a lower acetic acid content in stored cocoa compared with un-stored cocoa pods. As mentioned previously from the literature, acetic acid is produced by oxidation of the ethanol during the second fermentation phase (aerobic). These results were consistent with those obtained on ethanol (section 3.2.3) which was low after pod storage of 8 days. Therefore, acetic acid content, which is volatile, could vary according to the ethanol content and time.

2. 6. Variation of others volatile compounds

Pyrazines: Tetramethylpyrazine was the unique compound belonging to pyrazine family that was detected (Fig. 5.f.l.r and Fig. 6.f.l). In most cases, tetramethylpyrazine compound has been reported to produce characteristics notes of roasted cocoa that is important for flavor quality [START_REF] Afoakwa | Flavor formation and character in cocoa and chocolate: a critical review. Critical reviews in food science and nutrition, 48(9), 840-857. pre-conditioning and fermentation[END_REF]Serra-Bonvehi, 2005). Furthermore, various studies reported on the microbial origin of pyrazines in fermented food as cocoa and soybeans [START_REF] Selamat | Formation of methyl pyrazine during cocoa bean fermentation[END_REF][START_REF] Besson | Pyrazine production by Bacillus subtilis in solid-state fermentation on soybeans[END_REF]. In our case, tetramethylpyrazine was probably produced by Bacillus as Bacillus megaterium species that was detected by PCR-DGGE (Fig. 1). They play an important role in producing alkylypyrasines in association with B. subtilis during cocoa fermentation. Our results showed that tetramethylpyrazine level had no significant difference during fermentation times, pods storage times and between fermentation processing. Pyrazine compounds were not affected by any of the three processes.

Methoxyphenol: This volatile phenolic compound was only detected in fermented cocoa beans obtained from pods stored for 8 days and it was significantly (p<0.05) higher in the case of fermentation with turning (Fig. 5.f.l.r, Fig. 6.f.l). Some phenolic compounds were previously detected at the end of cocoa fermentation (6 and 8 days) [START_REF] Rodriguez-Campos | Effect of fermentation time and drying temperature on volatile compounds in cocoa[END_REF].

The presence of this compound only in fermented cocoa beans obtained from long pods storage when more microbial diversity was detected, could be explained by the increase of nutrient degradation, including phenolic acids, thus 2-methoxyphenol elaboration. We also connected the production of this compound to the natural heating during sun drying. In general, phenolic acids are degraded thermically or decomposed by microorganisms into phenols, as 2-methoxyphenol that results from a heat degradation of ferulic acid during coffee and cocoa roasting or kiln drying (Berlitz, 2009).

Organic acids composition

Five non-volatile organic acids were detected and quantified: citric, lactic, malic, oxalic and tartric acids. The total of non-volatile acids was significantly (p <0.05) higher on cocoa beans fermented without turning (when compared to turned cocoa), especially at the end of fermentation (5, 6 and 7 days) (Fig. 8). This confirmed the results obtained on volatile acids, which are less abundant during fermentation with aeration. The amount of acids is significantly greater (p <0.05) on cocoa beans with short pod storage durations (2 days) (Fig. 8). This is consistent with the results obtained on volatile acids by Afoakwa et al. (2013a;2013b) who showed consistent decreases in non-volatile acidity during pod storage.

The concentration of volatile and non-volatile acids identified was affected during fermentation probably under microbial activities. Those naturally present in cocoa beans were degraded, mainly citric acid, while others were produced during fermentation, such as acetic and lactic acids, which could be considered as cocoa quality markers.

4. Correlations between microbial ecology and volatile compounds produced in cocoa fermentation

In order to measure the impacts of turning during fermentation (fermentation time and pods storage time), PCA was applied on all volatiles compounds detected in fermented cocoa beans (Fig. 9). Statistical analysis showed a global (on the volatile groups) and specific (on volatile compounds) effects. We showed that ketones, alcohols, esters and acids contents were higher in cocoa fermented without turning, while aldehydes were significantly abundant in the process with turning. Pod storage duration had an effect on all chemical families except for aldehydes and pyrazines. ACP separated 4 samples groups according to fermentation time, fermentation process and pod storage time. Principal axis F1 (38.86%) separated cocoa beans samples fermented (5, 6 and 7 days) without turning from the beans fermented with turning (Fig. 9). ACP analysis allowed identifying discriminant compounds that were affected by studied parameters. Discriminant and significant compounds in fermentation without turning were acetic acid, isobutyric acid, 2,3-butanediol, acetoin, phenethyl acetate, isoamyl acetate.

In the case of turning, all aldehydes compounds detected and some esters were identified as discriminant compounds. The same axis F1 regrouped samples according to fermentation time, the higher level of acetic acid, ethyl acetate, isobutyl acetate and ethanol at the first stage of fermentation (2 and 4 days) discriminated beans fermented at short term from the other samples. F2 axis (18.45%), separated the two pods storage times starting from 5 days of fermentation, by 2-methoxyphenol and acetophenone that were significant (abundant) in long storage (8 days).

Variations in volatile composition were compared with literature and with microbial ecology analyses carried out in our study. Some hypotheses were given about the relationship that could exist between microbial community structures and volatile composition of cocoa beans.

CAC was applied on all data obtained in order to determine these relationships (Fig. 10).

Microbial taxa detection by DGGE and their correlation with volatile compounds had a similar evolution whatever the fermentation time and the processing applied. It was not possible or rather difficult to relate each volatile group and/or compound to a given microbial taxa, which is explained by the fact that several yeasts or bacteria could produce the same compounds [START_REF] Ho | Yeasts are essential for cocoa bean fermentation[END_REF][START_REF] Koné | Contribution of predominant yeasts to the occurrence of aroma compounds during cocoa bean fermentation[END_REF]. Moreover, the CAC described the relation between A. pasteurianus (V, W) acetic acid, and acetoin and between H. opuntiae to ethyl acetate and ethanol (Fig. 10). This suggested that acetic acid level was related to A. pasteurianus abundance. This hypothesis could be verified by the identification of volatile profile for each microbial species detected during fermentation.

Conclusion

This study measured the effect of three cocoa process parameters on both microbial community structures and volatile composition of cocoa beans, and showed that fermentation time has a major impact on the aroma, followed by turning. In our study, pod storage duration (2 days vs 8 days) had a minor impact on cocoa microbial communities. Contrariwise, the effect was pronounced on volatile composition when it was associated to turning. Fermented cocoa beans with desired flavors were obtained with short pod storage times, which promoted a significant presence of esters. Consequently, as expected a short fermentation without aeration could have a negative impact on the cocoa volatile composition, because of the high levels of undesirable compounds such as acetic acid and acetoin.

The molecular approach coupled to volatile and statistical analysis provided insight into the relationship between the microbial communities' structure and aroma compound production in fermenting cocoa beans. To better describe the relationship, it would be interesting to carry out the study in a specific way by identifying the aromatic profile of each microbial strains involved in the processes. Consequently, this global analysis allowed the identification of molecular and volatile markers, which could be good indicators of cocoa post-harvest processing. Acetic acid, acetoin, benzaldehyde, 2 and 3-methylbutanal content are related to cocoa fermentation time and aeration type, whereas, the evaluation of esters concentration could be used to distinguish pods storage duration of fermented beans. At this stage, the impact of post-harvest processing was measured on cocoa volatile composition but not specifically on aroma quality. That aspect yet remains to be explored in the future. multivariate direct gradient analysis. Ecology, 67(5), 1167-1179. Tomlins, K. I., Baker, D. M., Daplyn, P., & Adomako, D. (1993). Effect of fermentation and drying practices on the chemical and physical profiles of Ghana cocoa. Table 1 a,g,m), ketones (b,h,n), alcohols (c,i,o), esters (d,j,p), acids (e,k,q) and other volatile compounds (f,l,r) produced in cocoa beans previously stored during 2 and 8 days, fermented 2 days (2F), 4 days (4F) , 5 days (5F) with turning (+T) and without turning. indicates significant differences between fermentation with and without turning. indicates significant differences between the two pods storage duration. indicates significant differences between fermentation times.

. C (b) G D F H J K L M N O P Q W V T A B E Y J A B E S R Y C F K L M N P Q V W
7 6 5 4 2 A B C D E G F I H Y P Q W V N M L J C B A E F J L M P Q Y V W (a) S 2S2F 2S2F 2S2F T 2S2F T 2S4F 2S4F 2S4F T 2S4F T 2S5F 2S5F 2S5F T 2S5F T 2S6F 2S6F 2S6F T 2S6F T 2S7F 2S7F 2S7F T 2S7F T 8S2F 8S2F 8S2F T 8S2F T 8S4F 8S4F 8S4F T 8S4F T 8S5F 8S5F 8S5F T 8S5F T 8S6F 8S6F 8S6F T 8S6F T 8S7F 8S7F 8S7F T8S7F T -4 -3 -2 -1 0 1 2 3 4 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 F2 (
(a) v O (b) v O (c) v O 2S2F 2S2F 2S2F T 2S2F T 2S4F 2S4F 2S4F T 2S4F T 2S5F 2S5F 2S5F T 2S5F T 2S6F 2S6F 2S6F T 2S6F T 2S7F 2S7F 2S7F T 2S7F T 8S2F 8S2F 8S2F T 8S2F T 8S4F 8S4F 8S4F T 8S4F T 8S5F 8S5F 8S5F T 8S5F T 8S6F 8S6F 8S6F T 8S6F T 8S7F 8S7F 8S7F T 8S7F T -4 -3 -2 -1 0 1 2 3 4 5 -6 -5 -4 -3 -2 -1 0
T 2S2F T 2S4F 2S4F 2S4F T 2S4F T 2S5F 2S5F 2S5F T 2S5F T 2S6F 2S6F 2S6F T 2S6F T 2S7F 2S7F 2S7F T 2S7F T 8S2F 8S2F 8S2F T 8S2F T 8S4F 8S4F 8S4F T 8S4F T 4S5F 4S5F 8S5F T 8S5F T 8S6F 8S6F 8S6F T 8S6F T 8S7F 8S7F 8S7F T 8S7F T Isobutanal 2-Methylbutanal 3-Methylbutanal

  7.a.b) as 2,3-butanediol(Fig 5.c.i.o and Fig 6.c.i). This may be explained by the studies ofAfoakwa et al. (2013a; 2013b) that showed the conversion of total sugars with pod storage in alcohol.Furthermore, Meyer et al. (1989) observed that increasing pod storage time before fermentation improved subsequent microbial aeration, prevented LAB development/growth and increased yeasts abundance and activity. This result could be explained by the fact that alcoholic fermentation may begin during the long storage of the pods injured during the harvest, which are naturally or subsequently contaminated by yeasts.

  ., Alessandria, V., Valente, A., Dolci, P., & Cocolin, L. (2016). Molecular identification and physiological characterization of yeasts, lactic acid bacteria and acetic acid bacteria isolated from heap and box cocoa bean fermentations in West Africa. International journal of food microbiology, 216, 69-78. Yong, F. M., Lee, K. H., & Wong, H. A. (1981). The production of ethyl acetate by soy yeast Saccharomyces rouxii NRRL Y-1096. International Journal of Food Science & Technology, 16(2), 177-184.

Figure 1 :

 1 Figure 1: Bacterial DNA DGGE profiles obtained from fermented cocoa beans (with and without turning) during 2, 4, 5, 6 and 7 days fermentation, previously stored during 2 days (a) and 8 days (b). Identification of DGGE bands for bacterial species are given inTable 1.

Figure 2 :

 2 Figure 2: Principal component analysis (PCA) of bacterial 16S rDNA DGGE band profiles obtained from cocoa beans stored during 2 days, fermented with turning (2S2F T, 2S2F T, 2S4F T, 2S4F T, 2S5F T, 2S5F T, 2S6F T, 2S6F T, 2S7F T and 2S7F T) and without turning (2S2F , 2S2F , 2S4F , 2S4F , 2S5F , 2S5F , 2S6F , 2S6F , 2S7F and 2S7F) during 2, 4, 5, 6 and 7 days fermentation and from cocoa beans stored during 8 days, fermented with turning (8S2F T, 8S2F T, 8S4F T, 8S4F T, 8S5F T, 8S5F T, 8S6FT, 8S6F T, 8S7F T and 8S7F T) and without turning (8S2F , 8S2F , 8S4F , 8S4F , 8S5F , 8S5F , 8S6F , 8S6F , 8S7F and 8S7F) during 2, 4, 5, 6 and 7 days fermentation.

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Yeasts DNA DGGE profiles obtained from fermented cocoa beans (with and without turning) during 2, 4, 5, 6 and 7 days fermentation, previously stored during 2 days (a) and 8 days (b) . Identification of DGGE bands for yeast species are given inTable 2.

Figure 6 :Figure 7 :

 67 Figure6: Concentration of aldehydes (a,g,), ketones (b,h), alcohols (c,i), esters (d,j), acids (e,k) and other volatile compounds (f,l) produced in cocoa beans previously stored during 2 and 8 days, fermented 6 days (6F) and 7 days (7F) with turning (+T) and without turning. indicates significant differences between fermentation with and without turning. indicates significant differences between the two pods storage duration. indicates significant differences between fermentation times.

Figure 8 :

 8 Figure 8: Concentration of organic acids present in cocoa beans previously stored during 2 days (a) and 8 days (b), fermented with turning and without turning. In (c) the two storage durations (2 and 8 days) were compared. indicates significant differences between fermentation with and without turning. indicates significant differences between the two pods storage duration.

Figure 9 :Figure 10 :

 910 Figure 9 : Principal component analysis (PCA) of peak areas of volatiles compounds obtained from cocoa beans stored during 2 days, fermented with turning (2S2F T, 2S2F T, 2S4F T, 2S4F T, 2S5F T, 2S5F T, 2S6F T, 2S6F T, 2S7F T and 2S7F T) and without turning (2S2F , 2S2F , 2S4F , 2S4F , 2S5F , 2S5F , 2S6F, 2S6F , 2S7F and 2S7F) during 2, 4, 5, 6 and 7 days fermentation and from cocoa beans stored during 8 days, fermented with turning (8S2F T, 8S2F T, 8S4F T, 8S4F T, 8S5F T, 8S5F T, 8S6F T, 8S6F T, 8S7F T and 8S7F T) and without turning (8S2F , 8S2F , 8S4F , 8S4F , 8S5F , 8S5F , 8S6F , 8S6F , 8S7F and 8S7F) during 2, 4, 5, 6 and 7 days fermentation.

Table 1 : Identification of bacterial DNA bands isolated from DGGE experiment

 1

	Objects

Table 2 : Identification of DGGE bands for yeast species Band Family/ Genus/ species Percent identity Query cover E-value References

 2

	1	Hanseniaspora opuntiae	98%	100%	6e-93	KC111446
	2	Candida ethanolica/ Pichia deserticola	93%	100%	8e-56	KM234475
	3	Pichia sp	99%	100%	1e-79	JX408867
	4	Pichia galeiformis	93%	100%	3e-76	HM212622
	5	Hanseniaspora uvarum	93%	100%	4e-65	KM816746
	9	Pichia sp	95%	100%	4e-65	EU884437
	10	Pichia kudriavzevii	99%	100%	2e-47	KC494718
	11	Pichia manshurica	99%	100%	7e-11	JQ419868
	12	Candida ethanolica/ Pichia deserticola	93%	99%	2e-64	KM005182
	6 ,7, 13	NI

NI: not identified band

Table 3 : Volatiles compounds identified in cocoa beans fermented with and without turning

 3

	Group	RT (min)	RI	RI*	Compound
	Aldehydes	1.50	811	821	Isobutanal
		2.15	910	912	2-Methylbutanal
		2.19	913	910	3-Methylbutanal
		28.06	1497	1495	Benzaldehyde
	Ketones	2.94	971	984	2-Pentanone
		9.51	1153	1170	2-Heptanone
		21.06	1367	1388	2-Nonanone
		35.21	1629	1645	Acetophenone
		14.83	1252	1272	Acetoin
	Esters	1.94	886	885	Ethyl acetate
		2.28	920	916	Methyl isovalerate
		3.60	1011	1005	Isobutyl acetate
		3.72	1016	1015	Methyl isopentanoate
		6.92	1105	1117	Isoamyl acetate
		44.55	1802	1803	Phenylethyl acetate
	Alcohols	2.46	934	929	Ethanol

Acknowledgments

This work was supported by DESI grant from CIRAD France (support of PhD students) and Y. Hamdouche was supported by a PhD fellowship from HUBBARD company.

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Moens, F., Lefeber, T., & De Vuyst, L. (2014). Oxidation of metabolites highlights the microbial interactions and role of Acetobacter pasteurianus during cocoa bean fermentation. Systematic Bacteriology, 47, 1145-1156. Papalexandratou, Z., Camu, N., Falony, G., & De Vuyst, L. (2011). Comparison of the bacterial species diversity of spontaneous cocoa bean fermentations carried out at selected farms in Ivory Coast and Brazil. Food microbiology, 28(5), 964-973.